Left Termination proof of /tmp/tmpO4eoo6/pl8.2.1a.pl

Left Termination of the query pattern mergesort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 UndefinedPredicateInTriplesTransformerProof (⇐)

↳4 TRIPLES

↳5 TriplesToPiDPProof (⇐)

↳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 QDPSizeChangeProof (⇔)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇐)

        ↳34 QDP

        ↳35 MRRProof (⇔)

        ↳36 QDP

        ↳37 DependencyGraphProof (⇔)

        ↳38 QDP

        ↳39 UsableRulesProof (⇔)

        ↳40 QDP

        ↳41 QReductionProof (⇔)

        ↳42 QDP

        ↳43 QDPSizeChangeProof (⇔)

        ↳44 YES

    ↳45 PiDP

        ↳46 UsableRulesProof (⇔)

        ↳47 PiDP

        ↳48 PiDPToQDPProof (⇐)

        ↳49 QDP

        ↳50 QDPOrderProof (⇔)

        ↳51 QDP

        ↳52 DependencyGraphProof (⇔)

        ↳53 TRUE

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(E, []), .(E, [])).
mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V)))).
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)).
split([], [], []).
split(.(E, U), .(E, V), W) :- split(U, W, V).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Queries:

mergesort(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: mergesort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mergesort(T1, T2), SCOPE: 1, CLAUSE: 0 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])\nmergesort(T1, T2) !=? mergesort(.(X7, []), .(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(split(T12, X21, X22), ','(mergesort(.(T10, X21), X23), ','(mergesort(.(T11, X22), X24), merge(X23, X24, T14))))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(split(T12, X21, X22), ','(mergesort(.(T10, X21), X23), ','(mergesort(.(T11, X22), X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 7 | ','(split(T12, X21, X22), ','(mergesort(.(T10, X21), X23), ','(mergesort(.(T11, X22), X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(T12, X21, X22), ','(mergesort(.(T10, X21), X23), ','(mergesort(.(T11, X22), X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 7\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(split(T12, X21, X22), ','(mergesort(.(T10, X21), X23), ','(mergesort(.(T11, X22), X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(mergesort(.(T10, []), X23), ','(mergesort(.(T11, []), X24), merge(X23, X24, T14)))\n\nT10 is ground\nT11 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: mergesort(.(T10, []), X23)\n\nT10 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(mergesort(.(T11, []), X24), merge(T18, X24, T14))\n\nT11 is ground\nT18 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 0 | mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 1 | mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 1 | mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 1\n\nT10 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: mergesort(.(T10, []), X23), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: mergesort(.(T11, []), X24)\n\nT11 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: merge(T18, T29, T14)\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 3 | merge(T18, T29, T14), SCOPE: 4, CLAUSE: 4 | merge(T18, T29, T14), SCOPE: 4, CLAUSE: 5 | merge(T18, T29, T14), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 3\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 4 | merge(T18, T29, T14), SCOPE: 4, CLAUSE: 5 | merge(T18, T29, T14), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 4\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 5 | merge(T18, T29, T14), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nT29 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: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 5\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: merge(T18, T29, T14), SCOPE: 4, CLAUSE: 6\n\nT18 is ground\nT29 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(le(T63, T65), merge(T64, .(T65, T66), T68))\n\nT63 is ground\nT64 is ground\nT65 is ground\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: le(T63, T65)\n\nT63 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: merge(T64, .(T65, T66), T68)\n\nT64 is ground\nT65 is ground\nT66 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: le(T63, T65), SCOPE: 5, CLAUSE: 11 | le(T63, T65), SCOPE: 5, CLAUSE: 12 | le(T63, T65), SCOPE: 5, CLAUSE: 13\n\nT63 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: le(T63, T65), SCOPE: 5, CLAUSE: 11\n\nT63 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: le(T63, T65), SCOPE: 5, CLAUSE: 12 | le(T63, T65), SCOPE: 5, CLAUSE: 13\n\nT63 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: le(T81, T82)\n\nT81 is ground\nT82 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: le(T63, T65), SCOPE: 5, CLAUSE: 12\n\nT63 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: le(T63, T65), SCOPE: 5, CLAUSE: 13\n\nT63 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(gt(T106, T108), merge(.(T106, T107), T109, T111))\n\nT106 is ground\nT107 is ground\nT108 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: gt(T106, T108)\n\nT106 is ground\nT108 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: merge(.(T106, T107), T109, T111)\n\nT106 is ground\nT107 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: gt(T106, T108), SCOPE: 6, CLAUSE: 9 | gt(T106, T108), SCOPE: 6, CLAUSE: 10\n\nT106 is ground\nT108 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: gt(T106, T108), SCOPE: 6, CLAUSE: 9\n\nT106 is ground\nT108 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: gt(T106, T108), SCOPE: 6, CLAUSE: 10\n\nT106 is ground\nT108 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: gt(T124, T125)\n\nT124 is ground\nT125 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(split(T140, X180, X179), ','(mergesort(.(T10, .(T139, X179)), X23), ','(mergesort(.(T11, X180), X24), merge(X23, X24, T14))))\n\nT10 is ground\nT11 is ground\nT139 is ground\nT140 is ground\nX23 is free; \nX24 is free; \nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: split(T140, X180, X179)\n\nT140 is ground\nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(mergesort(.(T10, .(T139, T142)), X23), ','(mergesort(.(T11, T141), X24), merge(X23, X24, T14)))\n\nT10 is ground\nT11 is ground\nT139 is ground\nT141 is ground\nT142 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: split(T140, X180, X179), SCOPE: 7, CLAUSE: 7 | split(T140, X180, X179), SCOPE: 7, CLAUSE: 8\n\nT140 is ground\nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: split(T140, X180, X179), SCOPE: 7, CLAUSE: 7\n\nT140 is ground\nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: split(T140, X180, X179), SCOPE: 7, CLAUSE: 8\n\nT140 is ground\nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: split(T148, X202, X201)\n\nT148 is ground\nX201 is free; \nX202 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: mergesort(.(T10, .(T139, T142)), X23)\n\nT10 is ground\nT139 is ground\nT142 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(mergesort(.(T11, T141), X24), merge(T154, X24, T14))\n\nT11 is ground\nT141 is ground\nT154 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: mergesort(.(T11, T141), X24)\n\nT11 is ground\nT141 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: merge(T154, T162, T14)\n\nT154 is ground\nT162 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 88 [arrowhead = none ];
88 -> 2;

