(0) Obligation:

Clauses:

gcd(X, Y, D) :- ','(le(X, Y), gcd_le(X, Y, D)).
gcd(X, Y, D) :- ','(gt(X, Y), gcd_le(Y, X, D)).
gcd_le(0, Y, Y).
gcd_le(s(X), Y, D) :- ','(add(s(X), Z, Y), gcd(s(X), Z, D)).
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).
add(s(X), Y, s(Z)) :- add(X, Y, Z).
add(0, X, X).

Queries:

gcd(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

le9(s(T33), s(T34)) :- le9(T33, T34).
le9(0, s(T41)).
le9(0, 0).
add32(s(T87), X124, s(T88)) :- add32(T87, X124, T88).
add32(0, T93, T93).
gt66(s(T134), s(T135)) :- gt66(T134, T135).
gt66(s(T140), 0).
add27(T75, X100, T76) :- add32(T75, X100, T76).
gcd1(s(T19), s(T20), T10) :- le9(T19, T20).
gcd1(s(T54), s(T55), T57) :- ','(le9(T54, T55), add27(T54, X64, T55)).
gcd1(s(T54), s(T55), T57) :- ','(le9(T54, T55), ','(add27(T54, T60, T55), gcd1(s(T54), T60, T57))).
gcd1(0, s(T108), s(T108)).
gcd1(0, 0, 0).
gcd1(T118, T119, T121) :- gt66(T118, T119).
gcd1(T147, 0, T147) :- gt66(T147, 0).
gcd1(T155, s(T154), T157) :- ','(gt66(T155, s(T154)), add32(s(T154), X222, T155)).
gcd1(T155, s(T154), T157) :- ','(gt66(T155, s(T154)), ','(add32(s(T154), T160, T155), gcd1(s(T154), T160, T157))).

Queries:

gcd1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
gcd1_in: (b,b,f)
le9_in: (b,b)
add27_in: (b,f,b)
add32_in: (b,f,b)
gt66_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

GCD1_IN_GGA(s(T19), s(T20), T10) → U5_GGA(T19, T20, T10, le9_in_gg(T19, T20))
GCD1_IN_GGA(s(T19), s(T20), T10) → LE9_IN_GG(T19, T20)
LE9_IN_GG(s(T33), s(T34)) → U1_GG(T33, T34, le9_in_gg(T33, T34))
LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)
GCD1_IN_GGA(s(T54), s(T55), T57) → U6_GGA(T54, T55, T57, le9_in_gg(T54, T55))
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → U7_GGA(T54, T55, T57, add27_in_gag(T54, X64, T55))
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → ADD27_IN_GAG(T54, X64, T55)
ADD27_IN_GAG(T75, X100, T76) → U4_GAG(T75, X100, T76, add32_in_gag(T75, X100, T76))
ADD27_IN_GAG(T75, X100, T76) → ADD32_IN_GAG(T75, X100, T76)
ADD32_IN_GAG(s(T87), X124, s(T88)) → U2_GAG(T87, X124, T88, add32_in_gag(T87, X124, T88))
ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_GGA(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
U8_GGA(T54, T55, T57, add27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T118, T119, T121) → U10_GGA(T118, T119, T121, gt66_in_gg(T118, T119))
GCD1_IN_GGA(T118, T119, T121) → GT66_IN_GG(T118, T119)
GT66_IN_GG(s(T134), s(T135)) → U3_GG(T134, T135, gt66_in_gg(T134, T135))
GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)
GCD1_IN_GGA(T147, 0, T147) → U11_GGA(T147, gt66_in_gg(T147, 0))
GCD1_IN_GGA(T147, 0, T147) → GT66_IN_GG(T147, 0)
GCD1_IN_GGA(T155, s(T154), T157) → U12_GGA(T155, T154, T157, gt66_in_gg(T155, s(T154)))
GCD1_IN_GGA(T155, s(T154), T157) → GT66_IN_GG(T155, s(T154))
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → ADD32_IN_GAG(s(T154), X222, T155)
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_GGA(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_GGA(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_GGA(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U14_GGA(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
LE9_IN_GG(x1, x2)  =  LE9_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
ADD27_IN_GAG(x1, x2, x3)  =  ADD27_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x4)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x4)
GT66_IN_GG(x1, x2)  =  GT66_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U11_GGA(x1, x2)  =  U11_GGA(x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x4)

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

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T19), s(T20), T10) → U5_GGA(T19, T20, T10, le9_in_gg(T19, T20))
GCD1_IN_GGA(s(T19), s(T20), T10) → LE9_IN_GG(T19, T20)
LE9_IN_GG(s(T33), s(T34)) → U1_GG(T33, T34, le9_in_gg(T33, T34))
LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)
GCD1_IN_GGA(s(T54), s(T55), T57) → U6_GGA(T54, T55, T57, le9_in_gg(T54, T55))
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → U7_GGA(T54, T55, T57, add27_in_gag(T54, X64, T55))
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → ADD27_IN_GAG(T54, X64, T55)
ADD27_IN_GAG(T75, X100, T76) → U4_GAG(T75, X100, T76, add32_in_gag(T75, X100, T76))
ADD27_IN_GAG(T75, X100, T76) → ADD32_IN_GAG(T75, X100, T76)
ADD32_IN_GAG(s(T87), X124, s(T88)) → U2_GAG(T87, X124, T88, add32_in_gag(T87, X124, T88))
ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_GGA(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
U8_GGA(T54, T55, T57, add27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T118, T119, T121) → U10_GGA(T118, T119, T121, gt66_in_gg(T118, T119))
GCD1_IN_GGA(T118, T119, T121) → GT66_IN_GG(T118, T119)
GT66_IN_GG(s(T134), s(T135)) → U3_GG(T134, T135, gt66_in_gg(T134, T135))
GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)
GCD1_IN_GGA(T147, 0, T147) → U11_GGA(T147, gt66_in_gg(T147, 0))
GCD1_IN_GGA(T147, 0, T147) → GT66_IN_GG(T147, 0)
GCD1_IN_GGA(T155, s(T154), T157) → U12_GGA(T155, T154, T157, gt66_in_gg(T155, s(T154)))
GCD1_IN_GGA(T155, s(T154), T157) → GT66_IN_GG(T155, s(T154))
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_GGA(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → ADD32_IN_GAG(s(T154), X222, T155)
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_GGA(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_GGA(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_GGA(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U14_GGA(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
LE9_IN_GG(x1, x2)  =  LE9_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
ADD27_IN_GAG(x1, x2, x3)  =  ADD27_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4)  =  U4_GAG(x4)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_IN_GAG(x1, x3)
U2_GAG(x1, x2, x3, x4)  =  U2_GAG(x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x4)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x4)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x4)
GT66_IN_GG(x1, x2)  =  GT66_IN_GG(x1, x2)
U3_GG(x1, x2, x3)  =  U3_GG(x3)
U11_GGA(x1, x2)  =  U11_GGA(x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x2, x4)
U15_GGA(x1, x2, x3, x4)  =  U15_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 18 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)

The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
GT66_IN_GG(x1, x2)  =  GT66_IN_GG(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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:

  • GT66_IN_GG(s(T134), s(T135)) → GT66_IN_GG(T134, T135)
    The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)

The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_IN_GAG(x1, x3)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

ADD32_IN_GAG(s(T87), X124, s(T88)) → ADD32_IN_GAG(T87, X124, T88)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
ADD32_IN_GAG(x1, x2, x3)  =  ADD32_IN_GAG(x1, x3)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

ADD32_IN_GAG(s(T87), s(T88)) → ADD32_IN_GAG(T87, T88)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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:

  • ADD32_IN_GAG(s(T87), s(T88)) → ADD32_IN_GAG(T87, T88)
    The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)

The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
LE9_IN_GG(x1, x2)  =  LE9_IN_GG(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)

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

(26) PiDPToQDPProof (EQUIVALENT transformation)

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

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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:

  • LE9_IN_GG(s(T33), s(T34)) → LE9_IN_GG(T33, T34)
    The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T54), s(T55), T57) → U6_GGA(T54, T55, T57, le9_in_gg(T54, T55))
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, add27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T155, s(T154), T157) → U12_GGA(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_GGA(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_GGA(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

gcd1_in_gga(s(T19), s(T20), T10) → U5_gga(T19, T20, T10, le9_in_gg(T19, T20))
le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U5_gga(T19, T20, T10, le9_out_gg(T19, T20)) → gcd1_out_gga(s(T19), s(T20), T10)
gcd1_in_gga(s(T54), s(T55), T57) → U6_gga(T54, T55, T57, le9_in_gg(T54, T55))
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U7_gga(T54, T55, T57, add27_in_gag(T54, X64, T55))
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U7_gga(T54, T55, T57, add27_out_gag(T54, X64, T55)) → gcd1_out_gga(s(T54), s(T55), T57)
U6_gga(T54, T55, T57, le9_out_gg(T54, T55)) → U8_gga(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_gga(T54, T55, T57, add27_out_gag(T54, T60, T55)) → U9_gga(T54, T55, T57, gcd1_in_gga(s(T54), T60, T57))
gcd1_in_gga(0, s(T108), s(T108)) → gcd1_out_gga(0, s(T108), s(T108))
gcd1_in_gga(0, 0, 0) → gcd1_out_gga(0, 0, 0)
gcd1_in_gga(T118, T119, T121) → U10_gga(T118, T119, T121, gt66_in_gg(T118, T119))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U10_gga(T118, T119, T121, gt66_out_gg(T118, T119)) → gcd1_out_gga(T118, T119, T121)
gcd1_in_gga(T147, 0, T147) → U11_gga(T147, gt66_in_gg(T147, 0))
U11_gga(T147, gt66_out_gg(T147, 0)) → gcd1_out_gga(T147, 0, T147)
gcd1_in_gga(T155, s(T154), T157) → U12_gga(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U13_gga(T155, T154, T157, add32_in_gag(s(T154), X222, T155))
U13_gga(T155, T154, T157, add32_out_gag(s(T154), X222, T155)) → gcd1_out_gga(T155, s(T154), T157)
U12_gga(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_gga(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_gga(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → U15_gga(T155, T154, T157, gcd1_in_gga(s(T154), T160, T157))
U15_gga(T155, T154, T157, gcd1_out_gga(s(T154), T160, T157)) → gcd1_out_gga(T155, s(T154), T157)
U9_gga(T54, T55, T57, gcd1_out_gga(s(T54), T60, T57)) → gcd1_out_gga(s(T54), s(T55), T57)

The argument filtering Pi contains the following mapping:
gcd1_in_gga(x1, x2, x3)  =  gcd1_in_gga(x1, x2)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
gcd1_out_gga(x1, x2, x3)  =  gcd1_out_gga
U6_gga(x1, x2, x3, x4)  =  U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x4)
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
U11_gga(x1, x2)  =  U11_gga(x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x2, x4)
U15_gga(x1, x2, x3, x4)  =  U15_gga(x4)
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x2, x4)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T54), s(T55), T57) → U6_GGA(T54, T55, T57, le9_in_gg(T54, T55))
U6_GGA(T54, T55, T57, le9_out_gg(T54, T55)) → U8_GGA(T54, T55, T57, add27_in_gag(T54, T60, T55))
U8_GGA(T54, T55, T57, add27_out_gag(T54, T60, T55)) → GCD1_IN_GGA(s(T54), T60, T57)
GCD1_IN_GGA(T155, s(T154), T157) → U12_GGA(T155, T154, T157, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, T157, gt66_out_gg(T155, s(T154))) → U14_GGA(T155, T154, T157, add32_in_gag(s(T154), T160, T155))
U14_GGA(T155, T154, T157, add32_out_gag(s(T154), T160, T155)) → GCD1_IN_GGA(s(T154), T160, T157)

The TRS R consists of the following rules:

le9_in_gg(s(T33), s(T34)) → U1_gg(T33, T34, le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg(0, s(T41))
le9_in_gg(0, 0) → le9_out_gg(0, 0)
add27_in_gag(T75, X100, T76) → U4_gag(T75, X100, T76, add32_in_gag(T75, X100, T76))
gt66_in_gg(s(T134), s(T135)) → U3_gg(T134, T135, gt66_in_gg(T134, T135))
add32_in_gag(s(T87), X124, s(T88)) → U2_gag(T87, X124, T88, add32_in_gag(T87, X124, T88))
U1_gg(T33, T34, le9_out_gg(T33, T34)) → le9_out_gg(s(T33), s(T34))
U4_gag(T75, X100, T76, add32_out_gag(T75, X100, T76)) → add27_out_gag(T75, X100, T76)
U3_gg(T134, T135, gt66_out_gg(T134, T135)) → gt66_out_gg(s(T134), s(T135))
U2_gag(T87, X124, T88, add32_out_gag(T87, X124, T88)) → add32_out_gag(s(T87), X124, s(T88))
add32_in_gag(0, T93, T93) → add32_out_gag(0, T93, T93)
gt66_in_gg(s(T140), 0) → gt66_out_gg(s(T140), 0)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
le9_in_gg(x1, x2)  =  le9_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
0  =  0
le9_out_gg(x1, x2)  =  le9_out_gg
add27_in_gag(x1, x2, x3)  =  add27_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4)  =  U4_gag(x4)
add32_in_gag(x1, x2, x3)  =  add32_in_gag(x1, x3)
U2_gag(x1, x2, x3, x4)  =  U2_gag(x4)
add32_out_gag(x1, x2, x3)  =  add32_out_gag(x2)
add27_out_gag(x1, x2, x3)  =  add27_out_gag(x2)
gt66_in_gg(x1, x2)  =  gt66_in_gg(x1, x2)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
gt66_out_gg(x1, x2)  =  gt66_out_gg
GCD1_IN_GGA(x1, x2, x3)  =  GCD1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x4)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x2, x4)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T54), s(T55)) → U6_GGA(T54, T55, le9_in_gg(T54, T55))
U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, add27_in_gag(T54, T55))
U8_GGA(T54, add27_out_gag(T60)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)

