Left Termination proof of /tmp/tmpO8DAYD/mergesort

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 436 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 801 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 24 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇒, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔, 0 ms)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇒, 0 ms)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔, 0 ms)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔, 0 ms)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇒, 11 ms)

        ↳39 QDP

        ↳40 Rewriting (⇔, 289 ms)

        ↳41 QDP

        ↳42 Rewriting (⇔, 0 ms)

        ↳43 QDP

        ↳44 QDPOrderProof (⇔, 2214 ms)

        ↳45 QDP

        ↳46 DependencyGraphProof (⇔, 0 ms)

        ↳47 TRUE

    ↳48 PiDP

        ↳49 UsableRulesProof (⇔, 0 ms)

        ↳50 PiDP

        ↳51 PiDPToQDPProof (⇒, 1 ms)

        ↳52 QDP

        ↳53 Rewriting (⇔, 0 ms)

        ↳54 QDP

        ↳55 UsableRulesProof (⇔, 0 ms)

        ↳56 QDP

        ↳57 QReductionProof (⇔, 0 ms)

        ↳58 QDP

        ↳59 QDPOrderProof (⇔, 0 ms)

        ↳60 QDP

        ↳61 DependencyGraphProof (⇔, 0 ms)

        ↳62 TRUE

(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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: mergesort(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: 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];
10 [label="10: true | mergesort([], T2, T3), SCOPE: 1, CLAUSE: 1 | mergesort([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: 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];
17 [label="17: mergesort([], T2, T3), SCOPE: 1, CLAUSE: 1 | mergesort([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: mergesort([], T2, T3), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | mergesort(.(T8, []), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: 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];
31 [label="31: mergesort(.(T8, []), T2, T3), SCOPE: 1, CLAUSE: 2\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(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];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(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];
42 [label="42: ','(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];
54 [label="B: ','(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];
57 [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];
58 [label="N: ','(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];
60 [label="60: 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];
61 [label="61: 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];
67 [label="C: split(T61, X92, X91, T64)\n\nT61 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: 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];
71 [label="71: 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];
73 [label="73: 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];
75 [label="75: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: split(T81, X123, X122, T84)\n\nT81 is ground\nX122 is free\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: mergesort(.(T39, T48), X35, T49)\n\nT39 is ground\nT48 is ground\nX35 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="O: ','(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];
97 [label="E: mergesort(T47, X36, T88)\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="G: merge(T87, T93, T94, .(T95, T96))\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: 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];
105 [label="105: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 0\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: 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];
109 [label="109: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 1\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: mergesort(T47, X36, T88), SCOPE: 5, CLAUSE: 2\n\nT47 is ground\nX36 is free", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
124 [label="F: ','(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];
125 [label="125: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: 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];
127 [label="P: ','(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];
132 [label="132: mergesort(T131, X175, T133)\n\nT131 is ground\nX175 is free", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="Q: ','(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];
136 [label="136: mergesort(T132, X176, T138)\n\nT132 is ground\nX176 is free", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: merge(T137, T144, X177, .(T145, T146))\n\nT137 is ground\nT144 is ground\nX177 is free", fontsize=16, style=filled, fillcolor=lightyellow];
149 [label="149: 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];
152 [label="152: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 5\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
153 [label="153: 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];
155 [label="155: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
157 [label="157: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
159 [label="159: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
160 [label="160: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 6\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
161 [label="161: 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];
162 [label="162: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
164 [label="164: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
166 [label="166: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
173 [label="173: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 7\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
174 [label="174: merge(T87, T93, T94, .(T95, T96)), SCOPE: 6, CLAUSE: 8\n\nT87 is ground\nT93 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
182 [label="H: ','(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];
184 [label="184: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
187 [label="J: le(T219, T221)\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
189 [label="L: merge(T220, .(T221, T222), T226, T227)\n\nT220 is ground\nT221 is ground\nT222 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
190 [label="190: 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];
199 [label="199: le(T219, T221), SCOPE: 7, CLAUSE: 11\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
200 [label="200: 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];
202 [label="202: le(T240, T241)\n\nT240 is ground\nT241 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
204 [label="204: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
208 [label="208: le(T219, T221), SCOPE: 7, CLAUSE: 12\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
209 [label="209: le(T219, T221), SCOPE: 7, CLAUSE: 13\n\nT219 is ground\nT221 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
210 [label="210: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
211 [label="211: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
212 [label="212: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
215 [label="215: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
216 [label="216: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
217 [label="217: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
221 [label="221: 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];
223 [label="223: 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];
224 [label="224: 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];
225 [label="225: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
226 [label="226: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
227 [label="227: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
230 [label="230: 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];
235 [label="235: 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];
236 [label="236: 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];
240 [label="240: ','(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];
241 [label="241: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
252 [label="I: ','(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];
253 [label="253: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
258 [label="K: gt(T319, T321)\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
259 [label="M: merge(.(T319, T320), T322, T326, T327)\n\nT319 is ground\nT320 is ground\nT322 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
264 [label="264: 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];
266 [label="266: gt(T319, T321), SCOPE: 9, CLAUSE: 9\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
267 [label="267: gt(T319, T321), SCOPE: 9, CLAUSE: 10\n\nT319 is ground\nT321 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
271 [label="271: gt(T340, T341)\n\nT340 is ground\nT341 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
273 [label="273: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
275 [label="275: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
276 [label="276: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
277 [label="277: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
280 [label="280: 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];
281 [label="281: 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];
282 [label="282: 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];
283 [label="283: 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];
289 [label="289: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
290 [label="290: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
291 [label="291: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
298 [label="298: 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];
299 [label="299: 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];
313 [label="313: ','(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];
314 [label="314: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
326 [label="326: ','(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];
327 [label="327: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
330 [label="330: ','(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];
331 [label="331: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
332 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 332 [arrowhead = none ];
332 -> 6;

333 [label="EVAL with clause\nmergesort([], [], X2).\nand substitutionT1 -> [],\nT2 -> [],\nT3 -> T5,\nX2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 333 [arrowhead = none ];
333 -> 10;

334 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 334 [arrowhead = none ];
334 -> 14;

335 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 335 [arrowhead = none ];
335 -> 17;

336 [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];
14 -> 336 [arrowhead = none ];
336 -> 26;

337 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 337 [arrowhead = none ];
337 -> 29;

338 [label="BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []), Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 338 [arrowhead = none ];
338 -> 19;

339 [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];
19 -> 339 [arrowhead = none ];
339 -> 23;

340 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 340 [arrowhead = none ];
340 -> 31;

341 [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];
29 -> 341 [arrowhead = none ];
341 -> 39;

342 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 342 [arrowhead = none ];
342 -> 40;

343 [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];
31 -> 343 [arrowhead = none ];
343 -> 32;

344 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 344 [arrowhead = none ];
344 -> 41;

345 [label="BACKTRACK\nfor clause: split([], [], [], Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 345 [arrowhead = none ];
345 -> 42;

346 [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];
42 -> 346 [arrowhead = none ];
346 -> 54;

347 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 347 [arrowhead = none ];
347 -> 57;

348 [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];
54 -> 348 [arrowhead = none ];
348 -> 58;

349 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 349 [arrowhead = none ];
349 -> 60;

350 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 350 [arrowhead = none ];
350 -> 86;

351 [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];
58 -> 351 [arrowhead = none ];
351 -> 87;

352 [label="BACKTRACK\nfor clause: split([], [], [], Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 352 [arrowhead = none ];
352 -> 61;

353 [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];
61 -> 353 [arrowhead = none ];
353 -> 67;

354 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 354 [arrowhead = none ];
354 -> 68;

355 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 355 [arrowhead = none ];
355 -> 69;

356 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 356 [arrowhead = none ];
356 -> 71;

357 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 357 [arrowhead = none ];
357 -> 73;

358 [label="EVAL with clause\nsplit([], [], [], X99).\nand substitutionT61 -> [],\nX92 -> [],\nX91 -> [],\nT64 -> T71,\nX99 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 358 [arrowhead = none ];
358 -> 75;

359 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 359 [arrowhead = none ];
359 -> 76;

360 [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];
73 -> 360 [arrowhead = none ];
360 -> 82;

361 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 361 [arrowhead = none ];
361 -> 83;

362 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 362 [arrowhead = none ];
362 -> 77;

363 [label="INSTANCE with matching:\nT61 -> T81\nX92 -> X123\nX91 -> X122\nT64 -> T84", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 363 [arrowhead = none , style=dashed];
363 -> 67[style=dashed];

364 [label="INSTANCE with matching:\nT1 -> .(T39, T48)\nT2 -> X35\nT3 -> T49", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 364 [arrowhead = none , style=dashed];
364 -> 2[style=dashed];

365 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
87 -> 365 [arrowhead = none ];
365 -> 97;

366 [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];
87 -> 366 [arrowhead = none ];
366 -> 98;

367 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
97 -> 367 [arrowhead = none ];
367 -> 101;

368 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
98 -> 368 [arrowhead = none ];
368 -> 149;

369 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
101 -> 369 [arrowhead = none ];
369 -> 105;

370 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
101 -> 370 [arrowhead = none ];
370 -> 106;

371 [label="EVAL with clause\nmergesort([], [], X138).\nand substitutionT47 -> [],\nX36 -> [],\nT88 -> T103,\nX138 -> T103", fontsize=14, style = filled, fillcolor = lightblue];
105 -> 371 [arrowhead = none ];
371 -> 109;

372 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 372 [arrowhead = none ];
372 -> 110;

373 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 373 [arrowhead = none ];
373 -> 112;

374 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
106 -> 374 [arrowhead = none ];
374 -> 113;

375 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 375 [arrowhead = none ];
375 -> 111;

376 [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];
112 -> 376 [arrowhead = none ];
376 -> 114;

377 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
112 -> 377 [arrowhead = none ];
377 -> 115;

378 [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];
113 -> 378 [arrowhead = none ];
378 -> 124;

379 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 379 [arrowhead = none ];
379 -> 125;

380 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 380 [arrowhead = none ];
380 -> 116;

381 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
124 -> 381 [arrowhead = none ];
381 -> 126;

382 [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];
124 -> 382 [arrowhead = none ];
382 -> 127;

383 [label="INSTANCE with matching:\nT40 -> T124\nT41 -> .(T125, T126)\nX63 -> X173\nX62 -> X174\nT44 -> .(T129, T130)", fontsize=14, style = filled, fillcolor = lightgrey];
126 -> 383 [arrowhead = none , style=dashed];
383 -> 57[style=dashed];

384 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
127 -> 384 [arrowhead = none ];
384 -> 132;

385 [label="SPLIT 2\nnew knowledge:\nT131 is ground\nT137 is ground\nreplacements:X175 -> T137,\nT133 -> T138,\nT134 -> T139", fontsize=14, style = filled, fillcolor = violet];
127 -> 385 [arrowhead = none ];
385 -> 133;

386 [label="INSTANCE with matching:\nT47 -> T131\nX36 -> X175\nT88 -> T133", fontsize=14, style = filled, fillcolor = lightgrey];
132 -> 386 [arrowhead = none , style=dashed];
386 -> 97[style=dashed];

387 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
133 -> 387 [arrowhead = none ];
387 -> 136;

388 [label="SPLIT 2\nnew knowledge:\nT132 is ground\nT144 is ground\nreplacements:X176 -> T144,\nT139 -> T145,\nT138 -> T146", fontsize=14, style = filled, fillcolor = violet];
133 -> 388 [arrowhead = none ];
388 -> 137;

389 [label="INSTANCE with matching:\nT47 -> T132\nX36 -> X176\nT88 -> T138", fontsize=14, style = filled, fillcolor = lightgrey];
136 -> 389 [arrowhead = none , style=dashed];
389 -> 97[style=dashed];

390 [label="INSTANCE with matching:\nT87 -> T137\nT93 -> T144\nT94 -> X177\nT95 -> T145\nT96 -> T146", fontsize=14, style = filled, fillcolor = lightgrey];
137 -> 390 [arrowhead = none , style=dashed];
390 -> 98[style=dashed];

391 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
149 -> 391 [arrowhead = none ];
391 -> 152;

392 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
149 -> 392 [arrowhead = none ];
392 -> 153;

393 [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];
152 -> 393 [arrowhead = none ];
393 -> 155;

394 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
152 -> 394 [arrowhead = none ];
394 -> 157;

395 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
153 -> 395 [arrowhead = none ];
395 -> 160;

396 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
153 -> 396 [arrowhead = none ];
396 -> 161;

397 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
155 -> 397 [arrowhead = none ];
397 -> 159;

398 [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];
160 -> 398 [arrowhead = none ];
398 -> 162;

399 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
160 -> 399 [arrowhead = none ];
399 -> 164;

400 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
161 -> 400 [arrowhead = none ];
400 -> 173;

401 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
161 -> 401 [arrowhead = none ];
401 -> 174;

402 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
162 -> 402 [arrowhead = none ];
402 -> 166;

403 [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];
173 -> 403 [arrowhead = none ];
403 -> 182;

404 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
173 -> 404 [arrowhead = none ];
404 -> 184;

405 [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];
174 -> 405 [arrowhead = none ];
405 -> 330;

406 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
174 -> 406 [arrowhead = none ];
406 -> 331;

407 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
182 -> 407 [arrowhead = none ];
407 -> 187;

408 [label="SPLIT 2\nnew knowledge:\nT219 is ground\nT221 is ground", fontsize=14, style = filled, fillcolor = violet];
182 -> 408 [arrowhead = none ];
408 -> 189;

409 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
187 -> 409 [arrowhead = none ];
409 -> 190;

410 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
189 -> 410 [arrowhead = none ];
410 -> 221;

411 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
190 -> 411 [arrowhead = none ];
411 -> 199;

412 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
190 -> 412 [arrowhead = none ];
412 -> 200;

413 [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];
199 -> 413 [arrowhead = none ];
413 -> 202;

414 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
199 -> 414 [arrowhead = none ];
414 -> 204;

415 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
200 -> 415 [arrowhead = none ];
415 -> 208;

416 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
200 -> 416 [arrowhead = none ];
416 -> 209;

417 [label="INSTANCE with matching:\nT219 -> T240\nT221 -> T241", fontsize=14, style = filled, fillcolor = lightgrey];
202 -> 417 [arrowhead = none , style=dashed];
417 -> 187[style=dashed];

418 [label="EVAL with clause\nle(0, s(0)).\nand substitutionT219 -> 0,\nT221 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
208 -> 418 [arrowhead = none ];
418 -> 210;

419 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
208 -> 419 [arrowhead = none ];
419 -> 211;

420 [label="EVAL with clause\nle(0, 0).\nand substitutionT219 -> 0,\nT221 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
209 -> 420 [arrowhead = none ];
420 -> 215;

421 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
209 -> 421 [arrowhead = none ];
421 -> 216;

422 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
210 -> 422 [arrowhead = none ];
422 -> 212;

423 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
215 -> 423 [arrowhead = none ];
423 -> 217;

424 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
221 -> 424 [arrowhead = none ];
424 -> 223;

425 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
221 -> 425 [arrowhead = none ];
425 -> 224;

426 [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];
223 -> 426 [arrowhead = none ];
426 -> 225;

427 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
223 -> 427 [arrowhead = none ];
427 -> 226;

428 [label="BACKTRACK\nfor clause: merge(Xs, [], Xs, Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
224 -> 428 [arrowhead = none ];
428 -> 230;

429 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
225 -> 429 [arrowhead = none ];
429 -> 227;

430 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
230 -> 430 [arrowhead = none ];
430 -> 235;

431 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
230 -> 431 [arrowhead = none ];
431 -> 236;

432 [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];
235 -> 432 [arrowhead = none ];
432 -> 240;

433 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
235 -> 433 [arrowhead = none ];
433 -> 241;

434 [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];
236 -> 434 [arrowhead = none ];
434 -> 252;

435 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
236 -> 435 [arrowhead = none ];
435 -> 253;

436 [label="INSTANCE with matching:\nT219 -> T294\nT221 -> T296\nT220 -> T295\nT222 -> T297\nT226 -> T301\nT227 -> T302", fontsize=14, style = filled, fillcolor = lightgrey];
240 -> 436 [arrowhead = none , style=dashed];
436 -> 182[style=dashed];

437 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
252 -> 437 [arrowhead = none ];
437 -> 258;

438 [label="SPLIT 2\nnew knowledge:\nT319 is ground\nT321 is ground", fontsize=14, style = filled, fillcolor = violet];
252 -> 438 [arrowhead = none ];
438 -> 259;

439 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
258 -> 439 [arrowhead = none ];
439 -> 264;

440 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
259 -> 440 [arrowhead = none ];
440 -> 280;

441 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
264 -> 441 [arrowhead = none ];
441 -> 266;

442 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
264 -> 442 [arrowhead = none ];
442 -> 267;

443 [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];
266 -> 443 [arrowhead = none ];
443 -> 271;

444 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
266 -> 444 [arrowhead = none ];
444 -> 273;

445 [label="EVAL with clause\ngt(s(0), 0).\nand substitutionT319 -> s(0),\nT321 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
267 -> 445 [arrowhead = none ];
445 -> 275;

446 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
267 -> 446 [arrowhead = none ];
446 -> 276;

447 [label="INSTANCE with matching:\nT319 -> T340\nT321 -> T341", fontsize=14, style = filled, fillcolor = lightgrey];
271 -> 447 [arrowhead = none , style=dashed];
447 -> 258[style=dashed];

448 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
275 -> 448 [arrowhead = none ];
448 -> 277;

449 [label="BACKTRACK\nfor clause: merge([], Xs, Xs, Ls)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
280 -> 449 [arrowhead = none ];
449 -> 281;

450 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
281 -> 450 [arrowhead = none ];
450 -> 282;

451 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
281 -> 451 [arrowhead = none ];
451 -> 283;

452 [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];
282 -> 452 [arrowhead = none ];
452 -> 289;

453 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
282 -> 453 [arrowhead = none ];
453 -> 290;

454 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
283 -> 454 [arrowhead = none ];
454 -> 298;

455 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
283 -> 455 [arrowhead = none ];
455 -> 299;

456 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
289 -> 456 [arrowhead = none ];
456 -> 291;

457 [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];
298 -> 457 [arrowhead = none ];
457 -> 313;

458 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
298 -> 458 [arrowhead = none ];
458 -> 314;

459 [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];
299 -> 459 [arrowhead = none ];
459 -> 326;

460 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
299 -> 460 [arrowhead = none ];
460 -> 327;

461 [label="INSTANCE with matching:\nT219 -> T387\nT221 -> T389\nT220 -> T388\nT222 -> T390\nT226 -> T394\nT227 -> T395", fontsize=14, style = filled, fillcolor = lightgrey];
313 -> 461 [arrowhead = none , style=dashed];
461 -> 182[style=dashed];

462 [label="INSTANCE with matching:\nT319 -> T412\nT321 -> T414\nT320 -> T413\nT322 -> T415\nT326 -> T419\nT327 -> T420", fontsize=14, style = filled, fillcolor = lightgrey];
326 -> 462 [arrowhead = none , style=dashed];
462 -> 252[style=dashed];

463 [label="INSTANCE with matching:\nT319 -> T437\nT321 -> T439\nT320 -> T438\nT322 -> T440\nT326 -> T444\nT327 -> T445", fontsize=14, style = filled, fillcolor = lightgrey];
330 -> 463 [arrowhead = none , style=dashed];
463 -> 252[style=dashed];

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORTA_IN_GAA(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_GAA(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
MERGESORTA_IN_GAA(.(T39, .(T40, T41)), T45, .(T46, T44)) → PB_IN_GGAAAGAAAA(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)
PB_IN_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
PB_IN_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → SPLITD_IN_GGAAA(T40, T41, T47, T48, T49)
SPLITD_IN_GGAAA(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_GGAAA(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
SPLITD_IN_GGAAA(T60, T61, .(T60, X91), X92, .(T62, T64)) → SPLITC_IN_GAAA(T61, X92, X91, T64)
SPLITC_IN_GAAA(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_GAAA(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
SPLITC_IN_GAAA(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → SPLITC_IN_GAAA(T81, X123, X122, T84)
U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → PN_IN_GGAAGAAA(T39, T48, X35, T49, T47, X36, T50, T51)
PN_IN_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90) → U15_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
PN_IN_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90) → MERGESORTA_IN_GAA(.(T39, T48), T87, T88)
U15_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
U15_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → PO_IN_GAAGAA(T47, X36, T88, T87, T89, T90)
PO_IN_GAAGAA(T47, T93, T96, T87, T94, T95) → U17_GAAGAA(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
PO_IN_GAAGAA(T47, T93, T96, T87, T94, T95) → MERGESORTE_IN_GAA(T47, T93, T96)
MERGESORTE_IN_GAA(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_GAA(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
MERGESORTE_IN_GAA(.(T124, .(T125, T126)), X177, .(T129, T130)) → PF_IN_GGGAAAAAAA(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)
PF_IN_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
PF_IN_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → SPLITD_IN_GGAAA(T124, .(T125, T126), T131, T132, .(T134, T133))
U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → PP_IN_GAAGAAA(T131, X175, T133, T132, X176, X177, T134)
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → MERGESORTE_IN_GAA(T131, T137, T138)
U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → PQ_IN_GAAGAA(T132, X176, T138, T137, X177, T139)
PQ_IN_GAAGAA(T132, T144, T146, T137, X177, T145) → U23_GAAGAA(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
PQ_IN_GAAGAA(T132, T144, T146, T137, X177, T145) → MERGESORTE_IN_GAA(T132, T144, T146)
U23_GAAGAA(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_GAAGAA(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
U23_GAAGAA(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → MERGEG_IN_GGAAA(T137, T144, X177, T145, T146)
MERGEG_IN_GGAAA(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_GGAAA(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
MERGEG_IN_GGAAA(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227)
PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227) → U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227) → LEJ_IN_GG(T219, T221)
LEJ_IN_GG(s(T240), s(T241)) → U7_GG(T240, T241, leJ_in_gg(T240, T241))
LEJ_IN_GG(s(T240), s(T241)) → LEJ_IN_GG(T240, T241)
U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_GGGGAA(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → MERGEL_IN_GGGAA(T220, T221, T222, T226, T227)
MERGEL_IN_GGGAA(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_GGGAA(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
MERGEL_IN_GGGAA(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → PH_IN_GGGGAA(T294, T296, T295, T297, T301, T302)
MERGEL_IN_GGGAA(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_GGGAA(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
MERGEL_IN_GGGAA(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327)
PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327) → U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327) → GTK_IN_GG(T319, T321)
GTK_IN_GG(s(T340), s(T341)) → U8_GG(T340, T341, gtK_in_gg(T340, T341))
GTK_IN_GG(s(T340), s(T341)) → GTK_IN_GG(T340, T341)
U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_GGGGAA(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → MERGEM_IN_GGGAA(T319, T320, T322, T326, T327)
MERGEM_IN_GGGAA(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_GGGAA(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
MERGEM_IN_GGGAA(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → PH_IN_GGGGAA(T387, T389, T388, T390, T394, T395)
MERGEM_IN_GGGAA(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_GGGAA(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
MERGEM_IN_GGGAA(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → PI_IN_GGGGAA(T412, T414, T413, T415, T419, T420)
MERGEG_IN_GGAAA(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_GGAAA(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
MERGEG_IN_GGAAA(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → PI_IN_GGGGAA(T437, T439, T438, T440, T444, T445)
U17_GAAGAA(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_GAAGAA(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U17_GAAGAA(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → MERGEG_IN_GGAAA(T87, T93, T94, T95, T96)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

The argument filtering Pi contains the following mapping:
mergesortA_in_gaa(x1, x2, x3) = mergesortA_in_gaa(x1)
[] = []
mergesortA_out_gaa(x1, x2, x3) = mergesortA_out_gaa(x1, x2)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6, x7) = U1_gaa(x1, x2, x3, x7)
pB_in_ggaaagaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = pB_in_ggaaagaaaa(x1, x2, x6)
U13_ggaaagaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U13_ggaaagaaaa(x1, x2, x6, x11)
splitD_in_ggaaa(x1, x2, x3, x4, x5) = splitD_in_ggaaa(x1, x2)
U3_ggaaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaaa(x1, x2, x7)
splitC_in_gaaa(x1, x2, x3, x4) = splitC_in_gaaa(x1)
splitC_out_gaaa(x1, x2, x3, x4) = splitC_out_gaaa(x1, x2, x3)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7) = U2_gaaa(x1, x2, x7)
splitD_out_ggaaa(x1, x2, x3, x4, x5) = splitD_out_ggaaa(x1, x2, x3, x4)
U14_ggaaagaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U14_ggaaagaaaa(x1, x2, x3, x4, x6, x11)
pN_in_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pN_in_ggaagaaa(x1, x2, x5)
U15_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U15_ggaagaaa(x1, x2, x5, x9)
U16_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_ggaagaaa(x1, x2, x3, x5, x9)
pO_in_gaagaa(x1, x2, x3, x4, x5, x6) = pO_in_gaagaa(x1, x4)
U17_gaagaa(x1, x2, x3, x4, x5, x6, x7) = U17_gaagaa(x1, x4, x7)
mergesortE_in_gaa(x1, x2, x3) = mergesortE_in_gaa(x1)
mergesortE_out_gaa(x1, x2, x3) = mergesortE_out_gaa(x1, x2)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x1, x2, x3, x7)
pF_in_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = pF_in_gggaaaaaaa(x1, x2, x3)
U19_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U19_gggaaaaaaa(x1, x2, x3, x11)
U20_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U20_gggaaaaaaa(x1, x2, x3, x4, x5, x11)
pP_in_gaagaaa(x1, x2, x3, x4, x5, x6, x7) = pP_in_gaagaaa(x1, x4)
U21_gaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U21_gaagaaa(x1, x4, x8)
U22_gaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U22_gaagaaa(x1, x2, x4, x8)
pQ_in_gaagaa(x1, x2, x3, x4, x5, x6) = pQ_in_gaagaa(x1, x4)
U23_gaagaa(x1, x2, x3, x4, x5, x6, x7) = U23_gaagaa(x1, x4, x7)
U24_gaagaa(x1, x2, x3, x4, x5, x6, x7) = U24_gaagaa(x1, x2, x4, x7)
mergeG_in_ggaaa(x1, x2, x3, x4, x5) = mergeG_in_ggaaa(x1, x2)
mergeG_out_ggaaa(x1, x2, x3, x4, x5) = mergeG_out_ggaaa(x1, x2, x3)
U5_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U5_ggaaa(x1, x2, x3, x4, x8)
pH_in_ggggaa(x1, x2, x3, x4, x5, x6) = pH_in_ggggaa(x1, x2, x3, x4)
U25_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U25_ggggaa(x1, x2, x3, x4, x7)
leJ_in_gg(x1, x2) = leJ_in_gg(x1, x2)
s(x1) = s(x1)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
0 = 0
leJ_out_gg(x1, x2) = leJ_out_gg(x1, x2)
U26_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U26_ggggaa(x1, x2, x3, x4, x7)
mergeL_in_gggaa(x1, x2, x3, x4, x5) = mergeL_in_gggaa(x1, x2, x3)
mergeL_out_gggaa(x1, x2, x3, x4, x5) = mergeL_out_gggaa(x1, x2, x3, x4)
U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_gggaa(x1, x2, x3, x4, x8)
pH_out_ggggaa(x1, x2, x3, x4, x5, x6) = pH_out_ggggaa(x1, x2, x3, x4, x5)
U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U10_gggaa(x1, x2, x3, x4, x8)
pI_in_ggggaa(x1, x2, x3, x4, x5, x6) = pI_in_ggggaa(x1, x2, x3, x4)
U27_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U27_ggggaa(x1, x2, x3, x4, x7)
gtK_in_gg(x1, x2) = gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2) = gtK_out_gg(x1, x2)
U28_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U28_ggggaa(x1, x2, x3, x4, x7)
mergeM_in_gggaa(x1, x2, x3, x4, x5) = mergeM_in_gggaa(x1, x2, x3)
mergeM_out_gggaa(x1, x2, x3, x4, x5) = mergeM_out_gggaa(x1, x2, x3, x4)
U11_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U11_gggaa(x1, x2, x3, x4, x8)
U12_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U12_gggaa(x1, x2, x3, x4, x8)
pI_out_ggggaa(x1, x2, x3, x4, x5, x6) = pI_out_ggggaa(x1, x2, x3, x4, x5)
U6_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaaa(x1, x2, x3, x4, x8)
pQ_out_gaagaa(x1, x2, x3, x4, x5, x6) = pQ_out_gaagaa(x1, x2, x4, x5)
pP_out_gaagaaa(x1, x2, x3, x4, x5, x6, x7) = pP_out_gaagaaa(x1, x2, x4, x5, x6)
pF_out_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = pF_out_gggaaaaaaa(x1, x2, x3, x4, x5, x8, x9, x10)
U18_gaagaa(x1, x2, x3, x4, x5, x6, x7) = U18_gaagaa(x1, x2, x4, x7)
pO_out_gaagaa(x1, x2, x3, x4, x5, x6) = pO_out_gaagaa(x1, x2, x4, x5)
pN_out_ggaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = pN_out_ggaagaaa(x1, x2, x3, x5, x6, x7)
pB_out_ggaaagaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = pB_out_ggaaagaaaa(x1, x2, x3, x4, x6, x7, x8, x9)
MERGESORTA_IN_GAA(x1, x2, x3) = MERGESORTA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6, x7) = U1_GAA(x1, x2, x3, x7)
PB_IN_GGAAAGAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = PB_IN_GGAAAGAAAA(x1, x2, x6)
U13_GGAAAGAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U13_GGAAAGAAAA(x1, x2, x6, x11)
SPLITD_IN_GGAAA(x1, x2, x3, x4, x5) = SPLITD_IN_GGAAA(x1, x2)
U3_GGAAA(x1, x2, x3, x4, x5, x6, x7) = U3_GGAAA(x1, x2, x7)
SPLITC_IN_GAAA(x1, x2, x3, x4) = SPLITC_IN_GAAA(x1)
U2_GAAA(x1, x2, x3, x4, x5, x6, x7) = U2_GAAA(x1, x2, x7)
U14_GGAAAGAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U14_GGAAAGAAAA(x1, x2, x3, x4, x6, x11)
PN_IN_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = PN_IN_GGAAGAAA(x1, x2, x5)
U15_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U15_GGAAGAAA(x1, x2, x5, x9)
U16_GGAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U16_GGAAGAAA(x1, x2, x3, x5, x9)
PO_IN_GAAGAA(x1, x2, x3, x4, x5, x6) = PO_IN_GAAGAA(x1, x4)
U17_GAAGAA(x1, x2, x3, x4, x5, x6, x7) = U17_GAAGAA(x1, x4, 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)
PF_IN_GGGAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = PF_IN_GGGAAAAAAA(x1, x2, x3)
U19_GGGAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U19_GGGAAAAAAA(x1, x2, x3, x11)
U20_GGGAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U20_GGGAAAAAAA(x1, x2, x3, x4, x5, x11)
PP_IN_GAAGAAA(x1, x2, x3, x4, x5, x6, x7) = PP_IN_GAAGAAA(x1, x4)
U21_GAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U21_GAAGAAA(x1, x4, x8)
U22_GAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U22_GAAGAAA(x1, x2, x4, x8)
PQ_IN_GAAGAA(x1, x2, x3, x4, x5, x6) = PQ_IN_GAAGAA(x1, x4)
U23_GAAGAA(x1, x2, x3, x4, x5, x6, x7) = U23_GAAGAA(x1, x4, x7)
U24_GAAGAA(x1, x2, x3, x4, x5, x6, x7) = U24_GAAGAA(x1, x2, x4, x7)
MERGEG_IN_GGAAA(x1, x2, x3, x4, x5) = MERGEG_IN_GGAAA(x1, x2)
U5_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U5_GGAAA(x1, x2, x3, x4, x8)
PH_IN_GGGGAA(x1, x2, x3, x4, x5, x6) = PH_IN_GGGGAA(x1, x2, x3, x4)
U25_GGGGAA(x1, x2, x3, x4, x5, x6, x7) = U25_GGGGAA(x1, x2, x3, x4, x7)
LEJ_IN_GG(x1, x2) = LEJ_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3)
U26_GGGGAA(x1, x2, x3, x4, x5, x6, x7) = U26_GGGGAA(x1, x2, x3, x4, x7)
MERGEL_IN_GGGAA(x1, x2, x3, x4, x5) = MERGEL_IN_GGGAA(x1, x2, x3)
U9_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U9_GGGAA(x1, x2, x3, x4, x8)
U10_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U10_GGGAA(x1, x2, x3, x4, x8)
PI_IN_GGGGAA(x1, x2, x3, x4, x5, x6) = PI_IN_GGGGAA(x1, x2, x3, x4)
U27_GGGGAA(x1, x2, x3, x4, x5, x6, x7) = U27_GGGGAA(x1, x2, x3, x4, x7)
GTK_IN_GG(x1, x2) = GTK_IN_GG(x1, x2)
U8_GG(x1, x2, x3) = U8_GG(x1, x2, x3)
U28_GGGGAA(x1, x2, x3, x4, x5, x6, x7) = U28_GGGGAA(x1, x2, x3, x4, x7)
MERGEM_IN_GGGAA(x1, x2, x3, x4, x5) = MERGEM_IN_GGGAA(x1, x2, x3)
U11_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U11_GGGAA(x1, x2, x3, x4, x8)
U12_GGGAA(x1, x2, x3, x4, x5, x6, x7, x8) = U12_GGGAA(x1, x2, x3, x4, x8)
U6_GGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U6_GGAAA(x1, x2, x3, x4, x8)
U18_GAAGAA(x1, x2, x3, x4, x5, x6, x7) = U18_GAAGAA(x1, x2, x4, x7)

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

(4) Obligation:

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

MERGESORTA_IN_GAA(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_GAA(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
MERGESORTA_IN_GAA(.(T39, .(T40, T41)), T45, .(T46, T44)) → PB_IN_GGAAAGAAAA(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)
PB_IN_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
PB_IN_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → SPLITD_IN_GGAAA(T40, T41, T47, T48, T49)
SPLITD_IN_GGAAA(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_GGAAA(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
SPLITD_IN_GGAAA(T60, T61, .(T60, X91), X92, .(T62, T64)) → SPLITC_IN_GAAA(T61, X92, X91, T64)
SPLITC_IN_GAAA(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_GAAA(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
SPLITC_IN_GAAA(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → SPLITC_IN_GAAA(T81, X123, X122, T84)
U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → PN_IN_GGAAGAAA(T39, T48, X35, T49, T47, X36, T50, T51)
PN_IN_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90) → U15_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
PN_IN_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90) → MERGESORTA_IN_GAA(.(T39, T48), T87, T88)
U15_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
U15_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → PO_IN_GAAGAA(T47, X36, T88, T87, T89, T90)
PO_IN_GAAGAA(T47, T93, T96, T87, T94, T95) → U17_GAAGAA(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
PO_IN_GAAGAA(T47, T93, T96, T87, T94, T95) → MERGESORTE_IN_GAA(T47, T93, T96)
MERGESORTE_IN_GAA(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_GAA(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
MERGESORTE_IN_GAA(.(T124, .(T125, T126)), X177, .(T129, T130)) → PF_IN_GGGAAAAAAA(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)
PF_IN_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
PF_IN_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → SPLITD_IN_GGAAA(T124, .(T125, T126), T131, T132, .(T134, T133))
U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → PP_IN_GAAGAAA(T131, X175, T133, T132, X176, X177, T134)
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → MERGESORTE_IN_GAA(T131, T137, T138)
U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → PQ_IN_GAAGAA(T132, X176, T138, T137, X177, T139)
PQ_IN_GAAGAA(T132, T144, T146, T137, X177, T145) → U23_GAAGAA(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
PQ_IN_GAAGAA(T132, T144, T146, T137, X177, T145) → MERGESORTE_IN_GAA(T132, T144, T146)
U23_GAAGAA(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_GAAGAA(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
U23_GAAGAA(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → MERGEG_IN_GGAAA(T137, T144, X177, T145, T146)
MERGEG_IN_GGAAA(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_GGAAA(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
MERGEG_IN_GGAAA(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227)
PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227) → U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227) → LEJ_IN_GG(T219, T221)
LEJ_IN_GG(s(T240), s(T241)) → U7_GG(T240, T241, leJ_in_gg(T240, T241))
LEJ_IN_GG(s(T240), s(T241)) → LEJ_IN_GG(T240, T241)
U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_GGGGAA(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → MERGEL_IN_GGGAA(T220, T221, T222, T226, T227)
MERGEL_IN_GGGAA(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_GGGAA(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
MERGEL_IN_GGGAA(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → PH_IN_GGGGAA(T294, T296, T295, T297, T301, T302)
MERGEL_IN_GGGAA(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_GGGAA(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
MERGEL_IN_GGGAA(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327)
PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327) → U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327) → GTK_IN_GG(T319, T321)
GTK_IN_GG(s(T340), s(T341)) → U8_GG(T340, T341, gtK_in_gg(T340, T341))
GTK_IN_GG(s(T340), s(T341)) → GTK_IN_GG(T340, T341)
U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_GGGGAA(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → MERGEM_IN_GGGAA(T319, T320, T322, T326, T327)
MERGEM_IN_GGGAA(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_GGGAA(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
MERGEM_IN_GGGAA(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → PH_IN_GGGGAA(T387, T389, T388, T390, T394, T395)
MERGEM_IN_GGGAA(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_GGGAA(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
MERGEM_IN_GGGAA(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → PI_IN_GGGGAA(T412, T414, T413, T415, T419, T420)
MERGEG_IN_GGAAA(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_GGAAA(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
MERGEG_IN_GGAAA(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → PI_IN_GGGGAA(T437, T439, T438, T440, T444, T445)
U17_GAAGAA(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_GAAGAA(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U17_GAAGAA(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → MERGEG_IN_GGAAA(T87, T93, T94, T95, T96)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 34 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GTK_IN_GG(s(T340), s(T341)) → GTK_IN_GG(T340, T341)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(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:

GTK_IN_GG(s(T340), s(T341)) → GTK_IN_GG(T340, T341)

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:

GTK_IN_GG(s(T340), s(T341)) → GTK_IN_GG(T340, T341)

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:

GTK_IN_GG(s(T340), s(T341)) → GTK_IN_GG(T340, T341)
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:

LEJ_IN_GG(s(T240), s(T241)) → LEJ_IN_GG(T240, T241)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(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:

