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:

foldl#3(x2, Nil) → x2
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6)
main(x1) → foldl#3(Nil, x1)

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:

foldl#3(x2, Nil) → x2 [1]
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6) [1]
main(x1) → foldl#3(Nil, x1) [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:

foldl#3(x2, Nil) → x2 [1]
foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6) [1]
main(x1) → foldl#3(Nil, x1) [1]

The TRS has the following type information:

foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons Nil :: Nil:Cons Cons :: a → Nil:Cons → Nil:Cons main :: Nil:Cons → Nil: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:

foldl#3
main

(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: foldl#3(x2, Nil) → x2 [1] foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6) [1] main(x1) → foldl#3(Nil, x1) [1] The TRS has the following type information: foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons Nil :: Nil:Cons Cons :: a → Nil:Cons → Nil:Cons main :: Nil:Cons → Nil:Cons 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: foldl#3(x2, Nil) → x2 [1] foldl#3(x16, Cons(x24, x6)) → foldl#3(Cons(x24, x16), x6) [1] main(x1) → foldl#3(Nil, x1) [1] The TRS has the following type information: foldl#3 :: Nil:Cons → Nil:Cons → Nil:Cons Nil :: Nil:Cons Cons :: a → Nil:Cons → Nil:Cons main :: Nil:Cons → Nil:Cons 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
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

foldl#3(z, z') -{ 1 }→ x2 :|: z = x2, x2 >= 0, z' = 0
foldl#3(z, z') -{ 1 }→ foldl#3(1 + x24 + x16, x6) :|: z' = 1 + x24 + x6, z = x16, x6 >= 0, x24 >= 0, x16 >= 0
main(z) -{ 1 }→ foldl#3(0, x1) :|: x1 >= 0, z = x1

(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:

foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
foldl#3(z, z') -{ 1 }→ foldl#3(1 + x24 + z, x6) :|: z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
main(z) -{ 1 }→ foldl#3(0, z) :|: z >= 0

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

Found the following analysis order by SCC decomposition:

{ foldl#3 } { main }

(14) Obligation:

Complexity RNTS consisting of the following rules:

foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
foldl#3(z, z') -{ 1 }→ foldl#3(1 + x24 + z, x6) :|: z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
main(z) -{ 1 }→ foldl#3(0, z) :|: z >= 0

Function symbols to be analyzed: {foldl#3}, {main}

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

Computed SIZE bound using CoFloCo for: foldl#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z + z'

(16) Obligation:

Complexity RNTS consisting of the following rules:

foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
foldl#3(z, z') -{ 1 }→ foldl#3(1 + x24 + z, x6) :|: z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
main(z) -{ 1 }→ foldl#3(0, z) :|: z >= 0

Function symbols to be analyzed: {foldl#3}, {main}
Previous analysis results are:

foldl#3: runtime: ?, size: O(n1) [z + z']

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

Computed RUNTIME bound using CoFloCo for: foldl#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(18) Obligation:

Complexity RNTS consisting of the following rules:

foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
foldl#3(z, z') -{ 1 }→ foldl#3(1 + x24 + z, x6) :|: z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
main(z) -{ 1 }→ foldl#3(0, z) :|: z >= 0

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

foldl#3: runtime: O(n1) [1 + z'], size: O(n1) [z + z']

(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:

foldl#3(z, z') -{ 2 + x6 }→ s :|: s >= 0, s <= 1 * (1 + x24 + z) + 1 * x6, z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
main(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * z, z >= 0

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

foldl#3: runtime: O(n1) [1 + z'], size: O(n1) [z + z']

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

foldl#3(z, z') -{ 2 + x6 }→ s :|: s >= 0, s <= 1 * (1 + x24 + z) + 1 * x6, z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
main(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * z, z >= 0

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

foldl#3: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] main: runtime: ?, size: O(n1) [z]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

foldl#3(z, z') -{ 2 + x6 }→ s :|: s >= 0, s <= 1 * (1 + x24 + z) + 1 * x6, z' = 1 + x24 + x6, x6 >= 0, x24 >= 0, z >= 0
foldl#3(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
main(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * z, z >= 0

Function symbols to be analyzed:
Previous analysis results are:

foldl#3: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] main: runtime: O(n1) [2 + z], size: O(n1) [z]

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)