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:

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
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:

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

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

The TRS has the following type information:

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

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:

first
from
activate

(c) The following functions are completely defined:
none

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: first(0, X) → nil [1] first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1] from(X) → cons(X, n__from(s(X))) [1] first(X1, X2) → n__first(X1, X2) [1] from(X) → n__from(X) [1] activate(n__first(X1, X2)) → first(X1, X2) [1] activate(n__from(X)) → from(X) [1] activate(X) → X [1] The TRS has the following type information: first :: 0:s → nil:cons:n__first:n__from → nil:cons:n__first:n__from 0 :: 0:s nil :: nil:cons:n__first:n__from s :: 0:s → 0:s cons :: 0:s → nil:cons:n__first:n__from → nil:cons:n__first:n__from n__first :: 0:s → nil:cons:n__first:n__from → nil:cons:n__first:n__from activate :: nil:cons:n__first:n__from → nil:cons:n__first:n__from from :: 0:s → nil:cons:n__first:n__from n__from :: 0:s → nil:cons:n__first:n__from 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: first(0, X) → nil [1] first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z))) [1] from(X) → cons(X, n__from(s(X))) [1] first(X1, X2) → n__first(X1, X2) [1] from(X) → n__from(X) [1] activate(n__first(X1, X2)) → first(X1, X2) [1] activate(n__from(X)) → from(X) [1] activate(X) → X [1] The TRS has the following type information: first :: 0:s → nil:cons:n__first:n__from → nil:cons:n__first:n__from 0 :: 0:s nil :: nil:cons:n__first:n__from s :: 0:s → 0:s cons :: 0:s → nil:cons:n__first:n__from → nil:cons:n__first:n__from n__first :: 0:s → nil:cons:n__first:n__from → nil:cons:n__first:n__from activate :: nil:cons:n__first:n__from → nil:cons:n__first:n__from from :: 0:s → nil:cons:n__first:n__from n__from :: 0:s → nil:cons:n__first:n__from 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' = X, X >= 0, 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

(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' = X, X >= 0, 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

(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

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

Found the following analysis order by SCC decomposition:

{ from } { first, activate }

(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

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

(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:

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

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

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

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

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

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

(24) 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

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

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

(25) 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

(26) 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

Function symbols to be analyzed:
Previous analysis results are:

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

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)