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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog 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 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 2 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2 | ?_1\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(.(T2, [])), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX7 is free; \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(T3, X10), ','(tail(T3, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13)))))))\n\nT3 is ground\nX7 is free; \nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nordered(T3) !=? ordered([])\nordered(T3) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T3, X10), ','(tail(T3, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))), SCOPE: 2, CLAUSE: 3 | ','(head(T3, X10), ','(tail(T3, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX7 is free; \nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nordered(T3) !=? ordered([])\nordered(T3) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T3, X10), ','(tail(T3, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nX7 is free; \nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nX15 is free; \nordered(T3) !=? ordered([])\nordered(T3) !=? ordered(.(X7, []))\nhead(T3, X10) !=? head([], X15)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail(.(T4, T5), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T4, s(X12)), ordered(.(X12, X13))))))\n\nT4 is ground\nT5 is ground\nX7 is free; \nX10 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nX15 is free; \nordered(.(T4, T5)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\nX7 is free\nX10 is free; \nX15 is free; \nX18 is free; \nX19 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(.(T4, T5), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T4, s(X12)), ordered(.(X12, X13)))))), SCOPE: 3, CLAUSE: 5 | ','(tail(.(T4, T5), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T4, s(X12)), ordered(.(X12, X13)))))), SCOPE: 3, CLAUSE: 6\n\nT4 is ground\nT5 is ground\nX7 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nordered(.(T4, T5)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail(.(T4, T5), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T4, s(X12)), ordered(.(X12, X13)))))), SCOPE: 3, CLAUSE: 6\n\nT4 is ground\nT5 is ground\nX7 is free; \nX11 is free; \nX12 is free; \nX13 is free; \nordered(.(T4, T5)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(head(T7, X12), ','(tail(T7, X13), ','(less(T6, s(X12)), ordered(.(X12, X13)))))\n\nT6 is ground\nT7 is ground\nX7 is free; \nX12 is free; \nX13 is free; \nordered(.(T6, T7)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T7, X12), ','(tail(T7, X13), ','(less(T6, s(X12)), ordered(.(X12, X13))))), SCOPE: 4, CLAUSE: 3 | ','(head(T7, X12), ','(tail(T7, X13), ','(less(T6, s(X12)), ordered(.(X12, X13))))), SCOPE: 4, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX7 is free; \nX12 is free; \nX13 is free; \nordered(.(T6, T7)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T7, X12), ','(tail(T7, X13), ','(less(T6, s(X12)), ordered(.(X12, X13))))), SCOPE: 4, CLAUSE: 4\n\nT6 is ground\nT7 is ground\nX7 is free; \nX12 is free; \nX13 is free; \nX25 is free; \nordered(.(T6, T7)) !=? ordered(.(X7, []))\nhead(T7, X12) !=? head([], X25)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(.(T8, T9), X13), ','(less(T6, s(T8)), ordered(.(T8, X13))))\n\nT6 is ground\nT8 is ground\nT9 is ground\nX7 is free; \nX12 is free; \nX13 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\nX7 is free\nX12 is free; \nX25 is free; \nX28 is free; \nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail(.(T8, T9), X13), ','(less(T6, s(T8)), ordered(.(T8, X13)))), SCOPE: 5, CLAUSE: 5 | ','(tail(.(T8, T9), X13), ','(less(T6, s(T8)), ordered(.(T8, X13)))), SCOPE: 5, CLAUSE: 6\n\nT6 is ground\nT8 is ground\nT9 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(.(T8, T9), X13), ','(less(T6, s(T8)), ordered(.(T8, X13)))), SCOPE: 5, CLAUSE: 6\n\nT6 is ground\nT8 is ground\nT9 is ground\nX13 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(less(T6, s(T10)), ordered(.(T10, T11)))\n\nT6 is ground\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T6, s(T10))\n\nT6 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ordered(.(T10, T11))\n\nT10 is ground\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T6, s(T10)), SCOPE: 6, CLAUSE: 7 | less(T6, s(T10)), SCOPE: 6, CLAUSE: 8 | ?_6\n\nT6 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: !_6 | less(0, s(T12)), SCOPE: 6, CLAUSE: 8 | ?_6\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T6, s(T10)), SCOPE: 6, CLAUSE: 8\n\nT6 is ground\nT10 is ground\nX35 is free; \nless(T6, s(T10)) !=? less(0, s(X35))", 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(T13, X40), ','(p(s(T14), X41), less(X40, X41)))\n\nT13 is ground\nT14 is ground\nX35 is free; \nX40 is free; \nX41 is free; \nless(T13, s(T14)) !=? less(0, s(X35))", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T13, X40), ','(p(s(T14), X41), less(X40, X41))), SCOPE: 7, CLAUSE: 9 | ','(p(T13, X40), ','(p(s(T14), X41), less(X40, X41))), SCOPE: 7, CLAUSE: 10\n\nT13 is ground\nT14 is ground\nX35 is free; \nX40 is free; \nX41 is free; \nless(T13, s(T14)) !=? less(0, s(X35))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T13, X40), ','(p(s(T14), X41), less(X40, X41))), SCOPE: 7, CLAUSE: 10\n\nT13 is ground\nT14 is ground\nX35 is free; \nX40 is free; \nX41 is free; \nless(T13, s(T14)) !=? less(0, s(X35))\np(T13, X40) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(s(T14), X41), less(T15, X41))\n\nT14 is ground\nT15 is ground\nX35 is free; \nX40 is free; \nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX35 is free\nX40 is free; \nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(s(T14), X41), less(T15, X41)), SCOPE: 8, CLAUSE: 9 | ','(p(s(T14), X41), less(T15, X41)), SCOPE: 8, CLAUSE: 10\n\nT14 is ground\nT15 is ground\nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(s(T14), X41), less(T15, X41)), SCOPE: 8, CLAUSE: 10\n\nT14 is ground\nT15 is ground\nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T15, T16)\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: less(T15, T16), SCOPE: 9, CLAUSE: 7 | less(T15, T16), SCOPE: 9, CLAUSE: 8 | ?_9\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: !_9 | less(0, s(T17)), SCOPE: 9, CLAUSE: 8 | ?_9\n\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(T15, T16), SCOPE: 9, CLAUSE: 8\n\nT15 is ground\nT16 is ground\nX47 is free; \nless(T15, T16) !=? less(0, s(X47))", 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(T18, X52), ','(p(T19, X53), less(X52, X53)))\n\nT18 is ground\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nless(T18, T19) !=? less(0, s(X47))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(T18, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 9 | ','(p(T18, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nT18 is ground\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nless(T18, T19) !=? less(0, s(X47))", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(p(T19, X53), less(0, X53)) | ','(p(0, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nless(0, T19) !=? less(0, s(X47))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(T18, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nT18 is ground\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nless(T18, T19) !=? less(0, s(X47))\np(T18, X52) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(p(T19, X53), less(0, X53)), SCOPE: 11, CLAUSE: 9 | ','(p(T19, X53), less(0, X53)), SCOPE: 11, CLAUSE: 10 | ','(p(0, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nless(0, T19) !=? less(0, s(X47))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(0, 0) | ','(p(0, X53), less(0, X53)), SCOPE: 11, CLAUSE: 10 | ','(p(0, X52), ','(p(0, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nX47 is free\nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T19, X53), less(0, X53)), SCOPE: 11, CLAUSE: 10 | ','(p(0, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nless(0, T19) !=? less(0, s(X47))\np(T19, X53) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(0, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(0, X53), less(0, X53)), SCOPE: 11, CLAUSE: 10 | ','(p(0, X52), ','(p(0, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nX52 is free\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(0, 0), SCOPE: 12, CLAUSE: 7 | less(0, 0), SCOPE: 12, CLAUSE: 8 | ?_12\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: less(0, 0), SCOPE: 12, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(0, X59), ','(p(0, X60), less(X59, X60)))\n\nX59 is free\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(0, X59), ','(p(0, X60), less(X59, X60))), SCOPE: 13, CLAUSE: 9 | ','(p(0, X59), ','(p(0, X60), less(X59, X60))), SCOPE: 13, CLAUSE: 10\n\nX59 is free\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(p(0, X60), less(0, X60)) | ','(p(0, X59), ','(p(0, X60), less(X59, X60))), SCOPE: 13, CLAUSE: 10\n\nX59 is free\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(p(0, X60), less(0, X60)), SCOPE: 14, CLAUSE: 9 | ','(p(0, X60), less(0, X60)), SCOPE: 14, CLAUSE: 10 | ','(p(0, X59), ','(p(0, X60), less(X59, X60))), SCOPE: 13, CLAUSE: 10\n\nX59 is free\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(0, 0) | ','(p(0, X60), less(0, X60)), SCOPE: 14, CLAUSE: 10 | ','(p(0, X59), ','(p(0, X60), less(X59, X60))), SCOPE: 13, CLAUSE: 10\n\nX59 is free\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(0, X52), ','(p(0, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nX52 is free\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(p(0, X52), ','(p(T19, X53), less(X52, X53))), SCOPE: 10, CLAUSE: 10\n\nT19 is ground\nX47 is free; \nX52 is free; \nX53 is free; \nX64 is free; \nless(0, T19) !=? less(0, s(X47))\np(T19, X53) !=? p(0, 0)\np(T19, X53) !=? p(s(X64), X64)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: \n\nX47 is free\nX53 is free; \nX64 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(p(T19, X53), less(T20, X53))\n\nT19 is ground\nT20 is ground\nX47 is free; \nX52 is free; \nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\nX47 is free\nX52 is free; \nX67 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(p(T19, X53), less(T20, X53)), SCOPE: 15, CLAUSE: 9 | ','(p(T19, X53), less(T20, X53)), SCOPE: 15, CLAUSE: 10\n\nT19 is ground\nT20 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: less(T20, 0) | ','(p(0, X53), less(T20, X53)), SCOPE: 15, CLAUSE: 10\n\nT20 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(p(T19, X53), less(T20, X53)), SCOPE: 15, CLAUSE: 10\n\nT19 is ground\nT20 is ground\nX53 is free; \np(T19, X53) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T20, 0)\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(p(0, X53), less(T20, X53)), SCOPE: 15, CLAUSE: 10\n\nT20 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T20, 0), SCOPE: 16, CLAUSE: 7 | less(T20, 0), SCOPE: 16, CLAUSE: 8 | ?_16\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: less(T20, 0), SCOPE: 16, CLAUSE: 8\n\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(p(T21, X73), ','(p(0, X74), less(X73, X74)))\n\nT21 is ground\nX73 is free; \nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(p(T21, X73), ','(p(0, X74), less(X73, X74))), SCOPE: 17, CLAUSE: 9 | ','(p(T21, X73), ','(p(0, X74), less(X73, X74))), SCOPE: 17, CLAUSE: 10\n\nT21 is ground\nX73 is free; \nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(p(0, X74), less(0, X74)) | ','(p(0, X73), ','(p(0, X74), less(X73, X74))), SCOPE: 17, CLAUSE: 10\n\nX73 is free\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(p(T21, X73), ','(p(0, X74), less(X73, X74))), SCOPE: 17, CLAUSE: 10\n\nT21 is ground\nX73 is free; \nX74 is free; \np(T21, X73) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: ','(p(0, X74), less(0, X74)), SCOPE: 18, CLAUSE: 9 | ','(p(0, X74), less(0, X74)), SCOPE: 18, CLAUSE: 10 | ','(p(0, X73), ','(p(0, X74), less(X73, X74))), SCOPE: 17, CLAUSE: 10\n\nX73 is free\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: less(0, 0) | ','(p(0, X74), less(0, X74)), SCOPE: 18, CLAUSE: 10 | ','(p(0, X73), ','(p(0, X74), less(X73, X74))), SCOPE: 17, CLAUSE: 10\n\nX73 is free\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ','(p(0, X74), less(T22, X74))\n\nT22 is ground\nX73 is free; \nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\nX73 is free\nX76 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(p(0, X74), less(T22, X74)), SCOPE: 19, CLAUSE: 9 | ','(p(0, X74), less(T22, X74)), SCOPE: 19, CLAUSE: 10\n\nT22 is ground\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: less(T22, 0) | ','(p(0, X74), less(T22, X74)), SCOPE: 19, CLAUSE: 10\n\nT22 is ground\nX74 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: less(T20, T23)\n\nT20 is ground\nT23 is ground\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\nX53 is free\nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 88 [arrowhead = none ];
88 -> 2;

