Runtime Complexity of the query pattern
goal()
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(1)).

(0) Obligation:

Clauses:

goal :- ','(lte(X, s(s(s(s(0))))), even(X)).
lte(0, Y).
lte(X, s(Y)) :- ','(no(zero(X)), ','(p(X, P), lte(P, Y))).
even(0).
even(s(s(X))) :- even(X).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: goal()

(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: goal\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: goal, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(lte(X4, s(s(s(s(0))))), even(X4))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: lte(X4, s(s(s(s(0)))))\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: even(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: lte(X4, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 1 | lte(X4, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 2\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | lte(X4, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 2\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: lte(X4, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 2\n\nX4 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0))))))\n\nX21 is free\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0)))))), SCOPE: 3, CLAUSE: 8 | ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0)))))), SCOPE: 3, CLAUSE: 9\n\nX21 is free\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(','(call(zero(X28)), ','(!_3, failure(a))), ','(p(X28, X20), lte(X20, s(s(s(0)))))) | ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0)))))), SCOPE: 3, CLAUSE: 9\n\nX21 is free\nX20 is free\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(zero(X28), ','(','(!_3, failure(a)), ','(p(X28, X20), lte(X20, s(s(s(0))))))) | ?_4 | ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0)))))), SCOPE: 3, CLAUSE: 9\n\nX21 is free\nX20 is free\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(zero(X28), ','(','(!_3, failure(a)), ','(p(X28, X20), lte(X20, s(s(s(0))))))), SCOPE: 5, CLAUSE: 7 | ?_5 | ?_4 | ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0)))))), SCOPE: 3, CLAUSE: 9\n\nX21 is free\nX20 is free\nX28 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(','(!_3, failure(a)), ','(p(0, X20), lte(X20, s(s(s(0)))))) | ?_5 | ?_4 | ','(no(zero(X21)), ','(p(X21, X20), lte(X20, s(s(s(0)))))), SCOPE: 3, CLAUSE: 9\n\nX21 is free\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(failure(a), ','(p(0, X20), lte(X20, s(s(s(0))))))\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(failure(a), ','(p(0, X20), lte(X20, s(s(s(0)))))), SCOPE: 6, CLAUSE: 10\n\nX20 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(T1), SCOPE: 7, CLAUSE: 3 | even(T1), SCOPE: 7, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true | even(T1), SCOPE: 7, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: even(T1), SCOPE: 7, CLAUSE: 4\n\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: even(T1), SCOPE: 7, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: even(T5)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: even(T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 46 [arrowhead = none ];
46 -> 4;

47 [label="ONLY EVAL with clause\ngoal :- ','(lte(X4, s(s(s(s(0))))), even(X4)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 47 [arrowhead = none ];
47 -> 7;

48 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 48 [arrowhead = none ];
48 -> 15;

49 [label="SPLIT 2\nreplacements:X4 -> T1", fontsize=14, style = filled, fillcolor = violet];
7 -> 49 [arrowhead = none ];
49 -> 16;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 50 [arrowhead = none ];
50 -> 17;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 51 [arrowhead = none ];
51 -> 38;

52 [label="ONLY EVAL with clause\nlte(0, X9).\nand substitutionX4 -> 0,\nX9 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 52 [arrowhead = none ];
52 -> 19;

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 53 [arrowhead = none ];
53 -> 20;

54 [label="ONLY EVAL with clause\nlte(X18, s(X19)) :- ','(no(zero(X18)), ','(p(X18, X20), lte(X20, X19))).\nand substitutionX4 -> X21,\nX18 -> X21,\nX19 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 54 [arrowhead = none ];
54 -> 24;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 55 [arrowhead = none ];
55 -> 25;

56 [label="ONLY EVAL with clause\nno(X27) :- ','(call(X27), ','(!_3, failure(a))).\nand substitutionX21 -> X28,\nX27 -> zero(X28)", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 56 [arrowhead = none ];
56 -> 29;

57 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 57 [arrowhead = none ];
57 -> 30;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 58 [arrowhead = none ];
58 -> 31;

59 [label="ONLY EVAL with clause\nzero(0).\nand substitutionX28 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 59 [arrowhead = none ];
59 -> 34;

