Left Termination of the query pattern transpose(f,b) w.r.t. the given Prolog program could successfully be proven:



↳ PROLOG
  ↳ PrologToPiTRSProof

transpose2(A, B) :- transposeaux3(A, {}0, B).
transposeaux3(.2(R, Rs), underscore, .2(C, Cs)) :- row2col4(R, .2(C, Cs), Cols1, Accm), transposeaux3(Rs, Accm, Cols1).
transposeaux3({}0, X, X).
row2col4(.2(X, Xs), .2(.2(X, Ys), Cols), .2(Ys, Cols1), .2({}0, As)) :- row2col4(Xs, Cols, Cols1, As).
row2col4({}0, {}0, {}0, {}0).


With regard to the inferred argument filtering the predicates were used in the following modes:
transpose2: (f,b)
transpose_aux3: (f,b,b)
row2col4: (f,b,f,f)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:


transpose_2_in_ag2(A, B) -> if_transpose_2_in_1_ag3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
transpose_aux_3_in_agg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
transpose_aux_3_in_agg3([]_0, X, X) -> transpose_aux_3_out_agg3([]_0, X, X)
if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_agg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_agg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_ag3(A, B, transpose_aux_3_out_agg3(A, []_0, B)) -> transpose_2_out_ag2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_ag2(x1, x2)  =  transpose_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_ag3(x1, x2, x3)  =  if_transpose_2_in_1_ag1(x3)
transpose_aux_3_in_agg3(x1, x2, x3)  =  transpose_aux_3_in_agg2(x2, x3)
if_transpose_aux_3_in_1_agg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_agg1(x6)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
if_transpose_aux_3_in_2_agg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_agg2(x1, x8)
transpose_aux_3_out_agg3(x1, x2, x3)  =  transpose_aux_3_out_agg1(x1)
transpose_2_out_ag2(x1, x2)  =  transpose_2_out_ag1(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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

transpose_2_in_ag2(A, B) -> if_transpose_2_in_1_ag3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
transpose_aux_3_in_agg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
transpose_aux_3_in_agg3([]_0, X, X) -> transpose_aux_3_out_agg3([]_0, X, X)
if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_agg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_agg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_ag3(A, B, transpose_aux_3_out_agg3(A, []_0, B)) -> transpose_2_out_ag2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_ag2(x1, x2)  =  transpose_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_ag3(x1, x2, x3)  =  if_transpose_2_in_1_ag1(x3)
transpose_aux_3_in_agg3(x1, x2, x3)  =  transpose_aux_3_in_agg2(x2, x3)
if_transpose_aux_3_in_1_agg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_agg1(x6)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
if_transpose_aux_3_in_2_agg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_agg2(x1, x8)
transpose_aux_3_out_agg3(x1, x2, x3)  =  transpose_aux_3_out_agg1(x1)
transpose_2_out_ag2(x1, x2)  =  transpose_2_out_ag1(x1)


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

TRANSPOSE_2_IN_AG2(A, B) -> IF_TRANSPOSE_2_IN_1_AG3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
TRANSPOSE_2_IN_AG2(A, B) -> TRANSPOSE_AUX_3_IN_AGG3(A, []_0, B)
TRANSPOSE_AUX_3_IN_AGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_3_IN_AGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> ROW2COL_4_IN_AGAA4(R, ._22(C, Cs), Cols1, Accm)
ROW2COL_4_IN_AGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> IF_ROW2COL_4_IN_1_AGAA7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
ROW2COL_4_IN_AGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_AGAA4(Xs, Cols, Cols1, As)
IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> IF_TRANSPOSE_AUX_3_IN_2_AGG8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_2_in_ag2(A, B) -> if_transpose_2_in_1_ag3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
transpose_aux_3_in_agg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
transpose_aux_3_in_agg3([]_0, X, X) -> transpose_aux_3_out_agg3([]_0, X, X)
if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_agg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_agg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_ag3(A, B, transpose_aux_3_out_agg3(A, []_0, B)) -> transpose_2_out_ag2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_ag2(x1, x2)  =  transpose_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_ag3(x1, x2, x3)  =  if_transpose_2_in_1_ag1(x3)
transpose_aux_3_in_agg3(x1, x2, x3)  =  transpose_aux_3_in_agg2(x2, x3)
if_transpose_aux_3_in_1_agg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_agg1(x6)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
if_transpose_aux_3_in_2_agg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_agg2(x1, x8)
transpose_aux_3_out_agg3(x1, x2, x3)  =  transpose_aux_3_out_agg1(x1)
transpose_2_out_ag2(x1, x2)  =  transpose_2_out_ag1(x1)
IF_TRANSPOSE_2_IN_1_AG3(x1, x2, x3)  =  IF_TRANSPOSE_2_IN_1_AG1(x3)
IF_TRANSPOSE_AUX_3_IN_2_AGG8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_TRANSPOSE_AUX_3_IN_2_AGG2(x1, x8)
IF_TRANSPOSE_AUX_3_IN_1_AGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_AGG1(x6)
TRANSPOSE_2_IN_AG2(x1, x2)  =  TRANSPOSE_2_IN_AG1(x2)
ROW2COL_4_IN_AGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_AGAA1(x2)
TRANSPOSE_AUX_3_IN_AGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_AGG2(x2, x3)
IF_ROW2COL_4_IN_1_AGAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_ROW2COL_4_IN_1_AGAA3(x1, x3, x7)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

TRANSPOSE_2_IN_AG2(A, B) -> IF_TRANSPOSE_2_IN_1_AG3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
TRANSPOSE_2_IN_AG2(A, B) -> TRANSPOSE_AUX_3_IN_AGG3(A, []_0, B)
TRANSPOSE_AUX_3_IN_AGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_3_IN_AGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> ROW2COL_4_IN_AGAA4(R, ._22(C, Cs), Cols1, Accm)
ROW2COL_4_IN_AGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> IF_ROW2COL_4_IN_1_AGAA7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
ROW2COL_4_IN_AGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_AGAA4(Xs, Cols, Cols1, As)
IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> IF_TRANSPOSE_AUX_3_IN_2_AGG8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_2_in_ag2(A, B) -> if_transpose_2_in_1_ag3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
transpose_aux_3_in_agg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
transpose_aux_3_in_agg3([]_0, X, X) -> transpose_aux_3_out_agg3([]_0, X, X)
if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_agg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_agg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_ag3(A, B, transpose_aux_3_out_agg3(A, []_0, B)) -> transpose_2_out_ag2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_ag2(x1, x2)  =  transpose_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_ag3(x1, x2, x3)  =  if_transpose_2_in_1_ag1(x3)
transpose_aux_3_in_agg3(x1, x2, x3)  =  transpose_aux_3_in_agg2(x2, x3)
if_transpose_aux_3_in_1_agg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_agg1(x6)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
if_transpose_aux_3_in_2_agg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_agg2(x1, x8)
transpose_aux_3_out_agg3(x1, x2, x3)  =  transpose_aux_3_out_agg1(x1)
transpose_2_out_ag2(x1, x2)  =  transpose_2_out_ag1(x1)
IF_TRANSPOSE_2_IN_1_AG3(x1, x2, x3)  =  IF_TRANSPOSE_2_IN_1_AG1(x3)
IF_TRANSPOSE_AUX_3_IN_2_AGG8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_TRANSPOSE_AUX_3_IN_2_AGG2(x1, x8)
IF_TRANSPOSE_AUX_3_IN_1_AGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_AGG1(x6)
TRANSPOSE_2_IN_AG2(x1, x2)  =  TRANSPOSE_2_IN_AG1(x2)
ROW2COL_4_IN_AGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_AGAA1(x2)
TRANSPOSE_AUX_3_IN_AGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_AGG2(x2, x3)
IF_ROW2COL_4_IN_1_AGAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_ROW2COL_4_IN_1_AGAA3(x1, x3, x7)

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

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

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

ROW2COL_4_IN_AGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_AGAA4(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_2_in_ag2(A, B) -> if_transpose_2_in_1_ag3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
transpose_aux_3_in_agg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
transpose_aux_3_in_agg3([]_0, X, X) -> transpose_aux_3_out_agg3([]_0, X, X)
if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_agg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_agg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_ag3(A, B, transpose_aux_3_out_agg3(A, []_0, B)) -> transpose_2_out_ag2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_ag2(x1, x2)  =  transpose_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_ag3(x1, x2, x3)  =  if_transpose_2_in_1_ag1(x3)
transpose_aux_3_in_agg3(x1, x2, x3)  =  transpose_aux_3_in_agg2(x2, x3)
if_transpose_aux_3_in_1_agg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_agg1(x6)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
if_transpose_aux_3_in_2_agg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_agg2(x1, x8)
transpose_aux_3_out_agg3(x1, x2, x3)  =  transpose_aux_3_out_agg1(x1)
transpose_2_out_ag2(x1, x2)  =  transpose_2_out_ag1(x1)
ROW2COL_4_IN_AGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_AGAA1(x2)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
              ↳ PiDP

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

ROW2COL_4_IN_AGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_AGAA4(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
ROW2COL_4_IN_AGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_AGAA1(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPSizeChangeProof
              ↳ PiDP

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

ROW2COL_4_IN_AGAA1(._22(._22(X, Ys), Cols)) -> ROW2COL_4_IN_AGAA1(Cols)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ROW2COL_4_IN_AGAA1}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



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

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

IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG3(Rs, Accm, Cols1)
TRANSPOSE_AUX_3_IN_AGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_2_in_ag2(A, B) -> if_transpose_2_in_1_ag3(A, B, transpose_aux_3_in_agg3(A, []_0, B))
transpose_aux_3_in_agg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_agg6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_agg3(Rs, Accm, Cols1))
transpose_aux_3_in_agg3([]_0, X, X) -> transpose_aux_3_out_agg3([]_0, X, X)
if_transpose_aux_3_in_2_agg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_agg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_agg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_ag3(A, B, transpose_aux_3_out_agg3(A, []_0, B)) -> transpose_2_out_ag2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_ag2(x1, x2)  =  transpose_2_in_ag1(x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_ag3(x1, x2, x3)  =  if_transpose_2_in_1_ag1(x3)
transpose_aux_3_in_agg3(x1, x2, x3)  =  transpose_aux_3_in_agg2(x2, x3)
if_transpose_aux_3_in_1_agg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_agg1(x6)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
if_transpose_aux_3_in_2_agg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_agg2(x1, x8)
transpose_aux_3_out_agg3(x1, x2, x3)  =  transpose_aux_3_out_agg1(x1)
transpose_2_out_ag2(x1, x2)  =  transpose_2_out_ag1(x1)
IF_TRANSPOSE_AUX_3_IN_1_AGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_AGG1(x6)
TRANSPOSE_AUX_3_IN_AGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_AGG2(x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting we can delete all non-usable rules from R.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof

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

IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_out_agaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG3(Rs, Accm, Cols1)
TRANSPOSE_AUX_3_IN_AGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_AGG6(R, Rs, underscore, C, Cs, row2col_4_in_agaa4(R, ._22(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_4_in_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_agaa4(Xs, Cols, Cols1, As))
if_row2col_4_in_1_agaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_agaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_agaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
row2col_4_in_agaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_agaa4([]_0, []_0, []_0, []_0)

The argument filtering Pi contains the following mapping:
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
row2col_4_in_agaa4(x1, x2, x3, x4)  =  row2col_4_in_agaa1(x2)
if_row2col_4_in_1_agaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_agaa3(x1, x3, x7)
row2col_4_out_agaa4(x1, x2, x3, x4)  =  row2col_4_out_agaa3(x1, x3, x4)
IF_TRANSPOSE_AUX_3_IN_1_AGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_AGG1(x6)
TRANSPOSE_AUX_3_IN_AGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_AGG2(x2, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPPoloProof

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

IF_TRANSPOSE_AUX_3_IN_1_AGG1(row2col_4_out_agaa3(R, Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG2(Accm, Cols1)
TRANSPOSE_AUX_3_IN_AGG2(underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_AGG1(row2col_4_in_agaa1(._22(C, Cs)))

The TRS R consists of the following rules:

row2col_4_in_agaa1(._22(._22(X, Ys), Cols)) -> if_row2col_4_in_1_agaa3(X, Ys, row2col_4_in_agaa1(Cols))
if_row2col_4_in_1_agaa3(X, Ys, row2col_4_out_agaa3(Xs, Cols1, As)) -> row2col_4_out_agaa3(._22(X, Xs), ._22(Ys, Cols1), ._22([]_0, As))
row2col_4_in_agaa1([]_0) -> row2col_4_out_agaa3([]_0, []_0, []_0)

The set Q consists of the following terms:

row2col_4_in_agaa1(x0)
if_row2col_4_in_1_agaa3(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {TRANSPOSE_AUX_3_IN_AGG2, IF_TRANSPOSE_AUX_3_IN_1_AGG1}.
By using a polynomial ordering, the following set of Dependency Pairs of this DP problem can be strictly oriented.

TRANSPOSE_AUX_3_IN_AGG2(underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_AGG1(row2col_4_in_agaa1(._22(C, Cs)))
The remaining Dependency Pairs were at least non-strictly be oriented.

IF_TRANSPOSE_AUX_3_IN_1_AGG1(row2col_4_out_agaa3(R, Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG2(Accm, Cols1)
With the implicit AFS we had to orient the following set of usable rules non-strictly.

row2col_4_in_agaa1([]_0) -> row2col_4_out_agaa3([]_0, []_0, []_0)
row2col_4_in_agaa1(._22(._22(X, Ys), Cols)) -> if_row2col_4_in_1_agaa3(X, Ys, row2col_4_in_agaa1(Cols))
if_row2col_4_in_1_agaa3(X, Ys, row2col_4_out_agaa3(Xs, Cols1, As)) -> row2col_4_out_agaa3(._22(X, Xs), ._22(Ys, Cols1), ._22([]_0, As))
Used ordering: POLO with Polynomial interpretation:

POL(._22(x1, x2)) = 1 + x1 + 2·x2   
POL(if_row2col_4_in_1_agaa3(x1, x2, x3)) = 2 + 2·x2 + 2·x3   
POL(row2col_4_out_agaa3(x1, x2, x3)) = 2·x2   
POL(TRANSPOSE_AUX_3_IN_AGG2(x1, x2)) = 2·x2   
POL(IF_TRANSPOSE_AUX_3_IN_1_AGG1(x1)) = x1   
POL(row2col_4_in_agaa1(x1)) = x1   
POL([]_0) = 0   



↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ PiDP
              ↳ PiDP
                ↳ UsableRulesProof
                  ↳ PiDP
                    ↳ PiDPToQDPProof
                      ↳ QDP
                        ↳ QDPPoloProof
                          ↳ QDP
                            ↳ DependencyGraphProof

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

IF_TRANSPOSE_AUX_3_IN_1_AGG1(row2col_4_out_agaa3(R, Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_AGG2(Accm, Cols1)

The TRS R consists of the following rules:

row2col_4_in_agaa1(._22(._22(X, Ys), Cols)) -> if_row2col_4_in_1_agaa3(X, Ys, row2col_4_in_agaa1(Cols))
if_row2col_4_in_1_agaa3(X, Ys, row2col_4_out_agaa3(Xs, Cols1, As)) -> row2col_4_out_agaa3(._22(X, Xs), ._22(Ys, Cols1), ._22([]_0, As))
row2col_4_in_agaa1([]_0) -> row2col_4_out_agaa3([]_0, []_0, []_0)

The set Q consists of the following terms:

row2col_4_in_agaa1(x0)
if_row2col_4_in_1_agaa3(x0, x1, x2)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {TRANSPOSE_AUX_3_IN_AGG2, IF_TRANSPOSE_AUX_3_IN_1_AGG1}.
The approximation of the Dependency Graph contains 0 SCCs with 1 less node.