89 [label="EVAL with clause\nordered([]) :- !.\nand substitution\nT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 89 [arrowhead = none ];
89 -> 3;

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

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

92 [label="EVAL with clause\nordered(.(X7, [])) :- !.\nand substitution\nX7 -> T2\nT1 -> .(T2, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 92 [arrowhead = none ];
92 -> 7;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 93 [arrowhead = none ];
93 -> 8;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 94 [arrowhead = none ];
94 -> 6;

95 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 95 [arrowhead = none ];
95 -> 9;

96 [label="EVAL with clause\nordered(X9) :- ','(head(X9, X10), ','(tail(X9, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))).\nand substitution\nT1 -> T3\nX9 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 96 [arrowhead = none ];
96 -> 11;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 97 [arrowhead = none ];
97 -> 10;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 98 [arrowhead = none ];
98 -> 12;

99 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(T3), ordered([]))", fontsize=14, style = filled, fillcolor = white];
12 -> 99 [arrowhead = none ];
99 -> 13;

100 [label="EVAL with clause\nhead(.(X18, X19), X18).\nand substitution\nX18 -> T4\nX19 -> T5\nT3 -> .(T4, T5)\nX10 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 100 [arrowhead = none ];
100 -> 14;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 101 [arrowhead = none ];
101 -> 15;

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

103 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 103 [arrowhead = none ];
103 -> 17;

104 [label="EVAL with clause\ntail(.(X22, X23), X23).\nand substitution\nT4 -> T6\nX22 -> T6\nT5 -> T7\nX23 -> T7\nX11 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 104 [arrowhead = none ];
104 -> 18;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 105 [arrowhead = none ];
105 -> 19;

106 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(.(T6, T7)), ordered(.(X7, [])))", fontsize=14, style = filled, fillcolor = white];
19 -> 106 [arrowhead = none ];
106 -> 20;

107 [label="EVAL with clause\nhead(.(X28, X29), X28).\nand substitution\nX28 -> T8\nX29 -> T9\nT7 -> .(T8, T9)\nX12 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 107 [arrowhead = none ];
107 -> 21;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 108 [arrowhead = none ];
108 -> 22;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 109 [arrowhead = none ];
109 -> 23;

110 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 110 [arrowhead = none ];
110 -> 24;

111 [label="EVAL with clause\ntail(.(X32, X33), X33).\nand substitution\nT8 -> T10\nX32 -> T10\nT9 -> T11\nX33 -> T11\nX13 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 111 [arrowhead = none ];
111 -> 25;

112 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 112 [arrowhead = none ];
112 -> 26;

113 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
25 -> 113 [arrowhead = none ];
113 -> 27;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 114 [arrowhead = none ];
114 -> 28;

115 [label="INSTANCE with matching:\nT1 -> .(T10, T11)", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 115 [arrowhead = none , style=dashed];
115 -> 1[style=dashed];

116 [label="EVAL with clause\nless(0, s(X35)) :- !.\nand substitution\nT6 -> 0\nT10 -> T12\nX35 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 116 [arrowhead = none ];
116 -> 29;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 117 [arrowhead = none ];
117 -> 30;

118 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 118 [arrowhead = none ];
118 -> 31;

119 [label="EVAL with clause\nless(X38, X39) :- ','(p(X38, X40), ','(p(X39, X41), less(X40, X41))).\nand substitution\nT6 -> T13\nX38 -> T13\nT10 -> T14\nX39 -> s(T14)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 119 [arrowhead = none ];
119 -> 33;

120 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 120 [arrowhead = none ];
120 -> 32;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 121 [arrowhead = none ];
121 -> 34;

122 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T13, s(T14)), less(0, s(X35)))", fontsize=14, style = filled, fillcolor = white];
34 -> 122 [arrowhead = none ];
122 -> 35;

123 [label="EVAL with clause\np(s(X43), X43).\nand substitution\nX43 -> T15\nT13 -> s(T15)\nX40 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 123 [arrowhead = none ];
123 -> 36;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 124 [arrowhead = none ];
124 -> 37;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 125 [arrowhead = none ];
125 -> 38;

126 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
38 -> 126 [arrowhead = none ];
126 -> 39;

127 [label="EVAL with clause\np(s(X45), X45).\nand substitution\nT14 -> T16\nX45 -> T16\nX41 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 127 [arrowhead = none ];
127 -> 40;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 128 [arrowhead = none ];
128 -> 41;

129 [label="EVAL with clause\nless(0, s(X47)) :- !.\nand substitution\nT15 -> 0\nX47 -> T17\nT16 -> s(T17)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 129 [arrowhead = none ];
129 -> 42;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 130 [arrowhead = none ];
130 -> 43;

131 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 131 [arrowhead = none ];
131 -> 44;

132 [label="EVAL with clause\nless(X50, X51) :- ','(p(X50, X52), ','(p(X51, X53), less(X52, X53))).\nand substitution\nT15 -> T18\nX50 -> T18\nT16 -> T19\nX51 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 132 [arrowhead = none ];
132 -> 46;

133 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 133 [arrowhead = none ];
133 -> 45;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 134 [arrowhead = none ];
134 -> 47;

135 [label="EVAL with clause\np(0, 0).\nand substitution\nT18 -> 0\nX52 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 135 [arrowhead = none ];
135 -> 48;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 136 [arrowhead = none ];
136 -> 49;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 137 [arrowhead = none ];
137 -> 50;

138 [label="EVAL with clause\np(s(X67), X67).\nand substitution\nX67 -> T20\nT18 -> s(T20)\nX52 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 138 [arrowhead = none ];
138 -> 66;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 139 [arrowhead = none ];
139 -> 67;

140 [label="EVAL with clause\np(0, 0).\nand substitution\nT19 -> 0\nX53 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 140 [arrowhead = none ];
140 -> 51;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 141 [arrowhead = none ];
141 -> 52;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 142 [arrowhead = none ];
142 -> 53;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 143 [arrowhead = none ];
143 -> 54;

144 [label="BACKTRACK\nfor clause: p(s(X), X)\nwith clash: (less(0, T19), less(0, s(X47)))", fontsize=14, style = filled, fillcolor = white];
52 -> 144 [arrowhead = none ];
144 -> 64;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 145 [arrowhead = none ];
145 -> 55;

146 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
54 -> 146 [arrowhead = none ];
146 -> 62;

147 [label="BACKTRACK\nfor clause: less(0, s(X5)) :- !because of non-unification", fontsize=14, style = filled, fillcolor = white];
55 -> 147 [arrowhead = none ];
147 -> 56;

148 [label="EVAL with clause\nless(X57, X58) :- ','(p(X57, X59), ','(p(X58, X60), less(X59, X60))).\nand substitution\nX57 -> 0\nX58 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 148 [arrowhead = none ];
148 -> 57;

149 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 149 [arrowhead = none ];
149 -> 58;

150 [label="EVAL with clause\np(0, 0).\nand substitution\nX59 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 150 [arrowhead = none ];
150 -> 59;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 151 [arrowhead = none ];
151 -> 60;

152 [label="EVAL with clause\np(0, 0).\nand substitution\nX60 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 152 [arrowhead = none ];
152 -> 61;

153 [label="INSTANCE with matching:\nX53 -> X60\nX52 -> X59\nX47 -> X59", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 153 [arrowhead = none , style=dashed];
153 -> 51[style=dashed];

154 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 154 [arrowhead = none ];
154 -> 63;

155 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
64 -> 155 [arrowhead = none ];
155 -> 65;

156 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 156 [arrowhead = none ];
156 -> 68;

157 [label="EVAL with clause\np(0, 0).\nand substitution\nT19 -> 0\nX53 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 157 [arrowhead = none ];
157 -> 69;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 158 [arrowhead = none ];
158 -> 70;

159 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 159 [arrowhead = none ];
159 -> 71;

160 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
69 -> 160 [arrowhead = none ];
160 -> 72;

161 [label="EVAL with clause\np(s(X79), X79).\nand substitution\nX79 -> T23\nT19 -> s(T23)\nX53 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 161 [arrowhead = none ];
161 -> 86;

