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

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

0 Prolog

↳1 PredefinedPredicateTransformerProof (⇐)

↳2 Prolog

↳3 BuiltinConflictTransformerProof (⇐)

↳4 Prolog

↳5 PrologToDTProblemTransformerProof (⇐)

↳6 TRIPLES

↳7 TriplesToPiDPProof (⇐)

↳8 PiDP

↳9 DependencyGraphProof (⇔)

↳10 AND

    ↳11 PiDP

        ↳12 UsableRulesProof (⇔)

        ↳13 PiDP

        ↳14 PiDPToQDPProof (⇐)

        ↳15 QDP

        ↳16 QDPSizeChangeProof (⇔)

        ↳17 YES

    ↳18 PiDP

        ↳19 UsableRulesProof (⇔)

        ↳20 PiDP

        ↳21 PiDPToQDPProof (⇐)

        ↳22 QDP

        ↳23 QDPSizeChangeProof (⇔)

        ↳24 YES

    ↳25 PiDP

        ↳26 UsableRulesProof (⇔)

        ↳27 PiDP

        ↳28 PiDPToQDPProof (⇐)

        ↳29 QDP

        ↳30 Rewriting (⇔)

        ↳31 QDP

        ↳32 Rewriting (⇔)

        ↳33 QDP

        ↳34 QDPOrderProof (⇔)

        ↳35 QDP

        ↳36 DependencyGraphProof (⇔)

        ↳37 QDP

        ↳38 UsableRulesProof (⇔)

        ↳39 QDP

        ↳40 QReductionProof (⇔)

        ↳41 QDP

        ↳42 QDPOrderProof (⇔)

        ↳43 QDP

        ↳44 DependencyGraphProof (⇔)

        ↳45 TRUE

    ↳46 PiDP

        ↳47 UsableRulesProof (⇔)

        ↳48 PiDP

        ↳49 PiDPToQDPProof (⇐)

        ↳50 QDP

        ↳51 Rewriting (⇔)

        ↳52 QDP

        ↳53 UsableRulesProof (⇔)

        ↳54 QDP

        ↳55 QReductionProof (⇔)

        ↳56 QDP

        ↳57 QDPOrderProof (⇔)

        ↳58 QDP

        ↳59 DependencyGraphProof (⇔)

        ↳60 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)) :- ','(=(X, Y), merge(.(X, Xs), Ys, Zs)).

Queries:

mergesort(g,a).

(1) PredefinedPredicateTransformerProof (SOUND transformation)

Added definitions of predefined predicates [PROLOG].

(2) Obligation:

(3) BuiltinConflictTransformerProof (SOUND transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(4) 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)) :- ','(user_defined_=(X, Y), merge(.(X, Xs), Ys, Zs)).
user_defined_=(X, X).

Queries:

mergesort(g,a).

(5) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: mergesort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: mergesort(T1, T2), SCOPE: 1, CLAUSE: 0 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: mergesort(T1, T2), SCOPE: 1, CLAUSE: 1 | mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: mergesort([], T2), SCOPE: 1, CLAUSE: 1 | mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: mergesort([], T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: mergesort(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nmergesort(T1, T2) !=? mergesort([], [])\nmergesort(T1, T2) !=? mergesort(.(X7, []), .(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: mergesort(.(T4, []), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(split(.(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(T72, T73, X137), SCOPE: 6, CLAUSE: 5 | merge(T72, T73, X137), SCOPE: 6, CLAUSE: 6 | merge(T72, T73, X137), SCOPE: 6, CLAUSE: 7\n\nT72 is ground\nT73 is ground\nX137 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: merge(T72, T73, X137), SCOPE: 6, CLAUSE: 5\n\nT72 is ground\nT73 is ground\nX137 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: merge(T72, T73, X137), SCOPE: 6, CLAUSE: 6 | merge(T72, T73, X137), SCOPE: 6, CLAUSE: 7\n\nT72 is ground\nT73 is ground\nX137 is free; ", 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(T72, T73, X137), SCOPE: 6, CLAUSE: 6\n\nT72 is ground\nT73 is ground\nX137 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: merge(T72, T73, X137), SCOPE: 6, CLAUSE: 7\n\nT72 is ground\nT73 is ground\nX137 is free; ", 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: ','(user_defined_=(T94, T96), merge(.(T94, T95), T97, X185))\n\nT94 is ground\nT95 is ground\nT96 is ground\nT97 is ground\nX185 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(user_defined_=(T94, T96), merge(.(T94, T95), T97, X185)), SCOPE: 7, CLAUSE: 8\n\nT94 is ground\nT95 is ground\nT96 is ground\nT97 is ground\nX185 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: merge(.(T104, T95), T97, X185)\n\nT95 is ground\nT97 is ground\nT104 is ground\nX185 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: merge(T46, T49, T14), SCOPE: 8, CLAUSE: 5 | merge(T46, T49, T14), SCOPE: 8, CLAUSE: 6 | merge(T46, T49, T14), SCOPE: 8, CLAUSE: 7\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: merge(T46, T49, T14), SCOPE: 8, CLAUSE: 5\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: merge(T46, T49, T14), SCOPE: 8, CLAUSE: 6 | merge(T46, T49, T14), SCOPE: 8, CLAUSE: 7\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: merge(T46, T49, T14), SCOPE: 8, CLAUSE: 6\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: merge(T46, T49, T14), SCOPE: 8, CLAUSE: 7\n\nT46 is ground\nT49 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: ','(user_defined_=(T131, T133), merge(.(T131, T132), T134, T136))\n\nT131 is ground\nT132 is ground\nT133 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(user_defined_=(T131, T133), merge(.(T131, T132), T134, T136)), SCOPE: 9, CLAUSE: 8\n\nT131 is ground\nT132 is ground\nT133 is ground\nT134 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: merge(.(T143, T132), T134, T136)\n\nT132 is ground\nT134 is ground\nT143 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 85 [arrowhead = none ];
85 -> 2;

86 [label="EVAL with clause\nmergesort([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 86 [arrowhead = none ];
86 -> 3;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 87 [arrowhead = none ];
87 -> 4;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 88 [arrowhead = none ];
88 -> 5;

89 [label="EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitution\nX7 -> T4\nT1 -> .(T4, [])\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 89 [arrowhead = none ];
89 -> 8;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 90 [arrowhead = none ];
90 -> 9;

91 [label="BACKTRACK\nfor clause: mergesort(.(X, []), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 91 [arrowhead = none ];
91 -> 6;

92 [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 -> 92 [arrowhead = none ];
92 -> 7;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 93 [arrowhead = none ];
93 -> 10;

94 [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 -> 94 [arrowhead = none ];
94 -> 12;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 95 [arrowhead = none ];
95 -> 13;

96 [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 -> 96 [arrowhead = none ];
96 -> 11;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 97 [arrowhead = none ];
97 -> 14;

98 [label="BACKTRACK\nfor clause: split([], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 98 [arrowhead = none ];
98 -> 15;

99 [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 -> 99 [arrowhead = none ];
99 -> 16;

100 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 100 [arrowhead = none ];
100 -> 17;

101 [label="SPLIT 2\nnew knowledge:\nT28 is ground\nT29 is ground\nreplacements:\nX52 -> T28\nX51 -> T29", fontsize=14, style = filled, fillcolor = violet];
16 -> 101 [arrowhead = none ];
101 -> 18;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 102 [arrowhead = none ];
102 -> 19;

103 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 103 [arrowhead = none ];
103 -> 30;

104 [label="SPLIT 2\nnew knowledge:\nT46 is ground\nreplacements:\nX23 -> T46", fontsize=14, style = filled, fillcolor = violet];
18 -> 104 [arrowhead = none ];
104 -> 31;

105 [label="BACKTRACK\nfor clause: split([], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 105 [arrowhead = none ];
105 -> 20;

106 [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 -> 106 [arrowhead = none ];
106 -> 21;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 107 [arrowhead = none ];
107 -> 22;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 108 [arrowhead = none ];
108 -> 23;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 109 [arrowhead = none ];
109 -> 24;

110 [label="EVAL with clause\nsplit([], [], []).\nand substitution\nT37 -> []\nX82 -> []\nX81 -> []", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 110 [arrowhead = none ];
110 -> 25;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 111 [arrowhead = none ];
111 -> 26;

112 [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 -> 112 [arrowhead = none ];
112 -> 28;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 113 [arrowhead = none ];
113 -> 29;

114 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 114 [arrowhead = none ];
114 -> 27;

115 [label="INSTANCE with matching:\nT37 -> T43\nX82 -> X100\nX81 -> X99", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 115 [arrowhead = none , style=dashed];
115 -> 21[style=dashed];

116 [label="INSTANCE with matching:\nT1 -> .(T23, T29)\nT2 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 116 [arrowhead = none , style=dashed];
116 -> 1[style=dashed];

117 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
31 -> 117 [arrowhead = none ];
117 -> 32;

118 [label="SPLIT 2\nnew knowledge:\nT49 is ground\nreplacements:\nX24 -> T49", fontsize=14, style = filled, fillcolor = violet];
31 -> 118 [arrowhead = none ];
118 -> 33;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 119 [arrowhead = none ];
119 -> 34;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 120 [arrowhead = none ];
120 -> 69;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 121 [arrowhead = none ];
121 -> 35;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 122 [arrowhead = none ];
122 -> 36;

123 [label="EVAL with clause\nmergesort([], []).\nand substitution\nT28 -> []\nX24 -> []", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 123 [arrowhead = none ];
123 -> 37;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 124 [arrowhead = none ];
124 -> 38;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 125 [arrowhead = none ];
125 -> 40;

