(0) Obligation:
Clauses:at(X, fido) :- ','(at(X, mary), near(X)).
at(ta, mary).
at(jm, mary).
near(jm).
Queries:
at(a,a).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:at1(jm, fido).
at1(ta, mary).
at1(jm, mary).
at1(ta, mary).
at1(jm, mary).
at1(jm, mary).
Queries:
at1(a,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:
at1_in_aa(jm, fido) → at1_out_aa(jm, fido)
at1_in_aa(ta, mary) → at1_out_aa(ta, mary)
at1_in_aa(jm, mary) → at1_out_aa(jm, mary)
The argument filtering Pi contains the following mapping:
at1_in_aa(x1, x2) = at1_in_aa
at1_out_aa(x1, x2) = at1_out_aa(x1, 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:
at1_in_aa(jm, fido) → at1_out_aa(jm, fido)
at1_in_aa(ta, mary) → at1_out_aa(ta, mary)
at1_in_aa(jm, mary) → at1_out_aa(jm, mary)
The argument filtering Pi contains the following mapping:
at1_in_aa(x1, x2) = at1_in_aa
at1_out_aa(x1, x2) = at1_out_aa(x1, 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:
at1_in_aa(jm, fido) → at1_out_aa(jm, fido)
at1_in_aa(ta, mary) → at1_out_aa(ta, mary)
at1_in_aa(jm, mary) → at1_out_aa(jm, mary)
The argument filtering Pi contains the following mapping:
at1_in_aa(x1, x2) = at1_in_aa
at1_out_aa(x1, x2) = at1_out_aa(x1, 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:
at1_in_aa(jm, fido) → at1_out_aa(jm, fido)
at1_in_aa(ta, mary) → at1_out_aa(ta, mary)
at1_in_aa(jm, mary) → at1_out_aa(jm, mary)
The argument filtering Pi contains the following mapping:
at1_in_aa(x1, x2) = at1_in_aa
at1_out_aa(x1, x2) = at1_out_aa(x1, x2)
We have to consider all (P,R,Pi)-chains