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

(0) Obligation:

Clauses:

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

Queries:

div(g,g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

Clauses:

minus(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
minus(X, Y, Z) :- ','(=1(Y, 0), ','(!, =1(Z, X))).
minus(X, Y, Z) :- ','(=1(X, s(A)), ','(=1(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=1(Y, 0), ','(!, fail)).
div(X, Y, Z) :- ','(=1(X, 0), ','(!, =1(Z, 0))).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =1(Z, s(V)))).
=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:

minus(X, Y, Z) :- ','(=1(X, 0), =1(Z, 0)).
minus(X, Y, Z) :- ','(=1(Y, 0), =1(Z, X)).
minus(X, Y, Z) :- ','(=1(X, s(A)), ','(=1(Y, s(B)), minus(A, B, Z))).
div(X, Y, Z) :- ','(=1(Y, 0), fail).
div(X, Y, Z) :- ','(=1(X, 0), =1(Z, 0)).
div(X, Y, Z) :- ','(minus(X, Y, U), ','(div(U, Y, V), =1(Z, s(V)))).
=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, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa

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, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa

(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, Y, Z) → U8_GGA(X, Y, Z, =1_in_gg(Y, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U9_GGA(X, Y, Z, fail_in_)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → FAIL_IN_
FAIL_IN_ → U15_¹(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(X, Y, Z) → U10_GGA(X, Y, Z, =1_in_gg(X, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → U11_GGA(X, Y, Z, =1_in_ag(Z, 0))
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, Z) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, =1_in_gg(X, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → U2_GGA(X, Y, Z, =1_in_ag(Z, 0))
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
MINUS_IN_GGA(X, Y, Z) → U3_GGA(X, Y, Z, =1_in_gg(Y, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U4_GGA(X, Y, Z, =1_in_ag(Z, X))
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → =1_IN_AG(Z, X)
MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GA(X, s(A))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → =1_IN_GA(Y, s(B))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_GGA(X, Y, Z, minus_in_gga(A, B, Z))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U13_GGA(X, Y, Z, div_in_gga(U, Y, V))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → U14_GGA(X, Y, Z, =1_in_aa(Z, s(V)))
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → =1_IN_AA(Z, s(V))

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
=1_IN_GG(x1, x2)  =  =1_IN_GG(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
FAIL_IN_  =  FAIL_IN_
U15_¹(x1)  =  U15_¹(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
=1_IN_GA(x1, x2)  =  =1_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x4, x5)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
=1_IN_AA(x1, x2)  =  =1_IN_AA

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, Y, Z) → U8_GGA(X, Y, Z, =1_in_gg(Y, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U9_GGA(X, Y, Z, fail_in_)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → FAIL_IN_
FAIL_IN_ → U15_¹(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(X, Y, Z) → U10_GGA(X, Y, Z, =1_in_gg(X, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → U11_GGA(X, Y, Z, =1_in_ag(Z, 0))
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, Z) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, =1_in_gg(X, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → U2_GGA(X, Y, Z, =1_in_ag(Z, 0))
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
MINUS_IN_GGA(X, Y, Z) → U3_GGA(X, Y, Z, =1_in_gg(Y, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U4_GGA(X, Y, Z, =1_in_ag(Z, X))
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → =1_IN_AG(Z, X)
MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GA(X, s(A))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → =1_IN_GA(Y, s(B))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_GGA(X, Y, Z, minus_in_gga(A, B, Z))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U13_GGA(X, Y, Z, div_in_gga(U, Y, V))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → U14_GGA(X, Y, Z, =1_in_aa(Z, s(V)))
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → =1_IN_AA(Z, s(V))

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x1, x2, x4)
=1_IN_GG(x1, x2)  =  =1_IN_GG(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x1, x2, x4)
FAIL_IN_  =  FAIL_IN_
U15_¹(x1)  =  U15_¹(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x1, x2, x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
=1_IN_GA(x1, x2)  =  =1_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x4, x5)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x1, x2, x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x1, x2, x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x1, x2, x4)
=1_IN_AA(x1, x2)  =  =1_IN_AA

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 25 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x4, x5)

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(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)

The TRS R consists of the following rules:

=1_in_ga(X, X) → =1_out_ga(X, X)

The argument filtering Pi contains the following mapping:
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x4, x5)

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(X, Y) → U5_GGA(X, Y, =1_in_ga(X))
U5_GGA(X, Y, =1_out_ga(X, s(A))) → U6_GGA(X, Y, A, =1_in_ga(Y))
U6_GGA(X, Y, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B)

The TRS R consists of the following rules:

=1_in_ga(X) → =1_out_ga(X, X)

The set Q consists of the following terms:

=1_in_ga(x0)

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MINUS_IN_GGA(X, Y) → U5_GGA(X, Y, =1_in_ga(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(=1_in_ga(x₁)) = x₁   
POL(=1_out_ga(x₁, x₂)) = x₂   
POL(MINUS_IN_GGA(x₁, x₂)) = 1 + x₂   
POL(U5_GGA(x₁, x₂, x₃)) = x₂   
POL(U6_GGA(x₁, x₂, x₃, x₄)) = x₄   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

=1_in_ga(X) → =1_out_ga(X, X)

(17) Obligation:

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

U5_GGA(X, Y, =1_out_ga(X, s(A))) → U6_GGA(X, Y, A, =1_in_ga(Y))
U6_GGA(X, Y, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B)

The TRS R consists of the following rules:

=1_in_ga(X) → =1_out_ga(X, X)

The set Q consists of the following terms:

=1_in_ga(x0)

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

(18) DependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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

DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x1, x2, x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x1, x2, x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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)
U10_gga(x1, x2, x3, x4)  =  U10_gga(x1, x2, x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x1, x2, x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x1, x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x1, x2, x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x1, x2, x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)

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

(21) UsableRulesProof (EQUIVALENT transformation)

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

(22) Obligation:

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

DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)

The TRS R consists of the following rules:

minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X, X) → =1_out_ga(X, X)
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
=1_in_ag(X, X) → =1_out_ag(X, X)
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg(x1, x2)
0  =  0
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1, x2)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x2, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x1, x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x1, x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x1, x2, x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x1, x2, x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x1, x2, x4)

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

(23) PiDPToQDPProof (SOUND transformation)

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

(24) Obligation:

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

DIV_IN_GGA(X, Y) → U12_GGA(X, Y, minus_in_gga(X, Y))
U12_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(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(25) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(X, Y) → U12_GGA(X, Y, minus_in_gga(X, Y)) at position [2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U1_gga(x0, x1, =1_in_gg(x0, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_in_ga(x0)))

(26) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U1_gga(x0, x1, =1_in_gg(x0, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_in_ga(x0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(27) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_in_ga(x0))) at position [2,2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))

(28) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U1_gga(x0, x1, =1_in_gg(x0, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(29) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U1_gga(x0, x1, =1_in_gg(x0, 0))) at position [2,2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U1_gga(0, y1, =1_out_gg(0, 0)))

(30) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U1_gga(0, y1, =1_out_gg(0, 0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(31) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U1_gga(0, y1, =1_out_gg(0, 0))) at position [2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U2_gga(0, y1, =1_in_ag(0)))

(32) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U2_gga(0, y1, =1_in_ag(0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(33) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U2_gga(0, y1, =1_in_ag(0))) at position [2,2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U2_gga(0, y1, =1_out_ag(0, 0)))

(34) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U2_gga(0, y1, =1_out_ag(0, 0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(35) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(0, y1) → U12_GGA(0, y1, U2_gga(0, y1, =1_out_ag(0, 0))) at position [2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))

(36) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(37) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U3_gga(x0, x1, =1_in_gg(x1, 0))) at position [2,2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U3_gga(y0, 0, =1_out_gg(0, 0)))

(38) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U3_gga(y0, 0, =1_out_gg(0, 0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(39) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U3_gga(y0, 0, =1_out_gg(0, 0))) at position [2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U4_gga(y0, 0, =1_in_ag(y0)))

(40) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U4_gga(y0, 0, =1_in_ag(y0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(41) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U4_gga(y0, 0, =1_in_ag(y0))) at position [2,2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U4_gga(y0, 0, =1_out_ag(y0, y0)))

(42) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U4_gga(y0, 0, =1_out_ag(y0, y0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(43) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, U4_gga(y0, 0, =1_out_ag(y0, y0))) at position [2] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))

(44) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

DIV_IN_GGA(x0, x1) → U12_GGA(x0, x1, U5_gga(x0, x1, =1_out_ga(x0, x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(=1_in_ag(x₁)) = x₁   
POL(=1_in_ga(x₁)) = x₁   
POL(=1_in_gg(x₁, x₂)) = 1 + x₂   
POL(=1_out_ag(x₁, x₂)) = x₁   
POL(=1_out_ga(x₁, x₂)) = x₂   
POL(=1_out_gg(x₁, x₂)) = 1   
POL(DIV_IN_GGA(x₁, x₂)) = 1 + x₁   
POL(U12_GGA(x₁, x₂, x₃)) = x₃   
POL(U1_gga(x₁, x₂, x₃)) = 1   
POL(U2_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(U3_gga(x₁, x₂, x₃)) = x₁ + x₃   
POL(U4_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(U5_gga(x₁, x₂, x₃)) = x₃   
POL(U6_gga(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U7_gga(x₁, x₂, x₃)) = x₃   
POL(minus_in_gga(x₁, x₂)) = 1 + x₁   
POL(minus_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
=1_in_gg(X, X) → =1_out_gg(X, X)
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
=1_in_ag(X) → =1_out_ag(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)

(46) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(X, Y, =1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, Y, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(X, Y, =1_in_ga(X))
U1_gga(X, Y, =1_out_gg(X, 0)) → U2_gga(X, Y, =1_in_ag(0))
U3_gga(X, Y, =1_out_gg(Y, 0)) → U4_gga(X, Y, =1_in_ag(X))
U5_gga(X, Y, =1_out_ga(X, s(A))) → U6_gga(X, Y, A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X) → =1_out_ga(X, X)
U6_gga(X, Y, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X, X)
U7_gga(X, Y, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(47) 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.

(48) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))

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

minus_in_gga(x0, x1)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

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

(49) 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)
U1_gga(x0, x1, x2)
U3_gga(x0, x1, x2)
U5_gga(x0, x1, x2)
=1_in_gg(x0, x1)
U2_gga(x0, x1, x2)
U4_gga(x0, x1, x2)
=1_in_ga(x0)
U6_gga(x0, x1, x2, x3)
=1_in_ag(x0)
U7_gga(x0, x1, x2)

(50) Obligation:

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

U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))

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

(51) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y) we obtained the following new rules [LPAR04]:

U12_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
U12_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)

(52) Obligation:

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

DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))
U12_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
U12_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)

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

(53) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GGA(X, Y, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y) we obtained the following new rules [LPAR04]:

U12_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
U12_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)

(54) Obligation:

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

DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))
DIV_IN_GGA(y0, 0) → U12_GGA(y0, 0, minus_out_gga(y0, 0, y0))
U12_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
U12_GGA(z0, 0, minus_out_gga(z0, 0, z0)) → DIV_IN_GGA(z0, 0)

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

(55) 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 = U12_GGA(0, z0, minus_out_gga(0, z0, 0)) evaluates to t =U12_GGA(0, z0, minus_out_gga(0, z0, 0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U12_GGA(0, z0, minus_out_gga(0, z0, 0)) → DIV_IN_GGA(0, z0)
with rule U12_GGA(0, z0', minus_out_gga(0, z0', 0)) → DIV_IN_GGA(0, z0') at position [] and matcher [z0' / z0]

DIV_IN_GGA(0, z0) → U12_GGA(0, z0, minus_out_gga(0, z0, 0))
with rule DIV_IN_GGA(0, y1) → U12_GGA(0, y1, minus_out_gga(0, y1, 0))

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.

(56) FALSE

(57) 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, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(58) Obligation:

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

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa

(59) 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, Y, Z) → U8_GGA(X, Y, Z, =1_in_gg(Y, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U9_GGA(X, Y, Z, fail_in_)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → FAIL_IN_
FAIL_IN_ → U15_¹(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(X, Y, Z) → U10_GGA(X, Y, Z, =1_in_gg(X, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → U11_GGA(X, Y, Z, =1_in_ag(Z, 0))
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, Z) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, =1_in_gg(X, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → U2_GGA(X, Y, Z, =1_in_ag(Z, 0))
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
MINUS_IN_GGA(X, Y, Z) → U3_GGA(X, Y, Z, =1_in_gg(Y, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U4_GGA(X, Y, Z, =1_in_ag(Z, X))
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → =1_IN_AG(Z, X)
MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GA(X, s(A))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → =1_IN_GA(Y, s(B))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_GGA(X, Y, Z, minus_in_gga(A, B, Z))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U13_GGA(X, Y, Z, div_in_gga(U, Y, V))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → U14_GGA(X, Y, Z, =1_in_aa(Z, s(V)))
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → =1_IN_AA(Z, s(V))

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x4)
=1_IN_GG(x1, x2)  =  =1_IN_GG(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x4)
FAIL_IN_  =  FAIL_IN_
U15_¹(x1)  =  U15_¹(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x4)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x2, x4)
=1_IN_GA(x1, x2)  =  =1_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x4, x5)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x4)
=1_IN_AA(x1, x2)  =  =1_IN_AA

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

(60) Obligation:

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

DIV_IN_GGA(X, Y, Z) → U8_GGA(X, Y, Z, =1_in_gg(Y, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U9_GGA(X, Y, Z, fail_in_)
U8_GGA(X, Y, Z, =1_out_gg(Y, 0)) → FAIL_IN_
FAIL_IN_ → U15_¹(fail_in_g(b))
FAIL_IN_ → FAIL_IN_G(b)
DIV_IN_GGA(X, Y, Z) → U10_GGA(X, Y, Z, =1_in_gg(X, 0))
DIV_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → U11_GGA(X, Y, Z, =1_in_ag(Z, 0))
U10_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
DIV_IN_GGA(X, Y, Z) → MINUS_IN_GGA(X, Y, U)
MINUS_IN_GGA(X, Y, Z) → U1_GGA(X, Y, Z, =1_in_gg(X, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(X, 0)
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → U2_GGA(X, Y, Z, =1_in_ag(Z, 0))
U1_GGA(X, Y, Z, =1_out_gg(X, 0)) → =1_IN_AG(Z, 0)
MINUS_IN_GGA(X, Y, Z) → U3_GGA(X, Y, Z, =1_in_gg(Y, 0))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GG(Y, 0)
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → U4_GGA(X, Y, Z, =1_in_ag(Z, X))
U3_GGA(X, Y, Z, =1_out_gg(Y, 0)) → =1_IN_AG(Z, X)
MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
MINUS_IN_GGA(X, Y, Z) → =1_IN_GA(X, s(A))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → =1_IN_GA(Y, s(B))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_GGA(X, Y, Z, minus_in_gga(A, B, Z))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → U13_GGA(X, Y, Z, div_in_gga(U, Y, V))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → U14_GGA(X, Y, Z, =1_in_aa(Z, s(V)))
U13_GGA(X, Y, Z, div_out_gga(U, Y, V)) → =1_IN_AA(Z, s(V))

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4)  =  U8_GGA(x4)
=1_IN_GG(x1, x2)  =  =1_IN_GG(x1, x2)
U9_GGA(x1, x2, x3, x4)  =  U9_GGA(x4)
FAIL_IN_  =  FAIL_IN_
U15_¹(x1)  =  U15_¹(x1)
FAIL_IN_G(x1)  =  FAIL_IN_G(x1)
U10_GGA(x1, x2, x3, x4)  =  U10_GGA(x4)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x4)
=1_IN_AG(x1, x2)  =  =1_IN_AG(x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x2, x4)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x4)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x4)
U3_GGA(x1, x2, x3, x4)  =  U3_GGA(x1, x4)
U4_GGA(x1, x2, x3, x4)  =  U4_GGA(x4)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x2, x4)
=1_IN_GA(x1, x2)  =  =1_IN_GA(x1)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x4, x5)
U7_GGA(x1, x2, x3, x4)  =  U7_GGA(x4)
U13_GGA(x1, x2, x3, x4)  =  U13_GGA(x4)
U14_GGA(x1, x2, x3, x4)  =  U14_GGA(x4)
=1_IN_AA(x1, x2)  =  =1_IN_AA

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

(61) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 25 less nodes.

(62) Complex Obligation (AND)

(63) Obligation:

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

MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x2, x4)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x4, x5)

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

(64) UsableRulesProof (EQUIVALENT transformation)

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

(65) Obligation:

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

MINUS_IN_GGA(X, Y, Z) → U5_GGA(X, Y, Z, =1_in_ga(X, s(A)))
U5_GGA(X, Y, Z, =1_out_ga(X, s(A))) → U6_GGA(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_GGA(X, Y, Z, A, =1_out_ga(Y, s(B))) → MINUS_IN_GGA(A, B, Z)

The TRS R consists of the following rules:

=1_in_ga(X, X) → =1_out_ga(X, X)

The argument filtering Pi contains the following mapping:
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
MINUS_IN_GGA(x1, x2, x3)  =  MINUS_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4)  =  U5_GGA(x2, x4)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x4, x5)

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

(66) PiDPToQDPProof (SOUND transformation)

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

(67) Obligation:

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

MINUS_IN_GGA(X, Y) → U5_GGA(Y, =1_in_ga(X))
U5_GGA(Y, =1_out_ga(s(A))) → U6_GGA(A, =1_in_ga(Y))
U6_GGA(A, =1_out_ga(s(B))) → MINUS_IN_GGA(A, B)

The TRS R consists of the following rules:

=1_in_ga(X) → =1_out_ga(X)

The set Q consists of the following terms:

=1_in_ga(x0)

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

(68) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

U5_GGA(Y, =1_out_ga(s(A))) → U6_GGA(A, =1_in_ga(Y))
U6_GGA(A, =1_out_ga(s(B))) → MINUS_IN_GGA(A, B)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(=1_in_ga(x₁)) = 1 + x₁   
POL(=1_out_ga(x₁)) = 1 + x₁   
POL(MINUS_IN_GGA(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(U5_GGA(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(U6_GGA(x₁, x₂)) = 1 + x₁ + x₂   
POL(s(x₁)) = 1 + 2·x₁   

(69) Obligation:

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

MINUS_IN_GGA(X, Y) → U5_GGA(Y, =1_in_ga(X))

The TRS R consists of the following rules:

=1_in_ga(X) → =1_out_ga(X)

The set Q consists of the following terms:

=1_in_ga(x0)

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

(70) DependencyGraphProof (EQUIVALENT transformation)

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

(71) TRUE

(72) Obligation:

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

DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)

The TRS R consists of the following rules:

div_in_gga(X, Y, Z) → U8_gga(X, Y, Z, =1_in_gg(Y, 0))
=1_in_gg(X, X) → =1_out_gg(X, X)
U8_gga(X, Y, Z, =1_out_gg(Y, 0)) → U9_gga(X, Y, Z, fail_in_)
fail_in_ → U15_(fail_in_g(b))
fail_in_g(a) → fail_out_g(a)
U15_(fail_out_g(b)) → fail_out_
U9_gga(X, Y, Z, fail_out_) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U10_gga(X, Y, Z, =1_in_gg(X, 0))
U10_gga(X, Y, Z, =1_out_gg(X, 0)) → U11_gga(X, Y, Z, =1_in_ag(Z, 0))
=1_in_ag(X, X) → =1_out_ag(X, X)
U11_gga(X, Y, Z, =1_out_ag(Z, 0)) → div_out_gga(X, Y, Z)
div_in_gga(X, Y, Z) → U12_gga(X, Y, Z, minus_in_gga(X, Y, U))
minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
=1_in_ga(X, X) → =1_out_ga(X, X)
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)
U12_gga(X, Y, Z, minus_out_gga(X, Y, U)) → U13_gga(X, Y, Z, div_in_gga(U, Y, V))
U13_gga(X, Y, Z, div_out_gga(U, Y, V)) → U14_gga(X, Y, Z, =1_in_aa(Z, s(V)))
=1_in_aa(X, X) → =1_out_aa(X, X)
U14_gga(X, Y, Z, =1_out_aa(Z, s(V))) → div_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
div_in_gga(x1, x2, x3)  =  div_in_gga(x1, x2)
U8_gga(x1, x2, x3, x4)  =  U8_gga(x4)
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
U9_gga(x1, x2, x3, x4)  =  U9_gga(x4)
fail_in_  =  fail_in_
U15_(x1)  =  U15_(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
U10_gga(x1, x2, x3, x4)  =  U10_gga(x4)
U11_gga(x1, x2, x3, x4)  =  U11_gga(x4)
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
U12_gga(x1, x2, x3, x4)  =  U12_gga(x2, x4)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
U13_gga(x1, x2, x3, x4)  =  U13_gga(x4)
U14_gga(x1, x2, x3, x4)  =  U14_gga(x4)
=1_in_aa(x1, x2)  =  =1_in_aa
=1_out_aa(x1, x2)  =  =1_out_aa
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x2, x4)

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

(73) UsableRulesProof (EQUIVALENT transformation)

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

(74) Obligation:

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

DIV_IN_GGA(X, Y, Z) → U12_GGA(X, Y, Z, minus_in_gga(X, Y, U))
U12_GGA(X, Y, Z, minus_out_gga(X, Y, U)) → DIV_IN_GGA(U, Y, V)

