(0) Obligation:

Clauses:

p(X, g(X)).
p(X, f(X)) :- p(X, g(Y)).

Queries:

p(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

p1(T4, g(T4)).
p1(T11, f(T11)).
p1(T19, f(T19)).

Queries:

p1(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:

p1_in_ga(T4, g(T4)) → p1_out_ga(T4, g(T4))
p1_in_ga(T11, f(T11)) → p1_out_ga(T11, f(T11))

The argument filtering Pi contains the following mapping:
p1_in_ga(x1, x2)  =  p1_in_ga(x1)
p1_out_ga(x1, x2)  =  p1_out_ga(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

p1_in_ga(T4, g(T4)) → p1_out_ga(T4, g(T4))
p1_in_ga(T11, f(T11)) → p1_out_ga(T11, f(T11))

The argument filtering Pi contains the following mapping:
p1_in_ga(x1, x2)  =  p1_in_ga(x1)
p1_out_ga(x1, x2)  =  p1_out_ga(x2)

(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:

p1_in_ga(T4, g(T4)) → p1_out_ga(T4, g(T4))
p1_in_ga(T11, f(T11)) → p1_out_ga(T11, f(T11))

The argument filtering Pi contains the following mapping:
p1_in_ga(x1, x2)  =  p1_in_ga(x1)
p1_out_ga(x1, x2)  =  p1_out_ga(x2)

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:

p1_in_ga(T4, g(T4)) → p1_out_ga(T4, g(T4))
p1_in_ga(T11, f(T11)) → p1_out_ga(T11, f(T11))

The argument filtering Pi contains the following mapping:
p1_in_ga(x1, x2)  =  p1_in_ga(x1)
p1_out_ga(x1, x2)  =  p1_out_ga(x2)

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