The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

div2(S(S(x))) → +(S(0), div2(x))
div2(S(0)) → 0
div2(0) → 0

The (relative) TRS S consists of the following rules:

+(x, S(0)) → S(x)
+(S(0), y) → S(y)

Rewrite Strategy: INNERMOST

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

Transformed relative 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:

div2(S(S(x))) → +(S(0), div2(x)) [1]
div2(S(0)) → 0 [1]
div2(0) → 0 [1]
+(x, S(0)) → S(x) [0]
+(S(0), y) → S(y) [0]

Rewrite Strategy: INNERMOST

(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

(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:

div2(S(S(x))) → plus(S(0), div2(x)) [1]
div2(S(0)) → 0 [1]
div2(0) → 0 [1]
plus(x, S(0)) → S(x) [0]
plus(S(0), y) → S(y) [0]

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:

div2(S(S(x))) → plus(S(0), div2(x)) [1]
div2(S(0)) → 0 [1]
div2(0) → 0 [1]
plus(x, S(0)) → S(x) [0]
plus(S(0), y) → S(y) [0]

The TRS has the following type information:

div2 :: S:0 → S:0 S :: S:0 → S:0 plus :: S:0 → S:0 → S:0 0 :: S: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:
none

(c) The following functions are completely defined:

div2
plus

Due to the following rules being added:

plus(v0, v1) → 0 [0]

And the following fresh constants: none

(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: div2(S(S(x))) → plus(S(0), div2(x)) [1] div2(S(0)) → 0 [1] div2(0) → 0 [1] plus(x, S(0)) → S(x) [0] plus(S(0), y) → S(y) [0] plus(v0, v1) → 0 [0] The TRS has the following type information: div2 :: S:0 → S:0 S :: S:0 → S:0 plus :: S:0 → S:0 → S:0 0 :: S:0 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: div2(S(S(S(S(x'))))) → plus(S(0), plus(S(0), div2(x'))) [2] div2(S(S(S(0)))) → plus(S(0), 0) [2] div2(S(S(0))) → plus(S(0), 0) [2] div2(S(0)) → 0 [1] div2(0) → 0 [1] plus(x, S(0)) → S(x) [0] plus(S(0), y) → S(y) [0] plus(v0, v1) → 0 [0] The TRS has the following type information: div2 :: S:0 → S:0 S :: S:0 → S:0 plus :: S:0 → S:0 → S:0 0 :: S:0 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

(12) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(x'))) :|: z = 1 + (1 + (1 + (1 + x'))), x' >= 0
div2(z) -{ 2 }→ plus(1 + 0, 0) :|: z = 1 + (1 + (1 + 0))
div2(z) -{ 2 }→ plus(1 + 0, 0) :|: z = 1 + (1 + 0)
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 0 }→ 1 + x :|: x >= 0, z' = 1 + 0, z = x
plus(z, z') -{ 0 }→ 1 + y :|: z = 1 + 0, y >= 0, z' = y

(13) InliningProof (UPPER BOUND(ID) transformation)

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(x'))) :|: z = 1 + (1 + (1 + (1 + x'))), x' >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 0 }→ 1 + x :|: x >= 0, z' = 1 + 0, z = x
plus(z, z') -{ 0 }→ 1 + y :|: z = 1 + 0, y >= 0, z' = y

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(16) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ plus } { div2 }

(18) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {div2}

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {div2}
Previous analysis results are:

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

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: plus
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:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {div2}
Previous analysis results are:

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

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(24) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {div2}
Previous analysis results are:

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

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

Computed SIZE bound using KoAT for: div2
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

(26) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {div2}
Previous analysis results are:

plus: runtime: O(1) [0], size: O(n1) [z + z'] div2: runtime: ?, size: O(n1) [z]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: div2
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + 2·z

(28) Obligation:

Complexity RNTS consisting of the following rules:

div2(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, div2(z - 4))) :|: z - 4 >= 0
div2(z) -{ 1 }→ 0 :|: z = 1 + 0
div2(z) -{ 1 }→ 0 :|: z = 0
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + (1 + 0)), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 0 :|: z = 1 + (1 + 0), v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + (1 + 0)), 1 + 0 = 1 + 0, y >= 0, 0 = y
div2(z) -{ 2 }→ 1 + y :|: z = 1 + (1 + 0), 1 + 0 = 1 + 0, y >= 0, 0 = y
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

plus: runtime: O(1) [0], size: O(n1) [z + z'] div2: runtime: O(n1) [2 + 2·z], size: O(n1) [z]

(29) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(30) BOUNDS(1, n^1)