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

0 CpxTRS
↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 InnermostUnusableRulesProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxWeightedTrs
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CpxTypedWeightedTrs
↳7 CompletionProof (UPPER BOUND(ID), 0 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CpxTypedWeightedCompleteTrs
↳11 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳12 CpxRNTS
↳13 InliningProof (UPPER BOUND(ID), 111 ms)
↳14 CpxRNTS
↳15 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CpxRNTS
↳17 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 124 ms)
↳20 CpxRNTS
↳21 IntTrsBoundProof (UPPER BOUND(ID), 57 ms)
↳22 CpxRNTS
↳23 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 65 ms)
↳26 CpxRNTS
↳27 IntTrsBoundProof (UPPER BOUND(ID), 7 ms)
↳28 CpxRNTS
↳29 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 104 ms)
↳32 CpxRNTS
↳33 IntTrsBoundProof (UPPER BOUND(ID), 4 ms)
↳34 CpxRNTS
↳35 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳36 CpxRNTS
↳37 IntTrsBoundProof (UPPER BOUND(ID), 374 ms)
↳38 CpxRNTS
↳39 IntTrsBoundProof (UPPER BOUND(ID), 67 ms)
↳40 CpxRNTS
↳41 FinalProof (⇔, 0 ms)
↳42 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(f(a)) → c(n__f(n__g(n__f(n__a))))
f(X) → n__f(X)
g(X) → n__g(X)
a → n__a
activate(n__f(X)) → f(activate(X))
activate(n__g(X)) → g(activate(X))
activate(n__a) → a
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(f(a)) → c(n__f(n__g(n__f(n__a)))) [1]
f(X) → n__f(X) [1]
g(X) → n__g(X) [1]
a → n__a [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(n__a) → a [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

(3) 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(f(a)) → c(n__f(n__g(n__f(n__a)))) [1]

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

f(X) → n__f(X) [1]

(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(X) → n__f(X) [1]
g(X) → n__g(X) [1]
a → n__a [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(n__a) → a [1]
activate(X) → 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:

f(X) → n__f(X) [1]
g(X) → n__g(X) [1]
a → n__a [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(n__a) → a [1]
activate(X) → X [1]

The TRS has the following type information:
f :: n__f:n__g:n__a → n__f:n__g:n__a
n__f :: n__f:n__g:n__a → n__f:n__g:n__a
g :: n__f:n__g:n__a → n__f:n__g:n__a
n__g :: n__f:n__g:n__a → n__f:n__g:n__a
a :: n__f:n__g:n__a
n__a :: n__f:n__g:n__a
activate :: n__f:n__g:n__a → n__f:n__g:n__a

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:
none

(c) The following functions are completely defined:


activate
g
a
f

Due to the following rules being added:
none

And the following fresh constants: none

(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:

f(X) → n__f(X) [1]
g(X) → n__g(X) [1]
a → n__a [1]
activate(n__f(X)) → f(activate(X)) [1]
activate(n__g(X)) → g(activate(X)) [1]
activate(n__a) → a [1]
activate(X) → X [1]

The TRS has the following type information:
f :: n__f:n__g:n__a → n__f:n__g:n__a
n__f :: n__f:n__g:n__a → n__f:n__g:n__a
g :: n__f:n__g:n__a → n__f:n__g:n__a
n__g :: n__f:n__g:n__a → n__f:n__g:n__a
a :: n__f:n__g:n__a
n__a :: n__f:n__g:n__a
activate :: n__f:n__g:n__a → n__f:n__g:n__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:

f(X) → n__f(X) [1]
g(X) → n__g(X) [1]
a → n__a [1]
activate(n__f(n__f(X'))) → f(f(activate(X'))) [2]
activate(n__f(n__g(X''))) → f(g(activate(X''))) [2]
activate(n__f(n__a)) → f(a) [2]
activate(n__f(X)) → f(X) [2]
activate(n__g(n__f(X1))) → g(f(activate(X1))) [2]
activate(n__g(n__g(X2))) → g(g(activate(X2))) [2]
activate(n__g(n__a)) → g(a) [2]
activate(n__g(X)) → g(X) [2]
activate(n__a) → a [1]
activate(X) → X [1]

The TRS has the following type information:
f :: n__f:n__g:n__a → n__f:n__g:n__a
n__f :: n__f:n__g:n__a → n__f:n__g:n__a
g :: n__f:n__g:n__a → n__f:n__g:n__a
n__g :: n__f:n__g:n__a → n__f:n__g:n__a
a :: n__f:n__g:n__a
n__a :: n__f:n__g:n__a
activate :: n__f:n__g:n__a → n__f:n__g:n__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:

n__a => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ g(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ g(g(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0
activate(z) -{ 2 }→ g(f(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 2 }→ g(a) :|: z = 1 + 0
activate(z) -{ 2 }→ f(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ f(g(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(a) :|: z = 1 + 0
activate(z) -{ 1 }→ a :|: z = 0
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
g(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(13) 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
g(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
a -{ 1 }→ 0 :|:

(14) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ g(g(activate(X2))) :|: z = 1 + (1 + X2), X2 >= 0
activate(z) -{ 2 }→ g(f(activate(X1))) :|: X1 >= 0, z = 1 + (1 + X1)
activate(z) -{ 3 }→ g(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(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:

a -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ g(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ g }
{ f }
{ a }
{ activate }

(18) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ g(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {a}, {activate}

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


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

(20) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ g(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {f}, {a}, {activate}
Previous analysis results are:
g: runtime: ?, size: O(n1) [1 + z]

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


Computed RUNTIME bound using CoFloCo for: g
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:

a -{ 1 }→ 0 :|:
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 3 }→ g(0) :|: z = 1 + 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {f}, {a}, {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + 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:

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

Function symbols to be analyzed: {f}, {a}, {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]

(25) 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

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {f}, {a}, {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: ?, size: O(n1) [1 + z]

(27) 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

(28) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {a}, {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]

(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:

a -{ 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 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {a}, {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]

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


Computed SIZE bound using CoFloCo for: a
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:

a -{ 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 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {a}, {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]
a: runtime: ?, size: O(1) [0]

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


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

(34) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 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 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]
a: runtime: O(1) [1], 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:

a -{ 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 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]
a: runtime: O(1) [1], size: O(1) [0]

(37) 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

(38) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 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 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {activate}
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]
a: runtime: O(1) [1], size: O(1) [0]
activate: runtime: ?, size: O(n1) [z]

(39) 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

(40) Obligation:

Complexity RNTS consisting of the following rules:

a -{ 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 }→ g(g(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ g(f(activate(z - 2))) :|: z - 2 >= 0
activate(z) -{ 2 }→ f(g(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
g(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:
g: runtime: O(1) [1], size: O(n1) [1 + z]
f: runtime: O(1) [1], size: O(n1) [1 + z]
a: runtime: O(1) [1], size: O(1) [0]
activate: runtime: O(n1) [14 + 4·z], size: O(n1) [z]

(41) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(42) BOUNDS(1, n^1)