MAYBE

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proof of /hpcwork/ff862203/termcomp26/benchmarks/yGpnR.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

qs(a,g)

w.r.t. the given Prolog program could not be shown:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
    (3) DependencyPairsProof [EQUIVALENT, 34 ms]
    (4) PiDP
    (5) DependencyGraphProof [EQUIVALENT, 0 ms]
    (6) AND
        (7) PiDP
            (8) UsableRulesProof [EQUIVALENT, 0 ms]
            (9) PiDP
            (10) PiDPToQDPProof [SOUND, 0 ms]
            (11) QDP
            (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (13) YES
        (14) PiDP
            (15) UsableRulesProof [EQUIVALENT, 0 ms]
            (16) PiDP
            (17) PiDPToQDPProof [SOUND, 0 ms]
            (18) QDP
            (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (20) YES
        (21) PiDP
            (22) UsableRulesProof [EQUIVALENT, 0 ms]
            (23) PiDP
            (24) PiDPToQDPProof [SOUND, 0 ms]
            (25) QDP
            (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (27) YES
        (28) PiDP
            (29) UsableRulesProof [EQUIVALENT, 0 ms]
            (30) PiDP
            (31) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (32) QDP
            (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (34) YES
        (35) PiDP
            (36) UsableRulesProof [EQUIVALENT, 0 ms]
            (37) PiDP
            (38) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (39) QDP
            (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (41) YES
        (42) PiDP
            (43) UsableRulesProof [EQUIVALENT, 0 ms]
            (44) PiDP
            (45) PiDPToQDPProof [SOUND, 0 ms]
            (46) QDP
            (47) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (48) YES
        (49) PiDP
            (50) UsableRulesProof [EQUIVALENT, 0 ms]
            (51) PiDP
            (52) PiDPToQDPProof [SOUND, 0 ms]
            (53) QDP
            (54) QDPQMonotonicMRRProof [EQUIVALENT, 48 ms]
            (55) QDP
            (56) DependencyGraphProof [EQUIVALENT, 0 ms]
            (57) TRUE
        (58) PiDP
            (59) UsableRulesProof [EQUIVALENT, 0 ms]
            (60) PiDP
            (61) PiDPToQDPProof [SOUND, 0 ms]
            (62) QDP
            (63) NonTerminationLoopProof [COMPLETE, 0 ms]
            (64) NO
        (65) PiDP
            (66) UsableRulesProof [EQUIVALENT, 0 ms]
            (67) PiDP
            (68) PiDPToQDPProof [SOUND, 0 ms]
            (69) QDP
            (70) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (71) YES
        (72) PiDP
            (73) UsableRulesProof [EQUIVALENT, 0 ms]
            (74) PiDP
            (75) PiDPToQDPProof [SOUND, 0 ms]
            (76) QDP
            (77) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (78) YES
        (79) PiDP
            (80) UsableRulesProof [EQUIVALENT, 0 ms]
            (81) PiDP
            (82) PiDPToQDPProof [SOUND, 0 ms]
            (83) QDP
            (84) TransformationProof [SOUND, 0 ms]
            (85) QDP
            (86) TransformationProof [SOUND, 0 ms]
            (87) QDP
            (88) TransformationProof [EQUIVALENT, 0 ms]
            (89) QDP
            (90) TransformationProof [EQUIVALENT, 0 ms]
            (91) QDP
            (92) DependencyGraphProof [EQUIVALENT, 0 ms]
            (93) AND
                (94) QDP
                    (95) UsableRulesProof [EQUIVALENT, 0 ms]
                    (96) QDP
                    (97) QReductionProof [EQUIVALENT, 0 ms]
                    (98) QDP
                (99) QDP
        (100) PiDP
            (101) UsableRulesProof [EQUIVALENT, 0 ms]
            (102) PiDP
        (103) PiDP
            (104) UsableRulesProof [EQUIVALENT, 0 ms]
            (105) PiDP
(106) PrologToPiTRSProof [SOUND, 0 ms]
(107) PiTRS
    (108) DependencyPairsProof [EQUIVALENT, 20 ms]
    (109) PiDP
    (110) DependencyGraphProof [EQUIVALENT, 0 ms]
    (111) AND
        (112) PiDP
            (113) UsableRulesProof [EQUIVALENT, 0 ms]
            (114) PiDP
            (115) PiDPToQDPProof [SOUND, 0 ms]
            (116) QDP
            (117) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (118) YES
        (119) PiDP
            (120) UsableRulesProof [EQUIVALENT, 0 ms]
            (121) PiDP
            (122) PiDPToQDPProof [SOUND, 0 ms]
            (123) QDP
            (124) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (125) YES
        (126) PiDP
            (127) UsableRulesProof [EQUIVALENT, 0 ms]
            (128) PiDP
            (129) PiDPToQDPProof [SOUND, 0 ms]
            (130) QDP
            (131) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (132) YES
        (133) PiDP
            (134) UsableRulesProof [EQUIVALENT, 0 ms]
            (135) PiDP
            (136) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (137) QDP
            (138) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (139) YES
        (140) PiDP
            (141) UsableRulesProof [EQUIVALENT, 0 ms]
            (142) PiDP
            (143) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (144) QDP
            (145) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (146) YES
        (147) PiDP
            (148) UsableRulesProof [EQUIVALENT, 0 ms]
            (149) PiDP
            (150) PiDPToQDPProof [SOUND, 0 ms]
            (151) QDP
            (152) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (153) YES
        (154) PiDP
            (155) UsableRulesProof [EQUIVALENT, 0 ms]
            (156) PiDP
            (157) PiDPToQDPProof [SOUND, 0 ms]
            (158) QDP
            (159) QDPQMonotonicMRRProof [EQUIVALENT, 24 ms]
            (160) QDP
            (161) DependencyGraphProof [EQUIVALENT, 0 ms]
            (162) TRUE
        (163) PiDP
            (164) UsableRulesProof [EQUIVALENT, 0 ms]
            (165) PiDP
            (166) PiDPToQDPProof [SOUND, 0 ms]
            (167) QDP
            (168) NonTerminationLoopProof [COMPLETE, 0 ms]
            (169) NO
        (170) PiDP
            (171) UsableRulesProof [EQUIVALENT, 0 ms]
            (172) PiDP
            (173) PiDPToQDPProof [SOUND, 0 ms]
            (174) QDP
            (175) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (176) YES
        (177) PiDP
            (178) UsableRulesProof [EQUIVALENT, 0 ms]
            (179) PiDP
            (180) PiDPToQDPProof [SOUND, 0 ms]
            (181) QDP
            (182) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (183) YES
        (184) PiDP
            (185) UsableRulesProof [EQUIVALENT, 0 ms]
            (186) PiDP
            (187) PiDPToQDPProof [SOUND, 0 ms]
            (188) QDP
            (189) TransformationProof [SOUND, 0 ms]
            (190) QDP
            (191) TransformationProof [SOUND, 0 ms]
            (192) QDP
            (193) TransformationProof [EQUIVALENT, 0 ms]
            (194) QDP
            (195) TransformationProof [EQUIVALENT, 0 ms]
            (196) QDP
            (197) DependencyGraphProof [EQUIVALENT, 0 ms]
            (198) AND
                (199) QDP
                    (200) UsableRulesProof [EQUIVALENT, 0 ms]
                    (201) QDP
                    (202) QReductionProof [EQUIVALENT, 0 ms]
                    (203) QDP
                    (204) NonTerminationLoopProof [COMPLETE, 0 ms]
                    (205) NO
                (206) QDP
        (207) PiDP
            (208) UsableRulesProof [EQUIVALENT, 0 ms]
            (209) PiDP
        (210) PiDP
            (211) UsableRulesProof [EQUIVALENT, 0 ms]
            (212) PiDP
(213) PrologToTRSTransformerProof [SOUND, 99 ms]
(214) QTRS
    (215) DependencyPairsProof [EQUIVALENT, 0 ms]
    (216) QDP
    (217) DependencyGraphProof [EQUIVALENT, 0 ms]
    (218) AND
        (219) QDP
            (220) UsableRulesProof [EQUIVALENT, 0 ms]
            (221) QDP
            (222) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (223) YES
        (224) QDP
            (225) UsableRulesProof [EQUIVALENT, 0 ms]
            (226) QDP
            (227) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (228) YES
        (229) QDP
            (230) UsableRulesProof [EQUIVALENT, 0 ms]
            (231) QDP
            (232) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (233) YES
        (234) QDP
            (235) UsableRulesProof [EQUIVALENT, 0 ms]
            (236) QDP
            (237) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (238) YES
        (239) QDP
            (240) UsableRulesProof [EQUIVALENT, 0 ms]
            (241) QDP
            (242) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (243) YES
        (244) QDP
            (245) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (246) YES
        (247) QDP
            (248) QDPOrderProof [EQUIVALENT, 18 ms]
            (249) QDP
            (250) DependencyGraphProof [EQUIVALENT, 0 ms]
            (251) QDP
            (252) QDPOrderProof [EQUIVALENT, 20 ms]
            (253) QDP
            (254) DependencyGraphProof [EQUIVALENT, 0 ms]
            (255) TRUE
        (256) QDP
            (257) NonLoopProof [COMPLETE, 0 ms]
            (258) NO
        (259) QDP
            (260) UsableRulesProof [EQUIVALENT, 0 ms]
            (261) QDP
            (262) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (263) YES
        (264) QDP
            (265) UsableRulesProof [EQUIVALENT, 0 ms]
            (266) QDP
            (267) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (268) YES
        (269) QDP
            (270) NonLoopProof [COMPLETE, 387 ms]
            (271) NO
        (272) QDP
            (273) NonLoopProof [COMPLETE, 0 ms]
            (274) NO
        (275) QDP
            (276) NonLoopProof [COMPLETE, 4866 ms]
            (277) NO
(278) PrologToDTProblemTransformerProof [SOUND, 78 ms]
(279) TRIPLES
    (280) TriplesToPiDPProof [SOUND, 60 ms]
    (281) PiDP
    (282) DependencyGraphProof [EQUIVALENT, 0 ms]
    (283) AND
        (284) PiDP
            (285) UsableRulesProof [EQUIVALENT, 0 ms]
            (286) PiDP
            (287) PiDPToQDPProof [SOUND, 0 ms]
            (288) QDP
            (289) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (290) YES
        (291) PiDP
            (292) UsableRulesProof [EQUIVALENT, 0 ms]
            (293) PiDP
            (294) PiDPToQDPProof [SOUND, 0 ms]
            (295) QDP
            (296) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (297) YES
        (298) PiDP
            (299) UsableRulesProof [EQUIVALENT, 0 ms]
            (300) PiDP
            (301) PiDPToQDPProof [SOUND, 0 ms]
            (302) QDP
            (303) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (304) YES
        (305) PiDP
            (306) UsableRulesProof [EQUIVALENT, 0 ms]
            (307) PiDP
            (308) PiDPToQDPProof [SOUND, 0 ms]
            (309) QDP
            (310) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (311) YES
        (312) PiDP
            (313) UsableRulesProof [EQUIVALENT, 0 ms]
            (314) PiDP
            (315) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (316) QDP
            (317) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (318) YES
        (319) PiDP
            (320) UsableRulesProof [EQUIVALENT, 0 ms]
            (321) PiDP
            (322) PiDPToQDPProof [SOUND, 0 ms]
            (323) QDP
            (324) NonTerminationLoopProof [COMPLETE, 0 ms]
            (325) NO
        (326) PiDP
            (327) UsableRulesProof [EQUIVALENT, 0 ms]
            (328) PiDP
            (329) PiDPToQDPProof [SOUND, 0 ms]
            (330) QDP
            (331) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (332) YES
        (333) PiDP
            (334) UsableRulesProof [EQUIVALENT, 0 ms]
            (335) PiDP
            (336) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (337) QDP
            (338) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (339) YES
        (340) PiDP
            (341) UsableRulesProof [EQUIVALENT, 0 ms]
            (342) PiDP
            (343) PiDPToQDPProof [EQUIVALENT, 0 ms]
            (344) QDP
            (345) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (346) YES
        (347) PiDP
            (348) UsableRulesProof [EQUIVALENT, 0 ms]
            (349) PiDP
            (350) PiDPToQDPProof [SOUND, 0 ms]
            (351) QDP
            (352) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (353) YES
        (354) PiDP
            (355) UsableRulesProof [EQUIVALENT, 0 ms]
            (356) PiDP
            (357) PiDPToQDPProof [SOUND, 0 ms]
            (358) QDP
            (359) QDPQMonotonicMRRProof [EQUIVALENT, 32 ms]
            (360) QDP
            (361) DependencyGraphProof [EQUIVALENT, 0 ms]
            (362) TRUE
        (363) PiDP
            (364) UsableRulesProof [EQUIVALENT, 0 ms]
            (365) PiDP
            (366) PiDPToQDPProof [SOUND, 0 ms]
            (367) QDP
            (368) QDPOrderProof [EQUIVALENT, 16 ms]
            (369) QDP
            (370) DependencyGraphProof [EQUIVALENT, 0 ms]
            (371) TRUE
        (372) PiDP
            (373) UsableRulesProof [EQUIVALENT, 0 ms]
            (374) PiDP
            (375) PiDPToQDPProof [SOUND, 0 ms]
            (376) QDP
            (377) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (378) YES
        (379) PiDP
            (380) UsableRulesProof [EQUIVALENT, 0 ms]
            (381) PiDP
            (382) PiDPToQDPProof [SOUND, 0 ms]
            (383) QDP
            (384) QDPSizeChangeProof [EQUIVALENT, 0 ms]
            (385) YES
        (386) PiDP
            (387) UsableRulesProof [EQUIVALENT, 0 ms]
            (388) PiDP
            (389) PiDPToQDPProof [SOUND, 0 ms]
            (390) QDP
            (391) TransformationProof [SOUND, 0 ms]
            (392) QDP
            (393) TransformationProof [SOUND, 0 ms]
            (394) QDP
            (395) TransformationProof [EQUIVALENT, 0 ms]
            (396) QDP
            (397) TransformationProof [EQUIVALENT, 0 ms]
            (398) QDP
            (399) DependencyGraphProof [EQUIVALENT, 0 ms]
            (400) AND
                (401) QDP
                    (402) UsableRulesProof [EQUIVALENT, 0 ms]
                    (403) QDP
                    (404) QReductionProof [EQUIVALENT, 0 ms]
                    (405) QDP
                (406) QDP
        (407) PiDP
            (408) UsableRulesProof [EQUIVALENT, 0 ms]
            (409) PiDP
        (410) PiDP
            (411) UsableRulesProof [EQUIVALENT, 0 ms]
            (412) PiDP
(413) PrologToIRSwTTransformerProof [SOUND, 112 ms]
(414) AND
    (415) IRSwT
        (416) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (417) TRUE
    (418) IRSwT
        (419) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (420) TRUE
    (421) IRSwT
        (422) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (423) TRUE
    (424) IRSwT
        (425) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (426) TRUE
    (427) IRSwT
        (428) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (429) TRUE
    (430) IRSwT
        (431) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (432) TRUE
    (433) IRSwT
        (434) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (435) TRUE
    (436) IRSwT
        (437) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (438) IRSwT
        (439) IntTRSCompressionProof [EQUIVALENT, 9 ms]
        (440) IRSwT
        (441) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (442) IRSwT
        (443) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (444) IRSwT
        (445) FilterProof [EQUIVALENT, 0 ms]
        (446) IntTRS
        (447) IntTRSNonPeriodicNontermProof [COMPLETE, 5 ms]
        (448) NO
    (449) IRSwT
        (450) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (451) TRUE
    (452) IRSwT
        (453) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (454) TRUE
    (455) IRSwT
        (456) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (457) TRUE
    (458) IRSwT
        (459) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 0 ms]
        (460) IRSwT
        (461) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (462) IRSwT
        (463) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (464) IRSwT
        (465) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
        (466) IRSwT
        (467) FilterProof [EQUIVALENT, 0 ms]
        (468) IntTRS
        (469) IntTRSPeriodicNontermProof [COMPLETE, 0 ms]
        (470) NO
    (471) IRSwT
        (472) IRSwTSimpleDependencyGraphProof [EQUIVALENT, 13 ms]
        (473) IRSwT
        (474) IntTRSCompressionProof [EQUIVALENT, 78 ms]
        (475) IRSwT
        (476) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
        (477) IRSwT
        (478) IRSwTTerminationDigraphProof [EQUIVALENT, 1127 ms]
        (479) IRSwT


----------------------------------------

(0)
Obligation:
Clauses:

qs(.(X, Xs), Ys) :- ','(part(X, Xs, Littles, Bigs), ','(qs(Littles, Ls), ','(qs(Bigs, Bs), app(Ls, .(X, Bs), Ys)))).
qs([], []).
part(X, .(Y, Xs), .(Y, Ls), Bs) :- ','(gt(X, Y), part(X, Xs, Ls, Bs)).
part(X, .(Y, Xs), Ls, .(Y, Bs)) :- ','(le(X, Y), part(X, Xs, Ls, Bs)).
part(X, [], [], []).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(0), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(X)).
le(0, 0).


Query: qs(a,g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

qs_in_2: (f,b) (b,f) (f,f)

part_in_4: (f,f,f,f) (b,f,f,f) (b,b,f,f)

gt_in_2: (f,f) (b,f) (b,b)

le_in_2: (b,f) (f,f) (b,b)

app_in_3: (b,b,f) (b,f,f) (b,f,b)

Transforming Prolog into the following Term Rewriting System:

Pi-finite rewrite system:
The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)



----------------------------------------

(3) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_AA(X, Y)
   GT_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, gt_in_aa(X, Y))
   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GA(X, Y)
   GT_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, gt_in_ga(X, Y))
   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GA(X, Y)
   LE_IN_GA(s(X), s(Y)) -> U11_GA(X, Y, le_in_ga(X, Y))
   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_AA(X, Y)
   LE_IN_AA(s(X), s(Y)) -> U11_AA(X, Y, le_in_aa(X, Y))
   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y)
   GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y))
   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y)
   LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y))
   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys)
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAA(Ls, .(X, Bs), Ys)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAG(Ls, .(X, Bs), Ys)
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

GT_IN_AA(x1, x2)  =  GT_IN_AA

U10_AA(x1, x2, x3)  =  U10_AA(x3)

U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x2, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)

U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x1, x6)

U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)

LE_IN_AA(x1, x2)  =  LE_IN_AA

U11_AA(x1, x2, x3)  =  U11_AA(x3)

U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

U10_GG(x1, x2, x3)  =  U10_GG(x1, x2, x3)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)

U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)

U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)

U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)

U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x5)

U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x1, x2, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_AA(X, Y)
   GT_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, gt_in_aa(X, Y))
   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GA(X, Y)
   GT_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, gt_in_ga(X, Y))
   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GA(X, Y)
   LE_IN_GA(s(X), s(Y)) -> U11_GA(X, Y, le_in_ga(X, Y))
   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_AA(X, Y)
   LE_IN_AA(s(X), s(Y)) -> U11_AA(X, Y, le_in_aa(X, Y))
   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y)
   GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y))
   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y)
   LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y))
   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys)
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAA(Ls, .(X, Bs), Ys)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAG(Ls, .(X, Bs), Ys)
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

GT_IN_AA(x1, x2)  =  GT_IN_AA

U10_AA(x1, x2, x3)  =  U10_AA(x3)

U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

U10_GA(x1, x2, x3)  =  U10_GA(x1, x3)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x2, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

U11_GA(x1, x2, x3)  =  U11_GA(x1, x3)

U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x1, x6)

U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)

LE_IN_AA(x1, x2)  =  LE_IN_AA

U11_AA(x1, x2, x3)  =  U11_AA(x3)

U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

U10_GG(x1, x2, x3)  =  U10_GG(x1, x2, x3)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

U11_GG(x1, x2, x3)  =  U11_GG(x1, x2, x3)

U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x1, x2, x3, x6)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)

U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x2, x4, x5)

U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x2, x4)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x2, x3, x5)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)

U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x5)

U4_AG(x1, x2, x3, x4)  =  U4_AG(x3, x4)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x1, x2, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 13 SCCs with 42 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

----------------------------------------

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(8) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(9)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(10) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(11)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP_IN_GAG(.(X, Xs), .(X, Zs)) -> APP_IN_GAG(Xs, Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(12) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP_IN_GAG(.(X, Xs), .(X, Zs)) -> APP_IN_GAG(Xs, Zs)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(13)
YES

----------------------------------------

(14)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(15) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(16)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(17) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs)) -> APP_IN_GAA(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(19) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP_IN_GAA(.(X, Xs)) -> APP_IN_GAA(Xs)
The graph contains the following edges 1 > 1


----------------------------------------

(20)
YES

----------------------------------------

(21)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(22) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(23)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(24) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(26) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2


----------------------------------------

(27)
YES

----------------------------------------

(28)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(29) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(30)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(31) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(33) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(34)
YES

----------------------------------------

(35)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(36) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(37)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(38) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(39)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(40) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(41)
YES

----------------------------------------

(42)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(43) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(44)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

.(x1, x2)  =  .(x1, x2)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(45) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(46)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GGAA(X, Y, Xs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs)
   PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y))
   U7_GGAA(X, Y, Xs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs)

The TRS R consists of the following rules:

   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))

The set Q consists of the following terms:

   gt_in_gg(x0, x1)
   le_in_gg(x0, x1)
   U10_gg(x0, x1, x2)
   U11_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(47) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*U5_GGAA(X, Y, Xs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2


*U7_GGAA(X, Y, Xs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2


----------------------------------------

(48)
YES

----------------------------------------

(49)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(50) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(51)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)

The TRS R consists of the following rules:

   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   qs_in_ga([], []) -> qs_out_ga([], [])
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(52) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(53)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Bigs, qs_in_ga(Littles))
   U2_GA(X, Xs, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs)
   QS_IN_GA(.(X, Xs)) -> U1_GA(X, Xs, part_in_ggaa(X, Xs))
   U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles)

The TRS R consists of the following rules:

   qs_in_ga(.(X, Xs)) -> U1_ga(X, Xs, part_in_ggaa(X, Xs))
   qs_in_ga([]) -> qs_out_ga([], [])
   part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
   part_in_ggaa(X, []) -> part_out_ggaa(X, [], [], [])
   U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
   U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
   U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
   U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   app_in_gga(.(X, Xs), Ys) -> U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
   app_in_gga([], Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

   qs_in_ga(x0)
   part_in_ggaa(x0, x1)
   U1_ga(x0, x1, x2)
   U5_ggaa(x0, x1, x2, x3)
   U7_ggaa(x0, x1, x2, x3)
   U2_ga(x0, x1, x2, x3)
   gt_in_gg(x0, x1)
   U6_ggaa(x0, x1, x2, x3)
   le_in_gg(x0, x1)
   U8_ggaa(x0, x1, x2, x3)
   U3_ga(x0, x1, x2, x3)
   U10_gg(x0, x1, x2)
   U11_gg(x0, x1, x2)
   U4_ga(x0, x1, x2)
   app_in_gga(x0, x1)
   U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(54) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Bigs, qs_in_ga(Littles))
   QS_IN_GA(.(X, Xs)) -> U1_GA(X, Xs, part_in_ggaa(X, Xs))
   U1_GA(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles)


Used ordering: Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 2 + x_2
   POL(0) = 0
   POL(QS_IN_GA(x_1)) = x_1
   POL(U10_gg(x_1, x_2, x_3)) = 1
   POL(U11_gg(x_1, x_2, x_3)) = 0
   POL(U1_GA(x_1, x_2, x_3)) = 1 + x_3
   POL(U1_ga(x_1, x_2, x_3)) = 0
   POL(U2_GA(x_1, x_2, x_3, x_4)) = x_3
   POL(U2_ga(x_1, x_2, x_3, x_4)) = 0
   POL(U3_ga(x_1, x_2, x_3, x_4)) = 0
   POL(U4_ga(x_1, x_2, x_3)) = 0
   POL(U5_ggaa(x_1, x_2, x_3, x_4)) = 2 + x_3
   POL(U6_ggaa(x_1, x_2, x_3, x_4)) = 2 + x_4
   POL(U7_ggaa(x_1, x_2, x_3, x_4)) = 2 + x_3
   POL(U8_ggaa(x_1, x_2, x_3, x_4)) = 2 + x_4
   POL(U9_gga(x_1, x_2, x_3, x_4)) = 0
   POL([]) = 0
   POL(app_in_gga(x_1, x_2)) = x_1
   POL(app_out_gga(x_1, x_2, x_3)) = 0
   POL(gt_in_gg(x_1, x_2)) = 2*x_1
   POL(gt_out_gg(x_1, x_2)) = 0
   POL(le_in_gg(x_1, x_2)) = 0
   POL(le_out_gg(x_1, x_2)) = 0
   POL(part_in_ggaa(x_1, x_2)) = x_2
   POL(part_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(qs_in_ga(x_1)) = 2
   POL(qs_out_ga(x_1, x_2)) = 0
   POL(s(x_1)) = 1


----------------------------------------

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U2_GA(X, Xs, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs)

The TRS R consists of the following rules:

   qs_in_ga(.(X, Xs)) -> U1_ga(X, Xs, part_in_ggaa(X, Xs))
   qs_in_ga([]) -> qs_out_ga([], [])
   part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
   part_in_ggaa(X, []) -> part_out_ggaa(X, [], [], [])
   U1_ga(X, Xs, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Bigs, qs_in_ga(Littles))
   U5_ggaa(X, Y, Xs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
   U7_ggaa(X, Y, Xs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, part_in_ggaa(X, Xs))
   U2_ga(X, Xs, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ls, qs_in_ga(Bigs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U6_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U8_ggaa(X, Y, Xs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U3_ga(X, Xs, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, app_in_gga(Ls, .(X, Bs)))
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U4_ga(X, Xs, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   app_in_gga(.(X, Xs), Ys) -> U9_gga(X, Xs, Ys, app_in_gga(Xs, Ys))
   app_in_gga([], Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))

The set Q consists of the following terms:

   qs_in_ga(x0)
   part_in_ggaa(x0, x1)
   U1_ga(x0, x1, x2)
   U5_ggaa(x0, x1, x2, x3)
   U7_ggaa(x0, x1, x2, x3)
   U2_ga(x0, x1, x2, x3)
   gt_in_gg(x0, x1)
   U6_ggaa(x0, x1, x2, x3)
   le_in_gg(x0, x1)
   U8_ggaa(x0, x1, x2, x3)
   U3_ga(x0, x1, x2, x3)
   U10_gg(x0, x1, x2)
   U11_gg(x0, x1, x2)
   U4_ga(x0, x1, x2)
   app_in_gga(x0, x1)
   U9_gga(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(56) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(57)
TRUE

----------------------------------------

(58)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

LE_IN_AA(x1, x2)  =  LE_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(59) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(60)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

LE_IN_AA(x1, x2)  =  LE_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(61) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE_IN_AA -> LE_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(63) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LE_IN_AA evaluates to  t =LE_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE_IN_AA to LE_IN_AA.




----------------------------------------

(64)
NO

----------------------------------------

(65)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(66) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(67)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(68) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(69)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE_IN_GA(s(X)) -> LE_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(70) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LE_IN_GA(s(X)) -> LE_IN_GA(X)
The graph contains the following edges 1 > 1


----------------------------------------

(71)
YES

----------------------------------------

(72)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(73) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(74)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(75) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(76)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GT_IN_GA(s(X)) -> GT_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(77) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*GT_IN_GA(s(X)) -> GT_IN_GA(X)
The graph contains the following edges 1 > 1


----------------------------------------

(78)
YES

----------------------------------------

(79)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(80) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(81)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

.(x1, x2)  =  .(x1, x2)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(82) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(83)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, gt_out_ga(X, Y)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(X) -> U5_GAAA(X, gt_in_ga(X))
   PART_IN_GAAA(X) -> U7_GAAA(X, le_in_ga(X))
   U7_GAAA(X, le_out_ga(X)) -> PART_IN_GAAA(X)

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(84) TransformationProof (SOUND)
By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U5_GAAA(X, gt_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

   (PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0))))
   (PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0)),PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0)))


----------------------------------------

(85)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, gt_out_ga(X, Y)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(X) -> U7_GAAA(X, le_in_ga(X))
   U7_GAAA(X, le_out_ga(X)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(86) TransformationProof (SOUND)
By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U7_GAAA(X, le_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

   (PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0))))
   (PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0)),PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0)))


----------------------------------------

(87)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, gt_out_ga(X, Y)) -> PART_IN_GAAA(X)
   U7_GAAA(X, le_out_ga(X)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(88) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U5_GAAA(X, gt_out_ga(X, Y)) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]:

   (U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) -> PART_IN_GAAA(s(z0)),U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) -> PART_IN_GAAA(s(z0)))
   (U5_GAAA(s(0), gt_out_ga(s(0), 0)) -> PART_IN_GAAA(s(0)),U5_GAAA(s(0), gt_out_ga(s(0), 0)) -> PART_IN_GAAA(s(0)))


----------------------------------------

(89)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(X, le_out_ga(X)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0))
   U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) -> PART_IN_GAAA(s(z0))
   U5_GAAA(s(0), gt_out_ga(s(0), 0)) -> PART_IN_GAAA(s(0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(90) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U7_GAAA(X, le_out_ga(X)) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]:

   (U7_GAAA(s(z0), le_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)),U7_GAAA(s(z0), le_out_ga(s(z0))) -> PART_IN_GAAA(s(z0)))
   (U7_GAAA(0, le_out_ga(0)) -> PART_IN_GAAA(0),U7_GAAA(0, le_out_ga(0)) -> PART_IN_GAAA(0))


----------------------------------------

(91)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0))
   U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) -> PART_IN_GAAA(s(z0))
   U5_GAAA(s(0), gt_out_ga(s(0), 0)) -> PART_IN_GAAA(s(0))
   U7_GAAA(s(z0), le_out_ga(s(z0))) -> PART_IN_GAAA(s(z0))
   U7_GAAA(0, le_out_ga(0)) -> PART_IN_GAAA(0)

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(92) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(93)
Complex Obligation (AND)

----------------------------------------

(94)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(0, le_out_ga(0)) -> PART_IN_GAAA(0)
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(95) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(96)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(0, le_out_ga(0)) -> PART_IN_GAAA(0)
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0))

R is empty.
The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(97) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)


----------------------------------------

(98)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(0, le_out_ga(0)) -> PART_IN_GAAA(0)
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga(0))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(99)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(s(z0), gt_out_ga(s(z0), x1)) -> PART_IN_GAAA(s(z0))
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(x0, gt_in_ga(x0)))
   U5_GAAA(s(0), gt_out_ga(s(0), 0)) -> PART_IN_GAAA(s(0))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(s(0), 0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(x0, le_in_ga(x0)))
   U7_GAAA(s(z0), le_out_ga(s(z0))) -> PART_IN_GAAA(s(z0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(X, gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X)) -> U11_ga(X, le_in_ga(X))
   le_in_ga(0) -> le_out_ga(0)
   U10_ga(X, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, le_out_ga(X)) -> le_out_ga(s(X))

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0, x1)
   U11_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(100)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

GT_IN_AA(x1, x2)  =  GT_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(101) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(102)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

GT_IN_AA(x1, x2)  =  GT_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(103)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x1, x2, x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa(x1)

U4_ag(x1, x2, x3, x4)  =  U4_ag(x3, x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x1, x2, x4, x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x1, x2, x3)

qs_out_ag(x1, x2)  =  qs_out_ag(x2)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(104) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(105)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)

The TRS R consists of the following rules:

   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x1, x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x1, x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x1, x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x1, x3)

le_out_ga(x1, x2)  =  le_out_ga(x1)

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x1, x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x1, x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x1, x2, x3)

gt_out_gg(x1, x2)  =  gt_out_gg(x1, x2)

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x1, x2, x3, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x1, x2, x3)

le_out_gg(x1, x2)  =  le_out_gg(x1, x2)

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x1, x2, x3, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x1, x2, x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x1, x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x2, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x1, x2, x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x2, x3, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x1, x2, x3)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(106) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

qs_in_2: (f,b) (b,f) (f,f)

part_in_4: (f,f,f,f) (b,f,f,f) (b,b,f,f)

gt_in_2: (f,f) (b,f) (b,b)

le_in_2: (b,f) (f,f) (b,b)

app_in_3: (b,b,f) (b,f,f) (b,f,b)

Transforming Prolog into the following Term Rewriting System:

Pi-finite rewrite system:
The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



----------------------------------------

(107)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag



----------------------------------------

(108) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_AA(X, Y)
   GT_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, gt_in_aa(X, Y))
   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GA(X, Y)
   GT_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, gt_in_ga(X, Y))
   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GA(X, Y)
   LE_IN_GA(s(X), s(Y)) -> U11_GA(X, Y, le_in_ga(X, Y))
   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_AA(X, Y)
   LE_IN_AA(s(X), s(Y)) -> U11_AA(X, Y, le_in_aa(X, Y))
   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y)
   GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y))
   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y)
   LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y))
   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys)
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAA(Ls, .(X, Bs), Ys)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAG(Ls, .(X, Bs), Ys)
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

GT_IN_AA(x1, x2)  =  GT_IN_AA

U10_AA(x1, x2, x3)  =  U10_AA(x3)

U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

U10_GA(x1, x2, x3)  =  U10_GA(x3)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x2, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

U11_GA(x1, x2, x3)  =  U11_GA(x3)

U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x6)

U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)

LE_IN_AA(x1, x2)  =  LE_IN_AA

U11_AA(x1, x2, x3)  =  U11_AA(x3)

U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

U10_GG(x1, x2, x3)  =  U10_GG(x3)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x2, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

U11_GG(x1, x2, x3)  =  U11_GG(x3)

U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x2, x6)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)

U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)

U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x5)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)

U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)

U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(109)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AG(.(X, Xs), Ys) -> U1_AG(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AG(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_AAAA(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_AA(X, Y)
   GT_IN_AA(s(X), s(Y)) -> U10_AA(X, Y, gt_in_aa(X, Y))
   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_AAAA(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GA(X, Y)
   GT_IN_GA(s(X), s(Y)) -> U10_GA(X, Y, gt_in_ga(X, Y))
   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GA(X, Y)
   LE_IN_GA(s(X), s(Y)) -> U11_GA(X, Y, le_in_ga(X, Y))
   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_GAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_AAAA(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   PART_IN_AAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_AA(X, Y)
   LE_IN_AA(s(X), s(Y)) -> U11_AA(X, Y, le_in_aa(X, Y))
   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_AAAA(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_AAAA(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AG(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AG(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   QS_IN_GA(.(X, Xs), Ys) -> PART_IN_GGAA(X, Xs, Littles, Bigs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> GT_IN_GG(X, Y)
   GT_IN_GG(s(X), s(Y)) -> U10_GG(X, Y, gt_in_gg(X, Y))
   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> LE_IN_GG(X, Y)
   LE_IN_GG(s(X), s(Y)) -> U11_GG(X, Y, le_in_gg(X, Y))
   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_GGAA(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_GA(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_GA(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U3_GA(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> APP_IN_GGA(Ls, .(X, Bs), Ys)
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> U9_GGA(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AG(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AG(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   QS_IN_AA(.(X, Xs), Ys) -> PART_IN_AAAA(X, Xs, Littles, Bigs)
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_AA(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AA(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   U3_AA(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAA(Ls, .(X, Bs), Ys)
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> U9_GAA(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_AG(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   U3_AG(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> APP_IN_GAG(Ls, .(X, Bs), Ys)
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> U9_GAG(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

QS_IN_AG(x1, x2)  =  QS_IN_AG(x2)

U1_AG(x1, x2, x3, x4)  =  U1_AG(x3, x4)

PART_IN_AAAA(x1, x2, x3, x4)  =  PART_IN_AAAA

U5_AAAA(x1, x2, x3, x4, x5, x6)  =  U5_AAAA(x6)

GT_IN_AA(x1, x2)  =  GT_IN_AA

U10_AA(x1, x2, x3)  =  U10_AA(x3)

U6_AAAA(x1, x2, x3, x4, x5, x6)  =  U6_AAAA(x2, x6)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)

U10_GA(x1, x2, x3)  =  U10_GA(x3)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x2, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)

U11_GA(x1, x2, x3)  =  U11_GA(x3)

U8_GAAA(x1, x2, x3, x4, x5, x6)  =  U8_GAAA(x6)

U7_AAAA(x1, x2, x3, x4, x5, x6)  =  U7_AAAA(x6)

LE_IN_AA(x1, x2)  =  LE_IN_AA

U11_AA(x1, x2, x3)  =  U11_AA(x3)

U8_AAAA(x1, x2, x3, x4, x5, x6)  =  U8_AAAA(x6)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x3, x5)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)

U10_GG(x1, x2, x3)  =  U10_GG(x3)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x2, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)

U11_GG(x1, x2, x3)  =  U11_GG(x3)

U8_GGAA(x1, x2, x3, x4, x5, x6)  =  U8_GGAA(x2, x6)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)

U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x1, x4, x5)

U4_GA(x1, x2, x3, x4)  =  U4_GA(x4)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)

U9_GGA(x1, x2, x3, x4, x5)  =  U9_GGA(x1, x5)

U3_AG(x1, x2, x3, x4, x5)  =  U3_AG(x3, x4, x5)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)

U3_AA(x1, x2, x3, x4, x5)  =  U3_AA(x4, x5)

U4_AA(x1, x2, x3, x4)  =  U4_AA(x4)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)

U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x5)

U4_AG(x1, x2, x3, x4)  =  U4_AG(x4)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)

U9_GAG(x1, x2, x3, x4, x5)  =  U9_GAG(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(110) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 13 SCCs with 42 less nodes.
----------------------------------------

(111)
Complex Obligation (AND)

----------------------------------------

(112)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(113) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(114)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAG(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APP_IN_GAG(x1, x2, x3)  =  APP_IN_GAG(x1, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(115) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(116)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP_IN_GAG(.(X, Xs), .(X, Zs)) -> APP_IN_GAG(Xs, Zs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(117) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP_IN_GAG(.(X, Xs), .(X, Zs)) -> APP_IN_GAG(Xs, Zs)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(118)
YES

----------------------------------------

(119)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(120) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(121)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GAA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APP_IN_GAA(x1, x2, x3)  =  APP_IN_GAA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(122) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(123)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP_IN_GAA(.(X, Xs)) -> APP_IN_GAA(Xs)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(124) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP_IN_GAA(.(X, Xs)) -> APP_IN_GAA(Xs)
The graph contains the following edges 1 > 1


----------------------------------------

(125)
YES

----------------------------------------

(126)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(127) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(128)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APP_IN_GGA(.(X, Xs), Ys, .(X, Zs)) -> APP_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APP_IN_GGA(x1, x2, x3)  =  APP_IN_GGA(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(129) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(130)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(131) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APP_IN_GGA(.(X, Xs), Ys) -> APP_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2


----------------------------------------

(132)
YES

----------------------------------------

(133)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

LE_IN_GG(x1, x2)  =  LE_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(134) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(135)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(136) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(137)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(138) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LE_IN_GG(s(X), s(Y)) -> LE_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(139)
YES

----------------------------------------

(140)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

GT_IN_GG(x1, x2)  =  GT_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(141) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(142)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(143) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(144)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(145) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*GT_IN_GG(s(X), s(Y)) -> GT_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(146)
YES

----------------------------------------

(147)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(148) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(149)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GGAA(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)
   PART_IN_GGAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GGAA(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GGAA(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   U7_GGAA(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> PART_IN_GGAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

.(x1, x2)  =  .(x1, x2)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

PART_IN_GGAA(x1, x2, x3, x4)  =  PART_IN_GGAA(x1, x2)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(150) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(151)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GGAA(X, Y, Xs, gt_out_gg) -> PART_IN_GGAA(X, Xs)
   PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y))
   PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y))
   U7_GGAA(X, Y, Xs, le_out_gg) -> PART_IN_GGAA(X, Xs)

The TRS R consists of the following rules:

   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg
   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg
   le_in_gg(0, 0) -> le_out_gg
   U10_gg(gt_out_gg) -> gt_out_gg
   U11_gg(le_out_gg) -> le_out_gg

The set Q consists of the following terms:

   gt_in_gg(x0, x1)
   le_in_gg(x0, x1)
   U10_gg(x0)
   U11_gg(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(152) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*PART_IN_GGAA(X, .(Y, Xs)) -> U5_GGAA(X, Y, Xs, gt_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*PART_IN_GGAA(X, .(Y, Xs)) -> U7_GGAA(X, Y, Xs, le_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*U5_GGAA(X, Y, Xs, gt_out_gg) -> PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2


*U7_GGAA(X, Y, Xs, le_out_gg) -> PART_IN_GGAA(X, Xs)
The graph contains the following edges 1 >= 1, 3 >= 2


----------------------------------------

(153)
YES

----------------------------------------

(154)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(155) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(156)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_GA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_GA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_GA(Bigs, Bs)
   QS_IN_GA(.(X, Xs), Ys) -> U1_GA(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   U1_GA(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> QS_IN_GA(Littles, Ls)

The TRS R consists of the following rules:

   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   qs_in_ga([], []) -> qs_out_ga([], [])
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

QS_IN_GA(x1, x2)  =  QS_IN_GA(x1)

U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)

U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(157) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(158)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles))
   U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs)
   QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs))
   U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles)

The TRS R consists of the following rules:

   qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs))
   qs_in_ga([]) -> qs_out_ga([])
   part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
   part_in_ggaa(X, []) -> part_out_ggaa([], [])
   U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles))
   U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs))
   U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs))
   U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg
   U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg
   le_in_gg(0, 0) -> le_out_gg
   U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs))
   U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs)))
   U10_gg(gt_out_gg) -> gt_out_gg
   U11_gg(le_out_gg) -> le_out_gg
   U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys)
   app_in_gga(.(X, Xs), Ys) -> U9_gga(X, app_in_gga(Xs, Ys))
   app_in_gga([], Ys) -> app_out_gga(Ys)
   U9_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs))

The set Q consists of the following terms:

   qs_in_ga(x0)
   part_in_ggaa(x0, x1)
   U1_ga(x0, x1)
   U5_ggaa(x0, x1, x2, x3)
   U7_ggaa(x0, x1, x2, x3)
   U2_ga(x0, x1, x2)
   gt_in_gg(x0, x1)
   U6_ggaa(x0, x1)
   le_in_gg(x0, x1)
   U8_ggaa(x0, x1)
   U3_ga(x0, x1, x2)
   U10_gg(x0)
   U11_gg(x0)
   U4_ga(x0)
   app_in_gga(x0, x1)
   U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(159) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   QS_IN_GA(.(X, Xs)) -> U1_GA(X, part_in_ggaa(X, Xs))


Used ordering: Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(0) = 1
   POL(QS_IN_GA(x_1)) = x_1
   POL(U10_gg(x_1)) = 0
   POL(U11_gg(x_1)) = 2*x_1
   POL(U1_GA(x_1, x_2)) = 2*x_2
   POL(U1_ga(x_1, x_2)) = 2 + 2*x_1
   POL(U2_GA(x_1, x_2, x_3)) = 2*x_2
   POL(U2_ga(x_1, x_2, x_3)) = 2 + 2*x_1
   POL(U3_ga(x_1, x_2, x_3)) = 2 + 2*x_1
   POL(U4_ga(x_1)) = 0
   POL(U5_ggaa(x_1, x_2, x_3, x_4)) = 1 + 2*x_2 + 2*x_3
   POL(U6_ggaa(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U7_ggaa(x_1, x_2, x_3, x_4)) = 1 + 2*x_2 + 2*x_3
   POL(U8_ggaa(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(U9_gga(x_1, x_2)) = 2*x_1
   POL([]) = 0
   POL(app_in_gga(x_1, x_2)) = 2*x_1
   POL(app_out_gga(x_1)) = 0
   POL(gt_in_gg(x_1, x_2)) = 0
   POL(gt_out_gg) = 0
   POL(le_in_gg(x_1, x_2)) = 2*x_2
   POL(le_out_gg) = 1
   POL(part_in_ggaa(x_1, x_2)) = x_2
   POL(part_out_ggaa(x_1, x_2)) = x_1 + x_2
   POL(qs_in_ga(x_1)) = 2*x_1
   POL(qs_out_ga(x_1)) = 0
   POL(s(x_1)) = 1 + 2*x_1


----------------------------------------

(160)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U1_GA(X, part_out_ggaa(Littles, Bigs)) -> U2_GA(X, Bigs, qs_in_ga(Littles))
   U2_GA(X, Bigs, qs_out_ga(Ls)) -> QS_IN_GA(Bigs)
   U1_GA(X, part_out_ggaa(Littles, Bigs)) -> QS_IN_GA(Littles)

The TRS R consists of the following rules:

   qs_in_ga(.(X, Xs)) -> U1_ga(X, part_in_ggaa(X, Xs))
   qs_in_ga([]) -> qs_out_ga([])
   part_in_ggaa(X, .(Y, Xs)) -> U5_ggaa(X, Y, Xs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs)) -> U7_ggaa(X, Y, Xs, le_in_gg(X, Y))
   part_in_ggaa(X, []) -> part_out_ggaa([], [])
   U1_ga(X, part_out_ggaa(Littles, Bigs)) -> U2_ga(X, Bigs, qs_in_ga(Littles))
   U5_ggaa(X, Y, Xs, gt_out_gg) -> U6_ggaa(Y, part_in_ggaa(X, Xs))
   U7_ggaa(X, Y, Xs, le_out_gg) -> U8_ggaa(Y, part_in_ggaa(X, Xs))
   U2_ga(X, Bigs, qs_out_ga(Ls)) -> U3_ga(X, Ls, qs_in_ga(Bigs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg
   U6_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(.(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg
   le_in_gg(0, 0) -> le_out_gg
   U8_ggaa(Y, part_out_ggaa(Ls, Bs)) -> part_out_ggaa(Ls, .(Y, Bs))
   U3_ga(X, Ls, qs_out_ga(Bs)) -> U4_ga(app_in_gga(Ls, .(X, Bs)))
   U10_gg(gt_out_gg) -> gt_out_gg
   U11_gg(le_out_gg) -> le_out_gg
   U4_ga(app_out_gga(Ys)) -> qs_out_ga(Ys)
   app_in_gga(.(X, Xs), Ys) -> U9_gga(X, app_in_gga(Xs, Ys))
   app_in_gga([], Ys) -> app_out_gga(Ys)
   U9_gga(X, app_out_gga(Zs)) -> app_out_gga(.(X, Zs))

The set Q consists of the following terms:

   qs_in_ga(x0)
   part_in_ggaa(x0, x1)
   U1_ga(x0, x1)
   U5_ggaa(x0, x1, x2, x3)
   U7_ggaa(x0, x1, x2, x3)
   U2_ga(x0, x1, x2)
   gt_in_gg(x0, x1)
   U6_ggaa(x0, x1)
   le_in_gg(x0, x1)
   U8_ggaa(x0, x1)
   U3_ga(x0, x1, x2)
   U10_gg(x0)
   U11_gg(x0)
   U4_ga(x0)
   app_in_gga(x0, x1)
   U9_gga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(161) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
----------------------------------------

(162)
TRUE

----------------------------------------

(163)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

LE_IN_AA(x1, x2)  =  LE_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(164) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(165)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_AA(s(X), s(Y)) -> LE_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

LE_IN_AA(x1, x2)  =  LE_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(166) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(167)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE_IN_AA -> LE_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(168) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LE_IN_AA evaluates to  t =LE_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE_IN_AA to LE_IN_AA.




