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(x, y) → x
g(x, y) → y
f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

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(x, y) → x [1]
g(x, y) → y [1]
f(0, 1, x) → f(s(x), x, x) [1]
f(x, y, s(z)) → s(f(0, 1, z)) [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(x, y) → x [1]
g(x, y) → y [1]
f(0, 1, x) → f(s(x), x, x) [1]
f(x, y, s(z)) → s(f(0, 1, z)) [1]

The TRS has the following type information:

g :: g → g → g f :: 0:1:s → 0:1:s → 0:1:s → 0:1:s 0 :: 0:1:s 1 :: 0:1:s s :: 0:1:s → 0:1: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:

g
f

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const

(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(x, y) → x [1] g(x, y) → y [1] f(0, 1, x) → f(s(x), x, x) [1] f(x, y, s(z)) → s(f(0, 1, z)) [1] The TRS has the following type information: g :: g → g → g f :: 0:1:s → 0:1:s → 0:1:s → 0:1:s 0 :: 0:1:s 1 :: 0:1:s s :: 0:1:s → 0:1:s const :: g 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(x, y) → x [1] g(x, y) → y [1] f(0, 1, x) → f(s(x), x, x) [1] f(x, y, s(z)) → s(f(0, 1, z)) [1] The TRS has the following type information: g :: g → g → g f :: 0:1:s → 0:1:s → 0:1:s → 0:1:s 0 :: 0:1:s 1 :: 0:1:s s :: 0:1:s → 0:1:s const :: g 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
1 => 1
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'', z1) -{ 1 }→ f(1 + x, x, x) :|: x >= 0, z'' = 1, z1 = x, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z) :|: z >= 0, z' = x, z'' = y, x >= 0, y >= 0, z1 = 1 + z
g(z', z'') -{ 1 }→ x :|: z' = x, z'' = y, x >= 0, y >= 0
g(z', z'') -{ 1 }→ y :|: z' = x, z'' = y, x >= 0, y >= 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', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

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

Found the following analysis order by SCC decomposition:

{ g } { f }

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

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

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

Computed SIZE bound using CoFloCo for: g
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:

f(z', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

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

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

(17) 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: 1

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

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

g: runtime: O(1) [1], 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:

f(z', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

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

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

(21) 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

(22) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

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

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

(23) 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: 1 + 2·z1

(24) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'', z1) -{ 1 }→ f(1 + z1, z1, z1) :|: z1 >= 0, z'' = 1, z' = 0
f(z', z'', z1) -{ 1 }→ 1 + f(0, 1, z1 - 1) :|: z1 - 1 >= 0, z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z' :|: z' >= 0, z'' >= 0
g(z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0

Function symbols to be analyzed:
Previous analysis results are:

g: runtime: O(1) [1], size: O(n1) [z' + z''] f: runtime: O(n1) [1 + 2·z1], size: O(1) [0]

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)