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 CdtUnreachableProof (⇔, 0 ms)
↳4 CdtProblem
↳5 CdtLeafRemovalProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CdtProblem
↳9 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 88 ms)
↳10 CdtProblem
↳11 SIsEmptyProof (⇔, 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)))
2ndspos(0, Z) → rnil
2ndspos(s(N), cons(X, Z)) → 2ndspos(s(N), cons2(X, activate(Z)))
2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(Y), 2ndsneg(N, activate(Z)))
2ndsneg(0, Z) → rnil
2ndsneg(s(N), cons(X, Z)) → 2ndsneg(s(N), cons2(X, activate(Z)))
2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(Y), 2ndspos(N, activate(Z)))
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
plus(s(X), Y) → s(plus(X, Y))
times(0, Y) → 0
times(s(X), Y) → plus(Y, times(X, Y))
square(X) → times(X, X)
from(X) → n__from(X)
s(X) → n__s(X)
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:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
2ndspos(0, z0) → rnil
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2)))
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3)))
2ndsneg(0, z0) → rnil
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2)))
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3)))
pi(z0) → 2ndspos(z0, from(0))
plus(0, z0) → z0
plus(s(z0), z1) → s(plus(z0, z1))
times(0, z0) → 0
times(s(z0), z1) → plus(z1, times(z0, z1))
square(z0) → times(z0, z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2))
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3))
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2))
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3))
PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0))
PLUS(s(z0), z1) → c10(S(plus(z0, z1)), PLUS(z0, z1))
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
SQUARE(z0) → c13(TIMES(z0, z0))
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2))
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3))
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2))
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3))
PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0))
PLUS(s(z0), z1) → c10(S(plus(z0, z1)), PLUS(z0, z1))
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
SQUARE(z0) → c13(TIMES(z0, z0))
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

2NDSPOS, 2NDSNEG, PI, PLUS, TIMES, SQUARE, ACTIVATE

Compound Symbols:

c3, c4, c6, c7, c8, c10, c12, c13, c15, c16

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

2NDSPOS(s(z0), cons(z1, z2)) → c3(2NDSPOS(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2))
2NDSPOS(s(z0), cons2(z1, cons(z2, z3))) → c4(2NDSNEG(z0, activate(z3)), ACTIVATE(z3))
2NDSNEG(s(z0), cons(z1, z2)) → c6(2NDSNEG(s(z0), cons2(z1, activate(z2))), S(z0), ACTIVATE(z2))
2NDSNEG(s(z0), cons2(z1, cons(z2, z3))) → c7(2NDSPOS(z0, activate(z3)), ACTIVATE(z3))
PLUS(s(z0), z1) → c10(S(plus(z0, z1)), PLUS(z0, z1))
TIMES(s(z0), z1) → c12(PLUS(z1, times(z0, z1)), TIMES(z0, z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
2ndspos(0, z0) → rnil
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2)))
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3)))
2ndsneg(0, z0) → rnil
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2)))
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3)))
pi(z0) → 2ndspos(z0, from(0))
plus(0, z0) → z0
plus(s(z0), z1) → s(plus(z0, z1))
times(0, z0) → 0
times(s(z0), z1) → plus(z1, times(z0, z1))
square(z0) → times(z0, z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0))
SQUARE(z0) → c13(TIMES(z0, z0))
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0))
SQUARE(z0) → c13(TIMES(z0, z0))
ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

PI, SQUARE, ACTIVATE

Compound Symbols:

c8, c13, c15, c16

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

Removed 2 trailing nodes:

PI(z0) → c8(2NDSPOS(z0, from(0)), FROM(0))
SQUARE(z0) → c13(TIMES(z0, z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
2ndspos(0, z0) → rnil
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2)))
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3)))
2ndsneg(0, z0) → rnil
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2)))
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3)))
pi(z0) → 2ndspos(z0, from(0))
plus(0, z0) → z0
plus(s(z0), z1) → s(plus(z0, z1))
times(0, z0) → 0
times(s(z0), z1) → plus(z1, times(z0, z1))
square(z0) → times(z0, z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c15(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c16

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

Removed 2 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
2ndspos(0, z0) → rnil
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2)))
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3)))
2ndsneg(0, z0) → rnil
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2)))
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3)))
pi(z0) → 2ndspos(z0, from(0))
plus(0, z0) → z0
plus(s(z0), z1) → s(plus(z0, z1))
times(0, z0) → 0
times(s(z0), z1) → plus(z1, times(z0, z1))
square(z0) → times(z0, z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c16

(9) 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.

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = [4]x1   
POL(c15(x1)) = x1   
POL(c16(x1)) = x1   
POL(n__from(x1)) = [1] + x1   
POL(n__s(x1)) = [4] + x1   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
2ndspos(0, z0) → rnil
2ndspos(s(z0), cons(z1, z2)) → 2ndspos(s(z0), cons2(z1, activate(z2)))
2ndspos(s(z0), cons2(z1, cons(z2, z3))) → rcons(posrecip(z2), 2ndsneg(z0, activate(z3)))
2ndsneg(0, z0) → rnil
2ndsneg(s(z0), cons(z1, z2)) → 2ndsneg(s(z0), cons2(z1, activate(z2)))
2ndsneg(s(z0), cons2(z1, cons(z2, z3))) → rcons(negrecip(z2), 2ndspos(z0, activate(z3)))
pi(z0) → 2ndspos(z0, from(0))
plus(0, z0) → z0
plus(s(z0), z1) → s(plus(z0, z1))
times(0, z0) → 0
times(s(z0), z1) → plus(z1, times(z0, z1))
square(z0) → times(z0, z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
Defined Rule Symbols:

from, 2ndspos, 2ndsneg, pi, plus, times, square, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c15, c16

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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