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:

sum(0) → 0
sum(s(x)) → +(sum(x), s(x))
sum1(0) → 0
sum1(s(x)) → s(+(sum1(x), +(x, 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:

sum(0) → 0 [1]
sum(s(x)) → +(sum(x), s(x)) [1]
sum1(0) → 0 [1]
sum1(s(x)) → s(+(sum1(x), +(x, 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:

sum(0) → 0 [1]
sum(s(x)) → +(sum(x), s(x)) [1]
sum1(0) → 0 [1]
sum1(s(x)) → s(+(sum1(x), +(x, x))) [1]

The TRS has the following type information:

sum :: 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ sum1 :: 0:s:+ → 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:

sum
sum1

(c) The following functions are completely defined:
none

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: sum(0) → 0 [1] sum(s(x)) → +(sum(x), s(x)) [1] sum1(0) → 0 [1] sum1(s(x)) → s(+(sum1(x), +(x, x))) [1] The TRS has the following type information: sum :: 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ sum1 :: 0:s:+ → 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: sum(0) → 0 [1] sum(s(x)) → +(sum(x), s(x)) [1] sum1(0) → 0 [1] sum1(s(x)) → s(+(sum1(x), +(x, x))) [1] The TRS has the following type information: sum :: 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ sum1 :: 0:s:+ → 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

(10) Obligation:

Complexity RNTS consisting of the following rules:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(x) + (1 + x) :|: x >= 0, z = 1 + x
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 }→ 1 + (1 + sum1(x) + (1 + x + x)) :|: x >= 0, z = 1 + x

(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:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 }→ 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ sum1 } { sum }

(14) Obligation:

Complexity RNTS consisting of the following rules:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 }→ 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

Function symbols to be analyzed: {sum1}, {sum}

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

Computed SIZE bound using CoFloCo for: sum1
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + 2·z²

(16) Obligation:

Complexity RNTS consisting of the following rules:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 }→ 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

Function symbols to be analyzed: {sum1}, {sum}
Previous analysis results are:

sum1: runtime: ?, size: O(n2) [z + 2·z2]

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

Computed RUNTIME bound using CoFloCo for: sum1
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:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 }→ 1 + (1 + sum1(z - 1) + (1 + (z - 1) + (z - 1))) :|: z - 1 >= 0

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

sum1: runtime: O(n1) [1 + z], size: O(n2) [z + 2·z2]

(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:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 + z }→ 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + 1 * (z - 1), z - 1 >= 0

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

sum1: runtime: O(n1) [1 + z], size: O(n2) [z + 2·z2]

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

Computed SIZE bound using CoFloCo for: sum
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + z²

(22) Obligation:

Complexity RNTS consisting of the following rules:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 + z }→ 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + 1 * (z - 1), z - 1 >= 0

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

sum1: runtime: O(n1) [1 + z], size: O(n2) [z + 2·z2] sum: runtime: ?, size: O(n2) [z + z2]

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

Computed RUNTIME bound using CoFloCo for: sum
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:

sum(z) -{ 1 }→ 0 :|: z = 0
sum(z) -{ 1 }→ 1 + sum(z - 1) + (1 + (z - 1)) :|: z - 1 >= 0
sum1(z) -{ 1 }→ 0 :|: z = 0
sum1(z) -{ 1 + z }→ 1 + (1 + s + (1 + (z - 1) + (z - 1))) :|: s >= 0, s <= 2 * ((z - 1) * (z - 1)) + 1 * (z - 1), z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

sum1: runtime: O(n1) [1 + z], size: O(n2) [z + 2·z2] sum: runtime: O(n1) [1 + z], size: O(n2) [z + z2]

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)