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:

quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
quot(x, 0, s(z)) → s(quot(x, plus(z, s(0)), s(z)))

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:

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

quot(0, s(y), s(z)) → 0 [1]
quot(s(x), s(y), z) → quot(x, y, z) [1]
plus(0, y) → y [1]
plus(s(x), y) → s(plus(x, y)) [1]
quot(x, 0, s(z)) → s(quot(x, plus(z, s(0)), s(z))) [1]

The TRS has the following type information:

quot :: 0:s → 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:

quot

(c) The following functions are completely defined:

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: quot(0, s(y), s(z)) → 0 [1] quot(s(x), s(y), z) → quot(x, y, z) [1] plus(0, y) → y [1] plus(s(x), y) → s(plus(x, y)) [1] quot(x, 0, s(z)) → s(quot(x, plus(z, s(0)), s(z))) [1] The TRS has the following type information: quot :: 0:s → 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: quot(0, s(y), s(z)) → 0 [1] quot(s(x), s(y), z) → quot(x, y, z) [1] plus(0, y) → y [1] plus(s(x), y) → s(plus(x, y)) [1] quot(x, 0, s(0)) → s(quot(x, s(0), s(0))) [2] quot(x, 0, s(s(x'))) → s(quot(x, s(plus(x', s(0))), s(s(x')))) [2] The TRS has the following type information: quot :: 0:s → 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 }→ y :|: z'' = y, y >= 0, z' = 0
plus(z', z'') -{ 1 }→ 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
quot(z', z'', z1) -{ 1 }→ quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
quot(z', z'', z1) -{ 1 }→ 0 :|: z >= 0, y >= 0, z'' = 1 + y, z1 = 1 + z, z' = 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(x, 1 + plus(x', 1 + 0), 1 + (1 + x')) :|: z'' = 0, z' = x, x >= 0, x' >= 0, z1 = 1 + (1 + x')
quot(z', z'', z1) -{ 2 }→ 1 + quot(x, 1 + 0, 1 + 0) :|: z'' = 0, z' = x, z1 = 1 + 0, x >= 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 }→ 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ plus } { quot }

(14) Obligation:

Complexity RNTS consisting of the following rules:

plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 }→ 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

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

(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 }→ 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {quot}
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'

(18) Obligation:

Complexity RNTS consisting of the following rules:

plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 }→ 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + plus(z1 - 2, 1 + 0), 1 + (1 + (z1 - 2))) :|: z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

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

plus: runtime: O(n1) [1 + 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' }→ 1 + s :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 1 + z1 }→ 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= 1 * (z1 - 2) + 1 * (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 + z' }→ 1 + s :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 1 + z1 }→ 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= 1 * (z1 - 2) + 1 * (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

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

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

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

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5 + 5·z' + z'·z1 + z1

(24) Obligation:

Complexity RNTS consisting of the following rules:

plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 + z' }→ 1 + s :|: s >= 0, s <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 1 }→ quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
quot(z', z'', z1) -{ 1 }→ 0 :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 1 + z1 }→ 1 + quot(z', 1 + s', 1 + (1 + (z1 - 2))) :|: s' >= 0, s' <= 1 * (z1 - 2) + 1 * (1 + 0), z'' = 0, z' >= 0, z1 - 2 >= 0
quot(z', z'', z1) -{ 2 }→ 1 + quot(z', 1 + 0, 1 + 0) :|: z'' = 0, z1 = 1 + 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

plus: runtime: O(n1) [1 + z'], size: O(n1) [z' + z''] quot: runtime: O(n2) [5 + 5·z' + z'·z1 + z1], size: O(n1) [2 + 2·z']

(25) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(26) BOUNDS(1, n^2)