LEJ_IN_GG(s(T240), s(T241)) → LEJ_IN_GG(T240, T241)

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:

LEJ_IN_GG(s(T240), s(T241)) → LEJ_IN_GG(T240, T241)

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:

LEJ_IN_GG(s(T240), s(T241)) → LEJ_IN_GG(T240, T241)
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:

U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → MERGEL_IN_GGGAA(T220, T221, T222, T226, T227)
MERGEL_IN_GGGAA(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → PH_IN_GGGGAA(T294, T296, T295, T297, T301, T302)
PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227) → U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
MERGEL_IN_GGGAA(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327)
PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327) → U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → MERGEM_IN_GGGAA(T319, T320, T322, T326, T327)
MERGEM_IN_GGGAA(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → PH_IN_GGGGAA(T387, T389, T388, T390, T394, T395)
MERGEM_IN_GGGAA(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → PI_IN_GGGGAA(T412, T414, T413, T415, T419, T420)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(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:

U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → MERGEL_IN_GGGAA(T220, T221, T222, T226, T227)
MERGEL_IN_GGGAA(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → PH_IN_GGGGAA(T294, T296, T295, T297, T301, T302)
PH_IN_GGGGAA(T219, T221, T220, T222, T226, T227) → U25_GGGGAA(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
MERGEL_IN_GGGAA(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327)
PI_IN_GGGGAA(T319, T321, T320, T322, T326, T327) → U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
U27_GGGGAA(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → MERGEM_IN_GGGAA(T319, T320, T322, T326, T327)
MERGEM_IN_GGGAA(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → PH_IN_GGGGAA(T387, T389, T388, T390, T394, T395)
MERGEM_IN_GGGAA(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → PI_IN_GGGGAA(T412, T414, T413, T415, T419, T420)

The TRS R consists of the following rules:

leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
leJ_in_gg(x1, x2) = leJ_in_gg(x1, x2)
s(x1) = s(x1)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
0 = 0
leJ_out_gg(x1, x2) = leJ_out_gg(x1, x2)
gtK_in_gg(x1, x2) = gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2) = gtK_out_gg(x1, x2)
PH_IN_GGGGAA(x1, x2, x3, x4, x5, x6) = PH_IN_GGGGAA(x1, x2, x3, x4)
U25_GGGGAA(x1, x2, x3, x4, x5, x6, x7) = U25_GGGGAA(x1, x2, x3, x4, x7)
MERGEL_IN_GGGAA(x1, x2, x3, x4, x5) = MERGEL_IN_GGGAA(x1, x2, x3)
PI_IN_GGGGAA(x1, x2, x3, x4, x5, x6) = PI_IN_GGGGAA(x1, x2, x3, x4)
U27_GGGGAA(x1, x2, x3, x4, x5, x6, x7) = U27_GGGGAA(x1, x2, x3, x4, x7)
MERGEM_IN_GGGAA(x1, x2, x3, x4, x5) = MERGEM_IN_GGGAA(x1, x2, x3)

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:

U25_GGGGAA(T219, T221, T220, T222, leJ_out_gg(T219, T221)) → MERGEL_IN_GGGAA(T220, T221, T222)
MERGEL_IN_GGGAA(.(T294, T295), T296, T297) → PH_IN_GGGGAA(T294, T296, T295, T297)
PH_IN_GGGGAA(T219, T221, T220, T222) → U25_GGGGAA(T219, T221, T220, T222, leJ_in_gg(T219, T221))
MERGEL_IN_GGGAA(.(T319, T320), T321, T322) → PI_IN_GGGGAA(T319, T321, T320, T322)
PI_IN_GGGGAA(T319, T321, T320, T322) → U27_GGGGAA(T319, T321, T320, T322, gtK_in_gg(T319, T321))
U27_GGGGAA(T319, T321, T320, T322, gtK_out_gg(T319, T321)) → MERGEM_IN_GGGAA(T319, T320, T322)
MERGEM_IN_GGGAA(T387, T388, .(T389, T390)) → PH_IN_GGGGAA(T387, T389, T388, T390)
MERGEM_IN_GGGAA(T412, T413, .(T414, T415)) → PI_IN_GGGGAA(T412, T414, T413, T415)

The TRS R consists of the following rules:

leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))

The set Q consists of the following terms:

leJ_in_gg(x0, x1)
gtK_in_gg(x0, x1)
U7_gg(x0, x1, x2)
U8_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:

PH_IN_GGGGAA(T219, T221, T220, T222) → U25_GGGGAA(T219, T221, T220, T222, leJ_in_gg(T219, T221))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
U25_GGGGAA(T219, T221, T220, T222, leJ_out_gg(T219, T221)) → MERGEL_IN_GGGAA(T220, T221, T222)
The graph contains the following edges 3 >= 1, 2 >= 2, 5 > 2, 4 >= 3
MERGEL_IN_GGGAA(.(T294, T295), T296, T297) → PH_IN_GGGGAA(T294, T296, T295, T297)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3, 3 >= 4
MERGEL_IN_GGGAA(.(T319, T320), T321, T322) → PI_IN_GGGGAA(T319, T321, T320, T322)
The graph contains the following edges 1 > 1, 2 >= 2, 1 > 3, 3 >= 4
MERGEM_IN_GGGAA(T387, T388, .(T389, T390)) → PH_IN_GGGGAA(T387, T389, T388, T390)
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3, 3 > 4
PI_IN_GGGGAA(T319, T321, T320, T322) → U27_GGGGAA(T319, T321, T320, T322, gtK_in_gg(T319, T321))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4
U27_GGGGAA(T319, T321, T320, T322, gtK_out_gg(T319, T321)) → MERGEM_IN_GGGAA(T319, T320, T322)
The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2, 4 >= 3
MERGEM_IN_GGGAA(T412, T413, .(T414, T415)) → PI_IN_GGGGAA(T412, T414, T413, T415)
The graph contains the following edges 1 >= 1, 3 > 2, 2 >= 3, 3 > 4

(27) YES

(28) Obligation:

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

SPLITC_IN_GAAA(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → SPLITC_IN_GAAA(T81, X123, X122, T84)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(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:

SPLITC_IN_GAAA(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → SPLITC_IN_GAAA(T81, X123, X122, T84)

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

SPLITC_IN_GAAA(.(T80, T81)) → SPLITC_IN_GAAA(T81)

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:

SPLITC_IN_GAAA(.(T80, T81)) → SPLITC_IN_GAAA(T81)
The graph contains the following edges 1 > 1

(34) YES

(35) Obligation:

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

PF_IN_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → PP_IN_GAAGAAA(T131, X175, T133, T132, X176, X177, T134)
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → PQ_IN_GAAGAA(T132, X176, T138, T137, X177, T139)
PQ_IN_GAAGAA(T132, T144, T146, T137, X177, T145) → MERGESORTE_IN_GAA(T132, T144, T146)
MERGESORTE_IN_GAA(.(T124, .(T125, T126)), X177, .(T129, T130)) → PF_IN_GGGAAAAAAA(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → MERGESORTE_IN_GAA(T131, T137, T138)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(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:

PF_IN_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_GGGAAAAAAA(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → PP_IN_GAAGAAA(T131, X175, T133, T132, X176, X177, T134)
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_GAAGAAA(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → PQ_IN_GAAGAA(T132, X176, T138, T137, X177, T139)
PQ_IN_GAAGAA(T132, T144, T146, T137, X177, T145) → MERGESORTE_IN_GAA(T132, T144, T146)
MERGESORTE_IN_GAA(.(T124, .(T125, T126)), X177, .(T129, T130)) → PF_IN_GGGAAAAAAA(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)
PP_IN_GAAGAAA(T131, T137, T138, T132, X176, X177, T139) → MERGESORTE_IN_GAA(T131, T137, T138)

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
splitD_in_ggaaa(x1, x2, x3, x4, x5) = splitD_in_ggaaa(x1, x2)
U3_ggaaa(x1, x2, x3, x4, x5, x6, x7) = U3_ggaaa(x1, x2, x7)
splitC_in_gaaa(x1, x2, x3, x4) = splitC_in_gaaa(x1)
splitC_out_gaaa(x1, x2, x3, x4) = splitC_out_gaaa(x1, x2, x3)
U2_gaaa(x1, x2, x3, x4, x5, x6, x7) = U2_gaaa(x1, x2, x7)
splitD_out_ggaaa(x1, x2, x3, x4, x5) = splitD_out_ggaaa(x1, x2, x3, x4)
mergesortE_in_gaa(x1, x2, x3) = mergesortE_in_gaa(x1)
mergesortE_out_gaa(x1, x2, x3) = mergesortE_out_gaa(x1, x2)
U4_gaa(x1, x2, x3, x4, x5, x6, x7) = U4_gaa(x1, x2, x3, x7)
pF_in_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = pF_in_gggaaaaaaa(x1, x2, x3)
U19_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U19_gggaaaaaaa(x1, x2, x3, x11)
U20_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U20_gggaaaaaaa(x1, x2, x3, x4, x5, x11)
pP_in_gaagaaa(x1, x2, x3, x4, x5, x6, x7) = pP_in_gaagaaa(x1, x4)
U21_gaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U21_gaagaaa(x1, x4, x8)
U22_gaagaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U22_gaagaaa(x1, x2, x4, x8)
pQ_in_gaagaa(x1, x2, x3, x4, x5, x6) = pQ_in_gaagaa(x1, x4)
U23_gaagaa(x1, x2, x3, x4, x5, x6, x7) = U23_gaagaa(x1, x4, x7)
U24_gaagaa(x1, x2, x3, x4, x5, x6, x7) = U24_gaagaa(x1, x2, x4, x7)
mergeG_in_ggaaa(x1, x2, x3, x4, x5) = mergeG_in_ggaaa(x1, x2)
mergeG_out_ggaaa(x1, x2, x3, x4, x5) = mergeG_out_ggaaa(x1, x2, x3)
U5_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U5_ggaaa(x1, x2, x3, x4, x8)
pH_in_ggggaa(x1, x2, x3, x4, x5, x6) = pH_in_ggggaa(x1, x2, x3, x4)
U25_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U25_ggggaa(x1, x2, x3, x4, x7)
leJ_in_gg(x1, x2) = leJ_in_gg(x1, x2)
s(x1) = s(x1)
U7_gg(x1, x2, x3) = U7_gg(x1, x2, x3)
0 = 0
leJ_out_gg(x1, x2) = leJ_out_gg(x1, x2)
U26_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U26_ggggaa(x1, x2, x3, x4, x7)
mergeL_in_gggaa(x1, x2, x3, x4, x5) = mergeL_in_gggaa(x1, x2, x3)
mergeL_out_gggaa(x1, x2, x3, x4, x5) = mergeL_out_gggaa(x1, x2, x3, x4)
U9_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U9_gggaa(x1, x2, x3, x4, x8)
pH_out_ggggaa(x1, x2, x3, x4, x5, x6) = pH_out_ggggaa(x1, x2, x3, x4, x5)
U10_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U10_gggaa(x1, x2, x3, x4, x8)
pI_in_ggggaa(x1, x2, x3, x4, x5, x6) = pI_in_ggggaa(x1, x2, x3, x4)
U27_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U27_ggggaa(x1, x2, x3, x4, x7)
gtK_in_gg(x1, x2) = gtK_in_gg(x1, x2)
U8_gg(x1, x2, x3) = U8_gg(x1, x2, x3)
gtK_out_gg(x1, x2) = gtK_out_gg(x1, x2)
U28_ggggaa(x1, x2, x3, x4, x5, x6, x7) = U28_ggggaa(x1, x2, x3, x4, x7)
mergeM_in_gggaa(x1, x2, x3, x4, x5) = mergeM_in_gggaa(x1, x2, x3)
mergeM_out_gggaa(x1, x2, x3, x4, x5) = mergeM_out_gggaa(x1, x2, x3, x4)
U11_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U11_gggaa(x1, x2, x3, x4, x8)
U12_gggaa(x1, x2, x3, x4, x5, x6, x7, x8) = U12_gggaa(x1, x2, x3, x4, x8)
pI_out_ggggaa(x1, x2, x3, x4, x5, x6) = pI_out_ggggaa(x1, x2, x3, x4, x5)
U6_ggaaa(x1, x2, x3, x4, x5, x6, x7, x8) = U6_ggaaa(x1, x2, x3, x4, x8)
pQ_out_gaagaa(x1, x2, x3, x4, x5, x6) = pQ_out_gaagaa(x1, x2, x4, x5)
pP_out_gaagaaa(x1, x2, x3, x4, x5, x6, x7) = pP_out_gaagaaa(x1, x2, x4, x5, x6)
pF_out_gggaaaaaaa(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = pF_out_gggaaaaaaa(x1, x2, x3, x4, x5, x8, x9, x10)
MERGESORTE_IN_GAA(x1, x2, x3) = MERGESORTE_IN_GAA(x1)
PF_IN_GGGAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = PF_IN_GGGAAAAAAA(x1, x2, x3)
U19_GGGAAAAAAA(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11) = U19_GGGAAAAAAA(x1, x2, x3, x11)
PP_IN_GAAGAAA(x1, x2, x3, x4, x5, x6, x7) = PP_IN_GAAGAAA(x1, x4)
U21_GAAGAAA(x1, x2, x3, x4, x5, x6, x7, x8) = U21_GAAGAAA(x1, x4, x8)
PQ_IN_GAAGAA(x1, x2, x3, x4, x5, x6) = PQ_IN_GAAGAA(x1, x4)

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:

PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, splitD_in_ggaaa(T124, .(T125, T126)))
U19_GGGAAAAAAA(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → PP_IN_GAAGAAA(T131, T132)
PP_IN_GAAGAAA(T131, T132) → U21_GAAGAAA(T131, T132, mergesortE_in_gaa(T131))
U21_GAAGAAA(T131, T132, mergesortE_out_gaa(T131, T137)) → PQ_IN_GAAGAA(T132, T137)
PQ_IN_GAAGAA(T132, T137) → MERGESORTE_IN_GAA(T132)
MERGESORTE_IN_GAA(.(T124, .(T125, T126))) → PF_IN_GGGAAAAAAA(T124, T125, T126)
PP_IN_GAAGAAA(T131, T132) → MERGESORTE_IN_GAA(T131)

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61) → U3_ggaaa(T60, T61, splitC_in_gaaa(T61))
mergesortE_in_gaa([]) → mergesortE_out_gaa([], [])
mergesortE_in_gaa(.(T112, [])) → mergesortE_out_gaa(.(T112, []), .(T112, []))
mergesortE_in_gaa(.(T124, .(T125, T126))) → U4_gaa(T124, T125, T126, pF_in_gggaaaaaaa(T124, T125, T126))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U4_gaa(T124, T125, T126, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177)
splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
pF_in_gggaaaaaaa(T124, T125, T126) → U19_gggaaaaaaa(T124, T125, T126, splitD_in_ggaaa(T124, .(T125, T126)))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)
U19_gggaaaaaaa(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_in_gaagaaa(T131, T132))
U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_out_gaagaaa(T131, X175, T132, X176, X177)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, X175, X176, X177)
pP_in_gaagaaa(T131, T132) → U21_gaagaaa(T131, T132, mergesortE_in_gaa(T131))
U21_gaagaaa(T131, T132, mergesortE_out_gaa(T131, T137)) → U22_gaagaaa(T131, T137, T132, pQ_in_gaagaa(T132, T137))
U22_gaagaaa(T131, T137, T132, pQ_out_gaagaa(T132, X176, T137, X177)) → pP_out_gaagaaa(T131, T137, T132, X176, X177)
pQ_in_gaagaa(T132, T137) → U23_gaagaa(T132, T137, mergesortE_in_gaa(T132))
U23_gaagaa(T132, T137, mergesortE_out_gaa(T132, T144)) → U24_gaagaa(T132, T144, T137, mergeG_in_ggaaa(T137, T144))
U24_gaagaa(T132, T144, T137, mergeG_out_ggaaa(T137, T144, X177)) → pQ_out_gaagaa(T132, T144, T137, X177)
mergeG_in_ggaaa([], T173) → mergeG_out_ggaaa([], T173, T173)
mergeG_in_ggaaa(T188, []) → mergeG_out_ggaaa(T188, [], T188)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222)) → U5_ggaaa(T219, T220, T221, T222, pH_in_ggggaa(T219, T221, T220, T222))
mergeG_in_ggaaa(.(T437, T438), .(T439, T440)) → U6_ggaaa(T437, T438, T439, T440, pI_in_ggggaa(T437, T439, T438, T440))
U5_ggaaa(T219, T220, T221, T222, pH_out_ggggaa(T219, T221, T220, T222, T226)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226))
U6_ggaaa(T437, T438, T439, T440, pI_out_ggggaa(T437, T439, T438, T440, T444)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444))
pH_in_ggggaa(T219, T221, T220, T222) → U25_ggggaa(T219, T221, T220, T222, leJ_in_gg(T219, T221))
pI_in_ggggaa(T319, T321, T320, T322) → U27_ggggaa(T319, T321, T320, T322, gtK_in_gg(T319, T321))
U25_ggggaa(T219, T221, T220, T222, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, mergeL_in_gggaa(T220, T221, T222))
U27_ggggaa(T319, T321, T320, T322, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, mergeM_in_gggaa(T319, T320, T322))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U26_ggggaa(T219, T221, T220, T222, mergeL_out_gggaa(T220, T221, T222, T226)) → pH_out_ggggaa(T219, T221, T220, T222, T226)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U28_ggggaa(T319, T321, T320, T322, mergeM_out_gggaa(T319, T320, T322, T326)) → pI_out_ggggaa(T319, T321, T320, T322, T326)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
mergeL_in_gggaa([], T262, T263) → mergeL_out_gggaa([], T262, T263, .(T262, T263))
mergeL_in_gggaa(.(T294, T295), T296, T297) → U9_gggaa(T294, T295, T296, T297, pH_in_ggggaa(T294, T296, T295, T297))
mergeL_in_gggaa(.(T319, T320), T321, T322) → U10_gggaa(T319, T320, T321, T322, pI_in_ggggaa(T319, T321, T320, T322))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
mergeM_in_gggaa(T356, T357, []) → mergeM_out_gggaa(T356, T357, [], .(T356, T357))
mergeM_in_gggaa(T387, T388, .(T389, T390)) → U11_gggaa(T387, T388, T389, T390, pH_in_ggggaa(T387, T389, T388, T390))
mergeM_in_gggaa(T412, T413, .(T414, T415)) → U12_gggaa(T412, T413, T414, T415, pI_in_ggggaa(T412, T414, T413, T415))
U9_gggaa(T294, T295, T296, T297, pH_out_ggggaa(T294, T296, T295, T297, T301)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301))
U10_gggaa(T319, T320, T321, T322, pI_out_ggggaa(T319, T321, T320, T322, T326)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326))
U11_gggaa(T387, T388, T389, T390, pH_out_ggggaa(T387, T389, T388, T390, T394)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394))
U12_gggaa(T412, T413, T414, T415, pI_out_ggggaa(T412, T414, T413, T415, T419)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419))

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
mergesortE_in_gaa(x0)
U3_ggaaa(x0, x1, x2)
U4_gaa(x0, x1, x2, x3)
splitC_in_gaaa(x0)
pF_in_gggaaaaaaa(x0, x1, x2)
U2_gaaa(x0, x1, x2)
U19_gggaaaaaaa(x0, x1, x2, x3)
U20_gggaaaaaaa(x0, x1, x2, x3, x4, x5)
pP_in_gaagaaa(x0, x1)
U21_gaagaaa(x0, x1, x2)
U22_gaagaaa(x0, x1, x2, x3)
pQ_in_gaagaa(x0, x1)
U23_gaagaa(x0, x1, x2)
U24_gaagaa(x0, x1, x2, x3)
mergeG_in_ggaaa(x0, x1)
U5_ggaaa(x0, x1, x2, x3, x4)
U6_ggaaa(x0, x1, x2, x3, x4)
pH_in_ggggaa(x0, x1, x2, x3)
pI_in_ggggaa(x0, x1, x2, x3)
U25_ggggaa(x0, x1, x2, x3, x4)
U27_ggggaa(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U26_ggggaa(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U28_ggggaa(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
mergeL_in_gggaa(x0, x1, x2)
U8_gg(x0, x1, x2)
mergeM_in_gggaa(x0, x1, x2)
U9_gggaa(x0, x1, x2, x3, x4)
U10_gggaa(x0, x1, x2, x3, x4)
U11_gggaa(x0, x1, x2, x3, x4)
U12_gggaa(x0, x1, x2, x3, x4)

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

(40) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, splitD_in_ggaaa(T124, .(T125, T126))) at position [3] we obtained the following new rules [LPAR04]:

PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, U3_ggaaa(T124, .(T125, T126), splitC_in_gaaa(.(T125, T126))))

(41) Obligation:

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

U19_GGGAAAAAAA(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → PP_IN_GAAGAAA(T131, T132)
PP_IN_GAAGAAA(T131, T132) → U21_GAAGAAA(T131, T132, mergesortE_in_gaa(T131))
U21_GAAGAAA(T131, T132, mergesortE_out_gaa(T131, T137)) → PQ_IN_GAAGAA(T132, T137)
PQ_IN_GAAGAA(T132, T137) → MERGESORTE_IN_GAA(T132)
MERGESORTE_IN_GAA(.(T124, .(T125, T126))) → PF_IN_GGGAAAAAAA(T124, T125, T126)
PP_IN_GAAGAAA(T131, T132) → MERGESORTE_IN_GAA(T131)
PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, U3_ggaaa(T124, .(T125, T126), splitC_in_gaaa(.(T125, T126))))

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61) → U3_ggaaa(T60, T61, splitC_in_gaaa(T61))
mergesortE_in_gaa([]) → mergesortE_out_gaa([], [])
mergesortE_in_gaa(.(T112, [])) → mergesortE_out_gaa(.(T112, []), .(T112, []))
mergesortE_in_gaa(.(T124, .(T125, T126))) → U4_gaa(T124, T125, T126, pF_in_gggaaaaaaa(T124, T125, T126))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U4_gaa(T124, T125, T126, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177)
splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
pF_in_gggaaaaaaa(T124, T125, T126) → U19_gggaaaaaaa(T124, T125, T126, splitD_in_ggaaa(T124, .(T125, T126)))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)
U19_gggaaaaaaa(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_in_gaagaaa(T131, T132))
U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_out_gaagaaa(T131, X175, T132, X176, X177)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, X175, X176, X177)
pP_in_gaagaaa(T131, T132) → U21_gaagaaa(T131, T132, mergesortE_in_gaa(T131))
U21_gaagaaa(T131, T132, mergesortE_out_gaa(T131, T137)) → U22_gaagaaa(T131, T137, T132, pQ_in_gaagaa(T132, T137))
U22_gaagaaa(T131, T137, T132, pQ_out_gaagaa(T132, X176, T137, X177)) → pP_out_gaagaaa(T131, T137, T132, X176, X177)
pQ_in_gaagaa(T132, T137) → U23_gaagaa(T132, T137, mergesortE_in_gaa(T132))
U23_gaagaa(T132, T137, mergesortE_out_gaa(T132, T144)) → U24_gaagaa(T132, T144, T137, mergeG_in_ggaaa(T137, T144))
U24_gaagaa(T132, T144, T137, mergeG_out_ggaaa(T137, T144, X177)) → pQ_out_gaagaa(T132, T144, T137, X177)
mergeG_in_ggaaa([], T173) → mergeG_out_ggaaa([], T173, T173)
mergeG_in_ggaaa(T188, []) → mergeG_out_ggaaa(T188, [], T188)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222)) → U5_ggaaa(T219, T220, T221, T222, pH_in_ggggaa(T219, T221, T220, T222))
mergeG_in_ggaaa(.(T437, T438), .(T439, T440)) → U6_ggaaa(T437, T438, T439, T440, pI_in_ggggaa(T437, T439, T438, T440))
U5_ggaaa(T219, T220, T221, T222, pH_out_ggggaa(T219, T221, T220, T222, T226)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226))
U6_ggaaa(T437, T438, T439, T440, pI_out_ggggaa(T437, T439, T438, T440, T444)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444))
pH_in_ggggaa(T219, T221, T220, T222) → U25_ggggaa(T219, T221, T220, T222, leJ_in_gg(T219, T221))
pI_in_ggggaa(T319, T321, T320, T322) → U27_ggggaa(T319, T321, T320, T322, gtK_in_gg(T319, T321))
U25_ggggaa(T219, T221, T220, T222, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, mergeL_in_gggaa(T220, T221, T222))
U27_ggggaa(T319, T321, T320, T322, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, mergeM_in_gggaa(T319, T320, T322))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U26_ggggaa(T219, T221, T220, T222, mergeL_out_gggaa(T220, T221, T222, T226)) → pH_out_ggggaa(T219, T221, T220, T222, T226)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U28_ggggaa(T319, T321, T320, T322, mergeM_out_gggaa(T319, T320, T322, T326)) → pI_out_ggggaa(T319, T321, T320, T322, T326)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
mergeL_in_gggaa([], T262, T263) → mergeL_out_gggaa([], T262, T263, .(T262, T263))
mergeL_in_gggaa(.(T294, T295), T296, T297) → U9_gggaa(T294, T295, T296, T297, pH_in_ggggaa(T294, T296, T295, T297))
mergeL_in_gggaa(.(T319, T320), T321, T322) → U10_gggaa(T319, T320, T321, T322, pI_in_ggggaa(T319, T321, T320, T322))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
mergeM_in_gggaa(T356, T357, []) → mergeM_out_gggaa(T356, T357, [], .(T356, T357))
mergeM_in_gggaa(T387, T388, .(T389, T390)) → U11_gggaa(T387, T388, T389, T390, pH_in_ggggaa(T387, T389, T388, T390))
mergeM_in_gggaa(T412, T413, .(T414, T415)) → U12_gggaa(T412, T413, T414, T415, pI_in_ggggaa(T412, T414, T413, T415))
U9_gggaa(T294, T295, T296, T297, pH_out_ggggaa(T294, T296, T295, T297, T301)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301))
U10_gggaa(T319, T320, T321, T322, pI_out_ggggaa(T319, T321, T320, T322, T326)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326))
U11_gggaa(T387, T388, T389, T390, pH_out_ggggaa(T387, T389, T388, T390, T394)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394))
U12_gggaa(T412, T413, T414, T415, pI_out_ggggaa(T412, T414, T413, T415, T419)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419))

