Runtime Complexity proof of /tmp/tmpy8Qc5O/addneg.pl

Runtime Complexity of the query pattern
add(a,g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

0 Prolog

↳1 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 62 ms)

↳2 CdtProblem

    ↳3 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 91 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 62 ms)

↳10 CdtProblem

    ↳11 CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 90 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

add(X, 0, X).
add(X, Y, s(Z)) :- ','(\+(isZero(Y)), ','(p(Y, P), add(X, P, Z))).
p(0, 0).
p(s(X), X).
isZero(0).

Query: add(a,g,a)

(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: add(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: add(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX3 is free\nadd(T1, T2, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(\\+(isZero(0)), ','(p(0, X14), add(T13, X14, T14)))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(call(isZero(0)), ','(!_2, fail)) | ','(p(0, X14), add(T13, X14, T14))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(isZero(0), ','(!_2, fail)) | ?_3 | ','(p(0, X14), add(T13, X14, T14))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(isZero(0), ','(!_2, fail)), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(p(0, X14), add(T13, X14, T14))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_2, fail) | ?_4 | ?_3 | ','(p(0, X14), add(T13, X14, T14))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(\\+(isZero(T22)), ','(p(T22, X25), add(T24, X25, T25)))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(call(isZero(T22)), ','(!_5, fail)) | ','(p(T22, X25), add(T24, X25, T25))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(isZero(T22), ','(!_5, fail)) | ?_6 | ','(p(T22, X25), add(T24, X25, T25))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(isZero(T22), ','(!_5, fail)), SCOPE: 7, CLAUSE: 4 | ?_7 | ?_6 | ','(p(T22, X25), add(T24, X25, T25))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(!_5, fail) | ?_7 | ?_6 | ','(p(0, X25), add(T24, X25, T25))\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ?_7 | ?_6 | ','(p(T22, X25), add(T24, X25, T25))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)\nisZero(T22) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ?_6 | ','(p(T22, X25), add(T24, X25, T25))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)\nisZero(T22) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(p(T22, X25), add(T24, X25, T25))\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)\nisZero(T22) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(T22, X25), add(T24, X25, T25)), SCOPE: 8, CLAUSE: 2 | ','(p(T22, X25), add(T24, X25, T25)), SCOPE: 8, CLAUSE: 3\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)\nisZero(T22) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T22, X25), add(T24, X25, T25)), SCOPE: 8, CLAUSE: 3\n\nT22 is ground\nX3 is free\nX25 is free\nadd(T1, T22, T3) !=? add(X3, 0, X3)\nisZero(T22) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: add(T24, T30, T25)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 57 [arrowhead = none ];
57 -> 3;

58 [label="EVAL with clause\nadd(X3, 0, X3).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 58 [arrowhead = none ];
58 -> 5;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 60 [arrowhead = none ];
60 -> 12;

61 [label="EVAL with clause\nadd(X22, X23, s(X24)) :- ','(\\+(isZero(X23)), ','(p(X23, X25), add(X22, X25, X24))).\nand substitutionT1 -> T24,\nX22 -> T24,\nT2 -> T22,\nX23 -> T22,\nX24 -> T25,\nT3 -> s(T25),\nT21 -> T24,\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 61 [arrowhead = none ];
61 -> 27;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 62 [arrowhead = none ];
62 -> 28;

63 [label="EVAL with clause\nadd(X11, X12, s(X13)) :- ','(\\+(isZero(X12)), ','(p(X12, X14), add(X11, X14, X13))).\nand substitutionT1 -> T13,\nX11 -> T13,\nX12 -> 0,\nX13 -> T14,\nT3 -> s(T14),\nT11 -> T13,\nT12 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 63 [arrowhead = none ];
63 -> 16;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 64 [arrowhead = none ];
64 -> 17;

65 [label="NOT", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 65 [arrowhead = none ];
65 -> 19;

66 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 66 [arrowhead = none ];
66 -> 20;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="ONLY EVAL with clause\nisZero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 70 [arrowhead = none ];
70 -> 26;

71 [label="NOT", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 71 [arrowhead = none ];
71 -> 29;

72 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 72 [arrowhead = none ];
72 -> 31;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 73 [arrowhead = none ];
73 -> 33;

74 [label="EVAL with clause\nisZero(0).\nand substitutionT22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 74 [arrowhead = none ];
74 -> 35;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 75 [arrowhead = none ];
75 -> 37;

76 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 76 [arrowhead = none ];
76 -> 43;

77 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 77 [arrowhead = none ];
77 -> 46;

78 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 78 [arrowhead = none ];
78 -> 44;

79 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 79 [arrowhead = none ];
79 -> 48;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 80 [arrowhead = none ];
80 -> 51;

81 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (isZero(T22), isZero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 81 [arrowhead = none ];
81 -> 54;

82 [label="EVAL with clause\np(s(X30), X30).\nand substitutionX30 -> T30,\nT22 -> s(T30),\nX25 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 82 [arrowhead = none ];
82 -> 55;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 83 [arrowhead = none ];
83 -> 56;

84 [label="INSTANCE with matching:\nT1 -> T24\nT2 -> T30\nT3 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 84 [arrowhead = none , style=dashed];
84 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
U1(f1_out1, s(z0)) → f1_out1

Tuples:

F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(3) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
U1(f1_out1, s(z0)) → f1_out1

Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

S tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(5) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F1_IN(s(z0)) → c1(F1_IN(z0))

We considered the (Usable) Rules:none
And the Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F1_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
U1(f1_out1, s(z0)) → f1_out1

Tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

S tuples:none
K tuples:

F1_IN(s(z0)) → c1(F1_IN(z0))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c1

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))

(9) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: add(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: add(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: add(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX2 is free\nadd(T1, T2, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(\\+(isZero(0)), ','(p(0, X9), add(T10, X9, T11)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(call(isZero(0)), ','(!_2, fail)) | ','(p(0, X9), add(T10, X9, T11))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(isZero(0), ','(!_2, fail)) | ?_3 | ','(p(0, X9), add(T10, X9, T11))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(isZero(0), ','(!_2, fail)), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(p(0, X9), add(T10, X9, T11))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(!_2, fail) | ?_4 | ?_3 | ','(p(0, X9), add(T10, X9, T11))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(\\+(isZero(T16)), ','(p(T16, X16), add(T18, X16, T19)))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(call(isZero(T16)), ','(!_5, fail)) | ','(p(T16, X16), add(T18, X16, T19))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(isZero(T16), ','(!_5, fail)) | ?_6 | ','(p(T16, X16), add(T18, X16, T19))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(isZero(T16), ','(!_5, fail)), SCOPE: 7, CLAUSE: 4 | ?_7 | ?_6 | ','(p(T16, X16), add(T18, X16, T19))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(!_5, fail) | ?_7 | ?_6 | ','(p(0, X16), add(T18, X16, T19))\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ?_7 | ?_6 | ','(p(T16, X16), add(T18, X16, T19))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)\nisZero(T16) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_6 | ','(p(T16, X16), add(T18, X16, T19))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)\nisZero(T16) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(p(T16, X16), add(T18, X16, T19))\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)\nisZero(T16) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(T16, X16), add(T18, X16, T19)), SCOPE: 8, CLAUSE: 2 | ','(p(T16, X16), add(T18, X16, T19)), SCOPE: 8, CLAUSE: 3\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)\nisZero(T16) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(p(T16, X16), add(T18, X16, T19)), SCOPE: 8, CLAUSE: 3\n\nT16 is ground\nX2 is free\nX16 is free\nadd(T1, T16, T3) !=? add(X2, 0, X2)\nisZero(T16) !=? isZero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: add(T18, T24, T19)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="EVAL with clause\nadd(X2, 0, X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> 0,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 6;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 56 [arrowhead = none ];
56 -> 8;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 57 [arrowhead = none ];
57 -> 9;

58 [label="EVAL with clause\nadd(X13, X14, s(X15)) :- ','(\\+(isZero(X14)), ','(p(X14, X16), add(X13, X16, X15))).\nand substitutionT1 -> T18,\nX13 -> T18,\nT2 -> T16,\nX14 -> T16,\nX15 -> T19,\nT3 -> s(T19),\nT15 -> T18,\nT17 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 58 [arrowhead = none ];
58 -> 30;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 59 [arrowhead = none ];
59 -> 32;

60 [label="EVAL with clause\nadd(X6, X7, s(X8)) :- ','(\\+(isZero(X7)), ','(p(X7, X9), add(X6, X9, X8))).\nand substitutionT1 -> T10,\nX6 -> T10,\nX7 -> 0,\nX8 -> T11,\nT3 -> s(T11),\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="NOT", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 64 [arrowhead = none ];
64 -> 15;

65 [label="ONLY EVAL with clause\nisZero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 65 [arrowhead = none ];
65 -> 18;

66 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 66 [arrowhead = none ];
66 -> 21;

67 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 67 [arrowhead = none ];
67 -> 23;

68 [label="NOT", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 68 [arrowhead = none ];
68 -> 34;

69 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 69 [arrowhead = none ];
69 -> 36;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 70 [arrowhead = none ];
70 -> 38;

71 [label="EVAL with clause\nisZero(0).\nand substitutionT16 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 71 [arrowhead = none ];
71 -> 39;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 72 [arrowhead = none ];
72 -> 40;

73 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 73 [arrowhead = none ];
73 -> 41;

74 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 74 [arrowhead = none ];
74 -> 45;

75 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 75 [arrowhead = none ];
75 -> 42;

76 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 76 [arrowhead = none ];
76 -> 47;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 77 [arrowhead = none ];
77 -> 49;

78 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (isZero(T16), isZero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 78 [arrowhead = none ];
78 -> 50;

79 [label="EVAL with clause\np(s(X21), X21).\nand substitutionX21 -> T24,\nT16 -> s(T24),\nX16 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 79 [arrowhead = none ];
79 -> 52;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 80 [arrowhead = none ];
80 -> 53;

81 [label="INSTANCE with matching:\nT1 -> T18\nT2 -> T24\nT3 -> T19", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 81 [arrowhead = none , style=dashed];
81 -> 2[style=dashed];

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f2_in(z0), s(z0))
U1(f2_out1, s(z0)) → f2_out1

Tuples:

F2_IN(s(z0)) → c1(U1'(f2_in(z0), s(z0)), F2_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(U1'(f2_in(z0), s(z0)), F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(11) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f2_in(z0), s(z0))
U1(f2_out1, s(z0)) → f2_out1

Tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

S tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1

(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F2_IN(s(z0)) → c1(F2_IN(z0))

We considered the (Usable) Rules:none
And the Tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(F2_IN(x₁)) = x₁
POL(c1(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(z0)) → U1(f2_in(z0), s(z0))
U1(f2_out1, s(z0)) → f2_out1

Tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

S tuples:none
K tuples:

F2_IN(s(z0)) → c1(F2_IN(z0))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c1