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:

h(X, Z) → f(X, s(X), Z)
f(X, Y, g(X, Y)) → h(0, g(X, Y))
g(0, Y) → 0
g(X, s(Y)) → g(X, Y)

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(X, Y, g(X, Y)) → h(0, g(X, Y))

(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:

h(X, Z) → f(X, s(X), Z)
g(X, s(Y)) → g(X, Y)
g(0, Y) → 0

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:

h(X, Z) → f(X, s(X), Z) [1]
g(X, s(Y)) → g(X, Y) [1]
g(0, Y) → 0 [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:

h(X, Z) → f(X, s(X), Z) [1]
g(X, s(Y)) → g(X, Y) [1]
g(0, Y) → 0 [1]

The TRS has the following type information:

h :: s → a → f f :: s → s → a → f s :: s → s g :: 0 → s → 0 0 :: 0

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:

h
g

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const, const1, const2

(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: h(X, Z) → f(X, s(X), Z) [1] g(X, s(Y)) → g(X, Y) [1] g(0, Y) → 0 [1] The TRS has the following type information: h :: s → a → f f :: s → s → a → f s :: s → s g :: 0 → s → 0 0 :: 0 const :: f const1 :: s const2 :: a 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: h(X, Z) → f(X, s(X), Z) [1] g(X, s(Y)) → g(X, Y) [1] g(0, Y) → 0 [1] The TRS has the following type information: h :: s → a → f f :: s → s → a → f s :: s → s g :: 0 → s → 0 0 :: 0 const :: f const1 :: s const2 :: a 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
const1 => 0
const2 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

g(z, z') -{ 1 }→ g(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X
g(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0
h(z, z') -{ 1 }→ 1 + X + (1 + X) + Z :|: Z >= 0, X >= 0, z' = Z, z = X

(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:

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

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

Found the following analysis order by SCC decomposition:

{ g } { h }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z'

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(n1) [1 + 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:

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

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

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

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

Computed SIZE bound using CoFloCo for: h
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + 2·z + z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

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

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)