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:

not(true) → false
not(false) → true
odd(0) → false
odd(s(x)) → not(odd(x))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
+(s(x), y) → s(+(x, 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:

not(true) → false [1]
not(false) → true [1]
odd(0) → false [1]
odd(s(x)) → not(odd(x)) [1]
+(x, 0) → x [1]
+(x, s(y)) → s(+(x, y)) [1]
+(s(x), y) → s(+(x, y)) [1]

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:

not(true) → false [1]
not(false) → true [1]
odd(0) → false [1]
odd(s(x)) → not(odd(x)) [1]
plus(x, 0) → x [1]
plus(x, s(y)) → s(plus(x, y)) [1]
plus(s(x), y) → s(plus(x, y)) [1]

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:

not(true) → false [1]
not(false) → true [1]
odd(0) → false [1]
odd(s(x)) → not(odd(x)) [1]
plus(x, 0) → x [1]
plus(x, s(y)) → s(plus(x, y)) [1]
plus(s(x), y) → s(plus(x, y)) [1]

The TRS has the following type information:

not :: true:false → true:false true :: true:false false :: true:false odd :: 0:s → true:false 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 0:s → 0:s

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:

plus

(c) The following functions are completely defined:

odd
not

Due to the following rules being added:
none

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: not(true) → false [1] not(false) → true [1] odd(0) → false [1] odd(s(x)) → not(odd(x)) [1] plus(x, 0) → x [1] plus(x, s(y)) → s(plus(x, y)) [1] plus(s(x), y) → s(plus(x, y)) [1] The TRS has the following type information: not :: true:false → true:false true :: true:false false :: true:false odd :: 0:s → true:false 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 0:s → 0:s 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: not(true) → false [1] not(false) → true [1] odd(0) → false [1] odd(s(0)) → not(false) [2] odd(s(s(x'))) → not(not(odd(x'))) [2] plus(x, 0) → x [1] plus(x, s(y)) → s(plus(x, y)) [1] plus(s(x), y) → s(plus(x, y)) [1] The TRS has the following type information: not :: true:false → true:false true :: true:false false :: true:false odd :: 0:s → true:false 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 0:s → 0:s 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:

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(x'))) :|: x' >= 0, z = 1 + (1 + x')
odd(z) -{ 2 }→ not(0) :|: z = 1 + 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y

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

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

not(z) -{ 1 }→ 0 :|: z = 1
not(z) -{ 1 }→ 1 :|: z = 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(x'))) :|: x' >= 0, z = 1 + (1 + x')
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, 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:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ not } { plus } { odd }

(18) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {not}, {plus}, {odd}

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

Computed SIZE bound using CoFloCo for: not
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:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {not}, {plus}, {odd}
Previous analysis results are:

not: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using CoFloCo for: not
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(22) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

not: runtime: O(1) [1], size: O(1) [1]

(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:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

not: runtime: O(1) [1], size: O(1) [1]

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

not: runtime: O(1) [1], size: O(1) [1] plus: runtime: ?, size: O(n1) [z + z']

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

not: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n1) [1 + z + z'], size: O(n1) [z + z']

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 + z + z' }→ 1 + s :|: s >= 0, s <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 + z + z' }→ 1 + s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

not: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n1) [1 + z + z'], size: O(n1) [z + z']

(31) 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

(32) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 + z + z' }→ 1 + s :|: s >= 0, s <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 + z + z' }→ 1 + s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

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

not: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n1) [1 + z + z'], size: O(n1) [z + z'] odd: runtime: ?, size: O(1) [1]

(33) 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: 3 + 4·z

(34) Obligation:

Complexity RNTS consisting of the following rules:

not(z) -{ 1 }→ 1 :|: z = 0
not(z) -{ 1 }→ 0 :|: z = 1
odd(z) -{ 2 }→ not(not(odd(z - 2))) :|: z - 2 >= 0
odd(z) -{ 3 }→ 1 :|: z = 1 + 0, 0 = 0
odd(z) -{ 1 }→ 0 :|: z = 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 + z + z' }→ 1 + s :|: s >= 0, s <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0
plus(z, z') -{ 1 + z + z' }→ 1 + s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

not: runtime: O(1) [1], size: O(1) [1] plus: runtime: O(n1) [1 + z + z'], size: O(n1) [z + z'] odd: runtime: O(n1) [3 + 4·z], size: O(1) [1]

(35) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(36) BOUNDS(1, n^1)