YES
proof of /hpcwork/ff862203/termcomp26/benchmarks/MOOI2.pl
# AProVE Commit ID: 23a904c96b029b0a549cde0d0d17dbccf967db59 jckassing 20260626 unpublished dirty


Left Termination of the query pattern

times(g,g,a)

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

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 25 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 [SOUND, 0 ms]
        (32) QDP
        (33) MRRProof [EQUIVALENT, 36 ms]
        (34) QDP
        (35) DependencyGraphProof [EQUIVALENT, 0 ms]
        (36) TRUE


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

(0)
Obligation:
Clauses:

even(0, true).
even(s(0), false).
even(s(s(X)), B) :- even(X, B).
half(0, 0).
half(s(s(X)), s(Y)) :- half(X, Y).
plus(0, Y, Y).
plus(s(X), Y, s(Z)) :- plus(X, Y, Z).
times(0, Y, 0).
times(s(X), Y, Z) :- ','(even(s(X), B), if(B, s(X), Y, Z)).
if(true, s(X), Y, Z) :- ','(half(s(X), X1), ','(times(X1, Y, Y1), plus(Y1, Y1, Z))).
if(false, s(X), Y, Z) :- ','(times(X, Y, U), plus(Y, U, Z)).


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

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

times_in_3: (b,b,f)

even_in_2: (b,f)

if_in_4: (b,b,b,f)

half_in_2: (b,f)

plus_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:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

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

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)



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

(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:

   TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B))
   TIMES_IN_GGA(s(X), Y, Z) -> EVEN_IN_GA(s(X), B)
   EVEN_IN_GA(s(s(X)), B) -> U1_GA(X, B, even_in_ga(X, B))
   EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B)
   U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z)
   IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
   IF_IN_GGGA(true, s(X), Y, Z) -> HALF_IN_GA(s(X), X1)
   HALF_IN_GA(s(s(X)), s(Y)) -> U2_GA(X, Y, half_in_ga(X, Y))
   HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y)
   U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1))
   U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1)
   U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> PLUS_IN_GGA(Y1, Y1, Z)
   PLUS_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
   IF_IN_GGGA(false, s(X), Y, Z) -> U9_GGGA(X, Y, Z, times_in_gga(X, Y, U))
   IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U)
   U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z))
   U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z)

The TRS R consists of the following rules:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)

U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)

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

U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)

HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)

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

U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x4)

U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x4)

PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)

U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x2, x4)

U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x4)


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

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

   TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B))
   TIMES_IN_GGA(s(X), Y, Z) -> EVEN_IN_GA(s(X), B)
   EVEN_IN_GA(s(s(X)), B) -> U1_GA(X, B, even_in_ga(X, B))
   EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B)
   U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> U5_GGA(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z)
   IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
   IF_IN_GGGA(true, s(X), Y, Z) -> HALF_IN_GA(s(X), X1)
   HALF_IN_GA(s(s(X)), s(Y)) -> U2_GA(X, Y, half_in_ga(X, Y))
   HALF_IN_GA(s(s(X)), s(Y)) -> HALF_IN_GA(X, Y)
   U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> U7_GGGA(X, Y, Z, times_in_gga(X1, Y, Y1))
   U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1)
   U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_GGGA(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   U7_GGGA(X, Y, Z, times_out_gga(X1, Y, Y1)) -> PLUS_IN_GGA(Y1, Y1, Z)
   PLUS_IN_GGA(s(X), Y, s(Z)) -> U3_GGA(X, Y, Z, plus_in_gga(X, Y, Z))
   PLUS_IN_GGA(s(X), Y, s(Z)) -> PLUS_IN_GGA(X, Y, Z)
   IF_IN_GGGA(false, s(X), Y, Z) -> U9_GGGA(X, Y, Z, times_in_gga(X, Y, U))
   IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U)
   U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> U10_GGGA(X, Y, Z, plus_in_gga(Y, U, Z))
   U9_GGGA(X, Y, Z, times_out_gga(X, Y, U)) -> PLUS_IN_GGA(Y, U, Z)

The TRS R consists of the following rules:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)

U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)

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

U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)

HALF_IN_GA(x1, x2)  =  HALF_IN_GA(x1)

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

U7_GGGA(x1, x2, x3, x4)  =  U7_GGGA(x4)

U8_GGGA(x1, x2, x3, x4)  =  U8_GGGA(x4)

PLUS_IN_GGA(x1, x2, x3)  =  PLUS_IN_GGA(x1, x2)

U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)

U9_GGGA(x1, x2, x3, x4)  =  U9_GGGA(x2, x4)

U10_GGGA(x1, x2, x3, x4)  =  U10_GGGA(x4)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

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

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

The TRS R consists of the following rules:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

PLUS_IN_GGA(x1, x2, x3)  =  PLUS_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:

