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:

f(f(x)) → f(x)
f(s(x)) → f(x)
g(s(0)) → g(f(s(0)))

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of g: f

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(x)) → f(x)

(2) 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:

g(s(0)) → g(f(s(0)))
f(s(x)) → f(x)

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) 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:

g(s(0)) → g(f(s(0))) [1]
f(s(x)) → f(x) [1]

Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

g(s(0)) → g(f(s(0))) [1]
f(s(x)) → f(x) [1]

The TRS has the following type information:

g :: 0:s → g s :: 0:s → 0:s 0 :: 0:s f :: 0:s → 0:s

Rewrite Strategy: INNERMOST

(7) 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:

g

(c) The following functions are completely defined:

f

Due to the following rules being added:

f(v0) → 0 [0]

And the following fresh constants:

const

(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: g(s(0)) → g(f(s(0))) [1] f(s(x)) → f(x) [1] f(v0) → 0 [0] The TRS has the following type information: g :: 0:s → g s :: 0:s → 0:s 0 :: 0:s f :: 0:s → 0:s const :: g Rewrite Strategy: INNERMOST

(9) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(10) 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: g(s(0)) → g(f(0)) [2] g(s(0)) → g(0) [1] f(s(x)) → f(x) [1] f(v0) → 0 [0] The TRS has the following type information: g :: 0:s → g s :: 0:s → 0:s 0 :: 0:s f :: 0:s → 0:s const :: g Rewrite Strategy: INNERMOST

(11) 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
const => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

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

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(f(0)) :|: z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

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

Found the following analysis order by SCC decomposition:

{ f } { g }

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(f(0)) :|: z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

Function symbols to be analyzed: {f}, {g}

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

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(f(0)) :|: z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

Function symbols to be analyzed: {f}, {g}
Previous analysis results are:

f: runtime: ?, size: O(1) [0]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ 1 }→ f(z - 1) :|: z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(f(0)) :|: z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

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

f: runtime: O(n1) [z], size: O(1) [0]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ z }→ s' :|: s' >= 0, s' <= 0, z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(s) :|: s >= 0, s <= 0, z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

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

f: runtime: O(n1) [z], size: O(1) [0]

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

Computed SIZE bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(24) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ z }→ s' :|: s' >= 0, s' <= 0, z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(s) :|: s >= 0, s <= 0, z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

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

f: runtime: O(n1) [z], size: O(1) [0] g: runtime: ?, size: O(1) [0]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(26) Obligation:

Complexity RNTS consisting of the following rules:

f(z) -{ z }→ s' :|: s' >= 0, s' <= 0, z - 1 >= 0
f(z) -{ 0 }→ 0 :|: z >= 0
g(z) -{ 2 }→ g(s) :|: s >= 0, s <= 0, z = 1 + 0
g(z) -{ 1 }→ g(0) :|: z = 1 + 0

Function symbols to be analyzed:
Previous analysis results are:

f: runtime: O(n1) [z], size: O(1) [0] g: runtime: O(1) [2], size: O(1) [0]

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)