The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

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

from(X) → cons(X, n__from(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)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(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))
activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c7(FROM(z0))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c7(FROM(z0))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c7, c8

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 4 dangling nodes:

ACTIVATE(n__from(z0)) → c7(FROM(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(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))
activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

K tuples:none
Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

(5) 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.

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))

We considered the (Usable) Rules:

activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → 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)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)

And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

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

POL(0) = [4]   
POL(ACTIVATE(x₁)) = 0   
POL(FIRST(x₁, x₂)) = 0   
POL(SEL(x₁, x₂)) = x₁   
POL(activate(x₁)) = [2] + [3]x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(cons(x₁, x₂)) = [3] + x₁   
POL(first(x₁, x₂)) = [3] + x₁ + [2]x₂   
POL(from(x₁)) = [4] + [2]x₁   
POL(n__first(x₁, x₂)) = [2] + x₁ + x₂   
POL(n__from(x₁)) = [3] + x₁   
POL(nil) = [3]   
POL(s(x₁)) = [1] + x₁   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(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))
activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

K tuples:

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))

Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

(7) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

We considered the (Usable) Rules:

activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → 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)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)

And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

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

POL(0) = [2]   
POL(ACTIVATE(x₁)) = [1] + [2]x₁   
POL(FIRST(x₁, x₂)) = x₁ + [2]x₂   
POL(SEL(x₁, x₂)) = [2]x₁·x₂ + [3]x₁²   
POL(activate(x₁)) = [3] + x₁   
POL(c3(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c8(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₂   
POL(first(x₁, x₂)) = [2] + x₁ + x₂   
POL(from(x₁)) = [2]   
POL(n__first(x₁, x₂)) = x₁ + x₂   
POL(n__from(x₁)) = 0   
POL(nil) = [3]   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(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))
activate(n__from(z0)) → from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

S tuples:

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))

K tuples:

SEL(s(z0), cons(z1, z2)) → c6(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

Defined Rule Symbols:

from, first, sel, activate

Defined Pair Symbols:

FIRST, SEL, ACTIVATE

Compound Symbols:

c3, c6, c8

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

FIRST(s(z0), cons(z1, z2)) → c3(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c8(FIRST(z0, z1))

Now S is empty

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