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:

p(s(x)) → x
fac(0) → s(0)
fac(s(x)) → times(s(x), fac(p(s(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:

p(s(x)) → x [1]
fac(0) → s(0) [1]
fac(s(x)) → times(s(x), fac(p(s(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:

p(s(x)) → x [1]
fac(0) → s(0) [1]
fac(s(x)) → times(s(x), fac(p(s(x)))) [1]

The TRS has the following type information:

p :: s:0:times → s:0:times s :: s:0:times → s:0:times fac :: s:0:times → s:0:times 0 :: s:0:times times :: s:0:times → s:0:times → s:0:times

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:

fac

(c) The following functions are completely defined:

p

Due to the following rules being added:

p(v0) → 0 [0]

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: p(s(x)) → x [1] fac(0) → s(0) [1] fac(s(x)) → times(s(x), fac(p(s(x)))) [1] p(v0) → 0 [0] The TRS has the following type information: p :: s:0:times → s:0:times s :: s:0:times → s:0:times fac :: s:0:times → s:0:times 0 :: s:0:times times :: s:0:times → s:0:times → s:0:times 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: p(s(x)) → x [1] fac(0) → s(0) [1] fac(s(x)) → times(s(x), fac(x)) [2] fac(s(x)) → times(s(x), fac(0)) [1] p(v0) → 0 [0] The TRS has the following type information: p :: s:0:times → s:0:times s :: s:0:times → s:0:times fac :: s:0:times → s:0:times 0 :: s:0:times times :: s:0:times → s:0:times → s:0:times 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:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 2 }→ 1 + (1 + x) + fac(x) :|: x >= 0, z = 1 + x
fac(z) -{ 1 }→ 1 + (1 + x) + fac(0) :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0

(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:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 }→ 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ fac } { p }

(14) Obligation:

Complexity RNTS consisting of the following rules:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 }→ 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {fac}, {p}

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

Computed SIZE bound using PUBS for: fac
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + z + z²

(16) Obligation:

Complexity RNTS consisting of the following rules:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 }→ 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {fac}, {p}
Previous analysis results are:

fac: runtime: ?, size: O(n2) [1 + z + z2]

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

Computed RUNTIME bound using PUBS for: fac
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(18) Obligation:

Complexity RNTS consisting of the following rules:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 }→ 1 + (1 + (z - 1)) + fac(0) :|: z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + fac(z - 1) :|: z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 + 2·z }→ 1 + (1 + (z - 1)) + s :|: s >= 0, s <= 1 + 1 * (z - 1) + 1 * ((z - 1) * (z - 1)), z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + s' :|: s' >= 0, s' <= 1 + 1 * 0 + 1 * (0 * 0), z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

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

Computed SIZE bound using KoAT for: p
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:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 + 2·z }→ 1 + (1 + (z - 1)) + s :|: s >= 0, s <= 1 + 1 * (z - 1) + 1 * ((z - 1) * (z - 1)), z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + s' :|: s' >= 0, s' <= 1 + 1 * 0 + 1 * (0 * 0), z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

fac(z) -{ 1 }→ 1 + 0 :|: z = 0
fac(z) -{ 1 + 2·z }→ 1 + (1 + (z - 1)) + s :|: s >= 0, s <= 1 + 1 * (z - 1) + 1 * ((z - 1) * (z - 1)), z - 1 >= 0
fac(z) -{ 2 }→ 1 + (1 + (z - 1)) + s' :|: s' >= 0, s' <= 1 + 1 * 0 + 1 * (0 * 0), z - 1 >= 0
p(z) -{ 0 }→ 0 :|: z >= 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

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

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)