Left Termination proof of /tmp/tmpzrQq7m/palindrome.pl

Left Termination of the query pattern palindrome_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 PiDP

↳7 PiDPToQDPProof (⇔)

↳8 QDP

↳9 QDPSizeChangeProof (⇔)

↳10 YES

(0) Obligation:

Clauses:

palindrome(Xs) :- reverse(Xs, Xs).
reverse(X1s, X2s) :- reverse3(X1s, [], X2s).
reverse3(.(X, X1s), X2s, Ys) :- reverse3(X1s, .(X, X2s), Ys).
reverse3([], Xs, Xs).

Queries:

palindrome(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: palindrome(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: palindrome(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: reverse(T3, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: reverse(T3, T3), SCOPE: 2, CLAUSE: 1\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: reverse3(T8, [], T8)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: reverse3(T8, [], T8), SCOPE: 3, CLAUSE: 2 | reverse3(T8, [], T8), SCOPE: 3, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: reverse3(T8, [], T8), SCOPE: 3, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: reverse3(T8, [], T8), SCOPE: 3, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: reverse3(T20, .(T19, []), .(T19, T20))\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: reverse3(T20, .(T19, []), .(T19, T20)), SCOPE: 4, CLAUSE: 2 | reverse3(T20, .(T19, []), .(T19, T20)), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reverse3(T20, .(T19, []), .(T19, T20)), SCOPE: 4, CLAUSE: 2\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: reverse3(T20, .(T19, []), .(T19, T20)), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: reverse3(T37, .(T36, .(T38, [])), .(T38, .(T36, T37)))\n\nT36 is ground\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: reverse3(T37, .(T36, .(T38, [])), .(T38, .(T36, T37))), SCOPE: 5, CLAUSE: 2 | reverse3(T37, .(T36, .(T38, [])), .(T38, .(T36, T37))), SCOPE: 5, CLAUSE: 3\n\nT36 is ground\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: reverse3(T37, .(T36, .(T38, [])), .(T38, .(T36, T37))), SCOPE: 5, CLAUSE: 2\n\nT36 is ground\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: reverse3(T37, .(T36, .(T38, [])), .(T38, .(T36, T37))), SCOPE: 5, CLAUSE: 3\n\nT36 is ground\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: reverse3(T60, .(T59, .(T61, .(T62, []))), .(T62, .(T61, .(T59, T60))))\n\nT59 is ground\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: reverse3(T60, .(T59, .(T61, .(T62, []))), .(T62, .(T61, .(T59, T60)))), SCOPE: 6, CLAUSE: 2 | reverse3(T60, .(T59, .(T61, .(T62, []))), .(T62, .(T61, .(T59, T60)))), SCOPE: 6, CLAUSE: 3\n\nT59 is ground\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: reverse3(T60, .(T59, .(T61, .(T62, []))), .(T62, .(T61, .(T59, T60)))), SCOPE: 6, CLAUSE: 2\n\nT59 is ground\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: reverse3(T60, .(T59, .(T61, .(T62, []))), .(T62, .(T61, .(T59, T60)))), SCOPE: 6, CLAUSE: 3\n\nT59 is ground\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: reverse3(T89, .(T88, .(T90, .(T91, .(T92, [])))), .(T92, .(T91, .(T90, .(T88, T89)))))\n\nT88 is ground\nT89 is ground\nT90 is ground\nT91 is ground\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: reverse3(T89, .(T88, .(T90, .(T91, .(T92, [])))), .(T92, .(T91, .(T90, .(T88, T89))))), SCOPE: 7, CLAUSE: 2 | reverse3(T89, .(T88, .(T90, .(T91, .(T92, [])))), .(T92, .(T91, .(T90, .(T88, T89))))), SCOPE: 7, CLAUSE: 3\n\nT88 is ground\nT89 is ground\nT90 is ground\nT91 is ground\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: reverse3(T89, .(T88, .(T90, .(T91, .(T92, [])))), .(T92, .(T91, .(T90, .(T88, T89))))), SCOPE: 7, CLAUSE: 2\n\nT88 is ground\nT89 is ground\nT90 is ground\nT91 is ground\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: reverse3(T89, .(T88, .(T90, .(T91, .(T92, [])))), .(T92, .(T91, .(T90, .(T88, T89))))), SCOPE: 7, CLAUSE: 3\n\nT88 is ground\nT89 is ground\nT90 is ground\nT91 is ground\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: reverse3(T124, .(T123, .(T125, .(T126, .(T127, .(T128, []))))), .(T128, .(T127, .(T126, .(T125, .(T123, T124))))))\n\nT123 is ground\nT124 is ground\nT125 is ground\nT126 is ground\nT127 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: reverse3(T124, .(T123, .(T125, .(T126, .(T127, .(T128, []))))), .(T128, .(T127, .(T126, .(T125, .(T123, T124)))))), SCOPE: 8, CLAUSE: 2 | reverse3(T124, .(T123, .(T125, .(T126, .(T127, .(T128, []))))), .(T128, .(T127, .(T126, .(T125, .(T123, T124)))))), SCOPE: 8, CLAUSE: 3\n\nT123 is ground\nT124 is ground\nT125 is ground\nT126 is ground\nT127 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: reverse3(T124, .