Runtime Complexity proof of /tmp/tmpJ99n6Q/evenodd2.pl

Runtime Complexity of the query pattern
even(g)
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), 73 ms)

↳2 CdtProblem

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

    ↳4 CdtProblem

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

    ↳6 CdtProblem

    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 146 ms)

    ↳8 CdtProblem

    ↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

    ↳10 CdtProblem

    ↳11 SIsEmptyProof (⇔, 0 ms)

    ↳12 BOUNDS(O(1), O(1))

↳13 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 65 ms)

↳14 CdtProblem

    ↳15 CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳16 CdtProblem

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

    ↳18 CdtProblem

    ↳19 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 138 ms)

    ↳20 CdtProblem

    ↳21 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 0 ms)

    ↳22 CdtProblem

(0) Obligation:

Clauses:

even(0) :- !.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- !.
odd(N) :- ','(p(N, P), even(P)).
p(0, 0).
p(s(X), X).

Query: even(g)

(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: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: even(T1), SCOPE: 1, CLAUSE: 0 | even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: !_1 | even(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T4, X5), odd(X5))\n\nT4 is ground\nX5 is free\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(p(T4, X5), odd(X5)), SCOPE: 2, CLAUSE: 4 | ','(p(T4, X5), odd(X5)), SCOPE: 2, CLAUSE: 5\n\nT4 is ground\nX5 is free\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T4, X5), odd(X5)), SCOPE: 2, CLAUSE: 5\n\nT4 is ground\nX5 is free\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: odd(T7)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: odd(T7), SCOPE: 3, CLAUSE: 2 | odd(T7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: !_3 | odd(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: odd(T7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nodd(T7) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T10, X13), even(X13))\n\nT10 is ground\nX13 is free\nodd(T10) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(p(T10, X13), even(X13)), SCOPE: 4, CLAUSE: 4 | ','(p(T10, X13), even(X13)), SCOPE: 4, CLAUSE: 5\n\nT10 is ground\nX13 is free\nodd(T10) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(0) | ','(p(0, X13), even(X13)), SCOPE: 4, CLAUSE: 5\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(T10, X13), even(X13)), SCOPE: 4, CLAUSE: 5\n\nT10 is ground\nX13 is free\nodd(T10) !=? odd(s(0))\np(T10, X13) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(0, X13), even(X13)), SCOPE: 4, CLAUSE: 5\n\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: even(0), SCOPE: 5, CLAUSE: 0 | even(0), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: !_5 | even(0), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: even(T13)\n\nT13 is ground\nodd(s(T13)) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 59 [arrowhead = none ];
59 -> 3;

60 [label="EVAL with clause\neven(0) :- !_1.\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 60 [arrowhead = none ];
60 -> 5;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 61 [arrowhead = none ];
61 -> 7;

62 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 62 [arrowhead = none ];
62 -> 10;

63 [label="ONLY EVAL with clause\neven(X4) :- ','(p(X4, X5), odd(X5)).\nand substitutionT1 -> T4,\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 64 [arrowhead = none ];
64 -> 11;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 65 [arrowhead = none ];
65 -> 16;

66 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (even(T4), even(0))", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 66 [arrowhead = none ];
66 -> 18;

67 [label="EVAL with clause\np(s(X8), X8).\nand substitutionX8 -> T7,\nT4 -> s(T7),\nX5 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 67 [arrowhead = none ];
67 -> 19;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 68 [arrowhead = none ];
68 -> 21;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 69 [arrowhead = none ];
69 -> 24;

70 [label="EVAL with clause\nodd(s(0)) :- !_3.\nand substitutionT7 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 70 [arrowhead = none ];
70 -> 25;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 71 [arrowhead = none ];
71 -> 27;

72 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 72 [arrowhead = none ];
72 -> 29;

73 [label="ONLY EVAL with clause\nodd(X12) :- ','(p(X12, X13), even(X13)).\nand substitutionT7 -> T10,\nX12 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 73 [arrowhead = none ];
73 -> 34;

