Left Termination proof of /tmp/tmpVbTrcT/mergesort.pl

Left Termination of the query pattern mergesort_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 Narrowing (⇐)

                ↳29 QDP

                ↳30 Narrowing (⇐)

                ↳31 QDP

                ↳32 NonTerminationProof (⇔)

                ↳33 NO

            ↳34 PiDP

                ↳35 UsableRulesProof (⇔)

                ↳36 PiDP

                ↳37 PiDPToQDPProof (⇐)

                ↳38 QDP

                ↳39 QDPSizeChangeProof (⇔)

                ↳40 YES

            ↳41 PiDP

                ↳42 UsableRulesProof (⇔)

                ↳43 PiDP

                ↳44 PiDPToQDPProof (⇐)

                ↳45 QDP

                ↳46 Rewriting (⇔)

                ↳47 QDP

                ↳48 Rewriting (⇔)

                ↳49 QDP

                ↳50 QDPOrderProof (⇔)

                ↳51 QDP

                ↳52 DependencyGraphProof (⇔)

                ↳53 TRUE

            ↳54 PiDP

                ↳55 UsableRulesProof (⇔)

                ↳56 PiDP

                ↳57 PiDPToQDPProof (⇐)

                ↳58 QDP

                ↳59 Rewriting (⇔)

                ↳60 QDP

                ↳61 UsableRulesProof (⇔)

                ↳62 QDP

                ↳63 QReductionProof (⇔)

                ↳64 QDP

                ↳65 QDPOrderProof (⇔)

                ↳66 QDP

                ↳67 DependencyGraphProof (⇔)

                ↳68 TRUE

    ↳69 PrologToPiTRSProof (⇐)

        ↳70 PiTRS

        ↳71 DependencyPairsProof (⇔)

        ↳72 PiDP

        ↳73 DependencyGraphProof (⇔)

        ↳74 AND

            ↳75 PiDP

                ↳76 UsableRulesProof (⇔)

                ↳77 PiDP

                ↳78 PiDPToQDPProof (⇐)

                ↳79 QDP

                ↳80 NonTerminationProof (⇔)

                ↳81 NO

            ↳82 PiDP

                ↳83 UsableRulesProof (⇔)

                ↳84 PiDP

                ↳85 PiDPToQDPProof (⇐)

                ↳86 QDP

                ↳87 NonTerminationProof (⇔)

                ↳88 NO

            ↳89 PiDP

                ↳90 UsableRulesProof (⇔)

                ↳91 PiDP

                ↳92 PiDPToQDPProof (⇐)

                ↳93 QDP

                ↳94 Narrowing (⇐)

                ↳95 QDP

                ↳96 Narrowing (⇐)

                ↳97 QDP

                ↳98 NonTerminationProof (⇔)

                ↳99 NO

            ↳100 PiDP

                ↳101 UsableRulesProof (⇔)

                ↳102 PiDP

                ↳103 PiDPToQDPProof (⇐)

                ↳104 QDP

                ↳105 QDPSizeChangeProof (⇔)

                ↳106 YES

            ↳107 PiDP

                ↳108 UsableRulesProof (⇔)

                ↳109 PiDP

                ↳110 PiDPToQDPProof (⇐)

                ↳111 QDP

                ↳112 MRRProof (⇔)

                ↳113 QDP

                ↳114 MRRProof (⇔)

                ↳115 QDP

                ↳116 DependencyGraphProof (⇔)

                ↳117 TRUE

            ↳118 PiDP

                ↳119 UsableRulesProof (⇔)

                ↳120 PiDP

                ↳121 PiDPToQDPProof (⇐)

                ↳122 QDP

                ↳123 MRRProof (⇔)

                ↳124 QDP

                ↳125 DependencyGraphProof (⇔)

                ↳126 TRUE

(0) Obligation:

Clauses:

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

Queries:

mergesort(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: mergesort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mergesort(T1, T2), SCOPE: 1, CLAUSE: 0 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])\nmergesort(T1, T2) !=? mergesort(.(X7, []), .(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(split(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14))))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(split(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 3 | ','(split(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 4\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 4\n\nT10 is ground\nT11 is ground\nT12 is ground\nX21 is free; \nX22 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(split(.(T24, T25), X52, X51), ','(mergesort(.(T23, X51), X23), ','(mergesort(X52, X24), merge(X23, X24, T14))))\n\nT23 is ground\nT24 is ground\nT25 is ground\nX23 is free; \nX24 is free; \nX51 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: split(.(T24, T25), X52, X51)\n\nT24 is ground\nT25 is ground\nX51 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(mergesort(.(T23, T29), X23), ','(mergesort(T28, X24), merge(X23, X24, T14)))\n\nT23 is ground\nT28 is ground\nT29 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: split(.(T24, T25), X52, X51), SCOPE: 3, CLAUSE: 3 | split(.(T24, T25), X52, X51), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nT25 is ground\nX51 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: split(.(T24, T25), X52, X51), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nT25 is ground\nX51 is free; \nX52 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: split(T37, X82, X81)\n\nT37 is ground\nX81 is free; \nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: split(T37, X82, X81), SCOPE: 4, CLAUSE: 3 | split(T37, X82, X81), SCOPE: 4, CLAUSE: 4\n\nT37 is ground\nX81 is free; \nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: split(T37, X82, X81), SCOPE: 4, CLAUSE: 3\n\nT37 is ground\nX81 is free; \nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: split(T37, X82, X81), SCOPE: 4, CLAUSE: 4\n\nT37 is ground\nX81 is free; \nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: split(T43, X100, X99)\n\nT43 is ground\nX99 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: mergesort(.(T23, T29), X23)\n\nT23 is ground\nT29 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(mergesort(T28, X24), merge(T46, X24, T14))\n\nT28 is ground\nT46 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: mergesort(T28, X24)\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: merge(T46, T49, T14)\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: mergesort(T28, X24), SCOPE: 5, CLAUSE: 0 | mergesort(T28, X24), SCOPE: 5, CLAUSE: 1 | mergesort(T28, X24), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: mergesort(T28, X24), SCOPE: 5, CLAUSE: 0\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: mergesort(T28, X24), SCOPE: 5, CLAUSE: 1 | mergesort(T28, X24), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: mergesort(T28, X24), SCOPE: 5, CLAUSE: 1\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: mergesort(T28, X24), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(split(.(T61, .(T62, T63)), X133, X134), ','(mergesort(X133, X135), ','(mergesort(X134, X136), merge(X135, X136, X137))))\n\nT61 is ground\nT62 is ground\nT63 is ground\nX137 is free; \nX133 is free; \nX134 is free; \nX135 is free; \nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: split(.(T61, .(T62, T63)), X133, X134)\n\nT61 is ground\nT62 is ground\nT63 is ground\nX133 is free; \nX134 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(mergesort(T67, X135), ','(mergesort(T68, X136), merge(X135, X136, X137)))\n\nT67 is ground\nT68 is ground\nX137 is free; \nX135 is free; \nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: mergesort(T67, X135)\n\nT67 is ground\nX135 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(mergesort(T68, X136), merge(T72, X136, X137))\n\nT68 is ground\nT72 is ground\nX137 is free; \nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: mergesort(T68, X136)\n\nT68 is ground\nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: merge(T72, T73, X137)\n\nT72 is ground\nT73 is ground\nX137 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 5 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 7 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 8\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 5\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 7 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 8\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 7 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 8\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 7\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 8\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(le(T108, T110), merge(T109, .(T110, T111), T113))\n\nT108 is ground\nT109 is ground\nT110 is ground\nT111 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: le(T108, T110)\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: merge(T109, .(T110, T111), T113)\n\nT109 is ground\nT110 is ground\nT111 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: le(T108, T110), SCOPE: 7, CLAUSE: 11 | le(T108, T110), SCOPE: 7, CLAUSE: 12 | le(T108, T110), SCOPE: 7, CLAUSE: 13\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: le(T108, T110), SCOPE: 7, CLAUSE: 11\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: le(T108, T110), SCOPE: 7, CLAUSE: 12 | le(T108, T110), SCOPE: 7, CLAUSE: 13\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: le(T126, T127)\n\nT126 is ground\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: le(T108, T110), SCOPE: 7, CLAUSE: 12\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: le(T108, T110), SCOPE: 7, CLAUSE: 13\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(gt(T149, T151), merge(.(T149, T150), T152, T154))\n\nT149 is ground\nT150 is ground\nT151 is ground\nT152 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: gt(T149, T151)\n\nT149 is ground\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: merge(.(T149, T150), T152, T154)\n\nT149 is ground\nT150 is ground\nT152 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: gt(T149, T151), SCOPE: 8, CLAUSE: 9 | gt(T149, T151), SCOPE: 8, CLAUSE: 10\n\nT149 is ground\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: gt(T149, T151), SCOPE: 8, CLAUSE: 9\n\nT149 is ground\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: gt(T149, T151), SCOPE: 8, CLAUSE: 10\n\nT149 is ground\nT151 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: gt(T167, T168)\n\nT167 is ground\nT168 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 95 [arrowhead = none ];
95 -> 2;

