YES
Left Termination proof of ../tpdb/LP/SGST06/evenodd.pl
Left Termination of the query pattern
even_in_1(g)
w.r.t. the given Prolog program could successfully be proven:
↳ Prolog
↳ PrologToPiTRSProof
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).
We use the technique of [30]. 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(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))
The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
even_out_g(x1) = even_out_g
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
odd_in_g(x1) = odd_in_g(x1)
odd_out_g(x1) = odd_out_g
U2_g(x1, x2) = U2_g(x2)
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:
even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))
The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
even_out_g(x1) = even_out_g
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
odd_in_g(x1) = odd_in_g(x1)
odd_out_g(x1) = odd_out_g
U2_g(x1, x2) = U2_g(x2)
Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
EVEN_IN_G(s(s(s(X)))) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(s(s(X))))
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))
The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
even_out_g(x1) = even_out_g
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
odd_in_g(x1) = odd_in_g(x1)
odd_out_g(x1) = odd_out_g
U2_g(x1, x2) = U2_g(x2)
U2_G(x1, x2) = U2_G(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
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:
EVEN_IN_G(s(s(s(X)))) → U1_G(X, odd_in_g(X))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
ODD_IN_G(s(X)) → U2_G(X, even_in_g(s(s(X))))
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))
The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
even_out_g(x1) = even_out_g
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
odd_in_g(x1) = odd_in_g(x1)
odd_out_g(x1) = odd_out_g
U2_g(x1, x2) = U2_g(x2)
U2_G(x1, x2) = U2_G(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
EVEN_IN_G(x1) = EVEN_IN_G(x1)
We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 2 less nodes.
↳ Prolog
↳ PrologToPiTRSProof
↳ PiTRS
↳ DependencyPairsProof
↳ PiDP
↳ DependencyGraphProof
↳ PiDP
↳ UsableRulesProof
Pi DP problem:
The TRS P consists of the following rules:
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
The TRS R consists of the following rules:
even_in_g(0) → even_out_g(0)
even_in_g(s(s(0))) → even_out_g(s(s(0)))
even_in_g(s(s(s(X)))) → U1_g(X, odd_in_g(X))
odd_in_g(s(0)) → odd_out_g(s(0))
odd_in_g(s(X)) → U2_g(X, even_in_g(s(s(X))))
U2_g(X, even_out_g(s(s(X)))) → odd_out_g(s(X))
U1_g(X, odd_out_g(X)) → even_out_g(s(s(s(X))))
The argument filtering Pi contains the following mapping:
even_in_g(x1) = even_in_g(x1)
0 = 0
even_out_g(x1) = even_out_g
s(x1) = s(x1)
U1_g(x1, x2) = U1_g(x2)
odd_in_g(x1) = odd_in_g(x1)
odd_out_g(x1) = odd_out_g
U2_g(x1, x2) = U2_g(x2)
ODD_IN_G(x1) = ODD_IN_G(x1)
EVEN_IN_G(x1) = EVEN_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:
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))
EVEN_IN_G(s(s(s(X)))) → ODD_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
↳ RFCMatchBoundsDPProof
Q DP problem:
The TRS P consists of the following rules:
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
Termination of the TRS P cup R can be shown by a matchbound [6,7] of 1. This implies finiteness of the given DP problem.
The following rules (P cup R) were used to construct the certificate:
ODD_IN_G(s(X)) → EVEN_IN_G(s(s(X)))
EVEN_IN_G(s(s(s(X)))) → ODD_IN_G(X)
The certificate found is represented by the following graph.
The certificate consists of the following enumerated nodes:
50, 51, 52, 53, 54, 55
Node 50 is start node and node 51 is final node.
Those nodes are connect through the following edges:
- 50 to 52 labelled EVEN_IN_G_1(0)
- 50 to 51 labelled ODD_IN_G_1(0), ODD_IN_G_1(1)
- 50 to 54 labelled EVEN_IN_G_1(1)
- 51 to 51 labelled #_1(0)
- 52 to 53 labelled s_1(0)
- 53 to 51 labelled s_1(0)
- 54 to 55 labelled s_1(1)
- 55 to 51 labelled s_1(1)