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

(0) Obligation:

Clauses:

mergesort([], []).
mergesort(.(E, []), .(E, [])).
mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), 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).

Query: mergesort(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="F: 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\nX7 is free\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(T12, X20, X21), ','(mergesort(.(T10, X20), X22), ','(mergesort(.(T11, X21), X23), merge(X22, X23, T14))))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(split(T12, X20, X21), ','(mergesort(.(T10, X20), X22), ','(mergesort(.(T11, X21), X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 7 | ','(split(T12, X20, X21), ','(mergesort(.(T10, X20), X22), ','(mergesort(.(T11, X21), X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(split(T12, X20, X21), ','(mergesort(.(T10, X20), X22), ','(mergesort(.(T11, X21), X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 7\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(split(T12, X20, X21), ','(mergesort(.(T10, X20), X22), ','(mergesort(.(T11, X21), X23), merge(X22, X23, T14)))), SCOPE: 2, CLAUSE: 8\n\nT10 is ground\nT11 is ground\nT12 is ground\nX20 is free\nX21 is free\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(mergesort(.(T10, []), X22), ','(mergesort(.(T11, []), X23), merge(X22, X23, T14)))\n\nT10 is ground\nT11 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="A: mergesort(.(T10, []), X22)\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(mergesort(.(T11, []), X23), merge(T15, X23, T14))\n\nT11 is ground\nT15 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 0 | mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 1 | mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 1 | mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 1\n\nT10 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: mergesort(.(T10, []), X22), SCOPE: 3, CLAUSE: 2\n\nT10 is ground\nX22 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: mergesort(.(T11, []), X23)\n\nT11 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="B: merge(T15, T22, T14)\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 3 | merge(T15, T22, T14), SCOPE: 4, CLAUSE: 4 | merge(T15, T22, T14), SCOPE: 4, CLAUSE: 5 | merge(T15, T22, T14), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 3\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 4 | merge(T15, T22, T14), SCOPE: 4, CLAUSE: 5 | merge(T15, T22, T14), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 5 | merge(T15, T22, T14), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 5\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: merge(T15, T22, T14), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(le(T55, T57), merge(T56, .(T57, T58), T60))\n\nT55 is ground\nT56 is ground\nT57 is ground\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="C: le(T55, T57)\n\nT55 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: merge(T56, .(T57, T58), T60)\n\nT56 is ground\nT57 is ground\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: le(T55, T57), SCOPE: 5, CLAUSE: 11 | le(T55, T57), SCOPE: 5, CLAUSE: 12 | le(T55, T57), SCOPE: 5, CLAUSE: 13\n\nT55 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: le(T55, T57), SCOPE: 5, CLAUSE: 11\n\nT55 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: le(T55, T57), SCOPE: 5, CLAUSE: 12 | le(T55, T57), SCOPE: 5, CLAUSE: 13\n\nT55 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: le(T73, T74)\n\nT73 is ground\nT74 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: le(T55, T57), SCOPE: 5, CLAUSE: 12\n\nT55 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: le(T55, T57), SCOPE: 5, CLAUSE: 13\n\nT55 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(gt(T94, T96), merge(.(T94, T95), T97, T99))\n\nT94 is ground\nT95 is ground\nT96 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="D: gt(T94, T96)\n\nT94 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: merge(.(T94, T95), T97, T99)\n\nT94 is ground\nT95 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: gt(T94, T96), SCOPE: 6, CLAUSE: 9 | gt(T94, T96), SCOPE: 6, CLAUSE: 10\n\nT94 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: gt(T94, T96), SCOPE: 6, CLAUSE: 9\n\nT94 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: gt(T94, T96), SCOPE: 6, CLAUSE: 10\n\nT94 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: gt(T112, T113)\n\nT112 is ground\nT113 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(split(T128, X146, X145), ','(mergesort(.(T10, .(T127, X145)), X22), ','(mergesort(.(T11, X146), X23), merge(X22, X23, T14))))\n\nT10 is ground\nT11 is ground\nT127 is ground\nT128 is ground\nX22 is free\nX23 is free\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="E: split(T128, X146, X145)\n\nT128 is ground\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(mergesort(.(T10, .(T127, T130)), X22), ','(mergesort(.(T11, T129), X23), merge(X22, X23, T14)))\n\nT10 is ground\nT11 is ground\nT127 is ground\nT129 is ground\nT130 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: split(T128, X146, X145), SCOPE: 7, CLAUSE: 7 | split(T128, X146, X145), SCOPE: 7, CLAUSE: 8\n\nT128 is ground\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: split(T128, X146, X145), SCOPE: 7, CLAUSE: 7\n\nT128 is ground\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: split(T128, X146, X145), SCOPE: 7, CLAUSE: 8\n\nT128 is ground\nX145 is free\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: split(T136, X164, X163)\n\nT136 is ground\nX163 is free\nX164 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: mergesort(.(T10, .(T127, T130)), X22)\n\nT10 is ground\nT127 is ground\nT130 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(mergesort(.(T11, T129), X23), merge(T137, X23, T14))\n\nT11 is ground\nT129 is ground\nT137 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: mergesort(.(T11, T129), X23)\n\nT11 is ground\nT129 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: merge(T137, T138, T14)\n\nT137 is ground\nT138 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 88 [arrowhead = none ];
88 -> 2;

89 [label="EVAL with clause\nmergesort([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 89 [arrowhead = none ];
89 -> 3;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 90 [arrowhead = none ];
90 -> 4;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 91 [arrowhead = none ];
91 -> 5;

92 [label="EVAL with clause\nmergesort(.(X7, []), .(X7, [])).\nand substitutionX7 -> T4,\nT1 -> .(T4, []),\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 92 [arrowhead = none ];
92 -> 8;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 93 [arrowhead = none ];
93 -> 9;

94 [label="BACKTRACK\nfor clause: mergesort(.(E, []), .(E, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 94 [arrowhead = none ];
94 -> 6;

95 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 95 [arrowhead = none ];
95 -> 7;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 96 [arrowhead = none ];
96 -> 10;

97 [label="EVAL with clause\nmergesort(.(X16, .(X17, X18)), X19) :- ','(split(X18, X20, X21), ','(mergesort(.(X16, X20), X22), ','(mergesort(.(X17, X21), X23), merge(X22, X23, X19)))).\nand substitutionX16 -> T10,\nX17 -> T11,\nX18 -> T12,\nT1 -> .(T10, .(T11, T12)),\nT2 -> T14,\nX19 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 97 [arrowhead = none ];
97 -> 12;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 98 [arrowhead = none ];
98 -> 13;

99 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 99 [arrowhead = none ];
99 -> 11;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 100 [arrowhead = none ];
100 -> 14;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 101 [arrowhead = none ];
101 -> 15;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 102 [arrowhead = none ];
102 -> 16;

103 [label="EVAL with clause\nsplit([], [], []).\nand substitutionT12 -> [],\nX20 -> [],\nX21 -> []", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 103 [arrowhead = none ];
103 -> 17;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 104 [arrowhead = none ];
104 -> 18;

105 [label="EVAL with clause\nsplit(.(X141, X142), .(X141, X143), X144) :- split(X142, X144, X143).\nand substitutionX141 -> T127,\nX142 -> T128,\nT12 -> .(T127, T128),\nX143 -> X145,\nX20 -> .(T127, X145),\nX21 -> X146,\nX144 -> X146", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 105 [arrowhead = none ];
105 -> 72;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 106 [arrowhead = none ];
106 -> 73;

107 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 107 [arrowhead = none ];
107 -> 19;

108 [label="SPLIT 2\nnew knowledge:\nT10 is ground\nT15 is ground\nreplacements:X22 -> T15", fontsize=14, style = filled, fillcolor = violet];
17 -> 108 [arrowhead = none ];
108 -> 20;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 109 [arrowhead = none ];
109 -> 21;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
20 -> 110 [arrowhead = none ];
110 -> 28;

111 [label="SPLIT 2\nnew knowledge:\nT11 is ground\nT22 is ground\nreplacements:X23 -> T22", fontsize=14, style = filled, fillcolor = violet];
20 -> 111 [arrowhead = none ];
111 -> 29;

112 [label="BACKTRACK\nfor clause: mergesort([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 112 [arrowhead = none ];
112 -> 22;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 113 [arrowhead = none ];
113 -> 23;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 114 [arrowhead = none ];
114 -> 24;

115 [label="ONLY EVAL with clause\nmergesort(.(X28, []), .(X28, [])).\nand substitutionT10 -> T20,\nX28 -> T20,\nX22 -> .(T20, [])", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 115 [arrowhead = none ];
115 -> 25;

116 [label="BACKTRACK\nfor clause: mergesort(.(E, .(F, U)), V) :- ','(split(U, L2, L1), ','(mergesort(.(E, L2), X), ','(mergesort(.(F, L1), Z), merge(X, Z, V))))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 116 [arrowhead = none ];
116 -> 27;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 117 [arrowhead = none ];
117 -> 26;

118 [label="INSTANCE with matching:\nT10 -> T11\nX22 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
28 -> 118 [arrowhead = none , style=dashed];
118 -> 19[style=dashed];

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 119 [arrowhead = none ];
119 -> 30;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 120 [arrowhead = none ];
120 -> 31;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 121 [arrowhead = none ];
121 -> 32;

122 [label="EVAL with clause\nmerge(X39, [], X39).\nand substitutionT15 -> T29,\nX39 -> T29,\nT22 -> [],\nT14 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 122 [arrowhead = none ];
122 -> 33;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 123 [arrowhead = none ];
123 -> 34;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 124 [arrowhead = none ];
124 -> 36;

125 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 125 [arrowhead = none ];
125 -> 37;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 126 [arrowhead = none ];
126 -> 35;

127 [label="EVAL with clause\nmerge([], X44, X44).\nand substitutionT15 -> [],\nT22 -> T34,\nX44 -> T34,\nT14 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 127 [arrowhead = none ];
127 -> 38;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 128 [arrowhead = none ];
128 -> 39;

129 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 129 [arrowhead = none ];
129 -> 41;

130 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 130 [arrowhead = none ];
130 -> 42;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 131 [arrowhead = none ];
131 -> 40;

132 [label="EVAL with clause\nmerge(.(X65, X66), .(X67, X68), .(X65, X69)) :- ','(le(X65, X67), merge(X66, .(X67, X68), X69)).\nand substitutionX65 -> T55,\nX66 -> T56,\nT15 -> .(T55, T56),\nX67 -> T57,\nX68 -> T58,\nT22 -> .(T57, T58),\nX69 -> T60,\nT14 -> .(T55, T60),\nT59 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 132 [arrowhead = none ];
132 -> 43;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 133 [arrowhead = none ];
133 -> 44;

134 [label="EVAL with clause\nmerge(.(X103, X104), .(X105, X106), .(X105, X107)) :- ','(gt(X103, X105), merge(.(X103, X104), X106, X107)).\nand substitutionX103 -> T94,\nX104 -> T95,\nT15 -> .(T94, T95),\nX105 -> T96,\nX106 -> T97,\nT22 -> .(T96, T97),\nX107 -> T99,\nT14 -> .(T96, T99),\nT98 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 134 [arrowhead = none ];
134 -> 60;

135 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 135 [arrowhead = none ];
135 -> 61;

136 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
43 -> 136 [arrowhead = none ];
136 -> 45;

137 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT57 is ground", fontsize=14, style = filled, fillcolor = violet];
43 -> 137 [arrowhead = none ];
137 -> 46;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 138 [arrowhead = none ];
138 -> 47;

139 [label="INSTANCE with matching:\nT15 -> T56\nT22 -> .(T57, T58)\nT14 -> T60", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 139 [arrowhead = none , style=dashed];
139 -> 29[style=dashed];

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 140 [arrowhead = none ];
140 -> 48;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 141 [arrowhead = none ];
141 -> 49;

142 [label="EVAL with clause\nle(s(X82), s(X83)) :- le(X82, X83).\nand substitutionX82 -> T73,\nT55 -> s(T73),\nX83 -> T74,\nT57 -> s(T74)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 142 [arrowhead = none ];
142 -> 50;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 143 [arrowhead = none ];
143 -> 51;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 144 [arrowhead = none ];
144 -> 52;

145 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 145 [arrowhead = none ];
145 -> 53;

146 [label="INSTANCE with matching:\nT55 -> T73\nT57 -> T74", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 146 [arrowhead = none , style=dashed];
146 -> 45[style=dashed];

147 [label="EVAL with clause\nle(0, s(X90)).\nand substitutionT55 -> 0,\nX90 -> T81,\nT57 -> s(T81)", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 147 [arrowhead = none ];
147 -> 54;

148 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 148 [arrowhead = none ];
148 -> 55;

149 [label="EVAL with clause\nle(0, 0).\nand substitutionT55 -> 0,\nT57 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 149 [arrowhead = none ];
149 -> 57;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 150 [arrowhead = none ];
150 -> 58;

151 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 151 [arrowhead = none ];
151 -> 56;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 152 [arrowhead = none ];
152 -> 59;

153 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
60 -> 153 [arrowhead = none ];
153 -> 62;

154 [label="SPLIT 2\nnew knowledge:\nT94 is ground\nT96 is ground", fontsize=14, style = filled, fillcolor = violet];
60 -> 154 [arrowhead = none ];
154 -> 63;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 155 [arrowhead = none ];
155 -> 64;

156 [label="INSTANCE with matching:\nT15 -> .(T94, T95)\nT22 -> T97\nT14 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 156 [arrowhead = none , style=dashed];
156 -> 29[style=dashed];

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 157 [arrowhead = none ];
157 -> 65;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 158 [arrowhead = none ];
158 -> 66;

159 [label="EVAL with clause\ngt(s(X120), s(X121)) :- gt(X120, X121).\nand substitutionX120 -> T112,\nT94 -> s(T112),\nX121 -> T113,\nT96 -> s(T113)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 159 [arrowhead = none ];
159 -> 67;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 160 [arrowhead = none ];
160 -> 68;

161 [label="EVAL with clause\ngt(s(X126), 0).\nand substitutionX126 -> T118,\nT94 -> s(T118),\nT96 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 161 [arrowhead = none ];
161 -> 69;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 162 [arrowhead = none ];
162 -> 70;

163 [label="INSTANCE with matching:\nT94 -> T112\nT96 -> T113", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 163 [arrowhead = none , style=dashed];
163 -> 62[style=dashed];

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 164 [arrowhead = none ];
164 -> 71;

165 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
72 -> 165 [arrowhead = none ];
165 -> 74;

166 [label="SPLIT 2\nnew knowledge:\nT128 is ground\nT129 is ground\nT130 is ground\nreplacements:X146 -> T129,\nX145 -> T130", fontsize=14, style = filled, fillcolor = violet];
72 -> 166 [arrowhead = none ];
166 -> 75;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
74 -> 167 [arrowhead = none ];
167 -> 76;

168 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
75 -> 168 [arrowhead = none ];
168 -> 84;

169 [label="SPLIT 2\nnew knowledge:\nT10 is ground\nT127 is ground\nT130 is ground\nT137 is ground\nreplacements:X22 -> T137", fontsize=14, style = filled, fillcolor = violet];
75 -> 169 [arrowhead = none ];
169 -> 85;

170 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 170 [arrowhead = none ];
170 -> 77;

171 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 171 [arrowhead = none ];
171 -> 78;

172 [label="EVAL with clause\nsplit([], [], []).\nand substitutionT128 -> [],\nX146 -> [],\nX145 -> []", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 172 [arrowhead = none ];
172 -> 79;

173 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 173 [arrowhead = none ];
173 -> 80;

174 [label="EVAL with clause\nsplit(.(X159, X160), .(X159, X161), X162) :- split(X160, X162, X161).\nand substitutionX159 -> T135,\nX160 -> T136,\nT128 -> .(T135, T136),\nX161 -> X163,\nX146 -> .(T135, X163),\nX145 -> X164,\nX162 -> X164", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 174 [arrowhead = none ];
174 -> 82;

175 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 175 [arrowhead = none ];
175 -> 83;

176 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 176 [arrowhead = none ];
176 -> 81;

177 [label="INSTANCE with matching:\nT128 -> T136\nX146 -> X164\nX145 -> X163", fontsize=14, style = filled, fillcolor = lightgrey];
82 -> 177 [arrowhead = none , style=dashed];
177 -> 74[style=dashed];

178 [label="INSTANCE with matching:\nT1 -> .(T10, .(T127, T130))\nT2 -> X22", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 178 [arrowhead = none , style=dashed];
178 -> 1[style=dashed];

179 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
85 -> 179 [arrowhead = none ];
179 -> 86;

180 [label="SPLIT 2\nnew knowledge:\nT11 is ground\nT129 is ground\nT138 is ground\nreplacements:X23 -> T138", fontsize=14, style = filled, fillcolor = violet];
85 -> 180 [arrowhead = none ];
180 -> 87;

181 [label="INSTANCE with matching:\nT1 -> .(T11, T129)\nT2 -> X23", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 181 [arrowhead = none , style=dashed];
181 -> 1[style=dashed];

182 [label="INSTANCE with matching:\nT15 -> T137\nT22 -> T138", fontsize=14, style = filled, fillcolor = lightgrey];
87 -> 182 [arrowhead = none , style=dashed];
182 -> 29[style=dashed];

}

(2) Obligation:

Triples:

mergeB(.(X1, X2), .(X3, X4), .(X1, X5)) :- leC(X1, X3).
mergeB(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lecC(X1, X3), mergeB(X2, .(X3, X4), X5)).
mergeB(.(X1, X2), .(X3, X4), .(X3, X5)) :- gtD(X1, X3).
mergeB(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(gtcD(X1, X3), mergeB(.(X1, X2), X4, X5)).
leC(s(X1), s(X2)) :- leC(X1, X2).
gtD(s(X1), s(X2)) :- gtD(X1, X2).
splitE(.(X1, X2), .(X1, X3), X4) :- splitE(X2, X4, X3).
mergesortF(.(X1, .(X2, [])), X3) :- mergesortA(X1, X4).
mergesortF(.(X1, .(X2, [])), X3) :- ','(mergesortcA(X1, X4), mergesortA(X2, X5)).
mergesortF(.(X1, .(X2, [])), X3) :- ','(mergesortcA(X1, X4), ','(mergesortcA(X2, X5), mergeB(X4, X5, X3))).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- splitE(X4, X6, X7).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), mergesortF(.(X1, .(X3, X7)), X8)).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), ','(mergesortcF(.(X1, .(X3, X7)), X8), mergesortF(.(X2, X6), X9))).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), ','(mergesortcF(.(X1, .(X3, X7)), X8), ','(mergesortcF(.(X2, X6), X9), mergeB(X8, X9, X5)))).

Clauses:

mergesortcA(X1, .(X1, [])).
mergecB(X1, [], X1).
mergecB([], X1, X1).
mergecB(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lecC(X1, X3), mergecB(X2, .(X3, X4), X5)).
mergecB(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(gtcD(X1, X3), mergecB(.(X1, X2), X4, X5)).
lecC(s(X1), s(X2)) :- lecC(X1, X2).
lecC(0, s(X1)).
lecC(0, 0).
gtcD(s(X1), s(X2)) :- gtcD(X1, X2).
gtcD(s(X1), 0).
splitcE([], [], []).
splitcE(.(X1, X2), .(X1, X3), X4) :- splitcE(X2, X4, X3).
mergesortcF([], []).
mergesortcF(.(X1, []), .(X1, [])).
mergesortcF(.(X1, .(X2, [])), X3) :- ','(mergesortcA(X1, X4), ','(mergesortcA(X2, X5), mergecB(X4, X5, X3))).
mergesortcF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), ','(mergesortcF(.(X1, .(X3, X7)), X8), ','(mergesortcF(.(X2, X6), X9), mergecB(X8, X9, X5)))).

Afs:

mergesortF(x1, x2)  =  mergesortF(x1)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [DT09].

(4) Obligation:

Triples:

mergeB(.(X1, X2), .(X3, X4), .(X1, X5)) :- leC(X1, X3).
mergeB(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lecC(X1, X3), mergeB(X2, .(X3, X4), X5)).
mergeB(.(X1, X2), .(X3, X4), .(X3, X5)) :- gtD(X1, X3).
mergeB(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(gtcD(X1, X3), mergeB(.(X1, X2), X4, X5)).
leC(s(X1), s(X2)) :- leC(X1, X2).
gtD(s(X1), s(X2)) :- gtD(X1, X2).
splitE(.(X1, X2), .(X1, X3), X4) :- splitE(X2, X4, X3).
mergesortF(.(X1, .(X2, [])), X3) :- ','(mergesortcA(X1, X4), ','(mergesortcA(X2, X5), mergeB(X4, X5, X3))).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- splitE(X4, X6, X7).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), mergesortF(.(X1, .(X3, X7)), X8)).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), ','(mergesortcF(.(X1, .(X3, X7)), X8), mergesortF(.(X2, X6), X9))).
mergesortF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), ','(mergesortcF(.(X1, .(X3, X7)), X8), ','(mergesortcF(.(X2, X6), X9), mergeB(X8, X9, X5)))).

Clauses:

mergesortcA(X1, .(X1, [])).
mergecB(X1, [], X1).
mergecB([], X1, X1).
mergecB(.(X1, X2), .(X3, X4), .(X1, X5)) :- ','(lecC(X1, X3), mergecB(X2, .(X3, X4), X5)).
mergecB(.(X1, X2), .(X3, X4), .(X3, X5)) :- ','(gtcD(X1, X3), mergecB(.(X1, X2), X4, X5)).
lecC(s(X1), s(X2)) :- lecC(X1, X2).
lecC(0, s(X1)).
lecC(0, 0).
gtcD(s(X1), s(X2)) :- gtcD(X1, X2).
gtcD(s(X1), 0).
splitcE([], [], []).
splitcE(.(X1, X2), .(X1, X3), X4) :- splitcE(X2, X4, X3).
mergesortcF([], []).
mergesortcF(.(X1, []), .(X1, [])).
mergesortcF(.(X1, .(X2, [])), X3) :- ','(mergesortcA(X1, X4), ','(mergesortcA(X2, X5), mergecB(X4, X5, X3))).
mergesortcF(.(X1, .(X2, .(X3, X4))), X5) :- ','(splitcE(X4, X6, X7), ','(mergesortcF(.(X1, .(X3, X7)), X8), ','(mergesortcF(.(X2, X6), X9), mergecB(X8, X9, X5)))).

Afs:

mergesortF(x1, x2)  =  mergesortF(x1)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mergesortF_in: (b,f)
mergeB_in: (b,b,f)
leC_in: (b,b)
lecC_in: (b,b)
gtD_in: (b,b)
gtcD_in: (b,b)
splitE_in: (b,f,f)
splitcE_in: (b,f,f)
mergesortcF_in: (b,f)
mergecB_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MERGESORTF_IN_GA(.(X1, .(X2, [])), X3) → U10_GA(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U10_GA(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U11_GA(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U11_GA(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U12_GA(X1, X2, X3, mergeB_in_gga(X4, X5, X3))
U11_GA(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → MERGEB_IN_GGA(X4, X5, X3)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → U1_GGA(X1, X2, X3, X4, X5, leC_in_gg(X1, X3))
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → LEC_IN_GG(X1, X3)
LEC_IN_GG(s(X1), s(X2)) → U7_GG(X1, X2, leC_in_gg(X1, X2))
LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → U2_GGA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U2_GGA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U3_GGA(X1, X2, X3, X4, X5, mergeB_in_gga(X2, .(X3, X4), X5))
U2_GGA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → MERGEB_IN_GGA(X2, .(X3, X4), X5)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U4_GGA(X1, X2, X3, X4, X5, gtD_in_gg(X1, X3))
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → GTD_IN_GG(X1, X3)
GTD_IN_GG(s(X1), s(X2)) → U8_GG(X1, X2, gtD_in_gg(X1, X2))
GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U5_GGA(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U5_GGA(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U6_GGA(X1, X2, X3, X4, X5, mergeB_in_gga(.(X1, X2), X4, X5))
U5_GGA(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4, X5)
MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → U13_GA(X1, X2, X3, X4, X5, splitE_in_gaa(X4, X6, X7))
MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → SPLITE_IN_GAA(X4, X6, X7)
SPLITE_IN_GAA(.(X1, X2), .(X1, X3), X4) → U9_GAA(X1, X2, X3, X4, splitE_in_gaa(X2, X4, X3))
SPLITE_IN_GAA(.(X1, X2), .(X1, X3), X4) → SPLITE_IN_GAA(X2, X4, X3)
MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → U14_GA(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U15_GA(X1, X2, X3, X4, X5, mergesortF_in_ga(.(X1, .(X3, X7)), X8))
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → MERGESORTF_IN_GA(.(X1, .(X3, X7)), X8)
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U17_GA(X1, X2, X3, X4, X5, mergesortF_in_ga(.(X2, X6), X9))
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → MERGESORTF_IN_GA(.(X2, X6), X9)
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U18_GA(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U18_GA(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U19_GA(X1, X2, X3, X4, X5, mergeB_in_gga(X8, X9, X5))
U18_GA(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → MERGEB_IN_GGA(X8, X9, X5)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
mergesortF_in_ga(x1, x2)  =  mergesortF_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
mergeB_in_gga(x1, x2, x3)  =  mergeB_in_gga(x1, x2)
leC_in_gg(x1, x2)  =  leC_in_gg(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtD_in_gg(x1, x2)  =  gtD_in_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitE_in_gaa(x1, x2, x3)  =  splitE_in_gaa(x1)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
MERGESORTF_IN_GA(x1, x2)  =  MERGESORTF_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x4, x5)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
MERGEB_IN_GGA(x1, x2, x3)  =  MERGEB_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
LEC_IN_GG(x1, x2)  =  LEC_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
GTD_IN_GG(x1, x2)  =  GTD_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
U13_GA(x1, x2, x3, x4, x5, x6)  =  U13_GA(x1, x2, x3, x4, x6)
SPLITE_IN_GAA(x1, x2, x3)  =  SPLITE_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x5)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x4, x6)
U15_GA(x1, x2, x3, x4, x5, x6)  =  U15_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x1, x2, x3, x4, x6, x7)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x1, x2, x3, x4, x6)
U18_GA(x1, x2, x3, x4, x5, x6, x7)  =  U18_GA(x1, x2, x3, x4, x6, x7)
U19_GA(x1, x2, x3, x4, x5, x6)  =  U19_GA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

MERGESORTF_IN_GA(.(X1, .(X2, [])), X3) → U10_GA(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U10_GA(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U11_GA(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U11_GA(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U12_GA(X1, X2, X3, mergeB_in_gga(X4, X5, X3))
U11_GA(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → MERGEB_IN_GGA(X4, X5, X3)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → U1_GGA(X1, X2, X3, X4, X5, leC_in_gg(X1, X3))
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → LEC_IN_GG(X1, X3)
LEC_IN_GG(s(X1), s(X2)) → U7_GG(X1, X2, leC_in_gg(X1, X2))
LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → U2_GGA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U2_GGA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U3_GGA(X1, X2, X3, X4, X5, mergeB_in_gga(X2, .(X3, X4), X5))
U2_GGA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → MERGEB_IN_GGA(X2, .(X3, X4), X5)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U4_GGA(X1, X2, X3, X4, X5, gtD_in_gg(X1, X3))
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → GTD_IN_GG(X1, X3)
GTD_IN_GG(s(X1), s(X2)) → U8_GG(X1, X2, gtD_in_gg(X1, X2))
GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U5_GGA(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U5_GGA(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U6_GGA(X1, X2, X3, X4, X5, mergeB_in_gga(.(X1, X2), X4, X5))
U5_GGA(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4, X5)
MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → U13_GA(X1, X2, X3, X4, X5, splitE_in_gaa(X4, X6, X7))
MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → SPLITE_IN_GAA(X4, X6, X7)
SPLITE_IN_GAA(.(X1, X2), .(X1, X3), X4) → U9_GAA(X1, X2, X3, X4, splitE_in_gaa(X2, X4, X3))
SPLITE_IN_GAA(.(X1, X2), .(X1, X3), X4) → SPLITE_IN_GAA(X2, X4, X3)
MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → U14_GA(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U15_GA(X1, X2, X3, X4, X5, mergesortF_in_ga(.(X1, .(X3, X7)), X8))
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → MERGESORTF_IN_GA(.(X1, .(X3, X7)), X8)
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U17_GA(X1, X2, X3, X4, X5, mergesortF_in_ga(.(X2, X6), X9))
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → MERGESORTF_IN_GA(.(X2, X6), X9)
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U18_GA(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U18_GA(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U19_GA(X1, X2, X3, X4, X5, mergeB_in_gga(X8, X9, X5))
U18_GA(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → MERGEB_IN_GGA(X8, X9, X5)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
mergesortF_in_ga(x1, x2)  =  mergesortF_in_ga(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
mergeB_in_gga(x1, x2, x3)  =  mergeB_in_gga(x1, x2)
leC_in_gg(x1, x2)  =  leC_in_gg(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtD_in_gg(x1, x2)  =  gtD_in_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitE_in_gaa(x1, x2, x3)  =  splitE_in_gaa(x1)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
MERGESORTF_IN_GA(x1, x2)  =  MERGESORTF_IN_GA(x1)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x4, x5)
U12_GA(x1, x2, x3, x4)  =  U12_GA(x1, x2, x4)
MERGEB_IN_GGA(x1, x2, x3)  =  MERGEB_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5, x6)  =  U1_GGA(x1, x2, x3, x4, x6)
LEC_IN_GG(x1, x2)  =  LEC_IN_GG(x1, x2)
U7_GG(x1, x2, x3)  =  U7_GG(x1, x2, x3)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U3_GGA(x1, x2, x3, x4, x5, x6)  =  U3_GGA(x1, x2, x3, x4, x6)
U4_GGA(x1, x2, x3, x4, x5, x6)  =  U4_GGA(x1, x2, x3, x4, x6)
GTD_IN_GG(x1, x2)  =  GTD_IN_GG(x1, x2)
U8_GG(x1, x2, x3)  =  U8_GG(x1, x2, x3)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)
U6_GGA(x1, x2, x3, x4, x5, x6)  =  U6_GGA(x1, x2, x3, x4, x6)
U13_GA(x1, x2, x3, x4, x5, x6)  =  U13_GA(x1, x2, x3, x4, x6)
SPLITE_IN_GAA(x1, x2, x3)  =  SPLITE_IN_GAA(x1)
U9_GAA(x1, x2, x3, x4, x5)  =  U9_GAA(x1, x2, x5)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x4, x6)
U15_GA(x1, x2, x3, x4, x5, x6)  =  U15_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x1, x2, x3, x4, x6, x7)
U17_GA(x1, x2, x3, x4, x5, x6)  =  U17_GA(x1, x2, x3, x4, x6)
U18_GA(x1, x2, x3, x4, x5, x6, x7)  =  U18_GA(x1, x2, x3, x4, x6, x7)
U19_GA(x1, x2, x3, x4, x5, x6)  =  U19_GA(x1, x2, x3, x4, x6)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 20 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

SPLITE_IN_GAA(.(X1, X2), .(X1, X3), X4) → SPLITE_IN_GAA(X2, X4, X3)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
SPLITE_IN_GAA(x1, x2, x3)  =  SPLITE_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

SPLITE_IN_GAA(.(X1, X2), .(X1, X3), X4) → SPLITE_IN_GAA(X2, X4, X3)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

SPLITE_IN_GAA(.(X1, X2)) → SPLITE_IN_GAA(X2)

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

(14) 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:

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

(15) YES

(16) Obligation:

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

GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
GTD_IN_GG(x1, x2)  =  GTD_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

GTD_IN_GG(s(X1), s(X2)) → GTD_IN_GG(X1, X2)

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

(21) 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:

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

(22) YES

(23) Obligation:

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

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
LEC_IN_GG(x1, x2)  =  LEC_IN_GG(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)

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

(26) PiDPToQDPProof (EQUIVALENT transformation)

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

(27) Obligation:

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

LEC_IN_GG(s(X1), s(X2)) → LEC_IN_GG(X1, X2)

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

(28) 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:

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

(29) YES

(30) Obligation:

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

MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → U2_GGA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U2_GGA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → MERGEB_IN_GGA(X2, .(X3, X4), X5)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U5_GGA(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U5_GGA(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4, X5)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
MERGEB_IN_GGA(x1, x2, x3)  =  MERGEB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X1, X5)) → U2_GGA(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U2_GGA(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → MERGEB_IN_GGA(X2, .(X3, X4), X5)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4), .(X3, X5)) → U5_GGA(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U5_GGA(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4, X5)

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
MERGEB_IN_GGA(x1, x2, x3)  =  MERGEB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4, x5, x6)  =  U2_GGA(x1, x2, x3, x4, x6)
U5_GGA(x1, x2, x3, x4, x5, x6)  =  U5_GGA(x1, x2, x3, x4, x6)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U2_GGA(X1, X2, X3, X4, lecC_in_gg(X1, X3))
U2_GGA(X1, X2, X3, X4, lecC_out_gg(X1, X3)) → MERGEB_IN_GGA(X2, .(X3, X4))
MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U5_GGA(X1, X2, X3, X4, gtcD_in_gg(X1, X3))
U5_GGA(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4)

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
gtcD_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(35) 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:

MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U2_GGA(X1, X2, X3, X4, lecC_in_gg(X1, X3))

Strictly oriented rules of the TRS R:

gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + x₂   
POL(0) = 0   
POL(MERGEB_IN_GGA(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(U25_gg(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + x₃   
POL(U26_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U2_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + x₁ + 2·x₂ + 2·x₃ + 2·x₄ + x₅   
POL(U5_GGA(x₁, x₂, x₃, x₄, x₅)) = 2 + 2·x₁ + 2·x₂ + 2·x₃ + 2·x₄ + 2·x₅   
POL(gtcD_in_gg(x₁, x₂)) = 1 + x₁ + x₂   
POL(gtcD_out_gg(x₁, x₂)) = x₁ + x₂   
POL(lecC_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(lecC_out_gg(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(s(x₁)) = 2·x₁   

(36) Obligation:

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

U2_GGA(X1, X2, X3, X4, lecC_out_gg(X1, X3)) → MERGEB_IN_GGA(X2, .(X3, X4))
MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U5_GGA(X1, X2, X3, X4, gtcD_in_gg(X1, X3))
U5_GGA(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4)

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
gtcD_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

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

(38) Obligation:

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

U5_GGA(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U5_GGA(X1, X2, X3, X4, gtcD_in_gg(X1, X3))

The TRS R consists of the following rules:

lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
gtcD_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(39) 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.

(40) Obligation:

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

U5_GGA(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U5_GGA(X1, X2, X3, X4, gtcD_in_gg(X1, X3))

The TRS R consists of the following rules:

gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

lecC_in_gg(x0, x1)
gtcD_in_gg(x0, x1)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(41) 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].

lecC_in_gg(x0, x1)
U25_gg(x0, x1, x2)

(42) Obligation:

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

U5_GGA(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4)
MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U5_GGA(X1, X2, X3, X4, gtcD_in_gg(X1, X3))

The TRS R consists of the following rules:

gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

gtcD_in_gg(x0, x1)
U26_gg(x0, x1, x2)

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

(43) 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:

• MERGEB_IN_GGA(.(X1, X2), .(X3, X4)) → U5_GGA(X1, X2, X3, X4, gtcD_in_gg(X1, X3))
  The graph contains the following edges 1 > 1, 1 > 2, 2 > 3, 2 > 4

• U5_GGA(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → MERGEB_IN_GGA(.(X1, X2), X4)
  The graph contains the following edges 4 >= 2

(44) YES

(45) Obligation:

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

MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → U14_GA(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → MERGESORTF_IN_GA(.(X1, .(X3, X7)), X8)
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → MERGESORTF_IN_GA(.(X2, X6), X9)

The TRS R consists of the following rules:

mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))
splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
mergesortcF_in_ga([], []) → mergesortcF_out_ga([], [])
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
MERGESORTF_IN_GA(x1, x2)  =  MERGESORTF_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x1, x2, x3, x4, x6, x7)

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4))), X5) → U14_GA(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → MERGESORTF_IN_GA(.(X1, .(X3, X7)), X8)
U14_GA(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
U16_GA(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → MERGESORTF_IN_GA(.(X2, X6), X9)

The TRS R consists of the following rules:

splitcE_in_gaa([], [], []) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2), .(X1, X3), X4) → U27_gaa(X1, X2, X3, X4, splitcE_in_gaa(X2, X4, X3))
mergesortcF_in_ga(.(X1, .(X2, [])), X3) → U28_ga(X1, X2, X3, mergesortcA_in_ga(X1, X4))
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4))), X5) → U31_ga(X1, X2, X3, X4, X5, splitcE_in_gaa(X4, X6, X7))
U27_gaa(X1, X2, X3, X4, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
U28_ga(X1, X2, X3, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X3, X4, mergesortcA_in_ga(X2, X5))
U31_ga(X1, X2, X3, X4, X5, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_in_ga(.(X1, .(X3, X7)), X8))
mergesortcA_in_ga(X1, .(X1, [])) → mergesortcA_out_ga(X1, .(X1, []))
U29_ga(X1, X2, X3, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, X3, mergecB_in_gga(X4, X5, X3))
U32_ga(X1, X2, X3, X4, X5, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_in_ga(.(X2, X6), X9))
U30_ga(X1, X2, X3, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
U33_ga(X1, X2, X3, X4, X5, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, X5, mergecB_in_gga(X8, X9, X5))
mergecB_in_gga(X1, [], X1) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1, X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X1, X5)) → U21_gga(X1, X2, X3, X4, X5, lecC_in_gg(X1, X3))
mergecB_in_gga(.(X1, X2), .(X3, X4), .(X3, X5)) → U23_gga(X1, X2, X3, X4, X5, gtcD_in_gg(X1, X3))
mergesortcF_in_ga(.(X1, []), .(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
U34_ga(X1, X2, X3, X4, X5, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)
U21_gga(X1, X2, X3, X4, X5, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, X5, mergecB_in_gga(X2, .(X3, X4), X5))
U23_gga(X1, X2, X3, X4, X5, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, X5, mergecB_in_gga(.(X1, X2), X4, X5))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U22_gga(X1, X2, X3, X4, X5, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gga(X1, X2, X3, X4, X5, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
[]  =  []
mergesortcA_in_ga(x1, x2)  =  mergesortcA_in_ga(x1)
mergesortcA_out_ga(x1, x2)  =  mergesortcA_out_ga(x1, x2)
s(x1)  =  s(x1)
lecC_in_gg(x1, x2)  =  lecC_in_gg(x1, x2)
U25_gg(x1, x2, x3)  =  U25_gg(x1, x2, x3)
0  =  0
lecC_out_gg(x1, x2)  =  lecC_out_gg(x1, x2)
gtcD_in_gg(x1, x2)  =  gtcD_in_gg(x1, x2)
U26_gg(x1, x2, x3)  =  U26_gg(x1, x2, x3)
gtcD_out_gg(x1, x2)  =  gtcD_out_gg(x1, x2)
splitcE_in_gaa(x1, x2, x3)  =  splitcE_in_gaa(x1)
splitcE_out_gaa(x1, x2, x3)  =  splitcE_out_gaa(x1, x2, x3)
U27_gaa(x1, x2, x3, x4, x5)  =  U27_gaa(x1, x2, x5)
mergesortcF_in_ga(x1, x2)  =  mergesortcF_in_ga(x1)
mergesortcF_out_ga(x1, x2)  =  mergesortcF_out_ga(x1, x2)
U28_ga(x1, x2, x3, x4)  =  U28_ga(x1, x2, x4)
U29_ga(x1, x2, x3, x4, x5)  =  U29_ga(x1, x2, x4, x5)
U30_ga(x1, x2, x3, x4)  =  U30_ga(x1, x2, x4)
mergecB_in_gga(x1, x2, x3)  =  mergecB_in_gga(x1, x2)
mergecB_out_gga(x1, x2, x3)  =  mergecB_out_gga(x1, x2, x3)
U21_gga(x1, x2, x3, x4, x5, x6)  =  U21_gga(x1, x2, x3, x4, x6)
U22_gga(x1, x2, x3, x4, x5, x6)  =  U22_gga(x1, x2, x3, x4, x6)
U23_gga(x1, x2, x3, x4, x5, x6)  =  U23_gga(x1, x2, x3, x4, x6)
U24_gga(x1, x2, x3, x4, x5, x6)  =  U24_gga(x1, x2, x3, x4, x6)
U31_ga(x1, x2, x3, x4, x5, x6)  =  U31_ga(x1, x2, x3, x4, x6)
U32_ga(x1, x2, x3, x4, x5, x6, x7)  =  U32_ga(x1, x2, x3, x4, x6, x7)
U33_ga(x1, x2, x3, x4, x5, x6, x7)  =  U33_ga(x1, x2, x3, x4, x6, x7)
U34_ga(x1, x2, x3, x4, x5, x6)  =  U34_ga(x1, x2, x3, x4, x6)
MERGESORTF_IN_GA(x1, x2)  =  MERGESORTF_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5, x6)  =  U14_GA(x1, x2, x3, x4, x6)
U16_GA(x1, x2, x3, x4, x5, x6, x7)  =  U16_GA(x1, x2, x3, x4, x6, x7)

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4)))) → U14_GA(X1, X2, X3, X4, splitcE_in_gaa(X4))
U14_GA(X1, X2, X3, X4, splitcE_out_gaa(X4, X6, X7)) → MERGESORTF_IN_GA(.(X1, .(X3, X7)))
U14_GA(X1, X2, X3, X4, splitcE_out_gaa(X4, X6, X7)) → U16_GA(X1, X2, X3, X4, X6, mergesortcF_in_ga(.(X1, .(X3, X7))))
U16_GA(X1, X2, X3, X4, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → MERGESORTF_IN_GA(.(X2, X6))

The TRS R consists of the following rules:

splitcE_in_gaa([]) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2)) → U27_gaa(X1, X2, splitcE_in_gaa(X2))
mergesortcF_in_ga(.(X1, .(X2, []))) → U28_ga(X1, X2, mergesortcA_in_ga(X1))
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4)))) → U31_ga(X1, X2, X3, X4, splitcE_in_gaa(X4))
U27_gaa(X1, X2, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
U28_ga(X1, X2, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X4, mergesortcA_in_ga(X2))
U31_ga(X1, X2, X3, X4, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X6, mergesortcF_in_ga(.(X1, .(X3, X7))))
mergesortcA_in_ga(X1) → mergesortcA_out_ga(X1, .(X1, []))
U29_ga(X1, X2, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, mergecB_in_gga(X4, X5))
U32_ga(X1, X2, X3, X4, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X8, mergesortcF_in_ga(.(X2, X6)))
U30_ga(X1, X2, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
U33_ga(X1, X2, X3, X4, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, mergecB_in_gga(X8, X9))
mergecB_in_gga(X1, []) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4)) → U21_gga(X1, X2, X3, X4, lecC_in_gg(X1, X3))
mergecB_in_gga(.(X1, X2), .(X3, X4)) → U23_gga(X1, X2, X3, X4, gtcD_in_gg(X1, X3))
mergesortcF_in_ga(.(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
U34_ga(X1, X2, X3, X4, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)
U21_gga(X1, X2, X3, X4, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, mergecB_in_gga(X2, .(X3, X4)))
U23_gga(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, mergecB_in_gga(.(X1, X2), X4))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U22_gga(X1, X2, X3, X4, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gga(X1, X2, X3, X4, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

splitcE_in_gaa(x0)
mergesortcF_in_ga(x0)
U27_gaa(x0, x1, x2)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2, x3, x4)
mergesortcA_in_ga(x0)
U29_ga(x0, x1, x2, x3)
U32_ga(x0, x1, x2, x3, x4, x5)
U30_ga(x0, x1, x2)
U33_ga(x0, x1, x2, x3, x4, x5)
mergecB_in_gga(x0, x1)
U34_ga(x0, x1, x2, x3, x4)
U21_gga(x0, x1, x2, x3, x4)
U23_gga(x0, x1, x2, x3, x4)
lecC_in_gg(x0, x1)
U22_gga(x0, x1, x2, x3, x4)
gtcD_in_gg(x0, x1)
U24_gga(x0, x1, x2, x3, x4)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(50) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

MERGESORTF_IN_GA(.(X1, .(X2, .(X3, X4)))) → U14_GA(X1, X2, X3, X4, splitcE_in_gaa(X4))
U14_GA(X1, X2, X3, X4, splitcE_out_gaa(X4, X6, X7)) → U16_GA(X1, X2, X3, X4, X6, mergesortcF_in_ga(.(X1, .(X3, X7))))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(0) = 1   
POL(MERGESORTF_IN_GA(x₁)) = x₁   
POL(U14_GA(x₁, x₂, x₃, x₄, x₅)) = 1 + x₅   
POL(U16_GA(x₁, x₂, x₃, x₄, x₅, x₆)) = 1 + x₅   
POL(U21_gga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U22_gga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U23_gga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U24_gga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U25_gg(x₁, x₂, x₃)) = 1   
POL(U26_gg(x₁, x₂, x₃)) = 0   
POL(U27_gaa(x₁, x₂, x₃)) = 1 + x₃   
POL(U28_ga(x₁, x₂, x₃)) = 0   
POL(U29_ga(x₁, x₂, x₃, x₄)) = 0   
POL(U30_ga(x₁, x₂, x₃)) = 0   
POL(U31_ga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL(U32_ga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0   
POL(U33_ga(x₁, x₂, x₃, x₄, x₅, x₆)) = 0   
POL(U34_ga(x₁, x₂, x₃, x₄, x₅)) = 0   
POL([]) = 0   
POL(gtcD_in_gg(x₁, x₂)) = 0   
POL(gtcD_out_gg(x₁, x₂)) = 0   
POL(lecC_in_gg(x₁, x₂)) = 1 + x₁   
POL(lecC_out_gg(x₁, x₂)) = 1   
POL(mergecB_in_gga(x₁, x₂)) = 0   
POL(mergecB_out_gga(x₁, x₂, x₃)) = 0   
POL(mergesortcA_in_ga(x₁)) = 0   
POL(mergesortcA_out_ga(x₁, x₂)) = 0   
POL(mergesortcF_in_ga(x₁)) = 0   
POL(mergesortcF_out_ga(x₁, x₂)) = 0   
POL(s(x₁)) = 1 + x₁   
POL(splitcE_in_gaa(x₁)) = 1 + x₁   
POL(splitcE_out_gaa(x₁, x₂, x₃)) = 1 + x₂ + x₃   

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

splitcE_in_gaa([]) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2)) → U27_gaa(X1, X2, splitcE_in_gaa(X2))
U27_gaa(X1, X2, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)

(51) Obligation:

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

U14_GA(X1, X2, X3, X4, splitcE_out_gaa(X4, X6, X7)) → MERGESORTF_IN_GA(.(X1, .(X3, X7)))
U16_GA(X1, X2, X3, X4, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → MERGESORTF_IN_GA(.(X2, X6))

The TRS R consists of the following rules:

splitcE_in_gaa([]) → splitcE_out_gaa([], [], [])
splitcE_in_gaa(.(X1, X2)) → U27_gaa(X1, X2, splitcE_in_gaa(X2))
mergesortcF_in_ga(.(X1, .(X2, []))) → U28_ga(X1, X2, mergesortcA_in_ga(X1))
mergesortcF_in_ga(.(X1, .(X2, .(X3, X4)))) → U31_ga(X1, X2, X3, X4, splitcE_in_gaa(X4))
U27_gaa(X1, X2, splitcE_out_gaa(X2, X4, X3)) → splitcE_out_gaa(.(X1, X2), .(X1, X3), X4)
U28_ga(X1, X2, mergesortcA_out_ga(X1, X4)) → U29_ga(X1, X2, X4, mergesortcA_in_ga(X2))
U31_ga(X1, X2, X3, X4, splitcE_out_gaa(X4, X6, X7)) → U32_ga(X1, X2, X3, X4, X6, mergesortcF_in_ga(.(X1, .(X3, X7))))
mergesortcA_in_ga(X1) → mergesortcA_out_ga(X1, .(X1, []))
U29_ga(X1, X2, X4, mergesortcA_out_ga(X2, X5)) → U30_ga(X1, X2, mergecB_in_gga(X4, X5))
U32_ga(X1, X2, X3, X4, X6, mergesortcF_out_ga(.(X1, .(X3, X7)), X8)) → U33_ga(X1, X2, X3, X4, X8, mergesortcF_in_ga(.(X2, X6)))
U30_ga(X1, X2, mergecB_out_gga(X4, X5, X3)) → mergesortcF_out_ga(.(X1, .(X2, [])), X3)
U33_ga(X1, X2, X3, X4, X8, mergesortcF_out_ga(.(X2, X6), X9)) → U34_ga(X1, X2, X3, X4, mergecB_in_gga(X8, X9))
mergecB_in_gga(X1, []) → mergecB_out_gga(X1, [], X1)
mergecB_in_gga([], X1) → mergecB_out_gga([], X1, X1)
mergecB_in_gga(.(X1, X2), .(X3, X4)) → U21_gga(X1, X2, X3, X4, lecC_in_gg(X1, X3))
mergecB_in_gga(.(X1, X2), .(X3, X4)) → U23_gga(X1, X2, X3, X4, gtcD_in_gg(X1, X3))
mergesortcF_in_ga(.(X1, [])) → mergesortcF_out_ga(.(X1, []), .(X1, []))
U34_ga(X1, X2, X3, X4, mergecB_out_gga(X8, X9, X5)) → mergesortcF_out_ga(.(X1, .(X2, .(X3, X4))), X5)
U21_gga(X1, X2, X3, X4, lecC_out_gg(X1, X3)) → U22_gga(X1, X2, X3, X4, mergecB_in_gga(X2, .(X3, X4)))
U23_gga(X1, X2, X3, X4, gtcD_out_gg(X1, X3)) → U24_gga(X1, X2, X3, X4, mergecB_in_gga(.(X1, X2), X4))
lecC_in_gg(s(X1), s(X2)) → U25_gg(X1, X2, lecC_in_gg(X1, X2))
lecC_in_gg(0, s(X1)) → lecC_out_gg(0, s(X1))
lecC_in_gg(0, 0) → lecC_out_gg(0, 0)
U22_gga(X1, X2, X3, X4, mergecB_out_gga(X2, .(X3, X4), X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X1, X5))
gtcD_in_gg(s(X1), s(X2)) → U26_gg(X1, X2, gtcD_in_gg(X1, X2))
gtcD_in_gg(s(X1), 0) → gtcD_out_gg(s(X1), 0)
U24_gga(X1, X2, X3, X4, mergecB_out_gga(.(X1, X2), X4, X5)) → mergecB_out_gga(.(X1, X2), .(X3, X4), .(X3, X5))
U25_gg(X1, X2, lecC_out_gg(X1, X2)) → lecC_out_gg(s(X1), s(X2))
U26_gg(X1, X2, gtcD_out_gg(X1, X2)) → gtcD_out_gg(s(X1), s(X2))

The set Q consists of the following terms:

splitcE_in_gaa(x0)
mergesortcF_in_ga(x0)
U27_gaa(x0, x1, x2)
U28_ga(x0, x1, x2)
U31_ga(x0, x1, x2, x3, x4)
mergesortcA_in_ga(x0)
U29_ga(x0, x1, x2, x3)
U32_ga(x0, x1, x2, x3, x4, x5)
U30_ga(x0, x1, x2)
U33_ga(x0, x1, x2, x3, x4, x5)
mergecB_in_gga(x0, x1)
U34_ga(x0, x1, x2, x3, x4)
U21_gga(x0, x1, x2, x3, x4)
U23_gga(x0, x1, x2, x3, x4)
lecC_in_gg(x0, x1)
U22_gga(x0, x1, x2, x3, x4)
gtcD_in_gg(x0, x1)
U24_gga(x0, x1, x2, x3, x4)
U25_gg(x0, x1, x2)
U26_gg(x0, x1, x2)

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

(52) DependencyGraphProof (EQUIVALENT transformation)

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

(53) TRUE