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

(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).

Query: merge(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: 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\nX2 is free\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\nX4 is free\nmerge([], T5, T3) !=? merge(X4, [], X4)", 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\nX4 is free\nmerge([], T5, T3) !=? merge(X4, [], X4)", 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\nX2 is free\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\nX2 is free\nX26 is free\nmerge(T1, T2, T3) !=? merge([], X2, X2)\nmerge(T1, T2, T3) !=? merge(X26, [], X26)", 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\nX2 is free\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\nX2 is free\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\nX2 is free\nX26 is free\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(T1, T2, T3) !=? merge([], X2, X2)\nmerge(T1, T2, T3) !=? merge(X26, [], X26)\nmerge(T1, T2, T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", 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="B: 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="C: 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(T76, T74), merge(.(T74, T75), T77, T79))\n\nT74 is ground\nT75 is ground\nT76 is ground\nT77 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(T74, T75), .(T76, T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(less(T76, T74), merge(.(T74, T75), T77, T79)), SCOPE: 5, CLAUSE: 4 | ','(less(T76, T74), merge(.(T74, T75), T77, T79)), SCOPE: 5, CLAUSE: 5\n\nT74 is ground\nT75 is ground\nT76 is ground\nT77 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(T74, T75), .(T76, T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(less(T76, T74), merge(.(T74, T75), T77, T79)), SCOPE: 5, CLAUSE: 4\n\nT74 is ground\nT75 is ground\nT76 is ground\nT77 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(T74, T75), .(T76, T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(less(T76, T74), merge(.(T74, T75), T77, T79)), SCOPE: 5, CLAUSE: 5\n\nT74 is ground\nT75 is ground\nT76 is ground\nT77 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(T74, T75), .(T76, T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: merge(.(s(0), T75), T77, T79)\n\nT75 is ground\nT77 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(s(0), T75), .(0, T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(less(T84, T85), merge(.(s(T85), T75), T77, T79))\n\nT75 is ground\nT77 is ground\nT84 is ground\nT85 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(s(T85), T75), .(s(T84), T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: less(T84, T85)\n\nT75 is ground\nT77 is ground\nT84 is ground\nT85 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(s(T85), T75), .(s(T84), T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: merge(.(s(T85), T75), T77, T79)\n\nT75 is ground\nT77 is ground\nT84 is ground\nT85 is ground\nX42 is free\nX43 is free\nX44 is free\nX45 is free\nX46 is free\nmerge(.(s(T85), T75), .(s(T84), T77), T3) !=? merge(.(X42, X43), .(X44, X45), .(X42, X46))", 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 substitutionT1 -> [],\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 = lightgreen];
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(X26, [], X26).\nand substitutionT1 -> T7,\nX26 -> 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 = lightgreen];
4 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="EVAL with clause\nmerge(X4, [], X4).\nand substitutionX4 -> [],\nT5 -> [],\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 78 [arrowhead = none ];
78 -> 6;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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(.(X42, X43), .(X44, X45), .(X42, X46)) :- ','(leq(X42, X44), merge(X43, .(X44, X45), X46)).\nand substitutionX42 -> T17,\nX43 -> T18,\nT1 -> .(T17, T18),\nX44 -> T19,\nX45 -> T20,\nT2 -> .(T19, T20),\nX46 -> 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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
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(.(X97, X98), .(X99, X100), .(X99, X101)) :- ','(less(X99, X97), merge(.(X97, X98), X100, X101)).\nand substitutionX97 -> T74,\nX98 -> T75,\nT1 -> .(T74, T75),\nX99 -> T76,\nX100 -> T77,\nT2 -> .(T76, T77),\nX101 -> T79,\nT3 -> .(T76, T79),\nT78 -> T79", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 91 [arrowhead = none ];
91 -> 61;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT17 -> 0,\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 95 [arrowhead = none ];
95 -> 23;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT17 -> 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 = lightgreen];
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(X59), s(X60)) :- leq(X59, X60).\nand substitutionX59 -> T35,\nT17 -> s(T35),\nX60 -> 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 = lightgreen];
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\nnew knowledge:\nT35 is ground\nT36 is ground", 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 substitutionT35 -> 0,\nT36 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 114 [arrowhead = none ];
114 -> 38;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT35 -> 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 = lightgreen];
41 -> 120 [arrowhead = none ];
120 -> 44;

121 [label="EVAL with clause\nleq(s(X65), s(X66)) :- leq(X65, X66).\nand substitutionX65 -> T41,\nT35 -> s(T41),\nX66 -> 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 = lightgreen];
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(.(X79, X80), .(X81, X82), .(X81, X83)) :- ','(less(X81, X79), merge(.(X79, X80), X82, X83)).\nand substitutionT17 -> T57,\nX79 -> T57,\nT18 -> T58,\nX80 -> T58,\nT19 -> T59,\nX81 -> T59,\nT20 -> T60,\nX82 -> T60,\nX83 -> 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 = lightgreen];
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\nnew knowledge:\nT59 is ground\nT57 is ground", 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 substitutionT59 -> 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 = lightgreen];
54 -> 134 [arrowhead = none ];
134 -> 57;

135 [label="EVAL with clause\nless(s(X88), s(X89)) :- less(X88, X89).\nand substitutionX88 -> T67,\nT59 -> s(T67),\nX89 -> 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 = lightgreen];
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 substitutionT76 -> 0,\nT74 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 142 [arrowhead = none ];
142 -> 66;

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

144 [label="EVAL with clause\nless(s(X108), s(X109)) :- less(X108, X109).\nand substitutionX108 -> T84,\nT76 -> s(T84),\nX109 -> T85,\nT74 -> s(T85)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 144 [arrowhead = none ];
144 -> 68;

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

146 [label="INSTANCE with matching:\nT1 -> .(s(0), T75)\nT2 -> T77\nT3 -> T79", 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\nnew knowledge:\nT84 is ground\nT85 is ground", fontsize=14, style = filled, fillcolor = violet];
68 -> 148 [arrowhead = none ];
148 -> 71;

149 [label="INSTANCE with matching:\nT59 -> T84\nT57 -> T85", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 149 [arrowhead = none , style=dashed];
149 -> 51[style=dashed];

150 [label="INSTANCE with matching:\nT1 -> .(s(T85), T75)\nT2 -> T77\nT3 -> T79", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 150 [arrowhead = none , style=dashed];
150 -> 1[style=dashed];

}

(2) Obligation:

Triples:

leqB(s(X1), s(X2)) :- leqB(X1, X2).
lessC(s(X1), s(X2)) :- lessC(X1, X2).
mergeA(.(0, X1), .(0, X2), .(0, X3)) :- mergeA(X1, .(0, X2), X3).
mergeA(.(0, X1), .(s(0), X2), .(0, X3)) :- mergeA(X1, .(s(0), X2), X3).
mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) :- leqB(X1, X3).
mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) :- ','(leqcB(X1, X3), mergeA(X2, .(s(X3), X4), X5)).
mergeA(.(X1, X2), .(X3, X4), .(X3, X5)) :- lessC(X3, X1).
mergeA(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscC(X3, X1), mergeA(.(X1, X2), X4, X5)).
mergeA(.(s(0), X1), .(0, X2), .(0, X3)) :- mergeA(.(s(0), X1), X2, X3).
mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) :- lessC(X3, X1).
mergeA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) :- ','(lesscC(X3, X1), mergeA(.(s(X1), X2), X4, X5)).

Clauses:

mergecA([], X1, X1).
mergecA([], [], []).
mergecA(X1, [], X1).
mergecA(.(0, X1), .(0, X2), .(0, X3)) :- mergecA(X1, .(0, X2), X3).
mergecA(.(0, X1), .(s(0), X2), .(0, X3)) :- mergecA(X1, .(s(0), X2), X3).
mergecA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) :- ','(leqcB(X1, X3), mergecA(X2, .(s(X3), X4), X5)).
mergecA(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(lesscC(X3, X1), mergecA(.(X1, X2), X4, X5)).
mergecA(.(s(0), X1), .(0, X2), .(0, X3)) :- mergecA(.(s(0), X1), X2, X3).
mergecA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) :- ','(lesscC(X3, X1), mergecA(.(s(X1), X2), X4, X5)).
leqcB(0, 0).
leqcB(0, s(0)).
leqcB(s(X1), s(X2)) :- leqcB(X1, X2).
lesscC(0, s(0)).
lesscC(s(X1), s(X2)) :- lesscC(X1, X2).

