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

0 CpxTRS
↳1 NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxTRS
↳3 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxWeightedTrs
↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CpxTypedWeightedTrs
↳7 CompletionProof (UPPER BOUND(ID), 0 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳10 CpxTypedWeightedCompleteTrs
↳11 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)
↳12 CpxRNTS
↳13 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CpxRNTS
↳15 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 4 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 1192 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 618 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 550 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 117 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 1000 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 271 ms)
↳32 CpxRNTS
↳33 FinalProof (⇔, 0 ms)
↳34 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:

plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
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)))
div(div(x, y), z) → div(x, times(y, z))

Rewrite Strategy: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 1th argument of plus: plus, times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
div(div(x, y), z) → div(x, times(y, z))

(2) 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:

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

Rewrite Strategy: INNERMOST

(3) TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to weighted TRS

(4) 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:

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

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

The TRS has the following type information:
plus :: s:0 → s:0 → s:0
s :: s:0 → s:0
times :: s:0 → s:0 → s:0
quot :: s:0 → s:0 → s:0 → s:0
0 :: s:0
div :: s:0 → s:0 → s:0

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:


quot
div

(c) The following functions are completely defined:

times
plus

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:

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

The TRS has the following type information:
plus :: s:0 → s:0 → s:0
s :: s:0 → s:0
times :: s:0 → s:0 → s:0
quot :: s:0 → s:0 → s:0 → s:0
0 :: s:0
div :: s:0 → s:0 → s:0

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:

plus(s(x), y) → s(plus(x, y)) [1]
times(s(s(x')), y) → plus(y, plus(y, times(x', y))) [2]
times(s(s(0)), y) → plus(y, y) [2]
times(s(0), y) → plus(y, 0) [2]
quot(x, 0, s(z)) → s(div(x, s(z))) [1]
plus(x, 0) → x [1]
plus(0, y) → y [1]
div(0, y) → 0 [1]
div(x, y) → quot(x, y, y) [1]
times(s(0), y) → y [1]
quot(s(x), s(y), z) → quot(x, y, z) [1]
times(0, y) → 0 [1]
quot(0, s(y), z) → 0 [1]

The TRS has the following type information:
plus :: s:0 → s:0 → s:0
s :: s:0 → s:0
times :: s:0 → s:0 → s:0
quot :: s:0 → s:0 → s:0 → s:0
0 :: s:0
div :: s:0 → s:0 → s:0

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:

0 => 0

(12) 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
plus(z', z'') -{ 1 }→ x :|: z'' = 0, z' = x, x >= 0
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 :|: 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
times(z', z'') -{ 1 }→ y :|: z'' = y, y >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(y, y) :|: z' = 1 + (1 + 0), z'' = y, y >= 0
times(z', z'') -{ 2 }→ plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0
times(z', z'') -{ 2 }→ plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0
times(z', z'') -{ 1 }→ 0 :|: z'' = y, y >= 0, z' = 0

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

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

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

Found the following analysis order by SCC decomposition:

{ div, quot }
{ plus }
{ times }

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

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

(17) 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'

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

Function symbols to be analyzed: {div,quot}, {plus}, {times}
Previous analysis results are:
div: runtime: ?, size: O(n1) [z']
quot: runtime: ?, size: O(n1) [1 + z']

(19) 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''

(20) 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
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' - 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 >= 0, z'' - 1 >= 0, z' = 0
quot(z', z'', z1) -{ 1 }→ 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', 0) :|: 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:
div: runtime: O(n1) [3 + 3·z'], size: O(n1) [z']
quot: runtime: O(n1) [5 + 3·z' + z''], size: O(n1) [1 + z']

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'') -{ 6 + 3·z' + z'' }→ s' :|: s' >= 0, s' <= 1 * z' + 1, z' >= 0, z'' >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
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' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 2 + 3·z' + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1, 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) -{ 4 + 3·z' }→ 1 + s :|: s >= 0, s <= 1 * z', z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', 0) :|: 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:
div: runtime: O(n1) [3 + 3·z'], size: O(n1) [z']
quot: runtime: O(n1) [5 + 3·z' + z''], size: O(n1) [1 + z']

