Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog
↳1 PredefinedPredicateTransformerProof (⇐)
↳2 Prolog
↳3 CutEliminatorProof (⇐)
↳4 Prolog
  ↳5 PrologToPiTRSProof (⇐)
    ↳6 PiTRS
    ↳7 DependencyPairsProof (⇔)
    ↳8 PiDP
    ↳9 DependencyGraphProof (⇔)
    ↳10 AND
      ↳11 PiDP
        ↳12 UsableRulesProof (⇔)
        ↳13 PiDP
        ↳14 PiDPToQDPProof (⇐)
        ↳15 QDP
        ↳16 QDPSizeChangeProof (⇔)
        ↳17 TRUE
      ↳18 PiDP
        ↳19 UsableRulesProof (⇔)
        ↳20 PiDP
        ↳21 PiDPToQDPProof (⇐)
        ↳22 QDP
        ↳23 Narrowing (⇐)
        ↳24 QDP
        ↳25 Instantiation (⇔)
        ↳26 QDP
        ↳27 ForwardInstantiation (⇔)
        ↳28 QDP
        ↳29 DependencyGraphProof (⇔)
        ↳30 AND
          ↳31 QDP
            ↳32 UsableRulesProof (⇔)
            ↳33 QDP
            ↳34 QReductionProof (⇔)
            ↳35 QDP
            ↳36 NonTerminationProof (⇔)
            ↳37 FALSE
          ↳38 QDP
            ↳39 QDPOrderProof (⇔)
            ↳40 QDP
            ↳41 DependencyGraphProof (⇔)
            ↳42 TRUE
  ↳43 PrologToPiTRSProof (⇐)
    ↳44 PiTRS
    ↳45 DependencyPairsProof (⇔)
    ↳46 PiDP
    ↳47 DependencyGraphProof (⇔)
    ↳48 AND
      ↳49 PiDP
        ↳50 UsableRulesProof (⇔)
        ↳51 PiDP
        ↳52 PiDPToQDPProof (⇐)
        ↳53 QDP
        ↳54 QDPSizeChangeProof (⇔)
        ↳55 TRUE
      ↳56 PiDP
        ↳57 UsableRulesProof (⇔)
        ↳58 PiDP
        ↳59 PiDPToQDPProof (⇐)
        ↳60 QDP
        ↳61 Narrowing (⇐)
        ↳62 QDP
        ↳63 Instantiation (⇔)
        ↳64 QDP
        ↳65 DependencyGraphProof (⇔)
        ↳66 AND
          ↳67 QDP
            ↳68 UsableRulesProof (⇔)
            ↳69 QDP
            ↳70 QReductionProof (⇔)
            ↳71 QDP
            ↳72 NonTerminationProof (⇔)
            ↳73 FALSE
          ↳74 QDP
            ↳75 ForwardInstantiation (⇔)
            ↳76 QDP
            ↳77 QDPOrderProof (⇔)
            ↳78 QDP
            ↳79 DependencyGraphProof (⇔)
            ↳80 TRUE

(0) Obligation:

Clauses:

div(X, 0, Z) :- ','(!, fail).
div(0, Y, Z) :- ','(!, =(Z, 0)).
div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
=(X, X).

Queries:

div(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

Clauses:

div(X, 0, Z) :- ','(!, fail).
div(0, Y, Z) :- ','(!, =1(Z, 0)).
div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
=1(X, X).
fail :- fail(b).
fail(a).

Queries:

div(g,g,a).

(3) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(4) Obligation:

Clauses:

div(X, 0, Z) :- fail.
div(0, Y, Z) :- =1(Z, 0).
div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
=1(X, X).
fail :- fail(b).
fail(a).

Queries:

div(g,g,a).

(5) PrologToPiTRSProof (SOUND transformation)

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

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)

(7) 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:

DIV_IN_GGA(X, 0, Z) → U1_GGA(X, Z, fail_in_)
DIV_IN_GGA(X, 0, Z) → FAIL_IN_
FAIL_IN_ → U6_1(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(0, Y, Z) → U2_GGA(Y, Z, =1_in_ag(Z, 0))
DIV_IN_GGA(0, Y, Z) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U5_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U4_GGA(X, Y, Z, div_in_gga(U, Y, Z))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
FAIL_IN_  =  FAIL_IN_
U6_1(x1)  =  U6_1(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)

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

(8) Obligation:

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

DIV_IN_GGA(X, 0, Z) → U1_GGA(X, Z, fail_in_)
DIV_IN_GGA(X, 0, Z) → FAIL_IN_
FAIL_IN_ → U6_1(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(0, Y, Z) → U2_GGA(Y, Z, =1_in_ag(Z, 0))
DIV_IN_GGA(0, Y, Z) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U5_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U4_GGA(X, Y, Z, div_in_gga(U, Y, Z))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x3)
FAIL_IN_  =  FAIL_IN_
U6_1(x1)  =  U6_1(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)

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

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)

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

(16) 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:

  • MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(17) TRUE

(18) Obligation:

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

DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga
U2_gga(x1, x2, x3)  =  U2_gga(x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The argument filtering Pi contains the following mapping:
0  =  0
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x2, x4)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

DIV_IN_GGA(X, Y) → U3_GGA(Y, minus_in_gga(X, Y))
U3_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(23) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(X, Y) → U3_GGA(Y, minus_in_gga(X, Y)) at position [1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x1), U5_gga(minus_in_gga(x0, x1)))

(24) Obligation:

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

U3_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x1), U5_gga(minus_in_gga(x0, x1)))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(25) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y) we obtained the following new rules [LPAR04]:

U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U3_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)
U3_GGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(z1))

(26) Obligation:

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

DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x1), U5_gga(minus_in_gga(x0, x1)))
U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U3_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)
U3_GGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(z1))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(27) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U3_GGA(s(z1), minus_out_gga(x1)) → DIV_IN_GGA(x1, s(z1)) we obtained the following new rules [LPAR04]:

U3_GGA(s(x0), minus_out_gga(0)) → DIV_IN_GGA(0, s(x0))
U3_GGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(x0))

(28) Obligation:

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

DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x1), U5_gga(minus_in_gga(x0, x1)))
U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U3_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)
U3_GGA(s(x0), minus_out_gga(0)) → DIV_IN_GGA(0, s(x0))
U3_GGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(x0))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(29) DependencyGraphProof (EQUIVALENT transformation)

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

(30) Complex Obligation (AND)

(31) Obligation:

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

U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
U3_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
U3_GGA(s(x0), minus_out_gga(0)) → DIV_IN_GGA(0, s(x0))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(32) 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.

(33) Obligation:

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

U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
U3_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
U3_GGA(s(x0), minus_out_gga(0)) → DIV_IN_GGA(0, s(x0))

R is empty.
The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(34) 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].

minus_in_gga(x0, x1)
U5_gga(x0)

(35) Obligation:

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

U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
U3_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)
DIV_IN_GGA(x0, 0) → U3_GGA(0, minus_out_gga(x0))
U3_GGA(s(x0), minus_out_gga(0)) → DIV_IN_GGA(0, s(x0))

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

(36) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = DIV_IN_GGA(0, x0) evaluates to t =DIV_IN_GGA(0, x0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

DIV_IN_GGA(0, x0) → U3_GGA(x0, minus_out_gga(0))
with rule DIV_IN_GGA(0, x0') → U3_GGA(x0', minus_out_gga(0)) at position [] and matcher [x0' / x0]

U3_GGA(x0, minus_out_gga(0)) → DIV_IN_GGA(0, x0)
with rule U3_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(37) FALSE

(38) Obligation:

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

U3_GGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x1), U5_gga(minus_in_gga(x0, x1)))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


U3_GGA(s(x0), minus_out_gga(s(y_0))) → DIV_IN_GGA(s(y_0), s(x0))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(DIV_IN_GGA(x1, x2)) = x1 + x2   
POL(U3_GGA(x1, x2)) = x1 + x2   
POL(U5_gga(x1)) = x1   
POL(minus_in_gga(x1, x2)) = 1 + x1   
POL(minus_out_gga(x1)) = 1 + x1   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

(40) Obligation:

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

DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x1), U5_gga(minus_in_gga(x0, x1)))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0)
minus_in_gga(X, 0) → minus_out_gga(X)
minus_in_gga(s(X), s(Y)) → U5_gga(minus_in_gga(X, Y))
U5_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0)

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

(41) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(42) TRUE

(43) PrologToPiTRSProof (SOUND transformation)

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

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g(x1)
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2)
U2_gga(x1, x2, x3)  =  U2_gga(x1, x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(44) Obligation:

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

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g(x1)
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2)
U2_gga(x1, x2, x3)  =  U2_gga(x1, x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)

(45) 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:

DIV_IN_GGA(X, 0, Z) → U1_GGA(X, Z, fail_in_)
DIV_IN_GGA(X, 0, Z) → FAIL_IN_
FAIL_IN_ → U6_1(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(0, Y, Z) → U2_GGA(Y, Z, =1_in_ag(Z, 0))
DIV_IN_GGA(0, Y, Z) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U5_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U4_GGA(X, Y, Z, div_in_gga(U, Y, Z))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g(x1)
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2)
U2_gga(x1, x2, x3)  =  U2_gga(x1, x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
FAIL_IN_  =  FAIL_IN_
U6_1(x1)  =  U6_1(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U2_GGA(x1, x2, x3)  =  U2_GGA(x1, x3)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

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

(46) Obligation:

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

DIV_IN_GGA(X, 0, Z) → U1_GGA(X, Z, fail_in_)
DIV_IN_GGA(X, 0, Z) → FAIL_IN_
FAIL_IN_ → U6_1(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(0, Y, Z) → U2_GGA(Y, Z, =1_in_ag(Z, 0))
DIV_IN_GGA(0, Y, Z) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, s(Z)) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(s(X), s(Y), Z) → U5_GGA(X, Y, Z, minus_in_gga(X, Y, Z))
MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U4_GGA(X, Y, Z, div_in_gga(U, Y, Z))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g(x1)
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2)
U2_gga(x1, x2, x3)  =  U2_gga(x1, x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
FAIL_IN_  =  FAIL_IN_
U6_1(x1)  =  U6_1(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U2_GGA(x1, x2, x3)  =  U2_GGA(x1, x3)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)

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

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 9 less nodes.

(48) Complex Obligation (AND)

(49) Obligation:

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

MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g(x1)
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2)
U2_gga(x1, x2, x3)  =  U2_gga(x1, x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)

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

(50) UsableRulesProof (EQUIVALENT transformation)

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

(51) Obligation:

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

MINUS_IN_GGA(s(X), s(Y), Z) → MINUS_IN_GGA(X, Y, Z)

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

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

(52) PiDPToQDPProof (SOUND transformation)

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

(53) Obligation:

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

MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)

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

(54) 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:

  • MINUS_IN_GGA(s(X), s(Y)) → MINUS_IN_GGA(X, Y)
    The graph contains the following edges 1 > 1, 2 > 2

(55) TRUE

(56) Obligation:

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

DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

div_in_gga(X, 0, Z) → U1_gga(X, Z, fail_in_)
fail_in_ → U6_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U6_(fail_out_g(b)) → fail_out_
U1_gga(X, Z, fail_out_) → div_out_gga(X, 0, Z)
div_in_gga(0, Y, Z) → U2_gga(Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U2_gga(Y, Z, =1_out_ag(Z, 0)) → div_out_gga(0, Y, Z)
div_in_gga(X, Y, s(Z)) → U3_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)
U3_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U4_gga(X, Y, Z, div_in_gga(U, Y, Z))
U4_gga(X, Y, Z, div_out_gga(U, Y, Z)) → div_out_gga(X, Y, s(Z))

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
0  =  0
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
fail_in_  =  fail_in_
U6_(x1)  =  U6_(x1)
fail_in_g(x1)  =  fail_in_g(x1)
a  =  a
fail_out_g(x1)  =  fail_out_g(x1)
b  =  b
fail_out_  =  fail_out_
div_out_gga(x1, x2, x3)  =  div_out_gga(x1, x2)
U2_gga(x1, x2, x3)  =  U2_gga(x1, x3)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(57) UsableRulesProof (EQUIVALENT transformation)

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

(58) Obligation:

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

DIV_IN_GGA(X, Y, s(Z)) → U3_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U3_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, Z)

The TRS R consists of the following rules:

minus_in_gga(0, Y, 0) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0, X) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y), Z) → U5_gga(X, Y, Z, minus_in_gga(X, Y, Z))
U5_gga(X, Y, Z, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The argument filtering Pi contains the following mapping:
0  =  0
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)

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

(59) PiDPToQDPProof (SOUND transformation)

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

(60) Obligation:

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

DIV_IN_GGA(X, Y) → U3_GGA(X, Y, minus_in_gga(X, Y))
U3_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(61) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(X, Y) → U3_GGA(X, Y, minus_in_gga(X, Y)) at position [2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
DIV_IN_GGA(x0, 0) → U3_GGA(x0, 0, minus_out_gga(x0, 0, x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x0), s(x1), U5_gga(x0, x1, minus_in_gga(x0, x1)))