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
mergesortE_in_gaa(x0)
U3_ggaaa(x0, x1, x2)
U4_gaa(x0, x1, x2, x3)
splitC_in_gaaa(x0)
pF_in_gggaaaaaaa(x0, x1, x2)
U2_gaaa(x0, x1, x2)
U19_gggaaaaaaa(x0, x1, x2, x3)
U20_gggaaaaaaa(x0, x1, x2, x3, x4, x5)
pP_in_gaagaaa(x0, x1)
U21_gaagaaa(x0, x1, x2)
U22_gaagaaa(x0, x1, x2, x3)
pQ_in_gaagaa(x0, x1)
U23_gaagaa(x0, x1, x2)
U24_gaagaa(x0, x1, x2, x3)
mergeG_in_ggaaa(x0, x1)
U5_ggaaa(x0, x1, x2, x3, x4)
U6_ggaaa(x0, x1, x2, x3, x4)
pH_in_ggggaa(x0, x1, x2, x3)
pI_in_ggggaa(x0, x1, x2, x3)
U25_ggggaa(x0, x1, x2, x3, x4)
U27_ggggaa(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U26_ggggaa(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U28_ggggaa(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
mergeL_in_gggaa(x0, x1, x2)
U8_gg(x0, x1, x2)
mergeM_in_gggaa(x0, x1, x2)
U9_gggaa(x0, x1, x2, x3, x4)
U10_gggaa(x0, x1, x2, x3, x4)
U11_gggaa(x0, x1, x2, x3, x4)
U12_gggaa(x0, x1, x2, x3, x4)

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

(42) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, U3_ggaaa(T124, .(T125, T126), splitC_in_gaaa(.(T125, T126)))) at position [3,2] we obtained the following new rules [LPAR04]:

PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, U3_ggaaa(T124, .(T125, T126), U2_gaaa(T125, T126, splitC_in_gaaa(T126))))

(43) Obligation:

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

U19_GGGAAAAAAA(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → PP_IN_GAAGAAA(T131, T132)
PP_IN_GAAGAAA(T131, T132) → U21_GAAGAAA(T131, T132, mergesortE_in_gaa(T131))
U21_GAAGAAA(T131, T132, mergesortE_out_gaa(T131, T137)) → PQ_IN_GAAGAA(T132, T137)
PQ_IN_GAAGAA(T132, T137) → MERGESORTE_IN_GAA(T132)
MERGESORTE_IN_GAA(.(T124, .(T125, T126))) → PF_IN_GGGAAAAAAA(T124, T125, T126)
PP_IN_GAAGAAA(T131, T132) → MERGESORTE_IN_GAA(T131)
PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, U3_ggaaa(T124, .(T125, T126), U2_gaaa(T125, T126, splitC_in_gaaa(T126))))

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61) → U3_ggaaa(T60, T61, splitC_in_gaaa(T61))
mergesortE_in_gaa([]) → mergesortE_out_gaa([], [])
mergesortE_in_gaa(.(T112, [])) → mergesortE_out_gaa(.(T112, []), .(T112, []))
mergesortE_in_gaa(.(T124, .(T125, T126))) → U4_gaa(T124, T125, T126, pF_in_gggaaaaaaa(T124, T125, T126))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U4_gaa(T124, T125, T126, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177)
splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
pF_in_gggaaaaaaa(T124, T125, T126) → U19_gggaaaaaaa(T124, T125, T126, splitD_in_ggaaa(T124, .(T125, T126)))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)
U19_gggaaaaaaa(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_in_gaagaaa(T131, T132))
U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_out_gaagaaa(T131, X175, T132, X176, X177)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, X175, X176, X177)
pP_in_gaagaaa(T131, T132) → U21_gaagaaa(T131, T132, mergesortE_in_gaa(T131))
U21_gaagaaa(T131, T132, mergesortE_out_gaa(T131, T137)) → U22_gaagaaa(T131, T137, T132, pQ_in_gaagaa(T132, T137))
U22_gaagaaa(T131, T137, T132, pQ_out_gaagaa(T132, X176, T137, X177)) → pP_out_gaagaaa(T131, T137, T132, X176, X177)
pQ_in_gaagaa(T132, T137) → U23_gaagaa(T132, T137, mergesortE_in_gaa(T132))
U23_gaagaa(T132, T137, mergesortE_out_gaa(T132, T144)) → U24_gaagaa(T132, T144, T137, mergeG_in_ggaaa(T137, T144))
U24_gaagaa(T132, T144, T137, mergeG_out_ggaaa(T137, T144, X177)) → pQ_out_gaagaa(T132, T144, T137, X177)
mergeG_in_ggaaa([], T173) → mergeG_out_ggaaa([], T173, T173)
mergeG_in_ggaaa(T188, []) → mergeG_out_ggaaa(T188, [], T188)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222)) → U5_ggaaa(T219, T220, T221, T222, pH_in_ggggaa(T219, T221, T220, T222))
mergeG_in_ggaaa(.(T437, T438), .(T439, T440)) → U6_ggaaa(T437, T438, T439, T440, pI_in_ggggaa(T437, T439, T438, T440))
U5_ggaaa(T219, T220, T221, T222, pH_out_ggggaa(T219, T221, T220, T222, T226)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226))
U6_ggaaa(T437, T438, T439, T440, pI_out_ggggaa(T437, T439, T438, T440, T444)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444))
pH_in_ggggaa(T219, T221, T220, T222) → U25_ggggaa(T219, T221, T220, T222, leJ_in_gg(T219, T221))
pI_in_ggggaa(T319, T321, T320, T322) → U27_ggggaa(T319, T321, T320, T322, gtK_in_gg(T319, T321))
U25_ggggaa(T219, T221, T220, T222, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, mergeL_in_gggaa(T220, T221, T222))
U27_ggggaa(T319, T321, T320, T322, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, mergeM_in_gggaa(T319, T320, T322))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U26_ggggaa(T219, T221, T220, T222, mergeL_out_gggaa(T220, T221, T222, T226)) → pH_out_ggggaa(T219, T221, T220, T222, T226)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U28_ggggaa(T319, T321, T320, T322, mergeM_out_gggaa(T319, T320, T322, T326)) → pI_out_ggggaa(T319, T321, T320, T322, T326)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
mergeL_in_gggaa([], T262, T263) → mergeL_out_gggaa([], T262, T263, .(T262, T263))
mergeL_in_gggaa(.(T294, T295), T296, T297) → U9_gggaa(T294, T295, T296, T297, pH_in_ggggaa(T294, T296, T295, T297))
mergeL_in_gggaa(.(T319, T320), T321, T322) → U10_gggaa(T319, T320, T321, T322, pI_in_ggggaa(T319, T321, T320, T322))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
mergeM_in_gggaa(T356, T357, []) → mergeM_out_gggaa(T356, T357, [], .(T356, T357))
mergeM_in_gggaa(T387, T388, .(T389, T390)) → U11_gggaa(T387, T388, T389, T390, pH_in_ggggaa(T387, T389, T388, T390))
mergeM_in_gggaa(T412, T413, .(T414, T415)) → U12_gggaa(T412, T413, T414, T415, pI_in_ggggaa(T412, T414, T413, T415))
U9_gggaa(T294, T295, T296, T297, pH_out_ggggaa(T294, T296, T295, T297, T301)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301))
U10_gggaa(T319, T320, T321, T322, pI_out_ggggaa(T319, T321, T320, T322, T326)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326))
U11_gggaa(T387, T388, T389, T390, pH_out_ggggaa(T387, T389, T388, T390, T394)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394))
U12_gggaa(T412, T413, T414, T415, pI_out_ggggaa(T412, T414, T413, T415, T419)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419))

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
mergesortE_in_gaa(x0)
U3_ggaaa(x0, x1, x2)
U4_gaa(x0, x1, x2, x3)
splitC_in_gaaa(x0)
pF_in_gggaaaaaaa(x0, x1, x2)
U2_gaaa(x0, x1, x2)
U19_gggaaaaaaa(x0, x1, x2, x3)
U20_gggaaaaaaa(x0, x1, x2, x3, x4, x5)
pP_in_gaagaaa(x0, x1)
U21_gaagaaa(x0, x1, x2)
U22_gaagaaa(x0, x1, x2, x3)
pQ_in_gaagaa(x0, x1)
U23_gaagaa(x0, x1, x2)
U24_gaagaa(x0, x1, x2, x3)
mergeG_in_ggaaa(x0, x1)
U5_ggaaa(x0, x1, x2, x3, x4)
U6_ggaaa(x0, x1, x2, x3, x4)
pH_in_ggggaa(x0, x1, x2, x3)
pI_in_ggggaa(x0, x1, x2, x3)
U25_ggggaa(x0, x1, x2, x3, x4)
U27_ggggaa(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U26_ggggaa(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U28_ggggaa(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
mergeL_in_gggaa(x0, x1, x2)
U8_gg(x0, x1, x2)
mergeM_in_gggaa(x0, x1, x2)
U9_gggaa(x0, x1, x2, x3, x4)
U10_gggaa(x0, x1, x2, x3, x4)
U11_gggaa(x0, x1, x2, x3, x4)
U12_gggaa(x0, x1, x2, x3, x4)

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.

MERGESORTE_IN_GAA(.(T124, .(T125, T126))) → PF_IN_GGGAAAAAAA(T124, T125, T126)

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

POL( U21_GAAGAAA(x₁, ..., x₃) ) = x₂

POL( mergesortE_in_gaa(x₁) ) = 0

POL( [] ) = 0

POL( mergesortE_out_gaa(x₁, x₂) ) = 2x₂ + 2

POL( .(x₁, x₂) ) = 2x₂ + 1

POL( U4_gaa(x₁, ..., x₄) ) = max{0, -1}

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

POL( U19_GGGAAAAAAA(x₁, ..., x₄) ) = x₄ + 1

POL( U2_gaaa(x₁, ..., x₃) ) = 2x₃ + 1

POL( U3_ggaaa(x₁, ..., x₃) ) = x₃

POL( U19_gggaaaaaaa(x₁, ..., x₄) ) = max{0, -2}

POL( splitC_in_gaaa(x₁) ) = x₁

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

POL( splitD_out_ggaaa(x₁, ..., x₄) ) = max{0, x₃ + x₄ - 1}

POL( U20_gggaaaaaaa(x₁, ..., x₆) ) = max{0, 2x₁ + x₂ + x₄ + x₅ + 2x₆ - 1}

POL( pP_in_gaagaaa(x₁, x₂) ) = 2x₁

POL( pP_out_gaagaaa(x₁, ..., x₅) ) = 2x₂ + x₄ + x₅ + 2

POL( pF_out_gggaaaaaaa(x₁, ..., x₈) ) = 2x₁ + 2x₂ + 2x₃ + 2x₄ + 2x₅ + 2x₆ + 2x₇ + x₈

