(0) Obligation:

Clauses:

p(X, X, a) :- !.
p(X, Y, Z) :- ','(=(Z, a), ','(=(X, Y), p(X, Y, Z))).

Query: p(a,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

(2) Obligation:

Clauses:

pA(T5, T5, a).
pA(T22, T22, a).

Query: pA(a,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. 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:

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(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:

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(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:

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(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:

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(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