Left Termination proof of /tmp/tmph6IXhx/in-fb.pl

Left Termination of the query pattern in_in_2(a, g) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

in(X, tree(X, X1, X2)).
in(X, tree(Y, Left, X1)) :- ','(less(X, Y), in(X, Left)).
in(X, tree(Y, X1, Right)) :- ','(less(Y, X), in(X, Right)).
less(0, s(X)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

in(a,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_in: (f,b) (b,b)
less_in: (f,b) (f,f) (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

IN_IN_AG(X, tree(Y, Left, X1)) → U1_AG(X, Y, Left, X1, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X1)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X1, in_in_gg(X, Left))
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X1)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X1, in_in_gg(X, Left))
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X1, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X1, Right, in_in_gg(X, Right))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X1, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X1, Right, in_in_gg(X, Right))
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2) = in_in_ag(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
in_out_ag(x1, x2) = in_out_ag(x1)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x3, x5)
less_in_ag(x1, x2) = less_in_ag(x2)
s(x1) = s
less_out_ag(x1, x2) = less_out_ag(x1)
U5_ag(x1, x2, x3) = U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x5)
in_in_gg(x1, x2) = in_in_gg(x1, x2)
in_out_gg(x1, x2) = in_out_gg
U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x3, x5)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
less_out_gg(x1, x2) = less_out_gg
U5_gg(x1, x2, x3) = U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5) = U2_gg(x5)
U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x4, x5)
U4_gg(x1, x2, x3, x4, x5) = U4_gg(x5)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x4, x5)
less_in_ga(x1, x2) = less_in_ga(x1)
less_out_ga(x1, x2) = less_out_ga(x2)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U5_aa(x1, x2, x3) = U5_aa(x3)
U5_ga(x1, x2, x3) = U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x5)
U5_GG(x1, x2, x3) = U5_GG(x3)
LESS_IN_AA(x1, x2) = LESS_IN_AA
U5_GA(x1, x2, x3) = U5_GA(x3)
U5_AG(x1, x2, x3) = U5_AG(x3)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x5)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5)
LESS_IN_AG(x1, x2) = LESS_IN_AG(x2)
U4_GG(x1, x2, x3, x4, x5) = U4_GG(x5)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x4, x5)
U5_AA(x1, x2, x3) = U5_AA(x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x5)
U2_GG(x1, x2, x3, x4, x5) = U2_GG(x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x3, x5)
IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2)
IN_IN_AG(x1, x2) = IN_IN_AG(x2)

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

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

IN_IN_AG(X, tree(Y, Left, X1)) → U1_AG(X, Y, Left, X1, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X1)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X1, in_in_gg(X, Left))
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X1)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X1, in_in_gg(X, Left))
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X1, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X1, Right, in_in_gg(X, Right))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X1, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X1, Right, in_in_gg(X, Right))
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s
LESS_IN_AA(x1, x2) = LESS_IN_AA

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

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

LESS_IN_AA → LESS_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none

s = LESS_IN_AA evaluates to t =LESS_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree(x1, x2, x3)
s(x1) = s
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
less_out_gg(x1, x2) = less_out_gg
U5_gg(x1, x2, x3) = U5_gg(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U5_aa(x1, x2, x3) = U5_aa(x3)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x4, x5)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x3, x5)
IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

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

IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Left, less_in_gg(X, Y))
U3_GG(X, Right, less_out_gg) → IN_IN_GG(X, Right)
U1_GG(X, Left, less_out_gg) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Right, less_in_gg(Y, X))

The TRS R consists of the following rules:

less_in_gg(0, s) → less_out_gg
less_in_gg(s, s) → U5_gg(less_in_aa)
U5_gg(less_out_aa(X, Y)) → less_out_gg
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Left, less_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Right, less_in_gg(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2
U3_GG(X, Right, less_out_gg) → IN_IN_GG(X, Right)
The graph contains the following edges 1 >= 1, 2 >= 2
U1_GG(X, Left, less_out_gg) → IN_IN_GG(X, Left)
The graph contains the following edges 1 >= 1, 2 >= 2

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_in: (f,b) (b,b)
less_in: (f,b) (f,f) (b,b) (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

IN_IN_AG(X, tree(Y, Left, X1)) → U1_AG(X, Y, Left, X1, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X1)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X1, in_in_gg(X, Left))
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X1)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X1, in_in_gg(X, Left))
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X1, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X1, Right, in_in_gg(X, Right))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X1, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X1, Right, in_in_gg(X, Right))
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

The argument filtering Pi contains the following mapping:
in_in_ag(x1, x2) = in_in_ag(x2)
tree(x1, x2, x3) = tree(x1, x2, x3)
in_out_ag(x1, x2) = in_out_ag(x1, x2)
U1_ag(x1, x2, x3, x4, x5) = U1_ag(x2, x3, x4, x5)
less_in_ag(x1, x2) = less_in_ag(x2)
s(x1) = s
less_out_ag(x1, x2) = less_out_ag(x1, x2)
U5_ag(x1, x2, x3) = U5_ag(x3)
U2_ag(x1, x2, x3, x4, x5) = U2_ag(x1, x2, x3, x4, x5)
in_in_gg(x1, x2) = in_in_gg(x1, x2)
in_out_gg(x1, x2) = in_out_gg(x1, x2)
U1_gg(x1, x2, x3, x4, x5) = U1_gg(x1, x2, x3, x4, x5)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
less_out_gg(x1, x2) = less_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x3)
U2_gg(x1, x2, x3, x4, x5) = U2_gg(x1, x2, x3, x4, x5)
U3_gg(x1, x2, x3, x4, x5) = U3_gg(x1, x2, x3, x4, x5)
U4_gg(x1, x2, x3, x4, x5) = U4_gg(x1, x2, x3, x4, x5)
U3_ag(x1, x2, x3, x4, x5) = U3_ag(x2, x3, x4, x5)
less_in_ga(x1, x2) = less_in_ga(x1)
less_out_ga(x1, x2) = less_out_ga(x1, x2)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U5_aa(x1, x2, x3) = U5_aa(x3)
U5_ga(x1, x2, x3) = U5_ga(x3)
U4_ag(x1, x2, x3, x4, x5) = U4_ag(x1, x2, x3, x4, x5)
U5_GG(x1, x2, x3) = U5_GG(x3)
LESS_IN_AA(x1, x2) = LESS_IN_AA
U5_GA(x1, x2, x3) = U5_GA(x3)
U5_AG(x1, x2, x3) = U5_AG(x3)
U2_AG(x1, x2, x3, x4, x5) = U2_AG(x1, x2, x3, x4, x5)
LESS_IN_GA(x1, x2) = LESS_IN_GA(x1)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5)
LESS_IN_AG(x1, x2) = LESS_IN_AG(x2)
U4_GG(x1, x2, x3, x4, x5) = U4_GG(x1, x2, x3, x4, x5)
U3_AG(x1, x2, x3, x4, x5) = U3_AG(x2, x3, x4, x5)
U5_AA(x1, x2, x3) = U5_AA(x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x1, x2, x3, x4, x5)
U2_GG(x1, x2, x3, x4, x5) = U2_GG(x1, x2, x3, x4, x5)
U1_AG(x1, x2, x3, x4, x5) = U1_AG(x2, x3, x4, x5)
IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2)
IN_IN_AG(x1, x2) = IN_IN_AG(x2)

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

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

IN_IN_AG(X, tree(Y, Left, X1)) → U1_AG(X, Y, Left, X1, less_in_ag(X, Y))
IN_IN_AG(X, tree(Y, Left, X1)) → LESS_IN_AG(X, Y)
LESS_IN_AG(s(X), s(Y)) → U5_AG(X, Y, less_in_aa(X, Y))
LESS_IN_AG(s(X), s(Y)) → LESS_IN_AA(X, Y)
LESS_IN_AA(s(X), s(Y)) → U5_AA(X, Y, less_in_aa(X, Y))
LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → U2_AG(X, Y, Left, X1, in_in_gg(X, Left))
U1_AG(X, Y, Left, X1, less_out_ag(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
IN_IN_GG(X, tree(Y, Left, X1)) → LESS_IN_GG(X, Y)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_aa(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_AA(X, Y)
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → U2_GG(X, Y, Left, X1, in_in_gg(X, Left))
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, X1, Right)) → LESS_IN_GG(Y, X)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → U4_GG(X, Y, X1, Right, in_in_gg(X, Right))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_AG(X, tree(Y, X1, Right)) → U3_AG(X, Y, X1, Right, less_in_ga(Y, X))
IN_IN_AG(X, tree(Y, X1, Right)) → LESS_IN_GA(Y, X)
LESS_IN_GA(s(X), s(Y)) → U5_GA(X, Y, less_in_aa(X, Y))
LESS_IN_GA(s(X), s(Y)) → LESS_IN_AA(X, Y)
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → U4_AG(X, Y, X1, Right, in_in_gg(X, Right))
U3_AG(X, Y, X1, Right, less_out_ga(Y, X)) → IN_IN_GG(X, Right)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

LESS_IN_AA(s(X), s(Y)) → LESS_IN_AA(X, Y)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ NonTerminationProof

              ↳ PiDP

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

LESS_IN_AA → LESS_IN_AA

The TRS R consists of the following rules:none

s = LESS_IN_AA evaluates to t =LESS_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS_IN_AA to LESS_IN_AA.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))

The TRS R consists of the following rules:

in_in_ag(X, tree(X, X1, X2)) → in_out_ag(X, tree(X, X1, X2))
in_in_ag(X, tree(Y, Left, X1)) → U1_ag(X, Y, Left, X1, less_in_ag(X, Y))
less_in_ag(0, s(X)) → less_out_ag(0, s(X))
less_in_ag(s(X), s(Y)) → U5_ag(X, Y, less_in_aa(X, Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))
U5_ag(X, Y, less_out_aa(X, Y)) → less_out_ag(s(X), s(Y))
U1_ag(X, Y, Left, X1, less_out_ag(X, Y)) → U2_ag(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(X, X1, X2)) → in_out_gg(X, tree(X, X1, X2))
in_in_gg(X, tree(Y, Left, X1)) → U1_gg(X, Y, Left, X1, less_in_gg(X, Y))
less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
U1_gg(X, Y, Left, X1, less_out_gg(X, Y)) → U2_gg(X, Y, Left, X1, in_in_gg(X, Left))
in_in_gg(X, tree(Y, X1, Right)) → U3_gg(X, Y, X1, Right, less_in_gg(Y, X))
U3_gg(X, Y, X1, Right, less_out_gg(Y, X)) → U4_gg(X, Y, X1, Right, in_in_gg(X, Right))
U4_gg(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_gg(X, tree(Y, X1, Right))
U2_gg(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_gg(X, tree(Y, Left, X1))
U2_ag(X, Y, Left, X1, in_out_gg(X, Left)) → in_out_ag(X, tree(Y, Left, X1))
in_in_ag(X, tree(Y, X1, Right)) → U3_ag(X, Y, X1, Right, less_in_ga(Y, X))
less_in_ga(0, s(X)) → less_out_ga(0, s(X))
less_in_ga(s(X), s(Y)) → U5_ga(X, Y, less_in_aa(X, Y))
U5_ga(X, Y, less_out_aa(X, Y)) → less_out_ga(s(X), s(Y))
U3_ag(X, Y, X1, Right, less_out_ga(Y, X)) → U4_ag(X, Y, X1, Right, in_in_gg(X, Right))
U4_ag(X, Y, X1, Right, in_out_gg(X, Right)) → in_out_ag(X, tree(Y, X1, Right))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))

The TRS R consists of the following rules:

less_in_gg(0, s(X)) → less_out_gg(0, s(X))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_aa(X, Y))
U5_gg(X, Y, less_out_aa(X, Y)) → less_out_gg(s(X), s(Y))
less_in_aa(0, s(X)) → less_out_aa(0, s(X))
less_in_aa(s(X), s(Y)) → U5_aa(X, Y, less_in_aa(X, Y))
U5_aa(X, Y, less_out_aa(X, Y)) → less_out_aa(s(X), s(Y))

The argument filtering Pi contains the following mapping:
tree(x1, x2, x3) = tree(x1, x2, x3)
s(x1) = s
less_in_gg(x1, x2) = less_in_gg(x1, x2)
0 = 0
less_out_gg(x1, x2) = less_out_gg(x1, x2)
U5_gg(x1, x2, x3) = U5_gg(x3)
less_in_aa(x1, x2) = less_in_aa
less_out_aa(x1, x2) = less_out_aa(x1, x2)
U5_aa(x1, x2, x3) = U5_aa(x3)
U3_GG(x1, x2, x3, x4, x5) = U3_GG(x1, x2, x3, x4, x5)
U1_GG(x1, x2, x3, x4, x5) = U1_GG(x1, x2, x3, x4, x5)
IN_IN_GG(x1, x2) = IN_IN_GG(x1, x2)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))

The TRS R consists of the following rules:

less_in_gg(0, s) → less_out_gg(0, s)
less_in_gg(s, s) → U5_gg(less_in_aa)
U5_gg(less_out_aa(X, Y)) → less_out_gg(s, s)
less_in_aa → less_out_aa(0, s)
less_in_aa → U5_aa(less_in_aa)
U5_aa(less_out_aa(X, Y)) → less_out_aa(s, s)

The set Q consists of the following terms:

less_in_gg(x0, x1)
U5_gg(x0)
less_in_aa
U5_aa(x0)

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:

IN_IN_GG(X, tree(Y, Left, X1)) → U1_GG(X, Y, Left, X1, less_in_gg(X, Y))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4
IN_IN_GG(X, tree(Y, X1, Right)) → U3_GG(X, Y, X1, Right, less_in_gg(Y, X))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4
U3_GG(X, Y, X1, Right, less_out_gg(Y, X)) → IN_IN_GG(X, Right)
The graph contains the following edges 1 >= 1, 5 > 1, 4 >= 2
U1_GG(X, Y, Left, X1, less_out_gg(X, Y)) → IN_IN_GG(X, Left)
The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2