The TRS R consists of the following rules:

minus_in_gga(X, Y, Z) → U1_gga(X, Y, Z, =1_in_gg(X, 0))
minus_in_gga(X, Y, Z) → U3_gga(X, Y, Z, =1_in_gg(Y, 0))
minus_in_gga(X, Y, Z) → U5_gga(X, Y, Z, =1_in_ga(X, s(A)))
U1_gga(X, Y, Z, =1_out_gg(X, 0)) → U2_gga(X, Y, Z, =1_in_ag(Z, 0))
U3_gga(X, Y, Z, =1_out_gg(Y, 0)) → U4_gga(X, Y, Z, =1_in_ag(Z, X))
U5_gga(X, Y, Z, =1_out_ga(X, s(A))) → U6_gga(X, Y, Z, A, =1_in_ga(Y, s(B)))
=1_in_gg(X, X) → =1_out_gg(X, X)
U2_gga(X, Y, Z, =1_out_ag(Z, 0)) → minus_out_gga(X, Y, Z)
U4_gga(X, Y, Z, =1_out_ag(Z, X)) → minus_out_gga(X, Y, Z)
=1_in_ga(X, X) → =1_out_ga(X, X)
U6_gga(X, Y, Z, A, =1_out_ga(Y, s(B))) → U7_gga(X, Y, Z, minus_in_gga(A, B, Z))
=1_in_ag(X, X) → =1_out_ag(X, X)
U7_gga(X, Y, Z, minus_out_gga(A, B, Z)) → minus_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
=1_in_gg(x1, x2)  =  =1_in_gg(x1, x2)
=1_out_gg(x1, x2)  =  =1_out_gg
0  =  0
=1_in_ag(x1, x2)  =  =1_in_ag(x2)
=1_out_ag(x1, x2)  =  =1_out_ag(x1)
minus_in_gga(x1, x2, x3)  =  minus_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x4)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x4)
minus_out_gga(x1, x2, x3)  =  minus_out_gga(x3)
U3_gga(x1, x2, x3, x4)  =  U3_gga(x1, x4)
U4_gga(x1, x2, x3, x4)  =  U4_gga(x4)
U5_gga(x1, x2, x3, x4)  =  U5_gga(x2, x4)
=1_in_ga(x1, x2)  =  =1_in_ga(x1)
=1_out_ga(x1, x2)  =  =1_out_ga(x2)
s(x1)  =  s(x1)
U6_gga(x1, x2, x3, x4, x5)  =  U6_gga(x4, x5)
U7_gga(x1, x2, x3, x4)  =  U7_gga(x4)
DIV_IN_GGA(x1, x2, x3)  =  DIV_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4)  =  U12_GGA(x2, x4)

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

(75) PiDPToQDPProof (SOUND transformation)

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

(76) Obligation:

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

DIV_IN_GGA(X, Y) → U12_GGA(Y, minus_in_gga(X, Y))
U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(77) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(X, Y) → U12_GGA(Y, minus_in_gga(X, Y)) at position [1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(x0, x1) → U12_GGA(x1, U1_gga(=1_in_gg(x0, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_in_ga(x0)))

(78) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U1_gga(=1_in_gg(x0, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_in_ga(x0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(79) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_in_ga(x0))) at position [1,1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))

(80) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U1_gga(=1_in_gg(x0, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(81) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(x0, x1) → U12_GGA(x1, U1_gga(=1_in_gg(x0, 0))) at position [1,0] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(y1, U1_gga(=1_out_gg))

(82) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, U1_gga(=1_out_gg))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(83) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(0, y1) → U12_GGA(y1, U1_gga(=1_out_gg)) at position [1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(y1, U2_gga(=1_in_ag(0)))

(84) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, U2_gga(=1_in_ag(0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(85) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(0, y1) → U12_GGA(y1, U2_gga(=1_in_ag(0))) at position [1,0] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(y1, U2_gga(=1_out_ag(0)))

(86) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, U2_gga(=1_out_ag(0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(87) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(0, y1) → U12_GGA(y1, U2_gga(=1_out_ag(0))) at position [1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))

(88) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0)))
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(89) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV_IN_GGA(x0, x1) → U12_GGA(x1, U3_gga(x0, =1_in_gg(x1, 0))) at position [1,1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(0, U3_gga(y0, =1_out_gg))

(90) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, U3_gga(y0, =1_out_gg))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(91) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(y0, 0) → U12_GGA(0, U3_gga(y0, =1_out_gg)) at position [1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(0, U4_gga(=1_in_ag(y0)))

