(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).

Query: minus(g,g,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

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

Query: minus(g,g,a)

(3) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

(4) Obligation:

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

minusA_in_gga(0, T8, 0) → minusA_out_gga(0, T8, 0)

The argument filtering Pi contains the following mapping:
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
0  =  0
minusA_out_gga(x1, x2, x3)  =  minusA_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:

minusA_in_gga(0, T8, 0) → minusA_out_gga(0, T8, 0)

The argument filtering Pi contains the following mapping:
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
0  =  0
minusA_out_gga(x1, x2, x3)  =  minusA_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:

minusA_in_gga(0, T8, 0) → minusA_out_gga(0, T8, 0)

The argument filtering Pi contains the following mapping:
minusA_in_gga(x1, x2, x3)  =  minusA_in_gga(x1, x2)
0  =  0
minusA_out_gga(x1, x2, x3)  =  minusA_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) YES