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:

odd(Cons(x, xs)) → even(xs)
odd(Nil) → False
even(Cons(x, xs)) → odd(xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
even(Nil) → True
evenodd(x) → even(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:

odd(Cons(x, xs)) → even(xs) [1]
odd(Nil) → False [1]
even(Cons(x, xs)) → odd(xs) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
even(Nil) → True [1]
evenodd(x) → even(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:

odd(Cons(x, xs)) → even(xs) [1]
odd(Nil) → False [1]
even(Cons(x, xs)) → odd(xs) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
even(Nil) → True [1]
evenodd(x) → even(x) [1]

The TRS has the following type information:

odd :: Cons:Nil → False:True Cons :: a → Cons:Nil → Cons:Nil even :: Cons:Nil → False:True Nil :: Cons:Nil False :: False:True notEmpty :: Cons:Nil → False:True True :: False:True evenodd :: Cons:Nil → False:True

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:

odd
even
notEmpty
evenodd

(c) The following functions are completely defined:
none

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: odd(Cons(x, xs)) → even(xs) [1] odd(Nil) → False [1] even(Cons(x, xs)) → odd(xs) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] even(Nil) → True [1] evenodd(x) → even(x) [1] The TRS has the following type information: odd :: Cons:Nil → False:True Cons :: a → Cons:Nil → Cons:Nil even :: Cons:Nil → False:True Nil :: Cons:Nil False :: False:True notEmpty :: Cons:Nil → False:True True :: False:True evenodd :: Cons:Nil → False:True const :: a 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: odd(Cons(x, xs)) → even(xs) [1] odd(Nil) → False [1] even(Cons(x, xs)) → odd(xs) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] even(Nil) → True [1] evenodd(x) → even(x) [1] The TRS has the following type information: odd :: Cons:Nil → False:True Cons :: a → Cons:Nil → Cons:Nil even :: Cons:Nil → False:True Nil :: Cons:Nil False :: False:True notEmpty :: Cons:Nil → False:True True :: False:True evenodd :: Cons:Nil → False:True const :: a 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:

Nil => 0
False => 0
True => 1
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(x) :|: x >= 0, z = x
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

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

Found the following analysis order by SCC decomposition:

{ notEmpty } { even, odd } { evenodd }

(14) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {even,odd}, {evenodd}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {even,odd}, {evenodd}
Previous analysis results are:

notEmpty: runtime: ?, size: O(1) [1]

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {even,odd}, {evenodd}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {even,odd}, {evenodd}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {even,odd}, {evenodd}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] even: runtime: ?, size: O(1) [1] odd: runtime: ?, size: O(1) [1]

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

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

Computed RUNTIME bound using CoFloCo for: odd
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:

even(z) -{ 1 }→ odd(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 1 }→ even(z) :|: z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 1 }→ even(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

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

notEmpty: runtime: O(1) [1], size: O(1) [1] even: runtime: O(n1) [2 + 2·z], size: O(1) [1] odd: runtime: O(n1) [1 + 2·z], size: O(1) [1]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 2 + 2·xs }→ s' :|: s' >= 0, s' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 3 + 2·z }→ s'' :|: s'' >= 0, s'' <= 1, z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 3 + 2·xs }→ s :|: s >= 0, s <= 1, z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

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

notEmpty: runtime: O(1) [1], size: O(1) [1] even: runtime: O(n1) [2 + 2·z], size: O(1) [1] odd: runtime: O(n1) [1 + 2·z], size: O(1) [1]

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 2 + 2·xs }→ s' :|: s' >= 0, s' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 3 + 2·z }→ s'' :|: s'' >= 0, s'' <= 1, z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 3 + 2·xs }→ s :|: s >= 0, s <= 1, z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

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

notEmpty: runtime: O(1) [1], size: O(1) [1] even: runtime: O(n1) [2 + 2·z], size: O(1) [1] odd: runtime: O(n1) [1 + 2·z], size: O(1) [1] evenodd: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using CoFloCo for: evenodd
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 2·z

(30) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 2 + 2·xs }→ s' :|: s' >= 0, s' <= 1, z = 1 + x + xs, xs >= 0, x >= 0
even(z) -{ 1 }→ 1 :|: z = 0
evenodd(z) -{ 3 + 2·z }→ s'' :|: s'' >= 0, s'' <= 1, z >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0
odd(z) -{ 3 + 2·xs }→ s :|: s >= 0, s <= 1, z = 1 + x + xs, xs >= 0, x >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed:
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] even: runtime: O(n1) [2 + 2·z], size: O(1) [1] odd: runtime: O(n1) [1 + 2·z], size: O(1) [1] evenodd: runtime: O(n1) [3 + 2·z], size: O(1) [1]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)