(0) Obligation:
Clauses:
p(X, g(X)).
p(X, f(X)) :- p(X, g(Y)).
Query: p(g,a)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
pA(T4, g(T4)).
pA(T11, f(T11)).
pA(T19, f(T19)).
Query: pA(g,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_ga(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T11, f(T11)) → pA_out_ga(T11, f(T11))
The argument filtering Pi contains the following mapping:
pA_in_ga(
x1,
x2) =
pA_in_ga(
x1)
pA_out_ga(
x1,
x2) =
pA_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:
pA_in_ga(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T11, f(T11)) → pA_out_ga(T11, f(T11))
The argument filtering Pi contains the following mapping:
pA_in_ga(
x1,
x2) =
pA_in_ga(
x1)
pA_out_ga(
x1,
x2) =
pA_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:
pA_in_ga(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T11, f(T11)) → pA_out_ga(T11, f(T11))
The argument filtering Pi contains the following mapping:
pA_in_ga(
x1,
x2) =
pA_in_ga(
x1)
pA_out_ga(
x1,
x2) =
pA_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:
pA_in_ga(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T11, f(T11)) → pA_out_ga(T11, f(T11))
The argument filtering Pi contains the following mapping:
pA_in_ga(
x1,
x2) =
pA_in_ga(
x1)
pA_out_ga(
x1,
x2) =
pA_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