(23) 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''

(24) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'') -{ 6 + 3·z' + z'' }→ s' :|: s' >= 0, s' <= 1 * z' + 1, z' >= 0, z'' >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
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' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 2 + 3·z' + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1, 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) -{ 4 + 3·z' }→ 1 + s :|: s >= 0, s <= 1 * z', z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', 0) :|: 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:
div: runtime: O(n1) [3 + 3·z'], size: O(n1) [z']
quot: runtime: O(n1) [5 + 3·z' + z''], size: O(n1) [1 + z']
plus: runtime: ?, size: O(n1) [z' + z'']

(25) 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'

(26) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'') -{ 6 + 3·z' + z'' }→ s' :|: s' >= 0, s' <= 1 * z' + 1, z' >= 0, z'' >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
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' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 2 + 3·z' + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1, 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) -{ 4 + 3·z' }→ 1 + s :|: s >= 0, s <= 1 * z', z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 2 }→ plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}
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']
plus: runtime: O(n1) [1 + z'], size: O(n1) [z' + z'']

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'') -{ 6 + 3·z' + z'' }→ s' :|: s' >= 0, s' <= 1 * z' + 1, z' >= 0, z'' >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 + z' }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 2 + 3·z' + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1, 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) -{ 4 + 3·z' }→ 1 + s :|: s >= 0, s <= 1 * z', z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 3 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * z'' + 1 * z'', z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 3 + z'' }→ s3 :|: s3 >= 0, s3 <= 1 * z'' + 1 * 0, z'' >= 0, z' = 1 + 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}
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']
plus: runtime: O(n1) [1 + z'], size: O(n1) [z' + z'']

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


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

(30) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'') -{ 6 + 3·z' + z'' }→ s' :|: s' >= 0, s' <= 1 * z' + 1, z' >= 0, z'' >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 + z' }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 2 + 3·z' + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1, 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) -{ 4 + 3·z' }→ 1 + s :|: s >= 0, s <= 1 * z', z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 3 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * z'' + 1 * z'', z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 3 + z'' }→ s3 :|: s3 >= 0, s3 <= 1 * z'' + 1 * 0, z'' >= 0, z' = 1 + 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}
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']
plus: runtime: O(n1) [1 + z'], size: O(n1) [z' + z'']
times: runtime: ?, size: O(n2) [2·z'·z'' + 2·z'']

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using PUBS for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n2) with polynomial bound: 3 + 4·z' + 2·z'·z'' + z''

(32) Obligation:

Complexity RNTS consisting of the following rules:

div(z', z'') -{ 6 + 3·z' + z'' }→ s' :|: s' >= 0, s' <= 1 * z' + 1, z' >= 0, z'' >= 0
div(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
plus(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
plus(z', z'') -{ 1 + z' }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z' - 1) + 1 * z'', z' - 1 >= 0, z'' >= 0
quot(z', z'', z1) -{ 2 + 3·z' + z'' }→ s'' :|: s'' >= 0, s'' <= 1 * (z' - 1) + 1, 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) -{ 4 + 3·z' }→ 1 + s :|: s >= 0, s <= 1 * z', z'' = 0, z1 - 1 >= 0, z' >= 0
times(z', z'') -{ 3 + z'' }→ s2 :|: s2 >= 0, s2 <= 1 * z'' + 1 * z'', z' = 1 + (1 + 0), z'' >= 0
times(z', z'') -{ 3 + z'' }→ s3 :|: s3 >= 0, s3 <= 1 * z'' + 1 * 0, z'' >= 0, z' = 1 + 0
times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 1 + 0
times(z', z'') -{ 2 }→ plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
times(z', z'') -{ 1 }→ 0 :|: z'' >= 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']
plus: runtime: O(n1) [1 + z'], size: O(n1) [z' + z'']
times: runtime: O(n2) [3 + 4·z' + 2·z'·z'' + z''], size: O(n2) [2·z'·z'' + 2·z'']

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^2)