96 [label="EVAL with clause\nmergesort([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 96 [arrowhead = none ];
96 -> 3;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 97 [arrowhead = none ];
97 -> 4;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 98 [arrowhead = none ];
98 -> 5;

99 [label="EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitution\nX7 -> T4\nT1 -> .(T4, [])\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 99 [arrowhead = none ];
99 -> 8;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 100 [arrowhead = none ];
100 -> 9;

101 [label="BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 101 [arrowhead = none ];
101 -> 6;

102 [label="BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 102 [arrowhead = none ];
102 -> 7;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 103 [arrowhead = none ];
103 -> 10;

104 [label="EVAL with clause\nmergesort(.(X17, .(X18, X19)), X20) :- ','(split(.(X17, .(X18, X19)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, X20)))).\nand substitution\nX17 -> T10\nX18 -> T11\nX19 -> T12\nT1 -> .(T10, .(T11, T12))\nT2 -> T14\nX20 -> T14\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 104 [arrowhead = none ];
104 -> 12;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 105 [arrowhead = none ];
105 -> 13;

106 [label="BACKTRACK\nfor clause: mergesort(.(X, .(Y, Xs)), Ys) :- ','(split(.(X, .(Y, Xs)), X1s, X2s), ','(mergesort(X1s, Y1s), ','(mergesort(X2s, Y2s), merge(Y1s, Y2s, Ys))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 106 [arrowhead = none ];
106 -> 11;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 107 [arrowhead = none ];
107 -> 14;

108 [label="BACKTRACK\nfor clause: split([], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 108 [arrowhead = none ];
108 -> 15;

109 [label="ONLY EVAL with clause\nsplit(.(X47, X48), .(X47, X49), X50) :- split(X48, X50, X49).\nand substitution\nT10 -> T23\nX47 -> T23\nT11 -> T24\nT12 -> T25\nX48 -> .(T24, T25)\nX49 -> X51\nX21 -> .(T23, X51)\nX22 -> X52\nX50 -> X52", fontsize=14, style = filled, fillcolor = white];
15 -> 109 [arrowhead = none ];
109 -> 16;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 110 [arrowhead = none ];
110 -> 17;

111 [label="SPLIT 2\nnew knowledge:\nT28 is ground\nT29 is ground\nreplacements:\nX52 -> T28\nX51 -> T29", fontsize=14, style = filled, fillcolor = violet];
16 -> 111 [arrowhead = none ];
111 -> 18;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 112 [arrowhead = none ];
112 -> 19;

113 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 113 [arrowhead = none ];
113 -> 30;

114 [label="SPLIT 2\nnew knowledge:\nT46 is ground\nreplacements:\nX23 -> T46", fontsize=14, style = filled, fillcolor = violet];
18 -> 114 [arrowhead = none ];
114 -> 31;

115 [label="BACKTRACK\nfor clause: split([], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 115 [arrowhead = none ];
115 -> 20;

116 [label="ONLY EVAL with clause\nsplit(.(X77, X78), .(X77, X79), X80) :- split(X78, X80, X79).\nand substitution\nT24 -> T36\nX77 -> T36\nT25 -> T37\nX78 -> T37\nX79 -> X81\nX52 -> .(T36, X81)\nX51 -> X82\nX80 -> X82", fontsize=14, style = filled, fillcolor = white];
20 -> 116 [arrowhead = none ];
116 -> 21;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 117 [arrowhead = none ];
117 -> 22;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 118 [arrowhead = none ];
118 -> 23;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 119 [arrowhead = none ];
119 -> 24;

120 [label="EVAL with clause\nsplit([], [], []).\nand substitution\nT37 -> []\nX82 -> []\nX81 -> []", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 120 [arrowhead = none ];
120 -> 25;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 121 [arrowhead = none ];
121 -> 26;

122 [label="EVAL with clause\nsplit(.(X95, X96), .(X95, X97), X98) :- split(X96, X98, X97).\nand substitution\nX95 -> T42\nX96 -> T43\nT37 -> .(T42, T43)\nX97 -> X99\nX82 -> .(T42, X99)\nX81 -> X100\nX98 -> X100", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 122 [arrowhead = none ];
122 -> 28;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 123 [arrowhead = none ];
123 -> 29;

124 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 124 [arrowhead = none ];
124 -> 27;

125 [label="INSTANCE with matching:\nT37 -> T43\nX82 -> X100\nX81 -> X99", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 125 [arrowhead = none , style=dashed];
125 -> 21[style=dashed];

126 [label="INSTANCE with matching:\nT1 -> .(T23, T29)\nT2 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 126 [arrowhead = none , style=dashed];
126 -> 1[style=dashed];

127 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 127 [arrowhead = none ];
127 -> 32;

128 [label="SPLIT 2\nnew knowledge:\nT49 is ground\nreplacements:\nX24 -> T49", fontsize=14, style = filled, fillcolor = violet];
31 -> 128 [arrowhead = none ];
128 -> 33;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 129 [arrowhead = none ];
129 -> 34;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 130 [arrowhead = none ];
130 -> 53;

131 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 131 [arrowhead = none ];
131 -> 35;

132 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 132 [arrowhead = none ];
132 -> 36;

133 [label="EVAL with clause\nmergesort([], []).\nand substitution\nT28 -> []\nX24 -> []", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 133 [arrowhead = none ];
133 -> 37;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 134 [arrowhead = none ];
134 -> 38;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 135 [arrowhead = none ];
135 -> 40;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 136 [arrowhead = none ];
136 -> 41;

137 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 137 [arrowhead = none ];
137 -> 39;

138 [label="EVAL with clause\nmergesort(.(X113, []), .(X113, [])).\nand substitution\nX113 -> T54\nT28 -> .(T54, [])\nX24 -> .(T54, [])", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 138 [arrowhead = none ];
138 -> 42;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 139 [arrowhead = none ];
139 -> 43;

140 [label="EVAL with clause\nmergesort(.(X129, .(X130, X131)), X132) :- ','(split(.(X129, .(X130, X131)), X133, X134), ','(mergesort(X133, X135), ','(mergesort(X134, X136), merge(X135, X136, X132)))).\nand substitution\nX129 -> T61\nX130 -> T62\nX131 -> T63\nT28 -> .(T61, .(T62, T63))\nX24 -> X137\nX132 -> X137", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 140 [arrowhead = none ];
140 -> 45;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 141 [arrowhead = none ];
141 -> 46;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 142 [arrowhead = none ];
142 -> 44;

143 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 143 [arrowhead = none ];
143 -> 47;

144 [label="SPLIT 2\nnew knowledge:\nT67 is ground\nT68 is ground\nreplacements:\nX133 -> T67\nX134 -> T68", fontsize=14, style = filled, fillcolor = violet];
45 -> 144 [arrowhead = none ];
144 -> 48;

145 [label="INSTANCE with matching:\nT24 -> T61\nT25 -> .(T62, T63)\nX52 -> X133\nX51 -> X134", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 145 [arrowhead = none , style=dashed];
145 -> 17[style=dashed];

146 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
48 -> 146 [arrowhead = none ];
146 -> 49;

147 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nreplacements:\nX135 -> T72", fontsize=14, style = filled, fillcolor = violet];
48 -> 147 [arrowhead = none ];
147 -> 50;

148 [label="INSTANCE with matching:\nT28 -> T67\nX24 -> X135", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 148 [arrowhead = none , style=dashed];
148 -> 32[style=dashed];

149 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 149 [arrowhead = none ];
149 -> 51;

150 [label="SPLIT 2\nnew knowledge:\nT73 is ground\nreplacements:\nX136 -> T73", fontsize=14, style = filled, fillcolor = violet];
50 -> 150 [arrowhead = none ];
150 -> 52;

151 [label="INSTANCE with matching:\nT28 -> T68\nX24 -> X136", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 151 [arrowhead = none , style=dashed];
151 -> 32[style=dashed];

152 [label="INSTANCE with matching:\nT46 -> T72\nT49 -> T73\nT14 -> X137", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 152 [arrowhead = none , style=dashed];
152 -> 33[style=dashed];

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 153 [arrowhead = none ];
153 -> 54;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 154 [arrowhead = none ];
154 -> 55;

155 [label="EVAL with clause\nmerge([], X162, X162).\nand substitution\nT46 -> []\nT49 -> T82\nX162 -> T82\nT14 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 155 [arrowhead = none ];
155 -> 56;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 156 [arrowhead = none ];
156 -> 57;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 157 [arrowhead = none ];
157 -> 59;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 158 [arrowhead = none ];
158 -> 60;

159 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 159 [arrowhead = none ];
159 -> 58;

160 [label="EVAL with clause\nmerge(X168, [], X168).\nand substitution\nT46 -> T87\nX168 -> T87\nT49 -> []\nT14 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 160 [arrowhead = none ];
160 -> 61;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 161 [arrowhead = none ];
161 -> 62;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 162 [arrowhead = none ];
162 -> 64;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 163 [arrowhead = none ];
163 -> 65;

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 164 [arrowhead = none ];
164 -> 63;

165 [label="EVAL with clause\nmerge(.(X190, X191), .(X192, X193), .(X190, X194)) :- ','(le(X190, X192), merge(X191, .(X192, X193), X194)).\nand substitution\nX190 -> T108\nX191 -> T109\nT46 -> .(T108, T109)\nX192 -> T110\nX193 -> T111\nT49 -> .(T110, T111)\nX194 -> T113\nT14 -> .(T108, T113)\nT112 -> T113", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 165 [arrowhead = none ];
165 -> 66;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 166 [arrowhead = none ];
166 -> 67;

167 [label="EVAL with clause\nmerge(.(X236, X237), .(X238, X239), .(X238, X240)) :- ','(gt(X236, X238), merge(.(X236, X237), X239, X240)).\nand substitution\nX236 -> T149\nX237 -> T150\nT46 -> .(T149, T150)\nX238 -> T151\nX239 -> T152\nT49 -> .(T151, T152)\nX240 -> T154\nT14 -> .(T151, T154)\nT153 -> T154", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 167 [arrowhead = none ];
167 -> 83;

168 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 168 [arrowhead = none ];
168 -> 84;

169 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
66 -> 169 [arrowhead = none ];
169 -> 68;

170 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
66 -> 170 [arrowhead = none ];
170 -> 69;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
68 -> 171 [arrowhead = none ];
171 -> 70;

172 [label="INSTANCE with matching:\nT46 -> T109\nT49 -> .(T110, T111)\nT14 -> T113", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 172 [arrowhead = none , style=dashed];
172 -> 33[style=dashed];

173 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 173 [arrowhead = none ];
173 -> 71;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
70 -> 174 [arrowhead = none ];
174 -> 72;

175 [label="EVAL with clause\nle(s(X212), s(X213)) :- le(X212, X213).\nand substitution\nX212 -> T126\nT108 -> s(T126)\nX213 -> T127\nT110 -> s(T127)", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 175 [arrowhead = none ];
175 -> 73;

176 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
71 -> 176 [arrowhead = none ];
176 -> 74;

177 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 177 [arrowhead = none ];
177 -> 75;

178 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 178 [arrowhead = none ];
178 -> 76;

179 [label="INSTANCE with matching:\nT108 -> T126\nT110 -> T127", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 179 [arrowhead = none , style=dashed];
179 -> 68[style=dashed];

180 [label="EVAL with clause\nle(0, s(X222)).\nand substitution\nT108 -> 0\nX222 -> T134\nT110 -> s(T134)", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 180 [arrowhead = none ];
180 -> 77;

181 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 181 [arrowhead = none ];
181 -> 78;

182 [label="EVAL with clause\nle(0, 0).\nand substitution\nT108 -> 0\nT110 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 182 [arrowhead = none ];
182 -> 80;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 183 [arrowhead = none ];
183 -> 81;

184 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 184 [arrowhead = none ];
184 -> 79;

185 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 185 [arrowhead = none ];
185 -> 82;

186 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
83 -> 186 [arrowhead = none ];
186 -> 85;

187 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
83 -> 187 [arrowhead = none ];
187 -> 86;

188 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 188 [arrowhead = none ];
188 -> 87;

189 [label="INSTANCE with matching:\nT46 -> .(T149, T150)\nT49 -> T152\nT14 -> T154", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 189 [arrowhead = none , style=dashed];
189 -> 33[style=dashed];

190 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 190 [arrowhead = none ];
190 -> 88;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
87 -> 191 [arrowhead = none ];
191 -> 89;

192 [label="EVAL with clause\ngt(s(X258), s(X259)) :- gt(X258, X259).\nand substitution\nX258 -> T167\nT149 -> s(T167)\nX259 -> T168\nT151 -> s(T168)", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 192 [arrowhead = none ];
192 -> 90;

193 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 193 [arrowhead = none ];
193 -> 91;

194 [label="EVAL with clause\ngt(s(X266), 0).\nand substitution\nX266 -> T173\nT149 -> s(T173)\nT151 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 194 [arrowhead = none ];
194 -> 92;

195 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 195 [arrowhead = none ];
195 -> 93;

196 [label="INSTANCE with matching:\nT149 -> T167\nT151 -> T168", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 196 [arrowhead = none , style=dashed];
196 -> 85[style=dashed];

197 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 197 [arrowhead = none ];
197 -> 94;

}

(2) Obligation:

Clauses:

split21([], [], []).
split21(.(T42, T43), .(T42, X99), X100) :- split21(T43, X100, X99).
split17(T36, T37, .(T36, X81), X82) :- split21(T37, X82, X81).
mergesort32([], []).
mergesort32(.(T54, []), .(T54, [])).
mergesort32(.(T61, .(T62, T63)), X137) :- split17(T61, .(T62, T63), X133, X134).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(split17(T61, .(T62, T63), T67, T68), mergesort32(T67, X135)).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, T72), mergesort32(T68, X136))).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(split17(T61, .(T62, T63), T67, T68), ','(mergesort32(T67, T72), ','(mergesort32(T68, T73), merge33(T72, T73, X137)))).
merge33([], T82, T82).
merge33(T87, [], T87).
merge33(.(T108, T109), .(T110, T111), .(T108, T113)) :- le68(T108, T110).
merge33(.(T108, T109), .(T110, T111), .(T108, T113)) :- ','(le68(T108, T110), merge33(T109, .(T110, T111), T113)).
merge33(.(T149, T150), .(T151, T152), .(T151, T154)) :- gt85(T149, T151).
merge33(.(T149, T150), .(T151, T152), .(T151, T154)) :- ','(gt85(T149, T151), merge33(.(T149, T150), T152, T154)).
le68(s(T126), s(T127)) :- le68(T126, T127).
le68(0, s(T134)).
le68(0, 0).
gt85(s(T167), s(T168)) :- gt85(T167, T168).
gt85(s(T173), 0).
mergesort1([], []).
mergesort1(.(T4, []), .(T4, [])).
mergesort1(.(T23, .(T24, T25)), T14) :- split17(T24, T25, X52, X51).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(split17(T24, T25, T28, T29), mergesort1(.(T23, T29), X23)).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(split17(T24, T25, T28, T29), ','(mergesort1(.(T23, T29), T46), mergesort32(T28, X24))).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(split17(T24, T25, T28, T29), ','(mergesort1(.(T23, T29), T46), ','(mergesort32(T28, T49), merge33(T46, T49, T14)))).

