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

(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(g,a)

(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\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: transpose(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: transpose_aux(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 1 | transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: ','(row2col(T24, .(T28, T29), X35, X36), transpose_aux(T25, X36, X35))\n\nT24 is ground\nT25 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(T24, .(T28, T29), X35, X36)\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: transpose_aux(T25, T35, T34)\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: row2col(T24, .(T28, T29), X35, X36), SCOPE: 3, CLAUSE: 3 | row2col(T24, .(T28, T29), X35, X36), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: row2col(T24, .(T28, T29), X35, X36), SCOPE: 3, CLAUSE: 3\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: row2col(T24, .(T28, T29), X35, X36), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="A: row2col(T57, T60, X91, X92)\n\nT57 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(T57, T60, X91, X92), SCOPE: 4, CLAUSE: 3 | row2col(T57, T60, X91, X92), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: row2col(T57, T60, X91, X92), SCOPE: 4, CLAUSE: 3\n\nT57 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: row2col(T57, T60, X91, X92), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: row2col(T82, T85, X139, X140)\n\nT82 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(T25, T35, T34), SCOPE: 5, CLAUSE: 1 | transpose_aux(T25, T35, T34), SCOPE: 5, CLAUSE: 2\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: transpose_aux(T25, T35, T34), SCOPE: 5, CLAUSE: 1\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: transpose_aux(T25, T35, T34), SCOPE: 5, CLAUSE: 2\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(row2col(T115, .(T120, T121), X185, X186), transpose_aux(T116, X186, X185))\n\nT115 is ground\nT116 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, [], X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T7,\nX5 -> T7,\nT6 -> 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(.(X30, X31), X32, .(X33, X34)) :- ','(row2col(X30, .(X33, X34), X35, X36), transpose_aux(X31, X36, X35)).\nand substitutionX30 -> T24,\nX31 -> T25,\nT5 -> .(T24, T25),\nX32 -> [],\nX33 -> T28,\nX34 -> T29,\nT7 -> .(T28, T29),\nT26 -> T28,\nT27 -> 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([], X200, X200).\nand substitutionT5 -> [],\nX200 -> [],\nT7 -> []", 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:\nT24 is ground\nT35 is ground\nreplacements:X35 -> T34,\nX36 -> T35", 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(.(X85, X86), .(.(X85, X87), X88), .(X87, X89), .([], X90)) :- row2col(X86, X88, X89, X90).\nand substitutionX85 -> T56,\nX86 -> T57,\nT24 -> .(T56, T57),\nX87 -> T58,\nT28 -> .(T56, T58),\nT29 -> T60,\nX88 -> T60,\nX89 -> X91,\nX35 -> .(T58, X91),\nX90 -> X92,\nX36 -> .([], X92),\nT59 -> T60", 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([], [], [], [])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(.(X133, X134), .(.(X133, X135), X136), .(X135, X137), .([], X138)) :- row2col(X134, X136, X137, X138).\nand substitutionX133 -> T81,\nX134 -> T82,\nT57 -> .(T81, T82),\nX135 -> T83,\nX136 -> T85,\nT60 -> .(.(T81, T83), T85),\nX137 -> X139,\nX91 -> .(T83, X139),\nX138 -> X140,\nX92 -> .([], X140),\nT84 -> T85", 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([], [], [], []).\nand substitutionT57 -> [],\nT60 -> [],\nX91 -> [],\nX92 -> []", 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:\nT57 -> T82\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(.(X180, X181), X182, .(X183, X184)) :- ','(row2col(X180, .(X183, X184), X185, X186), transpose_aux(X181, X186, X185)).\nand substitutionX180 -> T115,\nX181 -> T116,\nT25 -> .(T115, T116),\nT35 -> T117,\nX182 -> T117,\nX183 -> T120,\nX184 -> T121,\nT34 -> .(T120, T121),\nT118 -> T120,\nT119 -> T121", 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([], X197, X197).\nand substitutionT25 -> [],\nT35 -> T128,\nX197 -> T128,\nT34 -> T128", 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:\nT24 -> T115\nT28 -> T120\nT29 -> T121\nX35 -> X185\nX36 -> X186\nT25 -> T116", 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(.(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(x1)

(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: (b,f)
pB_in: (b,f,f,f,f,b)
row2colA_in: (b,f,f,f)
row2colcC_in: (b,f,f,f,f)
row2colcA_in: (b,f,f,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → U5_GA(X1, X2, X3, X4, pB_in_gaaaag(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → PB_IN_GAAAAG(X1, X3, X4, X5, X6, X2)
PB_IN_GAAAAG(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → U2_GAAAAG(X1, X2, X3, X4, X5, X6, X7, row2colA_in_gaaa(X2, X4, X5, X6))
PB_IN_GAAAAG(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → ROW2COLA_IN_GAAA(X2, X4, X5, X6)
ROW2COLA_IN_GAAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U1_GAAA(X1, X2, X3, X4, X5, X6, row2colA_in_gaaa(X2, X4, X5, X6))
ROW2COLA_IN_GAAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_GAAA(X2, X4, X5, X6)
PB_IN_GAAAAG(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_gaaaa(X1, X2, X3, .(X4, X5), X6))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → U4_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_gaaaag(X7, X4, X5, X9, X10, X8))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_GAAAAG(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa([], [], [], []) → row2colcA_out_gaaa([], [], [], [])
U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcA_out_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcC_out_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
row2colA_in_gaaa(x1, x2, x3, x4)  =  row2colA_in_gaaa(x1)
row2colcC_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_gaaaa(x1, x2, x7)
row2colcA_in_gaaa(x1, x2, x3, x4)  =  row2colcA_in_gaaa(x1)
U7_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaaa(x1, x2, x7)
[]  =  []
row2colcA_out_gaaa(x1, x2, x3, x4)  =  row2colcA_out_gaaa(x1, x4)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_out_gaaaa(x1, x5)
TRANSPOSED_IN_GA(x1, x2)  =  TRANSPOSED_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U2_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GAAAAG(x1, x2, x7, x8)
ROW2COLA_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLA_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GAAA(x1, x2, x7)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GAAAAG(x1, x7, x8, x9)
U4_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_GAAAAG(x1, x6, x7, x8, 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_GA(.(X1, X2), .(X3, X4)) → U5_GA(X1, X2, X3, X4, pB_in_gaaaag(X1, X3, X4, X5, X6, X2))
TRANSPOSED_IN_GA(.(X1, X2), .(X3, X4)) → PB_IN_GAAAAG(X1, X3, X4, X5, X6, X2)
PB_IN_GAAAAG(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → U2_GAAAAG(X1, X2, X3, X4, X5, X6, X7, row2colA_in_gaaa(X2, X4, X5, X6))
PB_IN_GAAAAG(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6), X7) → ROW2COLA_IN_GAAA(X2, X4, X5, X6)
ROW2COLA_IN_GAAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U1_GAAA(X1, X2, X3, X4, X5, X6, row2colA_in_gaaa(X2, X4, X5, X6))
ROW2COLA_IN_GAAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_GAAA(X2, X4, X5, X6)
PB_IN_GAAAAG(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_gaaaa(X1, X2, X3, .(X4, X5), X6))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → U4_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, pB_in_gaaaag(X7, X4, X5, X9, X10, X8))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_GAAAAG(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa([], [], [], []) → row2colcA_out_gaaa([], [], [], [])
U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcA_out_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcC_out_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
row2colA_in_gaaa(x1, x2, x3, x4)  =  row2colA_in_gaaa(x1)
row2colcC_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_gaaaa(x1, x2, x7)
row2colcA_in_gaaa(x1, x2, x3, x4)  =  row2colcA_in_gaaa(x1)
U7_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaaa(x1, x2, x7)
[]  =  []
row2colcA_out_gaaa(x1, x2, x3, x4)  =  row2colcA_out_gaaa(x1, x4)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_out_gaaaa(x1, x5)
TRANSPOSED_IN_GA(x1, x2)  =  TRANSPOSED_IN_GA(x1)
U5_GA(x1, x2, x3, x4, x5)  =  U5_GA(x1, x2, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U2_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8)  =  U2_GAAAAG(x1, x2, x7, x8)
ROW2COLA_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLA_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GAAA(x1, x2, x7)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GAAAAG(x1, x7, x8, x9)
U4_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U4_GAAAAG(x1, x6, x7, x8, 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_GAAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_GAAA(X2, X4, X5, X6)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa([], [], [], []) → row2colcA_out_gaaa([], [], [], [])
U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcA_out_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcC_out_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2colcC_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_gaaaa(x1, x2, x7)
row2colcA_in_gaaa(x1, x2, x3, x4)  =  row2colcA_in_gaaa(x1)
U7_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaaa(x1, x2, x7)
[]  =  []
row2colcA_out_gaaa(x1, x2, x3, x4)  =  row2colcA_out_gaaa(x1, x4)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_out_gaaaa(x1, x5)
ROW2COLA_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLA_IN_GAAA(x1)

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_GAAA(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → ROW2COLA_IN_GAAA(X2, X4, X5, X6)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COLA_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLA_IN_GAAA(x1)

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_GAAA(.(X1, X2)) → ROW2COLA_IN_GAAA(X2)

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_GAAA(.(X1, X2)) → ROW2COLA_IN_GAAA(X2)
  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_GAAAAG(X1, X2, X3, .(X4, X5), X6, .(X7, X8)) → U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_in_gaaaa(X1, X2, X3, .(X4, X5), X6))
U3_GAAAAG(X1, X2, X3, X4, X5, X6, X7, X8, row2colcC_out_gaaaa(X1, X2, X3, .(X4, X5), X6)) → PB_IN_GAAAAG(X7, X4, X5, X9, X10, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6)) → U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6)) → U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_in_gaaa(X2, X4, X5, X6))
row2colcA_in_gaaa([], [], [], []) → row2colcA_out_gaaa([], [], [], [])
U7_gaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcA_out_gaaa(.(X1, X2), .(.(X1, X3), X4), .(X3, X5), .([], X6))
U11_gaaaa(X1, X2, X3, X4, X5, X6, row2colcA_out_gaaa(X2, X4, X5, X6)) → row2colcC_out_gaaaa(.(X1, X2), .(X1, X3), X4, .(X3, X5), .([], X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2colcC_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_in_gaaaa(x1)
U11_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U11_gaaaa(x1, x2, x7)
row2colcA_in_gaaa(x1, x2, x3, x4)  =  row2colcA_in_gaaa(x1)
U7_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U7_gaaa(x1, x2, x7)
[]  =  []
row2colcA_out_gaaa(x1, x2, x3, x4)  =  row2colcA_out_gaaa(x1, x4)
row2colcC_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colcC_out_gaaaa(x1, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U3_GAAAAG(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U3_GAAAAG(x1, x7, x8, 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_GAAAAG(X1, .(X7, X8)) → U3_GAAAAG(X1, X7, X8, row2colcC_in_gaaaa(X1))
U3_GAAAAG(X1, X7, X8, row2colcC_out_gaaaa(X1, X6)) → PB_IN_GAAAAG(X7, X8)

The TRS R consists of the following rules:

row2colcC_in_gaaaa(.(X1, X2)) → U11_gaaaa(X1, X2, row2colcA_in_gaaa(X2))
row2colcA_in_gaaa(.(X1, X2)) → U7_gaaa(X1, X2, row2colcA_in_gaaa(X2))
row2colcA_in_gaaa([]) → row2colcA_out_gaaa([], [])
U7_gaaa(X1, X2, row2colcA_out_gaaa(X2, X6)) → row2colcA_out_gaaa(.(X1, X2), .([], X6))
U11_gaaaa(X1, X2, row2colcA_out_gaaa(X2, X6)) → row2colcC_out_gaaaa(.(X1, X2), .([], X6))

The set Q consists of the following terms:

row2colcC_in_gaaaa(x0)
row2colcA_in_gaaa(x0)
U7_gaaa(x0, x1, x2)
U11_gaaaa(x0, x1, x2)

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

(17) 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:

• U3_GAAAAG(X1, X7, X8, row2colcC_out_gaaaa(X1, X6)) → PB_IN_GAAAAG(X7, X8)
  The graph contains the following edges 2 >= 1, 3 >= 2

• PB_IN_GAAAAG(X1, .(X7, X8)) → U3_GAAAAG(X1, X7, X8, row2colcC_in_gaaaa(X1))
  The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(18) YES