Left Termination proof of /tmp/tmp2rkXE6/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].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, 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(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))

MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3¹(X, Xs, Y, Ys, Zs, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U5¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U3¹(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4¹(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U3¹(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1¹(X, Xs, Y, Ys, Zs, leq_in(X, Y))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN(X, Y)
LEQ_IN(s(X), s(Y)) → U6¹(X, Y, leq_in(X, Y))
LEQ_IN(s(X), s(Y)) → LEQ_IN(X, Y)
U1¹(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2¹(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
U1¹(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → MERGE_IN(Xs, .(Y, Ys), Zs)

The TRS R consists of the following rules:

merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))

The argument filtering Pi contains the following mapping:
merge_in(x1, x2, x3) = merge_in(x1, x2)
.(x1, x2) = .(x1, x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x1, x2, x3, x4, x6)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U5(x1, x2, x3) = U5(x3)
0 = 0
less_out(x1, x2) = less_out
U4(x1, x2, x3, x4, x5, x6) = U4(x3, x6)
U1(x1, x2, x3, x4, x5, x6) = U1(x1, x2, x3, x4, x6)
leq_in(x1, x2) = leq_in(x1, x2)
U6(x1, x2, x3) = U6(x3)
leq_out(x1, x2) = leq_out
U2(x1, x2, x3, x4, x5, x6) = U2(x1, x6)
[] = []
merge_out(x1, x2, x3) = merge_out(x3)
LEQ_IN(x1, x2) = LEQ_IN(x1, x2)
U6¹(x1, x2, x3) = U6¹(x3)
U5¹(x1, x2, x3) = U5¹(x3)
LESS_IN(x1, x2) = LESS_IN(x1, x2)
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x1, x6)
U4¹(x1, x2, x3, x4, x5, x6) = U4¹(x3, x6)
U1¹(x1, x2, x3, x4, x5, x6) = U1¹(x1, x2, x3, x4, x6)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(x1, x2, x3, x4, x6)

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(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3¹(X, Xs, Y, Ys, Zs, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → LESS_IN(Y, X)
LESS_IN(s(X), s(Y)) → U5¹(X, Y, less_in(X, Y))
LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)
U3¹(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4¹(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
U3¹(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1¹(X, Xs, Y, Ys, Zs, leq_in(X, Y))
MERGE_IN(.(X, Xs), .(Y, Ys), .(X, Zs)) → LEQ_IN(X, Y)
LEQ_IN(s(X), s(Y)) → U6¹(X, Y, leq_in(X, Y))
LEQ_IN(s(X), s(Y)) → LEQ_IN(X, Y)
U1¹(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2¹(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
U1¹(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → MERGE_IN(Xs, .(Y, Ys), Zs)

The TRS R consists of the following rules:

merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

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

LEQ_IN(s(X), s(Y)) → LEQ_IN(X, Y)

The TRS R consists of the following rules:

merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, 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:

LEQ_IN(s(X), s(Y)) → LEQ_IN(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:

LEQ_IN(s(X), s(Y)) → LEQ_IN(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:

LEQ_IN(s(X), s(Y)) → LEQ_IN(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:

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

The TRS R consists of the following rules:

merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, 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:

LESS_IN(s(X), s(Y)) → LESS_IN(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:

LESS_IN(s(X), s(Y)) → LESS_IN(X, Y)

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

LESS_IN(s(X), s(Y)) → LESS_IN(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(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1¹(X, Xs, Y, Ys, Zs, leq_in(X, Y))
U3¹(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
U1¹(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → MERGE_IN(Xs, .(Y, Ys), Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3¹(X, Xs, Y, Ys, Zs, less_in(Y, X))

The TRS R consists of the following rules:

merge_in(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3(X, Xs, Y, Ys, Zs, less_in(Y, X))
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))
U3(X, Xs, Y, Ys, Zs, less_out(Y, X)) → U4(X, Xs, Y, Ys, Zs, merge_in(.(X, Xs), Ys, Zs))
merge_in(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1(X, Xs, Y, Ys, Zs, leq_in(X, Y))
leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U1(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → U2(X, Xs, Y, Ys, Zs, merge_in(Xs, .(Y, Ys), Zs))
merge_in(X, [], X) → merge_out(X, [], X)
merge_in([], X, X) → merge_out([], X, X)
U2(X, Xs, Y, Ys, Zs, merge_out(Xs, .(Y, Ys), Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(X, Zs))
U4(X, Xs, Y, Ys, Zs, merge_out(.(X, Xs), Ys, Zs)) → merge_out(.(X, Xs), .(Y, Ys), .(Y, Zs))

The argument filtering Pi contains the following mapping:
merge_in(x1, x2, x3) = merge_in(x1, x2)
.(x1, x2) = .(x1, x2)
U3(x1, x2, x3, x4, x5, x6) = U3(x1, x2, x3, x4, x6)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U5(x1, x2, x3) = U5(x3)
0 = 0
less_out(x1, x2) = less_out
U4(x1, x2, x3, x4, x5, x6) = U4(x3, x6)
U1(x1, x2, x3, x4, x5, x6) = U1(x1, x2, x3, x4, x6)
leq_in(x1, x2) = leq_in(x1, x2)
U6(x1, x2, x3) = U6(x3)
leq_out(x1, x2) = leq_out
U2(x1, x2, x3, x4, x5, x6) = U2(x1, x6)
[] = []
merge_out(x1, x2, x3) = merge_out(x3)
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U1¹(x1, x2, x3, x4, x5, x6) = U1¹(x1, x2, x3, x4, x6)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(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(.(X, Xs), .(Y, Ys), .(X, Zs)) → U1¹(X, Xs, Y, Ys, Zs, leq_in(X, Y))
U3¹(X, Xs, Y, Ys, Zs, less_out(Y, X)) → MERGE_IN(.(X, Xs), Ys, Zs)
U1¹(X, Xs, Y, Ys, Zs, leq_out(X, Y)) → MERGE_IN(Xs, .(Y, Ys), Zs)
MERGE_IN(.(X, Xs), .(Y, Ys), .(Y, Zs)) → U3¹(X, Xs, Y, Ys, Zs, less_in(Y, X))

The TRS R consists of the following rules:

leq_in(s(X), s(Y)) → U6(X, Y, leq_in(X, Y))
leq_in(0, s(0)) → leq_out(0, s(0))
leq_in(0, 0) → leq_out(0, 0)
less_in(s(X), s(Y)) → U5(X, Y, less_in(X, Y))
less_in(0, s(0)) → less_out(0, s(0))
U6(X, Y, leq_out(X, Y)) → leq_out(s(X), s(Y))
U5(X, Y, less_out(X, Y)) → less_out(s(X), s(Y))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
less_in(x1, x2) = less_in(x1, x2)
s(x1) = s(x1)
U5(x1, x2, x3) = U5(x3)
0 = 0
less_out(x1, x2) = less_out
leq_in(x1, x2) = leq_in(x1, x2)
U6(x1, x2, x3) = U6(x3)
leq_out(x1, x2) = leq_out
MERGE_IN(x1, x2, x3) = MERGE_IN(x1, x2)
U1¹(x1, x2, x3, x4, x5, x6) = U1¹(x1, x2, x3, x4, x6)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(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(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys)) → U1¹(X, Xs, Y, Ys, leq_in(X, Y))
U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)
U1¹(X, Xs, Y, Ys, leq_out) → MERGE_IN(Xs, .(Y, Ys))

The TRS R consists of the following rules:

leq_in(s(X), s(Y)) → U6(leq_in(X, Y))
leq_in(0, s(0)) → leq_out
leq_in(0, 0) → leq_out
less_in(s(X), s(Y)) → U5(less_in(X, Y))
less_in(0, s(0)) → less_out
U6(leq_out) → leq_out
U5(less_out) → less_out

The set Q consists of the following terms:

leq_in(x0, x1)
less_in(x0, x1)
U6(x0)
U5(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.

U1¹(X, Xs, Y, Ys, leq_out) → MERGE_IN(Xs, .(Y, Ys))

The remaining pairs can at least be oriented weakly.

MERGE_IN(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
MERGE_IN(.(X, Xs), .(Y, Ys)) → U1¹(X, Xs, Y, Ys, leq_in(X, Y))
U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)

Used ordering: Polynomial interpretation [25]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 0
POL(MERGE_IN(x₁, x₂)) = x₁
POL(U1¹(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂
POL(U3¹(x₁, x₂, x₃, x₄, x₅)) = 1 + x₂
POL(U5(x₁)) = 0
POL(U6(x₁)) = 0
POL(leq_in(x₁, x₂)) = 0
POL(leq_out) = 0
POL(less_in(x₁, x₂)) = 0
POL(less_out) = 0
POL(s(x₁)) = 0

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:

MERGE_IN(.(X, Xs), .(Y, Ys)) → U1¹(X, Xs, Y, Ys, leq_in(X, Y))
MERGE_IN(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)

The TRS R consists of the following rules:

leq_in(s(X), s(Y)) → U6(leq_in(X, Y))
leq_in(0, s(0)) → leq_out
leq_in(0, 0) → leq_out
less_in(s(X), s(Y)) → U5(less_in(X, Y))
less_in(0, s(0)) → less_out
U6(leq_out) → leq_out
U5(less_out) → less_out

The set Q consists of the following terms:

leq_in(x0, x1)
less_in(x0, x1)
U6(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

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

MERGE_IN(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)

The TRS R consists of the following rules:

leq_in(s(X), s(Y)) → U6(leq_in(X, Y))
leq_in(0, s(0)) → leq_out
leq_in(0, 0) → leq_out
less_in(s(X), s(Y)) → U5(less_in(X, Y))
less_in(0, s(0)) → less_out
U6(leq_out) → leq_out
U5(less_out) → less_out

The set Q consists of the following terms:

leq_in(x0, x1)
less_in(x0, x1)
U6(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

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

MERGE_IN(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U5(less_in(X, Y))
less_in(0, s(0)) → less_out
U5(less_out) → less_out

The set Q consists of the following terms:

leq_in(x0, x1)
less_in(x0, x1)
U6(x0)
U5(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

leq_in(x0, x1)
U6(x0)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ UsableRulesProof

                                  ↳ QDP

                                    ↳ QReductionProof

                                      ↳ QDP

                                        ↳ QDPSizeChangeProof

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

MERGE_IN(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)

The TRS R consists of the following rules:

less_in(s(X), s(Y)) → U5(less_in(X, Y))
less_in(0, s(0)) → less_out
U5(less_out) → less_out

The set Q consists of the following terms:

less_in(x0, x1)
U5(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:

U3¹(X, Xs, Y, Ys, less_out) → MERGE_IN(.(X, Xs), Ys)
The graph contains the following edges 4 >= 2
MERGE_IN(.(X, Xs), .(Y, Ys)) → U3¹(X, Xs, Y, Ys, less_in(Y, X))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4