126 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 126 [arrowhead = none ];
126 -> 41;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 127 [arrowhead = none ];
127 -> 39;

128 [label="EVAL with clause\nmergesort(.(X113, []), .(X113, [])).\nand substitution\nX113 -> T54\nT28 -> .(T54, [])\nX24 -> .(T54, [])", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 128 [arrowhead = none ];
128 -> 42;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 129 [arrowhead = none ];
129 -> 43;

130 [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 -> 130 [arrowhead = none ];
130 -> 45;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 131 [arrowhead = none ];
131 -> 46;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 132 [arrowhead = none ];
132 -> 44;

133 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 133 [arrowhead = none ];
133 -> 47;

134 [label="SPLIT 2\nnew knowledge:\nT67 is ground\nT68 is ground\nreplacements:\nX133 -> T67\nX134 -> T68", fontsize=14, style = filled, fillcolor = violet];
45 -> 134 [arrowhead = none ];
134 -> 48;

135 [label="INSTANCE with matching:\nT24 -> T61\nT25 -> .(T62, T63)\nX52 -> X133\nX51 -> X134", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 135 [arrowhead = none , style=dashed];
135 -> 17[style=dashed];

136 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
48 -> 136 [arrowhead = none ];
136 -> 49;

137 [label="SPLIT 2\nnew knowledge:\nT72 is ground\nreplacements:\nX135 -> T72", fontsize=14, style = filled, fillcolor = violet];
48 -> 137 [arrowhead = none ];
137 -> 50;

138 [label="INSTANCE with matching:\nT28 -> T67\nX24 -> X135", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 138 [arrowhead = none , style=dashed];
138 -> 32[style=dashed];

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
50 -> 139 [arrowhead = none ];
139 -> 51;

140 [label="SPLIT 2\nnew knowledge:\nT73 is ground\nreplacements:\nX136 -> T73", fontsize=14, style = filled, fillcolor = violet];
50 -> 140 [arrowhead = none ];
140 -> 52;

141 [label="INSTANCE with matching:\nT28 -> T68\nX24 -> X136", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 141 [arrowhead = none , style=dashed];
141 -> 32[style=dashed];

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 142 [arrowhead = none ];
142 -> 53;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 143 [arrowhead = none ];
143 -> 54;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 144 [arrowhead = none ];
144 -> 55;

145 [label="EVAL with clause\nmerge([], X160, X160).\nand substitution\nT72 -> []\nT73 -> T80\nX160 -> T80\nX137 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 145 [arrowhead = none ];
145 -> 56;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 146 [arrowhead = none ];
146 -> 57;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 147 [arrowhead = none ];
147 -> 59;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 149 [arrowhead = none ];
149 -> 58;

150 [label="EVAL with clause\nmerge(X166, [], X166).\nand substitution\nT72 -> T85\nX166 -> T85\nT73 -> []\nX137 -> T85", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 150 [arrowhead = none ];
150 -> 61;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 151 [arrowhead = none ];
151 -> 62;

152 [label="EVAL with clause\nmerge(.(X180, X181), .(X182, X183), .(X180, X184)) :- ','(user_defined_=(X180, X182), merge(.(X180, X181), X183, X184)).\nand substitution\nX180 -> T94\nX181 -> T95\nT72 -> .(T94, T95)\nX182 -> T96\nX183 -> T97\nT73 -> .(T96, T97)\nX184 -> X185\nX137 -> .(T94, X185)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 152 [arrowhead = none ];
152 -> 64;

153 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 153 [arrowhead = none ];
153 -> 65;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 154 [arrowhead = none ];
154 -> 63;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 155 [arrowhead = none ];
155 -> 66;

156 [label="EVAL with clause\nuser_defined_=(X197, X197).\nand substitution\nT94 -> T104\nX197 -> T104\nT96 -> T104", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 156 [arrowhead = none ];
156 -> 67;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 157 [arrowhead = none ];
157 -> 68;

158 [label="INSTANCE with matching:\nT46 -> .(T104, T95)\nT49 -> T97\nT14 -> X185", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 158 [arrowhead = none , style=dashed];
158 -> 33[style=dashed];

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 159 [arrowhead = none ];
159 -> 70;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 160 [arrowhead = none ];
160 -> 71;

161 [label="EVAL with clause\nmerge([], X209, X209).\nand substitution\nT46 -> []\nT49 -> T115\nX209 -> T115\nT14 -> T115", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 161 [arrowhead = none ];
161 -> 72;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 162 [arrowhead = none ];
162 -> 73;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 163 [arrowhead = none ];
163 -> 75;

164 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 164 [arrowhead = none ];
164 -> 76;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 165 [arrowhead = none ];
165 -> 74;

166 [label="EVAL with clause\nmerge(X215, [], X215).\nand substitution\nT46 -> T120\nX215 -> T120\nT49 -> []\nT14 -> T120", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 166 [arrowhead = none ];
166 -> 77;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 167 [arrowhead = none ];
167 -> 78;

168 [label="EVAL with clause\nmerge(.(X227, X228), .(X229, X230), .(X227, X231)) :- ','(user_defined_=(X227, X229), merge(.(X227, X228), X230, X231)).\nand substitution\nX227 -> T131\nX228 -> T132\nT46 -> .(T131, T132)\nX229 -> T133\nX230 -> T134\nT49 -> .(T133, T134)\nX231 -> T136\nT14 -> .(T131, T136)\nT135 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 168 [arrowhead = none ];
168 -> 80;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 169 [arrowhead = none ];
169 -> 81;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 170 [arrowhead = none ];
170 -> 79;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 171 [arrowhead = none ];
171 -> 82;

172 [label="EVAL with clause\nuser_defined_=(X243, X243).\nand substitution\nT131 -> T143\nX243 -> T143\nT133 -> T143", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 172 [arrowhead = none ];
172 -> 83;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 173 [arrowhead = none ];
173 -> 84;

174 [label="INSTANCE with matching:\nT46 -> .(T143, T132)\nT49 -> T134\nT14 -> T136", fontsize=14, style = filled, fillcolor = lightgrey];
83 -> 174 [arrowhead = none , style=dashed];
174 -> 33[style=dashed];

}

(6) Obligation:

Triples:

split21(.(T42, T43), .(T42, X99), X100) :- split21(T43, X100, X99).
split17(T36, T37, .(T36, X81), X82) :- split21(T37, X82, X81).
mergesort32(.(T61, .(T62, T63)), X137) :- split17(T61, .(T62, T63), X133, X134).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(splitc17(T61, .(T62, T63), T67, T68), mergesort32(T67, X135)).
mergesort32(.(T61, .(T62, T63)), X137) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, T72), mergesort32(T68, X136))).
mergesort32(.(T61, .(T62, T63)), .(T104, X185)) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, .(T104, T95)), ','(mergesortc32(T68, .(T104, T97)), merge33(.(T104, T95), T97, X185)))).
merge33(.(T143, T132), .(T143, T134), .(T143, T136)) :- merge33(.(T143, T132), T134, T136).
mergesort1(.(T23, .(T24, T25)), T14) :- split17(T24, T25, X52, X51).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(splitc17(T24, T25, T28, T29), mergesort1(.(T23, T29), X23)).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(splitc17(T24, T25, T28, T29), ','(mergesortc1(.(T23, T29), T46), mergesort32(T28, X24))).
mergesort1(.(T23, .(T24, T25)), T14) :- ','(splitc17(T24, T25, T28, T29), ','(mergesortc1(.(T23, T29), T46), ','(mergesortc32(T28, T49), merge33(T46, T49, T14)))).

Clauses:

splitc21([], [], []).
splitc21(.(T42, T43), .(T42, X99), X100) :- splitc21(T43, X100, X99).
mergesortc1([], []).
mergesortc1(.(T4, []), .(T4, [])).
mergesortc1(.(T23, .(T24, T25)), T14) :- ','(splitc17(T24, T25, T28, T29), ','(mergesortc1(.(T23, T29), T46), ','(mergesortc32(T28, T49), mergec33(T46, T49, T14)))).
splitc17(T36, T37, .(T36, X81), X82) :- splitc21(T37, X82, X81).
mergesortc32([], []).
mergesortc32(.(T54, []), .(T54, [])).
mergesortc32(.(T61, .(T62, T63)), T80) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, []), mergesortc32(T68, T80))).
mergesortc32(.(T61, .(T62, T63)), T85) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, T85), mergesortc32(T68, []))).
mergesortc32(.(T61, .(T62, T63)), .(T104, X185)) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, .(T104, T95)), ','(mergesortc32(T68, .(T104, T97)), mergec33(.(T104, T95), T97, X185)))).
mergec33([], T115, T115).
mergec33(T120, [], T120).
mergec33(.(T143, T132), .(T143, T134), .(T143, T136)) :- mergec33(.(T143, T132), T134, T136).

