Left Termination proof of /tmp/tmplHKzNt/rotate.pl

Left Termination of the query pattern rotate_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

rotate(X, Y) :- ','(append2(A, B, X), append1(B, A, Y)).
append1(.(X, Xs), Ys, .(X, Zs)) :- append1(Xs, Ys, Zs).
append1([], Ys, Ys).
append2(.(X, Xs), Ys, .(X, Zs)) :- append2(Xs, Ys, Zs).
append2([], Ys, Ys).

Queries:

rotate(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
rotate_in: (b,f)
append2_in: (f,f,b)
append1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

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:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)

Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

ROTATE_IN_GA(X, Y) → U1_GA(X, Y, append2_in_aag(A, B, X))
ROTATE_IN_GA(X, Y) → APPEND2_IN_AAG(A, B, X)
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)
U1_GA(X, Y, append2_out_aag(A, B, X)) → U2_GA(X, Y, append1_in_gga(B, A, Y))
U1_GA(X, Y, append2_out_aag(A, B, X)) → APPEND1_IN_GGA(B, A, Y)
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x5)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x1, x5)
ROTATE_IN_GA(x1, x2) = ROTATE_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
U1_GA(x1, x2, x3) = U1_GA(x3)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(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:

ROTATE_IN_GA(X, Y) → U1_GA(X, Y, append2_in_aag(A, B, X))
ROTATE_IN_GA(X, Y) → APPEND2_IN_AAG(A, B, X)
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → U4_AAG(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)
U1_GA(X, Y, append2_out_aag(A, B, X)) → U2_GA(X, Y, append1_in_gga(B, A, Y))
U1_GA(X, Y, append2_out_aag(A, B, X)) → APPEND1_IN_GGA(B, A, Y)
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → U3_GGA(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x5)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)
U4_AAG(x1, x2, x3, x4, x5) = U4_AAG(x1, x5)
ROTATE_IN_GA(x1, x2) = ROTATE_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
U1_GA(x1, x2, x3) = U1_GA(x3)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 6 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(x1, x2)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

APPEND1_IN_GGA(.(X, Xs), Ys, .(X, Zs)) → APPEND1_IN_GGA(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND1_IN_GGA(x1, x2, x3) = APPEND1_IN_GGA(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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

APPEND1_IN_GGA(.(X, Xs), Ys) → APPEND1_IN_GGA(Xs, Ys)

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:

APPEND1_IN_GGA(.(X, Xs), Ys) → APPEND1_IN_GGA(Xs, Ys)
The graph contains the following edges 1 > 1, 2 >= 2

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)

The TRS R consists of the following rules:

rotate_in_ga(X, Y) → U1_ga(X, Y, append2_in_aag(A, B, X))
append2_in_aag(.(X, Xs), Ys, .(X, Zs)) → U4_aag(X, Xs, Ys, Zs, append2_in_aag(Xs, Ys, Zs))
append2_in_aag([], Ys, Ys) → append2_out_aag([], Ys, Ys)
U4_aag(X, Xs, Ys, Zs, append2_out_aag(Xs, Ys, Zs)) → append2_out_aag(.(X, Xs), Ys, .(X, Zs))
U1_ga(X, Y, append2_out_aag(A, B, X)) → U2_ga(X, Y, append1_in_gga(B, A, Y))
append1_in_gga(.(X, Xs), Ys, .(X, Zs)) → U3_gga(X, Xs, Ys, Zs, append1_in_gga(Xs, Ys, Zs))
append1_in_gga([], Ys, Ys) → append1_out_gga([], Ys, Ys)
U3_gga(X, Xs, Ys, Zs, append1_out_gga(Xs, Ys, Zs)) → append1_out_gga(.(X, Xs), Ys, .(X, Zs))
U2_ga(X, Y, append1_out_gga(B, A, Y)) → rotate_out_ga(X, Y)

The argument filtering Pi contains the following mapping:
rotate_in_ga(x1, x2) = rotate_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
append2_in_aag(x1, x2, x3) = append2_in_aag(x3)
.(x1, x2) = .(x1, x2)
U4_aag(x1, x2, x3, x4, x5) = U4_aag(x1, x5)
append2_out_aag(x1, x2, x3) = append2_out_aag(x1, x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
append1_in_gga(x1, x2, x3) = append1_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
[] = []
append1_out_gga(x1, x2, x3) = append1_out_gga(x3)
rotate_out_ga(x1, x2) = rotate_out_ga(x2)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)

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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

APPEND2_IN_AAG(.(X, Xs), Ys, .(X, Zs)) → APPEND2_IN_AAG(Xs, Ys, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND2_IN_AAG(x1, x2, x3) = APPEND2_IN_AAG(x3)

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

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

APPEND2_IN_AAG(.(X, Zs)) → APPEND2_IN_AAG(Zs)

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

APPEND2_IN_AAG(.(X, Zs)) → APPEND2_IN_AAG(Zs)
The graph contains the following edges 1 > 1