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:

dbl(S(0), S(0)) → S(S(S(S(0))))
unsafe(S(x)) → dbl(unsafe(x), 0)
unsafe(0) → 0
dbl(0, y) → y

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:

dbl(S(0), S(0)) → S(S(S(S(0)))) [1]
unsafe(S(x)) → dbl(unsafe(x), 0) [1]
unsafe(0) → 0 [1]
dbl(0, y) → y [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:

dbl(S(0), S(0)) → S(S(S(S(0)))) [1]
unsafe(S(x)) → dbl(unsafe(x), 0) [1]
unsafe(0) → 0 [1]
dbl(0, y) → y [1]

The TRS has the following type information:

dbl :: 0:S → 0:S → 0:S S :: 0:S → 0:S 0 :: 0:S unsafe :: 0:S → 0: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:
none

(c) The following functions are completely defined:

unsafe
dbl

Due to the following rules being added:

dbl(v0, v1) → 0 [0]

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: dbl(S(0), S(0)) → S(S(S(S(0)))) [1] unsafe(S(x)) → dbl(unsafe(x), 0) [1] unsafe(0) → 0 [1] dbl(0, y) → y [1] dbl(v0, v1) → 0 [0] The TRS has the following type information: dbl :: 0:S → 0:S → 0:S S :: 0:S → 0:S 0 :: 0:S unsafe :: 0:S → 0:S 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: dbl(S(0), S(0)) → S(S(S(S(0)))) [1] unsafe(S(S(x'))) → dbl(dbl(unsafe(x'), 0), 0) [2] unsafe(S(0)) → dbl(0, 0) [2] unsafe(0) → 0 [1] dbl(0, y) → y [1] dbl(v0, v1) → 0 [0] The TRS has the following type information: dbl :: 0:S → 0:S → 0:S S :: 0:S → 0:S 0 :: 0:S unsafe :: 0:S → 0:S 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:

dbl(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
dbl(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(x'), 0), 0) :|: x' >= 0, z = 1 + (1 + x')
unsafe(z) -{ 2 }→ dbl(0, 0) :|: z = 1 + 0
unsafe(z) -{ 1 }→ 0 :|: z = 0

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
dbl(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
dbl(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

(12) Obligation:

Complexity RNTS consisting of the following rules:

dbl(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
dbl(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(x'), 0), 0) :|: x' >= 0, z = 1 + (1 + x')
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

(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:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

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

Found the following analysis order by SCC decomposition:

{ dbl } { unsafe }

(16) Obligation:

Complexity RNTS consisting of the following rules:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

Function symbols to be analyzed: {dbl}, {unsafe}

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

Computed SIZE bound using KoAT for: dbl
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + z'

(18) Obligation:

Complexity RNTS consisting of the following rules:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

Function symbols to be analyzed: {dbl}, {unsafe}
Previous analysis results are:

dbl: runtime: ?, size: O(n1) [3 + z']

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

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

dbl: runtime: O(1) [1], size: O(n1) [3 + z']

(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:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

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

dbl: runtime: O(1) [1], size: O(n1) [3 + z']

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

Computed SIZE bound using CoFloCo for: unsafe
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(24) Obligation:

Complexity RNTS consisting of the following rules:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

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

dbl: runtime: O(1) [1], size: O(n1) [3 + z'] unsafe: runtime: ?, size: O(1) [3]

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

Computed RUNTIME bound using PUBS for: unsafe
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 4·z

(26) Obligation:

Complexity RNTS consisting of the following rules:

dbl(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
dbl(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
dbl(z, z') -{ 1 }→ 1 + (1 + (1 + (1 + 0))) :|: z = 1 + 0, z' = 1 + 0
unsafe(z) -{ 3 }→ y :|: z = 1 + 0, y >= 0, 0 = 0, 0 = y
unsafe(z) -{ 2 }→ dbl(dbl(unsafe(z - 2), 0), 0) :|: z - 2 >= 0
unsafe(z) -{ 1 }→ 0 :|: z = 0
unsafe(z) -{ 2 }→ 0 :|: z = 1 + 0, v0 >= 0, v1 >= 0, 0 = v0, 0 = v1

Function symbols to be analyzed:
Previous analysis results are:

dbl: runtime: O(1) [1], size: O(n1) [3 + z'] unsafe: runtime: O(n1) [3 + 4·z], size: O(1) [3]

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)