Left Termination of the query pattern ordered_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(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, Y) :- ','(!, eq(Y, s(X5))).
less(X, Y) :- ','(p(X, Px), ','(p(Y, Py), less(Px, Py))).
p(0, 0).
p(s(X), X).
eq(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, eq(s(T12), s(X36))) | less(0, s(T12)), SCOPE: 6, CLAUSE: 8 | ?_6\n\nT12 is ground\nX36 is free; ", 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, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: eq(s(T12), s(X36))\n\nT12 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: eq(s(T12), s(X36)), SCOPE: 7, CLAUSE: 11\n\nT12 is ground\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T14, X43), ','(p(s(T15), X44), less(X43, X44)))\n\nT14 is ground\nT15 is ground\nX35 is free; \nX43 is free; \nX44 is free; \nless(T14, s(T15)) !=? less(0, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T14, X43), ','(p(s(T15), X44), less(X43, X44))), SCOPE: 8, CLAUSE: 9 | ','(p(T14, X43), ','(p(s(T15), X44), less(X43, X44))), SCOPE: 8, CLAUSE: 10\n\nT14 is ground\nT15 is ground\nX35 is free; \nX43 is free; \nX44 is free; \nless(T14, s(T15)) !=? less(0, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T14, X43), ','(p(s(T15), X44), less(X43, X44))), SCOPE: 8, CLAUSE: 10\n\nT14 is ground\nT15 is ground\nX35 is free; \nX43 is free; \nX44 is free; \nless(T14, s(T15)) !=? less(0, X35)\np(T14, X43) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(s(T15), X44), less(T16, X44))\n\nT15 is ground\nT16 is ground\nX35 is free; \nX43 is free; \nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\nX35 is free\nX43 is free; \nX46 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(s(T15), X44), less(T16, X44)), SCOPE: 9, CLAUSE: 9 | ','(p(s(T15), X44), less(T16, X44)), SCOPE: 9, CLAUSE: 10\n\nT15 is ground\nT16 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(s(T15), X44), less(T16, X44)), SCOPE: 9, CLAUSE: 10\n\nT15 is ground\nT16 is ground\nX44 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(T16, T17)\n\nT16 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(T16, T17), SCOPE: 10, CLAUSE: 7 | less(T16, T17), SCOPE: 10, CLAUSE: 8 | ?_10\n\nT16 is ground\nT17 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(!_10, eq(T18, s(X51))) | less(0, T18), SCOPE: 10, CLAUSE: 8 | ?_10\n\nT18 is ground\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T16, T17), SCOPE: 10, CLAUSE: 8\n\nT16 is ground\nT17 is ground\nX50 is free; \nless(T16, T17) !=? less(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: eq(T18, s(X51))\n\nT18 is ground\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: eq(T18, s(X51)), SCOPE: 11, CLAUSE: 11\n\nT18 is ground\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\nX51 is free\nX53 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(T21, X58), ','(p(T22, X59), less(X58, X59)))\n\nT21 is ground\nT22 is ground\nX50 is free; \nX58 is free; \nX59 is free; \nless(T21, T22) !=? less(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T21, X58), ','(p(T22, X59), less(X58, X59))), SCOPE: 12, CLAUSE: 9 | ','(p(T21, X58), ','(p(T22, X59), less(X58, X59))), SCOPE: 12, CLAUSE: 10\n\nT21 is ground\nT22 is ground\nX50 is free; \nX58 is free; \nX59 is free; \nless(T21, T22) !=? less(0, X50)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T21, X58), ','(p(T22, X59), less(X58, X59))), SCOPE: 12, CLAUSE: 10\n\nT21 is ground\nT22 is ground\nX50 is free; \nX58 is free; \nX59 is free; \nless(T21, T22) !=? less(0, X50)\np(T21, X58) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T22, X59), less(T23, X59))\n\nT22 is ground\nT23 is ground\nX50 is free; \nX58 is free; \nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\nX50 is free\nX58 is free; \nX61 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T22, X59), less(T23, X59)), SCOPE: 13, CLAUSE: 9 | ','(p(T22, X59), less(T23, X59)), SCOPE: 13, CLAUSE: 10\n\nT22 is ground\nT23 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: less(T23, 0) | ','(p(0, X59), less(T23, X59)), SCOPE: 13, CLAUSE: 10\n\nT23 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(T22, X59), less(T23, X59)), SCOPE: 13, CLAUSE: 10\n\nT22 is ground\nT23 is ground\nX59 is free; \np(T22, X59) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T23, 0)\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(p(0, X59), less(T23, X59)), SCOPE: 13, CLAUSE: 10\n\nT23 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(T23, 0), SCOPE: 14, CLAUSE: 7 | less(T23, 0), SCOPE: 14, CLAUSE: 8 | ?_14\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(!_14, eq(0, s(X64))) | less(0, 0), SCOPE: 14, CLAUSE: 8 | ?_14\n\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(T23, 0), SCOPE: 14, CLAUSE: 8\n\nT23 is ground\nX63 is free; \nless(T23, 0) !=? less(0, X63)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: eq(0, s(X64))\n\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: eq(0, s(X64)), SCOPE: 15, CLAUSE: 11\n\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(p(T24, X70), ','(p(0, X71), less(X70, X71)))\n\nT24 is ground\nX63 is free; \nX70 is free; \nX71 is free; \nless(T24, 0) !=? less(0, X63)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(p(T24, X70), ','(p(0, X71), less(X70, X71))), SCOPE: 16, CLAUSE: 9 | ','(p(T24, X70), ','(p(0, X71), less(X70, X71))), SCOPE: 16, CLAUSE: 10\n\nT24 is ground\nX63 is free; \nX70 is free; \nX71 is free; \nless(T24, 0) !=? less(0, X63)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(p(T24, X70), ','(p(0, X71), less(X70, X71))), SCOPE: 16, CLAUSE: 10\n\nT24 is ground\nX63 is free; \nX70 is free; \nX71 is free; \nless(T24, 0) !=? less(0, X63)\np(T24, X70) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(p(0, X71), less(T25, X71))\n\nT25 is ground\nX63 is free; \nX70 is free; \nX71 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\nX63 is free\nX70 is free; \nX73 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(p(0, X71), less(T25, X71)), SCOPE: 17, CLAUSE: 9 | ','(p(0, X71), less(T25, X71)), SCOPE: 17, CLAUSE: 10\n\nT25 is ground\nX71 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T25, 0) | ','(p(0, X71), less(T25, X71)), SCOPE: 17, CLAUSE: 10\n\nT25 is ground\nX71 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T23, T26)\n\nT23 is ground\nT26 is ground\nX59 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\nX59 is free\nX76 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 77 [arrowhead = none ];
77 -> 2;