74 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 74 [arrowhead = none ];
74 -> 31;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 75 [arrowhead = none ];
75 -> 36;

76 [label="EVAL with clause\np(0, 0).\nand substitutionT10 -> 0,\nX13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 76 [arrowhead = none ];
76 -> 38;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 77 [arrowhead = none ];
77 -> 40;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 78 [arrowhead = none ];
78 -> 41;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
38 -> 79 [arrowhead = none ];
79 -> 43;

80 [label="EVAL with clause\np(s(X17), X17).\nand substitutionX17 -> T13,\nT10 -> s(T13),\nX13 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 80 [arrowhead = none ];
80 -> 56;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 81 [arrowhead = none ];
81 -> 58;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 82 [arrowhead = none ];
82 -> 45;

83 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 83 [arrowhead = none ];
83 -> 54;

84 [label="ONLY EVAL with clause\neven(0) :- !_5.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 84 [arrowhead = none ];
84 -> 48;

85 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 85 [arrowhead = none ];
85 -> 50;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 86 [arrowhead = none ];
86 -> 52;

87 [label="INSTANCE with matching:\nT1 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 87 [arrowhead = none , style=dashed];
87 -> 1[style=dashed];

}

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(0)) → U1(f38_in, s(0))
f1_in(s(s(z0))) → U2(f1_in(z0), s(s(z0)))
U1(f38_out1, s(0)) → f1_out1
U1(f38_out2(z0), s(0)) → f1_out1
U2(f1_out1, s(s(z0))) → f1_out1
f41_in → f41_out1
f38_in → U3(f41_in, f43_in)
U3(f41_out1, z0) → f38_out1
U3(z0, f43_out1(z1)) → f38_out2(z1)

Tuples:

F1_IN(s(0)) → c2(U1'(f38_in, s(0)), F38_IN)
F1_IN(s(s(z0))) → c3(U2'(f1_in(z0), s(s(z0))), F1_IN(z0))
F38_IN → c8(U3'(f41_in, f43_in), F41_IN)

S tuples:

F1_IN(s(0)) → c2(U1'(f38_in, s(0)), F38_IN)
F1_IN(s(s(z0))) → c3(U2'(f1_in(z0), s(s(z0))), F1_IN(z0))
F38_IN → c8(U3'(f41_in, f43_in), F41_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f41_in, f38_in, U3

Defined Pair Symbols:

F1_IN, F38_IN

Compound Symbols:

c2, c3, c8

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

Split RHS of tuples not part of any SCC

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(0)) → U1(f38_in, s(0))
f1_in(s(s(z0))) → U2(f1_in(z0), s(s(z0)))
U1(f38_out1, s(0)) → f1_out1
U1(f38_out2(z0), s(0)) → f1_out1
U2(f1_out1, s(s(z0))) → f1_out1
f41_in → f41_out1
f38_in → U3(f41_in, f43_in)
U3(f41_out1, z0) → f38_out1
U3(z0, f43_out1(z1)) → f38_out2(z1)

Tuples:

F1_IN(s(s(z0))) → c3(U2'(f1_in(z0), s(s(z0))), F1_IN(z0))
F1_IN(s(0)) → c(U1'(f38_in, s(0)))
F1_IN(s(0)) → c(F38_IN)
F38_IN → c(U3'(f41_in, f43_in))
F38_IN → c(F41_IN)

S tuples:

F1_IN(s(s(z0))) → c3(U2'(f1_in(z0), s(s(z0))), F1_IN(z0))
F1_IN(s(0)) → c(U1'(f38_in, s(0)))
F1_IN(s(0)) → c(F38_IN)
F38_IN → c(U3'(f41_in, f43_in))
F38_IN → c(F41_IN)

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f41_in, f38_in, U3

Defined Pair Symbols:

F1_IN, F38_IN

Compound Symbols:

c3, c

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(0)) → U1(f38_in, s(0))
f1_in(s(s(z0))) → U2(f1_in(z0), s(s(z0)))
U1(f38_out1, s(0)) → f1_out1
U1(f38_out2(z0), s(0)) → f1_out1
U2(f1_out1, s(s(z0))) → f1_out1
f41_in → f41_out1
f38_in → U3(f41_in, f43_in)
U3(f41_out1, z0) → f38_out1
U3(z0, f43_out1(z1)) → f38_out2(z1)

Tuples:

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(s(z0))) → c3(F1_IN(z0))
F1_IN(s(0)) → c
F38_IN → c

S tuples:

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(s(z0))) → c3(F1_IN(z0))
F1_IN(s(0)) → c
F38_IN → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f41_in, f38_in, U3

Defined Pair Symbols:

F1_IN, F38_IN

Compound Symbols:

c, c3, c

(7) 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(0)) → c(F38_IN)
F1_IN(s(0)) → c
F38_IN → c

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

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(s(z0))) → c3(F1_IN(z0))
F1_IN(s(0)) → c
F38_IN → c

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

POL(0) = 0
POL(F1_IN(x₁)) = [3]
POL(F38_IN) = [2]
POL(c) = 0
POL(c(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(s(x₁)) = 0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(0)) → U1(f38_in, s(0))
f1_in(s(s(z0))) → U2(f1_in(z0), s(s(z0)))
U1(f38_out1, s(0)) → f1_out1
U1(f38_out2(z0), s(0)) → f1_out1
U2(f1_out1, s(s(z0))) → f1_out1
f41_in → f41_out1
f38_in → U3(f41_in, f43_in)
U3(f41_out1, z0) → f38_out1
U3(z0, f43_out1(z1)) → f38_out2(z1)

Tuples:

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(s(z0))) → c3(F1_IN(z0))
F1_IN(s(0)) → c
F38_IN → c

S tuples:

F1_IN(s(s(z0))) → c3(F1_IN(z0))

K tuples:

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(0)) → c
F38_IN → c

Defined Rule Symbols:

f1_in, U1, U2, f41_in, f38_in, U3

Defined Pair Symbols:

F1_IN, F38_IN

Compound Symbols:

c, c3, c

(9) 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(s(z0))) → c3(F1_IN(z0))

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

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(s(z0))) → c3(F1_IN(z0))
F1_IN(s(0)) → c
F38_IN → c

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

POL(0) = 0
POL(F1_IN(x₁)) = [3] + x₁
POL(F38_IN) = [3]
POL(c) = 0
POL(c(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(0) → f1_out1
f1_in(s(s(0))) → f1_out1
f1_in(s(0)) → U1(f38_in, s(0))
f1_in(s(s(z0))) → U2(f1_in(z0), s(s(z0)))
U1(f38_out1, s(0)) → f1_out1
U1(f38_out2(z0), s(0)) → f1_out1
U2(f1_out1, s(s(z0))) → f1_out1
f41_in → f41_out1
f38_in → U3(f41_in, f43_in)
U3(f41_out1, z0) → f38_out1
U3(z0, f43_out1(z1)) → f38_out2(z1)

Tuples:

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(s(z0))) → c3(F1_IN(z0))
F1_IN(s(0)) → c
F38_IN → c

S tuples:none
K tuples:

F1_IN(s(0)) → c(F38_IN)
F1_IN(s(0)) → c
F38_IN → c
F1_IN(s(s(z0))) → c3(F1_IN(z0))

Defined Rule Symbols:

f1_in, U1, U2, f41_in, f38_in, U3

Defined Pair Symbols:

F1_IN, F38_IN

Compound Symbols:

c, c3, c

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))

