Left Termination proof of /tmp/tmpF0ZG9T/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]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_in: (b,f)
transpose_aux_in: (b,b,f)
row2col_in: (b,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
U1_ga(A, B, transpose_aux_out_gga(A, [], B)) → transpose_out_ga(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_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
U1_ga(A, B, transpose_aux_out_gga(A, [], B)) → transpose_out_ga(A, B)

TRANSPOSE_IN_GA(A, B) → U1_GA(A, B, transpose_aux_in_gga(A, [], B))
TRANSPOSE_IN_GA(A, B) → TRANSPOSE_AUX_IN_GGA(A, [], B)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → U2_GGA(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_GAAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_GGA(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
U2_GGA(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
U1_ga(A, B, transpose_aux_out_gga(A, [], B)) → transpose_out_ga(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ga(x1, x2) = transpose_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
transpose_aux_in_gga(x1, x2, x3) = transpose_aux_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4) = row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaaa(x7)
[] = []
row2col_out_gaaa(x1, x2, x3, x4) = row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3) = transpose_aux_out_gga
transpose_out_ga(x1, x2) = transpose_out_ga
U4_GAAA(x1, x2, x3, x4, x5, x6, x7) = U4_GAAA(x7)
ROW2COL_IN_GAAA(x1, x2, x3, x4) = ROW2COL_IN_GAAA(x1)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3) = TRANSPOSE_AUX_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x6)
U1_GA(x1, x2, x3) = U1_GA(x3)
TRANSPOSE_IN_GA(x1, x2) = TRANSPOSE_IN_GA(x1)

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_GA(A, B) → U1_GA(A, B, transpose_aux_in_gga(A, [], B))
TRANSPOSE_IN_GA(A, B) → TRANSPOSE_AUX_IN_GGA(A, [], B)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → U2_GGA(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_GAAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_GGA(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
U2_GGA(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
U1_ga(A, B, transpose_aux_out_gga(A, [], B)) → transpose_out_ga(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_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
U1_ga(A, B, transpose_aux_out_gga(A, [], B)) → transpose_out_ga(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ga(x1, x2) = transpose_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
transpose_aux_in_gga(x1, x2, x3) = transpose_aux_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4) = row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaaa(x7)
[] = []
row2col_out_gaaa(x1, x2, x3, x4) = row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3) = transpose_aux_out_gga
transpose_out_ga(x1, x2) = transpose_out_ga
ROW2COL_IN_GAAA(x1, x2, x3, x4) = ROW2COL_IN_GAAA(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_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL_IN_GAAA(x1, x2, x3, x4) = ROW2COL_IN_GAAA(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_GAAA(.(X, Xs)) → ROW2COL_IN_GAAA(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_GAAA(.(X, Xs)) → ROW2COL_IN_GAAA(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:

U2_GGA(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → U2_GGA(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
U1_ga(A, B, transpose_aux_out_gga(A, [], B)) → transpose_out_ga(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ga(x1, x2) = transpose_in_ga(x1)
U1_ga(x1, x2, x3) = U1_ga(x3)
transpose_aux_in_gga(x1, x2, x3) = transpose_aux_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4) = row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaaa(x7)
[] = []
row2col_out_gaaa(x1, x2, x3, x4) = row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3) = transpose_aux_out_gga
transpose_out_ga(x1, x2) = transpose_out_ga
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3) = TRANSPOSE_AUX_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

U2_GGA(R, Rs, X, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → U2_GGA(R, Rs, X, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
row2col_in_gaaa(x1, x2, x3, x4) = row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaaa(x7)
[] = []
row2col_out_gaaa(x1, x2, x3, x4) = row2col_out_gaaa(x4)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x2, x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3) = TRANSPOSE_AUX_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

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

TRANSPOSE_AUX_IN_GGA(.(R, Rs), X) → U2_GGA(Rs, row2col_in_gaaa(R))
U2_GGA(Rs, row2col_out_gaaa(Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm)

The TRS R consists of the following rules:

row2col_in_gaaa(.(X, Xs)) → U4_gaaa(row2col_in_gaaa(Xs))
U4_gaaa(row2col_out_gaaa(As)) → row2col_out_gaaa(.([], As))
row2col_in_gaaa([]) → row2col_out_gaaa([])

The set Q consists of the following terms:

row2col_in_gaaa(x0)
U4_gaaa(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_GGA(.(R, Rs), X) → U2_GGA(Rs, row2col_in_gaaa(R))
The graph contains the following edges 1 > 1
U2_GGA(Rs, row2col_out_gaaa(Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm)
The graph contains the following edges 1 >= 1, 2 > 2