Left Termination of the query pattern mergesort_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

mergesort([], [], Ls).
mergesort(.(X, []), .(X, []), Ls).
mergesort(.(X, .(Y, Xs)), Ys, .(H, Ls)) :- ','(split(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)), ','(mergesort(X1s, Y1s, Ls), ','(mergesort(X2s, Y2s, Ls), merge(Y1s, Y2s, Ys, .(H, Ls))))).
split([], [], [], Ls).
split(.(X, Xs), .(X, Ys), Zs, .(H, Ls)) :- split(Xs, Zs, Ys, Ls).
merge([], Xs, Xs, Ls).
merge(Xs, [], Xs, Ls).
merge(.(X, Xs), .(Y, Ys), .(X, Zs), .(H, Ls)) :- ','(le(X, Y), merge(Xs, .(Y, Ys), Zs, Ls)).
merge(.(X, Xs), .(Y, Ys), .(Y, Zs), .(H, Ls)) :- ','(gt(X, Y), merge(.(X, Xs), Ys, Zs, Ls)).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(0), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(0)).
le(0, 0).

Queries:

mergesort(g,a,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, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mergesort(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | mergesort(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | mergesort([], T2, T3), SCOPE: 1, CLAUSE: 1 | mergesort([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mergesort(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2, T3) !=? mergesort([], [], X2)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: mergesort([], T2, T3), SCOPE: 1, CLAUSE: 1 | mergesort([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: mergesort([], T2, T3), 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(.(T8, []), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: mergesort(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2, T3) !=? mergesort([], [], X2)\nmergesort(T1, T2, T3) !=? mergesort(.(X14, []), .(X14, []), X15)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: mergesort(.(T8, []), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(split(.(T17, .(T18, T19)), X36, X37, .(T23, T24)), ','(mergesort(X36, X38, T24), ','(mergesort(X37, X39, T24), merge(X38, X39, T25, .(T23, T24)))))\n\nT17 is ground\nT18 is ground\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(split(.(T17, .(T18, T19)), X36, X37, .(T23, T24)), ','(mergesort(X36, X38, T24), ','(mergesort(X37, X39, T24), merge(X38, X39, T25, .(T23, T24))))), SCOPE: 2, CLAUSE: 3 | ','(split(.(T17, .(T18, T19)), X36, X37, .(T23, T24)), ','(mergesort(X36, X38, T24), ','(mergesort(X37, X39, T24), merge(X38, X39, T25, .(T23, T24))))), SCOPE: 2, CLAUSE: 4\n\nT17 is ground\nT18 is ground\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(.(T17, .(T18, T19)), X36, X37, .(T23, T24)), ','(mergesort(X36, X38, T24), ','(mergesort(X37, X39, T24), merge(X38, X39, T25, .(T23, T24))))), SCOPE: 2, CLAUSE: 4\n\nT17 is ground\nT18 is ground\nT19 is ground\nX36 is free; \nX37 is free; \nX38 is free; \nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(split(.(T44, T45), X80, X79, T48), ','(mergesort(.(T43, X79), X38, T48), ','(mergesort(X80, X39, T48), merge(X38, X39, T49, .(T50, T48)))))\n\nT43 is ground\nT44 is ground\nT45 is ground\nX38 is free; \nX39 is free; \nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: split(.(T44, T45), X80, X79, T48)\n\nT44 is ground\nT45 is ground\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(mergesort(.(T43, T56), X38, T57), ','(mergesort(T55, X39, T57), merge(X38, X39, T58, .(T59, T57))))\n\nT43 is ground\nT55 is ground\nT56 is ground\nX38 is free; \nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: split(.(T44, T45), X80, X79, T48), SCOPE: 3, CLAUSE: 3 | split(.(T44, T45), X80, X79, T48), SCOPE: 3, CLAUSE: 4\n\nT44 is ground\nT45 is ground\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: split(.(T44, T45), X80, X79, T48), SCOPE: 3, CLAUSE: 4\n\nT44 is ground\nT45 is ground\nX79 is free; \nX80 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: split(T73, X125, X124, T76)\n\nT73 is ground\nX124 is free; \nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: split(T73, X125, X124, T76), SCOPE: 4, CLAUSE: 3 | split(T73, X125, X124, T76), SCOPE: 4, CLAUSE: 4\n\nT73 is ground\nX124 is free; \nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: split(T73, X125, X124, T76), SCOPE: 4, CLAUSE: 3\n\nT73 is ground\nX124 is free; \nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: split(T73, X125, X124, T76), SCOPE: 4, CLAUSE: 4\n\nT73 is ground\nX124 is free; \nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: split(T93, X163, X162, T96)\n\nT93 is ground\nX162 is free; \nX163 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: mergesort(.(T43, T56), X38, T57)\n\nT43 is ground\nT56 is ground\nX38 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(mergesort(T55, X39, T104), merge(T103, X39, T105, .(T106, T104)))\n\nT55 is ground\nT103 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: mergesort(T55, X39, T104)\n\nT55 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: merge(T103, T113, T114, .(T115, T116))\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 0 | mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 1 | mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 2\n\nT55 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 0\n\nT55 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 1 | mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 2\n\nT55 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 1\n\nT55 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: mergesort(T55, X39, T104), SCOPE: 5, CLAUSE: 2\n\nT55 is ground\nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(split(.(T144, .(T145, T146)), X230, X231, .(T149, T150)), ','(mergesort(X230, X232, T150), ','(mergesort(X231, X233, T150), merge(X232, X233, X234, .(T149, T150)))))\n\nT144 is ground\nT145 is ground\nT146 is ground\nX234 is free; \nX230 is free; \nX231 is free; \nX232 is free; \nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: split(.(T144, .(T145, T146)), X230, X231, .(T149, T150))\n\nT144 is ground\nT145 is ground\nT146 is ground\nX230 is free; \nX231 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(mergesort(T156, X232, T158), ','(mergesort(T157, X233, T158), merge(X232, X233, X234, .(T159, T158))))\n\nT156 is ground\nT157 is ground\nX234 is free; \nX232 is free; \nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: mergesort(T156, X232, T158)\n\nT156 is ground\nX232 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(mergesort(T157, X233, T168), merge(T167, X233, X234, .(T169, T168)))\n\nT157 is ground\nT167 is ground\nX234 is free; \nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: mergesort(T157, X233, T168)\n\nT157 is ground\nX233 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: merge(T167, T174, X234, .(T175, T176))\n\nT167 is ground\nT174 is ground\nX234 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 5 | merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 6 | merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 7 | merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 8\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 5\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 6 | merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 7 | merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 8\n\nT103 is ground\nT113 is ground", 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: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 6\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 7 | merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 8\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 7\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: merge(T103, T113, T114, .(T115, T116)), SCOPE: 6, CLAUSE: 8\n\nT103 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(le(T249, T251), merge(T250, .(T251, T252), T256, T257))\n\nT249 is ground\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: le(T249, T251)\n\nT249 is ground\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: merge(T250, .(T251, T252), T256, T257)\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: le(T249, T251), SCOPE: 7, CLAUSE: 11 | le(T249, T251), SCOPE: 7, CLAUSE: 12 | le(T249, T251), SCOPE: 7, CLAUSE: 13\n\nT249 is ground\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: le(T249, T251), SCOPE: 7, CLAUSE: 11\n\nT249 is ground\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: le(T249, T251), SCOPE: 7, CLAUSE: 12 | le(T249, T251), SCOPE: 7, CLAUSE: 13\n\nT249 is ground\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: le(T270, T271)\n\nT270 is ground\nT271 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: le(T249, T251), SCOPE: 7, CLAUSE: 12\n\nT249 is ground\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: le(T249, T251), SCOPE: 7, CLAUSE: 13\n\nT249 is ground\nT251 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 5 | merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 6 | merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 7 | merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 8\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 5\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 6 | merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 7 | merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 8\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 7 | merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 8\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 7\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: merge(T250, .(T251, T252), T256, T257), SCOPE: 8, CLAUSE: 8\n\nT250 is ground\nT251 is ground\nT252 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(le(T324, T326), merge(T325, .(T326, T327), T331, T332))\n\nT324 is ground\nT325 is ground\nT326 is ground\nT327 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(gt(T349, T351), merge(.(T349, T350), T352, T356, T357))\n\nT349 is ground\nT350 is ground\nT351 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: gt(T349, T351)\n\nT349 is ground\nT351 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: merge(.(T349, T350), T352, T356, T357)\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: gt(T349, T351), SCOPE: 9, CLAUSE: 9 | gt(T349, T351), SCOPE: 9, CLAUSE: 10\n\nT349 is ground\nT351 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: gt(T349, T351), SCOPE: 9, CLAUSE: 9\n\nT349 is ground\nT351 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: gt(T349, T351), SCOPE: 9, CLAUSE: 10\n\nT349 is ground\nT351 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: gt(T370, T371)\n\nT370 is ground\nT371 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 5 | merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 6 | merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 7 | merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 8\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 6 | merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 7 | merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 8\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 6\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 7 | merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 8\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 7\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: merge(.(T349, T350), T352, T356, T357), SCOPE: 10, CLAUSE: 8\n\nT349 is ground\nT350 is ground\nT352 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(le(T423, T425), merge(T424, .(T425, T426), T430, T431))\n\nT423 is ground\nT424 is ground\nT425 is ground\nT426 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: ','(gt(T448, T450), merge(.(T448, T449), T451, T455, T456))\n\nT448 is ground\nT449 is ground\nT450 is ground\nT451 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(gt(T473, T475), merge(.(T473, T474), T476, T480, T481))\n\nT473 is ground\nT474 is ground\nT475 is ground\nT476 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 122 [arrowhead = none ];
122 -> 2;

123 [label="EVAL with clause\nmergesort([], [], X2).\nand substitution\nT1 -> []\nT2 -> []\nT3 -> T5\nX2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 123 [arrowhead = none ];
123 -> 3;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 124 [arrowhead = none ];
124 -> 4;

125 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 125 [arrowhead = none ];
125 -> 5;

126 [label="EVAL with clause\nmergesort(.(X14, []), .(X14, []), X15).\nand substitution\nX14 -> T8\nT1 -> .(T8, [])\nT2 -> .(T8, [])\nT3 -> T9\nX15 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 126 [arrowhead = none ];
126 -> 8;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 127 [arrowhead = none ];
127 -> 9;

128 [label="BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []), Ls)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 128 [arrowhead = none ];
128 -> 6;

129 [label="BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys, .(H, Ls)) :- ','(split(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)), ','(mergesort(X1s, Y1s, Ls), ','(mergesort(X2s, Y2s, Ls), merge(Y1s, Y2s, Ys, .(H, Ls)))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 129 [arrowhead = none ];
129 -> 7;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 130 [arrowhead = none ];
130 -> 10;

131 [label="EVAL with clause\nmergesort(.(X30, .(X31, X32)), X33, .(X34, X35)) :- ','(split(.(X30, .(X31, X32)), X36, X37, .(X34, X35)), ','(mergesort(X36, X38, X35), ','(mergesort(X37, X39, X35), merge(X38, X39, X33, .(X34, X35))))).\nand substitution\nX30 -> T17\nX31 -> T18\nX32 -> T19\nT1 -> .(T17, .(T18, T19))\nT2 -> T25\nX33 -> T25\nX34 -> T23\nX35 -> T24\nT3 -> .(T23, T24)\nT21 -> T23\nT22 -> T24\nT20 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 131 [arrowhead = none ];
131 -> 12;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 132 [arrowhead = none ];
132 -> 13;

133 [label="BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys, .(H, Ls)) :- ','(split(.(X, .(Y, Xs)), X1s, X2s, .(H, Ls)), ','(mergesort(X1s, Y1s, Ls), ','(mergesort(X2s, Y2s, Ls), merge(Y1s, Y2s, Ys, .(H, Ls)))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 133 [arrowhead = none ];
133 -> 11;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 134 [arrowhead = none ];
134 -> 14;

