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

(0) Obligation:

Clauses:

even(s(s(X))) :- even(X).
even(0).
lte(s(X), s(Y)) :- lte(X, Y).
lte(0, Y).
goal :- ','(lte(X, s(s(s(s(0))))), even(X)).

Query: goal()

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

even(0).
lte(0, Y).
even(s(s(X))) :- even(X).
lte(s(X), s(Y)) :- lte(X, Y).
goal :- ','(lte(X, s(s(s(s(0))))), even(X)).

Query: goal()

(3) 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];
4 [label="4: goal, SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(lte(X3, s(s(s(s(0))))), even(X3))\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: lte(X3, s(s(s(s(0)))))\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: lte(X3, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 1 | lte(X3, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 3\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true | lte(X3, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 3\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: lte(X3, s(s(s(s(0))))), SCOPE: 2, CLAUSE: 3\n\nX3 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: lte(X18, s(s(s(0))))\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: lte(X18, s(s(s(0)))), SCOPE: 3, CLAUSE: 1 | lte(X18, s(s(s(0)))), SCOPE: 3, CLAUSE: 3\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | lte(X18, s(s(s(0)))), SCOPE: 3, CLAUSE: 3\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: lte(X18, s(s(s(0)))), SCOPE: 3, CLAUSE: 3\n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: lte(X32, s(s(0)))\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: lte(X32, s(s(0))), SCOPE: 4, CLAUSE: 1 | lte(X32, s(s(0))), SCOPE: 4, CLAUSE: 3\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true | lte(X32, s(s(0))), SCOPE: 4, CLAUSE: 3\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: lte(X32, s(s(0))), SCOPE: 4, CLAUSE: 3\n\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: lte(X46, s(0))\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: lte(X46, s(0)), SCOPE: 5, CLAUSE: 1 | lte(X46, s(0)), SCOPE: 5, CLAUSE: 3\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: true | lte(X46, s(0)), SCOPE: 5, CLAUSE: 3\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: lte(X46, s(0)), SCOPE: 5, CLAUSE: 3\n\nX46 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: lte(X60, 0)\n\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: lte(X60, 0), SCOPE: 6, CLAUSE: 1 | lte(X60, 0), SCOPE: 6, CLAUSE: 3\n\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true | lte(X60, 0), SCOPE: 6, CLAUSE: 3\n\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: lte(X60, 0), SCOPE: 6, CLAUSE: 3\n\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: even(T1), SCOPE: 7, CLAUSE: 0 | even(T1), SCOPE: 7, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true | even(0), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: even(T1), SCOPE: 7, CLAUSE: 2\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: even(0), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: even(T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 70 [arrowhead = none ];
70 -> 4;

71 [label="ONLY EVAL with clause\ngoal :- ','(lte(X3, s(s(s(s(0))))), even(X3)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 71 [arrowhead = none ];
71 -> 6;

72 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
6 -> 72 [arrowhead = none ];
72 -> 15;

73 [label="SPLIT 2\nnew knowledge:\nT1 is ground\nreplacements:X3 -> T1", fontsize=14, style = filled, fillcolor = violet];
6 -> 73 [arrowhead = none ];
73 -> 16;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 74 [arrowhead = none ];
74 -> 17;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 75 [arrowhead = none ];
75 -> 59;

76 [label="ONLY EVAL with clause\nlte(0, X8).\nand substitutionX3 -> 0,\nX8 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 76 [arrowhead = none ];
76 -> 19;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 77 [arrowhead = none ];
77 -> 20;

78 [label="ONLY EVAL with clause\nlte(s(X16), s(X17)) :- lte(X16, X17).\nand substitutionX16 -> X18,\nX3 -> s(X18),\nX17 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 78 [arrowhead = none ];
78 -> 24;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 79 [arrowhead = none ];
79 -> 27;

80 [label="ONLY EVAL with clause\nlte(0, X22).\nand substitutionX18 -> 0,\nX22 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 80 [arrowhead = none ];
80 -> 28;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 81 [arrowhead = none ];
81 -> 29;

82 [label="ONLY EVAL with clause\nlte(s(X30), s(X31)) :- lte(X30, X31).\nand substitutionX30 -> X32,\nX18 -> s(X32),\nX31 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 82 [arrowhead = none ];
82 -> 32;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 83 [arrowhead = none ];
83 -> 35;

84 [label="ONLY EVAL with clause\nlte(0, X36).\nand substitutionX32 -> 0,\nX36 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 84 [arrowhead = none ];
84 -> 36;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 85 [arrowhead = none ];
85 -> 37;

86 [label="ONLY EVAL with clause\nlte(s(X44), s(X45)) :- lte(X44, X45).\nand substitutionX44 -> X46,\nX32 -> s(X46),\nX45 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 87 [arrowhead = none ];
87 -> 43;

88 [label="ONLY EVAL with clause\nlte(0, X50).\nand substitutionX46 -> 0,\nX50 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 88 [arrowhead = none ];
88 -> 44;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 89 [arrowhead = none ];
89 -> 45;

90 [label="ONLY EVAL with clause\nlte(s(X58), s(X59)) :- lte(X58, X59).\nand substitutionX58 -> X60,\nX46 -> s(X60),\nX59 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 90 [arrowhead = none ];
90 -> 48;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 91 [arrowhead = none ];
91 -> 51;

92 [label="ONLY EVAL with clause\nlte(0, X64).\nand substitutionX60 -> 0,\nX64 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 92 [arrowhead = none ];
92 -> 53;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 93 [arrowhead = none ];
93 -> 55;

94 [label="BACKTRACK\nfor clause: lte(s(X), s(Y)) :- lte(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 94 [arrowhead = none ];
94 -> 57;

95 [label="EVAL with clause\neven(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 95 [arrowhead = none ];
95 -> 61;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 96 [arrowhead = none ];
96 -> 62;

97 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 97 [arrowhead = none ];
97 -> 63;

98 [label="EVAL with clause\neven(s(s(X71))) :- even(X71).\nand substitutionX71 -> T4,\nT1 -> s(s(T4))", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 98 [arrowhead = none ];
98 -> 66;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 99 [arrowhead = none ];
99 -> 69;

100 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 100 [arrowhead = none ];
100 -> 64;

101 [label="INSTANCE with matching:\nT1 -> T4", fontsize=14, style = filled, fillcolor = lightgrey];
66 -> 101 [arrowhead = none , style=dashed];
101 -> 16[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1
f16_in(0) → f16_out1
f16_in(s(s(z0))) → U2(f16_in(z0), s(s(z0)))
U2(f16_out1, s(s(z0))) → f16_out1
f15_in → f15_out1(0)
f15_in → f15_out1(s(0))
f15_in → f15_out1(s(s(0)))
f15_in → f15_out1(s(s(s(0))))
f15_in → f15_out1(s(s(s(s(0)))))
f6_in → U3(f15_in)
U3(f15_out1(z0)) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f6_out1(z0)

Tuples:

F1_IN → c(U1'(f6_in), F6_IN)
F16_IN(s(s(z0))) → c3(U2'(f16_in(z0), s(s(z0))), F16_IN(z0))
F6_IN → c10(U3'(f15_in), F15_IN)
U3'(f15_out1(z0)) → c11(U4'(f16_in(z0), z0), F16_IN(z0))

S tuples:

F1_IN → c(U1'(f6_in), F6_IN)
F16_IN(s(s(z0))) → c3(U2'(f16_in(z0), s(s(z0))), F16_IN(z0))
F6_IN → c10(U3'(f15_in), F15_IN)
U3'(f15_out1(z0)) → c11(U4'(f16_in(z0), z0), F16_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f16_in, U2, f15_in, f6_in, U3, U4

Defined Pair Symbols:

F1_IN, F16_IN, F6_IN, U3'

Compound Symbols:

c, c3, c10, c11

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1
f16_in(0) → f16_out1
f16_in(s(s(z0))) → U2(f16_in(z0), s(s(z0)))
U2(f16_out1, s(s(z0))) → f16_out1
f15_in → f15_out1(0)
f15_in → f15_out1(s(0))
f15_in → f15_out1(s(s(0)))
f15_in → f15_out1(s(s(s(0))))
f15_in → f15_out1(s(s(s(s(0)))))
f6_in → U3(f15_in)
U3(f15_out1(z0)) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f6_out1(z0)

