Left Termination of the query pattern
rev(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))).

Query: rev(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="C: 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="A: 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(T81, T82, X83), ','(rev(X83, X84), rev(.(T80, X84), T84)))\n\nT80 is ground\nT81 is ground\nT82 is ground\nX83 is free\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="B: rev2(T81, T82, X83)\n\nT81 is ground\nT82 is ground\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(rev(T87, X84), rev(.(T80, X84), T84))\n\nT80 is ground\nT87 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: rev2(T81, T82, X83), SCOPE: 6, CLAUSE: 4 | rev2(T81, T82, X83), SCOPE: 6, CLAUSE: 5\n\nT81 is ground\nT82 is ground\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: rev2(T81, T82, X83), SCOPE: 6, CLAUSE: 4\n\nT81 is ground\nT82 is ground\nX83 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: rev2(T81, T82, X83), SCOPE: 6, CLAUSE: 5\n\nT81 is ground\nT82 is ground\nX83 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(T102, T103, X110), ','(rev(X110, X111), rev(.(T101, X111), X112)))\n\nT101 is ground\nT102 is ground\nT103 is ground\nX112 is free\nX110 is free\nX111 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: rev2(T102, T103, X110)\n\nT102 is ground\nT103 is ground\nX110 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(rev(T106, X111), rev(.(T101, X111), X112))\n\nT101 is ground\nT106 is ground\nX112 is free\nX111 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="D: rev(T106, X111)\n\nT106 is ground\nX111 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: rev(.(T101, T109), X112)\n\nT101 is ground\nT109 is ground\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: rev(T106, X111), SCOPE: 7, CLAUSE: 0 | rev(T106, X111), SCOPE: 7, CLAUSE: 1\n\nT106 is ground\nX111 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: rev(T106, X111), SCOPE: 7, CLAUSE: 0\n\nT106 is ground\nX111 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: rev(T106, X111), SCOPE: 7, CLAUSE: 1\n\nT106 is ground\nX111 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(T114, T115, X133), rev2(T114, T115, X134))\n\nT114 is ground\nT115 is ground\nX133 is free\nX134 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="E: rev1(T114, T115, X133)\n\nT114 is ground\nT115 is ground\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: rev2(T114, T115, X134)\n\nT114 is ground\nT115 is ground\nX134 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: rev1(T114, T115, X133), SCOPE: 8, CLAUSE: 2 | rev1(T114, T115, X133), SCOPE: 8, CLAUSE: 3\n\nT114 is ground\nT115 is ground\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: rev1(T114, T115, X133), SCOPE: 8, CLAUSE: 2\n\nT114 is ground\nT115 is ground\nX133 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: rev1(T114, T115, X133), SCOPE: 8, CLAUSE: 3\n\nT114 is ground\nT115 is ground\nX133 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(T133, T134, X158)\n\nT133 is ground\nT134 is ground\nX158 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: rev(T87, X84)\n\nT87 is ground\nX84 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: rev(.(T80, T139), T84)\n\nT80 is ground\nT139 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 substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 71 [arrowhead = none ];
71 -> 3;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionX9 -> 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 = lightgreen];
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 = lightgreen];
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(X17, [], X17).\nand substitutionT7 -> T17,\nX17 -> 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 = lightgreen];
10 -> 81 [arrowhead = none ];
81 -> 13;

82 [label="EVAL with clause\nrev1(X37, .(X38, X39), X40) :- rev1(X38, X39, X40).\nand substitutionT7 -> T35,\nX37 -> T35,\nX38 -> T36,\nX39 -> T37,\nT8 -> .(T36, T37),\nT11 -> T39,\nX40 -> 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 = lightgreen];
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(X24, [], []).\nand substitutionT17 -> T25,\nX24 -> T25,\nT18 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 87 [arrowhead = none ];
87 -> 17;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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:\nT36 is ground\nT37 is ground\nT39 is ground\nreplacements:T40 -> 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(X49, [], X49).\nand substitutionT36 -> T50,\nX49 -> 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 = lightgreen];
26 -> 98 [arrowhead = none ];
98 -> 29;

99 [label="EVAL with clause\nrev1(X58, .(X59, X60), X61) :- rev1(X59, X60, X61).\nand substitutionT36 -> T59,\nX58 -> T59,\nX59 -> T60,\nX60 -> T61,\nT37 -> .(T60, T61),\nT39 -> T63,\nX61 -> T63,\nT62 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 99 [arrowhead = none ];
99 -> 31;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
33 -> 103 [arrowhead = none ];
103 -> 34;

104 [label="ONLY EVAL with clause\nrev2(X79, .(X80, X81), X82) :- ','(rev2(X80, X81, X83), ','(rev(X83, X84), rev(.(X79, X84), X82))).\nand substitutionT35 -> T80,\nX79 -> T80,\nT36 -> T81,\nX80 -> T81,\nT37 -> T82,\nX81 -> T82,\nT43 -> T84,\nX82 -> T84,\nT83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT81 is ground\nT82 is ground\nT87 is ground\nreplacements:X83 -> T87", 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:\nT87 is ground\nT139 is ground\nreplacements:X84 -> T139", 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(X93, [], []).\nand substitutionT81 -> T94,\nX93 -> T94,\nT82 -> [],\nX83 -> []", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 112 [arrowhead = none ];
112 -> 41;

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

114 [label="EVAL with clause\nrev2(X106, .(X107, X108), X109) :- ','(rev2(X107, X108, X110), ','(rev(X110, X111), rev(.(X106, X111), X109))).\nand substitutionT81 -> T101,\nX106 -> T101,\nX107 -> T102,\nX108 -> T103,\nT82 -> .(T102, T103),\nX83 -> X112,\nX109 -> X112", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 114 [arrowhead = none ];
114 -> 44;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT102 is ground\nT103 is ground\nT106 is ground\nreplacements:X110 -> T106", fontsize=14, style = filled, fillcolor = violet];
44 -> 118 [arrowhead = none ];
118 -> 47;

119 [label="INSTANCE with matching:\nT81 -> T102\nT82 -> T103\nX83 -> X110", 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:\nT106 is ground\nT109 is ground\nreplacements:X111 -> T109", 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 -> .(T101, T109)\nT2 -> X112", 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 substitutionT106 -> [],\nX111 -> []", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 126 [arrowhead = none ];
126 -> 53;

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

128 [label="EVAL with clause\nrev(.(X129, X130), .(X131, X132)) :- ','(rev1(X129, X130, X131), rev2(X129, X130, X132)).\nand substitutionX129 -> T114,\nX130 -> T115,\nT106 -> .(T114, T115),\nX131 -> X133,\nX132 -> X134,\nX111 -> .(X133, X134)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 128 [arrowhead = none ];
128 -> 56;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT114 is ground\nT115 is ground\nT118 is ground\nreplacements:X133 -> T118", 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:\nT81 -> T114\nT82 -> T115\nX83 -> X134", 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(X143, [], X143).\nand substitutionT114 -> T125,\nX143 -> T125,\nT115 -> [],\nX133 -> T125", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 137 [arrowhead = none ];
137 -> 63;

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

139 [label="EVAL with clause\nrev1(X154, .(X155, X156), X157) :- rev1(X155, X156, X157).\nand substitutionT114 -> T132,\nX154 -> T132,\nX155 -> T133,\nX156 -> T134,\nT115 -> .(T133, T134),\nX133 -> X158,\nX157 -> X158", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 139 [arrowhead = none ];
139 -> 66;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT114 -> T133\nT115 -> T134\nX133 -> X158", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 142 [arrowhead = none , style=dashed];
142 -> 58[style=dashed];

143 [label="INSTANCE with matching:\nT106 -> T87\nX111 -> X84", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 143 [arrowhead = none , style=dashed];
143 -> 48[style=dashed];

144 [label="INSTANCE with matching:\nT1 -> .(T80, T139)\nT2 -> T84", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 144 [arrowhead = none , style=dashed];
144 -> 1[style=dashed];

}

(2) Obligation:

Triples:

rev1A(X1, .(X2, X3), X4) :- rev1A(X2, X3, X4).
rev2B(X1, .(X2, X3), X4) :- rev2B(X2, X3, X5).
rev2B(X1, .(X2, X3), X4) :- ','(rev2cB(X2, X3, X5), revD(X5, X6)).
rev2B(X1, .(X2, X3), X4) :- ','(rev2cB(X2, X3, X5), ','(revcD(X5, X6), revC(.(X1, X6), X4))).
rev1E(X1, .(X2, X3), X4) :- rev1E(X2, X3, X4).
revD(.(X1, X2), .(X3, X4)) :- rev1E(X1, X2, X3).
revD(.(X1, X2), .(X3, X4)) :- ','(rev1cE(X1, X2, X3), rev2B(X1, X2, X4)).
revC(.(X1, .(X2, X3)), .(X4, X5)) :- rev1A(X2, X3, X4).
revC(.(X1, .(X2, X3)), .(X4, X5)) :- ','(rev1cA(X2, X3, X4), rev2B(X2, X3, X6)).
revC(.(X1, .(X2, X3)), .(X4, X5)) :- ','(rev1cA(X2, X3, X4), ','(rev2cB(X2, X3, X6), revD(X6, X7))).
revC(.(X1, .(X2, X3)), .(X4, X5)) :- ','(rev1cA(X2, X3, X4), ','(rev2cB(X2, X3, X6), ','(revcD(X6, X7), revC(.(X1, X7), X5)))).

Clauses:

rev1cA(X1, [], X1).
rev1cA(X1, .(X2, X3), X4) :- rev1cA(X2, X3, X4).
rev2cB(X1, [], []).
rev2cB(X1, .(X2, X3), X4) :- ','(rev2cB(X2, X3, X5), ','(revcD(X5, X6), revcC(.(X1, X6), X4))).
revcC([], []).
revcC(.(X1, []), .(X1, [])).
revcC(.(X1, .(X2, X3)), .(X4, X5)) :- ','(rev1cA(X2, X3, X4), ','(rev2cB(X2, X3, X6), ','(revcD(X6, X7), revcC(.(X1, X7), X5)))).
rev1cE(X1, [], X1).
rev1cE(X1, .(X2, X3), X4) :- rev1cE(X2, X3, X4).
revcD([], []).
revcD(.(X1, X2), .(X3, X4)) :- ','(rev1cE(X1, X2, X3), rev2cB(X1, X2, X4)).

Afs:

revC(x1, x2)  =  revC(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
revC_in: (b,f)
rev1A_in: (b,b,f)
rev1cA_in: (b,b,f)
rev2B_in: (b,b,f)
rev2cB_in: (b,b,f)
revcD_in: (b,f)
rev1cE_in: (b,b,f)
revcC_in: (b,f)
revD_in: (b,f)
rev1E_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → U11_GA(X1, X2, X3, X4, X5, rev1A_in_gga(X2, X3, X4))
REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → REV1A_IN_GGA(X2, X3, X4)
REV1A_IN_GGA(X1, .(X2, X3), X4) → U1_GGA(X1, X2, X3, X4, rev1A_in_gga(X2, X3, X4))
REV1A_IN_GGA(X1, .(X2, X3), X4) → REV1A_IN_GGA(X2, X3, X4)
REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → U12_GA(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U13_GA(X1, X2, X3, X4, X5, rev2B_in_gga(X2, X3, X6))
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → REV2B_IN_GGA(X2, X3, X6)
REV2B_IN_GGA(X1, .(X2, X3), X4) → U2_GGA(X1, X2, X3, X4, rev2B_in_gga(X2, X3, X5))
REV2B_IN_GGA(X1, .(X2, X3), X4) → REV2B_IN_GGA(X2, X3, X5)
REV2B_IN_GGA(X1, .(X2, X3), X4) → U3_GGA(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U4_GGA(X1, X2, X3, X4, revD_in_ga(X5, X6))
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → REVD_IN_GA(X5, X6)
REVD_IN_GA(.(X1, X2), .(X3, X4)) → U8_GA(X1, X2, X3, X4, rev1E_in_gga(X1, X2, X3))
REVD_IN_GA(.(X1, X2), .(X3, X4)) → REV1E_IN_GGA(X1, X2, X3)
REV1E_IN_GGA(X1, .(X2, X3), X4) → U7_GGA(X1, X2, X3, X4, rev1E_in_gga(X2, X3, X4))
REV1E_IN_GGA(X1, .(X2, X3), X4) → REV1E_IN_GGA(X2, X3, X4)
REVD_IN_GA(.(X1, X2), .(X3, X4)) → U9_GA(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
U9_GA(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U10_GA(X1, X2, X3, X4, rev2B_in_gga(X1, X2, X4))
U9_GA(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → REV2B_IN_GGA(X1, X2, X4)
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U5_GGA(X1, X2, X3, X4, revcD_in_ga(X5, X6))
U5_GGA(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U6_GGA(X1, X2, X3, X4, revC_in_ga(.(X1, X6), X4))
U5_GGA(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → REVC_IN_GA(.(X1, X6), X4)
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U14_GA(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U15_GA(X1, X2, X3, X4, X5, revD_in_ga(X6, X7))
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → REVD_IN_GA(X6, X7)
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U16_GA(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U16_GA(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U17_GA(X1, X2, X3, X4, X5, revC_in_ga(.(X1, X7), X5))
U16_GA(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → REVC_IN_GA(.(X1, X7), X5)

The TRS R consists of the following rules:

rev1cA_in_gga(X1, [], X1) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3), X4) → U19_gga(X1, X2, X3, X4, rev1cA_in_gga(X2, X3, X4))
U19_gga(X1, X2, X3, X4, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
rev2cB_in_gga(X1, [], []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3), X4) → U20_gga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U20_gga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, X4, revcD_in_ga(X5, X6))
revcD_in_ga([], []) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2), .(X3, X4)) → U28_ga(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
rev1cE_in_gga(X1, [], X1) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3), X4) → U27_gga(X1, X2, X3, X4, rev1cE_in_gga(X2, X3, X4))
U27_gga(X1, X2, X3, X4, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, X4, rev2cB_in_gga(X1, X2, X4))
U29_ga(X1, X2, X3, X4, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U21_gga(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, X4, revcC_in_ga(.(X1, X6), X4))
revcC_in_ga([], []) → revcC_out_ga([], [])
revcC_in_ga(.(X1, []), .(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3)), .(X4, X5)) → U23_ga(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U23_ga(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U24_ga(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U25_ga(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, X5, revcC_in_ga(.(X1, X7), X5))
U26_ga(X1, X2, X3, X4, X5, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))
U22_gga(X1, X2, X3, X4, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)

The argument filtering Pi contains the following mapping:
revC_in_ga(x1, x2)  =  revC_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
rev1A_in_gga(x1, x2, x3)  =  rev1A_in_gga(x1, x2)
rev1cA_in_gga(x1, x2, x3)  =  rev1cA_in_gga(x1, x2)
[]  =  []
rev1cA_out_gga(x1, x2, x3)  =  rev1cA_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2B_in_gga(x1, x2, x3)  =  rev2B_in_gga(x1, x2)
rev2cB_in_gga(x1, x2, x3)  =  rev2cB_in_gga(x1, x2)
rev2cB_out_gga(x1, x2, x3)  =  rev2cB_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)
revcD_in_ga(x1, x2)  =  revcD_in_ga(x1)
revcD_out_ga(x1, x2)  =  revcD_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1cE_in_gga(x1, x2, x3)  =  rev1cE_in_gga(x1, x2)
rev1cE_out_gga(x1, x2, x3)  =  rev1cE_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)
revcC_in_ga(x1, x2)  =  revcC_in_ga(x1)
revcC_out_ga(x1, x2)  =  revcC_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)
revD_in_ga(x1, x2)  =  revD_in_ga(x1)
rev1E_in_gga(x1, x2, x3)  =  rev1E_in_gga(x1, x2)
REVC_IN_GA(x1, x2)  =  REVC_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5, x6)  =  U11_GA(x1, x2, x3, x6)
REV1A_IN_GGA(x1, x2, x3)  =  REV1A_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)
REV2B_IN_GGA(x1, x2, x3)  =  REV2B_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)
REVD_IN_GA(x1, x2)  =  REVD_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
REV1E_IN_GGA(x1, x2, x3)  =  REV1E_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:

REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → U11_GA(X1, X2, X3, X4, X5, rev1A_in_gga(X2, X3, X4))
REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → REV1A_IN_GGA(X2, X3, X4)
REV1A_IN_GGA(X1, .(X2, X3), X4) → U1_GGA(X1, X2, X3, X4, rev1A_in_gga(X2, X3, X4))
REV1A_IN_GGA(X1, .(X2, X3), X4) → REV1A_IN_GGA(X2, X3, X4)
REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → U12_GA(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U13_GA(X1, X2, X3, X4, X5, rev2B_in_gga(X2, X3, X6))
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → REV2B_IN_GGA(X2, X3, X6)
REV2B_IN_GGA(X1, .(X2, X3), X4) → U2_GGA(X1, X2, X3, X4, rev2B_in_gga(X2, X3, X5))
REV2B_IN_GGA(X1, .(X2, X3), X4) → REV2B_IN_GGA(X2, X3, X5)
REV2B_IN_GGA(X1, .(X2, X3), X4) → U3_GGA(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U4_GGA(X1, X2, X3, X4, revD_in_ga(X5, X6))
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → REVD_IN_GA(X5, X6)
REVD_IN_GA(.(X1, X2), .(X3, X4)) → U8_GA(X1, X2, X3, X4, rev1E_in_gga(X1, X2, X3))
REVD_IN_GA(.(X1, X2), .(X3, X4)) → REV1E_IN_GGA(X1, X2, X3)
REV1E_IN_GGA(X1, .(X2, X3), X4) → U7_GGA(X1, X2, X3, X4, rev1E_in_gga(X2, X3, X4))
REV1E_IN_GGA(X1, .(X2, X3), X4) → REV1E_IN_GGA(X2, X3, X4)
REVD_IN_GA(.(X1, X2), .(X3, X4)) → U9_GA(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
U9_GA(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U10_GA(X1, X2, X3, X4, rev2B_in_gga(X1, X2, X4))
U9_GA(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → REV2B_IN_GGA(X1, X2, X4)
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U5_GGA(X1, X2, X3, X4, revcD_in_ga(X5, X6))
U5_GGA(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U6_GGA(X1, X2, X3, X4, revC_in_ga(.(X1, X6), X4))
U5_GGA(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → REVC_IN_GA(.(X1, X6), X4)
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U14_GA(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U15_GA(X1, X2, X3, X4, X5, revD_in_ga(X6, X7))
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → REVD_IN_GA(X6, X7)
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U16_GA(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U16_GA(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U17_GA(X1, X2, X3, X4, X5, revC_in_ga(.(X1, X7), X5))
U16_GA(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → REVC_IN_GA(.(X1, X7), X5)

The TRS R consists of the following rules:

rev1cA_in_gga(X1, [], X1) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3), X4) → U19_gga(X1, X2, X3, X4, rev1cA_in_gga(X2, X3, X4))
U19_gga(X1, X2, X3, X4, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
rev2cB_in_gga(X1, [], []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3), X4) → U20_gga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U20_gga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, X4, revcD_in_ga(X5, X6))
revcD_in_ga([], []) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2), .(X3, X4)) → U28_ga(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
rev1cE_in_gga(X1, [], X1) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3), X4) → U27_gga(X1, X2, X3, X4, rev1cE_in_gga(X2, X3, X4))
U27_gga(X1, X2, X3, X4, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, X4, rev2cB_in_gga(X1, X2, X4))
U29_ga(X1, X2, X3, X4, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U21_gga(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, X4, revcC_in_ga(.(X1, X6), X4))
revcC_in_ga([], []) → revcC_out_ga([], [])
revcC_in_ga(.(X1, []), .(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3)), .(X4, X5)) → U23_ga(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U23_ga(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U24_ga(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U25_ga(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, X5, revcC_in_ga(.(X1, X7), X5))
U26_ga(X1, X2, X3, X4, X5, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))
U22_gga(X1, X2, X3, X4, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)

The argument filtering Pi contains the following mapping:
revC_in_ga(x1, x2)  =  revC_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
rev1A_in_gga(x1, x2, x3)  =  rev1A_in_gga(x1, x2)
rev1cA_in_gga(x1, x2, x3)  =  rev1cA_in_gga(x1, x2)
[]  =  []
rev1cA_out_gga(x1, x2, x3)  =  rev1cA_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2B_in_gga(x1, x2, x3)  =  rev2B_in_gga(x1, x2)
rev2cB_in_gga(x1, x2, x3)  =  rev2cB_in_gga(x1, x2)
rev2cB_out_gga(x1, x2, x3)  =  rev2cB_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)
revcD_in_ga(x1, x2)  =  revcD_in_ga(x1)
revcD_out_ga(x1, x2)  =  revcD_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1cE_in_gga(x1, x2, x3)  =  rev1cE_in_gga(x1, x2)
rev1cE_out_gga(x1, x2, x3)  =  rev1cE_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)
revcC_in_ga(x1, x2)  =  revcC_in_ga(x1)
revcC_out_ga(x1, x2)  =  revcC_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)
revD_in_ga(x1, x2)  =  revD_in_ga(x1)
rev1E_in_gga(x1, x2, x3)  =  rev1E_in_gga(x1, x2)
REVC_IN_GA(x1, x2)  =  REVC_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5, x6)  =  U11_GA(x1, x2, x3, x6)
REV1A_IN_GGA(x1, x2, x3)  =  REV1A_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)
REV2B_IN_GGA(x1, x2, x3)  =  REV2B_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)
REVD_IN_GA(x1, x2)  =  REVD_IN_GA(x1)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x5)
REV1E_IN_GGA(x1, x2, x3)  =  REV1E_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:

REV1E_IN_GGA(X1, .(X2, X3), X4) → REV1E_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

rev1cA_in_gga(X1, [], X1) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3), X4) → U19_gga(X1, X2, X3, X4, rev1cA_in_gga(X2, X3, X4))
U19_gga(X1, X2, X3, X4, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
rev2cB_in_gga(X1, [], []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3), X4) → U20_gga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U20_gga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, X4, revcD_in_ga(X5, X6))
revcD_in_ga([], []) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2), .(X3, X4)) → U28_ga(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
rev1cE_in_gga(X1, [], X1) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3), X4) → U27_gga(X1, X2, X3, X4, rev1cE_in_gga(X2, X3, X4))
U27_gga(X1, X2, X3, X4, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, X4, rev2cB_in_gga(X1, X2, X4))
U29_ga(X1, X2, X3, X4, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U21_gga(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, X4, revcC_in_ga(.(X1, X6), X4))
revcC_in_ga([], []) → revcC_out_ga([], [])
revcC_in_ga(.(X1, []), .(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3)), .(X4, X5)) → U23_ga(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U23_ga(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U24_ga(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U25_ga(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, X5, revcC_in_ga(.(X1, X7), X5))
U26_ga(X1, X2, X3, X4, X5, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))
U22_gga(X1, X2, X3, X4, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1cA_in_gga(x1, x2, x3)  =  rev1cA_in_gga(x1, x2)
[]  =  []
rev1cA_out_gga(x1, x2, x3)  =  rev1cA_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2cB_in_gga(x1, x2, x3)  =  rev2cB_in_gga(x1, x2)
rev2cB_out_gga(x1, x2, x3)  =  rev2cB_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)
revcD_in_ga(x1, x2)  =  revcD_in_ga(x1)
revcD_out_ga(x1, x2)  =  revcD_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1cE_in_gga(x1, x2, x3)  =  rev1cE_in_gga(x1, x2)
rev1cE_out_gga(x1, x2, x3)  =  rev1cE_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)
revcC_in_ga(x1, x2)  =  revcC_in_ga(x1)
revcC_out_ga(x1, x2)  =  revcC_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)
REV1E_IN_GGA(x1, x2, x3)  =  REV1E_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:

REV1E_IN_GGA(X1, .(X2, X3), X4) → REV1E_IN_GGA(X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV1E_IN_GGA(x1, x2, x3)  =  REV1E_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:

REV1E_IN_GGA(X1, .(X2, X3)) → REV1E_IN_GGA(X2, X3)

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:

• REV1E_IN_GGA(X1, .(X2, X3)) → REV1E_IN_GGA(X2, X3)
  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:

REV1A_IN_GGA(X1, .(X2, X3), X4) → REV1A_IN_GGA(X2, X3, X4)

The TRS R consists of the following rules:

rev1cA_in_gga(X1, [], X1) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3), X4) → U19_gga(X1, X2, X3, X4, rev1cA_in_gga(X2, X3, X4))
U19_gga(X1, X2, X3, X4, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
rev2cB_in_gga(X1, [], []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3), X4) → U20_gga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U20_gga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, X4, revcD_in_ga(X5, X6))
revcD_in_ga([], []) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2), .(X3, X4)) → U28_ga(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
rev1cE_in_gga(X1, [], X1) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3), X4) → U27_gga(X1, X2, X3, X4, rev1cE_in_gga(X2, X3, X4))
U27_gga(X1, X2, X3, X4, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, X4, rev2cB_in_gga(X1, X2, X4))
U29_ga(X1, X2, X3, X4, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U21_gga(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, X4, revcC_in_ga(.(X1, X6), X4))
revcC_in_ga([], []) → revcC_out_ga([], [])
revcC_in_ga(.(X1, []), .(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3)), .(X4, X5)) → U23_ga(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U23_ga(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U24_ga(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U25_ga(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, X5, revcC_in_ga(.(X1, X7), X5))
U26_ga(X1, X2, X3, X4, X5, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))
U22_gga(X1, X2, X3, X4, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1cA_in_gga(x1, x2, x3)  =  rev1cA_in_gga(x1, x2)
[]  =  []
rev1cA_out_gga(x1, x2, x3)  =  rev1cA_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2cB_in_gga(x1, x2, x3)  =  rev2cB_in_gga(x1, x2)
rev2cB_out_gga(x1, x2, x3)  =  rev2cB_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)
revcD_in_ga(x1, x2)  =  revcD_in_ga(x1)
revcD_out_ga(x1, x2)  =  revcD_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1cE_in_gga(x1, x2, x3)  =  rev1cE_in_gga(x1, x2)
rev1cE_out_gga(x1, x2, x3)  =  rev1cE_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)
revcC_in_ga(x1, x2)  =  revcC_in_ga(x1)
revcC_out_ga(x1, x2)  =  revcC_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)
REV1A_IN_GGA(x1, x2, x3)  =  REV1A_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:

REV1A_IN_GGA(X1, .(X2, X3), X4) → REV1A_IN_GGA(X2, X3, X4)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV1A_IN_GGA(x1, x2, x3)  =  REV1A_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:

REV1A_IN_GGA(X1, .(X2, X3)) → REV1A_IN_GGA(X2, X3)

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:

• REV1A_IN_GGA(X1, .(X2, X3)) → REV1A_IN_GGA(X2, X3)
  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:

REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → U12_GA(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → REV2B_IN_GGA(X2, X3, X6)
REV2B_IN_GGA(X1, .(X2, X3), X4) → REV2B_IN_GGA(X2, X3, X5)
REV2B_IN_GGA(X1, .(X2, X3), X4) → U3_GGA(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → REVD_IN_GA(X5, X6)
REVD_IN_GA(.(X1, X2), .(X3, X4)) → U9_GA(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
U9_GA(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → REV2B_IN_GGA(X1, X2, X4)
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U5_GGA(X1, X2, X3, X4, revcD_in_ga(X5, X6))
U5_GGA(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → REVC_IN_GA(.(X1, X6), X4)
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U14_GA(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → REVD_IN_GA(X6, X7)
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U16_GA(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U16_GA(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → REVC_IN_GA(.(X1, X7), X5)

The TRS R consists of the following rules:

rev1cA_in_gga(X1, [], X1) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3), X4) → U19_gga(X1, X2, X3, X4, rev1cA_in_gga(X2, X3, X4))
U19_gga(X1, X2, X3, X4, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
rev2cB_in_gga(X1, [], []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3), X4) → U20_gga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U20_gga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, X4, revcD_in_ga(X5, X6))
revcD_in_ga([], []) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2), .(X3, X4)) → U28_ga(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
rev1cE_in_gga(X1, [], X1) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3), X4) → U27_gga(X1, X2, X3, X4, rev1cE_in_gga(X2, X3, X4))
U27_gga(X1, X2, X3, X4, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, X4, rev2cB_in_gga(X1, X2, X4))
U29_ga(X1, X2, X3, X4, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U21_gga(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, X4, revcC_in_ga(.(X1, X6), X4))
revcC_in_ga([], []) → revcC_out_ga([], [])
revcC_in_ga(.(X1, []), .(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3)), .(X4, X5)) → U23_ga(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U23_ga(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U24_ga(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U25_ga(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, X5, revcC_in_ga(.(X1, X7), X5))
U26_ga(X1, X2, X3, X4, X5, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))
U22_gga(X1, X2, X3, X4, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1cA_in_gga(x1, x2, x3)  =  rev1cA_in_gga(x1, x2)
[]  =  []
rev1cA_out_gga(x1, x2, x3)  =  rev1cA_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2cB_in_gga(x1, x2, x3)  =  rev2cB_in_gga(x1, x2)
rev2cB_out_gga(x1, x2, x3)  =  rev2cB_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)
revcD_in_ga(x1, x2)  =  revcD_in_ga(x1)
revcD_out_ga(x1, x2)  =  revcD_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1cE_in_gga(x1, x2, x3)  =  rev1cE_in_gga(x1, x2)
rev1cE_out_gga(x1, x2, x3)  =  rev1cE_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)
revcC_in_ga(x1, x2)  =  revcC_in_ga(x1)
revcC_out_ga(x1, x2)  =  revcC_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)
REVC_IN_GA(x1, x2)  =  REVC_IN_GA(x1)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x1, x2, x3, x6)
REV2B_IN_GGA(x1, x2, x3)  =  REV2B_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
REVD_IN_GA(x1, x2)  =  REVD_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:

REVC_IN_GA(.(X1, .(X2, X3)), .(X4, X5)) → U12_GA(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → REV2B_IN_GGA(X2, X3, X6)
REV2B_IN_GGA(X1, .(X2, X3), X4) → REV2B_IN_GGA(X2, X3, X5)
REV2B_IN_GGA(X1, .(X2, X3), X4) → U3_GGA(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → REVD_IN_GA(X5, X6)
REVD_IN_GA(.(X1, X2), .(X3, X4)) → U9_GA(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
U9_GA(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → REV2B_IN_GGA(X1, X2, X4)
U3_GGA(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U5_GGA(X1, X2, X3, X4, revcD_in_ga(X5, X6))
U5_GGA(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → REVC_IN_GA(.(X1, X6), X4)
U12_GA(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U14_GA(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → REVD_IN_GA(X6, X7)
U14_GA(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U16_GA(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U16_GA(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → REVC_IN_GA(.(X1, X7), X5)

The TRS R consists of the following rules:

rev1cA_in_gga(X1, [], X1) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3), X4) → U19_gga(X1, X2, X3, X4, rev1cA_in_gga(X2, X3, X4))
rev2cB_in_gga(X1, [], []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3), X4) → U20_gga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3, X5))
rev1cE_in_gga(X1, [], X1) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3), X4) → U27_gga(X1, X2, X3, X4, rev1cE_in_gga(X2, X3, X4))
revcD_in_ga([], []) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2), .(X3, X4)) → U28_ga(X1, X2, X3, X4, rev1cE_in_gga(X1, X2, X3))
U19_gga(X1, X2, X3, X4, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
U20_gga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, X4, revcD_in_ga(X5, X6))
U27_gga(X1, X2, X3, X4, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, X3, X4, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, X4, rev2cB_in_gga(X1, X2, X4))
U21_gga(X1, X2, X3, X4, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, X4, revcC_in_ga(.(X1, X6), X4))
U29_ga(X1, X2, X3, X4, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U22_gga(X1, X2, X3, X4, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)
revcC_in_ga(.(X1, []), .(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3)), .(X4, X5)) → U23_ga(X1, X2, X3, X4, X5, rev1cA_in_gga(X2, X3, X4))
U23_ga(X1, X2, X3, X4, X5, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, X5, rev2cB_in_gga(X2, X3, X6))
U24_ga(X1, X2, X3, X4, X5, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, X5, revcD_in_ga(X6, X7))
U25_ga(X1, X2, X3, X4, X5, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, X5, revcC_in_ga(.(X1, X7), X5))
U26_ga(X1, X2, X3, X4, X5, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev1cA_in_gga(x1, x2, x3)  =  rev1cA_in_gga(x1, x2)
[]  =  []
rev1cA_out_gga(x1, x2, x3)  =  rev1cA_out_gga(x1, x2, x3)
U19_gga(x1, x2, x3, x4, x5)  =  U19_gga(x1, x2, x3, x5)
rev2cB_in_gga(x1, x2, x3)  =  rev2cB_in_gga(x1, x2)
rev2cB_out_gga(x1, x2, x3)  =  rev2cB_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)
revcD_in_ga(x1, x2)  =  revcD_in_ga(x1)
revcD_out_ga(x1, x2)  =  revcD_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4, x5)  =  U28_ga(x1, x2, x5)
rev1cE_in_gga(x1, x2, x3)  =  rev1cE_in_gga(x1, x2)
rev1cE_out_gga(x1, x2, x3)  =  rev1cE_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)
revcC_in_ga(x1, x2)  =  revcC_in_ga(x1)
revcC_out_ga(x1, x2)  =  revcC_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)
REVC_IN_GA(x1, x2)  =  REVC_IN_GA(x1)
U12_GA(x1, x2, x3, x4, x5, x6)  =  U12_GA(x1, x2, x3, x6)
REV2B_IN_GGA(x1, x2, x3)  =  REV2B_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)
REVD_IN_GA(x1, x2)  =  REVD_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:

REVC_IN_GA(.(X1, .(X2, X3))) → U12_GA(X1, X2, X3, rev1cA_in_gga(X2, X3))
U12_GA(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → REV2B_IN_GGA(X2, X3)
REV2B_IN_GGA(X1, .(X2, X3)) → REV2B_IN_GGA(X2, X3)
REV2B_IN_GGA(X1, .(X2, X3)) → U3_GGA(X1, X2, X3, rev2cB_in_gga(X2, X3))
U3_GGA(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → REVD_IN_GA(X5)
REVD_IN_GA(.(X1, X2)) → U9_GA(X1, X2, rev1cE_in_gga(X1, X2))
U9_GA(X1, X2, rev1cE_out_gga(X1, X2, X3)) → REV2B_IN_GGA(X1, X2)
U3_GGA(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → U5_GGA(X1, X2, X3, revcD_in_ga(X5))
U5_GGA(X1, X2, X3, revcD_out_ga(X5, X6)) → REVC_IN_GA(.(X1, X6))
U12_GA(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → U14_GA(X1, X2, X3, rev2cB_in_gga(X2, X3))
U14_GA(X1, X2, X3, rev2cB_out_gga(X2, X3, X6)) → REVD_IN_GA(X6)
U14_GA(X1, X2, X3, rev2cB_out_gga(X2, X3, X6)) → U16_GA(X1, X2, X3, revcD_in_ga(X6))
U16_GA(X1, X2, X3, revcD_out_ga(X6, X7)) → REVC_IN_GA(.(X1, X7))

The TRS R consists of the following rules:

rev1cA_in_gga(X1, []) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3)) → U19_gga(X1, X2, X3, rev1cA_in_gga(X2, X3))
rev2cB_in_gga(X1, []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3)) → U20_gga(X1, X2, X3, rev2cB_in_gga(X2, X3))
rev1cE_in_gga(X1, []) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3)) → U27_gga(X1, X2, X3, rev1cE_in_gga(X2, X3))
revcD_in_ga([]) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2)) → U28_ga(X1, X2, rev1cE_in_gga(X1, X2))
U19_gga(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
U20_gga(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, revcD_in_ga(X5))
U27_gga(X1, X2, X3, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, rev2cB_in_gga(X1, X2))
U21_gga(X1, X2, X3, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, revcC_in_ga(.(X1, X6)))
U29_ga(X1, X2, X3, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U22_gga(X1, X2, X3, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)
revcC_in_ga(.(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3))) → U23_ga(X1, X2, X3, rev1cA_in_gga(X2, X3))
U23_ga(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3))
U24_ga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, revcD_in_ga(X6))
U25_ga(X1, X2, X3, X4, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, revcC_in_ga(.(X1, X7)))
U26_ga(X1, X2, X3, X4, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))

The set Q consists of the following terms:

rev1cA_in_gga(x0, x1)
rev2cB_in_gga(x0, x1)
rev1cE_in_gga(x0, x1)
revcD_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)
revcC_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,JAR06].

The following pairs can be oriented strictly and are deleted.

REVC_IN_GA(.(X1, .(X2, X3))) → U12_GA(X1, X2, X3, rev1cA_in_gga(X2, X3))
REV2B_IN_GGA(X1, .(X2, X3)) → REV2B_IN_GGA(X2, X3)
U3_GGA(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → REVD_IN_GA(X5)
U3_GGA(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → U5_GGA(X1, X2, X3, revcD_in_ga(X5))
U14_GA(X1, X2, X3, rev2cB_out_gga(X2, X3, X6)) → REVD_IN_GA(X6)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(REV2B_IN_GGA(x₁, x₂)) = 1 + x₂   
POL(REVC_IN_GA(x₁)) = x₁   
POL(REVD_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(rev1cA_in_gga(x₁, x₂)) = 1   
POL(rev1cA_out_gga(x₁, x₂, x₃)) = 1   
POL(rev1cE_in_gga(x₁, x₂)) = 1   
POL(rev1cE_out_gga(x₁, x₂, x₃)) = 1   
POL(rev2cB_in_gga(x₁, x₂)) = 1 + x₂   
POL(rev2cB_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(revcC_in_ga(x₁)) = x₁   
POL(revcC_out_ga(x₁, x₂)) = x₂   
POL(revcD_in_ga(x₁)) = 1 + x₁   
POL(revcD_out_ga(x₁, x₂)) = 1 + x₂   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

rev1cA_in_gga(X1, []) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3)) → U19_gga(X1, X2, X3, rev1cA_in_gga(X2, X3))
rev2cB_in_gga(X1, []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3)) → U20_gga(X1, X2, X3, rev2cB_in_gga(X2, X3))
rev1cE_in_gga(X1, []) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3)) → U27_gga(X1, X2, X3, rev1cE_in_gga(X2, X3))
revcD_in_ga([]) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2)) → U28_ga(X1, X2, rev1cE_in_gga(X1, X2))
U23_ga(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3))
U20_gga(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, revcD_in_ga(X5))
U28_ga(X1, X2, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, rev2cB_in_gga(X1, X2))
U29_ga(X1, X2, X3, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U21_gga(X1, X2, X3, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, revcC_in_ga(.(X1, X6)))
revcC_in_ga(.(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
U22_gga(X1, X2, X3, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)
revcC_in_ga(.(X1, .(X2, X3))) → U23_ga(X1, X2, X3, rev1cA_in_gga(X2, X3))
U24_ga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, revcD_in_ga(X6))
U25_ga(X1, X2, X3, X4, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, revcC_in_ga(.(X1, X7)))
U26_ga(X1, X2, X3, X4, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))
U19_gga(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
U27_gga(X1, X2, X3, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)

(27) Obligation:

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

U12_GA(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → REV2B_IN_GGA(X2, X3)
REV2B_IN_GGA(X1, .(X2, X3)) → U3_GGA(X1, X2, X3, rev2cB_in_gga(X2, X3))
REVD_IN_GA(.(X1, X2)) → U9_GA(X1, X2, rev1cE_in_gga(X1, X2))
U9_GA(X1, X2, rev1cE_out_gga(X1, X2, X3)) → REV2B_IN_GGA(X1, X2)
U5_GGA(X1, X2, X3, revcD_out_ga(X5, X6)) → REVC_IN_GA(.(X1, X6))
U12_GA(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → U14_GA(X1, X2, X3, rev2cB_in_gga(X2, X3))
U14_GA(X1, X2, X3, rev2cB_out_gga(X2, X3, X6)) → U16_GA(X1, X2, X3, revcD_in_ga(X6))
U16_GA(X1, X2, X3, revcD_out_ga(X6, X7)) → REVC_IN_GA(.(X1, X7))

The TRS R consists of the following rules:

rev1cA_in_gga(X1, []) → rev1cA_out_gga(X1, [], X1)
rev1cA_in_gga(X1, .(X2, X3)) → U19_gga(X1, X2, X3, rev1cA_in_gga(X2, X3))
rev2cB_in_gga(X1, []) → rev2cB_out_gga(X1, [], [])
rev2cB_in_gga(X1, .(X2, X3)) → U20_gga(X1, X2, X3, rev2cB_in_gga(X2, X3))
rev1cE_in_gga(X1, []) → rev1cE_out_gga(X1, [], X1)
rev1cE_in_gga(X1, .(X2, X3)) → U27_gga(X1, X2, X3, rev1cE_in_gga(X2, X3))
revcD_in_ga([]) → revcD_out_ga([], [])
revcD_in_ga(.(X1, X2)) → U28_ga(X1, X2, rev1cE_in_gga(X1, X2))
U19_gga(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → rev1cA_out_gga(X1, .(X2, X3), X4)
U20_gga(X1, X2, X3, rev2cB_out_gga(X2, X3, X5)) → U21_gga(X1, X2, X3, revcD_in_ga(X5))
U27_gga(X1, X2, X3, rev1cE_out_gga(X2, X3, X4)) → rev1cE_out_gga(X1, .(X2, X3), X4)
U28_ga(X1, X2, rev1cE_out_gga(X1, X2, X3)) → U29_ga(X1, X2, X3, rev2cB_in_gga(X1, X2))
U21_gga(X1, X2, X3, revcD_out_ga(X5, X6)) → U22_gga(X1, X2, X3, revcC_in_ga(.(X1, X6)))
U29_ga(X1, X2, X3, rev2cB_out_gga(X1, X2, X4)) → revcD_out_ga(.(X1, X2), .(X3, X4))
U22_gga(X1, X2, X3, revcC_out_ga(.(X1, X6), X4)) → rev2cB_out_gga(X1, .(X2, X3), X4)
revcC_in_ga(.(X1, [])) → revcC_out_ga(.(X1, []), .(X1, []))
revcC_in_ga(.(X1, .(X2, X3))) → U23_ga(X1, X2, X3, rev1cA_in_gga(X2, X3))
U23_ga(X1, X2, X3, rev1cA_out_gga(X2, X3, X4)) → U24_ga(X1, X2, X3, X4, rev2cB_in_gga(X2, X3))
U24_ga(X1, X2, X3, X4, rev2cB_out_gga(X2, X3, X6)) → U25_ga(X1, X2, X3, X4, revcD_in_ga(X6))
U25_ga(X1, X2, X3, X4, revcD_out_ga(X6, X7)) → U26_ga(X1, X2, X3, X4, revcC_in_ga(.(X1, X7)))
U26_ga(X1, X2, X3, X4, revcC_out_ga(.(X1, X7), X5)) → revcC_out_ga(.(X1, .(X2, X3)), .(X4, X5))

The set Q consists of the following terms:

rev1cA_in_gga(x0, x1)
rev2cB_in_gga(x0, x1)
rev1cE_in_gga(x0, x1)
revcD_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)
revcC_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