Left Termination of the query pattern transpose(b,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: (b,b)
transpose_aux3: (b,b,b)
row2col4: (b,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_gg2(A, B) -> if_transpose_2_in_1_gg3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
transpose_aux_3_in_ggg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
transpose_aux_3_in_ggg3([]_0, X, X) -> transpose_aux_3_out_ggg3([]_0, X, X)
if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_ggg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_ggg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_gg3(A, B, transpose_aux_3_out_ggg3(A, []_0, B)) -> transpose_2_out_gg2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_gg2(x1, x2)  =  transpose_2_in_gg2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_gg3(x1, x2, x3)  =  if_transpose_2_in_1_gg1(x3)
transpose_aux_3_in_ggg3(x1, x2, x3)  =  transpose_aux_3_in_ggg3(x1, x2, x3)
if_transpose_aux_3_in_1_ggg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_ggg2(x2, x6)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
if_transpose_aux_3_in_2_ggg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_ggg1(x8)
transpose_aux_3_out_ggg3(x1, x2, x3)  =  transpose_aux_3_out_ggg
transpose_2_out_gg2(x1, x2)  =  transpose_2_out_gg

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_gg2(A, B) -> if_transpose_2_in_1_gg3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
transpose_aux_3_in_ggg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
transpose_aux_3_in_ggg3([]_0, X, X) -> transpose_aux_3_out_ggg3([]_0, X, X)
if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_ggg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_ggg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_gg3(A, B, transpose_aux_3_out_ggg3(A, []_0, B)) -> transpose_2_out_gg2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_gg2(x1, x2)  =  transpose_2_in_gg2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_gg3(x1, x2, x3)  =  if_transpose_2_in_1_gg1(x3)
transpose_aux_3_in_ggg3(x1, x2, x3)  =  transpose_aux_3_in_ggg3(x1, x2, x3)
if_transpose_aux_3_in_1_ggg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_ggg2(x2, x6)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
if_transpose_aux_3_in_2_ggg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_ggg1(x8)
transpose_aux_3_out_ggg3(x1, x2, x3)  =  transpose_aux_3_out_ggg
transpose_2_out_gg2(x1, x2)  =  transpose_2_out_gg


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

TRANSPOSE_2_IN_GG2(A, B) -> IF_TRANSPOSE_2_IN_1_GG3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
TRANSPOSE_2_IN_GG2(A, B) -> TRANSPOSE_AUX_3_IN_GGG3(A, []_0, B)
TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> ROW2COL_4_IN_GGAA4(R, ._22(C, Cs), Cols1, Accm)
ROW2COL_4_IN_GGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> IF_ROW2COL_4_IN_1_GGAA7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
ROW2COL_4_IN_GGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_GGAA4(Xs, Cols, Cols1, As)
IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> IF_TRANSPOSE_AUX_3_IN_2_GGG8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_GGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_2_in_gg2(A, B) -> if_transpose_2_in_1_gg3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
transpose_aux_3_in_ggg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
transpose_aux_3_in_ggg3([]_0, X, X) -> transpose_aux_3_out_ggg3([]_0, X, X)
if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_ggg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_ggg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_gg3(A, B, transpose_aux_3_out_ggg3(A, []_0, B)) -> transpose_2_out_gg2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_gg2(x1, x2)  =  transpose_2_in_gg2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_gg3(x1, x2, x3)  =  if_transpose_2_in_1_gg1(x3)
transpose_aux_3_in_ggg3(x1, x2, x3)  =  transpose_aux_3_in_ggg3(x1, x2, x3)
if_transpose_aux_3_in_1_ggg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_ggg2(x2, x6)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
if_transpose_aux_3_in_2_ggg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_ggg1(x8)
transpose_aux_3_out_ggg3(x1, x2, x3)  =  transpose_aux_3_out_ggg
transpose_2_out_gg2(x1, x2)  =  transpose_2_out_gg
IF_TRANSPOSE_AUX_3_IN_1_GGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_GGG2(x2, x6)
TRANSPOSE_2_IN_GG2(x1, x2)  =  TRANSPOSE_2_IN_GG2(x1, x2)
ROW2COL_4_IN_GGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_GGAA2(x1, x2)
IF_ROW2COL_4_IN_1_GGAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_ROW2COL_4_IN_1_GGAA2(x3, x7)
TRANSPOSE_AUX_3_IN_GGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_GGG3(x1, x2, x3)
IF_TRANSPOSE_AUX_3_IN_2_GGG8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_TRANSPOSE_AUX_3_IN_2_GGG1(x8)
IF_TRANSPOSE_2_IN_1_GG3(x1, x2, x3)  =  IF_TRANSPOSE_2_IN_1_GG1(x3)

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

↳ PROLOG
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
PiDP
          ↳ DependencyGraphProof

