YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/G2GE4.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

myis(a,g)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 24 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 [SOUND, 0 ms]
        (39) QDP
        (40) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (41) YES


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

(0)
Obligation:
Clauses:

myis(Z, X) :- evaluate(X, Z).
evaluate(+(X, Y), Z) :- ','(evaluate(X, X1), ','(evaluate(Y, Y1), add(X1, Y1, Z))).
evaluate(-(X, Y), Z) :- ','(evaluate(X, X1), ','(evaluate(Y, Y1), sub(X1, Y1, Z))).
evaluate(*(X, Y), Z) :- ','(evaluate(X, X1), ','(evaluate(Y, Y1), mult(X1, Y1, Z))).
evaluate(X, X) :- myinteger(X).
myinteger(s(X)) :- myinteger(X).
myinteger(0).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).
mult(s(X), Y, R) :- ','(mult(X, Y, Z), add(Y, Z, R)).
mult(0, Y, 0).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).
lt(s(X), s(Y)) :- lt(X, Y).
lt(0, s(Y)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).


Query: myis(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:

myis_in_2: (f,b)

evaluate_in_2: (b,f)

myinteger_in_1: (b)

mult_in_3: (b,b,f)

add_in_3: (b,b,f)

sub_in_3: (b,b,f)

Transforming Prolog into the following Term Rewriting System:

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

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)



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

(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:

   MYIS_IN_AG(Z, X) -> U1_AG(Z, X, evaluate_in_ga(X, Z))
   MYIS_IN_AG(Z, X) -> EVALUATE_IN_GA(X, Z)
   EVALUATE_IN_GA(+(X, Y), Z) -> U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(+(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(-(X, Y), Z) -> U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(-(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(*(X, Y), Z) -> U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(*(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(X, X) -> U11_GA(X, myinteger_in_g(X))
   EVALUATE_IN_GA(X, X) -> MYINTEGER_IN_G(X)
   MYINTEGER_IN_G(s(X)) -> U12_G(X, myinteger_in_g(X))
   MYINTEGER_IN_G(s(X)) -> MYINTEGER_IN_G(X)
   U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_GA(X, Y, Z, mult_in_gga(X1, Y1, Z))
   U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> MULT_IN_GGA(X1, Y1, Z)
   MULT_IN_GGA(s(X), Y, R) -> U15_GGA(X, Y, R, mult_in_gga(X, Y, Z))
   MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z)
   U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_GGA(X, Y, R, add_in_gga(Y, Z, R))
   U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> ADD_IN_GGA(Y, Z, R)
   ADD_IN_GGA(s(X), Y, s(Z)) -> U13_GGA(X, Y, Z, add_in_gga(X, Y, Z))
   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)
   U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_GA(X, Y, Z, sub_in_gga(X1, Y1, Z))
   U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> SUB_IN_GGA(X1, Y1, Z)
   SUB_IN_GGA(s(X), s(Y), Z) -> U14_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)
   U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_GA(X, Y, Z, add_in_gga(X1, Y1, Z))
   U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> ADD_IN_GGA(X1, Y1, Z)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

MYIS_IN_AG(x1, x2)  =  MYIS_IN_AG(x2)

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

EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)

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

U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)

U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)

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

MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)

U12_G(x1, x2)  =  U12_G(x2)

U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x4, x5)

U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)

MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x2, x4)

U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x4)

ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)

U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x4, x5)

U7_GA(x1, x2, x3, x4)  =  U7_GA(x4)

SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x4)

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

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


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

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

   MYIS_IN_AG(Z, X) -> U1_AG(Z, X, evaluate_in_ga(X, Z))
   MYIS_IN_AG(Z, X) -> EVALUATE_IN_GA(X, Z)
   EVALUATE_IN_GA(+(X, Y), Z) -> U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(+(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(-(X, Y), Z) -> U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(-(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(*(X, Y), Z) -> U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(*(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(X, X) -> U11_GA(X, myinteger_in_g(X))
   EVALUATE_IN_GA(X, X) -> MYINTEGER_IN_G(X)
   MYINTEGER_IN_G(s(X)) -> U12_G(X, myinteger_in_g(X))
   MYINTEGER_IN_G(s(X)) -> MYINTEGER_IN_G(X)
   U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_GA(X, Y, Z, mult_in_gga(X1, Y1, Z))
   U9_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> MULT_IN_GGA(X1, Y1, Z)
   MULT_IN_GGA(s(X), Y, R) -> U15_GGA(X, Y, R, mult_in_gga(X, Y, Z))
   MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z)
   U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_GGA(X, Y, R, add_in_gga(Y, Z, R))
   U15_GGA(X, Y, R, mult_out_gga(X, Y, Z)) -> ADD_IN_GGA(Y, Z, R)
   ADD_IN_GGA(s(X), Y, s(Z)) -> U13_GGA(X, Y, Z, add_in_gga(X, Y, Z))
   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)
   U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_GA(X, Y, Z, sub_in_gga(X1, Y1, Z))
   U6_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> SUB_IN_GGA(X1, Y1, Z)
   SUB_IN_GGA(s(X), s(Y), Z) -> U14_GGA(X, Y, Z, sub_in_gga(X, Y, Z))
   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)
   U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_GA(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_GA(X, Y, Z, add_in_gga(X1, Y1, Z))
   U3_GA(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> ADD_IN_GGA(X1, Y1, Z)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

MYIS_IN_AG(x1, x2)  =  MYIS_IN_AG(x2)

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

EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)

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

U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)

U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)

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

MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)

U12_G(x1, x2)  =  U12_G(x2)

U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x4, x5)

U10_GA(x1, x2, x3, x4)  =  U10_GA(x4)

MULT_IN_GGA(x1, x2, x3)  =  MULT_IN_GGA(x1, x2)

U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x2, x4)

U16_GGA(x1, x2, x3, x4)  =  U16_GGA(x4)

ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)

U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)

U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x4, x5)

U7_GA(x1, x2, x3, x4)  =  U7_GA(x4)

SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)

U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x4)

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

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


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 19 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)


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:

   SUB_IN_GGA(s(X), s(Y), Z) -> SUB_IN_GGA(X, Y, Z)

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

SUB_IN_GGA(x1, x2, x3)  =  SUB_IN_GGA(x1, x2)


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:

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

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:
*SUB_IN_GGA(s(X), s(Y)) -> SUB_IN_GGA(X, Y)
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:

   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)


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:

   ADD_IN_GGA(s(X), Y, s(Z)) -> ADD_IN_GGA(X, Y, Z)

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

ADD_IN_GGA(x1, x2, x3)  =  ADD_IN_GGA(x1, x2)


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:

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

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:
*ADD_IN_GGA(s(X), Y) -> ADD_IN_GGA(X, Y)
The graph contains the following edges 1 > 1, 2 >= 2


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

(20)
YES

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

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

   MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

MULT_IN_GGA(x1, x2, x3)  =  MULT_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:

   MULT_IN_GGA(s(X), Y, R) -> MULT_IN_GGA(X, Y, Z)

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

MULT_IN_GGA(x1, x2, x3)  =  MULT_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:

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

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:
*MULT_IN_GGA(s(X), Y) -> MULT_IN_GGA(X, Y)
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:

   MYINTEGER_IN_G(s(X)) -> MYINTEGER_IN_G(X)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

MYINTEGER_IN_G(x1)  =  MYINTEGER_IN_G(x1)


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:

   MYINTEGER_IN_G(s(X)) -> MYINTEGER_IN_G(X)

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:

   MYINTEGER_IN_G(s(X)) -> MYINTEGER_IN_G(X)

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:
*MYINTEGER_IN_G(s(X)) -> MYINTEGER_IN_G(X)
The graph contains the following edges 1 > 1


