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), y) → g(x, S(y))
f(y, S(x)) → f(S(y), x)
g(0, x2) → x2
f(x1, 0) → g(x1, 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), y) → g(x, S(y)) [1]
f(y, S(x)) → f(S(y), x) [1]
g(0, x2) → x2 [1]
f(x1, 0) → g(x1, 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), y) → g(x, S(y)) [1]
f(y, S(x)) → f(S(y), x) [1]
g(0, x2) → x2 [1]
f(x1, 0) → g(x1, 0) [1]

The TRS has the following type information:

g :: S:0 → S:0 → S:0 S :: S:0 → S:0 f :: S:0 → 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), y) → g(x, S(y)) [1] f(y, S(x)) → f(S(y), x) [1] g(0, x2) → x2 [1] f(x1, 0) → g(x1, 0) [1] The TRS has the following type information: g :: S:0 → S:0 → S:0 S :: S:0 → S:0 f :: S:0 → 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), y) → g(x, S(y)) [1] f(y, S(x)) → f(S(y), x) [1] g(0, x2) → x2 [1] f(x1, 0) → g(x1, 0) [1] The TRS has the following type information: g :: S:0 → S:0 → S:0 S :: S:0 → S:0 f :: S:0 → 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, z') -{ 1 }→ g(x1, 0) :|: x1 >= 0, z = x1, z' = 0
f(z, z') -{ 1 }→ f(1 + y, x) :|: z' = 1 + x, y >= 0, x >= 0, z = y
g(z, z') -{ 1 }→ x2 :|: z' = x2, z = 0, x2 >= 0
g(z, z') -{ 1 }→ g(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y

(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') -{ 1 }→ g(z, 0) :|: z >= 0, z' = 0
f(z, z') -{ 1 }→ f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
g(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
g(z, z') -{ 1 }→ g(z - 1, 1 + z') :|: z - 1 >= 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') -{ 1 }→ g(z, 0) :|: z >= 0, z' = 0
f(z, z') -{ 1 }→ f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
g(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
g(z, z') -{ 1 }→ g(z - 1, 1 + z') :|: z - 1 >= 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') -{ 1 }→ g(z, 0) :|: z >= 0, z' = 0
f(z, z') -{ 1 }→ f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
g(z, z') -{ 1 }→ z' :|: z = 0, z' >= 0
g(z, z') -{ 1 }→ g(z - 1, 1 + z') :|: z - 1 >= 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(n¹) with polynomial bound: 1 + z

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

g: runtime: O(n1) [1 + z], 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') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * 0, z >= 0, z' = 0
f(z, z') -{ 1 }→ f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
g(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 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(n1) [1 + z], 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(n¹) with polynomial bound: z + z'

(22) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * z + 1 * 0, z >= 0, z' = 0
f(z, z') -{ 1 }→ f(1 + z, z' - 1) :|: z >= 0, z' - 1 >= 0
g(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 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(n1) [1 + z], size: O(n1) [z + z'] f: runtime: ?, size: O(n1) [z + z']

(23) 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: 2 + z + 2·z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

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

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^1)