Left Termination of the query pattern
mergesort(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).

Query: mergesort(g,a,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="B: 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\nX2 is free\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\nX2 is free\nX13 is free\nX14 is free\nmergesort(T1, T2, T3) !=? mergesort([], [], X2)\nmergesort(T1, T2, T3) !=? mergesort(.(X13, []), .(X13, []), X14)", 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)), X33, X34, .(T23, T24)), ','(mergesort(X33, X35, T24), ','(mergesort(X34, X36, T24), merge(X35, X36, T25, .(T23, T24)))))\n\nT17 is ground\nT18 is ground\nT19 is ground\nX33 is free\nX34 is free\nX35 is free\nX36 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)), X33, X34, .(T23, T24)), ','(mergesort(X33, X35, T24), ','(mergesort(X34, X36, T24), merge(X35, X36, T25, .(T23, T24))))), SCOPE: 2, CLAUSE: 3 | ','(split(.(T17, .(T18, T19)), X33, X34, .(T23, T24)), ','(mergesort(X33, X35, T24), ','(mergesort(X34, X36, T24), merge(X35, X36, T25, .(T23, T24))))), SCOPE: 2, CLAUSE: 4\n\nT17 is ground\nT18 is ground\nT19 is ground\nX33 is free\nX34 is free\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(.(T17, .(T18, T19)), X33, X34, .(T23, T24)), ','(mergesort(X33, X35, T24), ','(mergesort(X34, X36, T24), merge(X35, X36, T25, .(T23, T24))))), SCOPE: 2, CLAUSE: 4\n\nT17 is ground\nT18 is ground\nT19 is ground\nX33 is free\nX34 is free\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(split(.(T40, T41), X63, X62, T44), ','(mergesort(.(T39, X62), X35, T44), ','(mergesort(X63, X36, T44), merge(X35, X36, T45, .(T46, T44)))))\n\nT39 is ground\nT40 is ground\nT41 is ground\nX35 is free\nX36 is free\nX62 is free\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="D: split(.(T40, T41), X63, X62, T44)\n\nT40 is ground\nT41 is ground\nX62 is free\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(mergesort(.(T39, T48), X35, T49), ','(mergesort(T47, X36, T49), merge(X35, X36, T50, .(T51, T49))))\n\nT39 is ground\nT47 is ground\nT48 is ground\nX35 is free\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: split(.(T40, T41), X63, X62, T44), SCOPE: 3, CLAUSE: 3 | split(.(T40, T41), X63, X62, T44), SCOPE: 3, CLAUSE: 4\n\nT40 is ground\nT41 is ground\nX62 is free\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: split(.(T40, T41), X63, X62, T44), SCOPE: 3, CLAUSE: 4\n\nT40 is ground\nT41 is ground\nX62 is free\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="A: split(T61, X92, X91, T64)\n\nT61 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: split(T61, X92, X91, T64), SCOPE: 4, CLAUSE: 3 | split(T61, X92, X91, T64), SCOPE: 4, CLAUSE: 4\n\nT61 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: split(T61, X92, X91, T64), SCOPE: 4, CLAUSE: 3\n\nT61 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: split(T61, X92, X91, T64), SCOPE: 4, CLAUSE: 4\n\nT61 is ground\nX91 is free\nX92 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(T81, X123, X122, T84)\n\nT81 is ground\nX122 is free\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: mergesort(.(T39, T48), X35, T49)\n\nT39 is ground\nT48 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(mergesort(T47, X36, T88), merge(T87, X36, T89, .(T90, T88)))\n\nT47 is ground\nT87 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="E: mergesort(T47, X36, T88)\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="C: merge(T87, T93, T94, .(T95, T96))\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 0 | mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 1 | mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 2\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 0\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 1 | mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 2\n\nT47 is ground\nX36 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(T47, X36, T88), SCOPE: 5, CLAUSE: 1\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 2\n\nT47 is ground\nX36 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(.(T124, .(T125, T126)), X173, X174, .(T129, T130)), ','(mergesort(X173, X175, T130), ','(mergesort(X174, X176, T130), merge(X175, X176, X177, .(T129, T130)))))\n\nT124 is ground\nT125 is ground\nT126 is ground\nX177 is free\nX173 is free\nX174 is free\nX175 is free\nX176 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: split(.(T124, .(T125, T126)), X173, X174, .(T129, T130))\n\nT124 is ground\nT125 is ground\nT126 is ground\nX173 is free\nX174 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(mergesort(T131, X175, T133), ','(mergesort(T132, X176, T133), merge(X175, X176, X177, .(T134, T133))))\n\nT131 is ground\nT132 is ground\nX177 is free\nX175 is free\nX176 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: mergesort(T131, X175, T133)\n\nT131 is ground\nX175 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(mergesort(T132, X176, T138), merge(T137, X176, X177, .(T139, T138)))\n\nT132 is ground\nT137 is ground\nX177 is free\nX176 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: mergesort(T132, X176, T138)\n\nT132 is ground\nX176 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: merge(T137, T144, X177, .(T145, T146))\n\nT137 is ground\nT144 is ground\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 5 | merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 6 | merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 7 | merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 8\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 5\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 6 | merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 7 | merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 8\n\nT87 is ground\nT93 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(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 6\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 7 | merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 8\n\nT87 is ground\nT93 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(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 7\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 8\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="F: ','(le(T219, T221), merge(T220, .(T221, T222), T226, T227))\n\nT219 is ground\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="H: le(T219, T221)\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: merge(T220, .(T221, T222), T226, T227)\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: le(T219, T221), SCOPE: 7, CLAUSE: 11 | le(T219, T221), SCOPE: 7, CLAUSE: 12 | le(T219, T221), SCOPE: 7, CLAUSE: 13\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: le(T219, T221), SCOPE: 7, CLAUSE: 11\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: le(T219, T221), SCOPE: 7, CLAUSE: 12 | le(T219, T221), SCOPE: 7, CLAUSE: 13\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: le(T240, T241)\n\nT240 is ground\nT241 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: le(T219, T221), SCOPE: 7, CLAUSE: 12\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: le(T219, T221), SCOPE: 7, CLAUSE: 13\n\nT219 is ground\nT221 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(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 5 | merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 6 | merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 7 | merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 8\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 5\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 6 | merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 7 | merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 8\n\nT220 is ground\nT221 is ground\nT222 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(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 7 | merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 8\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 7\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: merge(T220, .(T221, T222), T226, T227), SCOPE: 8, CLAUSE: 8\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(le(T294, T296), merge(T295, .(T296, T297), T301, T302))\n\nT294 is ground\nT295 is ground\nT296 is ground\nT297 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="G: ','(gt(T319, T321), merge(.(T319, T320), T322, T326, T327))\n\nT319 is ground\nT320 is ground\nT321 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="I: gt(T319, T321)\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: merge(.(T319, T320), T322, T326, T327)\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: gt(T319, T321), SCOPE: 9, CLAUSE: 9 | gt(T319, T321), SCOPE: 9, CLAUSE: 10\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: gt(T319, T321), SCOPE: 9, CLAUSE: 9\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: gt(T319, T321), SCOPE: 9, CLAUSE: 10\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: gt(T340, T341)\n\nT340 is ground\nT341 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(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 5 | merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 6 | merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 7 | merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 8\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 6 | merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 7 | merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 8\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 6\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 7 | merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 8\n\nT319 is ground\nT320 is ground\nT322 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(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 7\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: merge(.(T319, T320), T322, T326, T327), SCOPE: 10, CLAUSE: 8\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: ','(le(T387, T389), merge(T388, .(T389, T390), T394, T395))\n\nT387 is ground\nT388 is ground\nT389 is ground\nT390 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: ','(gt(T412, T414), merge(.(T412, T413), T415, T419, T420))\n\nT412 is ground\nT413 is ground\nT414 is ground\nT415 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
120 [label="120: ','(gt(T437, T439), merge(.(T437, T438), T440, T444, T445))\n\nT437 is ground\nT438 is ground\nT439 is ground\nT440 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 substitutionT1 -> [],\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 = lightgreen];
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(.(X13, []), .(X13, []), X14).\nand substitutionX13 -> T8,\nT1 -> .(T8, []),\nT2 -> .(T8, []),\nT3 -> T9,\nX14 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 126 [arrowhead = none ];
126 -> 8;

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

128 [label="BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []), Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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(.(X27, .(X28, X29)), X30, .(X31, X32)) :- ','(split(.(X27, .(X28, X29)), X33, X34, .(X31, X32)), ','(mergesort(X33, X35, X32), ','(mergesort(X34, X36, X32), merge(X35, X36, X30, .(X31, X32))))).\nand substitutionX27 -> T17,\nX28 -> T18,\nX29 -> T19,\nT1 -> .(T17, .(T18, T19)),\nT2 -> T25,\nX30 -> T25,\nX31 -> T23,\nX32 -> 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 = lightgreen];
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 = lightgreen];
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 = lightgreen];
14 -> 135 [arrowhead = none ];
135 -> 15;

136 [label="ONLY EVAL with clause\nsplit(.(X56, X57), .(X56, X58), X59, .(X60, X61)) :- split(X57, X59, X58, X61).\nand substitutionT17 -> T39,\nX56 -> T39,\nT18 -> T40,\nT19 -> T41,\nX57 -> .(T40, T41),\nX58 -> X62,\nX33 -> .(T39, X62),\nX34 -> X63,\nX59 -> X63,\nT23 -> T46,\nX60 -> T46,\nT24 -> T44,\nX61 -> T44,\nT43 -> T44,\nT25 -> T45,\nT42 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT40 is ground\nT41 is ground\nT47 is ground\nT48 is ground\nreplacements:X63 -> T47,\nX62 -> T48,\nT44 -> T49,\nT45 -> T50,\nT46 -> T51", 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:\nT39 is ground\nT48 is ground\nT87 is ground\nreplacements:X35 -> T87,\nT49 -> T88,\nT50 -> T89,\nT51 -> T90", 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 = lightgreen];
19 -> 142 [arrowhead = none ];
142 -> 20;

