Left Termination proof of /tmp/tmptHK3S5/pl4.4.3.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 MRRProof (⇔)

        ↳31 QDP

        ↳32 DependencyGraphProof (⇔)

        ↳33 QDP

        ↳34 UsableRulesProof (⇔)

        ↳35 QDP

        ↳36 QReductionProof (⇔)

        ↳37 QDP

        ↳38 QDPSizeChangeProof (⇔)

        ↳39 YES

(0) Obligation:

Clauses:

merge(X, [], X).
merge([], X, X).
merge(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z)).
merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), zero).
le(s(X), s(Y)) :- le(X, Y).
le(zero, s(Y)).
le(zero, zero).

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([], T11, T3), SCOPE: 1, CLAUSE: 2 | merge([], T11, T3), SCOPE: 1, CLAUSE: 3\n\nT11 is ground\nmerge([], T11, 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([], T11, T3), SCOPE: 1, CLAUSE: 2 | merge([], T11, T3), SCOPE: 1, CLAUSE: 3\n\nT11 is ground\nmerge([], T11, T3) !=? merge(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: merge([], T11, T3), SCOPE: 1, CLAUSE: 3\n\nT11 is ground\nmerge([], T11, T3) !=? merge(X2, [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(le(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: ','(le(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 6 | ','(le(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 7 | ','(le(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: ','(le(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: ','(le(T17, T19), merge(T18, .(T19, T20), T22)), SCOPE: 2, CLAUSE: 7 | ','(le(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: ','(le(T31, T32), merge(T18, .(s(T32), T20), T22))\n\nT18 is ground\nT20 is ground\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: le(T31, T32)\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: merge(T18, .(s(T32), T20), T22)\n\nT18 is ground\nT20 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: le(T31, T32), SCOPE: 3, CLAUSE: 6 | le(T31, T32), SCOPE: 3, CLAUSE: 7 | le(T31, T32), SCOPE: 3, CLAUSE: 8\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: le(T31, T32), SCOPE: 3, CLAUSE: 6\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: le(T31, T32), SCOPE: 3, CLAUSE: 7 | le(T31, T32), SCOPE: 3, CLAUSE: 8\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: le(T45, T46)\n\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: le(T31, T32), SCOPE: 3, CLAUSE: 7\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T31, T32), SCOPE: 3, CLAUSE: 8\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(le(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];
41 [label="41: ','(le(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];
42 [label="42: merge(T18, .(s(T64), T20), T22)\n\nT18 is ground\nT20 is ground\nT64 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(le(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];
45 [label="45: ?_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];
46 [label="46: merge(T18, .(zero, T20), T22)\n\nT18 is ground\nT20 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: ','(gt(T85, T87), merge(.(T85, T86), T88, T90))\n\nT85 is ground\nT86 is ground\nT87 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: gt(T85, T87)\n\nT85 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: merge(.(T85, T86), T88, T90)\n\nT85 is ground\nT86 is ground\nT88 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: gt(T85, T87), SCOPE: 4, CLAUSE: 4 | gt(T85, T87), SCOPE: 4, CLAUSE: 5\n\nT85 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: gt(T85, T87), SCOPE: 4, CLAUSE: 4\n\nT85 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: gt(T85, T87), SCOPE: 4, CLAUSE: 5\n\nT85 is ground\nT87 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: gt(T103, T104)\n\nT103 is ground\nT104 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(gt(T119, T121), merge(.(T119, T120), T122, T124))\n\nT119 is ground\nT120 is ground\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(gt(T119, T121), merge(.(T119, T120), T122, T124)), SCOPE: 5, CLAUSE: 4 | ','(gt(T119, T121), merge(.(T119, T120), T122, T124)), SCOPE: 5, CLAUSE: 5\n\nT119 is ground\nT120 is ground\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(gt(T119, T121), merge(.(T119, T120), T122, T124)), SCOPE: 5, CLAUSE: 4\n\nT119 is ground\nT120 is ground\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(gt(T119, T121), merge(.(T119, T120), T122, T124)), SCOPE: 5, CLAUSE: 5\n\nT119 is ground\nT120 is ground\nT121 is ground\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(gt(T133, T134), merge(.(s(T133), T120), T122, T124))\n\nT120 is ground\nT122 is ground\nT133 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: gt(T133, T134)\n\nT133 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: merge(.(s(T133), T120), T122, T124)\n\nT120 is ground\nT122 is ground\nT133 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: merge(.(s(T145), T120), T122, T124)\n\nT120 is ground\nT122 is ground\nT145 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", 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 -> T5\nX2 -> T5\nT2 -> []\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 -> []\nT2 -> T11\nX28 -> T11\nT3 -> T11", 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\nT5 -> []\nX5 -> []\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(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
7 -> 81 [arrowhead = none ];
81 -> 11;

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

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

84 [label="BACKTRACK\nfor clause: merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z))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)) :- ','(le(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(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 88 [arrowhead = none ];
88 -> 16;

89 [label="BACKTRACK\nfor clause: merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z))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(.(X158, X159), .(X160, X161), .(X160, X162)) :- ','(gt(X158, X160), merge(.(X158, X159), X161, X162)).\nand substitution\nX158 -> T119\nX159 -> T120\nT1 -> .(T119, T120)\nX160 -> T121\nX161 -> T122\nT2 -> .(T121, T122)\nX162 -> T124\nT3 -> .(T121, T124)\nT123 -> T124", 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\nle(s(X63), s(X64)) :- le(X63, X64).\nand substitution\nX63 -> T31\nT17 -> s(T31)\nX64 -> T32\nT19 -> s(T32)", 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 -> 40;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 98 [arrowhead = none ];
98 -> 41;

99 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 99 [arrowhead = none ];
99 -> 25;

100 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
23 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> .(s(T32), T20)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 102 [arrowhead = none , style=dashed];
102 -> 1[style=dashed];

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 103 [arrowhead = none ];
103 -> 28;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 104 [arrowhead = none ];
104 -> 29;

105 [label="EVAL with clause\nle(s(X79), s(X80)) :- le(X79, X80).\nand substitution\nX79 -> T45\nT31 -> s(T45)\nX80 -> T46\nT32 -> s(T46)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 105 [arrowhead = none ];
105 -> 30;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 106 [arrowhead = none ];
106 -> 31;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 107 [arrowhead = none ];
107 -> 32;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 108 [arrowhead = none ];
108 -> 33;

109 [label="INSTANCE with matching:\nT31 -> T45\nT32 -> T46", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 109 [arrowhead = none , style=dashed];
109 -> 25[style=dashed];

110 [label="EVAL with clause\nle(zero, s(X89)).\nand substitution\nT31 -> zero\nX89 -> T53\nT32 -> s(T53)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 110 [arrowhead = none ];
110 -> 34;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 111 [arrowhead = none ];
111 -> 35;

112 [label="EVAL with clause\nle(zero, zero).\nand substitution\nT31 -> zero\nT32 -> zero", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 112 [arrowhead = none ];
112 -> 37;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 113 [arrowhead = none ];
113 -> 38;

114 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 114 [arrowhead = none ];
114 -> 36;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 115 [arrowhead = none ];
115 -> 39;

116 [label="EVAL with clause\nle(zero, s(X99)).\nand substitution\nT17 -> zero\nX99 -> T64\nT19 -> s(T64)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 116 [arrowhead = none ];
116 -> 42;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 117 [arrowhead = none ];
117 -> 43;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 118 [arrowhead = none ];
118 -> 44;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
41 -> 119 [arrowhead = none ];
119 -> 45;

120 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> .(s(T64), T20)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 120 [arrowhead = none , style=dashed];
120 -> 1[style=dashed];

121 [label="EVAL with clause\nle(zero, zero).\nand substitution\nT17 -> zero\nT19 -> zero", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 121 [arrowhead = none ];
121 -> 46;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 122 [arrowhead = none ];
122 -> 47;

