The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

plus_x#1(0, x8) → x8
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14))
map#2(plus_x(x2), Nil) → Nil
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2))
main(x5, x12) → map#2(plus_x(x12), x5)

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^2).


The TRS R consists of the following rules:

plus_x#1(0, x8) → x8 [1]
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14)) [1]
map#2(plus_x(x2), Nil) → Nil [1]
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1]
main(x5, x12) → map#2(plus_x(x12), x5) [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:

plus_x#1(0, x8) → x8 [1]
plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14)) [1]
map#2(plus_x(x2), Nil) → Nil [1]
map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1]
main(x5, x12) → map#2(plus_x(x12), x5) [1]

The TRS has the following type information:

plus_x#1 :: 0:S → 0:S → 0:S 0 :: 0:S S :: 0:S → 0:S map#2 :: plus_x → Nil:Cons → Nil:Cons plus_x :: 0:S → plus_x Nil :: Nil:Cons Cons :: 0:S → Nil:Cons → Nil:Cons main :: Nil:Cons → 0:S → 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:

plus_x#1
map#2
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: plus_x#1(0, x8) → x8 [1] plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14)) [1] map#2(plus_x(x2), Nil) → Nil [1] map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1] main(x5, x12) → map#2(plus_x(x12), x5) [1] The TRS has the following type information: plus_x#1 :: 0:S → 0:S → 0:S 0 :: 0:S S :: 0:S → 0:S map#2 :: plus_x → Nil:Cons → Nil:Cons plus_x :: 0:S → plus_x Nil :: Nil:Cons Cons :: 0:S → Nil:Cons → Nil:Cons main :: Nil:Cons → 0:S → Nil:Cons const :: plus_x 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: plus_x#1(0, x8) → x8 [1] plus_x#1(S(x12), x14) → S(plus_x#1(x12, x14)) [1] map#2(plus_x(x2), Nil) → Nil [1] map#2(plus_x(x6), Cons(x4, x2)) → Cons(plus_x#1(x6, x4), map#2(plus_x(x6), x2)) [1] main(x5, x12) → map#2(plus_x(x12), x5) [1] The TRS has the following type information: plus_x#1 :: 0:S → 0:S → 0:S 0 :: 0:S S :: 0:S → 0:S map#2 :: plus_x → Nil:Cons → Nil:Cons plus_x :: 0:S → plus_x Nil :: Nil:Cons Cons :: 0:S → Nil:Cons → Nil:Cons main :: Nil:Cons → 0:S → Nil:Cons const :: plus_x 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
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

main(z, z') -{ 1 }→ map#2(1 + x12, x5) :|: x5 >= 0, x12 >= 0, z = x5, z' = x12
map#2(z, z') -{ 1 }→ 0 :|: z = 1 + x2, x2 >= 0, z' = 0
map#2(z, z') -{ 1 }→ 1 + plus_x#1(x6, x4) + map#2(1 + x6, x2) :|: x4 >= 0, z = 1 + x6, z' = 1 + x4 + x2, x6 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ x8 :|: x8 >= 0, z = 0, z' = x8
plus_x#1(z, z') -{ 1 }→ 1 + plus_x#1(x12, x14) :|: z = 1 + x12, x12 >= 0, x14 >= 0, z' = x14

(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:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ plus_x#1 } { map#2 } { main }

(14) Obligation:

Complexity RNTS consisting of the following rules:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus_x#1}, {map#2}, {main}

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

Computed SIZE bound using CoFloCo for: plus_x#1
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:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus_x#1}, {map#2}, {main}
Previous analysis results are:

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

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

Computed RUNTIME bound using CoFloCo for: plus_x#1
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:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, x4) + map#2(1 + (z - 1), x2) :|: x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 }→ 1 + plus_x#1(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {map#2}, {main}
Previous analysis results are:

plus_x#1: 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:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 + z }→ 1 + s' + map#2(1 + (z - 1), x2) :|: s' >= 0, s' <= 1 * (z - 1) + 1 * x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {map#2}, {main}
Previous analysis results are:

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

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

Computed SIZE bound using CoFloCo for: map#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z·z' + z' + z'²

(22) Obligation:

Complexity RNTS consisting of the following rules:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 + z }→ 1 + s' + map#2(1 + (z - 1), x2) :|: s' >= 0, s' <= 1 * (z - 1) + 1 * x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {map#2}, {main}
Previous analysis results are:

plus_x#1: runtime: O(n1) [1 + z], size: O(n1) [z + z'] map#2: runtime: ?, size: O(n2) [z·z' + z' + z'2]

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

Computed RUNTIME bound using CoFloCo for: map#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + z·z' + z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

main(z, z') -{ 1 }→ map#2(1 + z', z) :|: z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 1 + z }→ 1 + s' + map#2(1 + (z - 1), x2) :|: s' >= 0, s' <= 1 * (z - 1) + 1 * x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

plus_x#1: runtime: O(n1) [1 + z], size: O(n1) [z + z'] map#2: runtime: O(n2) [1 + z·z' + z'], size: O(n2) [z·z' + z' + z'2]

(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:

main(z, z') -{ 2 + 2·z + z·z' }→ s1 :|: s1 >= 0, s1 <= 1 * ((1 + z') * z) + 1 * (z * z) + 1 * z, z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 2 + x2 + x2·z + z }→ 1 + s' + s'' :|: s'' >= 0, s'' <= 1 * ((1 + (z - 1)) * x2) + 1 * (x2 * x2) + 1 * x2, s' >= 0, s' <= 1 * (z - 1) + 1 * x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

plus_x#1: runtime: O(n1) [1 + z], size: O(n1) [z + z'] map#2: runtime: O(n2) [1 + z·z' + z'], size: O(n2) [z·z' + z' + z'2]

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

Computed SIZE bound using KoAT for: main
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2·z + z·z' + z²

(28) Obligation:

Complexity RNTS consisting of the following rules:

main(z, z') -{ 2 + 2·z + z·z' }→ s1 :|: s1 >= 0, s1 <= 1 * ((1 + z') * z) + 1 * (z * z) + 1 * z, z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 2 + x2 + x2·z + z }→ 1 + s' + s'' :|: s'' >= 0, s'' <= 1 * ((1 + (z - 1)) * x2) + 1 * (x2 * x2) + 1 * x2, s' >= 0, s' <= 1 * (z - 1) + 1 * x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

plus_x#1: runtime: O(n1) [1 + z], size: O(n1) [z + z'] map#2: runtime: O(n2) [1 + z·z' + z'], size: O(n2) [z·z' + z' + z'2] main: runtime: ?, size: O(n2) [2·z + z·z' + z2]

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

Computed RUNTIME bound using KoAT for: main
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + 2·z + z·z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

main(z, z') -{ 2 + 2·z + z·z' }→ s1 :|: s1 >= 0, s1 <= 1 * ((1 + z') * z) + 1 * (z * z) + 1 * z, z >= 0, z' >= 0
map#2(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
map#2(z, z') -{ 2 + x2 + x2·z + z }→ 1 + s' + s'' :|: s'' >= 0, s'' <= 1 * ((1 + (z - 1)) * x2) + 1 * (x2 * x2) + 1 * x2, s' >= 0, s' <= 1 * (z - 1) + 1 * x4, x4 >= 0, z' = 1 + x4 + x2, z - 1 >= 0, x2 >= 0
plus_x#1(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus_x#1(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

plus_x#1: runtime: O(n1) [1 + z], size: O(n1) [z + z'] map#2: runtime: O(n2) [1 + z·z' + z'], size: O(n2) [z·z' + z' + z'2] main: runtime: O(n2) [2 + 2·z + z·z'], size: O(n2) [2·z + z·z' + z2]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^2)