(13) 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: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: even(T1), SCOPE: 1, CLAUSE: 0 | even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: !_1 | even(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T3, X3), odd(X3))\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 4 | ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: odd(T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: odd(T6), SCOPE: 3, CLAUSE: 2 | odd(T6), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: !_3 | odd(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: odd(T6), SCOPE: 3, CLAUSE: 3\n\nT6 is ground\nodd(T6) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(T9, X11), even(X11))\n\nT9 is ground\nX11 is free\nodd(T9) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T9, X11), even(X11)), SCOPE: 4, CLAUSE: 4 | ','(p(T9, X11), even(X11)), SCOPE: 4, CLAUSE: 5\n\nT9 is ground\nX11 is free\nodd(T9) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: even(0) | ','(p(0, X11), even(X11)), SCOPE: 4, CLAUSE: 5\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(T9, X11), even(X11)), SCOPE: 4, CLAUSE: 5\n\nT9 is ground\nX11 is free\nodd(T9) !=? odd(s(0))\np(T9, X11) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(0, X11), even(X11)), SCOPE: 4, CLAUSE: 5\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: even(0), SCOPE: 5, CLAUSE: 0 | even(0), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: !_5 | even(0), SCOPE: 5, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: even(T12)\n\nT12 is ground\nodd(s(T12)) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 58 [arrowhead = none ];
58 -> 4;

59 [label="EVAL with clause\neven(0) :- !_1.\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 59 [arrowhead = none ];
59 -> 6;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 61 [arrowhead = none ];
61 -> 9;

62 [label="ONLY EVAL with clause\neven(X2) :- ','(p(X2, X3), odd(X3)).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 64 [arrowhead = none ];
64 -> 15;

65 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (even(T3), even(0))", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 65 [arrowhead = none ];
65 -> 17;

66 [label="EVAL with clause\np(s(X6), X6).\nand substitutionX6 -> T6,\nT3 -> s(T6),\nX3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 66 [arrowhead = none ];
66 -> 20;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 68 [arrowhead = none ];
68 -> 23;

69 [label="EVAL with clause\nodd(s(0)) :- !_3.\nand substitutionT6 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 71 [arrowhead = none ];
71 -> 30;

72 [label="ONLY EVAL with clause\nodd(X10) :- ','(p(X10, X11), even(X11)).\nand substitutionT6 -> T9,\nX10 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 72 [arrowhead = none ];
72 -> 33;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 73 [arrowhead = none ];
73 -> 32;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 74 [arrowhead = none ];
74 -> 35;

75 [label="EVAL with clause\np(0, 0).\nand substitutionT9 -> 0,\nX11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 75 [arrowhead = none ];
75 -> 37;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 76 [arrowhead = none ];
76 -> 39;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 77 [arrowhead = none ];
77 -> 42;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 78 [arrowhead = none ];
78 -> 44;

79 [label="EVAL with clause\np(s(X15), X15).\nand substitutionX15 -> T12,\nT9 -> s(T12),\nX11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 79 [arrowhead = none ];
79 -> 55;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 80 [arrowhead = none ];
80 -> 57;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 81 [arrowhead = none ];
81 -> 46;

82 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 82 [arrowhead = none ];
82 -> 53;

83 [label="ONLY EVAL with clause\neven(0) :- !_5.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 83 [arrowhead = none ];
83 -> 47;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 84 [arrowhead = none ];
84 -> 49;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 85 [arrowhead = none ];
85 -> 51;

86 [label="INSTANCE with matching:\nT1 -> T12", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 86 [arrowhead = none , style=dashed];
86 -> 2[style=dashed];

}

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(0)) → U1(f37_in, s(0))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
U1(f37_out1, s(0)) → f2_out1
U1(f37_out2(z0), s(0)) → f2_out1
U2(f2_out1, s(s(z0))) → f2_out1
f42_in → f42_out1
f37_in → U3(f42_in, f44_in)
U3(f42_out1, z0) → f37_out1
U3(z0, f44_out1(z1)) → f37_out2(z1)

Tuples:

F2_IN(s(0)) → c2(U1'(f37_in, s(0)), F37_IN)
F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F37_IN → c8(U3'(f42_in, f44_in), F42_IN)

S tuples:

F2_IN(s(0)) → c2(U1'(f37_in, s(0)), F37_IN)
F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F37_IN → c8(U3'(f42_in, f44_in), F42_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f42_in, f37_in, U3

Defined Pair Symbols:

F2_IN, F37_IN

Compound Symbols:

c2, c3, c8

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(0)) → U1(f37_in, s(0))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
U1(f37_out1, s(0)) → f2_out1
U1(f37_out2(z0), s(0)) → f2_out1
U2(f2_out1, s(s(z0))) → f2_out1
f42_in → f42_out1
f37_in → U3(f42_in, f44_in)
U3(f42_out1, z0) → f37_out1
U3(z0, f44_out1(z1)) → f37_out2(z1)

Tuples:

F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(0)) → c(U1'(f37_in, s(0)))
F2_IN(s(0)) → c(F37_IN)
F37_IN → c(U3'(f42_in, f44_in))
F37_IN → c(F42_IN)

S tuples:

F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(0)) → c(U1'(f37_in, s(0)))
F2_IN(s(0)) → c(F37_IN)
F37_IN → c(U3'(f42_in, f44_in))
F37_IN → c(F42_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f42_in, f37_in, U3

Defined Pair Symbols:

F2_IN, F37_IN

Compound Symbols:

c3, c

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

Removed 4 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(0)) → U1(f37_in, s(0))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
U1(f37_out1, s(0)) → f2_out1
U1(f37_out2(z0), s(0)) → f2_out1
U2(f2_out1, s(s(z0))) → f2_out1
f42_in → f42_out1
f37_in → U3(f42_in, f44_in)
U3(f42_out1, z0) → f37_out1
U3(z0, f44_out1(z1)) → f37_out2(z1)

Tuples:

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(0)) → c
F37_IN → c

S tuples:

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(0)) → c
F37_IN → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f42_in, f37_in, U3

Defined Pair Symbols:

F2_IN, F37_IN

Compound Symbols:

c, c3, c

(19) 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(0)) → c(F37_IN)
F2_IN(s(0)) → c
F37_IN → c

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

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(0)) → c
F37_IN → c

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

POL(0) = 0
POL(F2_IN(x₁)) = [3]
POL(F37_IN) = [2]
POL(c) = 0
POL(c(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(s(x₁)) = 0

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(0)) → U1(f37_in, s(0))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
U1(f37_out1, s(0)) → f2_out1
U1(f37_out2(z0), s(0)) → f2_out1
U2(f2_out1, s(s(z0))) → f2_out1
f42_in → f42_out1
f37_in → U3(f42_in, f44_in)
U3(f42_out1, z0) → f37_out1
U3(z0, f44_out1(z1)) → f37_out2(z1)

Tuples:

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(0)) → c
F37_IN → c

S tuples:

F2_IN(s(s(z0))) → c3(F2_IN(z0))

K tuples:

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(0)) → c
F37_IN → c

Defined Rule Symbols:

f2_in, U1, U2, f42_in, f37_in, U3

Defined Pair Symbols:

F2_IN, F37_IN

Compound Symbols:

c, c3, c

(21) 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(s(z0))) → c3(F2_IN(z0))

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

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(0)) → c
F37_IN → c

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

POL(0) = 0
POL(F2_IN(x₁)) = [3] + x₁
POL(F37_IN) = [3]
POL(c) = 0
POL(c(x₁)) = x₁
POL(c3(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(0) → f2_out1
f2_in(s(s(0))) → f2_out1
f2_in(s(0)) → U1(f37_in, s(0))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
U1(f37_out1, s(0)) → f2_out1
U1(f37_out2(z0), s(0)) → f2_out1
U2(f2_out1, s(s(z0))) → f2_out1
f42_in → f42_out1
f37_in → U3(f42_in, f44_in)
U3(f42_out1, z0) → f37_out1
U3(z0, f44_out1(z1)) → f37_out2(z1)

Tuples:

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(0)) → c
F37_IN → c

S tuples:none
K tuples:

F2_IN(s(0)) → c(F37_IN)
F2_IN(s(0)) → c
F37_IN → c
F2_IN(s(s(z0))) → c3(F2_IN(z0))

Defined Rule Symbols:

f2_in, U1, U2, f42_in, f37_in, U3

Defined Pair Symbols:

F2_IN, F37_IN

Compound Symbols:

c, c3, c