89 [label="EVAL with clause\nmergesort([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 89 [arrowhead = none ];
89 -> 3;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 90 [arrowhead = none ];
90 -> 4;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 91 [arrowhead = none ];
91 -> 5;

92 [label="EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitution\nX7 -> T4\nT1 -> .(T4, [])\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 92 [arrowhead = none ];
92 -> 8;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 93 [arrowhead = none ];
93 -> 9;

94 [label="BACKTRACK\nfor clause: mergesort(.(E, []), .(E, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 94 [arrowhead = none ];
94 -> 6;

95 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 95 [arrowhead = none ];
95 -> 7;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 96 [arrowhead = none ];
96 -> 10;

97 [label="EVAL with clause\nmergesort(.(X17, .(X18, X19)), X20) :- ','(split(X19, X21, X22), ','(mergesort(.(X17, X21), X23), ','(mergesort(.(X18, X22), X24), merge(X23, X24, X20)))).\nand substitution\nX17 -> T10\nX18 -> T11\nX19 -> T12\nT1 -> .(T10, .(T11, T12))\nT2 -> T14\nX20 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 97 [arrowhead = none ];
97 -> 12;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 98 [arrowhead = none ];
98 -> 13;

99 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 99 [arrowhead = none ];
99 -> 11;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 100 [arrowhead = none ];
100 -> 14;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 101 [arrowhead = none ];
101 -> 15;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 102 [arrowhead = none ];
102 -> 16;

103 [label="EVAL with clause\nsplit([], [], []).\nand substitution\nT12 -> []\nX21 -> []\nX22 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 103 [arrowhead = none ];
103 -> 17;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 104 [arrowhead = none ];
104 -> 18;

105 [label="EVAL with clause\nsplit(.(X175, X176), .(X175, X177), X178) :- split(X176, X178, X177).\nand substitution\nX175 -> T139\nX176 -> T140\nT12 -> .(T139, T140)\nX177 -> X179\nX21 -> .(T139, X179)\nX22 -> X180\nX178 -> X180", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 105 [arrowhead = none ];
105 -> 72;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 106 [arrowhead = none ];
106 -> 73;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 107 [arrowhead = none ];
107 -> 19;

108 [label="SPLIT 2\nnew knowledge:\nT18 is ground\nreplacements:\nX23 -> T18", fontsize=14, style = filled, fillcolor = violet];
17 -> 108 [arrowhead = none ];
108 -> 20;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 109 [arrowhead = none ];
109 -> 21;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 110 [arrowhead = none ];
110 -> 28;

111 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:\nX24 -> T29", fontsize=14, style = filled, fillcolor = violet];
20 -> 111 [arrowhead = none ];
111 -> 29;

112 [label="BACKTRACK\nfor clause: mergesort([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
21 -> 112 [arrowhead = none ];
112 -> 22;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 113 [arrowhead = none ];
113 -> 23;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 114 [arrowhead = none ];
114 -> 24;

115 [label="ONLY EVAL with clause\nmergesort(.(X38, []), .(X38, [])).\nand substitution\nT10 -> T24\nX38 -> T24\nX23 -> .(T24, [])", fontsize=14, style = filled, fillcolor = white];
23 -> 115 [arrowhead = none ];
115 -> 25;

116 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
24 -> 116 [arrowhead = none ];
116 -> 27;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 117 [arrowhead = none ];
117 -> 26;

118 [label="INSTANCE with matching:\nT10 -> T11\nX23 -> X24", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 118 [arrowhead = none , style=dashed];
118 -> 19[style=dashed];

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 119 [arrowhead = none ];
119 -> 30;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 120 [arrowhead = none ];
120 -> 31;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 121 [arrowhead = none ];
121 -> 32;

122 [label="EVAL with clause\nmerge(X53, [], X53).\nand substitution\nT18 -> T37\nX53 -> T37\nT29 -> []\nT14 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 122 [arrowhead = none ];
122 -> 33;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 123 [arrowhead = none ];
123 -> 34;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 124 [arrowhead = none ];
124 -> 36;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 125 [arrowhead = none ];
125 -> 37;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 126 [arrowhead = none ];
126 -> 35;

127 [label="EVAL with clause\nmerge([], X59, X59).\nand substitution\nT18 -> []\nT29 -> T42\nX59 -> T42\nT14 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 127 [arrowhead = none ];
127 -> 38;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 128 [arrowhead = none ];
128 -> 39;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 129 [arrowhead = none ];
129 -> 41;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 130 [arrowhead = none ];
130 -> 42;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 131 [arrowhead = none ];
131 -> 40;

132 [label="EVAL with clause\nmerge(.(X81, X82), .(X83, X84), .(X81, X85)) :- ','(le(X81, X83), merge(X82, .(X83, X84), X85)).\nand substitution\nX81 -> T63\nX82 -> T64\nT18 -> .(T63, T64)\nX83 -> T65\nX84 -> T66\nT29 -> .(T65, T66)\nX85 -> T68\nT14 -> .(T63, T68)\nT67 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 132 [arrowhead = none ];
132 -> 43;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 133 [arrowhead = none ];
133 -> 44;

134 [label="EVAL with clause\nmerge(.(X129, X130), .(X131, X132), .(X131, X133)) :- ','(gt(X129, X131), merge(.(X129, X130), X132, X133)).\nand substitution\nX129 -> T106\nX130 -> T107\nT18 -> .(T106, T107)\nX131 -> T108\nX132 -> T109\nT29 -> .(T108, T109)\nX133 -> T111\nT14 -> .(T108, T111)\nT110 -> T111", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 134 [arrowhead = none ];
134 -> 60;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 135 [arrowhead = none ];
135 -> 61;

136 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 136 [arrowhead = none ];
136 -> 45;

137 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
43 -> 137 [arrowhead = none ];
137 -> 46;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 138 [arrowhead = none ];
138 -> 47;

139 [label="INSTANCE with matching:\nT18 -> T64\nT29 -> .(T65, T66)\nT14 -> T68", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 139 [arrowhead = none , style=dashed];
139 -> 29[style=dashed];

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 140 [arrowhead = none ];
140 -> 48;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 141 [arrowhead = none ];
141 -> 49;

142 [label="EVAL with clause\nle(s(X103), s(X104)) :- le(X103, X104).\nand substitution\nX103 -> T81\nT63 -> s(T81)\nX104 -> T82\nT65 -> s(T82)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 142 [arrowhead = none ];
142 -> 50;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 143 [arrowhead = none ];
143 -> 51;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 144 [arrowhead = none ];
144 -> 52;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 145 [arrowhead = none ];
145 -> 53;

146 [label="INSTANCE with matching:\nT63 -> T81\nT65 -> T82", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 146 [arrowhead = none , style=dashed];
146 -> 45[style=dashed];

147 [label="EVAL with clause\nle(0, s(X113)).\nand substitution\nT63 -> 0\nX113 -> T89\nT65 -> s(T89)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 147 [arrowhead = none ];
147 -> 54;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 148 [arrowhead = none ];
148 -> 55;

149 [label="EVAL with clause\nle(0, 0).\nand substitution\nT63 -> 0\nT65 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 149 [arrowhead = none ];
149 -> 57;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 150 [arrowhead = none ];
150 -> 58;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 151 [arrowhead = none ];
151 -> 56;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 152 [arrowhead = none ];
152 -> 59;

153 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
60 -> 153 [arrowhead = none ];
153 -> 62;

154 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
60 -> 154 [arrowhead = none ];
154 -> 63;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 155 [arrowhead = none ];
155 -> 64;

156 [label="INSTANCE with matching:\nT18 -> .(T106, T107)\nT29 -> T109\nT14 -> T111", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 156 [arrowhead = none , style=dashed];
156 -> 29[style=dashed];

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 157 [arrowhead = none ];
157 -> 65;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 158 [arrowhead = none ];
158 -> 66;

159 [label="EVAL with clause\ngt(s(X151), s(X152)) :- gt(X151, X152).\nand substitution\nX151 -> T124\nT106 -> s(T124)\nX152 -> T125\nT108 -> s(T125)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 159 [arrowhead = none ];
159 -> 67;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 160 [arrowhead = none ];
160 -> 68;

161 [label="EVAL with clause\ngt(s(X159), 0).\nand substitution\nX159 -> T130\nT106 -> s(T130)\nT108 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 161 [arrowhead = none ];
161 -> 69;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 162 [arrowhead = none ];
162 -> 70;

163 [label="INSTANCE with matching:\nT106 -> T124\nT108 -> T125", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 163 [arrowhead = none , style=dashed];
163 -> 62[style=dashed];

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 164 [arrowhead = none ];
164 -> 71;

165 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 165 [arrowhead = none ];
165 -> 74;

166 [label="SPLIT 2\nnew knowledge:\nT141 is ground\nT142 is ground\nreplacements:\nX180 -> T141\nX179 -> T142", fontsize=14, style = filled, fillcolor = violet];
72 -> 166 [arrowhead = none ];
166 -> 75;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 167 [arrowhead = none ];
167 -> 76;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
75 -> 168 [arrowhead = none ];
168 -> 84;

169 [label="SPLIT 2\nnew knowledge:\nT154 is ground\nreplacements:\nX23 -> T154", fontsize=14, style = filled, fillcolor = violet];
75 -> 169 [arrowhead = none ];
169 -> 85;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 170 [arrowhead = none ];
170 -> 77;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 171 [arrowhead = none ];
171 -> 78;

172 [label="EVAL with clause\nsplit([], [], []).\nand substitution\nT140 -> []\nX180 -> []\nX179 -> []", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 172 [arrowhead = none ];
172 -> 79;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 173 [arrowhead = none ];
173 -> 80;

174 [label="EVAL with clause\nsplit(.(X197, X198), .(X197, X199), X200) :- split(X198, X200, X199).\nand substitution\nX197 -> T147\nX198 -> T148\nT140 -> .(T147, T148)\nX199 -> X201\nX180 -> .(T147, X201)\nX179 -> X202\nX200 -> X202", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 174 [arrowhead = none ];
174 -> 82;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 175 [arrowhead = none ];
175 -> 83;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 176 [arrowhead = none ];
176 -> 81;

177 [label="INSTANCE with matching:\nT140 -> T148\nX180 -> X202\nX179 -> X201", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 177 [arrowhead = none , style=dashed];
177 -> 74[style=dashed];

178 [label="INSTANCE with matching:\nT1 -> .(T10, .(T139, T142))\nT2 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 178 [arrowhead = none , style=dashed];
178 -> 1[style=dashed];

179 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 179 [arrowhead = none ];
179 -> 86;

180 [label="SPLIT 2\nnew knowledge:\nT162 is ground\nreplacements:\nX24 -> T162", fontsize=14, style = filled, fillcolor = violet];
85 -> 180 [arrowhead = none ];
180 -> 87;

181 [label="INSTANCE with matching:\nT1 -> .(T11, T141)\nT2 -> X24", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 181 [arrowhead = none , style=dashed];
181 -> 1[style=dashed];

182 [label="INSTANCE with matching:\nT18 -> T154\nT29 -> T162", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 182 [arrowhead = none , style=dashed];
182 -> 29[style=dashed];

}

(2) Obligation:

Triples:

merge29(.(T63, T64), .(T65, T66), .(T63, T68)) :- le45(T63, T65).
merge29(.(T63, T64), .(T65, T66), .(T63, T68)) :- ','(lec45(T63, T65), merge29(T64, .(T65, T66), T68)).
merge29(.(T106, T107), .(T108, T109), .(T108, T111)) :- gt62(T106, T108).
merge29(.(T106, T107), .(T108, T109), .(T108, T111)) :- ','(gtc62(T106, T108), merge29(.(T106, T107), T109, T111)).
le45(s(T81), s(T82)) :- le45(T81, T82).
gt62(s(T124), s(T125)) :- gt62(T124, T125).
split74(.(T147, T148), .(T147, X201), X202) :- split74(T148, X202, X201).
mergesort1(.(T10, .(T11, [])), T14) :- mergesort19(T10, X23).
mergesort1(.(T10, .(T11, [])), T14) :- ','(mergesortc19(T10, T18), mergesort19(T11, X24)).
mergesort1(.(T10, .(T11, [])), T14) :- ','(mergesortc19(T10, T18), ','(mergesortc19(T11, T29), merge29(T18, T29, T14))).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- split74(T140, X180, X179).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), mergesort1(.(T10, .(T139, T142)), X23)).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), ','(mergesortc1(.(T10, .(T139, T142)), T154), mergesort1(.(T11, T141), X24))).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), ','(mergesortc1(.(T10, .(T139, T142)), T154), ','(mergesortc1(.(T11, T141), T162), merge29(T154, T162, T14)))).

Clauses:

mergesortc19(T24, .(T24, [])).
mergec29(T37, [], T37).
mergec29([], T42, T42).
mergec29(.(T63, T64), .(T65, T66), .(T63, T68)) :- ','(lec45(T63, T65), mergec29(T64, .(T65, T66), T68)).
mergec29(.(T106, T107), .(T108, T109), .(T108, T111)) :- ','(gtc62(T106, T108), mergec29(.(T106, T107), T109, T111)).
lec45(s(T81), s(T82)) :- lec45(T81, T82).
lec45(0, s(T89)).
lec45(0, 0).
gtc62(s(T124), s(T125)) :- gtc62(T124, T125).
gtc62(s(T130), 0).
splitc74([], [], []).
splitc74(.(T147, T148), .(T147, X201), X202) :- splitc74(T148, X202, X201).
mergesortc1([], []).
mergesortc1(.(T4, []), .(T4, [])).
mergesortc1(.(T10, .(T11, [])), T14) :- ','(mergesortc19(T10, T18), ','(mergesortc19(T11, T29), mergec29(T18, T29, T14))).
mergesortc1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), ','(mergesortc1(.(T10, .(T139, T142)), T154), ','(mergesortc1(.(T11, T141), T162), mergec29(T154, T162, T14)))).

Afs:

mergesort1(x1, x2) = mergesort1(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

merge29(.(T63, T64), .(T65, T66), .(T63, T68)) :- le45(T63, T65).
merge29(.(T63, T64), .(T65, T66), .(T63, T68)) :- ','(lec45(T63, T65), merge29(T64, .(T65, T66), T68)).
merge29(.(T106, T107), .(T108, T109), .(T108, T111)) :- gt62(T106, T108).
merge29(.(T106, T107), .(T108, T109), .(T108, T111)) :- ','(gtc62(T106, T108), merge29(.(T106, T107), T109, T111)).
le45(s(T81), s(T82)) :- le45(T81, T82).
gt62(s(T124), s(T125)) :- gt62(T124, T125).
split74(.(T147, T148), .(T147, X201), X202) :- split74(T148, X202, X201).
mergesort1(.(T10, .(T11, [])), T14) :- ','(mergesortc19(T10, T18), ','(mergesortc19(T11, T29), merge29(T18, T29, T14))).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- split74(T140, X180, X179).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), mergesort1(.(T10, .(T139, T142)), X23)).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), ','(mergesortc1(.(T10, .(T139, T142)), T154), mergesort1(.(T11, T141), X24))).
mergesort1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), ','(mergesortc1(.(T10, .(T139, T142)), T154), ','(mergesortc1(.(T11, T141), T162), merge29(T154, T162, T14)))).

Clauses:

mergesortc19(T24, .(T24, [])).
mergec29(T37, [], T37).
mergec29([], T42, T42).
mergec29(.(T63, T64), .(T65, T66), .(T63, T68)) :- ','(lec45(T63, T65), mergec29(T64, .(T65, T66), T68)).
mergec29(.(T106, T107), .(T108, T109), .(T108, T111)) :- ','(gtc62(T106, T108), mergec29(.(T106, T107), T109, T111)).
lec45(s(T81), s(T82)) :- lec45(T81, T82).
lec45(0, s(T89)).
lec45(0, 0).
gtc62(s(T124), s(T125)) :- gtc62(T124, T125).
gtc62(s(T130), 0).
splitc74([], [], []).
splitc74(.(T147, T148), .(T147, X201), X202) :- splitc74(T148, X202, X201).
mergesortc1([], []).
mergesortc1(.(T4, []), .(T4, [])).
mergesortc1(.(T10, .(T11, [])), T14) :- ','(mergesortc19(T10, T18), ','(mergesortc19(T11, T29), mergec29(T18, T29, T14))).
mergesortc1(.(T10, .(T11, .(T139, T140))), T14) :- ','(splitc74(T140, T141, T142), ','(mergesortc1(.(T10, .(T139, T142)), T154), ','(mergesortc1(.(T11, T141), T162), mergec29(T154, T162, T14)))).

Afs:

mergesort1(x1, x2) = mergesort1(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
merge29_in: (b,b,f)
le45_in: (b,b)
lec45_in: (b,b)
gt62_in: (b,b)
gtc62_in: (b,b)
split74_in: (b,f,f)
splitc74_in: (b,f,f)
mergesortc1_in: (b,f)
mergec29_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T10, .(T11, [])), T14) → U10_GA(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U10_GA(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U11_GA(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U11_GA(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U12_GA(T10, T11, T14, merge29_in_gga(T18, T29, T14))
U11_GA(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → MERGE29_IN_GGA(T18, T29, T14)
MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → U1_GGA(T63, T64, T65, T66, T68, le45_in_gg(T63, T65))
MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → LE45_IN_GG(T63, T65)
LE45_IN_GG(s(T81), s(T82)) → U7_GG(T81, T82, le45_in_gg(T81, T82))
LE45_IN_GG(s(T81), s(T82)) → LE45_IN_GG(T81, T82)
MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → U2_GGA(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U2_GGA(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U3_GGA(T63, T64, T65, T66, T68, merge29_in_gga(T64, .(T65, T66), T68))
U2_GGA(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → MERGE29_IN_GGA(T64, .(T65, T66), T68)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → U4_GGA(T106, T107, T108, T109, T111, gt62_in_gg(T106, T108))
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → GT62_IN_GG(T106, T108)
GT62_IN_GG(s(T124), s(T125)) → U8_GG(T124, T125, gt62_in_gg(T124, T125))
GT62_IN_GG(s(T124), s(T125)) → GT62_IN_GG(T124, T125)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → U5_GGA(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U5_GGA(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U6_GGA(T106, T107, T108, T109, T111, merge29_in_gga(.(T106, T107), T109, T111))
U5_GGA(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109, T111)
MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → U13_GA(T10, T11, T139, T140, T14, split74_in_gaa(T140, X180, X179))
MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → SPLIT74_IN_GAA(T140, X180, X179)
SPLIT74_IN_GAA(.(T147, T148), .(T147, X201), X202) → U9_GAA(T147, T148, X201, X202, split74_in_gaa(T148, X202, X201))
SPLIT74_IN_GAA(.(T147, T148), .(T147, X201), X202) → SPLIT74_IN_GAA(T148, X202, X201)
MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → U14_GA(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U15_GA(T10, T11, T139, T140, T14, mergesort1_in_ga(.(T10, .(T139, T142)), X23))
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → MERGESORT1_IN_GA(.(T10, .(T139, T142)), X23)
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U17_GA(T10, T11, T139, T140, T14, mergesort1_in_ga(.(T11, T141), X24))
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → MERGESORT1_IN_GA(.(T11, T141), X24)
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U18_GA(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U18_GA(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U19_GA(T10, T11, T139, T140, T14, merge29_in_gga(T154, T162, T14))
U18_GA(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → MERGE29_IN_GGA(T154, T162, T14)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2) = mergesort1_in_ga(x1)
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
merge29_in_gga(x1, x2, x3) = merge29_in_gga(x1, x2)
le45_in_gg(x1, x2) = le45_in_gg(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gt62_in_gg(x1, x2) = gt62_in_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
split74_in_gaa(x1, x2, x3) = split74_in_gaa(x1)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U10_GA(x1, x2, x3, x4) = U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x2, x4, x5)
U12_GA(x1, x2, x3, x4) = U12_GA(x1, x2, x4)
MERGE29_IN_GGA(x1, x2, x3) = MERGE29_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6) = U1_GGA(x1, x2, x3, x4, x6)
LE45_IN_GG(x1, x2) = LE45_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6) = U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6) = U4_GGA(x1, x2, x3, x4, x6)
GT62_IN_GG(x1, x2) = GT62_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x1, x2, x3)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6) = U6_GGA(x1, x2, x3, x4, x6)
U13_GA(x1, x2, x3, x4, x5, x6) = U13_GA(x1, x2, x3, x4, x6)
SPLIT74_IN_GAA(x1, x2, x3) = SPLIT74_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5) = U9_GAA(x1, x2, x5)
U14_GA(x1, x2, x3, x4, x5, x6) = U14_GA(x1, x2, x3, x4, x6)
U15_GA(x1, x2, x3, x4, x5, x6) = U15_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7) = U16_GA(x1, x2, x3, x4, x6, x7)
U17_GA(x1, x2, x3, x4, x5, x6) = U17_GA(x1, x2, x3, x4, x6)
U18_GA(x1, x2, x3, x4, x5, x6, x7) = U18_GA(x1, x2, x3, x4, x6, x7)
U19_GA(x1, x2, x3, x4, x5, x6) = U19_GA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

MERGESORT1_IN_GA(.(T10, .(T11, [])), T14) → U10_GA(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U10_GA(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U11_GA(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U11_GA(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U12_GA(T10, T11, T14, merge29_in_gga(T18, T29, T14))
U11_GA(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → MERGE29_IN_GGA(T18, T29, T14)
MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → U1_GGA(T63, T64, T65, T66, T68, le45_in_gg(T63, T65))
MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → LE45_IN_GG(T63, T65)
LE45_IN_GG(s(T81), s(T82)) → U7_GG(T81, T82, le45_in_gg(T81, T82))
LE45_IN_GG(s(T81), s(T82)) → LE45_IN_GG(T81, T82)
MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → U2_GGA(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U2_GGA(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U3_GGA(T63, T64, T65, T66, T68, merge29_in_gga(T64, .(T65, T66), T68))
U2_GGA(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → MERGE29_IN_GGA(T64, .(T65, T66), T68)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → U4_GGA(T106, T107, T108, T109, T111, gt62_in_gg(T106, T108))
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → GT62_IN_GG(T106, T108)
GT62_IN_GG(s(T124), s(T125)) → U8_GG(T124, T125, gt62_in_gg(T124, T125))
GT62_IN_GG(s(T124), s(T125)) → GT62_IN_GG(T124, T125)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → U5_GGA(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U5_GGA(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U6_GGA(T106, T107, T108, T109, T111, merge29_in_gga(.(T106, T107), T109, T111))
U5_GGA(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109, T111)
MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → U13_GA(T10, T11, T139, T140, T14, split74_in_gaa(T140, X180, X179))
MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → SPLIT74_IN_GAA(T140, X180, X179)
SPLIT74_IN_GAA(.(T147, T148), .(T147, X201), X202) → U9_GAA(T147, T148, X201, X202, split74_in_gaa(T148, X202, X201))
SPLIT74_IN_GAA(.(T147, T148), .(T147, X201), X202) → SPLIT74_IN_GAA(T148, X202, X201)
MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → U14_GA(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U15_GA(T10, T11, T139, T140, T14, mergesort1_in_ga(.(T10, .(T139, T142)), X23))
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → MERGESORT1_IN_GA(.(T10, .(T139, T142)), X23)
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U17_GA(T10, T11, T139, T140, T14, mergesort1_in_ga(.(T11, T141), X24))
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → MERGESORT1_IN_GA(.(T11, T141), X24)
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U18_GA(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U18_GA(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U19_GA(T10, T11, T139, T140, T14, merge29_in_gga(T154, T162, T14))
U18_GA(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → MERGE29_IN_GGA(T154, T162, T14)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 20 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SPLIT74_IN_GAA(.(T147, T148), .(T147, X201), X202) → SPLIT74_IN_GAA(T148, X202, X201)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
SPLIT74_IN_GAA(x1, x2, x3) = SPLIT74_IN_GAA(x1)

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

(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:

SPLIT74_IN_GAA(.(T147, T148), .(T147, X201), X202) → SPLIT74_IN_GAA(T148, X202, X201)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT74_IN_GAA(x1, x2, x3) = SPLIT74_IN_GAA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

SPLIT74_IN_GAA(.(T147, T148)) → SPLIT74_IN_GAA(T148)

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:

SPLIT74_IN_GAA(.(T147, T148)) → SPLIT74_IN_GAA(T148)
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

GT62_IN_GG(s(T124), s(T125)) → GT62_IN_GG(T124, T125)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
GT62_IN_GG(x1, x2) = GT62_IN_GG(x1, x2)

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

(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:

GT62_IN_GG(s(T124), s(T125)) → GT62_IN_GG(T124, T125)

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:

GT62_IN_GG(s(T124), s(T125)) → GT62_IN_GG(T124, T125)

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:

GT62_IN_GG(s(T124), s(T125)) → GT62_IN_GG(T124, T125)
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:

LE45_IN_GG(s(T81), s(T82)) → LE45_IN_GG(T81, T82)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
LE45_IN_GG(x1, x2) = LE45_IN_GG(x1, x2)

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:

LE45_IN_GG(s(T81), s(T82)) → LE45_IN_GG(T81, T82)

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

(26) PiDPToQDPProof (EQUIVALENT 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:

LE45_IN_GG(s(T81), s(T82)) → LE45_IN_GG(T81, T82)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

LE45_IN_GG(s(T81), s(T82)) → LE45_IN_GG(T81, T82)
The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → U2_GGA(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U2_GGA(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → MERGE29_IN_GGA(T64, .(T65, T66), T68)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → U5_GGA(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U5_GGA(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109, T111)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
MERGE29_IN_GGA(x1, x2, x3) = MERGE29_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MERGE29_IN_GGA(.(T63, T64), .(T65, T66), .(T63, T68)) → U2_GGA(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U2_GGA(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → MERGE29_IN_GGA(T64, .(T65, T66), T68)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109), .(T108, T111)) → U5_GGA(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U5_GGA(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109, T111)

The TRS R consists of the following rules:

lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
MERGE29_IN_GGA(x1, x2, x3) = MERGE29_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6) = U5_GGA(x1, x2, x3, x4, x6)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

MERGE29_IN_GGA(.(T63, T64), .(T65, T66)) → U2_GGA(T63, T64, T65, T66, lec45_in_gg(T63, T65))
U2_GGA(T63, T64, T65, T66, lec45_out_gg(T63, T65)) → MERGE29_IN_GGA(T64, .(T65, T66))
MERGE29_IN_GGA(.(T106, T107), .(T108, T109)) → U5_GGA(T106, T107, T108, T109, gtc62_in_gg(T106, T108))
U5_GGA(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109)

The TRS R consists of the following rules:

lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

lec45_in_gg(x0, x1)
gtc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(35) 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:

MERGE29_IN_GGA(.(T63, T64), .(T65, T66)) → U2_GGA(T63, T64, T65, T66, lec45_in_gg(T63, T65))

Strictly oriented rules of the TRS R:

gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(0) = 0
POL(MERGE29_IN_GGA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U25_gg(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U26_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U2_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U5_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅
POL(gtc62_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(gtc62_out_gg(x₁, x₂)) = x₁ + x₂
POL(lec45_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(lec45_out_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(s(x₁)) = 2·x₁

(36) Obligation:

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

U2_GGA(T63, T64, T65, T66, lec45_out_gg(T63, T65)) → MERGE29_IN_GGA(T64, .(T65, T66))
MERGE29_IN_GGA(.(T106, T107), .(T108, T109)) → U5_GGA(T106, T107, T108, T109, gtc62_in_gg(T106, T108))
U5_GGA(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109)

The TRS R consists of the following rules:

lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

lec45_in_gg(x0, x1)
gtc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

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

(38) Obligation:

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

U5_GGA(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109)) → U5_GGA(T106, T107, T108, T109, gtc62_in_gg(T106, T108))

The TRS R consists of the following rules:

lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

lec45_in_gg(x0, x1)
gtc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(39) 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.

(40) Obligation:

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

U5_GGA(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109)) → U5_GGA(T106, T107, T108, T109, gtc62_in_gg(T106, T108))

The TRS R consists of the following rules:

gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

lec45_in_gg(x0, x1)
gtc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(41) 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].

lec45_in_gg(x0, x1)
U25_gg(x0, x1, x2)

(42) Obligation:

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

U5_GGA(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109)
MERGE29_IN_GGA(.(T106, T107), .(T108, T109)) → U5_GGA(T106, T107, T108, T109, gtc62_in_gg(T106, T108))

The TRS R consists of the following rules:

gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

gtc62_in_gg(x0, x1)
U26_gg(x0, x1, x2)

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

(43) 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:

MERGE29_IN_GGA(.(T106, T107), .(T108, T109)) → U5_GGA(T106, T107, T108, T109, gtc62_in_gg(T106, T108))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
U5_GGA(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → MERGE29_IN_GGA(.(T106, T107), T109)
The graph contains the following edges 4 >= 2

(44) YES

(45) Obligation:

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

MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → U14_GA(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → MERGESORT1_IN_GA(.(T10, .(T139, T142)), X23)
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → MERGESORT1_IN_GA(.(T11, T141), X24)

The TRS R consists of the following rules:

mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))
splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5, x6) = U14_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7) = U16_GA(x1, x2, x3, x4, x6, x7)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140))), T14) → U14_GA(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → MERGESORT1_IN_GA(.(T10, .(T139, T142)), X23)
U14_GA(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
U16_GA(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → MERGESORT1_IN_GA(.(T11, T141), X24)

The TRS R consists of the following rules:

splitc74_in_gaa([], [], []) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148), .(T147, X201), X202) → U27_gaa(T147, T148, X201, X202, splitc74_in_gaa(T148, X202, X201))
mergesortc1_in_ga(.(T10, .(T11, [])), T14) → U28_ga(T10, T11, T14, mergesortc19_in_ga(T10, T18))
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140))), T14) → U31_ga(T10, T11, T139, T140, T14, splitc74_in_gaa(T140, T141, T142))
U27_gaa(T147, T148, X201, X202, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
U28_ga(T10, T11, T14, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T14, T18, mergesortc19_in_ga(T11, T29))
U31_ga(T10, T11, T139, T140, T14, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_in_ga(.(T10, .(T139, T142)), T154))
mergesortc19_in_ga(T24, .(T24, [])) → mergesortc19_out_ga(T24, .(T24, []))
U29_ga(T10, T11, T14, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, T14, mergec29_in_gga(T18, T29, T14))
U32_ga(T10, T11, T139, T140, T14, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_in_ga(.(T11, T141), T162))
U30_ga(T10, T11, T14, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
U33_ga(T10, T11, T139, T140, T14, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, T14, mergec29_in_gga(T154, T162, T14))
mergec29_in_gga(T37, [], T37) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42, T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66), .(T63, T68)) → U21_gga(T63, T64, T65, T66, T68, lec45_in_gg(T63, T65))
mergec29_in_gga(.(T106, T107), .(T108, T109), .(T108, T111)) → U23_gga(T106, T107, T108, T109, T111, gtc62_in_gg(T106, T108))
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
U34_ga(T10, T11, T139, T140, T14, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)
U21_gga(T63, T64, T65, T66, T68, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, T68, mergec29_in_gga(T64, .(T65, T66), T68))
U23_gga(T106, T107, T108, T109, T111, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, T111, mergec29_in_gga(.(T106, T107), T109, T111))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U22_gga(T63, T64, T65, T66, T68, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U24_gga(T106, T107, T108, T109, T111, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
[] = []
mergesortc19_in_ga(x1, x2) = mergesortc19_in_ga(x1)
mergesortc19_out_ga(x1, x2) = mergesortc19_out_ga(x1, x2)
s(x1) = s(x1)
lec45_in_gg(x1, x2) = lec45_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
0 = 0
lec45_out_gg(x1, x2) = lec45_out_gg(x1, x2)
gtc62_in_gg(x1, x2) = gtc62_in_gg(x1, x2)
U26_gg(x1, x2, x3) = U26_gg(x1, x2, x3)
gtc62_out_gg(x1, x2) = gtc62_out_gg(x1, x2)
splitc74_in_gaa(x1, x2, x3) = splitc74_in_gaa(x1)
splitc74_out_gaa(x1, x2, x3) = splitc74_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5) = U27_gaa(x1, x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4) = U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4) = U30_ga(x1, x2, x4)
mergec29_in_gga(x1, x2, x3) = mergec29_in_gga(x1, x2)
mergec29_out_gga(x1, x2, x3) = mergec29_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6) = U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6) = U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6) = U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6) = U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6) = U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7) = U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7) = U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x4, x6)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5, x6) = U14_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7) = U16_GA(x1, x2, x3, x4, x6, x7)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140)))) → U14_GA(T10, T11, T139, T140, splitc74_in_gaa(T140))
U14_GA(T10, T11, T139, T140, splitc74_out_gaa(T140, T141, T142)) → MERGESORT1_IN_GA(.(T10, .(T139, T142)))
U14_GA(T10, T11, T139, T140, splitc74_out_gaa(T140, T141, T142)) → U16_GA(T10, T11, T139, T140, T141, mergesortc1_in_ga(.(T10, .(T139, T142))))
U16_GA(T10, T11, T139, T140, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → MERGESORT1_IN_GA(.(T11, T141))

The TRS R consists of the following rules:

splitc74_in_gaa([]) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148)) → U27_gaa(T147, T148, splitc74_in_gaa(T148))
mergesortc1_in_ga(.(T10, .(T11, []))) → U28_ga(T10, T11, mergesortc19_in_ga(T10))
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140)))) → U31_ga(T10, T11, T139, T140, splitc74_in_gaa(T140))
U27_gaa(T147, T148, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
U28_ga(T10, T11, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T18, mergesortc19_in_ga(T11))
U31_ga(T10, T11, T139, T140, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T141, mergesortc1_in_ga(.(T10, .(T139, T142))))
mergesortc19_in_ga(T24) → mergesortc19_out_ga(T24, .(T24, []))
U29_ga(T10, T11, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, mergec29_in_gga(T18, T29))
U32_ga(T10, T11, T139, T140, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T154, mergesortc1_in_ga(.(T11, T141)))
U30_ga(T10, T11, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
U33_ga(T10, T11, T139, T140, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, mergec29_in_gga(T154, T162))
mergec29_in_gga(T37, []) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66)) → U21_gga(T63, T64, T65, T66, lec45_in_gg(T63, T65))
mergec29_in_gga(.(T106, T107), .(T108, T109)) → U23_gga(T106, T107, T108, T109, gtc62_in_gg(T106, T108))
mergesortc1_in_ga(.(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
U34_ga(T10, T11, T139, T140, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)
U21_gga(T63, T64, T65, T66, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, mergec29_in_gga(T64, .(T65, T66)))
U23_gga(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, mergec29_in_gga(.(T106, T107), T109))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U22_gga(T63, T64, T65, T66, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U24_gga(T106, T107, T108, T109, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

splitc74_in_gaa(x0)
mergesortc1_in_ga(x0)
U27_gaa(x0, x1, x2)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2, x3, x4)
mergesortc19_in_ga(x0)
U29_ga(x0, x1, x2, x3)
U32_ga(x0, x1, x2, x3, x4, x5)
U30_ga(x0, x1, x2)
U33_ga(x0, x1, x2, x3, x4, x5)
mergec29_in_gga(x0, x1)
U34_ga(x0, x1, x2, x3, x4)
U21_gga(x0, x1, x2, x3, x4)
U23_gga(x0, x1, x2, x3, x4)
lec45_in_gg(x0, x1)
U22_gga(x0, x1, x2, x3, x4)
gtc62_in_gg(x0, x1)
U24_gga(x0, x1, x2, x3, x4)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT1_IN_GA(.(T10, .(T11, .(T139, T140)))) → U14_GA(T10, T11, T139, T140, splitc74_in_gaa(T140))
U14_GA(T10, T11, T139, T140, splitc74_out_gaa(T140, T141, T142)) → U16_GA(T10, T11, T139, T140, T141, mergesortc1_in_ga(.(T10, .(T139, T142))))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂
POL(0) = 1
POL(MERGESORT1_IN_GA(x₁)) = x₁
POL(U14_GA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅
POL(U16_GA(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₅
POL(U21_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U22_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U23_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U24_gga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U25_gg(x₁, x₂, x₃)) = 1
POL(U26_gg(x₁, x₂, x₃)) = 0
POL(U27_gaa(x₁, x₂, x₃)) = 1 + x₃
POL(U28_ga(x₁, x₂, x₃)) = 0
POL(U29_ga(x₁, x₂, x₃, x₄)) = 0
POL(U30_ga(x₁, x₂, x₃)) = 0
POL(U31_ga(x₁, x₂, x₃, x₄, x₅)) = 0
POL(U32_ga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0
POL(U33_ga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0
POL(U34_ga(x₁, x₂, x₃, x₄, x₅)) = 0
POL([]) = 0
POL(gtc62_in_gg(x₁, x₂)) = 0
POL(gtc62_out_gg(x₁, x₂)) = 0
POL(lec45_in_gg(x₁, x₂)) = 1 + x₁
POL(lec45_out_gg(x₁, x₂)) = 1
POL(mergec29_in_gga(x₁, x₂)) = 0
POL(mergec29_out_gga(x₁, x₂, x₃)) = 0
POL(mergesortc19_in_ga(x₁)) = 0
POL(mergesortc19_out_ga(x₁, x₂)) = 0
POL(mergesortc1_in_ga(x₁)) = 0
POL(mergesortc1_out_ga(x₁, x₂)) = 0
POL(s(x₁)) = 1 + x₁
POL(splitc74_in_gaa(x₁)) = 1 + x₁
POL(splitc74_out_gaa(x₁, x₂, x₃)) = 1 + x₂ + x₃

The following usable rules [FROCOS05] were oriented:

splitc74_in_gaa([]) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148)) → U27_gaa(T147, T148, splitc74_in_gaa(T148))
U27_gaa(T147, T148, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)

(51) Obligation:

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

U14_GA(T10, T11, T139, T140, splitc74_out_gaa(T140, T141, T142)) → MERGESORT1_IN_GA(.(T10, .(T139, T142)))
U16_GA(T10, T11, T139, T140, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → MERGESORT1_IN_GA(.(T11, T141))

The TRS R consists of the following rules:

splitc74_in_gaa([]) → splitc74_out_gaa([], [], [])
splitc74_in_gaa(.(T147, T148)) → U27_gaa(T147, T148, splitc74_in_gaa(T148))
mergesortc1_in_ga(.(T10, .(T11, []))) → U28_ga(T10, T11, mergesortc19_in_ga(T10))
mergesortc1_in_ga(.(T10, .(T11, .(T139, T140)))) → U31_ga(T10, T11, T139, T140, splitc74_in_gaa(T140))
U27_gaa(T147, T148, splitc74_out_gaa(T148, X202, X201)) → splitc74_out_gaa(.(T147, T148), .(T147, X201), X202)
U28_ga(T10, T11, mergesortc19_out_ga(T10, T18)) → U29_ga(T10, T11, T18, mergesortc19_in_ga(T11))
U31_ga(T10, T11, T139, T140, splitc74_out_gaa(T140, T141, T142)) → U32_ga(T10, T11, T139, T140, T141, mergesortc1_in_ga(.(T10, .(T139, T142))))
mergesortc19_in_ga(T24) → mergesortc19_out_ga(T24, .(T24, []))
U29_ga(T10, T11, T18, mergesortc19_out_ga(T11, T29)) → U30_ga(T10, T11, mergec29_in_gga(T18, T29))
U32_ga(T10, T11, T139, T140, T141, mergesortc1_out_ga(.(T10, .(T139, T142)), T154)) → U33_ga(T10, T11, T139, T140, T154, mergesortc1_in_ga(.(T11, T141)))
U30_ga(T10, T11, mergec29_out_gga(T18, T29, T14)) → mergesortc1_out_ga(.(T10, .(T11, [])), T14)
U33_ga(T10, T11, T139, T140, T154, mergesortc1_out_ga(.(T11, T141), T162)) → U34_ga(T10, T11, T139, T140, mergec29_in_gga(T154, T162))
mergec29_in_gga(T37, []) → mergec29_out_gga(T37, [], T37)
mergec29_in_gga([], T42) → mergec29_out_gga([], T42, T42)
mergec29_in_gga(.(T63, T64), .(T65, T66)) → U21_gga(T63, T64, T65, T66, lec45_in_gg(T63, T65))
mergec29_in_gga(.(T106, T107), .(T108, T109)) → U23_gga(T106, T107, T108, T109, gtc62_in_gg(T106, T108))
mergesortc1_in_ga(.(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
U34_ga(T10, T11, T139, T140, mergec29_out_gga(T154, T162, T14)) → mergesortc1_out_ga(.(T10, .(T11, .(T139, T140))), T14)
U21_gga(T63, T64, T65, T66, lec45_out_gg(T63, T65)) → U22_gga(T63, T64, T65, T66, mergec29_in_gga(T64, .(T65, T66)))
U23_gga(T106, T107, T108, T109, gtc62_out_gg(T106, T108)) → U24_gga(T106, T107, T108, T109, mergec29_in_gga(.(T106, T107), T109))
lec45_in_gg(s(T81), s(T82)) → U25_gg(T81, T82, lec45_in_gg(T81, T82))
lec45_in_gg(0, s(T89)) → lec45_out_gg(0, s(T89))
lec45_in_gg(0, 0) → lec45_out_gg(0, 0)
U22_gga(T63, T64, T65, T66, mergec29_out_gga(T64, .(T65, T66), T68)) → mergec29_out_gga(.(T63, T64), .(T65, T66), .(T63, T68))
gtc62_in_gg(s(T124), s(T125)) → U26_gg(T124, T125, gtc62_in_gg(T124, T125))
gtc62_in_gg(s(T130), 0) → gtc62_out_gg(s(T130), 0)
U24_gga(T106, T107, T108, T109, mergec29_out_gga(.(T106, T107), T109, T111)) → mergec29_out_gga(.(T106, T107), .(T108, T109), .(T108, T111))
U25_gg(T81, T82, lec45_out_gg(T81, T82)) → lec45_out_gg(s(T81), s(T82))
U26_gg(T124, T125, gtc62_out_gg(T124, T125)) → gtc62_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

splitc74_in_gaa(x0)
mergesortc1_in_ga(x0)
U27_gaa(x0, x1, x2)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2, x3, x4)
mergesortc19_in_ga(x0)
U29_ga(x0, x1, x2, x3)
U32_ga(x0, x1, x2, x3, x4, x5)
U30_ga(x0, x1, x2)
U33_ga(x0, x1, x2, x3, x4, x5)
mergec29_in_gga(x0, x1)
U34_ga(x0, x1, x2, x3, x4)
U21_gga(x0, x1, x2, x3, x4)
U23_gga(x0, x1, x2, x3, x4)
lec45_in_gg(x0, x1)
U22_gga(x0, x1, x2, x3, x4)
gtc62_in_gg(x0, x1)
U24_gga(x0, x1, x2, x3, x4)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(52) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

(4) Obligation:

(5) TriplesToPiDPProof (SOUND transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (EQUIVALENT transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) PiDPToQDPProof (SOUND transformation)

(34) Obligation:

(35) MRRProof (EQUIVALENT transformation)

(36) Obligation:

(37) DependencyGraphProof (EQUIVALENT transformation)

(38) Obligation:

(39) UsableRulesProof (EQUIVALENT transformation)

(40) Obligation:

(41) QReductionProof (EQUIVALENT transformation)

(42) Obligation:

(43) QDPSizeChangeProof (EQUIVALENT transformation)

(44) YES

(45) Obligation:

(46) UsableRulesProof (EQUIVALENT transformation)

(47) Obligation:

(48) PiDPToQDPProof (SOUND transformation)

(49) Obligation:

(50) QDPOrderProof (EQUIVALENT transformation)

(51) Obligation:

(52) DependencyGraphProof (EQUIVALENT transformation)

(53) TRUE