Left Termination of the query pattern transpose_in_2(g, a) w.r.t. the given Prolog program could not be shown:



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ 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), A, B) :- row2col(Xs, Cols, Cols1, .([], A), B).
row2col([], [], [], A, A).

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) (f,b,b)
row2col_in: (f,b,f,b,f) (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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x2)

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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_AGAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_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)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_GGA(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_AGG(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x2)
U4_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AAAGA(x8)
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U3_AGG(x1, x2, x3, x4, x5, x6)  =  U3_AGG(x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
U4_AGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AGAGA(x8)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
ROW2COL_IN_AGAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AGAGA(x2, x4)
TRANSPOSE_IN_GA(x1, x2)  =  TRANSPOSE_IN_GA(x1)

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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_AGAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_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)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_GGA(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_AGG(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x2)
U4_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AAAGA(x8)
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U3_AGG(x1, x2, x3, x4, x5, x6)  =  U3_AGG(x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
U4_AGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AGAGA(x8)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
ROW2COL_IN_AGAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AGAGA(x2, x4)
TRANSPOSE_IN_GA(x1, x2)  =  TRANSPOSE_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 11 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP
  ↳ 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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(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
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP
  ↳ 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)  =  .
[]  =  []
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
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Instantiation
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.)

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(.) we obtained the following new rules:

ROW2COL_IN_AAAGA(.) → ROW2COL_IN_AAAGA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Instantiation
                          ↳ QDP
                            ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAGA(.) → ROW2COL_IN_AAAGA(.)

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(.) → ROW2COL_IN_AAAGA(.)

The TRS R consists of the following rules:none


s = ROW2COL_IN_AAAGA(.) evaluates to t =ROW2COL_IN_AAAGA(.)

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_AAAGA(.) to ROW2COL_IN_AAAGA(.).





↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
  ↳ PrologToPiTRSProof

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

U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x2)
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(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
  ↳ PrologToPiTRSProof

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

U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))

The TRS R consists of the following rules:

row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x3, x5)
[]  =  []
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(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
                        ↳ Narrowing
  ↳ PrologToPiTRSProof

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

TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(row2col_in_agaga(., []))
U2_AGG(row2col_out_agaga(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaga(., A) → U4_agaga(row2col_in_aaaga(.))
U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B)
row2col_in_aaaga(A) → U4_aaaga(row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A)
U4_aaaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_aaaga(., ., ., B)

The set Q consists of the following terms:

row2col_in_agaga(x0, x1)
U4_agaga(x0)
row2col_in_aaaga(x0)
U4_aaaga(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(row2col_in_agaga(., [])) at position [0] we obtained the following new rules:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))



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

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

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))
U2_AGG(row2col_out_agaga(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaga(., A) → U4_agaga(row2col_in_aaaga(.))
U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B)
row2col_in_aaaga(A) → U4_aaaga(row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A)
U4_aaaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_aaaga(., ., ., B)

The set Q consists of the following terms:

row2col_in_agaga(x0, x1)
U4_agaga(x0)
row2col_in_aaaga(x0)
U4_aaaga(x0)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
  ↳ PrologToPiTRSProof

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

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))
U2_AGG(row2col_out_agaga(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A)
U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B)
U4_aaaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_aaaga(., ., ., B)

The set Q consists of the following terms:

row2col_in_agaga(x0, x1)
U4_agaga(x0)
row2col_in_aaaga(x0)
U4_aaaga(x0)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

row2col_in_agaga(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ ForwardInstantiation
  ↳ PrologToPiTRSProof

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

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))
U2_AGG(row2col_out_agaga(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A)
U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B)
U4_aaaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_aaaga(., ., ., B)

The set Q consists of the following terms:

U4_agaga(x0)
row2col_in_aaaga(x0)
U4_aaaga(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_AGG(row2col_out_agaga(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1) we obtained the following new rules:

U2_AGG(row2col_out_agaga(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .)



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ ForwardInstantiation
                                      ↳ QDP
                                        ↳ NonTerminationProof
  ↳ PrologToPiTRSProof

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

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))
U2_AGG(row2col_out_agaga(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .)

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A)
U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B)
U4_aaaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_aaaga(., ., ., B)

The set Q consists of the following terms:

U4_agaga(x0)
row2col_in_aaaga(x0)
U4_aaaga(x0)

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 narrowing to the left:

The TRS P consists of the following rules:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))
U2_AGG(row2col_out_agaga(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .)

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A)
U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B)
U4_aaaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_aaaga(., ., ., B)


s = U2_AGG(U4_agaga(row2col_in_aaaga(A))) evaluates to t =U2_AGG(U4_agaga(row2col_in_aaaga(.)))

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



Rewriting sequence

U2_AGG(U4_agaga(row2col_in_aaaga(A))) → U2_AGG(U4_agaga(row2col_out_aaaga([], [], [], A)))
with rule row2col_in_aaaga(A') → row2col_out_aaaga([], [], [], A') at position [0,0] and matcher [A' / A]

U2_AGG(U4_agaga(row2col_out_aaaga([], [], [], A))) → U2_AGG(row2col_out_agaga(., ., A))
with rule U4_agaga(row2col_out_aaaga(Xs, Cols, Cols1, B)) → row2col_out_agaga(., ., B) at position [0] and matcher [Xs / [], Cols1 / [], Cols / [], B / A]

U2_AGG(row2col_out_agaga(., ., A)) → TRANSPOSE_AUX_IN_AGG(A, .)
with rule U2_AGG(row2col_out_agaga(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .) at position [] and matcher [x0 / ., x2 / A]

TRANSPOSE_AUX_IN_AGG(A, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))
with rule TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaga(row2col_in_aaaga(.)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.




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) (f,b,b)
row2col_in: (f,b,f,b,f) (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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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(x1, x3)
transpose_aux_in_gga(x1, x2, x3)  =  transpose_aux_in_gga(x1, x2)
.(x1, x2)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x3, x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x3, x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x3, x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x3, x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1, x2, x3)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2, x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1, 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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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(x1, x3)
transpose_aux_in_gga(x1, x2, x3)  =  transpose_aux_in_gga(x1, x2)
.(x1, x2)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x3, x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x3, x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x3, x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x3, x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1, x2, x3)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2, x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1, 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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_AGAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_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)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_GGA(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_AGG(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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(x1, x3)
transpose_aux_in_gga(x1, x2, x3)  =  transpose_aux_in_gga(x1, x2)
.(x1, x2)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x3, x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x3, x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x3, x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x3, x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1, x2, x3)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2, x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1, x2)
U4_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AAAGA(x6, x8)
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x3, x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x3, x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U3_AGG(x1, x2, x3, x4, x5, x6)  =  U3_AGG(x3, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x3, x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
U4_AGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AGAGA(x6, x8)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
ROW2COL_IN_AGAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AGAGA(x2, x4)
TRANSPOSE_IN_GA(x1, x2)  =  TRANSPOSE_IN_GA(x1)

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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_AGAGA(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
ROW2COL_IN_AGAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → ROW2COL_IN_AAAGA(Xs, Cols, Cols1, .([], A), B)
ROW2COL_IN_AAAGA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_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)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_GGA(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAGA(R, .(C, Cs), Cols1, [], Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_AGG(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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(x1, x3)
transpose_aux_in_gga(x1, x2, x3)  =  transpose_aux_in_gga(x1, x2)
.(x1, x2)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x3, x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x3, x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x3, x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x3, x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1, x2, x3)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2, x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1, x2)
U4_AAAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AAAGA(x6, x8)
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x3, x6)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x3, x6)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U3_AGG(x1, x2, x3, x4, x5, x6)  =  U3_AGG(x3, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x3, x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(x2, x3)
U4_AGAGA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_AGAGA(x6, x8)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
ROW2COL_IN_AAAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AAAGA(x4)
ROW2COL_IN_AGAGA(x1, x2, x3, x4, x5)  =  ROW2COL_IN_AGAGA(x2, x4)
TRANSPOSE_IN_GA(x1, x2)  =  TRANSPOSE_IN_GA(x1)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 2 SCCs with 11 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
              ↳ PiDP

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_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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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(x1, x3)
transpose_aux_in_gga(x1, x2, x3)  =  transpose_aux_in_gga(x1, x2)
.(x1, x2)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x3, x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x3, x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x3, x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x3, x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1, x2, x3)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2, x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1, 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
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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)  =  .
[]  =  []
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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Instantiation
              ↳ PiDP

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

ROW2COL_IN_AAAGA(A) → ROW2COL_IN_AAAGA(.)

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(.) we obtained the following new rules:

ROW2COL_IN_AAAGA(.) → ROW2COL_IN_AAAGA(.)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Instantiation
                          ↳ QDP
                            ↳ NonTerminationProof
              ↳ PiDP

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

ROW2COL_IN_AAAGA(.) → ROW2COL_IN_AAAGA(.)

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(.) → ROW2COL_IN_AAAGA(.)

The TRS R consists of the following rules:none


s = ROW2COL_IN_AAAGA(.) evaluates to t =ROW2COL_IN_AAAGA(.)

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_AAAGA(.) to ROW2COL_IN_AAAGA(.).





↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof

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

U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(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_agaga(R, .(C, Cs), Cols1, [], Accm))
row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(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) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_agaga([], [], [], A, A) → row2col_out_agaga([], [], [], A, A)
U2_gga(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_gga(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(.(R, Rs), X, .(C, Cs)) → U2_agg(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → U3_agg(R, Rs, X, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X, .(C, Cs))
U3_gga(R, Rs, X, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X, .(C, Cs))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
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(x1, x3)
transpose_aux_in_gga(x1, x2, x3)  =  transpose_aux_in_gga(x1, x2)
.(x1, x2)  =  .
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x3, x6)
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x3, x6)
transpose_aux_in_agg(x1, x2, x3)  =  transpose_aux_in_agg(x2, x3)
U2_agg(x1, x2, x3, x4, x5, x6)  =  U2_agg(x3, x6)
U3_agg(x1, x2, x3, x4, x5, x6)  =  U3_agg(x3, x6)
transpose_aux_out_agg(x1, x2, x3)  =  transpose_aux_out_agg(x1, x2, x3)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2, x3)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1, x2)
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x3, x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

U2_AGG(R, Rs, X, C, Cs, row2col_out_agaga(R, .(C, Cs), Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaga(R, .(C, Cs), Cols1, [], Accm))

The TRS R consists of the following rules:

row2col_in_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
U4_agaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_out_aaaga(Xs, Cols, Cols1, .([], A), B)) → row2col_out_agaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B)
row2col_in_aaaga(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) → U4_aaaga(X, Xs, Ys, Cols, Cols1, A, B, row2col_in_aaaga(Xs, Cols, Cols1, .([], A), B))
row2col_in_aaaga([], [], [], A, A) → row2col_out_aaaga([], [], [], A, A)
U4_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)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .
row2col_in_agaga(x1, x2, x3, x4, x5)  =  row2col_in_agaga(x2, x4)
U4_agaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_agaga(x6, x8)
row2col_in_aaaga(x1, x2, x3, x4, x5)  =  row2col_in_aaaga(x4)
U4_aaaga(x1, x2, x3, x4, x5, x6, x7, x8)  =  U4_aaaga(x6, x8)
row2col_out_aaaga(x1, x2, x3, x4, x5)  =  row2col_out_aaaga(x1, x2, x3, x4, x5)
row2col_out_agaga(x1, x2, x3, x4, x5)  =  row2col_out_agaga(x1, x2, x3, x4, x5)
[]  =  []
U2_AGG(x1, x2, x3, x4, x5, x6)  =  U2_AGG(x3, x6)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_AGG(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
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing

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

TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(X, row2col_in_agaga(., []))
U2_AGG(X, row2col_out_agaga(R, ., Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaga(., A) → U4_agaga(A, row2col_in_aaaga(.))
U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B)
row2col_in_aaaga(A) → U4_aaaga(A, row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A)
U4_aaaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_aaaga(., ., ., A, B)

The set Q consists of the following terms:

row2col_in_agaga(x0, x1)
U4_agaga(x0, x1)
row2col_in_aaaga(x0)
U4_aaaga(x0, x1)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(X, row2col_in_agaga(., [])) at position [1] we obtained the following new rules:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))



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

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

U2_AGG(X, row2col_out_agaga(R, ., Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))