123 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> .(zero, T20)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 124 [arrowhead = none , style=dashed];
124 -> 1[style=dashed];

125 [label="EVAL with clause\nmerge(.(X119, X120), .(X121, X122), .(X121, X123)) :- ','(gt(X119, X121), merge(.(X119, X120), X122, X123)).\nand substitution\nT17 -> T85\nX119 -> T85\nT18 -> T86\nX120 -> T86\nT19 -> T87\nX121 -> T87\nT20 -> T88\nX122 -> T88\nX123 -> T90\nT3 -> .(T87, T90)\nT89 -> T90", 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 -> .(T85, T86)\nT2 -> T88\nT3 -> T90", 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\ngt(s(X141), s(X142)) :- gt(X141, X142).\nand substitution\nX141 -> T103\nT85 -> s(T103)\nX142 -> T104\nT87 -> s(T104)", 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\ngt(s(X149), zero).\nand substitution\nX149 -> T109\nT85 -> s(T109)\nT87 -> zero", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 135 [arrowhead = none ];
135 -> 58;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 136 [arrowhead = none ];
136 -> 59;

137 [label="INSTANCE with matching:\nT85 -> T103\nT87 -> T104", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 137 [arrowhead = none , style=dashed];
137 -> 51[style=dashed];

138 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 138 [arrowhead = none ];
138 -> 60;

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\ngt(s(X176), s(X177)) :- gt(X176, X177).\nand substitution\nX176 -> T133\nT119 -> s(T133)\nX177 -> T134\nT121 -> s(T134)", 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\ngt(s(X188), zero).\nand substitution\nX188 -> T145\nT119 -> s(T145)\nT121 -> zero", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 144 [arrowhead = none ];
144 -> 70;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 145 [arrowhead = none ];
145 -> 71;

146 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 146 [arrowhead = none ];
146 -> 68;

147 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
66 -> 147 [arrowhead = none ];
147 -> 69;

148 [label="INSTANCE with matching:\nT85 -> T133\nT87 -> T134", fontsize=14, style = filled, fillcolor = lightgrey];
68 -> 148 [arrowhead = none , style=dashed];
148 -> 51[style=dashed];

149 [label="INSTANCE with matching:\nT1 -> .(s(T133), T120)\nT2 -> T122\nT3 -> T124", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 149 [arrowhead = none , style=dashed];
149 -> 1[style=dashed];

150 [label="INSTANCE with matching:\nT1 -> .(s(T145), T120)\nT2 -> T122\nT3 -> T124", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 150 [arrowhead = none , style=dashed];
150 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

le25(s(T45), s(T46)) :- le25(T45, T46).
le25(zero, s(T53)).
le25(zero, zero).
gt51(s(T103), s(T104)) :- gt51(T103, T104).
gt51(s(T109), zero).
merge1(T5, [], T5).
merge1([], [], []).
merge1([], T11, T11).
merge1(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) :- le25(T31, T32).
merge1(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) :- ','(le25(T31, T32), merge1(T18, .(s(T32), T20), T22)).
merge1(.(zero, T18), .(s(T64), T20), .(zero, T22)) :- merge1(T18, .(s(T64), T20), T22).
merge1(.(zero, T18), .(zero, T20), .(zero, T22)) :- merge1(T18, .(zero, T20), T22).
merge1(.(T85, T86), .(T87, T88), .(T87, T90)) :- gt51(T85, T87).
merge1(.(T85, T86), .(T87, T88), .(T87, T90)) :- ','(gt51(T85, T87), merge1(.(T85, T86), T88, T90)).
merge1(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) :- gt51(T133, T134).
merge1(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) :- ','(gt51(T133, T134), merge1(.(s(T133), T120), T122, T124)).
merge1(.(s(T145), T120), .(zero, T122), .(zero, T124)) :- merge1(.(s(T145), T120), T122, T124).

