Left Termination proof of /tmp/tmpRMF92o/pl8.3.1a.pl

Left Termination of the query pattern minsort_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇔)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔)

        ↳27 YES

    ↳28 PiDP

        ↳29 UsableRulesProof (⇔)

        ↳30 PiDP

        ↳31 PiDPToQDPProof (⇔)

        ↳32 QDP

        ↳33 QDPSizeChangeProof (⇔)

        ↳34 YES

    ↳35 PiDP

        ↳36 UsableRulesProof (⇔)

        ↳37 PiDP

        ↳38 PiDPToQDPProof (⇐)

        ↳39 QDP

        ↳40 QDPSizeChangeProof (⇔)

        ↳41 YES

    ↳42 PiDP

        ↳43 PiDPToQDPProof (⇐)

        ↳44 QDP

        ↳45 QDPOrderProof (⇔)

        ↳46 QDP

        ↳47 DependencyGraphProof (⇔)

        ↳48 TRUE

(0) Obligation:

Clauses:

minsort([], []).
minsort(L, .(X, L1)) :- ','(min1(X, L), ','(remove(X, L, L2), minsort(L2, L1))).
min1(M, .(X, L)) :- min2(X, M, L).
min2(X, X, []).
min2(X, A, .(M, L)) :- ','(min(X, M, B), min2(B, A, L)).
min(X, Y, X) :- le(X, Y).
min(X, Y, Y) :- gt(X, Y).
remove(N, .(N, L), L).
remove(N, .(M, L), .(M, L1)) :- ','(notEq(N, M), remove(N, L, L1)).
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).
notEq(s(X), s(Y)) :- notEq(X, Y).
notEq(s(X), 0).
notEq(0, s(X)).

Queries:

minsort(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: minsort(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: minsort(T1, T2), SCOPE: 1, CLAUSE: 0 | minsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | minsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: minsort(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nminsort(T1, T2) !=? minsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: minsort([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(min1(T7, []), ','(remove(T7, [], X7), minsort(X7, T8)))\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(min1(T7, []), ','(remove(T7, [], X7), minsort(X7, T8))), SCOPE: 2, CLAUSE: 2\n\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(min1(T16, T13), ','(remove(T16, T13, X20), minsort(X20, T17)))\n\nT13 is ground\nX20 is free; \nminsort(T13, T2) !=? minsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(min1(T16, T13), ','(remove(T16, T13, X20), minsort(X20, T17))), SCOPE: 3, CLAUSE: 2\n\nT13 is ground\nX20 is free; \nminsort(T13, T2) !=? minsort([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(min2(T25, T27, T26), ','(remove(T27, .(T25, T26), X20), minsort(X20, T28)))\n\nT25 is ground\nT26 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: min2(T25, T27, T26)\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(remove(T31, .(T25, T26), X20), minsort(X20, T32))\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: min2(T25, T27, T26), SCOPE: 4, CLAUSE: 3 | min2(T25, T27, T26), SCOPE: 4, CLAUSE: 4\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: min2(T25, T27, T26), SCOPE: 4, CLAUSE: 3\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: min2(T25, T27, T26), SCOPE: 4, CLAUSE: 4\n\nT25 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(min(T48, T50, X59), min2(X59, T52, T51))\n\nT48 is ground\nT50 is ground\nT51 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: min(T48, T50, X59)\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: min2(T55, T52, T51)\n\nT51 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: min(T48, T50, X59), SCOPE: 5, CLAUSE: 5 | min(T48, T50, X59), SCOPE: 5, CLAUSE: 6\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: min(T48, T50, X59), SCOPE: 5, CLAUSE: 5\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: min(T48, T50, X59), SCOPE: 5, CLAUSE: 6\n\nT48 is ground\nT50 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: le(T70, T71)\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: le(T70, T71), SCOPE: 6, CLAUSE: 11 | le(T70, T71), SCOPE: 6, CLAUSE: 12 | le(T70, T71), SCOPE: 6, CLAUSE: 13\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: le(T70, T71), SCOPE: 6, CLAUSE: 11\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T70, T71), SCOPE: 6, CLAUSE: 12 | le(T70, T71), SCOPE: 6, CLAUSE: 13\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: le(T82, T83)\n\nT82 is ground\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: le(T70, T71), SCOPE: 6, CLAUSE: 12\n\nT70 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T70, T71), SCOPE: 6, CLAUSE: 13\n\nT70 is ground\nT71 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: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: gt(T97, T98)\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: gt(T97, T98), SCOPE: 7, CLAUSE: 9 | gt(T97, T98), SCOPE: 7, CLAUSE: 10\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: gt(T97, T98), SCOPE: 7, CLAUSE: 9\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: gt(T97, T98), SCOPE: 7, CLAUSE: 10\n\nT97 is ground\nT98 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: gt(T109, T110)\n\nT109 is ground\nT110 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: remove(T31, .(T25, T26), X20)\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: minsort(T122, T32)\n\nT122 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 7 | remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 8\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 7\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: remove(T31, .(T25, T26), X20), SCOPE: 8, CLAUSE: 8\n\nT25 is ground\nT26 is ground\nT31 is ground\nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(notEq(T145, T146), remove(T145, T147, X170))\n\nT145 is ground\nT146 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: notEq(T145, T146)\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: remove(T145, T147, X170)\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: notEq(T145, T146), SCOPE: 9, CLAUSE: 14 | notEq(T145, T146), SCOPE: 9, CLAUSE: 15 | notEq(T145, T146), SCOPE: 9, CLAUSE: 16\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: notEq(T145, T146), SCOPE: 9, CLAUSE: 14\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: notEq(T145, T146), SCOPE: 9, CLAUSE: 15 | notEq(T145, T146), SCOPE: 9, CLAUSE: 16\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: notEq(T160, T161)\n\nT160 is ground\nT161 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: notEq(T145, T146), SCOPE: 9, CLAUSE: 15\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: notEq(T145, T146), SCOPE: 9, CLAUSE: 16\n\nT145 is ground\nT146 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: remove(T145, T147, X170), SCOPE: 10, CLAUSE: 7 | remove(T145, T147, X170), SCOPE: 10, CLAUSE: 8\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: remove(T145, T147, X170), SCOPE: 10, CLAUSE: 7\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: remove(T145, T147, X170), SCOPE: 10, CLAUSE: 8\n\nT145 is ground\nT147 is ground\nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(notEq(T192, T193), remove(T192, T194, X229))\n\nT192 is ground\nT193 is ground\nT194 is ground\nX229 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 85 [arrowhead = none ];
85 -> 2;