The TRS R consists of the following rules:

row2col_in_agaga(., A) → U4_agaga(A, row2col_in_aaaga(.))
U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B)
row2col_in_aaaga(A) → U4_aaaga(A, row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A)
U4_aaaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_aaaga(., ., ., A, B)

The set Q consists of the following terms:

row2col_in_agaga(x0, x1)
U4_agaga(x0, x1)
row2col_in_aaaga(x0)
U4_aaaga(x0, x1)

We have to consider all (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof

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

U2_AGG(X, row2col_out_agaga(R, ., Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(A, row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A)
U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B)
U4_aaaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_aaaga(., ., ., A, B)

The set Q consists of the following terms:

row2col_in_agaga(x0, x1)
U4_agaga(x0, x1)
row2col_in_aaaga(x0)
U4_aaaga(x0, x1)

We have to consider all (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

row2col_in_agaga(x0, x1)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ ForwardInstantiation

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

U2_AGG(X, row2col_out_agaga(R, ., Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(A, row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A)
U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B)
U4_aaaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_aaaga(., ., ., A, B)

The set Q consists of the following terms:

U4_agaga(x0, x1)
row2col_in_aaaga(x0)
U4_aaaga(x0, x1)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_AGG(X, row2col_out_agaga(R, ., Cols1, [], Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1) we obtained the following new rules:

U2_AGG(x0, row2col_out_agaga(x1, ., ., [], x3)) → TRANSPOSE_AUX_IN_AGG(x3, .)



↳ Prolog
  ↳ PrologToPiTRSProof
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ UsableRulesProof
                              ↳ QDP
                                ↳ QReductionProof
                                  ↳ QDP
                                    ↳ ForwardInstantiation
                                      ↳ QDP
                                        ↳ NonTerminationProof

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

U2_AGG(x0, row2col_out_agaga(x1, ., ., [], x3)) → TRANSPOSE_AUX_IN_AGG(x3, .)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(A, row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A)
U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B)
U4_aaaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_aaaga(., ., ., A, B)

The set Q consists of the following terms:

U4_agaga(x0, x1)
row2col_in_aaaga(x0)
U4_aaaga(x0, x1)

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 narrowing to the left:

The TRS P consists of the following rules:

U2_AGG(x0, row2col_out_agaga(x1, ., ., [], x3)) → TRANSPOSE_AUX_IN_AGG(x3, .)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))

The TRS R consists of the following rules:

row2col_in_aaaga(A) → U4_aaaga(A, row2col_in_aaaga(.))
row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A)
U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B)
U4_aaaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_aaaga(., ., ., A, B)


s = U2_AGG(x0, U4_agaga([], row2col_in_aaaga(.))) evaluates to t =U2_AGG(., U4_agaga([], row2col_in_aaaga(.)))

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



Rewriting sequence

U2_AGG(x0, U4_agaga([], row2col_in_aaaga(.))) → U2_AGG(x0, U4_agaga([], row2col_out_aaaga([], [], [], ., .)))
with rule row2col_in_aaaga(A) → row2col_out_aaaga([], [], [], A, A) at position [1,1] and matcher [A / .]

U2_AGG(x0, U4_agaga([], row2col_out_aaaga([], [], [], ., .))) → U2_AGG(x0, row2col_out_agaga(., ., ., [], .))
with rule U4_agaga(A, row2col_out_aaaga(Xs, Cols, Cols1, ., B)) → row2col_out_agaga(., ., ., A, B) at position [1] and matcher [Xs / [], Cols1 / [], Cols / [], B / ., A / []]

U2_AGG(x0, row2col_out_agaga(., ., ., [], .)) → TRANSPOSE_AUX_IN_AGG(., .)
with rule U2_AGG(x0', row2col_out_agaga(x1, ., ., [], x3)) → TRANSPOSE_AUX_IN_AGG(x3, .) at position [] and matcher [x1 / ., x3 / ., x0' / x0]

TRANSPOSE_AUX_IN_AGG(., .) → U2_AGG(., U4_agaga([], row2col_in_aaaga(.)))
with rule TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaga([], row2col_in_aaaga(.)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence


All these steps are and every following step will be a correct step w.r.t to Q.