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:

plus#2(0, x12) → x12
plus#2(S(x4), x2) → S(plus#2(x4, x2))
fold#3(Nil) → 0
fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2))
main(x1) → fold#3(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:

plus#2(0, x12) → x12 [1]
plus#2(S(x4), x2) → S(plus#2(x4, x2)) [1]
fold#3(Nil) → 0 [1]
fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2)) [1]
main(x1) → fold#3(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:

plus#2(0, x12) → x12 [1]
plus#2(S(x4), x2) → S(plus#2(x4, x2)) [1]
fold#3(Nil) → 0 [1]
fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2)) [1]
main(x1) → fold#3(x1) [1]

The TRS has the following type information:

plus#2 :: 0:S → 0:S → 0:S 0 :: 0:S S :: 0:S → 0:S fold#3 :: Nil:Cons → 0:S Nil :: Nil:Cons Cons :: 0:S → Nil:Cons → Nil:Cons main :: Nil:Cons → 0:S

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:

main

(c) The following functions are completely defined:

fold#3
plus#2

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: plus#2(0, x12) → x12 [1] plus#2(S(x4), x2) → S(plus#2(x4, x2)) [1] fold#3(Nil) → 0 [1] fold#3(Cons(x4, x2)) → plus#2(x4, fold#3(x2)) [1] main(x1) → fold#3(x1) [1] The TRS has the following type information: plus#2 :: 0:S → 0:S → 0:S 0 :: 0:S S :: 0:S → 0:S fold#3 :: Nil:Cons → 0:S Nil :: Nil:Cons Cons :: 0:S → Nil:Cons → Nil:Cons main :: Nil:Cons → 0:S 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#2(0, x12) → x12 [1] plus#2(S(x4), x2) → S(plus#2(x4, x2)) [1] fold#3(Nil) → 0 [1] fold#3(Cons(x4, Nil)) → plus#2(x4, 0) [2] fold#3(Cons(x4, Cons(x4', x2'))) → plus#2(x4, plus#2(x4', fold#3(x2'))) [2] main(x1) → fold#3(x1) [1] The TRS has the following type information: plus#2 :: 0:S → 0:S → 0:S 0 :: 0:S S :: 0:S → 0:S fold#3 :: Nil:Cons → 0:S Nil :: Nil:Cons Cons :: 0:S → Nil:Cons → Nil:Cons main :: Nil:Cons → 0:S 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:

fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 2 }→ plus#2(x4, 0) :|: z = 1 + x4 + 0, x4 >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(x1) :|: x1 >= 0, z = x1
plus#2(z, z') -{ 1 }→ x12 :|: x12 >= 0, z = 0, z' = x12
plus#2(z, z') -{ 1 }→ 1 + plus#2(x4, x2) :|: z' = x2, x4 >= 0, z = 1 + x4, x2 >= 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:

fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 2 }→ plus#2(z - 1, 0) :|: z - 1 >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(z, z') -{ 1 }→ 1 + plus#2(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ plus#2 } { fold#3 } { main }

(14) Obligation:

Complexity RNTS consisting of the following rules:

fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 2 }→ plus#2(z - 1, 0) :|: z - 1 >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(z, z') -{ 1 }→ 1 + plus#2(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus#2}, {fold#3}, {main}

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

Computed SIZE bound using CoFloCo for: plus#2
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:

fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 2 }→ plus#2(z - 1, 0) :|: z - 1 >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(z, z') -{ 1 }→ 1 + plus#2(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus#2}, {fold#3}, {main}
Previous analysis results are:

plus#2: runtime: ?, size: O(n1) [z + z']

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

Computed RUNTIME bound using CoFloCo for: plus#2
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:

fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 2 }→ plus#2(z - 1, 0) :|: z - 1 >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(z, z') -{ 1 }→ 1 + plus#2(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

plus#2: 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:

fold#3(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * 0, z - 1 >= 0
fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

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

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

Computed SIZE bound using KoAT for: fold#3
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:

fold#3(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * 0, z - 1 >= 0
fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(z, z') -{ 1 + z }→ 1 + s :|: s >= 0, s <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

plus#2: runtime: O(n1) [1 + z], size: O(n1) [z + z'] fold#3: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using CoFloCo for: fold#3
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 4 + 2·z

(24) Obligation:

Complexity RNTS consisting of the following rules:

fold#3(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * 0, z - 1 >= 0
fold#3(z) -{ 2 }→ plus#2(x4, plus#2(x4', fold#3(x2'))) :|: z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 1 }→ fold#3(z) :|: z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(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#2: runtime: O(n1) [1 + z], size: O(n1) [z + z'] fold#3: runtime: O(n1) [4 + 2·z], size: O(n1) [z]

(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:

fold#3(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * 0, z - 1 >= 0
fold#3(z) -{ 8 + 2·x2' + x4 + x4' }→ s2 :|: s'' >= 0, s'' <= 1 * x2', s1 >= 0, s1 <= 1 * x4' + 1 * s'', s2 >= 0, s2 <= 1 * x4 + 1 * s1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 5 + 2·z }→ s3 :|: s3 >= 0, s3 <= 1 * z, z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(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#2: runtime: O(n1) [1 + z], size: O(n1) [z + z'] fold#3: runtime: O(n1) [4 + 2·z], size: O(n1) [z]

(27) 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

(28) Obligation:

Complexity RNTS consisting of the following rules:

fold#3(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * 0, z - 1 >= 0
fold#3(z) -{ 8 + 2·x2' + x4 + x4' }→ s2 :|: s'' >= 0, s'' <= 1 * x2', s1 >= 0, s1 <= 1 * x4' + 1 * s'', s2 >= 0, s2 <= 1 * x4 + 1 * s1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 5 + 2·z }→ s3 :|: s3 >= 0, s3 <= 1 * z, z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(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#2: runtime: O(n1) [1 + z], size: O(n1) [z + z'] fold#3: runtime: O(n1) [4 + 2·z], size: O(n1) [z] main: runtime: ?, size: O(n1) [z]

(29) 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: 5 + 2·z

(30) Obligation:

Complexity RNTS consisting of the following rules:

fold#3(z) -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * 0, z - 1 >= 0
fold#3(z) -{ 8 + 2·x2' + x4 + x4' }→ s2 :|: s'' >= 0, s'' <= 1 * x2', s1 >= 0, s1 <= 1 * x4' + 1 * s'', s2 >= 0, s2 <= 1 * x4 + 1 * s1, z = 1 + x4 + (1 + x4' + x2'), x2' >= 0, x4 >= 0, x4' >= 0
fold#3(z) -{ 1 }→ 0 :|: z = 0
main(z) -{ 5 + 2·z }→ s3 :|: s3 >= 0, s3 <= 1 * z, z >= 0
plus#2(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus#2(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#2: runtime: O(n1) [1 + z], size: O(n1) [z + z'] fold#3: runtime: O(n1) [4 + 2·z], size: O(n1) [z] main: runtime: O(n1) [5 + 2·z], size: O(n1) [z]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)