----------------------------------------

(169)
NO

----------------------------------------

(170)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(171) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(172)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LE_IN_GA(s(X), s(Y)) -> LE_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

LE_IN_GA(x1, x2)  =  LE_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(173) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(174)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LE_IN_GA(s(X)) -> LE_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(175) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LE_IN_GA(s(X)) -> LE_IN_GA(X)
The graph contains the following edges 1 > 1


----------------------------------------

(176)
YES

----------------------------------------

(177)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(178) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(179)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_GA(s(X), s(Y)) -> GT_IN_GA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

GT_IN_GA(x1, x2)  =  GT_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(180) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(181)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GT_IN_GA(s(X)) -> GT_IN_GA(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(182) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*GT_IN_GA(s(X)) -> GT_IN_GA(X)
The graph contains the following edges 1 > 1


----------------------------------------

(183)
YES

----------------------------------------

(184)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(185) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(186)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)
   PART_IN_GAAA(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_GAAA(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   PART_IN_GAAA(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_GAAA(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   U7_GAAA(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> PART_IN_GAAA(X, Xs, Ls, Bs)

The TRS R consists of the following rules:

   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))

The argument filtering Pi contains the following mapping:
gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

.(x1, x2)  =  .(x1, x2)

PART_IN_GAAA(x1, x2, x3, x4)  =  PART_IN_GAAA(x1)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(187) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(188)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, gt_out_ga(Y)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(X) -> U5_GAAA(X, gt_in_ga(X))
   PART_IN_GAAA(X) -> U7_GAAA(X, le_in_ga(X))
   U7_GAAA(X, le_out_ga) -> PART_IN_GAAA(X)

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(189) TransformationProof (SOUND)
By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U5_GAAA(X, gt_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

   (PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0))))
   (PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0)),PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0)))


----------------------------------------

(190)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, gt_out_ga(Y)) -> PART_IN_GAAA(X)
   PART_IN_GAAA(X) -> U7_GAAA(X, le_in_ga(X))
   U7_GAAA(X, le_out_ga) -> PART_IN_GAAA(X)
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(191) TransformationProof (SOUND)
By narrowing [LPAR04] the rule PART_IN_GAAA(X) -> U7_GAAA(X, le_in_ga(X)) at position [1] we obtained the following new rules [LPAR04]:

   (PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(le_in_ga(x0))),PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(le_in_ga(x0))))
   (PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga),PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga))


----------------------------------------

(192)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(X, gt_out_ga(Y)) -> PART_IN_GAAA(X)
   U7_GAAA(X, le_out_ga) -> PART_IN_GAAA(X)
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(193) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U5_GAAA(X, gt_out_ga(Y)) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]:

   (U5_GAAA(s(z0), gt_out_ga(x1)) -> PART_IN_GAAA(s(z0)),U5_GAAA(s(z0), gt_out_ga(x1)) -> PART_IN_GAAA(s(z0)))
   (U5_GAAA(s(0), gt_out_ga(0)) -> PART_IN_GAAA(s(0)),U5_GAAA(s(0), gt_out_ga(0)) -> PART_IN_GAAA(s(0)))


----------------------------------------

(194)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(X, le_out_ga) -> PART_IN_GAAA(X)
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)
   U5_GAAA(s(z0), gt_out_ga(x1)) -> PART_IN_GAAA(s(z0))
   U5_GAAA(s(0), gt_out_ga(0)) -> PART_IN_GAAA(s(0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(195) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U7_GAAA(X, le_out_ga) -> PART_IN_GAAA(X) we obtained the following new rules [LPAR04]:

   (U7_GAAA(s(z0), le_out_ga) -> PART_IN_GAAA(s(z0)),U7_GAAA(s(z0), le_out_ga) -> PART_IN_GAAA(s(z0)))
   (U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0),U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0))


----------------------------------------

(196)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)
   U5_GAAA(s(z0), gt_out_ga(x1)) -> PART_IN_GAAA(s(z0))
   U5_GAAA(s(0), gt_out_ga(0)) -> PART_IN_GAAA(s(0))
   U7_GAAA(s(z0), le_out_ga) -> PART_IN_GAAA(s(z0))
   U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0)

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(197) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(198)
Complex Obligation (AND)

----------------------------------------

(199)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0)
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(200) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(201)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0)
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)

R is empty.
The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(202) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)


----------------------------------------

(203)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0)
   PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(204) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = PART_IN_GAAA(0) evaluates to  t =PART_IN_GAAA(0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga)
with rule PART_IN_GAAA(0) -> U7_GAAA(0, le_out_ga) at position [] and matcher [ ]

U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0)
with rule U7_GAAA(0, le_out_ga) -> PART_IN_GAAA(0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




----------------------------------------

(205)
NO

----------------------------------------

(206)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U5_GAAA(s(z0), gt_out_ga(x1)) -> PART_IN_GAAA(s(z0))
   PART_IN_GAAA(s(x0)) -> U5_GAAA(s(x0), U10_ga(gt_in_ga(x0)))
   U5_GAAA(s(0), gt_out_ga(0)) -> PART_IN_GAAA(s(0))
   PART_IN_GAAA(s(0)) -> U5_GAAA(s(0), gt_out_ga(0))
   PART_IN_GAAA(s(x0)) -> U7_GAAA(s(x0), U11_ga(le_in_ga(x0)))
   U7_GAAA(s(z0), le_out_ga) -> PART_IN_GAAA(s(z0))

The TRS R consists of the following rules:

   gt_in_ga(s(X)) -> U10_ga(gt_in_ga(X))
   gt_in_ga(s(0)) -> gt_out_ga(0)
   le_in_ga(s(X)) -> U11_ga(le_in_ga(X))
   le_in_ga(0) -> le_out_ga
   U10_ga(gt_out_ga(Y)) -> gt_out_ga(s(Y))
   U11_ga(le_out_ga) -> le_out_ga

The set Q consists of the following terms:

   gt_in_ga(x0)
   le_in_ga(x0)
   U10_ga(x0)
   U11_ga(x0)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(207)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

GT_IN_AA(x1, x2)  =  GT_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(208) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(209)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GT_IN_AA(s(X), s(Y)) -> GT_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

GT_IN_AA(x1, x2)  =  GT_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(210)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)

The TRS R consists of the following rules:

   qs_in_ag(.(X, Xs), Ys) -> U1_ag(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   U1_ag(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_ag(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   U2_ag(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ag(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa(.(X, Xs), Ys) -> U1_aa(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_aa(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_aa(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_aa(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_aa(X, Xs, Ys, Ls, qs_in_aa(Bigs, Bs))
   qs_in_aa([], []) -> qs_out_aa([], [])
   U3_aa(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_aa(X, Xs, Ys, app_in_gaa(Ls, .(X, Bs), Ys))
   app_in_gaa(.(X, Xs), Ys, .(X, Zs)) -> U9_gaa(X, Xs, Ys, Zs, app_in_gaa(Xs, Ys, Zs))
   app_in_gaa([], Ys, Ys) -> app_out_gaa([], Ys, Ys)
   U9_gaa(X, Xs, Ys, Zs, app_out_gaa(Xs, Ys, Zs)) -> app_out_gaa(.(X, Xs), Ys, .(X, Zs))
   U4_aa(X, Xs, Ys, app_out_gaa(Ls, .(X, Bs), Ys)) -> qs_out_aa(.(X, Xs), Ys)
   U3_ag(X, Xs, Ys, Ls, qs_out_aa(Bigs, Bs)) -> U4_ag(X, Xs, Ys, app_in_gag(Ls, .(X, Bs), Ys))
   app_in_gag(.(X, Xs), Ys, .(X, Zs)) -> U9_gag(X, Xs, Ys, Zs, app_in_gag(Xs, Ys, Zs))
   app_in_gag([], Ys, Ys) -> app_out_gag([], Ys, Ys)
   U9_gag(X, Xs, Ys, Zs, app_out_gag(Xs, Ys, Zs)) -> app_out_gag(.(X, Xs), Ys, .(X, Zs))
   U4_ag(X, Xs, Ys, app_out_gag(Ls, .(X, Bs), Ys)) -> qs_out_ag(.(X, Xs), Ys)
   qs_in_ag([], []) -> qs_out_ag([], [])

The argument filtering Pi contains the following mapping:
qs_in_ag(x1, x2)  =  qs_in_ag(x2)

U1_ag(x1, x2, x3, x4)  =  U1_ag(x3, x4)

part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x3, x5)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

U3_ag(x1, x2, x3, x4, x5)  =  U3_ag(x3, x4, x5)

qs_in_aa(x1, x2)  =  qs_in_aa

U1_aa(x1, x2, x3, x4)  =  U1_aa(x4)

U2_aa(x1, x2, x3, x4, x5)  =  U2_aa(x5)

U3_aa(x1, x2, x3, x4, x5)  =  U3_aa(x4, x5)

qs_out_aa(x1, x2)  =  qs_out_aa

U4_aa(x1, x2, x3, x4)  =  U4_aa(x4)

app_in_gaa(x1, x2, x3)  =  app_in_gaa(x1)

U9_gaa(x1, x2, x3, x4, x5)  =  U9_gaa(x5)

app_out_gaa(x1, x2, x3)  =  app_out_gaa

U4_ag(x1, x2, x3, x4)  =  U4_ag(x4)

app_in_gag(x1, x2, x3)  =  app_in_gag(x1, x3)

U9_gag(x1, x2, x3, x4, x5)  =  U9_gag(x5)

app_out_gag(x1, x2, x3)  =  app_out_gag(x2)

qs_out_ag(x1, x2)  =  qs_out_ag

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(211) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(212)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QS_IN_AA(.(X, Xs), Ys) -> U1_AA(X, Xs, Ys, part_in_aaaa(X, Xs, Littles, Bigs))
   U1_AA(X, Xs, Ys, part_out_aaaa(X, Xs, Littles, Bigs)) -> U2_AA(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   U2_AA(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> QS_IN_AA(Bigs, Bs)

The TRS R consists of the following rules:

   part_in_aaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_aaaa(X, Y, Xs, Ls, Bs, gt_in_aa(X, Y))
   part_in_aaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_aaaa(X, Y, Xs, Ls, Bs, le_in_aa(X, Y))
   part_in_aaaa(X, [], [], []) -> part_out_aaaa(X, [], [], [])
   qs_in_ga(.(X, Xs), Ys) -> U1_ga(X, Xs, Ys, part_in_ggaa(X, Xs, Littles, Bigs))
   qs_in_ga([], []) -> qs_out_ga([], [])
   U5_aaaa(X, Y, Xs, Ls, Bs, gt_out_aa(X, Y)) -> U6_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_aaaa(X, Y, Xs, Ls, Bs, le_out_aa(X, Y)) -> U8_aaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U1_ga(X, Xs, Ys, part_out_ggaa(X, Xs, Littles, Bigs)) -> U2_ga(X, Xs, Ys, Bigs, qs_in_ga(Littles, Ls))
   gt_in_aa(s(X), s(Y)) -> U10_aa(X, Y, gt_in_aa(X, Y))
   gt_in_aa(s(0), 0) -> gt_out_aa(s(0), 0)
   U6_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_aa(s(X), s(Y)) -> U11_aa(X, Y, le_in_aa(X, Y))
   le_in_aa(0, s(X)) -> le_out_aa(0, s(X))
   le_in_aa(0, 0) -> le_out_aa(0, 0)
   U8_aaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_aaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   part_in_ggaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_ggaa(X, Y, Xs, Ls, Bs, gt_in_gg(X, Y))
   part_in_ggaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_ggaa(X, Y, Xs, Ls, Bs, le_in_gg(X, Y))
   part_in_ggaa(X, [], [], []) -> part_out_ggaa(X, [], [], [])
   U2_ga(X, Xs, Ys, Bigs, qs_out_ga(Littles, Ls)) -> U3_ga(X, Xs, Ys, Ls, qs_in_ga(Bigs, Bs))
   U10_aa(X, Y, gt_out_aa(X, Y)) -> gt_out_aa(s(X), s(Y))
   part_in_gaaa(X, .(Y, Xs), .(Y, Ls), Bs) -> U5_gaaa(X, Y, Xs, Ls, Bs, gt_in_ga(X, Y))
   part_in_gaaa(X, .(Y, Xs), Ls, .(Y, Bs)) -> U7_gaaa(X, Y, Xs, Ls, Bs, le_in_ga(X, Y))
   part_in_gaaa(X, [], [], []) -> part_out_gaaa(X, [], [], [])
   U11_aa(X, Y, le_out_aa(X, Y)) -> le_out_aa(s(X), s(Y))
   U5_ggaa(X, Y, Xs, Ls, Bs, gt_out_gg(X, Y)) -> U6_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U7_ggaa(X, Y, Xs, Ls, Bs, le_out_gg(X, Y)) -> U8_ggaa(X, Y, Xs, Ls, Bs, part_in_ggaa(X, Xs, Ls, Bs))
   U3_ga(X, Xs, Ys, Ls, qs_out_ga(Bigs, Bs)) -> U4_ga(X, Xs, Ys, app_in_gga(Ls, .(X, Bs), Ys))
   U5_gaaa(X, Y, Xs, Ls, Bs, gt_out_ga(X, Y)) -> U6_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   U7_gaaa(X, Y, Xs, Ls, Bs, le_out_ga(X, Y)) -> U8_gaaa(X, Y, Xs, Ls, Bs, part_in_gaaa(X, Xs, Ls, Bs))
   gt_in_gg(s(X), s(Y)) -> U10_gg(X, Y, gt_in_gg(X, Y))
   gt_in_gg(s(0), 0) -> gt_out_gg(s(0), 0)
   U6_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_gg(s(X), s(Y)) -> U11_gg(X, Y, le_in_gg(X, Y))
   le_in_gg(0, s(X)) -> le_out_gg(0, s(X))
   le_in_gg(0, 0) -> le_out_gg(0, 0)
   U8_ggaa(X, Y, Xs, Ls, Bs, part_out_ggaa(X, Xs, Ls, Bs)) -> part_out_ggaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U4_ga(X, Xs, Ys, app_out_gga(Ls, .(X, Bs), Ys)) -> qs_out_ga(.(X, Xs), Ys)
   gt_in_ga(s(X), s(Y)) -> U10_ga(X, Y, gt_in_ga(X, Y))
   gt_in_ga(s(0), 0) -> gt_out_ga(s(0), 0)
   U6_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), .(Y, Ls), Bs)
   le_in_ga(s(X), s(Y)) -> U11_ga(X, Y, le_in_ga(X, Y))
   le_in_ga(0, s(X)) -> le_out_ga(0, s(X))
   le_in_ga(0, 0) -> le_out_ga(0, 0)
   U8_gaaa(X, Y, Xs, Ls, Bs, part_out_gaaa(X, Xs, Ls, Bs)) -> part_out_gaaa(X, .(Y, Xs), Ls, .(Y, Bs))
   U10_gg(X, Y, gt_out_gg(X, Y)) -> gt_out_gg(s(X), s(Y))
   U11_gg(X, Y, le_out_gg(X, Y)) -> le_out_gg(s(X), s(Y))
   app_in_gga(.(X, Xs), Ys, .(X, Zs)) -> U9_gga(X, Xs, Ys, Zs, app_in_gga(Xs, Ys, Zs))
   app_in_gga([], Ys, Ys) -> app_out_gga([], Ys, Ys)
   U10_ga(X, Y, gt_out_ga(X, Y)) -> gt_out_ga(s(X), s(Y))
   U11_ga(X, Y, le_out_ga(X, Y)) -> le_out_ga(s(X), s(Y))
   U9_gga(X, Xs, Ys, Zs, app_out_gga(Xs, Ys, Zs)) -> app_out_gga(.(X, Xs), Ys, .(X, Zs))

The argument filtering Pi contains the following mapping:
part_in_aaaa(x1, x2, x3, x4)  =  part_in_aaaa

U5_aaaa(x1, x2, x3, x4, x5, x6)  =  U5_aaaa(x6)

gt_in_aa(x1, x2)  =  gt_in_aa

U10_aa(x1, x2, x3)  =  U10_aa(x3)

gt_out_aa(x1, x2)  =  gt_out_aa(x1, x2)

U6_aaaa(x1, x2, x3, x4, x5, x6)  =  U6_aaaa(x2, x6)

part_in_gaaa(x1, x2, x3, x4)  =  part_in_gaaa(x1)

U5_gaaa(x1, x2, x3, x4, x5, x6)  =  U5_gaaa(x1, x6)

gt_in_ga(x1, x2)  =  gt_in_ga(x1)

s(x1)  =  s(x1)

U10_ga(x1, x2, x3)  =  U10_ga(x3)

0  =  0

gt_out_ga(x1, x2)  =  gt_out_ga(x2)

U6_gaaa(x1, x2, x3, x4, x5, x6)  =  U6_gaaa(x2, x6)

U7_gaaa(x1, x2, x3, x4, x5, x6)  =  U7_gaaa(x1, x6)

le_in_ga(x1, x2)  =  le_in_ga(x1)

U11_ga(x1, x2, x3)  =  U11_ga(x3)

le_out_ga(x1, x2)  =  le_out_ga

U8_gaaa(x1, x2, x3, x4, x5, x6)  =  U8_gaaa(x6)

part_out_gaaa(x1, x2, x3, x4)  =  part_out_gaaa(x3)

part_out_aaaa(x1, x2, x3, x4)  =  part_out_aaaa(x3)

U7_aaaa(x1, x2, x3, x4, x5, x6)  =  U7_aaaa(x6)

le_in_aa(x1, x2)  =  le_in_aa

U11_aa(x1, x2, x3)  =  U11_aa(x3)

le_out_aa(x1, x2)  =  le_out_aa(x1)

U8_aaaa(x1, x2, x3, x4, x5, x6)  =  U8_aaaa(x6)

qs_in_ga(x1, x2)  =  qs_in_ga(x1)

.(x1, x2)  =  .(x1, x2)

U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x4)

part_in_ggaa(x1, x2, x3, x4)  =  part_in_ggaa(x1, x2)

U5_ggaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaa(x1, x2, x3, x6)

gt_in_gg(x1, x2)  =  gt_in_gg(x1, x2)

U10_gg(x1, x2, x3)  =  U10_gg(x3)

gt_out_gg(x1, x2)  =  gt_out_gg

U6_ggaa(x1, x2, x3, x4, x5, x6)  =  U6_ggaa(x2, x6)

U7_ggaa(x1, x2, x3, x4, x5, x6)  =  U7_ggaa(x1, x2, x3, x6)

le_in_gg(x1, x2)  =  le_in_gg(x1, x2)

U11_gg(x1, x2, x3)  =  U11_gg(x3)

le_out_gg(x1, x2)  =  le_out_gg

U8_ggaa(x1, x2, x3, x4, x5, x6)  =  U8_ggaa(x2, x6)

[]  =  []

part_out_ggaa(x1, x2, x3, x4)  =  part_out_ggaa(x3, x4)

U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x4, x5)

qs_out_ga(x1, x2)  =  qs_out_ga(x2)

U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x1, x4, x5)

U4_ga(x1, x2, x3, x4)  =  U4_ga(x4)

app_in_gga(x1, x2, x3)  =  app_in_gga(x1, x2)

U9_gga(x1, x2, x3, x4, x5)  =  U9_gga(x1, x5)

app_out_gga(x1, x2, x3)  =  app_out_gga(x3)

QS_IN_AA(x1, x2)  =  QS_IN_AA

U1_AA(x1, x2, x3, x4)  =  U1_AA(x4)

U2_AA(x1, x2, x3, x4, x5)  =  U2_AA(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(213) PrologToTRSTransformerProof (SOUND)
Transformed Prolog program to TRS.

{
    "root": 1,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(qs (. X Xs) Ys)",
                "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))"
            ],
            [
                "(qs ([]) ([]))",
                null
            ],
            [
                "(part X (. Y Xs) (. Y Ls) Bs)",
                "(',' (gt X Y) (part X Xs Ls Bs))"
            ],
            [
                "(part X (. Y Xs) Ls (. Y Bs))",
                "(',' (le X Y) (part X Xs Ls Bs))"
            ],
            [
                "(part X ([]) ([]) ([]))",
                null
            ],
            [
                "(app (. X Xs) Ys (. X Zs))",
                "(app Xs Ys Zs)"
            ],
            [
                "(app ([]) Ys Ys)",
                null
            ],
            [
                "(gt (s X) (s Y))",
                "(gt X Y)"
            ],
            [
                "(gt (s (0)) (0))",
                null
            ],
            [
                "(le (s X) (s Y))",
                "(le X Y)"
            ],
            [
                "(le (0) (s X))",
                null
            ],
            [
                "(le (0) (0))",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "1187": {
                "goal": [{
                    "clause": 0,
                    "scope": 8,
                    "term": "(qs T23 X26)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T23"],
                    "free": ["X26"],
                    "exprvars": []
                }
            },
            "1186": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 8,
                        "term": "(qs T23 X26)"
                    },
                    {
                        "clause": 1,
                        "scope": 8,
                        "term": "(qs T23 X26)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T23"],
                    "free": ["X26"],
                    "exprvars": []
                }
            },
            "1185": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T24 X27) (app T205 (. T25 X27) T17))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T17",
                        "T205"
                    ],
                    "free": ["X27"],
                    "exprvars": []
                }
            },
            "1184": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T23 X26)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T23"],
                    "free": ["X26"],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "1183": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1182": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1181": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1180": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1337": {
                "goal": [{
                    "clause": 6,
                    "scope": 14,
                    "term": "(app T375 (. T381 T380) X593)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "1336": {
                "goal": [{
                    "clause": 5,
                    "scope": 14,
                    "term": "(app T375 (. T381 T380) X593)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "630": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T44 T45)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1335": {
                "goal": [
                    {
                        "clause": 5,
                        "scope": 14,
                        "term": "(app T375 (. T381 T380) X593)"
                    },
                    {
                        "clause": 6,
                        "scope": 14,
                        "term": "(app T375 (. T381 T380) X593)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "1334": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T375 (. T381 T380) X593)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "632": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T49 T50 X75 X76)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "1179": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1333": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T368 X592)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X592"],
                    "exprvars": []
                }
            },
            "1178": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "634": {
                "goal": [
                    {
                        "clause": 7,
                        "scope": 3,
                        "term": "(gt T44 T45)"
                    },
                    {
                        "clause": 8,
                        "scope": 3,
                        "term": "(gt T44 T45)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "997": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1177": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "998": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "636": {
                "goal": [{
                    "clause": 7,
                    "scope": 3,
                    "term": "(gt T44 T45)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "999": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "637": {
                "goal": [{
                    "clause": 8,
                    "scope": 3,
                    "term": "(gt T44 T45)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "638": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T63 T64)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "639": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1190": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1198": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    },
                    {
                        "clause": 3,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    },
                    {
                        "clause": 4,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
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                        "term": "(le T272 T273)"
                    },
                    {
                        "clause": 11,
                        "scope": 11,
                        "term": "(le T272 T273)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T272",
                        "T273"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "856": {
                "goal": [{
                    "clause": 7,
                    "scope": 5,
                    "term": "(gt T82 T85)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T82"],
                    "free": [],
                    "exprvars": []
                }
            },
            "857": {
                "goal": [{
                    "clause": 8,
                    "scope": 5,
                    "term": "(gt T82 T85)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T82"],
                    "free": [],
                    "exprvars": []
                }
            },
            "859": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T100 T102)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T100"],
                    "free": [],
                    "exprvars": []
                }
            },
            "1317": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 13,
                        "term": "(qs T24 X27)"
                    },
                    {
                        "clause": 1,
                        "scope": 13,
                        "term": "(qs T24 X27)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X27"],
                    "exprvars": []
                }
            },
            "1176": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1297": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T333 (. T334 T335) X537)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T333",
                        "T334",
                        "T335"
                    ],
                    "free": ["X537"],
                    "exprvars": []
                }
            },
            "1175": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1296": {
                "goal": [{
                    "clause": 6,
                    "scope": 12,
                    "term": "(app T304 (. T216 T309) X337)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T304",
                        "T309"
                    ],
                    "free": ["X337"],
                    "exprvars": []
                }
            },
            "1174": {
                "goal": [{
                    "clause": 11,
                    "scope": 7,
                    "term": "(le T169 T170)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1295": {
                "goal": [{
                    "clause": 5,
                    "scope": 12,
                    "term": "(app T304 (. T216 T309) X337)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T304",
                        "T309"
                    ],
                    "free": ["X337"],
                    "exprvars": []
                }
            },
            "1173": {
                "goal": [{
                    "clause": 10,
                    "scope": 7,
                    "term": "(le T169 T170)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1294": {
                "goal": [
                    {
                        "clause": 5,
                        "scope": 12,
                        "term": "(app T304 (. T216 T309) X337)"
                    },
                    {
                        "clause": 6,
                        "scope": 12,
                        "term": "(app T304 (. T216 T309) X337)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T304",
                        "T309"
                    ],
                    "free": ["X337"],
                    "exprvars": []
                }
            },
            "1172": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1293": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T304 (. T216 T309) X337)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T304",
                        "T309"
                    ],
                    "free": ["X337"],
                    "exprvars": []
                }
            },
            "1171": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(le T188 T189)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1292": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T222 X336)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T222"],
                    "free": ["X336"],
                    "exprvars": []
                }
            },
            "1170": {
                "goal": [
                    {
                        "clause": 10,
                        "scope": 7,
                        "term": "(le T169 T170)"
                    },
                    {
                        "clause": 11,
                        "scope": 7,
                        "term": "(le T169 T170)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1291": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T222 X336) (app T304 (. T216 X336) X337))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T222",
                        "T304"
                    ],
                    "free": [
                        "X337",
                        "X336"
                    ],
                    "exprvars": []
                }
            },
            "1290": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T221 X335)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T221"],
                    "free": ["X335"],
                    "exprvars": []
                }
            },
            "1327": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T368 X592) (app T375 (. T369 X592) X593))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": [
                        "X593",
                        "X592"
                    ],
                    "exprvars": []
                }
            },
            "1326": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T367 X591)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T367"],
                    "free": ["X591"],
                    "exprvars": []
                }
            },
            "1325": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T367 X591) (',' (qs T368 X592) (app X591 (. T369 X592) X593)))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T367"],
                    "free": [
                        "X593",
                        "X591",
                        "X592"
                    ],
                    "exprvars": []
                }
            },
            "861": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1324": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T362 T363 X589 X590)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X589",
                        "X590"
                    ],
                    "exprvars": []
                }
            },
            "1169": {
                "goal": [{
                    "clause": 9,
                    "scope": 7,
                    "term": "(le T169 T170)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1323": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1168": {
                "goal": [
                    {
                        "clause": 9,
                        "scope": 7,
                        "term": "(le T169 T170)"
                    },
                    {
                        "clause": 10,
                        "scope": 7,
                        "term": "(le T169 T170)"
                    },
                    {
                        "clause": 11,
                        "scope": 7,
                        "term": "(le T169 T170)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1289": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1322": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (part T362 T363 X589 X590) (',' (qs X589 X591) (',' (qs X590 X592) (app X591 (. T362 X592) X593))))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X593",
                        "X589",
                        "X590",
                        "X591",
                        "X592"
                    ],
                    "exprvars": []
                }
            },
            "1167": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T174 T175 X257 X258)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T174"],
                    "free": [
                        "X257",
                        "X258"
                    ],
                    "exprvars": []
                }
            },
            "1288": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1321": {
                "goal": [{
                    "clause": 1,
                    "scope": 13,
                    "term": "(qs T24 X27)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X27"],
                    "exprvars": []
                }
            },
            "1166": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(le T169 T170)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1287": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1320": {
                "goal": [{
                    "clause": 0,
                    "scope": 13,
                    "term": "(qs T24 X27)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X27"],
                    "exprvars": []
                }
            },
            "626": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (gt T44 T45) (part T44 T46 X75 X76))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "628": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 1,
                "to": 4,
                "label": "CASE"
            },
            {
                "from": 4,
                "to": 7,
                "label": "PARALLEL"
            },
            {
                "from": 4,
                "to": 8,
                "label": "PARALLEL"
            },
            {
                "from": 7,
                "to": 33,
                "label": "EVAL with clause\nqs(.(X21, X22), X23) :- ','(part(X21, X22, X24, X25), ','(qs(X24, X26), ','(qs(X25, X27), app(X26, .(X21, X27), X23)))).\nand substitutionX21 -> T18,\nX22 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX23 -> T17,\nT15 -> T18,\nT16 -> T19"
            },
            {
                "from": 7,
                "to": 34,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 8,
                "to": 1374,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []"
            },
            {
                "from": 8,
                "to": 1375,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 33,
                "to": 371,
                "label": "SPLIT 1"
            },
            {
                "from": 33,
                "to": 373,
                "label": "SPLIT 2\nnew knowledge:\nT23 is ground\nreplacements:X24 -> T23,\nX25 -> T24,\nT18 -> T25"
            },
            {
                "from": 371,
                "to": 389,
                "label": "CASE"
            },
            {
                "from": 373,
                "to": 1184,
                "label": "SPLIT 1"
            },
            {
                "from": 373,
                "to": 1185,
                "label": "SPLIT 2\nnew knowledge:\nT23 is ground\nT205 is ground\nreplacements:X26 -> T205"
            },
            {
                "from": 389,
                "to": 410,
                "label": "PARALLEL"
            },
            {
                "from": 389,
                "to": 411,
                "label": "PARALLEL"
            },
            {
                "from": 410,
                "to": 626,
                "label": "EVAL with clause\npart(X70, .(X71, X72), .(X71, X73), X74) :- ','(gt(X70, X71), part(X70, X72, X73, X74)).\nand substitutionT18 -> T44,\nX70 -> T44,\nX71 -> T45,\nX72 -> T46,\nT19 -> .(T45, T46),\nX73 -> X75,\nX24 -> .(T45, X75),\nX25 -> X76,\nX74 -> X76,\nT41 -> T44,\nT42 -> T45,\nT43 -> T46"
            },
            {
                "from": 410,
                "to": 628,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 411,
                "to": 1139,
                "label": "PARALLEL"
            },
            {
                "from": 411,
                "to": 1140,
                "label": "PARALLEL"
            },
            {
                "from": 626,
                "to": 630,
                "label": "SPLIT 1"
            },
            {
                "from": 626,
                "to": 632,
                "label": "SPLIT 2\nnew knowledge:\nT49 is ground\nT45 is ground\nreplacements:T44 -> T49,\nT46 -> T50"
            },
            {
                "from": 630,
                "to": 634,
                "label": "CASE"
            },
            {
                "from": 632,
                "to": 838,
                "label": "CASE"
            },
            {
                "from": 634,
                "to": 636,
                "label": "PARALLEL"
            },
            {
                "from": 634,
                "to": 637,
                "label": "PARALLEL"
            },
            {
                "from": 636,
                "to": 638,
                "label": "EVAL with clause\ngt(s(X89), s(X90)) :- gt(X89, X90).\nand substitutionX89 -> T63,\nT44 -> s(T63),\nX90 -> T64,\nT45 -> s(T64),\nT61 -> T63,\nT62 -> T64"
            },
            {
                "from": 636,
                "to": 639,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 637,
                "to": 835,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT44 -> s(0),\nT45 -> 0"
            },
            {
                "from": 637,
                "to": 836,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 638,
                "to": 630,
                "label": "INSTANCE with matching:\nT44 -> T63\nT45 -> T64"
            },
            {
                "from": 835,
                "to": 837,
                "label": "SUCCESS"
            },
            {
                "from": 838,
                "to": 841,
                "label": "PARALLEL"
            },
            {
                "from": 838,
                "to": 842,
                "label": "PARALLEL"
            },
            {
                "from": 841,
                "to": 847,
                "label": "EVAL with clause\npart(X128, .(X129, X130), .(X129, X131), X132) :- ','(gt(X128, X129), part(X128, X130, X131, X132)).\nand substitutionT49 -> T82,\nX128 -> T82,\nX129 -> T85,\nX130 -> T86,\nT50 -> .(T85, T86),\nX131 -> X133,\nX75 -> .(T85, X133),\nX76 -> X134,\nX132 -> X134,\nT83 -> T85,\nT84 -> T86"
            },
            {
                "from": 841,
                "to": 848,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 842,
                "to": 1000,
                "label": "PARALLEL"
            },
            {
                "from": 842,
                "to": 1001,
                "label": "PARALLEL"
            },
            {
                "from": 847,
                "to": 852,
                "label": "SPLIT 1"
            },
            {
                "from": 847,
                "to": 853,
                "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT85 is ground\nreplacements:T86 -> T89"
            },
            {
                "from": 852,
                "to": 854,
                "label": "CASE"
            },
            {
                "from": 853,
                "to": 632,
                "label": "INSTANCE with matching:\nT49 -> T82\nT50 -> T89\nX75 -> X133\nX76 -> X134"
            },
            {
                "from": 854,
                "to": 856,
                "label": "PARALLEL"
            },
            {
                "from": 854,
                "to": 857,
                "label": "PARALLEL"
            },
            {
                "from": 856,
                "to": 859,
                "label": "EVAL with clause\ngt(s(X147), s(X148)) :- gt(X147, X148).\nand substitutionX147 -> T100,\nT82 -> s(T100),\nX148 -> T102,\nT85 -> s(T102),\nT101 -> T102"
            },
            {
                "from": 856,
                "to": 861,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 857,
                "to": 997,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT82 -> s(0),\nT85 -> 0"
            },
            {
                "from": 857,
                "to": 998,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 859,
                "to": 852,
                "label": "INSTANCE with matching:\nT82 -> T100\nT85 -> T102"
            },
            {
                "from": 997,
                "to": 999,
                "label": "SUCCESS"
            },
            {
                "from": 1000,
                "to": 1002,
                "label": "EVAL with clause\npart(X186, .(X187, X188), X189, .(X187, X190)) :- ','(le(X186, X187), part(X186, X188, X189, X190)).\nand substitutionT49 -> T120,\nX186 -> T120,\nX187 -> T123,\nX188 -> T124,\nT50 -> .(T123, T124),\nX75 -> X191,\nX189 -> X191,\nX190 -> X192,\nX76 -> .(T123, X192),\nT121 -> T123,\nT122 -> T124"
            },
            {
                "from": 1000,
                "to": 1003,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1001,
                "to": 1133,
                "label": "EVAL with clause\npart(X223, [], [], []).\nand substitutionT49 -> T153,\nX223 -> T153,\nT50 -> [],\nX75 -> [],\nX76 -> []"
            },
            {
                "from": 1001,
                "to": 1135,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1002,
                "to": 1004,
                "label": "SPLIT 1"
            },
            {
                "from": 1002,
                "to": 1005,
                "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:T124 -> T127"
            },
            {
                "from": 1004,
                "to": 1006,
                "label": "CASE"
            },
            {
                "from": 1005,
                "to": 632,
                "label": "INSTANCE with matching:\nT49 -> T120\nT50 -> T127\nX75 -> X191\nX76 -> X192"
            },
            {
                "from": 1006,
                "to": 1007,
                "label": "PARALLEL"
            },
            {
                "from": 1006,
                "to": 1008,
                "label": "PARALLEL"
            },
            {
                "from": 1007,
                "to": 1095,
                "label": "EVAL with clause\nle(s(X205), s(X206)) :- le(X205, X206).\nand substitutionX205 -> T138,\nT120 -> s(T138),\nX206 -> T140,\nT123 -> s(T140),\nT139 -> T140"
            },
            {
                "from": 1007,
                "to": 1096,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1008,
                "to": 1097,
                "label": "PARALLEL"
            },
            {
                "from": 1008,
                "to": 1098,
                "label": "PARALLEL"
            },
            {
                "from": 1095,
                "to": 1004,
                "label": "INSTANCE with matching:\nT120 -> T138\nT123 -> T140"
            },
            {
                "from": 1097,
                "to": 1099,
                "label": "EVAL with clause\nle(0, s(X213)).\nand substitutionT120 -> 0,\nX213 -> T147,\nT123 -> s(T147)"
            },
            {
                "from": 1097,
                "to": 1100,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1098,
                "to": 1102,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT120 -> 0,\nT123 -> 0"
            },
            {
                "from": 1098,
                "to": 1103,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1099,
                "to": 1101,
                "label": "SUCCESS"
            },
            {
                "from": 1102,
                "to": 1104,
                "label": "SUCCESS"
            },
            {
                "from": 1133,
                "to": 1136,
                "label": "SUCCESS"
            },
            {
                "from": 1139,
                "to": 1164,
                "label": "EVAL with clause\npart(X252, .(X253, X254), X255, .(X253, X256)) :- ','(le(X252, X253), part(X252, X254, X255, X256)).\nand substitutionT18 -> T169,\nX252 -> T169,\nX253 -> T170,\nX254 -> T171,\nT19 -> .(T170, T171),\nX24 -> X257,\nX255 -> X257,\nX256 -> X258,\nX25 -> .(T170, X258),\nT166 -> T169,\nT167 -> T170,\nT168 -> T171"
            },
            {
                "from": 1139,
                "to": 1165,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1140,
                "to": 1181,
                "label": "EVAL with clause\npart(X289, [], [], []).\nand substitutionT18 -> T202,\nX289 -> T202,\nT19 -> [],\nX24 -> [],\nX25 -> []"
            },
            {
                "from": 1140,
                "to": 1182,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1164,
                "to": 1166,
                "label": "SPLIT 1"
            },
            {
                "from": 1164,
                "to": 1167,
                "label": "SPLIT 2\nnew knowledge:\nT174 is ground\nreplacements:T169 -> T174,\nT171 -> T175"
            },
            {
                "from": 1166,
                "to": 1168,
                "label": "CASE"
            },
            {
                "from": 1167,
                "to": 632,
                "label": "INSTANCE with matching:\nT49 -> T174\nT50 -> T175\nX75 -> X257\nX76 -> X258"
            },
            {
                "from": 1168,
                "to": 1169,
                "label": "PARALLEL"
            },
            {
                "from": 1168,
                "to": 1170,
                "label": "PARALLEL"
            },
            {
                "from": 1169,
                "to": 1171,
                "label": "EVAL with clause\nle(s(X271), s(X272)) :- le(X271, X272).\nand substitutionX271 -> T188,\nT169 -> s(T188),\nX272 -> T189,\nT170 -> s(T189),\nT186 -> T188,\nT187 -> T189"
            },
            {
                "from": 1169,
                "to": 1172,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1170,
                "to": 1173,
                "label": "PARALLEL"
            },
            {
                "from": 1170,
                "to": 1174,
                "label": "PARALLEL"
            },
            {
                "from": 1171,
                "to": 1166,
                "label": "INSTANCE with matching:\nT169 -> T188\nT170 -> T189"
            },
            {
                "from": 1173,
                "to": 1175,
                "label": "EVAL with clause\nle(0, s(X279)).\nand substitutionT169 -> 0,\nX279 -> T196,\nT170 -> s(T196)"
            },
            {
                "from": 1173,
                "to": 1176,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1174,
                "to": 1178,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT169 -> 0,\nT170 -> 0"
            },
            {
                "from": 1174,
                "to": 1179,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1175,
                "to": 1177,
                "label": "SUCCESS"
            },
            {
                "from": 1178,
                "to": 1180,
                "label": "SUCCESS"
            },
            {
                "from": 1181,
                "to": 1183,
                "label": "SUCCESS"
            },
            {
                "from": 1184,
                "to": 1186,
                "label": "CASE"
            },
            {
                "from": 1185,
                "to": 1313,
                "label": "SPLIT 1"
            },
            {
                "from": 1185,
                "to": 1314,
                "label": "SPLIT 2\nreplacements:X27 -> T348,\nT25 -> T349"
            },
            {
                "from": 1186,
                "to": 1187,
                "label": "PARALLEL"
            },
            {
                "from": 1186,
                "to": 1188,
                "label": "PARALLEL"
            },
            {
                "from": 1187,
                "to": 1189,
                "label": "EVAL with clause\nqs(.(X330, X331), X332) :- ','(part(X330, X331, X333, X334), ','(qs(X333, X335), ','(qs(X334, X336), app(X335, .(X330, X336), X332)))).\nand substitutionX330 -> T216,\nX331 -> T217,\nT23 -> .(T216, T217),\nX26 -> X337,\nX332 -> X337"
            },
            {
                "from": 1187,
                "to": 1190,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1188,
                "to": 1308,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT23 -> [],\nX26 -> []"
            },
            {
                "from": 1188,
                "to": 1309,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1189,
                "to": 1196,
                "label": "SPLIT 1"
            },
            {
                "from": 1189,
                "to": 1197,
                "label": "SPLIT 2\nnew knowledge:\nT216 is ground\nT217 is ground\nT221 is ground\nT222 is ground\nreplacements:X333 -> T221,\nX334 -> T222"
            },
            {
                "from": 1196,
                "to": 1198,
                "label": "CASE"
            },
            {
                "from": 1197,
                "to": 1290,
                "label": "SPLIT 1"
            },
            {
                "from": 1197,
                "to": 1291,
                "label": "SPLIT 2\nnew knowledge:\nT221 is ground\nT304 is ground\nreplacements:X335 -> T304"
            },
            {
                "from": 1198,
                "to": 1235,
                "label": "PARALLEL"
            },
            {
                "from": 1198,
                "to": 1236,
                "label": "PARALLEL"
            },
            {
                "from": 1235,
                "to": 1239,
                "label": "EVAL with clause\npart(X380, .(X381, X382), .(X381, X383), X384) :- ','(gt(X380, X381), part(X380, X382, X383, X384)).\nand substitutionT216 -> T238,\nX380 -> T238,\nX381 -> T239,\nX382 -> T240,\nT217 -> .(T239, T240),\nX383 -> X385,\nX333 -> .(T239, X385),\nX334 -> X386,\nX384 -> X386"
            },
            {
                "from": 1235,
                "to": 1240,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1236,
                "to": 1268,
                "label": "PARALLEL"
            },
            {
                "from": 1236,
                "to": 1269,
                "label": "PARALLEL"
            },
            {
                "from": 1239,
                "to": 1242,
                "label": "SPLIT 1"
            },
            {
                "from": 1239,
                "to": 1243,
                "label": "SPLIT 2\nnew knowledge:\nT238 is ground\nT239 is ground"
            },
            {
                "from": 1242,
                "to": 1246,
                "label": "CASE"
            },
            {
                "from": 1243,
                "to": 1196,
                "label": "INSTANCE with matching:\nT216 -> T238\nT217 -> T240\nX333 -> X385\nX334 -> X386"
            },
            {
                "from": 1246,
                "to": 1256,
                "label": "PARALLEL"
            },
            {
                "from": 1246,
                "to": 1257,
                "label": "PARALLEL"
            },
            {
                "from": 1256,
                "to": 1260,
                "label": "EVAL with clause\ngt(s(X399), s(X400)) :- gt(X399, X400).\nand substitutionX399 -> T253,\nT238 -> s(T253),\nX400 -> T254,\nT239 -> s(T254)"
            },
            {
                "from": 1256,
                "to": 1261,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1257,
                "to": 1262,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT238 -> s(0),\nT239 -> 0"
            },
            {
                "from": 1257,
                "to": 1263,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1260,
                "to": 1242,
                "label": "INSTANCE with matching:\nT238 -> T253\nT239 -> T254"
            },
            {
                "from": 1262,
                "to": 1264,
                "label": "SUCCESS"
            },
            {
                "from": 1268,
                "to": 1270,
                "label": "EVAL with clause\npart(X438, .(X439, X440), X441, .(X439, X442)) :- ','(le(X438, X439), part(X438, X440, X441, X442)).\nand substitutionT216 -> T272,\nX438 -> T272,\nX439 -> T273,\nX440 -> T274,\nT217 -> .(T273, T274),\nX333 -> X443,\nX441 -> X443,\nX442 -> X444,\nX334 -> .(T273, X444)"
            },
            {
                "from": 1268,
                "to": 1271,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1269,
                "to": 1287,
                "label": "EVAL with clause\npart(X475, [], [], []).\nand substitutionT216 -> T301,\nX475 -> T301,\nT217 -> [],\nX333 -> [],\nX334 -> []"
            },
            {
                "from": 1269,
                "to": 1288,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1270,
                "to": 1272,
                "label": "SPLIT 1"
            },
            {
                "from": 1270,
                "to": 1273,
                "label": "SPLIT 2\nnew knowledge:\nT272 is ground\nT273 is ground"
            },
            {
                "from": 1272,
                "to": 1274,
                "label": "CASE"
            },
            {
                "from": 1273,
                "to": 1196,
                "label": "INSTANCE with matching:\nT216 -> T272\nT217 -> T274\nX333 -> X443\nX334 -> X444"
            },
            {
                "from": 1274,
                "to": 1275,
                "label": "PARALLEL"
            },
            {
                "from": 1274,
                "to": 1276,
                "label": "PARALLEL"
            },
            {
                "from": 1275,
                "to": 1277,
                "label": "EVAL with clause\nle(s(X457), s(X458)) :- le(X457, X458).\nand substitutionX457 -> T287,\nT272 -> s(T287),\nX458 -> T288,\nT273 -> s(T288)"
            },
            {
                "from": 1275,
                "to": 1278,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1276,
                "to": 1279,
                "label": "PARALLEL"
            },
            {
                "from": 1276,
                "to": 1280,
                "label": "PARALLEL"
            },
            {
                "from": 1277,
                "to": 1272,
                "label": "INSTANCE with matching:\nT272 -> T287\nT273 -> T288"
            },
            {
                "from": 1279,
                "to": 1281,
                "label": "EVAL with clause\nle(0, s(X465)).\nand substitutionT272 -> 0,\nX465 -> T295,\nT273 -> s(T295)"
            },
            {
                "from": 1279,
                "to": 1282,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1280,
                "to": 1284,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT272 -> 0,\nT273 -> 0"
            },
            {
                "from": 1280,
                "to": 1285,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1281,
                "to": 1283,
                "label": "SUCCESS"
            },
            {
                "from": 1284,
                "to": 1286,
                "label": "SUCCESS"
            },
            {
                "from": 1287,
                "to": 1289,
                "label": "SUCCESS"
            },
            {
                "from": 1290,
                "to": 1184,
                "label": "INSTANCE with matching:\nT23 -> T221\nX26 -> X335"
            },
            {
                "from": 1291,
                "to": 1292,
                "label": "SPLIT 1"
            },
            {
                "from": 1291,
                "to": 1293,
                "label": "SPLIT 2\nnew knowledge:\nT222 is ground\nT309 is ground\nreplacements:X336 -> T309"
            },
            {
                "from": 1292,
                "to": 1184,
                "label": "INSTANCE with matching:\nT23 -> T222\nX26 -> X336"
            },
            {
                "from": 1293,
                "to": 1294,
                "label": "CASE"
            },
            {
                "from": 1294,
                "to": 1295,
                "label": "PARALLEL"
            },
            {
                "from": 1294,
                "to": 1296,
                "label": "PARALLEL"
            },
            {
                "from": 1295,
                "to": 1297,
                "label": "EVAL with clause\napp(.(X533, X534), X535, .(X533, X536)) :- app(X534, X535, X536).\nand substitutionX533 -> T332,\nX534 -> T333,\nT304 -> .(T332, T333),\nT216 -> T334,\nT309 -> T335,\nX535 -> .(T334, T335),\nX536 -> X537,\nX337 -> .(T332, X537)"
            },
            {
                "from": 1295,
                "to": 1300,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1296,
                "to": 1305,
                "label": "EVAL with clause\napp([], X545, X545).\nand substitutionT304 -> [],\nT216 -> T344,\nT309 -> T345,\nX545 -> .(T344, T345),\nX337 -> .(T344, T345)"
            },
            {
                "from": 1296,
                "to": 1306,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1297,
                "to": 1293,
                "label": "INSTANCE with matching:\nT304 -> T333\nT216 -> T334\nT309 -> T335\nX337 -> X537"
            },
            {
                "from": 1305,
                "to": 1307,
                "label": "SUCCESS"
            },
            {
                "from": 1308,
                "to": 1310,
                "label": "SUCCESS"
            },
            {
                "from": 1313,
                "to": 1317,
                "label": "CASE"
            },
            {
                "from": 1314,
                "to": 1358,
                "label": "CASE"
            },
            {
                "from": 1317,
                "to": 1320,
                "label": "PARALLEL"
            },
            {
                "from": 1317,
                "to": 1321,
                "label": "PARALLEL"
            },
            {
                "from": 1320,
                "to": 1322,
                "label": "EVAL with clause\nqs(.(X586, X587), X588) :- ','(part(X586, X587, X589, X590), ','(qs(X589, X591), ','(qs(X590, X592), app(X591, .(X586, X592), X588)))).\nand substitutionX586 -> T362,\nX587 -> T363,\nT24 -> .(T362, T363),\nX27 -> X593,\nX588 -> X593,\nT360 -> T362,\nT361 -> T363"
            },
            {
                "from": 1320,
                "to": 1323,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1321,
                "to": 1353,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT24 -> [],\nX27 -> []"
            },
            {
                "from": 1321,
                "to": 1354,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1322,
                "to": 1324,
                "label": "SPLIT 1"
            },
            {
                "from": 1322,
                "to": 1325,
                "label": "SPLIT 2\nnew knowledge:\nT367 is ground\nreplacements:X589 -> T367,\nX590 -> T368,\nT362 -> T369"
            },
            {
                "from": 1324,
                "to": 371,
                "label": "INSTANCE with matching:\nT18 -> T362\nT19 -> T363\nX24 -> X589\nX25 -> X590"
            },
            {
                "from": 1325,
                "to": 1326,
                "label": "SPLIT 1"
            },
            {
                "from": 1325,
                "to": 1327,
                "label": "SPLIT 2\nnew knowledge:\nT367 is ground\nT375 is ground\nreplacements:X591 -> T375"
            },
            {
                "from": 1326,
                "to": 1184,
                "label": "INSTANCE with matching:\nT23 -> T367\nX26 -> X591"
            },
            {
                "from": 1327,
                "to": 1333,
                "label": "SPLIT 1"
            },
            {
                "from": 1327,
                "to": 1334,
                "label": "SPLIT 2\nreplacements:X592 -> T380,\nT369 -> T381"
            },
            {
                "from": 1333,
                "to": 1313,
                "label": "INSTANCE with matching:\nT24 -> T368\nX27 -> X592"
            },
            {
                "from": 1334,
                "to": 1335,
                "label": "CASE"
            },
            {
                "from": 1335,
                "to": 1336,
                "label": "PARALLEL"
            },
            {
                "from": 1335,
                "to": 1337,
                "label": "PARALLEL"
            },
            {
                "from": 1336,
                "to": 1343,
                "label": "EVAL with clause\napp(.(X665, X666), X667, .(X665, X668)) :- app(X666, X667, X668).\nand substitutionX665 -> T404,\nX666 -> T405,\nT375 -> .(T404, T405),\nT381 -> T408,\nT380 -> T409,\nX667 -> .(T408, T409),\nX668 -> X669,\nX593 -> .(T404, X669),\nT406 -> T408,\nT407 -> T409"
            },
            {
                "from": 1336,
                "to": 1344,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1337,
                "to": 1348,
                "label": "EVAL with clause\napp([], X677, X677).\nand substitutionT375 -> [],\nT381 -> T418,\nT380 -> T419,\nX677 -> .(T418, T419),\nX593 -> .(T418, T419)"
            },
            {
                "from": 1337,
                "to": 1349,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1343,
                "to": 1334,
                "label": "INSTANCE with matching:\nT375 -> T405\nT381 -> T408\nT380 -> T409\nX593 -> X669"
            },
            {
                "from": 1348,
                "to": 1350,
                "label": "SUCCESS"
            },
            {
                "from": 1353,
                "to": 1355,
                "label": "SUCCESS"
            },
            {
                "from": 1358,
                "to": 1361,
                "label": "PARALLEL"
            },
            {
                "from": 1358,
                "to": 1362,
                "label": "PARALLEL"
            },
            {
                "from": 1361,
                "to": 1367,
                "label": "EVAL with clause\napp(.(X698, X699), X700, .(X698, X701)) :- app(X699, X700, X701).\nand substitutionX698 -> T445,\nX699 -> T446,\nT205 -> .(T445, T446),\nT349 -> T450,\nT348 -> T451,\nX700 -> .(T450, T451),\nX701 -> T449,\nT17 -> .(T445, T449),\nT447 -> T450,\nT448 -> T451"
            },
            {
                "from": 1361,
                "to": 1368,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1362,
                "to": 1371,
                "label": "EVAL with clause\napp([], X708, X708).\nand substitutionT205 -> [],\nT349 -> T461,\nT348 -> T462,\nX708 -> .(T461, T462),\nT17 -> .(T461, T462)"
            },
            {
                "from": 1362,
                "to": 1372,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1367,
                "to": 1314,
                "label": "INSTANCE with matching:\nT205 -> T446\nT349 -> T450\nT348 -> T451\nT17 -> T449"
            },
            {
                "from": 1371,
                "to": 1373,
                "label": "SUCCESS"
            },
            {
                "from": 1374,
                "to": 1376,
                "label": "SUCCESS"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(214)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.