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)
le25_in: (b,b)
gt51_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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_GGA(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → LE25_IN_GG(T31, T32)
LE25_IN_GG(s(T45), s(T46)) → U1_GG(T45, T46, le25_in_gg(T45, T46))
LE25_IN_GG(s(T45), s(T46)) → LE25_IN_GG(T45, T46)
U3_GGA(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_GGA(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
U3_GGA(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20), T22)
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_GGA(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(s(T64), T20), T22)
MERGE1_IN_GGA(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_GGA(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
MERGE1_IN_GGA(.(zero, T18), .(zero, T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(zero, T20), T22)
MERGE1_IN_GGA(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_GGA(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
MERGE1_IN_GGA(.(T85, T86), .(T87, T88), .(T87, T90)) → GT51_IN_GG(T85, T87)
GT51_IN_GG(s(T103), s(T104)) → U2_GG(T103, T104, gt51_in_gg(T103, T104))
GT51_IN_GG(s(T103), s(T104)) → GT51_IN_GG(T103, T104)
U7_GGA(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_GGA(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
U7_GGA(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88, T90)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_GGA(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → GT51_IN_GG(T133, T134)
U9_GGA(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_GGA(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
U9_GGA(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122, T124)
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_GGA(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122), .(zero, T124)) → MERGE1_IN_GGA(.(s(T145), T120), T122, T124)

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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6)
le25_in_gg(x1, x2) = le25_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
zero = zero
le25_out_gg(x1, x2) = le25_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x2, x3, x4, x6)
gt51_in_gg(x1, x2) = gt51_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
gt51_out_gg(x1, x2) = gt51_out_gg(x1, x2)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
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, x5) = U11_gga(x1, x2, x3, x5)
MERGE1_IN_GGA(x1, x2, x3) = MERGE1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)
LE25_IN_GG(x1, x2) = LE25_IN_GG(x1, x2)
U1_GG(x1, x2, x3) = U1_GG(x1, x2, x3)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5) = U5_GGA(x1, x2, x3, x5)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)
GT51_IN_GG(x1, x2) = GT51_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, 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, x5) = U11_GGA(x1, x2, x3, x5)

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(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_GGA(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → LE25_IN_GG(T31, T32)
LE25_IN_GG(s(T45), s(T46)) → U1_GG(T45, T46, le25_in_gg(T45, T46))
LE25_IN_GG(s(T45), s(T46)) → LE25_IN_GG(T45, T46)
U3_GGA(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_GGA(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
U3_GGA(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20), T22)
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_GGA(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(s(T64), T20), T22)
MERGE1_IN_GGA(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_GGA(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
MERGE1_IN_GGA(.(zero, T18), .(zero, T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(zero, T20), T22)
MERGE1_IN_GGA(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_GGA(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
MERGE1_IN_GGA(.(T85, T86), .(T87, T88), .(T87, T90)) → GT51_IN_GG(T85, T87)
GT51_IN_GG(s(T103), s(T104)) → U2_GG(T103, T104, gt51_in_gg(T103, T104))
GT51_IN_GG(s(T103), s(T104)) → GT51_IN_GG(T103, T104)
U7_GGA(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_GGA(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
U7_GGA(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88, T90)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_GGA(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → GT51_IN_GG(T133, T134)
U9_GGA(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_GGA(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
U9_GGA(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122, T124)
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_GGA(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122), .(zero, T124)) → MERGE1_IN_GGA(.(s(T145), T120), T122, T124)

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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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:

GT51_IN_GG(s(T103), s(T104)) → GT51_IN_GG(T103, T104)

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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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:

GT51_IN_GG(s(T103), s(T104)) → GT51_IN_GG(T103, T104)

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:

GT51_IN_GG(s(T103), s(T104)) → GT51_IN_GG(T103, T104)

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:

GT51_IN_GG(s(T103), s(T104)) → GT51_IN_GG(T103, T104)
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:

LE25_IN_GG(s(T45), s(T46)) → LE25_IN_GG(T45, T46)

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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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:

LE25_IN_GG(s(T45), s(T46)) → LE25_IN_GG(T45, T46)

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:

LE25_IN_GG(s(T45), s(T46)) → LE25_IN_GG(T45, T46)

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:

LE25_IN_GG(s(T45), s(T46)) → LE25_IN_GG(T45, T46)
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:

U3_GGA(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20), T22)
MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_GGA(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(s(T64), T20), T22)
MERGE1_IN_GGA(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_GGA(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88, T90)
MERGE1_IN_GGA(.(zero, T18), .(zero, T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(zero, T20), T22)
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122), .(zero, T124)) → MERGE1_IN_GGA(.(s(T145), T120), T122, T124)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_GGA(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122, T124)

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([], T11, T11) → merge1_out_gga([], T11, T11)
merge1_in_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_gga(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22))
U3_gga(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → U4_gga(T31, T18, T32, T20, T22, merge1_in_gga(T18, .(s(T32), T20), T22))
merge1_in_gga(.(zero, T18), .(s(T64), T20), .(zero, T22)) → U5_gga(T18, T64, T20, T22, merge1_in_gga(T18, .(s(T64), T20), T22))
merge1_in_gga(.(zero, T18), .(zero, T20), .(zero, T22)) → U6_gga(T18, T20, T22, merge1_in_gga(T18, .(zero, T20), T22))
merge1_in_gga(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_gga(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U7_gga(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → U8_gga(T85, T86, T87, T88, T90, merge1_in_gga(.(T85, T86), T88, T90))
merge1_in_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_gga(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U9_gga(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → U10_gga(T133, T120, T134, T122, T124, merge1_in_gga(.(s(T133), T120), T122, T124))
merge1_in_gga(.(s(T145), T120), .(zero, T122), .(zero, T124)) → U11_gga(T145, T120, T122, T124, merge1_in_gga(.(s(T145), T120), T122, T124))
U11_gga(T145, T120, T122, T124, merge1_out_gga(.(s(T145), T120), T122, T124)) → merge1_out_gga(.(s(T145), T120), .(zero, T122), .(zero, T124))
U10_gga(T133, T120, T134, T122, T124, merge1_out_gga(.(s(T133), T120), T122, T124)) → merge1_out_gga(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124))
U8_gga(T85, T86, T87, T88, T90, merge1_out_gga(.(T85, T86), T88, T90)) → merge1_out_gga(.(T85, T86), .(T87, T88), .(T87, T90))
U6_gga(T18, T20, T22, merge1_out_gga(T18, .(zero, T20), T22)) → merge1_out_gga(.(zero, T18), .(zero, T20), .(zero, T22))
U5_gga(T18, T64, T20, T22, merge1_out_gga(T18, .(s(T64), T20), T22)) → merge1_out_gga(.(zero, T18), .(s(T64), T20), .(zero, T22))
U4_gga(T31, T18, T32, T20, T22, merge1_out_gga(T18, .(s(T32), T20), T22)) → merge1_out_gga(.(s(T31), T18), .(s(T32), T20), .(s(T31), 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)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4, x5, x6) = U3_gga(x1, x2, x3, x4, x6)
le25_in_gg(x1, x2) = le25_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
zero = zero
le25_out_gg(x1, x2) = le25_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4, x5, x6) = U4_gga(x1, x2, x3, x4, x6)
U5_gga(x1, x2, x3, x4, x5) = U5_gga(x1, x2, x3, x5)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4, x5, x6) = U7_gga(x1, x2, x3, x4, x6)
gt51_in_gg(x1, x2) = gt51_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
gt51_out_gg(x1, x2) = gt51_out_gg(x1, x2)
U8_gga(x1, x2, x3, x4, x5, x6) = U8_gga(x1, x2, x3, x4, x6)
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, x5) = U11_gga(x1, x2, x3, x5)
MERGE1_IN_GGA(x1, x2, x3) = MERGE1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4, x5, x6) = U9_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:

U3_GGA(T31, T18, T32, T20, T22, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20), T22)
MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20), .(s(T31), T22)) → U3_GGA(T31, T18, T32, T20, T22, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(s(T64), T20), T22)
MERGE1_IN_GGA(.(T85, T86), .(T87, T88), .(T87, T90)) → U7_GGA(T85, T86, T87, T88, T90, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, T90, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88, T90)
MERGE1_IN_GGA(.(zero, T18), .(zero, T20), .(zero, T22)) → MERGE1_IN_GGA(T18, .(zero, T20), T22)
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122), .(zero, T124)) → MERGE1_IN_GGA(.(s(T145), T120), T122, T124)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122), .(s(T134), T124)) → U9_GGA(T133, T120, T134, T122, T124, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, T124, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122, T124)

The TRS R consists of the following rules:

le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
le25_in_gg(x1, x2) = le25_in_gg(x1, x2)
U1_gg(x1, x2, x3) = U1_gg(x1, x2, x3)
zero = zero
le25_out_gg(x1, x2) = le25_out_gg(x1, x2)
gt51_in_gg(x1, x2) = gt51_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
gt51_out_gg(x1, x2) = gt51_out_gg(x1, x2)
MERGE1_IN_GGA(x1, x2, x3) = MERGE1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)
U7_GGA(x1, x2, x3, x4, x5, x6) = U7_GGA(x1, x2, x3, x4, x6)
U9_GGA(x1, x2, x3, x4, x5, x6) = U9_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:

U3_GGA(T31, T18, T32, T20, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20))
MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20)) → U3_GGA(T31, T18, T32, T20, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(zero, T18), .(s(T64), T20)) → MERGE1_IN_GGA(T18, .(s(T64), T20))
MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
MERGE1_IN_GGA(.(zero, T18), .(zero, T20)) → MERGE1_IN_GGA(T18, .(zero, T20))
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122)) → MERGE1_IN_GGA(.(s(T145), T120), T122)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)

The TRS R consists of the following rules:

le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The set Q consists of the following terms:

le25_in_gg(x0, x1)
gt51_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U2_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(.(zero, T18), .(s(T64), T20)) → MERGE1_IN_GGA(T18, .(s(T64), T20))
MERGE1_IN_GGA(.(zero, T18), .(zero, T20)) → MERGE1_IN_GGA(T18, .(zero, T20))
MERGE1_IN_GGA(.(s(T145), T120), .(zero, T122)) → MERGE1_IN_GGA(.(s(T145), T120), T122)

Strictly oriented rules of the TRS R:

le25_in_gg(zero, s(T53)) → le25_out_gg(zero, s(T53))
le25_in_gg(zero, zero) → le25_out_gg(zero, zero)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + 2·x₂
POL(MERGE1_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(U1_gg(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃
POL(U2_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U3_GGA(x₁, x₂, x₃, x₄, x₅)) = 2·x₁ + x₂ + 2·x₃ + 2·x₄ + x₅
POL(U7_GGA(x₁, x₂, x₃, x₄, x₅)) = x₁ + 2·x₂ + x₃ + 2·x₄ + x₅
POL(U9_GGA(x₁, x₂, x₃, x₄, x₅)) = 2·x₁ + 2·x₂ + 2·x₃ + x₄ + 2·x₅
POL(gt51_in_gg(x₁, x₂)) = x₁ + x₂
POL(gt51_out_gg(x₁, x₂)) = x₁ + x₂
POL(le25_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(le25_out_gg(x₁, x₂)) = x₁ + 2·x₂
POL(s(x₁)) = 2·x₁
POL(zero) = 2

(29) Obligation:

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

U3_GGA(T31, T18, T32, T20, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20))
MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20)) → U3_GGA(T31, T18, T32, T20, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)

The TRS R consists of the following rules:

le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The set Q consists of the following terms:

le25_in_gg(x0, x1)
gt51_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U2_gg(x0, x1, x2)

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

(30) MRRProof (EQUIVALENT transformation)

U3_GGA(T31, T18, T32, T20, le25_out_gg(T31, T32)) → MERGE1_IN_GGA(T18, .(s(T32), T20))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + x₂
POL(MERGE1_IN_GGA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U1_gg(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U2_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U3_GGA(x₁, x₂, x₃, x₄, x₅)) = x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U7_GGA(x₁, x₂, x₃, x₄, x₅)) = x₁ + 2·x₂ + x₃ + 2·x₄ + x₅
POL(U9_GGA(x₁, x₂, x₃, x₄, x₅)) = 2·x₁ + 2·x₂ + x₃ + 2·x₄ + 2·x₅
POL(gt51_in_gg(x₁, x₂)) = x₁ + x₂
POL(gt51_out_gg(x₁, x₂)) = x₁ + x₂
POL(le25_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(le25_out_gg(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(s(x₁)) = 2·x₁
POL(zero) = 0

(31) Obligation:

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

MERGE1_IN_GGA(.(s(T31), T18), .(s(T32), T20)) → U3_GGA(T31, T18, T32, T20, le25_in_gg(T31, T32))
MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)

The TRS R consists of the following rules:

le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The set Q consists of the following terms:

le25_in_gg(x0, x1)
gt51_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U2_gg(x0, x1, x2)

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

(32) DependencyGraphProof (EQUIVALENT transformation)

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

(33) Obligation:

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

MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)

The TRS R consists of the following rules:

le25_in_gg(s(T45), s(T46)) → U1_gg(T45, T46, le25_in_gg(T45, T46))
gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U1_gg(T45, T46, le25_out_gg(T45, T46)) → le25_out_gg(s(T45), s(T46))
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The set Q consists of the following terms:

le25_in_gg(x0, x1)
gt51_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U2_gg(x0, x1, x2)

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

(34) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(35) Obligation:

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

MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)

The TRS R consists of the following rules:

gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The set Q consists of the following terms:

le25_in_gg(x0, x1)
gt51_in_gg(x0, x1)
U1_gg(x0, x1, x2)
U2_gg(x0, x1, x2)

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

(36) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le25_in_gg(x0, x1)
U1_gg(x0, x1, x2)

(37) Obligation:

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

MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)

The TRS R consists of the following rules:

gt51_in_gg(s(T103), s(T104)) → U2_gg(T103, T104, gt51_in_gg(T103, T104))
gt51_in_gg(s(T109), zero) → gt51_out_gg(s(T109), zero)
U2_gg(T103, T104, gt51_out_gg(T103, T104)) → gt51_out_gg(s(T103), s(T104))

The set Q consists of the following terms:

gt51_in_gg(x0, x1)
U2_gg(x0, x1, x2)

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

(38) 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:

U7_GGA(T85, T86, T87, T88, gt51_out_gg(T85, T87)) → MERGE1_IN_GGA(.(T85, T86), T88)
The graph contains the following edges 4 >= 2
U9_GGA(T133, T120, T134, T122, gt51_out_gg(T133, T134)) → MERGE1_IN_GGA(.(s(T133), T120), T122)
The graph contains the following edges 4 >= 2
MERGE1_IN_GGA(.(T85, T86), .(T87, T88)) → U7_GGA(T85, T86, T87, T88, gt51_in_gg(T85, T87))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
MERGE1_IN_GGA(.(s(T133), T120), .(s(T134), T122)) → U9_GGA(T133, T120, T134, T122, gt51_in_gg(T133, T134))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4

(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) MRRProof (EQUIVALENT transformation)

(31) Obligation:

(32) DependencyGraphProof (EQUIVALENT transformation)

(33) Obligation:

(34) UsableRulesProof (EQUIVALENT transformation)

(35) Obligation:

(36) QReductionProof (EQUIVALENT transformation)

(37) Obligation:

(38) QDPSizeChangeProof (EQUIVALENT transformation)

(39) YES