Left Termination proof of /tmp/tmponVi01/transpose-bf.pl

Left Termination of the query pattern transpose_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

↳ Prolog

  ↳ PrologToPiTRSProof

Clauses:

transpose(A, B) :- transpose_aux(A, [], B).
transpose_aux(.(R, Rs), X, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)).
transpose_aux([], X, X).
row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) :- row2col(Xs, Cols, Cols1, As).
row2col([], [], [], []).

Queries:

transpose(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:

transpose_in(A, B) → U1(A, B, transpose_aux_in(A, [], B))
transpose_aux_in([], X, X) → transpose_aux_out([], X, X)
transpose_aux_in(.(R, Rs), X, .(C, Cs)) → U2(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
row2col_in([], [], [], []) → row2col_out([], [], [], [])
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U3(R, Rs, X, C, Cs, transpose_aux_out(Rs, Accm, Cols1)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))
U1(A, B, transpose_aux_out(A, [], B)) → transpose_out(A, B)

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:

transpose_in(A, B) → U1(A, B, transpose_aux_in(A, [], B))
transpose_aux_in([], X, X) → transpose_aux_out([], X, X)
transpose_aux_in(.(R, Rs), X, .(C, Cs)) → U2(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
row2col_in([], [], [], []) → row2col_out([], [], [], [])
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U3(R, Rs, X, C, Cs, transpose_aux_out(Rs, Accm, Cols1)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))
U1(A, B, transpose_aux_out(A, [], B)) → transpose_out(A, B)

TRANSPOSE_IN(A, B) → U1¹(A, B, transpose_aux_in(A, [], B))
TRANSPOSE_IN(A, B) → TRANSPOSE_AUX_IN(A, [], B)
TRANSPOSE_AUX_IN(.(R, Rs), X, .(C, Cs)) → U2¹(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4¹(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN(Xs, Cols, Cols1, As)
U2¹(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3¹(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U2¹(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in(A, B) → U1(A, B, transpose_aux_in(A, [], B))
transpose_aux_in([], X, X) → transpose_aux_out([], X, X)
transpose_aux_in(.(R, Rs), X, .(C, Cs)) → U2(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
row2col_in([], [], [], []) → row2col_out([], [], [], [])
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U3(R, Rs, X, C, Cs, transpose_aux_out(Rs, Accm, Cols1)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))
U1(A, B, transpose_aux_out(A, [], B)) → transpose_out(A, B)

The argument filtering Pi contains the following mapping:
transpose_in(x1, x2) = transpose_in(x1)
U1(x1, x2, x3) = U1(x3)
transpose_aux_in(x1, x2, x3) = transpose_aux_in(x1, x2)
[] = []
transpose_aux_out(x1, x2, x3) = transpose_aux_out
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x2, x6)
row2col_in(x1, x2, x3, x4) = row2col_in(x1)
row2col_out(x1, x2, x3, x4) = row2col_out(x4)
U4(x1, x2, x3, x4, x5, x6, x7) = U4(x7)
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
transpose_out(x1, x2) = transpose_out
TRANSPOSE_AUX_IN(x1, x2, x3) = TRANSPOSE_AUX_IN(x1, x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x2, x6)
TRANSPOSE_IN(x1, x2) = TRANSPOSE_IN(x1)
U4¹(x1, x2, x3, x4, x5, x6, x7) = U4¹(x7)
ROW2COL_IN(x1, x2, x3, x4) = ROW2COL_IN(x1)
U3¹(x1, x2, x3, x4, x5, x6) = U3¹(x6)
U1¹(x1, x2, x3) = U1¹(x3)

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:

TRANSPOSE_IN(A, B) → U1¹(A, B, transpose_aux_in(A, [], B))
TRANSPOSE_IN(A, B) → TRANSPOSE_AUX_IN(A, [], B)
TRANSPOSE_AUX_IN(.(R, Rs), X, .(C, Cs)) → U2¹(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4¹(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN(Xs, Cols, Cols1, As)
U2¹(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3¹(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U2¹(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in(A, B) → U1(A, B, transpose_aux_in(A, [], B))
transpose_aux_in([], X, X) → transpose_aux_out([], X, X)
transpose_aux_in(.(R, Rs), X, .(C, Cs)) → U2(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
row2col_in([], [], [], []) → row2col_out([], [], [], [])
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U3(R, Rs, X, C, Cs, transpose_aux_out(Rs, Accm, Cols1)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))
U1(A, B, transpose_aux_out(A, [], B)) → transpose_out(A, B)

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in(A, B) → U1(A, B, transpose_aux_in(A, [], B))
transpose_aux_in([], X, X) → transpose_aux_out([], X, X)
transpose_aux_in(.(R, Rs), X, .(C, Cs)) → U2(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
row2col_in([], [], [], []) → row2col_out([], [], [], [])
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U3(R, Rs, X, C, Cs, transpose_aux_out(Rs, Accm, Cols1)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))
U1(A, B, transpose_aux_out(A, [], B)) → transpose_out(A, B)

The argument filtering Pi contains the following mapping:
transpose_in(x1, x2) = transpose_in(x1)
U1(x1, x2, x3) = U1(x3)
transpose_aux_in(x1, x2, x3) = transpose_aux_in(x1, x2)
[] = []
transpose_aux_out(x1, x2, x3) = transpose_aux_out
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x2, x6)
row2col_in(x1, x2, x3, x4) = row2col_in(x1)
row2col_out(x1, x2, x3, x4) = row2col_out(x4)
U4(x1, x2, x3, x4, x5, x6, x7) = U4(x7)
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
transpose_out(x1, x2) = transpose_out
ROW2COL_IN(x1, x2, x3, x4) = ROW2COL_IN(x1)

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:

ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
ROW2COL_IN(x1, x2, x3, x4) = ROW2COL_IN(x1)

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:

ROW2COL_IN(.(X, Xs)) → ROW2COL_IN(Xs)

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:

ROW2COL_IN(.(X, Xs)) → ROW2COL_IN(Xs)
The graph contains the following edges 1 > 1

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

U2¹(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN(.(R, Rs), X, .(C, Cs)) → U2¹(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in(A, B) → U1(A, B, transpose_aux_in(A, [], B))
transpose_aux_in([], X, X) → transpose_aux_out([], X, X)
transpose_aux_in(.(R, Rs), X, .(C, Cs)) → U2(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))
row2col_in([], [], [], []) → row2col_out([], [], [], [])
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → U3(R, Rs, X, C, Cs, transpose_aux_in(Rs, Accm, Cols1))
U3(R, Rs, X, C, Cs, transpose_aux_out(Rs, Accm, Cols1)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))
U1(A, B, transpose_aux_out(A, [], B)) → transpose_out(A, B)

The argument filtering Pi contains the following mapping:
transpose_in(x1, x2) = transpose_in(x1)
U1(x1, x2, x3) = U1(x3)
transpose_aux_in(x1, x2, x3) = transpose_aux_in(x1, x2)
[] = []
transpose_aux_out(x1, x2, x3) = transpose_aux_out
.(x1, x2) = .(x1, x2)
U2(x1, x2, x3, x4, x5, x6) = U2(x2, x6)
row2col_in(x1, x2, x3, x4) = row2col_in(x1)
row2col_out(x1, x2, x3, x4) = row2col_out(x4)
U4(x1, x2, x3, x4, x5, x6, x7) = U4(x7)
U3(x1, x2, x3, x4, x5, x6) = U3(x6)
transpose_out(x1, x2) = transpose_out
TRANSPOSE_AUX_IN(x1, x2, x3) = TRANSPOSE_AUX_IN(x1, x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x2, 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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U2¹(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN(.(R, Rs), X, .(C, Cs)) → U2¹(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4(X, Xs, Ys, Cols, Cols1, As, row2col_in(Xs, Cols, Cols1, As))
U4(X, Xs, Ys, Cols, Cols1, As, row2col_out(Xs, Cols, Cols1, As)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in([], [], [], []) → row2col_out([], [], [], [])

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
row2col_in(x1, x2, x3, x4) = row2col_in(x1)
row2col_out(x1, x2, x3, x4) = row2col_out(x4)
U4(x1, x2, x3, x4, x5, x6, x7) = U4(x7)
TRANSPOSE_AUX_IN(x1, x2, x3) = TRANSPOSE_AUX_IN(x1, x2)
U2¹(x1, x2, x3, x4, x5, x6) = U2¹(x2, 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

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

TRANSPOSE_AUX_IN(.(R, Rs), X) → U2¹(Rs, row2col_in(R))
U2¹(Rs, row2col_out(Accm)) → TRANSPOSE_AUX_IN(Rs, Accm)

The TRS R consists of the following rules:

row2col_in(.(X, Xs)) → U4(row2col_in(Xs))
U4(row2col_out(As)) → row2col_out(.([], As))
row2col_in([]) → row2col_out([])

The set Q consists of the following terms:

row2col_in(x0)
U4(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:

TRANSPOSE_AUX_IN(.(R, Rs), X) → U2¹(Rs, row2col_in(R))
The graph contains the following edges 1 > 1
U2¹(Rs, row2col_out(Accm)) → TRANSPOSE_AUX_IN(Rs, Accm)
The graph contains the following edges 1 >= 1, 2 > 2