Runtime Complexity proof of /tmp/tmp5nMnJQ/add3.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), 79 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

    ↳5 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 82 ms)

    ↳6 CdtProblem

    ↳7 SIsEmptyProof (⇔, 0 ms)

    ↳8 BOUNDS(O(1), O(1))

↳9 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 63 ms)

↳10 CdtProblem

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

    ↳12 CdtProblem

    ↳13 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 79 ms)

    ↳14 CdtProblem

(0) Obligation:

Clauses:

add(X, 0, X).
add(X, Y, s(Z)) :- ','(no(zero(Y)), ','(p(Y, P), add(X, P, Z))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X).
failure(b).

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];
2 [label="2: 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];
10 [label="10: add(T1, 0, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14)))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 2, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(call(zero(0)), ','(!_2, failure(a))), ','(p(0, X14), add(T13, X14, T14))) | ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 2, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X14), add(T13, X14, T14)))) | ?_3 | ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 2, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X14), add(T13, X14, T14)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 2, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(','(!_2, failure(a)), ','(p(0, X14), add(T13, X14, T14))) | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 2, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(p(0, X14), add(T13, X14, T14)))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(failure(a), ','(p(0, X14), add(T13, X14, T14))), SCOPE: 5, CLAUSE: 7\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(no(zero(T26)), ','(p(T26, X32), add(T28, X32, T29)))\n\nT26 is ground\nX3 is free\nX32 is free\nadd(T1, T26, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(no(zero(T26)), ','(p(T26, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(T26)), ','(p(T26, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT26 is ground\nX3 is free\nX32 is free\nadd(T1, T26, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(','(call(zero(T34)), ','(!_6, failure(a))), ','(p(T34, X32), add(T28, X32, T29))) | ','(no(zero(T34)), ','(p(T34, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT34 is ground\nX3 is free\nX32 is free\nadd(T1, T34, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(zero(T34), ','(','(!_6, failure(a)), ','(p(T34, X32), add(T28, X32, T29)))) | ?_7 | ','(no(zero(T34)), ','(p(T34, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT34 is ground\nX3 is free\nX32 is free\nadd(T1, T34, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(zero(T34), ','(','(!_6, failure(a)), ','(p(T34, X32), add(T28, X32, T29)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(zero(T34)), ','(p(T34, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT34 is ground\nX3 is free\nX32 is free\nadd(T1, T34, T3) !=? add(X3, 0, X3)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(','(!_6, failure(a)), ','(p(0, X32), add(T28, X32, T29))) | ?_8 | ?_7 | ','(no(zero(0)), ','(p(0, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ?_8 | ?_7 | ','(no(zero(T34)), ','(p(T34, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT34 is ground\nX3 is free\nX32 is free\nadd(T1, T34, T3) !=? add(X3, 0, X3)\nzero(T34) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(failure(a), ','(p(0, X32), add(T28, X32, T29)))\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(failure(a), ','(p(0, X32), add(T28, X32, T29))), SCOPE: 9, CLAUSE: 7\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ?_7 | ','(no(zero(T34)), ','(p(T34, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT34 is ground\nX3 is free\nX32 is free\nadd(T1, T34, T3) !=? add(X3, 0, X3)\nzero(T34) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(zero(T34)), ','(p(T34, X32), add(T28, X32, T29))), SCOPE: 6, CLAUSE: 6\n\nT34 is ground\nX3 is free\nX32 is free\nadd(T1, T34, T3) !=? add(X3, 0, X3)\nzero(T34) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(T41, X32), add(T28, X32, T29))\n\nT41 is ground\nX3 is free\nX32 is free\nadd(T1, T41, T3) !=? add(X3, 0, X3)\nzero(T41) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(T41, X32), add(T28, X32, T29)), SCOPE: 10, CLAUSE: 2 | ','(p(T41, X32), add(T28, X32, T29)), SCOPE: 10, CLAUSE: 3\n\nT41 is ground\nX3 is free\nX32 is free\nadd(T1, T41, T3) !=? add(X3, 0, X3)\nzero(T41) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(p(T41, X32), add(T28, X32, T29)), SCOPE: 10, CLAUSE: 3\n\nT41 is ground\nX3 is free\nX32 is free\nadd(T1, T41, T3) !=? add(X3, 0, X3)\nzero(T41) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: add(T28, T46, T29)\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 67 [arrowhead = none ];
67 -> 3;

68 [label="EVAL with clause\nadd(X3, 0, X3).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 68 [arrowhead = none ];
68 -> 5;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 69 [arrowhead = none ];
69 -> 7;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 70 [arrowhead = none ];
70 -> 10;

71 [label="EVAL with clause\nadd(X29, X30, s(X31)) :- ','(no(zero(X30)), ','(p(X30, X32), add(X29, X32, X31))).\nand substitutionT1 -> T28,\nX29 -> T28,\nT2 -> T26,\nX30 -> T26,\nX31 -> T29,\nT3 -> s(T29),\nT25 -> T28,\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 71 [arrowhead = none ];
71 -> 33;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 72 [arrowhead = none ];
72 -> 35;

73 [label="EVAL with clause\nadd(X11, X12, s(X13)) :- ','(no(zero(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];
10 -> 73 [arrowhead = none ];
73 -> 13;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 74 [arrowhead = none ];
74 -> 15;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 75 [arrowhead = none ];
75 -> 16;

76 [label="ONLY EVAL with clause\nno(X19) :- ','(call(X19), ','(!_2, failure(a))).\nand substitutionX19 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 76 [arrowhead = none ];
76 -> 18;

77 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 77 [arrowhead = none ];
77 -> 20;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 78 [arrowhead = none ];
78 -> 21;

79 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 79 [arrowhead = none ];
79 -> 23;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 80 [arrowhead = none ];
80 -> 25;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 82 [arrowhead = none ];
82 -> 30;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 83 [arrowhead = none ];
83 -> 36;

84 [label="ONLY EVAL with clause\nno(X37) :- ','(call(X37), ','(!_6, failure(a))).\nand substitutionT26 -> T34,\nX37 -> zero(T34)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 84 [arrowhead = none ];
84 -> 40;

85 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 85 [arrowhead = none ];
85 -> 42;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 86 [arrowhead = none ];
86 -> 45;

87 [label="EVAL with clause\nzero(0).\nand substitutionT34 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 87 [arrowhead = none ];
87 -> 46;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 88 [arrowhead = none ];
88 -> 49;

89 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 89 [arrowhead = none ];
89 -> 50;

90 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 90 [arrowhead = none ];
90 -> 55;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
50 -> 91 [arrowhead = none ];
91 -> 52;

92 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 92 [arrowhead = none ];
92 -> 53;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 93 [arrowhead = none ];
93 -> 56;

94 [label="ONLY EVAL with clause\nno(X44).\nand substitutionT34 -> T41,\nX44 -> zero(T41)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 94 [arrowhead = none ];
94 -> 58;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 95 [arrowhead = none ];
95 -> 61;

96 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T41), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 96 [arrowhead = none ];
96 -> 62;

97 [label="EVAL with clause\np(s(X49), X49).\nand substitutionX49 -> T46,\nT41 -> s(T46),\nX32 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 97 [arrowhead = none ];
97 -> 65;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 98 [arrowhead = none ];
98 -> 66;

99 [label="INSTANCE with matching:\nT1 -> T28\nT2 -> T46\nT3 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 99 [arrowhead = none , style=dashed];
99 -> 2[style=dashed];

}

(2) 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

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

Removed 1 trailing tuple parts

(4) 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

(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.

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

(6) 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

(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];
1 [label="1: 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];
11 [label="11: ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(','(call(zero(0)), ','(!_2, failure(a))), ','(p(0, X9), add(T10, X9, T11))) | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X9), add(T10, X9, T11)))) | ?_3 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X9), add(T10, X9, T11)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(','(!_2, failure(a)), ','(p(0, X9), add(T10, X9, T11))) | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 2, CLAUSE: 6\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(failure(a), ','(p(0, X9), add(T10, X9, T11)))\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(failure(a), ','(p(0, X9), add(T10, X9, T11))), SCOPE: 5, CLAUSE: 7\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(no(zero(T18)), ','(p(T18, X21), add(T20, X21, T21)))\n\nT18 is ground\nX2 is free\nX21 is free\nadd(T1, T18, 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: ','(no(zero(T18)), ','(p(T18, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(T18)), ','(p(T18, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT18 is ground\nX2 is free\nX21 is free\nadd(T1, T18, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(','(call(zero(T24)), ','(!_6, failure(a))), ','(p(T24, X21), add(T20, X21, T21))) | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(zero(T24), ','(','(!_6, failure(a)), ','(p(T24, X21), add(T20, X21, T21)))) | ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(zero(T24), ','(','(!_6, failure(a)), ','(p(T24, X21), add(T20, X21, T21)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(','(!_6, failure(a)), ','(p(0, X21), add(T20, X21, T21))) | ?_8 | ?_7 | ','(no(zero(0)), ','(p(0, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ?_8 | ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)\nzero(T24) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(failure(a), ','(p(0, X21), add(T20, X21, T21)))\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(failure(a), ','(p(0, X21), add(T20, X21, T21))), SCOPE: 9, CLAUSE: 7\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ?_7 | ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)\nzero(T24) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(no(zero(T24)), ','(p(T24, X21), add(T20, X21, T21))), SCOPE: 6, CLAUSE: 6\n\nT24 is ground\nX2 is free\nX21 is free\nadd(T1, T24, T3) !=? add(X2, 0, X2)\nzero(T24) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(T31, X21), add(T20, X21, T21))\n\nT31 is ground\nX2 is free\nX21 is free\nadd(T1, T31, T3) !=? add(X2, 0, X2)\nzero(T31) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(p(T31, X21), add(T20, X21, T21)), SCOPE: 10, CLAUSE: 2 | ','(p(T31, X21), add(T20, X21, T21)), SCOPE: 10, CLAUSE: 3\n\nT31 is ground\nX2 is free\nX21 is free\nadd(T1, T31, T3) !=? add(X2, 0, X2)\nzero(T31) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(p(T31, X21), add(T20, X21, T21)), SCOPE: 10, CLAUSE: 3\n\nT31 is ground\nX2 is free\nX21 is free\nadd(T1, T31, T3) !=? add(X2, 0, X2)\nzero(T31) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: add(T20, T36, T21)\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 65 [arrowhead = none ];
65 -> 4;

66 [label="EVAL with clause\nadd(X2, 0, X2).\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> 0,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 66 [arrowhead = none ];
66 -> 6;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 67 [arrowhead = none ];
67 -> 8;

68 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 68 [arrowhead = none ];
68 -> 9;

69 [label="EVAL with clause\nadd(X18, X19, s(X20)) :- ','(no(zero(X19)), ','(p(X19, X21), add(X18, X21, X20))).\nand substitutionT1 -> T20,\nX18 -> T20,\nT2 -> T18,\nX19 -> T18,\nX20 -> T21,\nT3 -> s(T21),\nT17 -> T20,\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 69 [arrowhead = none ];
69 -> 31;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 70 [arrowhead = none ];
70 -> 32;

71 [label="EVAL with clause\nadd(X6, X7, s(X8)) :- ','(no(zero(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 -> 71 [arrowhead = none ];
71 -> 11;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 72 [arrowhead = none ];
72 -> 12;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 73 [arrowhead = none ];
73 -> 14;

74 [label="ONLY EVAL with clause\nno(X12) :- ','(call(X12), ','(!_2, failure(a))).\nand substitutionX12 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 74 [arrowhead = none ];
74 -> 17;

75 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 75 [arrowhead = none ];
75 -> 19;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 76 [arrowhead = none ];
76 -> 22;

77 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 77 [arrowhead = none ];
77 -> 24;

78 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 78 [arrowhead = none ];
78 -> 26;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 79 [arrowhead = none ];
79 -> 27;

80 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 80 [arrowhead = none ];
80 -> 29;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 81 [arrowhead = none ];
81 -> 34;

82 [label="ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_6, failure(a))).\nand substitutionT18 -> T24,\nX24 -> zero(T24)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 82 [arrowhead = none ];
82 -> 37;

83 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 83 [arrowhead = none ];
83 -> 38;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 84 [arrowhead = none ];
84 -> 39;

85 [label="EVAL with clause\nzero(0).\nand substitutionT24 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 85 [arrowhead = none ];
85 -> 41;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 86 [arrowhead = none ];
86 -> 43;

87 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 87 [arrowhead = none ];
87 -> 44;

88 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 88 [arrowhead = none ];
88 -> 51;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 89 [arrowhead = none ];
89 -> 47;

90 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 90 [arrowhead = none ];
90 -> 48;

91 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 91 [arrowhead = none ];
91 -> 54;

92 [label="ONLY EVAL with clause\nno(X31).\nand substitutionT24 -> T31,\nX31 -> zero(T31)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 92 [arrowhead = none ];
92 -> 57;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 93 [arrowhead = none ];
93 -> 59;

94 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T31), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 94 [arrowhead = none ];
94 -> 60;

95 [label="EVAL with clause\np(s(X36), X36).\nand substitutionX36 -> T36,\nT31 -> s(T36),\nX21 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 95 [arrowhead = none ];
95 -> 63;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 96 [arrowhead = none ];
96 -> 64;

97 [label="INSTANCE with matching:\nT1 -> T20\nT2 -> T36\nT3 -> T21", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 97 [arrowhead = none , style=dashed];
97 -> 1[style=dashed];

}

(10) 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

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

Removed 1 trailing tuple parts

(12) 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

(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.

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

(14) 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