Tuples:

F16_IN(s(s(z0))) → c3(U2'(f16_in(z0), s(s(z0))), F16_IN(z0))
F1_IN → c1(U1'(f6_in))
F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
F6_IN → c1(F15_IN)
U3'(f15_out1(z0)) → c1(U4'(f16_in(z0), z0))
U3'(f15_out1(z0)) → c1(F16_IN(z0))

S tuples:

F16_IN(s(s(z0))) → c3(U2'(f16_in(z0), s(s(z0))), F16_IN(z0))
F1_IN → c1(U1'(f6_in))
F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
F6_IN → c1(F15_IN)
U3'(f15_out1(z0)) → c1(U4'(f16_in(z0), z0))
U3'(f15_out1(z0)) → c1(F16_IN(z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f16_in, U2, f15_in, f6_in, U3, U4

Defined Pair Symbols:

F16_IN, F1_IN, F6_IN, U3'

Compound Symbols:

c3, c1

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1
f16_in(0) → f16_out1
f16_in(s(s(z0))) → U2(f16_in(z0), s(s(z0)))
U2(f16_out1, s(s(z0))) → f16_out1
f15_in → f15_out1(0)
f15_in → f15_out1(s(0))
f15_in → f15_out1(s(s(0)))
f15_in → f15_out1(s(s(s(0))))
f15_in → f15_out1(s(s(s(s(0)))))
f6_in → U3(f15_in)
U3(f15_out1(z0)) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f6_out1(z0)

Tuples:

F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
U3'(f15_out1(z0)) → c1(F16_IN(z0))
F16_IN(s(s(z0))) → c3(F16_IN(z0))
F1_IN → c1
F6_IN → c1
U3'(f15_out1(z0)) → c1

S tuples:

F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
U3'(f15_out1(z0)) → c1(F16_IN(z0))
F16_IN(s(s(z0))) → c3(F16_IN(z0))
F1_IN → c1
F6_IN → c1
U3'(f15_out1(z0)) → c1

K tuples:none
Defined Rule Symbols:

f1_in, U1, f16_in, U2, f15_in, f6_in, U3, U4

Defined Pair Symbols:

F1_IN, F6_IN, U3', F16_IN

Compound Symbols:

c1, c3, c1

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
U3'(f15_out1(z0)) → c1(F16_IN(z0))
F1_IN → c1
F6_IN → c1
U3'(f15_out1(z0)) → c1
F6_IN → c1(U3'(f15_in))
F6_IN → c1
U3'(f15_out1(z0)) → c1(F16_IN(z0))
U3'(f15_out1(z0)) → c1

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in → U1(f6_in)
U1(f6_out1(z0)) → f1_out1
f16_in(0) → f16_out1
f16_in(s(s(z0))) → U2(f16_in(z0), s(s(z0)))
U2(f16_out1, s(s(z0))) → f16_out1
f15_in → f15_out1(0)
f15_in → f15_out1(s(0))
f15_in → f15_out1(s(s(0)))
f15_in → f15_out1(s(s(s(0))))
f15_in → f15_out1(s(s(s(s(0)))))
f6_in → U3(f15_in)
U3(f15_out1(z0)) → U4(f16_in(z0), z0)
U4(f16_out1, z0) → f6_out1(z0)

Tuples:

F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
U3'(f15_out1(z0)) → c1(F16_IN(z0))
F16_IN(s(s(z0))) → c3(F16_IN(z0))
F1_IN → c1
F6_IN → c1
U3'(f15_out1(z0)) → c1

S tuples:

F16_IN(s(s(z0))) → c3(F16_IN(z0))

K tuples:

F1_IN → c1(F6_IN)
F6_IN → c1(U3'(f15_in))
U3'(f15_out1(z0)) → c1(F16_IN(z0))
F1_IN → c1
F6_IN → c1
U3'(f15_out1(z0)) → c1

