(0) Obligation:
Clauses:
p(X, g(X)).
p(X, f(X)) :- p(X, g(Y)).
Query: p(g,a)
(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)
Transformed Prolog program to (Pi-)TRS.
(2) 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(
x1,
x2)
(3) 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(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(4) 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(
x1,
x2)
We have to consider all (P,R,Pi)-chains
(5) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,R,Pi) chain.
(6) YES