(T123, .(T125, .(T126, .(T127, .(T128, []))))), .(T128, .(T127, .(T126, .(T125, .(T123, T124)))))), SCOPE: 8, CLAUSE: 2\n\nT123 is ground\nT124 is ground\nT125 is ground\nT126 is ground\nT127 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: reverse3(T124, .(T123, .(T125, .(T126, .(T127, .(T128, []))))), .(T128, .(T127, .(T126, .(T125, .(T123, T124)))))), SCOPE: 8, CLAUSE: 3\n\nT123 is ground\nT124 is ground\nT125 is ground\nT126 is ground\nT127 is ground\nT128 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: reverse3(T165, .(T164, .(T166, .(T167, .(T168, .(T169, .(T170, [])))))), .(T170, .(T169, .(T168, .(T167, .(T166, .(T164, T165)))))))\n\nT164 is ground\nT165 is ground\nT166 is ground\nT167 is ground\nT168 is ground\nT169 is ground\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: reverse3(T165, .(T164, .(T166, .(T167, .(T168, .(T169, .(T170, [])))))), .(T170, .(T169, .(T168, .(T167, .(T166, .(T164, T165))))))), SCOPE: 9, CLAUSE: 2 | reverse3(T165, .(T164, .(T166, .(T167, .(T168, .(T169, .(T170, [])))))), .(T170, .(T169, .(T168, .(T167, .(T166, .(T164, T165))))))), SCOPE: 9, CLAUSE: 3\n\nT164 is ground\nT165 is ground\nT166 is ground\nT167 is ground\nT168 is ground\nT169 is ground\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: reverse3(T165, .(T164, .(T166, .(T167, .(T168, .(T169, .(T170, [])))))), .(T170, .(T169, .(T168, .(T167, .(T166, .(T164, T165))))))), SCOPE: 9, CLAUSE: 2\n\nT164 is ground\nT165 is ground\nT166 is ground\nT167 is ground\nT168 is ground\nT169 is ground\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: reverse3(T165, .(T164, .(T166, .(T167, .(T168, .(T169, .(T170, [])))))), .(T170, .(T169, .(T168, .(T167, .(T166, .(T164, T165))))))), SCOPE: 9, CLAUSE: 3\n\nT164 is ground\nT165 is ground\nT166 is ground\nT167 is ground\nT168 is ground\nT169 is ground\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: reverse3(T212, .(T211, .(T213, .(T214, .(T215, .(T216, .(T217, .(T218, []))))))), .(T218, .(T217, .(T216, .(T215, .(T214, .(T213, .(T211, T212))))))))\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: reverse3(T212, .(T211, .(T213, .(T214, .(T215, .(T216, .(T217, .(T218, []))))))), .(T218, .(T217, .(T216, .(T215, .(T214, .(T213, .(T211, T212)))))))), SCOPE: 10, CLAUSE: 2 | reverse3(T212, .(T211, .(T213, .(T214, .(T215, .(T216, .(T217, .(T218, []))))))), .(T218, .(T217, .(T216, .(T215, .(T214, .(T213, .(T211, T212)))))))), SCOPE: 10, CLAUSE: 3\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: reverse3(T212, .(T211, .(T213, .(T214, .(T215, .(T216, .(T217, .(T218, []))))))), .(T218, .(T217, .(T216, .(T215, .(T214, .(T213, .(T211, T212)))))))), SCOPE: 10, CLAUSE: 2\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: reverse3(T212, .(T211, .(T213, .(T214, .(T215, .(T216, .(T217, .(T218, []))))))), .(T218, .(T217, .(T216, .(T215, .(T214, .(T213, .(T211, T212)))))))), SCOPE: 10, CLAUSE: 3\n\nT211 is ground\nT212 is ground\nT213 is ground\nT214 is ground\nT215 is ground\nT216 is ground\nT217 is ground\nT218 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: reverse3(T265, .(T264, .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, [])))))))), .(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265)))))))))\n\nT264 is ground\nT265 is ground\nT266 is ground\nT267 is ground\nT268 is ground\nT269 is ground\nT270 is ground\nT271 is ground\nT272 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: reverse3(T265, .(T264, T283), .(T272, T284))\n\nT264 is ground\nT265 is ground\nT272 is ground\nT283 is ground\nT284 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: reverse3(T265, .(T264, T283), .(T272, T284)), SCOPE: 11, CLAUSE: 2 | reverse3(T265, .(T264, T283), .(T272, T284)), SCOPE: 11, CLAUSE: 3\n\nT264 is ground\nT265 is ground\nT272 is ground\nT283 is ground\nT284 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: reverse3(T265, .(T264, T283), .(T272, T284)), SCOPE: 11, CLAUSE: 2\n\nT264 is ground\nT265 is ground\nT272 is ground\nT283 is ground\nT284 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: reverse3(T265, .(T264, T283), .(T272, T284)), SCOPE: 11, CLAUSE: 3\n\nT264 is ground\nT265 is ground\nT272 is ground\nT283 is ground\nT284 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: reverse3(T316, .(T315, .(T317, T318)), .(T319, T320))\n\nT315 is ground\nT316 is ground\nT317 is ground\nT318 is ground\nT319 is ground\nT320 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", 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: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 79 [arrowhead = none ];
79 -> 2;

