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].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out


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(.(R, Rs), X, .(C, Cs)) → U11(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), A, B) → U21(X, Xs, Ys, Cols, Cols1, A, B, row2col_in(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2)
U21(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21(x8)
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(x4)
U11(x1, x2, x3, x4, x5, x6)  =  U11(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(.(R, Rs), X, .(C, Cs)) → U11(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), A, B) → U21(X, Xs, Ys, Cols, Cols1, A, B, row2col_in(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2)
U21(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21(x8)
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(x4)
U11(x1, x2, x3, x4, x5, x6)  =  U11(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(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(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(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(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(A) → ROW2COL_IN(.([], A))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN(A) → ROW2COL_IN(.([], A)) we obtained the following new rules:

ROW2COL_IN(.([], z0)) → ROW2COL_IN(.([], .([], 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(.([], z0)) → ROW2COL_IN(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN(.([], z0)) → ROW2COL_IN(.([], .([], z0))) we obtained the following new rules:

ROW2COL_IN(.([], .([], z0))) → ROW2COL_IN(.([], .([], .([], 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(.([], .([], z0))) → ROW2COL_IN(.([], .([], .([], 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(.([], .([], z0))) → ROW2COL_IN(.([], .([], .([], z0))))

The TRS R consists of the following rules:none


s = ROW2COL_IN(.([], .([], z0))) evaluates to t =ROW2COL_IN(.([], .([], .([], z0))))

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN(.([], .([], z0))) to ROW2COL_IN(.([], .([], .([], z0)))).




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_aux_in(.(R, Rs), X, .(C, Cs)) → U1(R, Rs, X, C, Cs, row2col_in(R, .(C, Cs), Cols1, [], Accm))
row2col_in(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U2(X, Xs, Ys, Cols, Cols1, A, B, row2col_in(Xs, Cols, Cols1, .([], A), B))
U2(X, Xs, Ys, Cols, Cols1, A, B, row2col_out(Xs, Cols, Cols1, .([], A), B)) → row2col_out(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
U1(R, Rs, X, C, Cs, row2col_out(R, .(C, Cs), Cols1, [], Accm)) → transpose_aux_out(.(R, Rs), X, .(C, Cs))

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x3, x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x6, x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x4, x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x2)

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x3, x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x6, x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x4, x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(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(.(R, Rs), X, .(C, Cs)) → U11(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), A, B) → U21(X, Xs, Ys, Cols, Cols1, A, B, row2col_in(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x3, x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x6, x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x4, x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x2)
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2)
U21(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21(x6, x8)
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(x4)
U11(x1, x2, x3, x4, x5, x6)  =  U11(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(.(R, Rs), X, .(C, Cs)) → U11(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), A, B) → U21(X, Xs, Ys, Cols, Cols1, A, B, row2col_in(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x3, x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x6, x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x4, x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x2)
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2)
U21(x1, x2, x3, x4, x5, x6, x7, x8)  =  U21(x6, x8)
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(x4)
U11(x1, x2, x3, x4, x5, x6)  =  U11(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(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2)
.(x1, x2)  =  .(x1, x2)
U1(x1, x2, x3, x4, x5, x6)  =  U1(x3, x6)
row2col_in(x1, x2, x3, x4, x5)  =  row2col_in(x4)
[]  =  []
U2(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2(x6, x8)
row2col_out(x1, x2, x3, x4, x5)  =  row2col_out(x4, x5)
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x2)
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(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(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN(Xs, Cols, Cols1, .([], A), B)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN(x1, x2, x3, x4, x5)  =  ROW2COL_IN(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ Instantiation

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

ROW2COL_IN(A) → ROW2COL_IN(.([], A))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN(A) → ROW2COL_IN(.([], A)) we obtained the following new rules:

ROW2COL_IN(.([], z0)) → ROW2COL_IN(.([], .([], 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(.([], z0)) → ROW2COL_IN(.([], .([], z0)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By instantiating [15] the rule ROW2COL_IN(.([], z0)) → ROW2COL_IN(.([], .([], z0))) we obtained the following new rules:

ROW2COL_IN(.([], .([], z0))) → ROW2COL_IN(.([], .([], .([], 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(.([], .([], z0))) → ROW2COL_IN(.([], .([], .([], 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(.([], .([], z0))) → ROW2COL_IN(.([], .([], .([], z0))))

The TRS R consists of the following rules:none


s = ROW2COL_IN(.([], .([], z0))) evaluates to t =ROW2COL_IN(.([], .([], .([], z0))))

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



Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from ROW2COL_IN(.([], .([], z0))) to ROW2COL_IN(.([], .([], .([], z0)))).