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

TRANSPOSE_2_IN_GG2(A, B) -> IF_TRANSPOSE_2_IN_1_GG3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
TRANSPOSE_2_IN_GG2(A, B) -> TRANSPOSE_AUX_3_IN_GGG3(A, []_0, B)
TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> ROW2COL_4_IN_GGAA4(R, ._22(C, Cs), Cols1, Accm)
ROW2COL_4_IN_GGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> IF_ROW2COL_4_IN_1_GGAA7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
ROW2COL_4_IN_GGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_GGAA4(Xs, Cols, Cols1, As)
IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> IF_TRANSPOSE_AUX_3_IN_2_GGG8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_GGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_2_in_gg2(A, B) -> if_transpose_2_in_1_gg3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
transpose_aux_3_in_ggg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
transpose_aux_3_in_ggg3([]_0, X, X) -> transpose_aux_3_out_ggg3([]_0, X, X)
if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_ggg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_ggg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_gg3(A, B, transpose_aux_3_out_ggg3(A, []_0, B)) -> transpose_2_out_gg2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_gg2(x1, x2)  =  transpose_2_in_gg2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_gg3(x1, x2, x3)  =  if_transpose_2_in_1_gg1(x3)
transpose_aux_3_in_ggg3(x1, x2, x3)  =  transpose_aux_3_in_ggg3(x1, x2, x3)
if_transpose_aux_3_in_1_ggg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_ggg2(x2, x6)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
if_transpose_aux_3_in_2_ggg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_ggg1(x8)
transpose_aux_3_out_ggg3(x1, x2, x3)  =  transpose_aux_3_out_ggg
transpose_2_out_gg2(x1, x2)  =  transpose_2_out_gg
IF_TRANSPOSE_AUX_3_IN_1_GGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_GGG2(x2, x6)
TRANSPOSE_2_IN_GG2(x1, x2)  =  TRANSPOSE_2_IN_GG2(x1, x2)
ROW2COL_4_IN_GGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_GGAA2(x1, x2)
IF_ROW2COL_4_IN_1_GGAA7(x1, x2, x3, x4, x5, x6, x7)  =  IF_ROW2COL_4_IN_1_GGAA2(x3, x7)
TRANSPOSE_AUX_3_IN_GGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_GGG3(x1, x2, x3)
IF_TRANSPOSE_AUX_3_IN_2_GGG8(x1, x2, x3, x4, x5, x6, x7, x8)  =  IF_TRANSPOSE_AUX_3_IN_2_GGG1(x8)
IF_TRANSPOSE_2_IN_1_GG3(x1, x2, x3)  =  IF_TRANSPOSE_2_IN_1_GG1(x3)

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_GGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_GGAA4(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_2_in_gg2(A, B) -> if_transpose_2_in_1_gg3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
transpose_aux_3_in_ggg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
transpose_aux_3_in_ggg3([]_0, X, X) -> transpose_aux_3_out_ggg3([]_0, X, X)
if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_ggg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_ggg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_gg3(A, B, transpose_aux_3_out_ggg3(A, []_0, B)) -> transpose_2_out_gg2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_gg2(x1, x2)  =  transpose_2_in_gg2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_gg3(x1, x2, x3)  =  if_transpose_2_in_1_gg1(x3)
transpose_aux_3_in_ggg3(x1, x2, x3)  =  transpose_aux_3_in_ggg3(x1, x2, x3)
if_transpose_aux_3_in_1_ggg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_ggg2(x2, x6)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
if_transpose_aux_3_in_2_ggg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_ggg1(x8)
transpose_aux_3_out_ggg3(x1, x2, x3)  =  transpose_aux_3_out_ggg
transpose_2_out_gg2(x1, x2)  =  transpose_2_out_gg
ROW2COL_4_IN_GGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_GGAA2(x1, 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_GGAA4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> ROW2COL_4_IN_GGAA4(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_GGAA4(x1, x2, x3, x4)  =  ROW2COL_4_IN_GGAA2(x1, 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_GGAA2(._22(X, Xs), ._22(._22(X, Ys), Cols)) -> ROW2COL_4_IN_GGAA2(Xs, 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_GGAA2}.
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:

TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_GGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_2_in_gg2(A, B) -> if_transpose_2_in_1_gg3(A, B, transpose_aux_3_in_ggg3(A, []_0, B))
transpose_aux_3_in_ggg3(._22(R, Rs), underscore, ._22(C, Cs)) -> if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
if_transpose_aux_3_in_1_ggg6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_in_ggg3(Rs, Accm, Cols1))
transpose_aux_3_in_ggg3([]_0, X, X) -> transpose_aux_3_out_ggg3([]_0, X, X)
if_transpose_aux_3_in_2_ggg8(R, Rs, underscore, C, Cs, Cols1, Accm, transpose_aux_3_out_ggg3(Rs, Accm, Cols1)) -> transpose_aux_3_out_ggg3(._22(R, Rs), underscore, ._22(C, Cs))
if_transpose_2_in_1_gg3(A, B, transpose_aux_3_out_ggg3(A, []_0, B)) -> transpose_2_out_gg2(A, B)

