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:

list(Cons(x, xs)) → list(xs)
list(Nil) → True
list(Nil) → isEmpty[Match](Nil)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(x) → list(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:

list(Cons(x, xs)) → list(xs) [1]
list(Nil) → True [1]
list(Nil) → isEmpty[Match](Nil) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(x) → list(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:

list(Cons(x, xs)) → list(xs) [1]
list(Nil) → True [1]
list(Nil) → isEmpty[Match](Nil) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(x) → list(x) [1]

The TRS has the following type information:

list :: Cons:Nil → True:isEmpty[Match]:False Cons :: a → Cons:Nil → Cons:Nil Nil :: Cons:Nil True :: True:isEmpty[Match]:False isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False notEmpty :: Cons:Nil → True:isEmpty[Match]:False False :: True:isEmpty[Match]:False goal :: Cons:Nil → True:isEmpty[Match]:False

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:

list
notEmpty
goal

(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: list(Cons(x, xs)) → list(xs) [1] list(Nil) → True [1] list(Nil) → isEmpty[Match](Nil) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] goal(x) → list(x) [1] The TRS has the following type information: list :: Cons:Nil → True:isEmpty[Match]:False Cons :: a → Cons:Nil → Cons:Nil Nil :: Cons:Nil True :: True:isEmpty[Match]:False isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False notEmpty :: Cons:Nil → True:isEmpty[Match]:False False :: True:isEmpty[Match]:False goal :: Cons:Nil → True:isEmpty[Match]:False 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: list(Cons(x, xs)) → list(xs) [1] list(Nil) → True [1] list(Nil) → isEmpty[Match](Nil) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] goal(x) → list(x) [1] The TRS has the following type information: list :: Cons:Nil → True:isEmpty[Match]:False Cons :: a → Cons:Nil → Cons:Nil Nil :: Cons:Nil True :: True:isEmpty[Match]:False isEmpty[Match] :: Cons:Nil → True:isEmpty[Match]:False notEmpty :: Cons:Nil → True:isEmpty[Match]:False False :: True:isEmpty[Match]:False goal :: Cons:Nil → True:isEmpty[Match]:False 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
True => 1
False => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

goal(z) -{ 1 }→ list(x) :|: x >= 0, z = x
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(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:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

Found the following analysis order by SCC decomposition:

{ notEmpty } { list } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {list}, {goal}

(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:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {notEmpty}, {list}, {goal}
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:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {list}, {goal}
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:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {list}, {goal}
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: list
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:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {list}, {goal}
Previous analysis results are:

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

goal(z) -{ 1 }→ list(z) :|: z >= 0
list(z) -{ 1 }→ list(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

notEmpty: runtime: O(1) [1], size: O(1) [1] list: runtime: O(n1) [1 + 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:

goal(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1, z >= 0
list(z) -{ 2 + xs }→ s :|: s >= 0, s <= 1, z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

notEmpty: runtime: O(1) [1], size: O(1) [1] list: runtime: O(n1) [1 + z], size: O(1) [1]

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

Computed SIZE bound using CoFloCo for: goal
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:

goal(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1, z >= 0
list(z) -{ 2 + xs }→ s :|: s >= 0, s <= 1, z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

notEmpty: runtime: O(1) [1], size: O(1) [1] list: runtime: O(n1) [1 + z], size: O(1) [1] goal: runtime: ?, size: O(1) [1]

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

goal(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1, z >= 0
list(z) -{ 2 + xs }→ s :|: s >= 0, s <= 1, z = 1 + x + xs, xs >= 0, x >= 0
list(z) -{ 1 }→ 1 :|: z = 0
list(z) -{ 1 }→ 1 + 0 :|: z = 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed:
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] list: runtime: O(n1) [1 + z], size: O(1) [1] goal: runtime: O(n1) [2 + z], size: O(1) [1]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)