Left Termination proof of /tmp/tmpa8SK9i/merge.pl

Left Termination of the query pattern merge_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

merge([], X, X).
merge(X, [], X).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs)).
less(0, s(0)).
less(s(X), s(Y)) :- less(X, Y).
leq(0, 0).
leq(0, s(0)).
leq(s(X), s(Y)) :- leq(X, Y).

Queries:

merge(g,g,a).

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

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

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:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN_GG(X, Y)
LEQ_IN_GG(s(X), s(Y)) → U6_GG(X, Y, leq_in_gg(X, Y))
LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GG(Y, X)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
[] = []
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2) = leq_in_gg(x1, x2)
0 = 0
leq_out_gg(x1, x2) = leq_out_gg
s(x1) = s(x1)
U6_gg(x1, x2, x3) = U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
less_out_gg(x1, x2) = less_out_gg
U5_gg(x1, x2, x3) = U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x3, x6)
U5_GG(x1, x2, x3) = U5_GG(x3)
LESS_IN_GG(x1, x2) = LESS_IN_GG(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6)
U6_GG(x1, x2, x3) = U6_GG(x3)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x6)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x3, x6)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)
LEQ_IN_GG(x1, x2) = LEQ_IN_GG(x1, x2)

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:

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN_GG(X, Y)
LEQ_IN_GG(s(X), s(Y)) → U6_GG(X, Y, leq_in_gg(X, Y))
LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN_GG(Y, X)
LESS_IN_GG(s(X), s(Y)) → U5_GG(X, Y, less_in_gg(X, Y))
LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_GGA(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

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

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

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

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)

R is empty.
Q is empty.
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:

LESS_IN_GG(s(X), s(Y)) → LESS_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)

From the DPs we obtained the following set of size-change graphs:

LEQ_IN_GG(s(X), s(Y)) → LEQ_IN_GG(X, Y)
The graph contains the following edges 1 > 1, 2 > 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

merge_in_gga([], X, X) → merge_out_gga([], X, X)
merge_in_gga(X, [], X) → merge_out_gga(X, [], X)
merge_in_gga(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_gga(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))
U1_gga(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → U2_gga(X, Xs, Y, Ys, Zs, merge_in_gga(Xs, .(Y, Ys), Zs))
merge_in_gga(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_gga(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U3_gga(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → U4_gga(X, Xs, Y, Ys, Zs, merge_in_gga(.(X, Xs), Ys, Zs))
U4_gga(X, Xs, Y, Ys, Zs, merge_out_gga(.(X, Xs), Ys, Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(Y, Zs))
U2_gga(X, Xs, Y, Ys, Zs, merge_out_gga(Xs, .(Y, Ys), Zs)) → merge_out_gga(.(X, Xs), .(Y, Ys), .(X, Zs))

The argument filtering Pi contains the following mapping:
merge_in_gga(x1, x2, x3) = merge_in_gga(x1, x2)
[] = []
merge_out_gga(x1, x2, x3) = merge_out_gga(x3)
.(x1, x2) = .(x1, x2)
U1_gga(x1, x2, x3, x4, x5, x6) = U1_gga(x1, x2, x3, x4, x6)
leq_in_gg(x1, x2) = leq_in_gg(x1, x2)
0 = 0
leq_out_gg(x1, x2) = leq_out_gg
s(x1) = s(x1)
U6_gg(x1, x2, x3) = U6_gg(x3)
U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x6)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
less_out_gg(x1, x2) = less_out_gg
U5_gg(x1, x2, x3) = U5_gg(x3)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x3, x6)
U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)

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

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3_GGA(X, Xs, Y, Ys, Zs, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1_GGA(X, Xs, Y, Ys, Zs, leq_in_gg(X, Y))
U1_GGA(X, Xs, Y, Ys, Zs, leq_out_gg(X, Y)) → MERGE_IN_GGA(Xs, .(Y, Ys), Zs)
U3_GGA(X, Xs, Y, Ys, Zs, less_out_gg(Y, X)) → MERGE_IN_GGA(.(X, Xs), Ys, Zs)

The TRS R consists of the following rules:

less_in_gg(0, s(0)) → less_out_gg(0, s(0))
less_in_gg(s(X), s(Y)) → U5_gg(X, Y, less_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg(0, 0)
leq_in_gg(0, s(0)) → leq_out_gg(0, s(0))
leq_in_gg(s(X), s(Y)) → U6_gg(X, Y, leq_in_gg(X, Y))
U5_gg(X, Y, less_out_gg(X, Y)) → less_out_gg(s(X), s(Y))
U6_gg(X, Y, leq_out_gg(X, Y)) → leq_out_gg(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
leq_in_gg(x1, x2) = leq_in_gg(x1, x2)
0 = 0
leq_out_gg(x1, x2) = leq_out_gg
s(x1) = s(x1)
U6_gg(x1, x2, x3) = U6_gg(x3)
less_in_gg(x1, x2) = less_in_gg(x1, x2)
less_out_gg(x1, x2) = less_out_gg
U5_gg(x1, x2, x3) = U5_gg(x3)
U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6)
MERGE_IN_GGA(x1, x2, x3) = MERGE_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)

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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

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

MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U3_GGA(X, Xs, Y, Ys, less_in_gg(Y, X))
U3_GGA(X, Xs, Y, Ys, less_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U1_GGA(X, Xs, Y, Ys, leq_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U1_GGA(X, Xs, Y, Ys, leq_in_gg(X, Y))

The TRS R consists of the following rules:

less_in_gg(0, s(0)) → less_out_gg
less_in_gg(s(X), s(Y)) → U5_gg(less_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg
leq_in_gg(0, s(0)) → leq_out_gg
leq_in_gg(s(X), s(Y)) → U6_gg(leq_in_gg(X, Y))
U5_gg(less_out_gg) → less_out_gg
U6_gg(leq_out_gg) → leq_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
leq_in_gg(x0, x1)
U5_gg(x0)
U6_gg(x0)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U3_GGA(X, Xs, Y, Ys, less_in_gg(Y, X))
MERGE_IN_GGA(.(X, Xs), .(Y, Ys)) → U1_GGA(X, Xs, Y, Ys, leq_in_gg(X, Y))

The remaining pairs can at least be oriented weakly.

U3_GGA(X, Xs, Y, Ys, less_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U1_GGA(X, Xs, Y, Ys, leq_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(0) = 0
POL(MERGE_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(U1_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂ + x₃ + x₄
POL(U3_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₁ + x₂ + x₄
POL(U5_gg(x₁)) = 0
POL(U6_gg(x₁)) = 0
POL(leq_in_gg(x₁, x₂)) = 0
POL(leq_out_gg) = 0
POL(less_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(less_out_gg) = 0
POL(s(x₁)) = x₁

The following usable rules [17] were oriented: none

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

U3_GGA(X, Xs, Y, Ys, less_out_gg) → MERGE_IN_GGA(.(X, Xs), Ys)
U1_GGA(X, Xs, Y, Ys, leq_out_gg) → MERGE_IN_GGA(Xs, .(Y, Ys))

The TRS R consists of the following rules:

less_in_gg(0, s(0)) → less_out_gg
less_in_gg(s(X), s(Y)) → U5_gg(less_in_gg(X, Y))
leq_in_gg(0, 0) → leq_out_gg
leq_in_gg(0, s(0)) → leq_out_gg
leq_in_gg(s(X), s(Y)) → U6_gg(leq_in_gg(X, Y))
U5_gg(less_out_gg) → less_out_gg
U6_gg(leq_out_gg) → leq_out_gg

The set Q consists of the following terms:

less_in_gg(x0, x1)
leq_in_gg(x0, x1)
U5_gg(x0)
U6_gg(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.