Defined Rule Symbols:

f1_in, U1, f16_in, U2, f15_in, f6_in, U3, U4

Defined Pair Symbols:

F1_IN, F6_IN, U3', F16_IN

Compound Symbols:

c1, c3, c1

(11) 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];
3 [label="3: goal, SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(lte(X1, s(s(s(s(0))))), even(X1))\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 1 | ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 3\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: even(0) | ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 3\n\nX1 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(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 3\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: even(0), SCOPE: 3, CLAUSE: 0 | even(0), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true | even(0), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: even(0), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(lte(X15, s(s(s(0)))), even(s(X15)))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: lte(X15, s(s(s(0))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: even(s(T1))\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: lte(X15, s(s(s(0)))), SCOPE: 4, CLAUSE: 1 | lte(X15, s(s(s(0)))), SCOPE: 4, CLAUSE: 3\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true | lte(X15, s(s(s(0)))), SCOPE: 4, CLAUSE: 3\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: lte(X15, s(s(s(0)))), SCOPE: 4, CLAUSE: 3\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: lte(X30, s(s(0)))\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: lte(X30, s(s(0))), SCOPE: 5, CLAUSE: 1 | lte(X30, s(s(0))), SCOPE: 5, CLAUSE: 3\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true | lte(X30, s(s(0))), SCOPE: 5, CLAUSE: 3\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: lte(X30, s(s(0))), SCOPE: 5, CLAUSE: 3\n\nX30 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: lte(X44, s(0))\n\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: lte(X44, s(0)), SCOPE: 6, CLAUSE: 1 | lte(X44, s(0)), SCOPE: 6, CLAUSE: 3\n\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true | lte(X44, s(0)), SCOPE: 6, CLAUSE: 3\n\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: lte(X44, s(0)), SCOPE: 6, CLAUSE: 3\n\nX44 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: lte(X58, 0)\n\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: lte(X58, 0), SCOPE: 7, CLAUSE: 1 | lte(X58, 0), SCOPE: 7, CLAUSE: 3\n\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true | lte(X58, 0), SCOPE: 7, CLAUSE: 3\n\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: lte(X58, 0), SCOPE: 7, CLAUSE: 3\n\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: even(s(T1)), SCOPE: 8, CLAUSE: 0 | even(s(T1)), SCOPE: 8, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: even(s(T1)), SCOPE: 8, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: even(T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: even(T4), SCOPE: 9, CLAUSE: 0 | even(T4), SCOPE: 9, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: true | even(0), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: even(T4), SCOPE: 9, CLAUSE: 2\n\nT4 is ground\neven(T4) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: even(0), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: even(T7)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 74 [arrowhead = none ];
74 -> 3;

75 [label="ONLY EVAL with clause\ngoal :- ','(lte(X1, s(s(s(s(0))))), even(X1)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 75 [arrowhead = none ];
75 -> 5;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 76 [arrowhead = none ];
76 -> 7;

77 [label="ONLY EVAL with clause\nlte(0, X4).\nand substitutionX1 -> 0,\nX4 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 77 [arrowhead = none ];
77 -> 8;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 78 [arrowhead = none ];
78 -> 9;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 79 [arrowhead = none ];
79 -> 10;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 80 [arrowhead = none ];
80 -> 11;

81 [label="ONLY EVAL with clause\nlte(s(X13), s(X14)) :- lte(X13, X14).\nand substitutionX13 -> X15,\nX1 -> s(X15),\nX14 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 81 [arrowhead = none ];
81 -> 18;

82 [label="ONLY EVAL with clause\neven(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 82 [arrowhead = none ];
82 -> 12;

83 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 83 [arrowhead = none ];
83 -> 13;

84 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 84 [arrowhead = none ];
84 -> 14;

85 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 85 [arrowhead = none ];
85 -> 21;

86 [label="SPLIT 2\nnew knowledge:\nT1 is ground\nreplacements:X15 -> T1", fontsize=14, style = filled, fillcolor = violet];
18 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 87 [arrowhead = none ];
87 -> 23;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 88 [arrowhead = none ];
88 -> 54;

