The Runtime Complexity (innermost) of the given CpxTRS could be proven to be 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:

cond1(true, x, y, z) → cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z)
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^2).


The TRS R consists of the following rules:

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

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, z) → cond2(gr(y, z), x, y, z) [1] cond2(true, x, y, z) → cond2(gr(y, z), x, p(y), z) [1] cond2(false, x, y, z) → cond1(gr(x, z), p(x), y, z) [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 → 0:s → cond1:cond2 true :: true:false cond2 :: true:false → 0:s → 0:s → 0:s → cond1:cond2 gr :: 0:s → 0:s → true:false p :: 0:s → 0:s false :: true:false 0 :: 0:s 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, z) → cond2(false, x, 0, z) [2] cond1(true, x, s(x'), 0) → cond2(true, x, s(x'), 0) [2] cond1(true, x, s(x''), s(y')) → cond2(gr(x'', y'), x, s(x''), s(y')) [2] cond2(true, x, 0, z) → cond2(false, x, 0, z) [3] cond2(true, x, s(x1), 0) → cond2(true, x, x1, 0) [3] cond2(true, x, s(x2), s(y'')) → cond2(gr(x2, y''), x, x2, s(y'')) [3] cond2(false, 0, y, z) → cond1(false, 0, y, z) [3] cond2(false, s(x3), y, 0) → cond1(true, x3, y, 0) [3] cond2(false, s(x4), y, s(y1)) → cond1(gr(x4, y1), x4, y, s(y1)) [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 → 0:s → cond1:cond2 true :: true:false cond2 :: true:false → 0:s → 0:s → 0:s → cond1:cond2 gr :: 0:s → 0:s → true:false p :: 0:s → 0:s false :: true:false 0 :: 0:s 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
false => 0
0 => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(x'', y'), x, 1 + x'', 1 + y') :|: z1 = 1 + x'', x >= 0, z'' = x, y' >= 0, z2 = 1 + y', z' = 1, x'' >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, x, 1 + x', 0) :|: z2 = 0, x >= 0, x' >= 0, z'' = x, z' = 1, z1 = 1 + x'
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, x, 0, z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(x2, y''), x, x2, 1 + y'') :|: z2 = 1 + y'', x >= 0, z'' = x, y'' >= 0, z' = 1, x2 >= 0, z1 = 1 + x2
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, x, x1, 0) :|: x1 >= 0, z2 = 0, x >= 0, z'' = x, z' = 1, z1 = 1 + x1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, x, 0, z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(x4, y1), x4, y, 1 + y1) :|: y1 >= 0, z2 = 1 + y1, z1 = y, x4 >= 0, z'' = 1 + x4, y >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, x3, y, 0) :|: z1 = y, z'' = 1 + x3, z2 = 0, y >= 0, x3 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, y, z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' = 1 + x, x >= 0
gr(z', z'') -{ 1 }→ 0 :|: x >= 0, z'' = x, z' = 0
p(z') -{ 1 }→ x :|: z' = 1 + x, x >= 0
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'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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:

{ p } { gr } { cond2, cond1 }

(14) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}

(15) 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'

(16) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:

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

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:

p: runtime: O(1) [1], 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:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:

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

(21) 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

(22) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:

p: runtime: O(1) [1], size: O(n1) [z'] gr: runtime: ?, size: O(1) [1]

(23) 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''

(24) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 }→ cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 }→ gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}
Previous analysis results are:

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

(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'', z1, z2) -{ 2 + z2 }→ cond2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 + z2 }→ cond2(s', z'', z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 + z2 }→ cond1(s'', z'' - 1, z1, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 + z'' }→ s1 :|: s1 >= 0, s1 <= 1, z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}
Previous analysis results are:

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

(27) 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

(28) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 + z2 }→ cond2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 + z2 }→ cond2(s', z'', z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 + z2 }→ cond1(s'', z'' - 1, z1, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 + z'' }→ s1 :|: s1 >= 0, s1 <= 1, z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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}
Previous analysis results are:

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

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

Computed RUNTIME bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 21 + 5·z'' + 2·z''·z2 + 8·z1 + 3·z1·z2 + 3·z2

Computed RUNTIME bound using PUBS for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 23 + 5·z'' + 2·z''·z2 + 8·z1 + 3·z1·z2 + 4·z2

(30) Obligation:

Complexity RNTS consisting of the following rules:

cond1(z', z'', z1, z2) -{ 2 + z2 }→ cond2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond1(z', z'', z1, z2) -{ 2 }→ cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
cond1(z', z'', z1, z2) -{ 2 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 + z2 }→ cond2(s', z'', z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
cond2(z', z'', z1, z2) -{ 3 }→ cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 }→ cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
cond2(z', z'', z1, z2) -{ 3 + z2 }→ cond1(s'', z'' - 1, z1, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
cond2(z', z'', z1, z2) -{ 3 }→ cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
gr(z', z'') -{ 1 + z'' }→ s1 :|: s1 >= 0, s1 <= 1, z' - 1 >= 0, z'' - 1 >= 0
gr(z', z'') -{ 1 }→ 1 :|: z'' = 0, z' - 1 >= 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:

p: runtime: O(1) [1], size: O(n1) [z'] gr: runtime: O(n1) [1 + z''], size: O(1) [1] cond2: runtime: O(n2) [21 + 5·z'' + 2·z''·z2 + 8·z1 + 3·z1·z2 + 3·z2], size: O(1) [0] cond1: runtime: O(n2) [23 + 5·z'' + 2·z''·z2 + 8·z1 + 3·z1·z2 + 4·z2], size: O(1) [0]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^2)