Runtime Complexity (innermost) proof of /tmp/tmpqfUnfI/Ex1_Luc02b

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

0 CpxTRS

↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)

↳2 CpxWeightedTrs

↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CpxTypedWeightedTrs

↳5 CompletionProof (UPPER BOUND(ID), 0 ms)

↳6 CpxTypedWeightedCompleteTrs

↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CpxTypedWeightedCompleteTrs

↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)

↳10 CpxRNTS

↳11 InliningProof (UPPER BOUND(ID), 149 ms)

↳12 CpxRNTS

↳13 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CpxRNTS

↳15 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)

↳16 CpxRNTS

↳17 IntTrsBoundProof (UPPER BOUND(ID), 1027 ms)

↳18 CpxRNTS

↳19 IntTrsBoundProof (UPPER BOUND(ID), 147 ms)

↳20 CpxRNTS

↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳22 CpxRNTS

↳23 IntTrsBoundProof (UPPER BOUND(ID), 234 ms)

↳24 CpxRNTS

↳25 IntTrsBoundProof (UPPER BOUND(ID), 63 ms)

↳26 CpxRNTS

↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)

↳28 CpxRNTS

↳29 IntTrsBoundProof (UPPER BOUND(ID), 672 ms)

↳30 CpxRNTS

↳31 IntTrsBoundProof (UPPER BOUND(ID), 275 ms)

↳32 CpxRNTS

↳33 FinalProof (⇔, 0 ms)

↳34 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)))
first(0, Z) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
sel(0, cons(X, Z)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
first(X1, X2) → n__first(X1, X2)
activate(n__from(X)) → from(X)
activate(n__first(X1, X2)) → first(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]
first(0, Z) → nil [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
sel(0, cons(X, Z)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [1]
from(X) → n__from(X) [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(X) [1]
activate(n__first(X1, X2)) → first(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]
first(0, Z) → nil [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
sel(0, cons(X, Z)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [1]
from(X) → n__from(X) [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(X) [1]
activate(n__first(X1, X2)) → first(X1, X2) [1]
activate(X) → X [1]

The TRS has the following type information:

from :: s:0 → n__from:cons:nil:n__first
cons :: s:0 → n__from:cons:nil:n__first → n__from:cons:nil:n__first
n__from :: s:0 → n__from:cons:nil:n__first
s :: s:0 → s:0
first :: s:0 → n__from:cons:nil:n__first → n__from:cons:nil:n__first
0 :: s:0
nil :: n__from:cons:nil:n__first
n__first :: s:0 → n__from:cons:nil:n__first → n__from:cons:nil:n__first
activate :: n__from:cons:nil:n__first → n__from:cons:nil:n__first
sel :: s:0 → n__from:cons:nil:n__first → 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:

sel

activate
from
first

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]
first(0, Z) → nil [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
sel(0, cons(X, Z)) → X [1]
sel(s(X), cons(Y, Z)) → sel(X, activate(Z)) [1]
from(X) → n__from(X) [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(X) [1]
activate(n__first(X1, X2)) → first(X1, X2) [1]
activate(X) → X [1]

The TRS has the following type information:

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]
first(0, Z) → nil [1]
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1]
sel(0, cons(X, Z)) → X [1]
sel(s(X), cons(Y, n__from(X'))) → sel(X, from(X')) [2]
sel(s(X), cons(Y, n__first(X1', X2'))) → sel(X, first(X1', X2')) [2]
sel(s(X), cons(Y, Z)) → sel(X, Z) [2]
from(X) → n__from(X) [1]
first(X1, X2) → n__first(X1, X2) [1]
activate(n__from(X)) → from(X) [1]
activate(n__first(X1, X2)) → first(X1, X2) [1]
activate(X) → X [1]

The TRS has the following type information:

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:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ from(X) :|: z = 1 + X, X >= 0
activate(z) -{ 1 }→ first(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
first(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
first(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
first(z, z') -{ 1 }→ 1 + Y + (1 + X + activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
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, from(X')) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0
sel(z, z') -{ 2 }→ sel(X, first(X1', X2')) :|: z = 1 + X, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), X >= 0

(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

(12) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ first(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'
first(z, z') -{ 1 }→ 0 :|: Z >= 0, z' = Z, z = 0
first(z, z') -{ 1 }→ 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
first(z, z') -{ 1 }→ 1 + Y + (1 + X + activate(Z)) :|: Z >= 0, z = 1 + X, Y >= 0, X >= 0, z' = 1 + Y + Z
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
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, first(X1', X2')) :|: z = 1 + X, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), X >= 0
sel(z, z') -{ 3 }→ sel(X, 1 + X'') :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(X, 1 + X'' + (1 + (1 + X''))) :|: z = 1 + X, Y >= 0, z' = 1 + Y + (1 + X'), X >= 0, X' >= 0, X'' >= 0, X' = X''

(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:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
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, first(X1', X2')) :|: X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

Found the following analysis order by SCC decomposition:

{ first, activate }
{ from }
{ sel }

(16) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
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, first(X1', X2')) :|: X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed: {first,activate}, {from}, {sel}

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

Computed SIZE bound using KoAT for: first
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

(18) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
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, first(X1', X2')) :|: X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

first: runtime: ?, size: O(n¹) [1 + z + 2·z']
activate: runtime: ?, size: O(n¹) [1 + 2·z]

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

Computed RUNTIME bound using KoAT for: first
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

(20) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ z :|: z >= 0
activate(z) -{ 1 }→ first(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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 1 }→ 1 + Y + (1 + (z - 1) + activate(Z)) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
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, first(X1', X2')) :|: X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

first: 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]

(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:

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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 10 + 2·Z }→ 1 + Y + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * Z + 1, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 7 + 2·X2' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1' + 2 * X2' + 1, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

first: 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]

(23) 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

(24) Obligation:

Complexity RNTS consisting of the following rules:

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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 10 + 2·Z }→ 1 + Y + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * Z + 1, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 7 + 2·X2' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1' + 2 * X2' + 1, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

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

(25) 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

(26) Obligation:

Complexity RNTS consisting of the following rules:

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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 10 + 2·Z }→ 1 + Y + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * Z + 1, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 7 + 2·X2' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1' + 2 * X2' + 1, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

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

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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 10 + 2·Z }→ 1 + Y + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * Z + 1, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 7 + 2·X2' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1' + 2 * X2' + 1, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

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

(29) 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: ?

(30) Obligation:

Complexity RNTS consisting of the following rules:

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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 10 + 2·Z }→ 1 + Y + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * Z + 1, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 7 + 2·X2' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1' + 2 * X2' + 1, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

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

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

(31) 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: ?

(32) Obligation:

Complexity RNTS consisting of the following rules:

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'
first(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
first(z, z') -{ 10 + 2·Z }→ 1 + Y + (1 + (z - 1) + s) :|: s >= 0, s <= 2 * Z + 1, Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
first(z, z') -{ 1 }→ 1 + z + z' :|: z >= 0, z' >= 0
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0
sel(z, z') -{ 1 }→ X :|: Z >= 0, X >= 0, z = 0, z' = 1 + X + Z
sel(z, z') -{ 2 }→ sel(z - 1, Z) :|: Z >= 0, Y >= 0, z - 1 >= 0, z' = 1 + Y + Z
sel(z, z') -{ 7 + 2·X2' }→ sel(z - 1, s') :|: s' >= 0, s' <= 1 * X1' + 2 * X2' + 1, X2' >= 0, Y >= 0, X1' >= 0, z' = 1 + Y + (1 + X1' + X2'), z - 1 >= 0
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'') :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''
sel(z, z') -{ 3 }→ sel(z - 1, 1 + X'' + (1 + (1 + X''))) :|: Y >= 0, z' = 1 + Y + (1 + X'), z - 1 >= 0, X' >= 0, X'' >= 0, X' = X''

Function symbols to be analyzed:
Previous analysis results are:

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

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(0) Obligation:

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

(2) Obligation:

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

(4) Obligation:

(5) CompletionProof (UPPER BOUND(ID) transformation)

(6) Obligation:

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

(8) Obligation:

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

(10) Obligation:

(11) InliningProof (UPPER BOUND(ID) transformation)

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

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

(20) Obligation:

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

(22) Obligation:

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

(24) Obligation:

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

(26) Obligation:

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

(28) Obligation:

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

(30) Obligation:

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

(32) Obligation:

(33) FinalProof (EQUIVALENT transformation)

(34) BOUNDS(1, EXP)