Left Termination proof of /tmp/tmp4d5381/naive

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 NonTerminationProof (⇔)

        ↳27 NO

(0) Obligation:

Clauses:

app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
app([], Ys, Ys).
reverse(.(X, Xs), Ys) :- ','(reverse(Xs, Zs), app(Zs, .(X, []), Ys)).
reverse([], []).

Queries:

reverse(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: reverse(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: reverse(T1, T2), SCOPE: 1, CLAUSE: 2 | reverse(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(reverse(T9, X7), app(X7, .(T10, []), T8)) | reverse(T1, T8), SCOPE: 1, CLAUSE: 3\n\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: reverse(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT2 is ground\nreverse(T1, T2) !=? reverse(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(reverse(T9, X7), app(X7, .(T10, []), T8)), SCOPE: 2, CLAUSE: 2 | ','(reverse(T9, X7), app(X7, .(T10, []), T8)), SCOPE: 2, CLAUSE: 3 | ?_2 | reverse(T1, T8), SCOPE: 1, CLAUSE: 3\n\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(reverse(T9, X7), app(X7, .(T10, []), T8)), SCOPE: 2, CLAUSE: 2\n\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(reverse(T9, X7), app(X7, .(T10, []), T8)), SCOPE: 2, CLAUSE: 3 | ?_2 | reverse(T1, T8), SCOPE: 1, CLAUSE: 3\n\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(reverse(T21, X32), ','(app(X32, .(T22, []), X33), app(X33, .(T23, []), T8)))\n\nT8 is ground\nX33 is free; \nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: reverse(T21, X32)\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(T26, .(T27, []), X33), app(X33, .(T28, []), T8))\n\nT8 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reverse(T21, X32), SCOPE: 3, CLAUSE: 2 | reverse(T21, X32), SCOPE: 3, CLAUSE: 3\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: reverse(T21, X32), SCOPE: 3, CLAUSE: 2\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: reverse(T21, X32), SCOPE: 3, CLAUSE: 3\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(reverse(T41, X68), app(X68, .(T42, []), X69))\n\nX69 is free\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: reverse(T41, X68)\n\nX68 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(T45, .(T46, []), X69)\n\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T45, .(T46, []), X69), SCOPE: 4, CLAUSE: 0 | app(T45, .(T46, []), X69), SCOPE: 4, CLAUSE: 1\n\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T45, .(T46, []), X69), SCOPE: 4, CLAUSE: 0\n\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T45, .(T46, []), X69), SCOPE: 4, CLAUSE: 1\n\nX69 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(T67, .(T68, []), X112)\n\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T26, .(T27, []), X33)\n\nX33 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T78, .(T79, []), T8)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T78, .(T79, []), T8), SCOPE: 5, CLAUSE: 0 | app(T78, .(T79, []), T8), SCOPE: 5, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(T78, .(T79, []), T8), SCOPE: 5, CLAUSE: 0\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T78, .(T79, []), T8), SCOPE: 5, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app(T107, .(T108, []), T106)\n\nT106 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(reverse(T9, X7), app(X7, .(T10, []), T8)), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_2 | reverse(T1, T8), SCOPE: 1, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app([], .(T116, []), T8)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app([], .(T116, []), T8), SCOPE: 6, CLAUSE: 0 | app([], .(T116, []), T8), SCOPE: 6, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app([], .(T116, []), T8), SCOPE: 6, CLAUSE: 1\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: reverse(T1, T8), SCOPE: 1, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 56 [arrowhead = none ];
56 -> 2;

57 [label="EVAL with clause\nreverse(.(X4, X5), X6) :- ','(reverse(X5, X7), app(X7, .(X4, []), X6)).\nand substitution\nX4 -> T10\nX5 -> T9\nT1 -> .(T10, T9)\nT2 -> T8\nX6 -> T8\nT7 -> T9\nT6 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 57 [arrowhead = none ];
57 -> 3;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 58 [arrowhead = none ];
58 -> 4;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 59 [arrowhead = none ];
59 -> 5;

60 [label="EVAL with clause\nreverse([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 60 [arrowhead = none ];
60 -> 53;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 61 [arrowhead = none ];
61 -> 54;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 62 [arrowhead = none ];
62 -> 6;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 63 [arrowhead = none ];
63 -> 7;

64 [label="EVAL with clause\nreverse(.(X29, X30), X31) :- ','(reverse(X30, X32), app(X32, .(X29, []), X31)).\nand substitution\nX29 -> T22\nX30 -> T21\nT9 -> .(T22, T21)\nX7 -> X33\nX31 -> X33\nT20 -> T21\nT19 -> T22\nT10 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 64 [arrowhead = none ];
64 -> 8;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 65 [arrowhead = none ];
65 -> 9;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 66 [arrowhead = none ];
66 -> 40;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 67 [arrowhead = none ];
67 -> 41;

68 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 68 [arrowhead = none ];
68 -> 10;

69 [label="SPLIT 2\nreplacements:\nX32 -> T26\nT22 -> T27\nT23 -> T28", fontsize=14, style = filled, fillcolor = violet];
8 -> 69 [arrowhead = none ];
69 -> 11;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 70 [arrowhead = none ];
70 -> 12;

71 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 71 [arrowhead = none ];
71 -> 30;

72 [label="SPLIT 2\nreplacements:\nX33 -> T78\nT28 -> T79", fontsize=14, style = filled, fillcolor = violet];
11 -> 72 [arrowhead = none ];
72 -> 31;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 73 [arrowhead = none ];
73 -> 13;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 74 [arrowhead = none ];
74 -> 14;

75 [label="EVAL with clause\nreverse(.(X65, X66), X67) :- ','(reverse(X66, X68), app(X68, .(X65, []), X67)).\nand substitution\nX65 -> T42\nX66 -> T41\nT21 -> .(T42, T41)\nX32 -> X69\nX67 -> X69\nT40 -> T41\nT39 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 75 [arrowhead = none ];
75 -> 15;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 76 [arrowhead = none ];
76 -> 16;

77 [label="EVAL with clause\nreverse([], []).\nand substitution\nT21 -> []\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 77 [arrowhead = none ];
77 -> 27;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 78 [arrowhead = none ];
78 -> 28;

79 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 79 [arrowhead = none ];
79 -> 17;

80 [label="SPLIT 2\nreplacements:\nX68 -> T45\nT42 -> T46", fontsize=14, style = filled, fillcolor = violet];
15 -> 80 [arrowhead = none ];
80 -> 18;

81 [label="INSTANCE with matching:\nT21 -> T41\nX32 -> X68", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 81 [arrowhead = none , style=dashed];
81 -> 10[style=dashed];

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 82 [arrowhead = none ];
82 -> 19;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 84 [arrowhead = none ];
84 -> 21;

85 [label="EVAL with clause\napp(.(X108, X109), X110, .(X108, X111)) :- app(X109, X110, X111).\nand substitution\nX108 -> T64\nX109 -> T67\nT45 -> .(T64, T67)\nT46 -> T68\nX110 -> .(T68, [])\nX111 -> X112\nX69 -> .(T64, X112)\nT65 -> T67\nT66 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 85 [arrowhead = none ];
85 -> 22;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 86 [arrowhead = none ];
86 -> 23;

87 [label="EVAL with clause\napp([], X124, X124).\nand substitution\nT45 -> []\nT46 -> T74\nX124 -> .(T74, [])\nX69 -> .(T74, [])", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 87 [arrowhead = none ];
87 -> 24;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 88 [arrowhead = none ];
88 -> 25;

89 [label="INSTANCE with matching:\nT45 -> T67\nT46 -> T68\nX69 -> X112", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 89 [arrowhead = none , style=dashed];
89 -> 18[style=dashed];

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 90 [arrowhead = none ];
90 -> 26;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 91 [arrowhead = none ];
91 -> 29;

92 [label="INSTANCE with matching:\nT45 -> T26\nT46 -> T27\nX69 -> X33", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 92 [arrowhead = none , style=dashed];
92 -> 18[style=dashed];

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 93 [arrowhead = none ];
93 -> 32;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 94 [arrowhead = none ];
94 -> 33;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 95 [arrowhead = none ];
95 -> 34;

96 [label="EVAL with clause\napp(.(X156, X157), X158, .(X156, X159)) :- app(X157, X158, X159).\nand substitution\nX156 -> T103\nX157 -> T107\nT78 -> .(T103, T107)\nT79 -> T108\nX158 -> .(T108, [])\nX159 -> T106\nT8 -> .(T103, T106)\nT104 -> T107\nT105 -> T108", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 96 [arrowhead = none ];
96 -> 35;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 97 [arrowhead = none ];
97 -> 36;

98 [label="EVAL with clause\napp([], X170, X170).\nand substitution\nT78 -> []\nT79 -> T115\nX170 -> .(T115, [])\nT8 -> .(T115, [])", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 98 [arrowhead = none ];
98 -> 37;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 99 [arrowhead = none ];
99 -> 38;

100 [label="INSTANCE with matching:\nT78 -> T107\nT79 -> T108\nT8 -> T106", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 100 [arrowhead = none , style=dashed];
100 -> 31[style=dashed];

101 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 101 [arrowhead = none ];
101 -> 39;

102 [label="EVAL with clause\nreverse([], []).\nand substitution\nT9 -> []\nX7 -> []\nT10 -> T116", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 102 [arrowhead = none ];
102 -> 42;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 103 [arrowhead = none ];
103 -> 43;

104 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 104 [arrowhead = none ];
104 -> 49;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 105 [arrowhead = none ];
105 -> 44;

106 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
44 -> 106 [arrowhead = none ];
106 -> 45;

107 [label="EVAL with clause\napp([], X184, X184).\nand substitution\nT116 -> T121\nX184 -> .(T121, [])\nT8 -> .(T121, [])", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 107 [arrowhead = none ];
107 -> 46;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 108 [arrowhead = none ];
108 -> 47;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 109 [arrowhead = none ];
109 -> 48;

110 [label="EVAL with clause\nreverse([], []).\nand substitution\nT1 -> []\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 110 [arrowhead = none ];
110 -> 50;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 111 [arrowhead = none ];
111 -> 51;

112 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 112 [arrowhead = none ];
112 -> 52;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 113 [arrowhead = none ];
113 -> 55;

}

(2) Obligation:

Triples:

reverse10(.(T42, T41), X69) :- reverse10(T41, X68).
reverse10(.(T46, T41), X69) :- ','(reversec10(T41, T45), app18(T45, T46, X69)).
app18(.(T64, T67), T68, .(T64, X112)) :- app18(T67, T68, X112).
app31(.(T103, T107), T108, .(T103, T106)) :- app31(T107, T108, T106).
reverse1(.(T23, .(T22, T21)), T8) :- reverse10(T21, X32).
reverse1(.(T28, .(T27, T21)), T8) :- ','(reversec10(T21, T26), app18(T26, T27, X33)).
reverse1(.(T79, .(T27, T21)), T8) :- ','(reversec10(T21, T26), ','(appc18(T26, T27, T78), app31(T78, T79, T8))).

Clauses:

reversec10(.(T46, T41), X69) :- ','(reversec10(T41, T45), appc18(T45, T46, X69)).
reversec10([], []).
appc18(.(T64, T67), T68, .(T64, X112)) :- appc18(T67, T68, X112).
appc18([], T74, .(T74, [])).
appc31(.(T103, T107), T108, .(T103, T106)) :- appc31(T107, T108, T106).
appc31([], T115, .(T115, [])).

Afs:

reverse1(x1, x2) = reverse1(x2)

(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:
reverse1_in: (f,b)
reverse10_in: (f,f)
reversec10_in: (f,f)
appc18_in: (b,f,f)
app18_in: (b,f,f)
app31_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REVERSE1_IN_AG(.(T23, .(T22, T21)), T8) → U6_AG(T23, T22, T21, T8, reverse10_in_aa(T21, X32))
REVERSE1_IN_AG(.(T23, .(T22, T21)), T8) → REVERSE10_IN_AA(T21, X32)
REVERSE10_IN_AA(.(T42, T41), X69) → U1_AA(T42, T41, X69, reverse10_in_aa(T41, X68))
REVERSE10_IN_AA(.(T42, T41), X69) → REVERSE10_IN_AA(T41, X68)
REVERSE10_IN_AA(.(T46, T41), X69) → U2_AA(T46, T41, X69, reversec10_in_aa(T41, T45))
U2_AA(T46, T41, X69, reversec10_out_aa(T41, T45)) → U3_AA(T46, T41, X69, app18_in_gaa(T45, T46, X69))
U2_AA(T46, T41, X69, reversec10_out_aa(T41, T45)) → APP18_IN_GAA(T45, T46, X69)
APP18_IN_GAA(.(T64, T67), T68, .(T64, X112)) → U4_GAA(T64, T67, T68, X112, app18_in_gaa(T67, T68, X112))
APP18_IN_GAA(.(T64, T67), T68, .(T64, X112)) → APP18_IN_GAA(T67, T68, X112)
REVERSE1_IN_AG(.(T28, .(T27, T21)), T8) → U7_AG(T28, T27, T21, T8, reversec10_in_aa(T21, T26))
U7_AG(T28, T27, T21, T8, reversec10_out_aa(T21, T26)) → U8_AG(T28, T27, T21, T8, app18_in_gaa(T26, T27, X33))
U7_AG(T28, T27, T21, T8, reversec10_out_aa(T21, T26)) → APP18_IN_GAA(T26, T27, X33)
REVERSE1_IN_AG(.(T79, .(T27, T21)), T8) → U9_AG(T79, T27, T21, T8, reversec10_in_aa(T21, T26))
U9_AG(T79, T27, T21, T8, reversec10_out_aa(T21, T26)) → U10_AG(T79, T27, T21, T8, appc18_in_gaa(T26, T27, T78))
U10_AG(T79, T27, T21, T8, appc18_out_gaa(T26, T27, T78)) → U11_AG(T79, T27, T21, T8, app31_in_gag(T78, T79, T8))
U10_AG(T79, T27, T21, T8, appc18_out_gaa(T26, T27, T78)) → APP31_IN_GAG(T78, T79, T8)
APP31_IN_GAG(.(T103, T107), T108, .(T103, T106)) → U5_GAG(T103, T107, T108, T106, app31_in_gag(T107, T108, T106))
APP31_IN_GAG(.(T103, T107), T108, .(T103, T106)) → APP31_IN_GAG(T107, T108, T106)

The TRS R consists of the following rules:

reversec10_in_aa(.(T46, T41), X69) → U13_aa(T46, T41, X69, reversec10_in_aa(T41, T45))
reversec10_in_aa([], []) → reversec10_out_aa([], [])
U13_aa(T46, T41, X69, reversec10_out_aa(T41, T45)) → U14_aa(T46, T41, X69, appc18_in_gaa(T45, T46, X69))
appc18_in_gaa(.(T64, T67), T68, .(T64, X112)) → U15_gaa(T64, T67, T68, X112, appc18_in_gaa(T67, T68, X112))
appc18_in_gaa([], T74, .(T74, [])) → appc18_out_gaa([], T74, .(T74, []))
U15_gaa(T64, T67, T68, X112, appc18_out_gaa(T67, T68, X112)) → appc18_out_gaa(.(T64, T67), T68, .(T64, X112))
U14_aa(T46, T41, X69, appc18_out_gaa(T45, T46, X69)) → reversec10_out_aa(.(T46, T41), X69)

The argument filtering Pi contains the following mapping:
reverse10_in_aa(x1, x2) = reverse10_in_aa
reversec10_in_aa(x1, x2) = reversec10_in_aa
U13_aa(x1, x2, x3, x4) = U13_aa(x4)
reversec10_out_aa(x1, x2) = reversec10_out_aa(x1, x2)
U14_aa(x1, x2, x3, x4) = U14_aa(x2, x4)
appc18_in_gaa(x1, x2, x3) = appc18_in_gaa(x1)
.(x1, x2) = .(x2)
U15_gaa(x1, x2, x3, x4, x5) = U15_gaa(x2, x5)
[] = []
appc18_out_gaa(x1, x2, x3) = appc18_out_gaa(x1, x3)
app18_in_gaa(x1, x2, x3) = app18_in_gaa(x1)
app31_in_gag(x1, x2, x3) = app31_in_gag(x1, x3)
REVERSE1_IN_AG(x1, x2) = REVERSE1_IN_AG(x2)
U6_AG(x1, x2, x3, x4, x5) = U6_AG(x4, x5)
REVERSE10_IN_AA(x1, x2) = REVERSE10_IN_AA
U1_AA(x1, x2, x3, x4) = U1_AA(x4)
U2_AA(x1, x2, x3, x4) = U2_AA(x4)
U3_AA(x1, x2, x3, x4) = U3_AA(x2, x4)
APP18_IN_GAA(x1, x2, x3) = APP18_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x2, x5)
U7_AG(x1, x2, x3, x4, x5) = U7_AG(x4, x5)
U8_AG(x1, x2, x3, x4, x5) = U8_AG(x3, x4, x5)
U9_AG(x1, x2, x3, x4, x5) = U9_AG(x4, x5)
U10_AG(x1, x2, x3, x4, x5) = U10_AG(x3, x4, x5)
U11_AG(x1, x2, x3, x4, x5) = U11_AG(x3, x4, x5)
APP31_IN_GAG(x1, x2, x3) = APP31_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5) = U5_GAG(x2, x4, x5)

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:

REVERSE1_IN_AG(.(T23, .(T22, T21)), T8) → U6_AG(T23, T22, T21, T8, reverse10_in_aa(T21, X32))
REVERSE1_IN_AG(.(T23, .(T22, T21)), T8) → REVERSE10_IN_AA(T21, X32)
REVERSE10_IN_AA(.(T42, T41), X69) → U1_AA(T42, T41, X69, reverse10_in_aa(T41, X68))
REVERSE10_IN_AA(.(T42, T41), X69) → REVERSE10_IN_AA(T41, X68)
REVERSE10_IN_AA(.(T46, T41), X69) → U2_AA(T46, T41, X69, reversec10_in_aa(T41, T45))
U2_AA(T46, T41, X69, reversec10_out_aa(T41, T45)) → U3_AA(T46, T41, X69, app18_in_gaa(T45, T46, X69))
U2_AA(T46, T41, X69, reversec10_out_aa(T41, T45)) → APP18_IN_GAA(T45, T46, X69)
APP18_IN_GAA(.(T64, T67), T68, .(T64, X112)) → U4_GAA(T64, T67, T68, X112, app18_in_gaa(T67, T68, X112))
APP18_IN_GAA(.(T64, T67), T68, .(T64, X112)) → APP18_IN_GAA(T67, T68, X112)
REVERSE1_IN_AG(.(T28, .(T27, T21)), T8) → U7_AG(T28, T27, T21, T8, reversec10_in_aa(T21, T26))
U7_AG(T28, T27, T21, T8, reversec10_out_aa(T21, T26)) → U8_AG(T28, T27, T21, T8, app18_in_gaa(T26, T27, X33))
U7_AG(T28, T27, T21, T8, reversec10_out_aa(T21, T26)) → APP18_IN_GAA(T26, T27, X33)
REVERSE1_IN_AG(.(T79, .(T27, T21)), T8) → U9_AG(T79, T27, T21, T8, reversec10_in_aa(T21, T26))
U9_AG(T79, T27, T21, T8, reversec10_out_aa(T21, T26)) → U10_AG(T79, T27, T21, T8, appc18_in_gaa(T26, T27, T78))
U10_AG(T79, T27, T21, T8, appc18_out_gaa(T26, T27, T78)) → U11_AG(T79, T27, T21, T8, app31_in_gag(T78, T79, T8))
U10_AG(T79, T27, T21, T8, appc18_out_gaa(T26, T27, T78)) → APP31_IN_GAG(T78, T79, T8)
APP31_IN_GAG(.(T103, T107), T108, .(T103, T106)) → U5_GAG(T103, T107, T108, T106, app31_in_gag(T107, T108, T106))
APP31_IN_GAG(.(T103, T107), T108, .(T103, T106)) → APP31_IN_GAG(T107, T108, T106)

The TRS R consists of the following rules:

reversec10_in_aa(.(T46, T41), X69) → U13_aa(T46, T41, X69, reversec10_in_aa(T41, T45))
reversec10_in_aa([], []) → reversec10_out_aa([], [])
U13_aa(T46, T41, X69, reversec10_out_aa(T41, T45)) → U14_aa(T46, T41, X69, appc18_in_gaa(T45, T46, X69))
appc18_in_gaa(.(T64, T67), T68, .(T64, X112)) → U15_gaa(T64, T67, T68, X112, appc18_in_gaa(T67, T68, X112))
appc18_in_gaa([], T74, .(T74, [])) → appc18_out_gaa([], T74, .(T74, []))
U15_gaa(T64, T67, T68, X112, appc18_out_gaa(T67, T68, X112)) → appc18_out_gaa(.(T64, T67), T68, .(T64, X112))
U14_aa(T46, T41, X69, appc18_out_gaa(T45, T46, X69)) → reversec10_out_aa(.(T46, T41), X69)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APP31_IN_GAG(.(T103, T107), T108, .(T103, T106)) → APP31_IN_GAG(T107, T108, T106)

The TRS R consists of the following rules:

reversec10_in_aa(.(T46, T41), X69) → U13_aa(T46, T41, X69, reversec10_in_aa(T41, T45))
reversec10_in_aa([], []) → reversec10_out_aa([], [])
U13_aa(T46, T41, X69, reversec10_out_aa(T41, T45)) → U14_aa(T46, T41, X69, appc18_in_gaa(T45, T46, X69))
appc18_in_gaa(.(T64, T67), T68, .(T64, X112)) → U15_gaa(T64, T67, T68, X112, appc18_in_gaa(T67, T68, X112))
appc18_in_gaa([], T74, .(T74, [])) → appc18_out_gaa([], T74, .(T74, []))
U15_gaa(T64, T67, T68, X112, appc18_out_gaa(T67, T68, X112)) → appc18_out_gaa(.(T64, T67), T68, .(T64, X112))
U14_aa(T46, T41, X69, appc18_out_gaa(T45, T46, X69)) → reversec10_out_aa(.(T46, T41), X69)

The argument filtering Pi contains the following mapping:
reversec10_in_aa(x1, x2) = reversec10_in_aa
U13_aa(x1, x2, x3, x4) = U13_aa(x4)
reversec10_out_aa(x1, x2) = reversec10_out_aa(x1, x2)
U14_aa(x1, x2, x3, x4) = U14_aa(x2, x4)
appc18_in_gaa(x1, x2, x3) = appc18_in_gaa(x1)
.(x1, x2) = .(x2)
U15_gaa(x1, x2, x3, x4, x5) = U15_gaa(x2, x5)
[] = []
appc18_out_gaa(x1, x2, x3) = appc18_out_gaa(x1, x3)
APP31_IN_GAG(x1, x2, x3) = APP31_IN_GAG(x1, x3)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APP31_IN_GAG(.(T103, T107), T108, .(T103, T106)) → APP31_IN_GAG(T107, T108, T106)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP31_IN_GAG(x1, x2, x3) = APP31_IN_GAG(x1, x3)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APP31_IN_GAG(.(T107), .(T106)) → APP31_IN_GAG(T107, T106)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP31_IN_GAG(.(T107), .(T106)) → APP31_IN_GAG(T107, T106)
The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

APP18_IN_GAA(.(T64, T67), T68, .(T64, X112)) → APP18_IN_GAA(T67, T68, X112)

The TRS R consists of the following rules:

reversec10_in_aa(.(T46, T41), X69) → U13_aa(T46, T41, X69, reversec10_in_aa(T41, T45))
reversec10_in_aa([], []) → reversec10_out_aa([], [])
U13_aa(T46, T41, X69, reversec10_out_aa(T41, T45)) → U14_aa(T46, T41, X69, appc18_in_gaa(T45, T46, X69))
appc18_in_gaa(.(T64, T67), T68, .(T64, X112)) → U15_gaa(T64, T67, T68, X112, appc18_in_gaa(T67, T68, X112))
appc18_in_gaa([], T74, .(T74, [])) → appc18_out_gaa([], T74, .(T74, []))
U15_gaa(T64, T67, T68, X112, appc18_out_gaa(T67, T68, X112)) → appc18_out_gaa(.(T64, T67), T68, .(T64, X112))
U14_aa(T46, T41, X69, appc18_out_gaa(T45, T46, X69)) → reversec10_out_aa(.(T46, T41), X69)

The argument filtering Pi contains the following mapping:
reversec10_in_aa(x1, x2) = reversec10_in_aa
U13_aa(x1, x2, x3, x4) = U13_aa(x4)
reversec10_out_aa(x1, x2) = reversec10_out_aa(x1, x2)
U14_aa(x1, x2, x3, x4) = U14_aa(x2, x4)
appc18_in_gaa(x1, x2, x3) = appc18_in_gaa(x1)
.(x1, x2) = .(x2)
U15_gaa(x1, x2, x3, x4, x5) = U15_gaa(x2, x5)
[] = []
appc18_out_gaa(x1, x2, x3) = appc18_out_gaa(x1, x3)
APP18_IN_GAA(x1, x2, x3) = APP18_IN_GAA(x1)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APP18_IN_GAA(.(T64, T67), T68, .(T64, X112)) → APP18_IN_GAA(T67, T68, X112)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP18_IN_GAA(x1, x2, x3) = APP18_IN_GAA(x1)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APP18_IN_GAA(.(T67)) → APP18_IN_GAA(T67)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

APP18_IN_GAA(.(T67)) → APP18_IN_GAA(T67)
The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

REVERSE10_IN_AA(.(T42, T41), X69) → REVERSE10_IN_AA(T41, X68)

The TRS R consists of the following rules:

reversec10_in_aa(.(T46, T41), X69) → U13_aa(T46, T41, X69, reversec10_in_aa(T41, T45))
reversec10_in_aa([], []) → reversec10_out_aa([], [])
U13_aa(T46, T41, X69, reversec10_out_aa(T41, T45)) → U14_aa(T46, T41, X69, appc18_in_gaa(T45, T46, X69))
appc18_in_gaa(.(T64, T67), T68, .(T64, X112)) → U15_gaa(T64, T67, T68, X112, appc18_in_gaa(T67, T68, X112))
appc18_in_gaa([], T74, .(T74, [])) → appc18_out_gaa([], T74, .(T74, []))
U15_gaa(T64, T67, T68, X112, appc18_out_gaa(T67, T68, X112)) → appc18_out_gaa(.(T64, T67), T68, .(T64, X112))
U14_aa(T46, T41, X69, appc18_out_gaa(T45, T46, X69)) → reversec10_out_aa(.(T46, T41), X69)

The argument filtering Pi contains the following mapping:
reversec10_in_aa(x1, x2) = reversec10_in_aa
U13_aa(x1, x2, x3, x4) = U13_aa(x4)
reversec10_out_aa(x1, x2) = reversec10_out_aa(x1, x2)
U14_aa(x1, x2, x3, x4) = U14_aa(x2, x4)
appc18_in_gaa(x1, x2, x3) = appc18_in_gaa(x1)
.(x1, x2) = .(x2)
U15_gaa(x1, x2, x3, x4, x5) = U15_gaa(x2, x5)
[] = []
appc18_out_gaa(x1, x2, x3) = appc18_out_gaa(x1, x3)
REVERSE10_IN_AA(x1, x2) = REVERSE10_IN_AA

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

REVERSE10_IN_AA(.(T42, T41), X69) → REVERSE10_IN_AA(T41, X68)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REVERSE10_IN_AA(x1, x2) = REVERSE10_IN_AA

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

REVERSE10_IN_AA → REVERSE10_IN_AA

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

(26) NonTerminationProof (EQUIVALENT transformation)

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

s = REVERSE10_IN_AA evaluates to t =REVERSE10_IN_AA

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from REVERSE10_IN_AA to REVERSE10_IN_AA.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) NonTerminationProof (EQUIVALENT transformation)

(27) NO