Left Termination proof of /tmp/tmplMlkR4/naive

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 QDPSizeChangeProof (⇔)

                ↳15 YES

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 QDPSizeChangeProof (⇔)

                ↳29 YES

    ↳30 PrologToPiTRSProof (⇐)

        ↳31 PiTRS

        ↳32 DependencyPairsProof (⇔)

        ↳33 PiDP

        ↳34 DependencyGraphProof (⇔)

        ↳35 AND

            ↳36 PiDP

                ↳37 UsableRulesProof (⇔)

                ↳38 PiDP

                ↳39 PiDPToQDPProof (⇐)

                ↳40 QDP

                ↳41 QDPSizeChangeProof (⇔)

                ↳42 YES

            ↳43 PiDP

                ↳44 UsableRulesProof (⇔)

                ↳45 PiDP

                ↳46 PiDPToQDPProof (⇐)

                ↳47 QDP

                ↳48 NonTerminationProof (⇔)

                ↳49 NO

            ↳50 PiDP

                ↳51 UsableRulesProof (⇔)

                ↳52 PiDP

                ↳53 PiDPToQDPProof (⇐)

                ↳54 QDP

                ↳55 QDPSizeChangeProof (⇔)

                ↳56 YES

(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(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: reverse(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: reverse(T1, T2), SCOPE: 1, CLAUSE: 2 | reverse(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(reverse(T7, X7), app(X7, .(T6, []), T9)) | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: reverse(T1, T2), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nreverse(T1, T2) !=? reverse(.(X4, X5), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2 | ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 3 | ?_2 | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(reverse(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 3 | ?_2 | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(reverse(T19, X32), ','(app(X32, .(T18, []), X33), app(X33, .(T6, []), T9)))\n\nT6 is ground\nT18 is ground\nT19 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(T19, X32)\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(app(T22, .(T18, []), X33), app(X33, .(T6, []), T9))\n\nT6 is ground\nT18 is ground\nT22 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: reverse(T19, X32), SCOPE: 3, CLAUSE: 2 | reverse(T19, X32), SCOPE: 3, CLAUSE: 3\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: reverse(T19, X32), SCOPE: 3, CLAUSE: 2\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: reverse(T19, X32), SCOPE: 3, CLAUSE: 3\n\nT19 is ground\nX32 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(reverse(T34, X68), app(X68, .(T33, []), X69))\n\nT33 is ground\nT34 is ground\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(T34, X68)\n\nT34 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(T37, .(T33, []), X69)\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 0 | app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 0\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T37, .(T33, []), X69), SCOPE: 4, CLAUSE: 1\n\nT33 is ground\nT37 is ground\nX69 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(T56, .(T57, []), X112)\n\nT56 is ground\nT57 is ground\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(T22, .(T18, []), X33)\n\nT18 is ground\nT22 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T67, .(T6, []), T9)\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 0 | app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 1\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 0\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T67, .(T6, []), T9), SCOPE: 5, CLAUSE: 1\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app(T92, .(T93, []), T95)\n\nT92 is ground\nT93 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(T7, X7), app(X7, .(T6, []), T9)), SCOPE: 2, CLAUSE: 3\n\nT6 is ground\nT7 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ?_2 | reverse(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: app([], .(T6, []), T9)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app([], .(T6, []), T9), SCOPE: 6, CLAUSE: 0 | app([], .(T6, []), T9), SCOPE: 6, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app([], .(T6, []), T9), SCOPE: 6, CLAUSE: 1\n\nT6 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(.(T6, T7), T2), SCOPE: 1, CLAUSE: 3\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 54 [arrowhead = none ];
54 -> 2;

55 [label="EVAL with clause\nreverse(.(X4, X5), X6) :- ','(reverse(X5, X7), app(X7, .(X4, []), X6)).\nand substitution\nX4 -> T6\nX5 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX6 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 55 [arrowhead = none ];
55 -> 3;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 56 [arrowhead = none ];
56 -> 4;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 57 [arrowhead = none ];
57 -> 5;

58 [label="EVAL with clause\nreverse([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 58 [arrowhead = none ];
58 -> 51;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 59 [arrowhead = none ];
59 -> 52;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 60 [arrowhead = none ];
60 -> 6;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 61 [arrowhead = none ];
61 -> 7;

62 [label="EVAL with clause\nreverse(.(X29, X30), X31) :- ','(reverse(X30, X32), app(X32, .(X29, []), X31)).\nand substitution\nX29 -> T18\nX30 -> T19\nT7 -> .(T18, T19)\nX7 -> X33\nX31 -> X33", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 62 [arrowhead = none ];
62 -> 8;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 64 [arrowhead = none ];
64 -> 40;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 65 [arrowhead = none ];
65 -> 41;

66 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 66 [arrowhead = none ];
66 -> 10;

67 [label="SPLIT 2\nnew knowledge:\nT22 is ground\nreplacements:\nX32 -> T22", fontsize=14, style = filled, fillcolor = violet];
8 -> 67 [arrowhead = none ];
67 -> 11;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 68 [arrowhead = none ];
68 -> 12;

69 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 69 [arrowhead = none ];
69 -> 30;

70 [label="SPLIT 2\nnew knowledge:\nT67 is ground\nreplacements:\nX33 -> T67", fontsize=14, style = filled, fillcolor = violet];
11 -> 70 [arrowhead = none ];
70 -> 31;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 71 [arrowhead = none ];
71 -> 13;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 72 [arrowhead = none ];
72 -> 14;

73 [label="EVAL with clause\nreverse(.(X65, X66), X67) :- ','(reverse(X66, X68), app(X68, .(X65, []), X67)).\nand substitution\nX65 -> T33\nX66 -> T34\nT19 -> .(T33, T34)\nX32 -> X69\nX67 -> X69", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 73 [arrowhead = none ];
73 -> 15;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 74 [arrowhead = none ];
74 -> 16;

75 [label="EVAL with clause\nreverse([], []).\nand substitution\nT19 -> []\nX32 -> []", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 75 [arrowhead = none ];
75 -> 27;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 76 [arrowhead = none ];
76 -> 28;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 77 [arrowhead = none ];
77 -> 17;

78 [label="SPLIT 2\nnew knowledge:\nT37 is ground\nreplacements:\nX68 -> T37", fontsize=14, style = filled, fillcolor = violet];
15 -> 78 [arrowhead = none ];
78 -> 18;

79 [label="INSTANCE with matching:\nT19 -> T34\nX32 -> X68", fontsize=14, style = filled, fillcolor = lightgrey];
17 -> 79 [arrowhead = none , style=dashed];
79 -> 10[style=dashed];

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 80 [arrowhead = none ];
80 -> 19;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 81 [arrowhead = none ];
81 -> 20;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="EVAL with clause\napp(.(X108, X109), X110, .(X108, X111)) :- app(X109, X110, X111).\nand substitution\nX108 -> T55\nX109 -> T56\nT37 -> .(T55, T56)\nT33 -> T57\nX110 -> .(T57, [])\nX111 -> X112\nX69 -> .(T55, X112)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 84 [arrowhead = none ];
84 -> 23;

85 [label="EVAL with clause\napp([], X124, X124).\nand substitution\nT37 -> []\nT33 -> T63\nX124 -> .(T63, [])\nX69 -> .(T63, [])", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 85 [arrowhead = none ];
85 -> 24;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 86 [arrowhead = none ];
86 -> 25;

87 [label="INSTANCE with matching:\nT37 -> T56\nT33 -> T57\nX69 -> X112", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 87 [arrowhead = none , style=dashed];
87 -> 18[style=dashed];

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 88 [arrowhead = none ];
88 -> 26;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 89 [arrowhead = none ];
89 -> 29;

90 [label="INSTANCE with matching:\nT37 -> T22\nT33 -> T18\nX69 -> X33", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 90 [arrowhead = none , style=dashed];
90 -> 18[style=dashed];

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 91 [arrowhead = none ];
91 -> 32;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 92 [arrowhead = none ];
92 -> 33;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 93 [arrowhead = none ];
93 -> 34;

94 [label="EVAL with clause\napp(.(X156, X157), X158, .(X156, X159)) :- app(X157, X158, X159).\nand substitution\nX156 -> T91\nX157 -> T92\nT67 -> .(T91, T92)\nT6 -> T93\nX158 -> .(T93, [])\nX159 -> T95\nT9 -> .(T91, T95)\nT94 -> T95", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 94 [arrowhead = none ];
94 -> 35;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 95 [arrowhead = none ];
95 -> 36;

96 [label="EVAL with clause\napp([], X170, X170).\nand substitution\nT67 -> []\nT6 -> T102\nX170 -> .(T102, [])\nT9 -> .(T102, [])", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 96 [arrowhead = none ];
96 -> 37;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 97 [arrowhead = none ];
97 -> 38;

98 [label="INSTANCE with matching:\nT67 -> T92\nT6 -> T93\nT9 -> T95", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 98 [arrowhead = none , style=dashed];
98 -> 31[style=dashed];

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 99 [arrowhead = none ];
99 -> 39;

100 [label="EVAL with clause\nreverse([], []).\nand substitution\nT7 -> []\nX7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 100 [arrowhead = none ];
100 -> 42;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 101 [arrowhead = none ];
101 -> 43;

102 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 102 [arrowhead = none ];
102 -> 49;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 103 [arrowhead = none ];
103 -> 44;

104 [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 -> 104 [arrowhead = none ];
104 -> 45;

105 [label="EVAL with clause\napp([], X184, X184).\nand substitution\nT6 -> T107\nX184 -> .(T107, [])\nT9 -> .(T107, [])", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 105 [arrowhead = none ];
105 -> 46;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 106 [arrowhead = none ];
106 -> 47;

107 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 107 [arrowhead = none ];
107 -> 48;

108 [label="BACKTRACK\nfor clause: reverse([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
49 -> 108 [arrowhead = none ];
108 -> 50;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 109 [arrowhead = none ];
109 -> 53;

}

(2) Obligation:

Clauses:

reverse10(.(T33, T34), X69) :- reverse10(T34, X68).
reverse10(.(T33, T34), X69) :- ','(reverse10(T34, T37), app18(T37, T33, X69)).
reverse10([], []).
app18(.(T55, T56), T57, .(T55, X112)) :- app18(T56, T57, X112).
app18([], T63, .(T63, [])).
app31(.(T91, T92), T93, .(T91, T95)) :- app31(T92, T93, T95).
app31([], T102, .(T102, [])).
reverse1(.(T6, .(T18, T19)), T9) :- reverse10(T19, X32).
reverse1(.(T6, .(T18, T19)), T9) :- ','(reverse10(T19, T22), app18(T22, T18, X33)).
reverse1(.(T6, .(T18, T19)), T9) :- ','(reverse10(T19, T22), ','(app18(T22, T18, T67), app31(T67, T6, T9))).
reverse1(.(T107, []), .(T107, [])).
reverse1([], []).

Queries:

reverse1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
reverse1_in: (b,f)
reverse10_in: (b,f)
app18_in: (f,f,f)
app31_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(5) DependencyPairsProof (EQUIVALENT transformation)

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

REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U6_GA(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → REVERSE10_IN_GA(T19, X32)
REVERSE10_IN_GA(.(T33, T34), X69) → U1_GA(T33, T34, X69, reverse10_in_ga(T34, X68))
REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)
REVERSE10_IN_GA(.(T33, T34), X69) → U2_GA(T33, T34, X69, reverse10_in_ga(T34, T37))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_GA(T33, T34, X69, app18_in_aaa(T37, T33, X69))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → APP18_IN_AAA(T37, T33, X69)
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → U4_AAA(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U7_GA(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → APP18_IN_AAA(T22, T18, X33)
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_GA(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → APP31_IN_GAA(T67, T6, T9)
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → U5_GAA(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

The argument filtering Pi contains the following mapping:
reverse1_in_ga(x1, x2) = reverse1_in_ga(x1)
.(x1, x2) = .(x2)
U6_ga(x1, x2, x3, x4, x5) = U6_ga(x5)
reverse10_in_ga(x1, x2) = reverse10_in_ga(x1)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
U2_ga(x1, x2, x3, x4) = U2_ga(x4)
[] = []
reverse10_out_ga(x1, x2) = reverse10_out_ga
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
app18_in_aaa(x1, x2, x3) = app18_in_aaa
U4_aaa(x1, x2, x3, x4, x5) = U4_aaa(x5)
app18_out_aaa(x1, x2, x3) = app18_out_aaa(x1, x3)
reverse1_out_ga(x1, x2) = reverse1_out_ga
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x5)
U10_ga(x1, x2, x3, x4, x5) = U10_ga(x5)
app31_in_gaa(x1, x2, x3) = app31_in_gaa(x1)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
app31_out_gaa(x1, x2, x3) = app31_out_gaa(x3)
REVERSE1_IN_GA(x1, x2) = REVERSE1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5) = U6_GA(x5)
REVERSE10_IN_GA(x1, x2) = REVERSE10_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
APP18_IN_AAA(x1, x2, x3) = APP18_IN_AAA
U4_AAA(x1, x2, x3, x4, x5) = U4_AAA(x5)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x5)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x5)
U10_GA(x1, x2, x3, x4, x5) = U10_GA(x5)
APP31_IN_GAA(x1, x2, x3) = APP31_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x5)

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

(6) Obligation:

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

REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U6_GA(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → REVERSE10_IN_GA(T19, X32)
REVERSE10_IN_GA(.(T33, T34), X69) → U1_GA(T33, T34, X69, reverse10_in_ga(T34, X68))
REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)
REVERSE10_IN_GA(.(T33, T34), X69) → U2_GA(T33, T34, X69, reverse10_in_ga(T34, T37))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_GA(T33, T34, X69, app18_in_aaa(T37, T33, X69))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → APP18_IN_AAA(T37, T33, X69)
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → U4_AAA(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U7_GA(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → APP18_IN_AAA(T22, T18, X33)
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_GA(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → APP31_IN_GAA(T67, T6, T9)
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → U5_GAA(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP31_IN_GAA(x1, x2, x3) = APP31_IN_GAA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP31_IN_GAA(.(T92)) → APP31_IN_GAA(T92)

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

(14) 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_GAA(.(T92)) → APP31_IN_GAA(T92)
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP18_IN_AAA(x1, x2, x3) = APP18_IN_AAA

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP18_IN_AAA → APP18_IN_AAA

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

(21) 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 = APP18_IN_AAA evaluates to t =APP18_IN_AAA

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 APP18_IN_AAA to APP18_IN_AAA.

(22) NO

(23) Obligation:

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

REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REVERSE10_IN_GA(x1, x2) = REVERSE10_IN_GA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

REVERSE10_IN_GA(.(T34)) → REVERSE10_IN_GA(T34)

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

(28) 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:

REVERSE10_IN_GA(.(T34)) → REVERSE10_IN_GA(T34)
The graph contains the following edges 1 > 1

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

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

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(32) DependencyPairsProof (EQUIVALENT transformation)

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

REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U6_GA(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → REVERSE10_IN_GA(T19, X32)
REVERSE10_IN_GA(.(T33, T34), X69) → U1_GA(T33, T34, X69, reverse10_in_ga(T34, X68))
REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)
REVERSE10_IN_GA(.(T33, T34), X69) → U2_GA(T33, T34, X69, reverse10_in_ga(T34, T37))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_GA(T33, T34, X69, app18_in_aaa(T37, T33, X69))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → APP18_IN_AAA(T37, T33, X69)
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → U4_AAA(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U7_GA(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → APP18_IN_AAA(T22, T18, X33)
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_GA(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → APP31_IN_GAA(T67, T6, T9)
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → U5_GAA(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

The argument filtering Pi contains the following mapping:
reverse1_in_ga(x1, x2) = reverse1_in_ga(x1)
.(x1, x2) = .(x2)
U6_ga(x1, x2, x3, x4, x5) = U6_ga(x3, x5)
reverse10_in_ga(x1, x2) = reverse10_in_ga(x1)
U1_ga(x1, x2, x3, x4) = U1_ga(x2, x4)
U2_ga(x1, x2, x3, x4) = U2_ga(x2, x4)
[] = []
reverse10_out_ga(x1, x2) = reverse10_out_ga(x1)
U3_ga(x1, x2, x3, x4) = U3_ga(x2, x4)
app18_in_aaa(x1, x2, x3) = app18_in_aaa
U4_aaa(x1, x2, x3, x4, x5) = U4_aaa(x5)
app18_out_aaa(x1, x2, x3) = app18_out_aaa(x1, x3)
reverse1_out_ga(x1, x2) = reverse1_out_ga(x1)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x3, x5)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x3, x5)
U10_ga(x1, x2, x3, x4, x5) = U10_ga(x3, x5)
app31_in_gaa(x1, x2, x3) = app31_in_gaa(x1)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x2, x5)
app31_out_gaa(x1, x2, x3) = app31_out_gaa(x1, x3)
REVERSE1_IN_GA(x1, x2) = REVERSE1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5) = U6_GA(x3, x5)
REVERSE10_IN_GA(x1, x2) = REVERSE10_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x2, x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x2, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x2, x4)
APP18_IN_AAA(x1, x2, x3) = APP18_IN_AAA
U4_AAA(x1, x2, x3, x4, x5) = U4_AAA(x5)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x3, x5)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x3, x5)
U10_GA(x1, x2, x3, x4, x5) = U10_GA(x3, x5)
APP31_IN_GAA(x1, x2, x3) = APP31_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x2, x5)

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

(33) Obligation:

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

REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U6_GA(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → REVERSE10_IN_GA(T19, X32)
REVERSE10_IN_GA(.(T33, T34), X69) → U1_GA(T33, T34, X69, reverse10_in_ga(T34, X68))
REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)
REVERSE10_IN_GA(.(T33, T34), X69) → U2_GA(T33, T34, X69, reverse10_in_ga(T34, T37))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_GA(T33, T34, X69, app18_in_aaa(T37, T33, X69))
U2_GA(T33, T34, X69, reverse10_out_ga(T34, T37)) → APP18_IN_AAA(T37, T33, X69)
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → U4_AAA(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)
REVERSE1_IN_GA(.(T6, .(T18, T19)), T9) → U7_GA(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → APP18_IN_AAA(T22, T18, X33)
U7_GA(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_GA(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_GA(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
U9_GA(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → APP31_IN_GAA(T67, T6, T9)
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → U5_GAA(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(35) Complex Obligation (AND)

(36) Obligation:

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

APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

APP31_IN_GAA(.(T91, T92), T93, .(T91, T95)) → APP31_IN_GAA(T92, T93, T95)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP31_IN_GAA(x1, x2, x3) = APP31_IN_GAA(x1)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

APP31_IN_GAA(.(T92)) → APP31_IN_GAA(T92)

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

(41) 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_GAA(.(T92)) → APP31_IN_GAA(T92)
The graph contains the following edges 1 > 1

(42) YES

(43) Obligation:

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

APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

APP18_IN_AAA(.(T55, T56), T57, .(T55, X112)) → APP18_IN_AAA(T56, T57, X112)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
APP18_IN_AAA(x1, x2, x3) = APP18_IN_AAA

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

APP18_IN_AAA → APP18_IN_AAA

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

(48) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP18_IN_AAA to APP18_IN_AAA.

(49) NO

(50) Obligation:

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

REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)

The TRS R consists of the following rules:

reverse1_in_ga(.(T6, .(T18, T19)), T9) → U6_ga(T6, T18, T19, T9, reverse10_in_ga(T19, X32))
reverse10_in_ga(.(T33, T34), X69) → U1_ga(T33, T34, X69, reverse10_in_ga(T34, X68))
reverse10_in_ga(.(T33, T34), X69) → U2_ga(T33, T34, X69, reverse10_in_ga(T34, T37))
reverse10_in_ga([], []) → reverse10_out_ga([], [])
U2_ga(T33, T34, X69, reverse10_out_ga(T34, T37)) → U3_ga(T33, T34, X69, app18_in_aaa(T37, T33, X69))
app18_in_aaa(.(T55, T56), T57, .(T55, X112)) → U4_aaa(T55, T56, T57, X112, app18_in_aaa(T56, T57, X112))
app18_in_aaa([], T63, .(T63, [])) → app18_out_aaa([], T63, .(T63, []))
U4_aaa(T55, T56, T57, X112, app18_out_aaa(T56, T57, X112)) → app18_out_aaa(.(T55, T56), T57, .(T55, X112))
U3_ga(T33, T34, X69, app18_out_aaa(T37, T33, X69)) → reverse10_out_ga(.(T33, T34), X69)
U1_ga(T33, T34, X69, reverse10_out_ga(T34, X68)) → reverse10_out_ga(.(T33, T34), X69)
U6_ga(T6, T18, T19, T9, reverse10_out_ga(T19, X32)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T6, .(T18, T19)), T9) → U7_ga(T6, T18, T19, T9, reverse10_in_ga(T19, T22))
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U8_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, X33))
U8_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, X33)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
U7_ga(T6, T18, T19, T9, reverse10_out_ga(T19, T22)) → U9_ga(T6, T18, T19, T9, app18_in_aaa(T22, T18, T67))
U9_ga(T6, T18, T19, T9, app18_out_aaa(T22, T18, T67)) → U10_ga(T6, T18, T19, T9, app31_in_gaa(T67, T6, T9))
app31_in_gaa(.(T91, T92), T93, .(T91, T95)) → U5_gaa(T91, T92, T93, T95, app31_in_gaa(T92, T93, T95))
app31_in_gaa([], T102, .(T102, [])) → app31_out_gaa([], T102, .(T102, []))
U5_gaa(T91, T92, T93, T95, app31_out_gaa(T92, T93, T95)) → app31_out_gaa(.(T91, T92), T93, .(T91, T95))
U10_ga(T6, T18, T19, T9, app31_out_gaa(T67, T6, T9)) → reverse1_out_ga(.(T6, .(T18, T19)), T9)
reverse1_in_ga(.(T107, []), .(T107, [])) → reverse1_out_ga(.(T107, []), .(T107, []))
reverse1_in_ga([], []) → reverse1_out_ga([], [])

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

REVERSE10_IN_GA(.(T33, T34), X69) → REVERSE10_IN_GA(T34, X68)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
REVERSE10_IN_GA(x1, x2) = REVERSE10_IN_GA(x1)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

REVERSE10_IN_GA(.(T34)) → REVERSE10_IN_GA(T34)

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

(55) 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:

REVERSE10_IN_GA(.(T34)) → REVERSE10_IN_GA(T34)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) NonTerminationProof (EQUIVALENT transformation)

(22) NO

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

(31) Obligation:

(32) DependencyPairsProof (EQUIVALENT transformation)

(33) Obligation:

(34) DependencyGraphProof (EQUIVALENT transformation)

(35) Complex Obligation (AND)

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) YES

(43) Obligation:

(44) UsableRulesProof (EQUIVALENT transformation)

(45) Obligation:

(46) PiDPToQDPProof (SOUND transformation)

(47) Obligation:

(48) NonTerminationProof (EQUIVALENT transformation)

(49) NO

(50) Obligation:

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

(53) PiDPToQDPProof (SOUND transformation)

(54) Obligation:

(55) QDPSizeChangeProof (EQUIVALENT transformation)

(56) YES