----------------------------------------

(215) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(216)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1_IN(T17) -> U1^1(f33_in(T17), T17)
   F1_IN(T17) -> F33_IN(T17)
   F630_IN -> U2^1(f630_in)
   F630_IN -> F630_IN
   F632_IN(T82) -> U3^1(f847_in(T82), T82)
   F632_IN(T82) -> F847_IN(T82)
   F632_IN(T120) -> U4^1(f1002_in(T120), T120)
   F632_IN(T120) -> F1002_IN(T120)
   F852_IN(s(T100)) -> U5^1(f852_in(T100), s(T100))
   F852_IN(s(T100)) -> F852_IN(T100)
   F1004_IN(s(T138)) -> U6^1(f1004_in(T138), s(T138))
   F1004_IN(s(T138)) -> F1004_IN(T138)
   F1166_IN -> U7^1(f1166_in)
   F1166_IN -> F1166_IN
   F1196_IN(T238, .(T239, T240)) -> U8^1(f1239_in(T238, T239, T240), T238, .(T239, T240))
   F1196_IN(T238, .(T239, T240)) -> F1239_IN(T238, T239, T240)
   F1196_IN(T272, .(T273, T274)) -> U9^1(f1270_in(T272, T273, T274), T272, .(T273, T274))
   F1196_IN(T272, .(T273, T274)) -> F1270_IN(T272, T273, T274)
   F1242_IN(s(T253), s(T254)) -> U10^1(f1242_in(T253, T254), s(T253), s(T254))
   F1242_IN(s(T253), s(T254)) -> F1242_IN(T253, T254)
   F1272_IN(s(T287), s(T288)) -> U11^1(f1272_in(T287, T288), s(T287), s(T288))
   F1272_IN(s(T287), s(T288)) -> F1272_IN(T287, T288)
   F1184_IN(.(T216, T217)) -> U12^1(f1189_in(T216, T217), .(T216, T217))
   F1184_IN(.(T216, T217)) -> F1189_IN(T216, T217)
   F1293_IN(.(T332, T333), T334, T335) -> U13^1(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   F1293_IN(.(T332, T333), T334, T335) -> F1293_IN(T333, T334, T335)
   F371_IN -> U14^1(f626_in)
   F371_IN -> F626_IN
   F371_IN -> U15^1(f1164_in)
   F371_IN -> F1164_IN
   F1313_IN -> U16^1(f1322_in)
   F1313_IN -> F1322_IN
   F1334_IN(.(T404, T405)) -> U17^1(f1334_in(T405), .(T404, T405))
   F1334_IN(.(T404, T405)) -> F1334_IN(T405)
   F1314_IN(.(T445, T446), .(T445, T449)) -> U18^1(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   F1314_IN(.(T445, T446), .(T445, T449)) -> F1314_IN(T446, T449)
   F33_IN(T17) -> U19^1(f371_in, T17)
   F33_IN(T17) -> F371_IN
   U19^1(f371_out1(T23), T17) -> U20^1(f373_in(T23, T17), T17, T23)
   U19^1(f371_out1(T23), T17) -> F373_IN(T23, T17)
   F373_IN(T23, T17) -> U21^1(f1184_in(T23), T23, T17)
   F373_IN(T23, T17) -> F1184_IN(T23)
   U21^1(f1184_out1(T205), T23, T17) -> U22^1(f1185_in(T205, T17), T23, T17, T205)
   U21^1(f1184_out1(T205), T23, T17) -> F1185_IN(T205, T17)
   F626_IN -> U23^1(f630_in)
   F626_IN -> F630_IN
   U23^1(f630_out1(T49, T45)) -> U24^1(f632_in(T49), T49, T45)
   U23^1(f630_out1(T49, T45)) -> F632_IN(T49)
   F847_IN(T82) -> U25^1(f852_in(T82), T82)
   F847_IN(T82) -> F852_IN(T82)
   U25^1(f852_out1(T85), T82) -> U26^1(f632_in(T82), T82, T85)
   U25^1(f852_out1(T85), T82) -> F632_IN(T82)
   F1002_IN(T120) -> U27^1(f1004_in(T120), T120)
   F1002_IN(T120) -> F1004_IN(T120)
   U27^1(f1004_out1, T120) -> U28^1(f632_in(T120), T120)
   U27^1(f1004_out1, T120) -> F632_IN(T120)
   F1164_IN -> U29^1(f1166_in)
   F1164_IN -> F1166_IN
   U29^1(f1166_out1(T174)) -> U30^1(f632_in(T174), T174)
   U29^1(f1166_out1(T174)) -> F632_IN(T174)
   F1185_IN(T205, T17) -> U31^1(f1313_in, T205, T17)
   F1185_IN(T205, T17) -> F1313_IN
   U31^1(f1313_out1, T205, T17) -> U32^1(f1314_in(T205, T17), T205, T17)
   U31^1(f1313_out1, T205, T17) -> F1314_IN(T205, T17)
   F1189_IN(T216, T217) -> U33^1(f1196_in(T216, T217), T216, T217)
   F1189_IN(T216, T217) -> F1196_IN(T216, T217)
   U33^1(f1196_out1(T221, T222), T216, T217) -> U34^1(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U33^1(f1196_out1(T221, T222), T216, T217) -> F1197_IN(T221, T222, T216)
   F1197_IN(T221, T222, T216) -> U35^1(f1184_in(T221), T221, T222, T216)
   F1197_IN(T221, T222, T216) -> F1184_IN(T221)
   U35^1(f1184_out1(T304), T221, T222, T216) -> U36^1(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U35^1(f1184_out1(T304), T221, T222, T216) -> F1291_IN(T222, T304, T216)
   F1239_IN(T238, T239, T240) -> U37^1(f1242_in(T238, T239), T238, T239, T240)
   F1239_IN(T238, T239, T240) -> F1242_IN(T238, T239)
   U37^1(f1242_out1, T238, T239, T240) -> U38^1(f1196_in(T238, T240), T238, T239, T240)
   U37^1(f1242_out1, T238, T239, T240) -> F1196_IN(T238, T240)
   F1270_IN(T272, T273, T274) -> U39^1(f1272_in(T272, T273), T272, T273, T274)
   F1270_IN(T272, T273, T274) -> F1272_IN(T272, T273)
   U39^1(f1272_out1, T272, T273, T274) -> U40^1(f1196_in(T272, T274), T272, T273, T274)
   U39^1(f1272_out1, T272, T273, T274) -> F1196_IN(T272, T274)
   F1291_IN(T222, T304, T216) -> U41^1(f1184_in(T222), T222, T304, T216)
   F1291_IN(T222, T304, T216) -> F1184_IN(T222)
   U41^1(f1184_out1(T309), T222, T304, T216) -> U42^1(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U41^1(f1184_out1(T309), T222, T304, T216) -> F1293_IN(T304, T216, T309)
   F1322_IN -> U43^1(f371_in)
   F1322_IN -> F371_IN
   U43^1(f371_out1(T367)) -> U44^1(f1325_in(T367), T367)
   U43^1(f371_out1(T367)) -> F1325_IN(T367)
   F1325_IN(T367) -> U45^1(f1184_in(T367), T367)
   F1325_IN(T367) -> F1184_IN(T367)
   U45^1(f1184_out1(T375), T367) -> U46^1(f1327_in(T375), T367, T375)
   U45^1(f1184_out1(T375), T367) -> F1327_IN(T375)
   F1327_IN(T375) -> U47^1(f1313_in, T375)
   F1327_IN(T375) -> F1313_IN
   U47^1(f1313_out1, T375) -> U48^1(f1334_in(T375), T375)
   U47^1(f1313_out1, T375) -> F1334_IN(T375)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(217) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 13 SCCs with 62 less nodes.
----------------------------------------

(218)
Complex Obligation (AND)

----------------------------------------

(219)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1314_IN(.(T445, T446), .(T445, T449)) -> F1314_IN(T446, T449)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(220) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(221)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1314_IN(.(T445, T446), .(T445, T449)) -> F1314_IN(T446, T449)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(222) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1314_IN(.(T445, T446), .(T445, T449)) -> F1314_IN(T446, T449)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(223)
YES

----------------------------------------

(224)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1334_IN(.(T404, T405)) -> F1334_IN(T405)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(225) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(226)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1334_IN(.(T404, T405)) -> F1334_IN(T405)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(227) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1334_IN(.(T404, T405)) -> F1334_IN(T405)
The graph contains the following edges 1 > 1


----------------------------------------

(228)
YES

----------------------------------------

(229)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1293_IN(.(T332, T333), T334, T335) -> F1293_IN(T333, T334, T335)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(230) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(231)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1293_IN(.(T332, T333), T334, T335) -> F1293_IN(T333, T334, T335)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(232) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1293_IN(.(T332, T333), T334, T335) -> F1293_IN(T333, T334, T335)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


----------------------------------------

(233)
YES

----------------------------------------

(234)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1272_IN(s(T287), s(T288)) -> F1272_IN(T287, T288)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(235) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(236)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1272_IN(s(T287), s(T288)) -> F1272_IN(T287, T288)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(237) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1272_IN(s(T287), s(T288)) -> F1272_IN(T287, T288)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(238)
YES

----------------------------------------

(239)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1242_IN(s(T253), s(T254)) -> F1242_IN(T253, T254)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(240) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(241)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1242_IN(s(T253), s(T254)) -> F1242_IN(T253, T254)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(242) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1242_IN(s(T253), s(T254)) -> F1242_IN(T253, T254)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(243)
YES

----------------------------------------

(244)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1196_IN(T238, .(T239, T240)) -> F1239_IN(T238, T239, T240)
   F1239_IN(T238, T239, T240) -> U37^1(f1242_in(T238, T239), T238, T239, T240)
   U37^1(f1242_out1, T238, T239, T240) -> F1196_IN(T238, T240)
   F1196_IN(T272, .(T273, T274)) -> F1270_IN(T272, T273, T274)
   F1270_IN(T272, T273, T274) -> U39^1(f1272_in(T272, T273), T272, T273, T274)
   U39^1(f1272_out1, T272, T273, T274) -> F1196_IN(T272, T274)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(245) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1239_IN(T238, T239, T240) -> U37^1(f1242_in(T238, T239), T238, T239, T240)
The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4


*U37^1(f1242_out1, T238, T239, T240) -> F1196_IN(T238, T240)
The graph contains the following edges 2 >= 1, 4 >= 2


*U39^1(f1272_out1, T272, T273, T274) -> F1196_IN(T272, T274)
The graph contains the following edges 2 >= 1, 4 >= 2


*F1196_IN(T238, .(T239, T240)) -> F1239_IN(T238, T239, T240)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*F1196_IN(T272, .(T273, T274)) -> F1270_IN(T272, T273, T274)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*F1270_IN(T272, T273, T274) -> U39^1(f1272_in(T272, T273), T272, T273, T274)
The graph contains the following edges 1 >= 2, 2 >= 3, 3 >= 4


----------------------------------------

(246)
YES

----------------------------------------

(247)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1189_IN(T216, T217) -> U33^1(f1196_in(T216, T217), T216, T217)
   U33^1(f1196_out1(T221, T222), T216, T217) -> F1197_IN(T221, T222, T216)
   F1197_IN(T221, T222, T216) -> U35^1(f1184_in(T221), T221, T222, T216)
   U35^1(f1184_out1(T304), T221, T222, T216) -> F1291_IN(T222, T304, T216)
   F1291_IN(T222, T304, T216) -> F1184_IN(T222)
   F1184_IN(.(T216, T217)) -> F1189_IN(T216, T217)
   F1197_IN(T221, T222, T216) -> F1184_IN(T221)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(248) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   U35^1(f1184_out1(T304), T221, T222, T216) -> F1291_IN(T222, T304, T216)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U33^1_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( U33_3(x_1, ..., x_3) ) = 0
POL( U35^1_4(x_1, ..., x_4) ) = x_1 + 2x_3
POL( f1196_in_2(x_1, x_2) ) = x_2
POL( ._2(x_1, x_2) ) = x_2 + 1
POL( U8_3(x_1, ..., x_3) ) = x_1
POL( f1239_in_3(x_1, ..., x_3) ) = x_3 + 1
POL( U9_3(x_1, ..., x_3) ) = x_1
POL( f1270_in_3(x_1, ..., x_3) ) = x_3 + 1
POL( [] ) = 1
POL( f1196_out1_2(x_1, x_2) ) = max{0, x_1 + x_2 - 1}
POL( f1184_in_1(x_1) ) = 2x_1
POL( U12_2(x_1, x_2) ) = 2
POL( f1189_in_2(x_1, x_2) ) = 2
POL( f1184_out1_1(x_1) ) = 1
POL( U34_5(x_1, ..., x_5) ) = 0
POL( f1197_in_3(x_1, ..., x_3) ) = 2
POL( f1197_out1_3(x_1, ..., x_3) ) = 0
POL( f1189_out1_5(x_1, ..., x_5) ) = 0
POL( U35_4(x_1, ..., x_4) ) = 0
POL( U36_5(x_1, ..., x_5) ) = max{0, -2}
POL( f1291_in_3(x_1, ..., x_3) ) = 0
POL( f1291_out1_2(x_1, x_2) ) = 0
POL( U41_4(x_1, ..., x_4) ) = max{0, -2}
POL( U42_5(x_1, ..., x_5) ) = 0
POL( f1293_in_3(x_1, ..., x_3) ) = 2x_3
POL( U37_4(x_1, ..., x_4) ) = x_4 + 1
POL( f1242_in_2(x_1, x_2) ) = 0
POL( s_1(x_1) ) = 0
POL( U10_3(x_1, ..., x_3) ) = 2
POL( 0 ) = 0
POL( f1242_out1 ) = 0
POL( U38_4(x_1, ..., x_4) ) = x_1 + 1
POL( f1239_out1_2(x_1, x_2) ) = x_1 + x_2
POL( f1270_out1_2(x_1, x_2) ) = x_1 + x_2
POL( U39_4(x_1, ..., x_4) ) = x_4 + 1
POL( f1272_in_2(x_1, x_2) ) = 0
POL( U11_3(x_1, ..., x_3) ) = max{0, -2}
POL( f1272_out1 ) = 0
POL( U40_4(x_1, ..., x_4) ) = x_1 + 1
POL( U13_4(x_1, ..., x_4) ) = 2x_2
POL( f1293_out1_1(x_1) ) = 2
POL( F1189_IN_2(x_1, x_2) ) = 2x_2 + 2
POL( F1197_IN_3(x_1, ..., x_3) ) = 2x_1 + 2x_2
POL( F1291_IN_3(x_1, ..., x_3) ) = 2x_1
POL( F1184_IN_1(x_1) ) = 2x_1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   f1184_in([]) -> f1184_out1([])
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)


----------------------------------------

(249)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1189_IN(T216, T217) -> U33^1(f1196_in(T216, T217), T216, T217)
   U33^1(f1196_out1(T221, T222), T216, T217) -> F1197_IN(T221, T222, T216)
   F1197_IN(T221, T222, T216) -> U35^1(f1184_in(T221), T221, T222, T216)
   F1291_IN(T222, T304, T216) -> F1184_IN(T222)
   F1184_IN(.(T216, T217)) -> F1189_IN(T216, T217)
   F1197_IN(T221, T222, T216) -> F1184_IN(T221)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(250) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(251)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U33^1(f1196_out1(T221, T222), T216, T217) -> F1197_IN(T221, T222, T216)
   F1197_IN(T221, T222, T216) -> F1184_IN(T221)
   F1184_IN(.(T216, T217)) -> F1189_IN(T216, T217)
   F1189_IN(T216, T217) -> U33^1(f1196_in(T216, T217), T216, T217)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(252) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   F1184_IN(.(T216, T217)) -> F1189_IN(T216, T217)
   F1189_IN(T216, T217) -> U33^1(f1196_in(T216, T217), T216, T217)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U33^1_3(x_1, ..., x_3) ) = 2x_1
POL( f1196_in_2(x_1, x_2) ) = x_2
POL( ._2(x_1, x_2) ) = 2x_2 + 2
POL( U8_3(x_1, ..., x_3) ) = 2x_1
POL( f1239_in_3(x_1, ..., x_3) ) = x_3 + 1
POL( U9_3(x_1, ..., x_3) ) = x_1
POL( f1270_in_3(x_1, ..., x_3) ) = 2x_3
POL( [] ) = 0
POL( f1196_out1_2(x_1, x_2) ) = x_1
POL( U37_4(x_1, ..., x_4) ) = x_4 + 1
POL( f1242_in_2(x_1, x_2) ) = 0
POL( s_1(x_1) ) = 2x_1
POL( U10_3(x_1, ..., x_3) ) = 2
POL( 0 ) = 0
POL( f1242_out1 ) = 0
POL( U38_4(x_1, ..., x_4) ) = x_1 + 1
POL( f1239_out1_2(x_1, x_2) ) = x_1 + 1
POL( f1270_out1_2(x_1, x_2) ) = 2x_1
POL( U39_4(x_1, ..., x_4) ) = 2x_4
POL( f1272_in_2(x_1, x_2) ) = 0
POL( U11_3(x_1, ..., x_3) ) = max{0, 2x_3 - 2}
POL( f1272_out1 ) = 0
POL( U40_4(x_1, ..., x_4) ) = 2x_1
POL( F1197_IN_3(x_1, ..., x_3) ) = 2x_1
POL( F1184_IN_1(x_1) ) = x_1
POL( F1189_IN_2(x_1, x_2) ) = 2x_2 + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)


----------------------------------------

(253)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U33^1(f1196_out1(T221, T222), T216, T217) -> F1197_IN(T221, T222, T216)
   F1197_IN(T221, T222, T216) -> F1184_IN(T221)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(254) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(255)
TRUE

----------------------------------------

(256)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1166_IN -> F1166_IN

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(257) NonLoopProof (COMPLETE)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0, 
?' = [ ], and ?' = [ ] on the rule
F1166_IN[ ]^n[ ] -> F1166_IN[ ]^n[ ]
This rule is correct for the QDP as the following derivation shows:

F1166_IN[ ]^n[ ] -> F1166_IN[ ]^n[ ]
    by Rule from TRS P
----------------------------------------

(258)
NO

----------------------------------------

(259)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1004_IN(s(T138)) -> F1004_IN(T138)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(260) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(261)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1004_IN(s(T138)) -> F1004_IN(T138)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(262) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F1004_IN(s(T138)) -> F1004_IN(T138)
The graph contains the following edges 1 > 1


----------------------------------------

(263)
YES

----------------------------------------

(264)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F852_IN(s(T100)) -> F852_IN(T100)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(265) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(266)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F852_IN(s(T100)) -> F852_IN(T100)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(267) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*F852_IN(s(T100)) -> F852_IN(T100)
The graph contains the following edges 1 > 1


----------------------------------------

(268)
YES

----------------------------------------

(269)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F632_IN(T82) -> F847_IN(T82)
   F847_IN(T82) -> U25^1(f852_in(T82), T82)
   U25^1(f852_out1(T85), T82) -> F632_IN(T82)
   F632_IN(T120) -> F1002_IN(T120)
   F1002_IN(T120) -> U27^1(f1004_in(T120), T120)
   U27^1(f1004_out1, T120) -> F632_IN(T120)

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(270) NonLoopProof (COMPLETE)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0, 
?' = [ ], and ?' = [x0 / 0] on the rule
U25^1(f852_out1(0), s(0))[ ]^n[ ] -> U25^1(f852_out1(0), s(0))[ ]^n[x0 / 0]
This rule is correct for the QDP as the following derivation shows:

U25^1(f852_out1(0), s(0))[ ]^n[ ] -> U25^1(f852_out1(0), s(0))[ ]^n[x0 / 0]
    by Equivalency by Simplifying Mu with mu1: [x0 / 0] mu2: [ ]
    intermediate steps: Instantiate mu
    U25^1(f852_out1(x0), s(0))[ ]^n[ ] -> U25^1(f852_out1(0), s(0))[ ]^n[ ]
        by Narrowing at position: []
        intermediate steps: Instantiation - Instantiation
        U25^1(f852_out1(T85), T82)[ ]^n[ ] -> F632_IN(T82)[ ]^n[ ]
            by Rule from TRS P

        F632_IN(s(0))[ ]^n[ ] -> U25^1(f852_out1(0), s(0))[ ]^n[ ]
            by Narrowing at position: [0]
            intermediate steps: Instantiation
            F632_IN(x0)[ ]^n[ ] -> U25^1(f852_in(x0), x0)[ ]^n[ ]
                by Narrowing at position: []
                intermediate steps: Instantiation
                F632_IN(T82)[ ]^n[ ] -> F847_IN(T82)[ ]^n[ ]
                    by Rule from TRS P

                intermediate steps: Instantiation - Instantiation
                F847_IN(T82)[ ]^n[ ] -> U25^1(f852_in(T82), T82)[ ]^n[ ]
                    by Rule from TRS P

            f852_in(s(0))[ ]^n[ ] -> f852_out1(0)[ ]^n[ ]
                by Rule from TRS R
----------------------------------------

(271)
NO

----------------------------------------

(272)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F630_IN -> F630_IN

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(273) NonLoopProof (COMPLETE)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0, 
?' = [ ], and ?' = [ ] on the rule
F630_IN[ ]^n[ ] -> F630_IN[ ]^n[ ]
This rule is correct for the QDP as the following derivation shows:

F630_IN[ ]^n[ ] -> F630_IN[ ]^n[ ]
    by Rule from TRS P
----------------------------------------

(274)
NO

----------------------------------------

(275)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   F1322_IN -> U43^1(f371_in)
   U43^1(f371_out1(T367)) -> F1325_IN(T367)
   F1325_IN(T367) -> U45^1(f1184_in(T367), T367)
   U45^1(f1184_out1(T375), T367) -> F1327_IN(T375)
   F1327_IN(T375) -> F1313_IN
   F1313_IN -> F1322_IN

The TRS R consists of the following rules:

   f1_in(T17) -> U1(f33_in(T17), T17)
   U1(f33_out1(X24, X26, T18, X27), T17) -> f1_out1
   f1_in([]) -> f1_out1
   f630_in -> U2(f630_in)
   U2(f630_out1(T63, T64)) -> f630_out1(s(T63), s(T64))
   f630_in -> f630_out1(s(0), 0)
   f632_in(T82) -> U3(f847_in(T82), T82)
   U3(f847_out1(T85, X133), T82) -> f632_out1(.(T85, X133))
   f632_in(T120) -> U4(f1002_in(T120), T120)
   U4(f1002_out1(X191), T120) -> f632_out1(X191)
   f632_in(T153) -> f632_out1([])
   f852_in(s(T100)) -> U5(f852_in(T100), s(T100))
   U5(f852_out1(T102), s(T100)) -> f852_out1(s(T102))
   f852_in(s(0)) -> f852_out1(0)
   f1004_in(s(T138)) -> U6(f1004_in(T138), s(T138))
   U6(f1004_out1, s(T138)) -> f1004_out1
   f1004_in(0) -> f1004_out1
   f1166_in -> U7(f1166_in)
   U7(f1166_out1(T188)) -> f1166_out1(s(T188))
   f1166_in -> f1166_out1(0)
   f1196_in(T238, .(T239, T240)) -> U8(f1239_in(T238, T239, T240), T238, .(T239, T240))
   U8(f1239_out1(X385, X386), T238, .(T239, T240)) -> f1196_out1(.(T239, X385), X386)
   f1196_in(T272, .(T273, T274)) -> U9(f1270_in(T272, T273, T274), T272, .(T273, T274))
   U9(f1270_out1(X443, X444), T272, .(T273, T274)) -> f1196_out1(X443, .(T273, X444))
   f1196_in(T301, []) -> f1196_out1([], [])
   f1242_in(s(T253), s(T254)) -> U10(f1242_in(T253, T254), s(T253), s(T254))
   U10(f1242_out1, s(T253), s(T254)) -> f1242_out1
   f1242_in(s(0), 0) -> f1242_out1
   f1272_in(s(T287), s(T288)) -> U11(f1272_in(T287, T288), s(T287), s(T288))
   U11(f1272_out1, s(T287), s(T288)) -> f1272_out1
   f1272_in(0, s(T295)) -> f1272_out1
   f1272_in(0, 0) -> f1272_out1
   f1184_in(.(T216, T217)) -> U12(f1189_in(T216, T217), .(T216, T217))
   U12(f1189_out1(X333, X334, X335, X336, X337), .(T216, T217)) -> f1184_out1(X337)
   f1184_in([]) -> f1184_out1([])
   f1293_in(.(T332, T333), T334, T335) -> U13(f1293_in(T333, T334, T335), .(T332, T333), T334, T335)
   U13(f1293_out1(X537), .(T332, T333), T334, T335) -> f1293_out1(.(T332, X537))
   f1293_in([], T344, T345) -> f1293_out1(.(T344, T345))
   f371_in -> U14(f626_in)
   U14(f626_out1(T44, T45, X75)) -> f371_out1(.(T45, X75))
   f371_in -> U15(f1164_in)
   U15(f1164_out1(T169, X257)) -> f371_out1(X257)
   f371_in -> f371_out1([])
   f1313_in -> U16(f1322_in)
   U16(f1322_out1(X589, X591)) -> f1313_out1
   f1313_in -> f1313_out1
   f1334_in(.(T404, T405)) -> U17(f1334_in(T405), .(T404, T405))
   U17(f1334_out1, .(T404, T405)) -> f1334_out1
   f1334_in([]) -> f1334_out1
   f1314_in(.(T445, T446), .(T445, T449)) -> U18(f1314_in(T446, T449), .(T445, T446), .(T445, T449))
   U18(f1314_out1(T450, T451), .(T445, T446), .(T445, T449)) -> f1314_out1(T450, T451)
   f1314_in([], .(T461, T462)) -> f1314_out1(T461, T462)
   f33_in(T17) -> U19(f371_in, T17)
   U19(f371_out1(T23), T17) -> U20(f373_in(T23, T17), T17, T23)
   U20(f373_out1(X26, T25, X27), T17, T23) -> f33_out1(T23, X26, T25, X27)
   f373_in(T23, T17) -> U21(f1184_in(T23), T23, T17)
   U21(f1184_out1(T205), T23, T17) -> U22(f1185_in(T205, T17), T23, T17, T205)
   U22(f1185_out1(T25, X27), T23, T17, T205) -> f373_out1(T205, T25, X27)
   f626_in -> U23(f630_in)
   U23(f630_out1(T49, T45)) -> U24(f632_in(T49), T49, T45)
   U24(f632_out1(X75), T49, T45) -> f626_out1(T49, T45, X75)
   f847_in(T82) -> U25(f852_in(T82), T82)
   U25(f852_out1(T85), T82) -> U26(f632_in(T82), T82, T85)
   U26(f632_out1(X133), T82, T85) -> f847_out1(T85, X133)
   f1002_in(T120) -> U27(f1004_in(T120), T120)
   U27(f1004_out1, T120) -> U28(f632_in(T120), T120)
   U28(f632_out1(X191), T120) -> f1002_out1(X191)
   f1164_in -> U29(f1166_in)
   U29(f1166_out1(T174)) -> U30(f632_in(T174), T174)
   U30(f632_out1(X257), T174) -> f1164_out1(T174, X257)
   f1185_in(T205, T17) -> U31(f1313_in, T205, T17)
   U31(f1313_out1, T205, T17) -> U32(f1314_in(T205, T17), T205, T17)
   U32(f1314_out1(T349, T348), T205, T17) -> f1185_out1(T349, T348)
   f1189_in(T216, T217) -> U33(f1196_in(T216, T217), T216, T217)
   U33(f1196_out1(T221, T222), T216, T217) -> U34(f1197_in(T221, T222, T216), T216, T217, T221, T222)
   U34(f1197_out1(X335, X336, X337), T216, T217, T221, T222) -> f1189_out1(T221, T222, X335, X336, X337)
   f1197_in(T221, T222, T216) -> U35(f1184_in(T221), T221, T222, T216)
   U35(f1184_out1(T304), T221, T222, T216) -> U36(f1291_in(T222, T304, T216), T221, T222, T216, T304)
   U36(f1291_out1(X336, X337), T221, T222, T216, T304) -> f1197_out1(T304, X336, X337)
   f1239_in(T238, T239, T240) -> U37(f1242_in(T238, T239), T238, T239, T240)
   U37(f1242_out1, T238, T239, T240) -> U38(f1196_in(T238, T240), T238, T239, T240)
   U38(f1196_out1(X385, X386), T238, T239, T240) -> f1239_out1(X385, X386)
   f1270_in(T272, T273, T274) -> U39(f1272_in(T272, T273), T272, T273, T274)
   U39(f1272_out1, T272, T273, T274) -> U40(f1196_in(T272, T274), T272, T273, T274)
   U40(f1196_out1(X443, X444), T272, T273, T274) -> f1270_out1(X443, X444)
   f1291_in(T222, T304, T216) -> U41(f1184_in(T222), T222, T304, T216)
   U41(f1184_out1(T309), T222, T304, T216) -> U42(f1293_in(T304, T216, T309), T222, T304, T216, T309)
   U42(f1293_out1(X337), T222, T304, T216, T309) -> f1291_out1(T309, X337)
   f1322_in -> U43(f371_in)
   U43(f371_out1(T367)) -> U44(f1325_in(T367), T367)
   U44(f1325_out1(X591), T367) -> f1322_out1(T367, X591)
   f1325_in(T367) -> U45(f1184_in(T367), T367)
   U45(f1184_out1(T375), T367) -> U46(f1327_in(T375), T367, T375)
   U46(f1327_out1, T367, T375) -> f1325_out1(T375)
   f1327_in(T375) -> U47(f1313_in, T375)
   U47(f1313_out1, T375) -> U48(f1334_in(T375), T375)
   U48(f1334_out1, T375) -> f1327_out1

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(276) NonLoopProof (COMPLETE)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 0, 
?' = [ ], and ?' = [ ] on the rule
F1313_IN[ ]^n[ ] -> F1313_IN[ ]^n[ ]
This rule is correct for the QDP as the following derivation shows:

F1313_IN[ ]^n[ ] -> F1313_IN[ ]^n[ ]
    by Narrowing at position: []
    F1313_IN[ ]^n[ ] -> F1322_IN[ ]^n[ ]
        by Rule from TRS P

    F1322_IN[ ]^n[ ] -> F1313_IN[ ]^n[ ]
        by Narrowing at position: []
        F1322_IN[ ]^n[ ] -> F1327_IN([])[ ]^n[ ]
            by Narrowing at position: []
            F1322_IN[ ]^n[ ] -> U45^1(f1184_out1([]), [])[ ]^n[ ]
                by Narrowing at position: [0]
                F1322_IN[ ]^n[ ] -> U45^1(f1184_in([]), [])[ ]^n[ ]
                    by Narrowing at position: []
                    F1322_IN[ ]^n[ ] -> F1325_IN([])[ ]^n[ ]
                        by Narrowing at position: []
                        F1322_IN[ ]^n[ ] -> U43^1(f371_out1([]))[ ]^n[ ]
                            by Narrowing at position: [0]
                            F1322_IN[ ]^n[ ] -> U43^1(f371_in)[ ]^n[ ]
                                by Rule from TRS P

                            f371_in[ ]^n[ ] -> f371_out1([])[ ]^n[ ]
                                by Rule from TRS R

                        intermediate steps: Instantiation - Instantiation
                        U43^1(f371_out1(T367))[ ]^n[ ] -> F1325_IN(T367)[ ]^n[ ]
                            by Rule from TRS P

                    intermediate steps: Instantiation - Instantiation
                    F1325_IN(T367)[ ]^n[ ] -> U45^1(f1184_in(T367), T367)[ ]^n[ ]
                        by Rule from TRS P

                f1184_in([])[ ]^n[ ] -> f1184_out1([])[ ]^n[ ]
                    by Rule from TRS R

            intermediate steps: Instantiation - Instantiation - Instantiation - Instantiation
            U45^1(f1184_out1(T375), T367)[ ]^n[ ] -> F1327_IN(T375)[ ]^n[ ]
                by Rule from TRS P

        intermediate steps: Instantiation - Instantiation
        F1327_IN(T375)[ ]^n[ ] -> F1313_IN[ ]^n[ ]
            by Rule from TRS P
----------------------------------------

(277)
NO

----------------------------------------

(278) PrologToDTProblemTransformerProof (SOUND)
Built DT problem from termination graph DT10.

{
    "root": 5,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(qs (. X Xs) Ys)",
                "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))"
            ],
            [
                "(qs ([]) ([]))",
                null
            ],
            [
                "(part X (. Y Xs) (. Y Ls) Bs)",
                "(',' (gt X Y) (part X Xs Ls Bs))"
            ],
            [
                "(part X (. Y Xs) Ls (. Y Bs))",
                "(',' (le X Y) (part X Xs Ls Bs))"
            ],
            [
                "(part X ([]) ([]) ([]))",
                null
            ],
            [
                "(app (. X Xs) Ys (. X Zs))",
                "(app Xs Ys Zs)"
            ],
            [
                "(app ([]) Ys Ys)",
                null
            ],
            [
                "(gt (s X) (s Y))",
                "(gt X Y)"
            ],
            [
                "(gt (s (0)) (0))",
                null
            ],
            [
                "(le (s X) (s Y))",
                "(le X Y)"
            ],
            [
                "(le (0) (s X))",
                null
            ],
            [
                "(le (0) (0))",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "1220": {
                "goal": [{
                    "clause": 5,
                    "scope": 8,
                    "term": "(app T160 (. T148 T178) X249)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T148",
                        "T160",
                        "T178"
                    ],
                    "free": ["X249"],
                    "exprvars": []
                }
            },
            "1583": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T453 (. T452 T454) T8)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T8",
                        "T453",
                        "T454"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1461": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1582": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs ([]) X10)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X10"],
                    "exprvars": []
                }
            },
            "1460": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T349 (. T352 T353) X608)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T349"],
                    "free": ["X608"],
                    "exprvars": []
                }
            },
            "1581": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs ([]) X10) (app T453 (. T452 X10) T8))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T8",
                        "T453"
                    ],
                    "free": ["X10"],
                    "exprvars": []
                }
            },
            "1580": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs ([]) X9)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
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                        "T148",
                        "T149"
                    ],
                    "free": [
                        "X249",
                        "X245",
                        "X246",
                        "X247",
                        "X248"
                    ],
                    "exprvars": []
                }
            },
            "1446": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T236 X419)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X419"],
                    "exprvars": []
                }
            },
            "109": {
                "goal": [
                    {
                        "clause": -1,
                        "scope": -1,
                        "term": "(',' (part T9 T10 X7 X8) (',' (qs X7 X9) (',' (qs X8 X10) (app X9 (. T9 X10) T8))))"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(qs T1 T8)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T8"],
                    "free": [
                        "X7",
                        "X8",
                        "X9",
                        "X10"
                    ],
                    "exprvars": []
                }
            },
            "1209": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 7,
                        "term": "(qs T153 X247)"
                    },
                    {
                        "clause": 1,
                        "scope": 7,
                        "term": "(qs T153 X247)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T153"],
                    "free": ["X247"],
                    "exprvars": []
                }
            },
            "1208": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T154 X248) (app T160 (. T148 X248) X249))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T148",
                        "T154",
                        "T160"
                    ],
                    "free": [
                        "X249",
                        "X248"
                    ],
                    "exprvars": []
                }
            },
            "1207": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T153 X247)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T153"],
                    "free": ["X247"],
                    "exprvars": []
                }
            },
            "1449": {
                "goal": [
                    {
                        "clause": 5,
                        "scope": 12,
                        "term": "(app T319 (. T325 T324) X420)"
                    },
                    {
                        "clause": 6,
                        "scope": 12,
                        "term": "(app T319 (. T325 T324) X420)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T319"],
                    "free": ["X420"],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 5,
                "to": 6,
                "label": "CASE"
            },
            {
                "from": 6,
                "to": 109,
                "label": "EVAL with clause\nqs(.(X4, X5), X6) :- ','(part(X4, X5, X7, X8), ','(qs(X7, X9), ','(qs(X8, X10), app(X9, .(X4, X10), X6)))).\nand substitutionX4 -> T9,\nX5 -> T10,\nT1 -> .(T9, T10),\nT2 -> T8,\nX6 -> T8,\nT6 -> T9,\nT7 -> T10"
            },
            {
                "from": 6,
                "to": 129,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 109,
                "to": 152,
                "label": "CASE"
            },
            {
                "from": 129,
                "to": 1596,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []"
            },
            {
                "from": 129,
                "to": 1597,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 152,
                "to": 212,
                "label": "PARALLEL"
            },
            {
                "from": 152,
                "to": 213,
                "label": "PARALLEL"
            },
            {
                "from": 212,
                "to": 218,
                "label": "EVAL with clause\npart(X39, .(X40, X41), .(X40, X42), X43) :- ','(gt(X39, X40), part(X39, X41, X42, X43)).\nand substitutionT9 -> T26,\nX39 -> T26,\nX40 -> T27,\nX41 -> T28,\nT10 -> .(T27, T28),\nX42 -> X44,\nX7 -> .(T27, X44),\nX8 -> X45,\nX43 -> X45,\nT23 -> T26,\nT24 -> T27,\nT25 -> T28"
            },
            {
                "from": 212,
                "to": 221,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 213,
                "to": 1527,
                "label": "PARALLEL"
            },
            {
                "from": 213,
                "to": 1528,
                "label": "PARALLEL"
            },
            {
                "from": 218,
                "to": 230,
                "label": "SPLIT 1"
            },
            {
                "from": 218,
                "to": 231,
                "label": "SPLIT 2\nnew knowledge:\nT31 is ground\nT33 is ground\nreplacements:T26 -> T31,\nT28 -> T32,\nT27 -> T33"
            },
            {
                "from": 230,
                "to": 250,
                "label": "CASE"
            },
            {
                "from": 231,
                "to": 826,
                "label": "SPLIT 1"
            },
            {
                "from": 231,
                "to": 827,
                "label": "SPLIT 2\nnew knowledge:\nT31 is ground\nT53 is ground\nreplacements:X44 -> T53,\nX45 -> T54"
            },
            {
                "from": 250,
                "to": 261,
                "label": "PARALLEL"
            },
            {
                "from": 250,
                "to": 263,
                "label": "PARALLEL"
            },
            {
                "from": 261,
                "to": 408,
                "label": "EVAL with clause\ngt(s(X58), s(X59)) :- gt(X58, X59).\nand substitutionX58 -> T46,\nT26 -> s(T46),\nX59 -> T47,\nT27 -> s(T47),\nT44 -> T46,\nT45 -> T47"
            },
            {
                "from": 261,
                "to": 409,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 263,
                "to": 823,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT26 -> s(0),\nT27 -> 0"
            },
            {
                "from": 263,
                "to": 824,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 408,
                "to": 230,
                "label": "INSTANCE with matching:\nT26 -> T46\nT27 -> T47"
            },
            {
                "from": 823,
                "to": 825,
                "label": "SUCCESS"
            },
            {
                "from": 826,
                "to": 828,
                "label": "CASE"
            },
            {
                "from": 827,
                "to": 1159,
                "label": "SPLIT 1"
            },
            {
                "from": 827,
                "to": 1160,
                "label": "SPLIT 2\nnew knowledge:\nT33 is ground\nT53 is ground\nT131 is ground\nreplacements:X9 -> T131"
            },
            {
                "from": 828,
                "to": 829,
                "label": "PARALLEL"
            },
            {
                "from": 828,
                "to": 830,
                "label": "PARALLEL"
            },
            {
                "from": 829,
                "to": 831,
                "label": "EVAL with clause\npart(X104, .(X105, X106), .(X105, X107), X108) :- ','(gt(X104, X105), part(X104, X106, X107, X108)).\nand substitutionT31 -> T70,\nX104 -> T70,\nX105 -> T73,\nX106 -> T74,\nT32 -> .(T73, T74),\nX107 -> X109,\nX44 -> .(T73, X109),\nX45 -> X110,\nX108 -> X110,\nT71 -> T73,\nT72 -> T74"
            },
            {
                "from": 829,
                "to": 832,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 830,
                "to": 839,
                "label": "PARALLEL"
            },
            {
                "from": 830,
                "to": 840,
                "label": "PARALLEL"
            },
            {
                "from": 831,
                "to": 833,
                "label": "SPLIT 1"
            },
            {
                "from": 831,
                "to": 834,
                "label": "SPLIT 2\nnew knowledge:\nT70 is ground\nT73 is ground\nreplacements:T74 -> T77"
            },
            {
                "from": 833,
                "to": 230,
                "label": "INSTANCE with matching:\nT26 -> T70\nT27 -> T73"
            },
            {
                "from": 834,
                "to": 826,
                "label": "INSTANCE with matching:\nT31 -> T70\nT32 -> T77\nX44 -> X109\nX45 -> X110"
            },
            {
                "from": 839,
                "to": 1129,
                "label": "EVAL with clause\npart(X150, .(X151, X152), X153, .(X151, X154)) :- ','(le(X150, X151), part(X150, X152, X153, X154)).\nand substitutionT31 -> T95,\nX150 -> T95,\nX151 -> T98,\nX152 -> T99,\nT32 -> .(T98, T99),\nX44 -> X155,\nX153 -> X155,\nX154 -> X156,\nX45 -> .(T98, X156),\nT96 -> T98,\nT97 -> T99"
            },
            {
                "from": 839,
                "to": 1130,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 840,
                "to": 1156,
                "label": "EVAL with clause\npart(X187, [], [], []).\nand substitutionT31 -> T128,\nX187 -> T128,\nT32 -> [],\nX44 -> [],\nX45 -> []"
            },
            {
                "from": 840,
                "to": 1157,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1129,
                "to": 1131,
                "label": "SPLIT 1"
            },
            {
                "from": 1129,
                "to": 1132,
                "label": "SPLIT 2\nnew knowledge:\nT95 is ground\nreplacements:T99 -> T102"
            },
            {
                "from": 1131,
                "to": 1134,
                "label": "CASE"
            },
            {
                "from": 1132,
                "to": 826,
                "label": "INSTANCE with matching:\nT31 -> T95\nT32 -> T102\nX44 -> X155\nX45 -> X156"
            },
            {
                "from": 1134,
                "to": 1137,
                "label": "PARALLEL"
            },
            {
                "from": 1134,
                "to": 1138,
                "label": "PARALLEL"
            },
            {
                "from": 1137,
                "to": 1146,
                "label": "EVAL with clause\nle(s(X169), s(X170)) :- le(X169, X170).\nand substitutionX169 -> T113,\nT95 -> s(T113),\nX170 -> T115,\nT98 -> s(T115),\nT114 -> T115"
            },
            {
                "from": 1137,
                "to": 1147,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1138,
                "to": 1148,
                "label": "PARALLEL"
            },
            {
                "from": 1138,
                "to": 1149,
                "label": "PARALLEL"
            },
            {
                "from": 1146,
                "to": 1131,
                "label": "INSTANCE with matching:\nT95 -> T113\nT98 -> T115"
            },
            {
                "from": 1148,
                "to": 1150,
                "label": "EVAL with clause\nle(0, s(X177)).\nand substitutionT95 -> 0,\nX177 -> T122,\nT98 -> s(T122)"
            },
            {
                "from": 1148,
                "to": 1151,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1149,
                "to": 1153,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT95 -> 0,\nT98 -> 0"
            },
            {
                "from": 1149,
                "to": 1154,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1150,
                "to": 1152,
                "label": "SUCCESS"
            },
            {
                "from": 1153,
                "to": 1155,
                "label": "SUCCESS"
            },
            {
                "from": 1156,
                "to": 1158,
                "label": "SUCCESS"
            },
            {
                "from": 1159,
                "to": 1161,
                "label": "CASE"
            },
            {
                "from": 1160,
                "to": 1237,
                "label": "SPLIT 1"
            },
            {
                "from": 1160,
                "to": 1238,
                "label": "SPLIT 2\nreplacements:X10 -> T217"
            },
            {
                "from": 1161,
                "to": 1162,
                "label": "PARALLEL"
            },
            {
                "from": 1161,
                "to": 1163,
                "label": "PARALLEL"
            },
            {
                "from": 1162,
                "to": 1204,
                "label": "ONLY EVAL with clause\nqs(.(X242, X243), X244) :- ','(part(X242, X243, X245, X246), ','(qs(X245, X247), ','(qs(X246, X248), app(X247, .(X242, X248), X244)))).\nand substitutionT33 -> T148,\nX242 -> T148,\nT53 -> T149,\nX243 -> T149,\nX9 -> X249,\nX244 -> X249"
            },
            {
                "from": 1163,
                "to": 1227,
                "label": "BACKTRACK\nfor clause: qs([], [])because of non-unification"
            },
            {
                "from": 1204,
                "to": 1205,
                "label": "SPLIT 1"
            },
            {
                "from": 1204,
                "to": 1206,
                "label": "SPLIT 2\nnew knowledge:\nT148 is ground\nT149 is ground\nT153 is ground\nT154 is ground\nreplacements:X245 -> T153,\nX246 -> T154"
            },
            {
                "from": 1205,
                "to": 826,
                "label": "INSTANCE with matching:\nT31 -> T148\nT32 -> T149\nX44 -> X245\nX45 -> X246"
            },
            {
                "from": 1206,
                "to": 1207,
                "label": "SPLIT 1"
            },
            {
                "from": 1206,
                "to": 1208,
                "label": "SPLIT 2\nnew knowledge:\nT153 is ground\nT160 is ground\nreplacements:X247 -> T160"
            },
            {
                "from": 1207,
                "to": 1209,
                "label": "CASE"
            },
            {
                "from": 1208,
                "to": 1217,
                "label": "SPLIT 1"
            },
            {
                "from": 1208,
                "to": 1218,
                "label": "SPLIT 2\nnew knowledge:\nT154 is ground\nT178 is ground\nreplacements:X248 -> T178"
            },
            {
                "from": 1209,
                "to": 1210,
                "label": "PARALLEL"
            },
            {
                "from": 1209,
                "to": 1211,
                "label": "PARALLEL"
            },
            {
                "from": 1210,
                "to": 1212,
                "label": "EVAL with clause\nqs(.(X304, X305), X306) :- ','(part(X304, X305, X307, X308), ','(qs(X307, X309), ','(qs(X308, X310), app(X309, .(X304, X310), X306)))).\nand substitutionX304 -> T171,\nX305 -> T172,\nT153 -> .(T171, T172),\nX247 -> X311,\nX306 -> X311"
            },
            {
                "from": 1210,
                "to": 1213,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1211,
                "to": 1214,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT153 -> [],\nX247 -> []"
            },
            {
                "from": 1211,
                "to": 1215,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1212,
                "to": 1204,
                "label": "INSTANCE with matching:\nT148 -> T171\nT149 -> T172\nX245 -> X307\nX246 -> X308\nX247 -> X309\nX248 -> X310\nX249 -> X311"
            },
            {
                "from": 1214,
                "to": 1216,
                "label": "SUCCESS"
            },
            {
                "from": 1217,
                "to": 1207,
                "label": "INSTANCE with matching:\nT153 -> T154\nX247 -> X248"
            },
            {
                "from": 1218,
                "to": 1219,
                "label": "CASE"
            },
            {
                "from": 1219,
                "to": 1220,
                "label": "PARALLEL"
            },
            {
                "from": 1219,
                "to": 1221,
                "label": "PARALLEL"
            },
            {
                "from": 1220,
                "to": 1222,
                "label": "EVAL with clause\napp(.(X360, X361), X362, .(X360, X363)) :- app(X361, X362, X363).\nand substitutionX360 -> T201,\nX361 -> T202,\nT160 -> .(T201, T202),\nT148 -> T203,\nT178 -> T204,\nX362 -> .(T203, T204),\nX363 -> X364,\nX249 -> .(T201, X364)"
            },
            {
                "from": 1220,
                "to": 1223,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1221,
                "to": 1224,
                "label": "EVAL with clause\napp([], X372, X372).\nand substitutionT160 -> [],\nT148 -> T213,\nT178 -> T214,\nX372 -> .(T213, T214),\nX249 -> .(T213, T214)"
            },
            {
                "from": 1221,
                "to": 1225,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1222,
                "to": 1218,
                "label": "INSTANCE with matching:\nT160 -> T202\nT148 -> T203\nT178 -> T204\nX249 -> X364"
            },
            {
                "from": 1224,
                "to": 1226,
                "label": "SUCCESS"
            },
            {
                "from": 1237,
                "to": 1241,
                "label": "CASE"
            },
            {
                "from": 1238,
                "to": 1480,
                "label": "CASE"
            },
            {
                "from": 1241,
                "to": 1244,
                "label": "PARALLEL"
            },
            {
                "from": 1241,
                "to": 1245,
                "label": "PARALLEL"
            },
            {
                "from": 1244,
                "to": 1258,
                "label": "EVAL with clause\nqs(.(X413, X414), X415) :- ','(part(X413, X414, X416, X417), ','(qs(X416, X418), ','(qs(X417, X419), app(X418, .(X413, X419), X415)))).\nand substitutionX413 -> T230,\nX414 -> T231,\nT54 -> .(T230, T231),\nX10 -> X420,\nX415 -> X420,\nT228 -> T230,\nT229 -> T231"
            },
            {
                "from": 1244,
                "to": 1259,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1245,
                "to": 1468,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT54 -> [],\nX10 -> []"
            },
            {
                "from": 1245,
                "to": 1476,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1258,
                "to": 1265,
                "label": "SPLIT 1"
            },
            {
                "from": 1258,
                "to": 1266,
                "label": "SPLIT 2\nnew knowledge:\nT235 is ground\nreplacements:X416 -> T235,\nX417 -> T236,\nT230 -> T237"
            },
            {
                "from": 1265,
                "to": 1267,
                "label": "CASE"
            },
            {
                "from": 1266,
                "to": 1438,
                "label": "SPLIT 1"
            },
            {
                "from": 1266,
                "to": 1439,
                "label": "SPLIT 2\nnew knowledge:\nT235 is ground\nT319 is ground\nreplacements:X418 -> T319"
            },
            {
                "from": 1267,
                "to": 1298,
                "label": "PARALLEL"
            },
            {
                "from": 1267,
                "to": 1299,
                "label": "PARALLEL"
            },
            {
                "from": 1298,
                "to": 1301,
                "label": "EVAL with clause\npart(X463, .(X464, X465), .(X464, X466), X467) :- ','(gt(X463, X464), part(X463, X465, X466, X467)).\nand substitutionT230 -> T256,\nX463 -> T256,\nX464 -> T257,\nX465 -> T258,\nT231 -> .(T257, T258),\nX466 -> X468,\nX416 -> .(T257, X468),\nX417 -> X469,\nX467 -> X469,\nT253 -> T256,\nT254 -> T257,\nT255 -> T258"
            },
            {
                "from": 1298,
                "to": 1302,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1299,
                "to": 1311,
                "label": "PARALLEL"
            },
            {
                "from": 1299,
                "to": 1312,
                "label": "PARALLEL"
            },
            {
                "from": 1301,
                "to": 1303,
                "label": "SPLIT 1"
            },
            {
                "from": 1301,
                "to": 1304,
                "label": "SPLIT 2\nnew knowledge:\nT261 is ground\nT257 is ground\nreplacements:T256 -> T261,\nT258 -> T262"
            },
            {
                "from": 1303,
                "to": 230,
                "label": "INSTANCE with matching:\nT26 -> T256\nT27 -> T257"
            },
            {
                "from": 1304,
                "to": 826,
                "label": "INSTANCE with matching:\nT31 -> T261\nT32 -> T262\nX44 -> X468\nX45 -> X469"
            },
            {
                "from": 1311,
                "to": 1397,
                "label": "EVAL with clause\npart(X509, .(X510, X511), X512, .(X510, X513)) :- ','(le(X509, X510), part(X509, X511, X512, X513)).\nand substitutionT230 -> T283,\nX509 -> T283,\nX510 -> T284,\nX511 -> T285,\nT231 -> .(T284, T285),\nX416 -> X514,\nX512 -> X514,\nX513 -> X515,\nX417 -> .(T284, X515),\nT280 -> T283,\nT281 -> T284,\nT282 -> T285"
            },
            {
                "from": 1311,
                "to": 1398,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1312,
                "to": 1427,
                "label": "EVAL with clause\npart(X546, [], [], []).\nand substitutionT230 -> T316,\nX546 -> T316,\nT231 -> [],\nX416 -> [],\nX417 -> []"
            },
            {
                "from": 1312,
                "to": 1432,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1397,
                "to": 1401,
                "label": "SPLIT 1"
            },
            {
                "from": 1397,
                "to": 1402,
                "label": "SPLIT 2\nnew knowledge:\nT288 is ground\nreplacements:T283 -> T288,\nT285 -> T289"
            },
            {
                "from": 1401,
                "to": 1404,
                "label": "CASE"
            },
            {
                "from": 1402,
                "to": 826,
                "label": "INSTANCE with matching:\nT31 -> T288\nT32 -> T289\nX44 -> X514\nX45 -> X515"
            },
            {
                "from": 1404,
                "to": 1406,
                "label": "PARALLEL"
            },
            {
                "from": 1404,
                "to": 1407,
                "label": "PARALLEL"
            },
            {
                "from": 1406,
                "to": 1410,
                "label": "EVAL with clause\nle(s(X528), s(X529)) :- le(X528, X529).\nand substitutionX528 -> T302,\nT283 -> s(T302),\nX529 -> T303,\nT284 -> s(T303),\nT300 -> T302,\nT301 -> T303"
            },
            {
                "from": 1406,
                "to": 1411,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1407,
                "to": 1412,
                "label": "PARALLEL"
            },
            {
                "from": 1407,
                "to": 1413,
                "label": "PARALLEL"
            },
            {
                "from": 1410,
                "to": 1401,
                "label": "INSTANCE with matching:\nT283 -> T302\nT284 -> T303"
            },
            {
                "from": 1412,
                "to": 1414,
                "label": "EVAL with clause\nle(0, s(X536)).\nand substitutionT283 -> 0,\nX536 -> T310,\nT284 -> s(T310)"
            },
            {
                "from": 1412,
                "to": 1415,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1413,
                "to": 1417,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT283 -> 0,\nT284 -> 0"
            },
            {
                "from": 1413,
                "to": 1418,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1414,
                "to": 1416,
                "label": "SUCCESS"
            },
            {
                "from": 1417,
                "to": 1419,
                "label": "SUCCESS"
            },
            {
                "from": 1427,
                "to": 1433,
                "label": "SUCCESS"
            },
            {
                "from": 1438,
                "to": 1237,
                "label": "INSTANCE with matching:\nT54 -> T235\nX10 -> X418"
            },
            {
                "from": 1439,
                "to": 1446,
                "label": "SPLIT 1"
            },
            {
                "from": 1439,
                "to": 1448,
                "label": "SPLIT 2\nreplacements:X419 -> T324,\nT237 -> T325"
            },
            {
                "from": 1446,
                "to": 1237,
                "label": "INSTANCE with matching:\nT54 -> T236\nX10 -> X419"
            },
            {
                "from": 1448,
                "to": 1449,
                "label": "CASE"
            },
            {
                "from": 1449,
                "to": 1452,
                "label": "PARALLEL"
            },
            {
                "from": 1449,
                "to": 1453,
                "label": "PARALLEL"
            },
            {
                "from": 1452,
                "to": 1460,
                "label": "EVAL with clause\napp(.(X604, X605), X606, .(X604, X607)) :- app(X605, X606, X607).\nand substitutionX604 -> T348,\nX605 -> T349,\nT319 -> .(T348, T349),\nT325 -> T352,\nT324 -> T353,\nX606 -> .(T352, T353),\nX607 -> X608,\nX420 -> .(T348, X608),\nT350 -> T352,\nT351 -> T353"
            },
            {
                "from": 1452,
                "to": 1461,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1453,
                "to": 1465,
                "label": "EVAL with clause\napp([], X616, X616).\nand substitutionT319 -> [],\nT325 -> T362,\nT324 -> T363,\nX616 -> .(T362, T363),\nX420 -> .(T362, T363)"
            },
            {
                "from": 1453,
                "to": 1466,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1460,
                "to": 1448,
                "label": "INSTANCE with matching:\nT319 -> T349\nT325 -> T352\nT324 -> T353\nX420 -> X608"
            },
            {
                "from": 1465,
                "to": 1467,
                "label": "SUCCESS"
            },
            {
                "from": 1468,
                "to": 1477,
                "label": "SUCCESS"
            },
            {
                "from": 1480,
                "to": 1486,
                "label": "PARALLEL"
            },
            {
                "from": 1480,
                "to": 1487,
                "label": "PARALLEL"
            },
            {
                "from": 1486,
                "to": 1493,
                "label": "EVAL with clause\napp(.(X637, X638), X639, .(X637, X640)) :- app(X638, X639, X640).\nand substitutionX637 -> T389,\nX638 -> T390,\nT131 -> .(T389, T390),\nT31 -> T391,\nT217 -> T394,\nX639 -> .(T391, T394),\nX640 -> T393,\nT8 -> .(T389, T393),\nT392 -> T394"
            },
            {
                "from": 1486,
                "to": 1494,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1487,
                "to": 1501,
                "label": "EVAL with clause\napp([], X647, X647).\nand substitutionT131 -> [],\nT31 -> T404,\nT217 -> T405,\nX647 -> .(T404, T405),\nT8 -> .(T404, T405)"
            },
            {
                "from": 1487,
                "to": 1502,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1493,
                "to": 1238,
                "label": "INSTANCE with matching:\nT131 -> T390\nT31 -> T391\nT217 -> T394\nT8 -> T393"
            },
            {
                "from": 1501,
                "to": 1503,
                "label": "SUCCESS"
            },
            {
                "from": 1527,
                "to": 1568,
                "label": "EVAL with clause\npart(X676, .(X677, X678), X679, .(X677, X680)) :- ','(le(X676, X677), part(X676, X678, X679, X680)).\nand substitutionT9 -> T421,\nX676 -> T421,\nX677 -> T422,\nX678 -> T423,\nT10 -> .(T422, T423),\nX7 -> X681,\nX679 -> X681,\nX680 -> X682,\nX8 -> .(T422, X682),\nT418 -> T421,\nT419 -> T422,\nT420 -> T423"
            },
            {
                "from": 1527,
                "to": 1569,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1528,
                "to": 1576,
                "label": "PARALLEL"
            },
            {
                "from": 1528,
                "to": 1577,
                "label": "PARALLEL"
            },
            {
                "from": 1568,
                "to": 1570,
                "label": "SPLIT 1"
            },
            {
                "from": 1568,
                "to": 1571,
                "label": "SPLIT 2\nnew knowledge:\nT426 is ground\nreplacements:T421 -> T426,\nT423 -> T427,\nT422 -> T428"
            },
            {
                "from": 1570,
                "to": 1401,
                "label": "INSTANCE with matching:\nT283 -> T421\nT284 -> T422"
            },
            {
                "from": 1571,
                "to": 1572,
                "label": "SPLIT 1"
            },
            {
                "from": 1571,
                "to": 1573,
                "label": "SPLIT 2\nnew knowledge:\nT426 is ground\nT434 is ground\nreplacements:X681 -> T434,\nX682 -> T435,\nT428 -> T436"
            },
            {
                "from": 1572,
                "to": 826,
                "label": "INSTANCE with matching:\nT31 -> T426\nT32 -> T427\nX44 -> X681\nX45 -> X682"
            },
            {
                "from": 1573,
                "to": 1574,
                "label": "SPLIT 1"
            },
            {
                "from": 1573,
                "to": 1575,
                "label": "SPLIT 2\nnew knowledge:\nT434 is ground\nT442 is ground\nreplacements:X9 -> T442"
            },
            {
                "from": 1574,
                "to": 1237,
                "label": "INSTANCE with matching:\nT54 -> T434\nX10 -> X9"
            },
            {
                "from": 1575,
                "to": 1160,
                "label": "INSTANCE with matching:\nT54 -> .(T436, T435)\nT131 -> T442\nT31 -> T426"
            },
            {
                "from": 1576,
                "to": 1578,
                "label": "EVAL with clause\npart(X729, [], [], []).\nand substitutionT9 -> T452,\nX729 -> T452,\nT10 -> [],\nX7 -> [],\nX8 -> [],\nT451 -> T452"
            },
            {
                "from": 1576,
                "to": 1579,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1577,
                "to": 1592,
                "label": "FAILURE"
            },
            {
                "from": 1578,
                "to": 1580,
                "label": "SPLIT 1"
            },
            {
                "from": 1578,
                "to": 1581,
                "label": "SPLIT 2\nnew knowledge:\nT453 is ground\nreplacements:X9 -> T453"
            },
            {
                "from": 1580,
                "to": 1237,
                "label": "INSTANCE with matching:\nT54 -> []\nX10 -> X9"
            },
            {
                "from": 1581,
                "to": 1582,
                "label": "SPLIT 1"
            },
            {
                "from": 1581,
                "to": 1583,
                "label": "SPLIT 2\nnew knowledge:\nT454 is ground\nreplacements:X10 -> T454"
            },
            {
                "from": 1582,
                "to": 1237,
                "label": "INSTANCE with matching:\nT54 -> []"
            },
            {
                "from": 1583,
                "to": 1584,
                "label": "CASE"
            },
            {
                "from": 1584,
                "to": 1585,
                "label": "PARALLEL"
            },
            {
                "from": 1584,
                "to": 1586,
                "label": "PARALLEL"
            },
            {
                "from": 1585,
                "to": 1587,
                "label": "EVAL with clause\napp(.(X778, X779), X780, .(X778, X781)) :- app(X779, X780, X781).\nand substitutionX778 -> T480,\nX779 -> T481,\nT453 -> .(T480, T481),\nT452 -> T485,\nT454 -> T483,\nX780 -> .(T485, T483),\nX781 -> T484,\nT8 -> .(T480, T484),\nT482 -> T485"
            },
            {
                "from": 1585,
                "to": 1588,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1586,
                "to": 1589,
                "label": "EVAL with clause\napp([], X788, X788).\nand substitutionT453 -> [],\nT452 -> T495,\nT454 -> T496,\nX788 -> .(T495, T496),\nT8 -> .(T495, T496)"
            },
            {
                "from": 1586,
                "to": 1590,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1587,
                "to": 1583,
                "label": "INSTANCE with matching:\nT453 -> T481\nT452 -> T485\nT454 -> T483\nT8 -> T484"
            },
            {
                "from": 1589,
                "to": 1591,
                "label": "SUCCESS"
            },
            {
                "from": 1592,
                "to": 1593,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT8 -> []"
            },
            {
                "from": 1592,
                "to": 1594,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1593,
                "to": 1595,
                "label": "SUCCESS"
            },
            {
                "from": 1596,
                "to": 1598,
                "label": "SUCCESS"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(279)
Obligation:
Triples:

gtA(s(X1), s(X2)) :- gtA(X1, X2).
partB(X1, .(X2, X3), .(X2, X4), X5) :- gtA(X1, X2).
partB(X1, .(X2, X3), .(X2, X4), X5) :- ','(gtcA(X1, X2), partB(X1, X3, X4, X5)).
partB(X1, .(X2, X3), X4, .(X2, X5)) :- leC(X1, X2).
partB(X1, .(X2, X3), X4, .(X2, X5)) :- ','(lecC(X1, X2), partB(X1, X3, X4, X5)).
leC(s(X1), s(X2)) :- leC(X1, X2).
pD(X1, X2, X3, X4, X5, X6, X7) :- partB(X1, X2, X3, X4).
pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcB(X1, X2, X3, X4), qsF(X3, X5)).
pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcB(X1, X2, X3, X4), ','(qscF(X3, X5), qsF(X4, X6))).
pD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcB(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appE(X5, X1, X6, X7)))).
qsF(.(X1, X2), X3) :- pD(X1, X2, X4, X5, X6, X7, X3).
appE(.(X1, X2), X3, X4, .(X1, X5)) :- appE(X2, X3, X4, X5).
leG(s(X1), s(X2)) :- leG(X1, X2).
qsH(.(X1, .(X2, X3)), X4) :- gtA(X1, X2).
qsH(.(X1, .(X2, X3)), X4) :- ','(gtcA(X1, X2), partB(X1, X3, X5, X6)).
qsH(.(X1, .(X2, X3)), X4) :- leG(X1, X2).
qsH(.(X1, .(X2, X3)), X4) :- ','(lecG(X1, X2), partB(X1, X3, X5, X6)).
qsH(.(X1, X2), X3) :- ','(partcJ(X1, X2, X4, X5), qsH(X4, X6)).
qsH(.(X1, X2), X3) :- ','(partcJ(X1, X2, X4, X5), ','(qscH(X4, X6), qsH(X5, X7))).
qsH(.(X1, X2), X3) :- ','(partcJ(X1, X2, X4, X5), ','(qscH(X4, X6), ','(qscH(X5, X7), appI(X6, X1, X7, X3)))).
appI(.(X1, X2), X3, X4, .(X1, X5)) :- appI(X2, X3, X4, X5).
appK(.(X1, X2), X3, X4, .(X1, X5)) :- appK(X2, X3, X4, X5).
pL(X1, X2, X3, X4, X5) :- qsH(X1, X2).
pL(X1, X2, X3, X4, X5) :- ','(qscH(X1, X2), appK(X3, X4, X2, X5)).
appM(.(X1, X2), X3, X4, .(X1, X5)) :- appM(X2, X3, X4, X5).
qsO(.(X1, .(X2, X3)), X4) :- gtA(X1, X2).
qsO(.(X1, .(X2, X3)), X4) :- ','(gtcA(X1, X2), partB(X1, X3, X5, X6)).
qsO(.(X1, .(X2, X3)), X4) :- ','(gtcA(X1, X2), ','(partcB(X1, X3, X5, X6), pD(X2, X5, X7, X8, X9, X10, X11))).
qsO(.(X1, .(X2, X3)), X4) :- ','(gtcA(X1, X2), ','(partcB(X1, X3, X5, X6), ','(qscN(X2, X5, X7), pL(X6, X8, X7, X1, X4)))).
qsO(.(X1, .(X2, X3)), X4) :- leG(X1, X2).
qsO(.(X1, .(X2, X3)), X4) :- ','(lecG(X1, X2), partB(X1, X3, X5, X6)).
qsO(.(X1, .(X2, X3)), X4) :- ','(lecG(X1, X2), ','(partcB(X1, X3, X5, X6), qsH(X5, X7))).
qsO(.(X1, .(X2, X3)), X4) :- ','(lecG(X1, X2), ','(partcB(X1, X3, X5, X6), ','(qscH(X5, X7), pL(.(X2, X6), X8, X7, X1, X4)))).
qsO(.(X1, []), X2) :- qsH([], X3).
qsO(.(X1, []), X2) :- ','(qscH([], X3), qsH([], X4)).
qsO(.(X1, []), X2) :- ','(qscH([], X3), ','(qscH([], X4), appM(X3, X1, X4, X2))).

Clauses:

gtcA(s(X1), s(X2)) :- gtcA(X1, X2).
gtcA(s(0), 0).
partcB(X1, .(X2, X3), .(X2, X4), X5) :- ','(gtcA(X1, X2), partcB(X1, X3, X4, X5)).
partcB(X1, .(X2, X3), X4, .(X2, X5)) :- ','(lecC(X1, X2), partcB(X1, X3, X4, X5)).
partcB(X1, [], [], []).
lecC(s(X1), s(X2)) :- lecC(X1, X2).
lecC(0, s(X1)).
lecC(0, 0).
qcD(X1, X2, X3, X4, X5, X6, X7) :- ','(partcB(X1, X2, X3, X4), ','(qscF(X3, X5), ','(qscF(X4, X6), appcE(X5, X1, X6, X7)))).
qscF(.(X1, X2), X3) :- qcD(X1, X2, X4, X5, X6, X7, X3).
qscF([], []).
appcE(.(X1, X2), X3, X4, .(X1, X5)) :- appcE(X2, X3, X4, X5).
appcE([], X1, X2, .(X1, X2)).
lecG(s(X1), s(X2)) :- lecG(X1, X2).
lecG(0, s(X1)).
lecG(0, 0).
qscH(.(X1, X2), X3) :- ','(partcJ(X1, X2, X4, X5), ','(qscH(X4, X6), ','(qscH(X5, X7), appcI(X6, X1, X7, X3)))).
qscH([], []).
appcI(.(X1, X2), X3, X4, .(X1, X5)) :- appcI(X2, X3, X4, X5).
appcI([], X1, X2, .(X1, X2)).
appcK(.(X1, X2), X3, X4, .(X1, X5)) :- appcK(X2, X3, X4, X5).
appcK([], X1, X2, .(X1, X2)).
qcL(X1, X2, X3, X4, X5) :- ','(qscH(X1, X2), appcK(X3, X4, X2, X5)).
appcM(.(X1, X2), X3, X4, .(X1, X5)) :- appcM(X2, X3, X4, X5).
appcM([], X1, X2, .(X1, X2)).
qscN(X1, X2, X3) :- qcD(X1, X2, X4, X5, X6, X7, X3).
partcJ(X1, .(X2, X3), .(X2, X4), X5) :- ','(gtcA(X1, X2), partcB(X1, X3, X4, X5)).
partcJ(X1, .(X2, X3), X4, .(X2, X5)) :- ','(lecG(X1, X2), partcB(X1, X3, X4, X5)).
partcJ(X1, [], [], []).

Afs:

qsO(x1, x2)  =  qsO(x2)


----------------------------------------

(280) TriplesToPiDPProof (SOUND)
We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:

qsO_in_2: (f,b)

gtA_in_2: (f,f) (b,f) (b,b)

gtcA_in_2: (f,f) (b,f) (b,b)

partB_in_4: (b,f,f,f) (b,b,f,f)

leC_in_2: (b,f) (b,b)

lecC_in_2: (b,f) (b,b)

partcB_in_4: (b,f,f,f) (b,b,f,f)

pD_in_7: (b,b,f,f,f,f,f)

qsF_in_2: (b,f)

qscF_in_2: (b,f)

qcD_in_7: (b,b,f,f,f,f,f)

appcE_in_4: (b,b,b,f)

appE_in_4: (b,b,b,f)

qscN_in_3: (b,b,f)

pL_in_5: (f,f,b,b,b)

qsH_in_2: (f,f) (b,f)

leG_in_2: (f,f) (b,b)

lecG_in_2: (f,f) (b,b)

partcJ_in_4: (f,f,f,f) (b,b,f,f)

qscH_in_2: (b,f) (f,f)

appcI_in_4: (b,b,b,f) (b,f,f,f)

appI_in_4: (b,b,b,f) (b,f,f,f)

appK_in_4: (b,b,f,b)

appM_in_4: (b,f,b,b)

Transforming TRIPLES into the following Term Rewriting System:

Pi DP problem:
The TRS P consists of the following rules:

   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U37_AG(X1, X2, X3, X4, gtA_in_aa(X1, X2))
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> GTA_IN_AA(X1, X2)
   GTA_IN_AA(s(X1), s(X2)) -> U1_AA(X1, X2, gtA_in_aa(X1, X2))
   GTA_IN_AA(s(X1), s(X2)) -> GTA_IN_AA(X1, X2)
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U38_AG(X1, X2, X3, X4, gtcA_in_aa(X1, X2))
   U38_AG(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> U39_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U38_AG(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GAAA(X1, X2, X3, X4, X5, gtA_in_ga(X1, X2))
   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> GTA_IN_GA(X1, X2)
   GTA_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, gtA_in_ga(X1, X2))
   GTA_IN_GA(s(X1), s(X2)) -> GTA_IN_GA(X1, X2)
   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U3_GAAA(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U4_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5))
   U3_GAAA(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GAAA(X1, X2, X3, X4, X5, leC_in_ga(X1, X2))
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> LEC_IN_GA(X1, X2)
   LEC_IN_GA(s(X1), s(X2)) -> U8_GA(X1, X2, leC_in_ga(X1, X2))
   LEC_IN_GA(s(X1), s(X2)) -> LEC_IN_GA(X1, X2)
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GAAA(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U6_GAAA(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U7_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5))
   U6_GAAA(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)
   U38_AG(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> U40_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6))
   U40_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U41_AG(X1, X2, X3, X4, pD_in_ggaaaaa(X2, X5, X7, X8, X9, X10, X11))
   U40_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> PD_IN_GGAAAAA(X2, X5, X7, X8, X9, X10, X11)
   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U9_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partB_in_ggaa(X1, X2, X3, X4))
   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTB_IN_GGAA(X1, X2, X3, X4)
   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GGAA(X1, X2, X3, X4, X5, gtA_in_gg(X1, X2))
   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> GTA_IN_GG(X1, X2)
   GTA_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, gtA_in_gg(X1, X2))
   GTA_IN_GG(s(X1), s(X2)) -> GTA_IN_GG(X1, X2)
   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U3_GGAA(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U4_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5))
   U3_GGAA(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GGAA(X1, X2, X3, X4, X5, leC_in_gg(X1, X2))
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> LEC_IN_GG(X1, X2)
   LEC_IN_GG(s(X1), s(X2)) -> U8_GG(X1, X2, leC_in_gg(X1, X2))
   LEC_IN_GG(s(X1), s(X2)) -> LEC_IN_GG(X1, X2)
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U6_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U7_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5))
   U6_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)
   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X3, X5))
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5)
   QSF_IN_GA(.(X1, X2), X3) -> U16_GA(X1, X2, X3, pD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3)
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X4, X6))
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6)
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, appE_in_ggga(X5, X1, X6, X7))
   U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> APPE_IN_GGGA(X5, X1, X6, X7)
   APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U17_GGGA(X1, X2, X3, X4, X5, appE_in_ggga(X2, X3, X4, X5))
   APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5)
   U40_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U42_AG(X1, X2, X3, X4, X6, qscN_in_gga(X2, X5, X7))
   U42_AG(X1, X2, X3, X4, X6, qscN_out_gga(X2, X5, X7)) -> U43_AG(X1, X2, X3, X4, pL_in_aaggg(X6, X8, X7, X1, X4))
   U42_AG(X1, X2, X3, X4, X6, qscN_out_gga(X2, X5, X7)) -> PL_IN_AAGGG(X6, X8, X7, X1, X4)
   PL_IN_AAGGG(X1, X2, X3, X4, X5) -> U33_AAGGG(X1, X2, X3, X4, X5, qsH_in_aa(X1, X2))
   PL_IN_AAGGG(X1, X2, X3, X4, X5) -> QSH_IN_AA(X1, X2)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U19_AA(X1, X2, X3, X4, gtA_in_aa(X1, X2))
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> GTA_IN_AA(X1, X2)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U20_AA(X1, X2, X3, X4, gtcA_in_aa(X1, X2))
   U20_AA(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> U21_AA(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U20_AA(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U22_AA(X1, X2, X3, X4, leG_in_aa(X1, X2))
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> LEG_IN_AA(X1, X2)
   LEG_IN_AA(s(X1), s(X2)) -> U18_AA(X1, X2, leG_in_aa(X1, X2))
   LEG_IN_AA(s(X1), s(X2)) -> LEG_IN_AA(X1, X2)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U23_AA(X1, X2, X3, X4, lecG_in_aa(X1, X2))
   U23_AA(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> U24_AA(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U23_AA(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   QSH_IN_AA(.(X1, X2), X3) -> U25_AA(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U26_AA(X1, X2, X3, qsH_in_ga(X4, X6))
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4, X6)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U19_GA(X1, X2, X3, X4, gtA_in_gg(X1, X2))
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> GTA_IN_GG(X1, X2)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U20_GA(X1, X2, X3, X4, gtcA_in_gg(X1, X2))
   U20_GA(X1, X2, X3, X4, gtcA_out_gg(X1, X2)) -> U21_GA(X1, X2, X3, X4, partB_in_ggaa(X1, X3, X5, X6))
   U20_GA(X1, X2, X3, X4, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X5, X6)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U22_GA(X1, X2, X3, X4, leG_in_gg(X1, X2))
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> LEG_IN_GG(X1, X2)
   LEG_IN_GG(s(X1), s(X2)) -> U18_GG(X1, X2, leG_in_gg(X1, X2))
   LEG_IN_GG(s(X1), s(X2)) -> LEG_IN_GG(X1, X2)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U23_GA(X1, X2, X3, X4, lecG_in_gg(X1, X2))
   U23_GA(X1, X2, X3, X4, lecG_out_gg(X1, X2)) -> U24_GA(X1, X2, X3, X4, partB_in_ggaa(X1, X3, X5, X6))
   U23_GA(X1, X2, X3, X4, lecG_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X5, X6)
   QSH_IN_GA(.(X1, X2), X3) -> U25_GA(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U26_GA(X1, X2, X3, qsH_in_ga(X4, X6))
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4, X6)
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U27_GA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U28_GA(X1, X2, X3, qsH_in_ga(X5, X7))
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_GA(X5, X7)
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U29_GA(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U29_GA(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U30_GA(X1, X2, X3, appI_in_ggga(X6, X1, X7, X3))
   U29_GA(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> APPI_IN_GGGA(X6, X1, X7, X3)
   APPI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U31_GGGA(X1, X2, X3, X4, X5, appI_in_ggga(X2, X3, X4, X5))
   APPI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GGGA(X2, X3, X4, X5)
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U27_AA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U28_AA(X1, X2, X3, qsH_in_aa(X5, X7))
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_AA(X5, X7)
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U29_AA(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   U29_AA(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U30_AA(X1, X2, X3, appI_in_gaaa(X6, X1, X7, X3))
   U29_AA(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> APPI_IN_GAAA(X6, X1, X7, X3)
   APPI_IN_GAAA(.(X1, X2), X3, X4, .(X1, X5)) -> U31_GAAA(X1, X2, X3, X4, X5, appI_in_gaaa(X2, X3, X4, X5))
   APPI_IN_GAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAAA(X2, X3, X4, X5)
   PL_IN_AAGGG(X1, X2, X3, X4, X5) -> U34_AAGGG(X1, X2, X3, X4, X5, qscH_in_aa(X1, X2))
   U34_AAGGG(X1, X2, X3, X4, X5, qscH_out_aa(X1, X2)) -> U35_AAGGG(X1, X2, X3, X4, X5, appK_in_ggag(X3, X4, X2, X5))
   U34_AAGGG(X1, X2, X3, X4, X5, qscH_out_aa(X1, X2)) -> APPK_IN_GGAG(X3, X4, X2, X5)
   APPK_IN_GGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U32_GGAG(X1, X2, X3, X4, X5, appK_in_ggag(X2, X3, X4, X5))
   APPK_IN_GGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPK_IN_GGAG(X2, X3, X4, X5)
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U44_AG(X1, X2, X3, X4, leG_in_aa(X1, X2))
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> LEG_IN_AA(X1, X2)
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U45_AG(X1, X2, X3, X4, lecG_in_aa(X1, X2))
   U45_AG(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> U46_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U45_AG(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   U45_AG(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> U47_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6))
   U47_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U48_AG(X1, X2, X3, X4, qsH_in_ga(X5, X7))
   U47_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> QSH_IN_GA(X5, X7)
   U47_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U49_AG(X1, X2, X3, X4, X6, qscH_in_ga(X5, X7))
   U49_AG(X1, X2, X3, X4, X6, qscH_out_ga(X5, X7)) -> U50_AG(X1, X2, X3, X4, pL_in_aaggg(.(X2, X6), X8, X7, X1, X4))
   U49_AG(X1, X2, X3, X4, X6, qscH_out_ga(X5, X7)) -> PL_IN_AAGGG(.(X2, X6), X8, X7, X1, X4)
   QSO_IN_AG(.(X1, []), X2) -> U51_AG(X1, X2, qsH_in_ga([], X3))
   QSO_IN_AG(.(X1, []), X2) -> QSH_IN_GA([], X3)
   QSO_IN_AG(.(X1, []), X2) -> U52_AG(X1, X2, qscH_in_ga([], X3))
   U52_AG(X1, X2, qscH_out_ga([], X3)) -> U53_AG(X1, X2, qsH_in_ga([], X4))
   U52_AG(X1, X2, qscH_out_ga([], X3)) -> QSH_IN_GA([], X4)
   U52_AG(X1, X2, qscH_out_ga([], X3)) -> U54_AG(X1, X2, X3, qscH_in_ga([], X4))
   U54_AG(X1, X2, X3, qscH_out_ga([], X4)) -> U55_AG(X1, X2, appM_in_gagg(X3, X1, X4, X2))
   U54_AG(X1, X2, X3, qscH_out_ga([], X4)) -> APPM_IN_GAGG(X3, X1, X4, X2)
   APPM_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U36_GAGG(X1, X2, X3, X4, X5, appM_in_gagg(X2, X3, X4, X5))
   APPM_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPM_IN_GAGG(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtA_in_aa(x1, x2)  =  gtA_in_aa

gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

partB_in_gaaa(x1, x2, x3, x4)  =  partB_in_gaaa(x1)

gtA_in_ga(x1, x2)  =  gtA_in_ga(x1)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

leC_in_ga(x1, x2)  =  leC_in_ga(x1)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

pD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_ggaaaaa(x1, x2)

partB_in_ggaa(x1, x2, x3, x4)  =  partB_in_ggaa(x1, x2)

.(x1, x2)  =  .(x1, x2)

gtA_in_gg(x1, x2)  =  gtA_in_gg(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

leC_in_gg(x1, x2)  =  leC_in_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qsF_in_ga(x1, x2)  =  qsF_in_ga(x1)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

appE_in_ggga(x1, x2, x3, x4)  =  appE_in_ggga(x1, x2, x3)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

pL_in_aaggg(x1, x2, x3, x4, x5)  =  pL_in_aaggg(x3, x4, x5)

qsH_in_aa(x1, x2)  =  qsH_in_aa

leG_in_aa(x1, x2)  =  leG_in_aa

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)

leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

appI_in_ggga(x1, x2, x3, x4)  =  appI_in_ggga(x1, x2, x3)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

appI_in_gaaa(x1, x2, x3, x4)  =  appI_in_gaaa(x1)

appK_in_ggag(x1, x2, x3, x4)  =  appK_in_ggag(x1, x2, x4)

appM_in_gagg(x1, x2, x3, x4)  =  appM_in_gagg(x1, x3, x4)

QSO_IN_AG(x1, x2)  =  QSO_IN_AG(x2)

U37_AG(x1, x2, x3, x4, x5)  =  U37_AG(x4, x5)

GTA_IN_AA(x1, x2)  =  GTA_IN_AA

U1_AA(x1, x2, x3)  =  U1_AA(x3)

U38_AG(x1, x2, x3, x4, x5)  =  U38_AG(x4, x5)

U39_AG(x1, x2, x3, x4, x5)  =  U39_AG(x4, x5)

PARTB_IN_GAAA(x1, x2, x3, x4)  =  PARTB_IN_GAAA(x1)

U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)

GTA_IN_GA(x1, x2)  =  GTA_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)

U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

LEC_IN_GA(x1, x2)  =  LEC_IN_GA(x1)

U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

U40_AG(x1, x2, x3, x4, x5)  =  U40_AG(x2, x4, x5)

U41_AG(x1, x2, x3, x4, x5)  =  U41_AG(x4, x5)

PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_GGAAAAA(x1, x2)

U9_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_GGAAAAA(x1, x2, x8)

PARTB_IN_GGAA(x1, x2, x3, x4)  =  PARTB_IN_GGAA(x1, x2)

U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)

GTA_IN_GG(x1, x2)  =  GTA_IN_GG(x1, x2)

U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)

U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)

U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

LEC_IN_GG(x1, x2)  =  LEC_IN_GG(x1, x2)

U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GGAAAAA(x1, x2, x8)

U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GGAAAAA(x1, x2, x8)

QSF_IN_GA(x1, x2)  =  QSF_IN_GA(x1)

U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x2, x4)

U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GGAAAAA(x1, x2, x4, x8)

U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GGAAAAA(x1, x2, x8)

U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GGAAAAA(x1, x2, x5, x8)

U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GGAAAAA(x1, x2, x8)

APPE_IN_GGGA(x1, x2, x3, x4)  =  APPE_IN_GGGA(x1, x2, x3)

U17_GGGA(x1, x2, x3, x4, x5, x6)  =  U17_GGGA(x1, x2, x3, x4, x6)

U42_AG(x1, x2, x3, x4, x5, x6)  =  U42_AG(x1, x4, x6)

U43_AG(x1, x2, x3, x4, x5)  =  U43_AG(x4, x5)

PL_IN_AAGGG(x1, x2, x3, x4, x5)  =  PL_IN_AAGGG(x3, x4, x5)

U33_AAGGG(x1, x2, x3, x4, x5, x6)  =  U33_AAGGG(x3, x4, x5, x6)

QSH_IN_AA(x1, x2)  =  QSH_IN_AA

U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)

U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(x5)

U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x5)

U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)

LEG_IN_AA(x1, x2)  =  LEG_IN_AA

U18_AA(x1, x2, x3)  =  U18_AA(x3)

U23_AA(x1, x2, x3, x4, x5)  =  U23_AA(x5)

U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)

U25_AA(x1, x2, x3, x4)  =  U25_AA(x4)

U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)

QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)

U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x2, x3, x5)

U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)

U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x2, x3, x5)

U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)

LEG_IN_GG(x1, x2)  =  LEG_IN_GG(x1, x2)

U18_GG(x1, x2, x3)  =  U18_GG(x1, x2, x3)

U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x1, x2, x3, x5)

U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)

U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)

U26_GA(x1, x2, x3, x4)  =  U26_GA(x1, x2, x4)

U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x4, x5)

U28_GA(x1, x2, x3, x4)  =  U28_GA(x1, x2, x4)

U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x1, x2, x4, x5)

U30_GA(x1, x2, x3, x4)  =  U30_GA(x1, x2, x4)

APPI_IN_GGGA(x1, x2, x3, x4)  =  APPI_IN_GGGA(x1, x2, x3)

U31_GGGA(x1, x2, x3, x4, x5, x6)  =  U31_GGGA(x1, x2, x3, x4, x6)

U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)

U28_AA(x1, x2, x3, x4)  =  U28_AA(x4)

U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x4, x5)

U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)

APPI_IN_GAAA(x1, x2, x3, x4)  =  APPI_IN_GAAA(x1)

U31_GAAA(x1, x2, x3, x4, x5, x6)  =  U31_GAAA(x1, x2, x6)

U34_AAGGG(x1, x2, x3, x4, x5, x6)  =  U34_AAGGG(x3, x4, x5, x6)

U35_AAGGG(x1, x2, x3, x4, x5, x6)  =  U35_AAGGG(x3, x4, x5, x6)

APPK_IN_GGAG(x1, x2, x3, x4)  =  APPK_IN_GGAG(x1, x2, x4)

U32_GGAG(x1, x2, x3, x4, x5, x6)  =  U32_GGAG(x1, x2, x3, x5, x6)

U44_AG(x1, x2, x3, x4, x5)  =  U44_AG(x4, x5)

U45_AG(x1, x2, x3, x4, x5)  =  U45_AG(x4, x5)

U46_AG(x1, x2, x3, x4, x5)  =  U46_AG(x4, x5)

U47_AG(x1, x2, x3, x4, x5)  =  U47_AG(x4, x5)

U48_AG(x1, x2, x3, x4, x5)  =  U48_AG(x4, x5)

U49_AG(x1, x2, x3, x4, x5, x6)  =  U49_AG(x1, x4, x6)

U50_AG(x1, x2, x3, x4, x5)  =  U50_AG(x4, x5)

U51_AG(x1, x2, x3)  =  U51_AG(x2, x3)

U52_AG(x1, x2, x3)  =  U52_AG(x2, x3)

U53_AG(x1, x2, x3)  =  U53_AG(x2, x3)

U54_AG(x1, x2, x3, x4)  =  U54_AG(x2, x3, x4)

U55_AG(x1, x2, x3)  =  U55_AG(x2, x3)

APPM_IN_GAGG(x1, x2, x3, x4)  =  APPM_IN_GAGG(x1, x3, x4)

U36_GAGG(x1, x2, x3, x4, x5, x6)  =  U36_GAGG(x1, x2, x4, x5, x6)


We have to consider all (P,R,Pi)-chains


Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES



----------------------------------------

(281)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U37_AG(X1, X2, X3, X4, gtA_in_aa(X1, X2))
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> GTA_IN_AA(X1, X2)
   GTA_IN_AA(s(X1), s(X2)) -> U1_AA(X1, X2, gtA_in_aa(X1, X2))
   GTA_IN_AA(s(X1), s(X2)) -> GTA_IN_AA(X1, X2)
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U38_AG(X1, X2, X3, X4, gtcA_in_aa(X1, X2))
   U38_AG(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> U39_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U38_AG(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GAAA(X1, X2, X3, X4, X5, gtA_in_ga(X1, X2))
   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> GTA_IN_GA(X1, X2)
   GTA_IN_GA(s(X1), s(X2)) -> U1_GA(X1, X2, gtA_in_ga(X1, X2))
   GTA_IN_GA(s(X1), s(X2)) -> GTA_IN_GA(X1, X2)
   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U3_GAAA(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U4_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5))
   U3_GAAA(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GAAA(X1, X2, X3, X4, X5, leC_in_ga(X1, X2))
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> LEC_IN_GA(X1, X2)
   LEC_IN_GA(s(X1), s(X2)) -> U8_GA(X1, X2, leC_in_ga(X1, X2))
   LEC_IN_GA(s(X1), s(X2)) -> LEC_IN_GA(X1, X2)
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GAAA(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U6_GAAA(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U7_GAAA(X1, X2, X3, X4, X5, partB_in_gaaa(X1, X3, X4, X5))
   U6_GAAA(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)
   U38_AG(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> U40_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6))
   U40_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U41_AG(X1, X2, X3, X4, pD_in_ggaaaaa(X2, X5, X7, X8, X9, X10, X11))
   U40_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> PD_IN_GGAAAAA(X2, X5, X7, X8, X9, X10, X11)
   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U9_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partB_in_ggaa(X1, X2, X3, X4))
   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> PARTB_IN_GGAA(X1, X2, X3, X4)
   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U2_GGAA(X1, X2, X3, X4, X5, gtA_in_gg(X1, X2))
   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> GTA_IN_GG(X1, X2)
   GTA_IN_GG(s(X1), s(X2)) -> U1_GG(X1, X2, gtA_in_gg(X1, X2))
   GTA_IN_GG(s(X1), s(X2)) -> GTA_IN_GG(X1, X2)
   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U3_GGAA(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U4_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5))
   U3_GGAA(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U5_GGAA(X1, X2, X3, X4, X5, leC_in_gg(X1, X2))
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> LEC_IN_GG(X1, X2)
   LEC_IN_GG(s(X1), s(X2)) -> U8_GG(X1, X2, leC_in_gg(X1, X2))
   LEC_IN_GG(s(X1), s(X2)) -> LEC_IN_GG(X1, X2)
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U6_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U7_GGAA(X1, X2, X3, X4, X5, partB_in_ggaa(X1, X3, X4, X5))
   U6_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)
   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U11_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X3, X5))
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5)
   QSF_IN_GA(.(X1, X2), X3) -> U16_GA(X1, X2, X3, pD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3)
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U13_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qsF_in_ga(X4, X6))
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6)
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U15_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, appE_in_ggga(X5, X1, X6, X7))
   U14_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> APPE_IN_GGGA(X5, X1, X6, X7)
   APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U17_GGGA(X1, X2, X3, X4, X5, appE_in_ggga(X2, X3, X4, X5))
   APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5)
   U40_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U42_AG(X1, X2, X3, X4, X6, qscN_in_gga(X2, X5, X7))
   U42_AG(X1, X2, X3, X4, X6, qscN_out_gga(X2, X5, X7)) -> U43_AG(X1, X2, X3, X4, pL_in_aaggg(X6, X8, X7, X1, X4))
   U42_AG(X1, X2, X3, X4, X6, qscN_out_gga(X2, X5, X7)) -> PL_IN_AAGGG(X6, X8, X7, X1, X4)
   PL_IN_AAGGG(X1, X2, X3, X4, X5) -> U33_AAGGG(X1, X2, X3, X4, X5, qsH_in_aa(X1, X2))
   PL_IN_AAGGG(X1, X2, X3, X4, X5) -> QSH_IN_AA(X1, X2)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U19_AA(X1, X2, X3, X4, gtA_in_aa(X1, X2))
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> GTA_IN_AA(X1, X2)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U20_AA(X1, X2, X3, X4, gtcA_in_aa(X1, X2))
   U20_AA(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> U21_AA(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U20_AA(X1, X2, X3, X4, gtcA_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U22_AA(X1, X2, X3, X4, leG_in_aa(X1, X2))
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> LEG_IN_AA(X1, X2)
   LEG_IN_AA(s(X1), s(X2)) -> U18_AA(X1, X2, leG_in_aa(X1, X2))
   LEG_IN_AA(s(X1), s(X2)) -> LEG_IN_AA(X1, X2)
   QSH_IN_AA(.(X1, .(X2, X3)), X4) -> U23_AA(X1, X2, X3, X4, lecG_in_aa(X1, X2))
   U23_AA(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> U24_AA(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U23_AA(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   QSH_IN_AA(.(X1, X2), X3) -> U25_AA(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U26_AA(X1, X2, X3, qsH_in_ga(X4, X6))
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4, X6)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U19_GA(X1, X2, X3, X4, gtA_in_gg(X1, X2))
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> GTA_IN_GG(X1, X2)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U20_GA(X1, X2, X3, X4, gtcA_in_gg(X1, X2))
   U20_GA(X1, X2, X3, X4, gtcA_out_gg(X1, X2)) -> U21_GA(X1, X2, X3, X4, partB_in_ggaa(X1, X3, X5, X6))
   U20_GA(X1, X2, X3, X4, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X5, X6)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U22_GA(X1, X2, X3, X4, leG_in_gg(X1, X2))
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> LEG_IN_GG(X1, X2)
   LEG_IN_GG(s(X1), s(X2)) -> U18_GG(X1, X2, leG_in_gg(X1, X2))
   LEG_IN_GG(s(X1), s(X2)) -> LEG_IN_GG(X1, X2)
   QSH_IN_GA(.(X1, .(X2, X3)), X4) -> U23_GA(X1, X2, X3, X4, lecG_in_gg(X1, X2))
   U23_GA(X1, X2, X3, X4, lecG_out_gg(X1, X2)) -> U24_GA(X1, X2, X3, X4, partB_in_ggaa(X1, X3, X5, X6))
   U23_GA(X1, X2, X3, X4, lecG_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X5, X6)
   QSH_IN_GA(.(X1, X2), X3) -> U25_GA(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U26_GA(X1, X2, X3, qsH_in_ga(X4, X6))
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4, X6)
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U27_GA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U28_GA(X1, X2, X3, qsH_in_ga(X5, X7))
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_GA(X5, X7)
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U29_GA(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U29_GA(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U30_GA(X1, X2, X3, appI_in_ggga(X6, X1, X7, X3))
   U29_GA(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> APPI_IN_GGGA(X6, X1, X7, X3)
   APPI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> U31_GGGA(X1, X2, X3, X4, X5, appI_in_ggga(X2, X3, X4, X5))
   APPI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GGGA(X2, X3, X4, X5)
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U27_AA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U28_AA(X1, X2, X3, qsH_in_aa(X5, X7))
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_AA(X5, X7)
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> U29_AA(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   U29_AA(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U30_AA(X1, X2, X3, appI_in_gaaa(X6, X1, X7, X3))
   U29_AA(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> APPI_IN_GAAA(X6, X1, X7, X3)
   APPI_IN_GAAA(.(X1, X2), X3, X4, .(X1, X5)) -> U31_GAAA(X1, X2, X3, X4, X5, appI_in_gaaa(X2, X3, X4, X5))
   APPI_IN_GAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAAA(X2, X3, X4, X5)
   PL_IN_AAGGG(X1, X2, X3, X4, X5) -> U34_AAGGG(X1, X2, X3, X4, X5, qscH_in_aa(X1, X2))
   U34_AAGGG(X1, X2, X3, X4, X5, qscH_out_aa(X1, X2)) -> U35_AAGGG(X1, X2, X3, X4, X5, appK_in_ggag(X3, X4, X2, X5))
   U34_AAGGG(X1, X2, X3, X4, X5, qscH_out_aa(X1, X2)) -> APPK_IN_GGAG(X3, X4, X2, X5)
   APPK_IN_GGAG(.(X1, X2), X3, X4, .(X1, X5)) -> U32_GGAG(X1, X2, X3, X4, X5, appK_in_ggag(X2, X3, X4, X5))
   APPK_IN_GGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPK_IN_GGAG(X2, X3, X4, X5)
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U44_AG(X1, X2, X3, X4, leG_in_aa(X1, X2))
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> LEG_IN_AA(X1, X2)
   QSO_IN_AG(.(X1, .(X2, X3)), X4) -> U45_AG(X1, X2, X3, X4, lecG_in_aa(X1, X2))
   U45_AG(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> U46_AG(X1, X2, X3, X4, partB_in_gaaa(X1, X3, X5, X6))
   U45_AG(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X5, X6)
   U45_AG(X1, X2, X3, X4, lecG_out_aa(X1, X2)) -> U47_AG(X1, X2, X3, X4, partcB_in_gaaa(X1, X3, X5, X6))
   U47_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U48_AG(X1, X2, X3, X4, qsH_in_ga(X5, X7))
   U47_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> QSH_IN_GA(X5, X7)
   U47_AG(X1, X2, X3, X4, partcB_out_gaaa(X1, X3, X5, X6)) -> U49_AG(X1, X2, X3, X4, X6, qscH_in_ga(X5, X7))
   U49_AG(X1, X2, X3, X4, X6, qscH_out_ga(X5, X7)) -> U50_AG(X1, X2, X3, X4, pL_in_aaggg(.(X2, X6), X8, X7, X1, X4))
   U49_AG(X1, X2, X3, X4, X6, qscH_out_ga(X5, X7)) -> PL_IN_AAGGG(.(X2, X6), X8, X7, X1, X4)
   QSO_IN_AG(.(X1, []), X2) -> U51_AG(X1, X2, qsH_in_ga([], X3))
   QSO_IN_AG(.(X1, []), X2) -> QSH_IN_GA([], X3)
   QSO_IN_AG(.(X1, []), X2) -> U52_AG(X1, X2, qscH_in_ga([], X3))
   U52_AG(X1, X2, qscH_out_ga([], X3)) -> U53_AG(X1, X2, qsH_in_ga([], X4))
   U52_AG(X1, X2, qscH_out_ga([], X3)) -> QSH_IN_GA([], X4)
   U52_AG(X1, X2, qscH_out_ga([], X3)) -> U54_AG(X1, X2, X3, qscH_in_ga([], X4))
   U54_AG(X1, X2, X3, qscH_out_ga([], X4)) -> U55_AG(X1, X2, appM_in_gagg(X3, X1, X4, X2))
   U54_AG(X1, X2, X3, qscH_out_ga([], X4)) -> APPM_IN_GAGG(X3, X1, X4, X2)
   APPM_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> U36_GAGG(X1, X2, X3, X4, X5, appM_in_gagg(X2, X3, X4, X5))
   APPM_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPM_IN_GAGG(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtA_in_aa(x1, x2)  =  gtA_in_aa

gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

partB_in_gaaa(x1, x2, x3, x4)  =  partB_in_gaaa(x1)

gtA_in_ga(x1, x2)  =  gtA_in_ga(x1)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

leC_in_ga(x1, x2)  =  leC_in_ga(x1)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

pD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  pD_in_ggaaaaa(x1, x2)

partB_in_ggaa(x1, x2, x3, x4)  =  partB_in_ggaa(x1, x2)

.(x1, x2)  =  .(x1, x2)

gtA_in_gg(x1, x2)  =  gtA_in_gg(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

leC_in_gg(x1, x2)  =  leC_in_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qsF_in_ga(x1, x2)  =  qsF_in_ga(x1)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

appE_in_ggga(x1, x2, x3, x4)  =  appE_in_ggga(x1, x2, x3)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

pL_in_aaggg(x1, x2, x3, x4, x5)  =  pL_in_aaggg(x3, x4, x5)

qsH_in_aa(x1, x2)  =  qsH_in_aa

leG_in_aa(x1, x2)  =  leG_in_aa

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

qsH_in_ga(x1, x2)  =  qsH_in_ga(x1)

leG_in_gg(x1, x2)  =  leG_in_gg(x1, x2)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

appI_in_ggga(x1, x2, x3, x4)  =  appI_in_ggga(x1, x2, x3)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

appI_in_gaaa(x1, x2, x3, x4)  =  appI_in_gaaa(x1)

appK_in_ggag(x1, x2, x3, x4)  =  appK_in_ggag(x1, x2, x4)

appM_in_gagg(x1, x2, x3, x4)  =  appM_in_gagg(x1, x3, x4)

QSO_IN_AG(x1, x2)  =  QSO_IN_AG(x2)

U37_AG(x1, x2, x3, x4, x5)  =  U37_AG(x4, x5)

GTA_IN_AA(x1, x2)  =  GTA_IN_AA

U1_AA(x1, x2, x3)  =  U1_AA(x3)

U38_AG(x1, x2, x3, x4, x5)  =  U38_AG(x4, x5)

U39_AG(x1, x2, x3, x4, x5)  =  U39_AG(x4, x5)

PARTB_IN_GAAA(x1, x2, x3, x4)  =  PARTB_IN_GAAA(x1)

U2_GAAA(x1, x2, x3, x4, x5, x6)  =  U2_GAAA(x1, x6)

GTA_IN_GA(x1, x2)  =  GTA_IN_GA(x1)

U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)

U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)

U4_GAAA(x1, x2, x3, x4, x5, x6)  =  U4_GAAA(x1, x6)

U5_GAAA(x1, x2, x3, x4, x5, x6)  =  U5_GAAA(x1, x6)

LEC_IN_GA(x1, x2)  =  LEC_IN_GA(x1)

U8_GA(x1, x2, x3)  =  U8_GA(x1, x3)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)

U7_GAAA(x1, x2, x3, x4, x5, x6)  =  U7_GAAA(x1, x6)

U40_AG(x1, x2, x3, x4, x5)  =  U40_AG(x2, x4, x5)

U41_AG(x1, x2, x3, x4, x5)  =  U41_AG(x4, x5)

PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_GGAAAAA(x1, x2)

U9_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U9_GGAAAAA(x1, x2, x8)

PARTB_IN_GGAA(x1, x2, x3, x4)  =  PARTB_IN_GGAA(x1, x2)

U2_GGAA(x1, x2, x3, x4, x5, x6)  =  U2_GGAA(x1, x2, x3, x6)

GTA_IN_GG(x1, x2)  =  GTA_IN_GG(x1, x2)

U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)

U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)

U4_GGAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAA(x1, x2, x3, x6)

U5_GGAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAA(x1, x2, x3, x6)

LEC_IN_GG(x1, x2)  =  LEC_IN_GG(x1, x2)

U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)

U7_GGAA(x1, x2, x3, x4, x5, x6)  =  U7_GGAA(x1, x2, x3, x6)

U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GGAAAAA(x1, x2, x8)

U11_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GGAAAAA(x1, x2, x8)

QSF_IN_GA(x1, x2)  =  QSF_IN_GA(x1)

U16_GA(x1, x2, x3, x4)  =  U16_GA(x1, x2, x4)

U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GGAAAAA(x1, x2, x4, x8)

U13_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GGAAAAA(x1, x2, x8)

U14_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GGAAAAA(x1, x2, x5, x8)

U15_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U15_GGAAAAA(x1, x2, x8)

APPE_IN_GGGA(x1, x2, x3, x4)  =  APPE_IN_GGGA(x1, x2, x3)

U17_GGGA(x1, x2, x3, x4, x5, x6)  =  U17_GGGA(x1, x2, x3, x4, x6)

U42_AG(x1, x2, x3, x4, x5, x6)  =  U42_AG(x1, x4, x6)

U43_AG(x1, x2, x3, x4, x5)  =  U43_AG(x4, x5)

PL_IN_AAGGG(x1, x2, x3, x4, x5)  =  PL_IN_AAGGG(x3, x4, x5)

U33_AAGGG(x1, x2, x3, x4, x5, x6)  =  U33_AAGGG(x3, x4, x5, x6)

QSH_IN_AA(x1, x2)  =  QSH_IN_AA

U19_AA(x1, x2, x3, x4, x5)  =  U19_AA(x5)

U20_AA(x1, x2, x3, x4, x5)  =  U20_AA(x5)

U21_AA(x1, x2, x3, x4, x5)  =  U21_AA(x5)

U22_AA(x1, x2, x3, x4, x5)  =  U22_AA(x5)

LEG_IN_AA(x1, x2)  =  LEG_IN_AA

U18_AA(x1, x2, x3)  =  U18_AA(x3)

U23_AA(x1, x2, x3, x4, x5)  =  U23_AA(x5)

U24_AA(x1, x2, x3, x4, x5)  =  U24_AA(x5)

U25_AA(x1, x2, x3, x4)  =  U25_AA(x4)

U26_AA(x1, x2, x3, x4)  =  U26_AA(x4)

QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)

U19_GA(x1, x2, x3, x4, x5)  =  U19_GA(x1, x2, x3, x5)

U20_GA(x1, x2, x3, x4, x5)  =  U20_GA(x1, x2, x3, x5)

U21_GA(x1, x2, x3, x4, x5)  =  U21_GA(x1, x2, x3, x5)

U22_GA(x1, x2, x3, x4, x5)  =  U22_GA(x1, x2, x3, x5)

LEG_IN_GG(x1, x2)  =  LEG_IN_GG(x1, x2)

U18_GG(x1, x2, x3)  =  U18_GG(x1, x2, x3)

U23_GA(x1, x2, x3, x4, x5)  =  U23_GA(x1, x2, x3, x5)

U24_GA(x1, x2, x3, x4, x5)  =  U24_GA(x1, x2, x3, x5)

U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)

U26_GA(x1, x2, x3, x4)  =  U26_GA(x1, x2, x4)

U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x4, x5)

U28_GA(x1, x2, x3, x4)  =  U28_GA(x1, x2, x4)

U29_GA(x1, x2, x3, x4, x5)  =  U29_GA(x1, x2, x4, x5)

U30_GA(x1, x2, x3, x4)  =  U30_GA(x1, x2, x4)

APPI_IN_GGGA(x1, x2, x3, x4)  =  APPI_IN_GGGA(x1, x2, x3)

U31_GGGA(x1, x2, x3, x4, x5, x6)  =  U31_GGGA(x1, x2, x3, x4, x6)

U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)

U28_AA(x1, x2, x3, x4)  =  U28_AA(x4)

U29_AA(x1, x2, x3, x4, x5)  =  U29_AA(x4, x5)

U30_AA(x1, x2, x3, x4)  =  U30_AA(x4)

APPI_IN_GAAA(x1, x2, x3, x4)  =  APPI_IN_GAAA(x1)

U31_GAAA(x1, x2, x3, x4, x5, x6)  =  U31_GAAA(x1, x2, x6)

U34_AAGGG(x1, x2, x3, x4, x5, x6)  =  U34_AAGGG(x3, x4, x5, x6)

U35_AAGGG(x1, x2, x3, x4, x5, x6)  =  U35_AAGGG(x3, x4, x5, x6)

APPK_IN_GGAG(x1, x2, x3, x4)  =  APPK_IN_GGAG(x1, x2, x4)

U32_GGAG(x1, x2, x3, x4, x5, x6)  =  U32_GGAG(x1, x2, x3, x5, x6)

U44_AG(x1, x2, x3, x4, x5)  =  U44_AG(x4, x5)

U45_AG(x1, x2, x3, x4, x5)  =  U45_AG(x4, x5)

U46_AG(x1, x2, x3, x4, x5)  =  U46_AG(x4, x5)

U47_AG(x1, x2, x3, x4, x5)  =  U47_AG(x4, x5)

U48_AG(x1, x2, x3, x4, x5)  =  U48_AG(x4, x5)

U49_AG(x1, x2, x3, x4, x5, x6)  =  U49_AG(x1, x4, x6)

U50_AG(x1, x2, x3, x4, x5)  =  U50_AG(x4, x5)

U51_AG(x1, x2, x3)  =  U51_AG(x2, x3)

U52_AG(x1, x2, x3)  =  U52_AG(x2, x3)

U53_AG(x1, x2, x3)  =  U53_AG(x2, x3)

U54_AG(x1, x2, x3, x4)  =  U54_AG(x2, x3, x4)

U55_AG(x1, x2, x3)  =  U55_AG(x2, x3)

APPM_IN_GAGG(x1, x2, x3, x4)  =  APPM_IN_GAGG(x1, x3, x4)

U36_GAGG(x1, x2, x3, x4, x5, x6)  =  U36_GAGG(x1, x2, x4, x5, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(282) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 17 SCCs with 98 less nodes.
----------------------------------------

(283)
Complex Obligation (AND)

----------------------------------------

(284)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPM_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPM_IN_GAGG(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

APPM_IN_GAGG(x1, x2, x3, x4)  =  APPM_IN_GAGG(x1, x3, x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(285) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(286)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPM_IN_GAGG(.(X1, X2), X3, X4, .(X1, X5)) -> APPM_IN_GAGG(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APPM_IN_GAGG(x1, x2, x3, x4)  =  APPM_IN_GAGG(x1, x3, x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(287) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(288)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPM_IN_GAGG(.(X1, X2), X4, .(X1, X5)) -> APPM_IN_GAGG(X2, X4, X5)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(289) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APPM_IN_GAGG(.(X1, X2), X4, .(X1, X5)) -> APPM_IN_GAGG(X2, X4, X5)
The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3


----------------------------------------

(290)
YES

----------------------------------------

(291)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPK_IN_GGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPK_IN_GGAG(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

APPK_IN_GGAG(x1, x2, x3, x4)  =  APPK_IN_GGAG(x1, x2, x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(292) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(293)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPK_IN_GGAG(.(X1, X2), X3, X4, .(X1, X5)) -> APPK_IN_GGAG(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APPK_IN_GGAG(x1, x2, x3, x4)  =  APPK_IN_GGAG(x1, x2, x4)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(294) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(295)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPK_IN_GGAG(.(X1, X2), X3, .(X1, X5)) -> APPK_IN_GGAG(X2, X3, X5)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(296) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APPK_IN_GGAG(.(X1, X2), X3, .(X1, X5)) -> APPK_IN_GGAG(X2, X3, X5)
The graph contains the following edges 1 > 1, 2 >= 2, 3 > 3


----------------------------------------

(297)
YES

----------------------------------------

(298)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPI_IN_GAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAAA(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

APPI_IN_GAAA(x1, x2, x3, x4)  =  APPI_IN_GAAA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(299) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(300)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPI_IN_GAAA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GAAA(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APPI_IN_GAAA(x1, x2, x3, x4)  =  APPI_IN_GAAA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(301) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(302)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPI_IN_GAAA(.(X1, X2)) -> APPI_IN_GAAA(X2)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(303) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APPI_IN_GAAA(.(X1, X2)) -> APPI_IN_GAAA(X2)
The graph contains the following edges 1 > 1


----------------------------------------

(304)
YES

----------------------------------------

(305)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

APPI_IN_GGGA(x1, x2, x3, x4)  =  APPI_IN_GGGA(x1, x2, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(306) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(307)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPI_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPI_IN_GGGA(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APPI_IN_GGGA(x1, x2, x3, x4)  =  APPI_IN_GGGA(x1, x2, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(308) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(309)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPI_IN_GGGA(.(X1, X2), X3, X4) -> APPI_IN_GGGA(X2, X3, X4)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(310) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APPI_IN_GGGA(.(X1, X2), X3, X4) -> APPI_IN_GGGA(X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


----------------------------------------

(311)
YES

----------------------------------------

(312)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEG_IN_GG(s(X1), s(X2)) -> LEG_IN_GG(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

LEG_IN_GG(x1, x2)  =  LEG_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(313) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(314)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEG_IN_GG(s(X1), s(X2)) -> LEG_IN_GG(X1, X2)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(315) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(316)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LEG_IN_GG(s(X1), s(X2)) -> LEG_IN_GG(X1, X2)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(317) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LEG_IN_GG(s(X1), s(X2)) -> LEG_IN_GG(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(318)
YES

----------------------------------------

(319)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEG_IN_AA(s(X1), s(X2)) -> LEG_IN_AA(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

LEG_IN_AA(x1, x2)  =  LEG_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(320) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(321)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEG_IN_AA(s(X1), s(X2)) -> LEG_IN_AA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

LEG_IN_AA(x1, x2)  =  LEG_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(322) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(323)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LEG_IN_AA -> LEG_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(324) NonTerminationLoopProof (COMPLETE)
We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LEG_IN_AA evaluates to  t =LEG_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
* Matcher: [ ]
* Semiunifier: [ ]

--------------------------------------------------------------------------------
Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LEG_IN_AA to LEG_IN_AA.




