The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(0, x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(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^3).


The TRS R consists of the following rules:

times(x, 0) → 0 [1]
times(x, s(y)) → plus(times(x, y), x) [1]
plus(x, 0) → x [1]
plus(0, x) → x [1]
plus(x, s(y)) → s(plus(x, y)) [1]
plus(s(x), y) → s(plus(x, 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:

times(x, 0) → 0 [1]
times(x, s(y)) → plus(times(x, y), x) [1]
plus(x, 0) → x [1]
plus(0, x) → 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:

times :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s plus :: 0: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:
none

(c) The following functions are completely defined:

times
plus

Due to the following rules being added:
none

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: times(x, 0) → 0 [1] times(x, s(y)) → plus(times(x, y), x) [1] plus(x, 0) → x [1] plus(0, x) → 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: times :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 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: times(x, 0) → 0 [1] times(x, s(0)) → plus(0, x) [2] times(x, s(s(y'))) → plus(plus(times(x, y'), x), x) [2] plus(x, 0) → x [1] plus(0, x) → 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: times :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 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:

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

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 2 }→ plus(0, z) :|: z >= 0, z' = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

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

Found the following analysis order by SCC decomposition:

{ plus } { times }

(14) Obligation:

Complexity RNTS consisting of the following rules:

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 2 }→ plus(0, z) :|: z >= 0, z' = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

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

(15) 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'

(16) Obligation:

Complexity RNTS consisting of the following rules:

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 2 }→ plus(0, z) :|: z >= 0, z' = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

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

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

(17) 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'

(18) Obligation:

Complexity RNTS consisting of the following rules:

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 2 }→ plus(0, z) :|: z >= 0, z' = 1 + 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

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

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

(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:

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * 0 + 1 * z, z >= 0, z' = 1 + 0
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * 0 + 1 * z, z >= 0, z' = 1 + 0
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

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

plus: runtime: O(n1) [1 + z + z'], size: O(n1) [z + z'] times: runtime: ?, size: O(n2) [z + 2·z·z']

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

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 4 + z + 4·z·z'² + 4·z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

plus(z, z') -{ 1 }→ z :|: z >= 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
times(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * 0 + 1 * z, z >= 0, z' = 1 + 0
times(z, z') -{ 2 }→ plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
times(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0

Function symbols to be analyzed:
Previous analysis results are:

plus: runtime: O(n1) [1 + z + z'], size: O(n1) [z + z'] times: runtime: O(n3) [4 + z + 4·z·z'2 + 4·z'], size: O(n2) [z + 2·z·z']

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^3)