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:

prime(0) → false
prime(s(0)) → false
prime(s(s(x))) → prime1(s(s(x)), s(x))
prime1(x, 0) → false
prime1(x, s(0)) → true
prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y)))
divp(x, y) → =(rem(x, y), 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:

prime(0) → false [1]
prime(s(0)) → false [1]
prime(s(s(x))) → prime1(s(s(x)), s(x)) [1]
prime1(x, 0) → false [1]
prime1(x, s(0)) → true [1]
prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
divp(x, y) → =(rem(x, y), 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:

prime(0) → false [1]
prime(s(0)) → false [1]
prime(s(s(x))) → prime1(s(s(x)), s(x)) [1]
prime1(x, 0) → false [1]
prime1(x, s(0)) → true [1]
prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1]
divp(x, y) → =(rem(x, y), 0) [1]

The TRS has the following type information:

prime :: 0:s → false:true:and 0 :: 0:s false :: false:true:and s :: 0:s → 0:s prime1 :: 0:s → 0:s → false:true:and true :: false:true:and and :: not → false:true:and → false:true:and not :: = → not divp :: 0:s → 0:s → = = :: rem → 0:s → = rem :: 0:s → 0:s → rem

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:

prime
prime1
divp

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const, const1, const2

(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: prime(0) → false [1] prime(s(0)) → false [1] prime(s(s(x))) → prime1(s(s(x)), s(x)) [1] prime1(x, 0) → false [1] prime1(x, s(0)) → true [1] prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1] divp(x, y) → =(rem(x, y), 0) [1] The TRS has the following type information: prime :: 0:s → false:true:and 0 :: 0:s false :: false:true:and s :: 0:s → 0:s prime1 :: 0:s → 0:s → false:true:and true :: false:true:and and :: not → false:true:and → false:true:and not :: = → not divp :: 0:s → 0:s → = = :: rem → 0:s → = rem :: 0:s → 0:s → rem const :: not const1 :: = const2 :: rem 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: prime(0) → false [1] prime(s(0)) → false [1] prime(s(s(x))) → prime1(s(s(x)), s(x)) [1] prime1(x, 0) → false [1] prime1(x, s(0)) → true [1] prime1(x, s(s(y))) → and(not(divp(s(s(y)), x)), prime1(x, s(y))) [1] divp(x, y) → =(rem(x, y), 0) [1] The TRS has the following type information: prime :: 0:s → false:true:and 0 :: 0:s false :: false:true:and s :: 0:s → 0:s prime1 :: 0:s → 0:s → false:true:and true :: false:true:and and :: not → false:true:and → false:true:and not :: = → not divp :: 0:s → 0:s → = = :: rem → 0:s → = rem :: 0:s → 0:s → rem const :: not const1 :: = const2 :: rem 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
false => 0
true => 1
const => 0
const1 => 0
const2 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y
prime(z) -{ 1 }→ prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x)
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: x >= 0, z' = 1 + 0, z = x
prime1(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
prime1(z, z') -{ 1 }→ 1 + (1 + divp(1 + (1 + y), x)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x

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

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + x + y) + 0 :|: x >= 0, y >= 0, z = x, z' = y
prime(z) -{ 1 }→ prime1(1 + (1 + x), 1 + x) :|: x >= 0, z = 1 + (1 + x)
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: x >= 0, z' = 1 + 0, z = x
prime1(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + y') + 0)) + prime1(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x, x' >= 0, y' >= 0, 1 + (1 + y) = x', x = y'

(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:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

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

Found the following analysis order by SCC decomposition:

{ divp } { prime1 } { prime }

(16) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {divp}, {prime1}, {prime}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {divp}, {prime1}, {prime}
Previous analysis results are:

divp: runtime: ?, size: O(n1) [2 + z + z']

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

Computed RUNTIME bound using CoFloCo for: divp
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:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime1}, {prime}
Previous analysis results are:

divp: runtime: O(1) [1], size: O(n1) [2 + z + 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:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime1}, {prime}
Previous analysis results are:

divp: runtime: O(1) [1], size: O(n1) [2 + z + z']

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

Computed SIZE bound using KoAT for: prime1
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + z·z' + 4·z' + z'²

(24) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed: {prime1}, {prime}
Previous analysis results are:

divp: runtime: O(1) [1], size: O(n1) [2 + z + z'] prime1: runtime: ?, size: O(n2) [1 + z·z' + 4·z' + z'2]

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

Computed RUNTIME bound using KoAT for: prime1
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + 2·z'

(26) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 }→ prime1(1 + (1 + (z - 2)), 1 + (z - 2)) :|: z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + prime1(z, 1 + (z' - 2)) :|: z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

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

divp: runtime: O(1) [1], size: O(n1) [2 + z + z'] prime1: runtime: O(n1) [2 + 2·z'], size: O(n2) [1 + z·z' + 4·z' + z'2]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 + 2·z }→ s :|: s >= 0, s <= 1 * ((1 + (z - 2)) * (1 + (z - 2))) + 1 * ((1 + (z - 2)) * (1 + (1 + (z - 2)))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 + 2·z' }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * (1 + (z' - 2))) + 1 * ((1 + (z' - 2)) * z) + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

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

divp: runtime: O(1) [1], size: O(n1) [2 + z + z'] prime1: runtime: O(n1) [2 + 2·z'], size: O(n2) [1 + z·z' + 4·z' + z'2]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: prime
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + z + 2·z²

(30) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 + 2·z }→ s :|: s >= 0, s <= 1 * ((1 + (z - 2)) * (1 + (z - 2))) + 1 * ((1 + (z - 2)) * (1 + (1 + (z - 2)))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 + 2·z' }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * (1 + (z' - 2))) + 1 * ((1 + (z' - 2)) * z) + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

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

divp: runtime: O(1) [1], size: O(n1) [2 + z + z'] prime1: runtime: O(n1) [2 + 2·z'], size: O(n2) [1 + z·z' + 4·z' + z'2] prime: runtime: ?, size: O(n2) [2 + z + 2·z2]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: prime
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(32) Obligation:

Complexity RNTS consisting of the following rules:

divp(z, z') -{ 1 }→ 1 + (1 + z + z') + 0 :|: z >= 0, z' >= 0
prime(z) -{ 1 + 2·z }→ s :|: s >= 0, s <= 1 * ((1 + (z - 2)) * (1 + (z - 2))) + 1 * ((1 + (z - 2)) * (1 + (1 + (z - 2)))) + 4 * (1 + (z - 2)) + 1, z - 2 >= 0
prime(z) -{ 1 }→ 0 :|: z = 0
prime(z) -{ 1 }→ 0 :|: z = 1 + 0
prime1(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 1 + 0
prime1(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
prime1(z, z') -{ 2 + 2·z' }→ 1 + (1 + (1 + (1 + x' + z) + 0)) + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * (1 + (z' - 2))) + 1 * ((1 + (z' - 2)) * z) + 4 * (1 + (z' - 2)) + 1, z >= 0, z' - 2 >= 0, x' >= 0, 1 + (1 + (z' - 2)) = x'

Function symbols to be analyzed:
Previous analysis results are:

divp: runtime: O(1) [1], size: O(n1) [2 + z + z'] prime1: runtime: O(n1) [2 + 2·z'], size: O(n2) [1 + z·z' + 4·z' + z'2] prime: runtime: O(n1) [1 + 2·z], size: O(n2) [2 + z + 2·z2]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^1)