Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 AND
    ↳9 PiDP
        ↳10 UsableRulesProof (⇔)
        ↳11 PiDP
        ↳12 PiDPToQDPProof (⇐)
        ↳13 QDP
        ↳14 QDPSizeChangeProof (⇔)
        ↳15 YES
    ↳16 PiDP
        ↳17 UsableRulesProof (⇔)
        ↳18 PiDP
        ↳19 PiDPToQDPProof (⇔)
        ↳20 QDP
        ↳21 QDPSizeChangeProof (⇔)
        ↳22 YES
    ↳23 PiDP
        ↳24 UsableRulesProof (⇔)
        ↳25 PiDP
        ↳26 PiDPToQDPProof (⇐)
        ↳27 QDP
        ↳28 QDPOrderProof (⇔)
        ↳29 QDP
        ↳30 PisEmptyProof (⇔)
        ↳31 YES

(0) Obligation:

Clauses:

div(X, s(Y), Z) :- div_s(X, Y, Z).
div_s(0, Y, 0).
div_s(s(X), Y, 0) :- lss(X, Y).
div_s(s(X), Y, s(Z)) :- ','(sub(X, Y, R), div_s(R, Y, Z)).
lss(s(X), s(Y)) :- lss(X, Y).
lss(0, s(Y)).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).

Queries:

div(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

lss13(s(T36), s(T37)) :- lss13(T36, T37).
lss13(0, s(T42)).
div_s3(0, T15, 0).
div_s3(s(T24), T25, 0) :- lss13(T24, T25).
div_s3(s(T49), T50, s(T52)) :- sub25(T49, T50, X58).
div_s3(s(T49), T50, s(T52)) :- ','(sub25(T49, T50, T55), div_s3(T55, T50, T52)).
sub25(s(T66), s(T67), X89) :- sub25(T66, T67, X89).
sub25(T72, 0, T72).
div1(T7, s(T8), T10) :- div_s3(T7, T8, T10).

Queries:

div1(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:
div1_in: (b,b,f)
div_s3_in: (b,b,f)
lss13_in: (b,b)
sub25_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga

(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:

DIV1_IN_GGA(T7, s(T8), T10) → U7_GGA(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
DIV1_IN_GGA(T7, s(T8), T10) → DIV_S3_IN_GGA(T7, T8, T10)
DIV_S3_IN_GGA(s(T24), T25, 0) → U2_GGA(T24, T25, lss13_in_gg(T24, T25))
DIV_S3_IN_GGA(s(T24), T25, 0) → LSS13_IN_GG(T24, T25)
LSS13_IN_GG(s(T36), s(T37)) → U1_GG(T36, T37, lss13_in_gg(T36, T37))
LSS13_IN_GG(s(T36), s(T37)) → LSS13_IN_GG(T36, T37)
DIV_S3_IN_GGA(s(T49), T50, s(T52)) → U3_GGA(T49, T50, T52, sub25_in_gga(T49, T50, X58))
DIV_S3_IN_GGA(s(T49), T50, s(T52)) → SUB25_IN_GGA(T49, T50, X58)
SUB25_IN_GGA(s(T66), s(T67), X89) → U6_GGA(T66, T67, X89, sub25_in_gga(T66, T67, X89))
SUB25_IN_GGA(s(T66), s(T67), X89) → SUB25_IN_GGA(T66, T67, X89)
DIV_S3_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_GGA(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U4_GGA(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → DIV_S3_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
DIV_S3_IN_GGA(x1, x2, x3)  =  DIV_S3_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
LSS13_IN_GG(x1, x2)  =  LSS13_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUB25_IN_GGA(x1, x2, x3)  =  SUB25_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(6) Obligation:

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

DIV1_IN_GGA(T7, s(T8), T10) → U7_GGA(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
DIV1_IN_GGA(T7, s(T8), T10) → DIV_S3_IN_GGA(T7, T8, T10)
DIV_S3_IN_GGA(s(T24), T25, 0) → U2_GGA(T24, T25, lss13_in_gg(T24, T25))
DIV_S3_IN_GGA(s(T24), T25, 0) → LSS13_IN_GG(T24, T25)
LSS13_IN_GG(s(T36), s(T37)) → U1_GG(T36, T37, lss13_in_gg(T36, T37))
LSS13_IN_GG(s(T36), s(T37)) → LSS13_IN_GG(T36, T37)
DIV_S3_IN_GGA(s(T49), T50, s(T52)) → U3_GGA(T49, T50, T52, sub25_in_gga(T49, T50, X58))
DIV_S3_IN_GGA(s(T49), T50, s(T52)) → SUB25_IN_GGA(T49, T50, X58)
SUB25_IN_GGA(s(T66), s(T67), X89) → U6_GGA(T66, T67, X89, sub25_in_gga(T66, T67, X89))
SUB25_IN_GGA(s(T66), s(T67), X89) → SUB25_IN_GGA(T66, T67, X89)
DIV_S3_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_GGA(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U4_GGA(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → DIV_S3_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga
DIV1_IN_GGA(x1, x2, x3)  =  DIV1_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
DIV_S3_IN_GGA(x1, x2, x3)  =  DIV_S3_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
LSS13_IN_GG(x1, x2)  =  LSS13_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x4)
SUB25_IN_GGA(x1, x2, x3)  =  SUB25_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4)  =  U6_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SUB25_IN_GGA(s(T66), s(T67), X89) → SUB25_IN_GGA(T66, T67, X89)

The TRS R consists of the following rules:

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga
SUB25_IN_GGA(x1, x2, x3)  =  SUB25_IN_GGA(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:

SUB25_IN_GGA(s(T66), s(T67), X89) → SUB25_IN_GGA(T66, T67, X89)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUB25_IN_GGA(x1, x2, x3)  =  SUB25_IN_GGA(x1, x2)

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

(12) PiDPToQDPProof (SOUND 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:

SUB25_IN_GGA(s(T66), s(T67)) → SUB25_IN_GGA(T66, T67)

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:

  • SUB25_IN_GGA(s(T66), s(T67)) → SUB25_IN_GGA(T66, T67)
    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:

LSS13_IN_GG(s(T36), s(T37)) → LSS13_IN_GG(T36, T37)

The TRS R consists of the following rules:

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga
LSS13_IN_GG(x1, x2)  =  LSS13_IN_GG(x1, x2)

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:

LSS13_IN_GG(s(T36), s(T37)) → LSS13_IN_GG(T36, T37)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

LSS13_IN_GG(s(T36), s(T37)) → LSS13_IN_GG(T36, T37)

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:

  • LSS13_IN_GG(s(T36), s(T37)) → LSS13_IN_GG(T36, T37)
    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:

DIV_S3_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → DIV_S3_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

div1_in_gga(T7, s(T8), T10) → U7_gga(T7, T8, T10, div_s3_in_gga(T7, T8, T10))
div_s3_in_gga(0, T15, 0) → div_s3_out_gga(0, T15, 0)
div_s3_in_gga(s(T24), T25, 0) → U2_gga(T24, T25, lss13_in_gg(T24, T25))
lss13_in_gg(s(T36), s(T37)) → U1_gg(T36, T37, lss13_in_gg(T36, T37))
lss13_in_gg(0, s(T42)) → lss13_out_gg(0, s(T42))
U1_gg(T36, T37, lss13_out_gg(T36, T37)) → lss13_out_gg(s(T36), s(T37))
U2_gga(T24, T25, lss13_out_gg(T24, T25)) → div_s3_out_gga(s(T24), T25, 0)
div_s3_in_gga(s(T49), T50, s(T52)) → U3_gga(T49, T50, T52, sub25_in_gga(T49, T50, X58))
sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)
U3_gga(T49, T50, T52, sub25_out_gga(T49, T50, X58)) → div_s3_out_gga(s(T49), T50, s(T52))
div_s3_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_gga(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → U5_gga(T49, T50, T52, div_s3_in_gga(T55, T50, T52))
U5_gga(T49, T50, T52, div_s3_out_gga(T55, T50, T52)) → div_s3_out_gga(s(T49), T50, s(T52))
U7_gga(T7, T8, T10, div_s3_out_gga(T7, T8, T10)) → div1_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3)  =  div1_in_gga(x1, x2)
s(x1)  =  s(x1)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
div_s3_in_gga(x1, x2, x3)  =  div_s3_in_gga(x1, x2)
0  =  0
div_s3_out_gga(x1, x2, x3)  =  div_s3_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
lss13_in_gg(x1, x2)  =  lss13_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
lss13_out_gg(x1, x2)  =  lss13_out_gg
U3_gga(x1, x2, x3, x4)  =  U3_gga(x4)
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
div1_out_gga(x1, x2, x3)  =  div1_out_gga
DIV_S3_IN_GGA(x1, x2, x3)  =  DIV_S3_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)

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:

DIV_S3_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, sub25_in_gga(T49, T50, T55))
U4_GGA(T49, T50, T52, sub25_out_gga(T49, T50, T55)) → DIV_S3_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

sub25_in_gga(s(T66), s(T67), X89) → U6_gga(T66, T67, X89, sub25_in_gga(T66, T67, X89))
sub25_in_gga(T72, 0, T72) → sub25_out_gga(T72, 0, T72)
U6_gga(T66, T67, X89, sub25_out_gga(T66, T67, X89)) → sub25_out_gga(s(T66), s(T67), X89)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
sub25_in_gga(x1, x2, x3)  =  sub25_in_gga(x1, x2)
U6_gga(x1, x2, x3, x4)  =  U6_gga(x4)
sub25_out_gga(x1, x2, x3)  =  sub25_out_gga(x3)
DIV_S3_IN_GGA(x1, x2, x3)  =  DIV_S3_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x2, x4)

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

(26) PiDPToQDPProof (SOUND 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:

DIV_S3_IN_GGA(s(T49), T50) → U4_GGA(T50, sub25_in_gga(T49, T50))
U4_GGA(T50, sub25_out_gga(T55)) → DIV_S3_IN_GGA(T55, T50)

The TRS R consists of the following rules:

sub25_in_gga(s(T66), s(T67)) → U6_gga(sub25_in_gga(T66, T67))
sub25_in_gga(T72, 0) → sub25_out_gga(T72)
U6_gga(sub25_out_gga(X89)) → sub25_out_gga(X89)

The set Q consists of the following terms:

sub25_in_gga(x0, x1)
U6_gga(x0)

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

(28) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DIV_S3_IN_GGA(s(T49), T50) → U4_GGA(T50, sub25_in_gga(T49, T50))
U4_GGA(T50, sub25_out_gga(T55)) → DIV_S3_IN_GGA(T55, T50)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U4_GGA(x1, x2) ) = 2x2


POL( sub25_in_gga(x1, x2) ) = 2x1 + 2


POL( s(x1) ) = 2x1 + 2


POL( U6_gga(x1) ) = 2x1 + 2


POL( 0 ) = 0


POL( sub25_out_gga(x1) ) = 2x1 + 2


POL( DIV_S3_IN_GGA(x1, x2) ) = 2x1 + 2



The following usable rules [FROCOS05] were oriented:

sub25_in_gga(s(T66), s(T67)) → U6_gga(sub25_in_gga(T66, T67))
sub25_in_gga(T72, 0) → sub25_out_gga(T72)
U6_gga(sub25_out_gga(X89)) → sub25_out_gga(X89)

(29) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

sub25_in_gga(s(T66), s(T67)) → U6_gga(sub25_in_gga(T66, T67))
sub25_in_gga(T72, 0) → sub25_out_gga(T72)
U6_gga(sub25_out_gga(X89)) → sub25_out_gga(X89)

The set Q consists of the following terms:

sub25_in_gga(x0, x1)
U6_gga(x0)

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

(30) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(31) YES