143 [label="EVAL with clause\nsplit(.(X85, X86), .(X85, X87), X88, .(X89, X90)) :- split(X86, X88, X87, X90).\nand substitutionT40 -> T60,\nX85 -> T60,\nT41 -> T61,\nX86 -> T61,\nX87 -> X91,\nX63 -> .(T60, X91),\nX62 -> X92,\nX88 -> X92,\nX89 -> T62,\nX90 -> T64,\nT44 -> .(T62, T64),\nT63 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 143 [arrowhead = none ];
143 -> 21;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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([], [], [], X99).\nand substitutionT61 -> [],\nX92 -> [],\nX91 -> [],\nT64 -> T71,\nX99 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 148 [arrowhead = none ];
148 -> 26;

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

150 [label="EVAL with clause\nsplit(.(X116, X117), .(X116, X118), X119, .(X120, X121)) :- split(X117, X119, X118, X121).\nand substitutionX116 -> T80,\nX117 -> T81,\nT61 -> .(T80, T81),\nX118 -> X122,\nX92 -> .(T80, X122),\nX91 -> X123,\nX119 -> X123,\nX120 -> T82,\nX121 -> T84,\nT64 -> .(T82, T84),\nT83 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 150 [arrowhead = none ];
150 -> 29;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT61 -> T81\nX92 -> X123\nX91 -> X122\nT64 -> T84", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 153 [arrowhead = none , style=dashed];
153 -> 21[style=dashed];

154 [label="INSTANCE with matching:\nT1 -> .(T39, T48)\nT2 -> X35\nT3 -> T49", 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:\nT47 is ground\nT93 is ground\nreplacements:X36 -> T93,\nT89 -> T94,\nT90 -> T95,\nT88 -> T96", 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([], [], X138).\nand substitutionT47 -> [],\nX36 -> [],\nT88 -> T103,\nX138 -> T103", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 161 [arrowhead = none ];
161 -> 38;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X147, []), .(X147, []), X148).\nand substitutionX147 -> T112,\nT47 -> .(T112, []),\nX36 -> .(T112, []),\nT88 -> T113,\nX148 -> T113", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 166 [arrowhead = none ];
166 -> 43;

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

168 [label="EVAL with clause\nmergesort(.(X167, .(X168, X169)), X170, .(X171, X172)) :- ','(split(.(X167, .(X168, X169)), X173, X174, .(X171, X172)), ','(mergesort(X173, X175, X172), ','(mergesort(X174, X176, X172), merge(X175, X176, X170, .(X171, X172))))).\nand substitutionX167 -> T124,\nX168 -> T125,\nX169 -> T126,\nT47 -> .(T124, .(T125, T126)),\nX36 -> X177,\nX170 -> X177,\nX171 -> T129,\nX172 -> T130,\nT88 -> .(T129, T130),\nT127 -> T129,\nT128 -> T130", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 168 [arrowhead = none ];
168 -> 46;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT124 is ground\nT125 is ground\nT126 is ground\nT131 is ground\nT132 is ground\nreplacements:X173 -> T131,\nX174 -> T132,\nT130 -> T133,\nT129 -> T134", fontsize=14, style = filled, fillcolor = violet];
46 -> 172 [arrowhead = none ];
172 -> 49;

173 [label="INSTANCE with matching:\nT40 -> T124\nT41 -> .(T125, T126)\nX63 -> X173\nX62 -> X174\nT44 -> .(T129, T130)", 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:\nT131 is ground\nT137 is ground\nreplacements:X175 -> T137,\nT133 -> T138,\nT134 -> T139", fontsize=14, style = filled, fillcolor = violet];
49 -> 175 [arrowhead = none ];
175 -> 51;

176 [label="INSTANCE with matching:\nT47 -> T131\nX36 -> X175\nT88 -> T133", 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:\nT132 is ground\nT144 is ground\nreplacements:X176 -> T144,\nT139 -> T145,\nT138 -> T146", fontsize=14, style = filled, fillcolor = violet];
51 -> 178 [arrowhead = none ];
178 -> 53;

179 [label="INSTANCE with matching:\nT47 -> T132\nX36 -> X176\nT88 -> T138", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 179 [arrowhead = none , style=dashed];
179 -> 33[style=dashed];

180 [label="INSTANCE with matching:\nT87 -> T137\nT93 -> T144\nT94 -> X177\nT95 -> T145\nT96 -> T146", 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([], X206, X206, X207).\nand substitutionT87 -> [],\nT93 -> T173,\nX206 -> T173,\nT94 -> T173,\nT95 -> T174,\nT96 -> T175,\nX207 -> .(T174, T175)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 183 [arrowhead = none ];
183 -> 57;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X216, [], X216, X217).\nand substitutionT87 -> T188,\nX216 -> T188,\nT93 -> [],\nT94 -> T188,\nT95 -> T189,\nT96 -> T190,\nX217 -> .(T189, T190)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 188 [arrowhead = none ];
188 -> 62;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X246, X247), .(X248, X249), .(X246, X250), .(X251, X252)) :- ','(le(X246, X248), merge(X247, .(X248, X249), X250, X252)).\nand substitutionX246 -> T219,\nX247 -> T220,\nT87 -> .(T219, T220),\nX248 -> T221,\nX249 -> T222,\nT93 -> .(T221, T222),\nX250 -> T226,\nT94 -> .(T219, T226),\nT95 -> T224,\nX251 -> T224,\nT96 -> T227,\nX252 -> T227,\nT223 -> T226,\nT225 -> T227", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 193 [arrowhead = none ];
193 -> 67;

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

195 [label="EVAL with clause\nmerge(.(X449, X450), .(X451, X452), .(X451, X453), .(X454, X455)) :- ','(gt(X449, X451), merge(.(X449, X450), X452, X453, X455)).\nand substitutionX449 -> T437,\nX450 -> T438,\nT87 -> .(T437, T438),\nX451 -> T439,\nX452 -> T440,\nT93 -> .(T439, T440),\nX453 -> T444,\nT94 -> .(T439, T444),\nT95 -> T442,\nX454 -> T442,\nT96 -> T445,\nX455 -> T445,\nT441 -> T444,\nT443 -> T445", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 195 [arrowhead = none ];
195 -> 120;

196 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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\nnew knowledge:\nT219 is ground\nT221 is ground", 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(X265), s(X266)) :- le(X265, X266).\nand substitutionX265 -> T240,\nT219 -> s(T240),\nX266 -> T241,\nT221 -> s(T241)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 203 [arrowhead = none ];
203 -> 74;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT219 -> T240\nT221 -> T241", 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 substitutionT219 -> 0,\nT221 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 208 [arrowhead = none ];
208 -> 78;

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

210 [label="EVAL with clause\nle(0, 0).\nand substitutionT219 -> 0,\nT221 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 210 [arrowhead = none ];
210 -> 81;

211 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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([], X281, X281, X282).\nand substitutionT220 -> [],\nT221 -> T262,\nT222 -> T263,\nX281 -> .(T262, T263),\nT226 -> .(T262, T263),\nT227 -> T264,\nX282 -> T264", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 216 [arrowhead = none ];
216 -> 87;

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

218 [label="BACKTRACK\nfor clause: merge(Xs, [], Xs, Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X313, X314), .(X315, X316), .(X313, X317), .(X318, X319)) :- ','(le(X313, X315), merge(X314, .(X315, X316), X317, X319)).\nand substitutionX313 -> T294,\nX314 -> T295,\nT220 -> .(T294, T295),\nT221 -> T296,\nX315 -> T296,\nT222 -> T297,\nX316 -> T297,\nX317 -> T301,\nT226 -> .(T294, T301),\nX318 -> T299,\nX319 -> T302,\nT227 -> .(T299, T302),\nT298 -> T301,\nT300 -> T302", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 222 [arrowhead = none ];
222 -> 93;

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

224 [label="EVAL with clause\nmerge(.(X336, X337), .(X338, X339), .(X338, X340), .(X341, X342)) :- ','(gt(X336, X338), merge(.(X336, X337), X339, X340, X342)).\nand substitutionX336 -> T319,\nX337 -> T320,\nT220 -> .(T319, T320),\nT221 -> T321,\nX338 -> T321,\nT222 -> T322,\nX339 -> T322,\nX340 -> T326,\nT226 -> .(T321, T326),\nX341 -> T324,\nX342 -> T327,\nT227 -> .(T324, T327),\nT323 -> T326,\nT325 -> T327", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 224 [arrowhead = none ];
224 -> 95;

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

226 [label="INSTANCE with matching:\nT219 -> T294\nT221 -> T296\nT220 -> T295\nT222 -> T297\nT226 -> T301\nT227 -> T302", 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\nnew knowledge:\nT319 is ground\nT321 is ground", 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(X355), s(X356)) :- gt(X355, X356).\nand substitutionX355 -> T340,\nT319 -> s(T340),\nX356 -> T341,\nT321 -> s(T341)", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 233 [arrowhead = none ];
233 -> 102;

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

235 [label="EVAL with clause\ngt(s(0), 0).\nand substitutionT319 -> s(0),\nT321 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 235 [arrowhead = none ];
235 -> 104;

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

237 [label="INSTANCE with matching:\nT319 -> T340\nT321 -> T341", 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 = lightgreen];
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(X373, [], X373, X374).\nand substitutionT319 -> T356,\nT320 -> T357,\nX373 -> .(T356, T357),\nT322 -> [],\nT326 -> .(T356, T357),\nT327 -> T358,\nX374 -> T358", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 242 [arrowhead = none ];
242 -> 111;

243 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(.(X403, X404), .(X405, X406), .(X403, X407), .(X408, X409)) :- ','(le(X403, X405), merge(X404, .(X405, X406), X407, X409)).\nand substitutionT319 -> T387,\nX403 -> T387,\nT320 -> T388,\nX404 -> T388,\nX405 -> T389,\nX406 -> T390,\nT322 -> .(T389, T390),\nX407 -> T394,\nT326 -> .(T387, T394),\nX408 -> T392,\nX409 -> T395,\nT327 -> .(T392, T395),\nT391 -> T394,\nT393 -> T395", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 247 [arrowhead = none ];
247 -> 116;