135 [label="BACKTRACK\nfor clause: split([], [], [], Ls)because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 135 [arrowhead = none ];
135 -> 15;

136 [label="ONLY EVAL with clause\nsplit(.(X73, X74), .(X73, X75), X76, .(X77, X78)) :- split(X74, X76, X75, X78).\nand substitution\nT17 -> T43\nX73 -> T43\nT18 -> T44\nT19 -> T45\nX74 -> .(T44, T45)\nX75 -> X79\nX36 -> .(T43, X79)\nX37 -> X80\nX76 -> X80\nT23 -> T50\nX77 -> T50\nT24 -> T48\nX78 -> T48\nT47 -> T48\nT25 -> T49\nT46 -> T50", fontsize=14, style = filled, fillcolor = white];
15 -> 136 [arrowhead = none ];
136 -> 16;

137 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 137 [arrowhead = none ];
137 -> 17;

138 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT56 is ground\nreplacements:\nX80 -> T55\nX79 -> T56\nT48 -> T57\nT49 -> T58\nT50 -> T59", fontsize=14, style = filled, fillcolor = violet];
16 -> 138 [arrowhead = none ];
138 -> 18;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 139 [arrowhead = none ];
139 -> 19;

140 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 140 [arrowhead = none ];
140 -> 31;

141 [label="SPLIT 2\nnew knowledge:\nT103 is ground\nreplacements:\nX38 -> T103\nT57 -> T104\nT58 -> T105\nT59 -> T106", fontsize=14, style = filled, fillcolor = violet];
18 -> 141 [arrowhead = none ];
141 -> 32;

142 [label="BACKTRACK\nfor clause: split([], [], [], Ls)because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 142 [arrowhead = none ];
142 -> 20;

143 [label="EVAL with clause\nsplit(.(X118, X119), .(X118, X120), X121, .(X122, X123)) :- split(X119, X121, X120, X123).\nand substitution\nT44 -> T72\nX118 -> T72\nT45 -> T73\nX119 -> T73\nX120 -> X124\nX80 -> .(T72, X124)\nX79 -> X125\nX121 -> X125\nX122 -> T74\nX123 -> T76\nT48 -> .(T74, T76)\nT75 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 143 [arrowhead = none ];
143 -> 21;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 144 [arrowhead = none ];
144 -> 22;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 145 [arrowhead = none ];
145 -> 23;

146 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 146 [arrowhead = none ];
146 -> 24;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 147 [arrowhead = none ];
147 -> 25;

148 [label="EVAL with clause\nsplit([], [], [], X138).\nand substitution\nT73 -> []\nX125 -> []\nX124 -> []\nT76 -> T83\nX138 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 148 [arrowhead = none ];
148 -> 26;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 149 [arrowhead = none ];
149 -> 27;

150 [label="EVAL with clause\nsplit(.(X156, X157), .(X156, X158), X159, .(X160, X161)) :- split(X157, X159, X158, X161).\nand substitution\nX156 -> T92\nX157 -> T93\nT73 -> .(T92, T93)\nX158 -> X162\nX125 -> .(T92, X162)\nX124 -> X163\nX159 -> X163\nX160 -> T94\nX161 -> T96\nT76 -> .(T94, T96)\nT95 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 150 [arrowhead = none ];
150 -> 29;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 151 [arrowhead = none ];
151 -> 30;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 152 [arrowhead = none ];
152 -> 28;

153 [label="INSTANCE with matching:\nT73 -> T93\nX125 -> X163\nX124 -> X162\nT76 -> T96", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 153 [arrowhead = none , style=dashed];
153 -> 21[style=dashed];

154 [label="INSTANCE with matching:\nT1 -> .(T43, T56)\nT2 -> X38\nT3 -> T57", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 154 [arrowhead = none , style=dashed];
154 -> 1[style=dashed];

155 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
32 -> 155 [arrowhead = none ];
155 -> 33;

156 [label="SPLIT 2\nnew knowledge:\nT113 is ground\nreplacements:\nX39 -> T113\nT105 -> T114\nT106 -> T115\nT104 -> T116", fontsize=14, style = filled, fillcolor = violet];
32 -> 156 [arrowhead = none ];
156 -> 34;

157 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 157 [arrowhead = none ];
157 -> 35;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 158 [arrowhead = none ];
158 -> 54;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 159 [arrowhead = none ];
159 -> 36;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 160 [arrowhead = none ];
160 -> 37;

161 [label="EVAL with clause\nmergesort([], [], X192).\nand substitution\nT55 -> []\nX39 -> []\nT104 -> T123\nX192 -> T123", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 161 [arrowhead = none ];
161 -> 38;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 162 [arrowhead = none ];
162 -> 39;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 163 [arrowhead = none ];
163 -> 41;

164 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 164 [arrowhead = none ];
164 -> 42;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 165 [arrowhead = none ];
165 -> 40;

166 [label="EVAL with clause\nmergesort(.(X202, []), .(X202, []), X203).\nand substitution\nX202 -> T132\nT55 -> .(T132, [])\nX39 -> .(T132, [])\nT104 -> T133\nX203 -> T133", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 166 [arrowhead = none ];
166 -> 43;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 167 [arrowhead = none ];
167 -> 44;

168 [label="EVAL with clause\nmergesort(.(X224, .(X225, X226)), X227, .(X228, X229)) :- ','(split(.(X224, .(X225, X226)), X230, X231, .(X228, X229)), ','(mergesort(X230, X232, X229), ','(mergesort(X231, X233, X229), merge(X232, X233, X227, .(X228, X229))))).\nand substitution\nX224 -> T144\nX225 -> T145\nX226 -> T146\nT55 -> .(T144, .(T145, T146))\nX39 -> X234\nX227 -> X234\nX228 -> T149\nX229 -> T150\nT104 -> .(T149, T150)\nT147 -> T149\nT148 -> T150", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 168 [arrowhead = none ];
168 -> 46;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 169 [arrowhead = none ];
169 -> 47;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 170 [arrowhead = none ];
170 -> 45;

171 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 171 [arrowhead = none ];
171 -> 48;

172 [label="SPLIT 2\nnew knowledge:\nT156 is ground\nT157 is ground\nreplacements:\nX230 -> T156\nX231 -> T157\nT150 -> T158\nT149 -> T159", fontsize=14, style = filled, fillcolor = violet];
46 -> 172 [arrowhead = none ];
172 -> 49;

173 [label="INSTANCE with matching:\nT44 -> T144\nT45 -> .(T145, T146)\nX80 -> X230\nX79 -> X231\nT48 -> .(T149, T150)", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 173 [arrowhead = none , style=dashed];
173 -> 17[style=dashed];

174 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 174 [arrowhead = none ];
174 -> 50;

175 [label="SPLIT 2\nnew knowledge:\nT167 is ground\nreplacements:\nX232 -> T167\nT158 -> T168\nT159 -> T169", fontsize=14, style = filled, fillcolor = violet];
49 -> 175 [arrowhead = none ];
175 -> 51;

176 [label="INSTANCE with matching:\nT55 -> T156\nX39 -> X232\nT104 -> T158", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 176 [arrowhead = none , style=dashed];
176 -> 33[style=dashed];

177 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
51 -> 177 [arrowhead = none ];
177 -> 52;

178 [label="SPLIT 2\nnew knowledge:\nT174 is ground\nreplacements:\nX233 -> T174\nT169 -> T175\nT168 -> T176", fontsize=14, style = filled, fillcolor = violet];
51 -> 178 [arrowhead = none ];
178 -> 53;

179 [label="INSTANCE with matching:\nT55 -> T157\nX39 -> X233\nT104 -> T168", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 179 [arrowhead = none , style=dashed];
179 -> 33[style=dashed];

180 [label="INSTANCE with matching:\nT103 -> T167\nT113 -> T174\nT114 -> X234\nT115 -> T175\nT116 -> T176", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 180 [arrowhead = none , style=dashed];
180 -> 34[style=dashed];

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 181 [arrowhead = none ];
181 -> 55;

182 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 182 [arrowhead = none ];
182 -> 56;

183 [label="EVAL with clause\nmerge([], X285, X285, X286).\nand substitution\nT103 -> []\nT113 -> T203\nX285 -> T203\nT114 -> T203\nT115 -> T204\nT116 -> T205\nX286 -> .(T204, T205)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 183 [arrowhead = none ];
183 -> 57;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 184 [arrowhead = none ];
184 -> 58;

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 185 [arrowhead = none ];
185 -> 60;

186 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 186 [arrowhead = none ];
186 -> 61;

187 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 187 [arrowhead = none ];
187 -> 59;

188 [label="EVAL with clause\nmerge(X297, [], X297, X298).\nand substitution\nT103 -> T218\nX297 -> T218\nT113 -> []\nT114 -> T218\nT115 -> T219\nT116 -> T220\nX298 -> .(T219, T220)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 188 [arrowhead = none ];
188 -> 62;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 189 [arrowhead = none ];
189 -> 63;

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 190 [arrowhead = none ];
190 -> 65;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
61 -> 191 [arrowhead = none ];
191 -> 66;

192 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 192 [arrowhead = none ];
192 -> 64;

193 [label="EVAL with clause\nmerge(.(X329, X330), .(X331, X332), .(X329, X333), .(X334, X335)) :- ','(le(X329, X331), merge(X330, .(X331, X332), X333, X335)).\nand substitution\nX329 -> T249\nX330 -> T250\nT103 -> .(T249, T250)\nX331 -> T251\nX332 -> T252\nT113 -> .(T251, T252)\nX333 -> T256\nT114 -> .(T249, T256)\nT115 -> T254\nX334 -> T254\nT116 -> T257\nX335 -> T257\nT253 -> T256\nT255 -> T257", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 193 [arrowhead = none ];
193 -> 67;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 194 [arrowhead = none ];
194 -> 68;

195 [label="EVAL with clause\nmerge(.(X579, X580), .(X581, X582), .(X581, X583), .(X584, X585)) :- ','(gt(X579, X581), merge(.(X579, X580), X582, X583, X585)).\nand substitution\nX579 -> T473\nX580 -> T474\nT103 -> .(T473, T474)\nX581 -> T475\nX582 -> T476\nT113 -> .(T475, T476)\nX583 -> T480\nT114 -> .(T475, T480)\nT115 -> T478\nX584 -> T478\nT116 -> T481\nX585 -> T481\nT477 -> T480\nT479 -> T481", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 195 [arrowhead = none ];
195 -> 120;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 196 [arrowhead = none ];
196 -> 121;

197 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
67 -> 197 [arrowhead = none ];
197 -> 69;

198 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
67 -> 198 [arrowhead = none ];
198 -> 70;

199 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 199 [arrowhead = none ];
199 -> 71;

200 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 200 [arrowhead = none ];
200 -> 84;

201 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 201 [arrowhead = none ];
201 -> 72;

202 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 202 [arrowhead = none ];
202 -> 73;

203 [label="EVAL with clause\nle(s(X355), s(X356)) :- le(X355, X356).\nand substitution\nX355 -> T270\nT249 -> s(T270)\nX356 -> T271\nT251 -> s(T271)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 203 [arrowhead = none ];
203 -> 74;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 204 [arrowhead = none ];
204 -> 75;

205 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 205 [arrowhead = none ];
205 -> 76;

206 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 206 [arrowhead = none ];
206 -> 77;

207 [label="INSTANCE with matching:\nT249 -> T270\nT251 -> T271", fontsize=14, style = filled, fillcolor = lightgrey];
74 -> 207 [arrowhead = none , style=dashed];
207 -> 69[style=dashed];

208 [label="EVAL with clause\nle(0, s(0)).\nand substitution\nT249 -> 0\nT251 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 208 [arrowhead = none ];
208 -> 78;

209 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 209 [arrowhead = none ];
209 -> 79;

210 [label="EVAL with clause\nle(0, 0).\nand substitution\nT249 -> 0\nT251 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 210 [arrowhead = none ];
210 -> 81;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 211 [arrowhead = none ];
211 -> 82;

212 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 212 [arrowhead = none ];
212 -> 80;

213 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 213 [arrowhead = none ];
213 -> 83;

214 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
84 -> 214 [arrowhead = none ];
214 -> 85;

215 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
84 -> 215 [arrowhead = none ];
215 -> 86;

216 [label="EVAL with clause\nmerge([], X373, X373, X374).\nand substitution\nT250 -> []\nT251 -> T292\nT252 -> T293\nX373 -> .(T292, T293)\nT256 -> .(T292, T293)\nT257 -> T294\nX374 -> T294", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 216 [arrowhead = none ];
216 -> 87;

217 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 217 [arrowhead = none ];
217 -> 88;

218 [label="BACKTRACK\nfor clause: merge(Xs, [], Xs, Ls)because of non-unification", fontsize=14, style = filled, fillcolor = white];
86 -> 218 [arrowhead = none ];
218 -> 90;

219 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 219 [arrowhead = none ];
219 -> 89;

220 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 220 [arrowhead = none ];
220 -> 91;

221 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
90 -> 221 [arrowhead = none ];
221 -> 92;

222 [label="EVAL with clause\nmerge(.(X407, X408), .(X409, X410), .(X407, X411), .(X412, X413)) :- ','(le(X407, X409), merge(X408, .(X409, X410), X411, X413)).\nand substitution\nX407 -> T324\nX408 -> T325\nT250 -> .(T324, T325)\nT251 -> T326\nX409 -> T326\nT252 -> T327\nX410 -> T327\nX411 -> T331\nT256 -> .(T324, T331)\nX412 -> T329\nX413 -> T332\nT257 -> .(T329, T332)\nT328 -> T331\nT330 -> T332", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 222 [arrowhead = none ];
222 -> 93;

223 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 223 [arrowhead = none ];
223 -> 94;

224 [label="EVAL with clause\nmerge(.(X437, X438), .(X439, X440), .(X439, X441), .(X442, X443)) :- ','(gt(X437, X439), merge(.(X437, X438), X440, X441, X443)).\nand substitution\nX437 -> T349\nX438 -> T350\nT250 -> .(T349, T350)\nT251 -> T351\nX439 -> T351\nT252 -> T352\nX440 -> T352\nX441 -> T356\nT256 -> .(T351, T356)\nX442 -> T354\nX443 -> T357\nT257 -> .(T354, T357)\nT353 -> T356\nT355 -> T357", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 224 [arrowhead = none ];
224 -> 95;

225 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 225 [arrowhead = none ];
225 -> 96;

226 [label="INSTANCE with matching:\nT249 -> T324\nT251 -> T326\nT250 -> T325\nT252 -> T327\nT256 -> T331\nT257 -> T332", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 226 [arrowhead = none , style=dashed];
226 -> 67[style=dashed];

227 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
95 -> 227 [arrowhead = none ];
227 -> 97;

228 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
95 -> 228 [arrowhead = none ];
228 -> 98;

229 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
97 -> 229 [arrowhead = none ];
229 -> 99;

230 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
98 -> 230 [arrowhead = none ];
230 -> 107;

231 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
99 -> 231 [arrowhead = none ];
231 -> 100;

232 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
99 -> 232 [arrowhead = none ];
232 -> 101;

233 [label="EVAL with clause\ngt(s(X463), s(X464)) :- gt(X463, X464).\nand substitution\nX463 -> T370\nT349 -> s(T370)\nX464 -> T371\nT351 -> s(T371)", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 233 [arrowhead = none ];
233 -> 102;

234 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 234 [arrowhead = none ];
234 -> 103;

235 [label="EVAL with clause\ngt(s(0), 0).\nand substitution\nT349 -> s(0)\nT351 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 235 [arrowhead = none ];
235 -> 104;

236 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 236 [arrowhead = none ];
236 -> 105;

237 [label="INSTANCE with matching:\nT349 -> T370\nT351 -> T371", fontsize=14, style = filled, fillcolor = lightgrey];
102 -> 237 [arrowhead = none , style=dashed];
237 -> 97[style=dashed];

238 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 238 [arrowhead = none ];
238 -> 106;

239 [label="BACKTRACK\nfor clause: merge([], Xs, Xs, Ls)because of non-unification", fontsize=14, style = filled, fillcolor = white];
107 -> 239 [arrowhead = none ];
239 -> 108;

240 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
108 -> 240 [arrowhead = none ];
240 -> 109;

241 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
108 -> 241 [arrowhead = none ];
241 -> 110;

242 [label="EVAL with clause\nmerge(X487, [], X487, X488).\nand substitution\nT349 -> T392\nT350 -> T393\nX487 -> .(T392, T393)\nT352 -> []\nT356 -> .(T392, T393)\nT357 -> T394\nX488 -> T394", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 242 [arrowhead = none ];
242 -> 111;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 243 [arrowhead = none ];
243 -> 112;

244 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
110 -> 244 [arrowhead = none ];
244 -> 114;

245 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
110 -> 245 [arrowhead = none ];
245 -> 115;

246 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
111 -> 246 [arrowhead = none ];
246 -> 113;

247 [label="EVAL with clause\nmerge(.(X519, X520), .(X521, X522), .(X519, X523), .(X524, X525)) :- ','(le(X519, X521), merge(X520, .(X521, X522), X523, X525)).\nand substitution\nT349 -> T423\nX519 -> T423\nT350 -> T424\nX520 -> T424\nX521 -> T425\nX522 -> T426\nT352 -> .(T425, T426)\nX523 -> T430\nT356 -> .(T423, T430)\nX524 -> T428\nX525 -> T431\nT357 -> .(T428, T431)\nT427 -> T430\nT429 -> T431", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 247 [arrowhead = none ];
247 -> 116;

248 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 248 [arrowhead = none ];
248 -> 117;

249 [label="EVAL with clause\nmerge(.(X549, X550), .(X551, X552), .(X551, X553), .(X554, X555)) :- ','(gt(X549, X551), merge(.(X549, X550), X552, X553, X555)).\nand substitution\nT349 -> T448\nX549 -> T448\nT350 -> T449\nX550 -> T449\nX551 -> T450\nX552 -> T451\nT352 -> .(T450, T451)\nX553 -> T455\nT356 -> .(T450, T455)\nX554 -> T453\nX555 -> T456\nT357 -> .(T453, T456)\nT452 -> T455\nT454 -> T456", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 249 [arrowhead = none ];
249 -> 118;

250 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 250 [arrowhead = none ];
250 -> 119;

251 [label="INSTANCE with matching:\nT249 -> T423\nT251 -> T425\nT250 -> T424\nT252 -> T426\nT256 -> T430\nT257 -> T431", fontsize=14, style = filled, fillcolor = lightgrey];
116 -> 251 [arrowhead = none , style=dashed];
251 -> 67[style=dashed];

252 [label="INSTANCE with matching:\nT349 -> T448\nT351 -> T450\nT350 -> T449\nT352 -> T451\nT356 -> T455\nT357 -> T456", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 252 [arrowhead = none , style=dashed];
252 -> 95[style=dashed];

253 [label="INSTANCE with matching:\nT349 -> T473\nT351 -> T475\nT350 -> T474\nT352 -> T476\nT356 -> T480\nT357 -> T481", fontsize=14, style = filled, fillcolor = lightgrey];
120 -> 253 [arrowhead = none , style=dashed];
253 -> 95[style=dashed];

}

(2) Obligation:

Triples:

split21(.(T92, T93), .(T92, X162), X163, .(T94, T96)) :- split21(T93, X163, X162, T96).
split17(T72, T73, .(T72, X124), X125, .(T74, T76)) :- split21(T73, X125, X124, T76).
mergesort33(.(T144, .(T145, T146)), X234, .(T149, T150)) :- split17(T144, .(T145, T146), X230, X231, .(T149, T150)).
mergesort33(.(T144, .(T145, T146)), X234, .(T159, T158)) :- ','(splitc17(T144, .(T145, T146), T156, T157, .(T159, T158)), mergesort33(T156, X232, T158)).
mergesort33(.(T144, .(T145, T146)), X234, .(T169, T168)) :- ','(splitc17(T144, .(T145, T146), T156, T157, .(T169, T168)), ','(mergesortc33(T156, T167, T168), mergesort33(T157, X233, T168))).
mergesort33(.(T144, .(T145, T146)), X234, .(T175, T176)) :- ','(splitc17(T144, .(T145, T146), T156, T157, .(T175, T176)), ','(mergesortc33(T156, T167, T176), ','(mergesortc33(T157, T174, T176), merge34(T167, T174, X234, T175, T176)))).
merge34(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) :- p67(T249, T251, T250, T252, T256, T257).
merge34(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) :- p95(T473, T475, T474, T476, T480, T481).
le69(s(T270), s(T271)) :- le69(T270, T271).
p67(T249, T251, T250, T252, T256, T257) :- le69(T249, T251).
p67(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) :- ','(lec69(T249, T326), p67(T324, T326, T325, T327, T331, T332)).
p67(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) :- ','(lec69(T249, T351), p95(T349, T351, T350, T352, T356, T357)).
gt97(s(T370), s(T371)) :- gt97(T370, T371).
p95(T349, T351, T350, T352, T356, T357) :- gt97(T349, T351).
p95(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) :- ','(gtc97(T423, T351), p67(T423, T425, T424, T426, T430, T431)).
p95(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) :- ','(gtc97(T448, T351), p95(T448, T450, T449, T451, T455, T456)).
mergesort1(.(T43, .(T44, T45)), T49, .(T50, T48)) :- split17(T44, T45, X80, X79, T48).
mergesort1(.(T43, .(T44, T45)), T58, .(T59, T57)) :- ','(splitc17(T44, T45, T55, T56, T57), mergesort1(.(T43, T56), X38, T57)).
mergesort1(.(T43, .(T44, T45)), T105, .(T106, T104)) :- ','(splitc17(T44, T45, T55, T56, T104), ','(mergesortc1(.(T43, T56), T103, T104), mergesort33(T55, X39, T104))).
mergesort1(.(T43, .(T44, T45)), T114, .(T115, T116)) :- ','(splitc17(T44, T45, T55, T56, T116), ','(mergesortc1(.(T43, T56), T103, T116), ','(mergesortc33(T55, T113, T116), merge34(T103, T113, T114, T115, T116)))).

