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:

minus(x, x) → 0
minus(s(x), s(y)) → minus(x, y)
minus(0, x) → 0
minus(x, 0) → x
div(s(x), s(y)) → s(div(minus(x, y), s(y)))
div(0, s(y)) → 0
f(x, 0, b) → x
f(x, s(y), b) → div(f(x, minus(s(y), s(0)), b), b)

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:

minus(x, x) → 0 [1]
minus(s(x), s(y)) → minus(x, y) [1]
minus(0, x) → 0 [1]
minus(x, 0) → x [1]
div(s(x), s(y)) → s(div(minus(x, y), s(y))) [1]
div(0, s(y)) → 0 [1]
f(x, 0, b) → x [1]
f(x, s(y), b) → div(f(x, minus(s(y), s(0)), b), b) [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:

minus(x, x) → 0 [1]
minus(s(x), s(y)) → minus(x, y) [1]
minus(0, x) → 0 [1]
minus(x, 0) → x [1]
div(s(x), s(y)) → s(div(minus(x, y), s(y))) [1]
div(0, s(y)) → 0 [1]
f(x, 0, b) → x [1]
f(x, s(y), b) → div(f(x, minus(s(y), s(0)), b), b) [1]

The TRS has the following type information:

minus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s div :: 0:s → 0:s → 0:s f :: 0:s → 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:

f
minus
div

Due to the following rules being added:

div(v0, v1) → 0 [0]

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: minus(x, x) → 0 [1] minus(s(x), s(y)) → minus(x, y) [1] minus(0, x) → 0 [1] minus(x, 0) → x [1] div(s(x), s(y)) → s(div(minus(x, y), s(y))) [1] div(0, s(y)) → 0 [1] f(x, 0, b) → x [1] f(x, s(y), b) → div(f(x, minus(s(y), s(0)), b), b) [1] div(v0, v1) → 0 [0] The TRS has the following type information: minus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s div :: 0:s → 0:s → 0:s f :: 0:s → 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: minus(x, x) → 0 [1] minus(s(x), s(y)) → minus(x, y) [1] minus(0, x) → 0 [1] minus(x, 0) → x [1] div(s(x), s(x)) → s(div(0, s(x))) [2] div(s(s(x')), s(s(y'))) → s(div(minus(x', y'), s(s(y')))) [2] div(s(0), s(y)) → s(div(0, s(y))) [2] div(s(x), s(0)) → s(div(x, s(0))) [2] div(0, s(y)) → 0 [1] f(x, 0, b) → x [1] f(x, s(0), b) → div(f(x, 0, b), b) [2] f(x, s(y), b) → div(f(x, minus(y, 0), b), b) [2] div(v0, v1) → 0 [0] The TRS has the following type information: minus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s div :: 0:s → 0:s → 0:s f :: 0:s → 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:

div(z, z') -{ 1 }→ 0 :|: z' = 1 + y, y >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
div(z, z') -{ 2 }→ 1 + div(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
div(z, z') -{ 2 }→ 1 + div(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
div(z, z') -{ 2 }→ 1 + div(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 1 + x
div(z, z') -{ 2 }→ 1 + div(0, 1 + y) :|: z' = 1 + y, z = 1 + 0, y >= 0
f(z, z', z'') -{ 1 }→ x :|: b >= 0, z'' = b, x >= 0, z = x, z' = 0
f(z, z', z'') -{ 2 }→ div(f(x, minus(y, 0), b), b) :|: z' = 1 + y, b >= 0, z'' = b, x >= 0, y >= 0, z = x
f(z, z', z'') -{ 2 }→ div(f(x, 0, b), b) :|: b >= 0, z'' = b, x >= 0, z' = 1 + 0, z = x
minus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
minus(z, z') -{ 1 }→ minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
minus(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = x
minus(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, 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:

div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 2 }→ 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 2 }→ 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 2 }→ div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

Found the following analysis order by SCC decomposition:

{ minus } { div } { f }

(14) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 2 }→ 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 2 }→ 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 2 }→ div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus}, {div}, {f}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 2 }→ 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 2 }→ 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 2 }→ div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

minus: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using PUBS for: minus
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 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 2 }→ 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 }→ 1 + div(0, 1 + (z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 2 }→ 1 + div(z - 1, 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 2 }→ div(f(z, minus(z' - 1, 0), z''), z'') :|: z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

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

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

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

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

(21) 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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

Computed RUNTIME bound using PUBS for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + 2·z + z·z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + z'], size: O(n1) [z] div: runtime: O(n2) [1 + 2·z + z·z'], size: O(n1) [z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 3 }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * 0, z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 + 2·s' + s'·z' + z' }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * s', s' >= 0, s' <= 1 * (z - 2), z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 3 }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * 0, z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 3·z }→ 1 + s4 :|: s4 >= 0, s4 <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 3 }→ div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= 1 * (z' - 1), z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1 * (z - 1), z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

minus: runtime: O(n1) [1 + z'], size: O(n1) [z] div: runtime: O(n2) [1 + 2·z + z·z'], size: O(n1) [z]

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 3 }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * 0, z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 + 2·s' + s'·z' + z' }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * s', s' >= 0, s' <= 1 * (z - 2), z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 3 }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * 0, z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 3·z }→ 1 + s4 :|: s4 >= 0, s4 <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 3 }→ div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= 1 * (z' - 1), z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1 * (z - 1), z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

minus: runtime: O(n1) [1 + z'], size: O(n1) [z] div: runtime: O(n2) [1 + 2·z + z·z'], size: O(n1) [z] f: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using PUBS for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 1 + 2·z·z' + z·z'·z'' + 4·z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
div(z, z') -{ 3 }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * 0, z' - 1 >= 0, z = 1 + (z' - 1)
div(z, z') -{ 2 + 2·s' + s'·z' + z' }→ 1 + s2 :|: s2 >= 0, s2 <= 1 * s', s' >= 0, s' <= 1 * (z - 2), z - 2 >= 0, z' - 2 >= 0
div(z, z') -{ 3 }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * 0, z = 1 + 0, z' - 1 >= 0
div(z, z') -{ 3·z }→ 1 + s4 :|: s4 >= 0, s4 <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
f(z, z', z'') -{ 1 }→ z :|: z'' >= 0, z >= 0, z' = 0
f(z, z', z'') -{ 3 }→ div(f(z, s'', z''), z'') :|: s'' >= 0, s'' <= 1 * (z' - 1), z'' >= 0, z >= 0, z' - 1 >= 0
f(z, z', z'') -{ 2 }→ div(f(z, 0, z''), z'') :|: z'' >= 0, z >= 0, z' = 1 + 0
minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1 * (z - 1), z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = z'
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed:
Previous analysis results are:

minus: runtime: O(n1) [1 + z'], size: O(n1) [z] div: runtime: O(n2) [1 + 2·z + z·z'], size: O(n1) [z] f: runtime: O(n3) [1 + 2·z·z' + z·z'·z'' + 4·z'], size: O(n1) [z]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^3)