(0) Obligation:
Clauses:
p(X) :- ','(q(f(Y)), p(Y)).
p(g(X)) :- p(X).
q(g(Y)).
Query: p(g)
(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_g(g(T6)) → U1_g(T6, pA_in_g(T6))
U1_g(T6, pA_out_g(T6)) → pA_out_g(g(T6))
Pi is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_G(g(T6)) → U1_G(T6, pA_in_g(T6))
PA_IN_G(g(T6)) → PA_IN_G(T6)
The TRS R consists of the following rules:
pA_in_g(g(T6)) → U1_g(T6, pA_in_g(T6))
U1_g(T6, pA_out_g(T6)) → pA_out_g(g(T6))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_G(g(T6)) → U1_G(T6, pA_in_g(T6))
PA_IN_G(g(T6)) → PA_IN_G(T6)
The TRS R consists of the following rules:
pA_in_g(g(T6)) → U1_g(T6, pA_in_g(T6))
U1_g(T6, pA_out_g(T6)) → pA_out_g(g(T6))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_G(g(T6)) → PA_IN_G(T6)
The TRS R consists of the following rules:
pA_in_g(g(T6)) → U1_g(T6, pA_in_g(T6))
U1_g(T6, pA_out_g(T6)) → pA_out_g(g(T6))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(7) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PA_IN_G(g(T6)) → PA_IN_G(T6)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(9) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PA_IN_G(g(T6)) → PA_IN_G(T6)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(11) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:
- PA_IN_G(g(T6)) → PA_IN_G(T6)
The graph contains the following edges 1 > 1
(12) YES