Left Termination proof of /tmp/tmp3h1skY/merge.pl

Left Termination of the query pattern merge_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

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 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇐)

        ↳27 QDP

        ↳28 MRRProof (⇔)

        ↳29 QDP

        ↳30 DependencyGraphProof (⇔)

        ↳31 TRUE

(0) Obligation:

Clauses:

merge([], X, X).
merge(X, [], X).
merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs)).
less(0, s(0)).
less(s(X), s(Y)) :- less(X, Y).
leq(0, 0).
leq(0, s(0)).
leq(s(X), s(Y)) :- leq(X, Y).

Queries:

merge(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: merge(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: merge(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | merge(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | merge(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | merge(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | merge([], T5, T3), SCOPE: 1, CLAUSE: 1 | merge([], T5, T3), SCOPE: 1, CLAUSE: 2 | merge([], T5, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: merge(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | merge(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | merge(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nmerge(T1, T2, T3) !=? merge([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: merge([], T5, T3), SCOPE: 1, CLAUSE: 1 | merge([], T5, T3), SCOPE: 1, CLAUSE: 2 | merge([], T5, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | merge([], [], T3), SCOPE: 1, CLAUSE: 2 | merge([], [], T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: merge([], T5, T3), SCOPE: 1, CLAUSE: 2 | merge([], T5, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground\nmerge([], T5, T3) !=? merge(X5, [], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: merge([], [], T3), SCOPE: 1, CLAUSE: 2 | merge([], [], T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: merge([], [], T3), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: merge([], T5, T3), SCOPE: 1, CLAUSE: 3\n\nT5 is ground\nmerge([], T5, T3) !=? merge(X5, [], X5)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true | merge(T7, [], T3), SCOPE: 1, CLAUSE: 2 | merge(T7, [], T3), SCOPE: 1, CLAUSE: 3\n\nT7 is ground\nmerge(T7, [], T3) !=? merge([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: merge(T1, T2, T3), SCOPE: 1, CLAUSE: 2 | merge(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nmerge(T1, T2, T3) !=? merge([], X2, X2)\nmerge(T1, T2, T3) !=? merge(X28, [], X28)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: merge(T7, [], T3), SCOPE: 1, CLAUSE: 2 | merge(T7, [], T3), SCOPE: 1, CLAUSE: 3\n\nT7 is ground\nmerge(T7, [], T3) !=? merge([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: merge(T7, [], T3), SCOPE: 1, CLAUSE: 3\n\nT7 is ground\nmerge(T7, [], T3) !=? merge([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)) | merge(.(T17, T18), .(T19, T20), T3), SCOPE: 1, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: merge(T1, T2, T3), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nmerge(T1, T2, T3) !=? merge([], X2, X2)\nmerge(T1, T2, T3) !=? merge(X28, [], X28)\nmerge(T1, T2, T3) !=? merge(.(X45, X46), .(X47, X48), .(X45, X49))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 6 | ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 7 | ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 8 | ?_2 | merge(.(T17, T18), .(T19, T20), T3), SCOPE: 1, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 6\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 7 | ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 8 | ?_2 | merge(.(T17, T18), .(T19, T20), T3), SCOPE: 1, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: merge(T18, .(0, T20), T22)\n\nT18 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 7\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 8 | ?_2 | merge(.(T17, T18), .(T19, T20), T3), SCOPE: 1, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: merge(T18, .(s(0), T20), T22)\n\nT18 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(leq(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 8\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ?_2 | merge(.(T17, T18), .(T19, T20), T3), SCOPE: 1, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(leq(T35, T36), merge(T18, .(s(T36), T20), T22))\n\nT18 is ground\nT20 is ground\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: leq(T35, T36)\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: merge(T18, .(s(T36), T20), T22)\n\nT18 is ground\nT20 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: leq(T35, T36), SCOPE: 3, CLAUSE: 6 | leq(T35, T36), SCOPE: 3, CLAUSE: 7 | leq(T35, T36), SCOPE: 3, CLAUSE: 8\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: leq(T35, T36), SCOPE: 3, CLAUSE: 6\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: leq(T35, T36), SCOPE: 3, CLAUSE: 7 | leq(T35, T36), SCOPE: 3, CLAUSE: 8\n\nT35 is ground\nT36 is ground", 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: leq(T35, T36), SCOPE: 3, CLAUSE: 7\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: leq(T35, T36), SCOPE: 3, CLAUSE: 8\n\nT35 is ground\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: leq(T41, T42)\n\nT41 is ground\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: merge(.(T17, T18), .(T19, T20), T3), SCOPE: 1, CLAUSE: 3\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(less(T59, T57), merge(.(T57, T58), T60, T62))\n\nT57 is ground\nT58 is ground\nT59 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T59, T57)\n\nT57 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: merge(.(T57, T58), T60, T62)\n\nT57 is ground\nT58 is ground\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(T59, T57), SCOPE: 4, CLAUSE: 4 | less(T59, T57), SCOPE: 4, CLAUSE: 5\n\nT57 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: less(T59, T57), SCOPE: 4, CLAUSE: 4\n\nT57 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(T59, T57), SCOPE: 4, CLAUSE: 5\n\nT57 is ground\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T67, T68)\n\nT67 is ground\nT68 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(less(T80, T78), merge(.(T78, T79), T81, T83))\n\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(less(T80, T78), merge(.(T78, T79), T81, T83)), SCOPE: 5, CLAUSE: 4 | ','(less(T80, T78), merge(.(T78, T79), T81, T83)), SCOPE: 5, CLAUSE: 5\n\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(less(T80, T78), merge(.(T78, T79), T81, T83)), SCOPE: 5, CLAUSE: 4\n\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(less(T80, T78), merge(.(T78, T79), T81, T83)), SCOPE: 5, CLAUSE: 5\n\nT78 is ground\nT79 is ground\nT80 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: merge(.(s(0), T79), T81, T83)\n\nT79 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(less(T90, T91), merge(.(s(T91), T79), T81, T83))\n\nT79 is ground\nT81 is ground\nT90 is ground\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: less(T90, T91)\n\nT90 is ground\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: merge(.(s(T91), T79), T81, T83)\n\nT79 is ground\nT81 is ground\nT91 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 72 [arrowhead = none ];
72 -> 2;

73 [label="EVAL with clause\nmerge([], X2, X2).\nand substitution\nT1 -> []\nT2 -> T5\nX2 -> T5\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 73 [arrowhead = none ];
73 -> 3;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 74 [arrowhead = none ];
74 -> 4;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 75 [arrowhead = none ];
75 -> 5;

76 [label="EVAL with clause\nmerge(X28, [], X28).\nand substitution\nT1 -> T7\nX28 -> T7\nT2 -> []\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="EVAL with clause\nmerge(X5, [], X5).\nand substitution\nX5 -> []\nT5 -> []\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 78 [arrowhead = none ];
78 -> 6;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 79 [arrowhead = none ];
79 -> 7;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 80 [arrowhead = none ];
80 -> 8;

81 [label="BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs))because of non-unification", fontsize=14, style = filled, fillcolor = white];
7 -> 81 [arrowhead = none ];
81 -> 11;

82 [label="BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs))because of non-unification", fontsize=14, style = filled, fillcolor = white];
8 -> 82 [arrowhead = none ];
82 -> 9;

83 [label="BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification", fontsize=14, style = filled, fillcolor = white];
9 -> 83 [arrowhead = none ];
83 -> 10;

84 [label="BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 84 [arrowhead = none ];
84 -> 12;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 85 [arrowhead = none ];
85 -> 15;

86 [label="EVAL with clause\nmerge(.(X45, X46), .(X47, X48), .(X45, X49)) :- ','(leq(X45, X47), merge(X46, .(X47, X48), X49)).\nand substitution\nX45 -> T17\nX46 -> T18\nT1 -> .(T17, T18)\nX47 -> T19\nX48 -> T20\nT2 -> .(T19, T20)\nX49 -> T22\nT3 -> .(T17, T22)\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 86 [arrowhead = none ];
86 -> 18;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 87 [arrowhead = none ];
87 -> 19;

88 [label="BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(X, Zs)) :- ','(leq(X, Y), merge(Xs, .(Y, Ys), Zs))because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 88 [arrowhead = none ];
88 -> 16;

89 [label="BACKTRACK\nfor clause: merge(.(X, Xs), .(Y, Ys), .(Y, Zs)) :- ','(less(Y, X), merge(.(X, Xs), Ys, Zs))because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 89 [arrowhead = none ];
89 -> 17;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 90 [arrowhead = none ];
90 -> 20;

91 [label="EVAL with clause\nmerge(.(X118, X119), .(X120, X121), .(X120, X122)) :- ','(less(X120, X118), merge(.(X118, X119), X121, X122)).\nand substitution\nX118 -> T78\nX119 -> T79\nT1 -> .(T78, T79)\nX120 -> T80\nX121 -> T81\nT2 -> .(T80, T81)\nX122 -> T83\nT3 -> .(T80, T83)\nT82 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 91 [arrowhead = none ];
91 -> 61;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 92 [arrowhead = none ];
92 -> 62;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 93 [arrowhead = none ];
93 -> 21;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 94 [arrowhead = none ];
94 -> 22;

95 [label="EVAL with clause\nleq(0, 0).\nand substitution\nT17 -> 0\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 95 [arrowhead = none ];
95 -> 23;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 96 [arrowhead = none ];
96 -> 24;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 97 [arrowhead = none ];
97 -> 25;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 98 [arrowhead = none ];
98 -> 26;

99 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> .(0, T20)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 99 [arrowhead = none , style=dashed];
99 -> 1[style=dashed];

100 [label="EVAL with clause\nleq(0, s(0)).\nand substitution\nT17 -> 0\nT19 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 100 [arrowhead = none ];
100 -> 27;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 101 [arrowhead = none ];
101 -> 28;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 102 [arrowhead = none ];
102 -> 29;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 103 [arrowhead = none ];
103 -> 30;

104 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> .(s(0), T20)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 104 [arrowhead = none , style=dashed];
104 -> 1[style=dashed];

105 [label="EVAL with clause\nleq(s(X67), s(X68)) :- leq(X67, X68).\nand substitution\nX67 -> T35\nT17 -> s(T35)\nX68 -> T36\nT19 -> s(T36)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 105 [arrowhead = none ];
105 -> 31;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 32;

107 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 107 [arrowhead = none ];
107 -> 48;

108 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 108 [arrowhead = none ];
108 -> 33;

109 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
31 -> 109 [arrowhead = none ];
109 -> 34;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 110 [arrowhead = none ];
110 -> 35;

111 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> .(s(T36), T20)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 111 [arrowhead = none , style=dashed];
111 -> 1[style=dashed];

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 112 [arrowhead = none ];
112 -> 36;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 113 [arrowhead = none ];
113 -> 37;

114 [label="EVAL with clause\nleq(0, 0).\nand substitution\nT35 -> 0\nT36 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 114 [arrowhead = none ];
114 -> 38;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 115 [arrowhead = none ];
115 -> 39;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 116 [arrowhead = none ];
116 -> 41;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 117 [arrowhead = none ];
117 -> 42;

118 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 118 [arrowhead = none ];
118 -> 40;

119 [label="EVAL with clause\nleq(0, s(0)).\nand substitution\nT35 -> 0\nT36 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 119 [arrowhead = none ];
119 -> 43;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 120 [arrowhead = none ];
120 -> 44;

121 [label="EVAL with clause\nleq(s(X75), s(X76)) :- leq(X75, X76).\nand substitution\nX75 -> T41\nT35 -> s(T41)\nX76 -> T42\nT36 -> s(T42)", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 122 [arrowhead = none ];
122 -> 47;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 123 [arrowhead = none ];
123 -> 45;

124 [label="INSTANCE with matching:\nT35 -> T41\nT36 -> T42", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 124 [arrowhead = none , style=dashed];
124 -> 33[style=dashed];

125 [label="EVAL with clause\nmerge(.(X91, X92), .(X93, X94), .(X93, X95)) :- ','(less(X93, X91), merge(.(X91, X92), X94, X95)).\nand substitution\nT17 -> T57\nX91 -> T57\nT18 -> T58\nX92 -> T58\nT19 -> T59\nX93 -> T59\nT20 -> T60\nX94 -> T60\nX95 -> T62\nT3 -> .(T59, T62)\nT61 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 125 [arrowhead = none ];
125 -> 49;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 126 [arrowhead = none ];
126 -> 50;

127 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 127 [arrowhead = none ];
127 -> 51;

128 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
49 -> 128 [arrowhead = none ];
128 -> 52;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 129 [arrowhead = none ];
129 -> 53;

130 [label="INSTANCE with matching:\nT1 -> .(T57, T58)\nT2 -> T60\nT3 -> T62", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 130 [arrowhead = none , style=dashed];
130 -> 1[style=dashed];

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 131 [arrowhead = none ];
131 -> 54;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 132 [arrowhead = none ];
132 -> 55;

133 [label="EVAL with clause\nless(0, s(0)).\nand substitution\nT59 -> 0\nT57 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 133 [arrowhead = none ];
133 -> 56;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 134 [arrowhead = none ];
134 -> 57;

135 [label="EVAL with clause\nless(s(X105), s(X106)) :- less(X105, X106).\nand substitution\nX105 -> T67\nT59 -> s(T67)\nX106 -> T68\nT57 -> s(T68)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 135 [arrowhead = none ];
135 -> 59;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 136 [arrowhead = none ];
136 -> 60;

137 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 137 [arrowhead = none ];
137 -> 58;

138 [label="INSTANCE with matching:\nT59 -> T67\nT57 -> T68", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 138 [arrowhead = none , style=dashed];
138 -> 51[style=dashed];

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 139 [arrowhead = none ];
139 -> 63;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 140 [arrowhead = none ];
140 -> 64;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 141 [arrowhead = none ];
141 -> 65;

142 [label="EVAL with clause\nless(0, s(0)).\nand substitution\nT80 -> 0\nT78 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 142 [arrowhead = none ];
142 -> 66;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 143 [arrowhead = none ];
143 -> 67;

144 [label="EVAL with clause\nless(s(X136), s(X137)) :- less(X136, X137).\nand substitution\nX136 -> T90\nT80 -> s(T90)\nX137 -> T91\nT78 -> s(T91)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 144 [arrowhead = none ];
144 -> 68;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 145 [arrowhead = none ];
145 -> 69;

146 [label="INSTANCE with matching:\nT1 -> .(s(0), T79)\nT2 -> T81\nT3 -> T83", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 146 [arrowhead = none , style=dashed];
146 -> 1[style=dashed];

147 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
68 -> 147 [arrowhead = none ];
147 -> 70;

148 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
68 -> 148 [arrowhead = none ];
148 -> 71;

149 [label="INSTANCE with matching:\nT59 -> T90\nT57 -> T91", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 149 [arrowhead = none , style=dashed];
149 -> 51[style=dashed];

150 [label="INSTANCE with matching:\nT1 -> .(s(T91), T79)\nT2 -> T81\nT3 -> T83", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 150 [arrowhead = none , style=dashed];
150 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

leq33(0, 0).
leq33(0, s(0)).
leq33(s(T41), s(T42)) :- leq33(T41, T42).
less51(0, s(0)).
less51(s(T67), s(T68)) :- less51(T67, T68).
merge1([], T5, T5).
merge1([], [], []).
merge1(T7, [], T7).
merge1(.(0, T18), .(0, T20), .(0, T22)) :- merge1(T18, .(0, T20), T22).
merge1(.(0, T18), .(s(0), T20), .(0, T22)) :- merge1(T18, .(s(0), T20), T22).
merge1(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) :- leq33(T35, T36).
merge1(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) :- ','(leq33(T35, T36), merge1(T18, .(s(T36), T20), T22)).
merge1(.(T57, T58), .(T59, T60), .(T59, T62)) :- less51(T59, T57).
merge1(.(T57, T58), .(T59, T60), .(T59, T62)) :- ','(less51(T59, T57), merge1(.(T57, T58), T60, T62)).
merge1(.(s(0), T79), .(0, T81), .(0, T83)) :- merge1(.(s(0), T79), T81, T83).
merge1(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) :- less51(T90, T91).
merge1(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) :- ','(less51(T90, T91), merge1(.(s(T91), T79), T81, T83)).

Queries:

merge1(g,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:
merge1_in: (b,b,f)
leq33_in: (b,b)
less51_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

(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:

MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → U3_GGA(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → U4_GGA(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_GGA(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → LEQ33_IN_GG(T35, T36)
LEQ33_IN_GG(s(T41), s(T42)) → U1_GG(T41, T42, leq33_in_gg(T41, T42))
LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)
U5_GGA(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_GGA(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
U5_GGA(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_GGA(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → LESS51_IN_GG(T59, T57)
LESS51_IN_GG(s(T67), s(T68)) → U2_GG(T67, T68, less51_in_gg(T67, T68))
LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)
U7_GGA(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_GGA(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
U7_GGA(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → U9_GGA(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_GGA(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → LESS51_IN_GG(T90, T91)
U10_GGA(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_GGA(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U10_GGA(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

The argument filtering Pi contains the following mapping:
merge1_in_gga(x1, x2, x3) = merge1_in_gga(x1, x2)
[] = []
merge1_out_gga(x1, x2, x3) = merge1_out_gga(x1, x2)
.(x1, x2) = .(x1, x2)
0 = 0
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
s(x1) = s(x1)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6)
leq33_in_gg(x1, x2) = leq33_in_gg(x1, x2)
leq33_out_gg(x1, x2) = leq33_out_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x1, x2, x3, x4, x6)
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x2, x3, x4, x6)
less51_in_gg(x1, x2) = less51_in_gg(x1, x2)
less51_out_gg(x1, x2) = less51_out_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
U9_gga(x1, x2, x3, x4) = U9_gga(x1, x2, x4)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6)
U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6)
MERGE1_IN_GGA(x1, x2, x3) = MERGE1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6)
LEQ33_IN_GG(x1, x2) = LEQ33_IN_GG(x1, x2)
U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)
LESS51_IN_GG(x1, x2) = LESS51_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3)
U8_GGA(x1, x2, x3, x4, x5, x6) = U8_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x1, x2, x4)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)

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

(6) Obligation:

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

MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → U3_GGA(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → U4_GGA(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_GGA(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → LEQ33_IN_GG(T35, T36)
LEQ33_IN_GG(s(T41), s(T42)) → U1_GG(T41, T42, leq33_in_gg(T41, T42))
LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)
U5_GGA(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_GGA(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
U5_GGA(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_GGA(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → LESS51_IN_GG(T59, T57)
LESS51_IN_GG(s(T67), s(T68)) → U2_GG(T67, T68, less51_in_gg(T67, T68))
LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)
U7_GGA(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_GGA(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
U7_GGA(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → U9_GGA(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_GGA(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → LESS51_IN_GG(T90, T91)
U10_GGA(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_GGA(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U10_GGA(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)

The TRS R consists of the following rules:

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

(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:

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)

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:

LESS51_IN_GG(s(T67), s(T68)) → LESS51_IN_GG(T67, T68)
The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)

The TRS R consists of the following rules:

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

(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:

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)

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

(21) 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:

LEQ33_IN_GG(s(T41), s(T42)) → LEQ33_IN_GG(T41, T42)
The graph contains the following edges 1 > 1, 2 > 2

(22) YES

(23) Obligation:

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

MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_GGA(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
U7_GGA(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_GGA(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
U5_GGA(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_GGA(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_GGA(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

merge1_in_gga([], T5, T5) → merge1_out_gga([], T5, T5)
merge1_in_gga([], [], []) → merge1_out_gga([], [], [])
merge1_in_gga(T7, [], T7) → merge1_out_gga(T7, [], T7)
merge1_in_gga(.(0, T18), .(0, T20), .(0, T22)) → U3_gga(T18, T20, T22, merge1_in_gga(T18, .(0, T20), T22))
merge1_in_gga(.(0, T18), .(s(0), T20), .(0, T22)) → U4_gga(T18, T20, T22, merge1_in_gga(T18, .(s(0), T20), T22))
merge1_in_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_gga(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U5_gga(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → U6_gga(T35, T18, T36, T20, T22, merge1_in_gga(T18, .(s(T36), T20), T22))
merge1_in_gga(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_gga(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U7_gga(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → U8_gga(T57, T58, T59, T60, T62, merge1_in_gga(.(T57, T58), T60, T62))
merge1_in_gga(.(s(0), T79), .(0, T81), .(0, T83)) → U9_gga(T79, T81, T83, merge1_in_gga(.(s(0), T79), T81, T83))
merge1_in_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_gga(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U10_gga(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → U11_gga(T91, T79, T90, T81, T83, merge1_in_gga(.(s(T91), T79), T81, T83))
U11_gga(T91, T79, T90, T81, T83, merge1_out_gga(.(s(T91), T79), T81, T83)) → merge1_out_gga(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83))
U9_gga(T79, T81, T83, merge1_out_gga(.(s(0), T79), T81, T83)) → merge1_out_gga(.(s(0), T79), .(0, T81), .(0, T83))
U8_gga(T57, T58, T59, T60, T62, merge1_out_gga(.(T57, T58), T60, T62)) → merge1_out_gga(.(T57, T58), .(T59, T60), .(T59, T62))
U6_gga(T35, T18, T36, T20, T22, merge1_out_gga(T18, .(s(T36), T20), T22)) → merge1_out_gga(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22))
U4_gga(T18, T20, T22, merge1_out_gga(T18, .(s(0), T20), T22)) → merge1_out_gga(.(0, T18), .(s(0), T20), .(0, T22))
U3_gga(T18, T20, T22, merge1_out_gga(T18, .(0, T20), T22)) → merge1_out_gga(.(0, T18), .(0, T20), .(0, T22))

The argument filtering Pi contains the following mapping:
merge1_in_gga(x1, x2, x3) = merge1_in_gga(x1, x2)
[] = []
merge1_out_gga(x1, x2, x3) = merge1_out_gga(x1, x2)
.(x1, x2) = .(x1, x2)
0 = 0
U3_gga(x1, x2, x3, x4) = U3_gga(x1, x2, x4)
s(x1) = s(x1)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
U5_gga(x1, x2, x3, x4, x5, x6) = U5_gga(x1, x2, x3, x4, x6)
leq33_in_gg(x1, x2) = leq33_in_gg(x1, x2)
leq33_out_gg(x1, x2) = leq33_out_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
U6_gga(x1, x2, x3, x4, x5, x6) = U6_gga(x1, x2, x3, x4, x6)
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x2, x3, x4, x6)
less51_in_gg(x1, x2) = less51_in_gg(x1, x2)
less51_out_gg(x1, x2) = less51_out_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
U9_gga(x1, x2, x3, x4) = U9_gga(x1, x2, x4)
U10_gga(x1, x2, x3, x4, x5, x6) = U10_gga(x1, x2, x3, x4, x6)
U11_gga(x1, x2, x3, x4, x5, x6) = U11_gga(x1, x2, x3, x4, x6)
MERGE1_IN_GGA(x1, x2, x3) = MERGE1_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6)

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

(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:

MERGE1_IN_GGA(.(T57, T58), .(T59, T60), .(T59, T62)) → U7_GGA(T57, T58, T59, T60, T62, less51_in_gg(T59, T57))
U7_GGA(T57, T58, T59, T60, T62, less51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60, T62)
MERGE1_IN_GGA(.(0, T18), .(0, T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(0, T20), T22)
MERGE1_IN_GGA(.(s(0), T79), .(0, T81), .(0, T83)) → MERGE1_IN_GGA(.(s(0), T79), T81, T83)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20), .(s(T35), T22)) → U5_GGA(T35, T18, T36, T20, T22, leq33_in_gg(T35, T36))
U5_GGA(T35, T18, T36, T20, T22, leq33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20), T22)
MERGE1_IN_GGA(.(0, T18), .(s(0), T20), .(0, T22)) → MERGE1_IN_GGA(T18, .(s(0), T20), T22)
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81), .(s(T90), T83)) → U10_GGA(T91, T79, T90, T81, T83, less51_in_gg(T90, T91))
U10_GGA(T91, T79, T90, T81, T83, less51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81, T83)

The TRS R consists of the following rules:

less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
0 = 0
s(x1) = s(x1)
leq33_in_gg(x1, x2) = leq33_in_gg(x1, x2)
leq33_out_gg(x1, x2) = leq33_out_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
less51_in_gg(x1, x2) = less51_in_gg(x1, x2)
less51_out_gg(x1, x2) = less51_out_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
MERGE1_IN_GGA(x1, x2, x3) = MERGE1_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6)

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:

MERGE1_IN_GGA(.(T57, T58), .(T59, T60)) → U7_GGA(T57, T58, T59, T60, less51_in_gg(T59, T57))
U7_GGA(T57, T58, T59, T60, less51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60)
MERGE1_IN_GGA(.(0, T18), .(0, T20)) → MERGE1_IN_GGA(T18, .(0, T20))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81)) → MERGE1_IN_GGA(.(s(0), T79), T81)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20)) → U5_GGA(T35, T18, T36, T20, leq33_in_gg(T35, T36))
U5_GGA(T35, T18, T36, T20, leq33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20)) → MERGE1_IN_GGA(T18, .(s(0), T20))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81)) → U10_GGA(T91, T79, T90, T81, less51_in_gg(T90, T91))
U10_GGA(T91, T79, T90, T81, less51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81)

The TRS R consists of the following rules:

less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))

The set Q consists of the following terms:

less51_in_gg(x0, x1)
leq33_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U1_gg(x0, x1, x2)

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

(28) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MERGE1_IN_GGA(.(T57, T58), .(T59, T60)) → U7_GGA(T57, T58, T59, T60, less51_in_gg(T59, T57))
U7_GGA(T57, T58, T59, T60, less51_out_gg(T59, T57)) → MERGE1_IN_GGA(.(T57, T58), T60)
MERGE1_IN_GGA(.(0, T18), .(0, T20)) → MERGE1_IN_GGA(T18, .(0, T20))
MERGE1_IN_GGA(.(s(0), T79), .(0, T81)) → MERGE1_IN_GGA(.(s(0), T79), T81)
MERGE1_IN_GGA(.(s(T35), T18), .(s(T36), T20)) → U5_GGA(T35, T18, T36, T20, leq33_in_gg(T35, T36))
MERGE1_IN_GGA(.(0, T18), .(s(0), T20)) → MERGE1_IN_GGA(T18, .(s(0), T20))
MERGE1_IN_GGA(.(s(T91), T79), .(s(T90), T81)) → U10_GGA(T91, T79, T90, T81, less51_in_gg(T90, T91))
U10_GGA(T91, T79, T90, T81, less51_out_gg(T90, T91)) → MERGE1_IN_GGA(.(s(T91), T79), T81)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(0) = 0
POL(MERGE1_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(U10_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + 2·x₁ + 2·x₂ + x₃ + 2·x₄ + 2·x₅
POL(U1_gg(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U2_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U5_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U7_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + 2·x₂ + x₃ + x₄ + x₅
POL(leq33_in_gg(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(leq33_out_gg(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(less51_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(less51_out_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 2·x₁

(29) Obligation:

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

U5_GGA(T35, T18, T36, T20, leq33_out_gg(T35, T36)) → MERGE1_IN_GGA(T18, .(s(T36), T20))

The TRS R consists of the following rules:

less51_in_gg(0, s(0)) → less51_out_gg(0, s(0))
less51_in_gg(s(T67), s(T68)) → U2_gg(T67, T68, less51_in_gg(T67, T68))
leq33_in_gg(0, 0) → leq33_out_gg(0, 0)
leq33_in_gg(0, s(0)) → leq33_out_gg(0, s(0))
leq33_in_gg(s(T41), s(T42)) → U1_gg(T41, T42, leq33_in_gg(T41, T42))
U2_gg(T67, T68, less51_out_gg(T67, T68)) → less51_out_gg(s(T67), s(T68))
U1_gg(T41, T42, leq33_out_gg(T41, T42)) → leq33_out_gg(s(T41), s(T42))

The set Q consists of the following terms:

less51_in_gg(x0, x1)
leq33_in_gg(x0, x1)
U2_gg(x0, x1, x2)
U1_gg(x0, x1, x2)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(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 (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) MRRProof (EQUIVALENT transformation)

(29) Obligation:

(30) DependencyGraphProof (EQUIVALENT transformation)

(31) TRUE