(0) Obligation:
Clauses:
p(X, X, a) :- !.
p(X, Y, Z) :- ','(eq(Z, a), ','(eq(X, Y), p(X, Y, Z))).
eq(X, X).
Query: p(a,a,a)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
pA(T5, T5, a).
pA(T28, T28, 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