Left Termination of the query pattern transpose_in_2(g, a) 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 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
        ↳32 QDPSizeChangeProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP

(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,a).

(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,f)
transpose_aux_in: (b,b,f)
row2col_in: (b,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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1)

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_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1)

(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_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), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GAAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_GGA(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1)
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)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x6)
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAAA(x1, x2, x7)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, 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_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), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GAAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_GGA(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1)
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)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x6)
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAAA(x1, x2, x7)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, 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_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1)
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)

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_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)

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_GAAA(.(X, Xs)) → ROW2COL_IN_GAAA(Xs)

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_GAAA(.(X, Xs)) → ROW2COL_IN_GAAA(Xs)
    The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x1, x2, x3, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x1, x2, x3, x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga(x1, x2)
transpose_out_ga(x1, x2)  =  transpose_out_ga(x1)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, 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_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x1, x2, x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x1, x4)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, 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_GGA(R, Rs, X1, row2col_out_gaaa(R, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1) → U2_GGA(R, Rs, X1, row2col_in_gaaa(R))

The TRS R consists of the following rules:

row2col_in_gaaa(.(X, Xs)) → U4_gaaa(X, Xs, row2col_in_gaaa(Xs))
U4_gaaa(X, Xs, row2col_out_gaaa(Xs, As)) → row2col_out_gaaa(.(X, Xs), .([], As))
row2col_in_gaaa([]) → row2col_out_gaaa([], [])

The set Q consists of the following terms:

row2col_in_gaaa(x0)
U4_gaaa(x0, x1, x2)

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

(19) 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_GGA(.(R, Rs), X1) → U2_GGA(R, Rs, X1, row2col_in_gaaa(R))
    The graph contains the following edges 1 > 1, 1 > 2, 2 >= 3

  • U2_GGA(R, Rs, X1, row2col_out_gaaa(R, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm)
    The graph contains the following edges 2 >= 1, 4 > 2

(20) TRUE

(21) 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,f)
transpose_aux_in: (b,b,f)
row2col_in: (b,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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga
transpose_out_ga(x1, x2)  =  transpose_out_ga

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga
transpose_out_ga(x1, x2)  =  transpose_out_ga

(23) 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_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), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GAAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_GGA(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga
transpose_out_ga(x1, x2)  =  transpose_out_ga
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)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x2, x6)
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAAA(x7)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)

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

(24) Obligation:

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), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_GAAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_GAAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_GGA(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga
transpose_out_ga(x1, x2)  =  transpose_out_ga
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)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x2, x6)
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)
U4_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAAA(x7)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x6)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

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

(26) Complex Obligation (AND)

(27) Obligation:

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

ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(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), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga
transpose_out_ga(x1, x2)  =  transpose_out_ga
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

ROW2COL_IN_GAAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_GAAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COL_IN_GAAA(x1, x2, x3, x4)  =  ROW2COL_IN_GAAA(x1)

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

ROW2COL_IN_GAAA(.(X, Xs)) → ROW2COL_IN_GAAA(Xs)

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

(32) 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_GAAA(.(X, Xs)) → ROW2COL_IN_GAAA(Xs)
    The graph contains the following edges 1 > 1

(33) TRUE

(34) Obligation:

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

U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_ga(A, B) → U1_ga(A, B, transpose_aux_in_gga(A, [], B))
transpose_aux_in_gga(.(R, Rs), X1, .(C, Cs)) → U2_gga(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))
row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_gga(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → U3_gga(R, Rs, X1, C, Cs, transpose_aux_in_gga(Rs, Accm, Cols1))
transpose_aux_in_gga([], X, X) → transpose_aux_out_gga([], X, X)
U3_gga(R, Rs, X1, C, Cs, transpose_aux_out_gga(Rs, Accm, Cols1)) → transpose_aux_out_gga(.(R, Rs), X1, .(C, Cs))
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)  =  .(x1, x2)
U2_gga(x1, x2, x3, x4, x5, x6)  =  U2_gga(x2, x6)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
U3_gga(x1, x2, x3, x4, x5, x6)  =  U3_gga(x6)
transpose_aux_out_gga(x1, x2, x3)  =  transpose_aux_out_gga
transpose_out_ga(x1, x2)  =  transpose_out_ga
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x2, x6)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

U2_GGA(R, Rs, X1, C, Cs, row2col_out_gaaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1, .(C, Cs)) → U2_GGA(R, Rs, X1, C, Cs, row2col_in_gaaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

row2col_in_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_gaaa(Xs, Cols, Cols1, As))
U4_gaaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_gaaa(Xs, Cols, Cols1, As)) → row2col_out_gaaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
row2col_in_gaaa([], [], [], []) → row2col_out_gaaa([], [], [], [])

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2col_in_gaaa(x1, x2, x3, x4)  =  row2col_in_gaaa(x1)
U4_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U4_gaaa(x7)
[]  =  []
row2col_out_gaaa(x1, x2, x3, x4)  =  row2col_out_gaaa(x4)
TRANSPOSE_AUX_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUX_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x2, x6)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

U2_GGA(Rs, row2col_out_gaaa(Accm)) → TRANSPOSE_AUX_IN_GGA(Rs, Accm)
TRANSPOSE_AUX_IN_GGA(.(R, Rs), X1) → U2_GGA(Rs, row2col_in_gaaa(R))

The TRS R consists of the following rules:

row2col_in_gaaa(.(X, Xs)) → U4_gaaa(row2col_in_gaaa(Xs))
U4_gaaa(row2col_out_gaaa(As)) → row2col_out_gaaa(.([], As))
row2col_in_gaaa([]) → row2col_out_gaaa([])

The set Q consists of the following terms:

row2col_in_gaaa(x0)
U4_gaaa(x0)

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