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:

minus(0, Y) → 0
minus(s(X), s(Y)) → minus(X, Y)
geq(X, 0) → true
geq(0, s(Y)) → false
geq(s(X), s(Y)) → geq(X, Y)
div(0, s(Y)) → 0
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
if(true, X, Y) → X
if(false, X, Y) → 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^1).


The TRS R consists of the following rules:

minus(0, Y) → 0 [1]
minus(s(X), s(Y)) → minus(X, Y) [1]
geq(X, 0) → true [1]
geq(0, s(Y)) → false [1]
geq(s(X), s(Y)) → geq(X, Y) [1]
div(0, s(Y)) → 0 [1]
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
if(true, X, Y) → X [1]
if(false, X, Y) → 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:

minus(0, Y) → 0 [1]
minus(s(X), s(Y)) → minus(X, Y) [1]
geq(X, 0) → true [1]
geq(0, s(Y)) → false [1]
geq(s(X), s(Y)) → geq(X, Y) [1]
div(0, s(Y)) → 0 [1]
div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
if(true, X, Y) → X [1]
if(false, X, Y) → Y [1]

The TRS has the following type information:

minus :: 0:s → 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s geq :: 0:s → 0:s → true:false true :: true:false false :: true:false div :: 0:s → 0:s → 0:s if :: true:false → 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:

geq
div
minus
if

Due to the following rules being added:

div(v0, v1) → 0 [0]
minus(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(0, Y) → 0 [1] minus(s(X), s(Y)) → minus(X, Y) [1] geq(X, 0) → true [1] geq(0, s(Y)) → false [1] geq(s(X), s(Y)) → geq(X, Y) [1] div(0, s(Y)) → 0 [1] div(s(X), s(Y)) → if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] if(true, X, Y) → X [1] if(false, X, Y) → Y [1] div(v0, v1) → 0 [0] minus(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 geq :: 0:s → 0:s → true:false true :: true:false false :: true:false div :: 0:s → 0:s → 0:s if :: true:false → 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(0, Y) → 0 [1] minus(s(X), s(Y)) → minus(X, Y) [1] geq(X, 0) → true [1] geq(0, s(Y)) → false [1] geq(s(X), s(Y)) → geq(X, Y) [1] div(0, s(Y)) → 0 [1] div(s(0), s(0)) → if(true, s(div(0, s(0))), 0) [3] div(s(X), s(0)) → if(true, s(div(0, s(0))), 0) [2] div(s(0), s(s(Y'))) → if(false, s(div(0, s(s(Y')))), 0) [3] div(s(0), s(s(Y'))) → if(false, s(div(0, s(s(Y')))), 0) [2] div(s(s(X')), s(s(Y''))) → if(geq(X', Y''), s(div(minus(X', Y''), s(s(Y'')))), 0) [3] div(s(s(X')), s(s(Y''))) → if(geq(X', Y''), s(div(0, s(s(Y'')))), 0) [2] if(true, X, Y) → X [1] if(false, X, Y) → Y [1] div(v0, v1) → 0 [0] minus(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 geq :: 0:s → 0:s → true:false true :: true:false false :: true:false div :: 0:s → 0:s → 0:s if :: true:false → 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
true => 1
false => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 3 }→ if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
div(z, z') -{ 2 }→ if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: Y >= 0, z' = 1 + Y, z = 0
div(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
geq(z, z') -{ 1 }→ geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
geq(z, z') -{ 1 }→ 1 :|: X >= 0, z = X, z' = 0
geq(z, z') -{ 1 }→ 0 :|: Y >= 0, z' = 1 + Y, z = 0
if(z, z', z'') -{ 1 }→ X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
if(z, z', z'') -{ 1 }→ Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
minus(z, z') -{ 1 }→ minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
minus(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0
minus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

if(z, z', z'') -{ 1 }→ X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
if(z, z', z'') -{ 1 }→ Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 3 }→ if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
div(z, z') -{ 2 }→ if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: Y >= 0, z' = 1 + Y, z = 0
div(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
geq(z, z') -{ 1 }→ geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
geq(z, z') -{ 1 }→ 1 :|: X >= 0, z = X, z' = 0
geq(z, z') -{ 1 }→ 0 :|: Y >= 0, z' = 1 + Y, z = 0
if(z, z', z'') -{ 1 }→ X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
if(z, z', z'') -{ 1 }→ Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
minus(z, z') -{ 1 }→ minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
minus(z, z') -{ 1 }→ 0 :|: z' = Y, Y >= 0, z = 0
minus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

(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') -{ 3 }→ if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 2 }→ if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
geq(z, z') -{ 1 }→ geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
geq(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
geq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
if(z, z', z'') -{ 1 }→ z' :|: z'' >= 0, z = 1, z' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z'' >= 0, 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 = 0
minus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ minus } { if } { geq } { div }

(16) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 3 }→ if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 2 }→ if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
geq(z, z') -{ 1 }→ geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
geq(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
geq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
if(z, z', z'') -{ 1 }→ z' :|: z'' >= 0, z = 1, z' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z'' >= 0, 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 = 0
minus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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

Computed SIZE bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(18) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 3 }→ if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 2 }→ if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
geq(z, z') -{ 1 }→ geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
geq(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
geq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
if(z, z', z'') -{ 1 }→ z' :|: z'' >= 0, z = 1, z' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z'' >= 0, 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 = 0
minus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0

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

minus: runtime: ?, size: O(1) [0]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

div(z, z') -{ 3 }→ if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 2 }→ if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
geq(z, z') -{ 1 }→ geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
geq(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
geq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
if(z, z', z'') -{ 1 }→ z' :|: z'' >= 0, z = 1, z' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z'' >= 0, 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 = 0
minus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0]

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

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0]

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

Computed SIZE bound using CoFloCo for: if
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') -{ 2 + z' }→ if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 2 }→ if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
geq(z, z') -{ 1 }→ geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
geq(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
geq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
if(z, z', z'') -{ 1 }→ z' :|: z'' >= 0, z = 1, z' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' >= 0, z = 0
minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
minus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: ?, size: O(n1) [z' + z'']

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], 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') -{ 2 + z' }→ if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 2 }→ if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
div(z, z') -{ 3 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
div(z, z') -{ 2 }→ if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
div(z, z') -{ 3 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 2 }→ if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
div(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
div(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
geq(z, z') -{ 1 }→ geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
geq(z, z') -{ 1 }→ 1 :|: z >= 0, z' = 0
geq(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0
if(z, z', z'') -{ 1 }→ z' :|: z'' >= 0, z = 1, z' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' >= 0, z = 0
minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
minus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n1) [z' + z'']

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

Computed SIZE bound using CoFloCo for: geq
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(30) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n1) [z' + z''] geq: runtime: ?, size: O(1) [1]

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

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n1) [z' + z''] geq: runtime: O(n1) [1 + z'], size: O(1) [1]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n1) [z' + z''] geq: runtime: O(n1) [1 + z'], size: O(1) [1]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

(36) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n1) [z' + z''] geq: runtime: O(n1) [1 + z'], size: O(1) [1] div: runtime: ?, size: O(1) [1]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 5 + 2·z'

(38) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

minus: runtime: O(n1) [1 + z'], size: O(1) [0] if: runtime: O(1) [1], size: O(n1) [z' + z''] geq: runtime: O(n1) [1 + z'], size: O(1) [1] div: runtime: O(n1) [5 + 2·z'], size: O(1) [1]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^1)