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

(0) Obligation:

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


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) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of from: s, from, activate
The following defined symbols can occur below the 0th argument of s: s, from, activate
The following defined symbols can occur below the 1th argument of 2ndspos: from

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
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(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)))
plus(s(X), Y) → s(plus(X, Y))
times(s(X), Y) → plus(Y, times(X, Y))

(2) Obligation:

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


The TRS R consists of the following rules:

square(X) → times(X, X)
activate(n__from(X)) → from(activate(X))
from(X) → cons(X, n__from(n__s(X)))
2ndspos(0, Z) → rnil
pi(X) → 2ndspos(X, from(0))
plus(0, Y) → Y
from(X) → n__from(X)
s(X) → n__s(X)
activate(X) → X
times(0, Y) → 0
2ndsneg(0, Z) → rnil
activate(n__s(X)) → s(activate(X))

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) Obligation:

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


The TRS R consists of the following rules:

square(X) → times(X, X) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
2ndspos(0, Z) → rnil [1]
pi(X) → 2ndspos(X, from(0)) [1]
plus(0, Y) → Y [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
times(0, Y) → 0 [1]
2ndsneg(0, Z) → rnil [1]
activate(n__s(X)) → s(activate(X)) [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

square(X) → times(X, X) [1]
activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
2ndspos(0, Z) → rnil [1]
pi(X) → 2ndspos(X, from(0)) [1]
plus(0, Y) → Y [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
times(0, Y) → 0 [1]
2ndsneg(0, Z) → rnil [1]
activate(n__s(X)) → s(activate(X)) [1]

The TRS has the following type information:

square :: n__from:n__s:cons:0 → n__from:n__s:cons:0 times :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → n__from:n__s:cons:0 activate :: n__from:n__s:cons:0 → n__from:n__s:cons:0 n__from :: n__from:n__s:cons:0 → n__from:n__s:cons:0 from :: n__from:n__s:cons:0 → n__from:n__s:cons:0 cons :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → n__from:n__s:cons:0 n__s :: n__from:n__s:cons:0 → n__from:n__s:cons:0 2ndspos :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → rnil 0 :: n__from:n__s:cons:0 rnil :: rnil pi :: n__from:n__s:cons:0 → rnil plus :: n__from:n__s:cons:0 → plus → plus s :: n__from:n__s:cons:0 → n__from:n__s:cons:0 2ndsneg :: n__from:n__s:cons:0 → a → rnil

Rewrite Strategy: INNERMOST

(7) CompletionProof (UPPER BOUND(ID) transformation)

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

square
2ndspos
pi
plus
times
2ndsneg

(c) The following functions are completely defined:

activate
from
s

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: square(X) → times(X, X) [1] activate(n__from(X)) → from(activate(X)) [1] from(X) → cons(X, n__from(n__s(X))) [1] 2ndspos(0, Z) → rnil [1] pi(X) → 2ndspos(X, from(0)) [1] plus(0, Y) → Y [1] from(X) → n__from(X) [1] s(X) → n__s(X) [1] activate(X) → X [1] times(0, Y) → 0 [1] 2ndsneg(0, Z) → rnil [1] activate(n__s(X)) → s(activate(X)) [1] The TRS has the following type information: square :: n__from:n__s:cons:0 → n__from:n__s:cons:0 times :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → n__from:n__s:cons:0 activate :: n__from:n__s:cons:0 → n__from:n__s:cons:0 n__from :: n__from:n__s:cons:0 → n__from:n__s:cons:0 from :: n__from:n__s:cons:0 → n__from:n__s:cons:0 cons :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → n__from:n__s:cons:0 n__s :: n__from:n__s:cons:0 → n__from:n__s:cons:0 2ndspos :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → rnil 0 :: n__from:n__s:cons:0 rnil :: rnil pi :: n__from:n__s:cons:0 → rnil plus :: n__from:n__s:cons:0 → plus → plus s :: n__from:n__s:cons:0 → n__from:n__s:cons:0 2ndsneg :: n__from:n__s:cons:0 → a → rnil const :: plus const1 :: a Rewrite Strategy: INNERMOST

(9) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(10) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: square(X) → times(X, X) [1] activate(n__from(n__from(X'))) → from(from(activate(X'))) [2] activate(n__from(X)) → from(X) [2] activate(n__from(n__s(X''))) → from(s(activate(X''))) [2] from(X) → cons(X, n__from(n__s(X))) [1] 2ndspos(0, Z) → rnil [1] pi(X) → 2ndspos(X, cons(0, n__from(n__s(0)))) [2] pi(X) → 2ndspos(X, n__from(0)) [2] plus(0, Y) → Y [1] from(X) → n__from(X) [1] s(X) → n__s(X) [1] activate(X) → X [1] times(0, Y) → 0 [1] 2ndsneg(0, Z) → rnil [1] activate(n__s(n__from(X1))) → s(from(activate(X1))) [2] activate(n__s(X)) → s(X) [2] activate(n__s(n__s(X2))) → s(s(activate(X2))) [2] The TRS has the following type information: square :: n__from:n__s:cons:0 → n__from:n__s:cons:0 times :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → n__from:n__s:cons:0 activate :: n__from:n__s:cons:0 → n__from:n__s:cons:0 n__from :: n__from:n__s:cons:0 → n__from:n__s:cons:0 from :: n__from:n__s:cons:0 → n__from:n__s:cons:0 cons :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → n__from:n__s:cons:0 n__s :: n__from:n__s:cons:0 → n__from:n__s:cons:0 2ndspos :: n__from:n__s:cons:0 → n__from:n__s:cons:0 → rnil 0 :: n__from:n__s:cons:0 rnil :: rnil pi :: n__from:n__s:cons:0 → rnil plus :: n__from:n__s:cons:0 → plus → plus s :: n__from:n__s:cons:0 → n__from:n__s:cons:0 2ndsneg :: n__from:n__s:cons:0 → a → rnil const :: plus const1 :: a Rewrite Strategy: INNERMOST

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0
rnil => 0
const => 0
const1 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ s(s(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0
activate(z) -{ 2 }→ s(from(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 2 }→ from(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ from(s(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ from(from(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
pi(z) -{ 2 }→ 2ndspos(X, 1 + 0) :|: X >= 0, z = X
pi(z) -{ 2 }→ 2ndspos(X, 1 + 0 + (1 + (1 + 0))) :|: X >= 0, z = X
plus(z, z') -{ 1 }→ Y :|: z' = Y, Y >= 0, z = 0
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
square(z) -{ 1 }→ times(X, X) :|: X >= 0, z = X
times(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0

(13) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
2ndspos(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
times(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(s(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0
activate(z) -{ 2 }→ s(from(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 2 }→ from(s(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ from(from(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 3 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
pi(z) -{ 3 }→ 0 :|: X >= 0, z = X, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, X = 0
pi(z) -{ 3 }→ 0 :|: X >= 0, z = X, Z >= 0, 1 + 0 = Z, X = 0
plus(z, z') -{ 1 }→ Y :|: z' = Y, Y >= 0, z = 0
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
square(z) -{ 2 }→ 0 :|: X >= 0, z = X, X = Y, Y >= 0, X = 0
times(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0

(15) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(16) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ from } { 2ndsneg } { pi } { 2ndspos } { s } { times } { square } { plus } { activate }

(18) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndsneg}, {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 2·z

(20) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {from}, {2ndsneg}, {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: ?, size: O(n1) [3 + 2·z]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(22) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z]

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(24) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: 2ndsneg
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(26) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndsneg}, {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: ?, size: O(1) [0]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: 2ndsneg
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(28) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0]

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(30) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: pi
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(32) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {pi}, {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: ?, size: O(1) [0]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: pi
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(34) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0]

(35) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(36) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: 2ndspos
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(38) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {2ndspos}, {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: ?, size: O(1) [0]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: 2ndspos
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(40) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0]

(41) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(42) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: s
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(44) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {s}, {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: ?, size: O(n1) [1 + z]

(45) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: s
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(46) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z]

(47) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(48) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z]

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(50) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}, {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: ?, size: O(1) [0]

(51) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(52) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0]

(53) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(54) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0]

(55) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: square
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(56) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {square}, {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: ?, size: O(1) [0]

(57) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: square
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(58) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0]

(59) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(60) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0]

(61) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z'

(62) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {plus}, {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0] plus: runtime: ?, size: O(n1) [z']

(63) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(64) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0] plus: runtime: O(1) [1], size: O(n1) [z']

(65) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(66) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0] plus: runtime: O(1) [1], size: O(n1) [z']

(67) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(68) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {activate}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0] plus: runtime: O(1) [1], size: O(n1) [z'] activate: runtime: ?, size: EXP

(69) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 4 + 4·z

(70) Obligation:

Complexity RNTS consisting of the following rules:

2ndsneg(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
2ndspos(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ from(from(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X' + (1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 + (1 + (1 + 0)) = Z, z = 0
pi(z) -{ 3 }→ 0 :|: z >= 0, Z >= 0, 1 + 0 = Z, z = 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
s(z) -{ 1 }→ 1 + z :|: z >= 0
square(z) -{ 2 }→ 0 :|: z >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed:
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] 2ndsneg: runtime: O(1) [1], size: O(1) [0] pi: runtime: O(1) [3], size: O(1) [0] 2ndspos: runtime: O(1) [1], size: O(1) [0] s: runtime: O(1) [1], size: O(n1) [1 + z] times: runtime: O(1) [1], size: O(1) [0] square: runtime: O(1) [2], size: O(1) [0] plus: runtime: O(1) [1], size: O(n1) [z'] activate: runtime: O(n1) [4 + 4·z], size: EXP

(71) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(72) BOUNDS(1, n^1)