Left Termination proof of /tmp/tmpeZzxXr/pl8.2.1.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 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇔)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 MRRProof (⇔)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔)

        ↳29 QDP

        ↳30 UsableRulesProof (⇔)

        ↳31 QDP

        ↳32 QReductionProof (⇔)

        ↳33 QDP

        ↳34 QDPSizeChangeProof (⇔)

        ↳35 YES

    ↳36 PiDP

        ↳37 UsableRulesProof (⇔)

        ↳38 PiDP

        ↳39 PiDPToQDPProof (⇐)

        ↳40 QDP

        ↳41 QDPSizeChangeProof (⇔)

        ↳42 YES

    ↳43 PiDP

        ↳44 UsableRulesProof (⇔)

        ↳45 PiDP

        ↳46 PiDPToQDPProof (⇐)

        ↳47 QDP

        ↳48 Rewriting (⇔)

        ↳49 QDP

        ↳50 Rewriting (⇔)

        ↳51 QDP

        ↳52 QDPOrderProof (⇔)

        ↳53 QDP

        ↳54 DependencyGraphProof (⇔)

        ↳55 TRUE

    ↳56 PiDP

        ↳57 UsableRulesProof (⇔)

        ↳58 PiDP

        ↳59 PiDPToQDPProof (⇐)

        ↳60 QDP

        ↳61 Rewriting (⇔)

        ↳62 QDP

        ↳63 UsableRulesProof (⇔)

        ↳64 QDP

        ↳65 QReductionProof (⇔)

        ↳66 QDP

        ↳67 QDPOrderProof (⇔)

        ↳68 QDP

        ↳69 DependencyGraphProof (⇔)

        ↳70 TRUE

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(E, []), .(E, [])).
mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V)))).
merge(X, [], X).
merge([], X, X).
merge(.(A, X), .(B, Y), .(A, Z)) :- ','(le(A, B), merge(X, .(B, Y), Z)).
merge(.(A, X), .(B, Y), .(B, Z)) :- ','(gt(A, B), merge(.(A, X), Y, Z)).
split([], [], []).
split(.(E, U), .(E, V), W) :- split(U, W, V).
gt(s(X), s(Y)) :- gt(X, Y).
gt(s(X), 0).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(Y)).
le(0, 0).

Queries:

mergesort(g,a).

(1) 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: 7 | ','(split(.(T10, .(T11, T12)), X21, X22), ','(mergesort(X21, X23), ','(mergesort(X22, X24), merge(X23, X24, T14)))), SCOPE: 2, CLAUSE: 8\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: 8\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: 7 | split(.(T24, T25), X52, X51), SCOPE: 3, CLAUSE: 8\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: 8\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: 7 | split(T37, X82, X81), SCOPE: 4, CLAUSE: 8\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: 7\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: 8\n\nT37 is ground\nX81 is free; \nX82 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: split(T43, X100, X99)\n\nT43 is ground\nX99 is free; \nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: mergesort(.(T23, T29), X23)\n\nT23 is ground\nT29 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(mergesort(T28, X24), merge(T46, X24, T14))\n\nT28 is ground\nT46 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: mergesort(T28, X24)\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: merge(T46, T49, T14)\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: mergesort(T28, X24), SCOPE: 5, CLAUSE: 0 | mergesort(T28, X24), SCOPE: 5, CLAUSE: 1 | mergesort(T28, X24), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: mergesort(T28, X24), SCOPE: 5, CLAUSE: 0\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: mergesort(T28, X24), SCOPE: 5, CLAUSE: 1 | mergesort(T28, X24), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: mergesort(T28, X24), SCOPE: 5, CLAUSE: 1\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: mergesort(T28, X24), SCOPE: 5, CLAUSE: 2\n\nT28 is ground\nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(split(.(T61, .(T62, T63)), X133, X134), ','(mergesort(X133, X135), ','(mergesort(X134, X136), merge(X135, X136, X137))))\n\nT61 is ground\nT62 is ground\nT63 is ground\nX137 is free; \nX133 is free; \nX134 is free; \nX135 is free; \nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: split(.(T61, .(T62, T63)), X133, X134)\n\nT61 is ground\nT62 is ground\nT63 is ground\nX133 is free; \nX134 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(mergesort(T67, X135), ','(mergesort(T68, X136), merge(X135, X136, X137)))\n\nT67 is ground\nT68 is ground\nX137 is free; \nX135 is free; \nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: mergesort(T67, X135)\n\nT67 is ground\nX135 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(mergesort(T68, X136), merge(T72, X136, X137))\n\nT68 is ground\nT72 is ground\nX137 is free; \nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: mergesort(T68, X136)\n\nT68 is ground\nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: merge(T72, T73, X137)\n\nT72 is ground\nT73 is ground\nX137 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 3 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 4 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 5 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 3\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 4 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 5 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 4\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 5 | merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 5\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: merge(T46, T49, T14), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(le(T108, T110), merge(T109, .(T110, T111), T113))\n\nT108 is ground\nT109 is ground\nT110 is ground\nT111 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: le(T108, T110)\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: merge(T109, .(T110, T111), T113)\n\nT109 is ground\nT110 is ground\nT111 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: le(T108, T110), SCOPE: 7, CLAUSE: 11 | le(T108, T110), SCOPE: 7, CLAUSE: 12 | le(T108, T110), SCOPE: 7, CLAUSE: 13\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: le(T108, T110), SCOPE: 7, CLAUSE: 11\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: le(T108, T110), SCOPE: 7, CLAUSE: 12 | le(T108, T110), SCOPE: 7, CLAUSE: 13\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: le(T126, T127)\n\nT126 is ground\nT127 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: le(T108, T110), SCOPE: 7, CLAUSE: 12\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: le(T108, T110), SCOPE: 7, CLAUSE: 13\n\nT108 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(gt(T151, T153), merge(.(T151, T152), T154, T156))\n\nT151 is ground\nT152 is ground\nT153 is ground\nT154 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: gt(T151, T153)\n\nT151 is ground\nT153 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: merge(.(T151, T152), T154, T156)\n\nT151 is ground\nT152 is ground\nT154 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: gt(T151, T153), SCOPE: 8, CLAUSE: 9 | gt(T151, T153), SCOPE: 8, CLAUSE: 10\n\nT151 is ground\nT153 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: gt(T151, T153), SCOPE: 8, CLAUSE: 9\n\nT151 is ground\nT153 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: gt(T151, T153), SCOPE: 8, CLAUSE: 10\n\nT151 is ground\nT153 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: gt(T169, T170)\n\nT169 is ground\nT170 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 95 [arrowhead = none ];
95 -> 2;

