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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 88 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 22 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 13 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇒, 0 ms)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔, 61 ms)
        ↳18 QDP
        ↳19 DependencyGraphProof (⇔, 0 ms)
        ↳20 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([], [], [], []).

Query: transpose(a,g)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

row2colA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) :- row2colA(X2, X4, X5, X6).
pB(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) :- row2colA(X2, X4, X5, X6).
pB(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) :- ','(row2colcC(X1, X2, X3, .(X4, X5), X6), pB(X7, X4, X5, X9, X10, X8)).
transposeD(.(X1, X2), .(X3, X4)) :- pB(X1, X3, X4, X5, X6, X2).

Clauses:

row2colcA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) :- row2colcA(X2, X4, X5, X6).
row2colcA([], [], [], []).
qcB(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) :- ','(row2colcC(X1, X2, X3, .(X4, X5), X6), qcB(X7, X4, X5, X9, X10, X8)).
qcB(X1, X2, X3, X4, X4, []) :- row2colcC(X1, X2, X3, X4, X4).
row2colcC(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) :- row2colcA(X2, X4, X5, X6).

Afs:

transposeD(x1, x2)  =  transposeD(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
transposeD_in: (f,b)
pB_in: (f,b,b,f,f,f)
row2colA_in: (f,b,f,f)
row2colcC_in: (f,b,b,f,f)
row2colcA_in: (f,b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSED_IN_AG(.(X1, X2), .(X3, X4)) → U5_AG(X1, X2, X3, X4, pB_in_aggaaa(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_AG(.(X1, X2), .(X3, X4)) → PB_IN_AGGAAA(X1, X3, X4, X5, X6, X2)
PB_IN_AGGAAA(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → U2_AGGAAA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_agaa(X2, X4, X5, X6))
PB_IN_AGGAAA(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
ROW2COLA_IN_AGAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U1_AGAA(X1, X2, X3, X4, X5, X6, row2colA_in_agaa(X2, X4, X5, X6))
ROW2COLA_IN_AGAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
PB_IN_AGGAAA(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_aggaa(X1, X2, X3, .(X4, X5), X6))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → U4_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_aggaaa(X7, X4, X5, X9, X10, X8))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_AGGAAA(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa([], [], [], []) → row2colcA_out_agaa([], [], [], [])
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
pB_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  pB_in_aggaaa(x2, x3)
row2colA_in_agaa(x1, x2, x3, x4)  =  row2colA_in_agaa(x2)
row2colcC_in_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colcA_in_agaa(x1, x2, x3, x4)  =  row2colcA_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
[]  =  []
row2colcA_out_agaa(x1, x2, x3, x4)  =  row2colcA_out_agaa(x1, x2, x3, x4)
row2colcC_out_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_out_aggaa(x1, x2, x3, x4, x5)
TRANSPOSED_IN_AG(x1, x2)  =  TRANSPOSED_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
PB_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_AGGAAA(x2, x3)
U2_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AGGAAA(x1, x3, x4, x8)
ROW2COLA_IN_AGAA(x1, x2, x3, x4)  =  ROW2COLA_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_AGAA(x1, x3, x4, x7)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, x9)
U4_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_AGGAAA(x1, x2, x3, x4, x5, x6, x9)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

TRANSPOSED_IN_AG(.(X1, X2), .(X3, X4)) → U5_AG(X1, X2, X3, X4, pB_in_aggaaa(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_AG(.(X1, X2), .(X3, X4)) → PB_IN_AGGAAA(X1, X3, X4, X5, X6, X2)
PB_IN_AGGAAA(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → U2_AGGAAA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_agaa(X2, X4, X5, X6))
PB_IN_AGGAAA(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
ROW2COLA_IN_AGAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U1_AGAA(X1, X2, X3, X4, X5, X6, row2colA_in_agaa(X2, X4, X5, X6))
ROW2COLA_IN_AGAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
PB_IN_AGGAAA(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_aggaa(X1, X2, X3, .(X4, X5), X6))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → U4_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_aggaaa(X7, X4, X5, X9, X10, X8))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_AGGAAA(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa([], [], [], []) → row2colcA_out_agaa([], [], [], [])
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
pB_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  pB_in_aggaaa(x2, x3)
row2colA_in_agaa(x1, x2, x3, x4)  =  row2colA_in_agaa(x2)
row2colcC_in_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colcA_in_agaa(x1, x2, x3, x4)  =  row2colcA_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
[]  =  []
row2colcA_out_agaa(x1, x2, x3, x4)  =  row2colcA_out_agaa(x1, x2, x3, x4)
row2colcC_out_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_out_aggaa(x1, x2, x3, x4, x5)
TRANSPOSED_IN_AG(x1, x2)  =  TRANSPOSED_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
PB_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_AGGAAA(x2, x3)
U2_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AGGAAA(x1, x3, x4, x8)
ROW2COLA_IN_AGAA(x1, x2, x3, x4)  =  ROW2COLA_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_AGAA(x1, x3, x4, x7)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, x9)
U4_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_AGGAAA(x1, x2, x3, x4, x5, x6, x9)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ROW2COLA_IN_AGAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)

