Left Termination proof of /tmp/tmpL2M1hf/p.pl

Left Termination of the query pattern p_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 PiDP

↳7 PiDPToQDPProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 DependencyGraphProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 DependencyGraphProof (⇔)

↳16 QDP

↳17 UsableRulesProof (⇔)

↳18 QDP

↳19 QReductionProof (⇔)

↳20 QDP

↳21 UsableRulesReductionPairsProof (⇔)

↳22 QDP

↳23 PisEmptyProof (⇔)

↳24 YES

(0) Obligation:

Clauses:

p(.(X, [])).
p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), p(.(s(s(s(s(Y)))), Xs))).
p(.(0, Xs)) :- p(Xs).

Queries:

p(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

p1(.(s(s(s(s(T24)))), .(T25, T26))) :- p1(.(T24, .(T25, T26))).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), p1(.(s(s(s(s(T25)))), T26))).
p1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), ','(pc1(.(s(s(s(s(T25)))), T26)), p1(.(s(s(s(s(T25)))), T26)))).
p1(.(s(s(0)), .(T51, T52))) :- p1(.(T51, T52)).
p1(.(s(s(0)), .(T51, T52))) :- ','(pc1(.(T51, T52)), p1(.(s(s(s(s(T51)))), T52))).
p1(.(0, .(s(s(T78)), .(T79, T80)))) :- p1(.(T78, .(T79, T80))).
p1(.(0, .(s(s(T78)), .(T79, T80)))) :- ','(pc1(.(T78, .(T79, T80))), p1(.(s(s(s(s(T79)))), T80))).
p1(.(0, .(0, T95))) :- p1(T95).

Clauses:

pc1(.(T3, [])).
pc1(.(s(s(s(s(T24)))), .(T25, T26))) :- ','(pc1(.(T24, .(T25, T26))), ','(pc1(.(s(s(s(s(T25)))), T26)), pc1(.(s(s(s(s(T25)))), T26)))).
pc1(.(s(s(0)), .(T51, T52))) :- ','(pc1(.(T51, T52)), pc1(.(s(s(s(s(T51)))), T52))).
pc1(.(0, .(T65, []))).
pc1(.(0, .(s(s(T78)), .(T79, T80)))) :- ','(pc1(.(T78, .(T79, T80))), pc1(.(s(s(s(s(T79)))), T80))).
pc1(.(0, .(0, T95))) :- pc1(T95).

Afs:

p1(x1) = p1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p1_in: (b)
pc1_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U1_G(T24, T25, T26, p1_in_g(.(T24, .(T25, T26))))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U3_G(T24, T25, T26, p1_in_g(.(s(s(s(s(T25)))), T26)))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U6_G(T51, T52, p1_in_g(.(T51, T52)))
P1_IN_G(.(s(s(0)), .(T51, T52))) → P1_IN_G(.(T51, T52))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U7_G(T51, T52, pc1_in_g(.(T51, T52)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → U8_G(T51, T52, p1_in_g(.(s(s(s(s(T51)))), T52)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → P1_IN_G(.(s(s(s(s(T51)))), T52))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U9_G(T78, T79, T80, p1_in_g(.(T78, .(T79, T80))))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → P1_IN_G(.(T78, .(T79, T80)))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U10_G(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U11_G(T78, T79, T80, p1_in_g(.(s(s(s(s(T79)))), T80)))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → P1_IN_G(.(s(s(s(s(T79)))), T80))
P1_IN_G(.(0, .(0, T95))) → U12_G(T95, p1_in_g(T95))
P1_IN_G(.(0, .(0, T95))) → P1_IN_G(T95)
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U5_G(T24, T25, T26, p1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

Pi is empty.
We have to consider all (P,R,Pi)-chains

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U1_G(T24, T25, T26, p1_in_g(.(T24, .(T25, T26))))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U3_G(T24, T25, T26, p1_in_g(.(s(s(s(s(T25)))), T26)))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U6_G(T51, T52, p1_in_g(.(T51, T52)))
P1_IN_G(.(s(s(0)), .(T51, T52))) → P1_IN_G(.(T51, T52))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U7_G(T51, T52, pc1_in_g(.(T51, T52)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → U8_G(T51, T52, p1_in_g(.(s(s(s(s(T51)))), T52)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → P1_IN_G(.(s(s(s(s(T51)))), T52))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U9_G(T78, T79, T80, p1_in_g(.(T78, .(T79, T80))))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → P1_IN_G(.(T78, .(T79, T80)))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U10_G(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U11_G(T78, T79, T80, p1_in_g(.(s(s(s(s(T79)))), T80)))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → P1_IN_G(.(s(s(s(s(T79)))), T80))
P1_IN_G(.(0, .(0, T95))) → U12_G(T95, p1_in_g(T95))
P1_IN_G(.(0, .(0, T95))) → P1_IN_G(T95)
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U5_G(T24, T25, T26, p1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

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 8 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(0)), .(T51, T52))) → P1_IN_G(.(T51, T52))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U7_G(T51, T52, pc1_in_g(.(T51, T52)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → P1_IN_G(.(s(s(s(s(T51)))), T52))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → P1_IN_G(.(T78, .(T79, T80)))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U10_G(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → P1_IN_G(.(s(s(s(s(T79)))), T80))
P1_IN_G(.(0, .(0, T95))) → P1_IN_G(T95)
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(7) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
P1_IN_G(.(s(s(0)), .(T51, T52))) → P1_IN_G(.(T51, T52))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U7_G(T51, T52, pc1_in_g(.(T51, T52)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → P1_IN_G(.(s(s(s(s(T51)))), T52))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → P1_IN_G(.(T78, .(T79, T80)))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U10_G(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → P1_IN_G(.(s(s(s(s(T79)))), T80))
P1_IN_G(.(0, .(0, T95))) → P1_IN_G(T95)
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P1_IN_G(.(s(s(0)), .(T51, T52))) → P1_IN_G(.(T51, T52))
P1_IN_G(.(s(s(0)), .(T51, T52))) → U7_G(T51, T52, pc1_in_g(.(T51, T52)))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → P1_IN_G(.(T78, .(T79, T80)))
P1_IN_G(.(0, .(s(s(T78)), .(T79, T80)))) → U10_G(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
P1_IN_G(.(0, .(0, T95))) → P1_IN_G(T95)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(0) = 1
POL(P1_IN_G(x₁)) = x₁
POL(U10_G(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U14_g(x₁, x₂, x₃, x₄)) = 0
POL(U15_g(x₁, x₂, x₃, x₄)) = 0
POL(U16_g(x₁, x₂, x₃, x₄)) = 0
POL(U17_g(x₁, x₂, x₃)) = 0
POL(U18_g(x₁, x₂, x₃)) = 0
POL(U19_g(x₁, x₂, x₃, x₄)) = 0
POL(U20_g(x₁, x₂, x₃, x₄)) = 0
POL(U21_g(x₁, x₂)) = 0
POL(U2_G(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U4_G(x₁, x₂, x₃, x₄)) = x₂ + x₃
POL(U7_G(x₁, x₂, x₃)) = x₁ + x₂
POL([]) = 1
POL(pc1_in_g(x₁)) = 1 + x₁
POL(pc1_out_g(x₁)) = 1
POL(s(x₁)) = x₁