162 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 162 [arrowhead = none ];
162 -> 87;

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 163 [arrowhead = none ];
163 -> 73;

164 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
72 -> 164 [arrowhead = none ];
164 -> 85;

165 [label="BACKTRACK\nfor clause: less(0, s(X5)) :- !because of non-unification", fontsize=14, style = filled, fillcolor = white];
73 -> 165 [arrowhead = none ];
165 -> 74;

166 [label="EVAL with clause\nless(X71, X72) :- ','(p(X71, X73), ','(p(X72, X74), less(X73, X74))).\nand substitution\nT20 -> T21\nX71 -> T21\nX72 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
74 -> 166 [arrowhead = none ];
166 -> 75;

167 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 167 [arrowhead = none ];
167 -> 76;

168 [label="EVAL with clause\np(0, 0).\nand substitution\nT21 -> 0\nX73 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 168 [arrowhead = none ];
168 -> 77;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 169 [arrowhead = none ];
169 -> 78;

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

171 [label="EVAL with clause\np(s(X76), X76).\nand substitution\nX76 -> T22\nT21 -> s(T22)\nX73 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 171 [arrowhead = none ];
171 -> 81;

172 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 172 [arrowhead = none ];
172 -> 82;

173 [label="EVAL with clause\np(0, 0).\nand substitution\nX74 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 173 [arrowhead = none ];
173 -> 80;

174 [label="INSTANCE with matching:\nX53 -> X74\nX52 -> X73\nX47 -> X73", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 174 [arrowhead = none , style=dashed];
174 -> 51[style=dashed];

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
81 -> 175 [arrowhead = none ];
175 -> 83;

176 [label="EVAL with clause\np(0, 0).\nand substitution\nX74 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 176 [arrowhead = none ];
176 -> 84;

177 [label="INSTANCE with matching:\nT20 -> T22\nX53 -> X74", fontsize=14, style = filled, fillcolor = lightgrey];
84 -> 177 [arrowhead = none , style=dashed];
177 -> 69[style=dashed];

178 [label="INSTANCE with matching:\nT15 -> T20\nT16 -> T23", fontsize=14, style = filled, fillcolor = lightgrey];
86 -> 178 [arrowhead = none , style=dashed];
178 -> 40[style=dashed];

}

(2) Obligation:

Clauses:

ordered1([]).
ordered1(.(T2, [])).
ordered1(.(T6, .(T10, T11))) :- less26(T6, T10).
ordered1(.(T6, .(T10, T11))) :- ','(less26(T6, T10), ordered1(.(T10, T11))).
less51(X53, X52) :- less51(X60, X59).
less69(0, X53) :- less51(X74, X73).
less69(s(T22), X53) :- less69(T22, X74).
less40(0, s(T17)).
less40(0, 0) :- less51(X60, X59).
less40(s(0), 0) :- less51(X74, X73).
less40(s(s(T22)), 0) :- less69(T22, X74).
less40(s(T20), s(T23)) :- less40(T20, T23).
less26(0, T12).
less26(s(0), s(T17)).
less26(s(0), 0) :- less51(X60, X59).
less26(s(s(0)), 0) :- less51(X74, X73).
less26(s(s(s(T22))), 0) :- less69(T22, X74).
less26(s(s(T20)), s(T23)) :- less40(T20, T23).

Queries:

ordered1(g).

(3) PrologToPiTRSProof (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)
less26_in: (b,b)
less51_in: (f,f)
less69_in: (b,f)
less40_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))
ORDERED1_IN_G(.(T6, .(T10, T11))) → LESS26_IN_GG(T6, T10)
LESS26_IN_GG(s(0), 0) → U10_GG(less51_in_aa(X60, X59))
LESS26_IN_GG(s(0), 0) → LESS51_IN_AA(X60, X59)
LESS51_IN_AA(X53, X52) → U3_AA(X53, X52, less51_in_aa(X60, X59))
LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)
LESS26_IN_GG(s(s(0)), 0) → U11_GG(less51_in_aa(X74, X73))
LESS26_IN_GG(s(s(0)), 0) → LESS51_IN_AA(X74, X73)
LESS26_IN_GG(s(s(s(T22))), 0) → U12_GG(T22, less69_in_ga(T22, X74))
LESS26_IN_GG(s(s(s(T22))), 0) → LESS69_IN_GA(T22, X74)
LESS69_IN_GA(0, X53) → U4_GA(X53, less51_in_aa(X74, X73))
LESS69_IN_GA(0, X53) → LESS51_IN_AA(X74, X73)
LESS69_IN_GA(s(T22), X53) → U5_GA(T22, X53, less69_in_ga(T22, X74))
LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)
LESS26_IN_GG(s(s(T20)), s(T23)) → U13_GG(T20, T23, less40_in_gg(T20, T23))
LESS26_IN_GG(s(s(T20)), s(T23)) → LESS40_IN_GG(T20, T23)
LESS40_IN_GG(0, 0) → U6_GG(less51_in_aa(X60, X59))
LESS40_IN_GG(0, 0) → LESS51_IN_AA(X60, X59)
LESS40_IN_GG(s(0), 0) → U7_GG(less51_in_aa(X74, X73))
LESS40_IN_GG(s(0), 0) → LESS51_IN_AA(X74, X73)
LESS40_IN_GG(s(s(T22)), 0) → U8_GG(T22, less69_in_ga(T22, X74))
LESS40_IN_GG(s(s(T22)), 0) → LESS69_IN_GA(T22, X74)
LESS40_IN_GG(s(T20), s(T23)) → U9_GG(T20, T23, less40_in_gg(T20, T23))
LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → U2_G(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)
LESS26_IN_GG(x1, x2)  =  LESS26_IN_GG(x1, x2)
U10_GG(x1)  =  U10_GG(x1)
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
U11_GG(x1)  =  U11_GG(x1)
U12_GG(x1, x2)  =  U12_GG(x1, x2)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)
U4_GA(x1, x2)  =  U4_GA(x2)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)
LESS40_IN_GG(x1, x2)  =  LESS40_IN_GG(x1, x2)
U6_GG(x1)  =  U6_GG(x1)
U7_GG(x1)  =  U7_GG(x1)
U8_GG(x1, x2)  =  U8_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U2_G(x1, x2, x3, x4)  =  U2_G(x1, x2, x3, x4)

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

(6) Obligation:

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

ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))
ORDERED1_IN_G(.(T6, .(T10, T11))) → LESS26_IN_GG(T6, T10)
LESS26_IN_GG(s(0), 0) → U10_GG(less51_in_aa(X60, X59))
LESS26_IN_GG(s(0), 0) → LESS51_IN_AA(X60, X59)
LESS51_IN_AA(X53, X52) → U3_AA(X53, X52, less51_in_aa(X60, X59))
LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)
LESS26_IN_GG(s(s(0)), 0) → U11_GG(less51_in_aa(X74, X73))
LESS26_IN_GG(s(s(0)), 0) → LESS51_IN_AA(X74, X73)
LESS26_IN_GG(s(s(s(T22))), 0) → U12_GG(T22, less69_in_ga(T22, X74))
LESS26_IN_GG(s(s(s(T22))), 0) → LESS69_IN_GA(T22, X74)
LESS69_IN_GA(0, X53) → U4_GA(X53, less51_in_aa(X74, X73))
LESS69_IN_GA(0, X53) → LESS51_IN_AA(X74, X73)
LESS69_IN_GA(s(T22), X53) → U5_GA(T22, X53, less69_in_ga(T22, X74))
LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)
LESS26_IN_GG(s(s(T20)), s(T23)) → U13_GG(T20, T23, less40_in_gg(T20, T23))
LESS26_IN_GG(s(s(T20)), s(T23)) → LESS40_IN_GG(T20, T23)
LESS40_IN_GG(0, 0) → U6_GG(less51_in_aa(X60, X59))
LESS40_IN_GG(0, 0) → LESS51_IN_AA(X60, X59)
LESS40_IN_GG(s(0), 0) → U7_GG(less51_in_aa(X74, X73))
LESS40_IN_GG(s(0), 0) → LESS51_IN_AA(X74, X73)
LESS40_IN_GG(s(s(T22)), 0) → U8_GG(T22, less69_in_ga(T22, X74))
LESS40_IN_GG(s(s(T22)), 0) → LESS69_IN_GA(T22, X74)
LESS40_IN_GG(s(T20), s(T23)) → U9_GG(T20, T23, less40_in_gg(T20, T23))
LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → U2_G(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)
LESS26_IN_GG(x1, x2)  =  LESS26_IN_GG(x1, x2)
U10_GG(x1)  =  U10_GG(x1)
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
U11_GG(x1)  =  U11_GG(x1)
U12_GG(x1, x2)  =  U12_GG(x1, x2)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)
U4_GA(x1, x2)  =  U4_GA(x2)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U13_GG(x1, x2, x3)  =  U13_GG(x1, x2, x3)
LESS40_IN_GG(x1, x2)  =  LESS40_IN_GG(x1, x2)
U6_GG(x1)  =  U6_GG(x1)
U7_GG(x1)  =  U7_GG(x1)
U8_GG(x1, x2)  =  U8_GG(x1, x2)
U9_GG(x1, x2, x3)  =  U9_GG(x1, x2, x3)
U2_G(x1, x2, x3, x4)  =  U2_G(x1, x2, x3, x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 21 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)

R is empty.
The argument filtering Pi contains the following mapping:
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

LESS51_IN_AA → LESS51_IN_AA

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

(14) 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 = LESS51_IN_AA evaluates to t =LESS51_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS51_IN_AA to LESS51_IN_AA.

(15) FALSE

(16) Obligation:

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

LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LESS69_IN_GA(s(T22)) → LESS69_IN_GA(T22)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS69_IN_GA(s(T22)) → LESS69_IN_GA(T22)
  The graph contains the following edges 1 > 1

(22) TRUE

(23) Obligation:

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

LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)
LESS40_IN_GG(x1, x2)  =  LESS40_IN_GG(x1, x2)

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

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)

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

(26) PiDPToQDPProof (EQUIVALENT transformation)

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

(27) Obligation:

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

LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)

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

(28) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)
  The graph contains the following edges 1 > 1, 2 > 2

(29) TRUE

(30) Obligation:

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

U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))
ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g(x1)
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x1, x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x1, x2, x3, x4)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)

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

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))
ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg(x1, x2)
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x1, x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga(x1)
U5_ga(x1, x2, x3)  =  U5_ga(x1, x3)
U13_gg(x1, x2, x3)  =  U13_gg(x1, x2, x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg(x1, x2)
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x1, x2)
U9_gg(x1, x2, x3)  =  U9_gg(x1, x2, x3)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x1, x2, x3, x4)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))
ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
U10_gg(less51_out_aa) → less26_out_gg(s(0), 0)
U11_gg(less51_out_aa) → less26_out_gg(s(s(0)), 0)
U12_gg(T22, less69_out_ga(T22)) → less26_out_gg(s(s(s(T22))), 0)
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
less51_in_aa → U3_aa(less51_in_aa)
less69_in_ga(0) → U4_ga(less51_in_aa)
less69_in_ga(s(T22)) → U5_ga(T22, less69_in_ga(T22))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22))
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U3_aa(less51_out_aa) → less51_out_aa
U4_ga(less51_out_aa) → less69_out_ga(0)
U5_ga(T22, less69_out_ga(T22)) → less69_out_ga(s(T22))
U6_gg(less51_out_aa) → less40_out_gg(0, 0)
U7_gg(less51_out_aa) → less40_out_gg(s(0), 0)
U8_gg(T22, less69_out_ga(T22)) → less40_out_gg(s(s(T22)), 0)
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))

The set Q consists of the following terms:

less26_in_gg(x0, x1)
U10_gg(x0)
U11_gg(x0)
U12_gg(x0, x1)
U13_gg(x0, x1, x2)
less51_in_aa
less69_in_ga(x0)
less40_in_gg(x0, x1)
U3_aa(x0)
U4_ga(x0)
U5_ga(x0, x1)
U6_gg(x0)
U7_gg(x0)
U8_gg(x0, x1)
U9_gg(x0, x1, x2)

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = 1   
POL(ORDERED1_IN_G(x₁)) = x₁   
POL(U10_gg(x₁)) = 1 + x₁   
POL(U11_gg(x₁)) = 1   
POL(U12_gg(x₁, x₂)) = 1   
POL(U13_gg(x₁, x₂, x₃)) = 1   
POL(U1_G(x₁, x₂, x₃, x₄)) = x₂ + x₃ + x₄   
POL(U3_aa(x₁)) = x₁   
POL(U4_ga(x₁)) = 0   
POL(U5_ga(x₁, x₂)) = 0   
POL(U6_gg(x₁)) = 0   
POL(U7_gg(x₁)) = 0   
POL(U8_gg(x₁, x₂)) = 0   
POL(U9_gg(x₁, x₂, x₃)) = 0   
POL(less26_in_gg(x₁, x₂)) = x₁   
POL(less26_out_gg(x₁, x₂)) = 1   
POL(less40_in_gg(x₁, x₂)) = 0   
POL(less40_out_gg(x₁, x₂)) = 0   
POL(less51_in_aa) = 0   
POL(less51_out_aa) = 1   
POL(less69_in_ga(x₁)) = 0   
POL(less69_out_ga(x₁)) = 0   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less51_in_aa → U3_aa(less51_in_aa)
U10_gg(less51_out_aa) → less26_out_gg(s(0), 0)
U11_gg(less51_out_aa) → less26_out_gg(s(s(0)), 0)
U12_gg(T22, less69_out_ga(T22)) → less26_out_gg(s(s(s(T22))), 0)
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U3_aa(less51_out_aa) → less51_out_aa

