(0) Obligation:
Clauses:
even(0).
even(s(X)) :- odd(X).
odd(s(X)) :- even(X).
Query: even(g)
(1) PrologToPiTRSProof (SOUND transformation)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
even_in: (b)
odd_in: (b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(X))
Pi is empty.
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(X))
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:
EVEN_IN_G(s(X)) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(X))
ODD_IN_G(s(X)) → EVEN_IN_G(X)
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(X))
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:
EVEN_IN_G(s(X)) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(X))
ODD_IN_G(s(X)) → EVEN_IN_G(X)
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(X))
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 2 less nodes.
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
EVEN_IN_G(s(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(X)
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(X)) → U1_g(X, odd_in_g(X))
odd_in_g(s(X)) → U2_g(X, even_in_g(X))
U2_g(X, even_out_g(X)) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(X))
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:
EVEN_IN_G(s(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(X)
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:
EVEN_IN_G(s(X)) → ODD_IN_G(X)
ODD_IN_G(s(X)) → EVEN_IN_G(X)
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:
- ODD_IN_G(s(X)) → EVEN_IN_G(X)
The graph contains the following edges 1 > 1
- EVEN_IN_G(s(X)) → ODD_IN_G(X)
The graph contains the following edges 1 > 1
(12) YES