The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^1)).

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtLeafRemovalProof (ComplexityIfPolyImplication, 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtUsableRulesProof (⇔, 0 ms)
↳8 CdtProblem
↳9 CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))), 24 ms)
↳10 CdtProblem
↳11 SIsEmptyProof (BOTH BOUNDS(ID, ID), 0 ms)
↳12 BOUNDS(O(1), O(1))

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

from(X) → cons(X, n__from(n__s(X)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
s(X) → n__s(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

FROM(z0) → c
FROM(z0) → c1
FIRST(0, z0) → c2
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
FIRST(z0, z1) → c4
SEL(0, cons(z0, z1)) → c5
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
S(z0) → c7
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c11
S tuples:

FROM(z0) → c
FROM(z0) → c1
FIRST(0, z0) → c2
FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
FIRST(z0, z1) → c4
SEL(0, cons(z0, z1)) → c5
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
S(z0) → c7
ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(z0) → c11
K tuples:none
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

FROM, FIRST, SEL, S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11

(3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 2 leading nodes:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
Removed 7 trailing nodes:

ACTIVATE(z0) → c11
FROM(z0) → c
FROM(z0) → c1
SEL(0, cons(z0, z1)) → c5
FIRST(0, z0) → c2
FIRST(z0, z1) → c4
S(z0) → c7

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__from(z0)) → c8(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

from, first, sel, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

(7) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(z0) → z0

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

(9) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [5] + [4]x1   
POL(c10(x1, x2)) = x1 + x2   
POL(c8(x1)) = x1   
POL(c9(x1)) = x1   
POL(n__first(x1, x2)) = [4] + x1 + x2   
POL(n__from(x1)) = [5] + x1   
POL(n__s(x1)) = [2] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c8(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c9(ACTIVATE(z0))
ACTIVATE(n__first(z0, z1)) → c10(ACTIVATE(z0), ACTIVATE(z1))
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c8, c9, c10

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

The set S is empty

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