(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