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), .([], As)) :- row2col(Xs, Cols, Cols1, As).
row2col([], [], [], []).

Queries:

transpose(g,a).

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

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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)
ROW2COL_IN_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA
U4_AAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AAAA(x7)
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_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AGAA(x7)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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)
ROW2COL_IN_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA
U4_AAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AAAA(x7)
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_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AGAA(x7)
U1_GA(x1, x2, x3)  =  U1_GA(x3)
ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
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_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA

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_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .
[]  =  []
ROW2COL_IN_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA

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
                        ↳ NonTerminationProof
              ↳ PiDP
  ↳ PrologToPiTRSProof

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

ROW2COL_IN_AAAA → ROW2COL_IN_AAAA

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_AAAA → ROW2COL_IN_AAAA

The TRS R consists of the following rules:none


s = ROW2COL_IN_AAAA evaluates to t =ROW2COL_IN_AAAA

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_AAAA to ROW2COL_IN_AAAA.





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

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

TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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:

TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .
row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x3, x4)
[]  =  []
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:

U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(row2col_in_agaa(.))

The TRS R consists of the following rules:

row2col_in_agaa(.) → U4_agaa(row2col_in_aaaa)
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .)
row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(row2col_in_agaa(.)) at position [0] we obtained the following new rules:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaa(row2col_in_aaaa))



↳ 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_agaa(row2col_in_aaaa))
U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaa(.) → U4_agaa(row2col_in_aaaa)
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .)
row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(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_agaa(row2col_in_aaaa))
U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(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_agaa(x0)



↳ 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:

U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaa(row2col_in_aaaa))

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1) we obtained the following new rules:

U2_AGG(row2col_out_agaa(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_agaa(row2col_in_aaaa))
U2_AGG(row2col_out_agaa(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .)

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(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_agaa(row2col_in_aaaa))
U2_AGG(row2col_out_agaa(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .)

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)


s = U2_AGG(U4_agaa(row2col_in_aaaa)) evaluates to t =U2_AGG(U4_agaa(row2col_in_aaaa))

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



Rewriting sequence

U2_AGG(U4_agaa(row2col_in_aaaa)) → U2_AGG(U4_agaa(row2col_out_aaaa([], [], [], [])))
with rule row2col_in_aaaa → row2col_out_aaaa([], [], [], []) at position [0,0] and matcher [ ]

U2_AGG(U4_agaa(row2col_out_aaaa([], [], [], []))) → U2_AGG(row2col_out_agaa(., ., .))
with rule U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., .) at position [0] and matcher [Xs / [], Cols1 / [], Cols / [], As / []]

U2_AGG(row2col_out_agaa(., ., .)) → TRANSPOSE_AUX_IN_AGG(., .)
with rule U2_AGG(row2col_out_agaa(x0, ., x2)) → TRANSPOSE_AUX_IN_AGG(x2, .) at position [] and matcher [x0 / ., x2 / .]

TRANSPOSE_AUX_IN_AGG(., .) → U2_AGG(U4_agaa(row2col_in_aaaa))
with rule TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(U4_agaa(row2col_in_aaaa))

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,f) (f,f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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)
ROW2COL_IN_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA
U4_AAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AAAA(x7)
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_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AGAA(x7)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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)
ROW2COL_IN_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA
U4_AAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AAAA(x7)
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_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_AGAA(x7)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
ROW2COL_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL_IN_AGAA(x2)
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_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X, .(C, Cs)) → U2_gga(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA

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_AAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AAAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .
[]  =  []
ROW2COL_IN_AAAA(x1, x2, x3, x4)  =  ROW2COL_IN_AAAA

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
                        ↳ NonTerminationProof
              ↳ PiDP

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

ROW2COL_IN_AAAA → ROW2COL_IN_AAAA

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_AAAA → ROW2COL_IN_AAAA

The TRS R consists of the following rules:none


s = ROW2COL_IN_AAAA evaluates to t =ROW2COL_IN_AAAA

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_AAAA to ROW2COL_IN_AAAA.