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

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

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

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

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

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

102 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
6 -> 102 [arrowhead = none ];
102 -> 7;

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

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

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

106 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(.(E, .(F, U)), W, Y), ','(mergesort(W, X), ','(mergesort(Y, Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 106 [arrowhead = none ];
106 -> 11;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

167 [label="EVAL with clause\nmerge(.(X238, X239), .(X240, X241), .(X240, X242)) :- ','(gt(X238, X240), merge(.(X238, X239), X241, X242)).\nand substitution\nX238 -> T151\nX239 -> T152\nT46 -> .(T151, T152)\nX240 -> T153\nX241 -> T154\nT49 -> .(T153, T154)\nX242 -> T156\nT14 -> .(T153, T156)\nT155 -> T156", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 167 [arrowhead = none ];
167 -> 83;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

189 [label="INSTANCE with matching:\nT46 -> .(T151, T152)\nT49 -> T154\nT14 -> T156", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 189 [arrowhead = none , style=dashed];
189 -> 33[style=dashed];

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

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

192 [label="EVAL with clause\ngt(s(X260), s(X261)) :- gt(X260, X261).\nand substitution\nX260 -> T169\nT151 -> s(T169)\nX261 -> T170\nT153 -> s(T170)", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 192 [arrowhead = none ];
192 -> 90;

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

194 [label="EVAL with clause\ngt(s(X268), 0).\nand substitution\nX268 -> T175\nT151 -> s(T175)\nT153 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 194 [arrowhead = none ];
194 -> 92;

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

196 [label="INSTANCE with matching:\nT151 -> T169\nT153 -> T170", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 196 [arrowhead = none , style=dashed];
196 -> 85[style=dashed];

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

}

(2) Obligation:

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)), X137) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, T72), ','(mergesortc32(T68, T73), merge33(T72, T73, X137)))).
merge33(.(T108, T109), .(T110, T111), .(T108, T113)) :- le68(T108, T110).
merge33(.(T108, T109), .(T110, T111), .(T108, T113)) :- ','(lec68(T108, T110), merge33(T109, .(T110, T111), T113)).
merge33(.(T151, T152), .(T153, T154), .(T153, T156)) :- gt85(T151, T153).
merge33(.(T151, T152), .(T153, T154), .(T153, T156)) :- ','(gtc85(T151, T153), merge33(.(T151, T152), T154, T156)).
le68(s(T126), s(T127)) :- le68(T126, T127).
gt85(s(T169), s(T170)) :- gt85(T169, T170).
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)), X137) :- ','(splitc17(T61, .(T62, T63), T67, T68), ','(mergesortc32(T67, T72), ','(mergesortc32(T68, T73), mergec33(T72, T73, X137)))).
mergec33(T82, [], T82).
mergec33([], T87, T87).
mergec33(.(T108, T109), .(T110, T111), .(T108, T113)) :- ','(lec68(T108, T110), mergec33(T109, .(T110, T111), T113)).
mergec33(.(T151, T152), .(T153, T154), .(T153, T156)) :- ','(gtc85(T151, T153), mergec33(.(T151, T152), T154, T156)).
lec68(s(T126), s(T127)) :- lec68(T126, T127).
lec68(0, s(T134)).
lec68(0, 0).
gtc85(s(T169), s(T170)) :- gtc85(T169, T170).
gtc85(s(T175), 0).

Afs:

mergesort1(x1, x2) = mergesort1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesort1_in: (b,f)
split17_in: (b,b,f,f)
split21_in: (b,f,f)
splitc17_in: (b,b,f,f)
splitc21_in: (b,f,f)
mergesortc1_in: (b,f)
mergesortc32_in: (b,f)
mergec33_in: (b,b,f)
lec68_in: (b,b)
gtc85_in: (b,b)
mergesort32_in: (b,f)
merge33_in: (b,b,f)
le68_in: (b,b)
gt85_in: (b,b)
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) → U18_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U20_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U21_GA(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U21_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U22_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U21_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_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(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)
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_gga(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → MERGE33_IN_GGA(T72, T73, X137)
MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_GGA(T108, T109, T110, T111, T113, le68_in_gg(T108, T110))
MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_GG(T108, T110)
LE68_IN_GG(s(T126), s(T127)) → U16_GG(T126, T127, le68_in_gg(T126, T127))
LE68_IN_GG(s(T126), s(T127)) → LE68_IN_GG(T126, T127)
MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → U11_GGA(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
U11_GGA(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U12_GGA(T108, T109, T110, T111, T113, merge33_in_gga(T109, .(T110, T111), T113))
U11_GGA(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → MERGE33_IN_GGA(T109, .(T110, T111), T113)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → U13_GGA(T151, T152, T153, T154, T156, gt85_in_gg(T151, T153))
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → GT85_IN_GG(T151, T153)
GT85_IN_GG(s(T169), s(T170)) → U17_GG(T169, T170, gt85_in_gg(T169, T170))
GT85_IN_GG(s(T169), s(T170)) → GT85_IN_GG(T169, T170)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → U14_GGA(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
U14_GGA(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U15_GGA(T151, T152, T153, T154, T156, merge33_in_gga(.(T151, T152), T154, T156))
U14_GGA(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154, T156)
U21_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U23_GA(T23, T24, T25, T14, T46, mergesortc32_in_ga(T28, T49))
U23_GA(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U24_GA(T23, T24, T25, T14, merge33_in_gga(T46, T49, T14))
U23_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_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
split17_in_ggaa(x1, x2, x3, x4) = split17_in_ggaa(x1, x2)
split21_in_gaa(x1, x2, x3) = split21_in_gaa(x1)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
mergesort32_in_ga(x1, x2) = mergesort32_in_ga(x1)
merge33_in_gga(x1, x2, x3) = merge33_in_gga(x1, x2)
le68_in_gg(x1, x2) = le68_in_gg(x1, x2)
gt85_in_gg(x1, x2) = gt85_in_gg(x1, x2)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U18_GA(x1, x2, x3, x4, x5) = U18_GA(x1, x2, x3, x5)
SPLIT17_IN_GGAA(x1, x2, x3, x4) = SPLIT17_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x1, x2, x5)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5) = U1_GAA(x1, x2, x5)
U19_GA(x1, x2, x3, x4, x5) = U19_GA(x1, x2, x3, x5)
U20_GA(x1, x2, x3, x4, x5) = U20_GA(x1, x2, x3, x5)
U21_GA(x1, x2, x3, x4, x5, x6) = U21_GA(x1, x2, x3, x5, x6)
U22_GA(x1, x2, x3, x4, x5) = U22_GA(x1, x2, x3, x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5)
U5_GA(x1, x2, x3, x4, x5) = U5_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5, x6) = U8_GA(x1, x2, x3, x5, x6)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x3, x5)
MERGE33_IN_GGA(x1, x2, x3) = MERGE33_IN_GGA(x1, x2)
U10_GGA(x1, x2, x3, x4, x5, x6) = U10_GGA(x1, x2, x3, x4, x6)
LE68_IN_GG(x1, x2) = LE68_IN_GG(x1, x2)
U16_GG(x1, x2, x3) = U16_GG(x1, x2, x3)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)
U12_GGA(x1, x2, x3, x4, x5, x6) = U12_GGA(x1, x2, x3, x4, x6)
U13_GGA(x1, x2, x3, x4, x5, x6) = U13_GGA(x1, x2, x3, x4, x6)
GT85_IN_GG(x1, x2) = GT85_IN_GG(x1, x2)
U17_GG(x1, x2, x3) = U17_GG(x1, x2, x3)
U14_GGA(x1, x2, x3, x4, x5, x6) = U14_GGA(x1, x2, x3, x4, x6)
U15_GGA(x1, x2, x3, x4, x5, x6) = U15_GGA(x1, x2, x3, x4, x6)
U23_GA(x1, x2, x3, x4, x5, x6) = U23_GA(x1, x2, x3, x5, x6)
U24_GA(x1, x2, x3, x4, x5) = U24_GA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U18_GA(T23, T24, T25, T14, split17_in_ggaa(T24, T25, X52, X51))
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → SPLIT17_IN_GGAA(T24, T25, X52, X51)
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → U2_GGAA(T36, T37, X81, X82, split21_in_gaa(T37, X82, X81))
SPLIT17_IN_GGAA(T36, T37, .(T36, X81), X82) → SPLIT21_IN_GAA(T37, X82, X81)
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → U1_GAA(T42, T43, X99, X100, split21_in_gaa(T43, X100, X99))
SPLIT21_IN_GAA(.(T42, T43), .(T42, X99), X100) → SPLIT21_IN_GAA(T43, X100, X99)
MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U20_GA(T23, T24, T25, T14, mergesort1_in_ga(.(T23, T29), X23))
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U21_GA(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U21_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U22_GA(T23, T24, T25, T14, mergesort32_in_ga(T28, X24))
U21_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_ggaa(T61, .(T62, T63), X133, X134))
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → SPLIT17_IN_GGAA(T61, .(T62, T63), X133, X134)
MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U5_GA(T61, T62, T63, X137, mergesort32_in_ga(T67, X135))
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(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)
U6_GA(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U8_GA(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U8_GA(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U9_GA(T61, T62, T63, X137, merge33_in_gga(T72, T73, X137))
U8_GA(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → MERGE33_IN_GGA(T72, T73, X137)
MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → U10_GGA(T108, T109, T110, T111, T113, le68_in_gg(T108, T110))
MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → LE68_IN_GG(T108, T110)
LE68_IN_GG(s(T126), s(T127)) → U16_GG(T126, T127, le68_in_gg(T126, T127))
LE68_IN_GG(s(T126), s(T127)) → LE68_IN_GG(T126, T127)
MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → U11_GGA(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
U11_GGA(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U12_GGA(T108, T109, T110, T111, T113, merge33_in_gga(T109, .(T110, T111), T113))
U11_GGA(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → MERGE33_IN_GGA(T109, .(T110, T111), T113)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → U13_GGA(T151, T152, T153, T154, T156, gt85_in_gg(T151, T153))
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → GT85_IN_GG(T151, T153)
GT85_IN_GG(s(T169), s(T170)) → U17_GG(T169, T170, gt85_in_gg(T169, T170))
GT85_IN_GG(s(T169), s(T170)) → GT85_IN_GG(T169, T170)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → U14_GGA(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
U14_GGA(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U15_GGA(T151, T152, T153, T154, T156, merge33_in_gga(.(T151, T152), T154, T156))
U14_GGA(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154, T156)
U21_GA(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U23_GA(T23, T24, T25, T14, T46, mergesortc32_in_ga(T28, T49))
U23_GA(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U24_GA(T23, T24, T25, T14, merge33_in_gga(T46, T49, T14))
U23_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_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_ga(T23, T24, T25, T14, mergec33_out_gga(T46, T49, T14)) → mergesortc1_out_ga(.(T23, .(T24, T25)), T14)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 27 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

GT85_IN_GG(s(T169), s(T170)) → GT85_IN_GG(T169, T170)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
GT85_IN_GG(x1, x2) = GT85_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

GT85_IN_GG(s(T169), s(T170)) → GT85_IN_GG(T169, T170)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

GT85_IN_GG(s(T169), s(T170)) → GT85_IN_GG(T169, T170)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

GT85_IN_GG(s(T169), s(T170)) → GT85_IN_GG(T169, T170)
The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LE68_IN_GG(s(T126), s(T127)) → LE68_IN_GG(T126, T127)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
LE68_IN_GG(x1, x2) = LE68_IN_GG(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LE68_IN_GG(s(T126), s(T127)) → LE68_IN_GG(T126, T127)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LE68_IN_GG(s(T126), s(T127)) → LE68_IN_GG(T126, T127)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

LE68_IN_GG(s(T126), s(T127)) → LE68_IN_GG(T126, T127)
The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → U11_GGA(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
U11_GGA(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → MERGE33_IN_GGA(T109, .(T110, T111), T113)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → U14_GGA(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
U14_GGA(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154, T156)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
MERGE33_IN_GGA(x1, x2, x3) = MERGE33_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)
U14_GGA(x1, x2, x3, x4, x5, x6) = U14_GGA(x1, x2, x3, x4, x6)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

MERGE33_IN_GGA(.(T108, T109), .(T110, T111), .(T108, T113)) → U11_GGA(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
U11_GGA(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → MERGE33_IN_GGA(T109, .(T110, T111), T113)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154), .(T153, T156)) → U14_GGA(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
U14_GGA(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154, T156)

The TRS R consists of the following rules:

lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
MERGE33_IN_GGA(x1, x2, x3) = MERGE33_IN_GGA(x1, x2)
U11_GGA(x1, x2, x3, x4, x5, x6) = U11_GGA(x1, x2, x3, x4, x6)
U14_GGA(x1, x2, x3, x4, x5, x6) = U14_GGA(x1, x2, x3, x4, x6)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

MERGE33_IN_GGA(.(T108, T109), .(T110, T111)) → U11_GGA(T108, T109, T110, T111, lec68_in_gg(T108, T110))
U11_GGA(T108, T109, T110, T111, lec68_out_gg(T108, T110)) → MERGE33_IN_GGA(T109, .(T110, T111))
MERGE33_IN_GGA(.(T151, T152), .(T153, T154)) → U14_GGA(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
U14_GGA(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154)

The TRS R consists of the following rules:

lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

lec68_in_gg(x0, x1)
gtc85_in_gg(x0, x1)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(26) MRRProof (EQUIVALENT transformation)

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

MERGE33_IN_GGA(.(T108, T109), .(T110, T111)) → U11_GGA(T108, T109, T110, T111, lec68_in_gg(T108, T110))

Strictly oriented rules of the TRS R:

gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(0) = 0
POL(MERGE33_IN_GGA(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U11_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅
POL(U14_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅
POL(U40_gg(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃
POL(U41_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(gtc85_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(gtc85_out_gg(x₁, x₂)) = x₁ + x₂
POL(lec68_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(lec68_out_gg(x₁, x₂)) = 2·x₁ + 2·x₂
POL(s(x₁)) = 2·x₁

(27) Obligation:

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

U11_GGA(T108, T109, T110, T111, lec68_out_gg(T108, T110)) → MERGE33_IN_GGA(T109, .(T110, T111))
MERGE33_IN_GGA(.(T151, T152), .(T153, T154)) → U14_GGA(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
U14_GGA(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154)

The TRS R consists of the following rules:

lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

lec68_in_gg(x0, x1)
gtc85_in_gg(x0, x1)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) Obligation:

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

U14_GGA(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154)) → U14_GGA(T151, T152, T153, T154, gtc85_in_gg(T151, T153))

The TRS R consists of the following rules:

lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

lec68_in_gg(x0, x1)
gtc85_in_gg(x0, x1)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(30) 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.

(31) Obligation:

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

U14_GGA(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154)) → U14_GGA(T151, T152, T153, T154, gtc85_in_gg(T151, T153))

The TRS R consists of the following rules:

gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

lec68_in_gg(x0, x1)
gtc85_in_gg(x0, x1)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(32) 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].

lec68_in_gg(x0, x1)
U40_gg(x0, x1, x2)

(33) Obligation:

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

U14_GGA(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154)
MERGE33_IN_GGA(.(T151, T152), .(T153, T154)) → U14_GGA(T151, T152, T153, T154, gtc85_in_gg(T151, T153))

The TRS R consists of the following rules:

gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

gtc85_in_gg(x0, x1)
U41_gg(x0, x1, x2)

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

(34) 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(.(T151, T152), .(T153, T154)) → U14_GGA(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4
U14_GGA(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → MERGE33_IN_GGA(.(T151, T152), T154)
The graph contains the following edges 4 >= 2

(35) YES

(36) Obligation:

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

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

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
SPLIT21_IN_GAA(x1, x2, x3) = SPLIT21_IN_GAA(x1)

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

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

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

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

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

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

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

(41) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

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

(42) YES

(43) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63)), X137) → U4_GA(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(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_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6)

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

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) 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_ggaa(T61, .(T62, T63), T67, T68))
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67, X135)
U4_GA(T61, T62, T63, X137, splitc17_out_ggaa(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_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
MERGESORT32_IN_GA(x1, x2) = MERGESORT32_IN_GA(x1)
U4_GA(x1, x2, x3, x4, x5) = U4_GA(x1, x2, x3, x5)
U6_GA(x1, x2, x3, x4, x5, x6) = U6_GA(x1, x2, x3, x5, x6)

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, splitc17_in_ggaa(T61, .(T62, T63)))
U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesortc32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37) → U31_ggaa(T36, T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63))) → U32_ga(T61, T62, T63, splitc17_in_ggaa(T61, .(T62, T63)))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U32_ga(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U33_ga(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, T72, mergesortc32_in_ga(T68))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U34_ga(T61, T62, T63, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, mergec33_in_gga(T72, T73))
U35_ga(T61, T62, T63, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
mergec33_in_gga(T82, []) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111)) → U36_gga(T108, T109, T110, T111, lec68_in_gg(T108, T110))
mergec33_in_gga(.(T151, T152), .(T153, T154)) → U38_gga(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
U36_gga(T108, T109, T110, T111, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, mergec33_in_gga(T109, .(T110, T111)))
U38_gga(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, mergec33_in_gga(.(T151, T152), T154))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U37_gga(T108, T109, T110, T111, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U39_gga(T151, T152, T153, T154, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
mergesortc32_in_ga(x0)
U31_ggaa(x0, x1, x2)
U32_ga(x0, x1, x2, x3)
splitc21_in_gaa(x0)
U33_ga(x0, x1, x2, x3, x4)
U26_gaa(x0, x1, x2)
U34_ga(x0, x1, x2, x3, x4)
U35_ga(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U36_gga(x0, x1, x2, x3, x4)
U38_gga(x0, x1, x2, x3, x4)
lec68_in_gg(x0, x1)
U37_gga(x0, x1, x2, x3, x4)
gtc85_in_gg(x0, x1)
U39_gga(x0, x1, x2, x3, x4)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(48) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, splitc17_in_ggaa(T61, .(T62, T63))) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U31_ggaa(T61, .(T62, T63), splitc21_in_gaa(.(T62, T63))))

(49) Obligation:

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

U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesortc32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U31_ggaa(T61, .(T62, T63), splitc21_in_gaa(.(T62, T63))))

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37) → U31_ggaa(T36, T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63))) → U32_ga(T61, T62, T63, splitc17_in_ggaa(T61, .(T62, T63)))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U32_ga(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U33_ga(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, T72, mergesortc32_in_ga(T68))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U34_ga(T61, T62, T63, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, mergec33_in_gga(T72, T73))
U35_ga(T61, T62, T63, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
mergec33_in_gga(T82, []) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111)) → U36_gga(T108, T109, T110, T111, lec68_in_gg(T108, T110))
mergec33_in_gga(.(T151, T152), .(T153, T154)) → U38_gga(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
U36_gga(T108, T109, T110, T111, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, mergec33_in_gga(T109, .(T110, T111)))
U38_gga(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, mergec33_in_gga(.(T151, T152), T154))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U37_gga(T108, T109, T110, T111, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U39_gga(T151, T152, T153, T154, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
mergesortc32_in_ga(x0)
U31_ggaa(x0, x1, x2)
U32_ga(x0, x1, x2, x3)
splitc21_in_gaa(x0)
U33_ga(x0, x1, x2, x3, x4)
U26_gaa(x0, x1, x2)
U34_ga(x0, x1, x2, x3, x4)
U35_ga(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U36_gga(x0, x1, x2, x3, x4)
U38_gga(x0, x1, x2, x3, x4)
lec68_in_gg(x0, x1)
U37_gga(x0, x1, x2, x3, x4)
gtc85_in_gg(x0, x1)
U39_gga(x0, x1, x2, x3, x4)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(50) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U31_ggaa(T61, .(T62, T63), splitc21_in_gaa(.(T62, T63)))) at position [3,2] we obtained the following new rules [LPAR04]:

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U31_ggaa(T61, .(T62, T63), U26_gaa(T62, T63, splitc21_in_gaa(T63))))

(51) Obligation:

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

U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesortc32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)
MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U31_ggaa(T61, .(T62, T63), U26_gaa(T62, T63, splitc21_in_gaa(T63))))

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37) → U31_ggaa(T36, T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63))) → U32_ga(T61, T62, T63, splitc17_in_ggaa(T61, .(T62, T63)))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U32_ga(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U33_ga(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, T72, mergesortc32_in_ga(T68))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U34_ga(T61, T62, T63, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, mergec33_in_gga(T72, T73))
U35_ga(T61, T62, T63, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
mergec33_in_gga(T82, []) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111)) → U36_gga(T108, T109, T110, T111, lec68_in_gg(T108, T110))
mergec33_in_gga(.(T151, T152), .(T153, T154)) → U38_gga(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
U36_gga(T108, T109, T110, T111, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, mergec33_in_gga(T109, .(T110, T111)))
U38_gga(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, mergec33_in_gga(.(T151, T152), T154))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U37_gga(T108, T109, T110, T111, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U39_gga(T151, T152, T153, T154, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
mergesortc32_in_ga(x0)
U31_ggaa(x0, x1, x2)
U32_ga(x0, x1, x2, x3)
splitc21_in_gaa(x0)
U33_ga(x0, x1, x2, x3, x4)
U26_gaa(x0, x1, x2)
U34_ga(x0, x1, x2, x3, x4)
U35_ga(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U36_gga(x0, x1, x2, x3, x4)
U38_gga(x0, x1, x2, x3, x4)
lec68_in_gg(x0, x1)
U37_gga(x0, x1, x2, x3, x4)
gtc85_in_gg(x0, x1)
U39_gga(x0, x1, x2, x3, x4)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT32_IN_GA(.(T61, .(T62, T63))) → U4_GA(T61, T62, T63, U31_ggaa(T61, .(T62, T63), U26_gaa(T62, T63, splitc21_in_gaa(T63))))

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

POL( U6_GA(x₁, ..., x₅) ) = 2x₃ + 2x₄

POL( mergesortc32_in_ga(x₁) ) = max{0, -1}

POL( [] ) = 0

POL( mergesortc32_out_ga(x₁, x₂) ) = max{0, x₂ - 2}

POL( .(x₁, x₂) ) = 2x₂ + 2

POL( U32_ga(x₁, ..., x₄) ) = max{0, x₁ + x₂ + x₃ + x₄ - 2}

POL( splitc17_in_ggaa(x₁, x₂) ) = max{0, -2}

POL( U26_gaa(x₁, ..., x₃) ) = 2x₃ + 1

POL( U31_ggaa(x₁, ..., x₃) ) = x₂ + x₃

POL( U4_GA(x₁, ..., x₄) ) = max{0, 2x₄ - 2}

POL( splitc21_in_gaa(x₁) ) = x₁ + 1

POL( splitc21_out_gaa(x₁, ..., x₃) ) = x₂ + 2x₃ + 1

POL( splitc17_out_ggaa(x₁, ..., x₄) ) = max{0, x₂ + x₃ + x₄ - 1}

POL( U33_ga(x₁, ..., x₅) ) = 2x₁ + 2x₂ + 1

POL( U34_ga(x₁, ..., x₅) ) = max{0, 2x₁ + x₂ + 2x₃ - 1}

POL( U35_ga(x₁, ..., x₄) ) = 2x₂ + 2

POL( mergec33_in_gga(x₁, x₂) ) = max{0, 2x₂ - 2}

POL( mergec33_out_gga(x₁, ..., x₃) ) = max{0, -2}

POL( U36_gga(x₁, ..., x₅) ) = 2x₂ + x₃ + 2x₄ + 2

POL( lec68_in_gg(x₁, x₂) ) = max{0, x₁ - 1}

POL( U38_gga(x₁, ..., x₅) ) = x₃ + x₄ + 2x₅ + 1

POL( gtc85_in_gg(x₁, x₂) ) = max{0, 2x₂ - 2}

POL( s(x₁) ) = max{0, 2x₁ - 2}

POL( U40_gg(x₁, ..., x₃) ) = max{0, x₂ - 2}

POL( 0 ) = 2

POL( lec68_out_gg(x₁, x₂) ) = max{0, -2}

POL( U37_gga(x₁, ..., x₅) ) = max{0, x₂ + 2x₃ + x₄ - 2}

POL( U41_gg(x₁, ..., x₃) ) = 2x₁ + 1

POL( gtc85_out_gg(x₁, x₂) ) = max{0, 2x₁ - 2}

POL( U39_gga(x₁, ..., x₅) ) = max{0, x₂ - 1}

POL( MERGESORT32_IN_GA(x₁) ) = max{0, 2x₁ - 1}

The following usable rules [FROCOS05] were oriented:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)

(53) Obligation:

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

U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → MERGESORT32_IN_GA(T67)
U4_GA(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U6_GA(T61, T62, T63, T68, mergesortc32_in_ga(T67))
U6_GA(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → MERGESORT32_IN_GA(T68)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37) → U31_ggaa(T36, T37, splitc21_in_gaa(T37))
mergesortc32_in_ga([]) → mergesortc32_out_ga([], [])
mergesortc32_in_ga(.(T54, [])) → mergesortc32_out_ga(.(T54, []), .(T54, []))
mergesortc32_in_ga(.(T61, .(T62, T63))) → U32_ga(T61, T62, T63, splitc17_in_ggaa(T61, .(T62, T63)))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U32_ga(T61, T62, T63, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, T68, mergesortc32_in_ga(T67))
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U33_ga(T61, T62, T63, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, T72, mergesortc32_in_ga(T68))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U34_ga(T61, T62, T63, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, mergec33_in_gga(T72, T73))
U35_ga(T61, T62, T63, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
mergec33_in_gga(T82, []) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111)) → U36_gga(T108, T109, T110, T111, lec68_in_gg(T108, T110))
mergec33_in_gga(.(T151, T152), .(T153, T154)) → U38_gga(T151, T152, T153, T154, gtc85_in_gg(T151, T153))
U36_gga(T108, T109, T110, T111, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, mergec33_in_gga(T109, .(T110, T111)))
U38_gga(T151, T152, T153, T154, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, mergec33_in_gga(.(T151, T152), T154))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U37_gga(T108, T109, T110, T111, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U39_gga(T151, T152, T153, T154, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
mergesortc32_in_ga(x0)
U31_ggaa(x0, x1, x2)
U32_ga(x0, x1, x2, x3)
splitc21_in_gaa(x0)
U33_ga(x0, x1, x2, x3, x4)
U26_gaa(x0, x1, x2)
U34_ga(x0, x1, x2, x3, x4)
U35_ga(x0, x1, x2, x3)
mergec33_in_gga(x0, x1)
U36_gga(x0, x1, x2, x3, x4)
U38_gga(x0, x1, x2, x3, x4)
lec68_in_gg(x0, x1)
U37_gga(x0, x1, x2, x3, x4)
gtc85_in_gg(x0, x1)
U39_gga(x0, x1, x2, x3, x4)
U40_gg(x0, x1, x2)
U41_gg(x0, x1, x2)

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

(54) DependencyGraphProof (EQUIVALENT transformation)

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

(55) TRUE

(56) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(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) → U27_ga(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U27_ga(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → U28_ga(T23, T24, T25, T14, T28, mergesortc1_in_ga(.(T23, T29), T46))
U28_ga(T23, T24, T25, T14, T28, mergesortc1_out_ga(.(T23, T29), T46)) → U29_ga(T23, T24, T25, T14, 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)), X137) → U32_ga(T61, T62, T63, X137, splitc17_in_ggaa(T61, .(T62, T63), T67, T68))
U32_ga(T61, T62, T63, X137, splitc17_out_ggaa(T61, .(T62, T63), T67, T68)) → U33_ga(T61, T62, T63, X137, T68, mergesortc32_in_ga(T67, T72))
U33_ga(T61, T62, T63, X137, T68, mergesortc32_out_ga(T67, T72)) → U34_ga(T61, T62, T63, X137, T72, mergesortc32_in_ga(T68, T73))
U34_ga(T61, T62, T63, X137, T72, mergesortc32_out_ga(T68, T73)) → U35_ga(T61, T62, T63, X137, mergec33_in_gga(T72, T73, X137))
mergec33_in_gga(T82, [], T82) → mergec33_out_gga(T82, [], T82)
mergec33_in_gga([], T87, T87) → mergec33_out_gga([], T87, T87)
mergec33_in_gga(.(T108, T109), .(T110, T111), .(T108, T113)) → U36_gga(T108, T109, T110, T111, T113, lec68_in_gg(T108, T110))
lec68_in_gg(s(T126), s(T127)) → U40_gg(T126, T127, lec68_in_gg(T126, T127))
lec68_in_gg(0, s(T134)) → lec68_out_gg(0, s(T134))
lec68_in_gg(0, 0) → lec68_out_gg(0, 0)
U40_gg(T126, T127, lec68_out_gg(T126, T127)) → lec68_out_gg(s(T126), s(T127))
U36_gga(T108, T109, T110, T111, T113, lec68_out_gg(T108, T110)) → U37_gga(T108, T109, T110, T111, T113, mergec33_in_gga(T109, .(T110, T111), T113))
mergec33_in_gga(.(T151, T152), .(T153, T154), .(T153, T156)) → U38_gga(T151, T152, T153, T154, T156, gtc85_in_gg(T151, T153))
gtc85_in_gg(s(T169), s(T170)) → U41_gg(T169, T170, gtc85_in_gg(T169, T170))
gtc85_in_gg(s(T175), 0) → gtc85_out_gg(s(T175), 0)
U41_gg(T169, T170, gtc85_out_gg(T169, T170)) → gtc85_out_gg(s(T169), s(T170))
U38_gga(T151, T152, T153, T154, T156, gtc85_out_gg(T151, T153)) → U39_gga(T151, T152, T153, T154, T156, mergec33_in_gga(.(T151, T152), T154, T156))
U39_gga(T151, T152, T153, T154, T156, mergec33_out_gga(.(T151, T152), T154, T156)) → mergec33_out_gga(.(T151, T152), .(T153, T154), .(T153, T156))
U37_gga(T108, T109, T110, T111, T113, mergec33_out_gga(T109, .(T110, T111), T113)) → mergec33_out_gga(.(T108, T109), .(T110, T111), .(T108, T113))
U35_ga(T61, T62, T63, X137, mergec33_out_gga(T72, T73, X137)) → mergesortc32_out_ga(.(T61, .(T62, T63)), X137)
U29_ga(T23, T24, T25, T14, T46, mergesortc32_out_ga(T28, T49)) → U30_ga(T23, T24, T25, T14, mergec33_in_gga(T46, T49, T14))
U30_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) = .(x1, x2)
splitc17_in_ggaa(x1, x2, x3, x4) = splitc17_in_ggaa(x1, x2)
U31_ggaa(x1, x2, x3, x4, x5) = U31_ggaa(x1, x2, x5)
splitc21_in_gaa(x1, x2, x3) = splitc21_in_gaa(x1)
[] = []
splitc21_out_gaa(x1, x2, x3) = splitc21_out_gaa(x1, x2, x3)
U26_gaa(x1, x2, x3, x4, x5) = U26_gaa(x1, x2, x5)
splitc17_out_ggaa(x1, x2, x3, x4) = splitc17_out_ggaa(x1, x2, x3, x4)
mergesortc1_in_ga(x1, x2) = mergesortc1_in_ga(x1)
mergesortc1_out_ga(x1, x2) = mergesortc1_out_ga(x1, x2)
U27_ga(x1, x2, x3, x4, x5) = U27_ga(x1, x2, x3, x5)
U28_ga(x1, x2, x3, x4, x5, x6) = U28_ga(x1, x2, x3, x5, x6)
U29_ga(x1, x2, x3, x4, x5, x6) = U29_ga(x1, x2, x3, x5, x6)
mergesortc32_in_ga(x1, x2) = mergesortc32_in_ga(x1)
mergesortc32_out_ga(x1, x2) = mergesortc32_out_ga(x1, x2)
U32_ga(x1, x2, x3, x4, x5) = U32_ga(x1, x2, x3, x5)
U33_ga(x1, x2, x3, x4, x5, x6) = U33_ga(x1, x2, x3, x5, x6)
U34_ga(x1, x2, x3, x4, x5, x6) = U34_ga(x1, x2, x3, x5, x6)
U35_ga(x1, x2, x3, x4, x5) = U35_ga(x1, x2, x3, x5)
mergec33_in_gga(x1, x2, x3) = mergec33_in_gga(x1, x2)
mergec33_out_gga(x1, x2, x3) = mergec33_out_gga(x1, x2, x3)
U36_gga(x1, x2, x3, x4, x5, x6) = U36_gga(x1, x2, x3, x4, x6)
lec68_in_gg(x1, x2) = lec68_in_gg(x1, x2)
s(x1) = s(x1)
U40_gg(x1, x2, x3) = U40_gg(x1, x2, x3)
0 = 0
lec68_out_gg(x1, x2) = lec68_out_gg(x1, x2)
U37_gga(x1, x2, x3, x4, x5, x6) = U37_gga(x1, x2, x3, x4, x6)
U38_gga(x1, x2, x3, x4, x5, x6) = U38_gga(x1, x2, x3, x4, x6)
gtc85_in_gg(x1, x2) = gtc85_in_gg(x1, x2)
U41_gg(x1, x2, x3) = U41_gg(x1, x2, x3)
gtc85_out_gg(x1, x2) = gtc85_out_gg(x1, x2)
U39_gga(x1, x2, x3, x4, x5, x6) = U39_gga(x1, x2, x3, x4, x6)
U30_ga(x1, x2, x3, x4, x5) = U30_ga(x1, x2, x3, x5)
MERGESORT1_IN_GA(x1, x2) = MERGESORT1_IN_GA(x1)
U19_GA(x1, x2, x3, x4, x5) = U19_GA(x1, x2, x3, x5)

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

(57) UsableRulesProof (EQUIVALENT transformation)

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

(58) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25)), T14) → U19_GA(T23, T24, T25, T14, splitc17_in_ggaa(T24, T25, T28, T29))
U19_GA(T23, T24, T25, T14, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29), X23)

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37, .(T36, X81), X82) → U31_ggaa(T36, T37, X81, X82, splitc21_in_gaa(T37, X82, X81))
U31_ggaa(T36, T37, X81, X82, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
splitc21_in_gaa([], [], []) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43), .(T42, X99), X100) → U26_gaa(T42, T43, X99, X100, splitc21_in_gaa(T43, X100, X99))
U26_gaa(T42, T43, X99, X100, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

(59) PiDPToQDPProof (SOUND transformation)

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

(60) Obligation:

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

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, splitc17_in_ggaa(T24, T25))
U19_GA(T23, T24, T25, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37) → U31_ggaa(T36, T37, splitc21_in_gaa(T37))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
U31_ggaa(x0, x1, x2)
splitc21_in_gaa(x0)
U26_gaa(x0, x1, x2)

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

(61) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, splitc17_in_ggaa(T24, T25)) at position [3] we obtained the following new rules [LPAR04]:

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, U31_ggaa(T24, T25, splitc21_in_gaa(T25)))

(62) Obligation:

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

U19_GA(T23, T24, T25, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, U31_ggaa(T24, T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc17_in_ggaa(T36, T37) → U31_ggaa(T36, T37, splitc21_in_gaa(T37))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
U31_ggaa(x0, x1, x2)
splitc21_in_gaa(x0)
U26_gaa(x0, x1, x2)

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

(63) 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.

(64) Obligation:

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

U19_GA(T23, T24, T25, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, U31_ggaa(T24, T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

splitc17_in_ggaa(x0, x1)
U31_ggaa(x0, x1, x2)
splitc21_in_gaa(x0)
U26_gaa(x0, x1, x2)

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

(65) 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_ggaa(x0, x1)

(66) Obligation:

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

U19_GA(T23, T24, T25, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))
MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, U31_ggaa(T24, T25, splitc21_in_gaa(T25)))

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

U31_ggaa(x0, x1, x2)
splitc21_in_gaa(x0)
U26_gaa(x0, x1, x2)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGESORT1_IN_GA(.(T23, .(T24, T25))) → U19_GA(T23, T24, T25, U31_ggaa(T24, T25, splitc21_in_gaa(T25)))

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

POL(.(x₁, x₂)) = 1 + x₂
POL(MERGESORT1_IN_GA(x₁)) = x₁
POL(U19_GA(x₁, x₂, x₃, x₄)) = x₄
POL(U26_gaa(x₁, x₂, x₃)) = 1 + x₃
POL(U31_ggaa(x₁, x₂, x₃)) = 1 + x₃
POL([]) = 0
POL(splitc17_out_ggaa(x₁, x₂, x₃, x₄)) = 1 + x₄
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(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

(68) Obligation:

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

U19_GA(T23, T24, T25, splitc17_out_ggaa(T24, T25, T28, T29)) → MERGESORT1_IN_GA(.(T23, T29))

The TRS R consists of the following rules:

splitc21_in_gaa([]) → splitc21_out_gaa([], [], [])
splitc21_in_gaa(.(T42, T43)) → U26_gaa(T42, T43, splitc21_in_gaa(T43))
U31_ggaa(T36, T37, splitc21_out_gaa(T37, X82, X81)) → splitc17_out_ggaa(T36, T37, .(T36, X81), X82)
U26_gaa(T42, T43, splitc21_out_gaa(T43, X100, X99)) → splitc21_out_gaa(.(T42, T43), .(T42, X99), X100)

The set Q consists of the following terms:

U31_ggaa(x0, x1, x2)
splitc21_in_gaa(x0)
U26_gaa(x0, x1, x2)

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

(69) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) MRRProof (EQUIVALENT transformation)

(27) Obligation:

(28) DependencyGraphProof (EQUIVALENT transformation)

(29) Obligation:

(30) UsableRulesProof (EQUIVALENT transformation)

(31) Obligation:

(32) QReductionProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPSizeChangeProof (EQUIVALENT transformation)

(35) YES

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) QDPSizeChangeProof (EQUIVALENT transformation)

(42) YES

(43) Obligation:

(44) UsableRulesProof (EQUIVALENT transformation)

(45) Obligation:

(46) PiDPToQDPProof (SOUND transformation)

(47) Obligation:

(48) Rewriting (EQUIVALENT transformation)

(49) Obligation:

(50) Rewriting (EQUIVALENT transformation)

(51) Obligation:

(52) QDPOrderProof (EQUIVALENT transformation)

(53) Obligation:

(54) DependencyGraphProof (EQUIVALENT transformation)

(55) TRUE

(56) Obligation:

(57) UsableRulesProof (EQUIVALENT transformation)

(58) Obligation:

(59) PiDPToQDPProof (SOUND transformation)

(60) Obligation:

(61) Rewriting (EQUIVALENT transformation)

(62) Obligation:

(63) UsableRulesProof (EQUIVALENT transformation)

(64) Obligation:

(65) QReductionProof (EQUIVALENT transformation)

(66) Obligation:

(67) QDPOrderProof (EQUIVALENT transformation)

(68) Obligation:

(69) DependencyGraphProof (EQUIVALENT transformation)

(70) TRUE