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

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 189 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 14 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 0 ms)

↳8 QDP

↳9 QDPSizeChangeProof (⇔, 0 ms)

↳10 YES

(0) Obligation:

Clauses:

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

Query: reverse(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="B: 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="A: 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 substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT5 -> [],\nT7 -> [],\nX9 -> [],\nT12 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 82 [arrowhead = none ];
82 -> 7;

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

84 [label="EVAL with clause\nrevacc(.(X18, X19), X20, X21) :- revacc(X19, X20, .(X18, X21)).\nand substitutionX18 -> T19,\nX19 -> T20,\nT5 -> .(T19, T20),\nT7 -> T22,\nX20 -> T22,\nX21 -> [],\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 84 [arrowhead = none ];
84 -> 10;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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([], X28, X28).\nand substitutionT20 -> [],\nT22 -> .(T36, []),\nX28 -> .(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 = lightgreen];
13 -> 91 [arrowhead = none ];
91 -> 16;

92 [label="EVAL with clause\nrevacc(.(X37, X38), X39, X40) :- revacc(X38, X39, .(X37, X40)).\nand substitutionX37 -> T45,\nX38 -> T46,\nT20 -> .(T45, T46),\nT22 -> T49,\nX39 -> T49,\nT19 -> T48,\nX40 -> .(T48, []),\nT47 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 92 [arrowhead = none ];
92 -> 18;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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([], X47, X47).\nand substitutionT46 -> [],\nT49 -> .(T69, .(T70, [])),\nX47 -> .(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 = lightgreen];
21 -> 99 [arrowhead = none ];
99 -> 24;

100 [label="EVAL with clause\nrevacc(.(X56, X57), X58, X59) :- revacc(X57, X58, .(X56, X59)).\nand substitutionX56 -> T81,\nX57 -> T82,\nT46 -> .(T81, T82),\nT49 -> T86,\nX58 -> T86,\nT45 -> T84,\nT48 -> T85,\nX59 -> .(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 = lightgreen];
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([], X66, X66).\nand substitutionT82 -> [],\nT86 -> .(T112, .(T113, .(T114, []))),\nX66 -> .(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 = lightgreen];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="EVAL with clause\nrevacc(.(X75, X76), X77, X78) :- revacc(X76, X77, .(X75, X78)).\nand substitutionX75 -> T127,\nX76 -> T128,\nT82 -> .(T127, T128),\nT86 -> T133,\nX77 -> T133,\nT81 -> T130,\nT84 -> T131,\nT85 -> T132,\nX78 -> .(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 = lightgreen];
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([], X85, X85).\nand substitutionT128 -> [],\nT133 -> .(T165, .(T166, .(T167, .(T168, [])))),\nX85 -> .(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 = lightgreen];
37 -> 115 [arrowhead = none ];
115 -> 40;

116 [label="EVAL with clause\nrevacc(.(X94, X95), X96, X97) :- revacc(X95, X96, .(X94, X97)).\nand substitutionX94 -> T183,\nX95 -> T184,\nT128 -> .(T183, T184),\nT133 -> T190,\nX96 -> T190,\nT127 -> T186,\nT130 -> T187,\nT131 -> T188,\nT132 -> T189,\nX97 -> .(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 = lightgreen];
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([], X104, X104).\nand substitutionT184 -> [],\nT190 -> .(T228, .(T229, .(T230, .(T231, .(T232, []))))),\nX104 -> .(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 = lightgreen];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\nrevacc(.(X113, X114), X115, X116) :- revacc(X114, X115, .(X113, X116)).\nand substitutionX113 -> T249,\nX114 -> T250,\nT184 -> .(T249, T250),\nT190 -> T257,\nX115 -> T257,\nT183 -> T252,\nT186 -> T253,\nT187 -> T254,\nT188 -> T255,\nT189 -> T256,\nX116 -> .(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 = lightgreen];
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([], X123, X123).\nand substitutionT250 -> [],\nT257 -> .(T301, .(T302, .(T303, .(T304, .(T305, .(T306, [])))))),\nX123 -> .(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 = lightgreen];
53 -> 131 [arrowhead = none ];
131 -> 56;

132 [label="EVAL with clause\nrevacc(.(X132, X133), X134, X135) :- revacc(X133, X134, .(X132, X135)).\nand substitutionX132 -> T325,\nX133 -> T326,\nT250 -> .(T325, T326),\nT257 -> T334,\nX134 -> T334,\nT249 -> T328,\nT252 -> T329,\nT253 -> T330,\nT254 -> T331,\nT255 -> T332,\nT256 -> T333,\nX135 -> .(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 = lightgreen];
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([], X142, X142).\nand substitutionT326 -> [],\nT334 -> .(T384, .(T385, .(T386, .(T387, .(T388, .(T389, .(T390, []))))))),\nX142 -> .(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 = lightgreen];
61 -> 139 [arrowhead = none ];
139 -> 64;

140 [label="EVAL with clause\nrevacc(.(X151, X152), X153, X154) :- revacc(X152, X153, .(X151, X154)).\nand substitutionX151 -> T411,\nX152 -> T412,\nT326 -> .(T411, T412),\nT334 -> T421,\nX153 -> T421,\nT325 -> T414,\nT328 -> T415,\nT329 -> T416,\nT330 -> T417,\nT331 -> T418,\nT332 -> T419,\nT333 -> T420,\nX154 -> .(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 = lightgreen];
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([], X163, X163).\nand substitutionT412 -> [],\nT421 -> .(T460, T461),\nX163 -> .(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 = lightgreen];
70 -> 148 [arrowhead = none ];
148 -> 73;

149 [label="EVAL with clause\nrevacc(.(X172, X173), X174, X175) :- revacc(X173, X174, .(X172, X175)).\nand substitutionX172 -> T472,\nX173 -> T473,\nT412 -> .(T472, T473),\nT421 -> T477,\nX174 -> T477,\nT411 -> T475,\nT440 -> T476,\nX175 -> .(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 = lightgreen];
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:

Triples:

revaccA(.(X1, X2), X3, X4, X5) :- revaccA(X2, X3, X1, .(X4, X5)).
reverseB(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) :- revaccA(X9, X10, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))).

Clauses:

revacccA([], .(X1, X2), X1, X2).
revacccA(.(X1, X2), X3, X4, X5) :- revacccA(X2, X3, X1, .(X4, X5)).

Afs:

reverseB(x1, x2) = reverseB(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
reverseB_in: (b,f)
revaccA_in: (b,f,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVERSEB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → U2_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revaccA_in_gagg(X9, X10, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))))
REVERSEB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → REVACCA_IN_GAGG(X9, X10, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))))
REVACCA_IN_GAGG(.(X1, X2), X3, X4, X5) → U1_GAGG(X1, X2, X3, X4, X5, revaccA_in_gagg(X2, X3, X1, .(X4, X5)))
REVACCA_IN_GAGG(.(X1, X2), X3, X4, X5) → REVACCA_IN_GAGG(X2, X3, X1, .(X4, X5))

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
revaccA_in_gagg(x1, x2, x3, x4) = revaccA_in_gagg(x1, x3, x4)
[] = []
REVERSEB_IN_GA(x1, x2) = REVERSEB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U2_GA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x11)
REVACCA_IN_GAGG(x1, x2, x3, x4) = REVACCA_IN_GAGG(x1, x3, x4)
U1_GAGG(x1, x2, x3, x4, x5, x6) = U1_GAGG(x1, x2, x4, x5, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

REVERSEB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → U2_GA(X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, revaccA_in_gagg(X9, X10, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, [])))))))))
REVERSEB_IN_GA(.(X1, .(X2, .(X3, .(X4, .(X5, .(X6, .(X7, .(X8, X9)))))))), X10) → REVACCA_IN_GAGG(X9, X10, X8, .(X7, .(X6, .(X5, .(X4, .(X3, .(X2, .(X1, []))))))))
REVACCA_IN_GAGG(.(X1, X2), X3, X4, X5) → U1_GAGG(X1, X2, X3, X4, X5, revaccA_in_gagg(X2, X3, X1, .(X4, X5)))
REVACCA_IN_GAGG(.(X1, X2), X3, X4, X5) → REVACCA_IN_GAGG(X2, X3, X1, .(X4, X5))

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

REVACCA_IN_GAGG(.(X1, X2), X3, X4, X5) → REVACCA_IN_GAGG(X2, X3, X1, .(X4, X5))

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

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

REVACCA_IN_GAGG(.(X1, X2), X4, X5) → REVACCA_IN_GAGG(X2, X1, .(X4, X5))

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

(9) 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:

REVACCA_IN_GAGG(.(X1, X2), X4, X5) → REVACCA_IN_GAGG(X2, X1, .(X4, X5))
The graph contains the following edges 1 > 1, 1 > 2