89 [label="ONLY EVAL with clause\nlte(0, X20).\nand substitutionX15 -> 0,\nX20 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 89 [arrowhead = none ];
89 -> 25;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 90 [arrowhead = none ];
90 -> 26;

91 [label="ONLY EVAL with clause\nlte(s(X28), s(X29)) :- lte(X28, X29).\nand substitutionX28 -> X30,\nX15 -> s(X30),\nX29 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 91 [arrowhead = none ];
91 -> 30;

92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 92 [arrowhead = none ];
92 -> 31;

93 [label="ONLY EVAL with clause\nlte(0, X34).\nand substitutionX30 -> 0,\nX34 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 93 [arrowhead = none ];
93 -> 33;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 94 [arrowhead = none ];
94 -> 34;

95 [label="ONLY EVAL with clause\nlte(s(X42), s(X43)) :- lte(X42, X43).\nand substitutionX42 -> X44,\nX30 -> s(X44),\nX43 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 95 [arrowhead = none ];
95 -> 38;

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 96 [arrowhead = none ];
96 -> 39;

97 [label="ONLY EVAL with clause\nlte(0, X48).\nand substitutionX44 -> 0,\nX48 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 97 [arrowhead = none ];
97 -> 41;

98 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 98 [arrowhead = none ];
98 -> 42;

99 [label="ONLY EVAL with clause\nlte(s(X56), s(X57)) :- lte(X56, X57).\nand substitutionX56 -> X58,\nX44 -> s(X58),\nX57 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 99 [arrowhead = none ];
99 -> 46;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 100 [arrowhead = none ];
100 -> 47;

101 [label="ONLY EVAL with clause\nlte(0, X62).\nand substitutionX58 -> 0,\nX62 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 101 [arrowhead = none ];
101 -> 49;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 102 [arrowhead = none ];
102 -> 50;

103 [label="BACKTRACK\nfor clause: lte(s(X), s(Y)) :- lte(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 103 [arrowhead = none ];
103 -> 52;

104 [label="BACKTRACK\nfor clause: even(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 104 [arrowhead = none ];
104 -> 56;

105 [label="EVAL with clause\neven(s(s(X68))) :- even(X68).\nand substitutionX68 -> T4,\nT1 -> s(T4)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 105 [arrowhead = none ];
105 -> 58;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 106 [arrowhead = none ];
106 -> 60;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 107 [arrowhead = none ];
107 -> 65;

108 [label="EVAL with clause\neven(0).\nand substitutionT4 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 108 [arrowhead = none ];
108 -> 67;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 109 [arrowhead = none ];
109 -> 68;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 110 [arrowhead = none ];
110 -> 70;

111 [label="EVAL with clause\neven(s(s(X72))) :- even(X72).\nand substitutionX72 -> T7,\nT4 -> s(s(T7))", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 111 [arrowhead = none ];
111 -> 72;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 112 [arrowhead = none ];
112 -> 73;

113 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 113 [arrowhead = none ];
113 -> 71;

114 [label="INSTANCE with matching:\nT4 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
72 -> 114 [arrowhead = none , style=dashed];
114 -> 58[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f8_in)
U1(f8_out1) → f2_out1
U1(f8_out2(z0)) → f2_out1
f58_in(0) → f58_out1
f58_in(s(s(z0))) → U2(f58_in(z0), s(s(z0)))
U2(f58_out1, s(s(z0))) → f58_out1
f21_in → f21_out1(0)
f21_in → f21_out1(s(0))
f21_in → f21_out1(s(s(0)))
f21_in → f21_out1(s(s(s(0))))
f22_in(s(z0)) → U3(f58_in(z0), s(z0))
U3(f58_out1, s(z0)) → f22_out1
f9_in → f9_out1
f10_in → U4(f18_in)
U4(f18_out1(z0)) → f10_out1(s(z0))
f18_in → U5(f21_in)
U5(f21_out1(z0)) → U6(f22_in(z0), z0)
U6(f22_out1, z0) → f18_out1(z0)
f8_in → U7(f9_in, f10_in)
U7(f9_out1, z0) → f8_out1
U7(z0, f10_out1(z1)) → f8_out2(z1)

Tuples:

F2_IN → c(U1'(f8_in), F8_IN)
F58_IN(s(s(z0))) → c4(U2'(f58_in(z0), s(s(z0))), F58_IN(z0))
F22_IN(s(z0)) → c10(U3'(f58_in(z0), s(z0)), F58_IN(z0))
F10_IN → c13(U4'(f18_in), F18_IN)
F18_IN → c15(U5'(f21_in), F21_IN)
U5'(f21_out1(z0)) → c16(U6'(f22_in(z0), z0), F22_IN(z0))
F8_IN → c18(U7'(f9_in, f10_in), F9_IN, F10_IN)

S tuples:

F2_IN → c(U1'(f8_in), F8_IN)
F58_IN(s(s(z0))) → c4(U2'(f58_in(z0), s(s(z0))), F58_IN(z0))
F22_IN(s(z0)) → c10(U3'(f58_in(z0), s(z0)), F58_IN(z0))
F10_IN → c13(U4'(f18_in), F18_IN)
F18_IN → c15(U5'(f21_in), F21_IN)
U5'(f21_out1(z0)) → c16(U6'(f22_in(z0), z0), F22_IN(z0))
F8_IN → c18(U7'(f9_in, f10_in), F9_IN, F10_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f58_in, U2, f21_in, f22_in, U3, f9_in, f10_in, U4, f18_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F2_IN, F58_IN, F22_IN, F10_IN, F18_IN, U5', F8_IN

Compound Symbols:

c, c4, c10, c13, c15, c16, c18

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

Split RHS of tuples not part of any SCC

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f8_in)
U1(f8_out1) → f2_out1
U1(f8_out2(z0)) → f2_out1
f58_in(0) → f58_out1
f58_in(s(s(z0))) → U2(f58_in(z0), s(s(z0)))
U2(f58_out1, s(s(z0))) → f58_out1
f21_in → f21_out1(0)
f21_in → f21_out1(s(0))
f21_in → f21_out1(s(s(0)))
f21_in → f21_out1(s(s(s(0))))
f22_in(s(z0)) → U3(f58_in(z0), s(z0))
U3(f58_out1, s(z0)) → f22_out1
f9_in → f9_out1
f10_in → U4(f18_in)
U4(f18_out1(z0)) → f10_out1(s(z0))
f18_in → U5(f21_in)
U5(f21_out1(z0)) → U6(f22_in(z0), z0)
U6(f22_out1, z0) → f18_out1(z0)
f8_in → U7(f9_in, f10_in)
U7(f9_out1, z0) → f8_out1
U7(z0, f10_out1(z1)) → f8_out2(z1)

Tuples:

F58_IN(s(s(z0))) → c4(U2'(f58_in(z0), s(s(z0))), F58_IN(z0))
F2_IN → c1(U1'(f8_in))
F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(U3'(f58_in(z0), s(z0)))
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(U4'(f18_in))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
F18_IN → c1(F21_IN)
U5'(f21_out1(z0)) → c1(U6'(f22_in(z0), z0))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(U7'(f9_in, f10_in))
F8_IN → c1(F9_IN)
F8_IN → c1(F10_IN)

S tuples:

F58_IN(s(s(z0))) → c4(U2'(f58_in(z0), s(s(z0))), F58_IN(z0))
F2_IN → c1(U1'(f8_in))
F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(U3'(f58_in(z0), s(z0)))
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(U4'(f18_in))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
F18_IN → c1(F21_IN)
U5'(f21_out1(z0)) → c1(U6'(f22_in(z0), z0))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(U7'(f9_in, f10_in))
F8_IN → c1(F9_IN)
F8_IN → c1(F10_IN)

K tuples:none
Defined Rule Symbols:

f2_in, U1, f58_in, U2, f21_in, f22_in, U3, f9_in, f10_in, U4, f18_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F58_IN, F2_IN, F22_IN, F10_IN, F18_IN, U5', F8_IN