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

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

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

89 [label="EVAL with clause\nminsort(X17, .(X18, X19)) :- ','(min1(X18, X17), ','(remove(X18, X17, X20), minsort(X20, X19))).\nand substitution\nT1 -> T13\nX17 -> T13\nX18 -> T16\nX19 -> T17\nT2 -> .(T16, T17)\nT14 -> T16\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 89 [arrowhead = none ];
89 -> 10;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 90 [arrowhead = none ];
90 -> 11;

91 [label="EVAL with clause\nminsort(X4, .(X5, X6)) :- ','(min1(X5, X4), ','(remove(X5, X4, X7), minsort(X7, X6))).\nand substitution\nX4 -> []\nX5 -> T7\nX6 -> T8\nT2 -> .(T7, T8)\nT5 -> T7\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 91 [arrowhead = none ];
91 -> 6;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 92 [arrowhead = none ];
92 -> 7;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 93 [arrowhead = none ];
93 -> 8;

94 [label="BACKTRACK\nfor clause: min1(M, .(X, L)) :- min2(X, M, L)because of non-unification", fontsize=14, style = filled, fillcolor = white];
8 -> 94 [arrowhead = none ];
94 -> 9;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 95 [arrowhead = none ];
95 -> 12;

96 [label="EVAL with clause\nmin1(X30, .(X31, X32)) :- min2(X31, X30, X32).\nand substitution\nT16 -> T27\nX30 -> T27\nX31 -> T25\nX32 -> T26\nT13 -> .(T25, T26)\nT24 -> T27\nT17 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 96 [arrowhead = none ];
96 -> 13;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 97 [arrowhead = none ];
97 -> 14;

98 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 98 [arrowhead = none ];
98 -> 15;

99 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nreplacements:\nT27 -> T31\nT28 -> T32", fontsize=14, style = filled, fillcolor = violet];
13 -> 99 [arrowhead = none ];
99 -> 16;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 100 [arrowhead = none ];
100 -> 17;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 101 [arrowhead = none ];
101 -> 53;

102 [label="SPLIT 2\nnew knowledge:\nT122 is ground\nreplacements:\nX20 -> T122", fontsize=14, style = filled, fillcolor = violet];
16 -> 102 [arrowhead = none ];
102 -> 54;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 103 [arrowhead = none ];
103 -> 18;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 104 [arrowhead = none ];
104 -> 19;

105 [label="EVAL with clause\nmin2(X44, X44, []).\nand substitution\nT25 -> T39\nX44 -> T39\nT27 -> T39\nT26 -> []", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 105 [arrowhead = none ];
105 -> 20;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 106 [arrowhead = none ];
106 -> 21;

107 [label="EVAL with clause\nmin2(X55, X56, .(X57, X58)) :- ','(min(X55, X57, X59), min2(X59, X56, X58)).\nand substitution\nT25 -> T48\nX55 -> T48\nT27 -> T52\nX56 -> T52\nX57 -> T50\nX58 -> T51\nT26 -> .(T50, T51)\nT49 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 107 [arrowhead = none ];
107 -> 23;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 108 [arrowhead = none ];
108 -> 24;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 109 [arrowhead = none ];
109 -> 22;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 110 [arrowhead = none ];
110 -> 25;

111 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nreplacements:\nX59 -> T55", fontsize=14, style = filled, fillcolor = violet];
23 -> 111 [arrowhead = none ];
111 -> 26;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 112 [arrowhead = none ];
112 -> 27;

113 [label="INSTANCE with matching:\nT25 -> T55\nT27 -> T52\nT26 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 113 [arrowhead = none , style=dashed];
113 -> 15[style=dashed];

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 114 [arrowhead = none ];
114 -> 28;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 115 [arrowhead = none ];
115 -> 29;

116 [label="ONLY EVAL with clause\nmin(X80, X81, X80) :- le(X80, X81).\nand substitution\nT48 -> T70\nX80 -> T70\nT50 -> T71\nX81 -> T71\nX59 -> T70", fontsize=14, style = filled, fillcolor = white];
28 -> 116 [arrowhead = none ];
116 -> 30;

117 [label="ONLY EVAL with clause\nmin(X110, X111, X111) :- gt(X110, X111).\nand substitution\nT48 -> T97\nX110 -> T97\nT50 -> T98\nX111 -> T98\nX59 -> T98", fontsize=14, style = filled, fillcolor = white];
29 -> 117 [arrowhead = none ];
117 -> 44;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 118 [arrowhead = none ];
118 -> 31;

119 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 119 [arrowhead = none ];
119 -> 32;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 120 [arrowhead = none ];
120 -> 33;

121 [label="EVAL with clause\nle(s(X92), s(X93)) :- le(X92, X93).\nand substitution\nX92 -> T82\nT70 -> s(T82)\nX93 -> T83\nT71 -> s(T83)", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 121 [arrowhead = none ];
121 -> 34;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 122 [arrowhead = none ];
122 -> 35;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 123 [arrowhead = none ];
123 -> 36;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 124 [arrowhead = none ];
124 -> 37;

125 [label="INSTANCE with matching:\nT70 -> T82\nT71 -> T83", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 125 [arrowhead = none , style=dashed];
125 -> 30[style=dashed];

126 [label="EVAL with clause\nle(0, s(X102)).\nand substitution\nT70 -> 0\nX102 -> T90\nT71 -> s(T90)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 126 [arrowhead = none ];
126 -> 38;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 127 [arrowhead = none ];
127 -> 39;

128 [label="EVAL with clause\nle(0, 0).\nand substitution\nT70 -> 0\nT71 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 128 [arrowhead = none ];
128 -> 41;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 129 [arrowhead = none ];
129 -> 42;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 130 [arrowhead = none ];
130 -> 40;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 131 [arrowhead = none ];
131 -> 43;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 132 [arrowhead = none ];
132 -> 45;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 133 [arrowhead = none ];
133 -> 46;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 134 [arrowhead = none ];
134 -> 47;

135 [label="EVAL with clause\ngt(s(X122), s(X123)) :- gt(X122, X123).\nand substitution\nX122 -> T109\nT97 -> s(T109)\nX123 -> T110\nT98 -> s(T110)", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 135 [arrowhead = none ];
135 -> 48;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 136 [arrowhead = none ];
136 -> 49;

137 [label="EVAL with clause\ngt(s(X130), 0).\nand substitution\nX130 -> T115\nT97 -> s(T115)\nT98 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 137 [arrowhead = none ];
137 -> 50;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 138 [arrowhead = none ];
138 -> 51;

139 [label="INSTANCE with matching:\nT97 -> T109\nT98 -> T110", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 139 [arrowhead = none , style=dashed];
139 -> 44[style=dashed];

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 140 [arrowhead = none ];
140 -> 52;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 141 [arrowhead = none ];
141 -> 55;

142 [label="INSTANCE with matching:\nT1 -> T122\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
54 -> 142 [arrowhead = none , style=dashed];
142 -> 1[style=dashed];

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 143 [arrowhead = none ];
143 -> 56;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
55 -> 144 [arrowhead = none ];
144 -> 57;

145 [label="EVAL with clause\nremove(X150, .(X150, X151), X151).\nand substitution\nT31 -> T135\nX150 -> T135\nT25 -> T135\nT26 -> T136\nX151 -> T136\nX20 -> T136", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 145 [arrowhead = none ];
145 -> 58;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 146 [arrowhead = none ];
146 -> 59;

147 [label="ONLY EVAL with clause\nremove(X166, .(X167, X168), .(X167, X169)) :- ','(notEq(X166, X167), remove(X166, X168, X169)).\nand substitution\nT31 -> T145\nX166 -> T145\nT25 -> T146\nX167 -> T146\nT26 -> T147\nX168 -> T147\nX169 -> X170\nX20 -> .(T146, X170)", fontsize=14, style = filled, fillcolor = white];
57 -> 147 [arrowhead = none ];
147 -> 61;

148 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 148 [arrowhead = none ];
148 -> 60;

149 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 149 [arrowhead = none ];
149 -> 62;

150 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
61 -> 150 [arrowhead = none ];
150 -> 63;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 151 [arrowhead = none ];
151 -> 64;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 152 [arrowhead = none ];
152 -> 77;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 153 [arrowhead = none ];
153 -> 65;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 154 [arrowhead = none ];
154 -> 66;

155 [label="EVAL with clause\nnotEq(s(X183), s(X184)) :- notEq(X183, X184).\nand substitution\nX183 -> T160\nT145 -> s(T160)\nX184 -> T161\nT146 -> s(T161)", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 155 [arrowhead = none ];
155 -> 67;

156 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 156 [arrowhead = none ];
156 -> 68;

157 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 157 [arrowhead = none ];
157 -> 69;

158 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 158 [arrowhead = none ];
158 -> 70;

159 [label="INSTANCE with matching:\nT145 -> T160\nT146 -> T161", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 159 [arrowhead = none , style=dashed];
159 -> 62[style=dashed];

160 [label="EVAL with clause\nnotEq(s(X193), 0).\nand substitution\nX193 -> T168\nT145 -> s(T168)\nT146 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 160 [arrowhead = none ];
160 -> 71;

161 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 161 [arrowhead = none ];
161 -> 72;

162 [label="EVAL with clause\nnotEq(0, s(X197)).\nand substitution\nT145 -> 0\nX197 -> T171\nT146 -> s(T171)", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 162 [arrowhead = none ];
162 -> 74;

163 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 163 [arrowhead = none ];
163 -> 75;

164 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 164 [arrowhead = none ];
164 -> 73;

165 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 165 [arrowhead = none ];
165 -> 76;

166 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 166 [arrowhead = none ];
166 -> 78;

167 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
77 -> 167 [arrowhead = none ];
167 -> 79;

168 [label="EVAL with clause\nremove(X211, .(X211, X212), X212).\nand substitution\nT145 -> T184\nX211 -> T184\nX212 -> T185\nT147 -> .(T184, T185)\nX170 -> T185", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 168 [arrowhead = none ];
168 -> 80;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 169 [arrowhead = none ];
169 -> 81;

170 [label="EVAL with clause\nremove(X225, .(X226, X227), .(X226, X228)) :- ','(notEq(X225, X226), remove(X225, X227, X228)).\nand substitution\nT145 -> T192\nX225 -> T192\nX226 -> T193\nX227 -> T194\nT147 -> .(T193, T194)\nX228 -> X229\nX170 -> .(T193, X229)", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 170 [arrowhead = none ];
170 -> 83;

171 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 171 [arrowhead = none ];
171 -> 84;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 172 [arrowhead = none ];
172 -> 82;

173 [label="INSTANCE with matching:\nT145 -> T192\nT146 -> T193\nT147 -> T194\nX170 -> X229", fontsize=14, style = filled, fillcolor = lightgrey];
83 -> 173 [arrowhead = none , style=dashed];
173 -> 61[style=dashed];

}

(2) Obligation:

Triples:

min215(T70, T52, .(T71, T51)) :- le30(T70, T71).
min215(T97, T52, .(T98, T51)) :- gt44(T97, T98).
min215(T48, T52, .(T50, T51)) :- ','(minc25(T48, T50, T55), min215(T55, T52, T51)).
le30(s(T82), s(T83)) :- le30(T82, T83).
gt44(s(T109), s(T110)) :- gt44(T109, T110).
notEq62(s(T160), s(T161)) :- notEq62(T160, T161).
p61(T145, T146, T147, X170) :- notEq62(T145, T146).
p61(T192, T146, .(T193, T194), .(T193, X229)) :- ','(notEqc62(T192, T146), p61(T192, T193, T194, X229)).
minsort1(.(T25, T26), .(T27, T28)) :- min215(T25, T27, T26).
minsort1(.(T146, T147), .(T145, T32)) :- ','(min2c15(T146, T145, T147), p61(T145, T146, T147, X170)).
minsort1(.(T25, T26), .(T31, T32)) :- ','(min2c15(T25, T31, T26), ','(removec53(T31, T25, T26, T122), minsort1(T122, T32))).

Clauses:

min2c15(T39, T39, []).
min2c15(T48, T52, .(T50, T51)) :- ','(minc25(T48, T50, T55), min2c15(T55, T52, T51)).
lec30(s(T82), s(T83)) :- lec30(T82, T83).
lec30(0, s(T90)).
lec30(0, 0).
gtc44(s(T109), s(T110)) :- gtc44(T109, T110).
gtc44(s(T115), 0).
minsortc1([], []).
minsortc1(.(T25, T26), .(T31, T32)) :- ','(min2c15(T25, T31, T26), ','(removec53(T31, T25, T26, T122), minsortc1(T122, T32))).
notEqc62(s(T160), s(T161)) :- notEqc62(T160, T161).
notEqc62(s(T168), 0).
notEqc62(0, s(T171)).
qc61(T184, T146, .(T184, T185), T185) :- notEqc62(T184, T146).
qc61(T192, T146, .(T193, T194), .(T193, X229)) :- ','(notEqc62(T192, T146), qc61(T192, T193, T194, X229)).
removec53(T135, T135, T136, T136).
removec53(T145, T146, T147, .(T146, X170)) :- qc61(T145, T146, T147, X170).
minc25(T70, T71, T70) :- lec30(T70, T71).
minc25(T97, T98, T98) :- gtc44(T97, T98).

Afs:

minsort1(x1, x2) = minsort1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minsort1_in: (b,f)
min215_in: (b,f,b)
le30_in: (b,b)
gt44_in: (b,b)
minc25_in: (b,b,f)
lec30_in: (b,b)
gtc44_in: (b,b)
min2c15_in: (b,f,b)
p61_in: (b,b,b,f)
notEq62_in: (b,b)
notEqc62_in: (b,b)
removec53_in: (b,b,b,f)
qc61_in: (b,b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U11_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T70, T52, .(T71, T51)) → U1_GAG(T70, T52, T71, T51, le30_in_gg(T70, T71))
MIN215_IN_GAG(T70, T52, .(T71, T51)) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U5_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN215_IN_GAG(T97, T52, .(T98, T51)) → U2_GAG(T97, T52, T98, T51, gt44_in_gg(T97, T98))
MIN215_IN_GAG(T97, T52, .(T98, T51)) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U6_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U4_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T146, T147), .(T145, T32)) → U12_GA(T146, T147, T145, T32, min2c15_in_gag(T146, T145, T147))
U12_GA(T146, T147, T145, T32, min2c15_out_gag(T146, T145, T147)) → U13_GA(T146, T147, T145, T32, p61_in_ggga(T145, T146, T147, X170))
U12_GA(T146, T147, T145, T32, min2c15_out_gag(T146, T145, T147)) → P61_IN_GGGA(T145, T146, T147, X170)
P61_IN_GGGA(T145, T146, T147, X170) → U8_GGGA(T145, T146, T147, X170, notEq62_in_gg(T145, T146))
P61_IN_GGGA(T145, T146, T147, X170) → NOTEQ62_IN_GG(T145, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U7_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_GGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_GGGA(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U9_GGGA(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U10_GGGA(T192, T146, T193, T194, X229, p61_in_ggga(T192, T193, T194, X229))
U9_GGGA(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → P61_IN_GGGA(T192, T193, T194, X229)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U14_GA(T25, T26, T31, T32, min2c15_in_gag(T25, T31, T26))
U14_GA(T25, T26, T31, T32, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, T31, T32, removec53_in_ggga(T31, T25, T26, T122))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T122)) → U16_GA(T25, T26, T31, T32, minsort1_in_ga(T122, T32))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T122)) → MINSORT1_IN_GA(T122, T32)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
minsort1_in_ga(x1, x2) = minsort1_in_ga(x1)
.(x1, x2) = .(x1, x2)
min215_in_gag(x1, x2, x3) = min215_in_gag(x1, x3)
le30_in_gg(x1, x2) = le30_in_gg(x1, x2)
s(x1) = s(x1)
gt44_in_gg(x1, x2) = gt44_in_gg(x1, x2)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
p61_in_ggga(x1, x2, x3, x4) = p61_in_ggga(x1, x2, x3)
notEq62_in_gg(x1, x2) = notEq62_in_gg(x1, x2)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
MINSORT1_IN_GA(x1, x2) = MINSORT1_IN_GA(x1)
U11_GA(x1, x2, x3, x4, x5) = U11_GA(x1, x2, x5)
MIN215_IN_GAG(x1, x2, x3) = MIN215_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5) = U1_GAG(x1, x3, x4, x5)
LE30_IN_GG(x1, x2) = LE30_IN_GG(x1, x2)
U5_GG(x1, x2, x3) = U5_GG(x1, x2, x3)
U2_GAG(x1, x2, x3, x4, x5) = U2_GAG(x1, x3, x4, x5)
GT44_IN_GG(x1, x2) = GT44_IN_GG(x1, x2)
U6_GG(x1, x2, x3) = U6_GG(x1, x2, x3)
U3_GAG(x1, x2, x3, x4, x5) = U3_GAG(x1, x3, x4, x5)
U4_GAG(x1, x2, x3, x4, x5) = U4_GAG(x1, x3, x4, x5)
U12_GA(x1, x2, x3, x4, x5) = U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5) = U13_GA(x1, x2, x5)
P61_IN_GGGA(x1, x2, x3, x4) = P61_IN_GGGA(x1, x2, x3)
U8_GGGA(x1, x2, x3, x4, x5) = U8_GGGA(x1, x2, x3, x5)
NOTEQ62_IN_GG(x1, x2) = NOTEQ62_IN_GG(x1, x2)
U7_GG(x1, x2, x3) = U7_GG(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6) = U9_GGGA(x1, x2, x3, x4, x6)
U10_GGGA(x1, x2, x3, x4, x5, x6) = U10_GGGA(x1, x2, x3, x4, x6)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x5)
U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x2, x5)
U16_GA(x1, x2, x3, x4, x5) = U16_GA(x1, x2, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → U11_GA(T25, T26, T27, T28, min215_in_gag(T25, T27, T26))
MINSORT1_IN_GA(.(T25, T26), .(T27, T28)) → MIN215_IN_GAG(T25, T27, T26)
MIN215_IN_GAG(T70, T52, .(T71, T51)) → U1_GAG(T70, T52, T71, T51, le30_in_gg(T70, T71))
MIN215_IN_GAG(T70, T52, .(T71, T51)) → LE30_IN_GG(T70, T71)
LE30_IN_GG(s(T82), s(T83)) → U5_GG(T82, T83, le30_in_gg(T82, T83))
LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
MIN215_IN_GAG(T97, T52, .(T98, T51)) → U2_GAG(T97, T52, T98, T51, gt44_in_gg(T97, T98))
MIN215_IN_GAG(T97, T52, .(T98, T51)) → GT44_IN_GG(T97, T98)
GT44_IN_GG(s(T109), s(T110)) → U6_GG(T109, T110, gt44_in_gg(T109, T110))
GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
MIN215_IN_GAG(T48, T52, .(T50, T51)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U4_GAG(T48, T52, T50, T51, min215_in_gag(T55, T52, T51))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)
MINSORT1_IN_GA(.(T146, T147), .(T145, T32)) → U12_GA(T146, T147, T145, T32, min2c15_in_gag(T146, T145, T147))
U12_GA(T146, T147, T145, T32, min2c15_out_gag(T146, T145, T147)) → U13_GA(T146, T147, T145, T32, p61_in_ggga(T145, T146, T147, X170))
U12_GA(T146, T147, T145, T32, min2c15_out_gag(T146, T145, T147)) → P61_IN_GGGA(T145, T146, T147, X170)
P61_IN_GGGA(T145, T146, T147, X170) → U8_GGGA(T145, T146, T147, X170, notEq62_in_gg(T145, T146))
P61_IN_GGGA(T145, T146, T147, X170) → NOTEQ62_IN_GG(T145, T146)
NOTEQ62_IN_GG(s(T160), s(T161)) → U7_GG(T160, T161, notEq62_in_gg(T160, T161))
NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
P61_IN_GGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_GGGA(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U9_GGGA(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U10_GGGA(T192, T146, T193, T194, X229, p61_in_ggga(T192, T193, T194, X229))
U9_GGGA(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → P61_IN_GGGA(T192, T193, T194, X229)
MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U14_GA(T25, T26, T31, T32, min2c15_in_gag(T25, T31, T26))
U14_GA(T25, T26, T31, T32, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, T31, T32, removec53_in_ggga(T31, T25, T26, T122))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T122)) → U16_GA(T25, T26, T31, T32, minsort1_in_ga(T122, T32))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T122)) → MINSORT1_IN_GA(T122, T32)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 6 SCCs with 17 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
NOTEQ62_IN_GG(x1, x2) = NOTEQ62_IN_GG(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

NOTEQ62_IN_GG(s(T160), s(T161)) → NOTEQ62_IN_GG(T160, T161)
The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

P61_IN_GGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_GGGA(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U9_GGGA(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → P61_IN_GGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
P61_IN_GGGA(x1, x2, x3, x4) = P61_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6) = U9_GGGA(x1, x2, x3, x4, x6)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

P61_IN_GGGA(T192, T146, .(T193, T194), .(T193, X229)) → U9_GGGA(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U9_GGGA(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → P61_IN_GGGA(T192, T193, T194, X229)

The TRS R consists of the following rules:

notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
0 = 0
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
P61_IN_GGGA(x1, x2, x3, x4) = P61_IN_GGGA(x1, x2, x3)
U9_GGGA(x1, x2, x3, x4, x5, x6) = U9_GGGA(x1, x2, x3, x4, x6)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

P61_IN_GGGA(T192, T146, .(T193, T194)) → U9_GGGA(T192, T146, T193, T194, notEqc62_in_gg(T192, T146))
U9_GGGA(T192, T146, T193, T194, notEqc62_out_gg(T192, T146)) → P61_IN_GGGA(T192, T193, T194)

The TRS R consists of the following rules:

notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))

The set Q consists of the following terms:

notEqc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

U9_GGGA(T192, T146, T193, T194, notEqc62_out_gg(T192, T146)) → P61_IN_GGGA(T192, T193, T194)
The graph contains the following edges 1 >= 1, 5 > 1, 3 >= 2, 4 >= 3
P61_IN_GGGA(T192, T146, .(T193, T194)) → U9_GGGA(T192, T146, T193, T194, notEqc62_in_gg(T192, T146))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 3 > 4

(20) YES

(21) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
GT44_IN_GG(x1, x2) = GT44_IN_GG(x1, x2)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(24) PiDPToQDPProof (EQUIVALENT transformation)

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

(25) Obligation:

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

GT44_IN_GG(s(T109), s(T110)) → GT44_IN_GG(T109, T110)
The graph contains the following edges 1 > 1, 2 > 2

(27) YES

(28) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
LE30_IN_GG(x1, x2) = LE30_IN_GG(x1, x2)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(31) PiDPToQDPProof (EQUIVALENT transformation)

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

(32) Obligation:

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)

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

(33) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

LE30_IN_GG(s(T82), s(T83)) → LE30_IN_GG(T82, T83)
The graph contains the following edges 1 > 1, 2 > 2

(34) YES

(35) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
MIN215_IN_GAG(x1, x2, x3) = MIN215_IN_GAG(x1, x3)
U3_GAG(x1, x2, x3, x4, x5) = U3_GAG(x1, x3, x4, x5)

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

(36) UsableRulesProof (EQUIVALENT transformation)

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

(37) Obligation:

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

MIN215_IN_GAG(T48, T52, .(T50, T51)) → U3_GAG(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U3_GAG(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T52, T51)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))

(38) PiDPToQDPProof (SOUND transformation)

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

(39) Obligation:

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

MIN215_IN_GAG(T48, .(T50, T51)) → U3_GAG(T48, T50, T51, minc25_in_gga(T48, T50))
U3_GAG(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
minc25_in_gga(T97, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
U30_gga(x0, x1, x2)
U31_gga(x0, x1, x2)
lec30_in_gg(x0, x1)
gtc44_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U21_gg(x0, x1, x2)

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

(40) 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:

U3_GAG(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → MIN215_IN_GAG(T55, T51)
The graph contains the following edges 4 > 1, 3 >= 2
MIN215_IN_GAG(T48, .(T50, T51)) → U3_GAG(T48, T50, T51, minc25_in_gga(T48, T50))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3

(41) YES

(42) Obligation:

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

MINSORT1_IN_GA(.(T25, T26), .(T31, T32)) → U14_GA(T25, T26, T31, T32, min2c15_in_gag(T25, T31, T26))
U14_GA(T25, T26, T31, T32, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, T31, T32, removec53_in_ggga(T31, T25, T26, T122))
U15_GA(T25, T26, T31, T32, removec53_out_ggga(T31, T25, T26, T122)) → MINSORT1_IN_GA(T122, T32)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71, T70) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, T52, .(T50, T51)) → U18_gag(T48, T52, T50, T51, minc25_in_gga(T48, T50, T55))
U18_gag(T48, T52, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T52, T50, T51, min2c15_in_gag(T55, T52, T51))
U19_gag(T48, T52, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147, .(T146, X170)) → U29_ggga(T145, T146, T147, X170, qc61_in_ggga(T145, T146, T147, X170))
qc61_in_ggga(T184, T146, .(T184, T185), T185) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194), .(T193, X229)) → U27_ggga(T192, T146, T193, T194, X229, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, X229, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, X229, qc61_in_ggga(T192, T193, T194, X229))
U28_ggga(T192, T146, T193, T194, X229, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, X170, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
s(x1) = s(x1)
minc25_in_gga(x1, x2, x3) = minc25_in_gga(x1, x2)
U30_gga(x1, x2, x3) = U30_gga(x1, x2, x3)
lec30_in_gg(x1, x2) = lec30_in_gg(x1, x2)
U20_gg(x1, x2, x3) = U20_gg(x1, x2, x3)
0 = 0
lec30_out_gg(x1, x2) = lec30_out_gg(x1, x2)
minc25_out_gga(x1, x2, x3) = minc25_out_gga(x1, x2, x3)
U31_gga(x1, x2, x3) = U31_gga(x1, x2, x3)
gtc44_in_gg(x1, x2) = gtc44_in_gg(x1, x2)
U21_gg(x1, x2, x3) = U21_gg(x1, x2, x3)
gtc44_out_gg(x1, x2) = gtc44_out_gg(x1, x2)
min2c15_in_gag(x1, x2, x3) = min2c15_in_gag(x1, x3)
[] = []
min2c15_out_gag(x1, x2, x3) = min2c15_out_gag(x1, x2, x3)
U18_gag(x1, x2, x3, x4, x5) = U18_gag(x1, x3, x4, x5)
U19_gag(x1, x2, x3, x4, x5) = U19_gag(x1, x3, x4, x5)
notEqc62_in_gg(x1, x2) = notEqc62_in_gg(x1, x2)
U25_gg(x1, x2, x3) = U25_gg(x1, x2, x3)
notEqc62_out_gg(x1, x2) = notEqc62_out_gg(x1, x2)
removec53_in_ggga(x1, x2, x3, x4) = removec53_in_ggga(x1, x2, x3)
removec53_out_ggga(x1, x2, x3, x4) = removec53_out_ggga(x1, x2, x3, x4)
U29_ggga(x1, x2, x3, x4, x5) = U29_ggga(x1, x2, x3, x5)
qc61_in_ggga(x1, x2, x3, x4) = qc61_in_ggga(x1, x2, x3)
U26_ggga(x1, x2, x3, x4) = U26_ggga(x1, x2, x3, x4)
qc61_out_ggga(x1, x2, x3, x4) = qc61_out_ggga(x1, x2, x3, x4)
U27_ggga(x1, x2, x3, x4, x5, x6) = U27_ggga(x1, x2, x3, x4, x6)
U28_ggga(x1, x2, x3, x4, x5, x6) = U28_ggga(x1, x2, x3, x4, x6)
MINSORT1_IN_GA(x1, x2) = MINSORT1_IN_GA(x1)
U14_GA(x1, x2, x3, x4, x5) = U14_GA(x1, x2, x5)
U15_GA(x1, x2, x3, x4, x5) = U15_GA(x1, x2, x5)

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

(43) PiDPToQDPProof (SOUND transformation)

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

(44) Obligation:

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

MINSORT1_IN_GA(.(T25, T26)) → U14_GA(T25, T26, min2c15_in_gag(T25, T26))
U14_GA(T25, T26, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, removec53_in_ggga(T31, T25, T26))
U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T122)) → MINSORT1_IN_GA(T122)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147) → U29_ggga(T145, T146, T147, qc61_in_ggga(T145, T146, T147))
qc61_in_ggga(T184, T146, .(T184, T185)) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194)) → U27_ggga(T192, T146, T193, T194, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, qc61_in_ggga(T192, T193, T194))
U28_ggga(T192, T146, T193, T194, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U30_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)
removec53_in_ggga(x0, x1, x2)
qc61_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2, x3)
U27_ggga(x0, x1, x2, x3, x4)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MINSORT1_IN_GA(.(T25, T26)) → U14_GA(T25, T26, min2c15_in_gag(T25, T26))

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

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(0) = 0
POL(MINSORT1_IN_GA(x₁)) = x₁
POL(U14_GA(x₁, x₂, x₃)) = x₁ + x₂
POL(U15_GA(x₁, x₂, x₃)) = x₃
POL(U18_gag(x₁, x₂, x₃, x₄)) = 0
POL(U19_gag(x₁, x₂, x₃, x₄)) = 0
POL(U20_gg(x₁, x₂, x₃)) = 0
POL(U21_gg(x₁, x₂, x₃)) = x₃
POL(U25_gg(x₁, x₂, x₃)) = 0
POL(U26_ggga(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(U27_ggga(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃ + x₄
POL(U28_ggga(x₁, x₂, x₃, x₄, x₅)) = 1 + x₃ + x₅
POL(U29_ggga(x₁, x₂, x₃, x₄)) = x₂ + x₄
POL(U30_gga(x₁, x₂, x₃)) = 0
POL(U31_gga(x₁, x₂, x₃)) = 0
POL([]) = 0
POL(gtc44_in_gg(x₁, x₂)) = 1
POL(gtc44_out_gg(x₁, x₂)) = x₂
POL(lec30_in_gg(x₁, x₂)) = 0
POL(lec30_out_gg(x₁, x₂)) = 0
POL(min2c15_in_gag(x₁, x₂)) = 0
POL(min2c15_out_gag(x₁, x₂, x₃)) = 0
POL(minc25_in_gga(x₁, x₂)) = 0
POL(minc25_out_gga(x₁, x₂, x₃)) = 0
POL(notEqc62_in_gg(x₁, x₂)) = 0
POL(notEqc62_out_gg(x₁, x₂)) = x₁
POL(qc61_in_ggga(x₁, x₂, x₃)) = x₃
POL(qc61_out_ggga(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(removec53_in_ggga(x₁, x₂, x₃)) = x₂ + x₃
POL(removec53_out_ggga(x₁, x₂, x₃, x₄)) = x₄
POL(s(x₁)) = 0

The following usable rules [FROCOS05] were oriented:

removec53_in_ggga(T135, T135, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147) → U29_ggga(T145, T146, T147, qc61_in_ggga(T145, T146, T147))
qc61_in_ggga(T184, T146, .(T184, T185)) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
qc61_in_ggga(T192, T146, .(T193, T194)) → U27_ggga(T192, T146, T193, T194, notEqc62_in_gg(T192, T146))
U29_ggga(T145, T146, T147, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U27_ggga(T192, T146, T193, T194, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, qc61_in_ggga(T192, T193, T194))
U28_ggga(T192, T146, T193, T194, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))

