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 PrologToDTProblemTransformerProof (⇐)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇐)
↳4 PiDP
↳5 DependencyGraphProof (⇔)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 YES
    ↳14 PiDP
        ↳15 PiDPToQDPProof (⇐)
        ↳16 QDP
        ↳17 QDPOrderProof (⇔)
        ↳18 QDP
        ↳19 DependencyGraphProof (⇔)
        ↳20 TRUE

(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).

Queries:

transpose(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

row2col14(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) :- row2col14(T86, T85, X150, X151).
p7(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97), T29) :- row2col14(T61, T60, X96, X97).
p7(T28, T26, T27, cons(T119, T120), T118, cons(T121, T122)) :- ','(row2colc9(T28, T26, T27, cons(T119, T120), T118), p7(T121, T119, T120, X202, X203, T122)).
transpose1(cons(T28, T29), cons(T26, T27)) :- p7(T28, T26, T27, X35, X36, T29).

Clauses:

row2colc14(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) :- row2colc14(T86, T85, X150, X151).
row2colc14(nil, nil, nil, nil).
qc7(T28, T26, T27, cons(T119, T120), T118, cons(T121, T122)) :- ','(row2colc9(T28, T26, T27, cons(T119, T120), T118), qc7(T121, T119, T120, X202, X203, T122)).
qc7(T28, T26, T27, T129, T129, nil) :- row2colc9(T28, T26, T27, T129, T129).
row2colc9(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97)) :- row2colc14(T61, T60, X96, X97).

Afs:

transpose1(x1, x2)  =  transpose1(x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
transpose1_in: (f,b)
p7_in: (f,b,b,f,f,f)
row2col14_in: (f,b,f,f)
row2colc9_in: (f,b,b,f,f)
row2colc14_in: (f,b,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSE1_IN_AG(cons(T28, T29), cons(T26, T27)) → U5_AG(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
TRANSPOSE1_IN_AG(cons(T28, T29), cons(T26, T27)) → P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29)
P7_IN_AGGAAA(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97), T29) → U2_AGGAAA(T57, T61, T59, T60, X96, X97, T29, row2col14_in_agaa(T61, T60, X96, X97))
P7_IN_AGGAAA(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97), T29) → ROW2COL14_IN_AGAA(T61, T60, X96, X97)
ROW2COL14_IN_AGAA(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → U1_AGAA(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
ROW2COL14_IN_AGAA(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)
P7_IN_AGGAAA(T28, T26, T27, cons(T119, T120), T118, cons(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_in_aggaa(T28, T26, T27, cons(T119, T120), T118))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → U4_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)

The TRS R consists of the following rules:

row2colc9_in_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97)) → U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_in_agaa(T61, T60, X96, X97))
row2colc14_in_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_in_agaa(T86, T85, X150, X151))
row2colc14_in_agaa(nil, nil, nil, nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))
U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
row2colc9_in_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colc14_in_agaa(x1, x2, x3, x4)  =  row2colc14_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
nil  =  nil
row2colc14_out_agaa(x1, x2, x3, x4)  =  row2colc14_out_agaa(x1, x2, x3, x4)
row2colc9_out_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_out_aggaa(x1, x2, x3, x4, x5)
TRANSPOSE1_IN_AG(x1, x2)  =  TRANSPOSE1_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_IN_AGGAAA(x2, x3)
U2_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AGGAAA(x1, x3, x4, x8)
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_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:

TRANSPOSE1_IN_AG(cons(T28, T29), cons(T26, T27)) → U5_AG(T28, T29, T26, T27, p7_in_aggaaa(T28, T26, T27, X35, X36, T29))
TRANSPOSE1_IN_AG(cons(T28, T29), cons(T26, T27)) → P7_IN_AGGAAA(T28, T26, T27, X35, X36, T29)
P7_IN_AGGAAA(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97), T29) → U2_AGGAAA(T57, T61, T59, T60, X96, X97, T29, row2col14_in_agaa(T61, T60, X96, X97))
P7_IN_AGGAAA(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97), T29) → ROW2COL14_IN_AGAA(T61, T60, X96, X97)
ROW2COL14_IN_AGAA(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → U1_AGAA(T82, T86, T84, T85, X150, X151, row2col14_in_agaa(T86, T85, X150, X151))
ROW2COL14_IN_AGAA(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)
P7_IN_AGGAAA(T28, T26, T27, cons(T119, T120), T118, cons(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_in_aggaa(T28, T26, T27, cons(T119, T120), T118))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → U4_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, p7_in_aggaaa(T121, T119, T120, X202, X203, T122))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)

The TRS R consists of the following rules:

row2colc9_in_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97)) → U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_in_agaa(T61, T60, X96, X97))
row2colc14_in_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_in_agaa(T86, T85, X150, X151))
row2colc14_in_agaa(nil, nil, nil, nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))
U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
p7_in_aggaaa(x1, x2, x3, x4, x5, x6)  =  p7_in_aggaaa(x2, x3)
row2col14_in_agaa(x1, x2, x3, x4)  =  row2col14_in_agaa(x2)
row2colc9_in_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colc14_in_agaa(x1, x2, x3, x4)  =  row2colc14_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
nil  =  nil
row2colc14_out_agaa(x1, x2, x3, x4)  =  row2colc14_out_agaa(x1, x2, x3, x4)
row2colc9_out_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_out_aggaa(x1, x2, x3, x4, x5)
TRANSPOSE1_IN_AG(x1, x2)  =  TRANSPOSE1_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5)  =  U5_AG(x3, x4, x5)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_IN_AGGAAA(x2, x3)
U2_AGGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_AGGAAA(x1, x3, x4, x8)
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_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:

ROW2COL14_IN_AGAA(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)