(36) Obligation:

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

ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
U10_gg(less51_out_aa) → less26_out_gg(s(0), 0)
U11_gg(less51_out_aa) → less26_out_gg(s(s(0)), 0)
U12_gg(T22, less69_out_ga(T22)) → less26_out_gg(s(s(s(T22))), 0)
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
less51_in_aa → U3_aa(less51_in_aa)
less69_in_ga(0) → U4_ga(less51_in_aa)
less69_in_ga(s(T22)) → U5_ga(T22, less69_in_ga(T22))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22))
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U3_aa(less51_out_aa) → less51_out_aa
U4_ga(less51_out_aa) → less69_out_ga(0)
U5_ga(T22, less69_out_ga(T22)) → less69_out_ga(s(T22))
U6_gg(less51_out_aa) → less40_out_gg(0, 0)
U7_gg(less51_out_aa) → less40_out_gg(s(0), 0)
U8_gg(T22, less69_out_ga(T22)) → less40_out_gg(s(s(T22)), 0)
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))

The set Q consists of the following terms:

less26_in_gg(x0, x1)
U10_gg(x0)
U11_gg(x0)
U12_gg(x0, x1)
U13_gg(x0, x1, x2)
less51_in_aa
less69_in_ga(x0)
less40_in_gg(x0, x1)
U3_aa(x0)
U4_ga(x0)
U5_ga(x0, x1)
U6_gg(x0)
U7_gg(x0)
U8_gg(x0, x1)
U9_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

(39) PrologToPiTRSProof (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)
less26_in: (b,b)
less51_in: (f,f)
less69_in: (b,f)
less40_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(40) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

(41) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))
ORDERED1_IN_G(.(T6, .(T10, T11))) → LESS26_IN_GG(T6, T10)
LESS26_IN_GG(s(0), 0) → U10_GG(less51_in_aa(X60, X59))
LESS26_IN_GG(s(0), 0) → LESS51_IN_AA(X60, X59)
LESS51_IN_AA(X53, X52) → U3_AA(X53, X52, less51_in_aa(X60, X59))
LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)
LESS26_IN_GG(s(s(0)), 0) → U11_GG(less51_in_aa(X74, X73))
LESS26_IN_GG(s(s(0)), 0) → LESS51_IN_AA(X74, X73)
LESS26_IN_GG(s(s(s(T22))), 0) → U12_GG(T22, less69_in_ga(T22, X74))
LESS26_IN_GG(s(s(s(T22))), 0) → LESS69_IN_GA(T22, X74)
LESS69_IN_GA(0, X53) → U4_GA(X53, less51_in_aa(X74, X73))
LESS69_IN_GA(0, X53) → LESS51_IN_AA(X74, X73)
LESS69_IN_GA(s(T22), X53) → U5_GA(T22, X53, less69_in_ga(T22, X74))
LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)
LESS26_IN_GG(s(s(T20)), s(T23)) → U13_GG(T20, T23, less40_in_gg(T20, T23))
LESS26_IN_GG(s(s(T20)), s(T23)) → LESS40_IN_GG(T20, T23)
LESS40_IN_GG(0, 0) → U6_GG(less51_in_aa(X60, X59))
LESS40_IN_GG(0, 0) → LESS51_IN_AA(X60, X59)
LESS40_IN_GG(s(0), 0) → U7_GG(less51_in_aa(X74, X73))
LESS40_IN_GG(s(0), 0) → LESS51_IN_AA(X74, X73)
LESS40_IN_GG(s(s(T22)), 0) → U8_GG(T22, less69_in_ga(T22, X74))
LESS40_IN_GG(s(s(T22)), 0) → LESS69_IN_GA(T22, X74)
LESS40_IN_GG(s(T20), s(T23)) → U9_GG(T20, T23, less40_in_gg(T20, T23))
LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → U2_G(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)
LESS26_IN_GG(x1, x2)  =  LESS26_IN_GG(x1, x2)
U10_GG(x1)  =  U10_GG(x1)
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
U11_GG(x1)  =  U11_GG(x1)
U12_GG(x1, x2)  =  U12_GG(x2)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)
U4_GA(x1, x2)  =  U4_GA(x2)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U13_GG(x1, x2, x3)  =  U13_GG(x3)
LESS40_IN_GG(x1, x2)  =  LESS40_IN_GG(x1, x2)
U6_GG(x1)  =  U6_GG(x1)
U7_GG(x1)  =  U7_GG(x1)
U8_GG(x1, x2)  =  U8_GG(x2)
U9_GG(x1, x2, x3)  =  U9_GG(x3)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(42) Obligation:

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

ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))
ORDERED1_IN_G(.(T6, .(T10, T11))) → LESS26_IN_GG(T6, T10)
LESS26_IN_GG(s(0), 0) → U10_GG(less51_in_aa(X60, X59))
LESS26_IN_GG(s(0), 0) → LESS51_IN_AA(X60, X59)
LESS51_IN_AA(X53, X52) → U3_AA(X53, X52, less51_in_aa(X60, X59))
LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)
LESS26_IN_GG(s(s(0)), 0) → U11_GG(less51_in_aa(X74, X73))
LESS26_IN_GG(s(s(0)), 0) → LESS51_IN_AA(X74, X73)
LESS26_IN_GG(s(s(s(T22))), 0) → U12_GG(T22, less69_in_ga(T22, X74))
LESS26_IN_GG(s(s(s(T22))), 0) → LESS69_IN_GA(T22, X74)
LESS69_IN_GA(0, X53) → U4_GA(X53, less51_in_aa(X74, X73))
LESS69_IN_GA(0, X53) → LESS51_IN_AA(X74, X73)
LESS69_IN_GA(s(T22), X53) → U5_GA(T22, X53, less69_in_ga(T22, X74))
LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)
LESS26_IN_GG(s(s(T20)), s(T23)) → U13_GG(T20, T23, less40_in_gg(T20, T23))
LESS26_IN_GG(s(s(T20)), s(T23)) → LESS40_IN_GG(T20, T23)
LESS40_IN_GG(0, 0) → U6_GG(less51_in_aa(X60, X59))
LESS40_IN_GG(0, 0) → LESS51_IN_AA(X60, X59)
LESS40_IN_GG(s(0), 0) → U7_GG(less51_in_aa(X74, X73))
LESS40_IN_GG(s(0), 0) → LESS51_IN_AA(X74, X73)
LESS40_IN_GG(s(s(T22)), 0) → U8_GG(T22, less69_in_ga(T22, X74))
LESS40_IN_GG(s(s(T22)), 0) → LESS69_IN_GA(T22, X74)
LESS40_IN_GG(s(T20), s(T23)) → U9_GG(T20, T23, less40_in_gg(T20, T23))
LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → U2_G(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)
LESS26_IN_GG(x1, x2)  =  LESS26_IN_GG(x1, x2)
U10_GG(x1)  =  U10_GG(x1)
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA
U3_AA(x1, x2, x3)  =  U3_AA(x3)
U11_GG(x1)  =  U11_GG(x1)
U12_GG(x1, x2)  =  U12_GG(x2)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)
U4_GA(x1, x2)  =  U4_GA(x2)
U5_GA(x1, x2, x3)  =  U5_GA(x3)
U13_GG(x1, x2, x3)  =  U13_GG(x3)
LESS40_IN_GG(x1, x2)  =  LESS40_IN_GG(x1, x2)
U6_GG(x1)  =  U6_GG(x1)
U7_GG(x1)  =  U7_GG(x1)
U8_GG(x1, x2)  =  U8_GG(x2)
U9_GG(x1, x2, x3)  =  U9_GG(x3)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(43) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 4 SCCs with 21 less nodes.

