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:

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

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:

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

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

The TRS has the following type information:

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

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:

g
f

(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: g(s(x)) → f(x) [1] f(0) → s(0) [1] f(s(x)) → s(s(g(x))) [1] g(0) → 0 [1] The TRS has the following type information: g :: s:0 → s:0 s :: s:0 → s:0 f :: s:0 → s:0 0 :: s:0 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: g(s(x)) → f(x) [1] f(0) → s(0) [1] f(s(x)) → s(s(g(x))) [1] g(0) → 0 [1] The TRS has the following type information: g :: s:0 → s:0 s :: s:0 → s:0 f :: s:0 → s:0 0 :: s:0 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:

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

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

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

Found the following analysis order by SCC decomposition:

{ f, g }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {f,g}

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

Computed SIZE bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

Computed RUNTIME bound using CoFloCo for: g
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:

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

Function symbols to be analyzed:
Previous analysis results are:

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

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)