Left Termination of the query pattern
transpose(g,g)
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,g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: transpose(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: transpose(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: transpose_aux(T5, [], T6)\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 1 | transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="B: ','(row2col(T23, .(T25, T26), X35, X36), transpose_aux(T24, X36, X35))\n\nT23 is ground\nT24 is ground\nT25 is ground\nT26 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="D: row2col(T23, .(T25, T26), X35, X36)\n\nT23 is ground\nT25 is ground\nT26 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="E: transpose_aux(T24, T32, T31)\n\nT24 is ground\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: row2col(T23, .(T25, T26), X35, X36), SCOPE: 3, CLAUSE: 3 | row2col(T23, .(T25, T26), X35, X36), SCOPE: 3, CLAUSE: 4\n\nT23 is ground\nT25 is ground\nT26 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: row2col(T23, .(T25, T26), X35, X36), SCOPE: 3, CLAUSE: 3\n\nT23 is ground\nT25 is ground\nT26 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: row2col(T23, .(T25, T26), X35, X36), SCOPE: 3, CLAUSE: 4\n\nT23 is ground\nT25 is ground\nT26 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="C: row2col(T54, T56, X91, X92)\n\nT54 is ground\nT56 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: row2col(T54, T56, X91, X92), SCOPE: 4, CLAUSE: 3 | row2col(T54, T56, X91, X92), SCOPE: 4, CLAUSE: 4\n\nT54 is ground\nT56 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: row2col(T54, T56, X91, X92), SCOPE: 4, CLAUSE: 3\n\nT54 is ground\nT56 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: row2col(T54, T56, X91, X92), SCOPE: 4, CLAUSE: 4\n\nT54 is ground\nT56 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: row2col(T78, T80, X139, X140)\n\nT78 is ground\nT80 is ground\nX139 is free\nX140 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: transpose_aux(T24, T32, T31), SCOPE: 5, CLAUSE: 1 | transpose_aux(T24, T32, T31), SCOPE: 5, CLAUSE: 2\n\nT24 is ground\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: transpose_aux(T24, T32, T31), SCOPE: 5, CLAUSE: 1\n\nT24 is ground\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: transpose_aux(T24, T32, T31), SCOPE: 5, CLAUSE: 2\n\nT24 is ground\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(row2col(T110, .(T113, T114), X185, X186), transpose_aux(T111, X186, X185))\n\nT110 is ground\nT111 is ground\nT113 is ground\nT114 is ground\nX185 is free\nX186 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 83 [arrowhead = none ];
83 -> 4;

84 [label="ONLY EVAL with clause\ntranspose(X4, X5) :- transpose_aux(X4, [], X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T6,\nX5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 84 [arrowhead = none ];
84 -> 8;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 85 [arrowhead = none ];
85 -> 9;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 86 [arrowhead = none ];
86 -> 12;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 87 [arrowhead = none ];
87 -> 13;

88 [label="EVAL with clause\ntranspose_aux(.(X30, X31), X32, .(X33, X34)) :- ','(row2col(X30, .(X33, X34), X35, X36), transpose_aux(X31, X36, X35)).\nand substitutionX30 -> T23,\nX31 -> T24,\nT5 -> .(T23, T24),\nX32 -> [],\nX33 -> T25,\nX34 -> T26,\nT6 -> .(T25, T26)", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 88 [arrowhead = none ];
88 -> 17;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 89 [arrowhead = none ];
89 -> 18;

90 [label="EVAL with clause\ntranspose_aux([], X200, X200).\nand substitutionT5 -> [],\nX200 -> [],\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 90 [arrowhead = none ];
90 -> 80;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 91 [arrowhead = none ];
91 -> 81;

92 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 92 [arrowhead = none ];
92 -> 23;

93 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nT25 is ground\nT26 is ground\nT31 is ground\nT32 is ground\nreplacements:X35 -> T31,\nX36 -> T32", fontsize=14, style = filled, fillcolor = violet];
17 -> 93 [arrowhead = none ];
93 -> 24;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 94 [arrowhead = none ];
94 -> 27;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 95 [arrowhead = none ];
95 -> 64;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 96 [arrowhead = none ];
96 -> 31;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 97 [arrowhead = none ];
97 -> 33;

