(0) Obligation:
Clauses:even(0).
even(s(s(0))).
even(s(s(s(X)))) :- odd(X).
odd(s(0)).
odd(s(X)) :- even(s(s(X))).
Queries:
even(g).
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph.
(2) Obligation:
Clauses:even1(0).
even1(s(s(0))).
even1(s(s(s(s(0))))).
even1(s(s(s(s(T6))))) :- even1(s(s(T6))).
Queries:
even1(g).
(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:even1_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))
Pi is empty.
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:The TRS R consists of the following rules:
even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))
Pi is empty.
(5) 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:
EVEN1_IN_G(s(s(s(s(T6))))) → U1_G(T6, even1_in_g(s(s(T6))))
EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))
The TRS R consists of the following rules:
even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:The TRS P consists of the following rules:
EVEN1_IN_G(s(s(s(s(T6))))) → U1_G(T6, even1_in_g(s(s(T6))))
EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))
The TRS R consists of the following rules:
even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.(8) Obligation:
Pi DP problem:The TRS P consists of the following rules:
EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))
The TRS R consists of the following rules:
even1_in_g(0) → even1_out_g(0)
even1_in_g(s(s(0))) → even1_out_g(s(s(0)))
even1_in_g(s(s(s(s(0))))) → even1_out_g(s(s(s(s(0)))))
even1_in_g(s(s(s(s(T6))))) → U1_g(T6, even1_in_g(s(s(T6))))
U1_g(T6, even1_out_g(s(s(T6)))) → even1_out_g(s(s(s(s(T6)))))
Pi is empty.
We have to consider all (P,R,Pi)-chains
(9) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.(10) Obligation:
Pi DP problem:The TRS P consists of the following rules:
EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.(12) Obligation:
Q DP problem:The TRS P consists of the following rules:
EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(13) 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:
- EVEN1_IN_G(s(s(s(s(T6))))) → EVEN1_IN_G(s(s(T6)))
The graph contains the following edges 1 > 1