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:

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(n__s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

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

The following rules are not reachable from basic terms in the dependency graph and can be removed:
2nd(cons(X, n__cons(Y, Z))) → activate(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:

activate(n__from(X)) → from(activate(X))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
activate(X) → X
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
cons(X1, X2) → n__cons(X1, X2)
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:

activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__cons(X1, X2)) → cons(activate(X1), X2) [1]
cons(X1, X2) → n__cons(X1, X2) [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:

activate(n__from(X)) → from(activate(X)) [1]
from(X) → cons(X, n__from(n__s(X))) [1]
from(X) → n__from(X) [1]
s(X) → n__s(X) [1]
activate(X) → X [1]
activate(n__cons(X1, X2)) → cons(activate(X1), X2) [1]
cons(X1, X2) → n__cons(X1, X2) [1]
activate(n__s(X)) → s(activate(X)) [1]

The TRS has the following type information:

activate :: n__from:n__s:n__cons → n__from:n__s:n__cons n__from :: n__from:n__s:n__cons → n__from:n__s:n__cons from :: n__from:n__s:n__cons → n__from:n__s:n__cons cons :: n__from:n__s:n__cons → n__from:n__s:n__cons → n__from:n__s:n__cons n__s :: n__from:n__s:n__cons → n__from:n__s:n__cons s :: n__from:n__s:n__cons → n__from:n__s:n__cons n__cons :: n__from:n__s:n__cons → n__from:n__s:n__cons → n__from:n__s:n__cons

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
from
cons
s

Due to the following rules being added:
none

And the following fresh constants:

const

(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: activate(n__from(X)) → from(activate(X)) [1] from(X) → cons(X, n__from(n__s(X))) [1] from(X) → n__from(X) [1] s(X) → n__s(X) [1] activate(X) → X [1] activate(n__cons(X1, X2)) → cons(activate(X1), X2) [1] cons(X1, X2) → n__cons(X1, X2) [1] activate(n__s(X)) → s(activate(X)) [1] The TRS has the following type information: activate :: n__from:n__s:n__cons → n__from:n__s:n__cons n__from :: n__from:n__s:n__cons → n__from:n__s:n__cons from :: n__from:n__s:n__cons → n__from:n__s:n__cons cons :: n__from:n__s:n__cons → n__from:n__s:n__cons → n__from:n__s:n__cons n__s :: n__from:n__s:n__cons → n__from:n__s:n__cons s :: n__from:n__s:n__cons → n__from:n__s:n__cons n__cons :: n__from:n__s:n__cons → n__from:n__s:n__cons → n__from:n__s:n__cons const :: n__from:n__s:n__cons 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: activate(n__from(n__from(X'))) → from(from(activate(X'))) [2] activate(n__from(X)) → from(X) [2] activate(n__from(n__cons(X1', X2'))) → from(cons(activate(X1'), X2')) [2] activate(n__from(n__s(X''))) → from(s(activate(X''))) [2] from(X) → cons(X, n__from(n__s(X))) [1] from(X) → n__from(X) [1] s(X) → n__s(X) [1] activate(X) → X [1] activate(n__cons(n__from(X3), X2)) → cons(from(activate(X3)), X2) [2] activate(n__cons(X1, X2)) → cons(X1, X2) [2] activate(n__cons(n__cons(X1'', X2''), X2)) → cons(cons(activate(X1''), X2''), X2) [2] activate(n__cons(n__s(X4), X2)) → cons(s(activate(X4)), X2) [2] cons(X1, X2) → n__cons(X1, X2) [1] activate(n__s(n__from(X5))) → s(from(activate(X5))) [2] activate(n__s(X)) → s(X) [2] activate(n__s(n__cons(X11, X21))) → s(cons(activate(X11), X21)) [2] activate(n__s(n__s(X6))) → s(s(activate(X6))) [2] The TRS has the following type information: activate :: n__from:n__s:n__cons → n__from:n__s:n__cons n__from :: n__from:n__s:n__cons → n__from:n__s:n__cons from :: n__from:n__s:n__cons → n__from:n__s:n__cons cons :: n__from:n__s:n__cons → n__from:n__s:n__cons → n__from:n__s:n__cons n__s :: n__from:n__s:n__cons → n__from:n__s:n__cons s :: n__from:n__s:n__cons → n__from:n__s:n__cons n__cons :: n__from:n__s:n__cons → n__from:n__s:n__cons → n__from:n__s:n__cons const :: n__from:n__s:n__cons 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:

const => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(X) :|: z = 1 + X, X >= 0
activate(z) -{ 2 }→ s(s(activate(X6))) :|: X6 >= 0, z = 1 + (1 + X6)
activate(z) -{ 2 }→ s(from(activate(X5))) :|: X5 >= 0, z = 1 + (1 + X5)
activate(z) -{ 2 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 0
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')
activate(z) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
cons(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
from(z) -{ 1 }→ cons(X, 1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
s(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:

s(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
cons(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
from(z) -{ 1 }→ cons(X, 1 + (1 + X)) :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X

(14) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 2 }→ s(s(activate(X6))) :|: X6 >= 0, z = 1 + (1 + X6)
activate(z) -{ 2 }→ s(from(activate(X5))) :|: X5 >= 0, z = 1 + (1 + X5)
activate(z) -{ 2 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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')
activate(z) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 2 }→ 1 + X1 + X2 :|: X >= 0, z = X, X1 >= 0, X2 >= 0, X = X1, 1 + (1 + X) = X2
s(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:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ from } { cons } { s } { activate }

(18) Obligation:

Complexity RNTS consisting of the following rules:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {from}, {cons}, {s}, {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:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed: {from}, {cons}, {s}, {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: 2

(22) Obligation:

Complexity RNTS consisting of the following rules:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

from: runtime: O(1) [2], 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:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

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

Computed RUNTIME bound using CoFloCo for: cons
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:

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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 3 }→ cons(X', 1 + (1 + X')) :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

(31) 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

(32) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

(33) 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

(34) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

(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:

activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

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

(37) 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: ?

(38) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

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

from: runtime: O(1) [2], size: O(n1) [3 + 2·z] cons: runtime: O(1) [1], size: O(n1) [1 + z + z'] s: runtime: O(1) [1], size: O(n1) [1 + z] activate: runtime: ?, size: EXP

(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: 11 + 37·z

(40) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 4 }→ s :|: s >= 0, s <= 1 * X' + 1 * (1 + (1 + X')) + 1, z - 1 >= 0, X' >= 0, z - 1 = X'
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 }→ s(cons(activate(X11), X21)) :|: z = 1 + (1 + X11 + X21), X11 >= 0, X21 >= 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) -{ 2 }→ from(cons(activate(X1'), X2')) :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0
activate(z) -{ 2 }→ cons(s(activate(X4)), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2
activate(z) -{ 2 }→ cons(from(activate(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0
activate(z) -{ 2 }→ cons(cons(activate(X1''), X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0
activate(z) -{ 3 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 3 }→ 1 + X1' + X2' :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2, X1' >= 0, X2' >= 0, X1 = X1', X2 = X2'
cons(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 2 }→ 1 + z + X2 :|: z >= 0, X2 >= 0, 1 + (1 + z) = X2
s(z) -{ 1 }→ 1 + z :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:

from: runtime: O(1) [2], size: O(n1) [3 + 2·z] cons: runtime: O(1) [1], size: O(n1) [1 + z + z'] s: runtime: O(1) [1], size: O(n1) [1 + z] activate: runtime: O(n1) [11 + 37·z], size: EXP

(41) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(42) BOUNDS(1, n^1)