Left Termination proof of /tmp/tmpj5pYip/transpose-bb.pl

Left Termination of the query pattern transpose_in_2(g, g) 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,g).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_in: (b,b)
transpose_aux_in: (b,b,b)
row2col_in: (b,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X, .(C, Cs)) → U2_ggg(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(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_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X, .(C, Cs)) → U2_ggg(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

TRANSPOSE_IN_GG(A, B) → U1_GG(A, B, transpose_aux_in_ggg(A, [], B))
TRANSPOSE_IN_GG(A, B) → TRANSPOSE_AUX_IN_GGG(A, [], B)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → U2_GGG(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)
U2_GGG(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_GGG(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
U2_GGG(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X, .(C, Cs)) → U2_ggg(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7)
[] = []
row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg
transpose_out_gg(x1, x2) = transpose_out_gg
U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6)
U3_GGG(x1, x2, x3, x4, x5, x6) = U3_GGG(x6)
ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(x1, x2)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U1_GG(x1, x2, x3) = U1_GG(x3)
U4_GGAA(x1, x2, x3, x4, x5, x6, x7) = U4_GGAA(x3, x7)
TRANSPOSE_IN_GG(x1, x2) = TRANSPOSE_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:

TRANSPOSE_IN_GG(A, B) → U1_GG(A, B, transpose_aux_in_ggg(A, [], B))
TRANSPOSE_IN_GG(A, B) → TRANSPOSE_AUX_IN_GGG(A, [], B)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → U2_GGG(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)
U2_GGG(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_GGG(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
U2_GGG(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X, .(C, Cs)) → U2_ggg(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(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_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X, .(C, Cs)) → U2_ggg(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7)
[] = []
row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg
transpose_out_gg(x1, x2) = transpose_out_gg
ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(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:

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL_IN_GGAA(x1, x2, x3, x4) = ROW2COL_IN_GGAA(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:

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) → ROW2COL_IN_GGAA(Xs, Cols)

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_GGAA(.(X, Xs), .(.(X, Ys), Cols)) → ROW2COL_IN_GGAA(Xs, Cols)
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:

TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → U2_GGG(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
U2_GGG(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X, .(C, Cs)) → U2_ggg(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2) = transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3) = transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6) = U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7)
[] = []
row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6) = U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3) = transpose_aux_out_ggg
transpose_out_gg(x1, x2) = transpose_out_gg
U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, 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:

TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → U2_GGG(R, Rs, X, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
U2_GGG(R, Rs, X, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
row2col_in_ggaa(x1, x2, x3, x4) = row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7) = U4_ggaa(x3, x7)
[] = []
row2col_out_ggaa(x1, x2, x3, x4) = row2col_out_ggaa(x3, x4)
U2_GGG(x1, x2, x3, x4, x5, x6) = U2_GGG(x2, x6)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3) = TRANSPOSE_AUX_IN_GGG(x1, x2, 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:

U2_GGG(Rs, row2col_out_ggaa(Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → U2_GGG(Rs, row2col_in_ggaa(R, .(C, Cs)))

The TRS R consists of the following rules:

row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols)) → U4_ggaa(Ys, row2col_in_ggaa(Xs, Cols))
U4_ggaa(Ys, row2col_out_ggaa(Cols1, As)) → row2col_out_ggaa(.(Ys, Cols1), .([], As))
row2col_in_ggaa([], []) → row2col_out_ggaa([], [])

The set Q consists of the following terms:

row2col_in_ggaa(x0, x1)
U4_ggaa(x0, x1)

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:

U2_GGG(Rs, row2col_out_ggaa(Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X, .(C, Cs)) → U2_GGG(Rs, row2col_in_ggaa(R, .(C, Cs)))
The graph contains the following edges 1 > 1