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))
U7_G(T51, T52, pc1_out_g(.(T51, T52))) → P1_IN_G(.(s(s(s(s(T51)))), T52))
U10_G(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → P1_IN_G(.(s(s(s(s(T79)))), T80))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → U2_G(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(P1_IN_G(x₁)) = x₁
POL(U14_g(x₁, x₂, x₃, x₄)) = 0
POL(U15_g(x₁, x₂, x₃, x₄)) = 0
POL(U16_g(x₁, x₂, x₃, x₄)) = 0
POL(U17_g(x₁, x₂, x₃)) = 0
POL(U18_g(x₁, x₂, x₃)) = 0
POL(U19_g(x₁, x₂, x₃, x₄)) = 0
POL(U20_g(x₁, x₂, x₃, x₄)) = 0
POL(U21_g(x₁, x₂)) = 0
POL(U2_G(x₁, x₂, x₃, x₄)) = 1 + x₃
POL(U4_G(x₁, x₂, x₃, x₄)) = 1 + x₃
POL([]) = 0
POL(pc1_in_g(x₁)) = x₁
POL(pc1_out_g(x₁)) = 0
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
U2_G(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U4_G(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U4_G(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → P1_IN_G(.(s(s(s(s(T25)))), T26))
P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(15) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

The TRS R consists of the following rules:

pc1_in_g(.(T3, [])) → pc1_out_g(.(T3, []))
pc1_in_g(.(s(s(s(s(T24)))), .(T25, T26))) → U14_g(T24, T25, T26, pc1_in_g(.(T24, .(T25, T26))))
pc1_in_g(.(s(s(0)), .(T51, T52))) → U17_g(T51, T52, pc1_in_g(.(T51, T52)))
pc1_in_g(.(0, .(T65, []))) → pc1_out_g(.(0, .(T65, [])))
pc1_in_g(.(0, .(s(s(T78)), .(T79, T80)))) → U19_g(T78, T79, T80, pc1_in_g(.(T78, .(T79, T80))))
pc1_in_g(.(0, .(0, T95))) → U21_g(T95, pc1_in_g(T95))
U21_g(T95, pc1_out_g(T95)) → pc1_out_g(.(0, .(0, T95)))
U19_g(T78, T79, T80, pc1_out_g(.(T78, .(T79, T80)))) → U20_g(T78, T79, T80, pc1_in_g(.(s(s(s(s(T79)))), T80)))
U20_g(T78, T79, T80, pc1_out_g(.(s(s(s(s(T79)))), T80))) → pc1_out_g(.(0, .(s(s(T78)), .(T79, T80))))
U17_g(T51, T52, pc1_out_g(.(T51, T52))) → U18_g(T51, T52, pc1_in_g(.(s(s(s(s(T51)))), T52)))
U18_g(T51, T52, pc1_out_g(.(s(s(s(s(T51)))), T52))) → pc1_out_g(.(s(s(0)), .(T51, T52)))
U14_g(T24, T25, T26, pc1_out_g(.(T24, .(T25, T26)))) → U15_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U15_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → U16_g(T24, T25, T26, pc1_in_g(.(s(s(s(s(T25)))), T26)))
U16_g(T24, T25, T26, pc1_out_g(.(s(s(s(s(T25)))), T26))) → pc1_out_g(.(s(s(s(s(T24)))), .(T25, T26)))

The set Q consists of the following terms:

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(17) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

R is empty.
The set Q consists of the following terms:

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(19) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

pc1_in_g(x0)
U21_g(x0, x1)
U19_g(x0, x1, x2, x3)
U20_g(x0, x1, x2, x3)
U17_g(x0, x1, x2)
U18_g(x0, x1, x2)
U14_g(x0, x1, x2, x3)
U15_g(x0, x1, x2, x3)
U16_g(x0, x1, x2, x3)

(20) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

P1_IN_G(.(s(s(s(s(T24)))), .(T25, T26))) → P1_IN_G(.(T24, .(T25, T26)))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(P1_IN_G(x₁)) = x₁
POL(s(x₁)) = x₁

(22) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) PiDPToQDPProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) DependencyGraphProof (EQUIVALENT transformation)

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) QReductionProof (EQUIVALENT transformation)

(20) Obligation:

(21) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(22) Obligation:

(23) PisEmptyProof (EQUIVALENT transformation)

(24) YES