Compound Symbols:

c4, c1

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

Removed 8 trailing tuple parts

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f8_in)
U1(f8_out1) → f2_out1
U1(f8_out2(z0)) → f2_out1
f58_in(0) → f58_out1
f58_in(s(s(z0))) → U2(f58_in(z0), s(s(z0)))
U2(f58_out1, s(s(z0))) → f58_out1
f21_in → f21_out1(0)
f21_in → f21_out1(s(0))
f21_in → f21_out1(s(s(0)))
f21_in → f21_out1(s(s(s(0))))
f22_in(s(z0)) → U3(f58_in(z0), s(z0))
U3(f58_out1, s(z0)) → f22_out1
f9_in → f9_out1
f10_in → U4(f18_in)
U4(f18_out1(z0)) → f10_out1(s(z0))
f18_in → U5(f21_in)
U5(f21_out1(z0)) → U6(f22_in(z0), z0)
U6(f22_out1, z0) → f18_out1(z0)
f8_in → U7(f9_in, f10_in)
U7(f9_out1, z0) → f8_out1
U7(z0, f10_out1(z1)) → f8_out2(z1)

Tuples:

F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(F10_IN)
F58_IN(s(s(z0))) → c4(F58_IN(z0))
F2_IN → c1
F22_IN(s(z0)) → c1
F10_IN → c1
F18_IN → c1
U5'(f21_out1(z0)) → c1
F8_IN → c1

S tuples:

F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(F10_IN)
F58_IN(s(s(z0))) → c4(F58_IN(z0))
F2_IN → c1
F22_IN(s(z0)) → c1
F10_IN → c1
F18_IN → c1
U5'(f21_out1(z0)) → c1
F8_IN → c1

K tuples:none
Defined Rule Symbols:

f2_in, U1, f58_in, U2, f21_in, f22_in, U3, f9_in, f10_in, U4, f18_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F2_IN, F22_IN, F10_IN, F18_IN, U5', F8_IN, F58_IN

Compound Symbols:

c1, c4, c1

(17) CdtKnowledgeProof (EQUIVALENT transformation)

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

F2_IN → c1(F8_IN)
F8_IN → c1(F10_IN)
F2_IN → c1
F10_IN → c1
F8_IN → c1
F8_IN → c1
F8_IN → c1(F10_IN)
F8_IN → c1
F8_IN → c1
F10_IN → c1(F18_IN)
F10_IN → c1
F18_IN → c1(U5'(f21_in))
F18_IN → c1
U5'(f21_out1(z0)) → c1(F22_IN(z0))
U5'(f21_out1(z0)) → c1
F22_IN(s(z0)) → c1(F58_IN(z0))
F22_IN(s(z0)) → c1

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f8_in)
U1(f8_out1) → f2_out1
U1(f8_out2(z0)) → f2_out1
f58_in(0) → f58_out1
f58_in(s(s(z0))) → U2(f58_in(z0), s(s(z0)))
U2(f58_out1, s(s(z0))) → f58_out1
f21_in → f21_out1(0)
f21_in → f21_out1(s(0))
f21_in → f21_out1(s(s(0)))
f21_in → f21_out1(s(s(s(0))))
f22_in(s(z0)) → U3(f58_in(z0), s(z0))
U3(f58_out1, s(z0)) → f22_out1
f9_in → f9_out1
f10_in → U4(f18_in)
U4(f18_out1(z0)) → f10_out1(s(z0))
f18_in → U5(f21_in)
U5(f21_out1(z0)) → U6(f22_in(z0), z0)
U6(f22_out1, z0) → f18_out1(z0)
f8_in → U7(f9_in, f10_in)
U7(f9_out1, z0) → f8_out1
U7(z0, f10_out1(z1)) → f8_out2(z1)

Tuples:

F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(F10_IN)
F58_IN(s(s(z0))) → c4(F58_IN(z0))
F2_IN → c1
F22_IN(s(z0)) → c1
F10_IN → c1
F18_IN → c1
U5'(f21_out1(z0)) → c1
F8_IN → c1

S tuples:

F58_IN(s(s(z0))) → c4(F58_IN(z0))

K tuples:

F2_IN → c1(F8_IN)
F8_IN → c1(F10_IN)
F2_IN → c1
F10_IN → c1
F8_IN → c1
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
F18_IN → c1
U5'(f21_out1(z0)) → c1(F22_IN(z0))
U5'(f21_out1(z0)) → c1
F22_IN(s(z0)) → c1(F58_IN(z0))
F22_IN(s(z0)) → c1

Defined Rule Symbols:

f2_in, U1, f58_in, U2, f21_in, f22_in, U3, f9_in, f10_in, U4, f18_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F2_IN, F22_IN, F10_IN, F18_IN, U5', F8_IN, F58_IN

Compound Symbols:

c1, c4, c1

(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.

