Runtime Complexity proof of /tmp/tmpE2rRAt/evenodd.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 LPReorderTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 60 ms)

↳4 CdtProblem

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

    ↳6 CdtProblem

    ↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 100 ms)

    ↳8 CdtProblem

    ↳9 SIsEmptyProof (⇔, 0 ms)

    ↳10 BOUNDS(O(1), O(1))

↳11 PrologToCdtProblemTransformerProof (UPPER BOUND (ID), 73 ms)

↳12 CdtProblem

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

    ↳14 CdtProblem

    ↳15 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 96 ms)

    ↳16 CdtProblem

(0) Obligation:

Clauses:

even(0).
even(s(s(0))).
even(s(s(s(X)))) :- odd(X).
odd(s(0)).
odd(s(X)) :- even(s(s(X))).

Query: even(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

even(0).
even(s(s(0))).
odd(s(0)).
even(s(s(s(X)))) :- odd(X).
odd(s(X)) :- even(s(s(X))).

Query: even(g)

(3) 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 | even(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | even(0), SCOPE: 1, CLAUSE: 1 | even(0), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: even(T1), SCOPE: 1, CLAUSE: 1 | even(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: even(0), SCOPE: 1, CLAUSE: 1 | even(0), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: even(0), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | even(s(s(0))), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: even(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\neven(T1) !=? even(0)\neven(T1) !=? even(s(s(0)))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: even(s(s(0))), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: odd(T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: odd(T4), SCOPE: 2, CLAUSE: 2 | odd(T4), SCOPE: 2, CLAUSE: 4\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true | odd(s(0)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: odd(T4), SCOPE: 2, CLAUSE: 4\n\nT4 is ground\nodd(T4) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: odd(s(0)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: even(s(s(0)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: even(s(s(0))), SCOPE: 3, CLAUSE: 0 | even(s(s(0))), SCOPE: 3, CLAUSE: 1 | even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: even(s(s(0))), SCOPE: 3, CLAUSE: 1 | even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true | even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: even(s(s(T7)))\n\nT7 is ground\nodd(s(T7)) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 51 [arrowhead = none ];
51 -> 4;

52 [label="EVAL with clause\neven(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 52 [arrowhead = none ];
52 -> 6;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 53 [arrowhead = none ];
53 -> 8;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 54 [arrowhead = none ];
54 -> 10;

55 [label="EVAL with clause\neven(s(s(0))).\nand substitutionT1 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 55 [arrowhead = none ];
55 -> 16;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="BACKTRACK\nfor clause: even(s(s(0)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 57 [arrowhead = none ];
57 -> 12;

58 [label="BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 58 [arrowhead = none ];
58 -> 14;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 59 [arrowhead = none ];
59 -> 20;

60 [label="EVAL with clause\neven(s(s(s(X5)))) :- odd(X5).\nand substitutionX5 -> T4,\nT1 -> s(s(s(T4)))", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 60 [arrowhead = none ];
60 -> 24;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 61 [arrowhead = none ];
61 -> 27;

62 [label="BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 62 [arrowhead = none ];
62 -> 22;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 63 [arrowhead = none ];
63 -> 33;

64 [label="EVAL with clause\nodd(s(0)).\nand substitutionT4 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 64 [arrowhead = none ];
64 -> 35;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 65 [arrowhead = none ];
65 -> 36;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 66 [arrowhead = none ];
66 -> 39;

67 [label="EVAL with clause\nodd(s(X12)) :- even(s(s(X12))).\nand substitutionX12 -> T7,\nT4 -> s(T7)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 67 [arrowhead = none ];
67 -> 49;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 68 [arrowhead = none ];
68 -> 50;

69 [label="ONLY EVAL with clause\nodd(s(X8)) :- even(s(s(X8))).\nand substitutionX8 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 69 [arrowhead = none ];
69 -> 41;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 70 [arrowhead = none ];
70 -> 44;

71 [label="BACKTRACK\nfor clause: even(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 71 [arrowhead = none ];
71 -> 45;

72 [label="ONLY EVAL with clause\neven(s(s(0))).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 72 [arrowhead = none ];
72 -> 46;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 73 [arrowhead = none ];
73 -> 47;

74 [label="BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 74 [arrowhead = none ];
74 -> 48;

75 [label="INSTANCE with matching:\nT1 -> s(s(T7))", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 75 [arrowhead = none , style=dashed];
75 -> 2[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(s(s(s(s(z0))))) → c3(U1'(f2_in(s(s(z0))), s(s(s(s(z0))))), F2_IN(s(s(z0))))

S tuples:

F2_IN(s(s(s(s(z0))))) → c3(U1'(f2_in(s(s(z0))), s(s(s(s(z0))))), F2_IN(s(s(z0))))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c3

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(s(s(s(s(z0))))) → c3(F2_IN(s(s(z0))))

S tuples:

F2_IN(s(s(s(s(z0))))) → c3(F2_IN(s(s(z0))))

K tuples:none
Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c3

(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.

F2_IN(s(s(s(s(z0))))) → c3(F2_IN(s(s(z0))))

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

F2_IN(s(s(s(s(z0))))) → c3(F2_IN(s(s(z0))))

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

POL(F2_IN(x₁)) = [2]x₁
POL(c3(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F2_IN(s(s(s(s(z0))))) → c3(F2_IN(s(s(z0))))

S tuples:none
K tuples:

F2_IN(s(s(s(s(z0))))) → c3(F2_IN(s(s(z0))))

Defined Rule Symbols:

f2_in, U1

Defined Pair Symbols:

F2_IN

Compound Symbols:

c3

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))

(11) 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 | even(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | even(0), SCOPE: 1, CLAUSE: 1 | even(0), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: even(T1), SCOPE: 1, CLAUSE: 1 | even(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: even(0), SCOPE: 1, CLAUSE: 1 | even(0), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: even(0), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | even(s(s(0))), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: even(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\neven(T1) !=? even(0)\neven(T1) !=? even(s(s(0)))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: even(s(s(0))), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: odd(T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: odd(T3), SCOPE: 2, CLAUSE: 2 | odd(T3), SCOPE: 2, CLAUSE: 4\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true | odd(s(0)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: odd(T3), SCOPE: 2, CLAUSE: 4\n\nT3 is ground\nodd(T3) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: odd(s(0)), SCOPE: 2, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: even(s(s(0)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: even(s(s(0))), SCOPE: 3, CLAUSE: 0 | even(s(s(0))), SCOPE: 3, CLAUSE: 1 | even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: even(s(s(0))), SCOPE: 3, CLAUSE: 1 | even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true | even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(s(s(0))), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: even(s(s(T6)))\n\nT6 is ground\nodd(s(T6)) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 44 [arrowhead = none ];
44 -> 3;

45 [label="EVAL with clause\neven(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 45 [arrowhead = none ];
45 -> 5;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 46 [arrowhead = none ];
46 -> 7;

47 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 47 [arrowhead = none ];
47 -> 9;

48 [label="EVAL with clause\neven(s(s(0))).\nand substitutionT1 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 48 [arrowhead = none ];
48 -> 15;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 49 [arrowhead = none ];
49 -> 17;

50 [label="BACKTRACK\nfor clause: even(s(s(0)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 50 [arrowhead = none ];
50 -> 11;

51 [label="BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 51 [arrowhead = none ];
51 -> 13;

52 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 52 [arrowhead = none ];
52 -> 19;

53 [label="EVAL with clause\neven(s(s(s(X4)))) :- odd(X4).\nand substitutionX4 -> T3,\nT1 -> s(s(s(T3)))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 53 [arrowhead = none ];
53 -> 23;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 54 [arrowhead = none ];
54 -> 25;

55 [label="BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 55 [arrowhead = none ];
55 -> 21;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 56 [arrowhead = none ];
56 -> 26;

57 [label="EVAL with clause\nodd(s(0)).\nand substitutionT3 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 57 [arrowhead = none ];
57 -> 28;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 58 [arrowhead = none ];
58 -> 29;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 59 [arrowhead = none ];
59 -> 30;

60 [label="EVAL with clause\nodd(s(X11)) :- even(s(s(X11))).\nand substitutionX11 -> T6,\nT3 -> s(T6)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 60 [arrowhead = none ];
60 -> 42;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 61 [arrowhead = none ];
61 -> 43;

62 [label="ONLY EVAL with clause\nodd(s(X7)) :- even(s(s(X7))).\nand substitutionX7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 62 [arrowhead = none ];
62 -> 31;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 63 [arrowhead = none ];
63 -> 32;

64 [label="BACKTRACK\nfor clause: even(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 64 [arrowhead = none ];
64 -> 34;

65 [label="ONLY EVAL with clause\neven(s(s(0))).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 65 [arrowhead = none ];
65 -> 37;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 66 [arrowhead = none ];
66 -> 38;

67 [label="BACKTRACK\nfor clause: even(s(s(s(X)))) :- odd(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 67 [arrowhead = none ];
67 -> 40;

68 [label="INSTANCE with matching:\nT1 -> s(s(T6))", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 68 [arrowhead = none , style=dashed];
68 -> 1[style=dashed];

}

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F1_IN(s(s(s(s(z0))))) → c3(U1'(f1_in(s(s(z0))), s(s(s(s(z0))))), F1_IN(s(s(z0))))

S tuples:

F1_IN(s(s(s(s(z0))))) → c3(U1'(f1_in(s(s(z0))), s(s(s(s(z0))))), F1_IN(s(s(z0))))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c3

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

Removed 1 trailing tuple parts

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F1_IN(s(s(s(s(z0))))) → c3(F1_IN(s(s(z0))))

S tuples:

F1_IN(s(s(s(s(z0))))) → c3(F1_IN(s(s(z0))))

K tuples:none
Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c3

(15) 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(s(s(z0))))) → c3(F1_IN(s(s(z0))))

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

F1_IN(s(s(s(s(z0))))) → c3(F1_IN(s(s(z0))))

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

POL(F1_IN(x₁)) = [2]x₁
POL(c3(x₁)) = x₁
POL(s(x₁)) = [1] + x₁

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

F1_IN(s(s(s(s(z0))))) → c3(F1_IN(s(s(z0))))

S tuples:none
K tuples:

F1_IN(s(s(s(s(z0))))) → c3(F1_IN(s(s(z0))))

Defined Rule Symbols:

f1_in, U1

Defined Pair Symbols:

F1_IN

Compound Symbols:

c3