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:

f(0) → cons(0, n__f(n__s(n__0)))
f(s(0)) → f(p(s(0)))
p(s(X)) → X
f(X) → n__f(X)
s(X) → n__s(X)
0 → n__0
activate(n__f(X)) → f(activate(X))
activate(n__s(X)) → s(activate(X))
activate(n__0) → 0
activate(X) → X

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) 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:

f(0) → cons(0, n__f(n__s(n__0))) [1]
f(s(0)) → f(p(s(0))) [1]
p(s(X)) → X [1]
f(X) → n__f(X) [1]
s(X) → n__s(X) [1]
0 → n__0 [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__s(X)) → s(activate(X)) [1]
activate(n__0) → 0 [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

0 => 0'

(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:

f(0') → cons(0', n__f(n__s(n__0))) [1]
f(s(0')) → f(p(s(0'))) [1]
p(s(X)) → X [1]
f(X) → n__f(X) [1]
s(X) → n__s(X) [1]
0' → n__0 [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__s(X)) → s(activate(X)) [1]
activate(n__0) → 0' [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Removed the following rules with non-basic left-hand side, as they cannot be used in innermost rewriting:

f(0') → cons(0', n__f(n__s(n__0))) [1]
f(s(0')) → f(p(s(0'))) [1]
p(s(X)) → X [1]

Due to the following rules that have to be used instead:

0' → n__0 [1]
s(X) → n__s(X) [1]

(6) 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:

f(X) → n__f(X) [1]
s(X) → n__s(X) [1]
0' → n__0 [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__s(X)) → s(activate(X)) [1]
activate(n__0) → 0' [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

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

f(X) → n__f(X) [1]
s(X) → n__s(X) [1]
0' → n__0 [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__s(X)) → s(activate(X)) [1]
activate(n__0) → 0' [1]
activate(X) → X [1]

The TRS has the following type information:

f :: n__f:n__s:n__0 → n__f:n__s:n__0 n__f :: n__f:n__s:n__0 → n__f:n__s:n__0 s :: n__f:n__s:n__0 → n__f:n__s:n__0 n__s :: n__f:n__s:n__0 → n__f:n__s:n__0 0' :: n__f:n__s:n__0 n__0 :: n__f:n__s:n__0 activate :: n__f:n__s:n__0 → n__f:n__s:n__0

Rewrite Strategy: INNERMOST

(9) 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:
none

(c) The following functions are completely defined:

activate
0'
f
s

Due to the following rules being added:
none

And the following fresh constants: none

(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: f(X) → n__f(X) [1] s(X) → n__s(X) [1] 0' → n__0 [1] activate(n__f(X)) → f(activate(X)) [1] activate(n__s(X)) → s(activate(X)) [1] activate(n__0) → 0' [1] activate(X) → X [1] The TRS has the following type information: f :: n__f:n__s:n__0 → n__f:n__s:n__0 n__f :: n__f:n__s:n__0 → n__f:n__s:n__0 s :: n__f:n__s:n__0 → n__f:n__s:n__0 n__s :: n__f:n__s:n__0 → n__f:n__s:n__0 0' :: n__f:n__s:n__0 n__0 :: n__f:n__s:n__0 activate :: n__f:n__s:n__0 → n__f:n__s:n__0 Rewrite Strategy: INNERMOST

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

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

(12) 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: f(X) → n__f(X) [1] s(X) → n__s(X) [1] 0' → n__0 [1] activate(n__f(n__f(X'))) → f(f(activate(X'))) [2] activate(n__f(n__s(X''))) → f(s(activate(X''))) [2] activate(n__f(n__0)) → f(0') [2] activate(n__f(X)) → f(X) [2] activate(n__s(n__f(X1))) → s(f(activate(X1))) [2] activate(n__s(n__s(X2))) → s(s(activate(X2))) [2] activate(n__s(n__0)) → s(0') [2] activate(n__s(X)) → s(X) [2] activate(n__0) → 0' [1] activate(X) → X [1] The TRS has the following type information: f :: n__f:n__s:n__0 → n__f:n__s:n__0 n__f :: n__f:n__s:n__0 → n__f:n__s:n__0 s :: n__f:n__s:n__0 → n__f:n__s:n__0 n__s :: n__f:n__s:n__0 → n__f:n__s:n__0 0' :: n__f:n__s:n__0 n__0 :: n__f:n__s:n__0 activate :: n__f:n__s:n__0 → n__f:n__s:n__0 Rewrite Strategy: INNERMOST

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

n__0 => 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 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(f(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 2 }→ s(0') :|: z = 1 + 0
activate(z) -{ 2 }→ f(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ f(s(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ f(f(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 2 }→ f(0') :|: z = 1 + 0
activate(z) -{ 1 }→ 0' :|: z = 0
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(15) InliningProof (UPPER BOUND(ID) transformation)

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

f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
0' -{ 1 }→ 0 :|:

(16) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 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(f(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(X''))) :|: z = 1 + (1 + X''), X'' >= 0
activate(z) -{ 2 }→ f(f(activate(X'))) :|: X' >= 0, z = 1 + (1 + X')
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ 0' } { f } { s } { activate }

(20) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {0'}, {f}, {s}, {activate}

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {0'}, {f}, {s}, {activate}
Previous analysis results are:

0': runtime: ?, size: O(1) [0]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0]

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: ?, size: O(n1) [1 + z]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ f(0) :|: z = 1 + 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z]

(33) 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

(34) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z] s: runtime: ?, size: O(n1) [1 + z]

(35) 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

(36) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ s(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z] s: runtime: O(1) [1], size: O(n1) [1 + z]

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z] s: runtime: O(1) [1], size: O(n1) [1 + z]

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

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

(40) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z] s: runtime: O(1) [1], size: O(n1) [1 + z] activate: runtime: ?, size: O(n1) [z]

(41) 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: 14 + 4·z

(42) Obligation:

Complexity RNTS consisting of the following rules:

0' -{ 1 }→ 0 :|:
activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 4 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1, z = 1 + 0
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ s(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ s(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(s(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ 0 :|: z = 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
f(z) -{ 1 }→ 1 + z :|: z >= 0
s(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:

0': runtime: O(1) [1], size: O(1) [0] f: runtime: O(1) [1], size: O(n1) [1 + z] s: runtime: O(1) [1], size: O(n1) [1 + z] activate: runtime: O(n1) [14 + 4·z], size: O(n1) [z]

(43) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(44) BOUNDS(1, n^1)