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) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ordered(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ordered(T1), SCOPE: 1, CLAUSE: 0 | ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nordered(T1) !=? ordered([])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: !_1 | ordered(.(T3, [])), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ordered(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14)))))))\n\nT5 is ground\nX11 is free; \nX12 is free; \nX13 is free; \nX14 is free; \nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))), SCOPE: 2, CLAUSE: 3 | ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; \nX12 is free; \nX13 is free; \nX14 is free; \nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T5, X11), ','(tail(T5, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX11 is free; \nX12 is free; \nX13 is free; \nX14 is free; \nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14))))))\n\nT10 is ground\nT11 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14)))))), SCOPE: 3, CLAUSE: 5 | ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14)))))), SCOPE: 3, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail(.(T10, T11), X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(T10, s(X13)), ordered(.(X13, X14)))))), SCOPE: 3, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nX12 is free; \nX13 is free; \nX14 is free; \nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14)))))\n\nT16 is ground\nT17 is ground\nX13 is free; \nX14 is free; \nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14))))), SCOPE: 4, CLAUSE: 3 | ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14))))), SCOPE: 4, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX13 is free; \nX14 is free; \nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T17, X13), ','(tail(T17, X14), ','(less(T16, s(X13)), ordered(.(X13, X14))))), SCOPE: 4, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX13 is free; \nX14 is free; \nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14))))\n\nT16 is ground\nT22 is ground\nT23 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14)))), SCOPE: 5, CLAUSE: 5 | ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14)))), SCOPE: 5, CLAUSE: 6\n\nT16 is ground\nT22 is ground\nT23 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(.(T22, T23), X14), ','(less(T16, s(T22)), ordered(.(T22, X14)))), SCOPE: 5, CLAUSE: 6\n\nT16 is ground\nT22 is ground\nT23 is ground\nX14 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(less(T16, s(T30)), ordered(.(T30, T31)))\n\nT16 is ground\nT30 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T16, s(T30))\n\nT16 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ordered(.(T30, T31))\n\nT30 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T16, s(T30)), SCOPE: 6, CLAUSE: 7 | less(T16, s(T30)), SCOPE: 6, CLAUSE: 8\n\nT16 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(!_6, eq(s(T36), s(X68))) | less(0, s(T36)), SCOPE: 6, CLAUSE: 8\n\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T16, s(T30)), SCOPE: 6, CLAUSE: 8\n\nT16 is ground\nT30 is ground\nless(T16, s(T30)) !=? less(0, X67)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: eq(s(T36), s(X68))\n\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: eq(s(T36), s(X68)), SCOPE: 7, CLAUSE: 11\n\nT36 is ground\nX68 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(T44, X84), ','(p(s(T45), X85), less(X84, X85)))\n\nT44 is ground\nT45 is ground\nX84 is free; \nX85 is free; \nless(T44, s(T45)) !=? less(0, X67)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T44, X84), ','(p(s(T45), X85), less(X84, X85))), SCOPE: 8, CLAUSE: 9 | ','(p(T44, X84), ','(p(s(T45), X85), less(X84, X85))), SCOPE: 8, CLAUSE: 10\n\nT44 is ground\nT45 is ground\nX84 is free; \nX85 is free; \nless(T44, s(T45)) !=? less(0, X67)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T44, X84), ','(p(s(T45), X85), less(X84, X85))), SCOPE: 8, CLAUSE: 10\n\nT44 is ground\nT45 is ground\nX84 is free; \nX85 is free; \nless(T44, s(T45)) !=? less(0, X67)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(s(T45), X85), less(T48, X85))\n\nT45 is ground\nT48 is ground\nX85 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(s(T45), X85), less(T48, X85)), SCOPE: 9, CLAUSE: 9 | ','(p(s(T45), X85), less(T48, X85)), SCOPE: 9, CLAUSE: 10\n\nT45 is ground\nT48 is ground\nX85 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(s(T45), X85), less(T48, X85)), SCOPE: 9, CLAUSE: 10\n\nT45 is ground\nT48 is ground\nX85 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(T48, T52)\n\nT48 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: less(T48, T52), SCOPE: 10, CLAUSE: 7 | less(T48, T52), SCOPE: 10, CLAUSE: 8\n\nT48 is ground\nT52 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(!_10, eq(T56, s(X107))) | less(0, T56), SCOPE: 10, CLAUSE: 8\n\nT56 is ground\nX107 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T48, T52), SCOPE: 10, CLAUSE: 8\n\nT48 is ground\nT52 is ground\nless(T48, T52) !=? less(0, X106)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: eq(T56, s(X107))\n\nT56 is ground\nX107 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: eq(T56, s(X107)), SCOPE: 11, CLAUSE: 11\n\nT56 is ground\nX107 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(T67, X124), ','(p(T68, X125), less(X124, X125)))\n\nT67 is ground\nT68 is ground\nX124 is free; \nX125 is free; \nless(T67, T68) !=? less(0, X106)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T67, X124), ','(p(T68, X125), less(X124, X125))), SCOPE: 12, CLAUSE: 9 | ','(p(T67, X124), ','(p(T68, X125), less(X124, X125))), SCOPE: 12, CLAUSE: 10\n\nT67 is ground\nT68 is ground\nX124 is free; \nX125 is free; \nless(T67, T68) !=? less(0, X106)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T67, X124), ','(p(T68, X125), less(X124, X125))), SCOPE: 12, CLAUSE: 10\n\nT67 is ground\nT68 is ground\nX124 is free; \nX125 is free; \nless(T67, T68) !=? less(0, X106)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T68, X125), less(T71, X125))\n\nT68 is ground\nT71 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T68, X125), less(T71, X125)), SCOPE: 13, CLAUSE: 9 | ','(p(T68, X125), less(T71, X125)), SCOPE: 13, CLAUSE: 10\n\nT68 is ground\nT71 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(T68, X125), less(T71, X125)), SCOPE: 13, CLAUSE: 9\n\nT68 is ground\nT71 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(T68, X125), less(T71, X125)), SCOPE: 13, CLAUSE: 10\n\nT68 is ground\nT71 is ground\nX125 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T71, 0)\n\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(T71, 0), SCOPE: 14, CLAUSE: 7 | less(T71, 0), SCOPE: 14, CLAUSE: 8\n\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(!_14, eq(0, s(X142))) | less(0, 0), SCOPE: 14, CLAUSE: 8\n\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(T71, 0), SCOPE: 14, CLAUSE: 8\n\nT71 is ground\nless(T71, 0) !=? less(0, X141)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: eq(0, s(X142))\n\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: eq(0, s(X142)), SCOPE: 15, CLAUSE: 11\n\nX142 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(p(T74, X156), ','(p(0, X157), less(X156, X157)))\n\nT74 is ground\nX156 is free; \nX157 is free; \nless(T74, 0) !=? less(0, X141)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(p(T74, X156), ','(p(0, X157), less(X156, X157))), SCOPE: 16, CLAUSE: 9 | ','(p(T74, X156), ','(p(0, X157), less(X156, X157))), SCOPE: 16, CLAUSE: 10\n\nT74 is ground\nX156 is free; \nX157 is free; \nless(T74, 0) !=? less(0, X141)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(p(T74, X156), ','(p(0, X157), less(X156, X157))), SCOPE: 16, CLAUSE: 10\n\nT74 is ground\nX156 is free; \nX157 is free; \nless(T74, 0) !=? less(0, X141)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(p(0, X157), less(T77, X157))\n\nT77 is ground\nX157 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(p(0, X157), less(T77, X157)), SCOPE: 17, CLAUSE: 9 | ','(p(0, X157), less(T77, X157)), SCOPE: 17, CLAUSE: 10\n\nT77 is ground\nX157 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(0, X157), less(T77, X157)), SCOPE: 17, CLAUSE: 9\n\nT77 is ground\nX157 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(p(0, X157), less(T77, X157)), SCOPE: 17, CLAUSE: 10\n\nT77 is ground\nX157 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: less(T77, 0)\n\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: less(T71, T80)\n\nT71 is ground\nT80 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 79 [arrowhead = none ];
79 -> 2;

80 [label="EVAL with clause\nordered([]) :- !_1.\nand substitution\nT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 81 [arrowhead = none ];
81 -> 4;

82 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="EVAL with clause\nordered(.(X7, [])) :- !_1.\nand substitution\nX7 -> T3\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 83 [arrowhead = none ];
83 -> 7;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 84 [arrowhead = none ];
84 -> 8;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 85 [arrowhead = none ];
85 -> 6;

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 86 [arrowhead = none ];
86 -> 9;

87 [label="ONLY EVAL with clause\nordered(X10) :- ','(head(X10, X11), ','(tail(X10, X12), ','(head(X12, X13), ','(tail(X12, X14), ','(less(X11, s(X13)), ordered(.(X13, X14))))))).\nand substitution\nT1 -> T5\nX10 -> T5", fontsize=14, style = filled, fillcolor = white];
8 -> 87 [arrowhead = none ];
87 -> 11;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 88 [arrowhead = none ];
88 -> 10;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 89 [arrowhead = none ];
89 -> 12;

90 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(T5), ordered([]))", fontsize=14, style = filled, fillcolor = white];
12 -> 90 [arrowhead = none ];
90 -> 13;

91 [label="EVAL with clause\nhead(.(X21, X22), X21).\nand substitution\nX21 -> T10\nX22 -> T11\nT5 -> .(T10, T11)\nX11 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 91 [arrowhead = none ];
91 -> 14;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 92 [arrowhead = none ];
92 -> 15;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 93 [arrowhead = none ];
93 -> 16;

94 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="ONLY EVAL with clause\ntail(.(X35, X36), X36).\nand substitution\nT10 -> T16\nX35 -> T16\nT11 -> T17\nX36 -> T17\nX12 -> T17", fontsize=14, style = filled, fillcolor = white];
17 -> 95 [arrowhead = none ];
95 -> 18;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 96 [arrowhead = none ];
96 -> 19;

97 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (ordered(.(T16, T17)), ordered(.(X7, [])))", fontsize=14, style = filled, fillcolor = white];
19 -> 97 [arrowhead = none ];
97 -> 20;

98 [label="EVAL with clause\nhead(.(X46, X47), X46).\nand substitution\nX46 -> T22\nX47 -> T23\nT17 -> .(T22, T23)\nX13 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 98 [arrowhead = none ];
98 -> 21;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 99 [arrowhead = none ];
99 -> 22;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 100 [arrowhead = none ];
100 -> 23;

101 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
23 -> 101 [arrowhead = none ];
101 -> 24;

102 [label="ONLY EVAL with clause\ntail(.(X58, X59), X59).\nand substitution\nT22 -> T30\nX58 -> T30\nT23 -> T31\nX59 -> T31\nX14 -> T31", fontsize=14, style = filled, fillcolor = white];
24 -> 102 [arrowhead = none ];
102 -> 25;

103 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 103 [arrowhead = none ];
103 -> 26;

104 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
25 -> 104 [arrowhead = none ];
104 -> 27;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 105 [arrowhead = none ];
105 -> 28;

106 [label="INSTANCE with matching:\nT1 -> .(T30, T31)", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 106 [arrowhead = none , style=dashed];
106 -> 1[style=dashed];

107 [label="EVAL with clause\nless(0, X67) :- ','(!_6, eq(X67, s(X68))).\nand substitution\nT16 -> 0\nT30 -> T36\nX67 -> s(T36)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 107 [arrowhead = none ];
107 -> 29;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 108 [arrowhead = none ];
108 -> 30;

109 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 109 [arrowhead = none ];
109 -> 31;

110 [label="ONLY EVAL with clause\nless(X82, X83) :- ','(p(X82, X84), ','(p(X83, X85), less(X84, X85))).\nand substitution\nT16 -> T44\nX82 -> T44\nT30 -> T45\nX83 -> s(T45)", fontsize=14, style = filled, fillcolor = white];
30 -> 110 [arrowhead = none ];
110 -> 35;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 111 [arrowhead = none ];
111 -> 32;

112 [label="ONLY EVAL with clause\neq(X72, X72).\nand substitution\nT36 -> T39\nX72 -> s(T39)\nX68 -> T39", fontsize=14, style = filled, fillcolor = white];
32 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 113 [arrowhead = none ];
113 -> 34;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 114 [arrowhead = none ];
114 -> 36;

115 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T44, s(T45)), less(0, X67))", fontsize=14, style = filled, fillcolor = white];
36 -> 115 [arrowhead = none ];
115 -> 37;

116 [label="EVAL with clause\np(s(X91), X91).\nand substitution\nX91 -> T48\nT44 -> s(T48)\nX84 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 116 [arrowhead = none ];
116 -> 38;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 117 [arrowhead = none ];
117 -> 39;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 118 [arrowhead = none ];
118 -> 40;

119 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
40 -> 119 [arrowhead = none ];
119 -> 41;

120 [label="ONLY EVAL with clause\np(s(X100), X100).\nand substitution\nT45 -> T52\nX100 -> T52\nX85 -> T52", fontsize=14, style = filled, fillcolor = white];
41 -> 120 [arrowhead = none ];
120 -> 42;

121 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 121 [arrowhead = none ];
121 -> 43;

122 [label="EVAL with clause\nless(0, X106) :- ','(!_10, eq(X106, s(X107))).\nand substitution\nT48 -> 0\nT52 -> T56\nX106 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 122 [arrowhead = none ];
122 -> 44;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 123 [arrowhead = none ];
123 -> 45;

124 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 124 [arrowhead = none ];
124 -> 46;

125 [label="ONLY EVAL with clause\nless(X122, X123) :- ','(p(X122, X124), ','(p(X123, X125), less(X124, X125))).\nand substitution\nT48 -> T67\nX122 -> T67\nT52 -> T68\nX123 -> T68", fontsize=14, style = filled, fillcolor = white];
45 -> 125 [arrowhead = none ];
125 -> 51;

126 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="EVAL with clause\neq(X111, X111).\nand substitution\nT56 -> s(T62)\nX111 -> s(T62)\nX107 -> T62\nT61 -> s(T62)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 127 [arrowhead = none ];
127 -> 48;

128 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 128 [arrowhead = none ];
128 -> 49;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 129 [arrowhead = none ];
129 -> 50;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 130 [arrowhead = none ];
130 -> 52;

131 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T67, T68), less(0, X106))", fontsize=14, style = filled, fillcolor = white];
52 -> 131 [arrowhead = none ];
131 -> 53;

132 [label="EVAL with clause\np(s(X131), X131).\nand substitution\nX131 -> T71\nT67 -> s(T71)\nX124 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 132 [arrowhead = none ];
132 -> 54;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 133 [arrowhead = none ];
133 -> 55;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 134 [arrowhead = none ];
134 -> 56;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 135 [arrowhead = none ];
135 -> 57;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 136 [arrowhead = none ];
136 -> 58;

137 [label="EVAL with clause\np(0, 0).\nand substitution\nT68 -> 0\nX125 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 137 [arrowhead = none ];
137 -> 59;

138 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 138 [arrowhead = none ];
138 -> 60;

139 [label="EVAL with clause\np(s(X177), X177).\nand substitution\nX177 -> T80\nT68 -> s(T80)\nX125 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 139 [arrowhead = none ];
139 -> 77;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 140 [arrowhead = none ];
140 -> 78;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 141 [arrowhead = none ];
141 -> 61;

142 [label="EVAL with clause\nless(0, X141) :- ','(!_14, eq(X141, s(X142))).\nand substitution\nT71 -> 0\nX141 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 142 [arrowhead = none ];
142 -> 62;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 143 [arrowhead = none ];
143 -> 63;

144 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 144 [arrowhead = none ];
144 -> 64;

145 [label="ONLY EVAL with clause\nless(X154, X155) :- ','(p(X154, X156), ','(p(X155, X157), less(X156, X157))).\nand substitution\nT71 -> T74\nX154 -> T74\nX155 -> 0", fontsize=14, style = filled, fillcolor = white];
63 -> 145 [arrowhead = none ];
145 -> 67;

146 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 146 [arrowhead = none ];
146 -> 65;

147 [label="BACKTRACK\nfor clause: eq(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 147 [arrowhead = none ];
147 -> 66;

148 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 148 [arrowhead = none ];
148 -> 68;

149 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T74, 0), less(0, X141))", fontsize=14, style = filled, fillcolor = white];
68 -> 149 [arrowhead = none ];
149 -> 69;

150 [label="EVAL with clause\np(s(X163), X163).\nand substitution\nX163 -> T77\nT74 -> s(T77)\nX156 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 150 [arrowhead = none ];
150 -> 70;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 151 [arrowhead = none ];
151 -> 71;

152 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
70 -> 152 [arrowhead = none ];
152 -> 72;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 153 [arrowhead = none ];
153 -> 73;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
72 -> 154 [arrowhead = none ];
154 -> 74;

155 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX157 -> 0", fontsize=14, style = filled, fillcolor = white];
73 -> 155 [arrowhead = none ];
155 -> 75;

156 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
74 -> 156 [arrowhead = none ];
156 -> 76;

157 [label="INSTANCE with matching:\nT71 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
75 -> 157 [arrowhead = none , style=dashed];
157 -> 59[style=dashed];

158 [label="INSTANCE with matching:\nT48 -> T71\nT52 -> T80", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 158 [arrowhead = none , style=dashed];
158 -> 42[style=dashed];

}

(2) Obligation:

Triples:

less59(s(T77)) :- less59(T77).
less42(s(T71), 0) :- less59(T71).
less42(s(T71), s(T80)) :- less42(T71, T80).
ordered1(.(s(T48), .(T52, T31))) :- less42(T48, T52).
ordered1(.(T16, .(T30, T31))) :- ','(lessc26(T16, T30), ordered1(.(T30, T31))).

Clauses:

orderedc1([]).
orderedc1(.(T3, [])).
orderedc1(.(T16, .(T30, T31))) :- ','(lessc26(T16, T30), orderedc1(.(T30, T31))).
lessc59(s(T77)) :- lessc59(T77).
lessc42(0, s(T62)).
lessc42(s(T71), 0) :- lessc59(T71).
lessc42(s(T71), s(T80)) :- lessc42(T71, T80).
lessc26(0, T39).
lessc26(s(T48), T52) :- lessc42(T48, T52).

Afs:

ordered1(x1)  =  ordered1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ordered1_in: (b)
less42_in: (b,b)
less59_in: (b)
lessc26_in: (b,b)
lessc42_in: (b,b)
lessc59_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

ORDERED1_IN_G(.(s(T48), .(T52, T31))) → U4_G(T48, T52, T31, less42_in_gg(T48, T52))
ORDERED1_IN_G(.(s(T48), .(T52, T31))) → LESS42_IN_GG(T48, T52)
LESS42_IN_GG(s(T71), 0) → U2_GG(T71, less59_in_g(T71))
LESS42_IN_GG(s(T71), 0) → LESS59_IN_G(T71)
LESS59_IN_G(s(T77)) → U1_G(T77, less59_in_g(T77))
LESS59_IN_G(s(T77)) → LESS59_IN_G(T77)
LESS42_IN_GG(s(T71), s(T80)) → U3_GG(T71, T80, less42_in_gg(T71, T80))
LESS42_IN_GG(s(T71), s(T80)) → LESS42_IN_GG(T71, T80)
ORDERED1_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → U6_G(T16, T30, T31, ordered1_in_g(.(T30, T31)))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

ORDERED1_IN_G(.(s(T48), .(T52, T31))) → U4_G(T48, T52, T31, less42_in_gg(T48, T52))
ORDERED1_IN_G(.(s(T48), .(T52, T31))) → LESS42_IN_GG(T48, T52)
LESS42_IN_GG(s(T71), 0) → U2_GG(T71, less59_in_g(T71))
LESS42_IN_GG(s(T71), 0) → LESS59_IN_G(T71)
LESS59_IN_G(s(T77)) → U1_G(T77, less59_in_g(T77))
LESS59_IN_G(s(T77)) → LESS59_IN_G(T77)
LESS42_IN_GG(s(T71), s(T80)) → U3_GG(T71, T80, less42_in_gg(T71, T80))
LESS42_IN_GG(s(T71), s(T80)) → LESS42_IN_GG(T71, T80)
ORDERED1_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → U6_G(T16, T30, T31, ordered1_in_g(.(T30, T31)))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LESS59_IN_G(s(T77)) → LESS59_IN_G(T77)

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LESS59_IN_G(s(T77)) → LESS59_IN_G(T77)

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

(10) PiDPToQDPProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS59_IN_G(s(T77)) → LESS59_IN_G(T77)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS59_IN_G(s(T77)) → LESS59_IN_G(T77)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