F58_IN(s(s(z0))) → c4(F58_IN(z0))

We considered the (Usable) Rules:

f21_in → f21_out1(0)
f21_in → f21_out1(s(0))
f21_in → f21_out1(s(s(0)))
f21_in → f21_out1(s(s(s(0))))

And the Tuples:

F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(F10_IN)
F58_IN(s(s(z0))) → c4(F58_IN(z0))
F2_IN → c1
F22_IN(s(z0)) → c1
F10_IN → c1
F18_IN → c1
U5'(f21_out1(z0)) → c1
F8_IN → c1

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

POL(0) = 0   
POL(F10_IN) = [3]   
POL(F18_IN) = [3]   
POL(F22_IN(x₁)) = x₁   
POL(F2_IN) = [3]   
POL(F58_IN(x₁)) = x₁   
POL(F8_IN) = [3]   
POL(U5'(x₁)) = x₁   
POL(c1) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(f21_in) = [3]   
POL(f21_out1(x₁)) = x₁   
POL(s(x₁)) = [1] + x₁   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in → U1(f8_in)
U1(f8_out1) → f2_out1
U1(f8_out2(z0)) → f2_out1
f58_in(0) → f58_out1
f58_in(s(s(z0))) → U2(f58_in(z0), s(s(z0)))
U2(f58_out1, s(s(z0))) → f58_out1
f21_in → f21_out1(0)
f21_in → f21_out1(s(0))
f21_in → f21_out1(s(s(0)))
f21_in → f21_out1(s(s(s(0))))
f22_in(s(z0)) → U3(f58_in(z0), s(z0))
U3(f58_out1, s(z0)) → f22_out1
f9_in → f9_out1
f10_in → U4(f18_in)
U4(f18_out1(z0)) → f10_out1(s(z0))
f18_in → U5(f21_in)
U5(f21_out1(z0)) → U6(f22_in(z0), z0)
U6(f22_out1, z0) → f18_out1(z0)
f8_in → U7(f9_in, f10_in)
U7(f9_out1, z0) → f8_out1
U7(z0, f10_out1(z1)) → f8_out2(z1)

Tuples:

F2_IN → c1(F8_IN)
F22_IN(s(z0)) → c1(F58_IN(z0))
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
U5'(f21_out1(z0)) → c1(F22_IN(z0))
F8_IN → c1(F10_IN)
F58_IN(s(s(z0))) → c4(F58_IN(z0))
F2_IN → c1
F22_IN(s(z0)) → c1
F10_IN → c1
F18_IN → c1
U5'(f21_out1(z0)) → c1
F8_IN → c1

S tuples:none
K tuples:

F2_IN → c1(F8_IN)
F8_IN → c1(F10_IN)
F2_IN → c1
F10_IN → c1
F8_IN → c1
F10_IN → c1(F18_IN)
F18_IN → c1(U5'(f21_in))
F18_IN → c1
U5'(f21_out1(z0)) → c1(F22_IN(z0))
U5'(f21_out1(z0)) → c1
F22_IN(s(z0)) → c1(F58_IN(z0))
F22_IN(s(z0)) → c1
F58_IN(s(s(z0))) → c4(F58_IN(z0))

Defined Rule Symbols:

f2_in, U1, f58_in, U2, f21_in, f22_in, U3, f9_in, f10_in, U4, f18_in, U5, U6, f8_in, U7

Defined Pair Symbols:

F2_IN, F22_IN, F10_IN, F18_IN, U5', F8_IN, F58_IN

Compound Symbols:

c1, c4, c1

(21) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(22) BOUNDS(O(1), O(1))