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:

from(X) → cons(X, n__from(s(X)))
head(cons(X, XS)) → X
2nd(cons(X, XS)) → head(activate(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
sel(0, cons(X, XS)) → X
sel(s(N), cons(X, XS)) → sel(N, activate(XS))
from(X) → n__from(X)
take(X1, X2) → n__take(X1, X2)
activate(n__from(X)) → from(X)
activate(n__take(X1, X2)) → take(X1, X2)
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:

from(X) → cons(X, n__from(s(X))) [1]
head(cons(X, XS)) → X [1]
2nd(cons(X, XS)) → head(activate(XS)) [1]
take(0, XS) → nil [1]
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS))) [1]
sel(0, cons(X, XS)) → X [1]
sel(s(N), cons(X, XS)) → sel(N, activate(XS)) [1]
from(X) → n__from(X) [1]
take(X1, X2) → n__take(X1, X2) [1]
activate(n__from(X)) → from(X) [1]
activate(n__take(X1, X2)) → take(X1, X2) [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:

from(X) → cons(X, n__from(s(X))) [1]
head(cons(X, XS)) → X [1]
2nd(cons(X, XS)) → head(activate(XS)) [1]
take(0, XS) → nil [1]
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS))) [1]
sel(0, cons(X, XS)) → X [1]
sel(s(N), cons(X, XS)) → sel(N, activate(XS)) [1]
from(X) → n__from(X) [1]
take(X1, X2) → n__take(X1, X2) [1]
activate(n__from(X)) → from(X) [1]
activate(n__take(X1, X2)) → take(X1, X2) [1]
activate(X) → X [1]

The TRS has the following type information:

from :: s:0 → n__from:cons:nil:n__take cons :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take n__from :: s:0 → n__from:cons:nil:n__take s :: s:0 → s:0 head :: n__from:cons:nil:n__take → s:0 2nd :: n__from:cons:nil:n__take → s:0 activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take take :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take 0 :: s:0 nil :: n__from:cons:nil:n__take n__take :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take sel :: s:0 → n__from:cons:nil:n__take → s:0

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:

head
2nd
sel

(c) The following functions are completely defined:

activate
take
from

Due to the following rules being added:
none

And the following fresh constants: none

(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: from(X) → cons(X, n__from(s(X))) [1] head(cons(X, XS)) → X [1] 2nd(cons(X, XS)) → head(activate(XS)) [1] take(0, XS) → nil [1] take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS))) [1] sel(0, cons(X, XS)) → X [1] sel(s(N), cons(X, XS)) → sel(N, activate(XS)) [1] from(X) → n__from(X) [1] take(X1, X2) → n__take(X1, X2) [1] activate(n__from(X)) → from(X) [1] activate(n__take(X1, X2)) → take(X1, X2) [1] activate(X) → X [1] The TRS has the following type information: from :: s:0 → n__from:cons:nil:n__take cons :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take n__from :: s:0 → n__from:cons:nil:n__take s :: s:0 → s:0 head :: n__from:cons:nil:n__take → s:0 2nd :: n__from:cons:nil:n__take → s:0 activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take take :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take 0 :: s:0 nil :: n__from:cons:nil:n__take n__take :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take sel :: s:0 → n__from:cons:nil:n__take → s:0 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: from(X) → cons(X, n__from(s(X))) [1] head(cons(X, XS)) → X [1] 2nd(cons(X, n__from(X'))) → head(from(X')) [2] 2nd(cons(X, n__take(X1', X2'))) → head(take(X1', X2')) [2] 2nd(cons(X, XS)) → head(XS) [2] take(0, XS) → nil [1] take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS))) [1] sel(0, cons(X, XS)) → X [1] sel(s(N), cons(X, n__from(X''))) → sel(N, from(X'')) [2] sel(s(N), cons(X, n__take(X1'', X2''))) → sel(N, take(X1'', X2'')) [2] sel(s(N), cons(X, XS)) → sel(N, XS) [2] from(X) → n__from(X) [1] take(X1, X2) → n__take(X1, X2) [1] activate(n__from(X)) → from(X) [1] activate(n__take(X1, X2)) → take(X1, X2) [1] activate(X) → X [1] The TRS has the following type information: from :: s:0 → n__from:cons:nil:n__take cons :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take n__from :: s:0 → n__from:cons:nil:n__take s :: s:0 → s:0 head :: n__from:cons:nil:n__take → s:0 2nd :: n__from:cons:nil:n__take → s:0 activate :: n__from:cons:nil:n__take → n__from:cons:nil:n__take take :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take 0 :: s:0 nil :: n__from:cons:nil:n__take n__take :: s:0 → n__from:cons:nil:n__take → n__from:cons:nil:n__take sel :: s:0 → n__from:cons:nil:n__take → s:0 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
nil => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 2 }→ head(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
2nd(z) -{ 2 }→ head(from(X')) :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ from(X) :|: z = 1 + X, X >= 0
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
sel(z, z') -{ 2 }→ sel(N, take(X1'', X2'')) :|: X1'' >= 0, z = 1 + N, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), N >= 0
sel(z, z') -{ 2 }→ sel(N, from(X'')) :|: z' = 1 + X + (1 + X''), z = 1 + N, X >= 0, X'' >= 0, N >= 0
take(z, z') -{ 1 }→ 0 :|: z' = XS, z = 0, XS >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + N + activate(XS)) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
take(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2