(44) Complex Obligation (AND)

(45) Obligation:

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

LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA

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

(46) UsableRulesProof (EQUIVALENT transformation)

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

(47) Obligation:

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

LESS51_IN_AA(X53, X52) → LESS51_IN_AA(X60, X59)

R is empty.
The argument filtering Pi contains the following mapping:
LESS51_IN_AA(x1, x2)  =  LESS51_IN_AA

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

(48) PiDPToQDPProof (SOUND transformation)

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

(49) Obligation:

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

LESS51_IN_AA → LESS51_IN_AA

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

(50) 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 = LESS51_IN_AA evaluates to t =LESS51_IN_AA

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

• Matcher: [ ]
• Semiunifier: [ ]

---

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from LESS51_IN_AA to LESS51_IN_AA.

(51) FALSE

(52) Obligation:

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

LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)

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

(53) UsableRulesProof (EQUIVALENT transformation)

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

(54) Obligation:

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

LESS69_IN_GA(s(T22), X53) → LESS69_IN_GA(T22, X74)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS69_IN_GA(x1, x2)  =  LESS69_IN_GA(x1)

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

(55) PiDPToQDPProof (SOUND transformation)

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

(56) Obligation:

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

LESS69_IN_GA(s(T22)) → LESS69_IN_GA(T22)

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

(57) 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:

• LESS69_IN_GA(s(T22)) → LESS69_IN_GA(T22)
  The graph contains the following edges 1 > 1

(58) TRUE

(59) Obligation:

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

LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
LESS40_IN_GG(x1, x2)  =  LESS40_IN_GG(x1, x2)

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

(60) UsableRulesProof (EQUIVALENT transformation)

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

(61) Obligation:

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

LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)

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

(62) PiDPToQDPProof (EQUIVALENT transformation)

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

(63) Obligation:

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

LESS40_IN_GG(s(T20), s(T23)) → LESS40_IN_GG(T20, T23)

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

(64) 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(T20), s(T23)) → LESS40_IN_GG(T20, T23)
  The graph contains the following edges 1 > 1, 2 > 2

(65) TRUE

(66) Obligation:

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

U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))
ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

ordered1_in_g([]) → ordered1_out_g([])
ordered1_in_g(.(T2, [])) → ordered1_out_g(.(T2, []))
ordered1_in_g(.(T6, .(T10, T11))) → U1_g(T6, T10, T11, less26_in_gg(T6, T10))
less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → ordered1_out_g(.(T6, .(T10, T11)))
U1_g(T6, T10, T11, less26_out_gg(T6, T10)) → U2_g(T6, T10, T11, ordered1_in_g(.(T10, T11)))
U2_g(T6, T10, T11, ordered1_out_g(.(T10, T11))) → ordered1_out_g(.(T6, .(T10, T11)))

The argument filtering Pi contains the following mapping:
ordered1_in_g(x1)  =  ordered1_in_g(x1)
[]  =  []
ordered1_out_g(x1)  =  ordered1_out_g
.(x1, x2)  =  .(x1, x2)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)

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

(67) UsableRulesProof (EQUIVALENT transformation)

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

(68) Obligation:

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

U1_G(T6, T10, T11, less26_out_gg(T6, T10)) → ORDERED1_IN_G(.(T10, T11))
ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T6, T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

less26_in_gg(0, T12) → less26_out_gg(0, T12)
less26_in_gg(s(0), s(T17)) → less26_out_gg(s(0), s(T17))
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa(X60, X59))
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa(X74, X73))
less26_in_gg(s(s(s(T22))), 0) → U12_gg(T22, less69_in_ga(T22, X74))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(T20, T23, less40_in_gg(T20, T23))
U10_gg(less51_out_aa(X60, X59)) → less26_out_gg(s(0), 0)
U11_gg(less51_out_aa(X74, X73)) → less26_out_gg(s(s(0)), 0)
U12_gg(T22, less69_out_ga(T22, X74)) → less26_out_gg(s(s(s(T22))), 0)
U13_gg(T20, T23, less40_out_gg(T20, T23)) → less26_out_gg(s(s(T20)), s(T23))
less51_in_aa(X53, X52) → U3_aa(X53, X52, less51_in_aa(X60, X59))
less69_in_ga(0, X53) → U4_ga(X53, less51_in_aa(X74, X73))
less69_in_ga(s(T22), X53) → U5_ga(T22, X53, less69_in_ga(T22, X74))
less40_in_gg(0, s(T17)) → less40_out_gg(0, s(T17))
less40_in_gg(0, 0) → U6_gg(less51_in_aa(X60, X59))
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa(X74, X73))
less40_in_gg(s(s(T22)), 0) → U8_gg(T22, less69_in_ga(T22, X74))
less40_in_gg(s(T20), s(T23)) → U9_gg(T20, T23, less40_in_gg(T20, T23))
U3_aa(X53, X52, less51_out_aa(X60, X59)) → less51_out_aa(X53, X52)
U4_ga(X53, less51_out_aa(X74, X73)) → less69_out_ga(0, X53)
U5_ga(T22, X53, less69_out_ga(T22, X74)) → less69_out_ga(s(T22), X53)
U6_gg(less51_out_aa(X60, X59)) → less40_out_gg(0, 0)
U7_gg(less51_out_aa(X74, X73)) → less40_out_gg(s(0), 0)
U8_gg(T22, less69_out_ga(T22, X74)) → less40_out_gg(s(s(T22)), 0)
U9_gg(T20, T23, less40_out_gg(T20, T23)) → less40_out_gg(s(T20), s(T23))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
less26_in_gg(x1, x2)  =  less26_in_gg(x1, x2)
0  =  0
less26_out_gg(x1, x2)  =  less26_out_gg
s(x1)  =  s(x1)
U10_gg(x1)  =  U10_gg(x1)
less51_in_aa(x1, x2)  =  less51_in_aa
U3_aa(x1, x2, x3)  =  U3_aa(x3)
less51_out_aa(x1, x2)  =  less51_out_aa
U11_gg(x1)  =  U11_gg(x1)
U12_gg(x1, x2)  =  U12_gg(x2)
less69_in_ga(x1, x2)  =  less69_in_ga(x1)
U4_ga(x1, x2)  =  U4_ga(x2)
less69_out_ga(x1, x2)  =  less69_out_ga
U5_ga(x1, x2, x3)  =  U5_ga(x3)
U13_gg(x1, x2, x3)  =  U13_gg(x3)
less40_in_gg(x1, x2)  =  less40_in_gg(x1, x2)
less40_out_gg(x1, x2)  =  less40_out_gg
U6_gg(x1)  =  U6_gg(x1)
U7_gg(x1)  =  U7_gg(x1)
U8_gg(x1, x2)  =  U8_gg(x2)
U9_gg(x1, x2, x3)  =  U9_gg(x3)
ORDERED1_IN_G(x1)  =  ORDERED1_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)

