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:

odd(S(x)) → even(x)
even(S(x)) → odd(x)
odd(0) → 0
even(0) → S(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:

odd(S(x)) → even(x) [1]
even(S(x)) → odd(x) [1]
odd(0) → 0 [1]
even(0) → S(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:

odd(S(x)) → even(x) [1]
even(S(x)) → odd(x) [1]
odd(0) → 0 [1]
even(0) → S(0) [1]

The TRS has the following type information:

odd :: S:0 → S:0 S :: S:0 → S:0 even :: 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:

odd
even

(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: odd(S(x)) → even(x) [1] even(S(x)) → odd(x) [1] odd(0) → 0 [1] even(0) → S(0) [1] The TRS has the following type information: odd :: S:0 → S:0 S :: S:0 → S:0 even :: 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: odd(S(x)) → even(x) [1] even(S(x)) → odd(x) [1] odd(0) → 0 [1] even(0) → S(0) [1] The TRS has the following type information: odd :: S:0 → S:0 S :: S:0 → S:0 even :: 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:

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

even(z) -{ 1 }→ odd(z - 1) :|: z - 1 >= 0
even(z) -{ 1 }→ 1 + 0 :|: z = 0
odd(z) -{ 1 }→ even(z - 1) :|: z - 1 >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

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

Found the following analysis order by SCC decomposition:

{ even, odd }

(14) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(z - 1) :|: z - 1 >= 0
even(z) -{ 1 }→ 1 + 0 :|: z = 0
odd(z) -{ 1 }→ even(z - 1) :|: z - 1 >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {even,odd}

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

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

even(z) -{ 1 }→ odd(z - 1) :|: z - 1 >= 0
even(z) -{ 1 }→ 1 + 0 :|: z = 0
odd(z) -{ 1 }→ even(z - 1) :|: z - 1 >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {even,odd}
Previous analysis results are:

even: runtime: ?, size: O(1) [1] odd: runtime: ?, size: O(1) [1]

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

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

Computed RUNTIME bound using CoFloCo for: odd
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:

even(z) -{ 1 }→ odd(z - 1) :|: z - 1 >= 0
even(z) -{ 1 }→ 1 + 0 :|: z = 0
odd(z) -{ 1 }→ even(z - 1) :|: z - 1 >= 0
odd(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed:
Previous analysis results are:

even: runtime: O(n1) [2 + 2·z], size: O(1) [1] odd: runtime: O(n1) [1 + 2·z], size: O(1) [1]

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)