The TRS R consists of the following rules:

le9_in_gg(s(T33), s(T34)) → U1_gg(le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg
le9_in_gg(0, 0) → le9_out_gg
add27_in_gag(T75, T76) → U4_gag(add32_in_gag(T75, T76))
gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
U1_gg(le9_out_gg) → le9_out_gg
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U3_gg(gt66_out_gg) → gt66_out_gg
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)
add32_in_gag(0, T93) → add32_out_gag(T93)
gt66_in_gg(s(T140), 0) → gt66_out_gg

The set Q consists of the following terms:

le9_in_gg(x0, x1)
add27_in_gag(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(35) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, add27_in_gag(T54, T55)) at position [1] we obtained the following new rules [LPAR04]:

U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, U4_gag(add32_in_gag(T54, T55)))

(36) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T54), s(T55)) → U6_GGA(T54, T55, le9_in_gg(T54, T55))
U8_GGA(T54, add27_out_gag(T60)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, U4_gag(add32_in_gag(T54, T55)))

The TRS R consists of the following rules:

le9_in_gg(s(T33), s(T34)) → U1_gg(le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg
le9_in_gg(0, 0) → le9_out_gg
add27_in_gag(T75, T76) → U4_gag(add32_in_gag(T75, T76))
gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
U1_gg(le9_out_gg) → le9_out_gg
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U3_gg(gt66_out_gg) → gt66_out_gg
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)
add32_in_gag(0, T93) → add32_out_gag(T93)
gt66_in_gg(s(T140), 0) → gt66_out_gg

The set Q consists of the following terms:

le9_in_gg(x0, x1)
add27_in_gag(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(37) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(38) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T54), s(T55)) → U6_GGA(T54, T55, le9_in_gg(T54, T55))
U8_GGA(T54, add27_out_gag(T60)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, U4_gag(add32_in_gag(T54, T55)))

The TRS R consists of the following rules:

add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)
gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
le9_in_gg(s(T33), s(T34)) → U1_gg(le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg
le9_in_gg(0, 0) → le9_out_gg
U1_gg(le9_out_gg) → le9_out_gg

The set Q consists of the following terms:

le9_in_gg(x0, x1)
add27_in_gag(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(39) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

add27_in_gag(x0, x1)

(40) Obligation:

Q DP problem:
The TRS P consists of the following rules:

GCD1_IN_GGA(s(T54), s(T55)) → U6_GGA(T54, T55, le9_in_gg(T54, T55))
U8_GGA(T54, add27_out_gag(T60)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, U4_gag(add32_in_gag(T54, T55)))

The TRS R consists of the following rules:

add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)
gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
le9_in_gg(s(T33), s(T34)) → U1_gg(le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg
le9_in_gg(0, 0) → le9_out_gg
U1_gg(le9_out_gg) → le9_out_gg

The set Q consists of the following terms:

le9_in_gg(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


GCD1_IN_GGA(s(T54), s(T55)) → U6_GGA(T54, T55, le9_in_gg(T54, T55))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(GCD1_IN_GGA(x1, x2)) = x1 + x2   
POL(U12_GGA(x1, x2, x3)) = 1 + x1 + x2   
POL(U14_GGA(x1, x2)) = 1 + x1 + x2   
POL(U1_gg(x1)) = 0   
POL(U2_gag(x1)) = x1   
POL(U3_gg(x1)) = 0   
POL(U4_gag(x1)) = 1 + x1   
POL(U6_GGA(x1, x2, x3)) = 1 + x1 + x2   
POL(U8_GGA(x1, x2)) = x1 + x2   
POL(add27_out_gag(x1)) = 1 + x1   
POL(add32_in_gag(x1, x2)) = x2   
POL(add32_out_gag(x1)) = x1   
POL(gt66_in_gg(x1, x2)) = 0   
POL(gt66_out_gg) = 0   
POL(le9_in_gg(x1, x2)) = 0   
POL(le9_out_gg) = 0   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)

(42) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U8_GGA(T54, add27_out_gag(T60)) → GCD1_IN_GGA(s(T54), T60)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))
U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
U6_GGA(T54, T55, le9_out_gg) → U8_GGA(T54, U4_gag(add32_in_gag(T54, T55)))

The TRS R consists of the following rules:

add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)
gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
le9_in_gg(s(T33), s(T34)) → U1_gg(le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg
le9_in_gg(0, 0) → le9_out_gg
U1_gg(le9_out_gg) → le9_out_gg

The set Q consists of the following terms:

le9_in_gg(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(44) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))

The TRS R consists of the following rules:

add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U4_gag(add32_out_gag(X100)) → add27_out_gag(X100)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)
gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
le9_in_gg(s(T33), s(T34)) → U1_gg(le9_in_gg(T33, T34))
le9_in_gg(0, s(T41)) → le9_out_gg
le9_in_gg(0, 0) → le9_out_gg
U1_gg(le9_out_gg) → le9_out_gg

The set Q consists of the following terms:

le9_in_gg(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(45) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(46) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))

The TRS R consists of the following rules:

gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)

The set Q consists of the following terms:

le9_in_gg(x0, x1)
gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U1_gg(x0)
U4_gag(x0)
U3_gg(x0)
U2_gag(x0)

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

(47) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le9_in_gg(x0, x1)
U1_gg(x0)
U4_gag(x0)

(48) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)
GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))

The TRS R consists of the following rules:

gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)

The set Q consists of the following terms:

gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U3_gg(x0)
U2_gag(x0)

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


GCD1_IN_GGA(T155, s(T154)) → U12_GGA(T155, T154, gt66_in_gg(T155, s(T154)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U12_GGA(x1, x2, x3)) = 0 +
[0,1]
·x1 +
[0,1]
·x2 +
[0,0]
·x3

POL(gt66_out_gg) =
/0\
\0/

POL(U14_GGA(x1, x2)) = 0 +
[0,1]
·x1 +
[1,0]
·x2

POL(add32_in_gag(x1, x2)) =
/0\
\1/
+
/10\
\00/
·x1 +
/01\
\01/
·x2

POL(s(x1)) =
/0\
\1/
+
/00\
\01/
·x1

POL(add32_out_gag(x1)) =
/1\
\1/
+
/01\
\01/
·x1

POL(GCD1_IN_GGA(x1, x2)) = 0 +
[0,1]
·x1 +
[0,1]
·x2

POL(gt66_in_gg(x1, x2)) =
/0\
\0/
+
/00\
\00/
·x1 +
/10\
\00/
·x2

POL(U2_gag(x1)) =
/0\
\0/
+
/01\
\01/
·x1

POL(U3_gg(x1)) =
/1\
\1/
+
/10\
\10/
·x1

POL(0) =
/1\
\0/

The following usable rules [FROCOS05] were oriented:

add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)

(50) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U12_GGA(T155, T154, gt66_out_gg) → U14_GGA(T154, add32_in_gag(s(T154), T155))
U14_GGA(T154, add32_out_gag(T160)) → GCD1_IN_GGA(s(T154), T160)

The TRS R consists of the following rules:

gt66_in_gg(s(T134), s(T135)) → U3_gg(gt66_in_gg(T134, T135))
gt66_in_gg(s(T140), 0) → gt66_out_gg
U3_gg(gt66_out_gg) → gt66_out_gg
add32_in_gag(s(T87), s(T88)) → U2_gag(add32_in_gag(T87, T88))
add32_in_gag(0, T93) → add32_out_gag(T93)
U2_gag(add32_out_gag(X124)) → add32_out_gag(X124)

The set Q consists of the following terms:

gt66_in_gg(x0, x1)
add32_in_gag(x0, x1)
U3_gg(x0)
U2_gag(x0)

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

(51) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(52) TRUE