(0) Obligation:
Clauses:
even(0) :- !.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- !.
odd(N) :- ','(p(N, P), even(P)).
p(0, 0).
p(s(X), X).
Query: even(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:
evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(0)) → evenA_out_g(s(0))
evenA_in_g(s(s(T12))) → U1_g(T12, evenA_in_g(T12))
U1_g(T12, evenA_out_g(T12)) → evenA_out_g(s(s(T12)))
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:
EVENA_IN_G(s(s(T12))) → U1_G(T12, evenA_in_g(T12))
EVENA_IN_G(s(s(T12))) → EVENA_IN_G(T12)
The TRS R consists of the following rules:
evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(0)) → evenA_out_g(s(0))
evenA_in_g(s(s(T12))) → U1_g(T12, evenA_in_g(T12))
U1_g(T12, evenA_out_g(T12)) → evenA_out_g(s(s(T12)))
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:
EVENA_IN_G(s(s(T12))) → U1_G(T12, evenA_in_g(T12))
EVENA_IN_G(s(s(T12))) → EVENA_IN_G(T12)
The TRS R consists of the following rules:
evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(0)) → evenA_out_g(s(0))
evenA_in_g(s(s(T12))) → U1_g(T12, evenA_in_g(T12))
U1_g(T12, evenA_out_g(T12)) → evenA_out_g(s(s(T12)))
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:
EVENA_IN_G(s(s(T12))) → EVENA_IN_G(T12)
The TRS R consists of the following rules:
evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(0)) → evenA_out_g(s(0))
evenA_in_g(s(s(T12))) → U1_g(T12, evenA_in_g(T12))
U1_g(T12, evenA_out_g(T12)) → evenA_out_g(s(s(T12)))
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:
EVENA_IN_G(s(s(T12))) → EVENA_IN_G(T12)
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:
EVENA_IN_G(s(s(T12))) → EVENA_IN_G(T12)
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:
- EVENA_IN_G(s(s(T12))) → EVENA_IN_G(T12)
The graph contains the following edges 1 > 1
(12) YES