The argument filtering Pi contains the following mapping:
transpose_2_in_gg2(x1, x2)  =  transpose_2_in_gg2(x1, x2)
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
if_transpose_2_in_1_gg3(x1, x2, x3)  =  if_transpose_2_in_1_gg1(x3)
transpose_aux_3_in_ggg3(x1, x2, x3)  =  transpose_aux_3_in_ggg3(x1, x2, x3)
if_transpose_aux_3_in_1_ggg6(x1, x2, x3, x4, x5, x6)  =  if_transpose_aux_3_in_1_ggg2(x2, x6)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
if_transpose_aux_3_in_2_ggg8(x1, x2, x3, x4, x5, x6, x7, x8)  =  if_transpose_aux_3_in_2_ggg1(x8)
transpose_aux_3_out_ggg3(x1, x2, x3)  =  transpose_aux_3_out_ggg
transpose_2_out_gg2(x1, x2)  =  transpose_2_out_gg
IF_TRANSPOSE_AUX_3_IN_1_GGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_GGG2(x2, x6)
TRANSPOSE_AUX_3_IN_GGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_GGG3(x1, 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:

TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_in_ggaa4(R, ._22(C, Cs), Cols1, Accm))
IF_TRANSPOSE_AUX_3_IN_1_GGG6(R, Rs, underscore, C, Cs, row2col_4_out_ggaa4(R, ._22(C, Cs), Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_GGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

row2col_4_in_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As)) -> if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_in_ggaa4(Xs, Cols, Cols1, As))
if_row2col_4_in_1_ggaa7(X, Xs, Ys, Cols, Cols1, As, row2col_4_out_ggaa4(Xs, Cols, Cols1, As)) -> row2col_4_out_ggaa4(._22(X, Xs), ._22(._22(X, Ys), Cols), ._22(Ys, Cols1), ._22([]_0, As))
row2col_4_in_ggaa4([]_0, []_0, []_0, []_0) -> row2col_4_out_ggaa4([]_0, []_0, []_0, []_0)

The argument filtering Pi contains the following mapping:
[]_0  =  []_0
._22(x1, x2)  =  ._22(x1, x2)
row2col_4_in_ggaa4(x1, x2, x3, x4)  =  row2col_4_in_ggaa2(x1, x2)
if_row2col_4_in_1_ggaa7(x1, x2, x3, x4, x5, x6, x7)  =  if_row2col_4_in_1_ggaa2(x3, x7)
row2col_4_out_ggaa4(x1, x2, x3, x4)  =  row2col_4_out_ggaa2(x3, x4)
IF_TRANSPOSE_AUX_3_IN_1_GGG6(x1, x2, x3, x4, x5, x6)  =  IF_TRANSPOSE_AUX_3_IN_1_GGG2(x2, x6)
TRANSPOSE_AUX_3_IN_GGG3(x1, x2, x3)  =  TRANSPOSE_AUX_3_IN_GGG3(x1, 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
                        ↳ QDPSizeChangeProof

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

TRANSPOSE_AUX_3_IN_GGG3(._22(R, Rs), underscore, ._22(C, Cs)) -> IF_TRANSPOSE_AUX_3_IN_1_GGG2(Rs, row2col_4_in_ggaa2(R, ._22(C, Cs)))
IF_TRANSPOSE_AUX_3_IN_1_GGG2(Rs, row2col_4_out_ggaa2(Cols1, Accm)) -> TRANSPOSE_AUX_3_IN_GGG3(Rs, Accm, Cols1)

The TRS R consists of the following rules:

row2col_4_in_ggaa2(._22(X, Xs), ._22(._22(X, Ys), Cols)) -> if_row2col_4_in_1_ggaa2(Ys, row2col_4_in_ggaa2(Xs, Cols))
if_row2col_4_in_1_ggaa2(Ys, row2col_4_out_ggaa2(Cols1, As)) -> row2col_4_out_ggaa2(._22(Ys, Cols1), ._22([]_0, As))
row2col_4_in_ggaa2([]_0, []_0) -> row2col_4_out_ggaa2([]_0, []_0)

The set Q consists of the following terms:

row2col_4_in_ggaa2(x0, x1)
if_row2col_4_in_1_ggaa2(x0, x1)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_TRANSPOSE_AUX_3_IN_1_GGG2, TRANSPOSE_AUX_3_IN_GGG3}.
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: