Left Termination proof of /tmp/tmp5GpLjK/ordered.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 166 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 51 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇔, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇔, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇒, 0 ms)

        ↳27 QDP

        ↳28 UsableRulesReductionPairsProof (⇔, 159 ms)

        ↳29 QDP

        ↳30 DependencyGraphProof (⇔, 0 ms)

        ↳31 TRUE

(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).

Query: ordered(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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\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(T5, X10), ','(tail(T5, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13)))))))\n\nT5 is ground\nX7 is free\nX10 is free\nX11 is free\nX12 is free\nX13 is free\nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T5, X10), ','(tail(T5, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))), SCOPE: 2, CLAUSE: 3 | ','(head(T5, X10), ','(tail(T5, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX7 is free\nX10 is free\nX11 is free\nX12 is free\nX13 is free\nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(head(T5, X10), ','(tail(T5, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))), SCOPE: 2, CLAUSE: 4\n\nT5 is ground\nX7 is free\nX10 is free\nX11 is free\nX12 is free\nX13 is free\nordered(T5) !=? ordered([])\nordered(T5) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(tail(.(T10, T11), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T10, s(X12)), ordered(.(X12, X13))))))\n\nT10 is ground\nT11 is ground\nX7 is free\nX11 is free\nX12 is free\nX13 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), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T10, s(X12)), ordered(.(X12, X13)))))), SCOPE: 3, CLAUSE: 5 | ','(tail(.(T10, T11), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T10, s(X12)), ordered(.(X12, X13)))))), SCOPE: 3, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nX7 is free\nX11 is free\nX12 is free\nX13 is free\nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(tail(.(T10, T11), X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(T10, s(X12)), ordered(.(X12, X13)))))), SCOPE: 3, CLAUSE: 6\n\nT10 is ground\nT11 is ground\nX7 is free\nX11 is free\nX12 is free\nX13 is free\nordered(.(T10, T11)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(head(T17, X12), ','(tail(T17, X13), ','(less(T16, s(X12)), ordered(.(X12, X13)))))\n\nT16 is ground\nT17 is ground\nX7 is free\nX12 is free\nX13 is free\nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(head(T17, X12), ','(tail(T17, X13), ','(less(T16, s(X12)), ordered(.(X12, X13))))), SCOPE: 4, CLAUSE: 3 | ','(head(T17, X12), ','(tail(T17, X13), ','(less(T16, s(X12)), ordered(.(X12, X13))))), SCOPE: 4, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX7 is free\nX12 is free\nX13 is free\nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(head(T17, X12), ','(tail(T17, X13), ','(less(T16, s(X12)), ordered(.(X12, X13))))), SCOPE: 4, CLAUSE: 4\n\nT16 is ground\nT17 is ground\nX7 is free\nX12 is free\nX13 is free\nordered(.(T16, T17)) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(.(T22, T23), X13), ','(less(T16, s(T22)), ordered(.(T22, X13))))\n\nT16 is ground\nT22 is ground\nT23 is ground\nX13 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), X13), ','(less(T16, s(T22)), ordered(.(T22, X13)))), SCOPE: 5, CLAUSE: 5 | ','(tail(.(T22, T23), X13), ','(less(T16, s(T22)), ordered(.(T22, X13)))), SCOPE: 5, CLAUSE: 6\n\nT16 is ground\nT22 is ground\nT23 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(.(T22, T23), X13), ','(less(T16, s(T22)), ordered(.(T22, X13)))), SCOPE: 5, CLAUSE: 6\n\nT16 is ground\nT22 is ground\nT23 is ground\nX13 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(X63))) | less(0, s(T36)), SCOPE: 6, CLAUSE: 8\n\nT36 is ground\nX63 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\nX62 is free\nless(T16, s(T30)) !=? less(0, X62)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: eq(s(T36), s(X63))\n\nT36 is ground\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: eq(s(T36), s(X63)), SCOPE: 7, CLAUSE: 11\n\nT36 is ground\nX63 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, X78), ','(p(s(T45), X79), less(X78, X79)))\n\nT44 is ground\nT45 is ground\nX62 is free\nX78 is free\nX79 is free\nless(T44, s(T45)) !=? less(0, X62)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T44, X78), ','(p(s(T45), X79), less(X78, X79))), SCOPE: 8, CLAUSE: 9 | ','(p(T44, X78), ','(p(s(T45), X79), less(X78, X79))), SCOPE: 8, CLAUSE: 10\n\nT44 is ground\nT45 is ground\nX62 is free\nX78 is free\nX79 is free\nless(T44, s(T45)) !=? less(0, X62)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T44, X78), ','(p(s(T45), X79), less(X78, X79))), SCOPE: 8, CLAUSE: 10\n\nT44 is ground\nT45 is ground\nX62 is free\nX78 is free\nX79 is free\nless(T44, s(T45)) !=? less(0, X62)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(s(T45), X79), less(T48, X79))\n\nT45 is ground\nT48 is ground\nX79 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), X79), less(T48, X79)), SCOPE: 9, CLAUSE: 9 | ','(p(s(T45), X79), less(T48, X79)), SCOPE: 9, CLAUSE: 10\n\nT45 is ground\nT48 is ground\nX79 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(s(T45), X79), less(T48, X79)), SCOPE: 9, CLAUSE: 10\n\nT45 is ground\nT48 is ground\nX79 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(X100))) | less(0, T56), SCOPE: 10, CLAUSE: 8\n\nT56 is ground\nX100 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\nX99 is free\nless(T48, T52) !=? less(0, X99)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: eq(T56, s(X100))\n\nT56 is ground\nX100 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: eq(T56, s(X100)), SCOPE: 11, CLAUSE: 11\n\nT56 is ground\nX100 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, X115), ','(p(T68, X116), less(X115, X116)))\n\nT67 is ground\nT68 is ground\nX99 is free\nX115 is free\nX116 is free\nless(T67, T68) !=? less(0, X99)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T67, X115), ','(p(T68, X116), less(X115, X116))), SCOPE: 12, CLAUSE: 9 | ','(p(T67, X115), ','(p(T68, X116), less(X115, X116))), SCOPE: 12, CLAUSE: 10\n\nT67 is ground\nT68 is ground\nX99 is free\nX115 is free\nX116 is free\nless(T67, T68) !=? less(0, X99)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T67, X115), ','(p(T68, X116), less(X115, X116))), SCOPE: 12, CLAUSE: 10\n\nT67 is ground\nT68 is ground\nX99 is free\nX115 is free\nX116 is free\nless(T67, T68) !=? less(0, X99)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T68, X116), less(T71, X116))\n\nT68 is ground\nT71 is ground\nX116 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T68, X116), less(T71, X116)), SCOPE: 13, CLAUSE: 9 | ','(p(T68, X116), less(T71, X116)), SCOPE: 13, CLAUSE: 10\n\nT68 is ground\nT71 is ground\nX116 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(T68, X116), less(T71, X116)), SCOPE: 13, CLAUSE: 9\n\nT68 is ground\nT71 is ground\nX116 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(T68, X116), less(T71, X116)), SCOPE: 13, CLAUSE: 10\n\nT68 is ground\nT71 is ground\nX116 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(X132))) | less(0, 0), SCOPE: 14, CLAUSE: 8\n\nX132 is free", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(T71, 0), SCOPE: 14, CLAUSE: 8\n\nT71 is ground\nX131 is free\nless(T71, 0) !=? less(0, X131)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: eq(0, s(X132))\n\nX132 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: eq(0, s(X132)), SCOPE: 15, CLAUSE: 11\n\nX132 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(p(T74, X145), ','(p(0, X146), less(X145, X146)))\n\nT74 is ground\nX131 is free\nX145 is free\nX146 is free\nless(T74, 0) !=? less(0, X131)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(p(T74, X145), ','(p(0, X146), less(X145, X146))), SCOPE: 16, CLAUSE: 9 | ','(p(T74, X145), ','(p(0, X146), less(X145, X146))), SCOPE: 16, CLAUSE: 10\n\nT74 is ground\nX131 is free\nX145 is free\nX146 is free\nless(T74, 0) !=? less(0, X131)", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ','(p(T74, X145), ','(p(0, X146), less(X145, X146))), SCOPE: 16, CLAUSE: 10\n\nT74 is ground\nX131 is free\nX145 is free\nX146 is free\nless(T74, 0) !=? less(0, X131)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: ','(p(0, X146), less(T77, X146))\n\nT77 is ground\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: ','(p(0, X146), less(T77, X146)), SCOPE: 17, CLAUSE: 9 | ','(p(0, X146), less(T77, X146)), SCOPE: 17, CLAUSE: 10\n\nT77 is ground\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(0, X146), less(T77, X146)), SCOPE: 17, CLAUSE: 9\n\nT77 is ground\nX146 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(p(0, X146), less(T77, X146)), SCOPE: 17, CLAUSE: 10\n\nT77 is ground\nX146 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 substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionX7 -> T3,\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 83 [arrowhead = none ];
83 -> 7;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X9) :- ','(head(X9, X10), ','(tail(X9, X11), ','(head(X11, X12), ','(tail(X11, X13), ','(less(X10, s(X12)), ordered(.(X12, X13))))))).\nand substitutionT1 -> T5,\nX9 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
12 -> 90 [arrowhead = none ];
90 -> 13;

91 [label="EVAL with clause\nhead(.(X20, X21), X20).\nand substitutionX20 -> T10,\nX21 -> T11,\nT5 -> .(T10, T11),\nX10 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 91 [arrowhead = none ];
91 -> 14;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
16 -> 94 [arrowhead = none ];
94 -> 17;

95 [label="ONLY EVAL with clause\ntail(.(X32, X33), X33).\nand substitutionT10 -> T16,\nX32 -> T16,\nT11 -> T17,\nX33 -> T17,\nX11 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
19 -> 97 [arrowhead = none ];
97 -> 20;

98 [label="EVAL with clause\nhead(.(X43, X44), X43).\nand substitutionX43 -> T22,\nX44 -> T23,\nT17 -> .(T22, T23),\nX12 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 98 [arrowhead = none ];
98 -> 21;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
23 -> 101 [arrowhead = none ];
101 -> 24;

102 [label="ONLY EVAL with clause\ntail(.(X53, X54), X54).\nand substitutionT22 -> T30,\nX53 -> T30,\nT23 -> T31,\nX54 -> T31,\nX13 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
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\nnew knowledge:\nT16 is ground\nT30 is ground", 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, X62) :- ','(!_6, eq(X62, s(X63))).\nand substitutionT16 -> 0,\nT30 -> T36,\nX62 -> s(T36)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 107 [arrowhead = none ];
107 -> 29;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X76, X77) :- ','(p(X76, X78), ','(p(X77, X79), less(X78, X79))).\nand substitutionT16 -> T44,\nX76 -> T44,\nT30 -> T45,\nX77 -> s(T45)", fontsize=14, style = filled, fillcolor = lightblue];
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(X66, X66).\nand substitutionT36 -> T39,\nX66 -> s(T39),\nX63 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
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, X62))", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 115 [arrowhead = none ];
115 -> 37;

116 [label="EVAL with clause\np(s(X85), X85).\nand substitutionX85 -> T48,\nT44 -> s(T48),\nX78 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 116 [arrowhead = none ];
116 -> 38;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
40 -> 119 [arrowhead = none ];
119 -> 41;

120 [label="ONLY EVAL with clause\np(s(X93), X93).\nand substitutionT45 -> T52,\nX93 -> T52,\nX79 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
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, X99) :- ','(!_10, eq(X99, s(X100))).\nand substitutionT48 -> 0,\nT52 -> T56,\nX99 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 122 [arrowhead = none ];
122 -> 44;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X113, X114) :- ','(p(X113, X115), ','(p(X114, X116), less(X115, X116))).\nand substitutionT48 -> T67,\nX113 -> T67,\nT52 -> T68,\nX114 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
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(X103, X103).\nand substitutionT56 -> s(T62),\nX103 -> s(T62),\nX100 -> 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 = lightgreen];
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, X99))", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 131 [arrowhead = none ];
131 -> 53;

132 [label="EVAL with clause\np(s(X122), X122).\nand substitutionX122 -> T71,\nT67 -> s(T71),\nX115 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 132 [arrowhead = none ];
132 -> 54;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT68 -> 0,\nX116 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 137 [arrowhead = none ];
137 -> 59;

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

139 [label="EVAL with clause\np(s(X165), X165).\nand substitutionX165 -> T80,\nT68 -> s(T80),\nX116 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 139 [arrowhead = none ];
139 -> 77;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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, X131) :- ','(!_14, eq(X131, s(X132))).\nand substitutionT71 -> 0,\nX131 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 142 [arrowhead = none ];
142 -> 62;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X143, X144) :- ','(p(X143, X145), ','(p(X144, X146), less(X145, X146))).\nand substitutionT71 -> T74,\nX143 -> T74,\nX144 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
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, X131))", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 149 [arrowhead = none ];
149 -> 69;

150 [label="EVAL with clause\np(s(X152), X152).\nand substitutionX152 -> T77,\nT74 -> s(T77),\nX145 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 150 [arrowhead = none ];
150 -> 70;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionX146 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 155 [arrowhead = none ];
155 -> 75;

156 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
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:

Clauses:

lessA(s(T77)) :- lessA(T77).
lessB(0, s(T62)).
lessB(s(T71), 0) :- lessA(T71).
lessB(s(T71), s(T80)) :- lessB(T71, T80).
lessC(0, T39).
lessC(s(T48), T52) :- lessB(T48, T52).
orderedD([]).
orderedD(.(T3, [])).
orderedD(.(T16, .(T30, T31))) :- lessC(T16, T30).
orderedD(.(T16, .(T30, T31))) :- ','(lessC(T16, T30), orderedD(.(T30, T31))).

Query: orderedD(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
orderedD_in: (b)
lessC_in: (b,b)
lessB_in: (b,b)
lessA_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

(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:

ORDEREDD_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessC_in_gg(T16, T30))
ORDEREDD_IN_G(.(T16, .(T30, T31))) → LESSC_IN_GG(T16, T30)
LESSC_IN_GG(s(T48), T52) → U4_GG(T48, T52, lessB_in_gg(T48, T52))
LESSC_IN_GG(s(T48), T52) → LESSB_IN_GG(T48, T52)
LESSB_IN_GG(s(T71), 0) → U2_GG(T71, lessA_in_g(T71))
LESSB_IN_GG(s(T71), 0) → LESSA_IN_G(T71)
LESSA_IN_G(s(T77)) → U1_G(T77, lessA_in_g(T77))
LESSA_IN_G(s(T77)) → LESSA_IN_G(T77)
LESSB_IN_GG(s(T71), s(T80)) → U3_GG(T71, T80, lessB_in_gg(T71, T80))
LESSB_IN_GG(s(T71), s(T80)) → LESSB_IN_GG(T71, T80)
U5_G(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_G(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U5_G(T16, T30, T31, lessC_out_gg(T16, T30)) → ORDEREDD_IN_G(.(T30, T31))

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

The argument filtering Pi contains the following mapping:
orderedD_in_g(x1) = orderedD_in_g(x1)
[] = []
orderedD_out_g(x1) = orderedD_out_g
.(x1, x2) = .(x1, x2)
U5_g(x1, x2, x3, x4) = U5_g(x2, x3, x4)
lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2)
0 = 0
lessC_out_gg(x1, x2) = lessC_out_gg
s(x1) = s(x1)
U4_gg(x1, x2, x3) = U4_gg(x3)
lessB_in_gg(x1, x2) = lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2) = lessB_out_gg
U2_gg(x1, x2) = U2_gg(x2)
lessA_in_g(x1) = lessA_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
lessA_out_g(x1) = lessA_out_g
U3_gg(x1, x2, x3) = U3_gg(x3)
U6_g(x1, x2, x3, x4) = U6_g(x4)
ORDEREDD_IN_G(x1) = ORDEREDD_IN_G(x1)
U5_G(x1, x2, x3, x4) = U5_G(x2, x3, x4)
LESSC_IN_GG(x1, x2) = LESSC_IN_GG(x1, x2)
U4_GG(x1, x2, x3) = U4_GG(x3)
LESSB_IN_GG(x1, x2) = LESSB_IN_GG(x1, x2)
U2_GG(x1, x2) = U2_GG(x2)
LESSA_IN_G(x1) = LESSA_IN_G(x1)
U1_G(x1, x2) = U1_G(x2)
U3_GG(x1, x2, x3) = U3_GG(x3)
U6_G(x1, x2, x3, x4) = U6_G(x4)

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

(6) Obligation:

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

ORDEREDD_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessC_in_gg(T16, T30))
ORDEREDD_IN_G(.(T16, .(T30, T31))) → LESSC_IN_GG(T16, T30)
LESSC_IN_GG(s(T48), T52) → U4_GG(T48, T52, lessB_in_gg(T48, T52))
LESSC_IN_GG(s(T48), T52) → LESSB_IN_GG(T48, T52)
LESSB_IN_GG(s(T71), 0) → U2_GG(T71, lessA_in_g(T71))
LESSB_IN_GG(s(T71), 0) → LESSA_IN_G(T71)
LESSA_IN_G(s(T77)) → U1_G(T77, lessA_in_g(T77))
LESSA_IN_G(s(T77)) → LESSA_IN_G(T77)
LESSB_IN_GG(s(T71), s(T80)) → U3_GG(T71, T80, lessB_in_gg(T71, T80))
LESSB_IN_GG(s(T71), s(T80)) → LESSB_IN_GG(T71, T80)
U5_G(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_G(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U5_G(T16, T30, T31, lessC_out_gg(T16, T30)) → ORDEREDD_IN_G(.(T30, T31))

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSA_IN_G(s(T77)) → LESSA_IN_G(T77)

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

(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:

LESSA_IN_G(s(T77)) → LESSA_IN_G(T77)

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

(12) PiDPToQDPProof (EQUIVALENT 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:

LESSA_IN_G(s(T77)) → LESSA_IN_G(T77)

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:

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

(15) YES

(16) Obligation:

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

LESSB_IN_GG(s(T71), s(T80)) → LESSB_IN_GG(T71, T80)

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

(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:

LESSB_IN_GG(s(T71), s(T80)) → LESSB_IN_GG(T71, T80)

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:

LESSB_IN_GG(s(T71), s(T80)) → LESSB_IN_GG(T71, T80)

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:

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

(22) YES

(23) Obligation:

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

U5_G(T16, T30, T31, lessC_out_gg(T16, T30)) → ORDEREDD_IN_G(.(T30, T31))
ORDEREDD_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessC_in_gg(T16, T30))

The TRS R consists of the following rules:

orderedD_in_g([]) → orderedD_out_g([])
orderedD_in_g(.(T3, [])) → orderedD_out_g(.(T3, []))
orderedD_in_g(.(T16, .(T30, T31))) → U5_g(T16, T30, T31, lessC_in_gg(T16, T30))
lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → orderedD_out_g(.(T16, .(T30, T31)))
U5_g(T16, T30, T31, lessC_out_gg(T16, T30)) → U6_g(T16, T30, T31, orderedD_in_g(.(T30, T31)))
U6_g(T16, T30, T31, orderedD_out_g(.(T30, T31))) → orderedD_out_g(.(T16, .(T30, T31)))

(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:

U5_G(T16, T30, T31, lessC_out_gg(T16, T30)) → ORDEREDD_IN_G(.(T30, T31))
ORDEREDD_IN_G(.(T16, .(T30, T31))) → U5_G(T16, T30, T31, lessC_in_gg(T16, T30))

The TRS R consists of the following rules:

lessC_in_gg(0, T39) → lessC_out_gg(0, T39)
lessC_in_gg(s(T48), T52) → U4_gg(T48, T52, lessB_in_gg(T48, T52))
U4_gg(T48, T52, lessB_out_gg(T48, T52)) → lessC_out_gg(s(T48), T52)
lessB_in_gg(0, s(T62)) → lessB_out_gg(0, s(T62))
lessB_in_gg(s(T71), 0) → U2_gg(T71, lessA_in_g(T71))
lessB_in_gg(s(T71), s(T80)) → U3_gg(T71, T80, lessB_in_gg(T71, T80))
U2_gg(T71, lessA_out_g(T71)) → lessB_out_gg(s(T71), 0)
U3_gg(T71, T80, lessB_out_gg(T71, T80)) → lessB_out_gg(s(T71), s(T80))
lessA_in_g(s(T77)) → U1_g(T77, lessA_in_g(T77))
U1_g(T77, lessA_out_g(T77)) → lessA_out_g(s(T77))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
lessC_in_gg(x1, x2) = lessC_in_gg(x1, x2)
0 = 0
lessC_out_gg(x1, x2) = lessC_out_gg
s(x1) = s(x1)
U4_gg(x1, x2, x3) = U4_gg(x3)
lessB_in_gg(x1, x2) = lessB_in_gg(x1, x2)
lessB_out_gg(x1, x2) = lessB_out_gg
U2_gg(x1, x2) = U2_gg(x2)
lessA_in_g(x1) = lessA_in_g(x1)
U1_g(x1, x2) = U1_g(x2)
lessA_out_g(x1) = lessA_out_g
U3_gg(x1, x2, x3) = U3_gg(x3)
ORDEREDD_IN_G(x1) = ORDEREDD_IN_G(x1)
U5_G(x1, x2, x3, x4) = U5_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:

U5_G(T30, T31, lessC_out_gg) → ORDEREDD_IN_G(.(T30, T31))
ORDEREDD_IN_G(.(T16, .(T30, T31))) → U5_G(T30, T31, lessC_in_gg(T16, T30))

The TRS R consists of the following rules:

lessC_in_gg(0, T39) → lessC_out_gg
lessC_in_gg(s(T48), T52) → U4_gg(lessB_in_gg(T48, T52))
U4_gg(lessB_out_gg) → lessC_out_gg
lessB_in_gg(0, s(T62)) → lessB_out_gg
lessB_in_gg(s(T71), 0) → U2_gg(lessA_in_g(T71))
lessB_in_gg(s(T71), s(T80)) → U3_gg(lessB_in_gg(T71, T80))
U2_gg(lessA_out_g) → lessB_out_gg
U3_gg(lessB_out_gg) → lessB_out_gg
lessA_in_g(s(T77)) → U1_g(lessA_in_g(T77))
U1_g(lessA_out_g) → lessA_out_g

The set Q consists of the following terms:

lessC_in_gg(x0, x1)
U4_gg(x0)
lessB_in_gg(x0, x1)
U2_gg(x0)
U3_gg(x0)
lessA_in_g(x0)
U1_g(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:

ORDEREDD_IN_G(.(T16, .(T30, T31))) → U5_G(T30, T31, lessC_in_gg(T16, T30))

The following rules are removed from R:

lessC_in_gg(0, T39) → lessC_out_gg
lessC_in_gg(s(T48), T52) → U4_gg(lessB_in_gg(T48, T52))
U4_gg(lessB_out_gg) → lessC_out_gg
lessB_in_gg(0, s(T62)) → lessB_out_gg
lessB_in_gg(s(T71), 0) → U2_gg(lessA_in_g(T71))
lessB_in_gg(s(T71), s(T80)) → U3_gg(lessB_in_gg(T71, T80))
U2_gg(lessA_out_g) → lessB_out_gg
U3_gg(lessB_out_gg) → lessB_out_gg
lessA_in_g(s(T77)) → U1_g(lessA_in_g(T77))

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + 2·x₂
POL(0) = 0
POL(ORDEREDD_IN_G(x₁)) = x₁
POL(U1_g(x₁)) = x₁
POL(U2_gg(x₁)) = 1 + 2·x₁
POL(U3_gg(x₁)) = 2 + 2·x₁
POL(U4_gg(x₁)) = 1 + x₁
POL(U5_G(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + x₃
POL(lessA_in_g(x₁)) = 1 + x₁
POL(lessA_out_g) = 2
POL(lessB_in_gg(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(lessB_out_gg) = 1
POL(lessC_in_gg(x₁, x₂)) = 1 + x₁ + x₂
POL(lessC_out_gg) = 0
POL(s(x₁)) = 1 + 2·x₁

(29) Obligation:

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

U5_G(T30, T31, lessC_out_gg) → ORDEREDD_IN_G(.(T30, T31))

The TRS R consists of the following rules:

U1_g(lessA_out_g) → lessA_out_g

The set Q consists of the following terms:

lessC_in_gg(x0, x1)
U4_gg(x0)
lessB_in_gg(x0, x1)
U2_gg(x0)
U3_gg(x0)
lessA_in_g(x0)
U1_g(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.

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) UsableRulesReductionPairsProof (EQUIVALENT transformation)

(29) Obligation:

(30) DependencyGraphProof (EQUIVALENT transformation)

(31) TRUE