Left Termination proof of /tmp/tmpykI

Left Termination of the query pattern transpose_aux_in_3(a, g, a) w.r.t. the given Prolog program could not be shown:

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

transpose_aux(a,g,a).

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

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga

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

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → U1_AGA(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga
TRANSPOSE_AUX_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX_IN_AGA(x2)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_AAAGA(x8)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)
U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x6)

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

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

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → U1_AGA(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga
TRANSPOSE_AUX_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX_IN_AGA(x2)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_AAAGA(x8)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)
U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)

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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)

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

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ Instantiation

  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ Instantiation

                        ↳ QDP

                          ↳ Instantiation

  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0))) we obtained the following new rules:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

↳ Prolog

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ Instantiation

                        ↳ QDP

                          ↳ Instantiation

                            ↳ QDP

                              ↳ NonTerminationProof

  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

The TRS R consists of the following rules:none

s = ROW2COL_IN_AAAGA(.([], .([], z0))) evaluates to t =ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [z0 / .([], z0)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN_AAAGA(.([], .([], z0))) to ROW2COL_IN_AAAGA(.([], .([], .([], z0)))).

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

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga(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:

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → U1_AGA(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga(x2)
TRANSPOSE_AUX_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX_IN_AGA(x2)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_AAAGA(x6, x8)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)
U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x3, x6)

We have to consider all (P,R,Pi)-chains

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

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

TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → U1_AGA(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AAAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_AAAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga(x2)
TRANSPOSE_AUX_IN_AGA(x1, x2, x3) = TRANSPOSE_AUX_IN_AGA(x2)
U2_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8) = U2_AAAGA(x6, x8)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)
U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x3, x6)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

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

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

transpose_aux_in_aga(.(R, Rs), X, .(C, Cs)) → U1_aga(R, Rs, X, C, Cs, row2col_in_aaaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U2_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1_aga(R, Rs, X, C, Cs, row2col_out_aaaga(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out_aga(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in_aga(x1, x2, x3) = transpose_aux_in_aga(x2)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x3, x6)
row2col_in_aaaga(x1, x2, x3, x4, x5) = row2col_in_aaaga(x4)
U2_aaaga(x1, x2, x3, x4, x5, x6, x7, x8) = U2_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5) = row2col_out_aaaga(x4, x5)
.(x1, x2) = .(x1, x2)
[] = []
transpose_aux_out_aga(x1, x2, x3) = transpose_aux_out_aga(x2)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5) = ROW2COL_IN_AAAGA(x4)

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

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

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

ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ Instantiation

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

ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.([], A)) we obtained the following new rules:

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ Instantiation

                        ↳ QDP

                          ↳ Instantiation

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

ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN_AAAGA(.([], z0)) → ROW2COL_IN_AAAGA(.([], .([], z0))) we obtained the following new rules:

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

↳ Prolog

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ PiDP

              ↳ UsableRulesProof

                ↳ PiDP

                  ↳ PiDPToQDPProof

                    ↳ QDP

                      ↳ Instantiation

                        ↳ QDP

                          ↳ Instantiation

                            ↳ QDP

                              ↳ NonTerminationProof

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

ROW2COL_IN_AAAGA(.([], .([], z0))) → ROW2COL_IN_AAAGA(.([], .([], .([], z0))))

Semiunifier: [ ]
Matcher: [z0 / .([], z0)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN_AAAGA(.([], .([], z0))) to ROW2COL_IN_AAAGA(.([], .([], .([], z0)))).