Afs:

mergeA(x1, x2, x3)  =  mergeA(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergeA_in: (b,b,f)
leqB_in: (b,b)
leqcB_in: (b,b)
lessC_in: (b,b)
lesscC_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) → U3_GGA(X1, X2, X3, mergeA_in_gga(X1, .(0, X2), X3))
MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) → MERGEA_IN_GGA(X1, .(0, X2), X3)
MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) → U4_GGA(X1, X2, X3, mergeA_in_gga(X1, .(s(0), X2), X3))
MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) → MERGEA_IN_GGA(X1, .(s(0), X2), X3)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → U5_GGA(X1, X2, X3, X4, X5, leqB_in_gg(X1, X3))
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → LEQB_IN_GG(X1, X3)
LEQB_IN_GG(s(X1), s(X2)) → U1_GG(X1, X2, leqB_in_gg(X1, X2))
LEQB_IN_GG(s(X1), s(X2)) → LEQB_IN_GG(X1, X2)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → U6_GGA(X1, X2, X3, X4, X5, leqcB_in_gg(X1, X3))
U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) → U7_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(X2, .(s(X3), X4), X5))
U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) → MERGEA_IN_GGA(X2, .(s(X3), X4), X5)
MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U8_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1))
MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → LESSC_IN_GG(X3, X1)
LESSC_IN_GG(s(X1), s(X2)) → U2_GG(X1, X2, lessC_in_gg(X1, X2))
LESSC_IN_GG(s(X1), s(X2)) → LESSC_IN_GG(X1, X2)
MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U9_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1))
U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → U10_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(X1, X2), X4, X5))
U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(X1, X2), X4, X5)
MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) → U11_GGA(X1, X2, X3, mergeA_in_gga(.(s(0), X1), X2, X3))
MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) → MERGEA_IN_GGA(.(s(0), X1), X2, X3)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → U12_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1))
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → LESSC_IN_GG(X3, X1)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → U13_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1))
U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → U14_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(s(X1), X2), X4, X5))
U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(s(X1), X2), X4, X5)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
mergeA_in_gga(x1, x2, x3)  =  mergeA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
leqB_in_gg(x1, x2)  =  leqB_in_gg(x1, x2)
leqcB_in_gg(x1, x2)  =  leqcB_in_gg(x1, x2)
leqcB_out_gg(x1, x2)  =  leqcB_out_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
lesscC_in_gg(x1, x2)  =  lesscC_in_gg(x1, x2)
lesscC_out_gg(x1, x2)  =  lesscC_out_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
MERGEA_IN_GGA(x1, x2, x3)  =  MERGEA_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)
LEQB_IN_GG(x1, x2)  =  LEQB_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)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x2, x3, x4, x6)
LESSC_IN_GG(x1, x2)  =  LESSC_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U9_GGA(x1, x2, x3, x4, x5, x6)  =  U9_GGA(x1, x2, x3, x4, x6)
U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)

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:

MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) → U3_GGA(X1, X2, X3, mergeA_in_gga(X1, .(0, X2), X3))
MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) → MERGEA_IN_GGA(X1, .(0, X2), X3)
MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) → U4_GGA(X1, X2, X3, mergeA_in_gga(X1, .(s(0), X2), X3))
MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) → MERGEA_IN_GGA(X1, .(s(0), X2), X3)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → U5_GGA(X1, X2, X3, X4, X5, leqB_in_gg(X1, X3))
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → LEQB_IN_GG(X1, X3)
LEQB_IN_GG(s(X1), s(X2)) → U1_GG(X1, X2, leqB_in_gg(X1, X2))
LEQB_IN_GG(s(X1), s(X2)) → LEQB_IN_GG(X1, X2)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → U6_GGA(X1, X2, X3, X4, X5, leqcB_in_gg(X1, X3))
U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) → U7_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(X2, .(s(X3), X4), X5))
U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) → MERGEA_IN_GGA(X2, .(s(X3), X4), X5)
MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U8_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1))
MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → LESSC_IN_GG(X3, X1)
LESSC_IN_GG(s(X1), s(X2)) → U2_GG(X1, X2, lessC_in_gg(X1, X2))
LESSC_IN_GG(s(X1), s(X2)) → LESSC_IN_GG(X1, X2)
MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U9_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1))
U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → U10_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(X1, X2), X4, X5))
U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(X1, X2), X4, X5)
MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) → U11_GGA(X1, X2, X3, mergeA_in_gga(.(s(0), X1), X2, X3))
MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) → MERGEA_IN_GGA(.(s(0), X1), X2, X3)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → U12_GGA(X1, X2, X3, X4, X5, lessC_in_gg(X3, X1))
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → LESSC_IN_GG(X3, X1)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → U13_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1))
U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → U14_GGA(X1, X2, X3, X4, X5, mergeA_in_gga(.(s(X1), X2), X4, X5))
U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(s(X1), X2), X4, X5)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
mergeA_in_gga(x1, x2, x3)  =  mergeA_in_gga(x1, x2)
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
leqB_in_gg(x1, x2)  =  leqB_in_gg(x1, x2)
leqcB_in_gg(x1, x2)  =  leqcB_in_gg(x1, x2)
leqcB_out_gg(x1, x2)  =  leqcB_out_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
lessC_in_gg(x1, x2)  =  lessC_in_gg(x1, x2)
lesscC_in_gg(x1, x2)  =  lesscC_in_gg(x1, x2)
lesscC_out_gg(x1, x2)  =  lesscC_out_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
MERGEA_IN_GGA(x1, x2, x3)  =  MERGEA_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)
LEQB_IN_GG(x1, x2)  =  LEQB_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)
U8_GGA(x1, x2, x3, x4, x5, x6)  =  U8_GGA(x1, x2, x3, x4, x6)
LESSC_IN_GG(x1, x2)  =  LESSC_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x2, x3)
U9_GGA(x1, x2, x3, x4, x5, x6)  =  U9_GGA(x1, x2, x3, x4, x6)
U10_GGA(x1, x2, x3, x4, x5, x6)  =  U10_GGA(x1, x2, x3, x4, x6)
U11_GGA(x1, x2, x3, x4)  =  U11_GGA(x1, x2, x4)
U12_GGA(x1, x2, x3, x4, x5, x6)  =  U12_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)
U14_GGA(x1, x2, x3, x4, x5, x6)  =  U14_GGA(x1, x2, x3, x4, x6)

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:

LESSC_IN_GG(s(X1), s(X2)) → LESSC_IN_GG(X1, X2)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

Pi is empty.
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:

LESSC_IN_GG(s(X1), s(X2)) → LESSC_IN_GG(X1, X2)

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

(10) PiDPToQDPProof (EQUIVALENT 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:

LESSC_IN_GG(s(X1), s(X2)) → LESSC_IN_GG(X1, X2)

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:

• LESSC_IN_GG(s(X1), s(X2)) → LESSC_IN_GG(X1, X2)
  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:

LEQB_IN_GG(s(X1), s(X2)) → LEQB_IN_GG(X1, X2)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

Pi is empty.
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:

LEQB_IN_GG(s(X1), s(X2)) → LEQB_IN_GG(X1, X2)

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

(17) PiDPToQDPProof (EQUIVALENT 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:

LEQB_IN_GG(s(X1), s(X2)) → LEQB_IN_GG(X1, X2)

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:

• LEQB_IN_GG(s(X1), s(X2)) → LEQB_IN_GG(X1, X2)
  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:

MERGEA_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U9_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1))
U9_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(X1, X2), X4, X5)
MERGEA_IN_GGA(.(0, X1), .(0, X2), .(0, X3)) → MERGEA_IN_GGA(X1, .(0, X2), X3)
MERGEA_IN_GGA(.(s(0), X1), .(0, X2), .(0, X3)) → MERGEA_IN_GGA(.(s(0), X1), X2, X3)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X1), X5)) → U6_GGA(X1, X2, X3, X4, X5, leqcB_in_gg(X1, X3))
U6_GGA(X1, X2, X3, X4, X5, leqcB_out_gg(X1, X3)) → MERGEA_IN_GGA(X2, .(s(X3), X4), X5)
MERGEA_IN_GGA(.(0, X1), .(s(0), X2), .(0, X3)) → MERGEA_IN_GGA(X1, .(s(0), X2), X3)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4), .(s(X3), X5)) → U13_GGA(X1, X2, X3, X4, X5, lesscC_in_gg(X3, X1))
U13_GGA(X1, X2, X3, X4, X5, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(s(X1), X2), X4, X5)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
0  =  0
s(x1)  =  s(x1)
leqcB_in_gg(x1, x2)  =  leqcB_in_gg(x1, x2)
leqcB_out_gg(x1, x2)  =  leqcB_out_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
lesscC_in_gg(x1, x2)  =  lesscC_in_gg(x1, x2)
lesscC_out_gg(x1, x2)  =  lesscC_out_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
MERGEA_IN_GGA(x1, x2, x3)  =  MERGEA_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4, x5, x6)  =  U9_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6)  =  U13_GGA(x1, x2, x3, x4, x6)

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

(22) PiDPToQDPProof (SOUND transformation)

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

(23) Obligation:

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

MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) → U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1))
U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(X1, X2), X4)
MERGEA_IN_GGA(.(0, X1), .(0, X2)) → MERGEA_IN_GGA(X1, .(0, X2))
MERGEA_IN_GGA(.(s(0), X1), .(0, X2)) → MERGEA_IN_GGA(.(s(0), X1), X2)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) → U6_GGA(X1, X2, X3, X4, leqcB_in_gg(X1, X3))
U6_GGA(X1, X2, X3, X4, leqcB_out_gg(X1, X3)) → MERGEA_IN_GGA(X2, .(s(X3), X4))
MERGEA_IN_GGA(.(0, X1), .(s(0), X2)) → MERGEA_IN_GGA(X1, .(s(0), X2))
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) → U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1))
U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(s(X1), X2), X4)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

leqcB_in_gg(x0, x1)
U25_gg(x0, x1, x2)
lesscC_in_gg(x0, x1)
U26_gg(x0, x1, x2)

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

(24) 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:

MERGEA_IN_GGA(.(X1, X2), .(X3, X4)) → U9_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1))
MERGEA_IN_GGA(.(0, X1), .(0, X2)) → MERGEA_IN_GGA(X1, .(0, X2))
MERGEA_IN_GGA(.(s(0), X1), .(0, X2)) → MERGEA_IN_GGA(.(s(0), X1), X2)
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) → U6_GGA(X1, X2, X3, X4, leqcB_in_gg(X1, X3))
U6_GGA(X1, X2, X3, X4, leqcB_out_gg(X1, X3)) → MERGEA_IN_GGA(X2, .(s(X3), X4))
MERGEA_IN_GGA(.(0, X1), .(s(0), X2)) → MERGEA_IN_GGA(X1, .(s(0), X2))
MERGEA_IN_GGA(.(s(X1), X2), .(s(X3), X4)) → U13_GGA(X1, X2, X3, X4, lesscC_in_gg(X3, X1))
U13_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(s(X1), X2), X4)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2 + x₁ + x₂   
POL(0) = 0   
POL(MERGEA_IN_GGA(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(U13_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅   
POL(U25_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U26_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U6_GGA(x₁, x₂, x₃, x₄, x₅)) = 1 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅   
POL(U9_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + 2·x₂ + x₃ + 2·x₄ + x₅   
POL(leqcB_in_gg(x₁, x₂)) = 2 + x₁ + x₂   
POL(leqcB_out_gg(x₁, x₂)) = 2 + x₁ + x₂   
POL(lesscC_in_gg(x₁, x₂)) = 2 + x₁ + x₂   
POL(lesscC_out_gg(x₁, x₂)) = 2 + x₁ + x₂   
POL(s(x₁)) = 2·x₁   

(25) Obligation:

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

U9_GGA(X1, X2, X3, X4, lesscC_out_gg(X3, X1)) → MERGEA_IN_GGA(.(X1, X2), X4)

The TRS R consists of the following rules:

leqcB_in_gg(0, 0) → leqcB_out_gg(0, 0)
leqcB_in_gg(0, s(0)) → leqcB_out_gg(0, s(0))
leqcB_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, leqcB_in_gg(X1, X2))
U25_gg(X1, X2, leqcB_out_gg(X1, X2)) → leqcB_out_gg(s(X1), s(X2))
lesscC_in_gg(0, s(0)) → lesscC_out_gg(0, s(0))
lesscC_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, lesscC_in_gg(X1, X2))
U26_gg(X1, X2, lesscC_out_gg(X1, X2)) → lesscC_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

leqcB_in_gg(x0, x1)
U25_gg(x0, x1, x2)
lesscC_in_gg(x0, x1)
U26_gg(x0, x1, x2)

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

(26) DependencyGraphProof (EQUIVALENT transformation)

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

(27) TRUE