Clauses:

splitc21([], [], [], T83).
splitc21(.(T92, T93), .(T92, X162), X163, .(T94, T96)) :- splitc21(T93, X163, X162, T96).
mergesortc1([], [], T5).
mergesortc1(.(T8, []), .(T8, []), T9).
mergesortc1(.(T43, .(T44, T45)), T114, .(T115, T116)) :- ','(splitc17(T44, T45, T55, T56, T116), ','(mergesortc1(.(T43, T56), T103, T116), ','(mergesortc33(T55, T113, T116), mergec34(T103, T113, T114, T115, T116)))).
splitc17(T72, T73, .(T72, X124), X125, .(T74, T76)) :- splitc21(T73, X125, X124, T76).
mergesortc33([], [], T123).
mergesortc33(.(T132, []), .(T132, []), T133).
mergesortc33(.(T144, .(T145, T146)), X234, .(T175, T176)) :- ','(splitc17(T144, .(T145, T146), T156, T157, .(T175, T176)), ','(mergesortc33(T156, T167, T176), ','(mergesortc33(T157, T174, T176), mergec34(T167, T174, X234, T175, T176)))).
mergec34([], T203, T203, T204, T205).
mergec34(T218, [], T218, T219, T220).
mergec34(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) :- qc67(T249, T251, T250, T252, T256, T257).
mergec34(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) :- qc95(T473, T475, T474, T476, T480, T481).
lec69(s(T270), s(T271)) :- lec69(T270, T271).
lec69(0, s(0)).
lec69(0, 0).
qc67(T249, T292, [], T293, .(T292, T293), T294) :- lec69(T249, T292).
qc67(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) :- ','(lec69(T249, T326), qc67(T324, T326, T325, T327, T331, T332)).
qc67(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) :- ','(lec69(T249, T351), qc95(T349, T351, T350, T352, T356, T357)).
gtc97(s(T370), s(T371)) :- gtc97(T370, T371).
gtc97(s(0), 0).
qc95(T392, T351, T393, [], .(T392, T393), T394) :- gtc97(T392, T351).
qc95(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) :- ','(gtc97(T423, T351), qc67(T423, T425, T424, T426, T430, T431)).
qc95(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) :- ','(gtc97(T448, T351), qc95(T448, T450, T449, T451, T455, T456)).

Afs:

mergesort1(x1, x2, x3)  =  mergesort1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f,f)
split17_in: (b,b,f,f,f)
split21_in: (b,f,f,f)
splitc17_in: (b,b,f,f,f)
splitc21_in: (b,f,f,f)
mergesortc1_in: (b,f,f)
mergesortc33_in: (b,f,f)
mergec34_in: (b,b,f,f,f)
qc67_in: (b,b,b,b,f,f)
lec69_in: (b,b)
qc95_in: (b,b,b,b,f,f)
gtc97_in: (b,b)
mergesort33_in: (b,f,f)
merge34_in: (b,b,f,f,f)
p67_in: (b,b,b,b,f,f)
le69_in: (b,b)
p95_in: (b,b,b,b,f,f)
gt97_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T49, .(T50, T48)) → U27_GAA(T43, T44, T45, T49, T50, T48, split17_in_ggaaa(T44, T45, X80, X79, T48))
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T49, .(T50, T48)) → SPLIT17_IN_GGAAA(T44, T45, X80, X79, T48)
SPLIT17_IN_GGAAA(T72, T73, .(T72, X124), X125, .(T74, T76)) → U2_GGAAA(T72, T73, X124, X125, T74, T76, split21_in_gaaa(T73, X125, X124, T76))
SPLIT17_IN_GGAAA(T72, T73, .(T72, X124), X125, .(T74, T76)) → SPLIT21_IN_GAAA(T73, X125, X124, T76)
SPLIT21_IN_GAAA(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U1_GAAA(T92, T93, X162, X163, T94, T96, split21_in_gaaa(T93, X163, X162, T96))
SPLIT21_IN_GAAA(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → SPLIT21_IN_GAAA(T93, X163, X162, T96)
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T58, .(T59, T57)) → U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_in_ggaaa(T44, T45, T55, T56, T57))
U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_out_ggaaa(T44, T45, T55, T56, T57)) → U29_GAA(T43, T44, T45, T58, T59, T57, mergesort1_in_gaa(.(T43, T56), X38, T57))
U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_out_ggaaa(T44, T45, T55, T56, T57)) → MERGESORT1_IN_GAA(.(T43, T56), X38, T57)
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T105, .(T106, T104)) → U30_GAA(T43, T44, T45, T105, T106, T104, splitc17_in_ggaaa(T44, T45, T55, T56, T104))
U30_GAA(T43, T44, T45, T105, T106, T104, splitc17_out_ggaaa(T44, T45, T55, T56, T104)) → U31_GAA(T43, T44, T45, T105, T106, T104, T55, mergesortc1_in_gaa(.(T43, T56), T103, T104))
U31_GAA(T43, T44, T45, T105, T106, T104, T55, mergesortc1_out_gaa(.(T43, T56), T103, T104)) → U32_GAA(T43, T44, T45, T105, T106, T104, mergesort33_in_gaa(T55, X39, T104))
U31_GAA(T43, T44, T45, T105, T106, T104, T55, mergesortc1_out_gaa(.(T43, T56), T103, T104)) → MERGESORT33_IN_GAA(T55, X39, T104)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T149, T150)) → U3_GAA(T144, T145, T146, X234, T149, T150, split17_in_ggaaa(T144, .(T145, T146), X230, X231, .(T149, T150)))
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T149, T150)) → SPLIT17_IN_GGAAA(T144, .(T145, T146), X230, X231, .(T149, T150))
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T159, T158)) → U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158)))
U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158))) → U5_GAA(T144, T145, T146, X234, T159, T158, mergesort33_in_gaa(T156, X232, T158))
U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158))) → MERGESORT33_IN_GAA(T156, X232, T158)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T169, T168)) → U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168)))
U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168))) → U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_in_gaa(T156, T167, T168))
U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_out_gaa(T156, T167, T168)) → U8_GAA(T144, T145, T146, X234, T169, T168, mergesort33_in_gaa(T157, X233, T168))
U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_out_gaa(T156, T167, T168)) → MERGESORT33_IN_GAA(T157, X233, T168)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T175, T176)) → U9_GAA(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U9_GAA(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U10_GAA(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U10_GAA(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U11_GAA(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U11_GAA(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U12_GAA(T144, T145, T146, X234, T175, T176, merge34_in_ggaaa(T167, T174, X234, T175, T176))
U11_GAA(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → MERGE34_IN_GGAAA(T167, T174, X234, T175, T176)
MERGE34_IN_GGAAA(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U13_GGAAA(T249, T250, T251, T252, T256, T254, T257, p67_in_ggggaa(T249, T251, T250, T252, T256, T257))
MERGE34_IN_GGAAA(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → P67_IN_GGGGAA(T249, T251, T250, T252, T256, T257)
P67_IN_GGGGAA(T249, T251, T250, T252, T256, T257) → U16_GGGGAA(T249, T251, T250, T252, T256, T257, le69_in_gg(T249, T251))
P67_IN_GGGGAA(T249, T251, T250, T252, T256, T257) → LE69_IN_GG(T249, T251)
LE69_IN_GG(s(T270), s(T271)) → U15_GG(T270, T271, le69_in_gg(T270, T271))
LE69_IN_GG(s(T270), s(T271)) → LE69_IN_GG(T270, T271)
P67_IN_GGGGAA(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U18_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, p67_in_ggggaa(T324, T326, T325, T327, T331, T332))
U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → P67_IN_GGGGAA(T324, T326, T325, T327, T331, T332)
P67_IN_GGGGAA(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U20_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, p95_in_ggggaa(T349, T351, T350, T352, T356, T357))
U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357)
P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357) → U22_GGGGAA(T349, T351, T350, T352, T356, T357, gt97_in_gg(T349, T351))
P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357) → GT97_IN_GG(T349, T351)
GT97_IN_GG(s(T370), s(T371)) → U21_GG(T370, T371, gt97_in_gg(T370, T371))
GT97_IN_GG(s(T370), s(T371)) → GT97_IN_GG(T370, T371)
P95_IN_GGGGAA(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U24_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, p67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → P67_IN_GGGGAA(T423, T425, T424, T426, T430, T431)
P95_IN_GGGGAA(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U26_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, p95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → P95_IN_GGGGAA(T448, T450, T449, T451, T455, T456)
MERGE34_IN_GGAAA(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U14_GGAAA(T473, T474, T475, T476, T480, T478, T481, p95_in_ggggaa(T473, T475, T474, T476, T480, T481))
MERGE34_IN_GGAAA(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → P95_IN_GGGGAA(T473, T475, T474, T476, T480, T481)
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T114, .(T115, T116)) → U33_GAA(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U33_GAA(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U34_GAA(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U34_GAA(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U35_GAA(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
U35_GAA(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U36_GAA(T43, T44, T45, T114, T115, T116, merge34_in_ggaaa(T103, T113, T114, T115, T116))
U35_GAA(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → MERGE34_IN_GGAAA(T103, T113, T114, T115, T116)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
mergesort1_in_gaa(x1, x2, x3)  =  mergesort1_in_gaa(x1)
.(x1, x2)  =  .(x1, x2)
split17_in_ggaaa(x1, x2, x3, x4, x5)  =  split17_in_ggaaa(x1, x2)
split21_in_gaaa(x1, x2, x3, x4)  =  split21_in_gaaa(x1)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
mergesort33_in_gaa(x1, x2, x3)  =  mergesort33_in_gaa(x1)
merge34_in_ggaaa(x1, x2, x3, x4, x5)  =  merge34_in_ggaaa(x1, x2)
p67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  p67_in_ggggaa(x1, x2, x3, x4)
le69_in_gg(x1, x2)  =  le69_in_gg(x1, x2)
p95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  p95_in_ggggaa(x1, x2, x3, x4)
gt97_in_gg(x1, x2)  =  gt97_in_gg(x1, x2)
MERGESORT1_IN_GAA(x1, x2, x3)  =  MERGESORT1_IN_GAA(x1)
U27_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U27_GAA(x1, x2, x3, x7)
SPLIT17_IN_GGAAA(x1, x2, x3, x4, x5)  =  SPLIT17_IN_GGAAA(x1, x2)
U2_GGAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAAA(x1, x2, x7)
SPLIT21_IN_GAAA(x1, x2, x3, x4)  =  SPLIT21_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GAAA(x1, x2, x7)
U28_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U28_GAA(x1, x2, x3, x7)
U29_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U29_GAA(x1, x2, x3, x7)
U30_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U30_GAA(x1, x2, x3, x7)
U31_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U31_GAA(x1, x2, x3, x7, x8)
U32_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U32_GAA(x1, x2, x3, x7)
MERGESORT33_IN_GAA(x1, x2, x3)  =  MERGESORT33_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAA(x1, x2, x3, x7)
U4_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAA(x1, x2, x3, x7)
U5_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U5_GAA(x1, x2, x3, x7)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U7_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_GAA(x1, x2, x3, x7, x8)
U8_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U8_GAA(x1, x2, x3, x7)
U9_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GAA(x1, x2, x3, x7)
U10_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GAA(x1, x2, x3, x7, x8)
U11_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GAA(x1, x2, x3, x7, x8)
U12_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GAA(x1, x2, x3, x7)
MERGE34_IN_GGAAA(x1, x2, x3, x4, x5)  =  MERGE34_IN_GGAAA(x1, x2)
U13_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GGAAA(x1, x2, x3, x4, x8)
P67_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P67_IN_GGGGAA(x1, x2, x3, x4)
U16_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GGGGAA(x1, x2, x3, x4, x7)
LE69_IN_GG(x1, x2)  =  LE69_IN_GG(x1, x2)
U15_GG(x1, x2, x3)  =  U15_GG(x1, x2, x3)
U17_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGAA(x1, x2, x3, x4, x5, x9)
U18_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U18_GGGGAA(x1, x2, x3, x4, x5, x9)
U19_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U19_GGGGAA(x1, x2, x3, x4, x5, x9)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U20_GGGGAA(x1, x2, x3, x4, x5, x9)
P95_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P95_IN_GGGGAA(x1, x2, x3, x4)
U22_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U22_GGGGAA(x1, x2, x3, x4, x7)
GT97_IN_GG(x1, x2)  =  GT97_IN_GG(x1, x2)
U21_GG(x1, x2, x3)  =  U21_GG(x1, x2, x3)
U23_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U23_GGGGAA(x1, x2, x3, x4, x5, x9)
U24_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U24_GGGGAA(x1, x2, x3, x4, x5, x9)
U25_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U25_GGGGAA(x1, x2, x3, x4, x5, x9)
U26_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_GGGGAA(x1, x2, x3, x4, x5, x9)
U14_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GGAAA(x1, x2, x3, x4, x8)
U33_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U33_GAA(x1, x2, x3, x7)
U34_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_GAA(x1, x2, x3, x7, x8)
U35_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U35_GAA(x1, x2, x3, x7, x8)
U36_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U36_GAA(x1, x2, x3, x7)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T49, .(T50, T48)) → U27_GAA(T43, T44, T45, T49, T50, T48, split17_in_ggaaa(T44, T45, X80, X79, T48))
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T49, .(T50, T48)) → SPLIT17_IN_GGAAA(T44, T45, X80, X79, T48)
SPLIT17_IN_GGAAA(T72, T73, .(T72, X124), X125, .(T74, T76)) → U2_GGAAA(T72, T73, X124, X125, T74, T76, split21_in_gaaa(T73, X125, X124, T76))
SPLIT17_IN_GGAAA(T72, T73, .(T72, X124), X125, .(T74, T76)) → SPLIT21_IN_GAAA(T73, X125, X124, T76)
SPLIT21_IN_GAAA(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U1_GAAA(T92, T93, X162, X163, T94, T96, split21_in_gaaa(T93, X163, X162, T96))
SPLIT21_IN_GAAA(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → SPLIT21_IN_GAAA(T93, X163, X162, T96)
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T58, .(T59, T57)) → U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_in_ggaaa(T44, T45, T55, T56, T57))
U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_out_ggaaa(T44, T45, T55, T56, T57)) → U29_GAA(T43, T44, T45, T58, T59, T57, mergesort1_in_gaa(.(T43, T56), X38, T57))
U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_out_ggaaa(T44, T45, T55, T56, T57)) → MERGESORT1_IN_GAA(.(T43, T56), X38, T57)
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T105, .(T106, T104)) → U30_GAA(T43, T44, T45, T105, T106, T104, splitc17_in_ggaaa(T44, T45, T55, T56, T104))
U30_GAA(T43, T44, T45, T105, T106, T104, splitc17_out_ggaaa(T44, T45, T55, T56, T104)) → U31_GAA(T43, T44, T45, T105, T106, T104, T55, mergesortc1_in_gaa(.(T43, T56), T103, T104))
U31_GAA(T43, T44, T45, T105, T106, T104, T55, mergesortc1_out_gaa(.(T43, T56), T103, T104)) → U32_GAA(T43, T44, T45, T105, T106, T104, mergesort33_in_gaa(T55, X39, T104))
U31_GAA(T43, T44, T45, T105, T106, T104, T55, mergesortc1_out_gaa(.(T43, T56), T103, T104)) → MERGESORT33_IN_GAA(T55, X39, T104)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T149, T150)) → U3_GAA(T144, T145, T146, X234, T149, T150, split17_in_ggaaa(T144, .(T145, T146), X230, X231, .(T149, T150)))
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T149, T150)) → SPLIT17_IN_GGAAA(T144, .(T145, T146), X230, X231, .(T149, T150))
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T159, T158)) → U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158)))
U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158))) → U5_GAA(T144, T145, T146, X234, T159, T158, mergesort33_in_gaa(T156, X232, T158))
U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158))) → MERGESORT33_IN_GAA(T156, X232, T158)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T169, T168)) → U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168)))
U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168))) → U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_in_gaa(T156, T167, T168))
U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_out_gaa(T156, T167, T168)) → U8_GAA(T144, T145, T146, X234, T169, T168, mergesort33_in_gaa(T157, X233, T168))
U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_out_gaa(T156, T167, T168)) → MERGESORT33_IN_GAA(T157, X233, T168)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T175, T176)) → U9_GAA(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U9_GAA(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U10_GAA(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U10_GAA(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U11_GAA(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U11_GAA(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U12_GAA(T144, T145, T146, X234, T175, T176, merge34_in_ggaaa(T167, T174, X234, T175, T176))
U11_GAA(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → MERGE34_IN_GGAAA(T167, T174, X234, T175, T176)
MERGE34_IN_GGAAA(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U13_GGAAA(T249, T250, T251, T252, T256, T254, T257, p67_in_ggggaa(T249, T251, T250, T252, T256, T257))
MERGE34_IN_GGAAA(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → P67_IN_GGGGAA(T249, T251, T250, T252, T256, T257)
P67_IN_GGGGAA(T249, T251, T250, T252, T256, T257) → U16_GGGGAA(T249, T251, T250, T252, T256, T257, le69_in_gg(T249, T251))
P67_IN_GGGGAA(T249, T251, T250, T252, T256, T257) → LE69_IN_GG(T249, T251)
LE69_IN_GG(s(T270), s(T271)) → U15_GG(T270, T271, le69_in_gg(T270, T271))
LE69_IN_GG(s(T270), s(T271)) → LE69_IN_GG(T270, T271)
P67_IN_GGGGAA(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U18_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, p67_in_ggggaa(T324, T326, T325, T327, T331, T332))
U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → P67_IN_GGGGAA(T324, T326, T325, T327, T331, T332)
P67_IN_GGGGAA(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U20_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, p95_in_ggggaa(T349, T351, T350, T352, T356, T357))
U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357)
P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357) → U22_GGGGAA(T349, T351, T350, T352, T356, T357, gt97_in_gg(T349, T351))
P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357) → GT97_IN_GG(T349, T351)
GT97_IN_GG(s(T370), s(T371)) → U21_GG(T370, T371, gt97_in_gg(T370, T371))
GT97_IN_GG(s(T370), s(T371)) → GT97_IN_GG(T370, T371)
P95_IN_GGGGAA(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U24_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, p67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → P67_IN_GGGGAA(T423, T425, T424, T426, T430, T431)
P95_IN_GGGGAA(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U26_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, p95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → P95_IN_GGGGAA(T448, T450, T449, T451, T455, T456)
MERGE34_IN_GGAAA(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U14_GGAAA(T473, T474, T475, T476, T480, T478, T481, p95_in_ggggaa(T473, T475, T474, T476, T480, T481))
MERGE34_IN_GGAAA(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → P95_IN_GGGGAA(T473, T475, T474, T476, T480, T481)
MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T114, .(T115, T116)) → U33_GAA(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U33_GAA(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U34_GAA(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U34_GAA(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U35_GAA(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
U35_GAA(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U36_GAA(T43, T44, T45, T114, T115, T116, merge34_in_ggaaa(T103, T113, T114, T115, T116))
U35_GAA(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → MERGE34_IN_GGAAA(T103, T113, T114, T115, T116)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
mergesort1_in_gaa(x1, x2, x3)  =  mergesort1_in_gaa(x1)
.(x1, x2)  =  .(x1, x2)
split17_in_ggaaa(x1, x2, x3, x4, x5)  =  split17_in_ggaaa(x1, x2)
split21_in_gaaa(x1, x2, x3, x4)  =  split21_in_gaaa(x1)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
mergesort33_in_gaa(x1, x2, x3)  =  mergesort33_in_gaa(x1)
merge34_in_ggaaa(x1, x2, x3, x4, x5)  =  merge34_in_ggaaa(x1, x2)
p67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  p67_in_ggggaa(x1, x2, x3, x4)
le69_in_gg(x1, x2)  =  le69_in_gg(x1, x2)
p95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  p95_in_ggggaa(x1, x2, x3, x4)
gt97_in_gg(x1, x2)  =  gt97_in_gg(x1, x2)
MERGESORT1_IN_GAA(x1, x2, x3)  =  MERGESORT1_IN_GAA(x1)
U27_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U27_GAA(x1, x2, x3, x7)
SPLIT17_IN_GGAAA(x1, x2, x3, x4, x5)  =  SPLIT17_IN_GGAAA(x1, x2)
U2_GGAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAAA(x1, x2, x7)
SPLIT21_IN_GAAA(x1, x2, x3, x4)  =  SPLIT21_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GAAA(x1, x2, x7)
U28_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U28_GAA(x1, x2, x3, x7)
U29_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U29_GAA(x1, x2, x3, x7)
U30_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U30_GAA(x1, x2, x3, x7)
U31_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U31_GAA(x1, x2, x3, x7, x8)
U32_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U32_GAA(x1, x2, x3, x7)
MERGESORT33_IN_GAA(x1, x2, x3)  =  MERGESORT33_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U3_GAA(x1, x2, x3, x7)
U4_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAA(x1, x2, x3, x7)
U5_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U5_GAA(x1, x2, x3, x7)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U7_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_GAA(x1, x2, x3, x7, x8)
U8_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U8_GAA(x1, x2, x3, x7)
U9_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GAA(x1, x2, x3, x7)
U10_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GAA(x1, x2, x3, x7, x8)
U11_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GAA(x1, x2, x3, x7, x8)
U12_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U12_GAA(x1, x2, x3, x7)
MERGE34_IN_GGAAA(x1, x2, x3, x4, x5)  =  MERGE34_IN_GGAAA(x1, x2)
U13_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U13_GGAAA(x1, x2, x3, x4, x8)
P67_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P67_IN_GGGGAA(x1, x2, x3, x4)
U16_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GGGGAA(x1, x2, x3, x4, x7)
LE69_IN_GG(x1, x2)  =  LE69_IN_GG(x1, x2)
U15_GG(x1, x2, x3)  =  U15_GG(x1, x2, x3)
U17_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGAA(x1, x2, x3, x4, x5, x9)
U18_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U18_GGGGAA(x1, x2, x3, x4, x5, x9)
U19_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U19_GGGGAA(x1, x2, x3, x4, x5, x9)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U20_GGGGAA(x1, x2, x3, x4, x5, x9)
P95_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P95_IN_GGGGAA(x1, x2, x3, x4)
U22_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U22_GGGGAA(x1, x2, x3, x4, x7)
GT97_IN_GG(x1, x2)  =  GT97_IN_GG(x1, x2)
U21_GG(x1, x2, x3)  =  U21_GG(x1, x2, x3)
U23_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U23_GGGGAA(x1, x2, x3, x4, x5, x9)
U24_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U24_GGGGAA(x1, x2, x3, x4, x5, x9)
U25_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U25_GGGGAA(x1, x2, x3, x4, x5, x9)
U26_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U26_GGGGAA(x1, x2, x3, x4, x5, x9)
U14_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U14_GGAAA(x1, x2, x3, x4, x8)
U33_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U33_GAA(x1, x2, x3, x7)
U34_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U34_GAA(x1, x2, x3, x7, x8)
U35_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U35_GAA(x1, x2, x3, x7, x8)
U36_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U36_GAA(x1, x2, x3, x7)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 38 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GT97_IN_GG(s(T370), s(T371)) → GT97_IN_GG(T370, T371)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
GT97_IN_GG(x1, x2)  =  GT97_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

GT97_IN_GG(s(T370), s(T371)) → GT97_IN_GG(T370, T371)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GT97_IN_GG(s(T370), s(T371)) → GT97_IN_GG(T370, T371)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GT97_IN_GG(s(T370), s(T371)) → GT97_IN_GG(T370, T371)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LE69_IN_GG(s(T270), s(T271)) → LE69_IN_GG(T270, T271)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
LE69_IN_GG(x1, x2)  =  LE69_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LE69_IN_GG(s(T270), s(T271)) → LE69_IN_GG(T270, T271)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LE69_IN_GG(s(T270), s(T271)) → LE69_IN_GG(T270, T271)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LE69_IN_GG(s(T270), s(T271)) → LE69_IN_GG(T270, T271)
  The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

P67_IN_GGGGAA(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → P67_IN_GGGGAA(T324, T326, T325, T327, T331, T332)
P67_IN_GGGGAA(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357)
P95_IN_GGGGAA(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → P67_IN_GGGGAA(T423, T425, T424, T426, T430, T431)
P95_IN_GGGGAA(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → P95_IN_GGGGAA(T448, T450, T449, T451, T455, T456)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
P67_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P67_IN_GGGGAA(x1, x2, x3, x4)
U17_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGAA(x1, x2, x3, x4, x5, x9)
U19_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U19_GGGGAA(x1, x2, x3, x4, x5, x9)
P95_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P95_IN_GGGGAA(x1, x2, x3, x4)
U23_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U23_GGGGAA(x1, x2, x3, x4, x5, x9)
U25_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U25_GGGGAA(x1, x2, x3, x4, x5, x9)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

P67_IN_GGGGAA(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U17_GGGGAA(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → P67_IN_GGGGAA(T324, T326, T325, T327, T331, T332)
P67_IN_GGGGAA(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U19_GGGGAA(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → P95_IN_GGGGAA(T349, T351, T350, T352, T356, T357)
P95_IN_GGGGAA(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U23_GGGGAA(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → P67_IN_GGGGAA(T423, T425, T424, T426, T430, T431)
P95_IN_GGGGAA(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U25_GGGGAA(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → P95_IN_GGGGAA(T448, T450, T449, T451, T455, T456)

The TRS R consists of the following rules:

lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
P67_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P67_IN_GGGGAA(x1, x2, x3, x4)
U17_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGAA(x1, x2, x3, x4, x5, x9)
U19_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U19_GGGGAA(x1, x2, x3, x4, x5, x9)
P95_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  P95_IN_GGGGAA(x1, x2, x3, x4)
U23_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U23_GGGGAA(x1, x2, x3, x4, x5, x9)
U25_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U25_GGGGAA(x1, x2, x3, x4, x5, x9)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

P67_IN_GGGGAA(T249, T326, .(T324, T325), T327) → U17_GGGGAA(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
U17_GGGGAA(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → P67_IN_GGGGAA(T324, T326, T325, T327)
P67_IN_GGGGAA(T249, T351, .(T349, T350), T352) → U19_GGGGAA(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
U19_GGGGAA(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → P95_IN_GGGGAA(T349, T351, T350, T352)
P95_IN_GGGGAA(T423, T351, T424, .(T425, T426)) → U23_GGGGAA(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
U23_GGGGAA(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → P67_IN_GGGGAA(T423, T425, T424, T426)
P95_IN_GGGGAA(T448, T351, T449, .(T450, T451)) → U25_GGGGAA(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U25_GGGGAA(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → P95_IN_GGGGAA(T448, T450, T449, T451)

The TRS R consists of the following rules:

lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

lec69_in_gg(x0, x1)
gtc97_in_gg(x0, x1)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• U17_GGGGAA(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → P67_IN_GGGGAA(T324, T326, T325, T327)
  The graph contains the following edges 3 >= 1, 2 >= 2, 6 > 2, 4 >= 3, 5 >= 4

• U23_GGGGAA(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → P67_IN_GGGGAA(T423, T425, T424, T426)
  The graph contains the following edges 1 >= 1, 6 > 1, 4 >= 2, 3 >= 3, 5 >= 4

• P67_IN_GGGGAA(T249, T326, .(T324, T325), T327) → U17_GGGGAA(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5

• P67_IN_GGGGAA(T249, T351, .(T349, T350), T352) → U19_GGGGAA(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5

• U19_GGGGAA(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → P95_IN_GGGGAA(T349, T351, T350, T352)
  The graph contains the following edges 3 >= 1, 2 >= 2, 6 > 2, 4 >= 3, 5 >= 4

• U25_GGGGAA(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → P95_IN_GGGGAA(T448, T450, T449, T451)
  The graph contains the following edges 1 >= 1, 6 > 1, 4 >= 2, 3 >= 3, 5 >= 4

• P95_IN_GGGGAA(T423, T351, T424, .(T425, T426)) → U23_GGGGAA(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 4 > 5

• P95_IN_GGGGAA(T448, T351, T449, .(T450, T451)) → U25_GGGGAA(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 4 > 5

(27) YES

(28) Obligation:

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

SPLIT21_IN_GAAA(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → SPLIT21_IN_GAAA(T93, X163, X162, T96)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
SPLIT21_IN_GAAA(x1, x2, x3, x4)  =  SPLIT21_IN_GAAA(x1)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

SPLIT21_IN_GAAA(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → SPLIT21_IN_GAAA(T93, X163, X162, T96)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SPLIT21_IN_GAAA(x1, x2, x3, x4)  =  SPLIT21_IN_GAAA(x1)

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

(31) PiDPToQDPProof (SOUND transformation)

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

(32) Obligation:

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

SPLIT21_IN_GAAA(.(T92, T93)) → SPLIT21_IN_GAAA(T93)

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

(33) 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:

• SPLIT21_IN_GAAA(.(T92, T93)) → SPLIT21_IN_GAAA(T93)
  The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T159, T158)) → U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158)))
U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158))) → MERGESORT33_IN_GAA(T156, X232, T158)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T169, T168)) → U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168)))
U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168))) → U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_in_gaa(T156, T167, T168))
U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_out_gaa(T156, T167, T168)) → MERGESORT33_IN_GAA(T157, X233, T168)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
MERGESORT33_IN_GAA(x1, x2, x3)  =  MERGESORT33_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAA(x1, x2, x3, x7)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U7_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_GAA(x1, x2, x3, x7, x8)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T159, T158)) → U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158)))
U4_GAA(T144, T145, T146, X234, T159, T158, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T159, T158))) → MERGESORT33_IN_GAA(T156, X232, T158)
MERGESORT33_IN_GAA(.(T144, .(T145, T146)), X234, .(T169, T168)) → U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168)))
U6_GAA(T144, T145, T146, X234, T169, T168, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T169, T168))) → U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_in_gaa(T156, T167, T168))
U7_GAA(T144, T145, T146, X234, T169, T168, T157, mergesortc33_out_gaa(T156, T167, T168)) → MERGESORT33_IN_GAA(T157, X233, T168)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
MERGESORT33_IN_GAA(x1, x2, x3)  =  MERGESORT33_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U4_GAA(x1, x2, x3, x7)
U6_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U6_GAA(x1, x2, x3, x7)
U7_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U7_GAA(x1, x2, x3, x7, x8)

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

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U6_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U7_GAA(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
U7_GAA(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → MERGESORT33_IN_GAA(T157)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(40) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146))) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146))))

(41) Obligation:

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

U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U6_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U7_GAA(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
U7_GAA(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → MERGESORT33_IN_GAA(T157)
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146))))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(42) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146))) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146))))

(43) Obligation:

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

U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)
U6_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U7_GAA(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
U7_GAA(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → MERGESORT33_IN_GAA(T157)
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146))))
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146))))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(44) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146)))) at position [3,2] we obtained the following new rules [LPAR04]:

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))

(45) Obligation:

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

U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)
U6_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U7_GAA(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
U7_GAA(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → MERGESORT33_IN_GAA(T157)
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146))))
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(46) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), splitc21_in_gaaa(.(T145, T146)))) at position [3,2] we obtained the following new rules [LPAR04]:

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))

(47) Obligation:

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

U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)
U6_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U7_GAA(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
U7_GAA(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → MERGESORT33_IN_GAA(T157)
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U7_GAA(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → MERGESORT33_IN_GAA(T157)

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

POL(U4_GAA(x1, x2, x3, x4)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 0, 0 ]

·

x3

+

[ 1, 0 ]

·

x4

POL(splitc17_out_ggaaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 0 /

·

x3

+

/ 0 0 \ \ 0 1 /

·

x4

POL(.(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

POL(MERGESORT33_IN_GAA(x1)) =

0

+

[ 0, 1 ]

·

x1

POL(U6_GAA(x1, x2, x3, x4)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 0, 0 ]

·

x3

+

[ 0, 1 ]

·

x4

POL(U7_GAA(x1, x2, x3, x4, x5)) =

0

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 0, 0 ]

·

x3

+

[ 0, 1 ]

·

x4

+

[ 0, 1 ]

·

x5

POL(mergesortc33_in_gaa(x1)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 0 /

·

x1

POL(mergesortc33_out_gaa(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U43_ggaaa(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 1 /

·

x3

POL(U38_gaaa(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 1 /

·

x3

POL(splitc21_in_gaaa(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL([]) =

/ 1 \ \ 0 /

POL(U44_gaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 1 /

·

x4

POL(splitc17_in_ggaaa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

POL(splitc21_out_gaaa(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 1 1 /

·

x3

POL(U45_gaa(x1, x2, x3, x4, x5)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

+

/ 0 0 \ \ 0 1 /

·

x5

POL(U46_gaa(x1, x2, x3, x4, x5)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

+

/ 0 0 \ \ 0 0 /

·

x5

POL(U47_gaa(x1, x2, x3, x4)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL(mergec34_in_ggaaa(x1, x2)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(mergec34_out_ggaaa(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 1 1 \ \ 1 0 /

·

x2

+

/ 0 1 \ \ 0 1 /

·

x3

POL(U48_ggaaa(x1, x2, x3, x4, x5)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

+

/ 0 1 \ \ 0 0 /

·

x5

POL(qc67_in_ggggaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 1 0 \ \ 1 0 /

·

x4

POL(U49_ggaaa(x1, x2, x3, x4, x5)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

+

/ 0 1 \ \ 1 1 /

·

x4

+

/ 0 0 \ \ 0 1 /

·

x5

POL(qc95_in_ggggaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(U51_ggggaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

POL(lec69_in_gg(x1, x2)) =

/ 1 \ \ 1 /

+

/ 1 0 \ \ 1 0 /

·

x1

+

/ 0 1 \ \ 0 1 /

·

x2

POL(U52_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 1 \ \ 0 0 /

·

x4

+

/ 0 0 \ \ 1 1 /

·

x5

+

/ 0 0 \ \ 1 1 /

·

x6

POL(U54_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 0 /

·

x2

+

/ 1 0 \ \ 1 0 /

·

x3

+

/ 1 0 \ \ 0 0 /

·

x4

+

/ 1 0 \ \ 0 1 /

·

x5

+

/ 0 0 \ \ 1 1 /

·

x6

POL(qc67_out_ggggaa(x1, x2, x3, x4, x5)) =

/ 1 \ \ 1 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 0 1 /

·

x2

+

/ 0 0 \ \ 0 1 /

·

x3

+

/ 0 1 \ \ 0 0 /

·

x4

+

/ 0 0 \ \ 0 0 /

·

x5

POL(U57_ggggaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

+

/ 0 1 \ \ 0 1 /

·

x3

+

/ 1 1 \ \ 1 1 /

·

x4

POL(gtc97_in_gg(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 1 1 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U58_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 1 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 0 /

·

x3

+

/ 1 0 \ \ 0 1 /

·

x4

+

/ 0 0 \ \ 1 1 /

·

x5

+

/ 0 0 \ \ 1 1 /

·

x6

POL(U60_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 1 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 1 0 \ \ 0 0 /

·

x4

+

/ 0 1 \ \ 1 0 /

·

x5

+

/ 1 1 \ \ 0 0 /

·

x6

POL(qc95_out_ggggaa(x1, x2, x3, x4, x5)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

+

/ 0 1 \ \ 1 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 1 1 /

·

x4

+

/ 1 1 \ \ 1 0 /

·

x5

POL(s(x1)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 0 /

·

x1

POL(U50_gg(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 1 1 \ \ 0 0 /

·

x3

POL(0) =

/ 0 \ \ 0 /

POL(lec69_out_gg(x1, x2)) =

/ 1 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

POL(U53_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 0 /

+

/ 1 1 \ \ 0 1 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 1 1 \ \ 0 0 /

·

x4

+

/ 0 0 \ \ 1 0 /

·

x5

+

/ 0 1 \ \ 0 0 /

·

x6

POL(U55_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 1 /

+

/ 0 1 \ \ 1 1 /

·

x1

+

/ 0 1 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 0 /

·

x3

+

/ 1 1 \ \ 0 1 /

·

x4

+

/ 1 0 \ \ 0 0 /

·

x5

+

/ 1 0 \ \ 1 0 /

·

x6

POL(U56_gg(x1, x2, x3)) =

/ 1 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(gtc97_out_gg(x1, x2)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 0 /

·

x1

+

/ 1 0 \ \ 1 0 /

·

x2

POL(U59_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 0 1 /

·

x1

+

/ 1 0 \ \ 0 1 /

·

x2

+

/ 0 1 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 1 0 /

·

x4

+

/ 0 1 \ \ 0 0 /

·

x5

+

/ 1 1 \ \ 0 0 /

·

x6

POL(U61_ggggaa(x1, x2, x3, x4, x5, x6)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 1 1 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 0 0 /

·

x3

+

/ 0 0 \ \ 1 0 /

·

x4

+

/ 0 0 \ \ 0 1 /

·

x5

+

/ 0 0 \ \ 0 1 /

·

x6

The following usable rules [FROCOS05] were oriented:

mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)

(49) Obligation:

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

U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)
U6_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U7_GAA(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))
MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U6_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(50) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(51) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))
U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
mergesortc33_in_gaa([]) → mergesortc33_out_gaa([], [])
mergesortc33_in_gaa(.(T132, [])) → mergesortc33_out_gaa(.(T132, []), .(T132, []))
mergesortc33_in_gaa(.(T144, .(T145, T146))) → U44_gaa(T144, T145, T146, splitc17_in_ggaaa(T144, .(T145, T146)))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U44_gaa(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → U45_gaa(T144, T145, T146, T157, mergesortc33_in_gaa(T156))
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U45_gaa(T144, T145, T146, T157, mergesortc33_out_gaa(T156, T167)) → U46_gaa(T144, T145, T146, T167, mergesortc33_in_gaa(T157))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U46_gaa(T144, T145, T146, T167, mergesortc33_out_gaa(T157, T174)) → U47_gaa(T144, T145, T146, mergec34_in_ggaaa(T167, T174))
U47_gaa(T144, T145, T146, mergec34_out_ggaaa(T167, T174, X234)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234)
mergec34_in_ggaaa([], T203) → mergec34_out_ggaaa([], T203, T203)
mergec34_in_ggaaa(T218, []) → mergec34_out_ggaaa(T218, [], T218)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252)) → U48_ggaaa(T249, T250, T251, T252, qc67_in_ggggaa(T249, T251, T250, T252))
mergec34_in_ggaaa(.(T473, T474), .(T475, T476)) → U49_ggaaa(T473, T474, T475, T476, qc95_in_ggggaa(T473, T475, T474, T476))
U48_ggaaa(T249, T250, T251, T252, qc67_out_ggggaa(T249, T251, T250, T252, T256)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256))
U49_ggaaa(T473, T474, T475, T476, qc95_out_ggggaa(T473, T475, T474, T476, T480)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480))
qc67_in_ggggaa(T249, T292, [], T293) → U51_ggggaa(T249, T292, T293, lec69_in_gg(T249, T292))
qc67_in_ggggaa(T249, T326, .(T324, T325), T327) → U52_ggggaa(T249, T326, T324, T325, T327, lec69_in_gg(T249, T326))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352) → U54_ggggaa(T249, T351, T349, T350, T352, lec69_in_gg(T249, T351))
qc95_in_ggggaa(T392, T351, T393, []) → U57_ggggaa(T392, T351, T393, gtc97_in_gg(T392, T351))
qc95_in_ggggaa(T423, T351, T424, .(T425, T426)) → U58_ggggaa(T423, T351, T424, T425, T426, gtc97_in_gg(T423, T351))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451)) → U60_ggggaa(T448, T351, T449, T450, T451, gtc97_in_gg(T448, T351))
U51_ggggaa(T249, T292, T293, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293))
U52_ggggaa(T249, T326, T324, T325, T327, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, qc67_in_ggggaa(T324, T326, T325, T327))
U54_ggggaa(T249, T351, T349, T350, T352, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, qc95_in_ggggaa(T349, T351, T350, T352))
U57_ggggaa(T392, T351, T393, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393))
U58_ggggaa(T423, T351, T424, T425, T426, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, qc67_in_ggggaa(T423, T425, T424, T426))
U60_ggggaa(T448, T351, T449, T450, T451, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, qc95_in_ggggaa(T448, T450, T449, T451))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U53_ggggaa(T249, T326, T324, T325, T327, qc67_out_ggggaa(T324, T326, T325, T327, T331)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331))
U55_ggggaa(T249, T351, T349, T350, T352, qc95_out_ggggaa(T349, T351, T350, T352, T356)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U59_ggggaa(T423, T351, T424, T425, T426, qc67_out_ggggaa(T423, T425, T424, T426, T430)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430))
U61_ggggaa(T448, T351, T449, T450, T451, qc95_out_ggggaa(T448, T450, T449, T451, T455)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455))
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(52) 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.

(53) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))
U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)

The TRS R consists of the following rules:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U43_ggaaa(x0, x1, x2)
U44_gaa(x0, x1, x2, x3)
splitc21_in_gaaa(x0)
U45_gaa(x0, x1, x2, x3, x4)
U38_gaaa(x0, x1, x2)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

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

(54) 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].

splitc17_in_ggaaa(x0, x1)
mergesortc33_in_gaa(x0)
U44_gaa(x0, x1, x2, x3)
U45_gaa(x0, x1, x2, x3, x4)
U46_gaa(x0, x1, x2, x3, x4)
U47_gaa(x0, x1, x2, x3)
mergec34_in_ggaaa(x0, x1)
U48_ggaaa(x0, x1, x2, x3, x4)
U49_ggaaa(x0, x1, x2, x3, x4)
qc67_in_ggggaa(x0, x1, x2, x3)
qc95_in_ggggaa(x0, x1, x2, x3)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
U57_ggggaa(x0, x1, x2, x3)
U58_ggggaa(x0, x1, x2, x3, x4, x5)
U60_ggggaa(x0, x1, x2, x3, x4, x5)
lec69_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
gtc97_in_gg(x0, x1)
U59_ggggaa(x0, x1, x2, x3, x4, x5)
U61_ggggaa(x0, x1, x2, x3, x4, x5)
U50_gg(x0, x1, x2)
U56_gg(x0, x1, x2)

(55) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))
U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)

The TRS R consists of the following rules:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)

The set Q consists of the following terms:

U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U4_GAA(T144, T145, T146, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157)) → MERGESORT33_IN_GAA(T156)

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

POL(MERGESORT33_IN_GAA(x1)) =

0

+

[ 1, 0 ]

·

x1

POL(.(x1, x2)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 1 1 \ \ 1 0 /

·

x2

POL(U4_GAA(x1, x2, x3, x4)) =

1

+

[ 0, 0 ]

·

x1

+

[ 0, 0 ]

·

x2

+

[ 0, 0 ]

·

x3

+

[ 0, 1 ]

·

x4

POL(U43_ggaaa(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

POL(U38_gaaa(x1, x2, x3)) =

/ 0 \ \ 1 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 1 \ \ 1 1 /

·

x3

POL(splitc21_in_gaaa(x1)) =

/ 0 \ \ 0 /

+

/ 1 0 \ \ 1 1 /

·

x1

POL(splitc17_out_ggaaa(x1, x2, x3, x4)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 0 0 /

·

x2

+

/ 0 0 \ \ 1 0 /

·

x3

+

/ 0 0 \ \ 0 0 /

·

x4

POL([]) =

/ 0 \ \ 0 /

POL(splitc21_out_gaaa(x1, x2, x3)) =

/ 0 \ \ 0 /

+

/ 0 0 \ \ 0 0 /

·

x1

+

/ 0 0 \ \ 1 1 /

·

x2

+

/ 1 1 \ \ 1 0 /

·

x3

The following usable rules [FROCOS05] were oriented:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)

(57) Obligation:

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

MERGESORT33_IN_GAA(.(T144, .(T145, T146))) → U4_GAA(T144, T145, T146, U43_ggaaa(T144, .(T145, T146), U38_gaaa(T145, T146, splitc21_in_gaaa(T146))))

The TRS R consists of the following rules:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)

The set Q consists of the following terms:

U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(58) DependencyGraphProof (EQUIVALENT transformation)

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

(59) TRUE

(60) Obligation:

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

MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T58, .(T59, T57)) → U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_in_ggaaa(T44, T45, T55, T56, T57))
U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_out_ggaaa(T44, T45, T55, T56, T57)) → MERGESORT1_IN_GAA(.(T43, T56), X38, T57)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
mergesortc1_in_gaa([], [], T5) → mergesortc1_out_gaa([], [], T5)
mergesortc1_in_gaa(.(T8, []), .(T8, []), T9) → mergesortc1_out_gaa(.(T8, []), .(T8, []), T9)
mergesortc1_in_gaa(.(T43, .(T44, T45)), T114, .(T115, T116)) → U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_in_ggaaa(T44, T45, T55, T56, T116))
U39_gaa(T43, T44, T45, T114, T115, T116, splitc17_out_ggaaa(T44, T45, T55, T56, T116)) → U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_in_gaa(.(T43, T56), T103, T116))
U40_gaa(T43, T44, T45, T114, T115, T116, T55, mergesortc1_out_gaa(.(T43, T56), T103, T116)) → U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_in_gaa(T55, T113, T116))
mergesortc33_in_gaa([], [], T123) → mergesortc33_out_gaa([], [], T123)
mergesortc33_in_gaa(.(T132, []), .(T132, []), T133) → mergesortc33_out_gaa(.(T132, []), .(T132, []), T133)
mergesortc33_in_gaa(.(T144, .(T145, T146)), X234, .(T175, T176)) → U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_in_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176)))
U44_gaa(T144, T145, T146, X234, T175, T176, splitc17_out_ggaaa(T144, .(T145, T146), T156, T157, .(T175, T176))) → U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_in_gaa(T156, T167, T176))
U45_gaa(T144, T145, T146, X234, T175, T176, T157, mergesortc33_out_gaa(T156, T167, T176)) → U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_in_gaa(T157, T174, T176))
U46_gaa(T144, T145, T146, X234, T175, T176, T167, mergesortc33_out_gaa(T157, T174, T176)) → U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_in_ggaaa(T167, T174, X234, T175, T176))
mergec34_in_ggaaa([], T203, T203, T204, T205) → mergec34_out_ggaaa([], T203, T203, T204, T205)
mergec34_in_ggaaa(T218, [], T218, T219, T220) → mergec34_out_ggaaa(T218, [], T218, T219, T220)
mergec34_in_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257) → U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_in_ggggaa(T249, T251, T250, T252, T256, T257))
qc67_in_ggggaa(T249, T292, [], T293, .(T292, T293), T294) → U51_ggggaa(T249, T292, T293, T294, lec69_in_gg(T249, T292))
lec69_in_gg(s(T270), s(T271)) → U50_gg(T270, T271, lec69_in_gg(T270, T271))
lec69_in_gg(0, s(0)) → lec69_out_gg(0, s(0))
lec69_in_gg(0, 0) → lec69_out_gg(0, 0)
U50_gg(T270, T271, lec69_out_gg(T270, T271)) → lec69_out_gg(s(T270), s(T271))
U51_ggggaa(T249, T292, T293, T294, lec69_out_gg(T249, T292)) → qc67_out_ggggaa(T249, T292, [], T293, .(T292, T293), T294)
qc67_in_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332)) → U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_in_gg(T249, T326))
U52_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, lec69_out_gg(T249, T326)) → U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_in_ggggaa(T324, T326, T325, T327, T331, T332))
qc67_in_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357)) → U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_in_gg(T249, T351))
U54_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, lec69_out_gg(T249, T351)) → U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_in_ggggaa(T349, T351, T350, T352, T356, T357))
qc95_in_ggggaa(T392, T351, T393, [], .(T392, T393), T394) → U57_ggggaa(T392, T351, T393, T394, gtc97_in_gg(T392, T351))
gtc97_in_gg(s(T370), s(T371)) → U56_gg(T370, T371, gtc97_in_gg(T370, T371))
gtc97_in_gg(s(0), 0) → gtc97_out_gg(s(0), 0)
U56_gg(T370, T371, gtc97_out_gg(T370, T371)) → gtc97_out_gg(s(T370), s(T371))
U57_ggggaa(T392, T351, T393, T394, gtc97_out_gg(T392, T351)) → qc95_out_ggggaa(T392, T351, T393, [], .(T392, T393), T394)
qc95_in_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431)) → U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_in_gg(T423, T351))
U58_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, gtc97_out_gg(T423, T351)) → U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_in_ggggaa(T423, T425, T424, T426, T430, T431))
U59_ggggaa(T423, T351, T424, T425, T426, T430, T428, T431, qc67_out_ggggaa(T423, T425, T424, T426, T430, T431)) → qc95_out_ggggaa(T423, T351, T424, .(T425, T426), .(T423, T430), .(T428, T431))
qc95_in_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456)) → U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_in_gg(T448, T351))
U60_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, gtc97_out_gg(T448, T351)) → U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_in_ggggaa(T448, T450, T449, T451, T455, T456))
U61_ggggaa(T448, T351, T449, T450, T451, T455, T453, T456, qc95_out_ggggaa(T448, T450, T449, T451, T455, T456)) → qc95_out_ggggaa(T448, T351, T449, .(T450, T451), .(T450, T455), .(T453, T456))
U55_ggggaa(T249, T351, T349, T350, T352, T356, T354, T357, qc95_out_ggggaa(T349, T351, T350, T352, T356, T357)) → qc67_out_ggggaa(T249, T351, .(T349, T350), T352, .(T351, T356), .(T354, T357))
U53_ggggaa(T249, T326, T324, T325, T327, T331, T329, T332, qc67_out_ggggaa(T324, T326, T325, T327, T331, T332)) → qc67_out_ggggaa(T249, T326, .(T324, T325), T327, .(T324, T331), .(T329, T332))
U48_ggaaa(T249, T250, T251, T252, T256, T254, T257, qc67_out_ggggaa(T249, T251, T250, T252, T256, T257)) → mergec34_out_ggaaa(.(T249, T250), .(T251, T252), .(T249, T256), T254, T257)
mergec34_in_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481) → U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_in_ggggaa(T473, T475, T474, T476, T480, T481))
U49_ggaaa(T473, T474, T475, T476, T480, T478, T481, qc95_out_ggggaa(T473, T475, T474, T476, T480, T481)) → mergec34_out_ggaaa(.(T473, T474), .(T475, T476), .(T475, T480), T478, T481)
U47_gaa(T144, T145, T146, X234, T175, T176, mergec34_out_ggaaa(T167, T174, X234, T175, T176)) → mergesortc33_out_gaa(.(T144, .(T145, T146)), X234, .(T175, T176))
U41_gaa(T43, T44, T45, T114, T115, T116, T103, mergesortc33_out_gaa(T55, T113, T116)) → U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_in_ggaaa(T103, T113, T114, T115, T116))
U42_gaa(T43, T44, T45, T114, T115, T116, mergec34_out_ggaaa(T103, T113, T114, T115, T116)) → mergesortc1_out_gaa(.(T43, .(T44, T45)), T114, .(T115, T116))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
mergesortc1_in_gaa(x1, x2, x3)  =  mergesortc1_in_gaa(x1)
mergesortc1_out_gaa(x1, x2, x3)  =  mergesortc1_out_gaa(x1, x2)
U39_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U39_gaa(x1, x2, x3, x7)
U40_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U40_gaa(x1, x2, x3, x7, x8)
U41_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U41_gaa(x1, x2, x3, x7, x8)
mergesortc33_in_gaa(x1, x2, x3)  =  mergesortc33_in_gaa(x1)
mergesortc33_out_gaa(x1, x2, x3)  =  mergesortc33_out_gaa(x1, x2)
U44_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U44_gaa(x1, x2, x3, x7)
U45_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U45_gaa(x1, x2, x3, x7, x8)
U46_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U46_gaa(x1, x2, x3, x7, x8)
U47_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U47_gaa(x1, x2, x3, x7)
mergec34_in_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_in_ggaaa(x1, x2)
mergec34_out_ggaaa(x1, x2, x3, x4, x5)  =  mergec34_out_ggaaa(x1, x2, x3)
U48_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U48_ggaaa(x1, x2, x3, x4, x8)
qc67_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
lec69_in_gg(x1, x2)  =  lec69_in_gg(x1, x2)
s(x1)  =  s(x1)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
0  =  0
lec69_out_gg(x1, x2)  =  lec69_out_gg(x1, x2)
qc67_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc67_out_ggggaa(x1, x2, x3, x4, x5)
U52_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U52_ggggaa(x1, x2, x3, x4, x5, x9)
U53_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U53_ggggaa(x1, x2, x3, x4, x5, x9)
U54_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U54_ggggaa(x1, x2, x3, x4, x5, x9)
U55_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U55_ggggaa(x1, x2, x3, x4, x5, x9)
qc95_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_in_ggggaa(x1, x2, x3, x4)
U57_ggggaa(x1, x2, x3, x4, x5)  =  U57_ggggaa(x1, x2, x3, x5)
gtc97_in_gg(x1, x2)  =  gtc97_in_gg(x1, x2)
U56_gg(x1, x2, x3)  =  U56_gg(x1, x2, x3)
gtc97_out_gg(x1, x2)  =  gtc97_out_gg(x1, x2)
qc95_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qc95_out_ggggaa(x1, x2, x3, x4, x5)
U58_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U58_ggggaa(x1, x2, x3, x4, x5, x9)
U59_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U59_ggggaa(x1, x2, x3, x4, x5, x9)
U60_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U60_ggggaa(x1, x2, x3, x4, x5, x9)
U61_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U61_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U49_ggaaa(x1, x2, x3, x4, x8)
U42_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U42_gaa(x1, x2, x3, x7)
MERGESORT1_IN_GAA(x1, x2, x3)  =  MERGESORT1_IN_GAA(x1)
U28_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U28_GAA(x1, x2, x3, x7)

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

(61) UsableRulesProof (EQUIVALENT transformation)

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

(62) Obligation:

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

MERGESORT1_IN_GAA(.(T43, .(T44, T45)), T58, .(T59, T57)) → U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_in_ggaaa(T44, T45, T55, T56, T57))
U28_GAA(T43, T44, T45, T58, T59, T57, splitc17_out_ggaaa(T44, T45, T55, T56, T57)) → MERGESORT1_IN_GAA(.(T43, T56), X38, T57)

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76)) → U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_in_gaaa(T73, X125, X124, T76))
U43_ggaaa(T72, T73, X124, X125, T74, T76, splitc21_out_gaaa(T73, X125, X124, T76)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125, .(T74, T76))
splitc21_in_gaaa([], [], [], T83) → splitc21_out_gaaa([], [], [], T83)
splitc21_in_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96)) → U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_in_gaaa(T93, X163, X162, T96))
U38_gaaa(T92, T93, X162, X163, T94, T96, splitc21_out_gaaa(T93, X163, X162, T96)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163, .(T94, T96))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitc17_in_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_in_ggaaa(x1, x2)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U43_ggaaa(x1, x2, x7)
splitc21_in_gaaa(x1, x2, x3, x4)  =  splitc21_in_gaaa(x1)
[]  =  []
splitc21_out_gaaa(x1, x2, x3, x4)  =  splitc21_out_gaaa(x1, x2, x3)
U38_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaaa(x1, x2, x7)
splitc17_out_ggaaa(x1, x2, x3, x4, x5)  =  splitc17_out_ggaaa(x1, x2, x3, x4)
MERGESORT1_IN_GAA(x1, x2, x3)  =  MERGESORT1_IN_GAA(x1)
U28_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U28_GAA(x1, x2, x3, x7)

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

(63) PiDPToQDPProof (SOUND transformation)

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

(64) Obligation:

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

MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, splitc17_in_ggaaa(T44, T45))
U28_GAA(T43, T44, T45, splitc17_out_ggaaa(T44, T45, T55, T56)) → MERGESORT1_IN_GAA(.(T43, T56))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(65) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, splitc17_in_ggaaa(T44, T45)) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, U43_ggaaa(T44, T45, splitc21_in_gaaa(T45)))

(66) Obligation:

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

U28_GAA(T43, T44, T45, splitc17_out_ggaaa(T44, T45, T55, T56)) → MERGESORT1_IN_GAA(.(T43, T56))
MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, U43_ggaaa(T44, T45, splitc21_in_gaaa(T45)))

The TRS R consists of the following rules:

splitc17_in_ggaaa(T72, T73) → U43_ggaaa(T72, T73, splitc21_in_gaaa(T73))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(67) 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.

(68) Obligation:

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

U28_GAA(T43, T44, T45, splitc17_out_ggaaa(T44, T45, T55, T56)) → MERGESORT1_IN_GAA(.(T43, T56))
MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, U43_ggaaa(T44, T45, splitc21_in_gaaa(T45)))

The TRS R consists of the following rules:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)

The set Q consists of the following terms:

splitc17_in_ggaaa(x0, x1)
U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(69) 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].

splitc17_in_ggaaa(x0, x1)

(70) Obligation:

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

U28_GAA(T43, T44, T45, splitc17_out_ggaaa(T44, T45, T55, T56)) → MERGESORT1_IN_GAA(.(T43, T56))
MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, U43_ggaaa(T44, T45, splitc21_in_gaaa(T45)))

The TRS R consists of the following rules:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)

The set Q consists of the following terms:

U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(71) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U28_GAA(T43, T44, T45, splitc17_out_ggaaa(T44, T45, T55, T56)) → MERGESORT1_IN_GAA(.(T43, T56))
MERGESORT1_IN_GAA(.(T43, .(T44, T45))) → U28_GAA(T43, T44, T45, U43_ggaaa(T44, T45, splitc21_in_gaaa(T45)))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U28_GAA(x₁, ..., x₄) ) = max{0, 2x₁ + x₂ + 2x₄ - 2}

POL( U43_ggaaa(x₁, ..., x₃) ) = 2x₃ + 2

POL( splitc21_in_gaaa(x₁) ) = 2x₁

POL( [] ) = 0

POL( splitc21_out_gaaa(x₁, ..., x₃) ) = x₂ + x₃

POL( .(x₁, x₂) ) = x₁ + 2x₂ + 1

POL( U38_gaaa(x₁, ..., x₃) ) = x₁ + 2x₃ + 1

POL( splitc17_out_ggaaa(x₁, ..., x₄) ) = 2x₄ + 2

POL( MERGESORT1_IN_GAA(x₁) ) = max{0, 2x₁ - 1}

The following usable rules [FROCOS05] were oriented:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)

(72) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

splitc21_in_gaaa([]) → splitc21_out_gaaa([], [], [])
splitc21_in_gaaa(.(T92, T93)) → U38_gaaa(T92, T93, splitc21_in_gaaa(T93))
U43_ggaaa(T72, T73, splitc21_out_gaaa(T73, X125, X124)) → splitc17_out_ggaaa(T72, T73, .(T72, X124), X125)
U38_gaaa(T92, T93, splitc21_out_gaaa(T93, X163, X162)) → splitc21_out_gaaa(.(T92, T93), .(T92, X162), X163)

The set Q consists of the following terms:

U43_ggaaa(x0, x1, x2)
splitc21_in_gaaa(x0)
U38_gaaa(x0, x1, x2)

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

(73) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(74) YES