POL( U21_gaagaaa(x₁, ..., x₃) ) = 2x₂ + 2

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

POL( U22_gaagaaa(x₁, ..., x₄) ) = x₁ + x₂ + x₃

POL( pQ_in_gaagaa(x₁, x₂) ) = 2x₁ + x₂ + 1

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

POL( U23_gaagaa(x₁, ..., x₃) ) = x₁ + 2x₂ + 2x₃ + 2

POL( U24_gaagaa(x₁, ..., x₄) ) = 2x₁ + x₂ + 2x₃

POL( mergeG_in_ggaaa(x₁, x₂) ) = 0

POL( mergeG_out_ggaaa(x₁, ..., x₃) ) = max{0, x₂ - 2}

POL( U5_ggaaa(x₁, ..., x₅) ) = 2x₁ + x₃ + 2x₅ + 2

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

POL( U6_ggaaa(x₁, ..., x₅) ) = x₁ + 2x₅ + 2

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

POL( U25_ggggaa(x₁, ..., x₅) ) = max{0, x₁ + x₃ + x₄ - 1}

POL( leJ_in_gg(x₁, x₂) ) = 2x₂

POL( pH_out_ggggaa(x₁, ..., x₅) ) = x₂ + 2x₄

POL( U27_ggggaa(x₁, ..., x₅) ) = max{0, -2}

POL( gtK_in_gg(x₁, x₂) ) = 0

POL( pI_out_ggggaa(x₁, ..., x₅) ) = 2x₃ + 2x₄ + x₅

POL( leJ_out_gg(x₁, x₂) ) = 1

POL( U26_ggggaa(x₁, ..., x₅) ) = max{0, 2x₂ + x₄ - 2}

POL( mergeL_in_gggaa(x₁, ..., x₃) ) = x₂

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

POL( U9_gggaa(x₁, ..., x₅) ) = 2x₂ + 2x₃ + x₄ + 2x₅ + 1

POL( s(x₁) ) = 2x₁ + 1

POL( U7_gg(x₁, ..., x₃) ) = max{0, -2}

POL( 0 ) = 0

POL( U10_gggaa(x₁, ..., x₅) ) = max{0, x₄ - 2}

POL( U8_gg(x₁, ..., x₃) ) = x₂ + 2

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

POL( U28_ggggaa(x₁, ..., x₅) ) = x₁ + 2x₂ + x₃ + 2x₄ + x₅ + 2

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

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

POL( U11_gggaa(x₁, ..., x₅) ) = max{0, 2x₁ + x₂ + 2x₄ - 2}

POL( U12_gggaa(x₁, ..., x₅) ) = max{0, x₁ + x₂ + x₃ - 2}

POL( PP_IN_GAAGAAA(x₁, x₂) ) = x₁ + x₂

POL( PQ_IN_GAAGAA(x₁, x₂) ) = x₁

POL( MERGESORTE_IN_GAA(x₁) ) = x₁

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

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

splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)

(45) Obligation:

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

U19_GGGAAAAAAA(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → PP_IN_GAAGAAA(T131, T132)
PP_IN_GAAGAAA(T131, T132) → U21_GAAGAAA(T131, T132, mergesortE_in_gaa(T131))
U21_GAAGAAA(T131, T132, mergesortE_out_gaa(T131, T137)) → PQ_IN_GAAGAA(T132, T137)
PQ_IN_GAAGAA(T132, T137) → MERGESORTE_IN_GAA(T132)
PP_IN_GAAGAAA(T131, T132) → MERGESORTE_IN_GAA(T131)
PF_IN_GGGAAAAAAA(T124, T125, T126) → U19_GGGAAAAAAA(T124, T125, T126, U3_ggaaa(T124, .(T125, T126), U2_gaaa(T125, T126, splitC_in_gaaa(T126))))

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61) → U3_ggaaa(T60, T61, splitC_in_gaaa(T61))
mergesortE_in_gaa([]) → mergesortE_out_gaa([], [])
mergesortE_in_gaa(.(T112, [])) → mergesortE_out_gaa(.(T112, []), .(T112, []))
mergesortE_in_gaa(.(T124, .(T125, T126))) → U4_gaa(T124, T125, T126, pF_in_gggaaaaaaa(T124, T125, T126))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U4_gaa(T124, T125, T126, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177)
splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
pF_in_gggaaaaaaa(T124, T125, T126) → U19_gggaaaaaaa(T124, T125, T126, splitD_in_ggaaa(T124, .(T125, T126)))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)
U19_gggaaaaaaa(T124, T125, T126, splitD_out_ggaaa(T124, .(T125, T126), T131, T132)) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_in_gaagaaa(T131, T132))
U20_gggaaaaaaa(T124, T125, T126, T131, T132, pP_out_gaagaaa(T131, X175, T132, X176, X177)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, X175, X176, X177)
pP_in_gaagaaa(T131, T132) → U21_gaagaaa(T131, T132, mergesortE_in_gaa(T131))
U21_gaagaaa(T131, T132, mergesortE_out_gaa(T131, T137)) → U22_gaagaaa(T131, T137, T132, pQ_in_gaagaa(T132, T137))
U22_gaagaaa(T131, T137, T132, pQ_out_gaagaa(T132, X176, T137, X177)) → pP_out_gaagaaa(T131, T137, T132, X176, X177)
pQ_in_gaagaa(T132, T137) → U23_gaagaa(T132, T137, mergesortE_in_gaa(T132))
U23_gaagaa(T132, T137, mergesortE_out_gaa(T132, T144)) → U24_gaagaa(T132, T144, T137, mergeG_in_ggaaa(T137, T144))
U24_gaagaa(T132, T144, T137, mergeG_out_ggaaa(T137, T144, X177)) → pQ_out_gaagaa(T132, T144, T137, X177)
mergeG_in_ggaaa([], T173) → mergeG_out_ggaaa([], T173, T173)
mergeG_in_ggaaa(T188, []) → mergeG_out_ggaaa(T188, [], T188)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222)) → U5_ggaaa(T219, T220, T221, T222, pH_in_ggggaa(T219, T221, T220, T222))
mergeG_in_ggaaa(.(T437, T438), .(T439, T440)) → U6_ggaaa(T437, T438, T439, T440, pI_in_ggggaa(T437, T439, T438, T440))
U5_ggaaa(T219, T220, T221, T222, pH_out_ggggaa(T219, T221, T220, T222, T226)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226))
U6_ggaaa(T437, T438, T439, T440, pI_out_ggggaa(T437, T439, T438, T440, T444)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444))
pH_in_ggggaa(T219, T221, T220, T222) → U25_ggggaa(T219, T221, T220, T222, leJ_in_gg(T219, T221))
pI_in_ggggaa(T319, T321, T320, T322) → U27_ggggaa(T319, T321, T320, T322, gtK_in_gg(T319, T321))
U25_ggggaa(T219, T221, T220, T222, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, mergeL_in_gggaa(T220, T221, T222))
U27_ggggaa(T319, T321, T320, T322, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, mergeM_in_gggaa(T319, T320, T322))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U26_ggggaa(T219, T221, T220, T222, mergeL_out_gggaa(T220, T221, T222, T226)) → pH_out_ggggaa(T219, T221, T220, T222, T226)
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U28_ggggaa(T319, T321, T320, T322, mergeM_out_gggaa(T319, T320, T322, T326)) → pI_out_ggggaa(T319, T321, T320, T322, T326)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
mergeL_in_gggaa([], T262, T263) → mergeL_out_gggaa([], T262, T263, .(T262, T263))
mergeL_in_gggaa(.(T294, T295), T296, T297) → U9_gggaa(T294, T295, T296, T297, pH_in_ggggaa(T294, T296, T295, T297))
mergeL_in_gggaa(.(T319, T320), T321, T322) → U10_gggaa(T319, T320, T321, T322, pI_in_ggggaa(T319, T321, T320, T322))
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
mergeM_in_gggaa(T356, T357, []) → mergeM_out_gggaa(T356, T357, [], .(T356, T357))
mergeM_in_gggaa(T387, T388, .(T389, T390)) → U11_gggaa(T387, T388, T389, T390, pH_in_ggggaa(T387, T389, T388, T390))
mergeM_in_gggaa(T412, T413, .(T414, T415)) → U12_gggaa(T412, T413, T414, T415, pI_in_ggggaa(T412, T414, T413, T415))
U9_gggaa(T294, T295, T296, T297, pH_out_ggggaa(T294, T296, T295, T297, T301)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301))
U10_gggaa(T319, T320, T321, T322, pI_out_ggggaa(T319, T321, T320, T322, T326)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326))
U11_gggaa(T387, T388, T389, T390, pH_out_ggggaa(T387, T389, T388, T390, T394)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394))
U12_gggaa(T412, T413, T414, T415, pI_out_ggggaa(T412, T414, T413, T415, T419)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419))

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
mergesortE_in_gaa(x0)
U3_ggaaa(x0, x1, x2)
U4_gaa(x0, x1, x2, x3)
splitC_in_gaaa(x0)
pF_in_gggaaaaaaa(x0, x1, x2)
U2_gaaa(x0, x1, x2)
U19_gggaaaaaaa(x0, x1, x2, x3)
U20_gggaaaaaaa(x0, x1, x2, x3, x4, x5)
pP_in_gaagaaa(x0, x1)
U21_gaagaaa(x0, x1, x2)
U22_gaagaaa(x0, x1, x2, x3)
pQ_in_gaagaa(x0, x1)
U23_gaagaa(x0, x1, x2)
U24_gaagaa(x0, x1, x2, x3)
mergeG_in_ggaaa(x0, x1)
U5_ggaaa(x0, x1, x2, x3, x4)
U6_ggaaa(x0, x1, x2, x3, x4)
pH_in_ggggaa(x0, x1, x2, x3)
pI_in_ggggaa(x0, x1, x2, x3)
U25_ggggaa(x0, x1, x2, x3, x4)
U27_ggggaa(x0, x1, x2, x3, x4)
leJ_in_gg(x0, x1)
U26_ggggaa(x0, x1, x2, x3, x4)
gtK_in_gg(x0, x1)
U28_ggggaa(x0, x1, x2, x3, x4)
U7_gg(x0, x1, x2)
mergeL_in_gggaa(x0, x1, x2)
U8_gg(x0, x1, x2)
mergeM_in_gggaa(x0, x1, x2)
U9_gggaa(x0, x1, x2, x3, x4)
U10_gggaa(x0, x1, x2, x3, x4)
U11_gggaa(x0, x1, x2, x3, x4)
U12_gggaa(x0, x1, x2, x3, x4)

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 6 less nodes.

(47) TRUE

(48) Obligation:

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

MERGESORTA_IN_GAA(.(T39, .(T40, T41)), T45, .(T46, T44)) → PB_IN_GGAAAGAAAA(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)
PB_IN_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → PN_IN_GGAAGAAA(T39, T48, X35, T49, T47, X36, T50, T51)
PN_IN_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90) → MERGESORTA_IN_GAA(.(T39, T48), T87, T88)

The TRS R consists of the following rules:

mergesortA_in_gaa([], [], T5) → mergesortA_out_gaa([], [], T5)
mergesortA_in_gaa(.(T8, []), .(T8, []), T9) → mergesortA_out_gaa(.(T8, []), .(T8, []), T9)
mergesortA_in_gaa(.(T39, .(T40, T41)), T45, .(T46, T44)) → U1_gaa(T39, T40, T41, T45, T46, T44, pB_in_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46))
pB_in_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
U13_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_in_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51))
pN_in_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90) → U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_in_gaa(.(T39, T48), T87, T88))
U15_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, mergesortA_out_gaa(.(T39, T48), T87, T88)) → U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_in_gaagaa(T47, X36, T88, T87, T89, T90))
pO_in_gaagaa(T47, T93, T96, T87, T94, T95) → U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_in_gaa(T47, T93, T96))
mergesortE_in_gaa([], [], T103) → mergesortE_out_gaa([], [], T103)
mergesortE_in_gaa(.(T112, []), .(T112, []), T113) → mergesortE_out_gaa(.(T112, []), .(T112, []), T113)
mergesortE_in_gaa(.(T124, .(T125, T126)), X177, .(T129, T130)) → U4_gaa(T124, T125, T126, X177, T129, T130, pF_in_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177))
pF_in_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177) → U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_in_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133)))
U19_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, splitD_out_ggaaa(T124, .(T125, T126), T131, T132, .(T134, T133))) → U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_in_gaagaaa(T131, X175, T133, T132, X176, X177, T134))
pP_in_gaagaaa(T131, T137, T138, T132, X176, X177, T139) → U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_in_gaa(T131, T137, T138))
U21_gaagaaa(T131, T137, T138, T132, X176, X177, T139, mergesortE_out_gaa(T131, T137, T138)) → U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_in_gaagaa(T132, X176, T138, T137, X177, T139))
pQ_in_gaagaa(T132, T144, T146, T137, X177, T145) → U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_in_gaa(T132, T144, T146))
U23_gaagaa(T132, T144, T146, T137, X177, T145, mergesortE_out_gaa(T132, T144, T146)) → U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_in_ggaaa(T137, T144, X177, T145, T146))
mergeG_in_ggaaa([], T173, T173, T174, T175) → mergeG_out_ggaaa([], T173, T173, T174, T175)
mergeG_in_ggaaa(T188, [], T188, T189, T190) → mergeG_out_ggaaa(T188, [], T188, T189, T190)
mergeG_in_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227) → U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_in_ggggaa(T219, T221, T220, T222, T226, T227))
pH_in_ggggaa(T219, T221, T220, T222, T226, T227) → U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_in_gg(T219, T221))
leJ_in_gg(s(T240), s(T241)) → U7_gg(T240, T241, leJ_in_gg(T240, T241))
leJ_in_gg(0, s(0)) → leJ_out_gg(0, s(0))
leJ_in_gg(0, 0) → leJ_out_gg(0, 0)
U7_gg(T240, T241, leJ_out_gg(T240, T241)) → leJ_out_gg(s(T240), s(T241))
U25_ggggaa(T219, T221, T220, T222, T226, T227, leJ_out_gg(T219, T221)) → U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_in_gggaa(T220, T221, T222, T226, T227))
mergeL_in_gggaa([], T262, T263, .(T262, T263), T264) → mergeL_out_gggaa([], T262, T263, .(T262, T263), T264)
mergeL_in_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302)) → U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_in_ggggaa(T294, T296, T295, T297, T301, T302))
U9_gggaa(T294, T295, T296, T297, T301, T299, T302, pH_out_ggggaa(T294, T296, T295, T297, T301, T302)) → mergeL_out_gggaa(.(T294, T295), T296, T297, .(T294, T301), .(T299, T302))
mergeL_in_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327)) → U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_in_ggggaa(T319, T321, T320, T322, T326, T327))
pI_in_ggggaa(T319, T321, T320, T322, T326, T327) → U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_in_gg(T319, T321))
gtK_in_gg(s(T340), s(T341)) → U8_gg(T340, T341, gtK_in_gg(T340, T341))
gtK_in_gg(s(0), 0) → gtK_out_gg(s(0), 0)
U8_gg(T340, T341, gtK_out_gg(T340, T341)) → gtK_out_gg(s(T340), s(T341))
U27_ggggaa(T319, T321, T320, T322, T326, T327, gtK_out_gg(T319, T321)) → U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_in_gggaa(T319, T320, T322, T326, T327))
mergeM_in_gggaa(T356, T357, [], .(T356, T357), T358) → mergeM_out_gggaa(T356, T357, [], .(T356, T357), T358)
mergeM_in_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395)) → U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_in_ggggaa(T387, T389, T388, T390, T394, T395))
U11_gggaa(T387, T388, T389, T390, T394, T392, T395, pH_out_ggggaa(T387, T389, T388, T390, T394, T395)) → mergeM_out_gggaa(T387, T388, .(T389, T390), .(T387, T394), .(T392, T395))
mergeM_in_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420)) → U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_in_ggggaa(T412, T414, T413, T415, T419, T420))
U12_gggaa(T412, T413, T414, T415, T419, T417, T420, pI_out_ggggaa(T412, T414, T413, T415, T419, T420)) → mergeM_out_gggaa(T412, T413, .(T414, T415), .(T414, T419), .(T417, T420))
U28_ggggaa(T319, T321, T320, T322, T326, T327, mergeM_out_gggaa(T319, T320, T322, T326, T327)) → pI_out_ggggaa(T319, T321, T320, T322, T326, T327)
U10_gggaa(T319, T320, T321, T322, T326, T324, T327, pI_out_ggggaa(T319, T321, T320, T322, T326, T327)) → mergeL_out_gggaa(.(T319, T320), T321, T322, .(T321, T326), .(T324, T327))
U26_ggggaa(T219, T221, T220, T222, T226, T227, mergeL_out_gggaa(T220, T221, T222, T226, T227)) → pH_out_ggggaa(T219, T221, T220, T222, T226, T227)
U5_ggaaa(T219, T220, T221, T222, T226, T224, T227, pH_out_ggggaa(T219, T221, T220, T222, T226, T227)) → mergeG_out_ggaaa(.(T219, T220), .(T221, T222), .(T219, T226), T224, T227)
mergeG_in_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445) → U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_in_ggggaa(T437, T439, T438, T440, T444, T445))
U6_ggaaa(T437, T438, T439, T440, T444, T442, T445, pI_out_ggggaa(T437, T439, T438, T440, T444, T445)) → mergeG_out_ggaaa(.(T437, T438), .(T439, T440), .(T439, T444), T442, T445)
U24_gaagaa(T132, T144, T146, T137, X177, T145, mergeG_out_ggaaa(T137, T144, X177, T145, T146)) → pQ_out_gaagaa(T132, T144, T146, T137, X177, T145)
U22_gaagaaa(T131, T137, T138, T132, X176, X177, T139, pQ_out_gaagaa(T132, X176, T138, T137, X177, T139)) → pP_out_gaagaaa(T131, T137, T138, T132, X176, X177, T139)
U20_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177, pP_out_gaagaaa(T131, X175, T133, T132, X176, X177, T134)) → pF_out_gggaaaaaaa(T124, T125, T126, T131, T132, T134, T133, X175, X176, X177)
U4_gaa(T124, T125, T126, X177, T129, T130, pF_out_gggaaaaaaa(T124, T125, T126, X173, X174, T129, T130, X175, X176, X177)) → mergesortE_out_gaa(.(T124, .(T125, T126)), X177, .(T129, T130))
U17_gaagaa(T47, T93, T96, T87, T94, T95, mergesortE_out_gaa(T47, T93, T96)) → U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_in_ggaaa(T87, T93, T94, T95, T96))
U18_gaagaa(T47, T93, T96, T87, T94, T95, mergeG_out_ggaaa(T87, T93, T94, T95, T96)) → pO_out_gaagaa(T47, T93, T96, T87, T94, T95)
U16_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90, pO_out_gaagaa(T47, X36, T88, T87, T89, T90)) → pN_out_ggaagaaa(T39, T48, T87, T88, T47, X36, T89, T90)
U14_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, pN_out_ggaagaaa(T39, T48, X35, T49, T47, X36, T50, T51)) → pB_out_ggaaagaaaa(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51)
U1_gaa(T39, T40, T41, T45, T46, T44, pB_out_ggaaagaaaa(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)) → mergesortA_out_gaa(.(T39, .(T40, T41)), T45, .(T46, T44))

(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:

MERGESORTA_IN_GAA(.(T39, .(T40, T41)), T45, .(T46, T44)) → PB_IN_GGAAAGAAAA(T40, T41, X63, X62, T44, T39, X35, X36, T45, T46)
PB_IN_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51) → U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_in_ggaaa(T40, T41, T47, T48, T49))
U13_GGAAAGAAAA(T40, T41, T47, T48, T49, T39, X35, X36, T50, T51, splitD_out_ggaaa(T40, T41, T47, T48, T49)) → PN_IN_GGAAGAAA(T39, T48, X35, T49, T47, X36, T50, T51)
PN_IN_GGAAGAAA(T39, T48, T87, T88, T47, X36, T89, T90) → MERGESORTA_IN_GAA(.(T39, T48), T87, T88)

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64)) → U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_in_gaaa(T61, X92, X91, T64))
U3_ggaaa(T60, T61, X91, X92, T62, T64, splitC_out_gaaa(T61, X92, X91, T64)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92, .(T62, T64))
splitC_in_gaaa([], [], [], T71) → splitC_out_gaaa([], [], [], T71)
splitC_in_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84)) → U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_in_gaaa(T81, X123, X122, T84))
U2_gaaa(T80, T81, X122, X123, T82, T84, splitC_out_gaaa(T81, X123, X122, T84)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123, .(T82, T84))

(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:

MERGESORTA_IN_GAA(.(T39, .(T40, T41))) → PB_IN_GGAAAGAAAA(T40, T41, T39)
PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, splitD_in_ggaaa(T40, T41))
U13_GGAAAGAAAA(T40, T41, T39, splitD_out_ggaaa(T40, T41, T47, T48)) → PN_IN_GGAAGAAA(T39, T48, T47)
PN_IN_GGAAGAAA(T39, T48, T47) → MERGESORTA_IN_GAA(.(T39, T48))

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61) → U3_ggaaa(T60, T61, splitC_in_gaaa(T61))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
U3_ggaaa(x0, x1, x2)
splitC_in_gaaa(x0)
U2_gaaa(x0, x1, x2)

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

(53) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, splitD_in_ggaaa(T40, T41)) at position [3] we obtained the following new rules [LPAR04]:

PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, U3_ggaaa(T40, T41, splitC_in_gaaa(T41)))

(54) Obligation:

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

MERGESORTA_IN_GAA(.(T39, .(T40, T41))) → PB_IN_GGAAAGAAAA(T40, T41, T39)
U13_GGAAAGAAAA(T40, T41, T39, splitD_out_ggaaa(T40, T41, T47, T48)) → PN_IN_GGAAGAAA(T39, T48, T47)
PN_IN_GGAAGAAA(T39, T48, T47) → MERGESORTA_IN_GAA(.(T39, T48))
PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, U3_ggaaa(T40, T41, splitC_in_gaaa(T41)))

The TRS R consists of the following rules:

splitD_in_ggaaa(T60, T61) → U3_ggaaa(T60, T61, splitC_in_gaaa(T61))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
U3_ggaaa(x0, x1, x2)
splitC_in_gaaa(x0)
U2_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:

MERGESORTA_IN_GAA(.(T39, .(T40, T41))) → PB_IN_GGAAAGAAAA(T40, T41, T39)
U13_GGAAAGAAAA(T40, T41, T39, splitD_out_ggaaa(T40, T41, T47, T48)) → PN_IN_GGAAGAAA(T39, T48, T47)
PN_IN_GGAAGAAA(T39, T48, T47) → MERGESORTA_IN_GAA(.(T39, T48))
PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, U3_ggaaa(T40, T41, splitC_in_gaaa(T41)))

The TRS R consists of the following rules:

splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)

The set Q consists of the following terms:

splitD_in_ggaaa(x0, x1)
U3_ggaaa(x0, x1, x2)
splitC_in_gaaa(x0)
U2_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].

splitD_in_ggaaa(x0, x1)

(58) Obligation:

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

MERGESORTA_IN_GAA(.(T39, .(T40, T41))) → PB_IN_GGAAAGAAAA(T40, T41, T39)
U13_GGAAAGAAAA(T40, T41, T39, splitD_out_ggaaa(T40, T41, T47, T48)) → PN_IN_GGAAGAAA(T39, T48, T47)
PN_IN_GGAAGAAA(T39, T48, T47) → MERGESORTA_IN_GAA(.(T39, T48))
PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, U3_ggaaa(T40, T41, splitC_in_gaaa(T41)))

The TRS R consists of the following rules:

splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)

The set Q consists of the following terms:

U3_ggaaa(x0, x1, x2)
splitC_in_gaaa(x0)
U2_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.

MERGESORTA_IN_GAA(.(T39, .(T40, T41))) → PB_IN_GGAAAGAAAA(T40, T41, T39)

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

POL(.(x₁, x₂)) = 1 + x₂
POL(MERGESORTA_IN_GAA(x₁)) = x₁
POL(PB_IN_GGAAAGAAAA(x₁, x₂, x₃)) = 1 + x₂
POL(PN_IN_GGAAGAAA(x₁, x₂, x₃)) = 1 + x₂
POL(U13_GGAAAGAAAA(x₁, x₂, x₃, x₄)) = x₄
POL(U2_gaaa(x₁, x₂, x₃)) = 1 + x₃
POL(U3_ggaaa(x₁, x₂, x₃)) = x₃
POL([]) = 0
POL(splitC_in_gaaa(x₁)) = 1 + x₁
POL(splitC_out_gaaa(x₁, x₂, x₃)) = 1 + x₂ + x₃
POL(splitD_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:

splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)

(60) Obligation:

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

U13_GGAAAGAAAA(T40, T41, T39, splitD_out_ggaaa(T40, T41, T47, T48)) → PN_IN_GGAAGAAA(T39, T48, T47)
PN_IN_GGAAGAAA(T39, T48, T47) → MERGESORTA_IN_GAA(.(T39, T48))
PB_IN_GGAAAGAAAA(T40, T41, T39) → U13_GGAAAGAAAA(T40, T41, T39, U3_ggaaa(T40, T41, splitC_in_gaaa(T41)))

The TRS R consists of the following rules:

splitC_in_gaaa([]) → splitC_out_gaaa([], [], [])
splitC_in_gaaa(.(T80, T81)) → U2_gaaa(T80, T81, splitC_in_gaaa(T81))
U3_ggaaa(T60, T61, splitC_out_gaaa(T61, X92, X91)) → splitD_out_ggaaa(T60, T61, .(T60, X91), X92)
U2_gaaa(T80, T81, splitC_out_gaaa(T81, X123, X122)) → splitC_out_gaaa(.(T80, T81), .(T80, X122), X123)

The set Q consists of the following terms:

U3_ggaaa(x0, x1, x2)
splitC_in_gaaa(x0)
U2_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 3 less nodes.

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (SOUND transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES

(35) Obligation:

(36) UsableRulesProof (EQUIVALENT transformation)

(37) Obligation:

(38) PiDPToQDPProof (SOUND transformation)

(39) Obligation:

(40) Rewriting (EQUIVALENT transformation)

(41) Obligation:

(42) Rewriting (EQUIVALENT transformation)

(43) Obligation:

(44) QDPOrderProof (EQUIVALENT transformation)

(45) Obligation:

(46) DependencyGraphProof (EQUIVALENT transformation)

(47) TRUE

(48) Obligation:

(49) UsableRulesProof (EQUIVALENT transformation)

(50) Obligation:

(51) PiDPToQDPProof (SOUND transformation)

(52) Obligation:

(53) Rewriting (EQUIVALENT transformation)

(54) Obligation:

(55) UsableRulesProof (EQUIVALENT transformation)

(56) Obligation:

(57) QReductionProof (EQUIVALENT transformation)

(58) Obligation:

(59) QDPOrderProof (EQUIVALENT transformation)

(60) Obligation:

(61) DependencyGraphProof (EQUIVALENT transformation)

(62) TRUE