Afs:

mergesort1(x1, x2) = mergesort1(x1)

(7) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split17_in: (f,b,f,f)
split21_in: (b,f,f)
splitc17_in: (f,b,f,f)
splitc21_in: (b,f,f)
mergesortc1_in: (b,f)
mergesortc32_in: (b,f) (b,b)
mergec33_in: (b,b,f) (b,b,b)
mergesort32_in: (b,f)
merge33_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U13_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_AGAA(T24, T25, X52, X51)
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → U2_AGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_AGAA(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) → U14_GA(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U15_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U16_GA(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U16_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U17_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U16_GA(T23, T24, T25, T14, T28, mergesortc1_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_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U8_GA(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T104, X185, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U9_GA(T61, T62, T63, T104, X185, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U10_GA(T61, T62, T63, T104, X185, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U10_GA(T61, T62, T63, T104, X185, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U11_GA(T61, T62, T63, T104, X185, merge33_in_gga(.(T104, T95), T97, X185))
U10_GA(T61, T62, T63, T104, X185, T95, mergesortc32_out_ga(T68, .(T104, T97))) → MERGE33_IN_GGA(.(T104, T95), T97, X185)
MERGE33_IN_GGA(.(T143, T132), .(T143, T134), .(T143, T136)) → U12_GGA(T143, T132, T134, T136, merge33_in_gga(.(T143, T132), T134, T136))
MERGE33_IN_GGA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_GGA(.(T143, T132), T134, T136)
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U16_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U18_GA(T23, T24, T25, T14, T46, mergesortc32_in_ga(T28, T49))
U18_GA(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U19_GA(T23, T24, T25, T14, merge33_in_gga(T46, T49, T14))
U18_GA(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → MERGE33_IN_GGA(T46, T49, T14)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T23, .(T24, T25)), T14) → U22_ga(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U22_ga(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_in_ga(.(T23, T29), T46))
U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_in_ga(T28, T49))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
mergec33_in_gga([], T115, T115) → mergec33_out_gga([], T115, T115)
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
mergec33_in_ggg([], T115, T115) → mergec33_out_ggg([], T115, T115)
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U25_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
mergesort1_in_ga(x1, x2) = mergesort1_in_ga(x1)
.(x1, x2) = .(x2)
split21_in_gaa(x1, x2, x3) = split21_in_gaa(x1)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U21_gaa(x1, x2, x3, x4, x5) = U21_gaa(x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x3, x5)
U23_ga(x1, x2, x3, x4, x5, x6, x7) = U23_ga(x3, x5, x7)
U24_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U24_ga(x3, x7, x8)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6, x7) = U28_ga(x3, x6, x7)
mergesortc32_in_gg(x1, x2) = mergesortc32_in_gg(x1, x2)
mergesortc32_out_gg(x1, x2) = mergesortc32_out_gg(x1, x2)
U27_gg(x1, x2, x3, x4, x5) = U27_gg(x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7) = U28_gg(x3, x4, x6, x7)
U30_gg(x1, x2, x3, x4, x5) = U30_gg(x3, x4, x5)
U31_gg(x1, x2, x3, x4, x5, x6, x7) = U31_gg(x3, x4, x6, x7)
U33_gg(x1, x2, x3, x4, x5, x6) = U33_gg(x3, x5, x6)
U34_gg(x1, x2, x3, x4, x5, x6, x7, x8) = U34_gg(x3, x5, x7, x8)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x3, x5)
U31_ga(x1, x2, x3, x4, x5, x6, x7) = U31_ga(x3, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x3, x6)
U34_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U34_ga(x3, x7, x8)
split17_in_agaa(x1, x2, x3, x4) = split17_in_agaa(x2)
splitc17_in_agaa(x1, x2, x3, x4) = splitc17_in_agaa(x2)
U26_agaa(x1, x2, x3, x4, x5) = U26_agaa(x2, x5)
splitc17_out_agaa(x1, x2, x3, x4) = splitc17_out_agaa(x2, x3, x4)
U35_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ga(x3, x8, x9)
U36_ga(x1, x2, x3, x4, x5, x6) = U36_ga(x3, x6)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U37_gga(x1, x2, x3, x4, x5) = U37_gga(x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x3, x4, x5)
U35_gg(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_gg(x3, x5, x8, x9)
U36_gg(x1, x2, x3, x4, x5, x6) = U36_gg(x3, x5, x6)
mergec33_in_ggg(x1, x2, x3) = mergec33_in_ggg(x1, x2, x3)
mergec33_out_ggg(x1, x2, x3) = mergec33_out_ggg(x1, x2, x3)
U37_ggg(x1, x2, x3, x4, x5) = U37_ggg(x2, x3, x4, x5)
U32_gg(x1, x2, x3, x4, x5) = U32_gg(x3, x4, x5)
U29_gg(x1, x2, x3, x4, x5) = U29_gg(x3, x4, x5)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x3, x5)
U25_ga(x1, x2, x3, x4, x5) = U25_ga(x3, x5)
mergesort32_in_ga(x1, x2) = mergesort32_in_ga(x1)
merge33_in_gga(x1, x2, x3) = merge33_in_gga(x1, x2)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U13_GA(x1, x2, x3, x4, x5) = U13_GA(x3, x5)
SPLIT17_IN_AGAA(x1, x2, x3, x4) = SPLIT17_IN_AGAA(x2)
U2_AGAA(x1, x2, x3, x4, x5) = U2_AGAA(x2, x5)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x2, x5)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x3, x5)
U15_GA(x1, x2, x3, x4, x5) = U15_GA(x3, x5)
U16_GA(x1, x2, x3, x4, x5, x6) = U16_GA(x3, x5, x6)
U17_GA(x1, x2, x3, x4, x5) = U17_GA(x3, x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x3, x5)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5, x6) = U8_GA(x3, x6)
U9_GA(x1, x2, x3, x4, x5, x6, x7) = U9_GA(x3, x6, x7)
U10_GA(x1, x2, x3, x4, x5, x6, x7) = U10_GA(x3, x6, x7)
U11_GA(x1, x2, x3, x4, x5, x6) = U11_GA(x3, x6)
MERGE33_IN_GGA(x1, x2, x3) = MERGE33_IN_GGA(x1, x2)
U12_GGA(x1, x2, x3, x4, x5) = U12_GGA(x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x3, x5)
U18_GA(x1, x2, x3, x4, x5, x6) = U18_GA(x3, x5, x6)
U19_GA(x1, x2, x3, x4, x5) = U19_GA(x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(8) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U13_GA(T23, T24, T25, T14, split17_in_agaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_AGAA(T24, T25, X52, X51)
SPLIT17_IN_AGAA(T36, T37, .(T36, X81), X82) → U2_AGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_AGAA(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) → U14_GA(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U15_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U16_GA(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U16_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U17_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U16_GA(T23, T24, T25, T14, T28, mergesortc1_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_agaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_AGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), .(T104, X185)) → U8_GA(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U8_GA(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U9_GA(T61, T62, T63, T104, X185, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U9_GA(T61, T62, T63, T104, X185, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U10_GA(T61, T62, T63, T104, X185, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U10_GA(T61, T62, T63, T104, X185, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U11_GA(T61, T62, T63, T104, X185, merge33_in_gga(.(T104, T95), T97, X185))
U10_GA(T61, T62, T63, T104, X185, T95, mergesortc32_out_ga(T68, .(T104, T97))) → MERGE33_IN_GGA(.(T104, T95), T97, X185)
MERGE33_IN_GGA(.(T143, T132), .(T143, T134), .(T143, T136)) → U12_GGA(T143, T132, T134, T136, merge33_in_gga(.(T143, T132), T134, T136))
MERGE33_IN_GGA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_GGA(.(T143, T132), T134, T136)
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U7_GA(T61, T62, T63, X137, mergesort32_in_ga(T68, X136))
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)
U16_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U18_GA(T23, T24, T25, T14, T46, mergesortc32_in_ga(T28, T49))
U18_GA(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U19_GA(T23, T24, T25, T14, merge33_in_gga(T46, T49, T14))
U18_GA(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → MERGE33_IN_GGA(T46, T49, T14)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T23, .(T24, T25)), T14) → U22_ga(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U22_ga(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_in_ga(.(T23, T29), T46))
U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_in_ga(T28, T49))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
mergec33_in_gga([], T115, T115) → mergec33_out_gga([], T115, T115)
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
mergec33_in_ggg([], T115, T115) → mergec33_out_ggg([], T115, T115)
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U25_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 22 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

MERGE33_IN_GGA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_GGA(.(T143, T132), T134, T136)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T23, .(T24, T25)), T14) → U22_ga(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U22_ga(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_in_ga(.(T23, T29), T46))
U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_in_ga(T28, T49))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
mergec33_in_gga([], T115, T115) → mergec33_out_gga([], T115, T115)
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
mergec33_in_ggg([], T115, T115) → mergec33_out_ggg([], T115, T115)
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U25_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U21_gaa(x1, x2, x3, x4, x5) = U21_gaa(x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x3, x5)
U23_ga(x1, x2, x3, x4, x5, x6, x7) = U23_ga(x3, x5, x7)
U24_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U24_ga(x3, x7, x8)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6, x7) = U28_ga(x3, x6, x7)
mergesortc32_in_gg(x1, x2) = mergesortc32_in_gg(x1, x2)
mergesortc32_out_gg(x1, x2) = mergesortc32_out_gg(x1, x2)
U27_gg(x1, x2, x3, x4, x5) = U27_gg(x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7) = U28_gg(x3, x4, x6, x7)
U30_gg(x1, x2, x3, x4, x5) = U30_gg(x3, x4, x5)
U31_gg(x1, x2, x3, x4, x5, x6, x7) = U31_gg(x3, x4, x6, x7)
U33_gg(x1, x2, x3, x4, x5, x6) = U33_gg(x3, x5, x6)
U34_gg(x1, x2, x3, x4, x5, x6, x7, x8) = U34_gg(x3, x5, x7, x8)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x3, x5)
U31_ga(x1, x2, x3, x4, x5, x6, x7) = U31_ga(x3, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x3, x6)
U34_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U34_ga(x3, x7, x8)
splitc17_in_agaa(x1, x2, x3, x4) = splitc17_in_agaa(x2)
U26_agaa(x1, x2, x3, x4, x5) = U26_agaa(x2, x5)
splitc17_out_agaa(x1, x2, x3, x4) = splitc17_out_agaa(x2, x3, x4)
U35_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ga(x3, x8, x9)
U36_ga(x1, x2, x3, x4, x5, x6) = U36_ga(x3, x6)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U37_gga(x1, x2, x3, x4, x5) = U37_gga(x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x3, x4, x5)
U35_gg(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_gg(x3, x5, x8, x9)
U36_gg(x1, x2, x3, x4, x5, x6) = U36_gg(x3, x5, x6)
mergec33_in_ggg(x1, x2, x3) = mergec33_in_ggg(x1, x2, x3)
mergec33_out_ggg(x1, x2, x3) = mergec33_out_ggg(x1, x2, x3)
U37_ggg(x1, x2, x3, x4, x5) = U37_ggg(x2, x3, x4, x5)
U32_gg(x1, x2, x3, x4, x5) = U32_gg(x3, x4, x5)
U29_gg(x1, x2, x3, x4, x5) = U29_gg(x3, x4, x5)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x3, x5)
U25_ga(x1, x2, x3, x4, x5) = U25_ga(x3, x5)
MERGE33_IN_GGA(x1, x2, x3) = MERGE33_IN_GGA(x1, x2)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MERGE33_IN_GGA(.(T143, T132), .(T143, T134), .(T143, T136)) → MERGE33_IN_GGA(.(T143, T132), T134, T136)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
MERGE33_IN_GGA(x1, x2, x3) = MERGE33_IN_GGA(x1, x2)

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

MERGE33_IN_GGA(.(T132), .(T134)) → MERGE33_IN_GGA(.(T132), T134)

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

(16) 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:

MERGE33_IN_GGA(.(T132), .(T134)) → MERGE33_IN_GGA(.(T132), T134)
The graph contains the following edges 1 >= 1, 2 > 2

(17) YES

(18) 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:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T23, .(T24, T25)), T14) → U22_ga(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U22_ga(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_in_ga(.(T23, T29), T46))
U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_in_ga(T28, T49))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
mergec33_in_gga([], T115, T115) → mergec33_out_gga([], T115, T115)
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
mergec33_in_ggg([], T115, T115) → mergec33_out_ggg([], T115, T115)
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U25_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U21_gaa(x1, x2, x3, x4, x5) = U21_gaa(x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x3, x5)
U23_ga(x1, x2, x3, x4, x5, x6, x7) = U23_ga(x3, x5, x7)
U24_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U24_ga(x3, x7, x8)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6, x7) = U28_ga(x3, x6, x7)
mergesortc32_in_gg(x1, x2) = mergesortc32_in_gg(x1, x2)
mergesortc32_out_gg(x1, x2) = mergesortc32_out_gg(x1, x2)
U27_gg(x1, x2, x3, x4, x5) = U27_gg(x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7) = U28_gg(x3, x4, x6, x7)
U30_gg(x1, x2, x3, x4, x5) = U30_gg(x3, x4, x5)
U31_gg(x1, x2, x3, x4, x5, x6, x7) = U31_gg(x3, x4, x6, x7)
U33_gg(x1, x2, x3, x4, x5, x6) = U33_gg(x3, x5, x6)
U34_gg(x1, x2, x3, x4, x5, x6, x7, x8) = U34_gg(x3, x5, x7, x8)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x3, x5)
U31_ga(x1, x2, x3, x4, x5, x6, x7) = U31_ga(x3, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x3, x6)
U34_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U34_ga(x3, x7, x8)
splitc17_in_agaa(x1, x2, x3, x4) = splitc17_in_agaa(x2)
U26_agaa(x1, x2, x3, x4, x5) = U26_agaa(x2, x5)
splitc17_out_agaa(x1, x2, x3, x4) = splitc17_out_agaa(x2, x3, x4)
U35_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ga(x3, x8, x9)
U36_ga(x1, x2, x3, x4, x5, x6) = U36_ga(x3, x6)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U37_gga(x1, x2, x3, x4, x5) = U37_gga(x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x3, x4, x5)
U35_gg(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_gg(x3, x5, x8, x9)
U36_gg(x1, x2, x3, x4, x5, x6) = U36_gg(x3, x5, x6)
mergec33_in_ggg(x1, x2, x3) = mergec33_in_ggg(x1, x2, x3)
mergec33_out_ggg(x1, x2, x3) = mergec33_out_ggg(x1, x2, x3)
U37_ggg(x1, x2, x3, x4, x5) = U37_ggg(x2, x3, x4, x5)
U32_gg(x1, x2, x3, x4, x5) = U32_gg(x3, x4, x5)
U29_gg(x1, x2, x3, x4, x5) = U29_gg(x3, x4, x5)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x3, x5)
U25_ga(x1, x2, x3, x4, x5) = U25_ga(x3, x5)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) 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) = .(x2)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

SPLIT21_IN_GAA(.(T43)) → SPLIT21_IN_GAA(T43)

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

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

(24) YES

(25) 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, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T23, .(T24, T25)), T14) → U22_ga(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U22_ga(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_in_ga(.(T23, T29), T46))
U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_in_ga(T28, T49))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
mergec33_in_gga([], T115, T115) → mergec33_out_gga([], T115, T115)
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
mergec33_in_ggg([], T115, T115) → mergec33_out_ggg([], T115, T115)
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U25_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U21_gaa(x1, x2, x3, x4, x5) = U21_gaa(x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x3, x5)
U23_ga(x1, x2, x3, x4, x5, x6, x7) = U23_ga(x3, x5, x7)
U24_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U24_ga(x3, x7, x8)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6, x7) = U28_ga(x3, x6, x7)
mergesortc32_in_gg(x1, x2) = mergesortc32_in_gg(x1, x2)
mergesortc32_out_gg(x1, x2) = mergesortc32_out_gg(x1, x2)
U27_gg(x1, x2, x3, x4, x5) = U27_gg(x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7) = U28_gg(x3, x4, x6, x7)
U30_gg(x1, x2, x3, x4, x5) = U30_gg(x3, x4, x5)
U31_gg(x1, x2, x3, x4, x5, x6, x7) = U31_gg(x3, x4, x6, x7)
U33_gg(x1, x2, x3, x4, x5, x6) = U33_gg(x3, x5, x6)
U34_gg(x1, x2, x3, x4, x5, x6, x7, x8) = U34_gg(x3, x5, x7, x8)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x3, x5)
U31_ga(x1, x2, x3, x4, x5, x6, x7) = U31_ga(x3, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x3, x6)
U34_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U34_ga(x3, x7, x8)
splitc17_in_agaa(x1, x2, x3, x4) = splitc17_in_agaa(x2)
U26_agaa(x1, x2, x3, x4, x5) = U26_agaa(x2, x5)
splitc17_out_agaa(x1, x2, x3, x4) = splitc17_out_agaa(x2, x3, x4)
U35_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ga(x3, x8, x9)
U36_ga(x1, x2, x3, x4, x5, x6) = U36_ga(x3, x6)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U37_gga(x1, x2, x3, x4, x5) = U37_gga(x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x3, x4, x5)
U35_gg(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_gg(x3, x5, x8, x9)
U36_gg(x1, x2, x3, x4, x5, x6) = U36_gg(x3, x5, x6)
mergec33_in_ggg(x1, x2, x3) = mergec33_in_ggg(x1, x2, x3)
mergec33_out_ggg(x1, x2, x3) = mergec33_out_ggg(x1, x2, x3)
U37_ggg(x1, x2, x3, x4, x5) = U37_ggg(x2, x3, x4, x5)
U32_gg(x1, x2, x3, x4, x5) = U32_gg(x3, x4, x5)
U29_gg(x1, x2, x3, x4, x5) = U29_gg(x3, x4, x5)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x3, x5)
U25_ga(x1, x2, x3, x4, x5) = U25_ga(x3, x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x3, x5, x6)

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

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) 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, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68, X136)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U21_gaa(x1, x2, x3, x4, x5) = U21_gaa(x2, x5)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6, x7) = U28_ga(x3, x6, x7)
mergesortc32_in_gg(x1, x2) = mergesortc32_in_gg(x1, x2)
mergesortc32_out_gg(x1, x2) = mergesortc32_out_gg(x1, x2)
U27_gg(x1, x2, x3, x4, x5) = U27_gg(x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7) = U28_gg(x3, x4, x6, x7)
U30_gg(x1, x2, x3, x4, x5) = U30_gg(x3, x4, x5)
U31_gg(x1, x2, x3, x4, x5, x6, x7) = U31_gg(x3, x4, x6, x7)
U33_gg(x1, x2, x3, x4, x5, x6) = U33_gg(x3, x5, x6)
U34_gg(x1, x2, x3, x4, x5, x6, x7, x8) = U34_gg(x3, x5, x7, x8)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x3, x5)
U31_ga(x1, x2, x3, x4, x5, x6, x7) = U31_ga(x3, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x3, x6)
U34_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U34_ga(x3, x7, x8)
splitc17_in_agaa(x1, x2, x3, x4) = splitc17_in_agaa(x2)
U26_agaa(x1, x2, x3, x4, x5) = U26_agaa(x2, x5)
splitc17_out_agaa(x1, x2, x3, x4) = splitc17_out_agaa(x2, x3, x4)
U35_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ga(x3, x8, x9)
U36_ga(x1, x2, x3, x4, x5, x6) = U36_ga(x3, x6)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U37_gga(x1, x2, x3, x4, x5) = U37_gga(x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x3, x4, x5)
U35_gg(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_gg(x3, x5, x8, x9)
U36_gg(x1, x2, x3, x4, x5, x6) = U36_gg(x3, x5, x6)
mergec33_in_ggg(x1, x2, x3) = mergec33_in_ggg(x1, x2, x3)
mergec33_out_ggg(x1, x2, x3) = mergec33_out_ggg(x1, x2, x3)
U37_ggg(x1, x2, x3, x4, x5) = U37_ggg(x2, x3, x4, x5)
U32_gg(x1, x2, x3, x4, x5) = U32_gg(x3, x4, x5)
U29_gg(x1, x2, x3, x4, x5) = U29_gg(x3, x4, x5)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x3, x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x3, x5, x6)

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, splitc17_in_agaa(.(T63)))
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesortc32_in_ga(T67))
U6_GA(T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.([])) → mergesortc32_out_ga(.([]), .([]))
mergesortc32_in_ga(.(.(T63))) → U27_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U30_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U33_ga(T63, splitc17_in_agaa(.(T63)))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U27_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U28_ga(T63, T68, mergesortc32_in_gg(T67, []))
U30_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U31_ga(T63, T68, mergesortc32_in_ga(T67))
U33_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U34_ga(T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U28_ga(T63, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T63, mergesortc32_in_ga(T68))
U31_ga(T63, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T63, T85, mergesortc32_in_gg(T68, []))
U34_ga(T63, T68, mergesortc32_out_ga(T67, .(T95))) → U35_ga(T63, T95, mergesortc32_in_ga(T68))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(.(T63)), T80) → U27_gg(T63, T80, splitc17_in_agaa(.(T63)))
mergesortc32_in_gg(.(.(T63)), T85) → U30_gg(T63, T85, splitc17_in_agaa(.(T63)))
U29_ga(T63, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(.(T63)), T80)
U32_ga(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(.(T63)), T85)
U35_ga(T63, T95, mergesortc32_out_ga(T68, .(T97))) → U36_ga(T63, mergec33_in_gga(.(T95), T97))
U27_gg(T63, T80, splitc17_out_agaa(.(T63), T67, T68)) → U28_gg(T63, T80, T68, mergesortc32_in_gg(T67, []))
U30_gg(T63, T85, splitc17_out_agaa(.(T63), T67, T68)) → U31_gg(T63, T85, T68, mergesortc32_in_gg(T67, T85))
U36_ga(T63, mergec33_out_gga(.(T95), T97, X185)) → mergesortc32_out_ga(.(.(T63)), .(X185))
U28_gg(T63, T80, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T63, T80, mergesortc32_in_gg(T68, T80))
U31_gg(T63, T85, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T63, T85, mergesortc32_in_gg(T68, []))
mergec33_in_gga(T120, []) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T132), .(T134)) → U37_gga(T132, T134, mergec33_in_gga(.(T132), T134))
U29_gg(T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(.(T63)), T80)
mergesortc32_in_gg(.([]), .([])) → mergesortc32_out_gg(.([]), .([]))
mergesortc32_in_gg(.(.(T63)), .(X185)) → U33_gg(T63, X185, splitc17_in_agaa(.(T63)))
U32_gg(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(.(T63)), T85)
U37_gga(T132, T134, mergec33_out_gga(.(T132), T134, T136)) → mergec33_out_gga(.(T132), .(T134), .(T136))
U33_gg(T63, X185, splitc17_out_agaa(.(T63), T67, T68)) → U34_gg(T63, X185, T68, mergesortc32_in_ga(T67))
U34_gg(T63, X185, T68, mergesortc32_out_ga(T67, .(T95))) → U35_gg(T63, X185, T95, mergesortc32_in_ga(T68))
U35_gg(T63, X185, T95, mergesortc32_out_ga(T68, .(T97))) → U36_gg(T63, X185, mergec33_in_ggg(.(T95), T97, X185))
U36_gg(T63, X185, mergec33_out_ggg(.(T95), T97, X185)) → mergesortc32_out_gg(.(.(T63)), .(X185))
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T132), .(T134), .(T136)) → U37_ggg(T132, T134, T136, mergec33_in_ggg(.(T132), T134, T136))
U37_ggg(T132, T134, T136, mergec33_out_ggg(.(T132), T134, T136)) → mergec33_out_ggg(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U26_agaa(x0, x1)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
splitc21_in_gaa(x0)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U21_gaa(x0, x1)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

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

(30) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, splitc17_in_agaa(.(T63))) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), splitc21_in_gaa(.(T63))))

(31) Obligation:

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

U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesortc32_in_ga(T67))
U6_GA(T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), splitc21_in_gaa(.(T63))))

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.([])) → mergesortc32_out_ga(.([]), .([]))
mergesortc32_in_ga(.(.(T63))) → U27_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U30_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U33_ga(T63, splitc17_in_agaa(.(T63)))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U27_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U28_ga(T63, T68, mergesortc32_in_gg(T67, []))
U30_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U31_ga(T63, T68, mergesortc32_in_ga(T67))
U33_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U34_ga(T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U28_ga(T63, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T63, mergesortc32_in_ga(T68))
U31_ga(T63, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T63, T85, mergesortc32_in_gg(T68, []))
U34_ga(T63, T68, mergesortc32_out_ga(T67, .(T95))) → U35_ga(T63, T95, mergesortc32_in_ga(T68))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(.(T63)), T80) → U27_gg(T63, T80, splitc17_in_agaa(.(T63)))
mergesortc32_in_gg(.(.(T63)), T85) → U30_gg(T63, T85, splitc17_in_agaa(.(T63)))
U29_ga(T63, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(.(T63)), T80)
U32_ga(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(.(T63)), T85)
U35_ga(T63, T95, mergesortc32_out_ga(T68, .(T97))) → U36_ga(T63, mergec33_in_gga(.(T95), T97))
U27_gg(T63, T80, splitc17_out_agaa(.(T63), T67, T68)) → U28_gg(T63, T80, T68, mergesortc32_in_gg(T67, []))
U30_gg(T63, T85, splitc17_out_agaa(.(T63), T67, T68)) → U31_gg(T63, T85, T68, mergesortc32_in_gg(T67, T85))
U36_ga(T63, mergec33_out_gga(.(T95), T97, X185)) → mergesortc32_out_ga(.(.(T63)), .(X185))
U28_gg(T63, T80, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T63, T80, mergesortc32_in_gg(T68, T80))
U31_gg(T63, T85, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T63, T85, mergesortc32_in_gg(T68, []))
mergec33_in_gga(T120, []) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T132), .(T134)) → U37_gga(T132, T134, mergec33_in_gga(.(T132), T134))
U29_gg(T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(.(T63)), T80)
mergesortc32_in_gg(.([]), .([])) → mergesortc32_out_gg(.([]), .([]))
mergesortc32_in_gg(.(.(T63)), .(X185)) → U33_gg(T63, X185, splitc17_in_agaa(.(T63)))
U32_gg(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(.(T63)), T85)
U37_gga(T132, T134, mergec33_out_gga(.(T132), T134, T136)) → mergec33_out_gga(.(T132), .(T134), .(T136))
U33_gg(T63, X185, splitc17_out_agaa(.(T63), T67, T68)) → U34_gg(T63, X185, T68, mergesortc32_in_ga(T67))
U34_gg(T63, X185, T68, mergesortc32_out_ga(T67, .(T95))) → U35_gg(T63, X185, T95, mergesortc32_in_ga(T68))
U35_gg(T63, X185, T95, mergesortc32_out_ga(T68, .(T97))) → U36_gg(T63, X185, mergec33_in_ggg(.(T95), T97, X185))
U36_gg(T63, X185, mergec33_out_ggg(.(T95), T97, X185)) → mergesortc32_out_gg(.(.(T63)), .(X185))
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T132), .(T134), .(T136)) → U37_ggg(T132, T134, T136, mergec33_in_ggg(.(T132), T134, T136))
U37_ggg(T132, T134, T136, mergec33_out_ggg(.(T132), T134, T136)) → mergec33_out_ggg(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U26_agaa(x0, x1)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
splitc21_in_gaa(x0)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U21_gaa(x0, x1)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

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

(32) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), splitc21_in_gaa(.(T63)))) at position [1,1] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))

(33) Obligation:

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

U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesortc32_in_ga(T67))
U6_GA(T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.([])) → mergesortc32_out_ga(.([]), .([]))
mergesortc32_in_ga(.(.(T63))) → U27_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U30_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U33_ga(T63, splitc17_in_agaa(.(T63)))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U27_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U28_ga(T63, T68, mergesortc32_in_gg(T67, []))
U30_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U31_ga(T63, T68, mergesortc32_in_ga(T67))
U33_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U34_ga(T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U28_ga(T63, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T63, mergesortc32_in_ga(T68))
U31_ga(T63, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T63, T85, mergesortc32_in_gg(T68, []))
U34_ga(T63, T68, mergesortc32_out_ga(T67, .(T95))) → U35_ga(T63, T95, mergesortc32_in_ga(T68))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(.(T63)), T80) → U27_gg(T63, T80, splitc17_in_agaa(.(T63)))
mergesortc32_in_gg(.(.(T63)), T85) → U30_gg(T63, T85, splitc17_in_agaa(.(T63)))
U29_ga(T63, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(.(T63)), T80)
U32_ga(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(.(T63)), T85)
U35_ga(T63, T95, mergesortc32_out_ga(T68, .(T97))) → U36_ga(T63, mergec33_in_gga(.(T95), T97))
U27_gg(T63, T80, splitc17_out_agaa(.(T63), T67, T68)) → U28_gg(T63, T80, T68, mergesortc32_in_gg(T67, []))
U30_gg(T63, T85, splitc17_out_agaa(.(T63), T67, T68)) → U31_gg(T63, T85, T68, mergesortc32_in_gg(T67, T85))
U36_ga(T63, mergec33_out_gga(.(T95), T97, X185)) → mergesortc32_out_ga(.(.(T63)), .(X185))
U28_gg(T63, T80, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T63, T80, mergesortc32_in_gg(T68, T80))
U31_gg(T63, T85, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T63, T85, mergesortc32_in_gg(T68, []))
mergec33_in_gga(T120, []) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T132), .(T134)) → U37_gga(T132, T134, mergec33_in_gga(.(T132), T134))
U29_gg(T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(.(T63)), T80)
mergesortc32_in_gg(.([]), .([])) → mergesortc32_out_gg(.([]), .([]))
mergesortc32_in_gg(.(.(T63)), .(X185)) → U33_gg(T63, X185, splitc17_in_agaa(.(T63)))
U32_gg(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(.(T63)), T85)
U37_gga(T132, T134, mergec33_out_gga(.(T132), T134, T136)) → mergec33_out_gga(.(T132), .(T134), .(T136))
U33_gg(T63, X185, splitc17_out_agaa(.(T63), T67, T68)) → U34_gg(T63, X185, T68, mergesortc32_in_ga(T67))
U34_gg(T63, X185, T68, mergesortc32_out_ga(T67, .(T95))) → U35_gg(T63, X185, T95, mergesortc32_in_ga(T68))
U35_gg(T63, X185, T95, mergesortc32_out_ga(T68, .(T97))) → U36_gg(T63, X185, mergec33_in_ggg(.(T95), T97, X185))
U36_gg(T63, X185, mergec33_out_ggg(.(T95), T97, X185)) → mergesortc32_out_gg(.(.(T63)), .(X185))
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T132), .(T134), .(T136)) → U37_ggg(T132, T134, T136, mergec33_in_ggg(.(T132), T134, T136))
U37_ggg(T132, T134, T136, mergec33_out_ggg(.(T132), T134, T136)) → mergec33_out_ggg(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U26_agaa(x0, x1)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
splitc21_in_gaa(x0)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U21_gaa(x0, x1)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U6_GA(T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)

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

POL(U4_GA(x₁, x₂)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(splitc17_out_agaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 1 0 /

· x₃

POL(.(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

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

· x₁

POL(U6_GA(x₁, x₂, x₃)) = 0 +
[ 0, 0 ]

· x₁ +
[ 1, 0 ]

· x₂ +
[ 0, 1 ]

· x₃

POL(mergesortc32_in_ga(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 1 /

· x₁

POL(mergesortc32_out_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U26_agaa(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 1 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(U21_gaa(x₁, x₂)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂

POL(splitc21_in_gaa(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL([]) =
/ 0 \

\ 1 /

POL(U27_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(splitc17_in_agaa(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(U30_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U33_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(splitc21_out_gaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 0 /

· x₂ +
/ 1 0 \

\ 1 0 /

· x₃

POL(U28_ga(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(mergesortc32_in_gg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂

POL(mergesortc32_out_gg(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 1 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(U27_gg(x₁, x₂, x₃)) =
/ 1 \

\ 1 /

+
/ 1 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U28_gg(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U30_gg(x₁, x₂, x₃)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U31_gg(x₁, x₂, x₃, x₄)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U33_gg(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U34_gg(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U31_ga(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U34_ga(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U35_ga(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U36_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(mergec33_in_gga(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(U32_ga(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U35_gg(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(U36_gg(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(mergec33_in_ggg(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U32_gg(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U29_gg(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 1 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U29_ga(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(mergec33_out_gga(x₁, x₂, x₃)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U37_gga(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 1 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 1 0 /

· x₃

POL(mergec33_out_ggg(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL(U37_ggg(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 1 1 \

\ 0 0 /

· x₄

The following usable rules [FROCOS05] were oriented:

mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.([])) → mergesortc32_out_ga(.([]), .([]))
mergesortc32_in_ga(.(.(T63))) → U27_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U30_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U33_ga(T63, splitc17_in_agaa(.(T63)))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U27_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U28_ga(T63, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(.(T63)), T80) → U27_gg(T63, T80, splitc17_in_agaa(.(T63)))
U27_gg(T63, T80, splitc17_out_agaa(.(T63), T67, T68)) → U28_gg(T63, T80, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(.(T63)), T85) → U30_gg(T63, T85, splitc17_in_agaa(.(T63)))
U30_gg(T63, T85, splitc17_out_agaa(.(T63), T67, T68)) → U31_gg(T63, T85, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(.(T63)), .(X185)) → U33_gg(T63, X185, splitc17_in_agaa(.(T63)))
U33_gg(T63, X185, splitc17_out_agaa(.(T63), T67, T68)) → U34_gg(T63, X185, T68, mergesortc32_in_ga(T67))
U30_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U31_ga(T63, T68, mergesortc32_in_ga(T67))
U33_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U34_ga(T63, T68, mergesortc32_in_ga(T67))
U34_ga(T63, T68, mergesortc32_out_ga(T67, .(T95))) → U35_ga(T63, T95, mergesortc32_in_ga(T68))
U35_ga(T63, T95, mergesortc32_out_ga(T68, .(T97))) → U36_ga(T63, mergec33_in_gga(.(T95), T97))
U31_ga(T63, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(.(T63)), T85)
U34_gg(T63, X185, T68, mergesortc32_out_ga(T67, .(T95))) → U35_gg(T63, X185, T95, mergesortc32_in_ga(T68))
U35_gg(T63, X185, T95, mergesortc32_out_ga(T68, .(T97))) → U36_gg(T63, X185, mergec33_in_ggg(.(T95), T97, X185))
U31_gg(T63, T85, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(.(T63)), T85)
U28_gg(T63, T80, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(.(T63)), T80)
U28_ga(T63, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T63, mergesortc32_in_ga(T68))
U29_ga(T63, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(.(T63)), T80)
U36_ga(T63, mergec33_out_gga(.(T95), T97, X185)) → mergesortc32_out_ga(.(.(T63)), .(X185))
U36_gg(T63, X185, mergec33_out_ggg(.(T95), T97, X185)) → mergesortc32_out_gg(.(.(T63)), .(X185))

(35) Obligation:

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

U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → U6_GA(T63, T68, mergesortc32_in_ga(T67))
MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.([])) → mergesortc32_out_ga(.([]), .([]))
mergesortc32_in_ga(.(.(T63))) → U27_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U30_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U33_ga(T63, splitc17_in_agaa(.(T63)))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U27_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U28_ga(T63, T68, mergesortc32_in_gg(T67, []))
U30_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U31_ga(T63, T68, mergesortc32_in_ga(T67))
U33_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U34_ga(T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U28_ga(T63, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T63, mergesortc32_in_ga(T68))
U31_ga(T63, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T63, T85, mergesortc32_in_gg(T68, []))
U34_ga(T63, T68, mergesortc32_out_ga(T67, .(T95))) → U35_ga(T63, T95, mergesortc32_in_ga(T68))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(.(T63)), T80) → U27_gg(T63, T80, splitc17_in_agaa(.(T63)))
mergesortc32_in_gg(.(.(T63)), T85) → U30_gg(T63, T85, splitc17_in_agaa(.(T63)))
U29_ga(T63, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(.(T63)), T80)
U32_ga(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(.(T63)), T85)
U35_ga(T63, T95, mergesortc32_out_ga(T68, .(T97))) → U36_ga(T63, mergec33_in_gga(.(T95), T97))
U27_gg(T63, T80, splitc17_out_agaa(.(T63), T67, T68)) → U28_gg(T63, T80, T68, mergesortc32_in_gg(T67, []))
U30_gg(T63, T85, splitc17_out_agaa(.(T63), T67, T68)) → U31_gg(T63, T85, T68, mergesortc32_in_gg(T67, T85))
U36_ga(T63, mergec33_out_gga(.(T95), T97, X185)) → mergesortc32_out_ga(.(.(T63)), .(X185))
U28_gg(T63, T80, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T63, T80, mergesortc32_in_gg(T68, T80))
U31_gg(T63, T85, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T63, T85, mergesortc32_in_gg(T68, []))
mergec33_in_gga(T120, []) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T132), .(T134)) → U37_gga(T132, T134, mergec33_in_gga(.(T132), T134))
U29_gg(T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(.(T63)), T80)
mergesortc32_in_gg(.([]), .([])) → mergesortc32_out_gg(.([]), .([]))
mergesortc32_in_gg(.(.(T63)), .(X185)) → U33_gg(T63, X185, splitc17_in_agaa(.(T63)))
U32_gg(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(.(T63)), T85)
U37_gga(T132, T134, mergec33_out_gga(.(T132), T134, T136)) → mergec33_out_gga(.(T132), .(T134), .(T136))
U33_gg(T63, X185, splitc17_out_agaa(.(T63), T67, T68)) → U34_gg(T63, X185, T68, mergesortc32_in_ga(T67))
U34_gg(T63, X185, T68, mergesortc32_out_ga(T67, .(T95))) → U35_gg(T63, X185, T95, mergesortc32_in_ga(T68))
U35_gg(T63, X185, T95, mergesortc32_out_ga(T68, .(T97))) → U36_gg(T63, X185, mergec33_in_ggg(.(T95), T97, X185))
U36_gg(T63, X185, mergec33_out_ggg(.(T95), T97, X185)) → mergesortc32_out_gg(.(.(T63)), .(X185))
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T132), .(T134), .(T136)) → U37_ggg(T132, T134, T136, mergec33_in_ggg(.(T132), T134, T136))
U37_ggg(T132, T134, T136, mergec33_out_ggg(.(T132), T134, T136)) → mergec33_out_ggg(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U26_agaa(x0, x1)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
splitc21_in_gaa(x0)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U21_gaa(x0, x1)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

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

(36) DependencyGraphProof (EQUIVALENT transformation)

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

(37) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.([])) → mergesortc32_out_ga(.([]), .([]))
mergesortc32_in_ga(.(.(T63))) → U27_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U30_ga(T63, splitc17_in_agaa(.(T63)))
mergesortc32_in_ga(.(.(T63))) → U33_ga(T63, splitc17_in_agaa(.(T63)))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U27_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U28_ga(T63, T68, mergesortc32_in_gg(T67, []))
U30_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U31_ga(T63, T68, mergesortc32_in_ga(T67))
U33_ga(T63, splitc17_out_agaa(.(T63), T67, T68)) → U34_ga(T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U28_ga(T63, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T63, mergesortc32_in_ga(T68))
U31_ga(T63, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T63, T85, mergesortc32_in_gg(T68, []))
U34_ga(T63, T68, mergesortc32_out_ga(T67, .(T95))) → U35_ga(T63, T95, mergesortc32_in_ga(T68))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(.(T63)), T80) → U27_gg(T63, T80, splitc17_in_agaa(.(T63)))
mergesortc32_in_gg(.(.(T63)), T85) → U30_gg(T63, T85, splitc17_in_agaa(.(T63)))
U29_ga(T63, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(.(T63)), T80)
U32_ga(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(.(T63)), T85)
U35_ga(T63, T95, mergesortc32_out_ga(T68, .(T97))) → U36_ga(T63, mergec33_in_gga(.(T95), T97))
U27_gg(T63, T80, splitc17_out_agaa(.(T63), T67, T68)) → U28_gg(T63, T80, T68, mergesortc32_in_gg(T67, []))
U30_gg(T63, T85, splitc17_out_agaa(.(T63), T67, T68)) → U31_gg(T63, T85, T68, mergesortc32_in_gg(T67, T85))
U36_ga(T63, mergec33_out_gga(.(T95), T97, X185)) → mergesortc32_out_ga(.(.(T63)), .(X185))
U28_gg(T63, T80, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T63, T80, mergesortc32_in_gg(T68, T80))
U31_gg(T63, T85, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T63, T85, mergesortc32_in_gg(T68, []))
mergec33_in_gga(T120, []) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T132), .(T134)) → U37_gga(T132, T134, mergec33_in_gga(.(T132), T134))
U29_gg(T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(.(T63)), T80)
mergesortc32_in_gg(.([]), .([])) → mergesortc32_out_gg(.([]), .([]))
mergesortc32_in_gg(.(.(T63)), .(X185)) → U33_gg(T63, X185, splitc17_in_agaa(.(T63)))
U32_gg(T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(.(T63)), T85)
U37_gga(T132, T134, mergec33_out_gga(.(T132), T134, T136)) → mergec33_out_gga(.(T132), .(T134), .(T136))
U33_gg(T63, X185, splitc17_out_agaa(.(T63), T67, T68)) → U34_gg(T63, X185, T68, mergesortc32_in_ga(T67))
U34_gg(T63, X185, T68, mergesortc32_out_ga(T67, .(T95))) → U35_gg(T63, X185, T95, mergesortc32_in_ga(T68))
U35_gg(T63, X185, T95, mergesortc32_out_ga(T68, .(T97))) → U36_gg(T63, X185, mergec33_in_ggg(.(T95), T97, X185))
U36_gg(T63, X185, mergec33_out_ggg(.(T95), T97, X185)) → mergesortc32_out_gg(.(.(T63)), .(X185))
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T132), .(T134), .(T136)) → U37_ggg(T132, T134, T136, mergec33_in_ggg(.(T132), T134, T136))
U37_ggg(T132, T134, T136, mergec33_out_ggg(.(T132), T134, T136)) → mergec33_out_ggg(.(T132), .(T134), .(T136))

The set Q consists of the following terms:

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U26_agaa(x0, x1)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
splitc21_in_gaa(x0)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U21_gaa(x0, x1)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

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

(38) 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.

(39) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)

The set Q consists of the following terms:

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U26_agaa(x0, x1)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
splitc21_in_gaa(x0)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
U21_gaa(x0, x1)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

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

(40) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

splitc17_in_agaa(x0)
mergesortc32_in_ga(x0)
U27_ga(x0, x1)
U30_ga(x0, x1)
U33_ga(x0, x1)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2)
U34_ga(x0, x1, x2)
mergesortc32_in_gg(x0, x1)
U29_ga(x0, x1)
U32_ga(x0, x1, x2)
U35_ga(x0, x1, x2)
U27_gg(x0, x1, x2)
U30_gg(x0, x1, x2)
U36_ga(x0, x1)
U28_gg(x0, x1, x2, x3)
U31_gg(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U29_gg(x0, x1, x2)
U32_gg(x0, x1, x2)
U37_gga(x0, x1, x2)
U33_gg(x0, x1, x2)
U34_gg(x0, x1, x2, x3)
U35_gg(x0, x1, x2, x3)
U36_gg(x0, x1, x2)
mergec33_in_ggg(x0, x1, x2)
U37_ggg(x0, x1, x2, x3)

(41) Obligation:

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

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))
U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)

The set Q consists of the following terms:

U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(42) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT32_IN_GA(.(.(T63))) → U4_GA(T63, U26_agaa(.(T63), U21_gaa(T63, splitc21_in_gaa(T63))))

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

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

· x₁

POL(.(x₁)) =
/ 1 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(U4_GA(x₁, x₂)) = 0 +
[ 0, 0 ]

· x₁ +
[ 0, 1 ]

· x₂

POL(U26_agaa(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂

POL(U21_gaa(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 0 /

· x₂

POL(splitc21_in_gaa(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 0 /

· x₁

POL(splitc17_out_agaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃

POL([]) =
/ 0 \

\ 0 /

POL(splitc21_out_gaa(x₁, x₂, x₃)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 0 /

· x₂ +
/ 1 0 \

\ 0 0 /

· x₃

The following usable rules [FROCOS05] were oriented:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)

(43) Obligation:

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

U4_GA(T63, splitc17_out_agaa(.(T63), T67, T68)) → MERGESORT32_IN_GA(T67)

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)

The set Q consists of the following terms:

U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(44) DependencyGraphProof (EQUIVALENT transformation)

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

(45) TRUE

(46) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U14_GA(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
mergesortc1_in_ga([], []) → mergesortc1_out_ga([], [])
mergesortc1_in_ga(.(T4, []), .(T4, [])) → mergesortc1_out_ga(.(T4, []), .(T4, []))
mergesortc1_in_ga(.(T23, .(T24, T25)), T14) → U22_ga(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U22_ga(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_in_ga(.(T23, T29), T46))
U23_ga(T23, T24, T25, T14, T28, T29, mergesortc1_out_ga(.(T23, T29), T46)) → U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_in_ga(T28, T49))
mergesortc32_in_ga([], []) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, []), .(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63)), T80) → U27_ga(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_ga(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg([], []) → mergesortc32_out_gg([], [])
mergesortc32_in_gg(.(T54, []), .(T54, [])) → mergesortc32_out_gg(.(T54, []), .(T54, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T80) → U27_gg(T61, T62, T63, T80, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U27_gg(T61, T62, T63, T80, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_in_gg(T67, []))
mergesortc32_in_gg(.(T61, .(T62, T63)), T85) → U30_gg(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_gg(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_in_gg(T67, T85))
mergesortc32_in_gg(.(T61, .(T62, T63)), .(T104, X185)) → U33_gg(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_gg(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
mergesortc32_in_ga(.(T61, .(T62, T63)), T85) → U30_ga(T61, T62, T63, T85, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U30_ga(T61, T62, T63, T85, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_in_ga(T67, T85))
mergesortc32_in_ga(.(T61, .(T62, T63)), .(T104, X185)) → U33_ga(T61, T62, T63, T104, X185, splitc17_in_agaa(T61, .(T62, T63), T67, T68))
U33_ga(T61, T62, T63, T104, X185, splitc17_out_agaa(T61, .(T62, T63), T67, T68)) → U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_in_ga(T67, .(T104, T95)))
U34_ga(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_ga(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_ga(T61, T62, T63, T104, X185, mergec33_in_gga(.(T104, T95), T97, X185))
mergec33_in_gga([], T115, T115) → mergec33_out_gga([], T115, T115)
mergec33_in_gga(T120, [], T120) → mergec33_out_gga(T120, [], T120)
mergec33_in_gga(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_gga(T143, T132, T134, T136, mergec33_in_gga(.(T143, T132), T134, T136))
U37_gga(T143, T132, T134, T136, mergec33_out_gga(.(T143, T132), T134, T136)) → mergec33_out_gga(.(T143, T132), .(T143, T134), .(T143, T136))
U36_ga(T61, T62, T63, T104, X185, mergec33_out_gga(.(T104, T95), T97, X185)) → mergesortc32_out_ga(.(T61, .(T62, T63)), .(T104, X185))
U31_ga(T61, T62, T63, T85, T67, T68, mergesortc32_out_ga(T67, T85)) → U32_ga(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_ga(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_ga(.(T61, .(T62, T63)), T85)
U34_gg(T61, T62, T63, T104, X185, T67, T68, mergesortc32_out_ga(T67, .(T104, T95))) → U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_in_ga(T68, .(T104, T97)))
U35_gg(T61, T62, T63, T104, X185, T67, T68, T95, mergesortc32_out_ga(T68, .(T104, T97))) → U36_gg(T61, T62, T63, T104, X185, mergec33_in_ggg(.(T104, T95), T97, X185))
mergec33_in_ggg([], T115, T115) → mergec33_out_ggg([], T115, T115)
mergec33_in_ggg(T120, [], T120) → mergec33_out_ggg(T120, [], T120)
mergec33_in_ggg(.(T143, T132), .(T143, T134), .(T143, T136)) → U37_ggg(T143, T132, T134, T136, mergec33_in_ggg(.(T143, T132), T134, T136))
U37_ggg(T143, T132, T134, T136, mergec33_out_ggg(.(T143, T132), T134, T136)) → mergec33_out_ggg(.(T143, T132), .(T143, T134), .(T143, T136))
U36_gg(T61, T62, T63, T104, X185, mergec33_out_ggg(.(T104, T95), T97, X185)) → mergesortc32_out_gg(.(T61, .(T62, T63)), .(T104, X185))
U31_gg(T61, T62, T63, T85, T67, T68, mergesortc32_out_gg(T67, T85)) → U32_gg(T61, T62, T63, T85, mergesortc32_in_gg(T68, []))
U32_gg(T61, T62, T63, T85, mergesortc32_out_gg(T68, [])) → mergesortc32_out_gg(.(T61, .(T62, T63)), T85)
U28_gg(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_gg(T61, T62, T63, T80, mergesortc32_in_gg(T68, T80))
U29_gg(T61, T62, T63, T80, mergesortc32_out_gg(T68, T80)) → mergesortc32_out_gg(.(T61, .(T62, T63)), T80)
U28_ga(T61, T62, T63, T80, T67, T68, mergesortc32_out_gg(T67, [])) → U29_ga(T61, T62, T63, T80, mergesortc32_in_ga(T68, T80))
U29_ga(T61, T62, T63, T80, mergesortc32_out_ga(T68, T80)) → mergesortc32_out_ga(.(T61, .(T62, T63)), T80)
U24_ga(T23, T24, T25, T14, T28, T29, T46, mergesortc32_out_ga(T28, T49)) → U25_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U25_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x2)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U21_gaa(x1, x2, x3, x4, x5) = U21_gaa(x2, x5)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U22_ga(x1, x2, x3, x4, x5) = U22_ga(x3, x5)
U23_ga(x1, x2, x3, x4, x5, x6, x7) = U23_ga(x3, x5, x7)
U24_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U24_ga(x3, x7, x8)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6, x7) = U28_ga(x3, x6, x7)
mergesortc32_in_gg(x1, x2) = mergesortc32_in_gg(x1, x2)
mergesortc32_out_gg(x1, x2) = mergesortc32_out_gg(x1, x2)
U27_gg(x1, x2, x3, x4, x5) = U27_gg(x3, x4, x5)
U28_gg(x1, x2, x3, x4, x5, x6, x7) = U28_gg(x3, x4, x6, x7)
U30_gg(x1, x2, x3, x4, x5) = U30_gg(x3, x4, x5)
U31_gg(x1, x2, x3, x4, x5, x6, x7) = U31_gg(x3, x4, x6, x7)
U33_gg(x1, x2, x3, x4, x5, x6) = U33_gg(x3, x5, x6)
U34_gg(x1, x2, x3, x4, x5, x6, x7, x8) = U34_gg(x3, x5, x7, x8)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x3, x5)
U31_ga(x1, x2, x3, x4, x5, x6, x7) = U31_ga(x3, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x3, x6)
U34_ga(x1, x2, x3, x4, x5, x6, x7, x8) = U34_ga(x3, x7, x8)
splitc17_in_agaa(x1, x2, x3, x4) = splitc17_in_agaa(x2)
U26_agaa(x1, x2, x3, x4, x5) = U26_agaa(x2, x5)
splitc17_out_agaa(x1, x2, x3, x4) = splitc17_out_agaa(x2, x3, x4)
U35_ga(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_ga(x3, x8, x9)
U36_ga(x1, x2, x3, x4, x5, x6) = U36_ga(x3, x6)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U37_gga(x1, x2, x3, x4, x5) = U37_gga(x2, x3, x5)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x3, x4, x5)
U35_gg(x1, x2, x3, x4, x5, x6, x7, x8, x9) = U35_gg(x3, x5, x8, x9)
U36_gg(x1, x2, x3, x4, x5, x6) = U36_gg(x3, x5, x6)
mergec33_in_ggg(x1, x2, x3) = mergec33_in_ggg(x1, x2, x3)
mergec33_out_ggg(x1, x2, x3) = mergec33_out_ggg(x1, x2, x3)
U37_ggg(x1, x2, x3, x4, x5) = U37_ggg(x2, x3, x4, x5)
U32_gg(x1, x2, x3, x4, x5) = U32_gg(x3, x4, x5)
U29_gg(x1, x2, x3, x4, x5) = U29_gg(x3, x4, x5)
U29_ga(x1, x2, x3, x4, x5) = U29_ga(x3, x5)
U25_ga(x1, x2, x3, x4, x5) = U25_ga(x3, x5)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x3, x5)

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

(47) UsableRulesProof (EQUIVALENT transformation)

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

(48) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U14_GA(T23, T24, T25, T14, splitc17_in_agaa(T24, T25, T28, T29))
U14_GA(T23, T24, T25, T14, splitc17_out_agaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

splitc17_in_agaa(T36, T37, .(T36, X81), X82) → U26_agaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
U26_agaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T36, T37, .(T36, X81), X82)
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U21_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U21_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

(49) PiDPToQDPProof (SOUND transformation)

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

(50) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, splitc17_in_agaa(T25))
U14_GA(T25, splitc17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

splitc17_in_agaa(x0)
U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(51) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, splitc17_in_agaa(T25)) at position [1] we obtained the following new rules [LPAR04]:

MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, U26_agaa(T25, splitc21_in_gaa(T25)))

(52) Obligation:

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

U14_GA(T25, splitc17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, U26_agaa(T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc17_in_agaa(T37) → U26_agaa(T37, splitc21_in_gaa(T37))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

splitc17_in_agaa(x0)
U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(53) 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.

(54) Obligation:

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

U14_GA(T25, splitc17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, U26_agaa(T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

splitc17_in_agaa(x0)
U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(55) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

splitc17_in_agaa(x0)

(56) Obligation:

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

U14_GA(T25, splitc17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))
MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, U26_agaa(T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U14_GA(T25, splitc17_out_agaa(T25, T28, T29)) → MERGESORT1_IN_GA(.(T29))

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

POL(.(x₁)) = 1 + x₁
POL(MERGESORT1_IN_GA(x₁)) = x₁
POL(U14_GA(x₁, x₂)) = 1 + x₂
POL(U21_gaa(x₁, x₂)) = 1 + x₂
POL(U26_agaa(x₁, x₂)) = 1 + x₂
POL([]) = 0
POL(splitc17_out_agaa(x₁, x₂, x₃)) = 1 + x₃
POL(splitc21_in_gaa(x₁)) = x₁
POL(splitc21_out_gaa(x₁, x₂, x₃)) = x₂ + x₃

The following usable rules [FROCOS05] were oriented:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)

(58) Obligation:

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

MERGESORT1_IN_GA(.(.(T25))) → U14_GA(T25, U26_agaa(T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T43)) → U21_gaa(T43, splitc21_in_gaa(T43))
U26_agaa(T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_agaa(T37, .(X81), X82)
U21_gaa(T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T43), .(X99), X100)

The set Q consists of the following terms:

U26_agaa(x0, x1)
splitc21_in_gaa(x0)
U21_gaa(x0, x1)

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

(59) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PredefinedPredicateTransformerProof (SOUND transformation)

(2) Obligation:

(3) BuiltinConflictTransformerProof (SOUND transformation)

(4) Obligation:

(5) PrologToDTProblemTransformerProof (SOUND transformation)

(6) Obligation:

(7) TriplesToPiDPProof (SOUND transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Complex Obligation (AND)

(11) Obligation:

(12) UsableRulesProof (EQUIVALENT transformation)

(13) Obligation:

(14) PiDPToQDPProof (SOUND transformation)

(15) Obligation:

(16) QDPSizeChangeProof (EQUIVALENT transformation)

(17) YES

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) PiDPToQDPProof (SOUND transformation)

(22) Obligation:

(23) QDPSizeChangeProof (EQUIVALENT transformation)

(24) YES

(25) Obligation:

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

(28) PiDPToQDPProof (SOUND transformation)

(29) Obligation:

(30) Rewriting (EQUIVALENT transformation)

(31) Obligation:

(32) Rewriting (EQUIVALENT transformation)

(33) Obligation:

(34) QDPOrderProof (EQUIVALENT transformation)

(35) Obligation:

(36) DependencyGraphProof (EQUIVALENT transformation)

(37) Obligation:

(38) UsableRulesProof (EQUIVALENT transformation)

(39) Obligation:

(40) QReductionProof (EQUIVALENT transformation)

(41) Obligation:

(42) QDPOrderProof (EQUIVALENT transformation)

(43) Obligation:

(44) DependencyGraphProof (EQUIVALENT transformation)

(45) TRUE

(46) Obligation:

(47) UsableRulesProof (EQUIVALENT transformation)

(48) Obligation:

(49) PiDPToQDPProof (SOUND transformation)

(50) Obligation:

(51) Rewriting (EQUIVALENT transformation)

(52) Obligation:

(53) UsableRulesProof (EQUIVALENT transformation)

(54) Obligation:

(55) QReductionProof (EQUIVALENT transformation)

(56) Obligation:

(57) QDPOrderProof (EQUIVALENT transformation)

(58) Obligation:

(59) DependencyGraphProof (EQUIVALENT transformation)

(60) TRUE