(92) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, U4_gga(=1_in_ag(y0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(93) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(y0, 0) → U12_GGA(0, U4_gga(=1_in_ag(y0))) at position [1,0] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(0, U4_gga(=1_out_ag(y0)))

(94) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, U4_gga(=1_out_ag(y0)))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(95) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule DIV_IN_GGA(y0, 0) → U12_GGA(0, U4_gga(=1_out_ag(y0))) at position [1] we obtained the following new rules [LPAR04]:

DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))

(96) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

DIV_IN_GGA(x0, x1) → U12_GGA(x1, U5_gga(x1, =1_out_ga(x0)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0   
POL(=1_in_ag(x₁)) = x₁   
POL(=1_in_ga(x₁)) = x₁   
POL(=1_in_gg(x₁, x₂)) = 0   
POL(=1_out_ag(x₁)) = x₁   
POL(=1_out_ga(x₁)) = x₁   
POL(=1_out_gg) = 0   
POL(DIV_IN_GGA(x₁, x₂)) = 1 + x₁   
POL(U12_GGA(x₁, x₂)) = x₂   
POL(U1_gga(x₁)) = 1   
POL(U2_gga(x₁)) = 1 + x₁   
POL(U3_gga(x₁, x₂)) = 1 + x₁   
POL(U4_gga(x₁)) = 1 + x₁   
POL(U5_gga(x₁, x₂)) = x₂   
POL(U6_gga(x₁, x₂)) = 1 + x₁   
POL(U7_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:

U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
=1_in_ag(X) → =1_out_ag(X)
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)

(98) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))

The TRS R consists of the following rules:

minus_in_gga(X, Y) → U1_gga(=1_in_gg(X, 0))
minus_in_gga(X, Y) → U3_gga(X, =1_in_gg(Y, 0))
minus_in_gga(X, Y) → U5_gga(Y, =1_in_ga(X))
U1_gga(=1_out_gg) → U2_gga(=1_in_ag(0))
U3_gga(X, =1_out_gg) → U4_gga(=1_in_ag(X))
U5_gga(Y, =1_out_ga(s(A))) → U6_gga(A, =1_in_ga(Y))
=1_in_gg(X, X) → =1_out_gg
U2_gga(=1_out_ag(Z)) → minus_out_gga(Z)
U4_gga(=1_out_ag(Z)) → minus_out_gga(Z)
=1_in_ga(X) → =1_out_ga(X)
U6_gga(A, =1_out_ga(s(B))) → U7_gga(minus_in_gga(A, B))
=1_in_ag(X) → =1_out_ag(X)
U7_gga(minus_out_gga(Z)) → minus_out_gga(Z)

The set Q consists of the following terms:

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(99) 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.

(100) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))

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

minus_in_gga(x0, x1)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

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

(101) 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)
U1_gga(x0)
U3_gga(x0, x1)
U5_gga(x0, x1)
=1_in_gg(x0, x1)
U2_gga(x0)
U4_gga(x0)
=1_in_ga(x0)
U6_gga(x0, x1)
=1_in_ag(x0)
U7_gga(x0)

(102) Obligation:

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

U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y)
DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))

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

(103) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y) we obtained the following new rules [LPAR04]:

U12_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U12_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)

(104) Obligation:

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

DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))
U12_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U12_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)

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

(105) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U12_GGA(Y, minus_out_gga(U)) → DIV_IN_GGA(U, Y) we obtained the following new rules [LPAR04]:

U12_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U12_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)

(106) Obligation:

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

DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))
DIV_IN_GGA(y0, 0) → U12_GGA(0, minus_out_gga(y0))
U12_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
U12_GGA(0, minus_out_gga(z0)) → DIV_IN_GGA(z0, 0)

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

(107) 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 = U12_GGA(z0, minus_out_gga(0)) evaluates to t =U12_GGA(z0, minus_out_gga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

U12_GGA(z0, minus_out_gga(0)) → DIV_IN_GGA(0, z0)
with rule U12_GGA(z0', minus_out_gga(0)) → DIV_IN_GGA(0, z0') at position [] and matcher [z0' / z0]

DIV_IN_GGA(0, z0) → U12_GGA(z0, minus_out_gga(0))
with rule DIV_IN_GGA(0, y1) → U12_GGA(y1, minus_out_gga(0))

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.

(108) FALSE