Queries:

mergesort1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split17_in: (b,b,f,f)
split21_in: (b,f,f)
mergesort32_in: (b,f)
merge33_in: (f,f,f)
le68_in: (f,f)
gt85_in: (f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(5) DependencyPairsProof (EQUIVALENT transformation)

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → GT85_IN_AA(T149, T151)
GT85_IN_AA(s(T167), s(T168)) → U15_AA(T167, T168, gt85_in_aa(T167, T168))
GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_AAA(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2) = mergesort1_in_ga(x1)
[] = []
mergesort1_out_ga(x1, x2) = mergesort1_out_ga(x1)
.(x1, x2) = .(x1, x2)
U16_ga(x1, x2, x3, x4, x5) = U16_ga(x1, x2, x3, x5)
split17_in_ggaa(x1, x2, x3, x4) = split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5)
split21_in_gaa(x1, x2, x3) = split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3) = split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4) = split17_out_ggaa(x1, x2, x3, x4)
U17_ga(x1, x2, x3, x4, x5) = U17_ga(x1, x2, x3, x5)
U18_ga(x1, x2, x3, x4, x5) = U18_ga(x1, x2, x3, x5)
U19_ga(x1, x2, x3, x4, x5, x6) = U19_ga(x1, x2, x3, x5, x6)
U20_ga(x1, x2, x3, x4, x5) = U20_ga(x1, x2, x3, x5)
mergesort32_in_ga(x1, x2) = mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2) = mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6) = U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6) = U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3) = merge33_in_aaa
merge33_out_aaa(x1, x2, x3) = merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6)
le68_in_aa(x1, x2) = le68_in_aa
U14_aa(x1, x2, x3) = U14_aa(x3)
le68_out_aa(x1, x2) = le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6) = U12_aaa(x6)
gt85_in_aa(x1, x2) = gt85_in_aa
U15_aa(x1, x2, x3) = U15_aa(x3)
gt85_out_aa(x1, x2) = gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6) = U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6) = U21_ga(x1, x2, x3, x6)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x1, x2, x3, x5)
s(x1) = s(x1)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U16_GA(x1, x2, x3, x4, x5) = U16_GA(x1, x2, x3, x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4) = SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x2, x5)
U17_GA(x1, x2, x3, x4, x5) = U17_GA(x1, x2, x3, x5)
U18_GA(x1, x2, x3, x4, x5) = U18_GA(x1, x2, x3, x5)
U19_GA(x1, x2, x3, x4, x5, x6) = U19_GA(x1, x2, x3, x5, x6)
U20_GA(x1, x2, x3, x4, x5) = U20_GA(x1, x2, x3, x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5, x6) = U8_GA(x1, x2, x3, x6)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x3, x5)
MERGE33_IN_AAA(x1, x2, x3) = MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6)
LE68_IN_AA(x1, x2) = LE68_IN_AA
U14_AA(x1, x2, x3) = U14_AA(x3)
U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6)
GT85_IN_AA(x1, x2) = GT85_IN_AA
U15_AA(x1, x2, x3) = U15_AA(x3)
U13_AAA(x1, x2, x3, x4, x5, x6) = U13_AAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6) = U21_GA(x1, x2, x3, x6)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x3, x5)

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

(6) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → GT85_IN_AA(T149, T151)
GT85_IN_AA(s(T167), s(T168)) → U15_AA(T167, T168, gt85_in_aa(T167, T168))
GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_AAA(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 25 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
GT85_IN_AA(x1, x2) = GT85_IN_AA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

GT85_IN_AA → GT85_IN_AA

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = GT85_IN_AA evaluates to t =GT85_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT85_IN_AA to GT85_IN_AA.

(15) NO

(16) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE68_IN_AA(x1, x2) = LE68_IN_AA

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LE68_IN_AA → LE68_IN_AA

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

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = LE68_IN_AA evaluates to t =LE68_IN_AA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE68_IN_AA to LE68_IN_AA.

(22) NO

(23) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)

The TRS R consists of the following rules:

le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
le68_in_aa(x1, x2) = le68_in_aa
U14_aa(x1, x2, x3) = U14_aa(x3)
le68_out_aa(x1, x2) = le68_out_aa(x1)
gt85_in_aa(x1, x2) = gt85_in_aa
U15_aa(x1, x2, x3) = U15_aa(x3)
gt85_out_aa(x1, x2) = gt85_out_aa(x2)
s(x1) = s(x1)
MERGE33_IN_AAA(x1, x2, x3) = MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U10_AAA(le68_in_aa)
MERGE33_IN_AAA → U12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T151)) → MERGE33_IN_AAA

The TRS R consists of the following rules:

le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(28) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAA → U10_AAA(le68_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAA → U10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))

(29) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T151)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))

The TRS R consists of the following rules:

le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(30) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAA → U12_AAA(gt85_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAA → U12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAA → U12_AAA(gt85_out_aa(0))

(31) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
U12_AAA(gt85_out_aa(T151)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))
MERGE33_IN_AAA → U12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAA → U12_AAA(gt85_out_aa(0))

The TRS R consists of the following rules:

le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(32) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MERGE33_IN_AAA evaluates to t =MERGE33_IN_AAA

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))
with rule MERGE33_IN_AAA → U10_AAA(le68_out_aa(0)) at position [] and matcher [ ]

U10_AAA(le68_out_aa(0)) → MERGE33_IN_AAA
with rule U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(33) NO

(34) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43)) → SPLIT21_IN_GAA(T43)

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

(39) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

SPLIT21_IN_GAA(.(T42, T43)) → SPLIT21_IN_GAA(T43)
The graph contains the following edges 1 > 1

(40) YES

(41) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(42) UsableRulesProof (EQUIVALENT transformation)

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

(43) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4) = split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x2, x5)
split21_in_gaa(x1, x2, x3) = split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3) = split21_out_gaa(x1, x2, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x2, x5)
split17_out_ggaa(x1, x2, x3, x4) = split17_out_ggaa(x1, x2, x3, x4)
mergesort32_in_ga(x1, x2) = mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2) = mergesort32_out_ga(x1)
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x1, x2, x3, x5)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x1, x2, x3, x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x1, x2, x3, x5)
U6_ga(x1, x2, x3, x4, x5, x6) = U6_ga(x1, x2, x3, x5, x6)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5, x6) = U8_ga(x1, x2, x3, x6)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x1, x2, x3, x5)
merge33_in_aaa(x1, x2, x3) = merge33_in_aaa
merge33_out_aaa(x1, x2, x3) = merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6)
le68_in_aa(x1, x2) = le68_in_aa
U14_aa(x1, x2, x3) = U14_aa(x3)
le68_out_aa(x1, x2) = le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6) = U12_aaa(x6)
gt85_in_aa(x1, x2) = gt85_in_aa
U15_aa(x1, x2, x3) = U15_aa(x3)
gt85_out_aa(x1, x2) = gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6) = U13_aaa(x6)
s(x1) = s(x1)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6)

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

(44) PiDPToQDPProof (SOUND transformation)

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

(45) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(46) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63))) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), split21_in_gaa(.(T62, T63))))

(47) Obligation:

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

U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), split21_in_gaa(.(T62, T63))))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(48) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), split21_in_gaa(.(T62, T63)))) at position [3,2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), U1_gaa(T62, T63, split21_in_gaa(T63))))

(49) Obligation:

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

U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), U1_gaa(T62, T63, split21_in_gaa(T63))))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U2_ggaa(T61, .(T62, T63), U1_gaa(T62, T63, split21_in_gaa(T63))))

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

POL(U4_GA(x₁, x₂, x₃, x₄)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 1, 0 ]

· x₄

POL(split17_out_ggaa(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 1 \

\ 0 0 /

· x₄

POL(.(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 1 0 /

· x₂

POL(MERGESORT32_IN_GA(x₁)) = 0 +
[ 0, 1 ]

· x₁

POL(U6_GA(x₁, x₂, x₃, x₄, x₅)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 0 ]

· x₂ +
[ 0, 0 ]

· x₃ +
[ 0, 1 ]

· x₄ +
[ 0, 0 ]

· x₅

POL(mergesort32_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(mergesort32_out_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(U2_ggaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃

POL(U1_gaa(x₁, x₂, x₃)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 1 1 \

\ 1 0 /

· x₃

POL(split21_in_gaa(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 0 /

· x₁

POL([]) =
/ 0 \

\ 0 /

POL(U3_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 1 0 /

· x₃ +
/ 1 0 \

\ 1 0 /

· x₄

POL(split17_in_ggaa(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂

POL(U4_ga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 1 1 /

· x₂ +
/ 0 1 \

\ 1 1 /

· x₃ +
/ 1 1 \

\ 0 1 /

· x₄

POL(split21_out_gaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 1 /

· x₂ +
/ 1 0 \

\ 1 1 /

· x₃

POL(U5_ga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂ +
/ 1 1 \

\ 1 1 /

· x₃ +
/ 1 1 \

\ 1 1 /

· x₄

POL(U6_ga(x₁, x₂, x₃, x₄, x₅)) =
/ 0 \

\ 1 /

+
/ 1 1 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 1 1 \

\ 1 1 /

· x₄ +
/ 1 1 \

\ 1 0 /

· x₅

POL(U7_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂ +
/ 1 1 \

\ 1 0 /

· x₃ +
/ 1 1 \

\ 1 0 /

· x₄

POL(U8_ga(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 1 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 1 1 /

· x₄

POL(U9_ga(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 1 1 /

· x₁ +
/ 1 1 \

\ 0 0 /

· x₂ +
/ 1 1 \

\ 1 0 /

· x₃ +
/ 0 0 \

\ 1 1 /

· x₄

POL(merge33_in_aaa) =
/ 1 \

\ 1 /

POL(merge33_out_aaa) =
/ 0 \

\ 0 /

POL(U10_aaa(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(le68_in_aa) =
/ 0 \

\ 0 /

POL(U12_aaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(gt85_in_aa) =
/ 0 \

\ 0 /

POL(U14_aa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(le68_out_aa(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 0 /

· x₁

POL(0) =
/ 0 \

\ 1 /

POL(U11_aaa(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(U15_aa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(gt85_out_aa(x₁)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(U13_aaa(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 0 /

· x₁

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)

(51) Obligation:

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

U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesort32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesort32_out_ga(T67)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga([])
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga(.(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(T61, T62, T63, split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, mergesort32_in_ga(T67))
U4_ga(T61, T62, T63, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U5_ga(T61, T62, T63, mergesort32_out_ga(T67)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U7_ga(T61, T62, T63, mergesort32_in_ga(T68))
U6_ga(T61, T62, T63, T68, mergesort32_out_ga(T67)) → U8_ga(T61, T62, T63, mergesort32_in_ga(T68))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, mergesort32_out_ga(T68)) → mergesort32_out_ga(.(T61, .(T62, T63)))
U8_ga(T61, T62, T63, mergesort32_out_ga(T68)) → U9_ga(T61, T62, T63, merge33_in_aaa)
U9_ga(T61, T62, T63, merge33_out_aaa) → mergesort32_out_ga(.(T61, .(T62, T63)))
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1, x2)
U3_ga(x0, x1, x2, x3)
U4_ga(x0, x1, x2, x3)
split21_in_gaa(x0)
U5_ga(x0, x1, x2, x3)
U6_ga(x0, x1, x2, x3, x4)
U1_gaa(x0, x1, x2)
U7_ga(x0, x1, x2, x3)
U8_ga(x0, x1, x2, x3)
U9_ga(x0, x1, x2, x3)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(52) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(53) TRUE

(54) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(55) UsableRulesProof (EQUIVALENT transformation)

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

(56) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

(57) PiDPToQDPProof (SOUND transformation)

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

(58) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, split17_in_ggaa(T24, T25))
U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(59) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, split17_in_ggaa(T24, T25)) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

(60) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, T37, split21_in_gaa(T37))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(61) 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.

(62) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(63) 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].

split17_in_ggaa(x0, x1)

(64) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, T24, T25, U2_ggaa(T24, T25, split21_in_gaa(T25)))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(MERGESORT1_IN_GA(x₁)) = x₁
POL(U17_GA(x₁, x₂, x₃, x₄)) = x₄
POL(U1_gaa(x₁, x₂, x₃)) = 1 + x₃
POL(U2_ggaa(x₁, x₂, x₃)) = 1 + x₃
POL([]) = 0
POL(split17_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(split21_in_gaa(x₁)) = x₁
POL(split21_out_gaa(x₁, x₂, x₃)) = x₂ + x₃

The following usable rules [FROCOS05] were oriented:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

(66) Obligation:

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

U17_GA(T23, T24, T25, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split21_in_gaa([]) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, T43, split21_in_gaa(T43))
U2_ggaa(T36, T37, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U1_gaa(T42, T43, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

U2_ggaa(x0, x1, x2)
split21_in_gaa(x0)
U1_gaa(x0, x1, x2)

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

(67) DependencyGraphProof (EQUIVALENT transformation)

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

(68) TRUE

(69) PrologToPiTRSProof (SOUND transformation)

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(70) Obligation:

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

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(71) 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:

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → GT85_IN_AA(T149, T151)
GT85_IN_AA(s(T167), s(T168)) → U15_AA(T167, T168, gt85_in_aa(T167, T168))
GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_AAA(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2) = mergesort1_in_ga(x1)
[] = []
mergesort1_out_ga(x1, x2) = mergesort1_out_ga
.(x1, x2) = .(x1, x2)
U16_ga(x1, x2, x3, x4, x5) = U16_ga(x5)
split17_in_ggaa(x1, x2, x3, x4) = split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x5)
split21_in_gaa(x1, x2, x3) = split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3) = split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5)
split17_out_ggaa(x1, x2, x3, x4) = split17_out_ggaa(x3, x4)
U17_ga(x1, x2, x3, x4, x5) = U17_ga(x1, x5)
U18_ga(x1, x2, x3, x4, x5) = U18_ga(x5)
U19_ga(x1, x2, x3, x4, x5, x6) = U19_ga(x5, x6)
U20_ga(x1, x2, x3, x4, x5) = U20_ga(x5)
mergesort32_in_ga(x1, x2) = mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2) = mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6) = U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6) = U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x5)
merge33_in_aaa(x1, x2, x3) = merge33_in_aaa
merge33_out_aaa(x1, x2, x3) = merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6)
le68_in_aa(x1, x2) = le68_in_aa
U14_aa(x1, x2, x3) = U14_aa(x3)
le68_out_aa(x1, x2) = le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6) = U12_aaa(x6)
gt85_in_aa(x1, x2) = gt85_in_aa
U15_aa(x1, x2, x3) = U15_aa(x3)
gt85_out_aa(x1, x2) = gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6) = U13_aaa(x6)
U21_ga(x1, x2, x3, x4, x5, x6) = U21_ga(x6)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x5)
s(x1) = s(x1)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U16_GA(x1, x2, x3, x4, x5) = U16_GA(x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4) = SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x5)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x5)
U17_GA(x1, x2, x3, x4, x5) = U17_GA(x1, x5)
U18_GA(x1, x2, x3, x4, x5) = U18_GA(x5)
U19_GA(x1, x2, x3, x4, x5, x6) = U19_GA(x5, x6)
U20_GA(x1, x2, x3, x4, x5) = U20_GA(x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x5)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x5, x6)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x5)
U8_GA(x1, x2, x3, x4, x5, x6) = U8_GA(x6)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x5)
MERGE33_IN_AAA(x1, x2, x3) = MERGE33_IN_AAA
U10_AAA(x1, x2, x3, x4, x5, x6) = U10_AAA(x6)
LE68_IN_AA(x1, x2) = LE68_IN_AA
U14_AA(x1, x2, x3) = U14_AA(x3)
U11_AAA(x1, x2, x3, x4, x5, x6) = U11_AAA(x6)
U12_AAA(x1, x2, x3, x4, x5, x6) = U12_AAA(x6)
GT85_IN_AA(x1, x2) = GT85_IN_AA
U15_AA(x1, x2, x3) = U15_AA(x3)
U13_AAA(x1, x2, x3, x4, x5, x6) = U13_AAA(x6)
U21_GA(x1, x2, x3, x4, x5, x6) = U21_GA(x6)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x5)

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

(72) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U16_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_GA(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → MERGESORT32_IN_GA(T28, X24)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U3_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → MERGE33_IN_AAA(T72, T73, X137)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_AA(T108, T110)
LE68_IN_AA(s(T126), s(T127)) → U14_AA(T126, T127, le68_in_aa(T126, T127))
LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_AAA(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → GT85_IN_AA(T149, T151)
GT85_IN_AA(s(T167), s(T168)) → U15_AA(T167, T168, gt85_in_aa(T167, T168))
GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_AAA(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)
U19_GA(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_GA(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_GA(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U21_GA(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → MERGE33_IN_AAA(T46, T49, T14)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(73) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 25 less nodes.

(74) Complex Obligation (AND)

(75) Obligation:

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

GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(76) UsableRulesProof (EQUIVALENT transformation)

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

(77) Obligation:

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

GT85_IN_AA(s(T167), s(T168)) → GT85_IN_AA(T167, T168)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
GT85_IN_AA(x1, x2) = GT85_IN_AA

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

(78) PiDPToQDPProof (SOUND transformation)

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

(79) Obligation:

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

GT85_IN_AA → GT85_IN_AA

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

(80) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from GT85_IN_AA to GT85_IN_AA.

(81) NO

(82) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(83) UsableRulesProof (EQUIVALENT transformation)

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

(84) Obligation:

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

LE68_IN_AA(s(T126), s(T127)) → LE68_IN_AA(T126, T127)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE68_IN_AA(x1, x2) = LE68_IN_AA

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

(85) PiDPToQDPProof (SOUND transformation)

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

(86) Obligation:

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

LE68_IN_AA → LE68_IN_AA

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

(87) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LE68_IN_AA to LE68_IN_AA.

(88) NO

(89) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(90) UsableRulesProof (EQUIVALENT transformation)

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

(91) Obligation:

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

U10_AAA(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → MERGE33_IN_AAA(T109, .(T110, T111), T113)
MERGE33_IN_AAA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_AAA(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
MERGE33_IN_AAA(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_AAA(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
U12_AAA(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → MERGE33_IN_AAA(.(T149, T150), T152, T154)

The TRS R consists of the following rules:

le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))

(92) PiDPToQDPProof (SOUND transformation)

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

(93) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U10_AAA(le68_in_aa)
MERGE33_IN_AAA → U12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T151)) → MERGE33_IN_AAA

The TRS R consists of the following rules:

le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(94) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAA → U10_AAA(le68_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAA → U10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))

(95) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U12_AAA(gt85_in_aa)
U12_AAA(gt85_out_aa(T151)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))

The TRS R consists of the following rules:

le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(96) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MERGE33_IN_AAA → U12_AAA(gt85_in_aa) at position [0] we obtained the following new rules [LPAR04]:

MERGE33_IN_AAA → U12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAA → U12_AAA(gt85_out_aa(0))

(97) Obligation:

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

U10_AAA(le68_out_aa(T108)) → MERGE33_IN_AAA
U12_AAA(gt85_out_aa(T151)) → MERGE33_IN_AAA
MERGE33_IN_AAA → U10_AAA(U14_aa(le68_in_aa))
MERGE33_IN_AAA → U10_AAA(le68_out_aa(0))
MERGE33_IN_AAA → U12_AAA(U15_aa(gt85_in_aa))
MERGE33_IN_AAA → U12_AAA(gt85_out_aa(0))

The TRS R consists of the following rules:

le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

le68_in_aa
gt85_in_aa
U14_aa(x0)
U15_aa(x0)

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

(98) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

(99) NO

(100) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(101) UsableRulesProof (EQUIVALENT transformation)

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

(102) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)

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

(103) PiDPToQDPProof (SOUND transformation)

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

(104) Obligation:

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

SPLIT21_IN_GAA(.(T42, T43)) → SPLIT21_IN_GAA(T43)

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

(105) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

SPLIT21_IN_GAA(.(T42, T43)) → SPLIT21_IN_GAA(T43)
The graph contains the following edges 1 > 1

(106) YES

(107) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(108) UsableRulesProof (EQUIVALENT transformation)

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

(109) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4) = split17_in_ggaa(x1, x2)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x1, x5)
split21_in_gaa(x1, x2, x3) = split21_in_gaa(x1)
split21_out_gaa(x1, x2, x3) = split21_out_gaa(x2, x3)
U1_gaa(x1, x2, x3, x4, x5) = U1_gaa(x1, x5)
split17_out_ggaa(x1, x2, x3, x4) = split17_out_ggaa(x3, x4)
mergesort32_in_ga(x1, x2) = mergesort32_in_ga(x1)
mergesort32_out_ga(x1, x2) = mergesort32_out_ga
U3_ga(x1, x2, x3, x4, x5) = U3_ga(x5)
U4_ga(x1, x2, x3, x4, x5) = U4_ga(x5)
U5_ga(x1, x2, x3, x4, x5) = U5_ga(x5)
U6_ga(x1, x2, x3, x4, x5, x6) = U6_ga(x5, x6)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x5)
U8_ga(x1, x2, x3, x4, x5, x6) = U8_ga(x6)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x5)
merge33_in_aaa(x1, x2, x3) = merge33_in_aaa
merge33_out_aaa(x1, x2, x3) = merge33_out_aaa
U10_aaa(x1, x2, x3, x4, x5, x6) = U10_aaa(x6)
le68_in_aa(x1, x2) = le68_in_aa
U14_aa(x1, x2, x3) = U14_aa(x3)
le68_out_aa(x1, x2) = le68_out_aa(x1)
U11_aaa(x1, x2, x3, x4, x5, x6) = U11_aaa(x6)
U12_aaa(x1, x2, x3, x4, x5, x6) = U12_aaa(x6)
gt85_in_aa(x1, x2) = gt85_in_aa
U15_aa(x1, x2, x3) = U15_aa(x3)
gt85_out_aa(x1, x2) = gt85_out_aa(x2)
U13_aaa(x1, x2, x3, x4, x5, x6) = U13_aaa(x6)
s(x1) = s(x1)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x5, x6)

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

(110) PiDPToQDPProof (SOUND transformation)

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

(111) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(split17_in_ggaa(T61, .(T62, T63)))
U4_GA(split17_out_ggaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(split17_out_ggaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
U3_ga(split17_out_ggaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_ggaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_ggaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U6_ga(T68, mergesort32_out_ga) → U8_ga(mergesort32_in_ga(T68))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U8_ga(mergesort32_out_ga) → U9_ga(merge33_in_aaa)
U9_ga(merge33_out_aaa) → mergesort32_out_ga
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1)
U3_ga(x0)
U4_ga(x0)
split21_in_gaa(x0)
U5_ga(x0)
U6_ga(x0, x1)
U1_gaa(x0, x1)
U7_ga(x0)
U8_ga(x0)
U9_ga(x0)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(112) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Strictly oriented rules of the TRS R:

mergesort32_in_ga(.(T54, [])) → mergesort32_out_ga

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(0) = 0
POL(MERGESORT32_IN_GA(x₁)) = x₁
POL(U10_aaa(x₁)) = x₁
POL(U11_aaa(x₁)) = x₁
POL(U12_aaa(x₁)) = x₁
POL(U13_aaa(x₁)) = x₁
POL(U14_aa(x₁)) = x₁
POL(U15_aa(x₁)) = x₁
POL(U1_gaa(x₁, x₂)) = 1 + x₁ + x₂
POL(U2_ggaa(x₁, x₂)) = x₁ + x₂
POL(U3_ga(x₁)) = x₁
POL(U4_GA(x₁)) = x₁
POL(U4_ga(x₁)) = x₁
POL(U5_ga(x₁)) = x₁
POL(U6_GA(x₁, x₂)) = x₁ + x₂
POL(U6_ga(x₁, x₂)) = x₁ + x₂
POL(U7_ga(x₁)) = x₁
POL(U8_ga(x₁)) = x₁
POL(U9_ga(x₁)) = x₁
POL([]) = 0
POL(gt85_in_aa) = 0
POL(gt85_out_aa(x₁)) = x₁
POL(le68_in_aa) = 0
POL(le68_out_aa(x₁)) = x₁
POL(merge33_in_aaa) = 0
POL(merge33_out_aaa) = 0
POL(mergesort32_in_ga(x₁)) = x₁
POL(mergesort32_out_ga) = 0
POL(s(x₁)) = x₁
POL(split17_in_ggaa(x₁, x₂)) = 1 + x₁ + x₂
POL(split17_out_ggaa(x₁, x₂)) = x₁ + x₂
POL(split21_in_gaa(x₁)) = 1 + x₁
POL(split21_out_gaa(x₁, x₂)) = 1 + x₁ + x₂

(113) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(split17_in_ggaa(T61, .(T62, T63)))
U4_GA(split17_out_ggaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(split17_out_ggaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(split17_in_ggaa(T61, .(T62, T63)))
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
U3_ga(split17_out_ggaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_ggaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U4_ga(split17_out_ggaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U6_ga(T68, mergesort32_out_ga) → U8_ga(mergesort32_in_ga(T68))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U8_ga(mergesort32_out_ga) → U9_ga(merge33_in_aaa)
U9_ga(merge33_out_aaa) → mergesort32_out_ga
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1)
U3_ga(x0)
U4_ga(x0)
split21_in_gaa(x0)
U5_ga(x0)
U6_ga(x0, x1)
U1_gaa(x0, x1)
U7_ga(x0)
U8_ga(x0)
U9_ga(x0)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(114) MRRProof (EQUIVALENT transformation)

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(split17_in_ggaa(T61, .(T62, T63)))
U4_GA(split17_out_ggaa(T67, T68)) → MERGESORT32_IN_GA(T67)
U6_GA(T68, mergesort32_out_ga) → MERGESORT32_IN_GA(T68)

Strictly oriented rules of the TRS R:

mergesort32_in_ga(.(T61, .(T62, T63))) → U3_ga(split17_in_ggaa(T61, .(T62, T63)))
U3_ga(split17_out_ggaa(X133, X134)) → mergesort32_out_ga
U4_ga(split17_out_ggaa(T67, T68)) → U5_ga(mergesort32_in_ga(T67))
U5_ga(mergesort32_out_ga) → mergesort32_out_ga
U6_ga(T68, mergesort32_out_ga) → U7_ga(mergesort32_in_ga(T68))
U6_ga(T68, mergesort32_out_ga) → U8_ga(mergesort32_in_ga(T68))
U7_ga(mergesort32_out_ga) → mergesort32_out_ga
U9_ga(merge33_out_aaa) → mergesort32_out_ga

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(0) = 0
POL(MERGESORT32_IN_GA(x₁)) = 2 + 2·x₁
POL(U10_aaa(x₁)) = 1 + 2·x₁
POL(U11_aaa(x₁)) = x₁
POL(U12_aaa(x₁)) = 1 + 2·x₁
POL(U13_aaa(x₁)) = x₁
POL(U14_aa(x₁)) = 2·x₁
POL(U15_aa(x₁)) = 2·x₁
POL(U1_gaa(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(U2_ggaa(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(U3_ga(x₁)) = 2 + x₁
POL(U4_GA(x₁)) = 1 + 2·x₁
POL(U4_ga(x₁)) = 2 + 2·x₁
POL(U5_ga(x₁)) = 1 + x₁
POL(U6_GA(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(U6_ga(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(U7_ga(x₁)) = 1 + x₁
POL(U8_ga(x₁)) = 1 + x₁
POL(U9_ga(x₁)) = 1 + 2·x₁
POL([]) = 0
POL(gt85_in_aa) = 0
POL(gt85_out_aa(x₁)) = x₁
POL(le68_in_aa) = 0
POL(le68_out_aa(x₁)) = 2·x₁
POL(merge33_in_aaa) = 1
POL(merge33_out_aaa) = 1
POL(mergesort32_in_ga(x₁)) = 2 + 2·x₁
POL(mergesort32_out_ga) = 2
POL(s(x₁)) = x₁
POL(split17_in_ggaa(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(split17_out_ggaa(x₁, x₂)) = 1 + x₁ + x₂
POL(split21_in_gaa(x₁)) = x₁
POL(split21_out_gaa(x₁, x₂)) = x₁ + 2·x₂

(115) Obligation:

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

U4_GA(split17_out_ggaa(T67, T68)) → U6_GA(T68, mergesort32_in_ga(T67))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
mergesort32_in_ga([]) → mergesort32_out_ga
mergesort32_in_ga(.(T61, .(T62, T63))) → U4_ga(split17_in_ggaa(T61, .(T62, T63)))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
U4_ga(split17_out_ggaa(T67, T68)) → U6_ga(T68, mergesort32_in_ga(T67))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)
U8_ga(mergesort32_out_ga) → U9_ga(merge33_in_aaa)
merge33_in_aaa → merge33_out_aaa
merge33_in_aaa → U10_aaa(le68_in_aa)
merge33_in_aaa → U12_aaa(gt85_in_aa)
U10_aaa(le68_out_aa(T108)) → merge33_out_aaa
U10_aaa(le68_out_aa(T108)) → U11_aaa(merge33_in_aaa)
U12_aaa(gt85_out_aa(T151)) → merge33_out_aaa
U12_aaa(gt85_out_aa(T151)) → U13_aaa(merge33_in_aaa)
le68_in_aa → U14_aa(le68_in_aa)
le68_in_aa → le68_out_aa(0)
U11_aaa(merge33_out_aaa) → merge33_out_aaa
gt85_in_aa → U15_aa(gt85_in_aa)
gt85_in_aa → gt85_out_aa(0)
U13_aaa(merge33_out_aaa) → merge33_out_aaa
U14_aa(le68_out_aa(T126)) → le68_out_aa(s(T126))
U15_aa(gt85_out_aa(T168)) → gt85_out_aa(s(T168))

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
mergesort32_in_ga(x0)
U2_ggaa(x0, x1)
U3_ga(x0)
U4_ga(x0)
split21_in_gaa(x0)
U5_ga(x0)
U6_ga(x0, x1)
U1_gaa(x0, x1)
U7_ga(x0)
U8_ga(x0)
U9_ga(x0)
merge33_in_aaa
U10_aaa(x0)
U12_aaa(x0)
le68_in_aa
U11_aaa(x0)
gt85_in_aa
U13_aaa(x0)
U14_aa(x0)
U15_aa(x0)

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

(116) DependencyGraphProof (EQUIVALENT transformation)

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

(117) TRUE

(118) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

mergesort1_in_ga([], []) → mergesort1_out_ga([], [])
mergesort1_in_ga(.(T4, []), .(T4, [])) → mergesort1_out_ga(.(T4, []), .(T4, []))
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U16_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
U16_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, X52, X51)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
mergesort1_in_ga(.(T23, .(T24, T25)), T14) → U17_ga(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U18_ga(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U18_ga(T23, T24, T25, T14, mergesort1_out_ga(.(T23, T29), X23)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U17_ga(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → U19_ga(T23, T24, T25, T14, T28, mergesort1_in_ga(.(T23, T29), T46))
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U20_ga(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
mergesort32_in_ga([], []) → mergesort32_out_ga([], [])
mergesort32_in_ga(.(T54, []), .(T54, [])) → mergesort32_out_ga(.(T54, []), .(T54, []))
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U3_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), X133, X134))
U3_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), X133, X134)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
mergesort32_in_ga(.(T61, .(T62, T63)), X137) → U4_ga(T61, T62, T63, X137, split17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_ga(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U5_ga(T61, T62, T63, X137, mergesort32_out_ga(T67, X135)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U4_ga(T61, T62, T63, X137, split17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_ga(T61, T62, T63, X137, T68, mergesort32_in_ga(T67, T72))
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U7_ga(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U7_ga(T61, T62, T63, X137, mergesort32_out_ga(T68, X136)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U6_ga(T61, T62, T63, X137, T68, mergesort32_out_ga(T67, T72)) → U8_ga(T61, T62, T63, X137, T72, mergesort32_in_ga(T68, T73))
U8_ga(T61, T62, T63, X137, T72, mergesort32_out_ga(T68, T73)) → U9_ga(T61, T62, T63, X137, merge33_in_aaa(T72, T73, X137))
merge33_in_aaa([], T82, T82) → merge33_out_aaa([], T82, T82)
merge33_in_aaa(T87, [], T87) → merge33_out_aaa(T87, [], T87)
merge33_in_aaa(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_aaa(T108, T109, T110, T111, T113, le68_in_aa(T108, T110))
le68_in_aa(s(T126), s(T127)) → U14_aa(T126, T127, le68_in_aa(T126, T127))
le68_in_aa(0, s(T134)) → le68_out_aa(0, s(T134))
le68_in_aa(0, 0) → le68_out_aa(0, 0)
U14_aa(T126, T127, le68_out_aa(T126, T127)) → le68_out_aa(s(T126), s(T127))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U10_aaa(T108, T109, T110, T111, T113, le68_out_aa(T108, T110)) → U11_aaa(T108, T109, T110, T111, T113, merge33_in_aaa(T109, .(T110, T111), T113))
merge33_in_aaa(.(T149, T150), .(T151, T152), .(T151, T154)) → U12_aaa(T149, T150, T151, T152, T154, gt85_in_aa(T149, T151))
gt85_in_aa(s(T167), s(T168)) → U15_aa(T167, T168, gt85_in_aa(T167, T168))
gt85_in_aa(s(T173), 0) → gt85_out_aa(s(T173), 0)
U15_aa(T167, T168, gt85_out_aa(T167, T168)) → gt85_out_aa(s(T167), s(T168))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U12_aaa(T149, T150, T151, T152, T154, gt85_out_aa(T149, T151)) → U13_aaa(T149, T150, T151, T152, T154, merge33_in_aaa(.(T149, T150), T152, T154))
U13_aaa(T149, T150, T151, T152, T154, merge33_out_aaa(.(T149, T150), T152, T154)) → merge33_out_aaa(.(T149, T150), .(T151, T152), .(T151, T154))
U11_aaa(T108, T109, T110, T111, T113, merge33_out_aaa(T109, .(T110, T111), T113)) → merge33_out_aaa(.(T108, T109), .(T110, T111), .(T108, T113))
U9_ga(T61, T62, T63, X137, merge33_out_aaa(T72, T73, X137)) → mergesort32_out_ga(.(T61, .(T62, T63)), X137)
U20_ga(T23, T24, T25, T14, mergesort32_out_ga(T28, X24)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)
U19_ga(T23, T24, T25, T14, T28, mergesort1_out_ga(.(T23, T29), T46)) → U21_ga(T23, T24, T25, T14, T46, mergesort32_in_ga(T28, T49))
U21_ga(T23, T24, T25, T14, T46, mergesort32_out_ga(T28, T49)) → U22_ga(T23, T24, T25, T14, merge33_in_aaa(T46, T49, T14))
U22_ga(T23, T24, T25, T14, merge33_out_aaa(T46, T49, T14)) → mergesort1_out_ga(.(T23, .(T24, T25)), T14)

(119) UsableRulesProof (EQUIVALENT transformation)

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

(120) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U17_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, T28, T29))
U17_GA(T23, T24, T25, T14, split17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37, .(T36, X81), X82) → U2_ggaa(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
U2_ggaa(T36, T37, X81, X82, split21_out_gaa(T37, X82, X81)) → split17_out_ggaa(T36, T37, .(T36, X81), X82)
split21_in_gaa([], [], []) → split21_out_gaa([], [], [])
split21_in_gaa(.(T42, T43), .(T42, X99), X100) → U1_gaa(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
U1_gaa(T42, T43, X99, X100, split21_out_gaa(T43, X100, X99)) → split21_out_gaa(.(T42, T43), .(T42, X99), X100)

(121) PiDPToQDPProof (SOUND transformation)

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

(122) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, split17_in_ggaa(T24, T25))
U17_GA(T23, split17_out_ggaa(T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(123) MRRProof (EQUIVALENT transformation)

U2_ggaa(T36, split21_out_gaa(X82, X81)) → split17_out_ggaa(.(T36, X81), X82)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(MERGESORT1_IN_GA(x₁)) = 2·x₁
POL(U17_GA(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(U1_gaa(x₁, x₂)) = x₁ + 2·x₂
POL(U2_ggaa(x₁, x₂)) = x₁ + 2·x₂
POL([]) = 0
POL(split17_in_ggaa(x₁, x₂)) = 2 + 2·x₁ + 2·x₂
POL(split17_out_ggaa(x₁, x₂)) = x₁ + 2·x₂
POL(split21_in_gaa(x₁)) = 1 + x₁
POL(split21_out_gaa(x₁, x₂)) = 1 + x₁ + 2·x₂

(124) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U17_GA(T23, split17_in_ggaa(T24, T25))
U17_GA(T23, split17_out_ggaa(T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

split17_in_ggaa(T36, T37) → U2_ggaa(T36, split21_in_gaa(T37))
split21_in_gaa([]) → split21_out_gaa([], [])
split21_in_gaa(.(T42, T43)) → U1_gaa(T42, split21_in_gaa(T43))
U1_gaa(T42, split21_out_gaa(X100, X99)) → split21_out_gaa(.(T42, X99), X100)

The set Q consists of the following terms:

split17_in_ggaa(x0, x1)
U2_ggaa(x0, x1)
split21_in_gaa(x0)
U1_gaa(x0, x1)

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

(125) DependencyGraphProof (EQUIVALENT transformation)

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