----------------------------------------

(325)
NO

----------------------------------------

(326)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

APPE_IN_GGGA(x1, x2, x3, x4)  =  APPE_IN_GGGA(x1, x2, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(327) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(328)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   APPE_IN_GGGA(.(X1, X2), X3, X4, .(X1, X5)) -> APPE_IN_GGGA(X2, X3, X4, X5)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)

APPE_IN_GGGA(x1, x2, x3, x4)  =  APPE_IN_GGGA(x1, x2, x3)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(329) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(330)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   APPE_IN_GGGA(.(X1, X2), X3, X4) -> APPE_IN_GGGA(X2, X3, X4)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(331) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*APPE_IN_GGGA(.(X1, X2), X3, X4) -> APPE_IN_GGGA(X2, X3, X4)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3


----------------------------------------

(332)
YES

----------------------------------------

(333)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEC_IN_GG(s(X1), s(X2)) -> LEC_IN_GG(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

LEC_IN_GG(x1, x2)  =  LEC_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(334) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(335)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEC_IN_GG(s(X1), s(X2)) -> LEC_IN_GG(X1, X2)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(336) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(337)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LEC_IN_GG(s(X1), s(X2)) -> LEC_IN_GG(X1, X2)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(338) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LEC_IN_GG(s(X1), s(X2)) -> LEC_IN_GG(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(339)
YES

----------------------------------------

(340)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GTA_IN_GG(s(X1), s(X2)) -> GTA_IN_GG(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

GTA_IN_GG(x1, x2)  =  GTA_IN_GG(x1, x2)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(341) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(342)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GTA_IN_GG(s(X1), s(X2)) -> GTA_IN_GG(X1, X2)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
----------------------------------------

(343) PiDPToQDPProof (EQUIVALENT)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(344)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GTA_IN_GG(s(X1), s(X2)) -> GTA_IN_GG(X1, X2)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(345) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*GTA_IN_GG(s(X1), s(X2)) -> GTA_IN_GG(X1, X2)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(346)
YES

----------------------------------------

(347)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U3_GGAA(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U6_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

PARTB_IN_GGAA(x1, x2, x3, x4)  =  PARTB_IN_GGAA(x1, x2)

U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(348) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(349)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GGAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GGAA(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U3_GGAA(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)
   PARTB_IN_GGAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GGAA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U6_GGAA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

PARTB_IN_GGAA(x1, x2, x3, x4)  =  PARTB_IN_GGAA(x1, x2)

U3_GGAA(x1, x2, x3, x4, x5, x6)  =  U3_GGAA(x1, x2, x3, x6)

U6_GGAA(x1, x2, x3, x4, x5, x6)  =  U6_GGAA(x1, x2, x3, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(350) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(351)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GGAA(X1, .(X2, X3)) -> U3_GGAA(X1, X2, X3, gtcA_in_gg(X1, X2))
   U3_GGAA(X1, X2, X3, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3)
   PARTB_IN_GGAA(X1, .(X2, X3)) -> U6_GGAA(X1, X2, X3, lecC_in_gg(X1, X2))
   U6_GGAA(X1, X2, X3, lecC_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3)

The TRS R consists of the following rules:

   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

   gtcA_in_gg(x0, x1)
   lecC_in_gg(x0, x1)
   U57_gg(x0, x1, x2)
   U62_gg(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(352) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*U3_GGAA(X1, X2, X3, gtcA_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2


*U6_GGAA(X1, X2, X3, lecC_out_gg(X1, X2)) -> PARTB_IN_GGAA(X1, X3)
The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2


*PARTB_IN_GGAA(X1, .(X2, X3)) -> U3_GGAA(X1, X2, X3, gtcA_in_gg(X1, X2))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


*PARTB_IN_GGAA(X1, .(X2, X3)) -> U6_GGAA(X1, X2, X3, lecC_in_gg(X1, X2))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3


----------------------------------------

(353)
YES

----------------------------------------

(354)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QSH_IN_GA(.(X1, X2), X3) -> U25_GA(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4, X6)
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U27_GA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_GA(X5, X7)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)

U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)

U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(355) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(356)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QSH_IN_GA(.(X1, X2), X3) -> U25_GA(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4, X6)
   U25_GA(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U27_GA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_GA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_GA(X5, X7)

The TRS R consists of the following rules:

   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

QSH_IN_GA(x1, x2)  =  QSH_IN_GA(x1)

U25_GA(x1, x2, x3, x4)  =  U25_GA(x1, x2, x4)

U27_GA(x1, x2, x3, x4, x5)  =  U27_GA(x1, x2, x4, x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(357) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(358)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   QSH_IN_GA(.(X1, X2)) -> U25_GA(X1, X2, partcJ_in_ggaa(X1, X2))
   U25_GA(X1, X2, partcJ_out_ggaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4)
   U25_GA(X1, X2, partcJ_out_ggaa(X1, X2, X4, X5)) -> U27_GA(X1, X2, X5, qscH_in_ga(X4))
   U27_GA(X1, X2, X5, qscH_out_ga(X4, X6)) -> QSH_IN_GA(X5)

The TRS R consists of the following rules:

   partcJ_in_ggaa(X1, .(X2, X3)) -> U80_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcJ_in_ggaa(X1, .(X2, X3)) -> U82_ggaa(X1, X2, X3, lecG_in_gg(X1, X2))
   partcJ_in_ggaa(X1, []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2)) -> U70_ga(X1, X2, partcJ_in_ggaa(X1, X2))
   qscH_in_ga([]) -> qscH_out_ga([], [])
   U80_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U82_ggaa(X1, X2, X3, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U70_ga(X1, X2, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X5, qscH_in_ga(X4))
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U81_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U83_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U71_ga(X1, X2, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X6, qscH_in_ga(X5))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U58_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U60_ggaa(X1, X2, X3, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], [])
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   U72_ga(X1, X2, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, appcI_in_ggga(X6, X1, X7))
   U58_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U60_ggaa(X1, X2, X3, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U73_ga(X1, X2, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   U59_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   appcI_in_ggga(.(X1, X2), X3, X4) -> U74_ggga(X1, X2, X3, X4, appcI_in_ggga(X2, X3, X4))
   appcI_in_ggga([], X1, X2) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   U74_ggga(X1, X2, X3, X4, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

   partcJ_in_ggaa(x0, x1)
   qscH_in_ga(x0)
   U80_ggaa(x0, x1, x2, x3)
   U82_ggaa(x0, x1, x2, x3)
   U70_ga(x0, x1, x2)
   gtcA_in_gg(x0, x1)
   U81_ggaa(x0, x1, x2, x3)
   lecG_in_gg(x0, x1)
   U83_ggaa(x0, x1, x2, x3)
   U71_ga(x0, x1, x2, x3)
   U57_gg(x0, x1, x2)
   partcB_in_ggaa(x0, x1)
   U69_gg(x0, x1, x2)
   U72_ga(x0, x1, x2, x3)
   U58_ggaa(x0, x1, x2, x3)
   U60_ggaa(x0, x1, x2, x3)
   U73_ga(x0, x1, x2)
   U59_ggaa(x0, x1, x2, x3)
   lecC_in_gg(x0, x1)
   U61_ggaa(x0, x1, x2, x3)
   appcI_in_ggga(x0, x1, x2)
   U62_gg(x0, x1, x2)
   U74_ggga(x0, x1, x2, x3, x4)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(359) QDPQMonotonicMRRProof (EQUIVALENT)
By using the Q-monotonic rule removal processor with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented such that it always occurs at a strongly monotonic position in a (P,Q,R)-chain.

Strictly oriented dependency pairs:

   QSH_IN_GA(.(X1, X2)) -> U25_GA(X1, X2, partcJ_in_ggaa(X1, X2))
   U25_GA(X1, X2, partcJ_out_ggaa(X1, X2, X4, X5)) -> QSH_IN_GA(X4)
   U27_GA(X1, X2, X5, qscH_out_ga(X4, X6)) -> QSH_IN_GA(X5)


Used ordering: Polynomial interpretation [POLO]:

   POL(.(x_1, x_2)) = 2 + 2*x_2
   POL(0) = 0
   POL(QSH_IN_GA(x_1)) = x_1
   POL(U25_GA(x_1, x_2, x_3)) = 1 + 2*x_3
   POL(U27_GA(x_1, x_2, x_3, x_4)) = 1 + 2*x_3
   POL(U57_gg(x_1, x_2, x_3)) = 0
   POL(U58_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3
   POL(U59_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4
   POL(U60_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3
   POL(U61_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4
   POL(U62_gg(x_1, x_2, x_3)) = 0
   POL(U69_gg(x_1, x_2, x_3)) = 0
   POL(U70_ga(x_1, x_2, x_3)) = 2
   POL(U71_ga(x_1, x_2, x_3, x_4)) = 0
   POL(U72_ga(x_1, x_2, x_3, x_4)) = 0
   POL(U73_ga(x_1, x_2, x_3)) = 0
   POL(U74_ggga(x_1, x_2, x_3, x_4, x_5)) = 0
   POL(U80_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3
   POL(U81_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4
   POL(U82_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_3
   POL(U83_ggaa(x_1, x_2, x_3, x_4)) = 2 + 2*x_4
   POL([]) = 0
   POL(appcI_in_ggga(x_1, x_2, x_3)) = 0
   POL(appcI_out_ggga(x_1, x_2, x_3, x_4)) = 0
   POL(gtcA_in_gg(x_1, x_2)) = 0
   POL(gtcA_out_gg(x_1, x_2)) = 0
   POL(lecC_in_gg(x_1, x_2)) = 1
   POL(lecC_out_gg(x_1, x_2)) = 0
   POL(lecG_in_gg(x_1, x_2)) = 0
   POL(lecG_out_gg(x_1, x_2)) = 0
   POL(partcB_in_ggaa(x_1, x_2)) = x_2
   POL(partcB_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(partcJ_in_ggaa(x_1, x_2)) = x_2
   POL(partcJ_out_ggaa(x_1, x_2, x_3, x_4)) = x_3 + x_4
   POL(qscH_in_ga(x_1)) = x_1
   POL(qscH_out_ga(x_1, x_2)) = 0
   POL(s(x_1)) = 0


----------------------------------------

(360)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U25_GA(X1, X2, partcJ_out_ggaa(X1, X2, X4, X5)) -> U27_GA(X1, X2, X5, qscH_in_ga(X4))

The TRS R consists of the following rules:

   partcJ_in_ggaa(X1, .(X2, X3)) -> U80_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcJ_in_ggaa(X1, .(X2, X3)) -> U82_ggaa(X1, X2, X3, lecG_in_gg(X1, X2))
   partcJ_in_ggaa(X1, []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2)) -> U70_ga(X1, X2, partcJ_in_ggaa(X1, X2))
   qscH_in_ga([]) -> qscH_out_ga([], [])
   U80_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U82_ggaa(X1, X2, X3, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U70_ga(X1, X2, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X5, qscH_in_ga(X4))
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U81_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U83_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U71_ga(X1, X2, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X6, qscH_in_ga(X5))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U58_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U60_ggaa(X1, X2, X3, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], [])
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   U72_ga(X1, X2, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, appcI_in_ggga(X6, X1, X7))
   U58_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U60_ggaa(X1, X2, X3, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U73_ga(X1, X2, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   U59_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   appcI_in_ggga(.(X1, X2), X3, X4) -> U74_ggga(X1, X2, X3, X4, appcI_in_ggga(X2, X3, X4))
   appcI_in_ggga([], X1, X2) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   U74_ggga(X1, X2, X3, X4, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

   partcJ_in_ggaa(x0, x1)
   qscH_in_ga(x0)
   U80_ggaa(x0, x1, x2, x3)
   U82_ggaa(x0, x1, x2, x3)
   U70_ga(x0, x1, x2)
   gtcA_in_gg(x0, x1)
   U81_ggaa(x0, x1, x2, x3)
   lecG_in_gg(x0, x1)
   U83_ggaa(x0, x1, x2, x3)
   U71_ga(x0, x1, x2, x3)
   U57_gg(x0, x1, x2)
   partcB_in_ggaa(x0, x1)
   U69_gg(x0, x1, x2)
   U72_ga(x0, x1, x2, x3)
   U58_ggaa(x0, x1, x2, x3)
   U60_ggaa(x0, x1, x2, x3)
   U73_ga(x0, x1, x2)
   U59_ggaa(x0, x1, x2, x3)
   lecC_in_gg(x0, x1)
   U61_ggaa(x0, x1, x2, x3)
   appcI_in_ggga(x0, x1, x2)
   U62_gg(x0, x1, x2)
   U74_ggga(x0, x1, x2, x3, x4)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(361) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
----------------------------------------

(362)
TRUE

----------------------------------------

(363)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5)
   QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3)
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_GGAAAAA(x1, x2)

U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GGAAAAA(x1, x2, x8)

QSF_IN_GA(x1, x2)  =  QSF_IN_GA(x1)

U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GGAAAAA(x1, x2, x4, x8)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(364) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(365)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PD_IN_GGAAAAA(X1, X2, X3, X4, X5, X6, X7) -> U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3, X5)
   QSF_IN_GA(.(X1, X2), X3) -> PD_IN_GGAAAAA(X1, X2, X4, X5, X6, X7, X3)
   U10_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   U12_GGAAAAA(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4, X6)

The TRS R consists of the following rules:

   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

0  =  0

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

PD_IN_GGAAAAA(x1, x2, x3, x4, x5, x6, x7)  =  PD_IN_GGAAAAA(x1, x2)

U10_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GGAAAAA(x1, x2, x8)

QSF_IN_GA(x1, x2)  =  QSF_IN_GA(x1)

U12_GGAAAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U12_GGAAAAA(x1, x2, x4, x8)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(366) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(367)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PD_IN_GGAAAAA(X1, X2) -> U10_GGAAAAA(X1, X2, partcB_in_ggaa(X1, X2))
   U10_GGAAAAA(X1, X2, partcB_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3)
   QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2)
   U10_GGAAAAA(X1, X2, partcB_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X4, qscF_in_ga(X3))
   U12_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4)

The TRS R consists of the following rules:

   partcB_in_ggaa(X1, .(X2, X3)) -> U58_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U60_ggaa(X1, X2, X3, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], [])
   qscF_in_ga(.(X1, X2)) -> U67_ga(X1, X2, qcD_in_ggaaaaa(X1, X2))
   qscF_in_ga([]) -> qscF_out_ga([], [])
   U58_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U60_ggaa(X1, X2, X3, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U67_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U59_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   qcD_in_ggaaaaa(X1, X2) -> U63_ggaaaaa(X1, X2, partcB_in_ggaa(X1, X2))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   U63_ggaaaaa(X1, X2, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3))
   U64_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4))
   U65_ggaaaaa(X1, X2, X3, X4, X5, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_in_ggga(X5, X1, X6))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   appcE_in_ggga(.(X1, X2), X3, X4) -> U68_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4))
   appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

   partcB_in_ggaa(x0, x1)
   qscF_in_ga(x0)
   U58_ggaa(x0, x1, x2, x3)
   U60_ggaa(x0, x1, x2, x3)
   U67_ga(x0, x1, x2)
   gtcA_in_gg(x0, x1)
   U59_ggaa(x0, x1, x2, x3)
   lecC_in_gg(x0, x1)
   U61_ggaa(x0, x1, x2, x3)
   qcD_in_ggaaaaa(x0, x1)
   U57_gg(x0, x1, x2)
   U62_gg(x0, x1, x2)
   U63_ggaaaaa(x0, x1, x2)
   U64_ggaaaaa(x0, x1, x2, x3, x4)
   U65_ggaaaaa(x0, x1, x2, x3, x4, x5)
   U66_ggaaaaa(x0, x1, x2, x3, x4, x5, x6)
   appcE_in_ggga(x0, x1, x2)
   U68_ggga(x0, x1, x2, x3, x4)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(368) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   U10_GGAAAAA(X1, X2, partcB_out_ggaa(X1, X2, X3, X4)) -> QSF_IN_GA(X3)
   QSF_IN_GA(.(X1, X2)) -> PD_IN_GGAAAAA(X1, X2)
   U12_GGAAAAA(X1, X2, X4, qscF_out_ga(X3, X5)) -> QSF_IN_GA(X4)
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U10_GGAAAAA_3(x_1, ..., x_3) ) = 2x_3 + 2
POL( U12_GGAAAAA_4(x_1, ..., x_4) ) = 2x_3 + 2
POL( U63_ggaaaaa_3(x_1, ..., x_3) ) = max{0, x_2 - 2}
POL( partcB_in_ggaa_2(x_1, x_2) ) = x_2
POL( ._2(x_1, x_2) ) = x_2 + 1
POL( U58_ggaa_4(x_1, ..., x_4) ) = x_3 + 1
POL( gtcA_in_gg_2(x_1, x_2) ) = 0
POL( U60_ggaa_4(x_1, ..., x_4) ) = x_3 + 1
POL( lecC_in_gg_2(x_1, x_2) ) = 0
POL( [] ) = 0
POL( partcB_out_ggaa_4(x_1, ..., x_4) ) = x_3 + x_4
POL( qscF_in_ga_1(x_1) ) = 2x_1
POL( U67_ga_3(x_1, ..., x_3) ) = 2x_1 + 2x_3
POL( qcD_in_ggaaaaa_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2
POL( qscF_out_ga_2(x_1, x_2) ) = max{0, x_2 - 2}
POL( U64_ggaaaaa_5(x_1, ..., x_5) ) = 2x_1 + x_2 + 2x_4 + 2x_5 + 2
POL( qcD_out_ggaaaaa_7(x_1, ..., x_7) ) = 2x_4 + 1
POL( U65_ggaaaaa_6(x_1, ..., x_6) ) = 2x_1 + 2x_3 + x_5 + 2
POL( U66_ggaaaaa_7(x_1, ..., x_7) ) = x_3 + 2x_4 + x_6 + 2
POL( appcE_in_ggga_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( gtcA_out_gg_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( U59_ggaa_4(x_1, ..., x_4) ) = x_4 + 1
POL( s_1(x_1) ) = max{0, -2}
POL( U57_gg_3(x_1, ..., x_3) ) = 2x_1 + 2
POL( 0 ) = 0
POL( U62_gg_3(x_1, ..., x_3) ) = max{0, 2x_2 - 2}
POL( lecC_out_gg_2(x_1, x_2) ) = max{0, 2x_2 - 2}
POL( U61_ggaa_4(x_1, ..., x_4) ) = x_4 + 1
POL( U68_ggga_5(x_1, ..., x_5) ) = max{0, 2x_1 - 2}
POL( appcE_out_ggga_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 - 2}
POL( PD_IN_GGAAAAA_2(x_1, x_2) ) = 2x_2 + 2
POL( QSF_IN_GA_1(x_1) ) = 2x_1 + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   partcB_in_ggaa(X1, .(X2, X3)) -> U58_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U60_ggaa(X1, X2, X3, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], [])
   U58_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U59_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   U60_ggaa(X1, X2, X3, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U61_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))


----------------------------------------

(369)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PD_IN_GGAAAAA(X1, X2) -> U10_GGAAAAA(X1, X2, partcB_in_ggaa(X1, X2))
   U10_GGAAAAA(X1, X2, partcB_out_ggaa(X1, X2, X3, X4)) -> U12_GGAAAAA(X1, X2, X4, qscF_in_ga(X3))

The TRS R consists of the following rules:

   partcB_in_ggaa(X1, .(X2, X3)) -> U58_ggaa(X1, X2, X3, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3)) -> U60_ggaa(X1, X2, X3, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, []) -> partcB_out_ggaa(X1, [], [], [])
   qscF_in_ga(.(X1, X2)) -> U67_ga(X1, X2, qcD_in_ggaaaaa(X1, X2))
   qscF_in_ga([]) -> qscF_out_ga([], [])
   U58_ggaa(X1, X2, X3, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U60_ggaa(X1, X2, X3, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, partcB_in_ggaa(X1, X3))
   U67_ga(X1, X2, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U59_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   qcD_in_ggaaaaa(X1, X2) -> U63_ggaaaaa(X1, X2, partcB_in_ggaa(X1, X2))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   U63_ggaaaaa(X1, X2, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, qscF_in_ga(X3))
   U64_ggaaaaa(X1, X2, X3, X4, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, qscF_in_ga(X4))
   U65_ggaaaaa(X1, X2, X3, X4, X5, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_in_ggga(X5, X1, X6))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   appcE_in_ggga(.(X1, X2), X3, X4) -> U68_ggga(X1, X2, X3, X4, appcE_in_ggga(X2, X3, X4))
   appcE_in_ggga([], X1, X2) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))

The set Q consists of the following terms:

   partcB_in_ggaa(x0, x1)
   qscF_in_ga(x0)
   U58_ggaa(x0, x1, x2, x3)
   U60_ggaa(x0, x1, x2, x3)
   U67_ga(x0, x1, x2)
   gtcA_in_gg(x0, x1)
   U59_ggaa(x0, x1, x2, x3)
   lecC_in_gg(x0, x1)
   U61_ggaa(x0, x1, x2, x3)
   qcD_in_ggaaaaa(x0, x1)
   U57_gg(x0, x1, x2)
   U62_gg(x0, x1, x2)
   U63_ggaaaaa(x0, x1, x2)
   U64_ggaaaaa(x0, x1, x2, x3, x4)
   U65_ggaaaaa(x0, x1, x2, x3, x4, x5)
   U66_ggaaaaa(x0, x1, x2, x3, x4, x5, x6)
   appcE_in_ggga(x0, x1, x2)
   U68_ggga(x0, x1, x2, x3, x4)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(370) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
----------------------------------------

(371)
TRUE

----------------------------------------

(372)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEC_IN_GA(s(X1), s(X2)) -> LEC_IN_GA(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

LEC_IN_GA(x1, x2)  =  LEC_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(373) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(374)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   LEC_IN_GA(s(X1), s(X2)) -> LEC_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

LEC_IN_GA(x1, x2)  =  LEC_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(375) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(376)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   LEC_IN_GA(s(X1)) -> LEC_IN_GA(X1)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(377) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*LEC_IN_GA(s(X1)) -> LEC_IN_GA(X1)
The graph contains the following edges 1 > 1


----------------------------------------

(378)
YES

----------------------------------------

(379)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GTA_IN_GA(s(X1), s(X2)) -> GTA_IN_GA(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

GTA_IN_GA(x1, x2)  =  GTA_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(380) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(381)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GTA_IN_GA(s(X1), s(X2)) -> GTA_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

GTA_IN_GA(x1, x2)  =  GTA_IN_GA(x1)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(382) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(383)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   GTA_IN_GA(s(X1)) -> GTA_IN_GA(X1)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(384) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*GTA_IN_GA(s(X1)) -> GTA_IN_GA(X1)
The graph contains the following edges 1 > 1


----------------------------------------

(385)
YES

----------------------------------------

(386)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U3_GAAA(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GAAA(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U6_GAAA(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

PARTB_IN_GAAA(x1, x2, x3, x4)  =  PARTB_IN_GAAA(x1)

U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(387) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(388)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GAAA(X1, .(X2, X3), .(X2, X4), X5) -> U3_GAAA(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U3_GAAA(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)
   PARTB_IN_GAAA(X1, .(X2, X3), X4, .(X2, X5)) -> U6_GAAA(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U6_GAAA(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1, X3, X4, X5)

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

.(x1, x2)  =  .(x1, x2)

PARTB_IN_GAAA(x1, x2, x3, x4)  =  PARTB_IN_GAAA(x1)

U3_GAAA(x1, x2, x3, x4, x5, x6)  =  U3_GAAA(x1, x6)

U6_GAAA(x1, x2, x3, x4, x5, x6)  =  U6_GAAA(x1, x6)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(389) PiDPToQDPProof (SOUND)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
----------------------------------------

(390)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GAAA(X1) -> U3_GAAA(X1, gtcA_in_ga(X1))
   U3_GAAA(X1, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1)
   PARTB_IN_GAAA(X1) -> U6_GAAA(X1, lecC_in_ga(X1))
   U6_GAAA(X1, lecC_out_ga(X1)) -> PARTB_IN_GAAA(X1)

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(391) TransformationProof (SOUND)
By narrowing [LPAR04] the rule PARTB_IN_GAAA(X1) -> U3_GAAA(X1, gtcA_in_ga(X1)) at position [1] we obtained the following new rules [LPAR04]:

   (PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0))),PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0))))
   (PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0)),PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0)))


----------------------------------------

(392)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U3_GAAA(X1, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1)
   PARTB_IN_GAAA(X1) -> U6_GAAA(X1, lecC_in_ga(X1))
   U6_GAAA(X1, lecC_out_ga(X1)) -> PARTB_IN_GAAA(X1)
   PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0)))
   PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0))

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(393) TransformationProof (SOUND)
By narrowing [LPAR04] the rule PARTB_IN_GAAA(X1) -> U6_GAAA(X1, lecC_in_ga(X1)) at position [1] we obtained the following new rules [LPAR04]:

   (PARTB_IN_GAAA(s(x0)) -> U6_GAAA(s(x0), U62_ga(x0, lecC_in_ga(x0))),PARTB_IN_GAAA(s(x0)) -> U6_GAAA(s(x0), U62_ga(x0, lecC_in_ga(x0))))
   (PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0)),PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0)))


----------------------------------------

(394)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U3_GAAA(X1, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1)
   U6_GAAA(X1, lecC_out_ga(X1)) -> PARTB_IN_GAAA(X1)
   PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0)))
   PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0))
   PARTB_IN_GAAA(s(x0)) -> U6_GAAA(s(x0), U62_ga(x0, lecC_in_ga(x0)))
   PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0))

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(395) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U3_GAAA(X1, gtcA_out_ga(X1, X2)) -> PARTB_IN_GAAA(X1) we obtained the following new rules [LPAR04]:

   (U3_GAAA(s(z0), gtcA_out_ga(s(z0), x1)) -> PARTB_IN_GAAA(s(z0)),U3_GAAA(s(z0), gtcA_out_ga(s(z0), x1)) -> PARTB_IN_GAAA(s(z0)))
   (U3_GAAA(s(0), gtcA_out_ga(s(0), 0)) -> PARTB_IN_GAAA(s(0)),U3_GAAA(s(0), gtcA_out_ga(s(0), 0)) -> PARTB_IN_GAAA(s(0)))


----------------------------------------

(396)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U6_GAAA(X1, lecC_out_ga(X1)) -> PARTB_IN_GAAA(X1)
   PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0)))
   PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0))
   PARTB_IN_GAAA(s(x0)) -> U6_GAAA(s(x0), U62_ga(x0, lecC_in_ga(x0)))
   PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0))
   U3_GAAA(s(z0), gtcA_out_ga(s(z0), x1)) -> PARTB_IN_GAAA(s(z0))
   U3_GAAA(s(0), gtcA_out_ga(s(0), 0)) -> PARTB_IN_GAAA(s(0))

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(397) TransformationProof (EQUIVALENT)
By instantiating [LPAR04] the rule U6_GAAA(X1, lecC_out_ga(X1)) -> PARTB_IN_GAAA(X1) we obtained the following new rules [LPAR04]:

   (U6_GAAA(s(z0), lecC_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0)),U6_GAAA(s(z0), lecC_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0)))
   (U6_GAAA(0, lecC_out_ga(0)) -> PARTB_IN_GAAA(0),U6_GAAA(0, lecC_out_ga(0)) -> PARTB_IN_GAAA(0))


----------------------------------------

(398)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0)))
   PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0))
   PARTB_IN_GAAA(s(x0)) -> U6_GAAA(s(x0), U62_ga(x0, lecC_in_ga(x0)))
   PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0))
   U3_GAAA(s(z0), gtcA_out_ga(s(z0), x1)) -> PARTB_IN_GAAA(s(z0))
   U3_GAAA(s(0), gtcA_out_ga(s(0), 0)) -> PARTB_IN_GAAA(s(0))
   U6_GAAA(s(z0), lecC_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0))
   U6_GAAA(0, lecC_out_ga(0)) -> PARTB_IN_GAAA(0)

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(399) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
----------------------------------------

(400)
Complex Obligation (AND)

----------------------------------------

(401)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U6_GAAA(0, lecC_out_ga(0)) -> PARTB_IN_GAAA(0)
   PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0))

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(402) UsableRulesProof (EQUIVALENT)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(403)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U6_GAAA(0, lecC_out_ga(0)) -> PARTB_IN_GAAA(0)
   PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0))

R is empty.
The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(404) QReductionProof (EQUIVALENT)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)


----------------------------------------

(405)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U6_GAAA(0, lecC_out_ga(0)) -> PARTB_IN_GAAA(0)
   PARTB_IN_GAAA(0) -> U6_GAAA(0, lecC_out_ga(0))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
----------------------------------------

(406)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U3_GAAA(s(z0), gtcA_out_ga(s(z0), x1)) -> PARTB_IN_GAAA(s(z0))
   PARTB_IN_GAAA(s(x0)) -> U3_GAAA(s(x0), U57_ga(x0, gtcA_in_ga(x0)))
   U3_GAAA(s(0), gtcA_out_ga(s(0), 0)) -> PARTB_IN_GAAA(s(0))
   PARTB_IN_GAAA(s(0)) -> U3_GAAA(s(0), gtcA_out_ga(s(0), 0))
   PARTB_IN_GAAA(s(x0)) -> U6_GAAA(s(x0), U62_ga(x0, lecC_in_ga(x0)))
   U6_GAAA(s(z0), lecC_out_ga(s(z0))) -> PARTB_IN_GAAA(s(z0))

The TRS R consists of the following rules:

   gtcA_in_ga(s(X1)) -> U57_ga(X1, gtcA_in_ga(X1))
   gtcA_in_ga(s(0)) -> gtcA_out_ga(s(0), 0)
   lecC_in_ga(s(X1)) -> U62_ga(X1, lecC_in_ga(X1))
   lecC_in_ga(0) -> lecC_out_ga(0)
   U57_ga(X1, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, lecC_out_ga(X1)) -> lecC_out_ga(s(X1))

The set Q consists of the following terms:

   gtcA_in_ga(x0)
   lecC_in_ga(x0)
   U57_ga(x0, x1)
   U62_ga(x0, x1)

We have to consider all (P,Q,R)-chains.
----------------------------------------

(407)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GTA_IN_AA(s(X1), s(X2)) -> GTA_IN_AA(X1, X2)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

GTA_IN_AA(x1, x2)  =  GTA_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(408) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(409)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   GTA_IN_AA(s(X1), s(X2)) -> GTA_IN_AA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)

GTA_IN_AA(x1, x2)  =  GTA_IN_AA


We have to consider all (P,R,Pi)-chains
----------------------------------------

(410)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QSH_IN_AA(.(X1, X2), X3) -> U25_AA(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U27_AA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_AA(X5, X7)

The TRS R consists of the following rules:

   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   qscF_in_ga(.(X1, X2), X3) -> U67_ga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   qcD_in_ggaaaaa(X1, X2, X3, X4, X5, X6, X7) -> U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_in_ggaa(X1, X2, X3, X4))
   U63_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, partcB_out_ggaa(X1, X2, X3, X4)) -> U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X3, X5))
   qscF_in_ga([], []) -> qscF_out_ga([], [])
   U64_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X3, X5)) -> U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_in_ga(X4, X6))
   U65_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, qscF_out_ga(X4, X6)) -> U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_in_ggga(X5, X1, X6, X7))
   appcE_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U68_ggga(X1, X2, X3, X4, X5, appcE_in_ggga(X2, X3, X4, X5))
   appcE_in_ggga([], X1, X2, .(X1, X2)) -> appcE_out_ggga([], X1, X2, .(X1, X2))
   U68_ggga(X1, X2, X3, X4, X5, appcE_out_ggga(X2, X3, X4, X5)) -> appcE_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U66_ggaaaaa(X1, X2, X3, X4, X5, X6, X7, appcE_out_ggga(X5, X1, X6, X7)) -> qcD_out_ggaaaaa(X1, X2, X3, X4, X5, X6, X7)
   U67_ga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscF_out_ga(.(X1, X2), X3)
   qscN_in_gga(X1, X2, X3) -> U79_gga(X1, X2, X3, qcD_in_ggaaaaa(X1, X2, X4, X5, X6, X7, X3))
   U79_gga(X1, X2, X3, qcD_out_ggaaaaa(X1, X2, X4, X5, X6, X7, X3)) -> qscN_out_gga(X1, X2, X3)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   qscH_in_aa(.(X1, X2), X3) -> U70_aa(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U70_aa(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U71_aa(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   U71_aa(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_aa(X1, X2, X3, X6, qscH_in_aa(X5, X7))
   qscH_in_aa([], []) -> qscH_out_aa([], [])
   U72_aa(X1, X2, X3, X6, qscH_out_aa(X5, X7)) -> U73_aa(X1, X2, X3, appcI_in_gaaa(X6, X1, X7, X3))
   appcI_in_gaaa(.(X1, X2), X3, X4, .(X1, X5)) -> U74_gaaa(X1, X2, X3, X4, X5, appcI_in_gaaa(X2, X3, X4, X5))
   appcI_in_gaaa([], X1, X2, .(X1, X2)) -> appcI_out_gaaa([], X1, X2, .(X1, X2))
   U74_gaaa(X1, X2, X3, X4, X5, appcI_out_gaaa(X2, X3, X4, X5)) -> appcI_out_gaaa(.(X1, X2), X3, X4, .(X1, X5))
   U73_aa(X1, X2, X3, appcI_out_gaaa(X6, X1, X7, X3)) -> qscH_out_aa(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

qscF_in_ga(x1, x2)  =  qscF_in_ga(x1)

U67_ga(x1, x2, x3, x4)  =  U67_ga(x1, x2, x4)

qcD_in_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_in_ggaaaaa(x1, x2)

U63_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U63_ggaaaaa(x1, x2, x8)

U64_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U64_ggaaaaa(x1, x2, x3, x4, x8)

qscF_out_ga(x1, x2)  =  qscF_out_ga(x1, x2)

U65_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U65_ggaaaaa(x1, x2, x3, x4, x5, x8)

U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U66_ggaaaaa(x1, x2, x3, x4, x5, x6, x8)

appcE_in_ggga(x1, x2, x3, x4)  =  appcE_in_ggga(x1, x2, x3)

U68_ggga(x1, x2, x3, x4, x5, x6)  =  U68_ggga(x1, x2, x3, x4, x6)

appcE_out_ggga(x1, x2, x3, x4)  =  appcE_out_ggga(x1, x2, x3, x4)

qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)  =  qcD_out_ggaaaaa(x1, x2, x3, x4, x5, x6, x7)

qscN_in_gga(x1, x2, x3)  =  qscN_in_gga(x1, x2)

U79_gga(x1, x2, x3, x4)  =  U79_gga(x1, x2, x4)

qscN_out_gga(x1, x2, x3)  =  qscN_out_gga(x1, x2, x3)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

qscH_in_aa(x1, x2)  =  qscH_in_aa

U70_aa(x1, x2, x3, x4)  =  U70_aa(x4)

U71_aa(x1, x2, x3, x4, x5, x6)  =  U71_aa(x6)

U72_aa(x1, x2, x3, x4, x5)  =  U72_aa(x4, x5)

qscH_out_aa(x1, x2)  =  qscH_out_aa

U73_aa(x1, x2, x3, x4)  =  U73_aa(x4)

appcI_in_gaaa(x1, x2, x3, x4)  =  appcI_in_gaaa(x1)

U74_gaaa(x1, x2, x3, x4, x5, x6)  =  U74_gaaa(x1, x2, x6)

appcI_out_gaaa(x1, x2, x3, x4)  =  appcI_out_gaaa(x1)

QSH_IN_AA(x1, x2)  =  QSH_IN_AA

U25_AA(x1, x2, x3, x4)  =  U25_AA(x4)

U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(411) UsableRulesProof (EQUIVALENT)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
----------------------------------------

(412)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   QSH_IN_AA(.(X1, X2), X3) -> U25_AA(X1, X2, X3, partcJ_in_aaaa(X1, X2, X4, X5))
   U25_AA(X1, X2, X3, partcJ_out_aaaa(X1, X2, X4, X5)) -> U27_AA(X1, X2, X3, X5, qscH_in_ga(X4, X6))
   U27_AA(X1, X2, X3, X5, qscH_out_ga(X4, X6)) -> QSH_IN_AA(X5, X7)

The TRS R consists of the following rules:

   partcJ_in_aaaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_aaaa(X1, X2, X3, X4, X5, gtcA_in_aa(X1, X2))
   partcJ_in_aaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_aaaa(X1, X2, X3, X4, X5, lecG_in_aa(X1, X2))
   partcJ_in_aaaa(X1, [], [], []) -> partcJ_out_aaaa(X1, [], [], [])
   qscH_in_ga(.(X1, X2), X3) -> U70_ga(X1, X2, X3, partcJ_in_ggaa(X1, X2, X4, X5))
   qscH_in_ga([], []) -> qscH_out_ga([], [])
   U80_aaaa(X1, X2, X3, X4, X5, gtcA_out_aa(X1, X2)) -> U81_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U82_aaaa(X1, X2, X3, X4, X5, lecG_out_aa(X1, X2)) -> U83_aaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U70_ga(X1, X2, X3, partcJ_out_ggaa(X1, X2, X4, X5)) -> U71_ga(X1, X2, X3, X4, X5, qscH_in_ga(X4, X6))
   gtcA_in_aa(s(X1), s(X2)) -> U57_aa(X1, X2, gtcA_in_aa(X1, X2))
   gtcA_in_aa(s(0), 0) -> gtcA_out_aa(s(0), 0)
   U81_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), .(X2, X4), X5)
   lecG_in_aa(s(X1), s(X2)) -> U69_aa(X1, X2, lecG_in_aa(X1, X2))
   lecG_in_aa(0, s(X1)) -> lecG_out_aa(0, s(X1))
   lecG_in_aa(0, 0) -> lecG_out_aa(0, 0)
   U83_aaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcJ_out_aaaa(X1, .(X2, X3), X4, .(X2, X5))
   partcJ_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U80_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   partcJ_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U82_ggaa(X1, X2, X3, X4, X5, lecG_in_gg(X1, X2))
   partcJ_in_ggaa(X1, [], [], []) -> partcJ_out_ggaa(X1, [], [], [])
   U71_ga(X1, X2, X3, X4, X5, qscH_out_ga(X4, X6)) -> U72_ga(X1, X2, X3, X6, qscH_in_ga(X5, X7))
   U57_aa(X1, X2, gtcA_out_aa(X1, X2)) -> gtcA_out_aa(s(X1), s(X2))
   partcB_in_gaaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_gaaa(X1, X2, X3, X4, X5, gtcA_in_ga(X1, X2))
   partcB_in_gaaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_gaaa(X1, X2, X3, X4, X5, lecC_in_ga(X1, X2))
   partcB_in_gaaa(X1, [], [], []) -> partcB_out_gaaa(X1, [], [], [])
   U69_aa(X1, X2, lecG_out_aa(X1, X2)) -> lecG_out_aa(s(X1), s(X2))
   U80_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U81_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U82_ggaa(X1, X2, X3, X4, X5, lecG_out_gg(X1, X2)) -> U83_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U72_ga(X1, X2, X3, X6, qscH_out_ga(X5, X7)) -> U73_ga(X1, X2, X3, appcI_in_ggga(X6, X1, X7, X3))
   U58_gaaa(X1, X2, X3, X4, X5, gtcA_out_ga(X1, X2)) -> U59_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   U60_gaaa(X1, X2, X3, X4, X5, lecC_out_ga(X1, X2)) -> U61_gaaa(X1, X2, X3, X4, X5, partcB_in_gaaa(X1, X3, X4, X5))
   gtcA_in_gg(s(X1), s(X2)) -> U57_gg(X1, X2, gtcA_in_gg(X1, X2))
   gtcA_in_gg(s(0), 0) -> gtcA_out_gg(s(0), 0)
   U81_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecG_in_gg(s(X1), s(X2)) -> U69_gg(X1, X2, lecG_in_gg(X1, X2))
   lecG_in_gg(0, s(X1)) -> lecG_out_gg(0, s(X1))
   lecG_in_gg(0, 0) -> lecG_out_gg(0, 0)
   U83_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcJ_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U73_ga(X1, X2, X3, appcI_out_ggga(X6, X1, X7, X3)) -> qscH_out_ga(.(X1, X2), X3)
   gtcA_in_ga(s(X1), s(X2)) -> U57_ga(X1, X2, gtcA_in_ga(X1, X2))
   gtcA_in_ga(s(0), 0) -> gtcA_out_ga(s(0), 0)
   U59_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_ga(s(X1), s(X2)) -> U62_ga(X1, X2, lecC_in_ga(X1, X2))
   lecC_in_ga(0, s(X1)) -> lecC_out_ga(0, s(X1))
   lecC_in_ga(0, 0) -> lecC_out_ga(0, 0)
   U61_gaaa(X1, X2, X3, X4, X5, partcB_out_gaaa(X1, X3, X4, X5)) -> partcB_out_gaaa(X1, .(X2, X3), X4, .(X2, X5))
   U57_gg(X1, X2, gtcA_out_gg(X1, X2)) -> gtcA_out_gg(s(X1), s(X2))
   partcB_in_ggaa(X1, .(X2, X3), .(X2, X4), X5) -> U58_ggaa(X1, X2, X3, X4, X5, gtcA_in_gg(X1, X2))
   partcB_in_ggaa(X1, .(X2, X3), X4, .(X2, X5)) -> U60_ggaa(X1, X2, X3, X4, X5, lecC_in_gg(X1, X2))
   partcB_in_ggaa(X1, [], [], []) -> partcB_out_ggaa(X1, [], [], [])
   U69_gg(X1, X2, lecG_out_gg(X1, X2)) -> lecG_out_gg(s(X1), s(X2))
   appcI_in_ggga(.(X1, X2), X3, X4, .(X1, X5)) -> U74_ggga(X1, X2, X3, X4, X5, appcI_in_ggga(X2, X3, X4, X5))
   appcI_in_ggga([], X1, X2, .(X1, X2)) -> appcI_out_ggga([], X1, X2, .(X1, X2))
   U57_ga(X1, X2, gtcA_out_ga(X1, X2)) -> gtcA_out_ga(s(X1), s(X2))
   U62_ga(X1, X2, lecC_out_ga(X1, X2)) -> lecC_out_ga(s(X1), s(X2))
   U58_ggaa(X1, X2, X3, X4, X5, gtcA_out_gg(X1, X2)) -> U59_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U60_ggaa(X1, X2, X3, X4, X5, lecC_out_gg(X1, X2)) -> U61_ggaa(X1, X2, X3, X4, X5, partcB_in_ggaa(X1, X3, X4, X5))
   U74_ggga(X1, X2, X3, X4, X5, appcI_out_ggga(X2, X3, X4, X5)) -> appcI_out_ggga(.(X1, X2), X3, X4, .(X1, X5))
   U59_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), .(X2, X4), X5)
   lecC_in_gg(s(X1), s(X2)) -> U62_gg(X1, X2, lecC_in_gg(X1, X2))
   lecC_in_gg(0, s(X1)) -> lecC_out_gg(0, s(X1))
   lecC_in_gg(0, 0) -> lecC_out_gg(0, 0)
   U61_ggaa(X1, X2, X3, X4, X5, partcB_out_ggaa(X1, X3, X4, X5)) -> partcB_out_ggaa(X1, .(X2, X3), X4, .(X2, X5))
   U62_gg(X1, X2, lecC_out_gg(X1, X2)) -> lecC_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
gtcA_in_aa(x1, x2)  =  gtcA_in_aa

U57_aa(x1, x2, x3)  =  U57_aa(x3)

gtcA_out_aa(x1, x2)  =  gtcA_out_aa(x1, x2)

s(x1)  =  s(x1)

gtcA_in_ga(x1, x2)  =  gtcA_in_ga(x1)

U57_ga(x1, x2, x3)  =  U57_ga(x1, x3)

0  =  0

gtcA_out_ga(x1, x2)  =  gtcA_out_ga(x1, x2)

lecC_in_ga(x1, x2)  =  lecC_in_ga(x1)

U62_ga(x1, x2, x3)  =  U62_ga(x1, x3)

lecC_out_ga(x1, x2)  =  lecC_out_ga(x1)

partcB_in_gaaa(x1, x2, x3, x4)  =  partcB_in_gaaa(x1)

U58_gaaa(x1, x2, x3, x4, x5, x6)  =  U58_gaaa(x1, x6)

U59_gaaa(x1, x2, x3, x4, x5, x6)  =  U59_gaaa(x1, x2, x6)

U60_gaaa(x1, x2, x3, x4, x5, x6)  =  U60_gaaa(x1, x6)

U61_gaaa(x1, x2, x3, x4, x5, x6)  =  U61_gaaa(x1, x6)

partcB_out_gaaa(x1, x2, x3, x4)  =  partcB_out_gaaa(x1, x3)

.(x1, x2)  =  .(x1, x2)

gtcA_in_gg(x1, x2)  =  gtcA_in_gg(x1, x2)

U57_gg(x1, x2, x3)  =  U57_gg(x1, x2, x3)

gtcA_out_gg(x1, x2)  =  gtcA_out_gg(x1, x2)

lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)

U62_gg(x1, x2, x3)  =  U62_gg(x1, x2, x3)

lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)

partcB_in_ggaa(x1, x2, x3, x4)  =  partcB_in_ggaa(x1, x2)

U58_ggaa(x1, x2, x3, x4, x5, x6)  =  U58_ggaa(x1, x2, x3, x6)

U59_ggaa(x1, x2, x3, x4, x5, x6)  =  U59_ggaa(x1, x2, x3, x6)

U60_ggaa(x1, x2, x3, x4, x5, x6)  =  U60_ggaa(x1, x2, x3, x6)

U61_ggaa(x1, x2, x3, x4, x5, x6)  =  U61_ggaa(x1, x2, x3, x6)

[]  =  []

partcB_out_ggaa(x1, x2, x3, x4)  =  partcB_out_ggaa(x1, x2, x3, x4)

lecG_in_aa(x1, x2)  =  lecG_in_aa

U69_aa(x1, x2, x3)  =  U69_aa(x3)

lecG_out_aa(x1, x2)  =  lecG_out_aa(x1)

partcJ_in_aaaa(x1, x2, x3, x4)  =  partcJ_in_aaaa

U80_aaaa(x1, x2, x3, x4, x5, x6)  =  U80_aaaa(x6)

U81_aaaa(x1, x2, x3, x4, x5, x6)  =  U81_aaaa(x2, x6)

partcJ_out_aaaa(x1, x2, x3, x4)  =  partcJ_out_aaaa(x3)

U82_aaaa(x1, x2, x3, x4, x5, x6)  =  U82_aaaa(x6)

U83_aaaa(x1, x2, x3, x4, x5, x6)  =  U83_aaaa(x6)

lecG_in_gg(x1, x2)  =  lecG_in_gg(x1, x2)

U69_gg(x1, x2, x3)  =  U69_gg(x1, x2, x3)

lecG_out_gg(x1, x2)  =  lecG_out_gg(x1, x2)

partcJ_in_ggaa(x1, x2, x3, x4)  =  partcJ_in_ggaa(x1, x2)

U80_ggaa(x1, x2, x3, x4, x5, x6)  =  U80_ggaa(x1, x2, x3, x6)

U81_ggaa(x1, x2, x3, x4, x5, x6)  =  U81_ggaa(x1, x2, x3, x6)

partcJ_out_ggaa(x1, x2, x3, x4)  =  partcJ_out_ggaa(x1, x2, x3, x4)

U82_ggaa(x1, x2, x3, x4, x5, x6)  =  U82_ggaa(x1, x2, x3, x6)

U83_ggaa(x1, x2, x3, x4, x5, x6)  =  U83_ggaa(x1, x2, x3, x6)

qscH_in_ga(x1, x2)  =  qscH_in_ga(x1)

U70_ga(x1, x2, x3, x4)  =  U70_ga(x1, x2, x4)

U71_ga(x1, x2, x3, x4, x5, x6)  =  U71_ga(x1, x2, x5, x6)

qscH_out_ga(x1, x2)  =  qscH_out_ga(x1, x2)

U72_ga(x1, x2, x3, x4, x5)  =  U72_ga(x1, x2, x4, x5)

U73_ga(x1, x2, x3, x4)  =  U73_ga(x1, x2, x4)

appcI_in_ggga(x1, x2, x3, x4)  =  appcI_in_ggga(x1, x2, x3)

U74_ggga(x1, x2, x3, x4, x5, x6)  =  U74_ggga(x1, x2, x3, x4, x6)

appcI_out_ggga(x1, x2, x3, x4)  =  appcI_out_ggga(x1, x2, x3, x4)

QSH_IN_AA(x1, x2)  =  QSH_IN_AA

U25_AA(x1, x2, x3, x4)  =  U25_AA(x4)

U27_AA(x1, x2, x3, x4, x5)  =  U27_AA(x5)


We have to consider all (P,R,Pi)-chains
----------------------------------------

(413) PrologToIRSwTTransformerProof (SOUND)
Transformed Prolog program to IRSwT according to method in Master Thesis of A. Weinert

{
    "root": 2,
    "program": {
        "directives": [],
        "clauses": [
            [
                "(qs (. X Xs) Ys)",
                "(',' (part X Xs Littles Bigs) (',' (qs Littles Ls) (',' (qs Bigs Bs) (app Ls (. X Bs) Ys))))"
            ],
            [
                "(qs ([]) ([]))",
                null
            ],
            [
                "(part X (. Y Xs) (. Y Ls) Bs)",
                "(',' (gt X Y) (part X Xs Ls Bs))"
            ],
            [
                "(part X (. Y Xs) Ls (. Y Bs))",
                "(',' (le X Y) (part X Xs Ls Bs))"
            ],
            [
                "(part X ([]) ([]) ([]))",
                null
            ],
            [
                "(app (. X Xs) Ys (. X Zs))",
                "(app Xs Ys Zs)"
            ],
            [
                "(app ([]) Ys Ys)",
                null
            ],
            [
                "(gt (s X) (s Y))",
                "(gt X Y)"
            ],
            [
                "(gt (s (0)) (0))",
                null
            ],
            [
                "(le (s X) (s Y))",
                "(le X Y)"
            ],
            [
                "(le (0) (s X))",
                null
            ],
            [
                "(le (0) (0))",
                null
            ]
        ]
    },
    "graph": {
        "nodes": {
            "1340": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "type": "Nodes",
            "1338": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1459": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1458": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1457": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1456": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "631": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T44 T45)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1455": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "478": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T23 X26) (',' (qs T24 X27) (app X26 (. T25 X27) T17)))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T17",
                        "T23"
                    ],
                    "free": [
                        "X26",
                        "X27"
                    ],
                    "exprvars": []
                }
            },
            "995": {
                "goal": [{
                    "clause": 3,
                    "scope": 4,
                    "term": "(part T49 T50 X75 X76)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "1454": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "633": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T49 T50 X75 X76)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "996": {
                "goal": [{
                    "clause": 4,
                    "scope": 4,
                    "term": "(part T49 T50 X75 X76)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "1332": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1331": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "635": {
                "goal": [
                    {
                        "clause": 7,
                        "scope": 3,
                        "term": "(gt T44 T45)"
                    },
                    {
                        "clause": 8,
                        "scope": 3,
                        "term": "(gt T44 T45)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "916": {
                "goal": [
                    {
                        "clause": 7,
                        "scope": 5,
                        "term": "(gt T82 T85)"
                    },
                    {
                        "clause": 8,
                        "scope": 5,
                        "term": "(gt T82 T85)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T82"],
                    "free": [],
                    "exprvars": []
                }
            },
            "1339": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1110": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T174 T175 X257 X258)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T174"],
                    "free": [
                        "X257",
                        "X258"
                    ],
                    "exprvars": []
                }
            },
            "1231": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1352": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T222 X336) (app T304 (. T216 X336) X337))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T222",
                        "T304"
                    ],
                    "free": [
                        "X337",
                        "X336"
                    ],
                    "exprvars": []
                }
            },
            "1230": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T253 T254)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T253",
                        "T254"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1351": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T221 X335)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T221"],
                    "free": ["X335"],
                    "exprvars": []
                }
            },
            "1195": {
                "goal": [
                    {
                        "clause": 3,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    },
                    {
                        "clause": 4,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T217"
                    ],
                    "free": [
                        "X333",
                        "X334"
                    ],
                    "exprvars": []
                }
            },
            "1194": {
                "goal": [{
                    "clause": 2,
                    "scope": 9,
                    "term": "(part T216 T217 X333 X334)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T217"
                    ],
                    "free": [
                        "X333",
                        "X334"
                    ],
                    "exprvars": []
                }
            },
            "1193": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    },
                    {
                        "clause": 3,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    },
                    {
                        "clause": 4,
                        "scope": 9,
                        "term": "(part T216 T217 X333 X334)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T217"
                    ],
                    "free": [
                        "X333",
                        "X334"
                    ],
                    "exprvars": []
                }
            },
            "1192": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T221 X335) (',' (qs T222 X336) (app X335 (. T216 X336) X337)))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T221",
                        "T222"
                    ],
                    "free": [
                        "X337",
                        "X335",
                        "X336"
                    ],
                    "exprvars": []
                }
            },
            "1191": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T216 T217 X333 X334)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T216",
                        "T217"
                    ],
                    "free": [
                        "X333",
                        "X334"
                    ],
                    "exprvars": []
                }
            },
            "1107": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (le T169 T170) (part T169 T171 X257 X258))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X257",
                        "X258"
                    ],
                    "exprvars": []
                }
            },
            "1228": {
                "goal": [{
                    "clause": 7,
                    "scope": 10,
                    "term": "(gt T238 T239)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T238",
                        "T239"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1106": {
                "goal": [{
                    "clause": 4,
                    "scope": 2,
                    "term": "(part T18 T19 X24 X25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X24",
                        "X25"
                    ],
                    "exprvars": []
                }
            },
            "2": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T1 T2)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T2"],
                    "free": [],
                    "exprvars": []
                }
            },
            "1105": {
                "goal": [{
                    "clause": 3,
                    "scope": 2,
                    "term": "(part T18 T19 X24 X25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X24",
                        "X25"
                    ],
                    "exprvars": []
                }
            },
            "1347": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "3": {
                "goal": [
                    {
                        "clause": 0,
                        "scope": 1,
                        "term": "(qs T1 T2)"
                    },
                    {
                        "clause": 1,
                        "scope": 1,
                        "term": "(qs T1 T2)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
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            },
            "855": {
                "goal": [
                    {
                        "clause": 2,
                        "scope": 4,
                        "term": "(part T49 T50 X75 X76)"
                    },
                    {
                        "clause": 3,
                        "scope": 4,
                        "term": "(part T49 T50 X75 X76)"
                    },
                    {
                        "clause": 4,
                        "scope": 4,
                        "term": "(part T49 T50 X75 X76)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "614": {
                "goal": [{
                    "clause": 2,
                    "scope": 2,
                    "term": "(part T18 T19 X24 X25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X24",
                        "X25"
                    ],
                    "exprvars": []
                }
            },
            "616": {
                "goal": [
                    {
                        "clause": 3,
                        "scope": 2,
                        "term": "(part T18 T19 X24 X25)"
                    },
                    {
                        "clause": 4,
                        "scope": 2,
                        "term": "(part T18 T19 X24 X25)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X24",
                        "X25"
                    ],
                    "exprvars": []
                }
            },
            "1319": {
                "goal": [{
                    "clause": 11,
                    "scope": 11,
                    "term": "(le T272 T273)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T272",
                        "T273"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1318": {
                "goal": [{
                    "clause": 10,
                    "scope": 11,
                    "term": "(le T272 T273)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T272",
                        "T273"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1330": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(true)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1451": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1450": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T405 (. T408 T409) X669)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T405"],
                    "free": ["X669"],
                    "exprvars": []
                }
            },
            "1447": {
                "goal": [{
                    "clause": 6,
                    "scope": 14,
                    "term": "(app T375 (. T381 T380) X593)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "860": {
                "goal": [{
                    "clause": 2,
                    "scope": 4,
                    "term": "(part T49 T50 X75 X76)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "1203": {
                "goal": [
                    {
                        "clause": 7,
                        "scope": 10,
                        "term": "(gt T238 T239)"
                    },
                    {
                        "clause": 8,
                        "scope": 10,
                        "term": "(gt T238 T239)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T238",
                        "T239"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1445": {
                "goal": [{
                    "clause": 5,
                    "scope": 14,
                    "term": "(app T375 (. T381 T380) X593)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "345": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (part T18 T19 X24 X25) (',' (qs X24 X26) (',' (qs X25 X27) (app X26 (. T18 X27) T17))))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T17"],
                    "free": [
                        "X24",
                        "X25",
                        "X26",
                        "X27"
                    ],
                    "exprvars": []
                }
            },
            "862": {
                "goal": [
                    {
                        "clause": 3,
                        "scope": 4,
                        "term": "(part T49 T50 X75 X76)"
                    },
                    {
                        "clause": 4,
                        "scope": 4,
                        "term": "(part T49 T50 X75 X76)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T49"],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "1202": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T238 T240 X385 X386)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T238",
                        "T240"
                    ],
                    "free": [
                        "X385",
                        "X386"
                    ],
                    "exprvars": []
                }
            },
            "1444": {
                "goal": [
                    {
                        "clause": 5,
                        "scope": 14,
                        "term": "(app T375 (. T381 T380) X593)"
                    },
                    {
                        "clause": 6,
                        "scope": 14,
                        "term": "(app T375 (. T381 T380) X593)"
                    }
                ],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "467": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T18 T19 X24 X25)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X24",
                        "X25"
                    ],
                    "exprvars": []
                }
            },
            "1201": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T238 T239)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [
                        "T238",
                        "T239"
                    ],
                    "free": [],
                    "exprvars": []
                }
            },
            "1443": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(app T375 (. T381 T380) X593)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": ["X593"],
                    "exprvars": []
                }
            },
            "347": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1200": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "1442": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(qs T368 X592)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": ["X592"],
                    "exprvars": []
                }
            },
            "1441": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (qs T368 X592) (app T375 (. T369 X592) X593))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T375"],
                    "free": [
                        "X593",
                        "X592"
                    ],
                    "exprvars": []
                }
            },
            "627": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(',' (gt T44 T45) (part T44 T46 X75 X76))"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [
                        "X75",
                        "X76"
                    ],
                    "exprvars": []
                }
            },
            "903": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(gt T82 T85)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T82"],
                    "free": [],
                    "exprvars": []
                }
            },
            "629": {
                "goal": [],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": [],
                    "free": [],
                    "exprvars": []
                }
            },
            "905": {
                "goal": [{
                    "clause": -1,
                    "scope": -1,
                    "term": "(part T82 T89 X133 X134)"
                }],
                "kb": {
                    "nonunifying": [],
                    "intvars": {},
                    "arithmetic": {
                        "type": "PlainIntegerRelationState",
                        "relations": []
                    },
                    "ground": ["T82"],
                    "free": [
                        "X133",
                        "X134"
                    ],
                    "exprvars": []
                }
            }
        },
        "edges": [
            {
                "from": 2,
                "to": 3,
                "label": "CASE"
            },
            {
                "from": 3,
                "to": 324,
                "label": "PARALLEL"
            },
            {
                "from": 3,
                "to": 326,
                "label": "PARALLEL"
            },
            {
                "from": 324,
                "to": 345,
                "label": "EVAL with clause\nqs(.(X21, X22), X23) :- ','(part(X21, X22, X24, X25), ','(qs(X24, X26), ','(qs(X25, X27), app(X26, .(X21, X27), X23)))).\nand substitutionX21 -> T18,\nX22 -> T19,\nT1 -> .(T18, T19),\nT2 -> T17,\nX23 -> T17,\nT15 -> T18,\nT16 -> T19"
            },
            {
                "from": 324,
                "to": 347,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 326,
                "to": 1498,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT1 -> [],\nT2 -> []"
            },
            {
                "from": 326,
                "to": 1499,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 345,
                "to": 467,
                "label": "SPLIT 1"
            },
            {
                "from": 345,
                "to": 478,
                "label": "SPLIT 2\nnew knowledge:\nT23 is ground\nreplacements:X24 -> T23,\nX25 -> T24,\nT18 -> T25"
            },
            {
                "from": 467,
                "to": 561,
                "label": "CASE"
            },
            {
                "from": 478,
                "to": 1127,
                "label": "SPLIT 1"
            },
            {
                "from": 478,
                "to": 1128,
                "label": "SPLIT 2\nnew knowledge:\nT23 is ground\nT205 is ground\nreplacements:X26 -> T205"
            },
            {
                "from": 561,
                "to": 614,
                "label": "PARALLEL"
            },
            {
                "from": 561,
                "to": 616,
                "label": "PARALLEL"
            },
            {
                "from": 614,
                "to": 627,
                "label": "EVAL with clause\npart(X70, .(X71, X72), .(X71, X73), X74) :- ','(gt(X70, X71), part(X70, X72, X73, X74)).\nand substitutionT18 -> T44,\nX70 -> T44,\nX71 -> T45,\nX72 -> T46,\nT19 -> .(T45, T46),\nX73 -> X75,\nX24 -> .(T45, X75),\nX25 -> X76,\nX74 -> X76,\nT41 -> T44,\nT42 -> T45,\nT43 -> T46"
            },
            {
                "from": 614,
                "to": 629,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 616,
                "to": 1105,
                "label": "PARALLEL"
            },
            {
                "from": 616,
                "to": 1106,
                "label": "PARALLEL"
            },
            {
                "from": 627,
                "to": 631,
                "label": "SPLIT 1"
            },
            {
                "from": 627,
                "to": 633,
                "label": "SPLIT 2\nnew knowledge:\nT49 is ground\nT45 is ground\nreplacements:T44 -> T49,\nT46 -> T50"
            },
            {
                "from": 631,
                "to": 635,
                "label": "CASE"
            },
            {
                "from": 633,
                "to": 855,
                "label": "CASE"
            },
            {
                "from": 635,
                "to": 843,
                "label": "PARALLEL"
            },
            {
                "from": 635,
                "to": 844,
                "label": "PARALLEL"
            },
            {
                "from": 843,
                "to": 845,
                "label": "EVAL with clause\ngt(s(X89), s(X90)) :- gt(X89, X90).\nand substitutionX89 -> T63,\nT44 -> s(T63),\nX90 -> T64,\nT45 -> s(T64),\nT61 -> T63,\nT62 -> T64"
            },
            {
                "from": 843,
                "to": 846,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 844,
                "to": 849,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT44 -> s(0),\nT45 -> 0"
            },
            {
                "from": 844,
                "to": 850,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 845,
                "to": 631,
                "label": "INSTANCE with matching:\nT44 -> T63\nT45 -> T64"
            },
            {
                "from": 849,
                "to": 851,
                "label": "SUCCESS"
            },
            {
                "from": 855,
                "to": 860,
                "label": "PARALLEL"
            },
            {
                "from": 855,
                "to": 862,
                "label": "PARALLEL"
            },
            {
                "from": 860,
                "to": 883,
                "label": "EVAL with clause\npart(X128, .(X129, X130), .(X129, X131), X132) :- ','(gt(X128, X129), part(X128, X130, X131, X132)).\nand substitutionT49 -> T82,\nX128 -> T82,\nX129 -> T85,\nX130 -> T86,\nT50 -> .(T85, T86),\nX131 -> X133,\nX75 -> .(T85, X133),\nX76 -> X134,\nX132 -> X134,\nT83 -> T85,\nT84 -> T86"
            },
            {
                "from": 860,
                "to": 888,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 862,
                "to": 995,
                "label": "PARALLEL"
            },
            {
                "from": 862,
                "to": 996,
                "label": "PARALLEL"
            },
            {
                "from": 883,
                "to": 903,
                "label": "SPLIT 1"
            },
            {
                "from": 883,
                "to": 905,
                "label": "SPLIT 2\nnew knowledge:\nT82 is ground\nT85 is ground\nreplacements:T86 -> T89"
            },
            {
                "from": 903,
                "to": 916,
                "label": "CASE"
            },
            {
                "from": 905,
                "to": 633,
                "label": "INSTANCE with matching:\nT49 -> T82\nT50 -> T89\nX75 -> X133\nX76 -> X134"
            },
            {
                "from": 916,
                "to": 920,
                "label": "PARALLEL"
            },
            {
                "from": 916,
                "to": 921,
                "label": "PARALLEL"
            },
            {
                "from": 920,
                "to": 938,
                "label": "EVAL with clause\ngt(s(X147), s(X148)) :- gt(X147, X148).\nand substitutionX147 -> T100,\nT82 -> s(T100),\nX148 -> T102,\nT85 -> s(T102),\nT101 -> T102"
            },
            {
                "from": 920,
                "to": 942,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 921,
                "to": 965,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT82 -> s(0),\nT85 -> 0"
            },
            {
                "from": 921,
                "to": 968,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 938,
                "to": 903,
                "label": "INSTANCE with matching:\nT82 -> T100\nT85 -> T102"
            },
            {
                "from": 965,
                "to": 972,
                "label": "SUCCESS"
            },
            {
                "from": 995,
                "to": 1009,
                "label": "EVAL with clause\npart(X186, .(X187, X188), X189, .(X187, X190)) :- ','(le(X186, X187), part(X186, X188, X189, X190)).\nand substitutionT49 -> T120,\nX186 -> T120,\nX187 -> T123,\nX188 -> T124,\nT50 -> .(T123, T124),\nX75 -> X191,\nX189 -> X191,\nX190 -> X192,\nX76 -> .(T123, X192),\nT121 -> T123,\nT122 -> T124"
            },
            {
                "from": 995,
                "to": 1010,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 996,
                "to": 1026,
                "label": "EVAL with clause\npart(X223, [], [], []).\nand substitutionT49 -> T153,\nX223 -> T153,\nT50 -> [],\nX75 -> [],\nX76 -> []"
            },
            {
                "from": 996,
                "to": 1027,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1009,
                "to": 1011,
                "label": "SPLIT 1"
            },
            {
                "from": 1009,
                "to": 1012,
                "label": "SPLIT 2\nnew knowledge:\nT120 is ground\nreplacements:T124 -> T127"
            },
            {
                "from": 1011,
                "to": 1013,
                "label": "CASE"
            },
            {
                "from": 1012,
                "to": 633,
                "label": "INSTANCE with matching:\nT49 -> T120\nT50 -> T127\nX75 -> X191\nX76 -> X192"
            },
            {
                "from": 1013,
                "to": 1014,
                "label": "PARALLEL"
            },
            {
                "from": 1013,
                "to": 1015,
                "label": "PARALLEL"
            },
            {
                "from": 1014,
                "to": 1016,
                "label": "EVAL with clause\nle(s(X205), s(X206)) :- le(X205, X206).\nand substitutionX205 -> T138,\nT120 -> s(T138),\nX206 -> T140,\nT123 -> s(T140),\nT139 -> T140"
            },
            {
                "from": 1014,
                "to": 1017,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1015,
                "to": 1018,
                "label": "PARALLEL"
            },
            {
                "from": 1015,
                "to": 1019,
                "label": "PARALLEL"
            },
            {
                "from": 1016,
                "to": 1011,
                "label": "INSTANCE with matching:\nT120 -> T138\nT123 -> T140"
            },
            {
                "from": 1018,
                "to": 1020,
                "label": "EVAL with clause\nle(0, s(X213)).\nand substitutionT120 -> 0,\nX213 -> T147,\nT123 -> s(T147)"
            },
            {
                "from": 1018,
                "to": 1021,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1019,
                "to": 1023,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT120 -> 0,\nT123 -> 0"
            },
            {
                "from": 1019,
                "to": 1024,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1020,
                "to": 1022,
                "label": "SUCCESS"
            },
            {
                "from": 1023,
                "to": 1025,
                "label": "SUCCESS"
            },
            {
                "from": 1026,
                "to": 1094,
                "label": "SUCCESS"
            },
            {
                "from": 1105,
                "to": 1107,
                "label": "EVAL with clause\npart(X252, .(X253, X254), X255, .(X253, X256)) :- ','(le(X252, X253), part(X252, X254, X255, X256)).\nand substitutionT18 -> T169,\nX252 -> T169,\nX253 -> T170,\nX254 -> T171,\nT19 -> .(T170, T171),\nX24 -> X257,\nX255 -> X257,\nX256 -> X258,\nX25 -> .(T170, X258),\nT166 -> T169,\nT167 -> T170,\nT168 -> T171"
            },
            {
                "from": 1105,
                "to": 1108,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1106,
                "to": 1124,
                "label": "EVAL with clause\npart(X289, [], [], []).\nand substitutionT18 -> T202,\nX289 -> T202,\nT19 -> [],\nX24 -> [],\nX25 -> []"
            },
            {
                "from": 1106,
                "to": 1125,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1107,
                "to": 1109,
                "label": "SPLIT 1"
            },
            {
                "from": 1107,
                "to": 1110,
                "label": "SPLIT 2\nnew knowledge:\nT174 is ground\nreplacements:T169 -> T174,\nT171 -> T175"
            },
            {
                "from": 1109,
                "to": 1111,
                "label": "CASE"
            },
            {
                "from": 1110,
                "to": 633,
                "label": "INSTANCE with matching:\nT49 -> T174\nT50 -> T175\nX75 -> X257\nX76 -> X258"
            },
            {
                "from": 1111,
                "to": 1112,
                "label": "PARALLEL"
            },
            {
                "from": 1111,
                "to": 1113,
                "label": "PARALLEL"
            },
            {
                "from": 1112,
                "to": 1114,
                "label": "EVAL with clause\nle(s(X271), s(X272)) :- le(X271, X272).\nand substitutionX271 -> T188,\nT169 -> s(T188),\nX272 -> T189,\nT170 -> s(T189),\nT186 -> T188,\nT187 -> T189"
            },
            {
                "from": 1112,
                "to": 1115,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1113,
                "to": 1116,
                "label": "PARALLEL"
            },
            {
                "from": 1113,
                "to": 1117,
                "label": "PARALLEL"
            },
            {
                "from": 1114,
                "to": 1109,
                "label": "INSTANCE with matching:\nT169 -> T188\nT170 -> T189"
            },
            {
                "from": 1116,
                "to": 1118,
                "label": "EVAL with clause\nle(0, s(X279)).\nand substitutionT169 -> 0,\nX279 -> T196,\nT170 -> s(T196)"
            },
            {
                "from": 1116,
                "to": 1119,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1117,
                "to": 1121,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT169 -> 0,\nT170 -> 0"
            },
            {
                "from": 1117,
                "to": 1122,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1118,
                "to": 1120,
                "label": "SUCCESS"
            },
            {
                "from": 1121,
                "to": 1123,
                "label": "SUCCESS"
            },
            {
                "from": 1124,
                "to": 1126,
                "label": "SUCCESS"
            },
            {
                "from": 1127,
                "to": 1141,
                "label": "CASE"
            },
            {
                "from": 1128,
                "to": 1390,
                "label": "SPLIT 1"
            },
            {
                "from": 1128,
                "to": 1391,
                "label": "SPLIT 2\nreplacements:X27 -> T348,\nT25 -> T349"
            },
            {
                "from": 1141,
                "to": 1142,
                "label": "PARALLEL"
            },
            {
                "from": 1141,
                "to": 1143,
                "label": "PARALLEL"
            },
            {
                "from": 1142,
                "to": 1144,
                "label": "EVAL with clause\nqs(.(X330, X331), X332) :- ','(part(X330, X331, X333, X334), ','(qs(X333, X335), ','(qs(X334, X336), app(X335, .(X330, X336), X332)))).\nand substitutionX330 -> T216,\nX331 -> T217,\nT23 -> .(T216, T217),\nX26 -> X337,\nX332 -> X337"
            },
            {
                "from": 1142,
                "to": 1145,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1143,
                "to": 1387,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT23 -> [],\nX26 -> []"
            },
            {
                "from": 1143,
                "to": 1388,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1144,
                "to": 1191,
                "label": "SPLIT 1"
            },
            {
                "from": 1144,
                "to": 1192,
                "label": "SPLIT 2\nnew knowledge:\nT216 is ground\nT217 is ground\nT221 is ground\nT222 is ground\nreplacements:X333 -> T221,\nX334 -> T222"
            },
            {
                "from": 1191,
                "to": 1193,
                "label": "CASE"
            },
            {
                "from": 1192,
                "to": 1351,
                "label": "SPLIT 1"
            },
            {
                "from": 1192,
                "to": 1352,
                "label": "SPLIT 2\nnew knowledge:\nT221 is ground\nT304 is ground\nreplacements:X335 -> T304"
            },
            {
                "from": 1193,
                "to": 1194,
                "label": "PARALLEL"
            },
            {
                "from": 1193,
                "to": 1195,
                "label": "PARALLEL"
            },
            {
                "from": 1194,
                "to": 1199,
                "label": "EVAL with clause\npart(X380, .(X381, X382), .(X381, X383), X384) :- ','(gt(X380, X381), part(X380, X382, X383, X384)).\nand substitutionT216 -> T238,\nX380 -> T238,\nX381 -> T239,\nX382 -> T240,\nT217 -> .(T239, T240),\nX383 -> X385,\nX333 -> .(T239, X385),\nX334 -> X386,\nX384 -> X386"
            },
            {
                "from": 1194,
                "to": 1200,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1195,
                "to": 1247,
                "label": "PARALLEL"
            },
            {
                "from": 1195,
                "to": 1248,
                "label": "PARALLEL"
            },
            {
                "from": 1199,
                "to": 1201,
                "label": "SPLIT 1"
            },
            {
                "from": 1199,
                "to": 1202,
                "label": "SPLIT 2\nnew knowledge:\nT238 is ground\nT239 is ground"
            },
            {
                "from": 1201,
                "to": 1203,
                "label": "CASE"
            },
            {
                "from": 1202,
                "to": 1191,
                "label": "INSTANCE with matching:\nT216 -> T238\nT217 -> T240\nX333 -> X385\nX334 -> X386"
            },
            {
                "from": 1203,
                "to": 1228,
                "label": "PARALLEL"
            },
            {
                "from": 1203,
                "to": 1229,
                "label": "PARALLEL"
            },
            {
                "from": 1228,
                "to": 1230,
                "label": "EVAL with clause\ngt(s(X399), s(X400)) :- gt(X399, X400).\nand substitutionX399 -> T253,\nT238 -> s(T253),\nX400 -> T254,\nT239 -> s(T254)"
            },
            {
                "from": 1228,
                "to": 1231,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1229,
                "to": 1232,
                "label": "EVAL with clause\ngt(s(0), 0).\nand substitutionT238 -> s(0),\nT239 -> 0"
            },
            {
                "from": 1229,
                "to": 1233,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1230,
                "to": 1201,
                "label": "INSTANCE with matching:\nT238 -> T253\nT239 -> T254"
            },
            {
                "from": 1232,
                "to": 1234,
                "label": "SUCCESS"
            },
            {
                "from": 1247,
                "to": 1249,
                "label": "EVAL with clause\npart(X438, .(X439, X440), X441, .(X439, X442)) :- ','(le(X438, X439), part(X438, X440, X441, X442)).\nand substitutionT216 -> T272,\nX438 -> T272,\nX439 -> T273,\nX440 -> T274,\nT217 -> .(T273, T274),\nX333 -> X443,\nX441 -> X443,\nX442 -> X444,\nX334 -> .(T273, X444)"
            },
            {
                "from": 1247,
                "to": 1250,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1248,
                "to": 1345,
                "label": "EVAL with clause\npart(X475, [], [], []).\nand substitutionT216 -> T301,\nX475 -> T301,\nT217 -> [],\nX333 -> [],\nX334 -> []"
            },
            {
                "from": 1248,
                "to": 1346,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1249,
                "to": 1251,
                "label": "SPLIT 1"
            },
            {
                "from": 1249,
                "to": 1252,
                "label": "SPLIT 2\nnew knowledge:\nT272 is ground\nT273 is ground"
            },
            {
                "from": 1251,
                "to": 1253,
                "label": "CASE"
            },
            {
                "from": 1252,
                "to": 1191,
                "label": "INSTANCE with matching:\nT216 -> T272\nT217 -> T274\nX333 -> X443\nX334 -> X444"
            },
            {
                "from": 1253,
                "to": 1254,
                "label": "PARALLEL"
            },
            {
                "from": 1253,
                "to": 1255,
                "label": "PARALLEL"
            },
            {
                "from": 1254,
                "to": 1315,
                "label": "EVAL with clause\nle(s(X457), s(X458)) :- le(X457, X458).\nand substitutionX457 -> T287,\nT272 -> s(T287),\nX458 -> T288,\nT273 -> s(T288)"
            },
            {
                "from": 1254,
                "to": 1316,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1255,
                "to": 1318,
                "label": "PARALLEL"
            },
            {
                "from": 1255,
                "to": 1319,
                "label": "PARALLEL"
            },
            {
                "from": 1315,
                "to": 1251,
                "label": "INSTANCE with matching:\nT272 -> T287\nT273 -> T288"
            },
            {
                "from": 1318,
                "to": 1330,
                "label": "EVAL with clause\nle(0, s(X465)).\nand substitutionT272 -> 0,\nX465 -> T295,\nT273 -> s(T295)"
            },
            {
                "from": 1318,
                "to": 1331,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1319,
                "to": 1338,
                "label": "EVAL with clause\nle(0, 0).\nand substitutionT272 -> 0,\nT273 -> 0"
            },
            {
                "from": 1319,
                "to": 1339,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1330,
                "to": 1332,
                "label": "SUCCESS"
            },
            {
                "from": 1338,
                "to": 1340,
                "label": "SUCCESS"
            },
            {
                "from": 1345,
                "to": 1347,
                "label": "SUCCESS"
            },
            {
                "from": 1351,
                "to": 1127,
                "label": "INSTANCE with matching:\nT23 -> T221\nX26 -> X335"
            },
            {
                "from": 1352,
                "to": 1377,
                "label": "SPLIT 1"
            },
            {
                "from": 1352,
                "to": 1378,
                "label": "SPLIT 2\nnew knowledge:\nT222 is ground\nT309 is ground\nreplacements:X336 -> T309"
            },
            {
                "from": 1377,
                "to": 1127,
                "label": "INSTANCE with matching:\nT23 -> T222\nX26 -> X336"
            },
            {
                "from": 1378,
                "to": 1379,
                "label": "CASE"
            },
            {
                "from": 1379,
                "to": 1380,
                "label": "PARALLEL"
            },
            {
                "from": 1379,
                "to": 1381,
                "label": "PARALLEL"
            },
            {
                "from": 1380,
                "to": 1382,
                "label": "EVAL with clause\napp(.(X533, X534), X535, .(X533, X536)) :- app(X534, X535, X536).\nand substitutionX533 -> T332,\nX534 -> T333,\nT304 -> .(T332, T333),\nT216 -> T334,\nT309 -> T335,\nX535 -> .(T334, T335),\nX536 -> X537,\nX337 -> .(T332, X537)"
            },
            {
                "from": 1380,
                "to": 1383,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1381,
                "to": 1384,
                "label": "EVAL with clause\napp([], X545, X545).\nand substitutionT304 -> [],\nT216 -> T344,\nT309 -> T345,\nX545 -> .(T344, T345),\nX337 -> .(T344, T345)"
            },
            {
                "from": 1381,
                "to": 1385,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1382,
                "to": 1378,
                "label": "INSTANCE with matching:\nT304 -> T333\nT216 -> T334\nT309 -> T335\nX337 -> X537"
            },
            {
                "from": 1384,
                "to": 1386,
                "label": "SUCCESS"
            },
            {
                "from": 1387,
                "to": 1389,
                "label": "SUCCESS"
            },
            {
                "from": 1390,
                "to": 1392,
                "label": "CASE"
            },
            {
                "from": 1391,
                "to": 1464,
                "label": "CASE"
            },
            {
                "from": 1392,
                "to": 1393,
                "label": "PARALLEL"
            },
            {
                "from": 1392,
                "to": 1394,
                "label": "PARALLEL"
            },
            {
                "from": 1393,
                "to": 1395,
                "label": "EVAL with clause\nqs(.(X586, X587), X588) :- ','(part(X586, X587, X589, X590), ','(qs(X589, X591), ','(qs(X590, X592), app(X591, .(X586, X592), X588)))).\nand substitutionX586 -> T362,\nX587 -> T363,\nT24 -> .(T362, T363),\nX27 -> X593,\nX588 -> X593,\nT360 -> T362,\nT361 -> T363"
            },
            {
                "from": 1393,
                "to": 1396,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1394,
                "to": 1457,
                "label": "EVAL with clause\nqs([], []).\nand substitutionT24 -> [],\nX27 -> []"
            },
            {
                "from": 1394,
                "to": 1458,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1395,
                "to": 1399,
                "label": "SPLIT 1"
            },
            {
                "from": 1395,
                "to": 1400,
                "label": "SPLIT 2\nnew knowledge:\nT367 is ground\nreplacements:X589 -> T367,\nX590 -> T368,\nT362 -> T369"
            },
            {
                "from": 1399,
                "to": 467,
                "label": "INSTANCE with matching:\nT18 -> T362\nT19 -> T363\nX24 -> X589\nX25 -> X590"
            },
            {
                "from": 1400,
                "to": 1440,
                "label": "SPLIT 1"
            },
            {
                "from": 1400,
                "to": 1441,
                "label": "SPLIT 2\nnew knowledge:\nT367 is ground\nT375 is ground\nreplacements:X591 -> T375"
            },
            {
                "from": 1440,
                "to": 1127,
                "label": "INSTANCE with matching:\nT23 -> T367\nX26 -> X591"
            },
            {
                "from": 1441,
                "to": 1442,
                "label": "SPLIT 1"
            },
            {
                "from": 1441,
                "to": 1443,
                "label": "SPLIT 2\nreplacements:X592 -> T380,\nT369 -> T381"
            },
            {
                "from": 1442,
                "to": 1390,
                "label": "INSTANCE with matching:\nT24 -> T368\nX27 -> X592"
            },
            {
                "from": 1443,
                "to": 1444,
                "label": "CASE"
            },
            {
                "from": 1444,
                "to": 1445,
                "label": "PARALLEL"
            },
            {
                "from": 1444,
                "to": 1447,
                "label": "PARALLEL"
            },
            {
                "from": 1445,
                "to": 1450,
                "label": "EVAL with clause\napp(.(X665, X666), X667, .(X665, X668)) :- app(X666, X667, X668).\nand substitutionX665 -> T404,\nX666 -> T405,\nT375 -> .(T404, T405),\nT381 -> T408,\nT380 -> T409,\nX667 -> .(T408, T409),\nX668 -> X669,\nX593 -> .(T404, X669),\nT406 -> T408,\nT407 -> T409"
            },
            {
                "from": 1445,
                "to": 1451,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1447,
                "to": 1454,
                "label": "EVAL with clause\napp([], X677, X677).\nand substitutionT375 -> [],\nT381 -> T418,\nT380 -> T419,\nX677 -> .(T418, T419),\nX593 -> .(T418, T419)"
            },
            {
                "from": 1447,
                "to": 1455,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1450,
                "to": 1443,
                "label": "INSTANCE with matching:\nT375 -> T405\nT381 -> T408\nT380 -> T409\nX593 -> X669"
            },
            {
                "from": 1454,
                "to": 1456,
                "label": "SUCCESS"
            },
            {
                "from": 1457,
                "to": 1459,
                "label": "SUCCESS"
            },
            {
                "from": 1464,
                "to": 1478,
                "label": "PARALLEL"
            },
            {
                "from": 1464,
                "to": 1479,
                "label": "PARALLEL"
            },
            {
                "from": 1478,
                "to": 1489,
                "label": "EVAL with clause\napp(.(X698, X699), X700, .(X698, X701)) :- app(X699, X700, X701).\nand substitutionX698 -> T445,\nX699 -> T446,\nT205 -> .(T445, T446),\nT349 -> T450,\nT348 -> T451,\nX700 -> .(T450, T451),\nX701 -> T449,\nT17 -> .(T445, T449),\nT447 -> T450,\nT448 -> T451"
            },
            {
                "from": 1478,
                "to": 1490,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1479,
                "to": 1495,
                "label": "EVAL with clause\napp([], X708, X708).\nand substitutionT205 -> [],\nT349 -> T461,\nT348 -> T462,\nX708 -> .(T461, T462),\nT17 -> .(T461, T462)"
            },
            {
                "from": 1479,
                "to": 1496,
                "label": "EVAL-BACKTRACK"
            },
            {
                "from": 1489,
                "to": 1391,
                "label": "INSTANCE with matching:\nT205 -> T446\nT349 -> T450\nT348 -> T451\nT17 -> T449"
            },
            {
                "from": 1495,
                "to": 1497,
                "label": "SUCCESS"
            },
            {
                "from": 1498,
                "to": 1500,
                "label": "SUCCESS"
            }
        ],
        "type": "Graph"
    }
}

----------------------------------------

(414)
Complex Obligation (AND)

----------------------------------------

(415)
Obligation:
Rules:
f1447_out(T375) -> f1444_out(T375) :|: TRUE
f1445_out(x) -> f1444_out(x) :|: TRUE
f1444_in(x1) -> f1447_in(x1) :|: TRUE
f1444_in(x2) -> f1445_in(x2) :|: TRUE
f1450_in(T405) -> f1443_in(T405) :|: TRUE
f1443_out(x3) -> f1450_out(x3) :|: TRUE
f1444_out(x4) -> f1443_out(x4) :|: TRUE
f1443_in(x5) -> f1444_in(x5) :|: TRUE
f1450_out(x6) -> f1445_out(.(x7, x6)) :|: TRUE
f1445_in(.(x8, x9)) -> f1450_in(x9) :|: TRUE
f1445_in(x10) -> f1451_in :|: TRUE
f1451_out -> f1445_out(x11) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x12) -> f2_out(x12) :|: TRUE
f3_in(x13) -> f326_in(x13) :|: TRUE
f324_out(x14) -> f3_out(x14) :|: TRUE
f326_out(x15) -> f3_out(x15) :|: TRUE
f3_in(x16) -> f324_in(x16) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x17) -> f347_in :|: TRUE
f347_out -> f324_out(x18) :|: TRUE
f324_in(x19) -> f345_in(x19) :|: TRUE
f345_in(x20) -> f467_in :|: TRUE
f478_out(x21, x22) -> f345_out(x22) :|: TRUE
f467_out -> f478_in(x23, x24) :|: TRUE
f478_in(x25, x26) -> f1127_in(x25) :|: TRUE
f1127_out(x27) -> f1128_in(x28, x29) :|: TRUE
f1128_out(x30, x31) -> f478_out(x32, x31) :|: TRUE
f1128_in(x33, x34) -> f1390_in :|: TRUE
f1391_out(x35, x36) -> f1128_out(x35, x36) :|: TRUE
f1390_out -> f1391_in(x37, x38) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x39) -> f1395_out :|: TRUE
f1440_out(x40) -> f1441_in(x41) :|: TRUE
f1441_out(x42) -> f1400_out(x43) :|: TRUE
f1400_in(x44) -> f1440_in(x44) :|: TRUE
f1441_in(x45) -> f1442_in :|: TRUE
f1443_out(x46) -> f1441_out(x46) :|: TRUE
f1442_out -> f1443_in(x47) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(416) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(417)
TRUE

----------------------------------------

(418)
Obligation:
Rules:
f1391_out(T446, T449) -> f1489_out(T446, T449) :|: TRUE
f1489_in(x, x1) -> f1391_in(x, x1) :|: TRUE
f1391_in(T205, T17) -> f1464_in(T205, T17) :|: TRUE
f1464_out(x2, x3) -> f1391_out(x2, x3) :|: TRUE
f1478_out(x4, x5) -> f1464_out(x4, x5) :|: TRUE
f1479_out(x6, x7) -> f1464_out(x6, x7) :|: TRUE
f1464_in(x8, x9) -> f1479_in(x8, x9) :|: TRUE
f1464_in(x10, x11) -> f1478_in(x10, x11) :|: TRUE
f1490_out -> f1478_out(x12, x13) :|: TRUE
f1478_in(x14, x15) -> f1490_in :|: TRUE
f1489_out(x16, x17) -> f1478_out(.(x18, x16), .(x18, x17)) :|: TRUE
f1478_in(.(x19, x20), .(x19, x21)) -> f1489_in(x20, x21) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x22) -> f2_out(x22) :|: TRUE
f3_in(x23) -> f326_in(x23) :|: TRUE
f324_out(x24) -> f3_out(x24) :|: TRUE
f326_out(x25) -> f3_out(x25) :|: TRUE
f3_in(x26) -> f324_in(x26) :|: TRUE
f345_out(x27) -> f324_out(x27) :|: TRUE
f324_in(x28) -> f347_in :|: TRUE
f347_out -> f324_out(x29) :|: TRUE
f324_in(x30) -> f345_in(x30) :|: TRUE
f345_in(x31) -> f467_in :|: TRUE
f478_out(x32, x33) -> f345_out(x33) :|: TRUE
f467_out -> f478_in(x34, x35) :|: TRUE
f478_in(x36, x37) -> f1127_in(x36) :|: TRUE
f1127_out(x38) -> f1128_in(x39, x40) :|: TRUE
f1128_out(x41, x42) -> f478_out(x43, x42) :|: TRUE
f1128_in(x44, x45) -> f1390_in :|: TRUE
f1391_out(x46, x47) -> f1128_out(x46, x47) :|: TRUE
f1390_out -> f1391_in(x48, x49) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(419) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(420)
TRUE

----------------------------------------

(421)
Obligation:
Rules:
f1379_in(T304, T216, T309) -> f1381_in(T304, T216, T309) :|: TRUE
f1381_out(x, x1, x2) -> f1379_out(x, x1, x2) :|: TRUE
f1380_out(x3, x4, x5) -> f1379_out(x3, x4, x5) :|: TRUE
f1379_in(x6, x7, x8) -> f1380_in(x6, x7, x8) :|: TRUE
f1382_in(T333, T334, T335) -> f1378_in(T333, T334, T335) :|: TRUE
f1378_out(x9, x10, x11) -> f1382_out(x9, x10, x11) :|: TRUE
f1379_out(x12, x13, x14) -> f1378_out(x12, x13, x14) :|: TRUE
f1378_in(x15, x16, x17) -> f1379_in(x15, x16, x17) :|: TRUE
f1380_in(.(x18, x19), x20, x21) -> f1382_in(x19, x20, x21) :|: TRUE
f1383_out -> f1380_out(x22, x23, x24) :|: TRUE
f1382_out(x25, x26, x27) -> f1380_out(.(x28, x25), x26, x27) :|: TRUE
f1380_in(x29, x30, x31) -> f1383_in :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x32) -> f2_out(x32) :|: TRUE
f3_in(x33) -> f326_in(x33) :|: TRUE
f324_out(x34) -> f3_out(x34) :|: TRUE
f326_out(x35) -> f3_out(x35) :|: TRUE
f3_in(x36) -> f324_in(x36) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x37) -> f347_in :|: TRUE
f347_out -> f324_out(x38) :|: TRUE
f324_in(x39) -> f345_in(x39) :|: TRUE
f345_in(x40) -> f467_in :|: TRUE
f478_out(x41, x42) -> f345_out(x42) :|: TRUE
f467_out -> f478_in(x43, x44) :|: TRUE
f478_in(x45, x46) -> f1127_in(x45) :|: TRUE
f1127_out(x47) -> f1128_in(x48, x49) :|: TRUE
f1128_out(x50, x51) -> f478_out(x52, x51) :|: TRUE
f1141_out(T23) -> f1127_out(T23) :|: TRUE
f1127_in(x53) -> f1141_in(x53) :|: TRUE
f1141_in(x54) -> f1143_in(x54) :|: TRUE
f1143_out(x55) -> f1141_out(x55) :|: TRUE
f1141_in(x56) -> f1142_in(x56) :|: TRUE
f1142_out(x57) -> f1141_out(x57) :|: TRUE
f1144_out(x58, x59) -> f1142_out(.(x58, x59)) :|: TRUE
f1142_in(x60) -> f1145_in :|: TRUE
f1145_out -> f1142_out(x61) :|: TRUE
f1142_in(.(x62, x63)) -> f1144_in(x62, x63) :|: TRUE
f1191_out(x64, x65) -> f1192_in(x66, x67, x64) :|: TRUE
f1144_in(x68, x69) -> f1191_in(x68, x69) :|: TRUE
f1192_out(x70, x71, x72) -> f1144_out(x72, x73) :|: TRUE
f1351_out(x74) -> f1352_in(x75, x76, x77) :|: TRUE
f1352_out(x78, x79, x80) -> f1192_out(x81, x78, x80) :|: TRUE
f1192_in(x82, x83, x84) -> f1351_in(x82) :|: TRUE
f1377_out(x85) -> f1378_in(x86, x87, x88) :|: TRUE
f1378_out(x89, x90, x91) -> f1352_out(x92, x89, x90) :|: TRUE
f1352_in(x93, x94, x95) -> f1377_in(x93) :|: TRUE
f1128_in(x96, x97) -> f1390_in :|: TRUE
f1391_out(x98, x99) -> f1128_out(x98, x99) :|: TRUE
f1390_out -> f1391_in(x100, x101) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x102) -> f1395_out :|: TRUE
f1440_out(x103) -> f1441_in(x104) :|: TRUE
f1441_out(x105) -> f1400_out(x106) :|: TRUE
f1400_in(x107) -> f1440_in(x107) :|: TRUE
f1127_out(x108) -> f1440_out(x108) :|: TRUE
f1440_in(x109) -> f1127_in(x109) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(422) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(423)
TRUE

----------------------------------------

(424)
Obligation:
Rules:
f1316_out -> f1254_out(T272, T273) :|: TRUE
f1315_out(T287, T288) -> f1254_out(s(T287), s(T288)) :|: TRUE
f1254_in(s(x), s(x1)) -> f1315_in(x, x1) :|: TRUE
f1254_in(x2, x3) -> f1316_in :|: TRUE
f1315_in(x4, x5) -> f1251_in(x4, x5) :|: TRUE
f1251_out(x6, x7) -> f1315_out(x6, x7) :|: TRUE
f1253_in(x8, x9) -> f1254_in(x8, x9) :|: TRUE
f1255_out(x10, x11) -> f1253_out(x10, x11) :|: TRUE
f1254_out(x12, x13) -> f1253_out(x12, x13) :|: TRUE
f1253_in(x14, x15) -> f1255_in(x14, x15) :|: TRUE
f1253_out(x16, x17) -> f1251_out(x16, x17) :|: TRUE
f1251_in(x18, x19) -> f1253_in(x18, x19) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x20) -> f2_out(x20) :|: TRUE
f3_in(x21) -> f326_in(x21) :|: TRUE
f324_out(x22) -> f3_out(x22) :|: TRUE
f326_out(x23) -> f3_out(x23) :|: TRUE
f3_in(x24) -> f324_in(x24) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x25) -> f347_in :|: TRUE
f347_out -> f324_out(x26) :|: TRUE
f324_in(x27) -> f345_in(x27) :|: TRUE
f345_in(x28) -> f467_in :|: TRUE
f478_out(x29, x30) -> f345_out(x30) :|: TRUE
f467_out -> f478_in(x31, x32) :|: TRUE
f478_in(x33, x34) -> f1127_in(x33) :|: TRUE
f1127_out(x35) -> f1128_in(x36, x37) :|: TRUE
f1128_out(x38, x39) -> f478_out(x40, x39) :|: TRUE
f1141_out(T23) -> f1127_out(T23) :|: TRUE
f1127_in(x41) -> f1141_in(x41) :|: TRUE
f1141_in(x42) -> f1143_in(x42) :|: TRUE
f1143_out(x43) -> f1141_out(x43) :|: TRUE
f1141_in(x44) -> f1142_in(x44) :|: TRUE
f1142_out(x45) -> f1141_out(x45) :|: TRUE
f1144_out(T216, T217) -> f1142_out(.(T216, T217)) :|: TRUE
f1142_in(x46) -> f1145_in :|: TRUE
f1145_out -> f1142_out(x47) :|: TRUE
f1142_in(.(x48, x49)) -> f1144_in(x48, x49) :|: TRUE
f1191_out(x50, x51) -> f1192_in(x52, x53, x50) :|: TRUE
f1144_in(x54, x55) -> f1191_in(x54, x55) :|: TRUE
f1192_out(x56, x57, x58) -> f1144_out(x58, x59) :|: TRUE
f1191_in(x60, x61) -> f1193_in(x60, x61) :|: TRUE
f1193_out(x62, x63) -> f1191_out(x62, x63) :|: TRUE
f1195_out(x64, x65) -> f1193_out(x64, x65) :|: TRUE
f1194_out(x66, x67) -> f1193_out(x66, x67) :|: TRUE
f1193_in(x68, x69) -> f1194_in(x68, x69) :|: TRUE
f1193_in(x70, x71) -> f1195_in(x70, x71) :|: TRUE
f1247_out(x72, x73) -> f1195_out(x72, x73) :|: TRUE
f1195_in(x74, x75) -> f1247_in(x74, x75) :|: TRUE
f1195_in(x76, x77) -> f1248_in(x76, x77) :|: TRUE
f1248_out(x78, x79) -> f1195_out(x78, x79) :|: TRUE
f1247_in(x80, x81) -> f1250_in :|: TRUE
f1247_in(x82, .(x83, x84)) -> f1249_in(x82, x83, x84) :|: TRUE
f1249_out(x85, x86, x87) -> f1247_out(x85, .(x86, x87)) :|: TRUE
f1250_out -> f1247_out(x88, x89) :|: TRUE
f1252_out(x90, x91) -> f1249_out(x90, x92, x91) :|: TRUE
f1249_in(x93, x94, x95) -> f1251_in(x93, x94) :|: TRUE
f1251_out(x96, x97) -> f1252_in(x96, x98) :|: TRUE
f1128_in(x99, x100) -> f1390_in :|: TRUE
f1391_out(x101, x102) -> f1128_out(x101, x102) :|: TRUE
f1390_out -> f1391_in(x103, x104) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x105) -> f1395_out :|: TRUE
f1440_out(x106) -> f1441_in(x107) :|: TRUE
f1441_out(x108) -> f1400_out(x109) :|: TRUE
f1400_in(x110) -> f1440_in(x110) :|: TRUE
f1127_out(x111) -> f1440_out(x111) :|: TRUE
f1440_in(x112) -> f1127_in(x112) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(425) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(426)
TRUE

----------------------------------------

(427)
Obligation:
Rules:
f1228_out(T238, T239) -> f1203_out(T238, T239) :|: TRUE
f1203_in(x, x1) -> f1228_in(x, x1) :|: TRUE
f1203_in(x2, x3) -> f1229_in(x2, x3) :|: TRUE
f1229_out(x4, x5) -> f1203_out(x4, x5) :|: TRUE
f1228_in(s(T253), s(T254)) -> f1230_in(T253, T254) :|: TRUE
f1230_out(x6, x7) -> f1228_out(s(x6), s(x7)) :|: TRUE
f1231_out -> f1228_out(x8, x9) :|: TRUE
f1228_in(x10, x11) -> f1231_in :|: TRUE
f1230_in(x12, x13) -> f1201_in(x12, x13) :|: TRUE
f1201_out(x14, x15) -> f1230_out(x14, x15) :|: TRUE
f1201_in(x16, x17) -> f1203_in(x16, x17) :|: TRUE
f1203_out(x18, x19) -> f1201_out(x18, x19) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x20) -> f2_out(x20) :|: TRUE
f3_in(x21) -> f326_in(x21) :|: TRUE
f324_out(x22) -> f3_out(x22) :|: TRUE
f326_out(x23) -> f3_out(x23) :|: TRUE
f3_in(x24) -> f324_in(x24) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x25) -> f347_in :|: TRUE
f347_out -> f324_out(x26) :|: TRUE
f324_in(x27) -> f345_in(x27) :|: TRUE
f345_in(x28) -> f467_in :|: TRUE
f478_out(x29, x30) -> f345_out(x30) :|: TRUE
f467_out -> f478_in(x31, x32) :|: TRUE
f478_in(x33, x34) -> f1127_in(x33) :|: TRUE
f1127_out(x35) -> f1128_in(x36, x37) :|: TRUE
f1128_out(x38, x39) -> f478_out(x40, x39) :|: TRUE
f1141_out(T23) -> f1127_out(T23) :|: TRUE
f1127_in(x41) -> f1141_in(x41) :|: TRUE
f1141_in(x42) -> f1143_in(x42) :|: TRUE
f1143_out(x43) -> f1141_out(x43) :|: TRUE
f1141_in(x44) -> f1142_in(x44) :|: TRUE
f1142_out(x45) -> f1141_out(x45) :|: TRUE
f1144_out(T216, T217) -> f1142_out(.(T216, T217)) :|: TRUE
f1142_in(x46) -> f1145_in :|: TRUE
f1145_out -> f1142_out(x47) :|: TRUE
f1142_in(.(x48, x49)) -> f1144_in(x48, x49) :|: TRUE
f1191_out(x50, x51) -> f1192_in(x52, x53, x50) :|: TRUE
f1144_in(x54, x55) -> f1191_in(x54, x55) :|: TRUE
f1192_out(x56, x57, x58) -> f1144_out(x58, x59) :|: TRUE
f1191_in(x60, x61) -> f1193_in(x60, x61) :|: TRUE
f1193_out(x62, x63) -> f1191_out(x62, x63) :|: TRUE
f1195_out(x64, x65) -> f1193_out(x64, x65) :|: TRUE
f1194_out(x66, x67) -> f1193_out(x66, x67) :|: TRUE
f1193_in(x68, x69) -> f1194_in(x68, x69) :|: TRUE
f1193_in(x70, x71) -> f1195_in(x70, x71) :|: TRUE
f1199_out(x72, x73, x74) -> f1194_out(x72, .(x73, x74)) :|: TRUE
f1200_out -> f1194_out(x75, x76) :|: TRUE
f1194_in(x77, .(x78, x79)) -> f1199_in(x77, x78, x79) :|: TRUE
f1194_in(x80, x81) -> f1200_in :|: TRUE
f1199_in(x82, x83, x84) -> f1201_in(x82, x83) :|: TRUE
f1202_out(x85, x86) -> f1199_out(x85, x87, x86) :|: TRUE
f1201_out(x88, x89) -> f1202_in(x88, x90) :|: TRUE
f1128_in(x91, x92) -> f1390_in :|: TRUE
f1391_out(x93, x94) -> f1128_out(x93, x94) :|: TRUE
f1390_out -> f1391_in(x95, x96) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x97) -> f1395_out :|: TRUE
f1440_out(x98) -> f1441_in(x99) :|: TRUE
f1441_out(x100) -> f1400_out(x101) :|: TRUE
f1400_in(x102) -> f1440_in(x102) :|: TRUE
f1127_out(x103) -> f1440_out(x103) :|: TRUE
f1440_in(x104) -> f1127_in(x104) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(428) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(429)
TRUE