80 [label="ONLY EVAL with clause\npalindrome(X2) :- reverse(X2, X2).\nand substitution\nT1 -> T3\nX2 -> T3", fontsize=14, style = filled, fillcolor = white];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 81 [arrowhead = none ];
81 -> 4;

82 [label="ONLY EVAL with clause\nreverse(X11, X12) :- reverse3(X11, [], X12).\nand substitution\nT3 -> T8\nX11 -> T8\nX12 -> T8", fontsize=14, style = filled, fillcolor = white];
4 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 85 [arrowhead = none ];
85 -> 8;

86 [label="EVAL with clause\nreverse3(.(X33, X34), X35, X36) :- reverse3(X34, .(X33, X35), X36).\nand substitution\nX33 -> T19\nX34 -> T20\nT8 -> .(T19, T20)\nX35 -> []\nX36 -> .(T19, T20)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 87 [arrowhead = none ];
87 -> 10;

88 [label="EVAL with clause\nreverse3([], X307, X307).\nand substitution\nT8 -> []\nX307 -> []", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 88 [arrowhead = none ];
88 -> 76;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 89 [arrowhead = none ];
89 -> 77;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 90 [arrowhead = none ];
90 -> 11;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 91 [arrowhead = none ];
91 -> 12;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 92 [arrowhead = none ];
92 -> 13;

93 [label="EVAL with clause\nreverse3(.(X61, X62), X63, X64) :- reverse3(X62, .(X61, X63), X64).\nand substitution\nX61 -> T36\nX62 -> T37\nT20 -> .(T36, T37)\nT19 -> T38\nX63 -> .(T38, [])\nX64 -> .(T38, .(T36, T37))", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 93 [arrowhead = none ];
93 -> 14;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 94 [arrowhead = none ];
94 -> 15;

95 [label="EVAL with clause\nreverse3([], X303, X303).\nand substitution\nT20 -> []\nT19 -> T417\nX303 -> .(T417, [])", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 95 [arrowhead = none ];
95 -> 73;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 96 [arrowhead = none ];
96 -> 74;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 97 [arrowhead = none ];
97 -> 16;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 98 [arrowhead = none ];
98 -> 17;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 99 [arrowhead = none ];
99 -> 18;

100 [label="EVAL with clause\nreverse3(.(X89, X90), X91, X92) :- reverse3(X90, .(X89, X91), X92).\nand substitution\nX89 -> T59\nX90 -> T60\nT37 -> .(T59, T60)\nT36 -> T61\nT38 -> T62\nX91 -> .(T61, .(T62, []))\nX92 -> .(T62, .(T61, .(T59, T60)))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 100 [arrowhead = none ];
100 -> 19;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 101 [arrowhead = none ];
101 -> 20;

102 [label="EVAL with clause\nreverse3([], X299, X299).\nand substitution\nT37 -> []\nT36 -> T414\nT38 -> T414\nX299 -> .(T414, .(T414, []))\nT413 -> T414", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 102 [arrowhead = none ];
102 -> 70;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 103 [arrowhead = none ];
103 -> 71;

104 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 104 [arrowhead = none ];
104 -> 21;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 105 [arrowhead = none ];
105 -> 22;

106 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 106 [arrowhead = none ];
106 -> 23;

107 [label="EVAL with clause\nreverse3(.(X117, X118), X119, X120) :- reverse3(X118, .(X117, X119), X120).\nand substitution\nX117 -> T88\nX118 -> T89\nT60 -> .(T88, T89)\nT59 -> T90\nT61 -> T91\nT62 -> T92\nX119 -> .(T90, .(T91, .(T92, [])))\nX120 -> .(T92, .(T91, .(T90, .(T88, T89))))", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 107 [arrowhead = none ];
107 -> 24;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 108 [arrowhead = none ];
108 -> 25;

109 [label="EVAL with clause\nreverse3([], X295, X295).\nand substitution\nT60 -> []\nT59 -> T408\nT61 -> T407\nT62 -> T408\nX295 -> .(T408, .(T407, .(T408, [])))\nT406 -> T408", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 109 [arrowhead = none ];
109 -> 67;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 110 [arrowhead = none ];
110 -> 68;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 111 [arrowhead = none ];
111 -> 26;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 112 [arrowhead = none ];
112 -> 27;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 113 [arrowhead = none ];
113 -> 28;

114 [label="EVAL with clause\nreverse3(.(X145, X146), X147, X148) :- reverse3(X146, .(X145, X147), X148).\nand substitution\nX145 -> T123\nX146 -> T124\nT89 -> .(T123, T124)\nT88 -> T125\nT90 -> T126\nT91 -> T127\nT92 -> T128\nX147 -> .(T125, .(T126, .(T127, .(T128, []))))\nX148 -> .(T128, .(T127, .(T126, .(T125, .(T123, T124)))))", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 114 [arrowhead = none ];
114 -> 29;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 115 [arrowhead = none ];
115 -> 30;

116 [label="EVAL with clause\nreverse3([], X291, X291).\nand substitution\nT89 -> []\nT88 -> T399\nT90 -> T398\nT91 -> T398\nT92 -> T399\nX291 -> .(T399, .(T398, .(T398, .(T399, []))))\nT396 -> T399\nT397 -> T398", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 116 [arrowhead = none ];
116 -> 64;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 117 [arrowhead = none ];
117 -> 65;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 118 [arrowhead = none ];
118 -> 31;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 119 [arrowhead = none ];
119 -> 32;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 120 [arrowhead = none ];
120 -> 33;

121 [label="EVAL with clause\nreverse3(.(X173, X174), X175, X176) :- reverse3(X174, .(X173, X175), X176).\nand substitution\nX173 -> T164\nX174 -> T165\nT124 -> .(T164, T165)\nT123 -> T166\nT125 -> T167\nT126 -> T168\nT127 -> T169\nT128 -> T170\nX175 -> .(T166, .(T167, .(T168, .(T169, .(T170, [])))))\nX176 -> .(T170, .(T169, .(T168, .(T167, .(T166, .(T164, T165))))))", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 121 [arrowhead = none ];
121 -> 34;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 122 [arrowhead = none ];
122 -> 35;

123 [label="EVAL with clause\nreverse3([], X287, X287).\nand substitution\nT124 -> []\nT123 -> T387\nT125 -> T386\nT126 -> T385\nT127 -> T386\nT128 -> T387\nX287 -> .(T387, .(T386, .(T385, .(T386, .(T387, [])))))\nT383 -> T387\nT384 -> T386", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 123 [arrowhead = none ];
123 -> 61;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 124 [arrowhead = none ];
124 -> 62;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 125 [arrowhead = none ];
125 -> 36;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 126 [arrowhead = none ];
126 -> 37;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 127 [arrowhead = none ];
127 -> 38;

128 [label="EVAL with clause\nreverse3(.(X201, X202), X203, X204) :- reverse3(X202, .(X201, X203), X204).\nand substitution\nX201 -> T211\nX202 -> T212\nT165 -> .(T211, T212)\nT164 -> T213\nT166 -> T214\nT167 -> T215\nT168 -> T216\nT169 -> T217\nT170 -> T218\nX203 -> .(T213, .(T214, .(T215, .(T216, .(T217, .(T218, []))))))\nX204 -> .(T218, .(T217, .(T216, .(T215, .(T214, .(T213, .(T211, T212)))))))", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 128 [arrowhead = none ];
128 -> 39;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 129 [arrowhead = none ];
129 -> 40;

130 [label="EVAL with clause\nreverse3([], X283, X283).\nand substitution\nT165 -> []\nT164 -> T372\nT166 -> T371\nT167 -> T370\nT168 -> T370\nT169 -> T371\nT170 -> T372\nX283 -> .(T372, .(T371, .(T370, .(T370, .(T371, .(T372, []))))))\nT367 -> T372\nT368 -> T371\nT369 -> T370", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 130 [arrowhead = none ];
130 -> 58;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 131 [arrowhead = none ];
131 -> 59;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 132 [arrowhead = none ];
132 -> 41;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 133 [arrowhead = none ];
133 -> 42;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 134 [arrowhead = none ];
134 -> 43;

135 [label="EVAL with clause\nreverse3(.(X229, X230), X231, X232) :- reverse3(X230, .(X229, X231), X232).\nand substitution\nX229 -> T264\nX230 -> T265\nT212 -> .(T264, T265)\nT211 -> T266\nT213 -> T267\nT214 -> T268\nT215 -> T269\nT216 -> T270\nT217 -> T271\nT218 -> T272\nX231 -> .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, [])))))))\nX232 -> .(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 135 [arrowhead = none ];
135 -> 44;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 136 [arrowhead = none ];
136 -> 45;

137 [label="EVAL with clause\nreverse3([], X279, X279).\nand substitution\nT212 -> []\nT211 -> T354\nT213 -> T353\nT214 -> T352\nT215 -> T351\nT216 -> T352\nT217 -> T353\nT218 -> T354\nX279 -> .(T354, .(T353, .(T352, .(T351, .(T352, .(T353, .(T354, [])))))))\nT348 -> T354\nT349 -> T353\nT350 -> T352", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 137 [arrowhead = none ];
137 -> 55;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 138 [arrowhead = none ];
138 -> 56;

139 [label="GENERALIZATION\nT283 <-- .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, [])))))))\nT284 <-- .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265)))))))\n\nNew Knowledge:\nT283 is ground\nT284 is ground", fontsize=14, style = filled, fillcolor = violet];
44 -> 139 [arrowhead = none ];
139 -> 46;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 140 [arrowhead = none ];
140 -> 47;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 141 [arrowhead = none ];
141 -> 48;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 142 [arrowhead = none ];
142 -> 49;

143 [label="EVAL with clause\nreverse3(.(X261, X262), X263, X264) :- reverse3(X262, .(X261, X263), X264).\nand substitution\nX261 -> T315\nX262 -> T316\nT265 -> .(T315, T316)\nT264 -> T317\nT283 -> T318\nX263 -> .(T317, T318)\nT272 -> T319\nT284 -> T320\nX264 -> .(T319, T320)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 143 [arrowhead = none ];
143 -> 50;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 144 [arrowhead = none ];
144 -> 51;

145 [label="EVAL with clause\nreverse3([], X275, X275).\nand substitution\nT265 -> []\nT264 -> T332\nT283 -> T333\nX275 -> .(T332, T333)\nT272 -> T332\nT284 -> T333", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 145 [arrowhead = none ];
145 -> 52;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 146 [arrowhead = none ];
146 -> 53;

147 [label="INSTANCE with matching:\nT265 -> T316\nT264 -> T315\nT283 -> .(T317, T318)\nT272 -> T319\nT284 -> T320", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 147 [arrowhead = none , style=dashed];
147 -> 46[style=dashed];

148 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 148 [arrowhead = none ];
148 -> 54;

149 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 149 [arrowhead = none ];
149 -> 57;

150 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 150 [arrowhead = none ];
150 -> 60;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 151 [arrowhead = none ];
151 -> 63;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 152 [arrowhead = none ];
152 -> 66;

153 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 153 [arrowhead = none ];
153 -> 69;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 154 [arrowhead = none ];
154 -> 72;

155 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 155 [arrowhead = none ];
155 -> 75;

156 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 156 [arrowhead = none ];
156 -> 78;

}

(2) Obligation:

Triples:

reverse346(.(T315, T316), T317, T318, T319, T320) :- reverse346(T316, T315, .(T317, T318), T319, T320).
palindrome1(.(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))) :- reverse346(T265, T264, .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, []))))))), T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265)))))))).

Clauses:

reverse3c46(.(T315, T316), T317, T318, T319, T320) :- reverse3c46(T316, T315, .(T317, T318), T319, T320).
reverse3c46([], T332, T333, T332, T333).

Afs:

palindrome1(x1) = palindrome1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
palindrome1_in: (b)
reverse346_in: (b,b,b,b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PALINDROME1_IN_G(.(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))) → U2_G(T272, T271, T270, T269, T268, T267, T266, T264, T265, reverse346_in_ggggg(T265, T264, .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, []))))))), T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265)))))))))
PALINDROME1_IN_G(.(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))) → REVERSE346_IN_GGGGG(T265, T264, .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, []))))))), T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))
REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → U1_GGGGG(T315, T316, T317, T318, T319, T320, reverse346_in_ggggg(T316, T315, .(T317, T318), T319, T320))
REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → REVERSE346_IN_GGGGG(T316, T315, .(T317, T318), T319, T320)

R is empty.
Pi is empty.
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:

PALINDROME1_IN_G(.(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))) → U2_G(T272, T271, T270, T269, T268, T267, T266, T264, T265, reverse346_in_ggggg(T265, T264, .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, []))))))), T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265)))))))))
PALINDROME1_IN_G(.(T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))) → REVERSE346_IN_GGGGG(T265, T264, .(T266, .(T267, .(T268, .(T269, .(T270, .(T271, .(T272, []))))))), T272, .(T271, .(T270, .(T269, .(T268, .(T267, .(T266, .(T264, T265))))))))
REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → U1_GGGGG(T315, T316, T317, T318, T319, T320, reverse346_in_ggggg(T316, T315, .(T317, T318), T319, T320))
REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → REVERSE346_IN_GGGGG(T316, T315, .(T317, T318), T319, T320)

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

(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:

REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → REVERSE346_IN_GGGGG(T316, T315, .(T317, T318), T319, T320)

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

(7) PiDPToQDPProof (EQUIVALENT 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:

REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → REVERSE346_IN_GGGGG(T316, T315, .(T317, T318), T319, T320)

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:

REVERSE346_IN_GGGGG(.(T315, T316), T317, T318, T319, T320) → REVERSE346_IN_GGGGG(T316, T315, .(T317, T318), T319, T320)
The graph contains the following edges 1 > 1, 1 > 2, 4 >= 4, 5 >= 5

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) PiDPToQDPProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPSizeChangeProof (EQUIVALENT transformation)

(10) YES