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

249 [label="EVAL with clause\nmerge(.(X426, X427), .(X428, X429), .(X428, X430), .(X431, X432)) :- ','(gt(X426, X428), merge(.(X426, X427), X429, X430, X432)).\nand substitutionT319 -> T412,\nX426 -> T412,\nT320 -> T413,\nX427 -> T413,\nX428 -> T414,\nX429 -> T415,\nT322 -> .(T414, T415),\nX430 -> T419,\nT326 -> .(T414, T419),\nX431 -> T417,\nX432 -> T420,\nT327 -> .(T417, T420),\nT416 -> T419,\nT418 -> T420", fontsize=14, style = filled, fillcolor = lightblue];
115 -> 249 [arrowhead = none ];
249 -> 118;

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

251 [label="INSTANCE with matching:\nT219 -> T387\nT221 -> T389\nT220 -> T388\nT222 -> T390\nT226 -> T394\nT227 -> T395", fontsize=14, style = filled, fillcolor = lightgrey];
116 -> 251 [arrowhead = none , style=dashed];
251 -> 67[style=dashed];

252 [label="INSTANCE with matching:\nT319 -> T412\nT321 -> T414\nT320 -> T413\nT322 -> T415\nT326 -> T419\nT327 -> T420", fontsize=14, style = filled, fillcolor = lightgrey];
118 -> 252 [arrowhead = none , style=dashed];
252 -> 95[style=dashed];

253 [label="INSTANCE with matching:\nT319 -> T437\nT321 -> T439\nT320 -> T438\nT322 -> T440\nT326 -> T444\nT327 -> T445", fontsize=14, style = filled, fillcolor = lightgrey];
120 -> 253 [arrowhead = none , style=dashed];
253 -> 95[style=dashed];

}

(2) Obligation:

Triples:

splitA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) :- splitA(X2, X4, X3, X6).
splitD(X1, X2, .(X1, X3), X4, .(X5, X6)) :- splitA(X2, X4, X3, X6).
mergesortE(.(X1, .(X2, X3)), X4, .(X5, X6)) :- splitD(X1, .(X2, X3), X7, X8, .(X5, X6)).
mergesortE(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X1, .(X2, X3), X7, X8, .(X5, X6)), mergesortE(X7, X9, X6)).
mergesortE(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X1, .(X2, X3), X7, X8, .(X5, X6)), ','(mergesortcE(X7, X9, X6), mergesortE(X8, X10, X6))).
mergesortE(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X1, .(X2, X3), X7, X8, .(X5, X6)), ','(mergesortcE(X7, X9, X6), ','(mergesortcE(X8, X10, X6), mergeC(X9, X10, X4, X5, X6)))).
mergeC(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) :- pF(X1, X3, X2, X4, X5, X7).
mergeC(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) :- pG(X1, X3, X2, X4, X5, X7).
leH(s(X1), s(X2)) :- leH(X1, X2).
pF(X1, X2, X3, X4, X5, X6) :- leH(X1, X2).
pF(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) :- ','(lecH(X1, X2), pF(X3, X2, X4, X5, X6, X8)).
pF(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) :- ','(lecH(X1, X2), pG(X3, X2, X4, X5, X6, X8)).
gtI(s(X1), s(X2)) :- gtI(X1, X2).
pG(X1, X2, X3, X4, X5, X6) :- gtI(X1, X2).
pG(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) :- ','(gtcI(X1, X2), pF(X1, X4, X3, X5, X6, X8)).
pG(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) :- ','(gtcI(X1, X2), pG(X1, X4, X3, X5, X6, X8)).
mergesortB(.(X1, .(X2, X3)), X4, .(X5, X6)) :- splitD(X2, X3, X7, X8, X6).
mergesortB(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X2, X3, X7, X8, X6), mergesortB(.(X1, X8), X9, X6)).
mergesortB(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X2, X3, X7, X8, X6), ','(mergesortcB(.(X1, X8), X9, X6), mergesortE(X7, X10, X6))).
mergesortB(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X2, X3, X7, X8, X6), ','(mergesortcB(.(X1, X8), X9, X6), ','(mergesortcE(X7, X10, X6), mergeC(X9, X10, X4, X5, X6)))).

Clauses:

splitcA([], [], [], X1).
splitcA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) :- splitcA(X2, X4, X3, X6).
mergesortcB([], [], X1).
mergesortcB(.(X1, []), .(X1, []), X2).
mergesortcB(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X2, X3, X7, X8, X6), ','(mergesortcB(.(X1, X8), X9, X6), ','(mergesortcE(X7, X10, X6), mergecC(X9, X10, X4, X5, X6)))).
splitcD(X1, X2, .(X1, X3), X4, .(X5, X6)) :- splitcA(X2, X4, X3, X6).
mergesortcE([], [], X1).
mergesortcE(.(X1, []), .(X1, []), X2).
mergesortcE(.(X1, .(X2, X3)), X4, .(X5, X6)) :- ','(splitcD(X1, .(X2, X3), X7, X8, .(X5, X6)), ','(mergesortcE(X7, X9, X6), ','(mergesortcE(X8, X10, X6), mergecC(X9, X10, X4, X5, X6)))).
mergecC([], X1, X1, X2, X3).
mergecC(X1, [], X1, X2, X3).
mergecC(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) :- qcF(X1, X3, X2, X4, X5, X7).
mergecC(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) :- qcG(X1, X3, X2, X4, X5, X7).
lecH(s(X1), s(X2)) :- lecH(X1, X2).
lecH(0, s(0)).
lecH(0, 0).
qcF(X1, X2, [], X3, .(X2, X3), X4) :- lecH(X1, X2).
qcF(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) :- ','(lecH(X1, X2), qcF(X3, X2, X4, X5, X6, X8)).
qcF(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) :- ','(lecH(X1, X2), qcG(X3, X2, X4, X5, X6, X8)).
gtcI(s(X1), s(X2)) :- gtcI(X1, X2).
gtcI(s(0), 0).
qcG(X1, X2, X3, [], .(X1, X3), X4) :- gtcI(X1, X2).
qcG(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) :- ','(gtcI(X1, X2), qcF(X1, X4, X3, X5, X6, X8)).
qcG(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) :- ','(gtcI(X1, X2), qcG(X1, X4, X3, X5, X6, X8)).

Afs:

mergesortB(x1, x2, x3)  =  mergesortB(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesortB_in: (b,f,f)
splitD_in: (b,b,f,f,f)
splitA_in: (b,f,f,f)
splitcD_in: (b,b,f,f,f)
splitcA_in: (b,f,f,f)
mergesortcB_in: (b,f,f)
mergesortcE_in: (b,f,f)
mergecC_in: (b,b,f,f,f)
qcF_in: (b,b,b,b,f,f)
lecH_in: (b,b)
qcG_in: (b,b,b,b,f,f)
gtcI_in: (b,b)
mergesortE_in: (b,f,f)
mergeC_in: (b,b,f,f,f)
pF_in: (b,b,b,b,f,f)
leH_in: (b,b)
pG_in: (b,b,b,b,f,f)
gtI_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U24_GAA(X1, X2, X3, X4, X5, X6, splitD_in_ggaaa(X2, X3, X7, X8, X6))
MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → SPLITD_IN_GGAAA(X2, X3, X7, X8, X6)
SPLITD_IN_GGAAA(X1, X2, .(X1, X3), X4, .(X5, X6)) → U2_GGAAA(X1, X2, X3, X4, X5, X6, splitA_in_gaaa(X2, X4, X3, X6))
SPLITD_IN_GGAAA(X1, X2, .(X1, X3), X4, .(X5, X6)) → SPLITA_IN_GAAA(X2, X4, X3, X6)
SPLITA_IN_GAAA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U1_GAAA(X1, X2, X3, X4, X5, X6, splitA_in_gaaa(X2, X4, X3, X6))
SPLITA_IN_GAAA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → SPLITA_IN_GAAA(X2, X4, X3, X6)
MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U26_GAA(X1, X2, X3, X4, X5, X6, mergesortB_in_gaa(.(X1, X8), X9, X6))
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → MERGESORTB_IN_GAA(.(X1, X8), X9, X6)
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U28_GAA(X1, X2, X3, X4, X5, X6, mergesortE_in_gaa(X7, X10, X6))
U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → MERGESORTE_IN_GAA(X7, X10, X6)
MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U3_GAA(X1, X2, X3, X4, X5, X6, splitD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → SPLITD_IN_GGAAA(X1, .(X2, X3), X7, X8, .(X5, X6))
MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U5_GAA(X1, X2, X3, X4, X5, X6, mergesortE_in_gaa(X7, X9, X6))
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → MERGESORTE_IN_GAA(X7, X9, X6)
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U7_GAA(X1, X2, X3, X4, X5, X6, mergesortE_in_gaa(X8, X10, X6))
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → MERGESORTE_IN_GAA(X8, X10, X6)
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U8_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U8_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U9_GAA(X1, X2, X3, X4, X5, X6, mergeC_in_ggaaa(X9, X10, X4, X5, X6))
U8_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → MERGEC_IN_GGAAA(X9, X10, X4, X5, X6)
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U10_GGAAA(X1, X2, X3, X4, X5, X6, X7, pF_in_ggggaa(X1, X3, X2, X4, X5, X7))
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → PF_IN_GGGGAA(X1, X3, X2, X4, X5, X7)
PF_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → U13_GGGGAA(X1, X2, X3, X4, X5, X6, leH_in_gg(X1, X2))
PF_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → LEH_IN_GG(X1, X2)
LEH_IN_GG(s(X1), s(X2)) → U12_GG(X1, X2, leH_in_gg(X1, X2))
LEH_IN_GG(s(X1), s(X2)) → LEH_IN_GG(X1, X2)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U15_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pF_in_ggggaa(X3, X2, X4, X5, X6, X8))
U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PF_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U17_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pG_in_ggggaa(X3, X2, X4, X5, X6, X8))
U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PG_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → U19_GGGGAA(X1, X2, X3, X4, X5, X6, gtI_in_gg(X1, X2))
PG_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → GTI_IN_GG(X1, X2)
GTI_IN_GG(s(X1), s(X2)) → U18_GG(X1, X2, gtI_in_gg(X1, X2))
GTI_IN_GG(s(X1), s(X2)) → GTI_IN_GG(X1, X2)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U21_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PF_IN_GGGGAA(X1, X4, X3, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U23_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PG_IN_GGGGAA(X1, X4, X3, X5, X6, X8)
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U11_GGAAA(X1, X2, X3, X4, X5, X6, X7, pG_in_ggggaa(X1, X3, X2, X4, X5, X7))
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → PG_IN_GGGGAA(X1, X3, X2, X4, X5, X7)
U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U29_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
U29_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U30_GAA(X1, X2, X3, X4, X5, X6, mergeC_in_ggaaa(X9, X10, X4, X5, X6))
U29_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → MERGEC_IN_GGAAA(X9, X10, X4, X5, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
mergesortB_in_gaa(x1, x2, x3)  =  mergesortB_in_gaa(x1)
.(x1, x2)  =  .(x1, x2)
splitD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitD_in_ggaaa(x1, x2)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
mergesortE_in_gaa(x1, x2, x3)  =  mergesortE_in_gaa(x1)
mergeC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergeC_in_ggaaa(x1, x2)
pF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pF_in_ggggaa(x1, x2, x3, x4)
leH_in_gg(x1, x2)  =  leH_in_gg(x1, x2)
pG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pG_in_ggggaa(x1, x2, x3, x4)
gtI_in_gg(x1, x2)  =  gtI_in_gg(x1, x2)
MERGESORTB_IN_GAA(x1, x2, x3)  =  MERGESORTB_IN_GAA(x1)
U24_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U24_GAA(x1, x2, x3, x7)
SPLITD_IN_GGAAA(x1, x2, x3, x4, x5)  =  SPLITD_IN_GGAAA(x1, x2)
U2_GGAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAAA(x1, x2, x7)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GAAA(x1, x2, x7)
U25_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAA(x1, x2, x3, x7)
U26_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U26_GAA(x1, x2, x3, x7)
U27_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U27_GAA(x1, x2, x3, x7, x8)
U28_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U28_GAA(x1, x2, x3, x7)
MERGESORTE_IN_GAA(x1, x2, x3)  =  MERGESORTE_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, x8)  =  U6_GAA(x1, x2, x3, x7, x8)
U7_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAA(x1, x2, x3, x7)
U8_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_GAA(x1, x2, x3, x7, x8)
U9_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GAA(x1, x2, x3, x7)
MERGEC_IN_GGAAA(x1, x2, x3, x4, x5)  =  MERGEC_IN_GGAAA(x1, x2)
U10_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GGAAA(x1, x2, x3, x4, x8)
PF_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PF_IN_GGGGAA(x1, x2, x3, x4)
U13_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGAA(x1, x2, x3, x4, x7)
LEH_IN_GG(x1, x2)  =  LEH_IN_GG(x1, x2)
U12_GG(x1, x2, x3)  =  U12_GG(x1, x2, x3)
U14_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_GGGGAA(x1, x2, x3, x4, x5, x9)
U15_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_GGGGAA(x1, x2, x3, x4, x5, x9)
U16_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGAA(x1, x2, x3, x4, x5, x9)
U17_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGAA(x1, x2, x3, x4, x5, x9)
PG_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PG_IN_GGGGAA(x1, x2, x3, x4)
U19_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U19_GGGGAA(x1, x2, x3, x4, x7)
GTI_IN_GG(x1, x2)  =  GTI_IN_GG(x1, x2)
U18_GG(x1, x2, x3)  =  U18_GG(x1, x2, x3)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U20_GGGGAA(x1, x2, x3, x4, x5, x9)
U21_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U21_GGGGAA(x1, x2, x3, x4, x5, x9)
U22_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U22_GGGGAA(x1, x2, x3, x4, x5, x9)
U23_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U23_GGGGAA(x1, x2, x3, x4, x5, x9)
U11_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GGAAA(x1, x2, x3, x4, x8)
U29_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_GAA(x1, x2, x3, x7, x8)
U30_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U30_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:

MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U24_GAA(X1, X2, X3, X4, X5, X6, splitD_in_ggaaa(X2, X3, X7, X8, X6))
MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → SPLITD_IN_GGAAA(X2, X3, X7, X8, X6)
SPLITD_IN_GGAAA(X1, X2, .(X1, X3), X4, .(X5, X6)) → U2_GGAAA(X1, X2, X3, X4, X5, X6, splitA_in_gaaa(X2, X4, X3, X6))
SPLITD_IN_GGAAA(X1, X2, .(X1, X3), X4, .(X5, X6)) → SPLITA_IN_GAAA(X2, X4, X3, X6)
SPLITA_IN_GAAA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U1_GAAA(X1, X2, X3, X4, X5, X6, splitA_in_gaaa(X2, X4, X3, X6))
SPLITA_IN_GAAA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → SPLITA_IN_GAAA(X2, X4, X3, X6)
MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U26_GAA(X1, X2, X3, X4, X5, X6, mergesortB_in_gaa(.(X1, X8), X9, X6))
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → MERGESORTB_IN_GAA(.(X1, X8), X9, X6)
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U28_GAA(X1, X2, X3, X4, X5, X6, mergesortE_in_gaa(X7, X10, X6))
U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → MERGESORTE_IN_GAA(X7, X10, X6)
MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U3_GAA(X1, X2, X3, X4, X5, X6, splitD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → SPLITD_IN_GGAAA(X1, .(X2, X3), X7, X8, .(X5, X6))
MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U5_GAA(X1, X2, X3, X4, X5, X6, mergesortE_in_gaa(X7, X9, X6))
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → MERGESORTE_IN_GAA(X7, X9, X6)
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U7_GAA(X1, X2, X3, X4, X5, X6, mergesortE_in_gaa(X8, X10, X6))
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → MERGESORTE_IN_GAA(X8, X10, X6)
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U8_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U8_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U9_GAA(X1, X2, X3, X4, X5, X6, mergeC_in_ggaaa(X9, X10, X4, X5, X6))
U8_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → MERGEC_IN_GGAAA(X9, X10, X4, X5, X6)
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U10_GGAAA(X1, X2, X3, X4, X5, X6, X7, pF_in_ggggaa(X1, X3, X2, X4, X5, X7))
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → PF_IN_GGGGAA(X1, X3, X2, X4, X5, X7)
PF_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → U13_GGGGAA(X1, X2, X3, X4, X5, X6, leH_in_gg(X1, X2))
PF_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → LEH_IN_GG(X1, X2)
LEH_IN_GG(s(X1), s(X2)) → U12_GG(X1, X2, leH_in_gg(X1, X2))
LEH_IN_GG(s(X1), s(X2)) → LEH_IN_GG(X1, X2)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U15_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pF_in_ggggaa(X3, X2, X4, X5, X6, X8))
U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PF_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U17_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pG_in_ggggaa(X3, X2, X4, X5, X6, X8))
U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PG_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → U19_GGGGAA(X1, X2, X3, X4, X5, X6, gtI_in_gg(X1, X2))
PG_IN_GGGGAA(X1, X2, X3, X4, X5, X6) → GTI_IN_GG(X1, X2)
GTI_IN_GG(s(X1), s(X2)) → U18_GG(X1, X2, gtI_in_gg(X1, X2))
GTI_IN_GG(s(X1), s(X2)) → GTI_IN_GG(X1, X2)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U21_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PF_IN_GGGGAA(X1, X4, X3, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U23_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, pG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PG_IN_GGGGAA(X1, X4, X3, X5, X6, X8)
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U11_GGAAA(X1, X2, X3, X4, X5, X6, X7, pG_in_ggggaa(X1, X3, X2, X4, X5, X7))
MERGEC_IN_GGAAA(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → PG_IN_GGGGAA(X1, X3, X2, X4, X5, X7)
U27_GAA(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U29_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
U29_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U30_GAA(X1, X2, X3, X4, X5, X6, mergeC_in_ggaaa(X9, X10, X4, X5, X6))
U29_GAA(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → MERGEC_IN_GGAAA(X9, X10, X4, X5, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
mergesortB_in_gaa(x1, x2, x3)  =  mergesortB_in_gaa(x1)
.(x1, x2)  =  .(x1, x2)
splitD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitD_in_ggaaa(x1, x2)
splitA_in_gaaa(x1, x2, x3, x4)  =  splitA_in_gaaa(x1)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
mergesortE_in_gaa(x1, x2, x3)  =  mergesortE_in_gaa(x1)
mergeC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergeC_in_ggaaa(x1, x2)
pF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pF_in_ggggaa(x1, x2, x3, x4)
leH_in_gg(x1, x2)  =  leH_in_gg(x1, x2)
pG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  pG_in_ggggaa(x1, x2, x3, x4)
gtI_in_gg(x1, x2)  =  gtI_in_gg(x1, x2)
MERGESORTB_IN_GAA(x1, x2, x3)  =  MERGESORTB_IN_GAA(x1)
U24_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U24_GAA(x1, x2, x3, x7)
SPLITD_IN_GGAAA(x1, x2, x3, x4, x5)  =  SPLITD_IN_GGAAA(x1, x2)
U2_GGAAA(x1, x2, x3, x4, x5, x6, x7)  =  U2_GGAAA(x1, x2, x7)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_IN_GAAA(x1)
U1_GAAA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GAAA(x1, x2, x7)
U25_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAA(x1, x2, x3, x7)
U26_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U26_GAA(x1, x2, x3, x7)
U27_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U27_GAA(x1, x2, x3, x7, x8)
U28_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U28_GAA(x1, x2, x3, x7)
MERGESORTE_IN_GAA(x1, x2, x3)  =  MERGESORTE_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, x8)  =  U6_GAA(x1, x2, x3, x7, x8)
U7_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U7_GAA(x1, x2, x3, x7)
U8_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U8_GAA(x1, x2, x3, x7, x8)
U9_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U9_GAA(x1, x2, x3, x7)
MERGEC_IN_GGAAA(x1, x2, x3, x4, x5)  =  MERGEC_IN_GGAAA(x1, x2)
U10_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U10_GGAAA(x1, x2, x3, x4, x8)
PF_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PF_IN_GGGGAA(x1, x2, x3, x4)
U13_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U13_GGGGAA(x1, x2, x3, x4, x7)
LEH_IN_GG(x1, x2)  =  LEH_IN_GG(x1, x2)
U12_GG(x1, x2, x3)  =  U12_GG(x1, x2, x3)
U14_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_GGGGAA(x1, x2, x3, x4, x5, x9)
U15_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U15_GGGGAA(x1, x2, x3, x4, x5, x9)
U16_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGAA(x1, x2, x3, x4, x5, x9)
U17_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U17_GGGGAA(x1, x2, x3, x4, x5, x9)
PG_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PG_IN_GGGGAA(x1, x2, x3, x4)
U19_GGGGAA(x1, x2, x3, x4, x5, x6, x7)  =  U19_GGGGAA(x1, x2, x3, x4, x7)
GTI_IN_GG(x1, x2)  =  GTI_IN_GG(x1, x2)
U18_GG(x1, x2, x3)  =  U18_GG(x1, x2, x3)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U20_GGGGAA(x1, x2, x3, x4, x5, x9)
U21_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U21_GGGGAA(x1, x2, x3, x4, x5, x9)
U22_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U22_GGGGAA(x1, x2, x3, x4, x5, x9)
U23_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U23_GGGGAA(x1, x2, x3, x4, x5, x9)
U11_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U11_GGAAA(x1, x2, x3, x4, x8)
U29_GAA(x1, x2, x3, x4, x5, x6, x7, x8)  =  U29_GAA(x1, x2, x3, x7, x8)
U30_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U30_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 33 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GTI_IN_GG(s(X1), s(X2)) → GTI_IN_GG(X1, X2)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
GTI_IN_GG(x1, x2)  =  GTI_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:

GTI_IN_GG(s(X1), s(X2)) → GTI_IN_GG(X1, X2)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GTI_IN_GG(s(X1), s(X2)) → GTI_IN_GG(X1, X2)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• GTI_IN_GG(s(X1), s(X2)) → GTI_IN_GG(X1, X2)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LEH_IN_GG(s(X1), s(X2)) → LEH_IN_GG(X1, X2)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
LEH_IN_GG(x1, x2)  =  LEH_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:

LEH_IN_GG(s(X1), s(X2)) → LEH_IN_GG(X1, X2)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LEH_IN_GG(s(X1), s(X2)) → LEH_IN_GG(X1, X2)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LEH_IN_GG(s(X1), s(X2)) → LEH_IN_GG(X1, X2)
  The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PF_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PG_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PF_IN_GGGGAA(X1, X4, X3, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PG_IN_GGGGAA(X1, X4, X3, X5, X6, X8)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
PF_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PF_IN_GGGGAA(x1, x2, x3, x4)
U14_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_GGGGAA(x1, x2, x3, x4, x5, x9)
U16_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGAA(x1, x2, x3, x4, x5, x9)
PG_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PG_IN_GGGGAA(x1, x2, x3, x4)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U20_GGGGAA(x1, x2, x3, x4, x5, x9)
U22_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U22_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:

PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U14_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PF_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U16_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → PG_IN_GGGGAA(X3, X2, X4, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U20_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PF_IN_GGGGAA(X1, X4, X3, X5, X6, X8)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U22_GGGGAA(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → PG_IN_GGGGAA(X1, X4, X3, X5, X6, X8)

The TRS R consists of the following rules:

lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
PF_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PF_IN_GGGGAA(x1, x2, x3, x4)
U14_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U14_GGGGAA(x1, x2, x3, x4, x5, x9)
U16_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U16_GGGGAA(x1, x2, x3, x4, x5, x9)
PG_IN_GGGGAA(x1, x2, x3, x4, x5, x6)  =  PG_IN_GGGGAA(x1, x2, x3, x4)
U20_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U20_GGGGAA(x1, x2, x3, x4, x5, x9)
U22_GGGGAA(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U22_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:

PF_IN_GGGGAA(X1, X2, .(X3, X4), X5) → U14_GGGGAA(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
U14_GGGGAA(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → PF_IN_GGGGAA(X3, X2, X4, X5)
PF_IN_GGGGAA(X1, X2, .(X3, X4), X5) → U16_GGGGAA(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
U16_GGGGAA(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → PG_IN_GGGGAA(X3, X2, X4, X5)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5)) → U20_GGGGAA(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
U20_GGGGAA(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → PF_IN_GGGGAA(X1, X4, X3, X5)
PG_IN_GGGGAA(X1, X2, X3, .(X4, X5)) → U22_GGGGAA(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
U22_GGGGAA(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → PG_IN_GGGGAA(X1, X4, X3, X5)

The TRS R consists of the following rules:

lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecH_in_gg(x0, x1)
gtcI_in_gg(x0, x1)
U44_gg(x0, x1, x2)
U50_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:

• U14_GGGGAA(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → PF_IN_GGGGAA(X3, X2, X4, X5)
  The graph contains the following edges 3 >= 1, 2 >= 2, 6 > 2, 4 >= 3, 5 >= 4

• U20_GGGGAA(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → PF_IN_GGGGAA(X1, X4, X3, X5)
  The graph contains the following edges 1 >= 1, 6 > 1, 4 >= 2, 3 >= 3, 5 >= 4

• PF_IN_GGGGAA(X1, X2, .(X3, X4), X5) → U14_GGGGAA(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5

• PF_IN_GGGGAA(X1, X2, .(X3, X4), X5) → U16_GGGGAA(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4, 4 >= 5

• U16_GGGGAA(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → PG_IN_GGGGAA(X3, X2, X4, X5)
  The graph contains the following edges 3 >= 1, 2 >= 2, 6 > 2, 4 >= 3, 5 >= 4

• U22_GGGGAA(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → PG_IN_GGGGAA(X1, X4, X3, X5)
  The graph contains the following edges 1 >= 1, 6 > 1, 4 >= 2, 3 >= 3, 5 >= 4

• PG_IN_GGGGAA(X1, X2, X3, .(X4, X5)) → U20_GGGGAA(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
  The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 4 > 5

• PG_IN_GGGGAA(X1, X2, X3, .(X4, X5)) → U22_GGGGAA(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
  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:

SPLITA_IN_GAAA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → SPLITA_IN_GAAA(X2, X4, X3, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
SPLITA_IN_GAAA(x1, x2, x3, x4)  =  SPLITA_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:

SPLITA_IN_GAAA(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → SPLITA_IN_GAAA(X2, X4, X3, X6)

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

SPLITA_IN_GAAA(.(X1, X2)) → SPLITA_IN_GAAA(X2)

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:

• SPLITA_IN_GAAA(.(X1, X2)) → SPLITA_IN_GAAA(X2)
  The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → MERGESORTE_IN_GAA(X7, X9, X6)
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → MERGESORTE_IN_GAA(X8, X10, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
MERGESORTE_IN_GAA(x1, x2, x3)  =  MERGESORTE_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, x8)  =  U6_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:

MERGESORTE_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → MERGESORTE_IN_GAA(X7, X9, X6)
U4_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U6_GAA(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → MERGESORTE_IN_GAA(X8, X10, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
MERGESORTE_IN_GAA(x1, x2, x3)  =  MERGESORTE_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, x8)  =  U6_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:

MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3)))
U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → MERGESORTE_IN_GAA(X7)
U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U6_GAA(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
U6_GAA(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → MERGESORTE_IN_GAA(X8)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2) → U37_ggaaa(X1, X2, splitcA_in_gaaa(X2))
mergesortcE_in_gaa([]) → mergesortcE_out_gaa([], [])
mergesortcE_in_gaa(.(X1, [])) → mergesortcE_out_gaa(.(X1, []), .(X1, []))
mergesortcE_in_gaa(.(X1, .(X2, X3))) → U38_gaa(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3)))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U38_gaa(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U39_gaa(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U39_gaa(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → U40_gaa(X1, X2, X3, X9, mergesortcE_in_gaa(X8))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)
U40_gaa(X1, X2, X3, X9, mergesortcE_out_gaa(X8, X10)) → U41_gaa(X1, X2, X3, mergecC_in_ggaaa(X9, X10))
U41_gaa(X1, X2, X3, mergecC_out_ggaaa(X9, X10, X4)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4)
mergecC_in_ggaaa([], X1) → mergecC_out_ggaaa([], X1, X1)
mergecC_in_ggaaa(X1, []) → mergecC_out_ggaaa(X1, [], X1)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U42_ggaaa(X1, X2, X3, X4, qcF_in_ggggaa(X1, X3, X2, X4))
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U43_ggaaa(X1, X2, X3, X4, qcG_in_ggggaa(X1, X3, X2, X4))
U42_ggaaa(X1, X2, X3, X4, qcF_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5))
U43_ggaaa(X1, X2, X3, X4, qcG_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5))
qcF_in_ggggaa(X1, X2, [], X3) → U45_ggggaa(X1, X2, X3, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U46_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U48_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, []) → U51_ggggaa(X1, X2, X3, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U52_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U54_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
U45_ggggaa(X1, X2, X3, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3))
U46_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X3, X2, X4, X5))
U48_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X3, X2, X4, X5))
U51_ggggaa(X1, X2, X3, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3))
U52_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X1, X4, X3, X5))
U54_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X1, X4, X3, X5))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U47_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6))
U49_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U53_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6))
U55_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6))
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
mergesortcE_in_gaa(x0)
U37_ggaaa(x0, x1, x2)
U38_gaa(x0, x1, x2, x3)
splitcA_in_gaaa(x0)
U39_gaa(x0, x1, x2, x3, x4)
U32_gaaa(x0, x1, x2)
U40_gaa(x0, x1, x2, x3, x4)
U41_gaa(x0, x1, x2, x3)
mergecC_in_ggaaa(x0, x1)
U42_ggaaa(x0, x1, x2, x3, x4)
U43_ggaaa(x0, x1, x2, x3, x4)
qcF_in_ggggaa(x0, x1, x2, x3)
qcG_in_ggggaa(x0, x1, x2, x3)
U45_ggggaa(x0, x1, x2, x3)
U46_ggggaa(x0, x1, x2, x3, x4, x5)
U48_ggggaa(x0, x1, x2, x3, x4, x5)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
lecH_in_gg(x0, x1)
U47_ggggaa(x0, x1, x2, x3, x4, x5)
U49_ggggaa(x0, x1, x2, x3, x4, x5)
gtcI_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
U44_gg(x0, x1, x2)
U50_gg(x0, x1, x2)

