(0) Obligation:

Clauses:

minus(X, Y, Z) :- ','(f(X, 0), ','(!, =(Z, 0))).
f(X, Y) :- ','(!, =(X, Y)).
f(X, Y) :- f(X, Y).
=(X, X).

Queries:

minus(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

minus1(0, T5, 0).

Queries:

minus1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

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

minus1_in_gga(0, T5, 0) → minus1_out_gga(0, T5, 0)

The argument filtering Pi contains the following mapping:
minus1_in_gga(x1, x2, x3)  =  minus1_in_gga(x1, x2)
0  =  0
minus1_out_gga(x1, x2, x3)  =  minus1_out_gga(x1, x2, x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

minus1_in_gga(0, T5, 0) → minus1_out_gga(0, T5, 0)

The argument filtering Pi contains the following mapping:
minus1_in_gga(x1, x2, x3)  =  minus1_in_gga(x1, x2)
0  =  0
minus1_out_gga(x1, x2, x3)  =  minus1_out_gga(x1, x2, x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

minus1_in_gga(0, T5, 0) → minus1_out_gga(0, T5, 0)

The argument filtering Pi contains the following mapping:
minus1_in_gga(x1, x2, x3)  =  minus1_in_gga(x1, x2)
0  =  0
minus1_out_gga(x1, x2, x3)  =  minus1_out_gga(x1, x2, x3)

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

(6) Obligation:

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

minus1_in_gga(0, T5, 0) → minus1_out_gga(0, T5, 0)

The argument filtering Pi contains the following mapping:
minus1_in_gga(x1, x2, x3)  =  minus1_in_gga(x1, x2)
0  =  0
minus1_out_gga(x1, x2, x3)  =  minus1_out_gga(x1, x2, x3)

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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE