Left Termination of the query pattern transpose_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
  ↳19 PrologToPiTRSProof (⇐)
    ↳20 PiTRS
    ↳21 DependencyPairsProof (⇔)
    ↳22 PiDP
    ↳23 DependencyGraphProof (⇔)
    ↳24 AND
      ↳25 PiDP
        ↳26 UsableRulesProof (⇔)
        ↳27 PiDP
        ↳28 PiDPToQDPProof (⇐)
        ↳29 QDP
        ↳30 QDPSizeChangeProof (⇔)
        ↳31 TRUE
      ↳32 PiDP
        ↳33 UsableRulesProof (⇔)
        ↳34 PiDP
        ↳35 PiDPToQDPProof (⇐)
        ↳36 QDP
        ↳37 QDPSizeChangeProof (⇔)
        ↳38 TRUE

(0) Obligation:

Clauses:

transpose(A, B) :- transpose_aux(A, [], B).
transpose_aux(.(R, Rs), X1, .(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,g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_in: (b,b)
transpose_aux_in: (b,b,b)
row2col_in: (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_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg
transpose_out_gg(x1, x2)  =  transpose_out_gg

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg
transpose_out_gg(x1, x2)  =  transpose_out_gg

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE_IN_GG(A, B) → U1_GG(A, B, transpose_aux_in_ggg(A, [], B))
TRANSPOSE_IN_GG(A, B) → TRANSPOSE_AUX_IN_GGG(A, [], B)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_GGG(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg
transpose_out_gg(x1, x2)  =  transpose_out_gg
TRANSPOSE_IN_GG(x1, x2)  =  TRANSPOSE_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x2, x6)
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGAA(x3, x7)
U3_GGG(x1, x2, x3, x4, x5, x6)  =  U3_GGG(x6)

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

(4) Obligation:

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

TRANSPOSE_IN_GG(A, B) → U1_GG(A, B, transpose_aux_in_ggg(A, [], B))
TRANSPOSE_IN_GG(A, B) → TRANSPOSE_AUX_IN_GGG(A, [], B)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_GGG(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg
transpose_out_gg(x1, x2)  =  transpose_out_gg
TRANSPOSE_IN_GG(x1, x2)  =  TRANSPOSE_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x2, x6)
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGAA(x3, x7)
U3_GGG(x1, x2, x3, x4, x5, x6)  =  U3_GGG(x6)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg
transpose_out_gg(x1, x2)  =  transpose_out_gg
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

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

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

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

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) → ROW2COL_IN_GGAA(Xs, Cols)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) → ROW2COL_IN_GGAA(Xs, Cols)
    The graph contains the following edges 1 > 1, 2 > 2

(13) TRUE

(14) Obligation:

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

U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x2, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg
transpose_out_gg(x1, x2)  =  transpose_out_gg
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x2, x6)

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

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

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

U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x3, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x3, x4)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x2, x6)

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

(17) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

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

U2_GGG(Rs, row2col_out_ggaa(Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(Rs, row2col_in_ggaa(R, .(C, Cs)))

The TRS R consists of the following rules:

row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols)) → U4_ggaa(Ys, row2col_in_ggaa(Xs, Cols))
U4_ggaa(Ys, row2col_out_ggaa(Cols1, As)) → row2col_out_ggaa(.(Ys, Cols1), .([], As))
row2col_in_ggaa([], []) → row2col_out_ggaa([], [])

The set Q consists of the following terms:

row2col_in_ggaa(x0, x1)
U4_ggaa(x0, x1)

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

(19) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose_in: (b,b)
transpose_aux_in: (b,b,b)
row2col_in: (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_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x1, x2, x3, x4, x5, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x1, x2, x3, x4, x5, x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg(x1, x2, x3)
transpose_out_gg(x1, x2)  =  transpose_out_gg(x1, x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

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

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x1, x2, x3, x4, x5, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x1, x2, x3, x4, x5, x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg(x1, x2, x3)
transpose_out_gg(x1, x2)  =  transpose_out_gg(x1, x2)

(21) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE_IN_GG(A, B) → U1_GG(A, B, transpose_aux_in_ggg(A, [], B))
TRANSPOSE_IN_GG(A, B) → TRANSPOSE_AUX_IN_GGG(A, [], B)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_GGG(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x1, x2, x3, x4, x5, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x1, x2, x3, x4, x5, x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg(x1, x2, x3)
transpose_out_gg(x1, x2)  =  transpose_out_gg(x1, x2)
TRANSPOSE_IN_GG(x1, x2)  =  TRANSPOSE_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x1, x2, x3, x4, x5, x6)
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGAA(x1, x2, x3, x4, x7)
U3_GGG(x1, x2, x3, x4, x5, x6)  =  U3_GGG(x1, x2, x3, x4, x5, x6)

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

(22) Obligation:

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

TRANSPOSE_IN_GG(A, B) → U1_GG(A, B, transpose_aux_in_ggg(A, [], B))
TRANSPOSE_IN_GG(A, B) → TRANSPOSE_AUX_IN_GGG(A, [], B)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_GGG(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x1, x2, x3, x4, x5, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x1, x2, x3, x4, x5, x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg(x1, x2, x3)
transpose_out_gg(x1, x2)  =  transpose_out_gg(x1, x2)
TRANSPOSE_IN_GG(x1, x2)  =  TRANSPOSE_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x1, x2, x3)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x1, x2, x3, x4, x5, x6)
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)
U4_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GGAA(x1, x2, x3, x4, x7)
U3_GGG(x1, x2, x3, x4, x5, x6)  =  U3_GGG(x1, x2, x3, x4, x5, x6)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(24) Complex Obligation (AND)

(25) Obligation:

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

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x1, x2, x3, x4, x5, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x1, x2, x3, x4, x5, x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg(x1, x2, x3)
transpose_out_gg(x1, x2)  =  transpose_out_gg(x1, x2)
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)

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

(26) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(27) Obligation:

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

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GGAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN_GGAA(x1, x2, x3, x4)  =  ROW2COL_IN_GGAA(x1, x2)

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

(28) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(29) Obligation:

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

ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) → ROW2COL_IN_GGAA(Xs, Cols)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(30) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • ROW2COL_IN_GGAA(.(X, Xs), .(.(X, Ys), Cols)) → ROW2COL_IN_GGAA(Xs, Cols)
    The graph contains the following edges 1 > 1, 2 > 2

(31) TRUE

(32) Obligation:

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

U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_gg(A, B) → U1_gg(A, B, transpose_aux_in_ggg(A, [], B))
transpose_aux_in_ggg(.(R, Rs), X1, .(C, Cs)) → U2_ggg(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))
row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_ggg(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → U3_ggg(R, Rs, X1, C, Cs, transpose_aux_in_ggg(Rs, Accm, Cols1))
transpose_aux_in_ggg([], X, X) → transpose_aux_out_ggg([], X, X)
U3_ggg(R, Rs, X1, C, Cs, transpose_aux_out_ggg(Rs, Accm, Cols1)) → transpose_aux_out_ggg(.(R, Rs), X1, .(C, Cs))
U1_gg(A, B, transpose_aux_out_ggg(A, [], B)) → transpose_out_gg(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_gg(x1, x2)  =  transpose_in_gg(x1, x2)
U1_gg(x1, x2, x3)  =  U1_gg(x1, x2, x3)
transpose_aux_in_ggg(x1, x2, x3)  =  transpose_aux_in_ggg(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U2_ggg(x1, x2, x3, x4, x5, x6)  =  U2_ggg(x1, x2, x3, x4, x5, x6)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
U3_ggg(x1, x2, x3, x4, x5, x6)  =  U3_ggg(x1, x2, x3, x4, x5, x6)
transpose_aux_out_ggg(x1, x2, x3)  =  transpose_aux_out_ggg(x1, x2, x3)
transpose_out_gg(x1, x2)  =  transpose_out_gg(x1, x2)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x1, x2, x3, x4, x5, x6)

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

(33) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(34) Obligation:

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

U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_ggaa(Xs, Cols, Cols1, As))
U4_ggaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_ggaa([], [], [], []) → row2col_out_ggaa([], [], [], [])

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2col_in_ggaa(x1, x2, x3, x4)  =  row2col_in_ggaa(x1, x2)
U4_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2col_out_ggaa(x1, x2, x3, x4)  =  row2col_out_ggaa(x1, x2, x3, x4)
TRANSPOSE_AUX_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGG(x1, x2, x3)
U2_GGG(x1, x2, x3, x4, x5, x6)  =  U2_GGG(x1, x2, x3, x4, x5, x6)

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

(35) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(36) Obligation:

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

U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs)))

The TRS R consists of the following rules:

row2col_in_ggaa(.(X, Xs), .(.(X, Ys), Cols)) → U4_ggaa(X, Xs, Ys, Cols, row2col_in_ggaa(Xs, Cols))
U4_ggaa(X, Xs, Ys, Cols, row2col_out_ggaa(Xs, Cols, Cols1, As)) → row2col_out_ggaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_ggaa([], []) → row2col_out_ggaa([], [], [], [])

The set Q consists of the following terms:

row2col_in_ggaa(x0, x1)
U4_ggaa(x0, x1, x2, x3, x4)

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

(37) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • TRANSPOSE_AUX_IN_GGG(.(R, Rs), X1, .(C, Cs)) → U2_GGG(R, Rs, X1, C, Cs, row2col_in_ggaa(R, .(C, Cs)))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3, 3 > 4, 3 > 5

  • U2_GGG(R, Rs, X1, C, Cs, row2col_out_ggaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGG(Rs, Accm, Cols1)
    The graph contains the following edges 2 >= 1, 6 > 2, 6 > 3

(38) TRUE