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

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

PLUS_IN_GGA(x1, x2, x3)  =  PLUS_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:

   PLUS_IN_GGA(s(X), Y) -> PLUS_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:
*PLUS_IN_GGA(s(X), Y) -> PLUS_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:

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

The TRS R consists of the following rules:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

HALF_IN_GA(x1, x2)  =  HALF_IN_GA(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:

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

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

HALF_IN_GA(x1, x2)  =  HALF_IN_GA(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:

   HALF_IN_GA(s(s(X))) -> HALF_IN_GA(X)

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


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

(20)
YES

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

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

   EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B)

The TRS R consists of the following rules:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)


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:

   EVEN_IN_GA(s(s(X)), B) -> EVEN_IN_GA(X, B)

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

EVEN_IN_GA(x1, x2)  =  EVEN_IN_GA(x1)


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:

   EVEN_IN_GA(s(s(X))) -> EVEN_IN_GA(X)

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


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

(27)
YES

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

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

   U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z)
   IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
   U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1)
   TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B))
   IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U)

The TRS R consists of the following rules:

   times_in_gga(0, Y, 0) -> times_out_gga(0, Y, 0)
   times_in_gga(s(X), Y, Z) -> U4_gga(X, Y, Z, even_in_ga(s(X), B))
   even_in_ga(0, true) -> even_out_ga(0, true)
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   U4_gga(X, Y, Z, even_out_ga(s(X), B)) -> U5_gga(X, Y, Z, if_in_ggga(B, s(X), Y, Z))
   if_in_ggga(true, s(X), Y, Z) -> U6_ggga(X, Y, Z, half_in_ga(s(X), X1))
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U6_ggga(X, Y, Z, half_out_ga(s(X), X1)) -> U7_ggga(X, Y, Z, times_in_gga(X1, Y, Y1))
   U7_ggga(X, Y, Z, times_out_gga(X1, Y, Y1)) -> U8_ggga(X, Y, Z, plus_in_gga(Y1, Y1, Z))
   plus_in_gga(0, Y, Y) -> plus_out_gga(0, Y, Y)
   plus_in_gga(s(X), Y, s(Z)) -> U3_gga(X, Y, Z, plus_in_gga(X, Y, Z))
   U3_gga(X, Y, Z, plus_out_gga(X, Y, Z)) -> plus_out_gga(s(X), Y, s(Z))
   U8_ggga(X, Y, Z, plus_out_gga(Y1, Y1, Z)) -> if_out_ggga(true, s(X), Y, Z)
   if_in_ggga(false, s(X), Y, Z) -> U9_ggga(X, Y, Z, times_in_gga(X, Y, U))
   U9_ggga(X, Y, Z, times_out_gga(X, Y, U)) -> U10_ggga(X, Y, Z, plus_in_gga(Y, U, Z))
   U10_ggga(X, Y, Z, plus_out_gga(Y, U, Z)) -> if_out_ggga(false, s(X), Y, Z)
   U5_gga(X, Y, Z, if_out_ggga(B, s(X), Y, Z)) -> times_out_gga(s(X), Y, Z)

The argument filtering Pi contains the following mapping:
times_in_gga(x1, x2, x3)  =  times_in_gga(x1, x2)

0  =  0

times_out_gga(x1, x2, x3)  =  times_out_gga(x3)

s(x1)  =  s(x1)

U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)

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

true  =  true

U6_ggga(x1, x2, x3, x4)  =  U6_ggga(x2, x4)

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

U7_ggga(x1, x2, x3, x4)  =  U7_ggga(x4)

U8_ggga(x1, x2, x3, x4)  =  U8_ggga(x4)

plus_in_gga(x1, x2, x3)  =  plus_in_gga(x1, x2)

plus_out_gga(x1, x2, x3)  =  plus_out_gga(x3)

U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)

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

false  =  false

U9_ggga(x1, x2, x3, x4)  =  U9_ggga(x2, x4)

U10_ggga(x1, x2, x3, x4)  =  U10_ggga(x4)

TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)

U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

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

U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)


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:

   U4_GGA(X, Y, Z, even_out_ga(s(X), B)) -> IF_IN_GGGA(B, s(X), Y, Z)
   IF_IN_GGGA(true, s(X), Y, Z) -> U6_GGGA(X, Y, Z, half_in_ga(s(X), X1))
   U6_GGGA(X, Y, Z, half_out_ga(s(X), X1)) -> TIMES_IN_GGA(X1, Y, Y1)
   TIMES_IN_GGA(s(X), Y, Z) -> U4_GGA(X, Y, Z, even_in_ga(s(X), B))
   IF_IN_GGGA(false, s(X), Y, Z) -> TIMES_IN_GGA(X, Y, U)

The TRS R consists of the following rules:

   half_in_ga(s(s(X)), s(Y)) -> U2_ga(X, Y, half_in_ga(X, Y))
   even_in_ga(s(0), false) -> even_out_ga(s(0), false)
   even_in_ga(s(s(X)), B) -> U1_ga(X, B, even_in_ga(X, B))
   U2_ga(X, Y, half_out_ga(X, Y)) -> half_out_ga(s(s(X)), s(Y))
   U1_ga(X, B, even_out_ga(X, B)) -> even_out_ga(s(s(X)), B)
   half_in_ga(0, 0) -> half_out_ga(0, 0)
   even_in_ga(0, true) -> even_out_ga(0, true)

The argument filtering Pi contains the following mapping:
0  =  0

s(x1)  =  s(x1)

even_in_ga(x1, x2)  =  even_in_ga(x1)

even_out_ga(x1, x2)  =  even_out_ga(x2)

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

true  =  true

half_in_ga(x1, x2)  =  half_in_ga(x1)

half_out_ga(x1, x2)  =  half_out_ga(x2)

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

false  =  false

TIMES_IN_GGA(x1, x2, x3)  =  TIMES_IN_GGA(x1, x2)

U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

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

U6_GGGA(x1, x2, x3, x4)  =  U6_GGGA(x2, x4)


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

(31) PiDPToQDPProof (SOUND)
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:

   U4_GGA(X, Y, even_out_ga(B)) -> IF_IN_GGGA(B, s(X), Y)
   IF_IN_GGGA(true, s(X), Y) -> U6_GGGA(Y, half_in_ga(s(X)))
   U6_GGGA(Y, half_out_ga(X1)) -> TIMES_IN_GGA(X1, Y)
   TIMES_IN_GGA(s(X), Y) -> U4_GGA(X, Y, even_in_ga(s(X)))
   IF_IN_GGGA(false, s(X), Y) -> TIMES_IN_GGA(X, Y)

The TRS R consists of the following rules:

   half_in_ga(s(s(X))) -> U2_ga(half_in_ga(X))
   even_in_ga(s(0)) -> even_out_ga(false)
   even_in_ga(s(s(X))) -> U1_ga(even_in_ga(X))
   U2_ga(half_out_ga(Y)) -> half_out_ga(s(Y))
   U1_ga(even_out_ga(B)) -> even_out_ga(B)
   half_in_ga(0) -> half_out_ga(0)
   even_in_ga(0) -> even_out_ga(true)

The set Q consists of the following terms:

   half_in_ga(x0)
   even_in_ga(x0)
   U2_ga(x0)
   U1_ga(x0)

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

(33) MRRProof (EQUIVALENT)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented dependency pairs:

   IF_IN_GGGA(false, s(X), Y) -> TIMES_IN_GGA(X, Y)

Strictly oriented rules of the TRS R:

   half_in_ga(s(s(X))) -> U2_ga(half_in_ga(X))
   even_in_ga(s(0)) -> even_out_ga(false)
   even_in_ga(s(s(X))) -> U1_ga(even_in_ga(X))

Used ordering: Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(IF_IN_GGGA(x_1, x_2, x_3)) = 2*x_1 + x_2 + x_3
   POL(TIMES_IN_GGA(x_1, x_2)) = 2*x_1 + x_2
   POL(U1_ga(x_1)) = x_1
   POL(U2_ga(x_1)) = 2 + 2*x_1
   POL(U4_GGA(x_1, x_2, x_3)) = 1 + 2*x_1 + x_2 + x_3
   POL(U6_GGGA(x_1, x_2)) = x_1 + x_2
   POL(even_in_ga(x_1)) = x_1
   POL(even_out_ga(x_1)) = 2*x_1
   POL(false) = 0
   POL(half_in_ga(x_1)) = x_1
   POL(half_out_ga(x_1)) = 2*x_1
   POL(s(x_1)) = 1 + 2*x_1
   POL(true) = 0


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

(34)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U4_GGA(X, Y, even_out_ga(B)) -> IF_IN_GGGA(B, s(X), Y)
   IF_IN_GGGA(true, s(X), Y) -> U6_GGGA(Y, half_in_ga(s(X)))
   U6_GGGA(Y, half_out_ga(X1)) -> TIMES_IN_GGA(X1, Y)
   TIMES_IN_GGA(s(X), Y) -> U4_GGA(X, Y, even_in_ga(s(X)))

The TRS R consists of the following rules:

   U2_ga(half_out_ga(Y)) -> half_out_ga(s(Y))
   U1_ga(even_out_ga(B)) -> even_out_ga(B)
   half_in_ga(0) -> half_out_ga(0)
   even_in_ga(0) -> even_out_ga(true)

The set Q consists of the following terms:

   half_in_ga(x0)
   even_in_ga(x0)
   U2_ga(x0)
   U1_ga(x0)

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

(35) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
----------------------------------------

(36)
TRUE
