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

(0) Obligation:

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


The TRS R consists of the following rules:

f(X) → cons(X, n__f(g(X)))
g(0) → s(0)
g(s(X)) → s(s(g(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
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, EXP).


The TRS R consists of the following rules:

f(X) → cons(X, n__f(g(X))) [1]
g(0) → s(0) [1]
g(s(X)) → s(s(g(X))) [1]
sel(0, cons(X, Y)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [1]
f(X) → n__f(X) [1]
activate(n__f(X)) → f(X) [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

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

f(X) → cons(X, n__f(g(X))) [1]
g(0) → s(0) [1]
g(s(X)) → s(s(g(X))) [1]
sel(0, cons(X, Y)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [1]
f(X) → n__f(X) [1]
activate(n__f(X)) → f(X) [1]
activate(X) → X [1]

The TRS has the following type information:

f :: 0:s → n__f:cons cons :: 0:s → n__f:cons → n__f:cons n__f :: 0:s → n__f:cons g :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s sel :: 0:s → n__f:cons → 0:s activate :: n__f:cons → n__f:cons

Rewrite Strategy: INNERMOST

(5) 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:

sel

(c) The following functions are completely defined:

activate
f
g

Due to the following rules being added:
none

And the following fresh constants:

const

(6) 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) → cons(X, n__f(g(X))) [1] g(0) → s(0) [1] g(s(X)) → s(s(g(X))) [1] sel(0, cons(X, Y)) → X [1] sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [1] f(X) → n__f(X) [1] activate(n__f(X)) → f(X) [1] activate(X) → X [1] The TRS has the following type information: f :: 0:s → n__f:cons cons :: 0:s → n__f:cons → n__f:cons n__f :: 0:s → n__f:cons g :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s sel :: 0:s → n__f:cons → 0:s activate :: n__f:cons → n__f:cons const :: n__f:cons Rewrite Strategy: INNERMOST

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

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

(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) → cons(X, n__f(g(X))) [1] g(0) → s(0) [1] g(s(X)) → s(s(g(X))) [1] sel(0, cons(X, Y)) → X [1] sel(s(X), cons(Y, n__f(X'))) → sel(X, f(X')) [2] sel(s(X), cons(Y, Z)) → sel(X, Z) [2] f(X) → n__f(X) [1] activate(n__f(X)) → f(X) [1] activate(X) → X [1] The TRS has the following type information: f :: 0:s → n__f:cons cons :: 0:s → n__f:cons → n__f:cons n__f :: 0:s → n__f:cons g :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s sel :: 0:s → n__f:cons → 0:s activate :: n__f:cons → n__f:cons const :: n__f:cons Rewrite Strategy: INNERMOST

(9) 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
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ f(X) :|: z = 1 + X, X >= 0
f(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
f(z) -{ 1 }→ 1 + X + (1 + g(X)) :|: X >= 0, z = X
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 }→ 1 + (1 + g(X)) :|: z = 1 + X, X >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(X, Z) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(X, f(X')) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + z + (1 + g(z)) :|: z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 }→ 1 + (1 + g(z - 1)) :|: z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

Found the following analysis order by SCC decomposition:

{ g } { f } { activate } { sel }

(14) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + z + (1 + g(z)) :|: z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 }→ 1 + (1 + g(z - 1)) :|: z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

(15) 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 + 2·z

(16) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + z + (1 + g(z)) :|: z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 }→ 1 + (1 + g(z - 1)) :|: z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

g: runtime: ?, size: O(n1) [1 + 2·z]

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 1 }→ 1 + z + (1 + g(z)) :|: z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 }→ 1 + (1 + g(z - 1)) :|: z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(21) 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: 3 + 3·z

(22) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 2 }→ sel(z - 1, f(X')) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

(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:

activate(z) -{ 2 + z }→ s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 4 + X' }→ sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 2 + z }→ s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 4 + X' }→ sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

g: runtime: O(n1) [1 + z], size: O(n1) [1 + 2·z] f: runtime: O(n1) [2 + z], size: O(n1) [3 + 3·z] activate: runtime: ?, size: O(n1) [3·z]

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 2 + z }→ s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 4 + X' }→ sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

activate(z) -{ 2 + z }→ s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 4 + X' }→ sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

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

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

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 2 + z }→ s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 4 + X' }→ sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

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

g: runtime: O(n1) [1 + z], size: O(n1) [1 + 2·z] f: runtime: O(n1) [2 + z], size: O(n1) [3 + 3·z] activate: runtime: O(n1) [2 + z], size: O(n1) [3·z] sel: runtime: ?, size: EXP

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 2 + z }→ s1 :|: s1 >= 0, s1 <= 3 * (z - 1) + 3, z - 1 >= 0
activate(z) -{ 1 }→ z :|: z >= 0
f(z) -{ 1 }→ 1 + z :|: z >= 0
f(z) -{ 2 + z }→ 1 + z + (1 + s) :|: s >= 0, s <= 2 * z + 1, z >= 0
g(z) -{ 1 }→ 1 + 0 :|: z = 0
g(z) -{ 1 + z }→ 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1) + 1, z - 1 >= 0
sel(z, z') -{ 1 }→ X :|: Y >= 0, X >= 0, z' = 1 + X + Y, z = 0
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 4 + X' }→ sel(z - 1, s'') :|: s'' >= 0, s'' <= 3 * X' + 3, Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0

Function symbols to be analyzed:
Previous analysis results are:

g: runtime: O(n1) [1 + z], size: O(n1) [1 + 2·z] f: runtime: O(n1) [2 + z], size: O(n1) [3 + 3·z] activate: runtime: O(n1) [2 + z], size: O(n1) [3·z] sel: runtime: EXP, size: EXP

(37) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(38) BOUNDS(1, EXP)