Left Termination proof of /tmp/tmpnwm98z/reverse.pl

Left Termination of the query pattern reverse_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

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 QDPSizeChangeProof (⇔)

↳14 YES

(0) Obligation:

Clauses:

reverse(L, LR) :- revacc(L, LR, []).
revacc([], L, L).
revacc(.(EL, T), R, A) :- revacc(T, R, .(EL, A)).

Queries:

reverse(g,a).

(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\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: reverse(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: revacc(T5, T7, [])\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: revacc(T5, T7, []), SCOPE: 2, CLAUSE: 1 | revacc(T5, T7, []), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: revacc(T5, T7, []), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: revacc(T5, T7, []), SCOPE: 2, CLAUSE: 2\n\nT5 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: revacc(T20, T22, .(T19, []))\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 1 | revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 1\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: revacc(T20, T22, .(T19, [])), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nT20 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: revacc(T46, T49, .(T45, .(T48, [])))\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 1 | revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 2\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 1\n\nT45 is ground\nT46 is ground\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: revacc(T46, T49, .(T45, .(T48, []))), SCOPE: 4, CLAUSE: 2\n\nT45 is ground\nT46 is ground\nT48 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: revacc(T82, T86, .(T81, .(T84, .(T85, []))))\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 1 | revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 2\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 1\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: revacc(T82, T86, .(T81, .(T84, .(T85, [])))), SCOPE: 5, CLAUSE: 2\n\nT81 is ground\nT82 is ground\nT84 is ground\nT85 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: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, [])))))\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 1 | revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 2\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 1\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: revacc(T128, T133, .(T127, .(T130, .(T131, .(T132, []))))), SCOPE: 6, CLAUSE: 2\n\nT127 is ground\nT128 is ground\nT130 is ground\nT131 is ground\nT132 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: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, []))))))\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 1 | revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 2\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 1\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: revacc(T184, T190, .(T183, .(T186, .(T187, .(T188, .(T189, [])))))), SCOPE: 7, CLAUSE: 2\n\nT183 is ground\nT184 is ground\nT186 is ground\nT187 is ground\nT188 is ground\nT189 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: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, [])))))))\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 1 | revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 2\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 1\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: revacc(T250, T257, .(T249, .(T252, .(T253, .(T254, .(T255, .(T256, []))))))), SCOPE: 8, CLAUSE: 2\n\nT249 is ground\nT250 is ground\nT252 is ground\nT253 is ground\nT254 is ground\nT255 is ground\nT256 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: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, []))))))))\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 1 | revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 2\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 1\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: revacc(T326, T334, .(T325, .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, [])))))))), SCOPE: 9, CLAUSE: 2\n\nT325 is ground\nT326 is ground\nT328 is ground\nT329 is ground\nT330 is ground\nT331 is ground\nT332 is ground\nT333 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: revacc(T412, T421, .(T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))\n\nT411 is ground\nT412 is ground\nT414 is ground\nT415 is ground\nT416 is ground\nT417 is ground\nT418 is ground\nT419 is ground\nT420 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: revacc(T412, T421, .(T411, T440))\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 1 | revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 2\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 1\n\nT411 is ground\nT412 is ground\nT440 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: revacc(T412, T421, .(T411, T440)), SCOPE: 10, CLAUSE: 2\n\nT411 is ground\nT412 is ground\nT440 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: revacc(T473, T477, .(T472, .(T475, T476)))\n\nT472 is ground\nT473 is ground\nT475 is ground\nT476 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) :- revacc(X3, X4, []).\nand substitution\nT1 -> T5\nX3 -> T5\nT2 -> T7\nX4 -> T7\nT6 -> 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\nrevacc([], X9, X9).\nand substitution\nT5 -> []\nT7 -> []\nX9 -> []\nT12 -> []", 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\nrevacc(.(X19, X20), X21, X22) :- revacc(X20, X21, .(X19, X22)).\nand substitution\nX19 -> T19\nX20 -> T20\nT5 -> .(T19, T20)\nT7 -> T22\nX21 -> T22\nX22 -> []\nT21 -> T22", 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\nrevacc([], X33, X33).\nand substitution\nT20 -> []\nT22 -> .(T36, [])\nX33 -> .(T36, [])\nT19 -> T36\nT35 -> .(T36, [])", 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\nrevacc(.(X43, X44), X45, X46) :- revacc(X44, X45, .(X43, X46)).\nand substitution\nX43 -> T45\nX44 -> T46\nT20 -> .(T45, T46)\nT22 -> T49\nX45 -> T49\nT19 -> T48\nX46 -> .(T48, [])\nT47 -> T49", 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\nrevacc([], X57, X57).\nand substitution\nT46 -> []\nT49 -> .(T69, .(T70, []))\nX57 -> .(T69, .(T70, []))\nT45 -> T69\nT48 -> T70\nT68 -> .(T69, .(T70, []))", 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\nrevacc(.(X67, X68), X69, X70) :- revacc(X68, X69, .(X67, X70)).\nand substitution\nX67 -> T81\nX68 -> T82\nT46 -> .(T81, T82)\nT49 -> T86\nX69 -> T86\nT45 -> T84\nT48 -> T85\nX70 -> .(T84, .(T85, []))\nT83 -> T86", 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\nrevacc([], X81, X81).\nand substitution\nT82 -> []\nT86 -> .(T112, .(T113, .(T114, [])))\nX81 -> .(T112, .(T113, .(T114, [])))\nT81 -> T112\nT84 -> T113\nT85 -> T114\nT111 -> .(T112, .(T113, .(T114, [])))", 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\nrevacc(.(X91, X92), X93, X94) :- revacc(X92, X93, .(X91, X94)).\nand substitution\nX91 -> T127\nX92 -> T128\nT82 -> .(T127, T128)\nT86 -> T133\nX93 -> T133\nT81 -> T130\nT84 -> T131\nT85 -> T132\nX94 -> .(T130, .(T131, .(T132, [])))\nT129 -> T133", 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\nrevacc([], X105, X105).\nand substitution\nT128 -> []\nT133 -> .(T165, .(T166, .(T167, .(T168, []))))\nX105 -> .(T165, .(T166, .(T167, .(T168, []))))\nT127 -> T165\nT130 -> T166\nT131 -> T167\nT132 -> T168\nT164 -> .(T165, .(T166, .(T167, .(T168, []))))", 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\nrevacc(.(X115, X116), X117, X118) :- revacc(X116, X117, .(X115, X118)).\nand substitution\nX115 -> T183\nX116 -> T184\nT128 -> .(T183, T184)\nT133 -> T190\nX117 -> T190\nT127 -> T186\nT130 -> T187\nT131 -> T188\nT132 -> T189\nX118 -> .(T186, .(T187, .(T188, .(T189, []))))\nT185 -> T190", 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\nrevacc([], X129, X129).\nand substitution\nT184 -> []\nT190 -> .(T228, .(T229, .(T230, .(T231, .(T232, [])))))\nX129 -> .(T228, .(T229, .(T230, .(T231, .(T232, [])))))\nT183 -> T228\nT186 -> T229\nT187 -> T230\nT188 -> T231\nT189 -> T232\nT227 -> .(T228, .(T229, .(T230, .(T231, .(T232, [])))))", 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\nrevacc(.(X139, X140), X141, X142) :- revacc(X140, X141, .(X139, X142)).\nand substitution\nX139 -> T249\nX140 -> T250\nT184 -> .(T249, T250)\nT190 -> T257\nX141 -> T257\nT183 -> T252\nT186 -> T253\nT187 -> T254\nT188 -> T255\nT189 -> T256\nX142 -> .(T252, .(T253, .(T254, .(T255, .(T256, [])))))\nT251 -> T257", 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\nrevacc([], X153, X153).\nand substitution\nT250 -> []\nT257 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))\nX153 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))\nT249 -> T301\nT252 -> T302\nT253 -> T303\nT254 -> T304\nT255 -> T305\nT256 -> T306\nT300 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))", 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\nrevacc(.(X163, X164), X165, X166) :- revacc(X164, X165, .(X163, X166)).\nand substitution\nX163 -> T325\nX164 -> T326\nT250 -> .(T325, T326)\nT257 -> T334\nX165 -> T334\nT249 -> T328\nT252 -> T329\nT253 -> T330\nT254 -> T331\nT255 -> T332\nT256 -> T333\nX166 -> .(T328, .(T329, .(T330, .(T331, .(T332, .(T333, []))))))\nT327 -> T334", 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\nrevacc([], X177, X177).\nand substitution\nT326 -> []\nT334 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))\nX177 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))\nT325 -> T384\nT328 -> T385\nT329 -> T386\nT330 -> T387\nT331 -> T388\nT332 -> T389\nT333 -> T390\nT383 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))", 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\nrevacc(.(X187, X188), X189, X190) :- revacc(X188, X189, .(X187, X190)).\nand substitution\nX187 -> T411\nX188 -> T412\nT326 -> .(T411, T412)\nT334 -> T421\nX189 -> T421\nT325 -> T414\nT328 -> T415\nT329 -> T416\nT330 -> T417\nT331 -> T418\nT332 -> T419\nT333 -> T420\nX190 -> .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))\nT413 -> T421", 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\nT440 <-- .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))\n\nNew Knowledge:\nT440 is ground", 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\nrevacc([], X203, X203).\nand substitution\nT412 -> []\nT421 -> .(T460, T461)\nX203 -> .(T460, T461)\nT411 -> T460\nT440 -> T461\nT459 -> .(T460, T461)", 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\nrevacc(.(X213, X214), X215, X216) :- revacc(X214, X215, .(X213, X216)).\nand substitution\nX213 -> T472\nX214 -> T473\nT412 -> .(T472, T473)\nT421 -> T477\nX215 -> T477\nT411 -> T475\nT440 -> T476\nX216 -> .(T475, T476)\nT474 -> T477", 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:\nT412 -> T473\nT421 -> T477\nT411 -> T472\nT440 -> .(T475, T476)", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 152 [arrowhead = none , style=dashed];
152 -> 68[style=dashed];

}

(2) Obligation:

Clauses:

revacc68([], .(T460, T461), T460, T461).
revacc68(.(T472, T473), T477, T475, T476) :- revacc68(T473, T477, T472, .(T475, T476)).
reverse1([], []).
reverse1(.(T36, []), .(T36, [])).
reverse1(.(T70, .(T69, [])), .(T69, .(T70, []))).
reverse1(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))).
reverse1(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))).
reverse1(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))).
reverse1(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))).
reverse1(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))).
reverse1(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) :- revacc68(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))).

Queries:

reverse1(g,a).

(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: (b,f)
revacc68_in: (b,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_ga([], []) → reverse1_out_ga([], [])
reverse1_in_ga(.(T36, []), .(T36, [])) → reverse1_out_ga(.(T36, []), .(T36, []))
reverse1_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverse1_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverse1_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverse1_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverse1_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverse1_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverse1_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverse1_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverse1_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverse1_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverse1_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverse1_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverse1_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revacc68_in_gagg([], .(T460, T461), T460, T461) → revacc68_out_gagg([], .(T460, T461), T460, T461)
revacc68_in_gagg(.(T472, T473), T477, T475, T476) → U1_gagg(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
U1_gagg(T472, T473, T477, T475, T476, revacc68_out_gagg(T473, T477, T472, .(T475, T476))) → revacc68_out_gagg(.(T472, T473), T477, T475, T476)
U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverse1_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

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_ga([], []) → reverse1_out_ga([], [])
reverse1_in_ga(.(T36, []), .(T36, [])) → reverse1_out_ga(.(T36, []), .(T36, []))
reverse1_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverse1_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverse1_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverse1_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverse1_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverse1_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverse1_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverse1_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverse1_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverse1_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverse1_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverse1_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverse1_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revacc68_in_gagg([], .(T460, T461), T460, T461) → revacc68_out_gagg([], .(T460, T461), T460, T461)
revacc68_in_gagg(.(T472, T473), T477, T475, T476) → U1_gagg(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
U1_gagg(T472, T473, T477, T475, T476, revacc68_out_gagg(T473, T477, T472, .(T475, T476))) → revacc68_out_gagg(.(T472, T473), T477, T475, T476)
U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverse1_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

(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_GA(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_GA(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
REVERSE1_IN_GA(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → REVACC68_IN_GAGG(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))
REVACC68_IN_GAGG(.(T472, T473), T477, T475, T476) → U1_GAGG(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
REVACC68_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACC68_IN_GAGG(T473, T477, T472, .(T475, T476))

The TRS R consists of the following rules:

reverse1_in_ga([], []) → reverse1_out_ga([], [])
reverse1_in_ga(.(T36, []), .(T36, [])) → reverse1_out_ga(.(T36, []), .(T36, []))
reverse1_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverse1_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverse1_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverse1_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverse1_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverse1_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverse1_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverse1_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverse1_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverse1_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverse1_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverse1_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverse1_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revacc68_in_gagg([], .(T460, T461), T460, T461) → revacc68_out_gagg([], .(T460, T461), T460, T461)
revacc68_in_gagg(.(T472, T473), T477, T475, T476) → U1_gagg(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
U1_gagg(T472, T473, T477, T475, T476, revacc68_out_gagg(T473, T477, T472, .(T475, T476))) → revacc68_out_gagg(.(T472, T473), T477, T475, T476)
U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverse1_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

The argument filtering Pi contains the following mapping:
reverse1_in_ga(x1, x2) = reverse1_in_ga(x1)
[] = []
reverse1_out_ga(x1, x2) = reverse1_out_ga(x2)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ga(x11)
revacc68_in_gagg(x1, x2, x3, x4) = revacc68_in_gagg(x1, x3, x4)
revacc68_out_gagg(x1, x2, x3, x4) = revacc68_out_gagg(x2)
U1_gagg(x1, x2, x3, x4, x5, x6) = U1_gagg(x6)
REVERSE1_IN_GA(x1, x2) = REVERSE1_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_GA(x11)
REVACC68_IN_GAGG(x1, x2, x3, x4) = REVACC68_IN_GAGG(x1, x3, x4)
U1_GAGG(x1, x2, x3, x4, x5, x6) = U1_GAGG(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_GA(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_GA(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
REVERSE1_IN_GA(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → REVACC68_IN_GAGG(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))
REVACC68_IN_GAGG(.(T472, T473), T477, T475, T476) → U1_GAGG(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
REVACC68_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACC68_IN_GAGG(T473, T477, T472, .(T475, T476))

The TRS R consists of the following rules:

reverse1_in_ga([], []) → reverse1_out_ga([], [])
reverse1_in_ga(.(T36, []), .(T36, [])) → reverse1_out_ga(.(T36, []), .(T36, []))
reverse1_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverse1_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverse1_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverse1_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverse1_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverse1_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverse1_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverse1_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverse1_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverse1_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverse1_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverse1_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverse1_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revacc68_in_gagg([], .(T460, T461), T460, T461) → revacc68_out_gagg([], .(T460, T461), T460, T461)
revacc68_in_gagg(.(T472, T473), T477, T475, T476) → U1_gagg(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
U1_gagg(T472, T473, T477, T475, T476, revacc68_out_gagg(T473, T477, T472, .(T475, T476))) → revacc68_out_gagg(.(T472, T473), T477, T475, T476)
U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverse1_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

(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:

REVACC68_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACC68_IN_GAGG(T473, T477, T472, .(T475, T476))

The TRS R consists of the following rules:

reverse1_in_ga([], []) → reverse1_out_ga([], [])
reverse1_in_ga(.(T36, []), .(T36, [])) → reverse1_out_ga(.(T36, []), .(T36, []))
reverse1_in_ga(.(T70, .(T69, [])), .(T69, .(T70, []))) → reverse1_out_ga(.(T70, .(T69, [])), .(T69, .(T70, [])))
reverse1_in_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, [])))) → reverse1_out_ga(.(T114, .(T113, .(T112, []))), .(T112, .(T113, .(T114, []))))
reverse1_in_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, []))))) → reverse1_out_ga(.(T168, .(T167, .(T166, .(T165, [])))), .(T165, .(T166, .(T167, .(T168, [])))))
reverse1_in_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, [])))))) → reverse1_out_ga(.(T232, .(T231, .(T230, .(T229, .(T228, []))))), .(T228, .(T229, .(T230, .(T231, .(T232, []))))))
reverse1_in_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, []))))))) → reverse1_out_ga(.(T306, .(T305, .(T304, .(T303, .(T302, .(T301, [])))))), .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))))
reverse1_in_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, [])))))))) → reverse1_out_ga(.(T390, .(T389, .(T388, .(T387, .(T386, .(T385, .(T384, []))))))), .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))))
reverse1_in_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421) → U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_in_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, [])))))))))
revacc68_in_gagg([], .(T460, T461), T460, T461) → revacc68_out_gagg([], .(T460, T461), T460, T461)
revacc68_in_gagg(.(T472, T473), T477, T475, T476) → U1_gagg(T472, T473, T477, T475, T476, revacc68_in_gagg(T473, T477, T472, .(T475, T476)))
U1_gagg(T472, T473, T477, T475, T476, revacc68_out_gagg(T473, T477, T472, .(T475, T476))) → revacc68_out_gagg(.(T472, T473), T477, T475, T476)
U2_ga(T420, T419, T418, T417, T416, T415, T414, T411, T412, T421, revacc68_out_gagg(T412, T421, T411, .(T414, .(T415, .(T416, .(T417, .(T418, .(T419, .(T420, []))))))))) → reverse1_out_ga(.(T420, .(T419, .(T418, .(T417, .(T416, .(T415, .(T414, .(T411, T412)))))))), T421)

The argument filtering Pi contains the following mapping:
reverse1_in_ga(x1, x2) = reverse1_in_ga(x1)
[] = []
reverse1_out_ga(x1, x2) = reverse1_out_ga(x2)
.(x1, x2) = .(x1, x2)
U2_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_ga(x11)
revacc68_in_gagg(x1, x2, x3, x4) = revacc68_in_gagg(x1, x3, x4)
revacc68_out_gagg(x1, x2, x3, x4) = revacc68_out_gagg(x2)
U1_gagg(x1, x2, x3, x4, x5, x6) = U1_gagg(x6)
REVACC68_IN_GAGG(x1, x2, x3, x4) = REVACC68_IN_GAGG(x1, 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:

REVACC68_IN_GAGG(.(T472, T473), T477, T475, T476) → REVACC68_IN_GAGG(T473, T477, T472, .(T475, T476))

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
REVACC68_IN_GAGG(x1, x2, x3, x4) = REVACC68_IN_GAGG(x1, 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:

REVACC68_IN_GAGG(.(T472, T473), T475, T476) → REVACC68_IN_GAGG(T473, T472, .(T475, T476))

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

(13) 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:

REVACC68_IN_GAGG(.(T472, T473), T475, T476) → REVACC68_IN_GAGG(T473, T472, .(T475, T476))
The graph contains the following edges 1 > 1, 1 > 2

(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) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES