Left Termination of the query pattern ordered_in_1(g) w.r.t. the given Prolog program could not be shown:

(0) Obligation:

Clauses:

ordered([]) :- !.
ordered(.(X1, [])) :- !.
ordered(Xs) :- ','(head(Xs, X), ','(tail(Xs, Ys), ','(head(Ys, Y), ','(tail(Ys, Zs), ','(less(X, s(Y)), ordered(.(Y, Zs))))))).
head([], X2).
head(.(X, X3), X).
tail([], []).
tail(.(X4, Xs), Xs).
less(0, s(X5)) :- !.
less(X, Y) :- ','(p(X, Px), ','(p(Y, Py), less(Px, Py))).
p(0, 0).
p(s(X), X).

Queries:

ordered(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ordered(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ordered(T1), SCOPE: 1, CLAUSE: 0 | ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nordered(T1) !=? ordered([])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: !_1 | ordered(.(T3, [])), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))\n\nT5 is ground\nX11 is free; \nX12 is free; \nX13 is free; \nX14 is free; \nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))), SCOPE: 2, CLAUSE: 3 | ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; \nX12 is free; \nX13 is free; \nX14 is free; \nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; \nX12 is free; \nX13 is free; \nX14 is free; \nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14))))))\n\nT10 is ground\nT11 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14)))))), SCOPE: 3, CLAUSE: 5 | ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14)))))), SCOPE: 3, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14)))))), SCOPE: 3, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14)))))\n\nT16 is ground\nT17 is ground\nX13 is free; \nX14 is free; \nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14))))), SCOPE: 4, CLAUSE: 3 | ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14))))), SCOPE: 4, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX13 is free; \nX14 is free; \nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14))))), SCOPE: 4, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX13 is free; \nX14 is free; \nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14))))\n\nT16 is ground\nT22 is ground\nT23 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14)))), SCOPE: 5, CLAUSE: 5 | ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14)))), SCOPE: 5, CLAUSE: 6\n\nT16 is ground\nT22 is ground\nT23 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14)))), SCOPE: 5, CLAUSE: 6\n\nT16 is ground\nT22 is ground\nT23 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(less(T16, s(T30)), ordered(.(T30, T31)))\n\nT16 is ground\nT30 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T16, s(T30))\n\nT16 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ordered(.(T30, T31))\n\nT30 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T16, s(T30)), SCOPE: 6, CLAUSE: 7 | less(T16, s(T30)), SCOPE: 6, CLAUSE: 8\n\nT16 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: !_6 | less(0, s(T36)), SCOPE: 6, CLAUSE: 8\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T16, s(T30)), SCOPE: 6, CLAUSE: 8\n\nT16 is ground\nT30 is ground\nless(T16, s(T30)) !=? less(0, s(X59))", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(T41, X71), ','(p(s(T42), X72), less(X71, X72)))\n\nT41 is ground\nT42 is ground\nX71 is free; \nX72 is free; \nless(T41, s(T42)) !=? less(0, s(X59))", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T41, X71), ','(p(s(T42), X72), less(X71, X72))), SCOPE: 7, CLAUSE: 9 | ','(p(T41, X71), ','(p(s(T42), X72), less(X71, X72))), SCOPE: 7, CLAUSE: 10\n\nT41 is ground\nT42 is ground\nX71 is free; \nX72 is free; \nless(T41, s(T42)) !=? less(0, s(X59))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T41, X71), ','(p(s(T42), X72), less(X71, X72))), SCOPE: 7, CLAUSE: 10\n\nT41 is ground\nT42 is ground\nX71 is free; \nX72 is free; \nless(T41, s(T42)) !=? less(0, s(X59))", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(s(T42), X72), less(T45, X72))\n\nT42 is ground\nT45 is ground\nX72 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(s(T42), X72), less(T45, X72)), SCOPE: 8, CLAUSE: 9 | ','(p(s(T42), X72), less(T45, X72)), SCOPE: 8, CLAUSE: 10\n\nT42 is ground\nT45 is ground\nX72 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(s(T42), X72), less(T45, X72)), SCOPE: 8, CLAUSE: 10\n\nT42 is ground\nT45 is ground\nX72 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T45, T49)\n\nT45 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: less(T45, T49), SCOPE: 9, CLAUSE: 7 | less(T45, T49), SCOPE: 9, CLAUSE: 8\n\nT45 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: !_9 | less(0, s(T53)), SCOPE: 9, CLAUSE: 8\n\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(T45, T49), SCOPE: 9, CLAUSE: 8\n\nT45 is ground\nT49 is ground\nless(T45, T49) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(p(T58, X100), ','(p(T59, X101), less(X100, X101)))\n\nT58 is ground\nT59 is ground\nX100 is free; \nX101 is free; \nless(T58, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(T58, X100), ','(p(T59, X101), less(X100, X101))), SCOPE: 10, CLAUSE: 9 | ','(p(T58, X100), ','(p(T59, X101), less(X100, X101))), SCOPE: 10, CLAUSE: 10\n\nT58 is ground\nT59 is ground\nX100 is free; \nX101 is free; \nless(T58, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(p(T58, X100), ','(p(T59, X101), less(X100, X101))), SCOPE: 10, CLAUSE: 9\n\nT58 is ground\nT59 is ground\nX100 is free; \nX101 is free; \nless(T58, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(T58, X100), ','(p(T59, X101), less(X100, X101))), SCOPE: 10, CLAUSE: 10\n\nT58 is ground\nT59 is ground\nX100 is free; \nX101 is free; \nless(T58, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(p(T59, X101), less(0, X101))\n\nT59 is ground\nX101 is free; \nless(0, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T59, X101), less(0, X101)), SCOPE: 11, CLAUSE: 9 | ','(p(T59, X101), less(0, X101)), SCOPE: 11, CLAUSE: 10\n\nT59 is ground\nX101 is free; \nless(0, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T59, X101), less(0, X101)), SCOPE: 11, CLAUSE: 9\n\nT59 is ground\nX101 is free; \nless(0, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T59, X101), less(0, X101)), SCOPE: 11, CLAUSE: 10\n\nT59 is ground\nX101 is free; \nless(0, T59) !=? less(0, s(X88))", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(0, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: less(0, 0), SCOPE: 12, CLAUSE: 7 | less(0, 0), SCOPE: 12, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: less(0, 0), SCOPE: 12, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(p(0, X122), ','(p(0, X123), less(X122, X123)))\n\nX122 is free\nX123 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(p(0, X122), ','(p(0, X123), less(X122, X123))), SCOPE: 13, CLAUSE: 9 | ','(p(0, X122), ','(p(0, X123), less(X122, X123))), SCOPE: 13, CLAUSE: 10\n\nX122 is free\nX123 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(0, X122), ','(p(0, X123), less(X122, X123))), SCOPE: 13, CLAUSE: 9\n\nX122 is free\nX123 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(0, X122), ','(p(0, X123), less(X122, X123))), SCOPE: 13, CLAUSE: 10\n\nX122 is free\nX123 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(p(0, X123), less(0, X123))\n\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(p(0, X123), less(0, X123)), SCOPE: 14, CLAUSE: 9 | ','(p(0, X123), less(0, X123)), SCOPE: 14, CLAUSE: 10\n\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(p(0, X123), less(0, X123)), SCOPE: 14, CLAUSE: 9\n\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(p(0, X123), less(0, X123)), SCOPE: 14, CLAUSE: 10\n\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(0, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(p(T59, X101), less(T63, X101))\n\nT59 is ground\nT63 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(T59, X101), less(T63, X101)), SCOPE: 15, CLAUSE: 9 | ','(p(T59, X101), less(T63, X101)), SCOPE: 15, CLAUSE: 10\n\nT59 is ground\nT63 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(p(T59, X101), less(T63, X101)), SCOPE: 15, CLAUSE: 9\n\nT59 is ground\nT63 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(p(T59, X101), less(T63, X101)), SCOPE: 15, CLAUSE: 10\n\nT59 is ground\nT63 is ground\nX101 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: less(T63, 0)\n\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: less(T63, 0), SCOPE: 16, CLAUSE: 7 | less(T63, 0), SCOPE: 16, CLAUSE: 8\n\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: less(T63, 0), SCOPE: 16, CLAUSE: 8\n\nT63 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(p(T67, X172), ','(p(0, X173), less(X172, X173)))\n\nT67 is ground\nX172 is free; \nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(T67, X172), ','(p(0, X173), less(X172, X173))), SCOPE: 17, CLAUSE: 9 | ','(p(T67, X172), ','(p(0, X173), less(X172, X173))), SCOPE: 17, CLAUSE: 10\n\nT67 is ground\nX172 is free; \nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(p(T67, X172), ','(p(0, X173), less(X172, X173))), SCOPE: 17, CLAUSE: 9\n\nT67 is ground\nX172 is free; \nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(p(T67, X172), ','(p(0, X173), less(X172, X173))), SCOPE: 17, CLAUSE: 10\n\nT67 is ground\nX172 is free; \nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(p(0, X173), less(0, X173))\n\nX173 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(p(0, X173), less(0, X173)), SCOPE: 18, CLAUSE: 9 | ','(p(0, X173), less(0, X173)), SCOPE: 18, CLAUSE: 10\n\nX173 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(p(0, X173), less(0, X173)), SCOPE: 18, CLAUSE: 9\n\nX173 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(p(0, X173), less(0, X173)), SCOPE: 18, CLAUSE: 10\n\nX173 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: less(0, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(p(0, X173), less(T70, X173))\n\nT70 is ground\nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(p(0, X173), less(T70, X173)), SCOPE: 19, CLAUSE: 9 | ','(p(0, X173), less(T70, X173)), SCOPE: 19, CLAUSE: 10\n\nT70 is ground\nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(p(0, X173), less(T70, X173)), SCOPE: 19, CLAUSE: 9\n\nT70 is ground\nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(p(0, X173), less(T70, X173)), SCOPE: 19, CLAUSE: 10\n\nT70 is ground\nX173 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: less(T70, 0)\n\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: less(T63, T76)\n\nT63 is ground\nT76 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 100 [arrowhead = none ];
100 -> 2;

101 [label="EVAL with clause\nordered([]) :- !_1.\nand substitution\nT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 101 [arrowhead = none ];
101 -> 3;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 102 [arrowhead = none ];
102 -> 4;

103 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 103 [arrowhead = none ];
103 -> 5;

104 [label="EVAL with clause\nordered(.(X7, [])) :- !_1.\nand substitution\nX7 -> T3\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 104 [arrowhead = none ];
104 -> 7;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 105 [arrowhead = none ];
105 -> 8;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 106 [arrowhead = none ];
106 -> 6;

107 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 107 [arrowhead = none ];
107 -> 9;

108 [label="ONLY EVAL with clause\nordered(X10) :- ','(head(X10, X11), ','(tail(X10, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))).\nand substitution\nT1 -> T5\nX10 -> T5", fontsize=14, style = filled, fillcolor = white];
8 -> 108 [arrowhead = none ];
108 -> 11;

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 109 [arrowhead = none ];
109 -> 10;

110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 110 [arrowhead = none ];
110 -> 12;

111 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(T5), ordered([]))", fontsize=14, style = filled, fillcolor = white];
12 -> 111 [arrowhead = none ];
111 -> 13;

112 [label="EVAL with clause\nhead(.(X21, X22), X21).\nand substitution\nX21 -> T10\nX22 -> T11\nT5 -> .(T10, T11)\nX11 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 112 [arrowhead = none ];
112 -> 14;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 113 [arrowhead = none ];
113 -> 15;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 114 [arrowhead = none ];
114 -> 16;

115 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 115 [arrowhead = none ];
115 -> 17;

116 [label="ONLY EVAL with clause\ntail(.(X33, X34), X34).\nand substitution\nT10 -> T16\nX33 -> T16\nT11 -> T17\nX34 -> T17\nX12 -> T17", fontsize=14, style = filled, fillcolor = white];
17 -> 116 [arrowhead = none ];
116 -> 18;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 117 [arrowhead = none ];
117 -> 19;

118 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(.(T16, T17)), ordered(.(X7, [])))", fontsize=14, style = filled, fillcolor = white];
19 -> 118 [arrowhead = none ];
118 -> 20;

119 [label="EVAL with clause\nhead(.(X43, X44), X43).\nand substitution\nX43 -> T22\nX44 -> T23\nT17 -> .(T22, T23)\nX13 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 119 [arrowhead = none ];
119 -> 21;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 120 [arrowhead = none ];
120 -> 22;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 121 [arrowhead = none ];
121 -> 23;

122 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 122 [arrowhead = none ];
122 -> 24;

123 [label="ONLY EVAL with clause\ntail(.(X53, X54), X54).\nand substitution\nT22 -> T30\nX53 -> T30\nT23 -> T31\nX54 -> T31\nX14 -> T31", fontsize=14, style = filled, fillcolor = white];
24 -> 123 [arrowhead = none ];
123 -> 25;

124 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 124 [arrowhead = none ];
124 -> 26;

125 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
25 -> 125 [arrowhead = none ];
125 -> 27;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 126 [arrowhead = none ];
126 -> 28;

127 [label="INSTANCE with matching:\nT1 -> .(T30, T31)", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 127 [arrowhead = none , style=dashed];
127 -> 1[style=dashed];

128 [label="EVAL with clause\nless(0, s(X59)) :- !_6.\nand substitution\nT16 -> 0\nT30 -> T36\nX59 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 128 [arrowhead = none ];
128 -> 29;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 129 [arrowhead = none ];
129 -> 30;

130 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 130 [arrowhead = none ];
130 -> 31;

131 [label="ONLY EVAL with clause\nless(X69, X70) :- ','(p(X69, X71), ','(p(X70, X72), less(X71, X72))).\nand substitution\nT16 -> T41\nX69 -> T41\nT30 -> T42\nX70 -> s(T42)", fontsize=14, style = filled, fillcolor = white];
30 -> 131 [arrowhead = none ];
131 -> 33;

132 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 132 [arrowhead = none ];
132 -> 32;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 133 [arrowhead = none ];
133 -> 34;

134 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T41, s(T42)), less(0, s(X59)))", fontsize=14, style = filled, fillcolor = white];
34 -> 134 [arrowhead = none ];
134 -> 35;

135 [label="EVAL with clause\np(s(X77), X77).\nand substitution\nX77 -> T45\nT41 -> s(T45)\nX71 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 135 [arrowhead = none ];
135 -> 36;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 136 [arrowhead = none ];
136 -> 37;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 137 [arrowhead = none ];
137 -> 38;

138 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
38 -> 138 [arrowhead = none ];
138 -> 39;

139 [label="ONLY EVAL with clause\np(s(X84), X84).\nand substitution\nT42 -> T49\nX84 -> T49\nX72 -> T49", fontsize=14, style = filled, fillcolor = white];
39 -> 139 [arrowhead = none ];
139 -> 40;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 140 [arrowhead = none ];
140 -> 41;

141 [label="EVAL with clause\nless(0, s(X88)) :- !_9.\nand substitution\nT45 -> 0\nX88 -> T53\nT49 -> s(T53)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 141 [arrowhead = none ];
141 -> 42;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 142 [arrowhead = none ];
142 -> 43;

143 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 143 [arrowhead = none ];
143 -> 44;

144 [label="ONLY EVAL with clause\nless(X98, X99) :- ','(p(X98, X100), ','(p(X99, X101), less(X100, X101))).\nand substitution\nT45 -> T58\nX98 -> T58\nT49 -> T59\nX99 -> T59", fontsize=14, style = filled, fillcolor = white];
43 -> 144 [arrowhead = none ];
144 -> 46;

145 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 145 [arrowhead = none ];
145 -> 45;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 146 [arrowhead = none ];
146 -> 47;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 147 [arrowhead = none ];
147 -> 48;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
47 -> 148 [arrowhead = none ];
148 -> 49;

149 [label="EVAL with clause\np(0, 0).\nand substitution\nT58 -> 0\nX100 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 149 [arrowhead = none ];
149 -> 50;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 150 [arrowhead = none ];
150 -> 51;

151 [label="EVAL with clause\np(s(X152), X152).\nand substitution\nX152 -> T63\nT58 -> s(T63)\nX100 -> T63", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 151 [arrowhead = none ];
151 -> 71;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 152 [arrowhead = none ];
152 -> 72;

153 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 153 [arrowhead = none ];
153 -> 52;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 154 [arrowhead = none ];
154 -> 53;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
52 -> 155 [arrowhead = none ];
155 -> 54;

156 [label="EVAL with clause\np(0, 0).\nand substitution\nT59 -> 0\nX101 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 156 [arrowhead = none ];
156 -> 55;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 157 [arrowhead = none ];
157 -> 56;

158 [label="BACKTRACK\nfor clause: p(s(X), X)\nwith clash: (less(0, T59), less(0, s(X88)))", fontsize=14, style = filled, fillcolor = white];
54 -> 158 [arrowhead = none ];
158 -> 70;

159 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 159 [arrowhead = none ];
159 -> 57;

160 [label="BACKTRACK\nfor clause: less(0, s(X5)) :- !because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 160 [arrowhead = none ];
160 -> 58;

161 [label="ONLY EVAL with clause\nless(X120, X121) :- ','(p(X120, X122), ','(p(X121, X123), less(X122, X123))).\nand substitution\nX120 -> 0\nX121 -> 0", fontsize=14, style = filled, fillcolor = white];
58 -> 161 [arrowhead = none ];
161 -> 59;

162 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 162 [arrowhead = none ];
162 -> 60;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 163 [arrowhead = none ];
163 -> 61;

164 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 164 [arrowhead = none ];
164 -> 62;

165 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX122 -> 0", fontsize=14, style = filled, fillcolor = white];
61 -> 165 [arrowhead = none ];
165 -> 63;

166 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 166 [arrowhead = none ];
166 -> 69;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 167 [arrowhead = none ];
167 -> 64;

168 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 168 [arrowhead = none ];
168 -> 65;

169 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
64 -> 169 [arrowhead = none ];
169 -> 66;

170 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX123 -> 0", fontsize=14, style = filled, fillcolor = white];
65 -> 170 [arrowhead = none ];
170 -> 67;

171 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
66 -> 171 [arrowhead = none ];
171 -> 68;

172 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 172 [arrowhead = none , style=dashed];
172 -> 55[style=dashed];

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 173 [arrowhead = none ];
173 -> 73;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 174 [arrowhead = none ];
174 -> 74;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 175 [arrowhead = none ];
175 -> 75;

176 [label="EVAL with clause\np(0, 0).\nand substitution\nT59 -> 0\nX101 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 176 [arrowhead = none ];
176 -> 76;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 177 [arrowhead = none ];
177 -> 77;

178 [label="EVAL with clause\np(s(X218), X218).\nand substitution\nX218 -> T76\nT59 -> s(T76)\nX101 -> T76", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 178 [arrowhead = none ];
178 -> 98;

179 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 179 [arrowhead = none ];
179 -> 99;

180 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 180 [arrowhead = none ];
180 -> 78;

181 [label="BACKTRACK\nfor clause: less(0, s(X5)) :- !because of non-unification", fontsize=14, style = filled, fillcolor = white];
78 -> 181 [arrowhead = none ];
181 -> 79;

182 [label="ONLY EVAL with clause\nless(X170, X171) :- ','(p(X170, X172), ','(p(X171, X173), less(X172, X173))).\nand substitution\nT63 -> T67\nX170 -> T67\nX171 -> 0", fontsize=14, style = filled, fillcolor = white];
79 -> 182 [arrowhead = none ];
182 -> 80;

183 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
80 -> 183 [arrowhead = none ];
183 -> 81;

184 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 184 [arrowhead = none ];
184 -> 82;

185 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
81 -> 185 [arrowhead = none ];
185 -> 83;

186 [label="EVAL with clause\np(0, 0).\nand substitution\nT67 -> 0\nX172 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 186 [arrowhead = none ];
186 -> 84;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 187 [arrowhead = none ];
187 -> 85;

188 [label="EVAL with clause\np(s(X196), X196).\nand substitution\nX196 -> T70\nT67 -> s(T70)\nX172 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 188 [arrowhead = none ];
188 -> 91;

189 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 189 [arrowhead = none ];
189 -> 92;

190 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 190 [arrowhead = none ];
190 -> 86;

191 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 191 [arrowhead = none ];
191 -> 87;

192 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 192 [arrowhead = none ];
192 -> 88;

193 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX173 -> 0", fontsize=14, style = filled, fillcolor = white];
87 -> 193 [arrowhead = none ];
193 -> 89;

194 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
88 -> 194 [arrowhead = none ];
194 -> 90;

195 [label="INSTANCE", fontsize=14, style = filled, fillcolor = lightgrey];
89 -> 195 [arrowhead = none , style=dashed];
195 -> 55[style=dashed];

196 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 196 [arrowhead = none ];
196 -> 93;

197 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
93 -> 197 [arrowhead = none ];
197 -> 94;

198 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
93 -> 198 [arrowhead = none ];
198 -> 95;

199 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX173 -> 0", fontsize=14, style = filled, fillcolor = white];
94 -> 199 [arrowhead = none ];
199 -> 96;

200 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
95 -> 200 [arrowhead = none ];
200 -> 97;

201 [label="INSTANCE with matching:\nT63 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
96 -> 201 [arrowhead = none , style=dashed];
201 -> 76[style=dashed];

202 [label="INSTANCE with matching:\nT45 -> T63\nT49 -> T76", fontsize=14, style = filled, fillcolor = lightgrey];
98 -> 202 [arrowhead = none , style=dashed];
202 -> 40[style=dashed];

}

(2) Obligation:

Triples:

less55 :- less55.
less76(0) :- less55.
less76(s(T70)) :- less76(T70).
less40(0, 0) :- less55.
less40(s(T63), 0) :- less76(T63).
less40(s(T63), s(T76)) :- less40(T63, T76).
ordered1(.(s(T45), .(T49, T31))) :- less40(T45, T49).
ordered1(.(T16, .(T30, T31))) :- ','(lessc26(T16, T30), ordered1(.(T30, T31))).

Clauses:

orderedc1([]).
orderedc1(.(T3, [])).
orderedc1(.(T16, .(T30, T31))) :- ','(lessc26(T16, T30), orderedc1(.(T30, T31))).
lessc55 :- lessc55.
lessc76(0) :- lessc55.
lessc76(s(T70)) :- lessc76(T70).
lessc40(0, s(T53)).
lessc40(0, 0) :- lessc55.
lessc40(s(T63), 0) :- lessc76(T63).
lessc40(s(T63), s(T76)) :- lessc40(T63, T76).
lessc26(0, T36).
lessc26(s(T45), T49) :- lessc40(T45, T49).

Afs:

ordered1(x1)  =  ordered1(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:
ordered1_in: (b)
less40_in: (b,b)
less76_in: (b)
lessc26_in: (b,b)
lessc40_in: (b,b)
lessc76_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

ORDERED1_IN_G(.(s(T45), .(T49, T31))) → U7_G(T45, T49, T31, less40_in_gg(T45, T49))
ORDERED1_IN_G(.(s(T45), .(T49, T31))) → LESS40_IN_GG(T45, T49)
LESS40_IN_GG(0, 0) → U4_GG(less55_in_)
LESS40_IN_GG(0, 0) → LESS55_IN_
LESS55_IN_ → U1_¹(less55_in_)
LESS55_IN_ → LESS55_IN_
LESS40_IN_GG(s(T63), 0) → U5_GG(T63, less76_in_g(T63))
LESS40_IN_GG(s(T63), 0) → LESS76_IN_G(T63)
LESS76_IN_G(0) → U2_G(less55_in_)
LESS76_IN_G(0) → LESS55_IN_
LESS76_IN_G(s(T70)) → U3_G(T70, less76_in_g(T70))
LESS76_IN_G(s(T70)) → LESS76_IN_G(T70)
LESS40_IN_GG(s(T63), s(T76)) → U6_GG(T63, T76, less40_in_gg(T63, T76))
LESS40_IN_GG(s(T63), s(T76)) → LESS40_IN_GG(T63, T76)
ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → U9_G(T16, T30, T31, ordered1_in_g(.(T30, T31)))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

Pi is empty.
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:

ORDERED1_IN_G(.(s(T45), .(T49, T31))) → U7_G(T45, T49, T31, less40_in_gg(T45, T49))
ORDERED1_IN_G(.(s(T45), .(T49, T31))) → LESS40_IN_GG(T45, T49)
LESS40_IN_GG(0, 0) → U4_GG(less55_in_)
LESS40_IN_GG(0, 0) → LESS55_IN_
LESS55_IN_ → U1_¹(less55_in_)
LESS55_IN_ → LESS55_IN_
LESS40_IN_GG(s(T63), 0) → U5_GG(T63, less76_in_g(T63))
LESS40_IN_GG(s(T63), 0) → LESS76_IN_G(T63)
LESS76_IN_G(0) → U2_G(less55_in_)
LESS76_IN_G(0) → LESS55_IN_
LESS76_IN_G(s(T70)) → U3_G(T70, less76_in_g(T70))
LESS76_IN_G(s(T70)) → LESS76_IN_G(T70)
LESS40_IN_GG(s(T63), s(T76)) → U6_GG(T63, T76, less40_in_gg(T63, T76))
LESS40_IN_GG(s(T63), s(T76)) → LESS40_IN_GG(T63, T76)
ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → U9_G(T16, T30, T31, ordered1_in_g(.(T30, T31)))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 12 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS55_IN_ → LESS55_IN_

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

Pi is empty.
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:

LESS55_IN_ → LESS55_IN_

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:

LESS55_IN_ → LESS55_IN_

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

(12) NonTerminationProof (EQUIVALENT transformation)

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

s = LESS55_IN_ evaluates to t =LESS55_IN_

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

• Semiunifier: [ ]
• Matcher: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS55_IN_ to LESS55_IN_.

(13) NO

(14) Obligation:

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

LESS76_IN_G(s(T70)) → LESS76_IN_G(T70)

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

Pi is empty.
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:

LESS76_IN_G(s(T70)) → LESS76_IN_G(T70)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESS76_IN_G(s(T70)) → LESS76_IN_G(T70)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS76_IN_G(s(T70)) → LESS76_IN_G(T70)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

LESS40_IN_GG(s(T63), s(T76)) → LESS40_IN_GG(T63, T76)

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

Pi is empty.
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:

LESS40_IN_GG(s(T63), s(T76)) → LESS40_IN_GG(T63, T76)

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:

LESS40_IN_GG(s(T63), s(T76)) → LESS40_IN_GG(T63, T76)

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:

• LESS40_IN_GG(s(T63), s(T76)) → LESS40_IN_GG(T63, T76)
  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:

ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

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

(29) PiDPToQDPProof (EQUIVALENT transformation)

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

(30) Obligation:

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc40_in_gg(x0, x1)
lessc55_in_
U13_(x0)
U16_gg(x0)
lessc76_in_g(x0)
U14_g(x0)
U15_g(x0, x1)
U17_gg(x0, x1)
U18_gg(x0, x1, x2)
U19_gg(x0, x1, x2)

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

(31) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

U13_(lessc55_out_) → lessc55_out_
U16_gg(lessc55_out_) → lessc40_out_gg(0, 0)
U17_gg(T63, lessc76_out_g(T63)) → lessc40_out_gg(s(T63), 0)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + x₂   
POL(0) = 0   
POL(ORDERED1_IN_G(x₁)) = 2·x₁   
POL(U13_(x₁)) = 2·x₁   
POL(U14_g(x₁)) = x₁   
POL(U15_g(x₁, x₂)) = x₁ + x₂   
POL(U16_gg(x₁)) = x₁   
POL(U17_gg(x₁, x₂)) = x₁ + x₂   
POL(U18_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U19_gg(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃   
POL(U8_G(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + x₄   
POL(lessc26_in_gg(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(lessc26_out_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc40_in_gg(x₁, x₂)) = 2·x₁ + x₂   
POL(lessc40_out_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc55_in_) = 0   
POL(lessc55_out_) = 1   
POL(lessc76_in_g(x₁)) = 2·x₁   
POL(lessc76_out_g(x₁)) = 1 + x₁   
POL(s(x₁)) = 2·x₁   

(32) Obligation:

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc55_in_ → U13_(lessc55_in_)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc76_in_g(0) → U14_g(lessc55_in_)
U14_g(lessc55_out_) → lessc76_out_g(0)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc40_in_gg(x0, x1)
lessc55_in_
U13_(x0)
U16_gg(x0)
lessc76_in_g(x0)
U14_g(x0)
U15_g(x0, x1)
U17_gg(x0, x1)
U18_gg(x0, x1, x2)
U19_gg(x0, x1, x2)

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

(33) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

U14_g(lessc55_out_) → lessc76_out_g(0)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + x₂   
POL(0) = 0   
POL(ORDERED1_IN_G(x₁)) = 2·x₁   
POL(U13_(x₁)) = x₁   
POL(U14_g(x₁)) = x₁   
POL(U15_g(x₁, x₂)) = x₁ + x₂   
POL(U16_gg(x₁)) = 2·x₁   
POL(U17_gg(x₁, x₂)) = x₁ + x₂   
POL(U18_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U19_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U8_G(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + x₄   
POL(lessc26_in_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc26_out_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc40_in_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc40_out_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc55_in_) = 0   
POL(lessc55_out_) = 2   
POL(lessc76_in_g(x₁)) = x₁   
POL(lessc76_out_g(x₁)) = x₁   
POL(s(x₁)) = 2·x₁   

(34) Obligation:

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
lessc76_in_g(0) → U14_g(lessc55_in_)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
lessc55_in_ → U13_(lessc55_in_)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc40_in_gg(x0, x1)
lessc55_in_
U13_(x0)
U16_gg(x0)
lessc76_in_g(x0)
U14_g(x0)
U15_g(x0, x1)
U17_gg(x0, x1)
U18_gg(x0, x1, x2)
U19_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:

ORDERED1_IN_G(.(T16, .(T30, T31))) → U8_G(T16, T30, T31, lessc26_in_gg(T16, T30))

Strictly oriented rules of the TRS R:

lessc26_in_gg(0, T36) → lessc26_out_gg(0, T36)
lessc26_in_gg(s(T45), T49) → U19_gg(T45, T49, lessc40_in_gg(T45, T49))
lessc40_in_gg(0, 0) → U16_gg(lessc55_in_)
lessc40_in_gg(s(T63), 0) → U17_gg(T63, lessc76_in_g(T63))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(0) = 1   
POL(ORDERED1_IN_G(x₁)) = 2 + x₁   
POL(U13_(x₁)) = x₁   
POL(U14_g(x₁)) = 1 + 2·x₁   
POL(U15_g(x₁, x₂)) = x₁ + x₂   
POL(U16_gg(x₁)) = 1 + x₁   
POL(U17_gg(x₁, x₂)) = x₁ + x₂   
POL(U18_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U19_gg(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃   
POL(U8_G(x₁, x₂, x₃, x₄)) = 2 + x₁ + x₂ + 2·x₃ + x₄   
POL(lessc26_in_gg(x₁, x₂)) = 2 + x₁ + 2·x₂   
POL(lessc26_out_gg(x₁, x₂)) = 1 + x₁ + x₂   
POL(lessc40_in_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc40_out_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc55_in_) = 0   
POL(lessc76_in_g(x₁)) = x₁   
POL(lessc76_out_g(x₁)) = x₁   
POL(s(x₁)) = 2·x₁   

(36) Obligation:

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

U8_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc40_in_gg(0, s(T53)) → lessc40_out_gg(0, s(T53))
lessc40_in_gg(s(T63), s(T76)) → U18_gg(T63, T76, lessc40_in_gg(T63, T76))
U19_gg(T45, T49, lessc40_out_gg(T45, T49)) → lessc26_out_gg(s(T45), T49)
U18_gg(T63, T76, lessc40_out_gg(T63, T76)) → lessc40_out_gg(s(T63), s(T76))
lessc76_in_g(0) → U14_g(lessc55_in_)
lessc76_in_g(s(T70)) → U15_g(T70, lessc76_in_g(T70))
U15_g(T70, lessc76_out_g(T70)) → lessc76_out_g(s(T70))
lessc55_in_ → U13_(lessc55_in_)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc40_in_gg(x0, x1)
lessc55_in_
U13_(x0)
U16_gg(x0)
lessc76_in_g(x0)
U14_g(x0)
U15_g(x0, x1)
U17_gg(x0, x1)
U18_gg(x0, x1, x2)
U19_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 0 SCCs with 1 less node.

(38) TRUE