(11) 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
sel(z, z') -{ 2 }→ sel(N, take(X1'', X2'')) :|: X1'' >= 0, z = 1 + N, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), N >= 0
sel(z, z') -{ 3 }→ sel(N, 1 + X') :|: z' = 1 + X + (1 + X''), z = 1 + N, X >= 0, X'' >= 0, N >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(N, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), z = 1 + N, X >= 0, X'' >= 0, N >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z' = XS, z = 0, XS >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + N + activate(XS)) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
take(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ from } { head } { take, activate } { 2nd } { sel }

(16) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {from}, {head}, {take,activate}, {2nd}, {sel}

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {from}, {head}, {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

(19) 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

(20) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {head}, {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {head}, {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {head}, {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

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

Computed SIZE bound using KoAT for: take
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z + 2·z'

Computed SIZE bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(30) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {take,activate}, {2nd}, {sel}
Previous analysis results are:

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

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

Computed RUNTIME bound using KoAT for: take
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 5 + 2·z'

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 9 + 2·z

(32) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 2 }→ head(take(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ take(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 2 }→ sel(z - 1, take(X1'', X2'')) :|: X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 1 }→ 1 + X + (1 + (z - 1) + activate(XS)) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {2nd}, {sel}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 8 + 2·X2' }→ s2 :|: s1 >= 0, s1 <= 1 * X1' + 2 * X2' + 1, s2 >= 0, s2 <= 1 * s1, X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 6 + 2·X2 }→ s'' :|: s'' >= 0, s'' <= 1 * X1 + 2 * X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 7 + 2·X2'' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1'' + 2 * X2'' + 1, X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 10 + 2·XS }→ 1 + X + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * XS + 1, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {2nd}, {sel}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z]

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

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 8 + 2·X2' }→ s2 :|: s1 >= 0, s1 <= 1 * X1' + 2 * X2' + 1, s2 >= 0, s2 <= 1 * s1, X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 6 + 2·X2 }→ s'' :|: s'' >= 0, s'' <= 1 * X1 + 2 * X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 7 + 2·X2'' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1'' + 2 * X2'' + 1, X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 10 + 2·XS }→ 1 + X + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * XS + 1, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed: {2nd}, {sel}
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z] 2nd: runtime: ?, size: O(n1) [2·z]

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

Computed RUNTIME bound using KoAT for: 2nd
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 19 + 2·z

(38) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 8 + 2·X2' }→ s2 :|: s1 >= 0, s1 <= 1 * X1' + 2 * X2' + 1, s2 >= 0, s2 <= 1 * s1, X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 6 + 2·X2 }→ s'' :|: s'' >= 0, s'' <= 1 * X1 + 2 * X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 7 + 2·X2'' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1'' + 2 * X2'' + 1, X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 10 + 2·XS }→ 1 + X + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * XS + 1, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

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

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z] 2nd: runtime: O(n1) [19 + 2·z], size: O(n1) [2·z]

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(40) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 8 + 2·X2' }→ s2 :|: s1 >= 0, s1 <= 1 * X1' + 2 * X2' + 1, s2 >= 0, s2 <= 1 * s1, X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 6 + 2·X2 }→ s'' :|: s'' >= 0, s'' <= 1 * X1 + 2 * X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 7 + 2·X2'' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1'' + 2 * X2'' + 1, X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 10 + 2·XS }→ 1 + X + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * XS + 1, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

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

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z] 2nd: runtime: O(n1) [19 + 2·z], size: O(n1) [2·z]

(41) 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: ?

(42) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 8 + 2·X2' }→ s2 :|: s1 >= 0, s1 <= 1 * X1' + 2 * X2' + 1, s2 >= 0, s2 <= 1 * s1, X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 6 + 2·X2 }→ s'' :|: s'' >= 0, s'' <= 1 * X1 + 2 * X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 7 + 2·X2'' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1'' + 2 * X2'' + 1, X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 10 + 2·XS }→ 1 + X + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * XS + 1, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

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

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z] 2nd: runtime: O(n1) [19 + 2·z], size: O(n1) [2·z] sel: runtime: ?, size: EXP

(43) 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: ?

(44) Obligation:

Complexity RNTS consisting of the following rules:

2nd(z) -{ 3 }→ X' :|: z = 1 + X + XS, X >= 0, XS >= 0, XS = 1 + X' + XS', X' >= 0, XS' >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' + (1 + (1 + X'')) = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 4 }→ X1 :|: z = 1 + X + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X'', 1 + X'' = 1 + X1 + XS, X1 >= 0, XS >= 0
2nd(z) -{ 8 + 2·X2' }→ s2 :|: s1 >= 0, s1 <= 1 * X1' + 2 * X2' + 1, s2 >= 0, s2 <= 1 * s1, X2' >= 0, X1' >= 0, X >= 0, z = 1 + X + (1 + X1' + X2')
activate(z) -{ 6 + 2·X2 }→ s'' :|: s'' >= 0, s'' <= 1 * X1 + 2 * X2 + 1, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 2 }→ 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
activate(z) -{ 2 }→ 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
head(z) -{ 1 }→ X :|: z = 1 + X + XS, X >= 0, XS >= 0
sel(z, z') -{ 1 }→ X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
sel(z, z') -{ 2 }→ sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
sel(z, z') -{ 7 + 2·X2'' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1'' + 2 * X2'' + 1, X1'' >= 0, X >= 0, X2'' >= 0, z' = 1 + X + (1 + X1'' + X2''), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X') :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X' + (1 + (1 + X'))) :|: z' = 1 + X + (1 + X''), X >= 0, X'' >= 0, z - 1 >= 0, X' >= 0, X'' = X'
take(z, z') -{ 1 }→ 0 :|: z = 0, z' >= 0
take(z, z') -{ 10 + 2·XS }→ 1 + X + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * XS + 1, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
take(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

from: runtime: O(1) [1], size: O(n1) [3 + 2·z] head: runtime: O(1) [1], size: O(n1) [z] take: runtime: O(n1) [5 + 2·z'], size: O(n1) [1 + z + 2·z'] activate: runtime: O(n1) [9 + 2·z], size: O(n1) [1 + 2·z] 2nd: runtime: O(n1) [19 + 2·z], size: O(n1) [2·z] sel: runtime: EXP, size: EXP

(45) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(46) BOUNDS(1, EXP)