98 [label="EVAL with clause\nrow2col(.(X85, X86), .(.(X85, X87), X88), .(X87, X89), .([], X90)) :- row2col(X86, X88, X89, X90).\nand substitutionX85 -> T53,\nX86 -> T54,\nT23 -> .(T53, T54),\nX87 -> T55,\nT25 -> .(T53, T55),\nT26 -> T56,\nX88 -> T56,\nX89 -> X91,\nX35 -> .(T55, X91),\nX90 -> X92,\nX36 -> .([], X92)", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 98 [arrowhead = none ];
98 -> 38;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 99 [arrowhead = none ];
99 -> 39;

100 [label="BACKTRACK\nfor clause: row2col([], [], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 100 [arrowhead = none ];
100 -> 59;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 101 [arrowhead = none ];
101 -> 40;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 102 [arrowhead = none ];
102 -> 43;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 103 [arrowhead = none ];
103 -> 44;

104 [label="EVAL with clause\nrow2col(.(X133, X134), .(.(X133, X135), X136), .(X135, X137), .([], X138)) :- row2col(X134, X136, X137, X138).\nand substitutionX133 -> T77,\nX134 -> T78,\nT54 -> .(T77, T78),\nX135 -> T79,\nX136 -> T80,\nT56 -> .(.(T77, T79), T80),\nX137 -> X139,\nX91 -> .(T79, X139),\nX138 -> X140,\nX92 -> .([], X140)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 104 [arrowhead = none ];
104 -> 50;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 105 [arrowhead = none ];
105 -> 51;

106 [label="EVAL with clause\nrow2col([], [], [], []).\nand substitutionT54 -> [],\nT56 -> [],\nX91 -> [],\nX92 -> []", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 106 [arrowhead = none ];
106 -> 55;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 107 [arrowhead = none ];
107 -> 56;

108 [label="INSTANCE with matching:\nT54 -> T78\nT56 -> T80\nX91 -> X139\nX92 -> X140", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 108 [arrowhead = none , style=dashed];
108 -> 38[style=dashed];

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 109 [arrowhead = none ];
109 -> 57;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 110 [arrowhead = none ];
110 -> 66;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 111 [arrowhead = none ];
111 -> 67;

112 [label="EVAL with clause\ntranspose_aux(.(X180, X181), X182, .(X183, X184)) :- ','(row2col(X180, .(X183, X184), X185, X186), transpose_aux(X181, X186, X185)).\nand substitutionX180 -> T110,\nX181 -> T111,\nT24 -> .(T110, T111),\nT32 -> T112,\nX182 -> T112,\nX183 -> T113,\nX184 -> T114,\nT31 -> .(T113, T114)", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 112 [arrowhead = none ];
112 -> 71;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 113 [arrowhead = none ];
113 -> 72;

114 [label="EVAL with clause\ntranspose_aux([], X197, X197).\nand substitutionT24 -> [],\nT32 -> T121,\nX197 -> T121,\nT31 -> T121", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 114 [arrowhead = none ];
114 -> 77;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 115 [arrowhead = none ];
115 -> 78;

116 [label="INSTANCE with matching:\nT23 -> T110\nT25 -> T113\nT26 -> T114\nX35 -> X185\nX36 -> X186\nT24 -> T111", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 116 [arrowhead = none , style=dashed];
116 -> 17[style=dashed];

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 117 [arrowhead = none ];
117 -> 79;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 118 [arrowhead = none ];
118 -> 82;

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

transposeA_in_gg(.(T23, T24), .(T25, T26)) → U1_gg(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
pB_in_gggaag(T23, T25, T26, T31, T32, T24) → U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
row2colC_in_ggaa([], [], [], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))
U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
transpose_auxE_in_ggg(.(T110, T111), T112, .(T113, T114)) → U4_ggg(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
U4_ggg(T110, T111, T112, T113, T114, pB_out_gggaag(T110, T113, T114, X185, X186, T111)) → transpose_auxE_out_ggg(.(T110, T111), T112, .(T113, T114))
transpose_auxE_in_ggg([], T121, T121) → transpose_auxE_out_ggg([], T121, T121)
U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_out_ggg(T24, T32, T31)) → pB_out_gggaag(T23, T25, T26, T31, T32, T24)
U1_gg(T23, T24, T25, T26, pB_out_gggaag(T23, T25, T26, X35, X36, T24)) → transposeA_out_gg(.(T23, T24), .(T25, T26))
transposeA_in_gg([], []) → transposeA_out_gg([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_gg(x1, x2)  =  transposeA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
pB_in_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gggaag(x1, x2, x3, x6)
U5_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gggaag(x1, x2, x3, x6, x7)
row2colD_in_gggaa(x1, x2, x3, x4, x5)  =  row2colD_in_gggaa(x1, x2, x3)
U3_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggaa(x1, x2, x3, x4, x7)
row2colC_in_ggaa(x1, x2, x3, x4)  =  row2colC_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2colC_out_ggaa(x1, x2, x3, x4)  =  row2colC_out_ggaa(x1, x2, x3, x4)
row2colD_out_gggaa(x1, x2, x3, x4, x5)  =  row2colD_out_gggaa(x1, x2, x3, x4, x5)
U6_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaag(x1, x2, x3, x4, x5, x6, x7)
transpose_auxE_in_ggg(x1, x2, x3)  =  transpose_auxE_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5, x6)  =  U4_ggg(x1, x2, x3, x4, x5, x6)
pB_out_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gggaag(x1, x2, x3, x4, x5, x6)
transpose_auxE_out_ggg(x1, x2, x3)  =  transpose_auxE_out_ggg(x1, x2, x3)
transposeA_out_gg(x1, x2)  =  transposeA_out_gg(x1, x2)

(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:

TRANSPOSEA_IN_GG(.(T23, T24), .(T25, T26)) → U1_GG(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
TRANSPOSEA_IN_GG(.(T23, T24), .(T25, T26)) → PB_IN_GGGAAG(T23, T25, T26, X35, X36, T24)
PB_IN_GGGAAG(T23, T25, T26, T31, T32, T24) → U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
PB_IN_GGGAAG(T23, T25, T26, T31, T32, T24) → ROW2COLD_IN_GGGAA(T23, T25, T26, T31, T32)
ROW2COLD_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_GGGAA(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
ROW2COLD_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → ROW2COLC_IN_GGAA(T54, T56, X91, X92)
ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_GGAA(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → ROW2COLC_IN_GGAA(T78, T80, X139, X140)
U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_GGGAAG(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → TRANSPOSE_AUXE_IN_GGG(T24, T32, T31)
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → U4_GGG(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → PB_IN_GGGAAG(T110, T113, T114, X185, X186, T111)

The TRS R consists of the following rules:

transposeA_in_gg(.(T23, T24), .(T25, T26)) → U1_gg(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
pB_in_gggaag(T23, T25, T26, T31, T32, T24) → U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
row2colC_in_ggaa([], [], [], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))
U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
transpose_auxE_in_ggg(.(T110, T111), T112, .(T113, T114)) → U4_ggg(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
U4_ggg(T110, T111, T112, T113, T114, pB_out_gggaag(T110, T113, T114, X185, X186, T111)) → transpose_auxE_out_ggg(.(T110, T111), T112, .(T113, T114))
transpose_auxE_in_ggg([], T121, T121) → transpose_auxE_out_ggg([], T121, T121)
U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_out_ggg(T24, T32, T31)) → pB_out_gggaag(T23, T25, T26, T31, T32, T24)
U1_gg(T23, T24, T25, T26, pB_out_gggaag(T23, T25, T26, X35, X36, T24)) → transposeA_out_gg(.(T23, T24), .(T25, T26))
transposeA_in_gg([], []) → transposeA_out_gg([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_gg(x1, x2)  =  transposeA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
pB_in_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gggaag(x1, x2, x3, x6)
U5_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gggaag(x1, x2, x3, x6, x7)
row2colD_in_gggaa(x1, x2, x3, x4, x5)  =  row2colD_in_gggaa(x1, x2, x3)
U3_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggaa(x1, x2, x3, x4, x7)
row2colC_in_ggaa(x1, x2, x3, x4)  =  row2colC_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2colC_out_ggaa(x1, x2, x3, x4)  =  row2colC_out_ggaa(x1, x2, x3, x4)
row2colD_out_gggaa(x1, x2, x3, x4, x5)  =  row2colD_out_gggaa(x1, x2, x3, x4, x5)
U6_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaag(x1, x2, x3, x4, x5, x6, x7)
transpose_auxE_in_ggg(x1, x2, x3)  =  transpose_auxE_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5, x6)  =  U4_ggg(x1, x2, x3, x4, x5, x6)
pB_out_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gggaag(x1, x2, x3, x4, x5, x6)
transpose_auxE_out_ggg(x1, x2, x3)  =  transpose_auxE_out_ggg(x1, x2, x3)
transposeA_out_gg(x1, x2)  =  transposeA_out_gg(x1, x2)
TRANSPOSEA_IN_GG(x1, x2)  =  TRANSPOSEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
PB_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGAAG(x1, x2, x3, x6)
U5_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GGGAAG(x1, x2, x3, x6, x7)
ROW2COLD_IN_GGGAA(x1, x2, x3, x4, x5)  =  ROW2COLD_IN_GGGAA(x1, x2, x3)
U3_GGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGGAA(x1, x2, x3, x4, x7)
ROW2COLC_IN_GGAA(x1, x2, x3, x4)  =  ROW2COLC_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAA(x1, x2, x3, x4, x7)
U6_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGAAG(x1, x2, x3, x4, x5, x6, x7)
TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4, x5, x6)  =  U4_GGG(x1, x2, x3, x4, x5, x6)

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

(4) Obligation:

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

TRANSPOSEA_IN_GG(.(T23, T24), .(T25, T26)) → U1_GG(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
TRANSPOSEA_IN_GG(.(T23, T24), .(T25, T26)) → PB_IN_GGGAAG(T23, T25, T26, X35, X36, T24)
PB_IN_GGGAAG(T23, T25, T26, T31, T32, T24) → U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
PB_IN_GGGAAG(T23, T25, T26, T31, T32, T24) → ROW2COLD_IN_GGGAA(T23, T25, T26, T31, T32)
ROW2COLD_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_GGGAA(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
ROW2COLD_IN_GGGAA(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → ROW2COLC_IN_GGAA(T54, T56, X91, X92)
ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_GGAA(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → ROW2COLC_IN_GGAA(T78, T80, X139, X140)
U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_GGGAAG(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → TRANSPOSE_AUXE_IN_GGG(T24, T32, T31)
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → U4_GGG(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → PB_IN_GGGAAG(T110, T113, T114, X185, X186, T111)

The TRS R consists of the following rules:

transposeA_in_gg(.(T23, T24), .(T25, T26)) → U1_gg(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
pB_in_gggaag(T23, T25, T26, T31, T32, T24) → U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
row2colC_in_ggaa([], [], [], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))
U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
transpose_auxE_in_ggg(.(T110, T111), T112, .(T113, T114)) → U4_ggg(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
U4_ggg(T110, T111, T112, T113, T114, pB_out_gggaag(T110, T113, T114, X185, X186, T111)) → transpose_auxE_out_ggg(.(T110, T111), T112, .(T113, T114))
transpose_auxE_in_ggg([], T121, T121) → transpose_auxE_out_ggg([], T121, T121)
U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_out_ggg(T24, T32, T31)) → pB_out_gggaag(T23, T25, T26, T31, T32, T24)
U1_gg(T23, T24, T25, T26, pB_out_gggaag(T23, T25, T26, X35, X36, T24)) → transposeA_out_gg(.(T23, T24), .(T25, T26))
transposeA_in_gg([], []) → transposeA_out_gg([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_gg(x1, x2)  =  transposeA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
pB_in_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gggaag(x1, x2, x3, x6)
U5_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gggaag(x1, x2, x3, x6, x7)
row2colD_in_gggaa(x1, x2, x3, x4, x5)  =  row2colD_in_gggaa(x1, x2, x3)
U3_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggaa(x1, x2, x3, x4, x7)
row2colC_in_ggaa(x1, x2, x3, x4)  =  row2colC_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2colC_out_ggaa(x1, x2, x3, x4)  =  row2colC_out_ggaa(x1, x2, x3, x4)
row2colD_out_gggaa(x1, x2, x3, x4, x5)  =  row2colD_out_gggaa(x1, x2, x3, x4, x5)
U6_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaag(x1, x2, x3, x4, x5, x6, x7)
transpose_auxE_in_ggg(x1, x2, x3)  =  transpose_auxE_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5, x6)  =  U4_ggg(x1, x2, x3, x4, x5, x6)
pB_out_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gggaag(x1, x2, x3, x4, x5, x6)
transpose_auxE_out_ggg(x1, x2, x3)  =  transpose_auxE_out_ggg(x1, x2, x3)
transposeA_out_gg(x1, x2)  =  transposeA_out_gg(x1, x2)
TRANSPOSEA_IN_GG(x1, x2)  =  TRANSPOSEA_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4, x5)  =  U1_GG(x1, x2, x3, x4, x5)
PB_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGAAG(x1, x2, x3, x6)
U5_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GGGAAG(x1, x2, x3, x6, x7)
ROW2COLD_IN_GGGAA(x1, x2, x3, x4, x5)  =  ROW2COLD_IN_GGGAA(x1, x2, x3)
U3_GGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GGGAA(x1, x2, x3, x4, x7)
ROW2COLC_IN_GGAA(x1, x2, x3, x4)  =  ROW2COLC_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAA(x1, x2, x3, x4, x7)
U6_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U6_GGGAAG(x1, x2, x3, x4, x5, x6, x7)
TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4, x5, x6)  =  U4_GGG(x1, x2, x3, x4, x5, 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 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → ROW2COLC_IN_GGAA(T78, T80, X139, X140)

The TRS R consists of the following rules:

transposeA_in_gg(.(T23, T24), .(T25, T26)) → U1_gg(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
pB_in_gggaag(T23, T25, T26, T31, T32, T24) → U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
row2colC_in_ggaa([], [], [], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))
U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
transpose_auxE_in_ggg(.(T110, T111), T112, .(T113, T114)) → U4_ggg(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
U4_ggg(T110, T111, T112, T113, T114, pB_out_gggaag(T110, T113, T114, X185, X186, T111)) → transpose_auxE_out_ggg(.(T110, T111), T112, .(T113, T114))
transpose_auxE_in_ggg([], T121, T121) → transpose_auxE_out_ggg([], T121, T121)
U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_out_ggg(T24, T32, T31)) → pB_out_gggaag(T23, T25, T26, T31, T32, T24)
U1_gg(T23, T24, T25, T26, pB_out_gggaag(T23, T25, T26, X35, X36, T24)) → transposeA_out_gg(.(T23, T24), .(T25, T26))
transposeA_in_gg([], []) → transposeA_out_gg([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_gg(x1, x2)  =  transposeA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
pB_in_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gggaag(x1, x2, x3, x6)
U5_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gggaag(x1, x2, x3, x6, x7)
row2colD_in_gggaa(x1, x2, x3, x4, x5)  =  row2colD_in_gggaa(x1, x2, x3)
U3_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggaa(x1, x2, x3, x4, x7)
row2colC_in_ggaa(x1, x2, x3, x4)  =  row2colC_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2colC_out_ggaa(x1, x2, x3, x4)  =  row2colC_out_ggaa(x1, x2, x3, x4)
row2colD_out_gggaa(x1, x2, x3, x4, x5)  =  row2colD_out_gggaa(x1, x2, x3, x4, x5)
U6_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaag(x1, x2, x3, x4, x5, x6, x7)
transpose_auxE_in_ggg(x1, x2, x3)  =  transpose_auxE_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5, x6)  =  U4_ggg(x1, x2, x3, x4, x5, x6)
pB_out_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gggaag(x1, x2, x3, x4, x5, x6)
transpose_auxE_out_ggg(x1, x2, x3)  =  transpose_auxE_out_ggg(x1, x2, x3)
transposeA_out_gg(x1, x2)  =  transposeA_out_gg(x1, x2)
ROW2COLC_IN_GGAA(x1, x2, x3, x4)  =  ROW2COLC_IN_GGAA(x1, 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:

ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → ROW2COLC_IN_GGAA(T78, T80, X139, X140)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COLC_IN_GGAA(x1, x2, x3, x4)  =  ROW2COLC_IN_GGAA(x1, 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:

ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80)) → ROW2COLC_IN_GGAA(T78, T80)

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:

• ROW2COLC_IN_GGAA(.(T77, T78), .(.(T77, T79), T80)) → ROW2COLC_IN_GGAA(T78, T80)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → TRANSPOSE_AUXE_IN_GGG(T24, T32, T31)
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → PB_IN_GGGAAG(T110, T113, T114, X185, X186, T111)
PB_IN_GGGAAG(T23, T25, T26, T31, T32, T24) → U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))

The TRS R consists of the following rules:

transposeA_in_gg(.(T23, T24), .(T25, T26)) → U1_gg(T23, T24, T25, T26, pB_in_gggaag(T23, T25, T26, X35, X36, T24))
pB_in_gggaag(T23, T25, T26, T31, T32, T24) → U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))
row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
row2colC_in_ggaa([], [], [], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))
U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
U5_gggaag(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_in_ggg(T24, T32, T31))
transpose_auxE_in_ggg(.(T110, T111), T112, .(T113, T114)) → U4_ggg(T110, T111, T112, T113, T114, pB_in_gggaag(T110, T113, T114, X185, X186, T111))
U4_ggg(T110, T111, T112, T113, T114, pB_out_gggaag(T110, T113, T114, X185, X186, T111)) → transpose_auxE_out_ggg(.(T110, T111), T112, .(T113, T114))
transpose_auxE_in_ggg([], T121, T121) → transpose_auxE_out_ggg([], T121, T121)
U6_gggaag(T23, T25, T26, T31, T32, T24, transpose_auxE_out_ggg(T24, T32, T31)) → pB_out_gggaag(T23, T25, T26, T31, T32, T24)
U1_gg(T23, T24, T25, T26, pB_out_gggaag(T23, T25, T26, X35, X36, T24)) → transposeA_out_gg(.(T23, T24), .(T25, T26))
transposeA_in_gg([], []) → transposeA_out_gg([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_gg(x1, x2)  =  transposeA_in_gg(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_gg(x1, x2, x3, x4, x5)  =  U1_gg(x1, x2, x3, x4, x5)
pB_in_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gggaag(x1, x2, x3, x6)
U5_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gggaag(x1, x2, x3, x6, x7)
row2colD_in_gggaa(x1, x2, x3, x4, x5)  =  row2colD_in_gggaa(x1, x2, x3)
U3_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggaa(x1, x2, x3, x4, x7)
row2colC_in_ggaa(x1, x2, x3, x4)  =  row2colC_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2colC_out_ggaa(x1, x2, x3, x4)  =  row2colC_out_ggaa(x1, x2, x3, x4)
row2colD_out_gggaa(x1, x2, x3, x4, x5)  =  row2colD_out_gggaa(x1, x2, x3, x4, x5)
U6_gggaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gggaag(x1, x2, x3, x4, x5, x6, x7)
transpose_auxE_in_ggg(x1, x2, x3)  =  transpose_auxE_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4, x5, x6)  =  U4_ggg(x1, x2, x3, x4, x5, x6)
pB_out_gggaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gggaag(x1, x2, x3, x4, x5, x6)
transpose_auxE_out_ggg(x1, x2, x3)  =  transpose_auxE_out_ggg(x1, x2, x3)
transposeA_out_gg(x1, x2)  =  transposeA_out_gg(x1, x2)
PB_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGAAG(x1, x2, x3, x6)
U5_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GGGAAG(x1, x2, x3, x6, x7)
TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)

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:

U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → TRANSPOSE_AUXE_IN_GGG(T24, T32, T31)
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → PB_IN_GGGAAG(T110, T113, T114, X185, X186, T111)
PB_IN_GGGAAG(T23, T25, T26, T31, T32, T24) → U5_GGGAAG(T23, T25, T26, T31, T32, T24, row2colD_in_gggaa(T23, T25, T26, T31, T32))

The TRS R consists of the following rules:

row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92)) → U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_in_ggaa(T54, T56, X91, X92))
U3_gggaa(T53, T54, T55, T56, X91, X92, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140)) → U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_in_ggaa(T78, T80, X139, X140))
row2colC_in_ggaa([], [], [], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, X139, X140, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2colD_in_gggaa(x1, x2, x3, x4, x5)  =  row2colD_in_gggaa(x1, x2, x3)
U3_gggaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gggaa(x1, x2, x3, x4, x7)
row2colC_in_ggaa(x1, x2, x3, x4)  =  row2colC_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_ggaa(x1, x2, x3, x4, x7)
[]  =  []
row2colC_out_ggaa(x1, x2, x3, x4)  =  row2colC_out_ggaa(x1, x2, x3, x4)
row2colD_out_gggaa(x1, x2, x3, x4, x5)  =  row2colD_out_gggaa(x1, x2, x3, x4, x5)
PB_IN_GGGAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GGGAAG(x1, x2, x3, x6)
U5_GGGAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GGGAAG(x1, x2, x3, x6, x7)
TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGG(x1, x2, x3)

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:

U5_GGGAAG(T23, T25, T26, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → TRANSPOSE_AUXE_IN_GGG(T24, T32, T31)
TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → PB_IN_GGGAAG(T110, T113, T114, T111)
PB_IN_GGGAAG(T23, T25, T26, T24) → U5_GGGAAG(T23, T25, T26, T24, row2colD_in_gggaa(T23, T25, T26))

The TRS R consists of the following rules:

row2colD_in_gggaa(.(T53, T54), .(T53, T55), T56) → U3_gggaa(T53, T54, T55, T56, row2colC_in_ggaa(T54, T56))
U3_gggaa(T53, T54, T55, T56, row2colC_out_ggaa(T54, T56, X91, X92)) → row2colD_out_gggaa(.(T53, T54), .(T53, T55), T56, .(T55, X91), .([], X92))
row2colC_in_ggaa(.(T77, T78), .(.(T77, T79), T80)) → U2_ggaa(T77, T78, T79, T80, row2colC_in_ggaa(T78, T80))
row2colC_in_ggaa([], []) → row2colC_out_ggaa([], [], [], [])
U2_ggaa(T77, T78, T79, T80, row2colC_out_ggaa(T78, T80, X139, X140)) → row2colC_out_ggaa(.(T77, T78), .(.(T77, T79), T80), .(T79, X139), .([], X140))

The set Q consists of the following terms:

row2colD_in_gggaa(x0, x1, x2)
U3_gggaa(x0, x1, x2, x3, x4)
row2colC_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2, x3, x4)

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

(19) 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:

• TRANSPOSE_AUXE_IN_GGG(.(T110, T111), T112, .(T113, T114)) → PB_IN_GGGAAG(T110, T113, T114, T111)
  The graph contains the following edges 1 > 1, 3 > 2, 3 > 3, 1 > 4

• PB_IN_GGGAAG(T23, T25, T26, T24) → U5_GGGAAG(T23, T25, T26, T24, row2colD_in_gggaa(T23, T25, T26))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4

• U5_GGGAAG(T23, T25, T26, T24, row2colD_out_gggaa(T23, T25, T26, T31, T32)) → TRANSPOSE_AUXE_IN_GGG(T24, T32, T31)
  The graph contains the following edges 4 >= 1, 5 > 2, 5 > 3

(20) YES