Left Termination of the query pattern
p1_in_1(g)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PrologToPiTRSProof
Clauses:
p1(f(X)) :- p1(X).
p2(f(X)) :- p2(X).
Queries:
p1(g).
We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))
The argument filtering Pi contains the following mapping:
p1_in_g(x1) = p1_in_g(x1)
f(x1) = f(x1)
U1_g(x1, x2) = U1_g(x2)
p1_out_g(x1) = p1_out_g
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
Pi-finite rewrite system:
The TRS R consists of the following rules:
p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))
The argument filtering Pi contains the following mapping:
p1_in_g(x1) = p1_in_g(x1)
f(x1) = f(x1)
U1_g(x1, x2) = U1_g(x2)
p1_out_g(x1) = p1_out_g
Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_G(f(X)) → U1_G(X, p1_in_g(X))
P1_IN_G(f(X)) → P1_IN_G(X)
The TRS R consists of the following rules:
p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))
The argument filtering Pi contains the following mapping:
p1_in_g(x1) = p1_in_g(x1)
f(x1) = f(x1)
U1_g(x1, x2) = U1_g(x2)
p1_out_g(x1) = p1_out_g
P1_IN_G(x1) = P1_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
We have to consider all (P,R,Pi)-chains
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_G(f(X)) → U1_G(X, p1_in_g(X))
P1_IN_G(f(X)) → P1_IN_G(X)
The TRS R consists of the following rules:
p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))
The argument filtering Pi contains the following mapping:
p1_in_g(x1) = p1_in_g(x1)
f(x1) = f(x1)
U1_g(x1, x2) = U1_g(x2)
p1_out_g(x1) = p1_out_g
P1_IN_G(x1) = P1_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 1 less node.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_G(f(X)) → P1_IN_G(X)
The TRS R consists of the following rules:
p1_in_g(f(X)) → U1_g(X, p1_in_g(X))
U1_g(X, p1_out_g(X)) → p1_out_g(f(X))
The argument filtering Pi contains the following mapping:
p1_in_g(x1) = p1_in_g(x1)
f(x1) = f(x1)
U1_g(x1, x2) = U1_g(x2)
p1_out_g(x1) = p1_out_g
P1_IN_G(x1) = P1_IN_G(x1)
We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
Pi DP problem:
The TRS P consists of the following rules:
P1_IN_G(f(X)) → P1_IN_G(X)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
↳ PiDP
↳ PiDPToQDPProof
↳ QDP
↳ QDPSizeChangeProof
Q DP problem:
The TRS P consists of the following rules:
P1_IN_G(f(X)) → P1_IN_G(X)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs:
- P1_IN_G(f(X)) → P1_IN_G(X)
The graph contains the following edges 1 > 1