60 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 60 [arrowhead = none ];
60 -> 35;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 61 [arrowhead = none ];
61 -> 36;

62 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 62 [arrowhead = none ];
62 -> 37;

63 [label="EVAL with clause\neven(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 63 [arrowhead = none ];
63 -> 39;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 64 [arrowhead = none ];
64 -> 40;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 65 [arrowhead = none ];
65 -> 41;

66 [label="EVAL with clause\neven(s(s(X35))) :- even(X35).\nand substitutionX35 -> T9,\nT1 -> s(s(T9)),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 66 [arrowhead = none ];
66 -> 44;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 67 [arrowhead = none ];
67 -> 45;

68 [label="EVAL with clause\neven(s(s(X32))) :- even(X32).\nand substitutionX32 -> T5,\nT1 -> s(s(T5)),\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 68 [arrowhead = none ];
68 -> 42;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 69 [arrowhead = none ];
69 -> 43;

70 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 70 [arrowhead = none , style=dashed];
70 -> 16[style=dashed];

71 [label="INSTANCE with matching:\nT1 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 71 [arrowhead = none , style=dashed];
71 -> 16[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f7_in)
U1(f7_out1(z0)) → f2_out1
f16_in → f16_out1(0)
f16_in → U2(f16_in)
f16_in → U3(f16_in)
U2(f16_out1(z0)) → f16_out1(s(s(z0)))
U3(f16_out1(z0)) → f16_out1(s(s(z0)))
f15_in → f15_out1
f7_in → U4(f15_in)
U4(f15_out1) → U5(f16_in)
U5(f16_out1(z0)) → f7_out1(z0)

Tuples:

F2_IN → c(U1'(f7_in), F7_IN)
F16_IN → c3(U2'(f16_in), F16_IN)
F16_IN → c4(U3'(f16_in), F16_IN)
F7_IN → c8(U4'(f15_in), F15_IN)
U4'(f15_out1) → c9(U5'(f16_in), F16_IN)

S tuples:

F2_IN → c(U1'(f7_in), F7_IN)
F16_IN → c3(U2'(f16_in), F16_IN)
F16_IN → c4(U3'(f16_in), F16_IN)
F7_IN → c8(U4'(f15_in), F15_IN)
U4'(f15_out1) → c9(U5'(f16_in), F16_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f16_in, U2, U3, f15_in, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F16_IN, F7_IN, U4'

Compound Symbols:

c, c3, c4, c8, c9

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f7_in)
U1(f7_out1(z0)) → f2_out1
f16_in → f16_out1(0)
f16_in → U2(f16_in)
f16_in → U3(f16_in)
U2(f16_out1(z0)) → f16_out1(s(s(z0)))
U3(f16_out1(z0)) → f16_out1(s(s(z0)))
f15_in → f15_out1
f7_in → U4(f15_in)
U4(f15_out1) → U5(f16_in)
U5(f16_out1(z0)) → f7_out1(z0)

Tuples:

F16_IN → c3(U2'(f16_in), F16_IN)
F16_IN → c4(U3'(f16_in), F16_IN)
F2_IN → c1(U1'(f7_in))
F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
F7_IN → c1(F15_IN)
U4'(f15_out1) → c1(U5'(f16_in))
U4'(f15_out1) → c1(F16_IN)

S tuples:

F16_IN → c3(U2'(f16_in), F16_IN)
F16_IN → c4(U3'(f16_in), F16_IN)
F2_IN → c1(U1'(f7_in))
F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
F7_IN → c1(F15_IN)
U4'(f15_out1) → c1(U5'(f16_in))
U4'(f15_out1) → c1(F16_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f16_in, U2, U3, f15_in, f7_in, U4, U5

Defined Pair Symbols:

F16_IN, F2_IN, F7_IN, U4'

Compound Symbols:

c3, c4, c1

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

Removed 5 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f7_in)
U1(f7_out1(z0)) → f2_out1
f16_in → f16_out1(0)
f16_in → U2(f16_in)
f16_in → U3(f16_in)
U2(f16_out1(z0)) → f16_out1(s(s(z0)))
U3(f16_out1(z0)) → f16_out1(s(s(z0)))
f15_in → f15_out1
f7_in → U4(f15_in)
U4(f15_out1) → U5(f16_in)
U5(f16_out1(z0)) → f7_out1(z0)

Tuples:

F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
U4'(f15_out1) → c1(F16_IN)
F16_IN → c3(F16_IN)
F16_IN → c4(F16_IN)
F2_IN → c1
F7_IN → c1
U4'(f15_out1) → c1

S tuples:

F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
U4'(f15_out1) → c1(F16_IN)
F16_IN → c3(F16_IN)
F16_IN → c4(F16_IN)
F2_IN → c1
F7_IN → c1
U4'(f15_out1) → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f16_in, U2, U3, f15_in, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F7_IN, U4', F16_IN

Compound Symbols:

c1, c3, c4, c1

(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
U4'(f15_out1) → c1(F16_IN)
F2_IN → c1
F7_IN → c1
U4'(f15_out1) → c1
F7_IN → c1(U4'(f15_in))
F7_IN → c1
U4'(f15_out1) → c1(F16_IN)
U4'(f15_out1) → c1

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f7_in)
U1(f7_out1(z0)) → f2_out1
f16_in → f16_out1(0)
f16_in → U2(f16_in)
f16_in → U3(f16_in)
U2(f16_out1(z0)) → f16_out1(s(s(z0)))
U3(f16_out1(z0)) → f16_out1(s(s(z0)))
f15_in → f15_out1
f7_in → U4(f15_in)
U4(f15_out1) → U5(f16_in)
U5(f16_out1(z0)) → f7_out1(z0)

Tuples:

F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
U4'(f15_out1) → c1(F16_IN)
F16_IN → c3(F16_IN)
F16_IN → c4(F16_IN)
F2_IN → c1
F7_IN → c1
U4'(f15_out1) → c1

S tuples:

F16_IN → c3(F16_IN)
F16_IN → c4(F16_IN)

K tuples:

F2_IN → c1(F7_IN)
F7_IN → c1(U4'(f15_in))
U4'(f15_out1) → c1(F16_IN)
F2_IN → c1
F7_IN → c1
U4'(f15_out1) → c1

Defined Rule Symbols:

f2_in, U1, f16_in, U2, U3, f15_in, f7_in, U4, U5

Defined Pair Symbols:

F2_IN, F7_IN, U4', F16_IN

Compound Symbols:

c1, c3, c4, c1

(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: goal\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: goal, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(lte(X2, s(s(s(s(0))))), even(X2))\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 1 | ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 2\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: even(0) | ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 2\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 2\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: even(0), SCOPE: 3, CLAUSE: 3 | even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true | even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18))\n\nX18 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18)), SCOPE: 4, CLAUSE: 8 | ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18)), SCOPE: 4, CLAUSE: 9\n\nX18 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(call(zero(X25)), ','(!_4, failure(a))), ','(','(p(X25, X17), lte(X17, s(s(s(0))))), even(X25))) | ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18)), SCOPE: 4, CLAUSE: 9\n\nX18 is free\nX17 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(zero(X25), ','(','(!_4, failure(a)), ','(','(p(X25, X17), lte(X17, s(s(s(0))))), even(X25)))) | ?_5 | ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18)), SCOPE: 4, CLAUSE: 9\n\nX18 is free\nX17 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(zero(X25), ','(','(!_4, failure(a)), ','(','(p(X25, X17), lte(X17, s(s(s(0))))), even(X25)))), SCOPE: 6, CLAUSE: 7 | ?_6 | ?_5 | ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18)), SCOPE: 4, CLAUSE: 9\n\nX18 is free\nX17 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(','(!_4, failure(a)), ','(','(p(0, X17), lte(X17, s(s(s(0))))), even(0))) | ?_6 | ?_5 | ','(','(no(zero(X18)), ','(p(X18, X17), lte(X17, s(s(s(0)))))), even(X18)), SCOPE: 4, CLAUSE: 9\n\nX18 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(failure(a), ','(','(p(0, X17), lte(X17, s(s(s(0))))), even(0)))\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(failure(a), ','(','(p(0, X17), lte(X17, s(s(s(0))))), even(0))), SCOPE: 7, CLAUSE: 10\n\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 34 [arrowhead = none ];
34 -> 3;