LESS42_IN_GG(s(T71), s(T80)) → LESS42_IN_GG(T71, T80)

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LESS42_IN_GG(s(T71), s(T80)) → LESS42_IN_GG(T71, T80)

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

(17) PiDPToQDPProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESS42_IN_GG(s(T71), s(T80)) → LESS42_IN_GG(T71, T80)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• LESS42_IN_GG(s(T71), s(T80)) → LESS42_IN_GG(T71, T80)
  The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

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

(22) PiDPToQDPProof (EQUIVALENT transformation)

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

(23) Obligation:

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc42_in_gg(x0, x1)
lessc59_in_g(x0)
U10_g(x0, x1)
U11_gg(x0, x1)
U12_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(24) MRRProof (EQUIVALENT transformation)

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

Strictly oriented rules of the TRS R:

U11_gg(T71, lessc59_out_g(T71)) → lessc42_out_gg(s(T71), 0)

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 2·x₁ + x₂   
POL(0) = 0   
POL(ORDERED1_IN_G(x₁)) = 2·x₁   
POL(U10_g(x₁, x₂)) = x₁ + x₂   
POL(U11_gg(x₁, x₂)) = x₁ + x₂   
POL(U12_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U13_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U5_G(x₁, x₂, x₃, x₄)) = 2·x₁ + 2·x₂ + 2·x₃ + x₄   
POL(lessc26_in_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc26_out_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc42_in_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc42_out_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc59_in_g(x₁)) = x₁   
POL(lessc59_out_g(x₁)) = 1 + x₁   
POL(s(x₁)) = 2·x₁   

(25) Obligation:

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessc26_in_gg(T16, T30))
U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc42_in_gg(x0, x1)
lessc59_in_g(x0)
U10_g(x0, x1)
U11_gg(x0, x1)
U12_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(26) MRRProof (EQUIVALENT transformation)

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

ORDERED1_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessc26_in_gg(T16, T30))

Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + 2·x₁ + 2·x₂   
POL(0) = 0   
POL(ORDERED1_IN_G(x₁)) = x₁   
POL(U10_g(x₁, x₂)) = x₁ + x₂   
POL(U11_gg(x₁, x₂)) = x₁ + x₂   
POL(U12_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U13_gg(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(U5_G(x₁, x₂, x₃, x₄)) = 1 + x₁ + 2·x₂ + 2·x₃ + x₄   
POL(lessc26_in_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc26_out_gg(x₁, x₂)) = x₁ + 2·x₂   
POL(lessc42_in_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc42_out_gg(x₁, x₂)) = x₁ + x₂   
POL(lessc59_in_g(x₁)) = x₁   
POL(lessc59_out_g(x₁)) = x₁   
POL(s(x₁)) = 2·x₁   

(27) Obligation:

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

U5_G(T16, T30, T31, lessc26_out_gg(T16, T30)) → ORDERED1_IN_G(.(T30, T31))

The TRS R consists of the following rules:

lessc26_in_gg(0, T39) → lessc26_out_gg(0, T39)
lessc26_in_gg(s(T48), T52) → U13_gg(T48, T52, lessc42_in_gg(T48, T52))
lessc42_in_gg(0, s(T62)) → lessc42_out_gg(0, s(T62))
lessc42_in_gg(s(T71), 0) → U11_gg(T71, lessc59_in_g(T71))
lessc59_in_g(s(T77)) → U10_g(T77, lessc59_in_g(T77))
U10_g(T77, lessc59_out_g(T77)) → lessc59_out_g(s(T77))
lessc42_in_gg(s(T71), s(T80)) → U12_gg(T71, T80, lessc42_in_gg(T71, T80))
U12_gg(T71, T80, lessc42_out_gg(T71, T80)) → lessc42_out_gg(s(T71), s(T80))
U13_gg(T48, T52, lessc42_out_gg(T48, T52)) → lessc26_out_gg(s(T48), T52)

The set Q consists of the following terms:

lessc26_in_gg(x0, x1)
lessc42_in_gg(x0, x1)
lessc59_in_g(x0)
U10_g(x0, x1)
U11_gg(x0, x1)
U12_gg(x0, x1, x2)
U13_gg(x0, x1, x2)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE