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:

cond1(true, x, y) → cond2(gr(y, 0), x, y)
cond2(true, x, y) → cond2(gr(y, 0), x, p(y))
cond2(false, x, y) → cond1(gr(x, 0), p(x), y)
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x

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:

cond1(true, x, y) → cond2(gr(y, 0), x, y) [1]
cond2(true, x, y) → cond2(gr(y, 0), x, p(y)) [1]
cond2(false, x, y) → cond1(gr(x, 0), p(x), y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [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:

cond1(true, x, y) → cond2(gr(y, 0), x, y) [1]
cond2(true, x, y) → cond2(gr(y, 0), x, p(y)) [1]
cond2(false, x, y) → cond1(gr(x, 0), p(x), y) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

The TRS has the following type information:

cond1 :: true:false → 0:s → 0:s → cond1:cond2 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2 gr :: 0:s → 0:s → true:false 0 :: 0:s p :: 0:s → 0:s false :: true:false 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:

cond1
cond2

(c) The following functions are completely defined:

gr
p

Due to the following rules being added:
none

And the following fresh constants:

const

(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: cond1(true, x, y) → cond2(gr(y, 0), x, y) [1] cond2(true, x, y) → cond2(gr(y, 0), x, p(y)) [1] cond2(false, x, y) → cond1(gr(x, 0), p(x), y) [1] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond1 :: true:false → 0:s → 0:s → cond1:cond2 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2 gr :: 0:s → 0:s → true:false 0 :: 0:s p :: 0:s → 0:s false :: true:false s :: 0:s → 0:s const :: cond1:cond2 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: cond1(true, x, 0) → cond2(false, x, 0) [2] cond1(true, x, s(x')) → cond2(true, x, s(x')) [2] cond2(true, x, 0) → cond2(false, x, 0) [3] cond2(true, x, s(x'')) → cond2(true, x, x'') [3] cond2(false, 0, y) → cond1(false, 0, y) [3] cond2(false, s(x1), y) → cond1(true, x1, y) [3] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond1 :: true:false → 0:s → 0:s → cond1:cond2 true :: true:false cond2 :: true:false → 0:s → 0:s → cond1:cond2 gr :: 0:s → 0:s → true:false 0 :: 0:s p :: 0:s → 0:s false :: true:false s :: 0:s → 0:s const :: cond1:cond2 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:

true => 1
0 => 0
false => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, x, 1 + x') :|: z' = x, z = 1, z'' = 1 + x', x >= 0, x' >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0
cond2(z, z', z'') -{ 3 }→ cond2(1, x, x'') :|: z' = x, z = 1, x >= 0, z'' = 1 + x'', x'' >= 0
cond2(z, z', z'') -{ 3 }→ cond2(0, x, 0) :|: z'' = 0, z' = x, z = 1, x >= 0
cond2(z, z', z'') -{ 3 }→ cond1(1, x1, y) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1, z = 0
cond2(z, z', z'') -{ 3 }→ cond1(0, 0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
gr(z, z') -{ 1 }→ gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
gr(z, z') -{ 1 }→ 1 :|: x >= 0, z = 1 + x, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 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:

cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
cond2(z, z', z'') -{ 3 }→ cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ cond2, cond1 } { p } { gr }

(14) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
cond2(z, z', z'') -{ 3 }→ cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}, {p}, {gr}

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

Computed SIZE bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(16) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
cond2(z, z', z'') -{ 3 }→ cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond2,cond1}, {p}, {gr}
Previous analysis results are:

cond2: runtime: ?, size: O(1) [0] cond1: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 21 + 5·z' + 3·z''

Computed RUNTIME bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 23 + 5·z' + 3·z''

(18) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 2 }→ cond2(1, z', 1 + (z'' - 1)) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond1(z, z', z'') -{ 2 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond2(1, z', z'' - 1) :|: z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 3 }→ cond2(0, z', 0) :|: z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 3 }→ cond1(1, z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 0
cond2(z, z', z'') -{ 3 }→ cond1(0, 0, z'') :|: z'' >= 0, z = 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}
Previous analysis results are:

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0]

(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:

cond1(z, z', z'') -{ 23 + 5·z' }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
cond1(z, z', z'') -{ 23 + 5·z' + 3·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 24 + 5·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 26 + 3·z'' }→ s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}
Previous analysis results are:

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0]

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

Computed SIZE bound using KoAT for: p
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:

cond1(z, z', z'') -{ 23 + 5·z' }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
cond1(z, z', z'') -{ 23 + 5·z' + 3·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 24 + 5·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 26 + 3·z'' }→ s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}
Previous analysis results are:

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0] p: runtime: ?, size: O(n1) [z]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 23 + 5·z' }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
cond1(z, z', z'') -{ 23 + 5·z' + 3·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 24 + 5·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 26 + 3·z'' }→ s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0] p: runtime: O(1) [1], 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:

cond1(z, z', z'') -{ 23 + 5·z' }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
cond1(z, z', z'') -{ 23 + 5·z' + 3·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 24 + 5·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 26 + 3·z'' }→ s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z]

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 23 + 5·z' }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
cond1(z, z', z'') -{ 23 + 5·z' + 3·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 24 + 5·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 26 + 3·z'' }→ s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: ?, size: O(1) [1]

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z, z', z'') -{ 23 + 5·z' }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' >= 0
cond1(z, z', z'') -{ 23 + 5·z' + 3·z'' }→ s' :|: s' >= 0, s' <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 24 + 5·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' >= 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' >= 0, z'' - 1 >= 0
cond2(z, z', z'') -{ 26 + 3·z'' }→ s2 :|: s2 >= 0, s2 <= 0, z'' >= 0, z = 0, z' = 0
cond2(z, z', z'') -{ 21 + 5·z' + 3·z'' }→ s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z'' >= 0, z = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

cond2: runtime: O(n1) [21 + 5·z' + 3·z''], size: O(1) [0] cond1: runtime: O(n1) [23 + 5·z' + 3·z''], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)