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

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)

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

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)

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


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_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(x2)
U1(x1, x2, x3)  =  U1(x3)
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2, x3)
[]  =  []
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x1)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
row2col_in(x1, x2, x3, x4)  =  row2col_in(x2)
row2col_out(x1, x2, x3, x4)  =  row2col_out(x1, x3, x4)
U4(x1, x2, x3, x4, x5, x6, x7)  =  U4(x1, x3, x7)
U3(x1, x2, x3, x4, x5, x6)  =  U3(x1, x6)
transpose_out(x1, x2)  =  transpose_out(x1)
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2, x3)
U21(x1, x2, x3, x4, x5, x6)  =  U21(x6)
TRANSPOSE_IN(x1, x2)  =  TRANSPOSE_IN(x2)
U41(x1, x2, x3, x4, x5, x6, x7)  =  U41(x1, x3, x7)
ROW2COL_IN(x1, x2, x3, x4)  =  ROW2COL_IN(x2)
U31(x1, x2, x3, x4, x5, x6)  =  U31(x1, 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)

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

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 5 less nodes.

↳ 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(x2)
U1(x1, x2, x3)  =  U1(x3)
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2, x3)
[]  =  []
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x1)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
row2col_in(x1, x2, x3, x4)  =  row2col_in(x2)
row2col_out(x1, x2, x3, x4)  =  row2col_out(x1, x3, x4)
U4(x1, x2, x3, x4, x5, x6, x7)  =  U4(x1, x3, x7)
U3(x1, x2, x3, x4, x5, x6)  =  U3(x1, x6)
transpose_out(x1, x2)  =  transpose_out(x1)
ROW2COL_IN(x1, x2, x3, x4)  =  ROW2COL_IN(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(.(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(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(.(.(X, Ys), Cols)) → ROW2COL_IN(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: 



↳ 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(x2)
U1(x1, x2, x3)  =  U1(x3)
transpose_aux_in(x1, x2, x3)  =  transpose_aux_in(x2, x3)
[]  =  []
transpose_aux_out(x1, x2, x3)  =  transpose_aux_out(x1)
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5, x6)  =  U2(x6)
row2col_in(x1, x2, x3, x4)  =  row2col_in(x2)
row2col_out(x1, x2, x3, x4)  =  row2col_out(x1, x3, x4)
U4(x1, x2, x3, x4, x5, x6, x7)  =  U4(x1, x3, x7)
U3(x1, x2, x3, x4, x5, x6)  =  U3(x1, x6)
transpose_out(x1, x2)  =  transpose_out(x1)
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2, x3)
U21(x1, x2, x3, x4, x5, x6)  =  U21(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(x2)
row2col_out(x1, x2, x3, x4)  =  row2col_out(x1, x3, x4)
U4(x1, x2, x3, x4, x5, x6, x7)  =  U4(x1, x3, x7)
TRANSPOSE_AUX_IN(x1, x2, x3)  =  TRANSPOSE_AUX_IN(x2, x3)
U21(x1, x2, x3, x4, x5, x6)  =  U21(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
                        ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

row2col_in(x0)
U4(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


U21(row2col_out(R, Cols1, Accm)) → TRANSPOSE_AUX_IN(Accm, Cols1)
The remaining pairs can at least be oriented weakly.

TRANSPOSE_AUX_IN(X, .(C, Cs)) → U21(row2col_in(.(C, Cs)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( U4(x1, ..., x3) ) =
/1\
\1/
+
/01\
\01/
·x1+
/10\
\00/
·x2+
/00\
\00/
·x3

M( row2col_in(x1) ) =
/0\
\0/
+
/01\
\01/
·x1

M( [] ) =
/0\
\1/

M( .(x1, x2) ) =
/1\
\0/
+
/01\
\10/
·x1+
/10\
\00/
·x2

M( row2col_out(x1, ..., x3) ) =
/1\
\0/
+
/00\
\00/
·x1+
/01\
\00/
·x2+
/00\
\00/
·x3

Tuple symbols:
M( TRANSPOSE_AUX_IN(x1, x2) ) = 0+
[0,0]
·x1+
[0,1]
·x2

M( U21(x1) ) = 0+
[1,0]
·x1


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ DependencyGraphProof

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

TRANSPOSE_AUX_IN(X, .(C, Cs)) → U21(row2col_in(.(C, Cs)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

row2col_in(x0)
U4(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.