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

(69) PiDPToQDPProof (SOUND transformation)

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

(70) Obligation:

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

U1_G(T10, T11, less26_out_gg) → ORDERED1_IN_G(.(T10, T11))
ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

less26_in_gg(0, T12) → less26_out_gg
less26_in_gg(s(0), s(T17)) → less26_out_gg
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(less69_in_ga(T22))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(less40_in_gg(T20, T23))
U10_gg(less51_out_aa) → less26_out_gg
U11_gg(less51_out_aa) → less26_out_gg
U12_gg(less69_out_ga) → less26_out_gg
U13_gg(less40_out_gg) → less26_out_gg
less51_in_aa → U3_aa(less51_in_aa)
less69_in_ga(0) → U4_ga(less51_in_aa)
less69_in_ga(s(T22)) → U5_ga(less69_in_ga(T22))
less40_in_gg(0, s(T17)) → less40_out_gg
less40_in_gg(0, 0) → U6_gg(less51_in_aa)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa)
less40_in_gg(s(s(T22)), 0) → U8_gg(less69_in_ga(T22))
less40_in_gg(s(T20), s(T23)) → U9_gg(less40_in_gg(T20, T23))
U3_aa(less51_out_aa) → less51_out_aa
U4_ga(less51_out_aa) → less69_out_ga
U5_ga(less69_out_ga) → less69_out_ga
U6_gg(less51_out_aa) → less40_out_gg
U7_gg(less51_out_aa) → less40_out_gg
U8_gg(less69_out_ga) → less40_out_gg
U9_gg(less40_out_gg) → less40_out_gg

The set Q consists of the following terms:

less26_in_gg(x0, x1)
U10_gg(x0)
U11_gg(x0)
U12_gg(x0)
U13_gg(x0)
less51_in_aa
less69_in_ga(x0)
less40_in_gg(x0, x1)
U3_aa(x0)
U4_ga(x0)
U5_ga(x0)
U6_gg(x0)
U7_gg(x0)
U8_gg(x0)
U9_gg(x0)

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

(71) 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.

The following dependency pairs can be deleted:

U1_G(T10, T11, less26_out_gg) → ORDERED1_IN_G(.(T10, T11))

The following rules are removed from R:

less26_in_gg(0, T12) → less26_out_gg
less26_in_gg(s(0), s(T17)) → less26_out_gg
less26_in_gg(s(0), 0) → U10_gg(less51_in_aa)
less26_in_gg(s(s(0)), 0) → U11_gg(less51_in_aa)
less26_in_gg(s(s(s(T22))), 0) → U12_gg(less69_in_ga(T22))
less26_in_gg(s(s(T20)), s(T23)) → U13_gg(less40_in_gg(T20, T23))
less40_in_gg(0, s(T17)) → less40_out_gg
less40_in_gg(0, 0) → U6_gg(less51_in_aa)
less40_in_gg(s(0), 0) → U7_gg(less51_in_aa)
less40_in_gg(s(s(T22)), 0) → U8_gg(less69_in_ga(T22))
less40_in_gg(s(T20), s(T23)) → U9_gg(less40_in_gg(T20, T23))
U13_gg(less40_out_gg) → less26_out_gg
U9_gg(less40_out_gg) → less40_out_gg
less69_in_ga(0) → U4_ga(less51_in_aa)
less69_in_ga(s(T22)) → U5_ga(less69_in_ga(T22))
U8_gg(less69_out_ga) → less40_out_gg
U5_ga(less69_out_ga) → less69_out_ga
U4_ga(less51_out_aa) → less69_out_ga
U3_aa(less51_out_aa) → less51_out_aa
U7_gg(less51_out_aa) → less40_out_gg
U6_gg(less51_out_aa) → less40_out_gg
U11_gg(less51_out_aa) → less26_out_gg
U10_gg(less51_out_aa) → less26_out_gg

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = x₁ + 2·x₂   
POL(0) = 2   
POL(ORDERED1_IN_G(x₁)) = 1 + x₁   
POL(U10_gg(x₁)) = 2·x₁   
POL(U11_gg(x₁)) = 2·x₁   
POL(U12_gg(x₁)) = 2·x₁   
POL(U13_gg(x₁)) = 2 + x₁   
POL(U1_G(x₁, x₂, x₃)) = x₁ + 2·x₂ + x₃   
POL(U3_aa(x₁)) = 2·x₁   
POL(U4_ga(x₁)) = 2·x₁   
POL(U5_ga(x₁)) = 2·x₁   
POL(U6_gg(x₁)) = 2 + 2·x₁   
POL(U7_gg(x₁)) = 2 + x₁   
POL(U8_gg(x₁)) = 2 + 2·x₁   
POL(U9_gg(x₁)) = 2 + 2·x₁   
POL(less26_in_gg(x₁, x₂)) = 1 + x₁ + x₂   
POL(less26_out_gg) = 2   
POL(less40_in_gg(x₁, x₂)) = 2 + x₁ + x₂   
POL(less40_out_gg) = 1   
POL(less51_in_aa) = 0   
POL(less51_out_aa) = 2   
POL(less69_in_ga(x₁)) = 1 + 2·x₁   
POL(less69_out_ga) = 1   
POL(s(x₁)) = 2 + 2·x₁   

(72) Obligation:

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

ORDERED1_IN_G(.(T6, .(T10, T11))) → U1_G(T10, T11, less26_in_gg(T6, T10))

The TRS R consists of the following rules:

less51_in_aa → U3_aa(less51_in_aa)
U12_gg(less69_out_ga) → less26_out_gg

The set Q consists of the following terms:

less26_in_gg(x0, x1)
U10_gg(x0)
U11_gg(x0)
U12_gg(x0)
U13_gg(x0)
less51_in_aa
less69_in_ga(x0)
less40_in_gg(x0, x1)
U3_aa(x0)
U4_ga(x0)
U5_ga(x0)
U6_gg(x0)
U7_gg(x0)
U8_gg(x0)
U9_gg(x0)

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

(73) DependencyGraphProof (EQUIVALENT transformation)

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

(74) TRUE