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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: transpose(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: transpose(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: transpose_aux(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: 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];
13 [label="13: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [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];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="D: row2col(T24, .(T28, T29), X35, X36)\n\nT24 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="E: transpose_aux(T25, T35, T34)\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: 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];
36 [label="36: 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];
37 [label="37: 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];
40 [label="C: row2col(T57, T60, X91, X92)\n\nT57 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: 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];
48 [label="48: 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];
49 [label="49: 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];
52 [label="52: row2col(T82, T85, X139, X140)\n\nT82 is ground\nX139 is free\nX140 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: 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];
67 [label="67: transpose_aux(T25, T35, T34), SCOPE: 5, CLAUSE: 1\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: transpose_aux(T25, T35, T34), SCOPE: 5, CLAUSE: 2\n\nT25 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(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];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 92 [arrowhead = none ];
92 -> 6;

93 [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];
6 -> 93 [arrowhead = none ];
93 -> 8;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 94 [arrowhead = none ];
94 -> 10;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 95 [arrowhead = none ];
95 -> 13;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 96 [arrowhead = none ];
96 -> 15;

97 [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];
13 -> 97 [arrowhead = none ];
97 -> 17;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 98 [arrowhead = none ];
98 -> 20;

99 [label="EVAL with clause\ntranspose_aux([], X200, X200).\nand substitutionT5 -> [],\nX200 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 99 [arrowhead = none ];
99 -> 89;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 100 [arrowhead = none ];
100 -> 90;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 101 [arrowhead = none ];
101 -> 24;

102 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT35 is ground\nreplacements:X35 -> T34,\nX36 -> T35", fontsize=14, style = filled, fillcolor = violet];
17 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 103 [arrowhead = none ];
103 -> 32;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 104 [arrowhead = none ];
104 -> 65;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 105 [arrowhead = none ];
105 -> 36;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 106 [arrowhead = none ];
106 -> 37;

107 [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];
36 -> 107 [arrowhead = none ];
107 -> 40;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="BACKTRACK\nfor clause: row2col([], [], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 109 [arrowhead = none ];
109 -> 62;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 110 [arrowhead = none ];
110 -> 45;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 111 [arrowhead = none ];
111 -> 48;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 112 [arrowhead = none ];
112 -> 49;

113 [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];
48 -> 113 [arrowhead = none ];
113 -> 52;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 114 [arrowhead = none ];
114 -> 53;

115 [label="EVAL with clause\nrow2col([], [], [], []).\nand substitutionT57 -> [],\nT60 -> [],\nX91 -> [],\nX92 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 115 [arrowhead = none ];
115 -> 58;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 116 [arrowhead = none ];
116 -> 59;

117 [label="INSTANCE with matching:\nT57 -> T82\nT60 -> T85\nX91 -> X139\nX92 -> X140", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 117 [arrowhead = none , style=dashed];
117 -> 40[style=dashed];

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 118 [arrowhead = none ];
118 -> 60;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 119 [arrowhead = none ];
119 -> 67;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 120 [arrowhead = none ];
120 -> 68;

121 [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];
67 -> 121 [arrowhead = none ];
121 -> 73;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 122 [arrowhead = none ];
122 -> 74;

123 [label="EVAL with clause\ntranspose_aux([], X197, X197).\nand substitutionT25 -> [],\nT35 -> T128,\nX197 -> T128,\nT34 -> T128", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 123 [arrowhead = none ];
123 -> 79;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 124 [arrowhead = none ];
124 -> 80;

125 [label="INSTANCE with matching:\nT24 -> T115\nT28 -> T120\nT29 -> T121\nX35 -> X185\nX36 -> X186\nT25 -> T116", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 125 [arrowhead = none , style=dashed];
125 -> 17[style=dashed];

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 126 [arrowhead = none ];
126 -> 82;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 127 [arrowhead = none ];
127 -> 91;

}

(2) Obligation:

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

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_in_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
row2colC_in_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
row2colC_in_gaaa([], [], [], []) → row2colC_out_gaaa([], [], [], [])
U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_out_gaaa(T82, T85, X139, X140)) → row2colC_out_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140))
U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_out_gaaa(T57, T60, X91, X92)) → row2colD_out_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T115, T116), T117, .(T120, T121)) → U4_gga(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
U4_gga(T115, T116, T117, T120, T121, pB_out_gaaaag(T115, T120, T121, X185, X186, T116)) → transpose_auxE_out_gga(.(T115, T116), T117, .(T120, T121))
transpose_auxE_in_gga([], T128, T128) → transpose_auxE_out_gga([], T128, T128)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2)  =  transposeA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gaaaa(x1, x2, x7)
row2colC_in_gaaa(x1, x2, x3, x4)  =  row2colC_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaaa(x1, x2, x7)
[]  =  []
row2colC_out_gaaa(x1, x2, x3, x4)  =  row2colC_out_gaaa(x1, x4)
row2colD_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3)  =  transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3)  =  transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2)  =  transposeA_out_ga(x1)

(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_GA(.(T24, T25), .(T28, T29)) → U1_GA(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
TRANSPOSEA_IN_GA(.(T24, T25), .(T28, T29)) → PB_IN_GAAAAG(T24, T28, T29, X35, X36, T25)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → ROW2COLD_IN_GAAAA(T24, T28, T29, T34, T35)
ROW2COLD_IN_GAAAA(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_GAAAA(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
ROW2COLD_IN_GAAAA(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → ROW2COLC_IN_GAAA(T57, T60, X91, X92)
ROW2COLC_IN_GAAA(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_GAAA(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
ROW2COLC_IN_GAAA(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → ROW2COLC_IN_GAAA(T82, T85, X139, X140)
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_GAAAAG(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117, .(T120, T121)) → U4_GGA(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117, .(T120, T121)) → PB_IN_GAAAAG(T115, T120, T121, X185, X186, T116)

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_in_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
row2colC_in_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
row2colC_in_gaaa([], [], [], []) → row2colC_out_gaaa([], [], [], [])
U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_out_gaaa(T82, T85, X139, X140)) → row2colC_out_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140))
U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_out_gaaa(T57, T60, X91, X92)) → row2colD_out_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T115, T116), T117, .(T120, T121)) → U4_gga(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
U4_gga(T115, T116, T117, T120, T121, pB_out_gaaaag(T115, T120, T121, X185, X186, T116)) → transpose_auxE_out_gga(.(T115, T116), T117, .(T120, T121))
transpose_auxE_in_gga([], T128, T128) → transpose_auxE_out_gga([], T128, T128)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2)  =  transposeA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gaaaa(x1, x2, x7)
row2colC_in_gaaa(x1, x2, x3, x4)  =  row2colC_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaaa(x1, x2, x7)
[]  =  []
row2colC_out_gaaa(x1, x2, x3, x4)  =  row2colC_out_gaaa(x1, x4)
row2colD_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3)  =  transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3)  =  transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2)  =  transposeA_out_ga(x1)
TRANSPOSEA_IN_GA(x1, x2)  =  TRANSPOSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GAAAAG(x1, x6, x7)
ROW2COLD_IN_GAAAA(x1, x2, x3, x4, x5)  =  ROW2COLD_IN_GAAAA(x1)
U3_GAAAA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAAAA(x1, x2, x7)
ROW2COLC_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLC_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GAAA(x1, x2, x7)
U6_GAAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAAAAG(x1, x5, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, 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_GA(.(T24, T25), .(T28, T29)) → U1_GA(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
TRANSPOSEA_IN_GA(.(T24, T25), .(T28, T29)) → PB_IN_GAAAAG(T24, T28, T29, X35, X36, T25)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → ROW2COLD_IN_GAAAA(T24, T28, T29, T34, T35)
ROW2COLD_IN_GAAAA(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_GAAAA(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
ROW2COLD_IN_GAAAA(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → ROW2COLC_IN_GAAA(T57, T60, X91, X92)
ROW2COLC_IN_GAAA(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_GAAA(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
ROW2COLC_IN_GAAA(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → ROW2COLC_IN_GAAA(T82, T85, X139, X140)
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_GAAAAG(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117, .(T120, T121)) → U4_GGA(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117, .(T120, T121)) → PB_IN_GAAAAG(T115, T120, T121, X185, X186, T116)

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_in_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
row2colC_in_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
row2colC_in_gaaa([], [], [], []) → row2colC_out_gaaa([], [], [], [])
U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_out_gaaa(T82, T85, X139, X140)) → row2colC_out_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140))
U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_out_gaaa(T57, T60, X91, X92)) → row2colD_out_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T115, T116), T117, .(T120, T121)) → U4_gga(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
U4_gga(T115, T116, T117, T120, T121, pB_out_gaaaag(T115, T120, T121, X185, X186, T116)) → transpose_auxE_out_gga(.(T115, T116), T117, .(T120, T121))
transpose_auxE_in_gga([], T128, T128) → transpose_auxE_out_gga([], T128, T128)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2)  =  transposeA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gaaaa(x1, x2, x7)
row2colC_in_gaaa(x1, x2, x3, x4)  =  row2colC_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaaa(x1, x2, x7)
[]  =  []
row2colC_out_gaaa(x1, x2, x3, x4)  =  row2colC_out_gaaa(x1, x4)
row2colD_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3)  =  transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3)  =  transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2)  =  transposeA_out_ga(x1)
TRANSPOSEA_IN_GA(x1, x2)  =  TRANSPOSEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x1, x2, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GAAAAG(x1, x6, x7)
ROW2COLD_IN_GAAAA(x1, x2, x3, x4, x5)  =  ROW2COLD_IN_GAAAA(x1)
U3_GAAAA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAAAA(x1, x2, x7)
ROW2COLC_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLC_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GAAA(x1, x2, x7)
U6_GAAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAAAAG(x1, x5, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, 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_GAAA(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → ROW2COLC_IN_GAAA(T82, T85, X139, X140)

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_in_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
row2colC_in_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
row2colC_in_gaaa([], [], [], []) → row2colC_out_gaaa([], [], [], [])
U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_out_gaaa(T82, T85, X139, X140)) → row2colC_out_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140))
U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_out_gaaa(T57, T60, X91, X92)) → row2colD_out_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T115, T116), T117, .(T120, T121)) → U4_gga(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
U4_gga(T115, T116, T117, T120, T121, pB_out_gaaaag(T115, T120, T121, X185, X186, T116)) → transpose_auxE_out_gga(.(T115, T116), T117, .(T120, T121))
transpose_auxE_in_gga([], T128, T128) → transpose_auxE_out_gga([], T128, T128)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2)  =  transposeA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gaaaa(x1, x2, x7)
row2colC_in_gaaa(x1, x2, x3, x4)  =  row2colC_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaaa(x1, x2, x7)
[]  =  []
row2colC_out_gaaa(x1, x2, x3, x4)  =  row2colC_out_gaaa(x1, x4)
row2colD_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3)  =  transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3)  =  transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2)  =  transposeA_out_ga(x1)
ROW2COLC_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLC_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:

ROW2COLC_IN_GAAA(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → ROW2COLC_IN_GAAA(T82, T85, X139, X140)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
ROW2COLC_IN_GAAA(x1, x2, x3, x4)  =  ROW2COLC_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:

ROW2COLC_IN_GAAA(.(T81, T82)) → ROW2COLC_IN_GAAA(T82)

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_GAAA(.(T81, T82)) → ROW2COLC_IN_GAAA(T82)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117, .(T120, T121)) → PB_IN_GAAAAG(T115, T120, T121, X185, X186, T116)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))

The TRS R consists of the following rules:

transposeA_in_ga(.(T24, T25), .(T28, T29)) → U1_ga(T24, T25, T28, T29, pB_in_gaaaag(T24, T28, T29, X35, X36, T25))
pB_in_gaaaag(T24, T28, T29, T34, T35, T25) → U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))
row2colD_in_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
row2colC_in_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
row2colC_in_gaaa([], [], [], []) → row2colC_out_gaaa([], [], [], [])
U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_out_gaaa(T82, T85, X139, X140)) → row2colC_out_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140))
U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_out_gaaa(T57, T60, X91, X92)) → row2colD_out_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92))
U5_gaaaag(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_in_gga(T25, T35, T34))
transpose_auxE_in_gga(.(T115, T116), T117, .(T120, T121)) → U4_gga(T115, T116, T117, T120, T121, pB_in_gaaaag(T115, T120, T121, X185, X186, T116))
U4_gga(T115, T116, T117, T120, T121, pB_out_gaaaag(T115, T120, T121, X185, X186, T116)) → transpose_auxE_out_gga(.(T115, T116), T117, .(T120, T121))
transpose_auxE_in_gga([], T128, T128) → transpose_auxE_out_gga([], T128, T128)
U6_gaaaag(T24, T28, T29, T34, T35, T25, transpose_auxE_out_gga(T25, T35, T34)) → pB_out_gaaaag(T24, T28, T29, T34, T35, T25)
U1_ga(T24, T25, T28, T29, pB_out_gaaaag(T24, T28, T29, X35, X36, T25)) → transposeA_out_ga(.(T24, T25), .(T28, T29))
transposeA_in_ga([], []) → transposeA_out_ga([], [])

The argument filtering Pi contains the following mapping:
transposeA_in_ga(x1, x2)  =  transposeA_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x1, x2, x5)
pB_in_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_in_gaaaag(x1, x6)
U5_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U5_gaaaag(x1, x6, x7)
row2colD_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gaaaa(x1, x2, x7)
row2colC_in_gaaa(x1, x2, x3, x4)  =  row2colC_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaaa(x1, x2, x7)
[]  =  []
row2colC_out_gaaa(x1, x2, x3, x4)  =  row2colC_out_gaaa(x1, x4)
row2colD_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_out_gaaaa(x1, x5)
U6_gaaaag(x1, x2, x3, x4, x5, x6, x7)  =  U6_gaaaag(x1, x5, x6, x7)
transpose_auxE_in_gga(x1, x2, x3)  =  transpose_auxE_in_gga(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6)  =  U4_gga(x1, x2, x3, x6)
pB_out_gaaaag(x1, x2, x3, x4, x5, x6)  =  pB_out_gaaaag(x1, x5, x6)
transpose_auxE_out_gga(x1, x2, x3)  =  transpose_auxE_out_gga(x1, x2)
transposeA_out_ga(x1, x2)  =  transposeA_out_ga(x1)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GAAAAG(x1, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGA(x1, x2)

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_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_out_gaaaa(T24, T28, T29, T34, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35, T34)
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117, .(T120, T121)) → PB_IN_GAAAAG(T115, T120, T121, X185, X186, T116)
PB_IN_GAAAAG(T24, T28, T29, T34, T35, T25) → U5_GAAAAG(T24, T28, T29, T34, T35, T25, row2colD_in_gaaaa(T24, T28, T29, T34, T35))

The TRS R consists of the following rules:

row2colD_in_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92)) → U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_in_gaaa(T57, T60, X91, X92))
U3_gaaaa(T56, T57, T58, T60, X91, X92, row2colC_out_gaaa(T57, T60, X91, X92)) → row2colD_out_gaaaa(.(T56, T57), .(T56, T58), T60, .(T58, X91), .([], X92))
row2colC_in_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140)) → U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_in_gaaa(T82, T85, X139, X140))
row2colC_in_gaaa([], [], [], []) → row2colC_out_gaaa([], [], [], [])
U2_gaaa(T81, T82, T83, T85, X139, X140, row2colC_out_gaaa(T82, T85, X139, X140)) → row2colC_out_gaaa(.(T81, T82), .(.(T81, T83), T85), .(T83, X139), .([], X140))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
row2colD_in_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_in_gaaaa(x1)
U3_gaaaa(x1, x2, x3, x4, x5, x6, x7)  =  U3_gaaaa(x1, x2, x7)
row2colC_in_gaaa(x1, x2, x3, x4)  =  row2colC_in_gaaa(x1)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U2_gaaa(x1, x2, x7)
[]  =  []
row2colC_out_gaaa(x1, x2, x3, x4)  =  row2colC_out_gaaa(x1, x4)
row2colD_out_gaaaa(x1, x2, x3, x4, x5)  =  row2colD_out_gaaaa(x1, x5)
PB_IN_GAAAAG(x1, x2, x3, x4, x5, x6)  =  PB_IN_GAAAAG(x1, x6)
U5_GAAAAG(x1, x2, x3, x4, x5, x6, x7)  =  U5_GAAAAG(x1, x6, x7)
TRANSPOSE_AUXE_IN_GGA(x1, x2, x3)  =  TRANSPOSE_AUXE_IN_GGA(x1, x2)

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_GAAAAG(T24, T25, row2colD_out_gaaaa(T24, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35)
TRANSPOSE_AUXE_IN_GGA(.(T115, T116), T117) → PB_IN_GAAAAG(T115, T116)
PB_IN_GAAAAG(T24, T25) → U5_GAAAAG(T24, T25, row2colD_in_gaaaa(T24))

The TRS R consists of the following rules:

row2colD_in_gaaaa(.(T56, T57)) → U3_gaaaa(T56, T57, row2colC_in_gaaa(T57))
U3_gaaaa(T56, T57, row2colC_out_gaaa(T57, X92)) → row2colD_out_gaaaa(.(T56, T57), .([], X92))
row2colC_in_gaaa(.(T81, T82)) → U2_gaaa(T81, T82, row2colC_in_gaaa(T82))
row2colC_in_gaaa([]) → row2colC_out_gaaa([], [])
U2_gaaa(T81, T82, row2colC_out_gaaa(T82, X140)) → row2colC_out_gaaa(.(T81, T82), .([], X140))

The set Q consists of the following terms:

row2colD_in_gaaaa(x0)
U3_gaaaa(x0, x1, x2)
row2colC_in_gaaa(x0)
U2_gaaa(x0, x1, x2)

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_GGA(.(T115, T116), T117) → PB_IN_GAAAAG(T115, T116)
  The graph contains the following edges 1 > 1, 1 > 2

• PB_IN_GAAAAG(T24, T25) → U5_GAAAAG(T24, T25, row2colD_in_gaaaa(T24))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U5_GAAAAG(T24, T25, row2colD_out_gaaaa(T24, T35)) → TRANSPOSE_AUXE_IN_GGA(T25, T35)
  The graph contains the following edges 2 >= 1, 3 > 2

(20) YES