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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="D: transpose(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: transpose(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: transpose_aux(T7, nil, T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: transpose_aux(T7, nil, T6), SCOPE: 2, CLAUSE: 1 | transpose_aux(T7, nil, T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: transpose_aux(T7, nil, T6), SCOPE: 2, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: transpose_aux(T7, nil, T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: ','(row2col(T28, cons(T26, T27), X35, X36), transpose_aux(T29, X36, X35))\n\nT26 is ground\nT27 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="C: row2col(T28, cons(T26, T27), X35, X36)\n\nT26 is ground\nT27 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: transpose_aux(T36, T35, T34)\n\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: row2col(T28, cons(T26, T27), X35, X36), SCOPE: 3, CLAUSE: 3 | row2col(T28, cons(T26, T27), X35, X36), SCOPE: 3, CLAUSE: 4\n\nT26 is ground\nT27 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: row2col(T28, cons(T26, T27), X35, X36), SCOPE: 3, CLAUSE: 3\n\nT26 is ground\nT27 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: row2col(T28, cons(T26, T27), X35, X36), SCOPE: 3, CLAUSE: 4\n\nT26 is ground\nT27 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="A: row2col(T61, T60, X91, X92)\n\nT60 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: row2col(T61, T60, X91, X92), SCOPE: 4, CLAUSE: 3 | row2col(T61, T60, X91, X92), SCOPE: 4, CLAUSE: 4\n\nT60 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: row2col(T61, T60, X91, X92), SCOPE: 4, CLAUSE: 3\n\nT60 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: row2col(T61, T60, X91, X92), SCOPE: 4, CLAUSE: 4\n\nT60 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: row2col(T86, T85, X139, X140)\n\nT85 is ground\nX139 is free\nX140 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: transpose_aux(T36, T35, T34), SCOPE: 5, CLAUSE: 1 | transpose_aux(T36, T35, T34), SCOPE: 5, CLAUSE: 2\n\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: transpose_aux(T36, T35, T34), SCOPE: 5, CLAUSE: 1\n\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: transpose_aux(T36, T35, T34), SCOPE: 5, CLAUSE: 2\n\nT34 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(row2col(T121, cons(T119, T120), X185, X186), transpose_aux(T122, X186, X185))\n\nT119 is ground\nT120 is ground\nX185 is free\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 2;

37 [label="ONLY EVAL with clause\ntranspose(X4, X5) :- transpose_aux(X4, nil, X5).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T6,\nX5 -> T6,\nT5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 39 [arrowhead = none ];
39 -> 5;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 40 [arrowhead = none ];
40 -> 6;

41 [label="EVAL with clause\ntranspose_aux(cons(X30, X31), X32, cons(X33, X34)) :- ','(row2col(X30, cons(X33, X34), X35, X36), transpose_aux(X31, X36, X35)).\nand substitutionX30 -> T28,\nX31 -> T29,\nT7 -> cons(T28, T29),\nX32 -> nil,\nX33 -> T26,\nX34 -> T27,\nT6 -> cons(T26, T27),\nT24 -> T28,\nT25 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 41 [arrowhead = none ];
41 -> 7;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 42 [arrowhead = none ];
42 -> 8;

43 [label="EVAL with clause\ntranspose_aux(nil, X200, X200).\nand substitutionT7 -> nil,\nX200 -> nil,\nT6 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 43 [arrowhead = none ];
43 -> 33;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 44 [arrowhead = none ];
44 -> 34;

45 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 45 [arrowhead = none ];
45 -> 9;

46 [label="SPLIT 2\nnew knowledge:\nT28 is ground\nT26 is ground\nT27 is ground\nT34 is ground\nT35 is ground\nreplacements:X35 -> T34,\nX36 -> T35,\nT29 -> T36", fontsize=14, style = filled, fillcolor = violet];
7 -> 46 [arrowhead = none ];
46 -> 10;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 47 [arrowhead = none ];
47 -> 11;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 48 [arrowhead = none ];
48 -> 25;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 50 [arrowhead = none ];
50 -> 13;

51 [label="EVAL with clause\nrow2col(cons(X85, X86), cons(cons(X85, X87), X88), cons(X87, X89), cons(nil, X90)) :- row2col(X86, X88, X89, X90).\nand substitutionX85 -> T57,\nX86 -> T61,\nT28 -> cons(T57, T61),\nX87 -> T59,\nT26 -> cons(T57, T59),\nT27 -> T60,\nX88 -> T60,\nX89 -> X91,\nX35 -> cons(T59, X91),\nX90 -> X92,\nX36 -> cons(nil, X92),\nT58 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 51 [arrowhead = none ];
51 -> 14;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 52 [arrowhead = none ];
52 -> 15;

53 [label="BACKTRACK\nfor clause: row2col(nil, nil, nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 53 [arrowhead = none ];
53 -> 24;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 54 [arrowhead = none ];
54 -> 16;

55 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 55 [arrowhead = none ];
55 -> 17;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="EVAL with clause\nrow2col(cons(X133, X134), cons(cons(X133, X135), X136), cons(X135, X137), cons(nil, X138)) :- row2col(X134, X136, X137, X138).\nand substitutionX133 -> T82,\nX134 -> T86,\nT61 -> cons(T82, T86),\nX135 -> T84,\nX136 -> T85,\nT60 -> cons(cons(T82, T84), T85),\nX137 -> X139,\nX91 -> cons(T84, X139),\nX138 -> X140,\nX92 -> cons(nil, X140),\nT83 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 57 [arrowhead = none ];
57 -> 19;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 58 [arrowhead = none ];
58 -> 20;

59 [label="EVAL with clause\nrow2col(nil, nil, nil, nil).\nand substitutionT61 -> nil,\nT60 -> nil,\nX91 -> nil,\nX92 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 59 [arrowhead = none ];
59 -> 21;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 60 [arrowhead = none ];
60 -> 22;

61 [label="INSTANCE with matching:\nT61 -> T86\nT60 -> T85\nX91 -> X139\nX92 -> X140", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 61 [arrowhead = none , style=dashed];
61 -> 14[style=dashed];

62 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 62 [arrowhead = none ];
62 -> 23;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 63 [arrowhead = none ];
63 -> 26;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 64 [arrowhead = none ];
64 -> 27;

65 [label="EVAL with clause\ntranspose_aux(cons(X180, X181), X182, cons(X183, X184)) :- ','(row2col(X180, cons(X183, X184), X185, X186), transpose_aux(X181, X186, X185)).\nand substitutionX180 -> T121,\nX181 -> T122,\nT36 -> cons(T121, T122),\nT35 -> T118,\nX182 -> T118,\nX183 -> T119,\nX184 -> T120,\nT34 -> cons(T119, T120),\nT116 -> T121,\nT117 -> T122", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 65 [arrowhead = none ];
65 -> 28;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 66 [arrowhead = none ];
66 -> 29;

67 [label="EVAL with clause\ntranspose_aux(nil, X197, X197).\nand substitutionT36 -> nil,\nT35 -> T129,\nX197 -> T129,\nT34 -> T129", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 67 [arrowhead = none ];
67 -> 30;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 68 [arrowhead = none ];
68 -> 31;

69 [label="INSTANCE with matching:\nT28 -> T121\nT26 -> T119\nT27 -> T120\nX35 -> X185\nX36 -> X186\nT29 -> T122", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 69 [arrowhead = none , style=dashed];
69 -> 7[style=dashed];

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 70 [arrowhead = none ];
70 -> 32;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 71 [arrowhead = none ];
71 -> 35;

}

(2) Obligation:

Triples:

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

Clauses:

row2colcA(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) :- row2colcA(X2, X4, X5, X6).
row2colcA(nil, nil, nil, nil).
qcB(X1, X2, X3, cons(X4, X5), X6, cons(X7, X8)) :- ','(row2colcC(X1, X2, X3, cons(X4, X5), X6), qcB(X7, X4, X5, X9, X10, X8)).
qcB(X1, X2, X3, X4, X4, nil) :- row2colcC(X1, X2, X3, X4, X4).
row2colcC(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, 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(cons(X1, X2), cons(X3, X4)) → U5_AG(X1, X2, X3, X4, pB_in_aggaaa(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_AG(cons(X1, X2), cons(X3, X4)) → PB_IN_AGGAAA(X1, X3, X4, X5, X6, X2)
PB_IN_AGGAAA(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6), X7) → U2_AGGAAA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_agaa(X2, X4, X5, X6))
PB_IN_AGGAAA(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6), X7) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
ROW2COLA_IN_AGAA(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → U1_AGAA(X1, X2, X3, X4, X5, X6, row2colA_in_agaa(X2, X4, X5, X6))
ROW2COLA_IN_AGAA(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
PB_IN_AGGAAA(X1, X2, X3, cons(X4, X5), X6, cons(X7, X8)) → U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_aggaa(X1, X2, X3, cons(X4, X5), X6))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, cons(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, cons(X4, X5), X6)) → PB_IN_AGGAAA(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(nil, nil, nil, nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(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)
nil  =  nil
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(cons(X1, X2), cons(X3, X4)) → U5_AG(X1, X2, X3, X4, pB_in_aggaaa(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_AG(cons(X1, X2), cons(X3, X4)) → PB_IN_AGGAAA(X1, X3, X4, X5, X6, X2)
PB_IN_AGGAAA(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6), X7) → U2_AGGAAA(X1, X2, X3, X4, X5, X6, X7, row2colA_in_agaa(X2, X4, X5, X6))
PB_IN_AGGAAA(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6), X7) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
ROW2COLA_IN_AGAA(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → U1_AGAA(X1, X2, X3, X4, X5, X6, row2colA_in_agaa(X2, X4, X5, X6))
ROW2COLA_IN_AGAA(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)
PB_IN_AGGAAA(X1, X2, X3, cons(X4, X5), X6, cons(X7, X8)) → U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_aggaa(X1, X2, X3, cons(X4, X5), X6))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, cons(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, cons(X4, X5), X6)) → PB_IN_AGGAAA(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(nil, nil, nil, nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(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)
nil  =  nil
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(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)

The TRS R consists of the following rules:

row2colcC_in_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(nil, nil, nil, nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(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)
nil  =  nil
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(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → ROW2COLA_IN_AGAA(X2, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(x1, x2)
nil  =  nil
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(cons(cons(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(cons(cons(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, cons(X4, X5), X6, cons(X7, X8)) → U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_aggaa(X1, X2, X3, cons(X4, X5), X6))
U3_AGGAAA(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_aggaa(X1, X2, X3, cons(X4, X5), X6)) → PB_IN_AGGAAA(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6)) → U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6)) → U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_in_agaa(X2, X4, X5, X6))
row2colcA_in_agaa(nil, nil, nil, nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U7_agaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))
U11_aggaa(X1, X2, X3, X4, X5, X6, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6))

The argument filtering Pi contains the following mapping:
cons(x1, x2)  =  cons(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)
nil  =  nil
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, cons(X4, X5), X6)) → PB_IN_AGGAAA(X4, X5)

The TRS R consists of the following rules:

row2colcC_in_aggaa(cons(X1, X3), X4) → U11_aggaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(cons(cons(X1, X3), X4)) → U7_agaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U7_agaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))
U11_aggaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, 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(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(cons(x₁, x₂)) = 1 + x₁ + x₂   
POL(nil) = 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(cons(X1, X3), X4) → U11_aggaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(cons(cons(X1, X3), X4)) → U7_agaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U11_aggaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, X6))
U7_agaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))

(18) Obligation:

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

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

The TRS R consists of the following rules:

row2colcC_in_aggaa(cons(X1, X3), X4) → U11_aggaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(cons(cons(X1, X3), X4)) → U7_agaa(X1, X3, X4, row2colcA_in_agaa(X4))
row2colcA_in_agaa(nil) → row2colcA_out_agaa(nil, nil, nil, nil)
U7_agaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcA_out_agaa(cons(X1, X2), cons(cons(X1, X3), X4), cons(X3, X5), cons(nil, X6))
U11_aggaa(X1, X3, X4, row2colcA_out_agaa(X2, X4, X5, X6)) → row2colcC_out_aggaa(cons(X1, X2), cons(X1, X3), X4, cons(X3, X5), cons(nil, 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