(46) Obligation:

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

U14_GA(T25, T26, min2c15_out_gag(T25, T31, T26)) → U15_GA(T25, T26, removec53_in_ggga(T31, T25, T26))
U15_GA(T25, T26, removec53_out_ggga(T31, T25, T26, T122)) → MINSORT1_IN_GA(T122)

The TRS R consists of the following rules:

minc25_in_gga(T70, T71) → U30_gga(T70, T71, lec30_in_gg(T70, T71))
lec30_in_gg(s(T82), s(T83)) → U20_gg(T82, T83, lec30_in_gg(T82, T83))
lec30_in_gg(0, s(T90)) → lec30_out_gg(0, s(T90))
lec30_in_gg(0, 0) → lec30_out_gg(0, 0)
U20_gg(T82, T83, lec30_out_gg(T82, T83)) → lec30_out_gg(s(T82), s(T83))
U30_gga(T70, T71, lec30_out_gg(T70, T71)) → minc25_out_gga(T70, T71, T70)
minc25_in_gga(T97, T98) → U31_gga(T97, T98, gtc44_in_gg(T97, T98))
gtc44_in_gg(s(T109), s(T110)) → U21_gg(T109, T110, gtc44_in_gg(T109, T110))
gtc44_in_gg(s(T115), 0) → gtc44_out_gg(s(T115), 0)
U21_gg(T109, T110, gtc44_out_gg(T109, T110)) → gtc44_out_gg(s(T109), s(T110))
U31_gga(T97, T98, gtc44_out_gg(T97, T98)) → minc25_out_gga(T97, T98, T98)
min2c15_in_gag(T39, []) → min2c15_out_gag(T39, T39, [])
min2c15_in_gag(T48, .(T50, T51)) → U18_gag(T48, T50, T51, minc25_in_gga(T48, T50))
U18_gag(T48, T50, T51, minc25_out_gga(T48, T50, T55)) → U19_gag(T48, T50, T51, min2c15_in_gag(T55, T51))
U19_gag(T48, T50, T51, min2c15_out_gag(T55, T52, T51)) → min2c15_out_gag(T48, T52, .(T50, T51))
notEqc62_in_gg(s(T160), s(T161)) → U25_gg(T160, T161, notEqc62_in_gg(T160, T161))
notEqc62_in_gg(s(T168), 0) → notEqc62_out_gg(s(T168), 0)
notEqc62_in_gg(0, s(T171)) → notEqc62_out_gg(0, s(T171))
U25_gg(T160, T161, notEqc62_out_gg(T160, T161)) → notEqc62_out_gg(s(T160), s(T161))
removec53_in_ggga(T135, T135, T136) → removec53_out_ggga(T135, T135, T136, T136)
removec53_in_ggga(T145, T146, T147) → U29_ggga(T145, T146, T147, qc61_in_ggga(T145, T146, T147))
qc61_in_ggga(T184, T146, .(T184, T185)) → U26_ggga(T184, T146, T185, notEqc62_in_gg(T184, T146))
U26_ggga(T184, T146, T185, notEqc62_out_gg(T184, T146)) → qc61_out_ggga(T184, T146, .(T184, T185), T185)
qc61_in_ggga(T192, T146, .(T193, T194)) → U27_ggga(T192, T146, T193, T194, notEqc62_in_gg(T192, T146))
U27_ggga(T192, T146, T193, T194, notEqc62_out_gg(T192, T146)) → U28_ggga(T192, T146, T193, T194, qc61_in_ggga(T192, T193, T194))
U28_ggga(T192, T146, T193, T194, qc61_out_ggga(T192, T193, T194, X229)) → qc61_out_ggga(T192, T146, .(T193, T194), .(T193, X229))
U29_ggga(T145, T146, T147, qc61_out_ggga(T145, T146, T147, X170)) → removec53_out_ggga(T145, T146, T147, .(T146, X170))

The set Q consists of the following terms:

minc25_in_gga(x0, x1)
lec30_in_gg(x0, x1)
U20_gg(x0, x1, x2)
U30_gga(x0, x1, x2)
gtc44_in_gg(x0, x1)
U21_gg(x0, x1, x2)
U31_gga(x0, x1, x2)
min2c15_in_gag(x0, x1)
U18_gag(x0, x1, x2, x3)
U19_gag(x0, x1, x2, x3)
notEqc62_in_gg(x0, x1)
U25_gg(x0, x1, x2)
removec53_in_ggga(x0, x1, x2)
qc61_in_ggga(x0, x1, x2)
U26_ggga(x0, x1, x2, x3)
U27_ggga(x0, x1, x2, x3, x4)
U28_ggga(x0, x1, x2, x3, x4)
U29_ggga(x0, x1, x2, x3)

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

(47) DependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (EQUIVALENT transformation)

(25) Obligation:

(26) QDPSizeChangeProof (EQUIVALENT transformation)

(27) YES

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) PiDPToQDPProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPSizeChangeProof (EQUIVALENT transformation)

(34) YES

(35) Obligation:

(36) UsableRulesProof (EQUIVALENT transformation)

(37) Obligation:

(38) PiDPToQDPProof (SOUND transformation)

(39) Obligation:

(40) QDPSizeChangeProof (EQUIVALENT transformation)

(41) YES

(42) Obligation:

(43) PiDPToQDPProof (SOUND transformation)

(44) Obligation:

(45) QDPOrderProof (EQUIVALENT transformation)

(46) Obligation:

(47) DependencyGraphProof (EQUIVALENT transformation)

(48) TRUE