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

(34)
YES

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

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

   U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   EVALUATE_IN_GA(+(X, Y), Z) -> U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(+(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(-(X, Y), Z) -> U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
   U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   EVALUATE_IN_GA(-(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(*(X, Y), Z) -> U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
   U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   EVALUATE_IN_GA(*(X, Y), Z) -> EVALUATE_IN_GA(X, X1)

The TRS R consists of the following rules:

   myis_in_ag(Z, X) -> U1_ag(Z, X, evaluate_in_ga(X, Z))
   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U1_ag(Z, X, evaluate_out_ga(X, Z)) -> myis_out_ag(Z, X)

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

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

evaluate_in_ga(x1, x2)  =  evaluate_in_ga(x1)

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

myis_out_ag(x1, x2)  =  myis_out_ag(x1)

EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)

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

U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)

U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)


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:

   U2_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   EVALUATE_IN_GA(+(X, Y), Z) -> U2_GA(X, Y, Z, evaluate_in_ga(X, X1))
   EVALUATE_IN_GA(+(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(-(X, Y), Z) -> U5_GA(X, Y, Z, evaluate_in_ga(X, X1))
   U5_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   EVALUATE_IN_GA(-(X, Y), Z) -> EVALUATE_IN_GA(X, X1)
   EVALUATE_IN_GA(*(X, Y), Z) -> U8_GA(X, Y, Z, evaluate_in_ga(X, X1))
   U8_GA(X, Y, Z, evaluate_out_ga(X, X1)) -> EVALUATE_IN_GA(Y, Y1)
   EVALUATE_IN_GA(*(X, Y), Z) -> EVALUATE_IN_GA(X, X1)

The TRS R consists of the following rules:

   evaluate_in_ga(+(X, Y), Z) -> U2_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(-(X, Y), Z) -> U5_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(*(X, Y), Z) -> U8_ga(X, Y, Z, evaluate_in_ga(X, X1))
   evaluate_in_ga(X, X) -> U11_ga(X, myinteger_in_g(X))
   U2_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U3_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U5_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U6_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U8_ga(X, Y, Z, evaluate_out_ga(X, X1)) -> U9_ga(X, Y, Z, X1, evaluate_in_ga(Y, Y1))
   U11_ga(X, myinteger_out_g(X)) -> evaluate_out_ga(X, X)
   U3_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U4_ga(X, Y, Z, add_in_gga(X1, Y1, Z))
   U6_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U7_ga(X, Y, Z, sub_in_gga(X1, Y1, Z))
   U9_ga(X, Y, Z, X1, evaluate_out_ga(Y, Y1)) -> U10_ga(X, Y, Z, mult_in_gga(X1, Y1, Z))
   myinteger_in_g(s(X)) -> U12_g(X, myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g(0)
   U4_ga(X, Y, Z, add_out_gga(X1, Y1, Z)) -> evaluate_out_ga(+(X, Y), Z)
   U7_ga(X, Y, Z, sub_out_gga(X1, Y1, Z)) -> evaluate_out_ga(-(X, Y), Z)
   U10_ga(X, Y, Z, mult_out_gga(X1, Y1, Z)) -> evaluate_out_ga(*(X, Y), Z)
   U12_g(X, myinteger_out_g(X)) -> myinteger_out_g(s(X))
   add_in_gga(s(X), Y, s(Z)) -> U13_gga(X, Y, Z, add_in_gga(X, Y, Z))
   add_in_gga(0, X, X) -> add_out_gga(0, X, X)
   sub_in_gga(s(X), s(Y), Z) -> U14_gga(X, Y, Z, sub_in_gga(X, Y, Z))
   sub_in_gga(X, 0, X) -> sub_out_gga(X, 0, X)
   mult_in_gga(s(X), Y, R) -> U15_gga(X, Y, R, mult_in_gga(X, Y, Z))
   mult_in_gga(0, Y, 0) -> mult_out_gga(0, Y, 0)
   U13_gga(X, Y, Z, add_out_gga(X, Y, Z)) -> add_out_gga(s(X), Y, s(Z))
   U14_gga(X, Y, Z, sub_out_gga(X, Y, Z)) -> sub_out_gga(s(X), s(Y), Z)
   U15_gga(X, Y, R, mult_out_gga(X, Y, Z)) -> U16_gga(X, Y, R, add_in_gga(Y, Z, R))
   U16_gga(X, Y, R, add_out_gga(Y, Z, R)) -> mult_out_gga(s(X), Y, R)

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

+(x1, x2)  =  +(x1, x2)

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

-(x1, x2)  =  -(x1, x2)

U5_ga(x1, x2, x3, x4)  =  U5_ga(x2, x4)

*(x1, x2)  =  *(x1, x2)

U8_ga(x1, x2, x3, x4)  =  U8_ga(x2, x4)

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

myinteger_in_g(x1)  =  myinteger_in_g(x1)

s(x1)  =  s(x1)

U12_g(x1, x2)  =  U12_g(x2)

0  =  0

myinteger_out_g(x1)  =  myinteger_out_g

evaluate_out_ga(x1, x2)  =  evaluate_out_ga(x2)

U9_ga(x1, x2, x3, x4, x5)  =  U9_ga(x4, x5)

U10_ga(x1, x2, x3, x4)  =  U10_ga(x4)

mult_in_gga(x1, x2, x3)  =  mult_in_gga(x1, x2)

U15_gga(x1, x2, x3, x4)  =  U15_gga(x2, x4)

mult_out_gga(x1, x2, x3)  =  mult_out_gga(x3)

U16_gga(x1, x2, x3, x4)  =  U16_gga(x4)

add_in_gga(x1, x2, x3)  =  add_in_gga(x1, x2)

U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)

add_out_gga(x1, x2, x3)  =  add_out_gga(x3)

U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x4, x5)

U7_ga(x1, x2, x3, x4)  =  U7_ga(x4)

sub_in_gga(x1, x2, x3)  =  sub_in_gga(x1, x2)

U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)

sub_out_gga(x1, x2, x3)  =  sub_out_gga(x3)

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

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

EVALUATE_IN_GA(x1, x2)  =  EVALUATE_IN_GA(x1)

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

U5_GA(x1, x2, x3, x4)  =  U5_GA(x2, x4)

U8_GA(x1, x2, x3, x4)  =  U8_GA(x2, x4)


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

(38) PiDPToQDPProof (SOUND)
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:

   U2_GA(Y, evaluate_out_ga(X1)) -> EVALUATE_IN_GA(Y)
   EVALUATE_IN_GA(+(X, Y)) -> U2_GA(Y, evaluate_in_ga(X))
   EVALUATE_IN_GA(+(X, Y)) -> EVALUATE_IN_GA(X)
   EVALUATE_IN_GA(-(X, Y)) -> U5_GA(Y, evaluate_in_ga(X))
   U5_GA(Y, evaluate_out_ga(X1)) -> EVALUATE_IN_GA(Y)
   EVALUATE_IN_GA(-(X, Y)) -> EVALUATE_IN_GA(X)
   EVALUATE_IN_GA(*(X, Y)) -> U8_GA(Y, evaluate_in_ga(X))
   U8_GA(Y, evaluate_out_ga(X1)) -> EVALUATE_IN_GA(Y)
   EVALUATE_IN_GA(*(X, Y)) -> EVALUATE_IN_GA(X)

The TRS R consists of the following rules:

   evaluate_in_ga(+(X, Y)) -> U2_ga(Y, evaluate_in_ga(X))
   evaluate_in_ga(-(X, Y)) -> U5_ga(Y, evaluate_in_ga(X))
   evaluate_in_ga(*(X, Y)) -> U8_ga(Y, evaluate_in_ga(X))
   evaluate_in_ga(X) -> U11_ga(X, myinteger_in_g(X))
   U2_ga(Y, evaluate_out_ga(X1)) -> U3_ga(X1, evaluate_in_ga(Y))
   U5_ga(Y, evaluate_out_ga(X1)) -> U6_ga(X1, evaluate_in_ga(Y))
   U8_ga(Y, evaluate_out_ga(X1)) -> U9_ga(X1, evaluate_in_ga(Y))
   U11_ga(X, myinteger_out_g) -> evaluate_out_ga(X)
   U3_ga(X1, evaluate_out_ga(Y1)) -> U4_ga(add_in_gga(X1, Y1))
   U6_ga(X1, evaluate_out_ga(Y1)) -> U7_ga(sub_in_gga(X1, Y1))
   U9_ga(X1, evaluate_out_ga(Y1)) -> U10_ga(mult_in_gga(X1, Y1))
   myinteger_in_g(s(X)) -> U12_g(myinteger_in_g(X))
   myinteger_in_g(0) -> myinteger_out_g
   U4_ga(add_out_gga(Z)) -> evaluate_out_ga(Z)
   U7_ga(sub_out_gga(Z)) -> evaluate_out_ga(Z)
   U10_ga(mult_out_gga(Z)) -> evaluate_out_ga(Z)
   U12_g(myinteger_out_g) -> myinteger_out_g
   add_in_gga(s(X), Y) -> U13_gga(add_in_gga(X, Y))
   add_in_gga(0, X) -> add_out_gga(X)
   sub_in_gga(s(X), s(Y)) -> U14_gga(sub_in_gga(X, Y))
   sub_in_gga(X, 0) -> sub_out_gga(X)
   mult_in_gga(s(X), Y) -> U15_gga(Y, mult_in_gga(X, Y))
   mult_in_gga(0, Y) -> mult_out_gga(0)
   U13_gga(add_out_gga(Z)) -> add_out_gga(s(Z))
   U14_gga(sub_out_gga(Z)) -> sub_out_gga(Z)
   U15_gga(Y, mult_out_gga(Z)) -> U16_gga(add_in_gga(Y, Z))
   U16_gga(add_out_gga(R)) -> mult_out_gga(R)

The set Q consists of the following terms:

   evaluate_in_ga(x0)
   U2_ga(x0, x1)
   U5_ga(x0, x1)
   U8_ga(x0, x1)
   U11_ga(x0, x1)
   U3_ga(x0, x1)
   U6_ga(x0, x1)
   U9_ga(x0, x1)
   myinteger_in_g(x0)
   U4_ga(x0)
   U7_ga(x0)
   U10_ga(x0)
   U12_g(x0)
   add_in_gga(x0, x1)
   sub_in_gga(x0, x1)
   mult_in_gga(x0, x1)
   U13_gga(x0)
   U14_gga(x0)
   U15_gga(x0, x1)
   U16_gga(x0)

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:
*EVALUATE_IN_GA(+(X, Y)) -> U2_GA(Y, evaluate_in_ga(X))
The graph contains the following edges 1 > 1


*U2_GA(Y, evaluate_out_ga(X1)) -> EVALUATE_IN_GA(Y)
The graph contains the following edges 1 >= 1


*U5_GA(Y, evaluate_out_ga(X1)) -> EVALUATE_IN_GA(Y)
The graph contains the following edges 1 >= 1


*U8_GA(Y, evaluate_out_ga(X1)) -> EVALUATE_IN_GA(Y)
The graph contains the following edges 1 >= 1


*EVALUATE_IN_GA(-(X, Y)) -> U5_GA(Y, evaluate_in_ga(X))
The graph contains the following edges 1 > 1


*EVALUATE_IN_GA(*(X, Y)) -> U8_GA(Y, evaluate_in_ga(X))
The graph contains the following edges 1 > 1


*EVALUATE_IN_GA(+(X, Y)) -> EVALUATE_IN_GA(X)
The graph contains the following edges 1 > 1


*EVALUATE_IN_GA(-(X, Y)) -> EVALUATE_IN_GA(X)
The graph contains the following edges 1 > 1


*EVALUATE_IN_GA(*(X, Y)) -> EVALUATE_IN_GA(X)
The graph contains the following edges 1 > 1


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

(41)
YES
