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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔)
↳4 CdtProblem
↳5 CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication)
↳6 CdtProblem
↳7 CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID))
↳8 CdtProblem
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳10 CdtProblem
↳11 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))))
↳12 CdtProblem
↳13 SIsEmptyProof (⇔)
↳14 BOUNDS(O(1), O(1))

(0) Obligation:

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

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
S tuples:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, cons, s, activate

Defined Pair Symbols:

2ND, FROM, ACTIVATE

Compound Symbols:

c, c1, c5, c6, c7

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
S tuples:

FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))
ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, cons, s, activate

Defined Pair Symbols:

FROM, ACTIVATE

Compound Symbols:

c1, c5, c6, c7

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 4 dangling nodes:

FROM(z0) → c1(CONS(z0, n__from(n__s(z0))))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__cons(z0, z1)) → c5(CONS(activate(z0), z1), ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, cons, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

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

Removed 3 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

2nd, from, cons, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

(9) 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__cons(z0, z1)) → c5(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = x12   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(n__cons(x1, x2)) = [1] + x1   
POL(n__from(x1)) = x1   
POL(n__s(x1)) = x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
K tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
Defined Rule Symbols:

2nd, from, cons, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

(11) 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__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [2]x1 + [3]x12   
POL(c5(x1)) = x1   
POL(c6(x1)) = x1   
POL(c7(x1)) = x1   
POL(n__cons(x1, x2)) = [1] + x1   
POL(n__from(x1)) = [2] + x1   
POL(n__s(x1)) = [2] + x1   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons(z0, n__cons(z1, z2))) → activate(z1)
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
cons(z0, z1) → n__cons(z0, z1)
s(z0) → n__s(z0)
activate(n__cons(z0, z1)) → cons(activate(z0), z1)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__cons(z0, z1)) → c5(ACTIVATE(z0))
ACTIVATE(n__from(z0)) → c6(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c7(ACTIVATE(z0))
Defined Rule Symbols:

2nd, from, cons, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6, c7

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))