78 [label="EVAL with clause\nordered([]) :- !.\nand substitution\nT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 78 [arrowhead = none ];
78 -> 3;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 79 [arrowhead = none ];
79 -> 4;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 80 [arrowhead = none ];
80 -> 5;

81 [label="EVAL with clause\nordered(.(X7, [])) :- !.\nand substitution\nX7 -> T2\nT1 -> .(T2, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 81 [arrowhead = none ];
81 -> 7;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 82 [arrowhead = none ];
82 -> 8;

83 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 84 [arrowhead = none ];
84 -> 9;

85 [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 -> 85 [arrowhead = none ];
85 -> 11;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 86 [arrowhead = none ];
86 -> 10;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 87 [arrowhead = none ];
87 -> 12;

88 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(T3), ordered([]))", fontsize=14, style = filled, fillcolor = white];
12 -> 88 [arrowhead = none ];
88 -> 13;

89 [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 -> 89 [arrowhead = none ];
89 -> 14;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 90 [arrowhead = none ];
90 -> 15;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 91 [arrowhead = none ];
91 -> 16;

92 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 92 [arrowhead = none ];
92 -> 17;

93 [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 -> 93 [arrowhead = none ];
93 -> 18;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 94 [arrowhead = none ];
94 -> 19;

95 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(.(T6, T7)), ordered(.(X7, [])))", fontsize=14, style = filled, fillcolor = white];
19 -> 95 [arrowhead = none ];
95 -> 20;

96 [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 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 97 [arrowhead = none ];
97 -> 22;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 98 [arrowhead = none ];
98 -> 23;

99 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 99 [arrowhead = none ];
99 -> 24;

100 [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 -> 100 [arrowhead = none ];
100 -> 25;

101 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 101 [arrowhead = none ];
101 -> 26;

102 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
25 -> 102 [arrowhead = none ];
102 -> 27;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 103 [arrowhead = none ];
103 -> 28;

104 [label="INSTANCE with matching:\nT1 -> .(T10, T11)", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 104 [arrowhead = none , style=dashed];
104 -> 1[style=dashed];

105 [label="EVAL with clause\nless(0, X35) :- ','(!, eq(X35, s(X36))).\nand substitution\nT6 -> 0\nT10 -> T12\nX35 -> s(T12)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 105 [arrowhead = none ];
105 -> 29;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 106 [arrowhead = none ];
106 -> 30;

107 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 107 [arrowhead = none ];
107 -> 31;

108 [label="EVAL with clause\nless(X41, X42) :- ','(p(X41, X43), ','(p(X42, X44), less(X43, X44))).\nand substitution\nT6 -> T14\nX41 -> T14\nT10 -> T15\nX42 -> s(T15)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 108 [arrowhead = none ];
108 -> 35;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 109 [arrowhead = none ];
109 -> 32;

110 [label="EVAL with clause\neq(X38, X38).\nand substitution\nT12 -> T13\nX38 -> s(T13)\nX36 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 110 [arrowhead = none ];
110 -> 33;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 111 [arrowhead = none ];
111 -> 34;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 112 [arrowhead = none ];
112 -> 36;

113 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T14, s(T15)), less(0, X35))", fontsize=14, style = filled, fillcolor = white];
36 -> 113 [arrowhead = none ];
113 -> 37;

114 [label="EVAL with clause\np(s(X46), X46).\nand substitution\nX46 -> T16\nT14 -> s(T16)\nX43 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 114 [arrowhead = none ];
114 -> 38;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 115 [arrowhead = none ];
115 -> 39;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 116 [arrowhead = none ];
116 -> 40;

117 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 117 [arrowhead = none ];
117 -> 41;

118 [label="EVAL with clause\np(s(X48), X48).\nand substitution\nT15 -> T17\nX48 -> T17\nX44 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 118 [arrowhead = none ];
118 -> 42;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 119 [arrowhead = none ];
119 -> 43;

120 [label="EVAL with clause\nless(0, X50) :- ','(!, eq(X50, s(X51))).\nand substitution\nT16 -> 0\nT17 -> T18\nX50 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 120 [arrowhead = none ];
120 -> 44;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 121 [arrowhead = none ];
121 -> 45;

122 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 122 [arrowhead = none ];
122 -> 46;

123 [label="EVAL with clause\nless(X56, X57) :- ','(p(X56, X58), ','(p(X57, X59), less(X58, X59))).\nand substitution\nT16 -> T21\nX56 -> T21\nT17 -> T22\nX57 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 123 [arrowhead = none ];
123 -> 51;

124 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 124 [arrowhead = none ];
124 -> 47;

125 [label="EVAL with clause\neq(X53, X53).\nand substitution\nT18 -> s(T20)\nX53 -> s(T20)\nX51 -> T20\nT19 -> s(T20)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 125 [arrowhead = none ];
125 -> 48;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 126 [arrowhead = none ];
126 -> 49;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 127 [arrowhead = none ];
127 -> 50;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 128 [arrowhead = none ];
128 -> 52;

129 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T21, T22), less(0, X50))", fontsize=14, style = filled, fillcolor = white];
52 -> 129 [arrowhead = none ];
129 -> 53;

130 [label="EVAL with clause\np(s(X61), X61).\nand substitution\nX61 -> T23\nT21 -> s(T23)\nX58 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 130 [arrowhead = none ];
130 -> 54;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 131 [arrowhead = none ];
131 -> 55;

132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 132 [arrowhead = none ];
132 -> 56;

133 [label="EVAL with clause\np(0, 0).\nand substitution\nT22 -> 0\nX59 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 133 [arrowhead = none ];
133 -> 57;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 134 [arrowhead = none ];
134 -> 58;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 135 [arrowhead = none ];
135 -> 59;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
57 -> 136 [arrowhead = none ];
136 -> 60;

137 [label="EVAL with clause\np(s(X76), X76).\nand substitution\nX76 -> T26\nT22 -> s(T26)\nX59 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 137 [arrowhead = none ];
137 -> 75;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 138 [arrowhead = none ];
138 -> 76;

139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 139 [arrowhead = none ];
139 -> 61;

140 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
60 -> 140 [arrowhead = none ];
140 -> 74;

141 [label="EVAL with clause\nless(0, X63) :- ','(!, eq(X63, s(X64))).\nand substitution\nT23 -> 0\nX63 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 141 [arrowhead = none ];
141 -> 62;

142 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 142 [arrowhead = none ];
142 -> 63;

143 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 143 [arrowhead = none ];
143 -> 64;

144 [label="EVAL with clause\nless(X68, X69) :- ','(p(X68, X70), ','(p(X69, X71), less(X70, X71))).\nand substitution\nT23 -> T24\nX68 -> T24\nX69 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 144 [arrowhead = none ];
144 -> 67;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 145 [arrowhead = none ];
145 -> 65;

146 [label="BACKTRACK\nfor clause: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 146 [arrowhead = none ];
146 -> 66;

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 147 [arrowhead = none ];
147 -> 68;

148 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T24, 0), less(0, X63))", fontsize=14, style = filled, fillcolor = white];
68 -> 148 [arrowhead = none ];
148 -> 69;

149 [label="EVAL with clause\np(s(X73), X73).\nand substitution\nX73 -> T25\nT24 -> s(T25)\nX70 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 149 [arrowhead = none ];
149 -> 70;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 150 [arrowhead = none ];
150 -> 71;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 151 [arrowhead = none ];
151 -> 72;

152 [label="EVAL with clause\np(0, 0).\nand substitution\nX71 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 152 [arrowhead = none ];
152 -> 73;

153 [label="INSTANCE with matching:\nT23 -> T25\nX59 -> X71", fontsize=14, style = filled, fillcolor = lightgrey];
73 -> 153 [arrowhead = none , style=dashed];
153 -> 57[style=dashed];

154 [label="INSTANCE with matching:\nT16 -> T23\nT17 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 154 [arrowhead = none , style=dashed];
154 -> 42[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))).
less57(s(T25), X59) :- less57(T25, X71).
less42(0, s(T20)).
less42(s(s(T25)), 0) :- less57(T25, X71).
less42(s(T23), s(T26)) :- less42(T23, T26).
less26(0, T13).
less26(s(0), s(T20)).
less26(s(s(s(T25))), 0) :- less57(T25, X71).
less26(s(s(T23)), s(T26)) :- less42(T23, T26).

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)
less57_in: (b,f)
less42_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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(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(s(s(T25))), 0) → U6_GG(T25, less57_in_ga(T25, X71))
LESS26_IN_GG(s(s(s(T25))), 0) → LESS57_IN_GA(T25, X71)
LESS57_IN_GA(s(T25), X59) → U3_GA(T25, X59, less57_in_ga(T25, X71))
LESS57_IN_GA(s(T25), X59) → LESS57_IN_GA(T25, X71)
LESS26_IN_GG(s(s(T23)), s(T26)) → U7_GG(T23, T26, less42_in_gg(T23, T26))
LESS26_IN_GG(s(s(T23)), s(T26)) → LESS42_IN_GG(T23, T26)
LESS42_IN_GG(s(s(T25)), 0) → U4_GG(T25, less57_in_ga(T25, X71))
LESS42_IN_GG(s(s(T25)), 0) → LESS57_IN_GA(T25, X71)
LESS42_IN_GG(s(T23), s(T26)) → U5_GG(T23, T26, less42_in_gg(T23, T26))
LESS42_IN_GG(s(T23), s(T26)) → LESS42_IN_GG(T23, T26)
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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_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)
U6_GG(x1, x2)  =  U6_GG(x2)
LESS57_IN_GA(x1, x2)  =  LESS57_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
LESS42_IN_GG(x1, x2)  =  LESS42_IN_GG(x1, x2)
U4_GG(x1, x2)  =  U4_GG(x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_G(x1, x2, x3, x4)  =  U2_G(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(s(s(T25))), 0) → U6_GG(T25, less57_in_ga(T25, X71))
LESS26_IN_GG(s(s(s(T25))), 0) → LESS57_IN_GA(T25, X71)
LESS57_IN_GA(s(T25), X59) → U3_GA(T25, X59, less57_in_ga(T25, X71))
LESS57_IN_GA(s(T25), X59) → LESS57_IN_GA(T25, X71)
LESS26_IN_GG(s(s(T23)), s(T26)) → U7_GG(T23, T26, less42_in_gg(T23, T26))
LESS26_IN_GG(s(s(T23)), s(T26)) → LESS42_IN_GG(T23, T26)
LESS42_IN_GG(s(s(T25)), 0) → U4_GG(T25, less57_in_ga(T25, X71))
LESS42_IN_GG(s(s(T25)), 0) → LESS57_IN_GA(T25, X71)
LESS42_IN_GG(s(T23), s(T26)) → U5_GG(T23, T26, less42_in_gg(T23, T26))
LESS42_IN_GG(s(T23), s(T26)) → LESS42_IN_GG(T23, T26)
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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_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)
U6_GG(x1, x2)  =  U6_GG(x2)
LESS57_IN_GA(x1, x2)  =  LESS57_IN_GA(x1)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U7_GG(x1, x2, x3)  =  U7_GG(x3)
LESS42_IN_GG(x1, x2)  =  LESS42_IN_GG(x1, x2)
U4_GG(x1, x2)  =  U4_GG(x2)
U5_GG(x1, x2, x3)  =  U5_GG(x3)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 10 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS57_IN_GA(s(T25), X59) → LESS57_IN_GA(T25, X71)

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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
LESS57_IN_GA(x1, x2)  =  LESS57_IN_GA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS57_IN_GA(s(T25), X59) → LESS57_IN_GA(T25, X71)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS57_IN_GA(x1, x2)  =  LESS57_IN_GA(x1)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

LESS57_IN_GA(s(T25)) → LESS57_IN_GA(T25)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS57_IN_GA(s(T25)) → LESS57_IN_GA(T25)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

LESS42_IN_GG(s(T23), s(T26)) → LESS42_IN_GG(T23, T26)

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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_gg(x3)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
LESS42_IN_GG(x1, x2)  =  LESS42_IN_GG(x1, x2)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESS42_IN_GG(s(T23), s(T26)) → LESS42_IN_GG(T23, T26)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

LESS42_IN_GG(s(T23), s(T26)) → LESS42_IN_GG(T23, T26)

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:

• LESS42_IN_GG(s(T23), s(T26)) → LESS42_IN_GG(T23, T26)
  The graph contains the following edges 1 > 1, 2 > 2

(22) TRUE

(23) 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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_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

(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:

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, T13) → less26_out_gg(0, T13)
less26_in_gg(s(0), s(T20)) → less26_out_gg(s(0), s(T20))
less26_in_gg(s(s(s(T25))), 0) → U6_gg(T25, less57_in_ga(T25, X71))
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(T23, T26, less42_in_gg(T23, T26))
U6_gg(T25, less57_out_ga(T25, X71)) → less26_out_gg(s(s(s(T25))), 0)
U7_gg(T23, T26, less42_out_gg(T23, T26)) → less26_out_gg(s(s(T23)), s(T26))
less57_in_ga(s(T25), X59) → U3_ga(T25, X59, less57_in_ga(T25, X71))
less42_in_gg(0, s(T20)) → less42_out_gg(0, s(T20))
less42_in_gg(s(s(T25)), 0) → U4_gg(T25, less57_in_ga(T25, X71))
less42_in_gg(s(T23), s(T26)) → U5_gg(T23, T26, less42_in_gg(T23, T26))
U3_ga(T25, X59, less57_out_ga(T25, X71)) → less57_out_ga(s(T25), X59)
U4_gg(T25, less57_out_ga(T25, X71)) → less42_out_gg(s(s(T25)), 0)
U5_gg(T23, T26, less42_out_gg(T23, T26)) → less42_out_gg(s(T23), s(T26))

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)
U6_gg(x1, x2)  =  U6_gg(x2)
less57_in_ga(x1, x2)  =  less57_in_ga(x1)
U3_ga(x1, x2, x3)  =  U3_ga(x3)
less57_out_ga(x1, x2)  =  less57_out_ga
U7_gg(x1, x2, x3)  =  U7_gg(x3)
less42_in_gg(x1, x2)  =  less42_in_gg(x1, x2)
less42_out_gg(x1, x2)  =  less42_out_gg
U4_gg(x1, x2)  =  U4_gg(x2)
U5_gg(x1, x2, x3)  =  U5_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

(26) PiDPToQDPProof (SOUND 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:

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, T13) → less26_out_gg
less26_in_gg(s(0), s(T20)) → less26_out_gg
less26_in_gg(s(s(s(T25))), 0) → U6_gg(less57_in_ga(T25))
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(less42_in_gg(T23, T26))
U6_gg(less57_out_ga) → less26_out_gg
U7_gg(less42_out_gg) → less26_out_gg
less57_in_ga(s(T25)) → U3_ga(less57_in_ga(T25))
less42_in_gg(0, s(T20)) → less42_out_gg
less42_in_gg(s(s(T25)), 0) → U4_gg(less57_in_ga(T25))
less42_in_gg(s(T23), s(T26)) → U5_gg(less42_in_gg(T23, T26))
U3_ga(less57_out_ga) → less57_out_ga
U4_gg(less57_out_ga) → less42_out_gg
U5_gg(less42_out_gg) → less42_out_gg

The set Q consists of the following terms:

less26_in_gg(x0, x1)
U6_gg(x0)
U7_gg(x0)
less57_in_ga(x0)
less42_in_gg(x0, x1)
U3_ga(x0)
U4_gg(x0)
U5_gg(x0)

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

(28) 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:

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

The following rules are removed from R:

less26_in_gg(0, T13) → less26_out_gg
less26_in_gg(s(0), s(T20)) → less26_out_gg
less26_in_gg(s(s(s(T25))), 0) → U6_gg(less57_in_ga(T25))
less26_in_gg(s(s(T23)), s(T26)) → U7_gg(less42_in_gg(T23, T26))
less42_in_gg(0, s(T20)) → less42_out_gg
less42_in_gg(s(s(T25)), 0) → U4_gg(less57_in_ga(T25))
less42_in_gg(s(T23), s(T26)) → U5_gg(less42_in_gg(T23, T26))
U7_gg(less42_out_gg) → less26_out_gg
U5_gg(less42_out_gg) → less42_out_gg
less57_in_ga(s(T25)) → U3_ga(less57_in_ga(T25))
U4_gg(less57_out_ga) → less42_out_gg
U3_ga(less57_out_ga) → less57_out_ga
U6_gg(less57_out_ga) → less26_out_gg

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2 + 2·x₁ + 2·x₂   
POL(0) = 0   
POL(ORDERED1_IN_G(x₁)) = 2 + x₁   
POL(U1_G(x₁, x₂, x₃)) = 2 + 2·x₁ + 2·x₂ + x₃   
POL(U3_ga(x₁)) = 1 + 2·x₁   
POL(U4_gg(x₁)) = 2 + x₁   
POL(U5_gg(x₁)) = 2 + 2·x₁   
POL(U6_gg(x₁)) = 1 + 2·x₁   
POL(U7_gg(x₁)) = 2 + x₁   
POL(less26_in_gg(x₁, x₂)) = 2 + 2·x₁ + 2·x₂   
POL(less26_out_gg) = 2   
POL(less42_in_gg(x₁, x₂)) = 1 + x₁ + x₂   
POL(less42_out_gg) = 1   
POL(less57_in_ga(x₁)) = x₁   
POL(less57_out_ga) = 2   
POL(s(x₁)) = 2 + 2·x₁   

(29) Obligation:

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

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

R is empty.
The set Q consists of the following terms:

less26_in_gg(x0, x1)
U6_gg(x0)
U7_gg(x0)
less57_in_ga(x0)
less42_in_gg(x0, x1)
U3_ga(x0)
U4_gg(x0)
U5_gg(x0)

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

(30) DependencyGraphProof (EQUIVALENT transformation)

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

(31) TRUE