The TRS R consists of the following rules:

row2colcC_in_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa([], [], [], []) → row2colcA_out_agaa([], [], [], [])
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2colcC_in_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colcA_in_agaa(x1, x2, x3, x4)  =  row2colcA_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
[]  =  []
row2colcA_out_agaa(x1, x2, x3, x4)  =  row2colcA_out_agaa(x1, x2, x3, x4)
row2colcC_out_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_out_aggaa(x1, x2, x3, x4, x5)
ROW2COLA_IN_AGAA(x1, x2, x3, x4)  =  ROW2COLA_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:

ROW2COLA_IN_AGAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COLA_IN_AGAA(x1, x2, x3, x4)  =  ROW2COLA_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:

ROW2COLA_IN_AGAA(.(.(X1, X3), X4)) → ROW2COLA_IN_AGAA(X4)

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:

  • ROW2COLA_IN_AGAA(.(.(X1, X3), X4)) → ROW2COLA_IN_AGAA(X4)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

PB_IN_AGGAAA(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_aggaa(X1, X2, X3, .(X4, X5), X6))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_AGGAAA(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa([], [], [], []) → row2colcA_out_agaa([], [], [], [])
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2colcC_in_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colcA_in_agaa(x1, x2, x3, x4)  =  row2colcA_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
[]  =  []
row2colcA_out_agaa(x1, x2, x3, x4)  =  row2colcA_out_agaa(x1, x2, x3, x4)
row2colcC_out_aggaa(x1, x2, x3, x4, x5)  =  row2colcC_out_aggaa(x1, x2, x3, x4, x5)
PB_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  PB_IN_AGGAAA(x2, x3)
U3_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_AGGAAA(x2, x3, x9)

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

(15) PiDPToQDPProof (SOUND transformation)

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

(16) Obligation:

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

PB_IN_AGGAAA(X2, X3) → U3_AGGAAA(X2, X3, row2colcC_in_aggaa(X2, X3))
U3_AGGAAA(X2, X3, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_AGGAAA(X4, X5)

The TRS R consists of the following rules:

row2colcC_in_aggaa(.(X1, X3), X4) → U11_aggaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(.(.(X1, X3), X4)) → U7_agaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa([]) → row2colcA_out_agaa([], [], [], [])
U7_agaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_aggaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The set Q consists of the following terms:

row2colcC_in_aggaa(x0, x1)
row2colcA_in_agaa(x0)
U7_agaa(x0, x1, x2, x3)
U11_aggaa(x0, x1, x2, x3)

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

(17) QDPOrderProof (EQUIVALENT transformation)

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


The following pairs can be oriented strictly and are deleted.


PB_IN_AGGAAA(X2, X3) → U3_AGGAAA(X2, X3, row2colcC_in_aggaa(X2, X3))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x1 + x2   
POL(PB_IN_AGGAAA(x1, x2)) = 1 + x1 + x2   
POL(U11_aggaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL(U3_AGGAAA(x1, x2, x3)) = x3   
POL(U7_agaa(x1, x2, x3, x4)) = 1 + x2 + x4   
POL([]) = 0   
POL(row2colcA_in_agaa(x1)) = x1   
POL(row2colcA_out_agaa(x1, x2, x3, x4)) = x3   
POL(row2colcC_in_aggaa(x1, x2)) = x1 + x2   
POL(row2colcC_out_aggaa(x1, x2, x3, x4, x5)) = x4   

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

row2colcC_in_aggaa(.(X1, X3), X4) → U11_aggaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(.(.(X1, X3), X4)) → U7_agaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa([]) → row2colcA_out_agaa([], [], [], [])
U11_aggaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))
U7_agaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))

(18) Obligation:

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

U3_AGGAAA(X2, X3, row2colcC_out_aggaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_AGGAAA(X4, X5)

The TRS R consists of the following rules:

row2colcC_in_aggaa(.(X1, X3), X4) → U11_aggaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(.(.(X1, X3), X4)) → U7_agaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa([]) → row2colcA_out_agaa([], [], [], [])
U7_agaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_aggaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The set Q consists of the following terms:

row2colcC_in_aggaa(x0, x1)
row2colcA_in_agaa(x0)
U7_agaa(x0, x1, x2, x3)
U11_aggaa(x0, x1, x2, x3)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE