Left Termination of the query pattern rev_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

rev([], []).
rev(.(X, XS), .(Y, YS)) :- ','(rev1(X, XS, Y), rev2(X, XS, YS)).
rev1(X, [], X).
rev1(X, .(Y, YS), Z) :- rev1(Y, YS, Z).
rev2(X, [], []).
rev2(X, .(Y, YS), ZS) :- ','(rev2(Y, YS, US), ','(rev(US, VS), rev(.(X, VS), ZS))).

Queries:

rev(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rev(T1, T2), SCOPE: 1, CLAUSE: 0 | rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(rev1(T7, T8, T11), rev2(T7, T8, T12))\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 2 | ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(rev1(T7, T8, T11), rev2(T7, T8, T12)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: rev2(T17, [], T18)\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: rev2(T17, [], T18), SCOPE: 3, CLAUSE: 4 | rev2(T17, [], T18), SCOPE: 3, CLAUSE: 5\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: rev2(T17, [], T18), SCOPE: 3, CLAUSE: 4\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: rev2(T17, [], T18), SCOPE: 3, CLAUSE: 5\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(rev1(T36, T37, T39), rev2(T35, .(T36, T37), T40))\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: rev1(T36, T37, T39)\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: rev2(T35, .(T36, T37), T43)\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 2 | rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 3\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 2\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: rev1(T36, T37, T39), SCOPE: 4, CLAUSE: 3\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: rev1(T60, T61, T63)\n\nT60 is ground\nT61 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: rev2(T35, .(T36, T37), T43), SCOPE: 5, CLAUSE: 4 | rev2(T35, .(T36, T37), T43), SCOPE: 5, CLAUSE: 5\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: rev2(T35, .(T36, T37), T43), SCOPE: 5, CLAUSE: 5\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(rev2(T85, T86, X104), ','(rev(X104, X105), rev(.(T84, X105), T88)))\n\nT84 is ground\nT85 is ground\nT86 is ground\nX104 is free; \nX105 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: rev2(T85, T86, X104)\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(rev(T91, X105), rev(.(T84, X105), T88))\n\nT84 is ground\nT91 is ground\nX105 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 4 | rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 5\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 4\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: rev2(T85, T86, X104), SCOPE: 6, CLAUSE: 5\n\nT85 is ground\nT86 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(rev2(T106, T107, X132), ','(rev(X132, X133), rev(.(T105, X133), X134)))\n\nT105 is ground\nT106 is ground\nT107 is ground\nX134 is free; \nX132 is free; \nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: rev2(T106, T107, X132)\n\nT106 is ground\nT107 is ground\nX132 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(rev(T110, X133), rev(.(T105, X133), X134))\n\nT105 is ground\nT110 is ground\nX134 is free; \nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: rev(T110, X133)\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: rev(.(T105, T113), X134)\n\nT105 is ground\nT113 is ground\nX134 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: rev(T110, X133), SCOPE: 7, CLAUSE: 0 | rev(T110, X133), SCOPE: 7, CLAUSE: 1\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: rev(T110, X133), SCOPE: 7, CLAUSE: 0\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: rev(T110, X133), SCOPE: 7, CLAUSE: 1\n\nT110 is ground\nX133 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(rev1(T118, T119, X159), rev2(T118, T119, X160))\n\nT118 is ground\nT119 is ground\nX159 is free; \nX160 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: rev1(T118, T119, X159)\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: rev2(T118, T119, X160)\n\nT118 is ground\nT119 is ground\nX160 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 2 | rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 3\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 2\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: rev1(T118, T119, X159), SCOPE: 8, CLAUSE: 3\n\nT118 is ground\nT119 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: rev1(T137, T138, X189)\n\nT137 is ground\nT138 is ground\nX189 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: rev(T91, X105)\n\nT91 is ground\nX105 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: rev(.(T84, T145), T88)\n\nT84 is ground\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 70 [arrowhead = none ];
70 -> 2;

71 [label="EVAL with clause\nrev([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 71 [arrowhead = none ];
71 -> 3;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 72 [arrowhead = none ];
72 -> 4;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 73 [arrowhead = none ];
73 -> 5;

74 [label="EVAL with clause\nrev(.(X9, X10), .(X11, X12)) :- ','(rev1(X9, X10, X11), rev2(X9, X10, X12)).\nand substitution\nX9 -> T7\nX10 -> T8\nT1 -> .(T7, T8)\nX11 -> T11\nX12 -> T12\nT2 -> .(T11, T12)\nT9 -> T11\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 74 [arrowhead = none ];
74 -> 7;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 75 [arrowhead = none ];
75 -> 8;

76 [label="BACKTRACK\nfor clause: rev(.(X, XS), .(Y, YS)) :- ','(rev1(X, XS, Y), rev2(X, XS, YS))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 76 [arrowhead = none ];
76 -> 6;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 77 [arrowhead = none ];
77 -> 9;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 78 [arrowhead = none ];
78 -> 10;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 79 [arrowhead = none ];
79 -> 11;

80 [label="EVAL with clause\nrev1(X21, [], X21).\nand substitution\nT7 -> T17\nX21 -> T17\nT8 -> []\nT11 -> T17\nT12 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 80 [arrowhead = none ];
80 -> 12;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="EVAL with clause\nrev1(X43, .(X44, X45), X46) :- rev1(X44, X45, X46).\nand substitution\nT7 -> T35\nX43 -> T35\nX44 -> T36\nX45 -> T37\nT8 -> .(T36, T37)\nT11 -> T39\nX46 -> T39\nT38 -> T39\nT12 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 84 [arrowhead = none ];
84 -> 14;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 85 [arrowhead = none ];
85 -> 15;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="EVAL with clause\nrev2(X29, [], []).\nand substitution\nT17 -> T25\nX29 -> T25\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 87 [arrowhead = none ];
87 -> 17;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 88 [arrowhead = none ];
88 -> 18;

89 [label="BACKTRACK\nfor clause: rev2(X, .(Y, YS), ZS) :- ','(rev2(Y, YS, US), ','(rev(US, VS), rev(.(X, VS), ZS)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 89 [arrowhead = none ];
89 -> 20;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 90 [arrowhead = none ];
90 -> 19;

91 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 91 [arrowhead = none ];
91 -> 23;

92 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nreplacements:\nT40 -> T43", fontsize=14, style = filled, fillcolor = violet];
21 -> 92 [arrowhead = none ];
92 -> 24;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 93 [arrowhead = none ];
93 -> 25;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 94 [arrowhead = none ];
94 -> 33;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 95 [arrowhead = none ];
95 -> 26;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 96 [arrowhead = none ];
96 -> 27;

97 [label="EVAL with clause\nrev1(X59, [], X59).\nand substitution\nT36 -> T50\nX59 -> T50\nT37 -> []\nT39 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 97 [arrowhead = none ];
97 -> 28;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 98 [arrowhead = none ];
98 -> 29;

99 [label="EVAL with clause\nrev1(X69, .(X70, X71), X72) :- rev1(X70, X71, X72).\nand substitution\nT36 -> T59\nX69 -> T59\nX70 -> T60\nX71 -> T61\nT37 -> .(T60, T61)\nT39 -> T63\nX72 -> T63\nT62 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 99 [arrowhead = none ];
99 -> 31;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 100 [arrowhead = none ];
100 -> 32;

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 101 [arrowhead = none ];
101 -> 30;

102 [label="INSTANCE with matching:\nT36 -> T60\nT37 -> T61\nT39 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 102 [arrowhead = none , style=dashed];
102 -> 23[style=dashed];

103 [label="BACKTRACK\nfor clause: rev2(X, [], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
33 -> 103 [arrowhead = none ];
103 -> 34;

104 [label="ONLY EVAL with clause\nrev2(X100, .(X101, X102), X103) :- ','(rev2(X101, X102, X104), ','(rev(X104, X105), rev(.(X100, X105), X103))).\nand substitution\nT35 -> T84\nX100 -> T84\nT36 -> T85\nX101 -> T85\nT37 -> T86\nX102 -> T86\nT43 -> T88\nX103 -> T88\nT87 -> T88", fontsize=14, style = filled, fillcolor = white];
34 -> 104 [arrowhead = none ];
104 -> 35;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 105 [arrowhead = none ];
105 -> 36;

106 [label="SPLIT 2\nnew knowledge:\nT91 is ground\nreplacements:\nX104 -> T91", fontsize=14, style = filled, fillcolor = violet];
35 -> 106 [arrowhead = none ];
106 -> 37;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 107 [arrowhead = none ];
107 -> 38;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
37 -> 108 [arrowhead = none ];
108 -> 68;

109 [label="SPLIT 2\nnew knowledge:\nT145 is ground\nreplacements:\nX105 -> T145", fontsize=14, style = filled, fillcolor = violet];
37 -> 109 [arrowhead = none ];
109 -> 69;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 110 [arrowhead = none ];
110 -> 39;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 111 [arrowhead = none ];
111 -> 40;

112 [label="EVAL with clause\nrev2(X114, [], []).\nand substitution\nT85 -> T98\nX114 -> T98\nT86 -> []\nX104 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 112 [arrowhead = none ];
112 -> 41;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 113 [arrowhead = none ];
113 -> 42;

114 [label="EVAL with clause\nrev2(X128, .(X129, X130), X131) :- ','(rev2(X129, X130, X132), ','(rev(X132, X133), rev(.(X128, X133), X131))).\nand substitution\nT85 -> T105\nX128 -> T105\nX129 -> T106\nX130 -> T107\nT86 -> .(T106, T107)\nX104 -> X134\nX131 -> X134", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 114 [arrowhead = none ];
114 -> 44;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 115 [arrowhead = none ];
115 -> 45;

116 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 116 [arrowhead = none ];
116 -> 43;

117 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
44 -> 117 [arrowhead = none ];
117 -> 46;

118 [label="SPLIT 2\nnew knowledge:\nT110 is ground\nreplacements:\nX132 -> T110", fontsize=14, style = filled, fillcolor = violet];
44 -> 118 [arrowhead = none ];
118 -> 47;

119 [label="INSTANCE with matching:\nT85 -> T106\nT86 -> T107\nX104 -> X132", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 119 [arrowhead = none , style=dashed];
119 -> 36[style=dashed];

120 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
47 -> 120 [arrowhead = none ];
120 -> 48;

121 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nreplacements:\nX133 -> T113", fontsize=14, style = filled, fillcolor = violet];
47 -> 121 [arrowhead = none ];
121 -> 49;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 122 [arrowhead = none ];
122 -> 50;

123 [label="INSTANCE with matching:\nT1 -> .(T105, T113)\nT2 -> X134", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 123 [arrowhead = none , style=dashed];
123 -> 1[style=dashed];

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 124 [arrowhead = none ];
124 -> 51;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 125 [arrowhead = none ];
125 -> 52;

126 [label="EVAL with clause\nrev([], []).\nand substitution\nT110 -> []\nX133 -> []", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 126 [arrowhead = none ];
126 -> 53;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 127 [arrowhead = none ];
127 -> 54;

128 [label="EVAL with clause\nrev(.(X155, X156), .(X157, X158)) :- ','(rev1(X155, X156, X157), rev2(X155, X156, X158)).\nand substitution\nX155 -> T118\nX156 -> T119\nT110 -> .(T118, T119)\nX157 -> X159\nX158 -> X160\nX133 -> .(X159, X160)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 128 [arrowhead = none ];
128 -> 56;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 129 [arrowhead = none ];
129 -> 57;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 130 [arrowhead = none ];
130 -> 55;

131 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
56 -> 131 [arrowhead = none ];
131 -> 58;

132 [label="SPLIT 2\nnew knowledge:\nT122 is ground\nreplacements:\nX159 -> T122", fontsize=14, style = filled, fillcolor = violet];
56 -> 132 [arrowhead = none ];
132 -> 59;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 133 [arrowhead = none ];
133 -> 60;

134 [label="INSTANCE with matching:\nT85 -> T118\nT86 -> T119\nX104 -> X160", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 134 [arrowhead = none , style=dashed];
134 -> 36[style=dashed];

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 135 [arrowhead = none ];
135 -> 61;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 136 [arrowhead = none ];
136 -> 62;

137 [label="EVAL with clause\nrev1(X173, [], X173).\nand substitution\nT118 -> T129\nX173 -> T129\nT119 -> []\nX159 -> T129", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 137 [arrowhead = none ];
137 -> 63;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 138 [arrowhead = none ];
138 -> 64;

139 [label="EVAL with clause\nrev1(X185, .(X186, X187), X188) :- rev1(X186, X187, X188).\nand substitution\nT118 -> T136\nX185 -> T136\nX186 -> T137\nX187 -> T138\nT119 -> .(T137, T138)\nX159 -> X189\nX188 -> X189", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 139 [arrowhead = none ];
139 -> 66;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 67;

141 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 141 [arrowhead = none ];
141 -> 65;

142 [label="INSTANCE with matching:\nT118 -> T137\nT119 -> T138\nX159 -> X189", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 142 [arrowhead = none , style=dashed];
142 -> 58[style=dashed];

143 [label="INSTANCE with matching:\nT110 -> T91\nX133 -> X105", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 143 [arrowhead = none , style=dashed];
143 -> 48[style=dashed];

144 [label="INSTANCE with matching:\nT1 -> .(T84, T145)\nT2 -> T88", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

}

(2) Obligation:

Triples:

rev123(T59, .(T60, T61), T63) :- rev123(T60, T61, T63).
rev236(T105, .(T106, T107), X134) :- rev236(T106, T107, X132).
rev236(T105, .(T106, T107), X134) :- ','(rev2c36(T106, T107, T110), rev48(T110, X133)).
rev236(T105, .(T106, T107), X134) :- ','(rev2c36(T106, T107, T110), ','(revc48(T110, T113), rev11(.(T105, T113), X134))).
rev158(T136, .(T137, T138), X189) :- rev158(T137, T138, X189).
rev48(.(T118, T119), .(X159, X160)) :- rev158(T118, T119, X159).
rev48(.(T118, T119), .(T122, X160)) :- ','(rev1c58(T118, T119, T122), rev236(T118, T119, X160)).
rev11(.(T35, .(T36, T37)), .(T39, T40)) :- rev123(T36, T37, T39).
rev11(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev1c23(T85, T86, T39), rev236(T85, T86, X104)).
rev11(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev1c23(T85, T86, T39), ','(rev2c36(T85, T86, T91), rev48(T91, X105))).
rev11(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev1c23(T85, T86, T39), ','(rev2c36(T85, T86, T91), ','(revc48(T91, T145), rev11(.(T84, T145), T88)))).

Clauses:

rev1c23(T50, [], T50).
rev1c23(T59, .(T60, T61), T63) :- rev1c23(T60, T61, T63).
rev2c36(T98, [], []).
rev2c36(T105, .(T106, T107), X134) :- ','(rev2c36(T106, T107, T110), ','(revc48(T110, T113), revc1(.(T105, T113), X134))).
revc1([], []).
revc1(.(T25, []), .(T25, [])).
revc1(.(T84, .(T85, T86)), .(T39, T88)) :- ','(rev1c23(T85, T86, T39), ','(rev2c36(T85, T86, T91), ','(revc48(T91, T145), revc1(.(T84, T145), T88)))).
rev1c58(T129, [], T129).
rev1c58(T136, .(T137, T138), X189) :- rev1c58(T137, T138, X189).
revc48([], []).
revc48(.(T118, T119), .(T122, X160)) :- ','(rev1c58(T118, T119, T122), rev2c36(T118, T119, X160)).

Afs:

rev11(x1, x2)  =  rev11(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev11_in: (b,f)
rev123_in: (b,b,f)
rev1c23_in: (b,b,f)
rev236_in: (b,b,f)
rev2c36_in: (b,b,f)
revc48_in: (b,f)
rev1c58_in: (b,b,f)
revc1_in: (b,f)
rev48_in: (b,f)
rev158_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → U11_GA(T35, T36, T37, T39, T40, rev123_in_gga(T36, T37, T39))
REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_GGA(T36, T37, T39)
REV123_IN_GGA(T59, .(T60, T61), T63) → U1_GGA(T59, T60, T61, T63, rev123_in_gga(T60, T61, T63))
REV123_IN_GGA(T59, .(T60, T61), T63) → REV123_IN_GGA(T60, T61, T63)
REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U13_GA(T84, T85, T86, T39, T88, rev236_in_gga(T85, T86, X104))
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → REV236_IN_GGA(T85, T86, X104)
REV236_IN_GGA(T105, .(T106, T107), X134) → U2_GGA(T105, T106, T107, X134, rev236_in_gga(T106, T107, X132))
REV236_IN_GGA(T105, .(T106, T107), X134) → REV236_IN_GGA(T106, T107, X132)
REV236_IN_GGA(T105, .(T106, T107), X134) → U3_GGA(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U4_GGA(T105, T106, T107, X134, rev48_in_ga(T110, X133))
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → REV48_IN_GA(T110, X133)
REV48_IN_GA(.(T118, T119), .(X159, X160)) → U8_GA(T118, T119, X159, X160, rev158_in_gga(T118, T119, X159))
REV48_IN_GA(.(T118, T119), .(X159, X160)) → REV158_IN_GGA(T118, T119, X159)
REV158_IN_GGA(T136, .(T137, T138), X189) → U7_GGA(T136, T137, T138, X189, rev158_in_gga(T137, T138, X189))
REV158_IN_GGA(T136, .(T137, T138), X189) → REV158_IN_GGA(T137, T138, X189)
REV48_IN_GA(.(T118, T119), .(T122, X160)) → U9_GA(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
U9_GA(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U10_GA(T118, T119, T122, X160, rev236_in_gga(T118, T119, X160))
U9_GA(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → REV236_IN_GGA(T118, T119, X160)
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U5_GGA(T105, T106, T107, X134, revc48_in_ga(T110, T113))
U5_GGA(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U6_GGA(T105, T106, T107, X134, rev11_in_ga(.(T105, T113), X134))
U5_GGA(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → REV11_IN_GA(.(T105, T113), X134)
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U15_GA(T84, T85, T86, T39, T88, rev48_in_ga(T91, X105))
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → REV48_IN_GA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U16_GA(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U17_GA(T84, T85, T86, T39, T88, rev11_in_ga(.(T84, T145), T88))
U16_GA(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → REV11_IN_GA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev1c23_in_gga(T50, [], T50) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61), T63) → U19_gga(T59, T60, T61, T63, rev1c23_in_gga(T60, T61, T63))
U19_gga(T59, T60, T61, T63, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
rev2c36_in_gga(T98, [], []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107), X134) → U20_gga(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U20_gga(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, X134, revc48_in_ga(T110, T113))
revc48_in_ga([], []) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119), .(T122, X160)) → U28_ga(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
rev1c58_in_gga(T129, [], T129) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138), X189) → U27_gga(T136, T137, T138, X189, rev1c58_in_gga(T137, T138, X189))
U27_gga(T136, T137, T138, X189, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, X160, rev2c36_in_gga(T118, T119, X160))
U29_ga(T118, T119, T122, X160, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U21_gga(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, X134, revc1_in_ga(.(T105, T113), X134))
revc1_in_ga([], []) → revc1_out_ga([], [])
revc1_in_ga(.(T25, []), .(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U23_ga(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U23_ga(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U24_ga(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U25_ga(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, T88, revc1_in_ga(.(T84, T145), T88))
U26_ga(T84, T85, T86, T39, T88, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U22_gga(T105, T106, T107, X134, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)

The argument filtering Pi contains the following mapping:
rev11_in_ga(x1, x2)  =  rev11_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
rev123_in_gga(x1, x2, x3)  =  rev123_in_gga(x1, x2)
rev1c23_in_gga(x1, x2, x3)  =  rev1c23_in_gga(x1, x2)
[]  =  []
rev1c23_out_gga(x1, x2, x3)  =  rev1c23_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev236_in_gga(x1, x2, x3)  =  rev236_in_gga(x1, x2)
rev2c36_in_gga(x1, x2, x3)  =  rev2c36_in_gga(x1, x2)
rev2c36_out_gga(x1, x2, x3)  =  rev2c36_out_gga(x1, x2, x3)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
revc48_in_ga(x1, x2)  =  revc48_in_ga(x1)
revc48_out_ga(x1, x2)  =  revc48_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1c58_in_gga(x1, x2, x3)  =  rev1c58_in_gga(x1, x2)
rev1c58_out_gga(x1, x2, x3)  =  rev1c58_out_gga(x1, x2, x3)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x1, x2, x3, x5)
revc1_in_ga(x1, x2)  =  revc1_in_ga(x1)
revc1_out_ga(x1, x2)  =  revc1_out_ga(x1, x2)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x1, x2, x3, x4, x6)
U25_ga(x1, x2, x3, x4, x5, x6)  =  U25_ga(x1, x2, x3, x4, x6)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x4, x6)
rev48_in_ga(x1, x2)  =  rev48_in_ga(x1)
rev158_in_gga(x1, x2, x3)  =  rev158_in_gga(x1, x2)
REV11_IN_GA(x1, x2)  =  REV11_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5, x6)  =  U11_GA(x1, x2, x3, x6)
REV123_IN_GGA(x1, x2, x3)  =  REV123_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x1, x2, x3, x6)
U13_GA(x1, x2, x3, x4, x5, x6)  =  U13_GA(x1, x2, x3, x6)
REV236_IN_GGA(x1, x2, x3)  =  REV236_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
REV48_IN_GA(x1, x2)  =  REV48_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
REV158_IN_GGA(x1, x2, x3)  =  REV158_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6)  =  U15_GA(x1, x2, x3, x6)
U16_GA(x1, x2, x3, x4, x5, x6)  =  U16_GA(x1, x2, x3, x6)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x1, x2, x3, x6)

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:

REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → U11_GA(T35, T36, T37, T39, T40, rev123_in_gga(T36, T37, T39))
REV11_IN_GA(.(T35, .(T36, T37)), .(T39, T40)) → REV123_IN_GGA(T36, T37, T39)
REV123_IN_GGA(T59, .(T60, T61), T63) → U1_GGA(T59, T60, T61, T63, rev123_in_gga(T60, T61, T63))
REV123_IN_GGA(T59, .(T60, T61), T63) → REV123_IN_GGA(T60, T61, T63)
REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U13_GA(T84, T85, T86, T39, T88, rev236_in_gga(T85, T86, X104))
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → REV236_IN_GGA(T85, T86, X104)
REV236_IN_GGA(T105, .(T106, T107), X134) → U2_GGA(T105, T106, T107, X134, rev236_in_gga(T106, T107, X132))
REV236_IN_GGA(T105, .(T106, T107), X134) → REV236_IN_GGA(T106, T107, X132)
REV236_IN_GGA(T105, .(T106, T107), X134) → U3_GGA(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U4_GGA(T105, T106, T107, X134, rev48_in_ga(T110, X133))
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → REV48_IN_GA(T110, X133)
REV48_IN_GA(.(T118, T119), .(X159, X160)) → U8_GA(T118, T119, X159, X160, rev158_in_gga(T118, T119, X159))
REV48_IN_GA(.(T118, T119), .(X159, X160)) → REV158_IN_GGA(T118, T119, X159)
REV158_IN_GGA(T136, .(T137, T138), X189) → U7_GGA(T136, T137, T138, X189, rev158_in_gga(T137, T138, X189))
REV158_IN_GGA(T136, .(T137, T138), X189) → REV158_IN_GGA(T137, T138, X189)
REV48_IN_GA(.(T118, T119), .(T122, X160)) → U9_GA(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
U9_GA(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U10_GA(T118, T119, T122, X160, rev236_in_gga(T118, T119, X160))
U9_GA(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → REV236_IN_GGA(T118, T119, X160)
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U5_GGA(T105, T106, T107, X134, revc48_in_ga(T110, T113))
U5_GGA(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U6_GGA(T105, T106, T107, X134, rev11_in_ga(.(T105, T113), X134))
U5_GGA(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → REV11_IN_GA(.(T105, T113), X134)
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U15_GA(T84, T85, T86, T39, T88, rev48_in_ga(T91, X105))
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → REV48_IN_GA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U16_GA(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U17_GA(T84, T85, T86, T39, T88, rev11_in_ga(.(T84, T145), T88))
U16_GA(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → REV11_IN_GA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev1c23_in_gga(T50, [], T50) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61), T63) → U19_gga(T59, T60, T61, T63, rev1c23_in_gga(T60, T61, T63))
U19_gga(T59, T60, T61, T63, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
rev2c36_in_gga(T98, [], []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107), X134) → U20_gga(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U20_gga(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, X134, revc48_in_ga(T110, T113))
revc48_in_ga([], []) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119), .(T122, X160)) → U28_ga(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
rev1c58_in_gga(T129, [], T129) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138), X189) → U27_gga(T136, T137, T138, X189, rev1c58_in_gga(T137, T138, X189))
U27_gga(T136, T137, T138, X189, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, X160, rev2c36_in_gga(T118, T119, X160))
U29_ga(T118, T119, T122, X160, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U21_gga(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, X134, revc1_in_ga(.(T105, T113), X134))
revc1_in_ga([], []) → revc1_out_ga([], [])
revc1_in_ga(.(T25, []), .(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U23_ga(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U23_ga(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U24_ga(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U25_ga(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, T88, revc1_in_ga(.(T84, T145), T88))
U26_ga(T84, T85, T86, T39, T88, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U22_gga(T105, T106, T107, X134, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)

The argument filtering Pi contains the following mapping:
rev11_in_ga(x1, x2)  =  rev11_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
rev123_in_gga(x1, x2, x3)  =  rev123_in_gga(x1, x2)
rev1c23_in_gga(x1, x2, x3)  =  rev1c23_in_gga(x1, x2)
[]  =  []
rev1c23_out_gga(x1, x2, x3)  =  rev1c23_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev236_in_gga(x1, x2, x3)  =  rev236_in_gga(x1, x2)
rev2c36_in_gga(x1, x2, x3)  =  rev2c36_in_gga(x1, x2)
rev2c36_out_gga(x1, x2, x3)  =  rev2c36_out_gga(x1, x2, x3)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
revc48_in_ga(x1, x2)  =  revc48_in_ga(x1)
revc48_out_ga(x1, x2)  =  revc48_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1c58_in_gga(x1, x2, x3)  =  rev1c58_in_gga(x1, x2)
rev1c58_out_gga(x1, x2, x3)  =  rev1c58_out_gga(x1, x2, x3)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x1, x2, x3, x5)
revc1_in_ga(x1, x2)  =  revc1_in_ga(x1)
revc1_out_ga(x1, x2)  =  revc1_out_ga(x1, x2)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x1, x2, x3, x4, x6)
U25_ga(x1, x2, x3, x4, x5, x6)  =  U25_ga(x1, x2, x3, x4, x6)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x4, x6)
rev48_in_ga(x1, x2)  =  rev48_in_ga(x1)
rev158_in_gga(x1, x2, x3)  =  rev158_in_gga(x1, x2)
REV11_IN_GA(x1, x2)  =  REV11_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5, x6)  =  U11_GA(x1, x2, x3, x6)
REV123_IN_GGA(x1, x2, x3)  =  REV123_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x2, x3, x5)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x1, x2, x3, x6)
U13_GA(x1, x2, x3, x4, x5, x6)  =  U13_GA(x1, x2, x3, x6)
REV236_IN_GGA(x1, x2, x3)  =  REV236_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x1, x2, x3, x5)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
REV48_IN_GA(x1, x2)  =  REV48_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
REV158_IN_GGA(x1, x2, x3)  =  REV158_IN_GGA(x1, x2)
U7_GGA(x1, x2, x3, x4, x5)  =  U7_GGA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U6_GGA(x1, x2, x3, x4, x5)  =  U6_GGA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x6)
U15_GA(x1, x2, x3, x4, x5, x6)  =  U15_GA(x1, x2, x3, x6)
U16_GA(x1, x2, x3, x4, x5, x6)  =  U16_GA(x1, x2, x3, x6)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(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 3 SCCs with 13 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

REV158_IN_GGA(T136, .(T137, T138), X189) → REV158_IN_GGA(T137, T138, X189)

The TRS R consists of the following rules:

rev1c23_in_gga(T50, [], T50) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61), T63) → U19_gga(T59, T60, T61, T63, rev1c23_in_gga(T60, T61, T63))
U19_gga(T59, T60, T61, T63, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
rev2c36_in_gga(T98, [], []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107), X134) → U20_gga(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U20_gga(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, X134, revc48_in_ga(T110, T113))
revc48_in_ga([], []) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119), .(T122, X160)) → U28_ga(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
rev1c58_in_gga(T129, [], T129) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138), X189) → U27_gga(T136, T137, T138, X189, rev1c58_in_gga(T137, T138, X189))
U27_gga(T136, T137, T138, X189, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, X160, rev2c36_in_gga(T118, T119, X160))
U29_ga(T118, T119, T122, X160, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U21_gga(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, X134, revc1_in_ga(.(T105, T113), X134))
revc1_in_ga([], []) → revc1_out_ga([], [])
revc1_in_ga(.(T25, []), .(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U23_ga(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U23_ga(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U24_ga(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U25_ga(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, T88, revc1_in_ga(.(T84, T145), T88))
U26_ga(T84, T85, T86, T39, T88, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U22_gga(T105, T106, T107, X134, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1c23_in_gga(x1, x2, x3)  =  rev1c23_in_gga(x1, x2)
[]  =  []
rev1c23_out_gga(x1, x2, x3)  =  rev1c23_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2c36_in_gga(x1, x2, x3)  =  rev2c36_in_gga(x1, x2)
rev2c36_out_gga(x1, x2, x3)  =  rev2c36_out_gga(x1, x2, x3)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
revc48_in_ga(x1, x2)  =  revc48_in_ga(x1)
revc48_out_ga(x1, x2)  =  revc48_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1c58_in_gga(x1, x2, x3)  =  rev1c58_in_gga(x1, x2)
rev1c58_out_gga(x1, x2, x3)  =  rev1c58_out_gga(x1, x2, x3)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x1, x2, x3, x5)
revc1_in_ga(x1, x2)  =  revc1_in_ga(x1)
revc1_out_ga(x1, x2)  =  revc1_out_ga(x1, x2)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x1, x2, x3, x4, x6)
U25_ga(x1, x2, x3, x4, x5, x6)  =  U25_ga(x1, x2, x3, x4, x6)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x4, x6)
REV158_IN_GGA(x1, x2, x3)  =  REV158_IN_GGA(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:

REV158_IN_GGA(T136, .(T137, T138), X189) → REV158_IN_GGA(T137, T138, X189)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV158_IN_GGA(x1, x2, x3)  =  REV158_IN_GGA(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:

REV158_IN_GGA(T136, .(T137, T138)) → REV158_IN_GGA(T137, T138)

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:

• REV158_IN_GGA(T136, .(T137, T138)) → REV158_IN_GGA(T137, T138)
  The graph contains the following edges 2 > 1, 2 > 2

(13) YES

(14) Obligation:

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

REV123_IN_GGA(T59, .(T60, T61), T63) → REV123_IN_GGA(T60, T61, T63)

The TRS R consists of the following rules:

rev1c23_in_gga(T50, [], T50) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61), T63) → U19_gga(T59, T60, T61, T63, rev1c23_in_gga(T60, T61, T63))
U19_gga(T59, T60, T61, T63, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
rev2c36_in_gga(T98, [], []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107), X134) → U20_gga(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U20_gga(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, X134, revc48_in_ga(T110, T113))
revc48_in_ga([], []) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119), .(T122, X160)) → U28_ga(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
rev1c58_in_gga(T129, [], T129) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138), X189) → U27_gga(T136, T137, T138, X189, rev1c58_in_gga(T137, T138, X189))
U27_gga(T136, T137, T138, X189, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, X160, rev2c36_in_gga(T118, T119, X160))
U29_ga(T118, T119, T122, X160, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U21_gga(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, X134, revc1_in_ga(.(T105, T113), X134))
revc1_in_ga([], []) → revc1_out_ga([], [])
revc1_in_ga(.(T25, []), .(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U23_ga(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U23_ga(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U24_ga(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U25_ga(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, T88, revc1_in_ga(.(T84, T145), T88))
U26_ga(T84, T85, T86, T39, T88, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U22_gga(T105, T106, T107, X134, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1c23_in_gga(x1, x2, x3)  =  rev1c23_in_gga(x1, x2)
[]  =  []
rev1c23_out_gga(x1, x2, x3)  =  rev1c23_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2c36_in_gga(x1, x2, x3)  =  rev2c36_in_gga(x1, x2)
rev2c36_out_gga(x1, x2, x3)  =  rev2c36_out_gga(x1, x2, x3)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
revc48_in_ga(x1, x2)  =  revc48_in_ga(x1)
revc48_out_ga(x1, x2)  =  revc48_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1c58_in_gga(x1, x2, x3)  =  rev1c58_in_gga(x1, x2)
rev1c58_out_gga(x1, x2, x3)  =  rev1c58_out_gga(x1, x2, x3)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x1, x2, x3, x5)
revc1_in_ga(x1, x2)  =  revc1_in_ga(x1)
revc1_out_ga(x1, x2)  =  revc1_out_ga(x1, x2)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x1, x2, x3, x4, x6)
U25_ga(x1, x2, x3, x4, x5, x6)  =  U25_ga(x1, x2, x3, x4, x6)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x4, x6)
REV123_IN_GGA(x1, x2, x3)  =  REV123_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:

REV123_IN_GGA(T59, .(T60, T61), T63) → REV123_IN_GGA(T60, T61, T63)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV123_IN_GGA(x1, x2, x3)  =  REV123_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:

REV123_IN_GGA(T59, .(T60, T61)) → REV123_IN_GGA(T60, T61)

R is empty.
Q is empty.
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:

• REV123_IN_GGA(T59, .(T60, T61)) → REV123_IN_GGA(T60, T61)
  The graph contains the following edges 2 > 1, 2 > 2

(20) YES

(21) Obligation:

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

REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → REV236_IN_GGA(T85, T86, X104)
REV236_IN_GGA(T105, .(T106, T107), X134) → REV236_IN_GGA(T106, T107, X132)
REV236_IN_GGA(T105, .(T106, T107), X134) → U3_GGA(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → REV48_IN_GA(T110, X133)
REV48_IN_GA(.(T118, T119), .(T122, X160)) → U9_GA(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
U9_GA(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → REV236_IN_GGA(T118, T119, X160)
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U5_GGA(T105, T106, T107, X134, revc48_in_ga(T110, T113))
U5_GGA(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → REV11_IN_GA(.(T105, T113), X134)
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → REV48_IN_GA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U16_GA(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → REV11_IN_GA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev1c23_in_gga(T50, [], T50) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61), T63) → U19_gga(T59, T60, T61, T63, rev1c23_in_gga(T60, T61, T63))
U19_gga(T59, T60, T61, T63, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
rev2c36_in_gga(T98, [], []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107), X134) → U20_gga(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U20_gga(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, X134, revc48_in_ga(T110, T113))
revc48_in_ga([], []) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119), .(T122, X160)) → U28_ga(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
rev1c58_in_gga(T129, [], T129) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138), X189) → U27_gga(T136, T137, T138, X189, rev1c58_in_gga(T137, T138, X189))
U27_gga(T136, T137, T138, X189, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, X160, rev2c36_in_gga(T118, T119, X160))
U29_ga(T118, T119, T122, X160, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U21_gga(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, X134, revc1_in_ga(.(T105, T113), X134))
revc1_in_ga([], []) → revc1_out_ga([], [])
revc1_in_ga(.(T25, []), .(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U23_ga(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U23_ga(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U24_ga(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U25_ga(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, T88, revc1_in_ga(.(T84, T145), T88))
U26_ga(T84, T85, T86, T39, T88, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U22_gga(T105, T106, T107, X134, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1c23_in_gga(x1, x2, x3)  =  rev1c23_in_gga(x1, x2)
[]  =  []
rev1c23_out_gga(x1, x2, x3)  =  rev1c23_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2c36_in_gga(x1, x2, x3)  =  rev2c36_in_gga(x1, x2)
rev2c36_out_gga(x1, x2, x3)  =  rev2c36_out_gga(x1, x2, x3)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
revc48_in_ga(x1, x2)  =  revc48_in_ga(x1)
revc48_out_ga(x1, x2)  =  revc48_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1c58_in_gga(x1, x2, x3)  =  rev1c58_in_gga(x1, x2)
rev1c58_out_gga(x1, x2, x3)  =  rev1c58_out_gga(x1, x2, x3)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x1, x2, x3, x5)
revc1_in_ga(x1, x2)  =  revc1_in_ga(x1)
revc1_out_ga(x1, x2)  =  revc1_out_ga(x1, x2)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x1, x2, x3, x4, x6)
U25_ga(x1, x2, x3, x4, x5, x6)  =  U25_ga(x1, x2, x3, x4, x6)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x4, x6)
REV11_IN_GA(x1, x2)  =  REV11_IN_GA(x1)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x1, x2, x3, x6)
REV236_IN_GGA(x1, x2, x3)  =  REV236_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
REV48_IN_GA(x1, x2)  =  REV48_IN_GA(x1)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x6)
U16_GA(x1, x2, x3, x4, x5, x6)  =  U16_GA(x1, x2, x3, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

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

REV11_IN_GA(.(T84, .(T85, T86)), .(T39, T88)) → U12_GA(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → REV236_IN_GGA(T85, T86, X104)
REV236_IN_GGA(T105, .(T106, T107), X134) → REV236_IN_GGA(T106, T107, X132)
REV236_IN_GGA(T105, .(T106, T107), X134) → U3_GGA(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → REV48_IN_GA(T110, X133)
REV48_IN_GA(.(T118, T119), .(T122, X160)) → U9_GA(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
U9_GA(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → REV236_IN_GGA(T118, T119, X160)
U3_GGA(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U5_GGA(T105, T106, T107, X134, revc48_in_ga(T110, T113))
U5_GGA(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → REV11_IN_GA(.(T105, T113), X134)
U12_GA(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U14_GA(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → REV48_IN_GA(T91, X105)
U14_GA(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U16_GA(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U16_GA(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → REV11_IN_GA(.(T84, T145), T88)

The TRS R consists of the following rules:

rev1c23_in_gga(T50, [], T50) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61), T63) → U19_gga(T59, T60, T61, T63, rev1c23_in_gga(T60, T61, T63))
rev2c36_in_gga(T98, [], []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107), X134) → U20_gga(T105, T106, T107, X134, rev2c36_in_gga(T106, T107, T110))
rev1c58_in_gga(T129, [], T129) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138), X189) → U27_gga(T136, T137, T138, X189, rev1c58_in_gga(T137, T138, X189))
revc48_in_ga([], []) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119), .(T122, X160)) → U28_ga(T118, T119, T122, X160, rev1c58_in_gga(T118, T119, T122))
U19_gga(T59, T60, T61, T63, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
U20_gga(T105, T106, T107, X134, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, X134, revc48_in_ga(T110, T113))
U27_gga(T136, T137, T138, X189, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, T122, X160, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, X160, rev2c36_in_gga(T118, T119, X160))
U21_gga(T105, T106, T107, X134, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, X134, revc1_in_ga(.(T105, T113), X134))
U29_ga(T118, T119, T122, X160, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U22_gga(T105, T106, T107, X134, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)
revc1_in_ga(.(T25, []), .(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86)), .(T39, T88)) → U23_ga(T84, T85, T86, T39, T88, rev1c23_in_gga(T85, T86, T39))
U23_ga(T84, T85, T86, T39, T88, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, T88, rev2c36_in_gga(T85, T86, T91))
U24_ga(T84, T85, T86, T39, T88, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, T88, revc48_in_ga(T91, T145))
U25_ga(T84, T85, T86, T39, T88, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, T88, revc1_in_ga(.(T84, T145), T88))
U26_ga(T84, T85, T86, T39, T88, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1c23_in_gga(x1, x2, x3)  =  rev1c23_in_gga(x1, x2)
[]  =  []
rev1c23_out_gga(x1, x2, x3)  =  rev1c23_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2c36_in_gga(x1, x2, x3)  =  rev2c36_in_gga(x1, x2)
rev2c36_out_gga(x1, x2, x3)  =  rev2c36_out_gga(x1, x2, x3)
U20_gga(x1, x2, x3, x4, x5)  =  U20_gga(x1, x2, x3, x5)
U21_gga(x1, x2, x3, x4, x5)  =  U21_gga(x1, x2, x3, x5)
revc48_in_ga(x1, x2)  =  revc48_in_ga(x1)
revc48_out_ga(x1, x2)  =  revc48_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1c58_in_gga(x1, x2, x3)  =  rev1c58_in_gga(x1, x2)
rev1c58_out_gga(x1, x2, x3)  =  rev1c58_out_gga(x1, x2, x3)
U27_gga(x1, x2, x3, x4, x5)  =  U27_gga(x1, x2, x3, x5)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x3, x5)
U22_gga(x1, x2, x3, x4, x5)  =  U22_gga(x1, x2, x3, x5)
revc1_in_ga(x1, x2)  =  revc1_in_ga(x1)
revc1_out_ga(x1, x2)  =  revc1_out_ga(x1, x2)
U23_ga(x1, x2, x3, x4, x5, x6)  =  U23_ga(x1, x2, x3, x6)
U24_ga(x1, x2, x3, x4, x5, x6)  =  U24_ga(x1, x2, x3, x4, x6)
U25_ga(x1, x2, x3, x4, x5, x6)  =  U25_ga(x1, x2, x3, x4, x6)
U26_ga(x1, x2, x3, x4, x5, x6)  =  U26_ga(x1, x2, x3, x4, x6)
REV11_IN_GA(x1, x2)  =  REV11_IN_GA(x1)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x1, x2, x3, x6)
REV236_IN_GGA(x1, x2, x3)  =  REV236_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
REV48_IN_GA(x1, x2)  =  REV48_IN_GA(x1)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x5)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x6)
U16_GA(x1, x2, x3, x4, x5, x6)  =  U16_GA(x1, x2, x3, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

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

REV11_IN_GA(.(T84, .(T85, T86))) → U12_GA(T84, T85, T86, rev1c23_in_gga(T85, T86))
U12_GA(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → REV236_IN_GGA(T85, T86)
REV236_IN_GGA(T105, .(T106, T107)) → REV236_IN_GGA(T106, T107)
REV236_IN_GGA(T105, .(T106, T107)) → U3_GGA(T105, T106, T107, rev2c36_in_gga(T106, T107))
U3_GGA(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → REV48_IN_GA(T110)
REV48_IN_GA(.(T118, T119)) → U9_GA(T118, T119, rev1c58_in_gga(T118, T119))
U9_GA(T118, T119, rev1c58_out_gga(T118, T119, T122)) → REV236_IN_GGA(T118, T119)
U3_GGA(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → U5_GGA(T105, T106, T107, revc48_in_ga(T110))
U5_GGA(T105, T106, T107, revc48_out_ga(T110, T113)) → REV11_IN_GA(.(T105, T113))
U12_GA(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → U14_GA(T84, T85, T86, rev2c36_in_gga(T85, T86))
U14_GA(T84, T85, T86, rev2c36_out_gga(T85, T86, T91)) → REV48_IN_GA(T91)
U14_GA(T84, T85, T86, rev2c36_out_gga(T85, T86, T91)) → U16_GA(T84, T85, T86, revc48_in_ga(T91))
U16_GA(T84, T85, T86, revc48_out_ga(T91, T145)) → REV11_IN_GA(.(T84, T145))

The TRS R consists of the following rules:

rev1c23_in_gga(T50, []) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61)) → U19_gga(T59, T60, T61, rev1c23_in_gga(T60, T61))
rev2c36_in_gga(T98, []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107)) → U20_gga(T105, T106, T107, rev2c36_in_gga(T106, T107))
rev1c58_in_gga(T129, []) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138)) → U27_gga(T136, T137, T138, rev1c58_in_gga(T137, T138))
revc48_in_ga([]) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119)) → U28_ga(T118, T119, rev1c58_in_gga(T118, T119))
U19_gga(T59, T60, T61, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
U20_gga(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, revc48_in_ga(T110))
U27_gga(T136, T137, T138, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, rev2c36_in_gga(T118, T119))
U21_gga(T105, T106, T107, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, revc1_in_ga(.(T105, T113)))
U29_ga(T118, T119, T122, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U22_gga(T105, T106, T107, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)
revc1_in_ga(.(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86))) → U23_ga(T84, T85, T86, rev1c23_in_gga(T85, T86))
U23_ga(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, rev2c36_in_gga(T85, T86))
U24_ga(T84, T85, T86, T39, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, revc48_in_ga(T91))
U25_ga(T84, T85, T86, T39, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, revc1_in_ga(.(T84, T145)))
U26_ga(T84, T85, T86, T39, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))

The set Q consists of the following terms:

rev1c23_in_gga(x0, x1)
rev2c36_in_gga(x0, x1)
rev1c58_in_gga(x0, x1)
revc48_in_ga(x0)
U19_gga(x0, x1, x2, x3)
U20_gga(x0, x1, x2, x3)
U27_gga(x0, x1, x2, x3)
U28_ga(x0, x1, x2)
U21_gga(x0, x1, x2, x3)
U29_ga(x0, x1, x2, x3)
U22_gga(x0, x1, x2, x3)
revc1_in_ga(x0)
U23_ga(x0, x1, x2, x3)
U24_ga(x0, x1, x2, x3, x4)
U25_ga(x0, x1, x2, x3, x4)
U26_ga(x0, x1, x2, x3, x4)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

REV11_IN_GA(.(T84, .(T85, T86))) → U12_GA(T84, T85, T86, rev1c23_in_gga(T85, T86))
REV236_IN_GGA(T105, .(T106, T107)) → REV236_IN_GGA(T106, T107)
U3_GGA(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → REV48_IN_GA(T110)
U3_GGA(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → U5_GGA(T105, T106, T107, revc48_in_ga(T110))
U14_GA(T84, T85, T86, rev2c36_out_gga(T85, T86, T91)) → REV48_IN_GA(T91)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂   
POL(REV11_IN_GA(x₁)) = x₁   
POL(REV236_IN_GGA(x₁, x₂)) = 1 + x₂   
POL(REV48_IN_GA(x₁)) = x₁   
POL(U12_GA(x₁, x₂, x₃, x₄)) = 1 + x₃   
POL(U14_GA(x₁, x₂, x₃, x₄)) = x₄   
POL(U16_GA(x₁, x₂, x₃, x₄)) = x₄   
POL(U19_gga(x₁, x₂, x₃, x₄)) = 1   
POL(U20_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U21_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U22_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U23_ga(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄   
POL(U24_ga(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅   
POL(U25_ga(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅   
POL(U26_ga(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅   
POL(U27_gga(x₁, x₂, x₃, x₄)) = 1   
POL(U28_ga(x₁, x₂, x₃)) = 1 + x₂ + x₃   
POL(U29_ga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U3_GGA(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U5_GGA(x₁, x₂, x₃, x₄)) = x₄   
POL(U9_GA(x₁, x₂, x₃)) = 1 + x₂   
POL([]) = 0   
POL(rev1c23_in_gga(x₁, x₂)) = 1   
POL(rev1c23_out_gga(x₁, x₂, x₃)) = 1   
POL(rev1c58_in_gga(x₁, x₂)) = 1   
POL(rev1c58_out_gga(x₁, x₂, x₃)) = 1   
POL(rev2c36_in_gga(x₁, x₂)) = 1 + x₂   
POL(rev2c36_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(revc1_in_ga(x₁)) = x₁   
POL(revc1_out_ga(x₁, x₂)) = x₂   
POL(revc48_in_ga(x₁)) = 1 + x₁   
POL(revc48_out_ga(x₁, x₂)) = 1 + x₂   

The following usable rules [FROCOS05] were oriented:

rev1c23_in_gga(T50, []) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61)) → U19_gga(T59, T60, T61, rev1c23_in_gga(T60, T61))
rev2c36_in_gga(T98, []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107)) → U20_gga(T105, T106, T107, rev2c36_in_gga(T106, T107))
rev1c58_in_gga(T129, []) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138)) → U27_gga(T136, T137, T138, rev1c58_in_gga(T137, T138))
revc48_in_ga([]) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119)) → U28_ga(T118, T119, rev1c58_in_gga(T118, T119))
U23_ga(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, rev2c36_in_gga(T85, T86))
U20_gga(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, revc48_in_ga(T110))
U28_ga(T118, T119, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, rev2c36_in_gga(T118, T119))
U29_ga(T118, T119, T122, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U21_gga(T105, T106, T107, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, revc1_in_ga(.(T105, T113)))
revc1_in_ga(.(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
U22_gga(T105, T106, T107, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)
revc1_in_ga(.(T84, .(T85, T86))) → U23_ga(T84, T85, T86, rev1c23_in_gga(T85, T86))
U24_ga(T84, T85, T86, T39, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, revc48_in_ga(T91))
U25_ga(T84, T85, T86, T39, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, revc1_in_ga(.(T84, T145)))
U26_ga(T84, T85, T86, T39, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))
U19_gga(T59, T60, T61, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
U27_gga(T136, T137, T138, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)

(27) Obligation:

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

U12_GA(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → REV236_IN_GGA(T85, T86)
REV236_IN_GGA(T105, .(T106, T107)) → U3_GGA(T105, T106, T107, rev2c36_in_gga(T106, T107))
REV48_IN_GA(.(T118, T119)) → U9_GA(T118, T119, rev1c58_in_gga(T118, T119))
U9_GA(T118, T119, rev1c58_out_gga(T118, T119, T122)) → REV236_IN_GGA(T118, T119)
U5_GGA(T105, T106, T107, revc48_out_ga(T110, T113)) → REV11_IN_GA(.(T105, T113))
U12_GA(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → U14_GA(T84, T85, T86, rev2c36_in_gga(T85, T86))
U14_GA(T84, T85, T86, rev2c36_out_gga(T85, T86, T91)) → U16_GA(T84, T85, T86, revc48_in_ga(T91))
U16_GA(T84, T85, T86, revc48_out_ga(T91, T145)) → REV11_IN_GA(.(T84, T145))

The TRS R consists of the following rules:

rev1c23_in_gga(T50, []) → rev1c23_out_gga(T50, [], T50)
rev1c23_in_gga(T59, .(T60, T61)) → U19_gga(T59, T60, T61, rev1c23_in_gga(T60, T61))
rev2c36_in_gga(T98, []) → rev2c36_out_gga(T98, [], [])
rev2c36_in_gga(T105, .(T106, T107)) → U20_gga(T105, T106, T107, rev2c36_in_gga(T106, T107))
rev1c58_in_gga(T129, []) → rev1c58_out_gga(T129, [], T129)
rev1c58_in_gga(T136, .(T137, T138)) → U27_gga(T136, T137, T138, rev1c58_in_gga(T137, T138))
revc48_in_ga([]) → revc48_out_ga([], [])
revc48_in_ga(.(T118, T119)) → U28_ga(T118, T119, rev1c58_in_gga(T118, T119))
U19_gga(T59, T60, T61, rev1c23_out_gga(T60, T61, T63)) → rev1c23_out_gga(T59, .(T60, T61), T63)
U20_gga(T105, T106, T107, rev2c36_out_gga(T106, T107, T110)) → U21_gga(T105, T106, T107, revc48_in_ga(T110))
U27_gga(T136, T137, T138, rev1c58_out_gga(T137, T138, X189)) → rev1c58_out_gga(T136, .(T137, T138), X189)
U28_ga(T118, T119, rev1c58_out_gga(T118, T119, T122)) → U29_ga(T118, T119, T122, rev2c36_in_gga(T118, T119))
U21_gga(T105, T106, T107, revc48_out_ga(T110, T113)) → U22_gga(T105, T106, T107, revc1_in_ga(.(T105, T113)))
U29_ga(T118, T119, T122, rev2c36_out_gga(T118, T119, X160)) → revc48_out_ga(.(T118, T119), .(T122, X160))
U22_gga(T105, T106, T107, revc1_out_ga(.(T105, T113), X134)) → rev2c36_out_gga(T105, .(T106, T107), X134)
revc1_in_ga(.(T25, [])) → revc1_out_ga(.(T25, []), .(T25, []))
revc1_in_ga(.(T84, .(T85, T86))) → U23_ga(T84, T85, T86, rev1c23_in_gga(T85, T86))
U23_ga(T84, T85, T86, rev1c23_out_gga(T85, T86, T39)) → U24_ga(T84, T85, T86, T39, rev2c36_in_gga(T85, T86))
U24_ga(T84, T85, T86, T39, rev2c36_out_gga(T85, T86, T91)) → U25_ga(T84, T85, T86, T39, revc48_in_ga(T91))
U25_ga(T84, T85, T86, T39, revc48_out_ga(T91, T145)) → U26_ga(T84, T85, T86, T39, revc1_in_ga(.(T84, T145)))
U26_ga(T84, T85, T86, T39, revc1_out_ga(.(T84, T145), T88)) → revc1_out_ga(.(T84, .(T85, T86)), .(T39, T88))

The set Q consists of the following terms:

rev1c23_in_gga(x0, x1)
rev2c36_in_gga(x0, x1)
rev1c58_in_gga(x0, x1)
revc48_in_ga(x0)
U19_gga(x0, x1, x2, x3)
U20_gga(x0, x1, x2, x3)
U27_gga(x0, x1, x2, x3)
U28_ga(x0, x1, x2)
U21_gga(x0, x1, x2, x3)
U29_ga(x0, x1, x2, x3)
U22_gga(x0, x1, x2, x3)
revc1_in_ga(x0)
U23_ga(x0, x1, x2, x3)
U24_ga(x0, x1, x2, x3, x4)
U25_ga(x0, x1, x2, x3, x4)
U26_ga(x0, x1, x2, x3, x4)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes.

(29) TRUE