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

(40) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3))) at position [3] we obtained the following new rules [LPAR04]:

MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, U37_ggaaa(X1, .(X2, X3), splitcA_in_gaaa(.(X2, X3))))

(41) Obligation:

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

U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → MERGESORTE_IN_GAA(X7)
U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U6_GAA(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
U6_GAA(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → MERGESORTE_IN_GAA(X8)
MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, U37_ggaaa(X1, .(X2, X3), splitcA_in_gaaa(.(X2, X3))))

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2) → U37_ggaaa(X1, X2, splitcA_in_gaaa(X2))
mergesortcE_in_gaa([]) → mergesortcE_out_gaa([], [])
mergesortcE_in_gaa(.(X1, [])) → mergesortcE_out_gaa(.(X1, []), .(X1, []))
mergesortcE_in_gaa(.(X1, .(X2, X3))) → U38_gaa(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3)))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U38_gaa(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U39_gaa(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U39_gaa(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → U40_gaa(X1, X2, X3, X9, mergesortcE_in_gaa(X8))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)
U40_gaa(X1, X2, X3, X9, mergesortcE_out_gaa(X8, X10)) → U41_gaa(X1, X2, X3, mergecC_in_ggaaa(X9, X10))
U41_gaa(X1, X2, X3, mergecC_out_ggaaa(X9, X10, X4)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4)
mergecC_in_ggaaa([], X1) → mergecC_out_ggaaa([], X1, X1)
mergecC_in_ggaaa(X1, []) → mergecC_out_ggaaa(X1, [], X1)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U42_ggaaa(X1, X2, X3, X4, qcF_in_ggggaa(X1, X3, X2, X4))
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U43_ggaaa(X1, X2, X3, X4, qcG_in_ggggaa(X1, X3, X2, X4))
U42_ggaaa(X1, X2, X3, X4, qcF_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5))
U43_ggaaa(X1, X2, X3, X4, qcG_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5))
qcF_in_ggggaa(X1, X2, [], X3) → U45_ggggaa(X1, X2, X3, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U46_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U48_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, []) → U51_ggggaa(X1, X2, X3, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U52_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U54_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
U45_ggggaa(X1, X2, X3, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3))
U46_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X3, X2, X4, X5))
U48_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X3, X2, X4, X5))
U51_ggggaa(X1, X2, X3, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3))
U52_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X1, X4, X3, X5))
U54_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X1, X4, X3, X5))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U47_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6))
U49_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U53_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6))
U55_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6))
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
mergesortcE_in_gaa(x0)
U37_ggaaa(x0, x1, x2)
U38_gaa(x0, x1, x2, x3)
splitcA_in_gaaa(x0)
U39_gaa(x0, x1, x2, x3, x4)
U32_gaaa(x0, x1, x2)
U40_gaa(x0, x1, x2, x3, x4)
U41_gaa(x0, x1, x2, x3)
mergecC_in_ggaaa(x0, x1)
U42_ggaaa(x0, x1, x2, x3, x4)
U43_ggaaa(x0, x1, x2, x3, x4)
qcF_in_ggggaa(x0, x1, x2, x3)
qcG_in_ggggaa(x0, x1, x2, x3)
U45_ggggaa(x0, x1, x2, x3)
U46_ggggaa(x0, x1, x2, x3, x4, x5)
U48_ggggaa(x0, x1, x2, x3, x4, x5)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
lecH_in_gg(x0, x1)
U47_ggggaa(x0, x1, x2, x3, x4, x5)
U49_ggggaa(x0, x1, x2, x3, x4, x5)
gtcI_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
U44_gg(x0, x1, x2)
U50_gg(x0, x1, x2)

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

(42) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, U37_ggaaa(X1, .(X2, X3), splitcA_in_gaaa(.(X2, X3)))) at position [3,2] we obtained the following new rules [LPAR04]:

MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, U37_ggaaa(X1, .(X2, X3), U32_gaaa(X2, X3, splitcA_in_gaaa(X3))))

(43) Obligation:

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

U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → MERGESORTE_IN_GAA(X7)
U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U6_GAA(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
U6_GAA(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → MERGESORTE_IN_GAA(X8)
MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, U37_ggaaa(X1, .(X2, X3), U32_gaaa(X2, X3, splitcA_in_gaaa(X3))))

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2) → U37_ggaaa(X1, X2, splitcA_in_gaaa(X2))
mergesortcE_in_gaa([]) → mergesortcE_out_gaa([], [])
mergesortcE_in_gaa(.(X1, [])) → mergesortcE_out_gaa(.(X1, []), .(X1, []))
mergesortcE_in_gaa(.(X1, .(X2, X3))) → U38_gaa(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3)))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U38_gaa(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U39_gaa(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U39_gaa(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → U40_gaa(X1, X2, X3, X9, mergesortcE_in_gaa(X8))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)
U40_gaa(X1, X2, X3, X9, mergesortcE_out_gaa(X8, X10)) → U41_gaa(X1, X2, X3, mergecC_in_ggaaa(X9, X10))
U41_gaa(X1, X2, X3, mergecC_out_ggaaa(X9, X10, X4)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4)
mergecC_in_ggaaa([], X1) → mergecC_out_ggaaa([], X1, X1)
mergecC_in_ggaaa(X1, []) → mergecC_out_ggaaa(X1, [], X1)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U42_ggaaa(X1, X2, X3, X4, qcF_in_ggggaa(X1, X3, X2, X4))
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U43_ggaaa(X1, X2, X3, X4, qcG_in_ggggaa(X1, X3, X2, X4))
U42_ggaaa(X1, X2, X3, X4, qcF_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5))
U43_ggaaa(X1, X2, X3, X4, qcG_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5))
qcF_in_ggggaa(X1, X2, [], X3) → U45_ggggaa(X1, X2, X3, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U46_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U48_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, []) → U51_ggggaa(X1, X2, X3, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U52_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U54_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
U45_ggggaa(X1, X2, X3, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3))
U46_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X3, X2, X4, X5))
U48_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X3, X2, X4, X5))
U51_ggggaa(X1, X2, X3, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3))
U52_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X1, X4, X3, X5))
U54_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X1, X4, X3, X5))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U47_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6))
U49_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U53_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6))
U55_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6))
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
mergesortcE_in_gaa(x0)
U37_ggaaa(x0, x1, x2)
U38_gaa(x0, x1, x2, x3)
splitcA_in_gaaa(x0)
U39_gaa(x0, x1, x2, x3, x4)
U32_gaaa(x0, x1, x2)
U40_gaa(x0, x1, x2, x3, x4)
U41_gaa(x0, x1, x2, x3)
mergecC_in_ggaaa(x0, x1)
U42_ggaaa(x0, x1, x2, x3, x4)
U43_ggaaa(x0, x1, x2, x3, x4)
qcF_in_ggggaa(x0, x1, x2, x3)
qcG_in_ggggaa(x0, x1, x2, x3)
U45_ggggaa(x0, x1, x2, x3)
U46_ggggaa(x0, x1, x2, x3, x4, x5)
U48_ggggaa(x0, x1, x2, x3, x4, x5)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
lecH_in_gg(x0, x1)
U47_ggggaa(x0, x1, x2, x3, x4, x5)
U49_ggggaa(x0, x1, x2, x3, x4, x5)
gtcI_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
U44_gg(x0, x1, x2)
U50_gg(x0, x1, x2)

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → MERGESORTE_IN_GAA(X7)
U4_GAA(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U6_GAA(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
MERGESORTE_IN_GAA(.(X1, .(X2, X3))) → U4_GAA(X1, X2, X3, U37_ggaaa(X1, .(X2, X3), U32_gaaa(X2, X3, splitcA_in_gaaa(X3))))

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

POL( U6_GAA(x₁, ..., x₅) ) = 2x₄ + 2x₅ + 1

POL( mergesortcE_in_gaa(x₁) ) = x₁

POL( [] ) = 0

POL( mergesortcE_out_gaa(x₁, x₂) ) = max{0, -2}

POL( .(x₁, x₂) ) = 2x₂ + 2

POL( U38_gaa(x₁, ..., x₄) ) = 2x₃ + 2

POL( splitcD_in_ggaaa(x₁, x₂) ) = max{0, 2x₁ - 2}

POL( U32_gaaa(x₁, ..., x₃) ) = 2x₃ + 2

POL( U37_ggaaa(x₁, ..., x₃) ) = x₃ + 2

POL( U4_GAA(x₁, ..., x₄) ) = 2x₄ + 2

POL( splitcA_in_gaaa(x₁) ) = x₁

POL( splitcA_out_gaaa(x₁, ..., x₃) ) = x₂ + 2x₃

POL( splitcD_out_ggaaa(x₁, ..., x₄) ) = x₃ + x₄

POL( U39_gaa(x₁, ..., x₅) ) = x₃ + 1

POL( U40_gaa(x₁, ..., x₅) ) = 1

POL( U41_gaa(x₁, ..., x₄) ) = max{0, -1}

POL( mergecC_in_ggaaa(x₁, x₂) ) = 1

POL( mergecC_out_ggaaa(x₁, ..., x₃) ) = x₁ + 2x₂

POL( U42_ggaaa(x₁, ..., x₅) ) = max{0, x₃ + x₄ + x₅ - 2}

POL( qcF_in_ggggaa(x₁, ..., x₄) ) = x₂ + x₃ + 1

POL( U43_ggaaa(x₁, ..., x₅) ) = max{0, 2x₁ + 2x₃ + x₄ - 2}

POL( qcG_in_ggggaa(x₁, ..., x₄) ) = 2x₁ + 2x₂ + 1

POL( U47_ggggaa(x₁, ..., x₆) ) = x₂ + x₃ + 1

POL( U53_ggggaa(x₁, ..., x₆) ) = max{0, 2x₂ + x₄ + 2x₅ - 2}

POL( U45_ggggaa(x₁, ..., x₄) ) = x₁ + 2x₂ + 2

POL( lecH_in_gg(x₁, x₂) ) = 2

POL( U46_ggggaa(x₁, ..., x₆) ) = x₁ + x₄ + 2x₅ + x₆ + 2

POL( U48_ggggaa(x₁, ..., x₆) ) = max{0, x₁ + x₃ + x₄ - 2}

POL( qcF_out_ggggaa(x₁, ..., x₅) ) = max{0, x₁ + x₄ - 1}

POL( U49_ggggaa(x₁, ..., x₆) ) = max{0, 2x₂ + x₅ - 1}

POL( U55_ggggaa(x₁, ..., x₆) ) = x₁ + 2x₂ + x₃ + 2x₄ + 2x₅ + 2x₆ + 2

POL( U51_ggggaa(x₁, ..., x₄) ) = x₁ + 2x₂ + x₃ + 2

POL( gtcI_in_gg(x₁, x₂) ) = 2

POL( U52_ggggaa(x₁, ..., x₆) ) = max{0, 2x₁ + x₂ + x₃ + 2x₄ + 2x₅ + 2x₆ - 2}

POL( U54_ggggaa(x₁, ..., x₆) ) = max{0, x₆ - 1}

POL( qcG_out_ggggaa(x₁, ..., x₅) ) = x₂ + 2x₅ + 2

POL( s(x₁) ) = max{0, 2x₁ - 2}

POL( U44_gg(x₁, ..., x₃) ) = x₁ + x₂ + 2x₃ + 2

POL( 0 ) = 0

POL( lecH_out_gg(x₁, x₂) ) = max{0, x₁ - 1}

POL( U50_gg(x₁, ..., x₃) ) = x₁ + 2

POL( gtcI_out_gg(x₁, x₂) ) = 0

POL( MERGESORTE_IN_GAA(x₁) ) = 2x₁ + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