----------------------------------------

(430)
Obligation:
Rules:
f1201_in(T238, T239) -> f1203_in(T238, T239) :|: TRUE
f1203_out(x, x1) -> f1201_out(x, x1) :|: TRUE
f1191_in(T216, T217) -> f1193_in(T216, T217) :|: TRUE
f1193_out(x2, x3) -> f1191_out(x2, x3) :|: TRUE
f1330_in -> f1330_out :|: TRUE
f1228_out(x4, x5) -> f1203_out(x4, x5) :|: TRUE
f1203_in(x6, x7) -> f1228_in(x6, x7) :|: TRUE
f1203_in(x8, x9) -> f1229_in(x8, x9) :|: TRUE
f1229_out(x10, x11) -> f1203_out(x10, x11) :|: TRUE
f1228_in(s(T253), s(T254)) -> f1230_in(T253, T254) :|: TRUE
f1230_out(x12, x13) -> f1228_out(s(x12), s(x13)) :|: TRUE
f1231_out -> f1228_out(x14, x15) :|: TRUE
f1228_in(x16, x17) -> f1231_in :|: TRUE
f1255_in(T272, T273) -> f1318_in(T272, T273) :|: TRUE
f1319_out(x18, x19) -> f1255_out(x18, x19) :|: TRUE
f1318_out(x20, x21) -> f1255_out(x20, x21) :|: TRUE
f1255_in(x22, x23) -> f1319_in(x22, x23) :|: TRUE
f1199_in(x24, x25, x26) -> f1201_in(x24, x25) :|: TRUE
f1202_out(x27, x28) -> f1199_out(x27, x29, x28) :|: TRUE
f1201_out(x30, x31) -> f1202_in(x30, x32) :|: TRUE
f1232_out -> f1229_out(s(0), 0) :|: TRUE
f1229_in(x33, x34) -> f1233_in :|: TRUE
f1229_in(s(0), 0) -> f1232_in :|: TRUE
f1233_out -> f1229_out(x35, x36) :|: TRUE
f1232_in -> f1232_out :|: TRUE
f1338_in -> f1338_out :|: TRUE
f1319_in(0, 0) -> f1338_in :|: TRUE
f1319_in(x37, x38) -> f1339_in :|: TRUE
f1339_out -> f1319_out(x39, x40) :|: TRUE
f1338_out -> f1319_out(0, 0) :|: TRUE
f1247_in(x41, x42) -> f1250_in :|: TRUE
f1247_in(x43, .(x44, x45)) -> f1249_in(x43, x44, x45) :|: TRUE
f1249_out(x46, x47, x48) -> f1247_out(x46, .(x47, x48)) :|: TRUE
f1250_out -> f1247_out(x49, x50) :|: TRUE
f1253_in(x51, x52) -> f1254_in(x51, x52) :|: TRUE
f1255_out(x53, x54) -> f1253_out(x53, x54) :|: TRUE
f1254_out(x55, x56) -> f1253_out(x55, x56) :|: TRUE
f1253_in(x57, x58) -> f1255_in(x57, x58) :|: TRUE
f1253_out(x59, x60) -> f1251_out(x59, x60) :|: TRUE
f1251_in(x61, x62) -> f1253_in(x61, x62) :|: TRUE
f1252_out(x63, x64) -> f1249_out(x63, x65, x64) :|: TRUE
f1249_in(x66, x67, x68) -> f1251_in(x66, x67) :|: TRUE
f1251_out(x69, x70) -> f1252_in(x69, x71) :|: TRUE
f1252_in(x72, x73) -> f1191_in(x72, x73) :|: TRUE
f1191_out(x74, x75) -> f1252_out(x74, x75) :|: TRUE
f1316_out -> f1254_out(x76, x77) :|: TRUE
f1315_out(T287, T288) -> f1254_out(s(T287), s(T288)) :|: TRUE
f1254_in(s(x78), s(x79)) -> f1315_in(x78, x79) :|: TRUE
f1254_in(x80, x81) -> f1316_in :|: TRUE
f1331_out -> f1318_out(x82, x83) :|: TRUE
f1318_in(0, s(T295)) -> f1330_in :|: TRUE
f1318_in(x84, x85) -> f1331_in :|: TRUE
f1330_out -> f1318_out(0, s(x86)) :|: TRUE
f1191_out(x87, x88) -> f1202_out(x87, x88) :|: TRUE
f1202_in(x89, x90) -> f1191_in(x89, x90) :|: TRUE
f1230_in(x91, x92) -> f1201_in(x91, x92) :|: TRUE
f1201_out(x93, x94) -> f1230_out(x93, x94) :|: TRUE
f1195_out(x95, x96) -> f1193_out(x95, x96) :|: TRUE
f1194_out(x97, x98) -> f1193_out(x97, x98) :|: TRUE
f1193_in(x99, x100) -> f1194_in(x99, x100) :|: TRUE
f1193_in(x101, x102) -> f1195_in(x101, x102) :|: TRUE
f1199_out(x103, x104, x105) -> f1194_out(x103, .(x104, x105)) :|: TRUE
f1200_out -> f1194_out(x106, x107) :|: TRUE
f1194_in(x108, .(x109, x110)) -> f1199_in(x108, x109, x110) :|: TRUE
f1194_in(x111, x112) -> f1200_in :|: TRUE
f1247_out(x113, x114) -> f1195_out(x113, x114) :|: TRUE
f1195_in(x115, x116) -> f1247_in(x115, x116) :|: TRUE
f1195_in(x117, x118) -> f1248_in(x117, x118) :|: TRUE
f1248_out(x119, x120) -> f1195_out(x119, x120) :|: TRUE
f1315_in(x121, x122) -> f1251_in(x121, x122) :|: TRUE
f1251_out(x123, x124) -> f1315_out(x123, x124) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x125) -> f2_out(x125) :|: TRUE
f3_in(x126) -> f326_in(x126) :|: TRUE
f324_out(x127) -> f3_out(x127) :|: TRUE
f326_out(x128) -> f3_out(x128) :|: TRUE
f3_in(x129) -> f324_in(x129) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x130) -> f347_in :|: TRUE
f347_out -> f324_out(x131) :|: TRUE
f324_in(x132) -> f345_in(x132) :|: TRUE
f345_in(x133) -> f467_in :|: TRUE
f478_out(x134, x135) -> f345_out(x135) :|: TRUE
f467_out -> f478_in(x136, x137) :|: TRUE
f478_in(x138, x139) -> f1127_in(x138) :|: TRUE
f1127_out(x140) -> f1128_in(x141, x142) :|: TRUE
f1128_out(x143, x144) -> f478_out(x145, x144) :|: TRUE
f1128_in(x146, x147) -> f1390_in :|: TRUE
f1391_out(x148, x149) -> f1128_out(x148, x149) :|: TRUE
f1390_out -> f1391_in(x150, x151) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x152) -> f1395_out :|: TRUE
f1440_out(x153) -> f1441_in(x154) :|: TRUE
f1441_out(x155) -> f1400_out(x156) :|: TRUE
f1400_in(x157) -> f1440_in(x157) :|: TRUE
f1127_out(x158) -> f1440_out(x158) :|: TRUE
f1440_in(x159) -> f1127_in(x159) :|: TRUE
f1141_out(T23) -> f1127_out(T23) :|: TRUE
f1127_in(x160) -> f1141_in(x160) :|: TRUE
f1141_in(x161) -> f1143_in(x161) :|: TRUE
f1143_out(x162) -> f1141_out(x162) :|: TRUE
f1141_in(x163) -> f1142_in(x163) :|: TRUE
f1142_out(x164) -> f1141_out(x164) :|: TRUE
f1144_out(x165, x166) -> f1142_out(.(x165, x166)) :|: TRUE
f1142_in(x167) -> f1145_in :|: TRUE
f1145_out -> f1142_out(x168) :|: TRUE
f1142_in(.(x169, x170)) -> f1144_in(x169, x170) :|: TRUE
f1191_out(x171, x172) -> f1192_in(x173, x174, x171) :|: TRUE
f1144_in(x175, x176) -> f1191_in(x175, x176) :|: TRUE
f1192_out(x177, x178, x179) -> f1144_out(x179, x180) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(431) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(432)
TRUE

----------------------------------------

(433)
Obligation:
Rules:
f1144_out(T216, T217) -> f1142_out(.(T216, T217)) :|: TRUE
f1142_in(T23) -> f1145_in :|: TRUE
f1145_out -> f1142_out(x) :|: TRUE
f1142_in(.(x1, x2)) -> f1144_in(x1, x2) :|: TRUE
f1191_in(x3, x4) -> f1193_in(x3, x4) :|: TRUE
f1193_out(x5, x6) -> f1191_out(x5, x6) :|: TRUE
f1330_in -> f1330_out :|: TRUE
f1199_in(T238, T239, T240) -> f1201_in(T238, T239) :|: TRUE
f1202_out(x7, x8) -> f1199_out(x7, x9, x8) :|: TRUE
f1201_out(x10, x11) -> f1202_in(x10, x12) :|: TRUE
f1338_in -> f1338_out :|: TRUE
f1319_in(0, 0) -> f1338_in :|: TRUE
f1319_in(T272, T273) -> f1339_in :|: TRUE
f1339_out -> f1319_out(x13, x14) :|: TRUE
f1338_out -> f1319_out(0, 0) :|: TRUE
f1253_out(x15, x16) -> f1251_out(x15, x16) :|: TRUE
f1251_in(x17, x18) -> f1253_in(x17, x18) :|: TRUE
f1127_out(T221) -> f1351_out(T221) :|: TRUE
f1351_in(x19) -> f1127_in(x19) :|: TRUE
f1316_out -> f1254_out(x20, x21) :|: TRUE
f1315_out(T287, T288) -> f1254_out(s(T287), s(T288)) :|: TRUE
f1254_in(s(x22), s(x23)) -> f1315_in(x22, x23) :|: TRUE
f1254_in(x24, x25) -> f1316_in :|: TRUE
f1379_out(x26, x27, x28) -> f1378_out(x26, x27, x28) :|: TRUE
f1378_in(x29, x30, x31) -> f1379_in(x29, x30, x31) :|: TRUE
f1331_out -> f1318_out(x32, x33) :|: TRUE
f1318_in(0, s(T295)) -> f1330_in :|: TRUE
f1318_in(x34, x35) -> f1331_in :|: TRUE
f1330_out -> f1318_out(0, s(x36)) :|: TRUE
f1377_in(T222) -> f1127_in(T222) :|: TRUE
f1127_out(x37) -> f1377_out(x37) :|: TRUE
f1345_in -> f1345_out :|: TRUE
f1191_out(x38, x39) -> f1202_out(x38, x39) :|: TRUE
f1202_in(x40, x41) -> f1191_in(x40, x41) :|: TRUE
f1195_out(x42, x43) -> f1193_out(x42, x43) :|: TRUE
f1194_out(x44, x45) -> f1193_out(x44, x45) :|: TRUE
f1193_in(x46, x47) -> f1194_in(x46, x47) :|: TRUE
f1193_in(x48, x49) -> f1195_in(x48, x49) :|: TRUE
f1199_out(x50, x51, x52) -> f1194_out(x50, .(x51, x52)) :|: TRUE
f1200_out -> f1194_out(x53, x54) :|: TRUE
f1194_in(x55, .(x56, x57)) -> f1199_in(x55, x56, x57) :|: TRUE
f1194_in(x58, x59) -> f1200_in :|: TRUE
f1247_out(x60, x61) -> f1195_out(x60, x61) :|: TRUE
f1195_in(x62, x63) -> f1247_in(x62, x63) :|: TRUE
f1195_in(x64, x65) -> f1248_in(x64, x65) :|: TRUE
f1248_out(x66, x67) -> f1195_out(x66, x67) :|: TRUE
f1315_in(x68, x69) -> f1251_in(x68, x69) :|: TRUE
f1251_out(x70, x71) -> f1315_out(x70, x71) :|: TRUE
f1201_in(x72, x73) -> f1203_in(x72, x73) :|: TRUE
f1203_out(x74, x75) -> f1201_out(x74, x75) :|: TRUE
f1141_in(x76) -> f1143_in(x76) :|: TRUE
f1143_out(x77) -> f1141_out(x77) :|: TRUE
f1141_in(x78) -> f1142_in(x78) :|: TRUE
f1142_out(x79) -> f1141_out(x79) :|: TRUE
f1384_in -> f1384_out :|: TRUE
f1255_in(x80, x81) -> f1318_in(x80, x81) :|: TRUE
f1319_out(x82, x83) -> f1255_out(x82, x83) :|: TRUE
f1318_out(x84, x85) -> f1255_out(x84, x85) :|: TRUE
f1255_in(x86, x87) -> f1319_in(x86, x87) :|: TRUE
f1228_out(x88, x89) -> f1203_out(x88, x89) :|: TRUE
f1203_in(x90, x91) -> f1228_in(x90, x91) :|: TRUE
f1203_in(x92, x93) -> f1229_in(x92, x93) :|: TRUE
f1229_out(x94, x95) -> f1203_out(x94, x95) :|: TRUE
f1228_in(s(T253), s(T254)) -> f1230_in(T253, T254) :|: TRUE
f1230_out(x96, x97) -> f1228_out(s(x96), s(x97)) :|: TRUE
f1231_out -> f1228_out(x98, x99) :|: TRUE
f1228_in(x100, x101) -> f1231_in :|: TRUE
f1379_in(x102, x103, x104) -> f1381_in(x102, x103, x104) :|: TRUE
f1381_out(x105, x106, x107) -> f1379_out(x105, x106, x107) :|: TRUE
f1380_out(x108, x109, x110) -> f1379_out(x108, x109, x110) :|: TRUE
f1379_in(x111, x112, x113) -> f1380_in(x111, x112, x113) :|: TRUE
f1232_out -> f1229_out(s(0), 0) :|: TRUE
f1229_in(x114, x115) -> f1233_in :|: TRUE
f1229_in(s(0), 0) -> f1232_in :|: TRUE
f1233_out -> f1229_out(x116, x117) :|: TRUE
f1232_in -> f1232_out :|: TRUE
f1247_in(x118, x119) -> f1250_in :|: TRUE
f1247_in(x120, .(x121, x122)) -> f1249_in(x120, x121, x122) :|: TRUE
f1249_out(x123, x124, x125) -> f1247_out(x123, .(x124, x125)) :|: TRUE
f1250_out -> f1247_out(x126, x127) :|: TRUE
f1377_out(x128) -> f1378_in(x129, x130, x131) :|: TRUE
f1378_out(x132, x133, x134) -> f1352_out(x135, x132, x133) :|: TRUE
f1352_in(x136, x137, x138) -> f1377_in(x136) :|: TRUE
f1346_out -> f1248_out(x139, x140) :|: TRUE
f1248_in(T301, []) -> f1345_in :|: TRUE
f1345_out -> f1248_out(x141, []) :|: TRUE
f1248_in(x142, x143) -> f1346_in :|: TRUE
f1141_out(x144) -> f1127_out(x144) :|: TRUE
f1127_in(x145) -> f1141_in(x145) :|: TRUE
f1253_in(x146, x147) -> f1254_in(x146, x147) :|: TRUE
f1255_out(x148, x149) -> f1253_out(x148, x149) :|: TRUE
f1254_out(x150, x151) -> f1253_out(x150, x151) :|: TRUE
f1253_in(x152, x153) -> f1255_in(x152, x153) :|: TRUE
f1252_out(x154, x155) -> f1249_out(x154, x156, x155) :|: TRUE
f1249_in(x157, x158, x159) -> f1251_in(x157, x158) :|: TRUE
f1251_out(x160, x161) -> f1252_in(x160, x162) :|: TRUE
f1252_in(x163, x164) -> f1191_in(x163, x164) :|: TRUE
f1191_out(x165, x166) -> f1252_out(x165, x166) :|: TRUE
f1191_out(x167, x168) -> f1192_in(x169, x170, x167) :|: TRUE
f1144_in(x171, x172) -> f1191_in(x171, x172) :|: TRUE
f1192_out(x173, x174, x175) -> f1144_out(x175, x176) :|: TRUE
f1230_in(x177, x178) -> f1201_in(x177, x178) :|: TRUE
f1201_out(x179, x180) -> f1230_out(x179, x180) :|: TRUE
f1382_in(T333, T334, T335) -> f1378_in(T333, T334, T335) :|: TRUE
f1378_out(x181, x182, x183) -> f1382_out(x181, x182, x183) :|: TRUE
f1380_in(.(x184, x185), x186, x187) -> f1382_in(x185, x186, x187) :|: TRUE
f1383_out -> f1380_out(x188, x189, x190) :|: TRUE
f1382_out(x191, x192, x193) -> f1380_out(.(x194, x191), x192, x193) :|: TRUE
f1380_in(x195, x196, x197) -> f1383_in :|: TRUE
f1351_out(x198) -> f1352_in(x199, x200, x201) :|: TRUE
f1352_out(x202, x203, x204) -> f1192_out(x205, x202, x204) :|: TRUE
f1192_in(x206, x207, x208) -> f1351_in(x206) :|: TRUE
f1385_out -> f1381_out(x209, x210, x211) :|: TRUE
f1381_in([], T344, T345) -> f1384_in :|: TRUE
f1384_out -> f1381_out([], x212, x213) :|: TRUE
f1381_in(x214, x215, x216) -> f1385_in :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x217) -> f2_out(x217) :|: TRUE
f3_in(x218) -> f326_in(x218) :|: TRUE
f324_out(x219) -> f3_out(x219) :|: TRUE
f326_out(x220) -> f3_out(x220) :|: TRUE
f3_in(x221) -> f324_in(x221) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x222) -> f347_in :|: TRUE
f347_out -> f324_out(x223) :|: TRUE
f324_in(x224) -> f345_in(x224) :|: TRUE
f345_in(x225) -> f467_in :|: TRUE
f478_out(x226, x227) -> f345_out(x227) :|: TRUE
f467_out -> f478_in(x228, x229) :|: TRUE
f478_in(x230, x231) -> f1127_in(x230) :|: TRUE
f1127_out(x232) -> f1128_in(x233, x234) :|: TRUE
f1128_out(x235, x236) -> f478_out(x237, x236) :|: TRUE
f1128_in(x238, x239) -> f1390_in :|: TRUE
f1391_out(x240, x241) -> f1128_out(x240, x241) :|: TRUE
f1390_out -> f1391_in(x242, x243) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x244) -> f1395_out :|: TRUE
f1440_out(x245) -> f1441_in(x246) :|: TRUE
f1441_out(x247) -> f1400_out(x248) :|: TRUE
f1400_in(x249) -> f1440_in(x249) :|: TRUE
f1127_out(x250) -> f1440_out(x250) :|: TRUE
f1440_in(x251) -> f1127_in(x251) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(434) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(435)
TRUE

----------------------------------------

(436)
Obligation:
Rules:
f1114_in -> f1109_in :|: TRUE
f1109_out -> f1114_out :|: TRUE
f1112_out -> f1111_out :|: TRUE
f1111_in -> f1113_in :|: TRUE
f1113_out -> f1111_out :|: TRUE
f1111_in -> f1112_in :|: TRUE
f1109_in -> f1111_in :|: TRUE
f1111_out -> f1109_out :|: TRUE
f1112_in -> f1115_in :|: TRUE
f1115_out -> f1112_out :|: TRUE
f1114_out -> f1112_out :|: TRUE
f1112_in -> f1114_in :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x) -> f2_out(x) :|: TRUE
f3_in(x1) -> f326_in(x1) :|: TRUE
f324_out(x2) -> f3_out(x2) :|: TRUE
f326_out(x3) -> f3_out(x3) :|: TRUE
f3_in(x4) -> f324_in(x4) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x5) -> f347_in :|: TRUE
f347_out -> f324_out(x6) :|: TRUE
f324_in(x7) -> f345_in(x7) :|: TRUE
f345_in(x8) -> f467_in :|: TRUE
f478_out(x9, x10) -> f345_out(x10) :|: TRUE
f467_out -> f478_in(x11, x12) :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1108_out -> f1105_out :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1105_in -> f1108_in :|: TRUE
f1110_out(T174) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x13) :|: TRUE
f478_in(x14, x15) -> f1127_in(x14) :|: TRUE
f1127_out(x16) -> f1128_in(x17, x18) :|: TRUE
f1128_out(x19, x20) -> f478_out(x21, x20) :|: TRUE
f1128_in(x22, x23) -> f1390_in :|: TRUE
f1391_out(x24, x25) -> f1128_out(x24, x25) :|: TRUE
f1390_out -> f1391_in(x26, x27) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x28) -> f1395_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(437) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:

intTRSProblem:
f1114_in -> f1109_in :|: TRUE
f1111_in -> f1112_in :|: TRUE
f1109_in -> f1111_in :|: TRUE
f1112_in -> f1114_in :|: TRUE


----------------------------------------

(438)
Obligation:
Rules:
f1114_in -> f1109_in :|: TRUE
f1111_in -> f1112_in :|: TRUE
f1109_in -> f1111_in :|: TRUE
f1112_in -> f1114_in :|: TRUE

----------------------------------------

(439) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(440)
Obligation:
Rules:
f1114_in -> f1114_in :|: TRUE

----------------------------------------

(441) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(442)
Obligation:
Rules:
f1114_in -> f1114_in :|: TRUE

----------------------------------------

(443) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1114_in -> f1114_in :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!
----------------------------------------

(444)
Obligation:

Termination digraph:
Nodes:
(1) f1114_in -> f1114_in :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!

----------------------------------------

(445) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f1114_in()
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(446)
Obligation:
Rules:
f1114_in -> f1114_in :|: TRUE

----------------------------------------

(447) IntTRSNonPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc) -> f(1) :|: pc = 1 && TRUE
Proved unsatisfiability of the following formula, indicating that the system is never left after entering:
((run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T)) and !(((run2_0 * 1)) = ((1 * 1)) and T))
Proved satisfiability of the following formula, indicating that the system is entered at least once:
(run2_0 = ((1 * 1)) and (((run1_0 * 1)) = ((1 * 1)) and T))

----------------------------------------

(448)
NO

----------------------------------------

(449)
Obligation:
Rules:
f1011_in(T120) -> f1013_in(T120) :|: TRUE
f1013_out(x) -> f1011_out(x) :|: TRUE
f1014_in(x1) -> f1017_in :|: TRUE
f1017_out -> f1014_out(x2) :|: TRUE
f1014_in(s(T138)) -> f1016_in(T138) :|: TRUE
f1016_out(x3) -> f1014_out(s(x3)) :|: TRUE
f1016_in(x4) -> f1011_in(x4) :|: TRUE
f1011_out(x5) -> f1016_out(x5) :|: TRUE
f1013_in(x6) -> f1015_in(x6) :|: TRUE
f1015_out(x7) -> f1013_out(x7) :|: TRUE
f1014_out(x8) -> f1013_out(x8) :|: TRUE
f1013_in(x9) -> f1014_in(x9) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x10) -> f2_out(x10) :|: TRUE
f3_in(x11) -> f326_in(x11) :|: TRUE
f324_out(x12) -> f3_out(x12) :|: TRUE
f326_out(x13) -> f3_out(x13) :|: TRUE
f3_in(x14) -> f324_in(x14) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x15) -> f347_in :|: TRUE
f347_out -> f324_out(x16) :|: TRUE
f324_in(x17) -> f345_in(x17) :|: TRUE
f345_in(x18) -> f467_in :|: TRUE
f478_out(x19, x20) -> f345_out(x20) :|: TRUE
f467_out -> f478_in(x21, x22) :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1108_out -> f1105_out :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1105_in -> f1108_in :|: TRUE
f1110_out(T174) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x23) :|: TRUE
f633_out(x24) -> f1110_out(x24) :|: TRUE
f1110_in(x25) -> f633_in(x25) :|: TRUE
f855_out(T49) -> f633_out(T49) :|: TRUE
f633_in(x26) -> f855_in(x26) :|: TRUE
f855_in(x27) -> f862_in(x27) :|: TRUE
f862_out(x28) -> f855_out(x28) :|: TRUE
f855_in(x29) -> f860_in(x29) :|: TRUE
f860_out(x30) -> f855_out(x30) :|: TRUE
f996_out(x31) -> f862_out(x31) :|: TRUE
f862_in(x32) -> f995_in(x32) :|: TRUE
f995_out(x33) -> f862_out(x33) :|: TRUE
f862_in(x34) -> f996_in(x34) :|: TRUE
f1010_out -> f995_out(x35) :|: TRUE
f1009_out(x36) -> f995_out(x36) :|: TRUE
f995_in(x37) -> f1010_in :|: TRUE
f995_in(x38) -> f1009_in(x38) :|: TRUE
f1009_in(x39) -> f1011_in(x39) :|: TRUE
f1012_out(x40) -> f1009_out(x40) :|: TRUE
f1011_out(x41) -> f1012_in(x41) :|: TRUE
f614_in -> f629_in :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f629_out -> f614_out :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(x42) -> f627_out :|: TRUE
f631_out -> f633_in(x43) :|: TRUE
f478_in(x44, x45) -> f1127_in(x44) :|: TRUE
f1127_out(x46) -> f1128_in(x47, x48) :|: TRUE
f1128_out(x49, x50) -> f478_out(x51, x50) :|: TRUE
f1128_in(x52, x53) -> f1390_in :|: TRUE
f1391_out(x54, x55) -> f1128_out(x54, x55) :|: TRUE
f1390_out -> f1391_in(x56, x57) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x58) -> f1395_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(450) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(451)
TRUE

----------------------------------------

(452)
Obligation:
Rules:
f920_in(T82) -> f942_in :|: TRUE
f938_out(T100) -> f920_out(s(T100)) :|: TRUE
f942_out -> f920_out(x) :|: TRUE
f920_in(s(x1)) -> f938_in(x1) :|: TRUE
f938_in(x2) -> f903_in(x2) :|: TRUE
f903_out(x3) -> f938_out(x3) :|: TRUE
f921_out(x4) -> f916_out(x4) :|: TRUE
f916_in(x5) -> f920_in(x5) :|: TRUE
f916_in(x6) -> f921_in(x6) :|: TRUE
f920_out(x7) -> f916_out(x7) :|: TRUE
f916_out(x8) -> f903_out(x8) :|: TRUE
f903_in(x9) -> f916_in(x9) :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x10) -> f2_out(x10) :|: TRUE
f3_in(x11) -> f326_in(x11) :|: TRUE
f324_out(x12) -> f3_out(x12) :|: TRUE
f326_out(x13) -> f3_out(x13) :|: TRUE
f3_in(x14) -> f324_in(x14) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x15) -> f347_in :|: TRUE
f347_out -> f324_out(x16) :|: TRUE
f324_in(x17) -> f345_in(x17) :|: TRUE
f345_in(x18) -> f467_in :|: TRUE
f478_out(x19, x20) -> f345_out(x20) :|: TRUE
f467_out -> f478_in(x21, x22) :|: TRUE
f478_in(x23, x24) -> f1127_in(x23) :|: TRUE
f1127_out(x25) -> f1128_in(x26, x27) :|: TRUE
f1128_out(x28, x29) -> f478_out(x30, x29) :|: TRUE
f1128_in(x31, x32) -> f1390_in :|: TRUE
f1391_out(x33, x34) -> f1128_out(x33, x34) :|: TRUE
f1390_out -> f1391_in(x35, x36) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x37) -> f1395_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f614_in -> f629_in :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f629_out -> f614_out :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(T49) -> f627_out :|: TRUE
f631_out -> f633_in(x38) :|: TRUE
f855_out(x39) -> f633_out(x39) :|: TRUE
f633_in(x40) -> f855_in(x40) :|: TRUE
f855_in(x41) -> f862_in(x41) :|: TRUE
f862_out(x42) -> f855_out(x42) :|: TRUE
f855_in(x43) -> f860_in(x43) :|: TRUE
f860_out(x44) -> f855_out(x44) :|: TRUE
f860_in(x45) -> f883_in(x45) :|: TRUE
f860_in(x46) -> f888_in :|: TRUE
f883_out(x47) -> f860_out(x47) :|: TRUE
f888_out -> f860_out(x48) :|: TRUE
f905_out(x49) -> f883_out(x49) :|: TRUE
f883_in(x50) -> f903_in(x50) :|: TRUE
f903_out(x51) -> f905_in(x51) :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1108_out -> f1105_out :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1105_in -> f1108_in :|: TRUE
f1110_out(T174) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x52) :|: TRUE
f633_out(x53) -> f1110_out(x53) :|: TRUE
f1110_in(x54) -> f633_in(x54) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(453) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(454)
TRUE

----------------------------------------

(455)
Obligation:
Rules:
f1019_in(T120) -> f1024_in :|: TRUE
f1023_out -> f1019_out(0) :|: TRUE
f1019_in(0) -> f1023_in :|: TRUE
f1024_out -> f1019_out(x) :|: TRUE
f1018_in(0) -> f1020_in :|: TRUE
f1018_in(x1) -> f1021_in :|: TRUE
f1020_out -> f1018_out(0) :|: TRUE
f1021_out -> f1018_out(x2) :|: TRUE
f1012_in(x3) -> f633_in(x3) :|: TRUE
f633_out(x4) -> f1012_out(x4) :|: TRUE
f920_in(T82) -> f942_in :|: TRUE
f938_out(T100) -> f920_out(s(T100)) :|: TRUE
f942_out -> f920_out(x5) :|: TRUE
f920_in(s(x6)) -> f938_in(x6) :|: TRUE
f938_in(x7) -> f903_in(x7) :|: TRUE
f903_out(x8) -> f938_out(x8) :|: TRUE
f1014_in(x9) -> f1017_in :|: TRUE
f1017_out -> f1014_out(x10) :|: TRUE
f1014_in(s(T138)) -> f1016_in(T138) :|: TRUE
f1016_out(x11) -> f1014_out(s(x11)) :|: TRUE
f855_in(T49) -> f862_in(T49) :|: TRUE
f862_out(x12) -> f855_out(x12) :|: TRUE
f855_in(x13) -> f860_in(x13) :|: TRUE
f860_out(x14) -> f855_out(x14) :|: TRUE
f921_out(x15) -> f916_out(x15) :|: TRUE
f916_in(x16) -> f920_in(x16) :|: TRUE
f916_in(x17) -> f921_in(x17) :|: TRUE
f920_out(x18) -> f916_out(x18) :|: TRUE
f1009_in(x19) -> f1011_in(x19) :|: TRUE
f1012_out(x20) -> f1009_out(x20) :|: TRUE
f1011_out(x21) -> f1012_in(x21) :|: TRUE
f1011_in(x22) -> f1013_in(x22) :|: TRUE
f1013_out(x23) -> f1011_out(x23) :|: TRUE
f1016_in(x24) -> f1011_in(x24) :|: TRUE
f1011_out(x25) -> f1016_out(x25) :|: TRUE
f905_out(x26) -> f883_out(x26) :|: TRUE
f883_in(x27) -> f903_in(x27) :|: TRUE
f903_out(x28) -> f905_in(x28) :|: TRUE
f905_in(x29) -> f633_in(x29) :|: TRUE
f633_out(x30) -> f905_out(x30) :|: TRUE
f965_out -> f921_out(s(0)) :|: TRUE
f921_in(s(0)) -> f965_in :|: TRUE
f968_out -> f921_out(x31) :|: TRUE
f921_in(x32) -> f968_in :|: TRUE
f855_out(x33) -> f633_out(x33) :|: TRUE
f633_in(x34) -> f855_in(x34) :|: TRUE
f996_out(x35) -> f862_out(x35) :|: TRUE
f862_in(x36) -> f995_in(x36) :|: TRUE
f995_out(x37) -> f862_out(x37) :|: TRUE
f862_in(x38) -> f996_in(x38) :|: TRUE
f916_out(x39) -> f903_out(x39) :|: TRUE
f903_in(x40) -> f916_in(x40) :|: TRUE
f1015_in(x41) -> f1018_in(x41) :|: TRUE
f1019_out(x42) -> f1015_out(x42) :|: TRUE
f1018_out(x43) -> f1015_out(x43) :|: TRUE
f1015_in(x44) -> f1019_in(x44) :|: TRUE
f1013_in(x45) -> f1015_in(x45) :|: TRUE
f1015_out(x46) -> f1013_out(x46) :|: TRUE
f1014_out(x47) -> f1013_out(x47) :|: TRUE
f1013_in(x48) -> f1014_in(x48) :|: TRUE
f1023_in -> f1023_out :|: TRUE
f1020_in -> f1020_out :|: TRUE
f1010_out -> f995_out(x49) :|: TRUE
f1009_out(x50) -> f995_out(x50) :|: TRUE
f995_in(x51) -> f1010_in :|: TRUE
f995_in(x52) -> f1009_in(x52) :|: TRUE
f860_in(x53) -> f883_in(x53) :|: TRUE
f860_in(x54) -> f888_in :|: TRUE
f883_out(x55) -> f860_out(x55) :|: TRUE
f888_out -> f860_out(x56) :|: TRUE
f965_in -> f965_out :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x57) -> f2_out(x57) :|: TRUE
f3_in(x58) -> f326_in(x58) :|: TRUE
f324_out(x59) -> f3_out(x59) :|: TRUE
f326_out(x60) -> f3_out(x60) :|: TRUE
f3_in(x61) -> f324_in(x61) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x62) -> f347_in :|: TRUE
f347_out -> f324_out(x63) :|: TRUE
f324_in(x64) -> f345_in(x64) :|: TRUE
f345_in(x65) -> f467_in :|: TRUE
f478_out(x66, x67) -> f345_out(x67) :|: TRUE
f467_out -> f478_in(x68, x69) :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f614_in -> f629_in :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f629_out -> f614_out :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(x70) -> f627_out :|: TRUE
f631_out -> f633_in(x71) :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1108_out -> f1105_out :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1105_in -> f1108_in :|: TRUE
f1110_out(T174) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x72) :|: TRUE
f633_out(x73) -> f1110_out(x73) :|: TRUE
f1110_in(x74) -> f633_in(x74) :|: TRUE
f478_in(x75, x76) -> f1127_in(x75) :|: TRUE
f1127_out(x77) -> f1128_in(x78, x79) :|: TRUE
f1128_out(x80, x81) -> f478_out(x82, x81) :|: TRUE
f1128_in(x83, x84) -> f1390_in :|: TRUE
f1391_out(x85, x86) -> f1128_out(x85, x86) :|: TRUE
f1390_out -> f1391_in(x87, x88) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x89) -> f1395_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(456) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:


----------------------------------------

(457)
TRUE

----------------------------------------

(458)
Obligation:
Rules:
f845_in -> f631_in :|: TRUE
f631_out -> f845_out :|: TRUE
f843_in -> f846_in :|: TRUE
f846_out -> f843_out :|: TRUE
f845_out -> f843_out :|: TRUE
f843_in -> f845_in :|: TRUE
f635_in -> f844_in :|: TRUE
f635_in -> f843_in :|: TRUE
f843_out -> f635_out :|: TRUE
f844_out -> f635_out :|: TRUE
f635_out -> f631_out :|: TRUE
f631_in -> f635_in :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x) -> f2_out(x) :|: TRUE
f3_in(x1) -> f326_in(x1) :|: TRUE
f324_out(x2) -> f3_out(x2) :|: TRUE
f326_out(x3) -> f3_out(x3) :|: TRUE
f3_in(x4) -> f324_in(x4) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x5) -> f347_in :|: TRUE
f347_out -> f324_out(x6) :|: TRUE
f324_in(x7) -> f345_in(x7) :|: TRUE
f345_in(x8) -> f467_in :|: TRUE
f478_out(x9, x10) -> f345_out(x10) :|: TRUE
f467_out -> f478_in(x11, x12) :|: TRUE
f478_in(x13, x14) -> f1127_in(x13) :|: TRUE
f1127_out(x15) -> f1128_in(x16, x17) :|: TRUE
f1128_out(x18, x19) -> f478_out(x20, x19) :|: TRUE
f1128_in(x21, x22) -> f1390_in :|: TRUE
f1391_out(x23, x24) -> f1128_out(x23, x24) :|: TRUE
f1390_out -> f1391_in(x25, x26) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1399_out -> f1400_in(T367) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x27) -> f1395_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f614_in -> f629_in :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f629_out -> f614_out :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(T49) -> f627_out :|: TRUE
f631_out -> f633_in(x28) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(459) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:

intTRSProblem:
f845_in -> f631_in :|: TRUE
f843_in -> f845_in :|: TRUE
f635_in -> f843_in :|: TRUE
f631_in -> f635_in :|: TRUE


----------------------------------------

(460)
Obligation:
Rules:
f845_in -> f631_in :|: TRUE
f843_in -> f845_in :|: TRUE
f635_in -> f843_in :|: TRUE
f631_in -> f635_in :|: TRUE

----------------------------------------

(461) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(462)
Obligation:
Rules:
f635_in -> f635_in :|: TRUE

----------------------------------------

(463) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(464)
Obligation:
Rules:
f635_in -> f635_in :|: TRUE

----------------------------------------

(465) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f635_in -> f635_in :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!
----------------------------------------

(466)
Obligation:

Termination digraph:
Nodes:
(1) f635_in -> f635_in :|: TRUE

Arcs:
(1) -> (1)

This digraph is fully evaluated!

----------------------------------------

(467) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f635_in()
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(468)
Obligation:
Rules:
f635_in -> f635_in :|: TRUE

----------------------------------------

(469) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc) -> f(1) :|: pc = 1 && TRUE
Witness term starting non-terminating reduction: f(1)
----------------------------------------

(470)
NO

----------------------------------------

(471)
Obligation:
Rules:
f1191_in(T216, T217) -> f1193_in(T216, T217) :|: TRUE
f1193_out(x, x1) -> f1191_out(x, x1) :|: TRUE
f1330_in -> f1330_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
f920_in(T82) -> f942_in :|: TRUE
f938_out(T100) -> f920_out(s(T100)) :|: TRUE
f942_out -> f920_out(x2) :|: TRUE
f920_in(s(x3)) -> f938_in(x3) :|: TRUE
f614_in -> f629_in :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f629_out -> f614_out :|: TRUE
f1440_out(T367) -> f1441_in(T375) :|: TRUE
f1441_out(x4) -> f1400_out(x5) :|: TRUE
f1400_in(x6) -> f1440_in(x6) :|: TRUE
f1319_in(0, 0) -> f1338_in :|: TRUE
f1319_in(T272, T273) -> f1339_in :|: TRUE
f1339_out -> f1319_out(x7, x8) :|: TRUE
f1338_out -> f1319_out(0, 0) :|: TRUE
f1009_in(T120) -> f1011_in(T120) :|: TRUE
f1012_out(x9) -> f1009_out(x9) :|: TRUE
f1011_out(x10) -> f1012_in(x10) :|: TRUE
f1253_out(x11, x12) -> f1251_out(x11, x12) :|: TRUE
f1251_in(x13, x14) -> f1253_in(x13, x14) :|: TRUE
f905_in(x15) -> f633_in(x15) :|: TRUE
f633_out(x16) -> f905_out(x16) :|: TRUE
f1113_in -> f1116_in :|: TRUE
f1113_in -> f1117_in :|: TRUE
f1116_out -> f1113_out :|: TRUE
f1117_out -> f1113_out :|: TRUE
f1114_in -> f1109_in :|: TRUE
f1109_out -> f1114_out :|: TRUE
f1112_in -> f1115_in :|: TRUE
f1115_out -> f1112_out :|: TRUE
f1114_out -> f1112_out :|: TRUE
f1112_in -> f1114_in :|: TRUE
f1331_out -> f1318_out(x17, x18) :|: TRUE
f1318_in(0, s(T295)) -> f1330_in :|: TRUE
f1318_in(x19, x20) -> f1331_in :|: TRUE
f1330_out -> f1318_out(0, s(x21)) :|: TRUE
f1377_in(T222) -> f1127_in(T222) :|: TRUE
f1127_out(x22) -> f1377_out(x22) :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f1143_in([]) -> f1387_in :|: TRUE
f1388_out -> f1143_out(T23) :|: TRUE
f1143_in(x23) -> f1388_in :|: TRUE
f1387_out -> f1143_out([]) :|: TRUE
f1199_out(T238, T239, T240) -> f1194_out(T238, .(T239, T240)) :|: TRUE
f1200_out -> f1194_out(x24, x25) :|: TRUE
f1194_in(x26, .(x27, x28)) -> f1199_in(x26, x27, x28) :|: TRUE
f1194_in(x29, x30) -> f1200_in :|: TRUE
f849_in -> f849_out :|: TRUE
f1247_out(x31, x32) -> f1195_out(x31, x32) :|: TRUE
f1195_in(x33, x34) -> f1247_in(x33, x34) :|: TRUE
f1195_in(x35, x36) -> f1248_in(x35, x36) :|: TRUE
f1248_out(x37, x38) -> f1195_out(x37, x38) :|: TRUE
f1315_in(T287, T288) -> f1251_in(T287, T288) :|: TRUE
f1251_out(x39, x40) -> f1315_out(x39, x40) :|: TRUE
f996_in(T153) -> f1026_in :|: TRUE
f1026_out -> f996_out(x41) :|: TRUE
f996_in(T49) -> f1027_in :|: TRUE
f1027_out -> f996_out(x42) :|: TRUE
f1201_in(x43, x44) -> f1203_in(x43, x44) :|: TRUE
f1203_out(x45, x46) -> f1201_out(x45, x46) :|: TRUE
f1019_in(x47) -> f1024_in :|: TRUE
f1023_out -> f1019_out(0) :|: TRUE
f1019_in(0) -> f1023_in :|: TRUE
f1024_out -> f1019_out(x48) :|: TRUE
f1108_out -> f1105_out :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1105_in -> f1108_in :|: TRUE
f1228_in(s(T253), s(T254)) -> f1230_in(T253, T254) :|: TRUE
f1230_out(x49, x50) -> f1228_out(s(x49), s(x50)) :|: TRUE
f1231_out -> f1228_out(x51, x52) :|: TRUE
f1228_in(x53, x54) -> f1231_in :|: TRUE
f1012_in(x55) -> f633_in(x55) :|: TRUE
f633_out(x56) -> f1012_out(x56) :|: TRUE
f1232_out -> f1229_out(s(0), 0) :|: TRUE
f1229_in(x57, x58) -> f1233_in :|: TRUE
f1229_in(s(0), 0) -> f1232_in :|: TRUE
f1233_out -> f1229_out(x59, x60) :|: TRUE
f635_in -> f844_in :|: TRUE
f635_in -> f843_in :|: TRUE
f843_out -> f635_out :|: TRUE
f844_out -> f635_out :|: TRUE
f1377_out(x61) -> f1378_in(x62, x63, x64) :|: TRUE
f1378_out(x65, x66, x67) -> f1352_out(x68, x65, x66) :|: TRUE
f1352_in(x69, x70, x71) -> f1377_in(x69) :|: TRUE
f1346_out -> f1248_out(x72, x73) :|: TRUE
f1248_in(T301, []) -> f1345_in :|: TRUE
f1345_out -> f1248_out(x74, []) :|: TRUE
f1248_in(x75, x76) -> f1346_in :|: TRUE
f1393_out -> f1392_out :|: TRUE
f1394_out -> f1392_out :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1392_in -> f1394_in :|: TRUE
f1141_out(x77) -> f1127_out(x77) :|: TRUE
f1127_in(x78) -> f1141_in(x78) :|: TRUE
f1252_out(x79, x80) -> f1249_out(x79, x81, x80) :|: TRUE
f1249_in(x82, x83, x84) -> f1251_in(x82, x83) :|: TRUE
f1251_out(x85, x86) -> f1252_in(x85, x87) :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f1016_in(T138) -> f1011_in(T138) :|: TRUE
f1011_out(x88) -> f1016_out(x88) :|: TRUE
f905_out(x89) -> f883_out(x89) :|: TRUE
f883_in(x90) -> f903_in(x90) :|: TRUE
f903_out(x91) -> f905_in(x91) :|: TRUE
f855_out(x92) -> f633_out(x92) :|: TRUE
f633_in(x93) -> f855_in(x93) :|: TRUE
f996_out(x94) -> f862_out(x94) :|: TRUE
f862_in(x95) -> f995_in(x95) :|: TRUE
f995_out(x96) -> f862_out(x96) :|: TRUE
f862_in(x97) -> f996_in(x97) :|: TRUE
f916_out(x98) -> f903_out(x98) :|: TRUE
f903_in(x99) -> f916_in(x99) :|: TRUE
f1013_in(x100) -> f1015_in(x100) :|: TRUE
f1015_out(x101) -> f1013_out(x101) :|: TRUE
f1014_out(x102) -> f1013_out(x102) :|: TRUE
f1013_in(x103) -> f1014_in(x103) :|: TRUE
f1230_in(x104, x105) -> f1201_in(x104, x105) :|: TRUE
f1201_out(x106, x107) -> f1230_out(x106, x107) :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(x108) -> f627_out :|: TRUE
f631_out -> f633_in(x109) :|: TRUE
f1109_in -> f1111_in :|: TRUE
f1111_out -> f1109_out :|: TRUE
f1020_in -> f1020_out :|: TRUE
f1351_out(x110) -> f1352_in(x111, x112, x113) :|: TRUE
f1352_out(x114, x115, x116) -> f1192_out(x117, x114, x116) :|: TRUE
f1192_in(x118, x119, x120) -> f1351_in(x118) :|: TRUE
f860_in(x121) -> f883_in(x121) :|: TRUE
f860_in(x122) -> f888_in :|: TRUE
f883_out(x123) -> f860_out(x123) :|: TRUE
f888_out -> f860_out(x124) :|: TRUE
f1441_in(x125) -> f1442_in :|: TRUE
f1443_out(x126) -> f1441_out(x126) :|: TRUE
f1442_out -> f1443_in(x127) :|: TRUE
f1144_out(x128, x129) -> f1142_out(.(x128, x129)) :|: TRUE
f1142_in(x130) -> f1145_in :|: TRUE
f1145_out -> f1142_out(x131) :|: TRUE
f1142_in(.(x132, x133)) -> f1144_in(x132, x133) :|: TRUE
f843_in -> f846_in :|: TRUE
f846_out -> f843_out :|: TRUE
f845_out -> f843_out :|: TRUE
f843_in -> f845_in :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1018_in(0) -> f1020_in :|: TRUE
f1018_in(x134) -> f1021_in :|: TRUE
f1020_out -> f1018_out(0) :|: TRUE
f1021_out -> f1018_out(x135) :|: TRUE
f1014_in(x136) -> f1017_in :|: TRUE
f1017_out -> f1014_out(x137) :|: TRUE
f1014_in(s(x138)) -> f1016_in(x138) :|: TRUE
f1016_out(x139) -> f1014_out(s(x139)) :|: TRUE
f938_in(x140) -> f903_in(x140) :|: TRUE
f903_out(x141) -> f938_out(x141) :|: TRUE
f1199_in(x142, x143, x144) -> f1201_in(x142, x143) :|: TRUE
f1202_out(x145, x146) -> f1199_out(x145, x147, x146) :|: TRUE
f1201_out(x148, x149) -> f1202_in(x148, x150) :|: TRUE
f1395_out -> f1393_out :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1393_in -> f1396_in :|: TRUE
f1396_out -> f1393_out :|: TRUE
f1338_in -> f1338_out :|: TRUE
f1450_out(T405) -> f1445_out(.(T404, T405)) :|: TRUE
f1445_in(.(x151, x152)) -> f1450_in(x152) :|: TRUE
f1445_in(x153) -> f1451_in :|: TRUE
f1451_out -> f1445_out(x154) :|: TRUE
f1127_out(T221) -> f1351_out(T221) :|: TRUE
f1351_in(x155) -> f1127_in(x155) :|: TRUE
f965_out -> f921_out(s(0)) :|: TRUE
f921_in(s(0)) -> f965_in :|: TRUE
f968_out -> f921_out(x156) :|: TRUE
f921_in(x157) -> f968_in :|: TRUE
f1316_out -> f1254_out(x158, x159) :|: TRUE
f1315_out(x160, x161) -> f1254_out(s(x160), s(x161)) :|: TRUE
f1254_in(s(x162), s(x163)) -> f1315_in(x162, x163) :|: TRUE
f1254_in(x164, x165) -> f1316_in :|: TRUE
f1116_in -> f1119_in :|: TRUE
f1116_in -> f1118_in :|: TRUE
f1118_out -> f1116_out :|: TRUE
f1119_out -> f1116_out :|: TRUE
f1379_out(x166, x167, x168) -> f1378_out(x166, x167, x168) :|: TRUE
f1378_in(x169, x170, x171) -> f1379_in(x169, x170, x171) :|: TRUE
f633_out(T174) -> f1110_out(T174) :|: TRUE
f1110_in(x172) -> f633_in(x172) :|: TRUE
f1345_in -> f1345_out :|: TRUE
f1191_out(x173, x174) -> f1202_out(x173, x174) :|: TRUE
f1202_in(x175, x176) -> f1191_in(x175, x176) :|: TRUE
f1454_in -> f1454_out :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1392_out -> f1390_out :|: TRUE
f1106_in -> f1125_in :|: TRUE
f1124_out -> f1106_out :|: TRUE
f1106_in -> f1124_in :|: TRUE
f1125_out -> f1106_out :|: TRUE
f1195_out(x177, x178) -> f1193_out(x177, x178) :|: TRUE
f1194_out(x179, x180) -> f1193_out(x179, x180) :|: TRUE
f1193_in(x181, x182) -> f1194_in(x181, x182) :|: TRUE
f1193_in(x183, x184) -> f1195_in(x183, x184) :|: TRUE
f844_in -> f850_in :|: TRUE
f849_out -> f844_out :|: TRUE
f844_in -> f849_in :|: TRUE
f850_out -> f844_out :|: TRUE
f1444_out(x185) -> f1443_out(x185) :|: TRUE
f1443_in(x186) -> f1444_in(x186) :|: TRUE
f1121_out -> f1117_out :|: TRUE
f1122_out -> f1117_out :|: TRUE
f1117_in -> f1122_in :|: TRUE
f1117_in -> f1121_in :|: TRUE
f1110_out(x187) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x188) :|: TRUE
f1387_in -> f1387_out :|: TRUE
f1141_in(x189) -> f1143_in(x189) :|: TRUE
f1143_out(x190) -> f1141_out(x190) :|: TRUE
f1141_in(x191) -> f1142_in(x191) :|: TRUE
f1142_out(x192) -> f1141_out(x192) :|: TRUE
f1450_in(x193) -> f1443_in(x193) :|: TRUE
f1443_out(x194) -> f1450_out(x194) :|: TRUE
f635_out -> f631_out :|: TRUE
f631_in -> f635_in :|: TRUE
f1121_in -> f1121_out :|: TRUE
f1384_in -> f1384_out :|: TRUE
f1379_in(x195, x196, x197) -> f1381_in(x195, x196, x197) :|: TRUE
f1381_out(x198, x199, x200) -> f1379_out(x198, x199, x200) :|: TRUE
f1380_out(x201, x202, x203) -> f1379_out(x201, x202, x203) :|: TRUE
f1379_in(x204, x205, x206) -> f1380_in(x204, x205, x206) :|: TRUE
f1255_in(x207, x208) -> f1318_in(x207, x208) :|: TRUE
f1319_out(x209, x210) -> f1255_out(x209, x210) :|: TRUE
f1318_out(x211, x212) -> f1255_out(x211, x212) :|: TRUE
f1255_in(x213, x214) -> f1319_in(x213, x214) :|: TRUE
f1228_out(x215, x216) -> f1203_out(x215, x216) :|: TRUE
f1203_in(x217, x218) -> f1228_in(x217, x218) :|: TRUE
f1203_in(x219, x220) -> f1229_in(x219, x220) :|: TRUE
f1229_out(x221, x222) -> f1203_out(x221, x222) :|: TRUE
f1232_in -> f1232_out :|: TRUE
f855_in(x223) -> f862_in(x223) :|: TRUE
f862_out(x224) -> f855_out(x224) :|: TRUE
f855_in(x225) -> f860_in(x225) :|: TRUE
f860_out(x226) -> f855_out(x226) :|: TRUE
f921_out(x227) -> f916_out(x227) :|: TRUE
f916_in(x228) -> f920_in(x228) :|: TRUE
f916_in(x229) -> f921_in(x229) :|: TRUE
f920_out(x230) -> f916_out(x230) :|: TRUE
f1447_out(x231) -> f1444_out(x231) :|: TRUE
f1445_out(x232) -> f1444_out(x232) :|: TRUE
f1444_in(x233) -> f1447_in(x233) :|: TRUE
f1444_in(x234) -> f1445_in(x234) :|: TRUE
f1112_out -> f1111_out :|: TRUE
f1111_in -> f1113_in :|: TRUE
f1113_out -> f1111_out :|: TRUE
f1111_in -> f1112_in :|: TRUE
f1247_in(x235, x236) -> f1250_in :|: TRUE
f1247_in(x237, .(x238, x239)) -> f1249_in(x237, x238, x239) :|: TRUE
f1249_out(x240, x241, x242) -> f1247_out(x240, .(x241, x242)) :|: TRUE
f1250_out -> f1247_out(x243, x244) :|: TRUE
f1124_in -> f1124_out :|: TRUE
f1253_in(x245, x246) -> f1254_in(x245, x246) :|: TRUE
f1255_out(x247, x248) -> f1253_out(x247, x248) :|: TRUE
f1254_out(x249, x250) -> f1253_out(x249, x250) :|: TRUE
f1253_in(x251, x252) -> f1255_in(x251, x252) :|: TRUE
f1127_out(x253) -> f1440_out(x253) :|: TRUE
f1440_in(x254) -> f1127_in(x254) :|: TRUE
f1447_in([]) -> f1454_in :|: TRUE
f1455_out -> f1447_out(x255) :|: TRUE
f1447_in(x256) -> f1455_in :|: TRUE
f1454_out -> f1447_out([]) :|: TRUE
f1011_in(x257) -> f1013_in(x257) :|: TRUE
f1013_out(x258) -> f1011_out(x258) :|: TRUE
f845_in -> f631_in :|: TRUE
f631_out -> f845_out :|: TRUE
f1442_in -> f1390_in :|: TRUE
f1390_out -> f1442_out :|: TRUE
f1252_in(x259, x260) -> f1191_in(x259, x260) :|: TRUE
f1191_out(x261, x262) -> f1252_out(x261, x262) :|: TRUE
f1026_in -> f1026_out :|: TRUE
f1191_out(x263, x264) -> f1192_in(x265, x266, x263) :|: TRUE
f1144_in(x267, x268) -> f1191_in(x267, x268) :|: TRUE
f1192_out(x269, x270, x271) -> f1144_out(x271, x272) :|: TRUE
f1015_in(x273) -> f1018_in(x273) :|: TRUE
f1019_out(x274) -> f1015_out(x274) :|: TRUE
f1018_out(x275) -> f1015_out(x275) :|: TRUE
f1015_in(x276) -> f1019_in(x276) :|: TRUE
f1118_in -> f1118_out :|: TRUE
f1023_in -> f1023_out :|: TRUE
f1399_out -> f1400_in(x277) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1400_out(x278) -> f1395_out :|: TRUE
f1382_in(T333, T334, T335) -> f1378_in(T333, T334, T335) :|: TRUE
f1378_out(x279, x280, x281) -> f1382_out(x279, x280, x281) :|: TRUE
f1380_in(.(x282, x283), x284, x285) -> f1382_in(x283, x284, x285) :|: TRUE
f1383_out -> f1380_out(x286, x287, x288) :|: TRUE
f1382_out(x289, x290, x291) -> f1380_out(.(x292, x289), x290, x291) :|: TRUE
f1380_in(x293, x294, x295) -> f1383_in :|: TRUE
f1010_out -> f995_out(x296) :|: TRUE
f1009_out(x297) -> f995_out(x297) :|: TRUE
f995_in(x298) -> f1010_in :|: TRUE
f995_in(x299) -> f1009_in(x299) :|: TRUE
f1385_out -> f1381_out(x300, x301, x302) :|: TRUE
f1381_in([], T344, T345) -> f1384_in :|: TRUE
f1384_out -> f1381_out([], x303, x304) :|: TRUE
f1381_in(x305, x306, x307) -> f1385_in :|: TRUE
f965_in -> f965_out :|: TRUE
f2_in(T2) -> f3_in(T2) :|: TRUE
f3_out(x308) -> f2_out(x308) :|: TRUE
f3_in(x309) -> f326_in(x309) :|: TRUE
f324_out(x310) -> f3_out(x310) :|: TRUE
f326_out(x311) -> f3_out(x311) :|: TRUE
f3_in(x312) -> f324_in(x312) :|: TRUE
f345_out(T17) -> f324_out(T17) :|: TRUE
f324_in(x313) -> f347_in :|: TRUE
f347_out -> f324_out(x314) :|: TRUE
f324_in(x315) -> f345_in(x315) :|: TRUE
f345_in(x316) -> f467_in :|: TRUE
f478_out(x317, x318) -> f345_out(x318) :|: TRUE
f467_out -> f478_in(x319, x320) :|: TRUE
f478_in(x321, x322) -> f1127_in(x321) :|: TRUE
f1127_out(x323) -> f1128_in(x324, x325) :|: TRUE
f1128_out(x326, x327) -> f478_out(x328, x327) :|: TRUE
f1128_in(x329, x330) -> f1390_in :|: TRUE
f1391_out(x331, x332) -> f1128_out(x331, x332) :|: TRUE
f1390_out -> f1391_in(x333, x334) :|: TRUE
Start term: f2_in(T2)

----------------------------------------

(472) IRSwTSimpleDependencyGraphProof (EQUIVALENT)
Constructed simple dependency graph.

Simplified to the following IRSwTs:

intTRSProblem:
f1191_in(T216, T217) -> f1193_in(T216, T217) :|: TRUE
f1193_out(x, x1) -> f1191_out(x, x1) :|: TRUE
f1330_in -> f1330_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
f938_out(T100) -> f920_out(s(T100)) :|: TRUE
f920_in(s(x3)) -> f938_in(x3) :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f1440_out(T367) -> f1441_in(T375) :|: TRUE
f1400_in(x6) -> f1440_in(x6) :|: TRUE
f1319_in(0, 0) -> f1338_in :|: TRUE
f1338_out -> f1319_out(0, 0) :|: TRUE
f1009_in(T120) -> f1011_in(T120) :|: TRUE
f1012_out(x9) -> f1009_out(x9) :|: TRUE
f1011_out(x10) -> f1012_in(x10) :|: TRUE
f1253_out(x11, x12) -> f1251_out(x11, x12) :|: TRUE
f1251_in(x13, x14) -> f1253_in(x13, x14) :|: TRUE
f905_in(x15) -> f633_in(x15) :|: TRUE
f633_out(x16) -> f905_out(x16) :|: TRUE
f1113_in -> f1116_in :|: TRUE
f1113_in -> f1117_in :|: TRUE
f1116_out -> f1113_out :|: TRUE
f1117_out -> f1113_out :|: TRUE
f1114_in -> f1109_in :|: TRUE
f1109_out -> f1114_out :|: TRUE
f1114_out -> f1112_out :|: TRUE
f1112_in -> f1114_in :|: TRUE
f1318_in(0, s(T295)) -> f1330_in :|: TRUE
f1330_out -> f1318_out(0, s(x21)) :|: TRUE
f1377_in(T222) -> f1127_in(T222) :|: TRUE
f1127_out(x22) -> f1377_out(x22) :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f1143_in([]) -> f1387_in :|: TRUE
f1387_out -> f1143_out([]) :|: TRUE
f1199_out(T238, T239, T240) -> f1194_out(T238, .(T239, T240)) :|: TRUE
f1194_in(x26, .(x27, x28)) -> f1199_in(x26, x27, x28) :|: TRUE
f849_in -> f849_out :|: TRUE
f1247_out(x31, x32) -> f1195_out(x31, x32) :|: TRUE
f1195_in(x33, x34) -> f1247_in(x33, x34) :|: TRUE
f1195_in(x35, x36) -> f1248_in(x35, x36) :|: TRUE
f1248_out(x37, x38) -> f1195_out(x37, x38) :|: TRUE
f1315_in(T287, T288) -> f1251_in(T287, T288) :|: TRUE
f1251_out(x39, x40) -> f1315_out(x39, x40) :|: TRUE
f996_in(T153) -> f1026_in :|: TRUE
f1026_out -> f996_out(x41) :|: TRUE
f1201_in(x43, x44) -> f1203_in(x43, x44) :|: TRUE
f1203_out(x45, x46) -> f1201_out(x45, x46) :|: TRUE
f1023_out -> f1019_out(0) :|: TRUE
f1019_in(0) -> f1023_in :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1228_in(s(T253), s(T254)) -> f1230_in(T253, T254) :|: TRUE
f1230_out(x49, x50) -> f1228_out(s(x49), s(x50)) :|: TRUE
f1012_in(x55) -> f633_in(x55) :|: TRUE
f633_out(x56) -> f1012_out(x56) :|: TRUE
f1232_out -> f1229_out(s(0), 0) :|: TRUE
f1229_in(s(0), 0) -> f1232_in :|: TRUE
f635_in -> f844_in :|: TRUE
f635_in -> f843_in :|: TRUE
f843_out -> f635_out :|: TRUE
f844_out -> f635_out :|: TRUE
f1377_out(x61) -> f1378_in(x62, x63, x64) :|: TRUE
f1378_out(x65, x66, x67) -> f1352_out(x68, x65, x66) :|: TRUE
f1352_in(x69, x70, x71) -> f1377_in(x69) :|: TRUE
f1248_in(T301, []) -> f1345_in :|: TRUE
f1345_out -> f1248_out(x74, []) :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1141_out(x77) -> f1127_out(x77) :|: TRUE
f1127_in(x78) -> f1141_in(x78) :|: TRUE
f1252_out(x79, x80) -> f1249_out(x79, x81, x80) :|: TRUE
f1249_in(x82, x83, x84) -> f1251_in(x82, x83) :|: TRUE
f1251_out(x85, x86) -> f1252_in(x85, x87) :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f1016_in(T138) -> f1011_in(T138) :|: TRUE
f1011_out(x88) -> f1016_out(x88) :|: TRUE
f905_out(x89) -> f883_out(x89) :|: TRUE
f883_in(x90) -> f903_in(x90) :|: TRUE
f903_out(x91) -> f905_in(x91) :|: TRUE
f855_out(x92) -> f633_out(x92) :|: TRUE
f633_in(x93) -> f855_in(x93) :|: TRUE
f996_out(x94) -> f862_out(x94) :|: TRUE
f862_in(x95) -> f995_in(x95) :|: TRUE
f995_out(x96) -> f862_out(x96) :|: TRUE
f862_in(x97) -> f996_in(x97) :|: TRUE
f916_out(x98) -> f903_out(x98) :|: TRUE
f903_in(x99) -> f916_in(x99) :|: TRUE
f1013_in(x100) -> f1015_in(x100) :|: TRUE
f1015_out(x101) -> f1013_out(x101) :|: TRUE
f1014_out(x102) -> f1013_out(x102) :|: TRUE
f1013_in(x103) -> f1014_in(x103) :|: TRUE
f1230_in(x104, x105) -> f1201_in(x104, x105) :|: TRUE
f1201_out(x106, x107) -> f1230_out(x106, x107) :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(x108) -> f627_out :|: TRUE
f631_out -> f633_in(x109) :|: TRUE
f1109_in -> f1111_in :|: TRUE
f1111_out -> f1109_out :|: TRUE
f1020_in -> f1020_out :|: TRUE
f1351_out(x110) -> f1352_in(x111, x112, x113) :|: TRUE
f1352_out(x114, x115, x116) -> f1192_out(x117, x114, x116) :|: TRUE
f1192_in(x118, x119, x120) -> f1351_in(x118) :|: TRUE
f860_in(x121) -> f883_in(x121) :|: TRUE
f883_out(x123) -> f860_out(x123) :|: TRUE
f1441_in(x125) -> f1442_in :|: TRUE
f1144_out(x128, x129) -> f1142_out(.(x128, x129)) :|: TRUE
f1142_in(.(x132, x133)) -> f1144_in(x132, x133) :|: TRUE
f845_out -> f843_out :|: TRUE
f843_in -> f845_in :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1018_in(0) -> f1020_in :|: TRUE
f1020_out -> f1018_out(0) :|: TRUE
f1014_in(s(x138)) -> f1016_in(x138) :|: TRUE
f1016_out(x139) -> f1014_out(s(x139)) :|: TRUE
f938_in(x140) -> f903_in(x140) :|: TRUE
f903_out(x141) -> f938_out(x141) :|: TRUE
f1199_in(x142, x143, x144) -> f1201_in(x142, x143) :|: TRUE
f1202_out(x145, x146) -> f1199_out(x145, x147, x146) :|: TRUE
f1201_out(x148, x149) -> f1202_in(x148, x150) :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1338_in -> f1338_out :|: TRUE
f1127_out(T221) -> f1351_out(T221) :|: TRUE
f1351_in(x155) -> f1127_in(x155) :|: TRUE
f965_out -> f921_out(s(0)) :|: TRUE
f921_in(s(0)) -> f965_in :|: TRUE
f1315_out(x160, x161) -> f1254_out(s(x160), s(x161)) :|: TRUE
f1254_in(s(x162), s(x163)) -> f1315_in(x162, x163) :|: TRUE
f1116_in -> f1118_in :|: TRUE
f1118_out -> f1116_out :|: TRUE
f1379_out(x166, x167, x168) -> f1378_out(x166, x167, x168) :|: TRUE
f1378_in(x169, x170, x171) -> f1379_in(x169, x170, x171) :|: TRUE
f633_out(T174) -> f1110_out(T174) :|: TRUE
f1110_in(x172) -> f633_in(x172) :|: TRUE
f1345_in -> f1345_out :|: TRUE
f1191_out(x173, x174) -> f1202_out(x173, x174) :|: TRUE
f1202_in(x175, x176) -> f1191_in(x175, x176) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1124_out -> f1106_out :|: TRUE
f1106_in -> f1124_in :|: TRUE
f1195_out(x177, x178) -> f1193_out(x177, x178) :|: TRUE
f1194_out(x179, x180) -> f1193_out(x179, x180) :|: TRUE
f1193_in(x181, x182) -> f1194_in(x181, x182) :|: TRUE
f1193_in(x183, x184) -> f1195_in(x183, x184) :|: TRUE
f849_out -> f844_out :|: TRUE
f844_in -> f849_in :|: TRUE
f1121_out -> f1117_out :|: TRUE
f1117_in -> f1121_in :|: TRUE
f1110_out(x187) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x188) :|: TRUE
f1387_in -> f1387_out :|: TRUE
f1141_in(x189) -> f1143_in(x189) :|: TRUE
f1143_out(x190) -> f1141_out(x190) :|: TRUE
f1141_in(x191) -> f1142_in(x191) :|: TRUE
f1142_out(x192) -> f1141_out(x192) :|: TRUE
f635_out -> f631_out :|: TRUE
f631_in -> f635_in :|: TRUE
f1121_in -> f1121_out :|: TRUE
f1384_in -> f1384_out :|: TRUE
f1379_in(x195, x196, x197) -> f1381_in(x195, x196, x197) :|: TRUE
f1381_out(x198, x199, x200) -> f1379_out(x198, x199, x200) :|: TRUE
f1380_out(x201, x202, x203) -> f1379_out(x201, x202, x203) :|: TRUE
f1379_in(x204, x205, x206) -> f1380_in(x204, x205, x206) :|: TRUE
f1255_in(x207, x208) -> f1318_in(x207, x208) :|: TRUE
f1319_out(x209, x210) -> f1255_out(x209, x210) :|: TRUE
f1318_out(x211, x212) -> f1255_out(x211, x212) :|: TRUE
f1255_in(x213, x214) -> f1319_in(x213, x214) :|: TRUE
f1228_out(x215, x216) -> f1203_out(x215, x216) :|: TRUE
f1203_in(x217, x218) -> f1228_in(x217, x218) :|: TRUE
f1203_in(x219, x220) -> f1229_in(x219, x220) :|: TRUE
f1229_out(x221, x222) -> f1203_out(x221, x222) :|: TRUE
f1232_in -> f1232_out :|: TRUE
f855_in(x223) -> f862_in(x223) :|: TRUE
f862_out(x224) -> f855_out(x224) :|: TRUE
f855_in(x225) -> f860_in(x225) :|: TRUE
f860_out(x226) -> f855_out(x226) :|: TRUE
f921_out(x227) -> f916_out(x227) :|: TRUE
f916_in(x228) -> f920_in(x228) :|: TRUE
f916_in(x229) -> f921_in(x229) :|: TRUE
f920_out(x230) -> f916_out(x230) :|: TRUE
f1112_out -> f1111_out :|: TRUE
f1111_in -> f1113_in :|: TRUE
f1113_out -> f1111_out :|: TRUE
f1111_in -> f1112_in :|: TRUE
f1247_in(x237, .(x238, x239)) -> f1249_in(x237, x238, x239) :|: TRUE
f1249_out(x240, x241, x242) -> f1247_out(x240, .(x241, x242)) :|: TRUE
f1124_in -> f1124_out :|: TRUE
f1253_in(x245, x246) -> f1254_in(x245, x246) :|: TRUE
f1255_out(x247, x248) -> f1253_out(x247, x248) :|: TRUE
f1254_out(x249, x250) -> f1253_out(x249, x250) :|: TRUE
f1253_in(x251, x252) -> f1255_in(x251, x252) :|: TRUE
f1127_out(x253) -> f1440_out(x253) :|: TRUE
f1440_in(x254) -> f1127_in(x254) :|: TRUE
f1011_in(x257) -> f1013_in(x257) :|: TRUE
f1013_out(x258) -> f1011_out(x258) :|: TRUE
f845_in -> f631_in :|: TRUE
f631_out -> f845_out :|: TRUE
f1442_in -> f1390_in :|: TRUE
f1252_in(x259, x260) -> f1191_in(x259, x260) :|: TRUE
f1191_out(x261, x262) -> f1252_out(x261, x262) :|: TRUE
f1026_in -> f1026_out :|: TRUE
f1191_out(x263, x264) -> f1192_in(x265, x266, x263) :|: TRUE
f1144_in(x267, x268) -> f1191_in(x267, x268) :|: TRUE
f1192_out(x269, x270, x271) -> f1144_out(x271, x272) :|: TRUE
f1015_in(x273) -> f1018_in(x273) :|: TRUE
f1019_out(x274) -> f1015_out(x274) :|: TRUE
f1018_out(x275) -> f1015_out(x275) :|: TRUE
f1015_in(x276) -> f1019_in(x276) :|: TRUE
f1118_in -> f1118_out :|: TRUE
f1023_in -> f1023_out :|: TRUE
f1399_out -> f1400_in(x277) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1382_in(T333, T334, T335) -> f1378_in(T333, T334, T335) :|: TRUE
f1378_out(x279, x280, x281) -> f1382_out(x279, x280, x281) :|: TRUE
f1380_in(.(x282, x283), x284, x285) -> f1382_in(x283, x284, x285) :|: TRUE
f1382_out(x289, x290, x291) -> f1380_out(.(x292, x289), x290, x291) :|: TRUE
f1009_out(x297) -> f995_out(x297) :|: TRUE
f995_in(x299) -> f1009_in(x299) :|: TRUE
f1381_in([], T344, T345) -> f1384_in :|: TRUE
f1384_out -> f1381_out([], x303, x304) :|: TRUE
f965_in -> f965_out :|: TRUE
f467_out -> f478_in(x319, x320) :|: TRUE
f478_in(x321, x322) -> f1127_in(x321) :|: TRUE
f1127_out(x323) -> f1128_in(x324, x325) :|: TRUE
f1128_in(x329, x330) -> f1390_in :|: TRUE


----------------------------------------

(473)
Obligation:
Rules:
f1191_in(T216, T217) -> f1193_in(T216, T217) :|: TRUE
f1193_out(x, x1) -> f1191_out(x, x1) :|: TRUE
f1330_in -> f1330_out :|: TRUE
f467_out -> f1399_out :|: TRUE
f1399_in -> f467_in :|: TRUE
f938_out(T100) -> f920_out(s(T100)) :|: TRUE
f920_in(s(x3)) -> f938_in(x3) :|: TRUE
f614_in -> f627_in :|: TRUE
f627_out -> f614_out :|: TRUE
f1440_out(T367) -> f1441_in(T375) :|: TRUE
f1400_in(x6) -> f1440_in(x6) :|: TRUE
f1319_in(0, 0) -> f1338_in :|: TRUE
f1338_out -> f1319_out(0, 0) :|: TRUE
f1009_in(T120) -> f1011_in(T120) :|: TRUE
f1012_out(x9) -> f1009_out(x9) :|: TRUE
f1011_out(x10) -> f1012_in(x10) :|: TRUE
f1253_out(x11, x12) -> f1251_out(x11, x12) :|: TRUE
f1251_in(x13, x14) -> f1253_in(x13, x14) :|: TRUE
f905_in(x15) -> f633_in(x15) :|: TRUE
f633_out(x16) -> f905_out(x16) :|: TRUE
f1113_in -> f1116_in :|: TRUE
f1113_in -> f1117_in :|: TRUE
f1116_out -> f1113_out :|: TRUE
f1117_out -> f1113_out :|: TRUE
f1114_in -> f1109_in :|: TRUE
f1109_out -> f1114_out :|: TRUE
f1114_out -> f1112_out :|: TRUE
f1112_in -> f1114_in :|: TRUE
f1318_in(0, s(T295)) -> f1330_in :|: TRUE
f1330_out -> f1318_out(0, s(x21)) :|: TRUE
f1377_in(T222) -> f1127_in(T222) :|: TRUE
f1127_out(x22) -> f1377_out(x22) :|: TRUE
f561_in -> f614_in :|: TRUE
f616_out -> f561_out :|: TRUE
f614_out -> f561_out :|: TRUE
f561_in -> f616_in :|: TRUE
f1143_in([]) -> f1387_in :|: TRUE
f1387_out -> f1143_out([]) :|: TRUE
f1199_out(T238, T239, T240) -> f1194_out(T238, .(T239, T240)) :|: TRUE
f1194_in(x26, .(x27, x28)) -> f1199_in(x26, x27, x28) :|: TRUE
f849_in -> f849_out :|: TRUE
f1247_out(x31, x32) -> f1195_out(x31, x32) :|: TRUE
f1195_in(x33, x34) -> f1247_in(x33, x34) :|: TRUE
f1195_in(x35, x36) -> f1248_in(x35, x36) :|: TRUE
f1248_out(x37, x38) -> f1195_out(x37, x38) :|: TRUE
f1315_in(T287, T288) -> f1251_in(T287, T288) :|: TRUE
f1251_out(x39, x40) -> f1315_out(x39, x40) :|: TRUE
f996_in(T153) -> f1026_in :|: TRUE
f1026_out -> f996_out(x41) :|: TRUE
f1201_in(x43, x44) -> f1203_in(x43, x44) :|: TRUE
f1203_out(x45, x46) -> f1201_out(x45, x46) :|: TRUE
f1023_out -> f1019_out(0) :|: TRUE
f1019_in(0) -> f1023_in :|: TRUE
f1105_in -> f1107_in :|: TRUE
f1107_out -> f1105_out :|: TRUE
f1228_in(s(T253), s(T254)) -> f1230_in(T253, T254) :|: TRUE
f1230_out(x49, x50) -> f1228_out(s(x49), s(x50)) :|: TRUE
f1012_in(x55) -> f633_in(x55) :|: TRUE
f633_out(x56) -> f1012_out(x56) :|: TRUE
f1232_out -> f1229_out(s(0), 0) :|: TRUE
f1229_in(s(0), 0) -> f1232_in :|: TRUE
f635_in -> f844_in :|: TRUE
f635_in -> f843_in :|: TRUE
f843_out -> f635_out :|: TRUE
f844_out -> f635_out :|: TRUE
f1377_out(x61) -> f1378_in(x62, x63, x64) :|: TRUE
f1378_out(x65, x66, x67) -> f1352_out(x68, x65, x66) :|: TRUE
f1352_in(x69, x70, x71) -> f1377_in(x69) :|: TRUE
f1248_in(T301, []) -> f1345_in :|: TRUE
f1345_out -> f1248_out(x74, []) :|: TRUE
f1392_in -> f1393_in :|: TRUE
f1141_out(x77) -> f1127_out(x77) :|: TRUE
f1127_in(x78) -> f1141_in(x78) :|: TRUE
f1252_out(x79, x80) -> f1249_out(x79, x81, x80) :|: TRUE
f1249_in(x82, x83, x84) -> f1251_in(x82, x83) :|: TRUE
f1251_out(x85, x86) -> f1252_in(x85, x87) :|: TRUE
f561_out -> f467_out :|: TRUE
f467_in -> f561_in :|: TRUE
f1016_in(T138) -> f1011_in(T138) :|: TRUE
f1011_out(x88) -> f1016_out(x88) :|: TRUE
f905_out(x89) -> f883_out(x89) :|: TRUE
f883_in(x90) -> f903_in(x90) :|: TRUE
f903_out(x91) -> f905_in(x91) :|: TRUE
f855_out(x92) -> f633_out(x92) :|: TRUE
f633_in(x93) -> f855_in(x93) :|: TRUE
f996_out(x94) -> f862_out(x94) :|: TRUE
f862_in(x95) -> f995_in(x95) :|: TRUE
f995_out(x96) -> f862_out(x96) :|: TRUE
f862_in(x97) -> f996_in(x97) :|: TRUE
f916_out(x98) -> f903_out(x98) :|: TRUE
f903_in(x99) -> f916_in(x99) :|: TRUE
f1013_in(x100) -> f1015_in(x100) :|: TRUE
f1015_out(x101) -> f1013_out(x101) :|: TRUE
f1014_out(x102) -> f1013_out(x102) :|: TRUE
f1013_in(x103) -> f1014_in(x103) :|: TRUE
f1230_in(x104, x105) -> f1201_in(x104, x105) :|: TRUE
f1201_out(x106, x107) -> f1230_out(x106, x107) :|: TRUE
f627_in -> f631_in :|: TRUE
f633_out(x108) -> f627_out :|: TRUE
f631_out -> f633_in(x109) :|: TRUE
f1109_in -> f1111_in :|: TRUE
f1111_out -> f1109_out :|: TRUE
f1020_in -> f1020_out :|: TRUE
f1351_out(x110) -> f1352_in(x111, x112, x113) :|: TRUE
f1352_out(x114, x115, x116) -> f1192_out(x117, x114, x116) :|: TRUE
f1192_in(x118, x119, x120) -> f1351_in(x118) :|: TRUE
f860_in(x121) -> f883_in(x121) :|: TRUE
f883_out(x123) -> f860_out(x123) :|: TRUE
f1441_in(x125) -> f1442_in :|: TRUE
f1144_out(x128, x129) -> f1142_out(.(x128, x129)) :|: TRUE
f1142_in(.(x132, x133)) -> f1144_in(x132, x133) :|: TRUE
f845_out -> f843_out :|: TRUE
f843_in -> f845_in :|: TRUE
f1106_out -> f616_out :|: TRUE
f616_in -> f1105_in :|: TRUE
f616_in -> f1106_in :|: TRUE
f1105_out -> f616_out :|: TRUE
f1018_in(0) -> f1020_in :|: TRUE
f1020_out -> f1018_out(0) :|: TRUE
f1014_in(s(x138)) -> f1016_in(x138) :|: TRUE
f1016_out(x139) -> f1014_out(s(x139)) :|: TRUE
f938_in(x140) -> f903_in(x140) :|: TRUE
f903_out(x141) -> f938_out(x141) :|: TRUE
f1199_in(x142, x143, x144) -> f1201_in(x142, x143) :|: TRUE
f1202_out(x145, x146) -> f1199_out(x145, x147, x146) :|: TRUE
f1201_out(x148, x149) -> f1202_in(x148, x150) :|: TRUE
f1393_in -> f1395_in :|: TRUE
f1338_in -> f1338_out :|: TRUE
f1127_out(T221) -> f1351_out(T221) :|: TRUE
f1351_in(x155) -> f1127_in(x155) :|: TRUE
f965_out -> f921_out(s(0)) :|: TRUE
f921_in(s(0)) -> f965_in :|: TRUE
f1315_out(x160, x161) -> f1254_out(s(x160), s(x161)) :|: TRUE
f1254_in(s(x162), s(x163)) -> f1315_in(x162, x163) :|: TRUE
f1116_in -> f1118_in :|: TRUE
f1118_out -> f1116_out :|: TRUE
f1379_out(x166, x167, x168) -> f1378_out(x166, x167, x168) :|: TRUE
f1378_in(x169, x170, x171) -> f1379_in(x169, x170, x171) :|: TRUE
f633_out(T174) -> f1110_out(T174) :|: TRUE
f1110_in(x172) -> f633_in(x172) :|: TRUE
f1345_in -> f1345_out :|: TRUE
f1191_out(x173, x174) -> f1202_out(x173, x174) :|: TRUE
f1202_in(x175, x176) -> f1191_in(x175, x176) :|: TRUE
f1390_in -> f1392_in :|: TRUE
f1124_out -> f1106_out :|: TRUE
f1106_in -> f1124_in :|: TRUE
f1195_out(x177, x178) -> f1193_out(x177, x178) :|: TRUE
f1194_out(x179, x180) -> f1193_out(x179, x180) :|: TRUE
f1193_in(x181, x182) -> f1194_in(x181, x182) :|: TRUE
f1193_in(x183, x184) -> f1195_in(x183, x184) :|: TRUE
f849_out -> f844_out :|: TRUE
f844_in -> f849_in :|: TRUE
f1121_out -> f1117_out :|: TRUE
f1117_in -> f1121_in :|: TRUE
f1110_out(x187) -> f1107_out :|: TRUE
f1107_in -> f1109_in :|: TRUE
f1109_out -> f1110_in(x188) :|: TRUE
f1387_in -> f1387_out :|: TRUE
f1141_in(x189) -> f1143_in(x189) :|: TRUE
f1143_out(x190) -> f1141_out(x190) :|: TRUE
f1141_in(x191) -> f1142_in(x191) :|: TRUE
f1142_out(x192) -> f1141_out(x192) :|: TRUE
f635_out -> f631_out :|: TRUE
f631_in -> f635_in :|: TRUE
f1121_in -> f1121_out :|: TRUE
f1384_in -> f1384_out :|: TRUE
f1379_in(x195, x196, x197) -> f1381_in(x195, x196, x197) :|: TRUE
f1381_out(x198, x199, x200) -> f1379_out(x198, x199, x200) :|: TRUE
f1380_out(x201, x202, x203) -> f1379_out(x201, x202, x203) :|: TRUE
f1379_in(x204, x205, x206) -> f1380_in(x204, x205, x206) :|: TRUE
f1255_in(x207, x208) -> f1318_in(x207, x208) :|: TRUE
f1319_out(x209, x210) -> f1255_out(x209, x210) :|: TRUE
f1318_out(x211, x212) -> f1255_out(x211, x212) :|: TRUE
f1255_in(x213, x214) -> f1319_in(x213, x214) :|: TRUE
f1228_out(x215, x216) -> f1203_out(x215, x216) :|: TRUE
f1203_in(x217, x218) -> f1228_in(x217, x218) :|: TRUE
f1203_in(x219, x220) -> f1229_in(x219, x220) :|: TRUE
f1229_out(x221, x222) -> f1203_out(x221, x222) :|: TRUE
f1232_in -> f1232_out :|: TRUE
f855_in(x223) -> f862_in(x223) :|: TRUE
f862_out(x224) -> f855_out(x224) :|: TRUE
f855_in(x225) -> f860_in(x225) :|: TRUE
f860_out(x226) -> f855_out(x226) :|: TRUE
f921_out(x227) -> f916_out(x227) :|: TRUE
f916_in(x228) -> f920_in(x228) :|: TRUE
f916_in(x229) -> f921_in(x229) :|: TRUE
f920_out(x230) -> f916_out(x230) :|: TRUE
f1112_out -> f1111_out :|: TRUE
f1111_in -> f1113_in :|: TRUE
f1113_out -> f1111_out :|: TRUE
f1111_in -> f1112_in :|: TRUE
f1247_in(x237, .(x238, x239)) -> f1249_in(x237, x238, x239) :|: TRUE
f1249_out(x240, x241, x242) -> f1247_out(x240, .(x241, x242)) :|: TRUE
f1124_in -> f1124_out :|: TRUE
f1253_in(x245, x246) -> f1254_in(x245, x246) :|: TRUE
f1255_out(x247, x248) -> f1253_out(x247, x248) :|: TRUE
f1254_out(x249, x250) -> f1253_out(x249, x250) :|: TRUE
f1253_in(x251, x252) -> f1255_in(x251, x252) :|: TRUE
f1127_out(x253) -> f1440_out(x253) :|: TRUE
f1440_in(x254) -> f1127_in(x254) :|: TRUE
f1011_in(x257) -> f1013_in(x257) :|: TRUE
f1013_out(x258) -> f1011_out(x258) :|: TRUE
f845_in -> f631_in :|: TRUE
f631_out -> f845_out :|: TRUE
f1442_in -> f1390_in :|: TRUE
f1252_in(x259, x260) -> f1191_in(x259, x260) :|: TRUE
f1191_out(x261, x262) -> f1252_out(x261, x262) :|: TRUE
f1026_in -> f1026_out :|: TRUE
f1191_out(x263, x264) -> f1192_in(x265, x266, x263) :|: TRUE
f1144_in(x267, x268) -> f1191_in(x267, x268) :|: TRUE
f1192_out(x269, x270, x271) -> f1144_out(x271, x272) :|: TRUE
f1015_in(x273) -> f1018_in(x273) :|: TRUE
f1019_out(x274) -> f1015_out(x274) :|: TRUE
f1018_out(x275) -> f1015_out(x275) :|: TRUE
f1015_in(x276) -> f1019_in(x276) :|: TRUE
f1118_in -> f1118_out :|: TRUE
f1023_in -> f1023_out :|: TRUE
f1399_out -> f1400_in(x277) :|: TRUE
f1395_in -> f1399_in :|: TRUE
f1382_in(T333, T334, T335) -> f1378_in(T333, T334, T335) :|: TRUE
f1378_out(x279, x280, x281) -> f1382_out(x279, x280, x281) :|: TRUE
f1380_in(.(x282, x283), x284, x285) -> f1382_in(x283, x284, x285) :|: TRUE
f1382_out(x289, x290, x291) -> f1380_out(.(x292, x289), x290, x291) :|: TRUE
f1009_out(x297) -> f995_out(x297) :|: TRUE
f995_in(x299) -> f1009_in(x299) :|: TRUE
f1381_in([], T344, T345) -> f1384_in :|: TRUE
f1384_out -> f1381_out([], x303, x304) :|: TRUE
f965_in -> f965_out :|: TRUE
f467_out -> f478_in(x319, x320) :|: TRUE
f478_in(x321, x322) -> f1127_in(x321) :|: TRUE
f1127_out(x323) -> f1128_in(x324, x325) :|: TRUE
f1128_in(x329, x330) -> f1390_in :|: TRUE

----------------------------------------

(474) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(475)
Obligation:
Rules:
f1251_in(cons_0, cons_01) -> f1253_out(0, 0) :|: TRUE && cons_0 = 0 && cons_01 = 0
f1379_out(x166:0, x167:0, x168:0) -> f1379_out(.(x292:0, x166:0), x167:0, x168:0) :|: TRUE
f1141_in([]) -> f1141_out([]) :|: TRUE
f916_in(s(x3:0)) -> f916_in(x3:0) :|: TRUE
f1141_out(x77:0) -> f1379_in(x62:0, x63:0, x64:0) :|: TRUE
f1399_in -> f467_out :|: TRUE
f855_out(x92:0) -> f855_out(x92:0) :|: TRUE
f855_in(x225:0) -> f916_in(x225:0) :|: TRUE
f916_in(s(x)) -> f916_out(s(0)) :|: TRUE && x = 0
f1111_out -> f1111_out :|: TRUE
f1111_in -> f1111_out :|: TRUE
f1379_in(.(x282:0, x283:0), x205:0, x206:0) -> f1379_in(x283:0, x205:0, x206:0) :|: TRUE
f1193_out(x:0, x1:0) -> f1193_out(x:0, .(x147:0, x1:0)) :|: TRUE
f1011_out(x10:0) -> f855_in(x10:0) :|: TRUE
f1399_in -> f1111_in :|: TRUE
f467_out -> f1141_in(x319:0) :|: TRUE
f631_out -> f631_out :|: TRUE
f916_out(x98:0) -> f855_in(x98:0) :|: TRUE
f635_in -> f635_in :|: TRUE
f1141_out(x1) -> f1141_in(x2) :|: TRUE
f1141_out(x3) -> f1399_in :|: TRUE
f855_in(x223:0) -> f855_out(x41:0) :|: TRUE
f1013_in(x4) -> f1011_out(0) :|: TRUE && x4 = 0
f1379_in([], x196:0, x197:0) -> f1379_out([], x303:0, x304:0) :|: TRUE
f1253_out(x11:0, x12:0) -> f1253_out(s(x11:0), s(x12:0)) :|: TRUE
f631_out -> f855_in(x109:0) :|: TRUE
f1203_out(x45:0, x46:0) -> f1191_in(x45:0, x150:0) :|: TRUE
f855_out(x5) -> f467_out :|: TRUE
f1111_in -> f1111_in :|: TRUE
f1011_out(x88:0) -> f1011_out(s(x88:0)) :|: TRUE
f1251_in(s(x162:0), s(x163:0)) -> f1251_in(x162:0, x163:0) :|: TRUE
f1191_in(T216:0, .(x238:0, x239:0)) -> f1251_in(T216:0, x238:0) :|: TRUE
f1379_out(x6, x7, x8) -> f1141_out(.(x7, x9)) :|: TRUE
f855_in(x10) -> f1013_in(x10) :|: TRUE
f635_in -> f631_out :|: TRUE
f916_out(x11) -> f916_out(s(x11)) :|: TRUE
f1399_in -> f635_in :|: TRUE
f1111_out -> f855_in(x188:0) :|: TRUE
f1013_in(s(x138:0)) -> f1013_in(x138:0) :|: TRUE
f1203_out(x12, x13) -> f1203_out(s(x12), s(x13)) :|: TRUE
f1201_in(s(x14), x15) -> f1203_out(s(0), 0) :|: TRUE && x14 = 0 && x15 = 0
f1193_out(x16, x17) -> f1141_in(x18) :|: TRUE
f1141_in(.(x132:0, x133:0)) -> f1191_in(x132:0, x133:0) :|: TRUE
f1201_in(s(T253:0), s(T254:0)) -> f1201_in(T253:0, T254:0) :|: TRUE
f1191_in(x19, .(x20, x21)) -> f1201_in(x19, x20) :|: TRUE
f1191_in(x22, []) -> f1193_out(x23, []) :|: TRUE
f1251_in(x24, s(x25)) -> f1253_out(0, s(x26)) :|: TRUE && x24 = 0
f1253_out(x27, x28) -> f1191_in(x27, x29) :|: TRUE

----------------------------------------

(476) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(477)
Obligation:
Rules:
f1251_in(cons_0, cons_01) -> f1253_out(0, 0) :|: TRUE && cons_0 = 0 && cons_01 = 0
f1379_out(x166:0, x167:0, x168:0) -> f1379_out(.(x292:0, x166:0), x167:0, x168:0) :|: TRUE
f1141_in([]) -> f1141_out([]) :|: TRUE
f916_in(s(x3:0)) -> f916_in(x3:0) :|: TRUE
f1141_out(x77:0) -> f1379_in(x62:0, x63:0, x64:0) :|: TRUE
f1399_in -> f467_out :|: TRUE
f855_out(x92:0) -> f855_out(x92:0) :|: TRUE
f855_in(x225:0) -> f916_in(x225:0) :|: TRUE
f916_in(s(x)) -> f916_out(s(0)) :|: TRUE && x = 0
f1111_out -> f1111_out :|: TRUE
f1111_in -> f1111_out :|: TRUE
f1379_in(.(x282:0, x283:0), x205:0, x206:0) -> f1379_in(x283:0, x205:0, x206:0) :|: TRUE
f1193_out(x:0, x1:0) -> f1193_out(x:0, .(x147:0, x1:0)) :|: TRUE
f1011_out(x10:0) -> f855_in(x10:0) :|: TRUE
f1399_in -> f1111_in :|: TRUE
f467_out -> f1141_in(x319:0) :|: TRUE
f631_out -> f631_out :|: TRUE
f916_out(x98:0) -> f855_in(x98:0) :|: TRUE
f635_in -> f635_in :|: TRUE
f1141_out(x1) -> f1141_in(x2) :|: TRUE
f1141_out(x3) -> f1399_in :|: TRUE
f855_in(x223:0) -> f855_out(x41:0) :|: TRUE
f1013_in(x4) -> f1011_out(0) :|: TRUE && x4 = 0
f1379_in([], x196:0, x197:0) -> f1379_out([], x303:0, x304:0) :|: TRUE
f1253_out(x11:0, x12:0) -> f1253_out(s(x11:0), s(x12:0)) :|: TRUE
f631_out -> f855_in(x109:0) :|: TRUE
f1203_out(x45:0, x46:0) -> f1191_in(x45:0, x150:0) :|: TRUE
f855_out(x5) -> f467_out :|: TRUE
f1111_in -> f1111_in :|: TRUE
f1011_out(x88:0) -> f1011_out(s(x88:0)) :|: TRUE
f1251_in(s(x162:0), s(x163:0)) -> f1251_in(x162:0, x163:0) :|: TRUE
f1191_in(T216:0, .(x238:0, x239:0)) -> f1251_in(T216:0, x238:0) :|: TRUE
f1379_out(x6, x7, x8) -> f1141_out(.(x7, x9)) :|: TRUE
f855_in(x10) -> f1013_in(x10) :|: TRUE
f635_in -> f631_out :|: TRUE
f916_out(x11) -> f916_out(s(x11)) :|: TRUE
f1399_in -> f635_in :|: TRUE
f1111_out -> f855_in(x188:0) :|: TRUE
f1013_in(s(x138:0)) -> f1013_in(x138:0) :|: TRUE
f1203_out(x12, x13) -> f1203_out(s(x12), s(x13)) :|: TRUE
f1201_in(s(x14), x15) -> f1203_out(s(0), 0) :|: TRUE && x14 = 0 && x15 = 0
f1193_out(x16, x17) -> f1141_in(x18) :|: TRUE
f1141_in(.(x132:0, x133:0)) -> f1191_in(x132:0, x133:0) :|: TRUE
f1201_in(s(T253:0), s(T254:0)) -> f1201_in(T253:0, T254:0) :|: TRUE
f1191_in(x19, .(x20, x21)) -> f1201_in(x19, x20) :|: TRUE
f1191_in(x22, []) -> f1193_out(x23, []) :|: TRUE
f1251_in(x24, s(x25)) -> f1253_out(0, s(x26)) :|: TRUE && x24 = 0
f1253_out(x27, x28) -> f1191_in(x27, x29) :|: TRUE

----------------------------------------

(478) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1251_in(cons_0, cons_01) -> f1253_out(0, 0) :|: TRUE && cons_0 = 0 && cons_01 = 0
(2) f1379_out(x166:0, x167:0, x168:0) -> f1379_out(.(x292:0, x166:0), x167:0, x168:0) :|: TRUE
(3) f1141_in([]) -> f1141_out([]) :|: TRUE
(4) f916_in(s(x3:0)) -> f916_in(x3:0) :|: TRUE
(5) f1141_out(x77:0) -> f1379_in(x62:0, x63:0, x64:0) :|: TRUE
(6) f1399_in -> f467_out :|: TRUE
(7) f855_out(x92:0) -> f855_out(x92:0) :|: TRUE
(8) f855_in(x225:0) -> f916_in(x225:0) :|: TRUE
(9) f916_in(s(x)) -> f916_out(s(0)) :|: TRUE && x = 0
(10) f1111_out -> f1111_out :|: TRUE
(11) f1111_in -> f1111_out :|: TRUE
(12) f1379_in(.(x282:0, x283:0), x205:0, x206:0) -> f1379_in(x283:0, x205:0, x206:0) :|: TRUE
(13) f1193_out(x:0, x1:0) -> f1193_out(x:0, .(x147:0, x1:0)) :|: TRUE
(14) f1011_out(x10:0) -> f855_in(x10:0) :|: TRUE
(15) f1399_in -> f1111_in :|: TRUE
(16) f467_out -> f1141_in(x319:0) :|: TRUE
(17) f631_out -> f631_out :|: TRUE
(18) f916_out(x98:0) -> f855_in(x98:0) :|: TRUE
(19) f635_in -> f635_in :|: TRUE
(20) f1141_out(x1) -> f1141_in(x2) :|: TRUE
(21) f1141_out(x3) -> f1399_in :|: TRUE
(22) f855_in(x223:0) -> f855_out(x41:0) :|: TRUE
(23) f1013_in(x4) -> f1011_out(0) :|: TRUE && x4 = 0
(24) f1379_in([], x196:0, x197:0) -> f1379_out([], x303:0, x304:0) :|: TRUE
(25) f1253_out(x11:0, x12:0) -> f1253_out(s(x11:0), s(x12:0)) :|: TRUE
(26) f631_out -> f855_in(x109:0) :|: TRUE
(27) f1203_out(x45:0, x46:0) -> f1191_in(x45:0, x150:0) :|: TRUE
(28) f855_out(x5) -> f467_out :|: TRUE
(29) f1111_in -> f1111_in :|: TRUE
(30) f1011_out(x88:0) -> f1011_out(s(x88:0)) :|: TRUE
(31) f1251_in(s(x162:0), s(x163:0)) -> f1251_in(x162:0, x163:0) :|: TRUE
(32) f1191_in(T216:0, .(x238:0, x239:0)) -> f1251_in(T216:0, x238:0) :|: TRUE
(33) f1379_out(x6, x7, x8) -> f1141_out(.(x7, x9)) :|: TRUE
(34) f855_in(x10) -> f1013_in(x10) :|: TRUE
(35) f635_in -> f631_out :|: TRUE
(36) f916_out(x11) -> f916_out(s(x11)) :|: TRUE
(37) f1399_in -> f635_in :|: TRUE
(38) f1111_out -> f855_in(x188:0) :|: TRUE
(39) f1013_in(s(x138:0)) -> f1013_in(x138:0) :|: TRUE
(40) f1203_out(x12, x13) -> f1203_out(s(x12), s(x13)) :|: TRUE
(41) f1201_in(s(x14), x15) -> f1203_out(s(0), 0) :|: TRUE && x14 = 0 && x15 = 0
(42) f1193_out(x16, x17) -> f1141_in(x18) :|: TRUE
(43) f1141_in(.(x132:0, x133:0)) -> f1191_in(x132:0, x133:0) :|: TRUE
(44) f1201_in(s(T253:0), s(T254:0)) -> f1201_in(T253:0, T254:0) :|: TRUE
(45) f1191_in(x19, .(x20, x21)) -> f1201_in(x19, x20) :|: TRUE
(46) f1191_in(x22, []) -> f1193_out(x23, []) :|: TRUE
(47) f1251_in(x24, s(x25)) -> f1253_out(0, s(x26)) :|: TRUE && x24 = 0
(48) f1253_out(x27, x28) -> f1191_in(x27, x29) :|: TRUE

Arcs:
(1) -> (25), (48)
(2) -> (2), (33)
(3) -> (5), (20), (21)
(4) -> (4), (9)
(5) -> (12), (24)
(6) -> (16)
(7) -> (7), (28)
(8) -> (4), (9)
(9) -> (18), (36)
(10) -> (10), (38)
(11) -> (10), (38)
(12) -> (12), (24)
(13) -> (13), (42)
(14) -> (8), (22), (34)
(15) -> (11), (29)
(16) -> (3), (43)
(17) -> (17), (26)
(18) -> (8), (22), (34)
(19) -> (19), (35)
(20) -> (3), (43)
(21) -> (6), (15), (37)
(22) -> (7), (28)
(23) -> (14), (30)
(24) -> (2), (33)
(25) -> (25), (48)
(26) -> (8), (22), (34)
(27) -> (32), (45), (46)
(28) -> (16)
(29) -> (11), (29)
(30) -> (14), (30)
(31) -> (1), (31), (47)
(32) -> (1), (31), (47)
(33) -> (5), (20), (21)
(34) -> (23), (39)
(35) -> (17), (26)
(36) -> (18), (36)
(37) -> (19), (35)
(38) -> (8), (22), (34)
(39) -> (23), (39)
(40) -> (27), (40)
(41) -> (27), (40)
(42) -> (3), (43)
(43) -> (32), (45), (46)
(44) -> (41), (44)
(45) -> (41), (44)
(46) -> (13), (42)
(47) -> (25), (48)
(48) -> (32), (45), (46)

This digraph is fully evaluated!
----------------------------------------

(479)
Obligation:

Termination digraph:
Nodes:
(1) f1251_in(cons_0, cons_01) -> f1253_out(0, 0) :|: TRUE && cons_0 = 0 && cons_01 = 0
(2) f1251_in(s(x162:0), s(x163:0)) -> f1251_in(x162:0, x163:0) :|: TRUE
(3) f1191_in(T216:0, .(x238:0, x239:0)) -> f1251_in(T216:0, x238:0) :|: TRUE
(4) f1203_out(x45:0, x46:0) -> f1191_in(x45:0, x150:0) :|: TRUE
(5) f1203_out(x12, x13) -> f1203_out(s(x12), s(x13)) :|: TRUE
(6) f1201_in(s(x14), x15) -> f1203_out(s(0), 0) :|: TRUE && x14 = 0 && x15 = 0
(7) f1201_in(s(T253:0), s(T254:0)) -> f1201_in(T253:0, T254:0) :|: TRUE
(8) f1191_in(x19, .(x20, x21)) -> f1201_in(x19, x20) :|: TRUE
(9) f1141_in(.(x132:0, x133:0)) -> f1191_in(x132:0, x133:0) :|: TRUE
(10) f467_out -> f1141_in(x319:0) :|: TRUE
(11) f855_out(x5) -> f467_out :|: TRUE
(12) f855_out(x92:0) -> f855_out(x92:0) :|: TRUE
(13) f855_in(x223:0) -> f855_out(x41:0) :|: TRUE
(14) f1011_out(x10:0) -> f855_in(x10:0) :|: TRUE
(15) f1011_out(x88:0) -> f1011_out(s(x88:0)) :|: TRUE
(16) f1013_in(x4) -> f1011_out(0) :|: TRUE && x4 = 0
(17) f1013_in(s(x138:0)) -> f1013_in(x138:0) :|: TRUE
(18) f855_in(x10) -> f1013_in(x10) :|: TRUE
(19) f916_out(x98:0) -> f855_in(x98:0) :|: TRUE
(20) f916_out(x11) -> f916_out(s(x11)) :|: TRUE
(21) f916_in(s(x)) -> f916_out(s(0)) :|: TRUE && x = 0
(22) f916_in(s(x3:0)) -> f916_in(x3:0) :|: TRUE
(23) f855_in(x225:0) -> f916_in(x225:0) :|: TRUE
(24) f1111_out -> f855_in(x188:0) :|: TRUE
(25) f1111_out -> f1111_out :|: TRUE
(26) f1111_in -> f1111_out :|: TRUE
(27) f1111_in -> f1111_in :|: TRUE
(28) f1399_in -> f1111_in :|: TRUE
(29) f631_out -> f855_in(x109:0) :|: TRUE
(30) f631_out -> f631_out :|: TRUE
(31) f635_in -> f631_out :|: TRUE
(32) f635_in -> f635_in :|: TRUE
(33) f1399_in -> f635_in :|: TRUE
(34) f1399_in -> f467_out :|: TRUE
(35) f1141_out(x3) -> f1399_in :|: TRUE
(36) f1141_in([]) -> f1141_out([]) :|: TRUE
(37) f1193_out(x16, x17) -> f1141_in(x18) :|: TRUE
(38) f1193_out(x:0, x1:0) -> f1193_out(x:0, .(x147:0, x1:0)) :|: TRUE
(39) f1191_in(x22, []) -> f1193_out(x23, []) :|: TRUE
(40) f1253_out(x27, x28) -> f1191_in(x27, x29) :|: TRUE
(41) f1253_out(x11:0, x12:0) -> f1253_out(s(x11:0), s(x12:0)) :|: TRUE
(42) f1251_in(x24, s(x25)) -> f1253_out(0, s(x26)) :|: TRUE && x24 = 0
(43) f1141_out(x1) -> f1141_in(x2) :|: TRUE
(44) f1379_out(x6, x7, x8) -> f1141_out(.(x7, x9)) :|: TRUE
(45) f1379_out(x166:0, x167:0, x168:0) -> f1379_out(.(x292:0, x166:0), x167:0, x168:0) :|: TRUE
(46) f1379_in([], x196:0, x197:0) -> f1379_out([], x303:0, x304:0) :|: TRUE
(47) f1379_in(.(x282:0, x283:0), x205:0, x206:0) -> f1379_in(x283:0, x205:0, x206:0) :|: TRUE
(48) f1141_out(x77:0) -> f1379_in(x62:0, x63:0, x64:0) :|: TRUE

Arcs:
(1) -> (40), (41)
(2) -> (1), (2), (42)
(3) -> (1), (2), (42)
(4) -> (3), (8), (39)
(5) -> (4), (5)
(6) -> (4), (5)
(7) -> (6), (7)
(8) -> (6), (7)
(9) -> (3), (8), (39)
(10) -> (9), (36)
(11) -> (10)
(12) -> (11), (12)
(13) -> (11), (12)
(14) -> (13), (18), (23)
(15) -> (14), (15)
(16) -> (14), (15)
(17) -> (16), (17)
(18) -> (16), (17)
(19) -> (13), (18), (23)
(20) -> (19), (20)
(21) -> (19), (20)
(22) -> (21), (22)
(23) -> (21), (22)
(24) -> (13), (18), (23)
(25) -> (24), (25)
(26) -> (24), (25)
(27) -> (26), (27)
(28) -> (26), (27)
(29) -> (13), (18), (23)
(30) -> (29), (30)
(31) -> (29), (30)
(32) -> (31), (32)
(33) -> (31), (32)
(34) -> (10)
(35) -> (28), (33), (34)
(36) -> (35), (43), (48)
(37) -> (9), (36)
(38) -> (37), (38)
(39) -> (37), (38)
(40) -> (3), (8), (39)
(41) -> (40), (41)
(42) -> (40), (41)
(43) -> (9), (36)
(44) -> (35), (43), (48)
(45) -> (44), (45)
(46) -> (44), (45)
(47) -> (46), (47)
(48) -> (46), (47)

This digraph is fully evaluated!