(62) Obligation:

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

U3_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
DIV_IN_GGA(x0, 0) → U3_GGA(x0, 0, minus_out_gga(x0, 0, x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x0), s(x1), U5_gga(x0, x1, minus_in_gga(x0, x1)))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(63) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U3_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y) we obtained the following new rules [LPAR04]:

U3_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
U3_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)
U3_GGA(s(z0), s(z1), minus_out_gga(s(z0), s(z1), x2)) → DIV_IN_GGA(x2, s(z1))

(64) Obligation:

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

DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
DIV_IN_GGA(x0, 0) → U3_GGA(x0, 0, minus_out_gga(x0, 0, x0))
DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x0), s(x1), U5_gga(x0, x1, minus_in_gga(x0, x1)))
U3_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
U3_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)
U3_GGA(s(z0), s(z1), minus_out_gga(s(z0), s(z1), x2)) → DIV_IN_GGA(x2, s(z1))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(65) DependencyGraphProof (EQUIVALENT transformation)

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

(66) Complex Obligation (AND)

(67) Obligation:

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

U3_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
U3_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)
DIV_IN_GGA(x0, 0) → U3_GGA(x0, 0, minus_out_gga(x0, 0, x0))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(68) 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.

(69) Obligation:

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

U3_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
U3_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)
DIV_IN_GGA(x0, 0) → U3_GGA(x0, 0, minus_out_gga(x0, 0, x0))

R is empty.
The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(70) 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].

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

(71) Obligation:

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

U3_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
U3_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)
DIV_IN_GGA(x0, 0) → U3_GGA(x0, 0, minus_out_gga(x0, 0, x0))

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

(72) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = DIV_IN_GGA(0, x0) evaluates to t =DIV_IN_GGA(0, x0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:
  • Semiunifier: [ ]
  • Matcher: [ ]



Rewriting sequence

DIV_IN_GGA(0, x0) → U3_GGA(0, x0, minus_out_gga(0, x0, 0))
with rule DIV_IN_GGA(0, x0') → U3_GGA(0, x0', minus_out_gga(0, x0', 0)) at position [] and matcher [x0' / x0]

U3_GGA(0, x0, minus_out_gga(0, x0, 0)) → DIV_IN_GGA(0, x0)
with rule U3_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.



(73) FALSE

(74) Obligation:

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

DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x0), s(x1), U5_gga(x0, x1, minus_in_gga(x0, x1)))
U3_GGA(s(z0), s(z1), minus_out_gga(s(z0), s(z1), x2)) → DIV_IN_GGA(x2, s(z1))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(75) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U3_GGA(s(z0), s(z1), minus_out_gga(s(z0), s(z1), x2)) → DIV_IN_GGA(x2, s(z1)) we obtained the following new rules [LPAR04]:

U3_GGA(s(x0), s(x1), minus_out_gga(s(x0), s(x1), s(y_0))) → DIV_IN_GGA(s(y_0), s(x1))

(76) Obligation:

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

DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x0), s(x1), U5_gga(x0, x1, minus_in_gga(x0, x1)))
U3_GGA(s(x0), s(x1), minus_out_gga(s(x0), s(x1), s(y_0))) → DIV_IN_GGA(s(y_0), s(x1))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(77) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


DIV_IN_GGA(s(x0), s(x1)) → U3_GGA(s(x0), s(x1), U5_gga(x0, x1, minus_in_gga(x0, x1)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(DIV_IN_GGA(x1, x2)) = 1 + x1   
POL(U3_GGA(x1, x2, x3)) = 1 + x3   
POL(U5_gga(x1, x2, x3)) = x3   
POL(minus_in_gga(x1, x2)) = x1   
POL(minus_out_gga(x1, x2, x3)) = x3   
POL(s(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

(78) Obligation:

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

U3_GGA(s(x0), s(x1), minus_out_gga(s(x0), s(x1), s(y_0))) → DIV_IN_GGA(s(y_0), s(x1))

The TRS R consists of the following rules:

minus_in_gga(0, Y) → minus_out_gga(0, Y, 0)
minus_in_gga(X, 0) → minus_out_gga(X, 0, X)
minus_in_gga(s(X), s(Y)) → U5_gga(X, Y, minus_in_gga(X, Y))
U5_gga(X, Y, minus_out_gga(X, Y, Z)) → minus_out_gga(s(X), s(Y), Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U5_gga(x0, x1, x2)

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

(79) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(80) TRUE