35 [label="ONLY EVAL with clause\ngoal :- ','(lte(X2, s(s(s(s(0))))), even(X2)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 35 [arrowhead = none ];
35 -> 5;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 36 [arrowhead = none ];
36 -> 6;

37 [label="ONLY EVAL with clause\nlte(0, X5).\nand substitutionX2 -> 0,\nX5 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 37 [arrowhead = none ];
37 -> 8;

38 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 38 [arrowhead = none ];
38 -> 9;

39 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 39 [arrowhead = none ];
39 -> 10;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 40 [arrowhead = none ];
40 -> 11;

41 [label="ONLY EVAL with clause\nlte(X15, s(X16)) :- ','(no(zero(X15)), ','(p(X15, X17), lte(X17, X16))).\nand substitutionX2 -> X18,\nX15 -> X18,\nX16 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 41 [arrowhead = none ];
41 -> 18;

42 [label="ONLY EVAL with clause\neven(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 42 [arrowhead = none ];
42 -> 12;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 43 [arrowhead = none ];
43 -> 13;

44 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 44 [arrowhead = none ];
44 -> 14;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 45 [arrowhead = none ];
45 -> 21;

46 [label="ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_4, failure(a))).\nand substitutionX18 -> X25,\nX24 -> zero(X25)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 46 [arrowhead = none ];
46 -> 22;

47 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 47 [arrowhead = none ];
47 -> 23;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 48 [arrowhead = none ];
48 -> 26;

49 [label="ONLY EVAL with clause\nzero(0).\nand substitutionX25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 49 [arrowhead = none ];
49 -> 27;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 50 [arrowhead = none ];
50 -> 28;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 51 [arrowhead = none ];
51 -> 32;

52 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 52 [arrowhead = none ];
52 -> 33;

}

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f8_in)
U1(f8_out1) → f1_out1
U1(f8_out2(z0)) → f1_out1
f9_in → f9_out1
f8_in → U2(f9_in, f10_in)
U2(f9_out1, z0) → f8_out1
U2(z0, f10_out1(z0)) → f8_out2(z0)

Tuples:

F1_IN → c(U1'(f8_in), F8_IN)
F8_IN → c4(U2'(f9_in, f10_in), F9_IN)

S tuples:

F1_IN → c(U1'(f8_in), F8_IN)
F8_IN → c4(U2'(f9_in, f10_in), F9_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f8_in, U2

Defined Pair Symbols:

F1_IN, F8_IN

Compound Symbols:

c, c4

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

Split RHS of tuples not part of any SCC

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f8_in)
U1(f8_out1) → f1_out1
U1(f8_out2(z0)) → f1_out1
f9_in → f9_out1
f8_in → U2(f9_in, f10_in)
U2(f9_out1, z0) → f8_out1
U2(z0, f10_out1(z0)) → f8_out2(z0)

Tuples:

F1_IN → c1(U1'(f8_in))
F1_IN → c1(F8_IN)
F8_IN → c1(U2'(f9_in, f10_in))
F8_IN → c1(F9_IN)

S tuples:

F1_IN → c1(U1'(f8_in))
F1_IN → c1(F8_IN)
F8_IN → c1(U2'(f9_in, f10_in))
F8_IN → c1(F9_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f8_in, U2

Defined Pair Symbols:

F1_IN, F8_IN

Compound Symbols:

c1

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

Removed 3 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f8_in)
U1(f8_out1) → f1_out1
U1(f8_out2(z0)) → f1_out1
f9_in → f9_out1
f8_in → U2(f9_in, f10_in)
U2(f9_out1, z0) → f8_out1
U2(z0, f10_out1(z0)) → f8_out2(z0)

Tuples:

F1_IN → c1(F8_IN)
F1_IN → c1
F8_IN → c1

S tuples:

F1_IN → c1(F8_IN)
F1_IN → c1
F8_IN → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f9_in, f8_in, U2

Defined Pair Symbols:

F1_IN, F8_IN

Compound Symbols:

c1, c1

(15) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN → c1(F8_IN)
F1_IN → c1
F8_IN → c1
F8_IN → c1
F8_IN → c1
F8_IN → c1

Now S is empty

(16) BOUNDS(O(1), O(1))