Left Termination proof of /tmp/tmphBJaix/reverse-fb.pl

Left Termination of the query pattern reverse_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 PiDP

        ↳9 UsableRulesProof (⇔)

        ↳10 PiDP

        ↳11 PiDPToQDPProof (⇐)

        ↳12 QDP

        ↳13 Instantiation (⇔)

        ↳14 QDP

        ↳15 Instantiation (⇔)

        ↳16 QDP

        ↳17 NonTerminationProof (⇔)

        ↳18 NO

    ↳19 PrologToPiTRSProof (⇐)

        ↳20 PiTRS

        ↳21 DependencyPairsProof (⇔)

        ↳22 PiDP

        ↳23 DependencyGraphProof (⇔)

        ↳24 PiDP

        ↳25 UsableRulesProof (⇔)

        ↳26 PiDP

        ↳27 PiDPToQDPProof (⇐)

        ↳28 QDP

        ↳29 Instantiation (⇔)

        ↳30 QDP

        ↳31 Instantiation (⇔)

        ↳32 QDP

        ↳33 NonTerminationProof (⇔)

        ↳34 NO

(0) Obligation:

Clauses:

reverse(X1s, X2s) :- reverse(X1s, [], X2s).
reverse([], Xs, Xs).
reverse(.(X, X1s), X2s, Ys) :- reverse(X1s, .(X, X2s), Ys).

Queries:

reverse(a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: reverse(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: reverse(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: reverse(T7, [], T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: reverse(T7, [], T6), SCOPE: 2, CLAUSE: 1 | reverse(T7, [], T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: reverse(T7, [], T6), SCOPE: 2, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: reverse(T7, [], T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: reverse(T17, .(T18, []), T16)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reverse(T17, .(T18, []), T16), SCOPE: 3, CLAUSE: 1 | reverse(T17, .(T18, []), T16), SCOPE: 3, CLAUSE: 2\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: reverse(T17, .(T18, []), T16), SCOPE: 3, CLAUSE: 1\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: reverse(T17, .(T18, []), T16), SCOPE: 3, CLAUSE: 2\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: reverse(T38, .(T39, .(T40, [])), T37)\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: reverse(T38, .(T39, .(T40, [])), T37), SCOPE: 4, CLAUSE: 1 | reverse(T38, .(T39, .(T40, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: reverse(T38, .(T39, .(T40, [])), T37), SCOPE: 4, CLAUSE: 1\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: reverse(T38, .(T39, .(T40, [])), T37), SCOPE: 4, CLAUSE: 2\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: reverse(T70, .(T71, .(T72, .(T73, []))), T69)\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: reverse(T70, .(T71, .(T72, .(T73, []))), T69), SCOPE: 5, CLAUSE: 1 | reverse(T70, .(T71, .(T72, .(T73, []))), T69), SCOPE: 5, CLAUSE: 2\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: reverse(T70, .(T71, .(T72, .(T73, []))), T69), SCOPE: 5, CLAUSE: 1\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: reverse(T70, .(T71, .(T72, .(T73, []))), T69), SCOPE: 5, CLAUSE: 2\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: reverse(T113, .(T114, .(T115, .(T116, .(T117, [])))), T112)\n\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: reverse(T113, .(T114, .(T115, .(T116, .(T117, [])))), T112), SCOPE: 6, CLAUSE: 1 | reverse(T113, .(T114, .(T115, .(T116, .(T117, [])))), T112), SCOPE: 6, CLAUSE: 2\n\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: reverse(T113, .(T114, .(T115, .(T116, .(T117, [])))), T112), SCOPE: 6, CLAUSE: 1\n\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: reverse(T113, .(T114, .(T115, .(T116, .(T117, [])))), T112), SCOPE: 6, CLAUSE: 2\n\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: reverse(T167, .(T168, .(T169, .(T170, .(T171, .(T172, []))))), T166)\n\nT166 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: reverse(T167, .(T168, .(T169, .(T170, .(T171, .(T172, []))))), T166), SCOPE: 7, CLAUSE: 1 | reverse(T167, .(T168, .(T169, .(T170, .(T171, .(T172, []))))), T166), SCOPE: 7, CLAUSE: 2\n\nT166 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: reverse(T167, .(T168, .(T169, .(T170, .(T171, .(T172, []))))), T166), SCOPE: 7, CLAUSE: 1\n\nT166 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: reverse(T167, .(T168, .(T169, .(T170, .(T171, .(T172, []))))), T166), SCOPE: 7, CLAUSE: 2\n\nT166 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: reverse(T232, .(T233, .(T234, .(T235, .(T236, .(T237, .(T238, [])))))), T231)\n\nT231 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: reverse(T232, .(T233, .(T234, .(T235, .(T236, .(T237, .(T238, [])))))), T231), SCOPE: 8, CLAUSE: 1 | reverse(T232, .(T233, .(T234, .(T235, .(T236, .(T237, .(T238, [])))))), T231), SCOPE: 8, CLAUSE: 2\n\nT231 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: reverse(T232, .(T233, .(T234, .(T235, .(T236, .(T237, .(T238, [])))))), T231), SCOPE: 8, CLAUSE: 1\n\nT231 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: reverse(T232, .(T233, .(T234, .(T235, .(T236, .(T237, .(T238, [])))))), T231), SCOPE: 8, CLAUSE: 2\n\nT231 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: reverse(T308, .(T309, .(T310, .(T311, .(T312, .(T313, .(T314, .(T315, []))))))), T307)\n\nT307 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: reverse(T308, .(T309, .(T310, .(T311, .(T312, .(T313, .(T314, .(T315, []))))))), T307), SCOPE: 9, CLAUSE: 1 | reverse(T308, .(T309, .(T310, .(T311, .(T312, .(T313, .(T314, .(T315, []))))))), T307), SCOPE: 9, CLAUSE: 2\n\nT307 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: reverse(T308, .(T309, .(T310, .(T311, .(T312, .(T313, .(T314, .(T315, []))))))), T307), SCOPE: 9, CLAUSE: 1\n\nT307 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: reverse(T308, .(T309, .(T310, .(T311, .(T312, .(T313, .(T314, .(T315, []))))))), T307), SCOPE: 9, CLAUSE: 2\n\nT307 is ground", 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: reverse(T395, .(T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, [])))))))), T394)\n\nT394 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: reverse(T395, .(T396, T420), T394)\n\nT394 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: reverse(T395, .(T396, T420), T394), SCOPE: 10, CLAUSE: 1 | reverse(T395, .(T396, T420), T394), SCOPE: 10, CLAUSE: 2\n\nT394 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: reverse(T395, .(T396, T420), T394), SCOPE: 10, CLAUSE: 1\n\nT394 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: reverse(T395, .(T396, T420), T394), SCOPE: 10, CLAUSE: 2\n\nT394 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: reverse(T450, .(T451, .(T452, T453)), T449)\n\nT449 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="ONLY EVAL with clause\nreverse(X3, X4) :- reverse(X3, [], X4).\nand substitution\nT1 -> T7\nX3 -> T7\nT2 -> T6\nX4 -> T6\nT5 -> T7", fontsize=14, style = filled, fillcolor = white];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 81 [arrowhead = none ];
81 -> 6;

82 [label="EVAL with clause\nreverse([], X9, X9).\nand substitution\nT7 -> []\nX9 -> []\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 82 [arrowhead = none ];
82 -> 7;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 83 [arrowhead = none ];
83 -> 8;

84 [label="EVAL with clause\nreverse(.(X19, X20), X21, X22) :- reverse(X20, .(X19, X21), X22).\nand substitution\nX19 -> T18\nX20 -> T17\nT7 -> .(T18, T17)\nX21 -> []\nT6 -> T16\nX22 -> T16\nT15 -> T17\nT14 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 10;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 85 [arrowhead = none ];
85 -> 11;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 87 [arrowhead = none ];
87 -> 12;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 88 [arrowhead = none ];
88 -> 13;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 89 [arrowhead = none ];
89 -> 14;

90 [label="EVAL with clause\nreverse([], X33, X33).\nand substitution\nT17 -> []\nT18 -> T25\nX33 -> .(T25, [])\nT16 -> .(T25, [])", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 90 [arrowhead = none ];
90 -> 15;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 91 [arrowhead = none ];
91 -> 16;

92 [label="EVAL with clause\nreverse(.(X43, X44), X45, X46) :- reverse(X44, .(X43, X45), X46).\nand substitution\nX43 -> T39\nX44 -> T38\nT17 -> .(T39, T38)\nT18 -> T40\nX45 -> .(T40, [])\nT16 -> T37\nX46 -> T37\nT35 -> T38\nT34 -> T39\nT36 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 92 [arrowhead = none ];
92 -> 18;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 93 [arrowhead = none ];
93 -> 19;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 95 [arrowhead = none ];
95 -> 20;

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 97 [arrowhead = none ];
97 -> 22;

98 [label="EVAL with clause\nreverse([], X57, X57).\nand substitution\nT38 -> []\nT39 -> T53\nT40 -> T54\nX57 -> .(T53, .(T54, []))\nT37 -> .(T53, .(T54, []))", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 98 [arrowhead = none ];
98 -> 23;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 99 [arrowhead = none ];
99 -> 24;

100 [label="EVAL with clause\nreverse(.(X67, X68), X69, X70) :- reverse(X68, .(X67, X69), X70).\nand substitution\nX67 -> T71\nX68 -> T70\nT38 -> .(T71, T70)\nT39 -> T72\nT40 -> T73\nX69 -> .(T72, .(T73, []))\nT37 -> T69\nX70 -> T69\nT66 -> T70\nT65 -> T71\nT67 -> T72\nT68 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 103 [arrowhead = none ];
103 -> 28;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 104 [arrowhead = none ];
104 -> 29;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 105 [arrowhead = none ];
105 -> 30;

106 [label="EVAL with clause\nreverse([], X81, X81).\nand substitution\nT70 -> []\nT71 -> T92\nT72 -> T93\nT73 -> T94\nX81 -> .(T92, .(T93, .(T94, [])))\nT69 -> .(T92, .(T93, .(T94, [])))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 31;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="EVAL with clause\nreverse(.(X91, X92), X93, X94) :- reverse(X92, .(X91, X93), X94).\nand substitution\nX91 -> T114\nX92 -> T113\nT70 -> .(T114, T113)\nT71 -> T115\nT72 -> T116\nT73 -> T117\nX93 -> .(T115, .(T116, .(T117, [])))\nT69 -> T112\nX94 -> T112\nT108 -> T113\nT107 -> T114\nT109 -> T115\nT110 -> T116\nT111 -> T117", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 108 [arrowhead = none ];
108 -> 34;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 109 [arrowhead = none ];
109 -> 35;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 111 [arrowhead = none ];
111 -> 36;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 112 [arrowhead = none ];
112 -> 37;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 113 [arrowhead = none ];
113 -> 38;

114 [label="EVAL with clause\nreverse([], X105, X105).\nand substitution\nT113 -> []\nT114 -> T142\nT115 -> T143\nT116 -> T144\nT117 -> T145\nX105 -> .(T142, .(T143, .(T144, .(T145, []))))\nT112 -> .(T142, .(T143, .(T144, .(T145, []))))", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 114 [arrowhead = none ];
114 -> 39;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="EVAL with clause\nreverse(.(X115, X116), X117, X118) :- reverse(X116, .(X115, X117), X118).\nand substitution\nX115 -> T168\nX116 -> T167\nT113 -> .(T168, T167)\nT114 -> T169\nT115 -> T170\nT116 -> T171\nT117 -> T172\nX117 -> .(T169, .(T170, .(T171, .(T172, []))))\nT112 -> T166\nX118 -> T166\nT161 -> T167\nT160 -> T168\nT162 -> T169\nT163 -> T170\nT164 -> T171\nT165 -> T172", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 116 [arrowhead = none ];
116 -> 42;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 117 [arrowhead = none ];
117 -> 43;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 118 [arrowhead = none ];
118 -> 41;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 119 [arrowhead = none ];
119 -> 44;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 120 [arrowhead = none ];
120 -> 45;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="EVAL with clause\nreverse([], X129, X129).\nand substitution\nT167 -> []\nT168 -> T203\nT169 -> T204\nT170 -> T205\nT171 -> T206\nT172 -> T207\nX129 -> .(T203, .(T204, .(T205, .(T206, .(T207, [])))))\nT166 -> .(T203, .(T204, .(T205, .(T206, .(T207, [])))))", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 122 [arrowhead = none ];
122 -> 47;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\nreverse(.(X139, X140), X141, X142) :- reverse(X140, .(X139, X141), X142).\nand substitution\nX139 -> T233\nX140 -> T232\nT167 -> .(T233, T232)\nT168 -> T234\nT169 -> T235\nT170 -> T236\nT171 -> T237\nT172 -> T238\nX141 -> .(T234, .(T235, .(T236, .(T237, .(T238, [])))))\nT166 -> T231\nX142 -> T231\nT225 -> T232\nT224 -> T233\nT226 -> T234\nT227 -> T235\nT228 -> T236\nT229 -> T237\nT230 -> T238", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 124 [arrowhead = none ];
124 -> 50;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 125 [arrowhead = none ];
125 -> 51;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 126 [arrowhead = none ];
126 -> 49;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 127 [arrowhead = none ];
127 -> 52;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 128 [arrowhead = none ];
128 -> 53;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 129 [arrowhead = none ];
129 -> 54;

130 [label="EVAL with clause\nreverse([], X153, X153).\nand substitution\nT232 -> []\nT233 -> T275\nT234 -> T276\nT235 -> T277\nT236 -> T278\nT237 -> T279\nT238 -> T280\nX153 -> .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))\nT231 -> .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 130 [arrowhead = none ];
130 -> 55;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 131 [arrowhead = none ];
131 -> 56;

132 [label="EVAL with clause\nreverse(.(X163, X164), X165, X166) :- reverse(X164, .(X163, X165), X166).\nand substitution\nX163 -> T309\nX164 -> T308\nT232 -> .(T309, T308)\nT233 -> T310\nT234 -> T311\nT235 -> T312\nT236 -> T313\nT237 -> T314\nT238 -> T315\nX165 -> .(T310, .(T311, .(T312, .(T313, .(T314, .(T315, []))))))\nT231 -> T307\nX166 -> T307\nT300 -> T308\nT299 -> T309\nT301 -> T310\nT302 -> T311\nT303 -> T312\nT304 -> T313\nT305 -> T314\nT306 -> T315", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 132 [arrowhead = none ];
132 -> 58;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 133 [arrowhead = none ];
133 -> 59;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 134 [arrowhead = none ];
134 -> 57;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 135 [arrowhead = none ];
135 -> 60;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 136 [arrowhead = none ];
136 -> 61;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 137 [arrowhead = none ];
137 -> 62;

138 [label="EVAL with clause\nreverse([], X177, X177).\nand substitution\nT308 -> []\nT309 -> T358\nT310 -> T359\nT311 -> T360\nT312 -> T361\nT313 -> T362\nT314 -> T363\nT315 -> T364\nX177 -> .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))\nT307 -> .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 138 [arrowhead = none ];
138 -> 63;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 139 [arrowhead = none ];
139 -> 64;

140 [label="EVAL with clause\nreverse(.(X187, X188), X189, X190) :- reverse(X188, .(X187, X189), X190).\nand substitution\nX187 -> T396\nX188 -> T395\nT308 -> .(T396, T395)\nT309 -> T397\nT310 -> T398\nT311 -> T399\nT312 -> T400\nT313 -> T401\nT314 -> T402\nT315 -> T403\nX189 -> .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, [])))))))\nT307 -> T394\nX190 -> T394\nT386 -> T395\nT385 -> T396\nT387 -> T397\nT388 -> T398\nT389 -> T399\nT390 -> T400\nT391 -> T401\nT392 -> T402\nT393 -> T403", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 140 [arrowhead = none ];
140 -> 66;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 141 [arrowhead = none ];
141 -> 67;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 142 [arrowhead = none ];
142 -> 65;

143 [label="GENERALIZATION\nT420 <-- .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, [])))))))", fontsize=14, style = filled, fillcolor = violet];
66 -> 143 [arrowhead = none ];
143 -> 68;

144 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 144 [arrowhead = none ];
144 -> 69;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 145 [arrowhead = none ];
145 -> 70;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 146 [arrowhead = none ];
146 -> 71;

147 [label="EVAL with clause\nreverse([], X203, X203).\nand substitution\nT395 -> []\nT396 -> T433\nT420 -> T434\nX203 -> .(T433, T434)\nT394 -> .(T433, T434)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 147 [arrowhead = none ];
147 -> 72;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 148 [arrowhead = none ];
148 -> 73;

149 [label="EVAL with clause\nreverse(.(X213, X214), X215, X216) :- reverse(X214, .(X213, X215), X216).\nand substitution\nX213 -> T451\nX214 -> T450\nT395 -> .(T451, T450)\nT396 -> T452\nT420 -> T453\nX215 -> .(T452, T453)\nT394 -> T449\nX216 -> T449\nT446 -> T450\nT445 -> T451\nT447 -> T452\nT448 -> T453", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 149 [arrowhead = none ];
149 -> 75;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 150 [arrowhead = none ];
150 -> 76;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 151 [arrowhead = none ];
151 -> 74;

152 [label="INSTANCE with matching:\nT395 -> T450\nT396 -> T451\nT420 -> .(T452, T453)\nT394 -> T449", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 152 [arrowhead = none , style=dashed];
152 -> 68[style=dashed];

}

(2) Obligation:

Clauses:

reverse68([], T433, T434, .(T433, T434)).
reverse68(.(T451, T450), T452, T453, T449) :- reverse68(T450, T451, .(T452, T453), T449).
reverse1([], []).
reverse1(.(T25, []), .(T25, [])).
reverse1(.(T54, .(T53, [])), .(T53, .(T54, []))).
reverse1(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))).
reverse1(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))).
reverse1(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))).
reverse1(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))).
reverse1(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))).
reverse1(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) :- reverse68(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394).

Queries:

reverse1(a,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
reverse1_in: (f,b)
reverse68_in: (f,f,b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_AG(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → REVERSE68_IN_AAGG(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → U1_AAGG(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

The argument filtering Pi contains the following mapping:
reverse1_in_ag(x1, x2) = reverse1_in_ag(x2)
[] = []
reverse1_out_ag(x1, x2) = reverse1_out_ag(x1)
.(x1, x2) = .(x2)
U2_ag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ag(x11)
reverse68_in_aagg(x1, x2, x3, x4) = reverse68_in_aagg(x3, x4)
reverse68_out_aagg(x1, x2, x3, x4) = reverse68_out_aagg(x1)
U1_aagg(x1, x2, x3, x4, x5, x6) = U1_aagg(x6)
REVERSE1_IN_AG(x1, x2) = REVERSE1_IN_AG(x2)
U2_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_AG(x11)
REVERSE68_IN_AAGG(x1, x2, x3, x4) = REVERSE68_IN_AAGG(x3, x4)
U1_AAGG(x1, x2, x3, x4, x5, x6) = U1_AAGG(x6)

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

(6) Obligation:

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

REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_AG(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → REVERSE68_IN_AAGG(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → U1_AAGG(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

The argument filtering Pi contains the following mapping:
reverse1_in_ag(x1, x2) = reverse1_in_ag(x2)
[] = []
reverse1_out_ag(x1, x2) = reverse1_out_ag(x1)
.(x1, x2) = .(x2)
U2_ag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ag(x11)
reverse68_in_aagg(x1, x2, x3, x4) = reverse68_in_aagg(x3, x4)
reverse68_out_aagg(x1, x2, x3, x4) = reverse68_out_aagg(x1)
U1_aagg(x1, x2, x3, x4, x5, x6) = U1_aagg(x6)
REVERSE68_IN_AAGG(x1, x2, x3, x4) = REVERSE68_IN_AAGG(x3, x4)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REVERSE68_IN_AAGG(x1, x2, x3, x4) = REVERSE68_IN_AAGG(x3, x4)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

REVERSE68_IN_AAGG(T453, T449) → REVERSE68_IN_AAGG(.(T453), T449)

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

(13) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REVERSE68_IN_AAGG(T453, T449) → REVERSE68_IN_AAGG(.(T453), T449) we obtained the following new rules [LPAR04]:

REVERSE68_IN_AAGG(.(z0), z1) → REVERSE68_IN_AAGG(.(.(z0)), z1)

(14) Obligation:

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

REVERSE68_IN_AAGG(.(z0), z1) → REVERSE68_IN_AAGG(.(.(z0)), z1)

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

(15) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REVERSE68_IN_AAGG(.(z0), z1) → REVERSE68_IN_AAGG(.(.(z0)), z1) we obtained the following new rules [LPAR04]:

REVERSE68_IN_AAGG(.(.(z0)), z1) → REVERSE68_IN_AAGG(.(.(.(z0))), z1)

(16) Obligation:

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

REVERSE68_IN_AAGG(.(.(z0)), z1) → REVERSE68_IN_AAGG(.(.(.(z0))), z1)

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

(17) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = REVERSE68_IN_AAGG(.(.(z0)), z1) evaluates to t =REVERSE68_IN_AAGG(.(.(.(z0))), z1)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [z0 / .(z0)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REVERSE68_IN_AAGG(.(.(z0)), z1) to REVERSE68_IN_AAGG(.(.(.(z0))), z1).

(18) NO

(19) PrologToPiTRSProof (SOUND transformation)

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(20) Obligation:

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

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

(21) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_AG(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → REVERSE68_IN_AAGG(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → U1_AAGG(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

The argument filtering Pi contains the following mapping:
reverse1_in_ag(x1, x2) = reverse1_in_ag(x2)
[] = []
reverse1_out_ag(x1, x2) = reverse1_out_ag(x1, x2)
.(x1, x2) = .(x2)
U2_ag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ag(x10, x11)
reverse68_in_aagg(x1, x2, x3, x4) = reverse68_in_aagg(x3, x4)
reverse68_out_aagg(x1, x2, x3, x4) = reverse68_out_aagg(x1, x3, x4)
U1_aagg(x1, x2, x3, x4, x5, x6) = U1_aagg(x4, x5, x6)
REVERSE1_IN_AG(x1, x2) = REVERSE1_IN_AG(x2)
U2_AG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_AG(x10, x11)
REVERSE68_IN_AAGG(x1, x2, x3, x4) = REVERSE68_IN_AAGG(x3, x4)
U1_AAGG(x1, x2, x3, x4, x5, x6) = U1_AAGG(x4, x5, x6)

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

(22) Obligation:

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

REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_AG(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
REVERSE1_IN_AG(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → REVERSE68_IN_AAGG(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → U1_AAGG(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(24) Obligation:

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

REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

The TRS R consists of the following rules:

reverse1_in_ag([], []) → reverse1_out_ag([], [])
reverse1_in_ag(.(T25, []), .(T25, [])) → reverse1_out_ag(.(T25, []), .(T25, []))
reverse1_in_ag(.(T54, .(T53, [])), .(T53, .(T54, []))) → reverse1_out_ag(.(T54, .(T53, [])), .(T53, .(T54, [])))
reverse1_in_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, [])))) → reverse1_out_ag(.(T94, .(T93, .(T92, []))), .(T92, .(T93, .(T94, []))))
reverse1_in_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, []))))) → reverse1_out_ag(.(T145, .(T144, .(T143, .(T142, [])))), .(T142, .(T143, .(T144, .(T145, [])))))
reverse1_in_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, [])))))) → reverse1_out_ag(.(T207, .(T206, .(T205, .(T204, .(T203, []))))), .(T203, .(T204, .(T205, .(T206, .(T207, []))))))
reverse1_in_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, []))))))) → reverse1_out_ag(.(T280, .(T279, .(T278, .(T277, .(T276, .(T275, [])))))), .(T275, .(T276, .(T277, .(T278, .(T279, .(T280, [])))))))
reverse1_in_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, [])))))))) → reverse1_out_ag(.(T364, .(T363, .(T362, .(T361, .(T360, .(T359, .(T358, []))))))), .(T358, .(T359, .(T360, .(T361, .(T362, .(T363, .(T364, []))))))))
reverse1_in_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394) → U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_in_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394))
reverse68_in_aagg([], T433, T434, .(T433, T434)) → reverse68_out_aagg([], T433, T434, .(T433, T434))
reverse68_in_aagg(.(T451, T450), T452, T453, T449) → U1_aagg(T451, T450, T452, T453, T449, reverse68_in_aagg(T450, T451, .(T452, T453), T449))
U1_aagg(T451, T450, T452, T453, T449, reverse68_out_aagg(T450, T451, .(T452, T453), T449)) → reverse68_out_aagg(.(T451, T450), T452, T453, T449)
U2_ag(T403, T402, T401, T400, T399, T398, T397, T396, T395, T394, reverse68_out_aagg(T395, T396, .(T397, .(T398, .(T399, .(T400, .(T401, .(T402, .(T403, []))))))), T394)) → reverse1_out_ag(.(T403, .(T402, .(T401, .(T400, .(T399, .(T398, .(T397, .(T396, T395)))))))), T394)

The argument filtering Pi contains the following mapping:
reverse1_in_ag(x1, x2) = reverse1_in_ag(x2)
[] = []
reverse1_out_ag(x1, x2) = reverse1_out_ag(x1, x2)
.(x1, x2) = .(x2)
U2_ag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ag(x10, x11)
reverse68_in_aagg(x1, x2, x3, x4) = reverse68_in_aagg(x3, x4)
reverse68_out_aagg(x1, x2, x3, x4) = reverse68_out_aagg(x1, x3, x4)
U1_aagg(x1, x2, x3, x4, x5, x6) = U1_aagg(x4, x5, x6)
REVERSE68_IN_AAGG(x1, x2, x3, x4) = REVERSE68_IN_AAGG(x3, x4)

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

(25) UsableRulesProof (EQUIVALENT transformation)

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

(26) Obligation:

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

REVERSE68_IN_AAGG(.(T451, T450), T452, T453, T449) → REVERSE68_IN_AAGG(T450, T451, .(T452, T453), T449)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REVERSE68_IN_AAGG(x1, x2, x3, x4) = REVERSE68_IN_AAGG(x3, x4)

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

(27) PiDPToQDPProof (SOUND transformation)

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

(28) Obligation:

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

REVERSE68_IN_AAGG(T453, T449) → REVERSE68_IN_AAGG(.(T453), T449)

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

(29) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REVERSE68_IN_AAGG(T453, T449) → REVERSE68_IN_AAGG(.(T453), T449) we obtained the following new rules [LPAR04]:

REVERSE68_IN_AAGG(.(z0), z1) → REVERSE68_IN_AAGG(.(.(z0)), z1)

(30) Obligation:

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

REVERSE68_IN_AAGG(.(z0), z1) → REVERSE68_IN_AAGG(.(.(z0)), z1)

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

(31) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule REVERSE68_IN_AAGG(.(z0), z1) → REVERSE68_IN_AAGG(.(.(z0)), z1) we obtained the following new rules [LPAR04]:

REVERSE68_IN_AAGG(.(.(z0)), z1) → REVERSE68_IN_AAGG(.(.(.(z0))), z1)

(32) Obligation:

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

REVERSE68_IN_AAGG(.(.(z0)), z1) → REVERSE68_IN_AAGG(.(.(.(z0))), z1)

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

(33) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [z0 / .(z0)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REVERSE68_IN_AAGG(.(.(z0)), z1) to REVERSE68_IN_AAGG(.(.(.(z0))), z1).

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) Instantiation (EQUIVALENT transformation)

(14) Obligation:

(15) Instantiation (EQUIVALENT transformation)

(16) Obligation:

(17) NonTerminationProof (EQUIVALENT transformation)

(18) NO

(19) PrologToPiTRSProof (SOUND transformation)

(20) Obligation:

(21) DependencyPairsProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) Obligation:

(25) UsableRulesProof (EQUIVALENT transformation)

(26) Obligation:

(27) PiDPToQDPProof (SOUND transformation)

(28) Obligation:

(29) Instantiation (EQUIVALENT transformation)

(30) Obligation:

(31) Instantiation (EQUIVALENT transformation)

(32) Obligation:

(33) NonTerminationProof (EQUIVALENT transformation)

(34) NO