mergesortcE_in_gaa([]) → mergesortcE_out_gaa([], [])
mergesortcE_in_gaa(.(X1, [])) → mergesortcE_out_gaa(.(X1, []), .(X1, []))
mergesortcE_in_gaa(.(X1, .(X2, X3))) → U38_gaa(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3)))
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U38_gaa(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U39_gaa(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
U39_gaa(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → U40_gaa(X1, X2, X3, X9, mergesortcE_in_gaa(X8))
U40_gaa(X1, X2, X3, X9, mergesortcE_out_gaa(X8, X10)) → U41_gaa(X1, X2, X3, mergecC_in_ggaaa(X9, X10))
U41_gaa(X1, X2, X3, mergecC_out_ggaaa(X9, X10, X4)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4)

(45) Obligation:

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

U6_GAA(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → MERGESORTE_IN_GAA(X8)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2) → U37_ggaaa(X1, X2, splitcA_in_gaaa(X2))
mergesortcE_in_gaa([]) → mergesortcE_out_gaa([], [])
mergesortcE_in_gaa(.(X1, [])) → mergesortcE_out_gaa(.(X1, []), .(X1, []))
mergesortcE_in_gaa(.(X1, .(X2, X3))) → U38_gaa(X1, X2, X3, splitcD_in_ggaaa(X1, .(X2, X3)))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U38_gaa(X1, X2, X3, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8)) → U39_gaa(X1, X2, X3, X8, mergesortcE_in_gaa(X7))
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U39_gaa(X1, X2, X3, X8, mergesortcE_out_gaa(X7, X9)) → U40_gaa(X1, X2, X3, X9, mergesortcE_in_gaa(X8))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)
U40_gaa(X1, X2, X3, X9, mergesortcE_out_gaa(X8, X10)) → U41_gaa(X1, X2, X3, mergecC_in_ggaaa(X9, X10))
U41_gaa(X1, X2, X3, mergecC_out_ggaaa(X9, X10, X4)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4)
mergecC_in_ggaaa([], X1) → mergecC_out_ggaaa([], X1, X1)
mergecC_in_ggaaa(X1, []) → mergecC_out_ggaaa(X1, [], X1)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U42_ggaaa(X1, X2, X3, X4, qcF_in_ggggaa(X1, X3, X2, X4))
mergecC_in_ggaaa(.(X1, X2), .(X3, X4)) → U43_ggaaa(X1, X2, X3, X4, qcG_in_ggggaa(X1, X3, X2, X4))
U42_ggaaa(X1, X2, X3, X4, qcF_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5))
U43_ggaaa(X1, X2, X3, X4, qcG_out_ggggaa(X1, X3, X2, X4, X5)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5))
qcF_in_ggggaa(X1, X2, [], X3) → U45_ggggaa(X1, X2, X3, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U46_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5) → U48_ggggaa(X1, X2, X3, X4, X5, lecH_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, []) → U51_ggggaa(X1, X2, X3, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U52_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5)) → U54_ggggaa(X1, X2, X3, X4, X5, gtcI_in_gg(X1, X2))
U45_ggggaa(X1, X2, X3, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3))
U46_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X3, X2, X4, X5))
U48_ggggaa(X1, X2, X3, X4, X5, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X3, X2, X4, X5))
U51_ggggaa(X1, X2, X3, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3))
U52_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, qcF_in_ggggaa(X1, X4, X3, X5))
U54_ggggaa(X1, X2, X3, X4, X5, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, qcG_in_ggggaa(X1, X4, X3, X5))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U47_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6))
U49_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X3, X2, X4, X5, X6)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U53_ggggaa(X1, X2, X3, X4, X5, qcF_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6))
U55_ggggaa(X1, X2, X3, X4, X5, qcG_out_ggggaa(X1, X4, X3, X5, X6)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6))
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
mergesortcE_in_gaa(x0)
U37_ggaaa(x0, x1, x2)
U38_gaa(x0, x1, x2, x3)
splitcA_in_gaaa(x0)
U39_gaa(x0, x1, x2, x3, x4)
U32_gaaa(x0, x1, x2)
U40_gaa(x0, x1, x2, x3, x4)
U41_gaa(x0, x1, x2, x3)
mergecC_in_ggaaa(x0, x1)
U42_ggaaa(x0, x1, x2, x3, x4)
U43_ggaaa(x0, x1, x2, x3, x4)
qcF_in_ggggaa(x0, x1, x2, x3)
qcG_in_ggggaa(x0, x1, x2, x3)
U45_ggggaa(x0, x1, x2, x3)
U46_ggggaa(x0, x1, x2, x3, x4, x5)
U48_ggggaa(x0, x1, x2, x3, x4, x5)
U51_ggggaa(x0, x1, x2, x3)
U52_ggggaa(x0, x1, x2, x3, x4, x5)
U54_ggggaa(x0, x1, x2, x3, x4, x5)
lecH_in_gg(x0, x1)
U47_ggggaa(x0, x1, x2, x3, x4, x5)
U49_ggggaa(x0, x1, x2, x3, x4, x5)
gtcI_in_gg(x0, x1)
U53_ggggaa(x0, x1, x2, x3, x4, x5)
U55_ggggaa(x0, x1, x2, x3, x4, x5)
U44_gg(x0, x1, x2)
U50_gg(x0, x1, x2)

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

(46) DependencyGraphProof (EQUIVALENT transformation)

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

(47) TRUE

(48) Obligation:

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

MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → MERGESORTB_IN_GAA(.(X1, X8), X9, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
mergesortcB_in_gaa([], [], X1) → mergesortcB_out_gaa([], [], X1)
mergesortcB_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcB_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcB_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U33_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_in_gaa(.(X1, X8), X9, X6))
U34_gaa(X1, X2, X3, X4, X5, X6, X7, mergesortcB_out_gaa(.(X1, X8), X9, X6)) → U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X7, X10, X6))
mergesortcE_in_gaa([], [], X1) → mergesortcE_out_gaa([], [], X1)
mergesortcE_in_gaa(.(X1, []), .(X1, []), X2) → mergesortcE_out_gaa(.(X1, []), .(X1, []), X2)
mergesortcE_in_gaa(.(X1, .(X2, X3)), X4, .(X5, X6)) → U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6)))
U38_gaa(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X1, .(X2, X3), X7, X8, .(X5, X6))) → U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_in_gaa(X7, X9, X6))
U39_gaa(X1, X2, X3, X4, X5, X6, X8, mergesortcE_out_gaa(X7, X9, X6)) → U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_in_gaa(X8, X10, X6))
U40_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X8, X10, X6)) → U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
mergecC_in_ggaaa([], X1, X1, X2, X3) → mergecC_out_ggaaa([], X1, X1, X2, X3)
mergecC_in_ggaaa(X1, [], X1, X2, X3) → mergecC_out_ggaaa(X1, [], X1, X2, X3)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7) → U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_in_ggggaa(X1, X3, X2, X4, X5, X7))
qcF_in_ggggaa(X1, X2, [], X3, .(X2, X3), X4) → U45_ggggaa(X1, X2, X3, X4, lecH_in_gg(X1, X2))
lecH_in_gg(s(X1), s(X2)) → U44_gg(X1, X2, lecH_in_gg(X1, X2))
lecH_in_gg(0, s(0)) → lecH_out_gg(0, s(0))
lecH_in_gg(0, 0) → lecH_out_gg(0, 0)
U44_gg(X1, X2, lecH_out_gg(X1, X2)) → lecH_out_gg(s(X1), s(X2))
U45_ggggaa(X1, X2, X3, X4, lecH_out_gg(X1, X2)) → qcF_out_ggggaa(X1, X2, [], X3, .(X2, X3), X4)
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8)) → U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U46_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcF_in_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8)) → U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_in_gg(X1, X2))
U48_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, lecH_out_gg(X1, X2)) → U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X3, X2, X4, X5, X6, X8))
qcG_in_ggggaa(X1, X2, X3, [], .(X1, X3), X4) → U51_ggggaa(X1, X2, X3, X4, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(X1), s(X2)) → U50_gg(X1, X2, gtcI_in_gg(X1, X2))
gtcI_in_gg(s(0), 0) → gtcI_out_gg(s(0), 0)
U50_gg(X1, X2, gtcI_out_gg(X1, X2)) → gtcI_out_gg(s(X1), s(X2))
U51_ggggaa(X1, X2, X3, X4, gtcI_out_gg(X1, X2)) → qcG_out_ggggaa(X1, X2, X3, [], .(X1, X3), X4)
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8)) → U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U52_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_in_ggggaa(X1, X4, X3, X5, X6, X8))
U53_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X1, X6), .(X7, X8))
qcG_in_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8)) → U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_in_gg(X1, X2))
U54_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, gtcI_out_gg(X1, X2)) → U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_in_ggggaa(X1, X4, X3, X5, X6, X8))
U55_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X1, X4, X3, X5, X6, X8)) → qcG_out_ggggaa(X1, X2, X3, .(X4, X5), .(X4, X6), .(X7, X8))
U49_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcG_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X2, X6), .(X7, X8))
U47_ggggaa(X1, X2, X3, X4, X5, X6, X7, X8, qcF_out_ggggaa(X3, X2, X4, X5, X6, X8)) → qcF_out_ggggaa(X1, X2, .(X3, X4), X5, .(X3, X6), .(X7, X8))
U42_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcF_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X1, X5), X6, X7)
mergecC_in_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7) → U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_in_ggggaa(X1, X3, X2, X4, X5, X7))
U43_ggaaa(X1, X2, X3, X4, X5, X6, X7, qcG_out_ggggaa(X1, X3, X2, X4, X5, X7)) → mergecC_out_ggaaa(.(X1, X2), .(X3, X4), .(X3, X5), X6, X7)
U41_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcE_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))
U35_gaa(X1, X2, X3, X4, X5, X6, X9, mergesortcE_out_gaa(X7, X10, X6)) → U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_in_ggaaa(X9, X10, X4, X5, X6))
U36_gaa(X1, X2, X3, X4, X5, X6, mergecC_out_ggaaa(X9, X10, X4, X5, X6)) → mergesortcB_out_gaa(.(X1, .(X2, X3)), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
mergesortcB_in_gaa(x1, x2, x3)  =  mergesortcB_in_gaa(x1)
mergesortcB_out_gaa(x1, x2, x3)  =  mergesortcB_out_gaa(x1, x2)
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)
mergesortcE_in_gaa(x1, x2, x3)  =  mergesortcE_in_gaa(x1)
mergesortcE_out_gaa(x1, x2, x3)  =  mergesortcE_out_gaa(x1, x2)
U38_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U38_gaa(x1, x2, x3, x7)
U39_gaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U39_gaa(x1, x2, x3, x7, x8)
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)  =  U41_gaa(x1, x2, x3, x7)
mergecC_in_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_in_ggaaa(x1, x2)
mergecC_out_ggaaa(x1, x2, x3, x4, x5)  =  mergecC_out_ggaaa(x1, x2, x3)
U42_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U42_ggaaa(x1, x2, x3, x4, x8)
qcF_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_in_ggggaa(x1, x2, x3, x4)
U45_ggggaa(x1, x2, x3, x4, x5)  =  U45_ggggaa(x1, x2, x3, x5)
lecH_in_gg(x1, x2)  =  lecH_in_gg(x1, x2)
s(x1)  =  s(x1)
U44_gg(x1, x2, x3)  =  U44_gg(x1, x2, x3)
0  =  0
lecH_out_gg(x1, x2)  =  lecH_out_gg(x1, x2)
qcF_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcF_out_ggggaa(x1, x2, x3, x4, x5)
U46_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U46_ggggaa(x1, x2, x3, x4, x5, x9)
U47_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U47_ggggaa(x1, x2, x3, x4, x5, x9)
U48_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U48_ggggaa(x1, x2, x3, x4, x5, x9)
U49_ggggaa(x1, x2, x3, x4, x5, x6, x7, x8, x9)  =  U49_ggggaa(x1, x2, x3, x4, x5, x9)
qcG_in_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_in_ggggaa(x1, x2, x3, x4)
U51_ggggaa(x1, x2, x3, x4, x5)  =  U51_ggggaa(x1, x2, x3, x5)
gtcI_in_gg(x1, x2)  =  gtcI_in_gg(x1, x2)
U50_gg(x1, x2, x3)  =  U50_gg(x1, x2, x3)
gtcI_out_gg(x1, x2)  =  gtcI_out_gg(x1, x2)
qcG_out_ggggaa(x1, x2, x3, x4, x5, x6)  =  qcG_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)
U43_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8)  =  U43_ggaaa(x1, x2, x3, x4, x8)
U36_gaa(x1, x2, x3, x4, x5, x6, x7)  =  U36_gaa(x1, x2, x3, x7)
MERGESORTB_IN_GAA(x1, x2, x3)  =  MERGESORTB_IN_GAA(x1)
U25_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAA(x1, x2, x3, x7)

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

(49) UsableRulesProof (EQUIVALENT transformation)

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

(50) Obligation:

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

MERGESORTB_IN_GAA(.(X1, .(X2, X3)), X4, .(X5, X6)) → U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_in_ggaaa(X2, X3, X7, X8, X6))
U25_GAA(X1, X2, X3, X4, X5, X6, splitcD_out_ggaaa(X2, X3, X7, X8, X6)) → MERGESORTB_IN_GAA(.(X1, X8), X9, X6)

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6)) → U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U37_ggaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4, .(X5, X6))
splitcA_in_gaaa([], [], [], X1) → splitcA_out_gaaa([], [], [], X1)
splitcA_in_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6)) → U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_in_gaaa(X2, X4, X3, X6))
U32_gaaa(X1, X2, X3, X4, X5, X6, splitcA_out_gaaa(X2, X4, X3, X6)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4, .(X5, X6))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
splitcD_in_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_in_ggaaa(x1, x2)
U37_ggaaa(x1, x2, x3, x4, x5, x6, x7)  =  U37_ggaaa(x1, x2, x7)
splitcA_in_gaaa(x1, x2, x3, x4)  =  splitcA_in_gaaa(x1)
[]  =  []
splitcA_out_gaaa(x1, x2, x3, x4)  =  splitcA_out_gaaa(x1, x2, x3)
U32_gaaa(x1, x2, x3, x4, x5, x6, x7)  =  U32_gaaa(x1, x2, x7)
splitcD_out_ggaaa(x1, x2, x3, x4, x5)  =  splitcD_out_ggaaa(x1, x2, x3, x4)
MERGESORTB_IN_GAA(x1, x2, x3)  =  MERGESORTB_IN_GAA(x1)
U25_GAA(x1, x2, x3, x4, x5, x6, x7)  =  U25_GAA(x1, x2, x3, x7)

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

(51) PiDPToQDPProof (SOUND transformation)

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

(52) Obligation:

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

MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, splitcD_in_ggaaa(X2, X3))
U25_GAA(X1, X2, X3, splitcD_out_ggaaa(X2, X3, X7, X8)) → MERGESORTB_IN_GAA(.(X1, X8))

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2) → U37_ggaaa(X1, X2, splitcA_in_gaaa(X2))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
U37_ggaaa(x0, x1, x2)
splitcA_in_gaaa(x0)
U32_gaaa(x0, x1, x2)

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

(53) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, splitcD_in_ggaaa(X2, X3)) at position [3] we obtained the following new rules [LPAR04]:

MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, U37_ggaaa(X2, X3, splitcA_in_gaaa(X3)))

(54) Obligation:

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

U25_GAA(X1, X2, X3, splitcD_out_ggaaa(X2, X3, X7, X8)) → MERGESORTB_IN_GAA(.(X1, X8))
MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, U37_ggaaa(X2, X3, splitcA_in_gaaa(X3)))

The TRS R consists of the following rules:

splitcD_in_ggaaa(X1, X2) → U37_ggaaa(X1, X2, splitcA_in_gaaa(X2))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
U37_ggaaa(x0, x1, x2)
splitcA_in_gaaa(x0)
U32_gaaa(x0, x1, x2)

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

(55) 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.

(56) Obligation:

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

U25_GAA(X1, X2, X3, splitcD_out_ggaaa(X2, X3, X7, X8)) → MERGESORTB_IN_GAA(.(X1, X8))
MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, U37_ggaaa(X2, X3, splitcA_in_gaaa(X3)))

The TRS R consists of the following rules:

splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)

The set Q consists of the following terms:

splitcD_in_ggaaa(x0, x1)
U37_ggaaa(x0, x1, x2)
splitcA_in_gaaa(x0)
U32_gaaa(x0, x1, x2)

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

(57) 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].

splitcD_in_ggaaa(x0, x1)

(58) Obligation:

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

U25_GAA(X1, X2, X3, splitcD_out_ggaaa(X2, X3, X7, X8)) → MERGESORTB_IN_GAA(.(X1, X8))
MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, U37_ggaaa(X2, X3, splitcA_in_gaaa(X3)))

The TRS R consists of the following rules:

splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)

The set Q consists of the following terms:

U37_ggaaa(x0, x1, x2)
splitcA_in_gaaa(x0)
U32_gaaa(x0, x1, x2)

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

(59) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

MERGESORTB_IN_GAA(.(X1, .(X2, X3))) → U25_GAA(X1, X2, X3, U37_ggaaa(X2, X3, splitcA_in_gaaa(X3)))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(MERGESORTB_IN_GAA(x₁)) = x₁   
POL(U25_GAA(x₁, x₂, x₃, x₄)) = x₄   
POL(U32_gaaa(x₁, x₂, x₃)) = 1 + x₃   
POL(U37_ggaaa(x₁, x₂, x₃)) = 1 + x₃   
POL([]) = 0   
POL(splitcA_in_gaaa(x₁)) = x₁   
POL(splitcA_out_gaaa(x₁, x₂, x₃)) = x₂ + x₃   
POL(splitcD_out_ggaaa(x₁, x₂, x₃, x₄)) = 1 + x₄   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)

(60) Obligation:

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

U25_GAA(X1, X2, X3, splitcD_out_ggaaa(X2, X3, X7, X8)) → MERGESORTB_IN_GAA(.(X1, X8))

The TRS R consists of the following rules:

splitcA_in_gaaa([]) → splitcA_out_gaaa([], [], [])
splitcA_in_gaaa(.(X1, X2)) → U32_gaaa(X1, X2, splitcA_in_gaaa(X2))
U37_ggaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcD_out_ggaaa(X1, X2, .(X1, X3), X4)
U32_gaaa(X1, X2, splitcA_out_gaaa(X2, X4, X3)) → splitcA_out_gaaa(.(X1, X2), .(X1, X3), X4)

The set Q consists of the following terms:

U37_ggaaa(x0, x1, x2)
splitcA_in_gaaa(x0)
U32_gaaa(x0, x1, x2)

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

(61) DependencyGraphProof (EQUIVALENT transformation)

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

(62) TRUE