The TRS R consists of the following rules:

row2colc9_in_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97)) → U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_in_agaa(T61, T60, X96, X97))
row2colc14_in_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_in_agaa(T86, T85, X150, X151))
row2colc14_in_agaa(nil, nil, nil, nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))
U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
row2colc9_in_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colc14_in_agaa(x1, x2, x3, x4)  =  row2colc14_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
nil  =  nil
row2colc14_out_agaa(x1, x2, x3, x4)  =  row2colc14_out_agaa(x1, x2, x3, x4)
row2colc9_out_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_out_aggaa(x1, x2, x3, x4, x5)
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_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:

ROW2COL14_IN_AGAA(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → ROW2COL14_IN_AGAA(T86, T85, X150, X151)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
ROW2COL14_IN_AGAA(x1, x2, x3, x4)  =  ROW2COL14_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:

ROW2COL14_IN_AGAA(cons(cons(T82, T84), T85)) → ROW2COL14_IN_AGAA(T85)

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:

  • ROW2COL14_IN_AGAA(cons(cons(T82, T84), T85)) → ROW2COL14_IN_AGAA(T85)
    The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

P7_IN_AGGAAA(T28, T26, T27, cons(T119, T120), T118, cons(T121, T122)) → U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_in_aggaa(T28, T26, T27, cons(T119, T120), T118))
U3_AGGAAA(T28, T26, T27, T119, T120, T118, T121, T122, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → P7_IN_AGGAAA(T121, T119, T120, X202, X203, T122)

The TRS R consists of the following rules:

row2colc9_in_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97)) → U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_in_agaa(T61, T60, X96, X97))
row2colc14_in_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151)) → U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_in_agaa(T86, T85, X150, X151))
row2colc14_in_agaa(nil, nil, nil, nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U7_agaa(T82, T86, T84, T85, X150, X151, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))
U11_aggaa(T57, T61, T59, T60, X96, X97, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
row2colc9_in_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_in_aggaa(x2, x3)
U11_aggaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_aggaa(x1, x3, x4, x7)
row2colc14_in_agaa(x1, x2, x3, x4)  =  row2colc14_in_agaa(x2)
U7_agaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_agaa(x1, x3, x4, x7)
nil  =  nil
row2colc14_out_agaa(x1, x2, x3, x4)  =  row2colc14_out_agaa(x1, x2, x3, x4)
row2colc9_out_aggaa(x1, x2, x3, x4, x5)  =  row2colc9_out_aggaa(x1, x2, x3, x4, x5)
P7_IN_AGGAAA(x1, x2, x3, x4, x5, x6)  =  P7_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:

P7_IN_AGGAAA(T26, T27) → U3_AGGAAA(T26, T27, row2colc9_in_aggaa(T26, T27))
U3_AGGAAA(T26, T27, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → P7_IN_AGGAAA(T119, T120)

The TRS R consists of the following rules:

row2colc9_in_aggaa(cons(T57, T59), T60) → U11_aggaa(T57, T59, T60, row2colc14_in_agaa(T60))
row2colc14_in_agaa(cons(cons(T82, T84), T85)) → U7_agaa(T82, T84, T85, row2colc14_in_agaa(T85))
row2colc14_in_agaa(nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U7_agaa(T82, T84, T85, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))
U11_aggaa(T57, T59, T60, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))

The set Q consists of the following terms:

row2colc9_in_aggaa(x0, x1)
row2colc14_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].


The following pairs can be oriented strictly and are deleted.


P7_IN_AGGAAA(T26, T27) → U3_AGGAAA(T26, T27, row2colc9_in_aggaa(T26, T27))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(P7_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(cons(x1, x2)) = 1 + x1 + x2   
POL(nil) = 0   
POL(row2colc14_in_agaa(x1)) = x1   
POL(row2colc14_out_agaa(x1, x2, x3, x4)) = x3   
POL(row2colc9_in_aggaa(x1, x2)) = x1 + x2   
POL(row2colc9_out_aggaa(x1, x2, x3, x4, x5)) = x4   

The following usable rules [FROCOS05] were oriented:

row2colc9_in_aggaa(cons(T57, T59), T60) → U11_aggaa(T57, T59, T60, row2colc14_in_agaa(T60))
row2colc14_in_agaa(cons(cons(T82, T84), T85)) → U7_agaa(T82, T84, T85, row2colc14_in_agaa(T85))
row2colc14_in_agaa(nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U11_aggaa(T57, T59, T60, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))
U7_agaa(T82, T84, T85, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))

(18) Obligation:

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

U3_AGGAAA(T26, T27, row2colc9_out_aggaa(T28, T26, T27, cons(T119, T120), T118)) → P7_IN_AGGAAA(T119, T120)

The TRS R consists of the following rules:

row2colc9_in_aggaa(cons(T57, T59), T60) → U11_aggaa(T57, T59, T60, row2colc14_in_agaa(T60))
row2colc14_in_agaa(cons(cons(T82, T84), T85)) → U7_agaa(T82, T84, T85, row2colc14_in_agaa(T85))
row2colc14_in_agaa(nil) → row2colc14_out_agaa(nil, nil, nil, nil)
U7_agaa(T82, T84, T85, row2colc14_out_agaa(T86, T85, X150, X151)) → row2colc14_out_agaa(cons(T82, T86), cons(cons(T82, T84), T85), cons(T84, X150), cons(nil, X151))
U11_aggaa(T57, T59, T60, row2colc14_out_agaa(T61, T60, X96, X97)) → row2colc9_out_aggaa(cons(T57, T61), cons(T57, T59), T60, cons(T59, X96), cons(nil, X97))

The set Q consists of the following terms:

row2colc9_in_aggaa(x0, x1)
row2colc14_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