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

(0) Obligation:

Clauses:

transpose(A, B) :- transpose_aux(A, nil, B).
transpose_aux(cons(R, Rs), X1, cons(C, Cs)) :- ','(row2col(R, cons(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1)).
transpose_aux(nil, X, X).
row2col(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) :- row2col(Xs, Cols, Cols1, As).
row2col(nil, nil, nil, nil).

Query: transpose(a,g)

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. 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, nil, B))
transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(nil, X, X) → transpose_aux_out_agg(nil, X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, nil, 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)
cons(x1, x2)  =  cons(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)
nil  =  nil
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)

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, nil, B))
transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(nil, X, X) → transpose_aux_out_agg(nil, X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, nil, 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)
cons(x1, x2)  =  cons(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)
nil  =  nil
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)

(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, nil, B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, nil, B)
TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) → ROW2COL_IN_AGAA(R, cons(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(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, cons(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, nil, B))
transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(nil, X, X) → transpose_aux_out_agg(nil, X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, nil, 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)
cons(x1, x2)  =  cons(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)
nil  =  nil
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

(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, nil, B))
TRANSPOSE_IN_AG(A, B) → TRANSPOSE_AUX_IN_AGG(A, nil, B)
TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) → ROW2COL_IN_AGAA(R, cons(C, Cs), Cols1, Accm)
ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_AGAA(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
ROW2COL_IN_AGAA(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)
U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(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, cons(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, nil, B))
transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(nil, X, X) → transpose_aux_out_agg(nil, X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, nil, 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)
cons(x1, x2)  =  cons(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)
nil  =  nil
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

(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(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, 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, nil, B))
transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(nil, X, X) → transpose_aux_out_agg(nil, X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, nil, 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)
cons(x1, x2)  =  cons(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)
nil  =  nil
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

(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(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → ROW2COL_IN_AGAA(Xs, Cols, Cols1, As)

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

(13) YES

(14) Obligation:

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

U2_AGG(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → TRANSPOSE_AUX_IN_AGG(Rs, Accm, Cols1)
TRANSPOSE_AUX_IN_AGG(cons(R, Rs), X1, cons(C, Cs)) → U2_AGG(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(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, nil, B))
transpose_aux_in_agg(cons(R, Rs), X1, cons(C, Cs)) → U2_agg(R, Rs, X1, C, Cs, row2col_in_agaa(R, cons(C, Cs), Cols1, Accm))
row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As)) → U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_in_agaa(Xs, Cols, Cols1, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)
U4_agaa(X, Xs, Ys, Cols, Cols1, As, row2col_out_agaa(Xs, Cols, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
U2_agg(R, Rs, X1, C, Cs, row2col_out_agaa(R, cons(C, Cs), Cols1, Accm)) → U3_agg(R, Rs, X1, C, Cs, transpose_aux_in_agg(Rs, Accm, Cols1))
transpose_aux_in_agg(nil, X, X) → transpose_aux_out_agg(nil, X, X)
U3_agg(R, Rs, X1, C, Cs, transpose_aux_out_agg(Rs, Accm, Cols1)) → transpose_aux_out_agg(cons(R, Rs), X1, cons(C, Cs))
U1_ag(A, B, transpose_aux_out_agg(A, nil, 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)
cons(x1, x2)  =  cons(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)
nil  =  nil
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

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

The TRS R consists of the following rules:

row2col_in_agaa(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, 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(cons(X, Xs), cons(cons(X, Ys), Cols), cons(Ys, Cols1), cons(nil, As))
row2col_in_agaa(nil, nil, nil, nil) → row2col_out_agaa(nil, nil, nil, nil)

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(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)
nil  =  nil
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

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

The TRS R consists of the following rules:

row2col_in_agaa(cons(cons(X, Ys), Cols)) → U4_agaa(X, Ys, row2col_in_agaa(Cols))
U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As))
row2col_in_agaa(nil) → row2col_out_agaa(nil, nil, nil)

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.

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

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: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U2_AGG(x₁) ) = x₁ + 2

POL( row2col_in_agaa(x₁) ) = max{0, 2x₁ - 2}

POL( cons(x₁, x₂) ) = x₁ + x₂ + 1

POL( U4_agaa(x₁, ..., x₃) ) = 2x₁ + 2x₂ + x₃ + 2

POL( nil ) = 0

POL( row2col_out_agaa(x₁, ..., x₃) ) = max{0, 2x₂ - 1}

POL( TRANSPOSE_AUX_IN_AGG(x₁, x₂) ) = 2x₂

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

row2col_in_agaa(cons(cons(X, Ys), Cols)) → U4_agaa(X, Ys, row2col_in_agaa(Cols))
row2col_in_agaa(nil) → row2col_out_agaa(nil, nil, nil)
U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As))

(20) Obligation:

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

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

The TRS R consists of the following rules:

row2col_in_agaa(cons(cons(X, Ys), Cols)) → U4_agaa(X, Ys, row2col_in_agaa(Cols))
U4_agaa(X, Ys, row2col_out_agaa(Xs, Cols1, As)) → row2col_out_agaa(cons(X, Xs), cons(Ys, Cols1), cons(nil, As))
row2col_in_agaa(nil) → row2col_out_agaa(nil, nil, nil)

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.

(21) DependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE