Left Termination proof of /tmp/tmp6WNQqA/transpose-fb.pl

Left Termination of the query pattern transpose_in_2(a, 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 QDPOrderProof (⇔)

        ↳38 QDP

        ↳39 DependencyGraphProof (⇔)

        ↳40 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(a,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: (f,b)
transpose_aux_in: (f,b,b)
row2col_in: (f,b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

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_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

(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_AG(A, B) → U1_AG(A, B, transpose_aux_in_agg(A, [], B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, [], B)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ag(x1, x2) = transpose_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3)
.(x1, x2) = .(x1, x2)
U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x3, x4, x5, x6)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x4, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x2, x3, x4)
U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x3, x4, x5, x6)
transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1, x2, x3)
transpose_out_ag(x1, x2) = transpose_out_ag(x1, x2)
TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x2, x3)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x3, x4, x5, x6)
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2)
U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x4, x7)
U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, x3, x4, x5, 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_AG(A, B) → U1_AG(A, B, transpose_aux_in_agg(A, [], B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, [], B)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

(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_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

(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_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(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_AGAA(.(.(X, Ys), Cols)) → ROW2COL_IN_AGAA(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_AGAA(.(.(X, Ys), Cols)) → ROW2COL_IN_AGAA(Cols)
The graph contains the following edges 1 > 1

(13) TRUE

(14) Obligation:

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

U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ag(x1, x2) = transpose_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3)
.(x1, x2) = .(x1, x2)
U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x3, x4, x5, x6)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x4, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x2, x3, x4)
U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x3, x4, x5, x6)
transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1, x2, x3)
transpose_out_ag(x1, x2) = transpose_out_ag(x1, x2)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x3, x4, x5, 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_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x4, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x2, x3, x4)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x3, x4, x5, 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_AGG(X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(X1, .(C, Cs)) → U2_AGG(X1, C, Cs, row2col_in_agaa(.(C, Cs)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0, x1, x2, x3)

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

(19) PrologToPiTRSProof (SOUND transformation)

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

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_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

(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_AG(A, B) → U1_AG(A, B, transpose_aux_in_agg(A, [], B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, [], B)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ag(x1, x2) = transpose_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x3)
transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3)
.(x1, x2) = .(x1, x2)
U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4)
U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6)
transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1)
transpose_out_ag(x1, x2) = transpose_out_ag(x1)
TRANSPOSE_IN_AG(x1, x2) = TRANSPOSE_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x3)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(x6)
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(x2)
U4_AGAA(x1, x2, x3, x4, x5, x6, x7) = U4_AGAA(x1, x3, x7)
U3_AGG(x1, x2, x3, x4, x5, x6) = U3_AGG(x1, 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_AG(A, B) → U1_AG(A, B, transpose_aux_in_agg(A, [], B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, [], B)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → ROW2COL_IN_AGAA(R, .(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_AGG(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

(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_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ag(x1, x2) = transpose_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x3)
transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3)
.(x1, x2) = .(x1, x2)
U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4)
U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6)
transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1)
transpose_out_ag(x1, x2) = transpose_out_ag(x1)
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(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_AGAA(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
ROW2COL_IN_AGAA(x1, x2, x3, x4) = ROW2COL_IN_AGAA(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_AGAA(.(.(X, Ys), Cols)) → ROW2COL_IN_AGAA(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_AGAA(.(.(X, Ys), Cols)) → ROW2COL_IN_AGAA(Cols)
The graph contains the following edges 1 > 1

(31) TRUE

(32) Obligation:

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

U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

transpose_in_ag(A, B) → U1_ag(A, B, transpose_aux_in_agg(A, [], B))
transpose_aux_in_agg(.(R, Rs), X1, .(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))
row2col_in_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa([], [], [], []) → row2col_out_agaa([], [], [], [])
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg([], X, X) → transpose_aux_out_agg([], X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(.(R, Rs), X1, .(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, [], B)) → transpose_out_ag(A, B)

The argument filtering Pi contains the following mapping:
transpose_in_ag(x1, x2) = transpose_in_ag(x2)
U1_ag(x1, x2, x3) = U1_ag(x3)
transpose_aux_in_agg(x1, x2, x3) = transpose_aux_in_agg(x2, x3)
.(x1, x2) = .(x1, x2)
U2_agg(x1, x2, x3, x4, x5, x6) = U2_agg(x6)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4)
U3_agg(x1, x2, x3, x4, x5, x6) = U3_agg(x1, x6)
transpose_aux_out_agg(x1, x2, x3) = transpose_aux_out_agg(x1)
transpose_out_ag(x1, x2) = transpose_out_ag(x1)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(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_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, .(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(.(R, Rs), X1, .(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, .(C, Cs), Cols1, Accm))

The TRS R consists of the following rules:

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

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
row2col_in_agaa(x1, x2, x3, x4) = row2col_in_agaa(x2)
U4_agaa(x1, x2, x3, x4, x5, x6, x7) = U4_agaa(x1, x3, x7)
[] = []
row2col_out_agaa(x1, x2, x3, x4) = row2col_out_agaa(x1, x3, x4)
TRANSPOSE_AUX_IN_AGG(x1, x2, x3) = TRANSPOSE_AUX_IN_AGG(x2, x3)
U2_AGG(x1, x2, x3, x4, x5, x6) = U2_AGG(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_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(X1, .(C, Cs)) → U2_AGG(row2col_in_agaa(.(C, Cs)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0, x1, x2)

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U2_AGG(row2col_out_agaa(R, Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Accm, Cols1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(U2_AGG(x₁)) = 1 +
[ 1, 0 ]

· x₁

POL(row2col_out_agaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(TRANSPOSE_AUX_IN_AGG(x₁, x₂)) = 0 +
[ 0, 0 ]

· x₁ +
[ 1, 1 ]

· x₂

POL(.(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 1 0 \

\ 0 1 /

· x₂

POL(row2col_in_agaa(x₁)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 1 0 /

· x₁

POL(U4_agaa(x₁, x₂, x₃)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂ +
/ 1 0 \

\ 0 0 /

· x₃

POL([]) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

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

(38) Obligation:

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

TRANSPOSE_AUX_IN_AGG(X1, .(C, Cs)) → U2_AGG(row2col_in_agaa(.(C, Cs)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

row2col_in_agaa(x0)
U4_agaa(x0, x1, x2)

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

(39) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) PrologToPiTRSProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) PrologToPiTRSProof (SOUND transformation)

(20) Obligation:

(21) DependencyPairsProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Complex Obligation (AND)

(25) Obligation:

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

(28) PiDPToQDPProof (SOUND transformation)

(29) Obligation:

(30) QDPSizeChangeProof (EQUIVALENT transformation)

(31) TRUE

(32) Obligation:

(33) UsableRulesProof (EQUIVALENT transformation)

(34) Obligation:

(35) PiDPToQDPProof (SOUND transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) DependencyGraphProof (EQUIVALENT transformation)

(40) TRUE