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:

div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, 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^1).


The TRS R consists of the following rules:

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

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

The TRS has the following type information:

div :: 0:s → 0:s → 0:s 0 :: 0:s quot :: 0:s → 0:s → 0:s → 0:s 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:

div
quot

(c) The following functions are completely defined:
none

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: div(0, y) → 0 [1] div(x, y) → quot(x, y, y) [1] quot(0, s(y), z) → 0 [1] quot(s(x), s(y), z) → quot(x, y, z) [1] quot(x, 0, s(z)) → s(div(x, s(z))) [1] The TRS has the following type information: div :: 0:s → 0:s → 0:s 0 :: 0:s quot :: 0:s → 0:s → 0:s → 0:s 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: div(0, y) → 0 [1] div(x, y) → quot(x, y, y) [1] quot(0, s(y), z) → 0 [1] quot(s(x), s(y), z) → quot(x, y, z) [1] quot(x, 0, s(z)) → s(div(x, s(z))) [1] The TRS has the following type information: div :: 0:s → 0:s → 0:s 0 :: 0:s quot :: 0:s → 0:s → 0:s → 0:s 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:

div(z', z'') -{ 1 }→ quot(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' = y, y >= 0, z' = 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 :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
quot(z', z'', z1) -{ 1 }→ 1 + div(x, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z

(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:

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

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

Found the following analysis order by SCC decomposition:

{ div, quot }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {div,quot}

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

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

Computed RUNTIME bound using KoAT for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 3·z'

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 5 + 3·z' + z''

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

div: runtime: O(n1) [3 + 3·z'], size: O(n1) [z'] quot: runtime: O(n1) [5 + 3·z' + z''], size: O(n1) [1 + z']

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)