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:

from(X) → cons(X, n__from(s(X)))
after(0, XS) → XS
after(s(N), cons(X, XS)) → after(N, activate(XS))
from(X) → n__from(X)
activate(n__from(X)) → from(X)
activate(X) → X

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) Obligation:

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


The TRS R consists of the following rules:

from(X) → cons(X, n__from(s(X))) [1]
after(0, XS) → XS [1]
after(s(N), cons(X, XS)) → after(N, activate(XS)) [1]
from(X) → n__from(X) [1]
activate(n__from(X)) → from(X) [1]
activate(X) → X [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(4) Obligation:

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

from(X) → cons(X, n__from(s(X))) [1]
after(0, XS) → XS [1]
after(s(N), cons(X, XS)) → after(N, activate(XS)) [1]
from(X) → n__from(X) [1]
activate(n__from(X)) → from(X) [1]
activate(X) → X [1]

The TRS has the following type information:

from :: s:0 → n__from:cons cons :: s:0 → n__from:cons → n__from:cons n__from :: s:0 → n__from:cons s :: s:0 → s:0 after :: s:0 → n__from:cons → n__from:cons 0 :: s:0 activate :: n__from:cons → n__from:cons

Rewrite Strategy: INNERMOST

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

The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

after

(c) The following functions are completely defined:

activate
from

Due to the following rules being added:
none

And the following fresh constants:

const

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) → cons(X, n__from(s(X))) [1] after(0, XS) → XS [1] after(s(N), cons(X, XS)) → after(N, activate(XS)) [1] from(X) → n__from(X) [1] activate(n__from(X)) → from(X) [1] activate(X) → X [1] The TRS has the following type information: from :: s:0 → n__from:cons cons :: s:0 → n__from:cons → n__from:cons n__from :: s:0 → n__from:cons s :: s:0 → s:0 after :: s:0 → n__from:cons → n__from:cons 0 :: s:0 activate :: n__from:cons → n__from:cons const :: n__from:cons Rewrite Strategy: INNERMOST

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

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

(8) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: from(X) → cons(X, n__from(s(X))) [1] after(0, XS) → XS [1] after(s(N), cons(X, n__from(X'))) → after(N, from(X')) [2] after(s(N), cons(X, XS)) → after(N, XS) [2] from(X) → n__from(X) [1] activate(n__from(X)) → from(X) [1] activate(X) → X [1] The TRS has the following type information: from :: s:0 → n__from:cons cons :: s:0 → n__from:cons → n__from:cons n__from :: s:0 → n__from:cons s :: s:0 → s:0 after :: s:0 → n__from:cons → n__from:cons 0 :: s:0 activate :: n__from:cons → n__from:cons const :: n__from:cons Rewrite Strategy: INNERMOST

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

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

0 => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

activate(z) -{ 1 }→ X :|: X >= 0, z = X
activate(z) -{ 1 }→ from(X) :|: z = 1 + X, X >= 0
after(z, z') -{ 1 }→ XS :|: z' = XS, z = 0, XS >= 0
after(z, z') -{ 2 }→ after(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
after(z, z') -{ 2 }→ after(N, from(X')) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0
from(z) -{ 1 }→ 1 + X :|: X >= 0, z = X
from(z) -{ 1 }→ 1 + X + (1 + (1 + X)) :|: X >= 0, z = X

(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) -{ 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'
after(z, z') -{ 1 }→ XS :|: z' = XS, z = 0, XS >= 0
after(z, z') -{ 2 }→ after(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
after(z, z') -{ 3 }→ after(N, 1 + X'') :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(N, 1 + X'' + (1 + (1 + X''))) :|: z = 1 + N, z' = 1 + X + (1 + X'), X >= 0, X' >= 0, N >= 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

(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) -{ 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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ activate } { from } { after }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {activate}, {from}, {after}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

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

Computed RUNTIME bound using KoAT for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(20) Obligation:

Complexity RNTS consisting of the following rules:

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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

activate: runtime: O(1) [3], 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) -{ 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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

activate: runtime: O(1) [3], 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) -{ 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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

activate: runtime: O(1) [3], 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) -{ 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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

activate: runtime: O(1) [3], 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) -{ 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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

activate: runtime: O(1) [3], 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: after
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) -{ 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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

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

Computed RUNTIME bound using KoAT for: after
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 8·z

(32) Obligation:

Complexity RNTS consisting of the following rules:

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'
after(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
after(z, z') -{ 2 }→ after(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
after(z, z') -{ 3 }→ after(z - 1, 1 + X'') :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
after(z, z') -{ 3 }→ after(z - 1, 1 + X'' + (1 + (1 + X''))) :|: z' = 1 + X + (1 + X'), X >= 0, X' >= 0, z - 1 >= 0, X'' >= 0, X' = X''
from(z) -{ 1 }→ 1 + z :|: z >= 0
from(z) -{ 1 }→ 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed:
Previous analysis results are:

activate: runtime: O(1) [3], size: O(n1) [1 + 2·z] from: runtime: O(1) [1], size: O(n1) [3 + 2·z] after: runtime: O(n1) [1 + 8·z], size: EXP

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)