The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

cond(true, x) → cond(odd(x), p(x))
odd(0) → false
odd(s(0)) → true
odd(s(s(x))) → odd(x)
p(0) → 0
p(s(x)) → x

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^2).


The TRS R consists of the following rules:

cond(true, x) → cond(odd(x), p(x)) [1]
odd(0) → false [1]
odd(s(0)) → true [1]
odd(s(s(x))) → odd(x) [1]
p(0) → 0 [1]
p(s(x)) → x [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:

cond(true, x) → cond(odd(x), p(x)) [1]
odd(0) → false [1]
odd(s(0)) → true [1]
odd(s(s(x))) → odd(x) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

The TRS has the following type information:

cond :: true:false → 0:s → cond true :: true:false odd :: 0:s → true:false p :: 0:s → 0:s 0 :: 0:s false :: true:false s :: 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:

cond

(c) The following functions are completely defined:

odd
p

Due to the following rules being added:
none

And the following fresh constants:

const

(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: cond(true, x) → cond(odd(x), p(x)) [1] odd(0) → false [1] odd(s(0)) → true [1] odd(s(s(x))) → odd(x) [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond :: true:false → 0:s → cond true :: true:false odd :: 0:s → true:false p :: 0:s → 0:s 0 :: 0:s false :: true:false s :: 0:s → 0:s const :: cond 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: cond(true, 0) → cond(false, 0) [3] cond(true, s(0)) → cond(true, 0) [3] cond(true, s(s(x'))) → cond(odd(x'), s(x')) [3] odd(0) → false [1] odd(s(0)) → true [1] odd(s(s(x))) → odd(x) [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond :: true:false → 0:s → cond true :: true:false odd :: 0:s → true:false p :: 0:s → 0:s 0 :: 0:s false :: true:false s :: 0:s → 0:s const :: cond 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:

true => 1
0 => 0
false => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 3 }→ cond(odd(x'), 1 + x') :|: z' = 1 + (1 + x'), z = 1, x' >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ 1 }→ odd(x) :|: x >= 0, z = 1 + (1 + x)
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(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:

cond(z, z') -{ 3 }→ cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ 1 }→ odd(z - 2) :|: z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ odd } { p } { cond }

(14) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 3 }→ cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ 1 }→ odd(z - 2) :|: z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {odd}, {p}, {cond}

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

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:

cond(z, z') -{ 3 }→ cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ 1 }→ odd(z - 2) :|: z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

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

Computed RUNTIME bound using PUBS for: odd
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:

cond(z, z') -{ 3 }→ cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ 1 }→ odd(z - 2) :|: z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {cond}
Previous analysis results are:

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

(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:

cond(z, z') -{ 2 + z' }→ cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ z }→ s' :|: s' >= 0, s' <= 1, z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {cond}
Previous analysis results are:

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 2 + z' }→ cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ z }→ s' :|: s' >= 0, s' <= 1, z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {cond}
Previous analysis results are:

odd: runtime: O(n1) [1 + z], size: O(1) [1] p: runtime: ?, size: O(n1) [z]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 2 + z' }→ cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ z }→ s' :|: s' >= 0, s' <= 1, z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

odd: runtime: O(n1) [1 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n1) [z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 2 + z' }→ cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ z }→ s' :|: s' >= 0, s' <= 1, z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

odd: runtime: O(n1) [1 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n1) [z]

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

Computed SIZE bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(28) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 2 + z' }→ cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ z }→ s' :|: s' >= 0, s' <= 1, z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

odd: runtime: O(n1) [1 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n1) [z] cond: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 8 + 3·z' + z'²

(30) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z') -{ 2 + z' }→ cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
cond(z, z') -{ 3 }→ cond(1, 0) :|: z = 1, z' = 1 + 0
cond(z, z') -{ 3 }→ cond(0, 0) :|: z = 1, z' = 0
odd(z) -{ z }→ s' :|: s' >= 0, s' <= 1, z - 2 >= 0
odd(z) -{ 1 }→ 1 :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

odd: runtime: O(n1) [1 + z], size: O(1) [1] p: runtime: O(1) [1], size: O(n1) [z] cond: runtime: O(n2) [8 + 3·z' + z'2], size: O(1) [0]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^2)