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

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

TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U2_gga(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(R, .(C, Cs), Cols1, Accm))
U2_agg(R, Rs, X, C, Cs, row2col_out_agaa(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_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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:

TRANSPOSE_AUX_IN_AGG(.(R, Rs), X, .(C, Cs)) → U2_AGG(R, Rs, X, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
U2_AGG(R, Rs, X, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_aaaa(Xs, Cols, Cols1, As))
row2col_in_aaaa([], [], [], []) → row2col_out_aaaa([], [], [], [])
U4_aaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .
row2col_in_agaa(x1, x2, x3, x4)  =  row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_agaa(x7)
row2col_in_aaaa(x1, x2, x3, x4)  =  row2col_in_aaaa
U4_aaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_aaaa(x7)
row2col_out_aaaa(x1, x2, x3, x4)  =  row2col_out_aaaa(x1, x2, x3, x4)
row2col_out_agaa(x1, x2, x3, x4)  =  row2col_out_agaa(x1, x2, x3, x4)
[]  =  []
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_agaa(.))
U2_AGG(X, row2col_out_agaa(R, ., Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaa(.) → U4_agaa(row2col_in_aaaa)
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .)
row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(x0)

We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule TRANSPOSE_AUX_IN_AGG(X, .) → U2_AGG(X, row2col_in_agaa(.)) at position [1] we obtained the following new rules:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))



↳ 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:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))
U2_AGG(X, row2col_out_agaa(R, ., Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_agaa(.) → U4_agaa(row2col_in_aaaa)
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .)
row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(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
  ↳ 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:

TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))
U2_AGG(X, row2col_out_agaa(R, ., Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(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_agaa(x0)



↳ 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_agaa(R, ., Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(x0)

We have to consider all (P,Q,R)-chains.
By forward instantiating [14] the rule U2_AGG(X, row2col_out_agaa(R, ., Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1) we obtained the following new rules:

U2_AGG(x0, row2col_out_agaa(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_agaa(x1, ., ., x3)) → TRANSPOSE_AUX_IN_AGG(x3, .)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)

The set Q consists of the following terms:

U4_agaa(x0)
row2col_in_aaaa
U4_aaaa(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:

U2_AGG(x0, row2col_out_agaa(x1, ., ., x3)) → TRANSPOSE_AUX_IN_AGG(x3, .)
TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))

The TRS R consists of the following rules:

row2col_in_aaaa → U4_aaaa(row2col_in_aaaa)
row2col_in_aaaa → row2col_out_aaaa([], [], [], [])
U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .)
U4_aaaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_aaaa(., ., ., .)


s = U2_AGG(x0, U4_agaa(row2col_in_aaaa)) evaluates to t =U2_AGG(., U4_agaa(row2col_in_aaaa))

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



Rewriting sequence

U2_AGG(x0, U4_agaa(row2col_in_aaaa)) → U2_AGG(x0, U4_agaa(row2col_out_aaaa([], [], [], [])))
with rule row2col_in_aaaa → row2col_out_aaaa([], [], [], []) at position [1,0] and matcher [ ]

U2_AGG(x0, U4_agaa(row2col_out_aaaa([], [], [], []))) → U2_AGG(x0, row2col_out_agaa(., ., ., .))
with rule U4_agaa(row2col_out_aaaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(., ., ., .) at position [1] and matcher [Xs / [], Cols1 / [], Cols / [], As / []]

U2_AGG(x0, row2col_out_agaa(., ., ., .)) → TRANSPOSE_AUX_IN_AGG(., .)
with rule U2_AGG(x0', row2col_out_agaa(x1, ., ., x3)) → TRANSPOSE_AUX_IN_AGG(x3, .) at position [] and matcher [x1 / ., x3 / ., x0' / x0]

TRANSPOSE_AUX_IN_AGG(., .) → U2_AGG(., U4_agaa(row2col_in_aaaa))
with rule TRANSPOSE_AUX_IN_AGG(y0, .) → U2_AGG(y0, U4_agaa(row2col_in_aaaa))

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.