Left Termination of the query pattern rev_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

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

Queries:

rev(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rev(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rev(T1, T2), SCOPE: 1, CLAUSE: 0 | rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: rev(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nrev(T1, T2) !=? rev([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: rev([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(rev(T7, X10), app(X10, .(T6, []), T9))\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 0 | ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 0\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(rev(T7, X10), app(X10, .(T6, []), T9)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: app([], .(T6, []), T9)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 2 | app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app([], .(T6, []), T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(rev(T22, X38), ','(app(X38, .(T21, []), X39), app(X39, .(T6, []), T9)))\n\nT6 is ground\nT21 is ground\nT22 is ground\nX39 is free; \nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: rev(T22, X38)\n\nT22 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(app(T23, .(T21, []), X39), app(X39, .(T6, []), T9))\n\nT6 is ground\nT21 is ground\nT23 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: rev(T22, X38), SCOPE: 4, CLAUSE: 0 | rev(T22, X38), SCOPE: 4, CLAUSE: 1\n\nT22 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: rev(T22, X38), SCOPE: 4, CLAUSE: 0\n\nT22 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: rev(T22, X38), SCOPE: 4, CLAUSE: 1\n\nT22 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(rev(T29, X55), app(X55, .(T28, []), X56))\n\nT28 is ground\nT29 is ground\nX56 is free; \nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: rev(T29, X55)\n\nT29 is ground\nX55 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T30, .(T28, []), X56)\n\nT28 is ground\nT30 is ground\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: app(T30, .(T28, []), X56), SCOPE: 5, CLAUSE: 2 | app(T30, .(T28, []), X56), SCOPE: 5, CLAUSE: 3\n\nT28 is ground\nT30 is ground\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T30, .(T28, []), X56), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nT30 is ground\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: app(T30, .(T28, []), X56), SCOPE: 5, CLAUSE: 3\n\nT28 is ground\nT30 is ground\nX56 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T45, .(T46, []), X82)\n\nT45 is ground\nT46 is ground\nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: app(T23, .(T21, []), X39)\n\nT21 is ground\nT23 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: app(T51, .(T6, []), T9)\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 2 | app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 2\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app(T51, .(T6, []), T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app(T70, .(T71, []), T73)\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 53 [arrowhead = none ];
53 -> 2;

54 [label="EVAL with clause\nrev([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 54 [arrowhead = none ];
54 -> 3;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 55 [arrowhead = none ];
55 -> 4;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 56 [arrowhead = none ];
56 -> 5;

57 [label="EVAL with clause\nrev(.(X7, X8), X9) :- ','(rev(X8, X10), app(X10, .(X7, []), X9)).\nand substitution\nX7 -> T6\nX8 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX9 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="BACKTRACK\nfor clause: rev(.(X, Xs), Ys) :- ','(rev(Xs, Zs), app(Zs, .(X, []), Ys))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 59 [arrowhead = none ];
59 -> 6;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 60 [arrowhead = none ];
60 -> 9;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 61 [arrowhead = none ];
61 -> 10;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="EVAL with clause\nrev([], []).\nand substitution\nT7 -> []\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="EVAL with clause\nrev(.(X35, X36), X37) :- ','(rev(X36, X38), app(X38, .(X35, []), X37)).\nand substitution\nX35 -> T21\nX36 -> T22\nT7 -> .(T21, T22)\nX10 -> X39\nX37 -> X39", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="EVAL with clause\napp([], X20, X20).\nand substitution\nT6 -> T16\nX20 -> .(T16, [])\nT9 -> .(T16, [])", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 72 [arrowhead = none ];
72 -> 20;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 73 [arrowhead = none ];
73 -> 19;

74 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 74 [arrowhead = none ];
74 -> 23;

75 [label="SPLIT 2\nnew knowledge:\nT23 is ground\nreplacements:\nX38 -> T23", fontsize=14, style = filled, fillcolor = violet];
21 -> 75 [arrowhead = none ];
75 -> 24;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 76 [arrowhead = none ];
76 -> 25;

77 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 77 [arrowhead = none ];
77 -> 43;

78 [label="SPLIT 2\nnew knowledge:\nT51 is ground\nreplacements:\nX39 -> T51", fontsize=14, style = filled, fillcolor = violet];
24 -> 78 [arrowhead = none ];
78 -> 44;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 80 [arrowhead = none ];
80 -> 27;

81 [label="EVAL with clause\nrev([], []).\nand substitution\nT22 -> []\nX38 -> []", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="EVAL with clause\nrev(.(X52, X53), X54) :- ','(rev(X53, X55), app(X55, .(X52, []), X54)).\nand substitution\nX52 -> T28\nX53 -> T29\nT22 -> .(T28, T29)\nX38 -> X56\nX54 -> X56", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 83 [arrowhead = none ];
83 -> 31;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 85 [arrowhead = none ];
85 -> 30;

86 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 86 [arrowhead = none ];
86 -> 33;

87 [label="SPLIT 2\nnew knowledge:\nT30 is ground\nreplacements:\nX55 -> T30", fontsize=14, style = filled, fillcolor = violet];
31 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="INSTANCE with matching:\nT22 -> T29\nX38 -> X55", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 88 [arrowhead = none , style=dashed];
88 -> 23[style=dashed];

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 90 [arrowhead = none ];
90 -> 36;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 91 [arrowhead = none ];
91 -> 37;

92 [label="EVAL with clause\napp([], X66, X66).\nand substitution\nT30 -> []\nT28 -> T37\nX66 -> .(T37, [])\nX56 -> .(T37, [])", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 92 [arrowhead = none ];
92 -> 38;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 93 [arrowhead = none ];
93 -> 39;

94 [label="EVAL with clause\napp(.(X78, X79), X80, .(X78, X81)) :- app(X79, X80, X81).\nand substitution\nX78 -> T44\nX79 -> T45\nT30 -> .(T44, T45)\nT28 -> T46\nX80 -> .(T46, [])\nX81 -> X82\nX56 -> .(T44, X82)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 94 [arrowhead = none ];
94 -> 41;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 95 [arrowhead = none ];
95 -> 42;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 96 [arrowhead = none ];
96 -> 40;

97 [label="INSTANCE with matching:\nT30 -> T45\nT28 -> T46\nX56 -> X82", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 97 [arrowhead = none , style=dashed];
97 -> 34[style=dashed];

98 [label="INSTANCE with matching:\nT30 -> T23\nT28 -> T21\nX56 -> X39", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 98 [arrowhead = none , style=dashed];
98 -> 34[style=dashed];

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 101 [arrowhead = none ];
101 -> 47;

102 [label="EVAL with clause\napp([], X99, X99).\nand substitution\nT51 -> []\nT6 -> T60\nX99 -> .(T60, [])\nT9 -> .(T60, [])", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 102 [arrowhead = none ];
102 -> 48;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 103 [arrowhead = none ];
103 -> 49;

104 [label="EVAL with clause\napp(.(X109, X110), X111, .(X109, X112)) :- app(X110, X111, X112).\nand substitution\nX109 -> T69\nX110 -> T70\nT51 -> .(T69, T70)\nT6 -> T71\nX111 -> .(T71, [])\nX112 -> T73\nT9 -> .(T69, T73)\nT72 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 104 [arrowhead = none ];
104 -> 51;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 105 [arrowhead = none ];
105 -> 52;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 106 [arrowhead = none ];
106 -> 50;

107 [label="INSTANCE with matching:\nT51 -> T70\nT6 -> T71\nT9 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 107 [arrowhead = none , style=dashed];
107 -> 44[style=dashed];

}

(2) Obligation:

Triples:

rev23(.(T28, T29), X56) :- rev23(T29, X55).
rev23(.(T28, T29), X56) :- ','(revc23(T29, T30), app34(T30, T28, X56)).
app34(.(T44, T45), T46, .(T44, X82)) :- app34(T45, T46, X82).
app44(.(T69, T70), T71, .(T69, T73)) :- app44(T70, T71, T73).
rev1(.(T6, .(T21, T22)), T9) :- rev23(T22, X38).
rev1(.(T6, .(T21, T22)), T9) :- ','(revc23(T22, T23), app34(T23, T21, X39)).
rev1(.(T6, .(T21, T22)), T9) :- ','(revc23(T22, T23), ','(appc34(T23, T21, T51), app44(T51, T6, T9))).

Clauses:

revc23([], []).
revc23(.(T28, T29), X56) :- ','(revc23(T29, T30), appc34(T30, T28, X56)).
appc34([], T37, .(T37, [])).
appc34(.(T44, T45), T46, .(T44, X82)) :- appc34(T45, T46, X82).
appc44([], T60, .(T60, [])).
appc44(.(T69, T70), T71, .(T69, T73)) :- appc44(T70, T71, T73).

Afs:

rev1(x1, x2)  =  rev1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
rev1_in: (b,f)
rev23_in: (b,f)
revc23_in: (b,f)
appc34_in: (b,b,f)
app34_in: (b,b,f)
app44_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

REV1_IN_GA(.(T6, .(T21, T22)), T9) → U6_GA(T6, T21, T22, T9, rev23_in_ga(T22, X38))
REV1_IN_GA(.(T6, .(T21, T22)), T9) → REV23_IN_GA(T22, X38)
REV23_IN_GA(.(T28, T29), X56) → U1_GA(T28, T29, X56, rev23_in_ga(T29, X55))
REV23_IN_GA(.(T28, T29), X56) → REV23_IN_GA(T29, X55)
REV23_IN_GA(.(T28, T29), X56) → U2_GA(T28, T29, X56, revc23_in_ga(T29, T30))
U2_GA(T28, T29, X56, revc23_out_ga(T29, T30)) → U3_GA(T28, T29, X56, app34_in_gga(T30, T28, X56))
U2_GA(T28, T29, X56, revc23_out_ga(T29, T30)) → APP34_IN_GGA(T30, T28, X56)
APP34_IN_GGA(.(T44, T45), T46, .(T44, X82)) → U4_GGA(T44, T45, T46, X82, app34_in_gga(T45, T46, X82))
APP34_IN_GGA(.(T44, T45), T46, .(T44, X82)) → APP34_IN_GGA(T45, T46, X82)
REV1_IN_GA(.(T6, .(T21, T22)), T9) → U7_GA(T6, T21, T22, T9, revc23_in_ga(T22, T23))
U7_GA(T6, T21, T22, T9, revc23_out_ga(T22, T23)) → U8_GA(T6, T21, T22, T9, app34_in_gga(T23, T21, X39))
U7_GA(T6, T21, T22, T9, revc23_out_ga(T22, T23)) → APP34_IN_GGA(T23, T21, X39)
U7_GA(T6, T21, T22, T9, revc23_out_ga(T22, T23)) → U9_GA(T6, T21, T22, T9, appc34_in_gga(T23, T21, T51))
U9_GA(T6, T21, T22, T9, appc34_out_gga(T23, T21, T51)) → U10_GA(T6, T21, T22, T9, app44_in_gga(T51, T6, T9))
U9_GA(T6, T21, T22, T9, appc34_out_gga(T23, T21, T51)) → APP44_IN_GGA(T51, T6, T9)
APP44_IN_GGA(.(T69, T70), T71, .(T69, T73)) → U5_GGA(T69, T70, T71, T73, app44_in_gga(T70, T71, T73))
APP44_IN_GGA(.(T69, T70), T71, .(T69, T73)) → APP44_IN_GGA(T70, T71, T73)

The TRS R consists of the following rules:

revc23_in_ga([], []) → revc23_out_ga([], [])
revc23_in_ga(.(T28, T29), X56) → U12_ga(T28, T29, X56, revc23_in_ga(T29, T30))
U12_ga(T28, T29, X56, revc23_out_ga(T29, T30)) → U13_ga(T28, T29, X56, appc34_in_gga(T30, T28, X56))
appc34_in_gga([], T37, .(T37, [])) → appc34_out_gga([], T37, .(T37, []))
appc34_in_gga(.(T44, T45), T46, .(T44, X82)) → U14_gga(T44, T45, T46, X82, appc34_in_gga(T45, T46, X82))
U14_gga(T44, T45, T46, X82, appc34_out_gga(T45, T46, X82)) → appc34_out_gga(.(T44, T45), T46, .(T44, X82))
U13_ga(T28, T29, X56, appc34_out_gga(T30, T28, X56)) → revc23_out_ga(.(T28, T29), X56)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev23_in_ga(x1, x2)  =  rev23_in_ga(x1)
revc23_in_ga(x1, x2)  =  revc23_in_ga(x1)
[]  =  []
revc23_out_ga(x1, x2)  =  revc23_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc34_in_gga(x1, x2, x3)  =  appc34_in_gga(x1, x2)
appc34_out_gga(x1, x2, x3)  =  appc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
app34_in_gga(x1, x2, x3)  =  app34_in_gga(x1, x2)
app44_in_gga(x1, x2, x3)  =  app44_in_gga(x1, x2)
REV1_IN_GA(x1, x2)  =  REV1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REV23_IN_GA(x1, x2)  =  REV23_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
APP34_IN_GGA(x1, x2, x3)  =  APP34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
APP44_IN_GGA(x1, x2, x3)  =  APP44_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, 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:

REV1_IN_GA(.(T6, .(T21, T22)), T9) → U6_GA(T6, T21, T22, T9, rev23_in_ga(T22, X38))
REV1_IN_GA(.(T6, .(T21, T22)), T9) → REV23_IN_GA(T22, X38)
REV23_IN_GA(.(T28, T29), X56) → U1_GA(T28, T29, X56, rev23_in_ga(T29, X55))
REV23_IN_GA(.(T28, T29), X56) → REV23_IN_GA(T29, X55)
REV23_IN_GA(.(T28, T29), X56) → U2_GA(T28, T29, X56, revc23_in_ga(T29, T30))
U2_GA(T28, T29, X56, revc23_out_ga(T29, T30)) → U3_GA(T28, T29, X56, app34_in_gga(T30, T28, X56))
U2_GA(T28, T29, X56, revc23_out_ga(T29, T30)) → APP34_IN_GGA(T30, T28, X56)
APP34_IN_GGA(.(T44, T45), T46, .(T44, X82)) → U4_GGA(T44, T45, T46, X82, app34_in_gga(T45, T46, X82))
APP34_IN_GGA(.(T44, T45), T46, .(T44, X82)) → APP34_IN_GGA(T45, T46, X82)
REV1_IN_GA(.(T6, .(T21, T22)), T9) → U7_GA(T6, T21, T22, T9, revc23_in_ga(T22, T23))
U7_GA(T6, T21, T22, T9, revc23_out_ga(T22, T23)) → U8_GA(T6, T21, T22, T9, app34_in_gga(T23, T21, X39))
U7_GA(T6, T21, T22, T9, revc23_out_ga(T22, T23)) → APP34_IN_GGA(T23, T21, X39)
U7_GA(T6, T21, T22, T9, revc23_out_ga(T22, T23)) → U9_GA(T6, T21, T22, T9, appc34_in_gga(T23, T21, T51))
U9_GA(T6, T21, T22, T9, appc34_out_gga(T23, T21, T51)) → U10_GA(T6, T21, T22, T9, app44_in_gga(T51, T6, T9))
U9_GA(T6, T21, T22, T9, appc34_out_gga(T23, T21, T51)) → APP44_IN_GGA(T51, T6, T9)
APP44_IN_GGA(.(T69, T70), T71, .(T69, T73)) → U5_GGA(T69, T70, T71, T73, app44_in_gga(T70, T71, T73))
APP44_IN_GGA(.(T69, T70), T71, .(T69, T73)) → APP44_IN_GGA(T70, T71, T73)

The TRS R consists of the following rules:

revc23_in_ga([], []) → revc23_out_ga([], [])
revc23_in_ga(.(T28, T29), X56) → U12_ga(T28, T29, X56, revc23_in_ga(T29, T30))
U12_ga(T28, T29, X56, revc23_out_ga(T29, T30)) → U13_ga(T28, T29, X56, appc34_in_gga(T30, T28, X56))
appc34_in_gga([], T37, .(T37, [])) → appc34_out_gga([], T37, .(T37, []))
appc34_in_gga(.(T44, T45), T46, .(T44, X82)) → U14_gga(T44, T45, T46, X82, appc34_in_gga(T45, T46, X82))
U14_gga(T44, T45, T46, X82, appc34_out_gga(T45, T46, X82)) → appc34_out_gga(.(T44, T45), T46, .(T44, X82))
U13_ga(T28, T29, X56, appc34_out_gga(T30, T28, X56)) → revc23_out_ga(.(T28, T29), X56)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
rev23_in_ga(x1, x2)  =  rev23_in_ga(x1)
revc23_in_ga(x1, x2)  =  revc23_in_ga(x1)
[]  =  []
revc23_out_ga(x1, x2)  =  revc23_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc34_in_gga(x1, x2, x3)  =  appc34_in_gga(x1, x2)
appc34_out_gga(x1, x2, x3)  =  appc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
app34_in_gga(x1, x2, x3)  =  app34_in_gga(x1, x2)
app44_in_gga(x1, x2, x3)  =  app44_in_gga(x1, x2)
REV1_IN_GA(x1, x2)  =  REV1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
REV23_IN_GA(x1, x2)  =  REV23_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
APP34_IN_GGA(x1, x2, x3)  =  APP34_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
APP44_IN_GGA(x1, x2, x3)  =  APP44_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Complex Obligation (AND)

(7) Obligation:

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

APP44_IN_GGA(.(T69, T70), T71, .(T69, T73)) → APP44_IN_GGA(T70, T71, T73)

The TRS R consists of the following rules:

revc23_in_ga([], []) → revc23_out_ga([], [])
revc23_in_ga(.(T28, T29), X56) → U12_ga(T28, T29, X56, revc23_in_ga(T29, T30))
U12_ga(T28, T29, X56, revc23_out_ga(T29, T30)) → U13_ga(T28, T29, X56, appc34_in_gga(T30, T28, X56))
appc34_in_gga([], T37, .(T37, [])) → appc34_out_gga([], T37, .(T37, []))
appc34_in_gga(.(T44, T45), T46, .(T44, X82)) → U14_gga(T44, T45, T46, X82, appc34_in_gga(T45, T46, X82))
U14_gga(T44, T45, T46, X82, appc34_out_gga(T45, T46, X82)) → appc34_out_gga(.(T44, T45), T46, .(T44, X82))
U13_ga(T28, T29, X56, appc34_out_gga(T30, T28, X56)) → revc23_out_ga(.(T28, T29), X56)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revc23_in_ga(x1, x2)  =  revc23_in_ga(x1)
[]  =  []
revc23_out_ga(x1, x2)  =  revc23_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc34_in_gga(x1, x2, x3)  =  appc34_in_gga(x1, x2)
appc34_out_gga(x1, x2, x3)  =  appc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
APP44_IN_GGA(x1, x2, x3)  =  APP44_IN_GGA(x1, x2)

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:

APP44_IN_GGA(.(T69, T70), T71, .(T69, T73)) → APP44_IN_GGA(T70, T71, T73)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP44_IN_GGA(x1, x2, x3)  =  APP44_IN_GGA(x1, x2)

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:

APP44_IN_GGA(.(T69, T70), T71) → APP44_IN_GGA(T70, T71)

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:

• APP44_IN_GGA(.(T69, T70), T71) → APP44_IN_GGA(T70, T71)
  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:

APP34_IN_GGA(.(T44, T45), T46, .(T44, X82)) → APP34_IN_GGA(T45, T46, X82)

The TRS R consists of the following rules:

revc23_in_ga([], []) → revc23_out_ga([], [])
revc23_in_ga(.(T28, T29), X56) → U12_ga(T28, T29, X56, revc23_in_ga(T29, T30))
U12_ga(T28, T29, X56, revc23_out_ga(T29, T30)) → U13_ga(T28, T29, X56, appc34_in_gga(T30, T28, X56))
appc34_in_gga([], T37, .(T37, [])) → appc34_out_gga([], T37, .(T37, []))
appc34_in_gga(.(T44, T45), T46, .(T44, X82)) → U14_gga(T44, T45, T46, X82, appc34_in_gga(T45, T46, X82))
U14_gga(T44, T45, T46, X82, appc34_out_gga(T45, T46, X82)) → appc34_out_gga(.(T44, T45), T46, .(T44, X82))
U13_ga(T28, T29, X56, appc34_out_gga(T30, T28, X56)) → revc23_out_ga(.(T28, T29), X56)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revc23_in_ga(x1, x2)  =  revc23_in_ga(x1)
[]  =  []
revc23_out_ga(x1, x2)  =  revc23_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc34_in_gga(x1, x2, x3)  =  appc34_in_gga(x1, x2)
appc34_out_gga(x1, x2, x3)  =  appc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
APP34_IN_GGA(x1, x2, x3)  =  APP34_IN_GGA(x1, x2)

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:

APP34_IN_GGA(.(T44, T45), T46, .(T44, X82)) → APP34_IN_GGA(T45, T46, X82)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP34_IN_GGA(x1, x2, x3)  =  APP34_IN_GGA(x1, x2)

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:

APP34_IN_GGA(.(T44, T45), T46) → APP34_IN_GGA(T45, T46)

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:

• APP34_IN_GGA(.(T44, T45), T46) → APP34_IN_GGA(T45, T46)
  The graph contains the following edges 1 > 1, 2 >= 2

(20) YES

(21) Obligation:

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

REV23_IN_GA(.(T28, T29), X56) → REV23_IN_GA(T29, X55)

The TRS R consists of the following rules:

revc23_in_ga([], []) → revc23_out_ga([], [])
revc23_in_ga(.(T28, T29), X56) → U12_ga(T28, T29, X56, revc23_in_ga(T29, T30))
U12_ga(T28, T29, X56, revc23_out_ga(T29, T30)) → U13_ga(T28, T29, X56, appc34_in_gga(T30, T28, X56))
appc34_in_gga([], T37, .(T37, [])) → appc34_out_gga([], T37, .(T37, []))
appc34_in_gga(.(T44, T45), T46, .(T44, X82)) → U14_gga(T44, T45, T46, X82, appc34_in_gga(T45, T46, X82))
U14_gga(T44, T45, T46, X82, appc34_out_gga(T45, T46, X82)) → appc34_out_gga(.(T44, T45), T46, .(T44, X82))
U13_ga(T28, T29, X56, appc34_out_gga(T30, T28, X56)) → revc23_out_ga(.(T28, T29), X56)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
revc23_in_ga(x1, x2)  =  revc23_in_ga(x1)
[]  =  []
revc23_out_ga(x1, x2)  =  revc23_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
appc34_in_gga(x1, x2, x3)  =  appc34_in_gga(x1, x2)
appc34_out_gga(x1, x2, x3)  =  appc34_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
REV23_IN_GA(x1, x2)  =  REV23_IN_GA(x1)

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:

REV23_IN_GA(.(T28, T29), X56) → REV23_IN_GA(T29, X55)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REV23_IN_GA(x1, x2)  =  REV23_IN_GA(x1)

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:

REV23_IN_GA(.(T28, T29)) → REV23_IN_GA(T29)

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

(26) 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:

• REV23_IN_GA(.(T28, T29)) → REV23_IN_GA(T29)
  The graph contains the following edges 1 > 1

(27) YES