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

(0) Obligation:

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


The TRS R consists of the following rules:

bsort(S(x'), Cons(x, xs)) → bsort(x', bubble(x, xs))
len(Cons(x, xs)) → +(S(0), len(xs))
bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
len(Nil) → 0
bubble(x, Nil) → Cons(x, Nil)
bsort(0, xs) → xs
bubblesort(xs) → bsort(len(xs), xs)

The (relative) TRS S consists of the following rules:

+(x, S(0)) → S(x)
+(S(0), y) → S(y)
<(S(x), S(y)) → <(x, y)
<(0, S(y)) → True
<(x, 0) → False
bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs))
bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs))

Rewrite Strategy: INNERMOST

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

Transformed relative 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:

bsort(S(x'), Cons(x, xs)) → bsort(x', bubble(x, xs)) [1]
len(Cons(x, xs)) → +(S(0), len(xs)) [1]
bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs)) [1]
len(Nil) → 0 [1]
bubble(x, Nil) → Cons(x, Nil) [1]
bsort(0, xs) → xs [1]
bubblesort(xs) → bsort(len(xs), xs) [1]
+(x, S(0)) → S(x) [0]
+(S(0), y) → S(y) [0]
<(S(x), S(y)) → <(x, y) [0]
<(0, S(y)) → True [0]
<(x, 0) → False [0]
bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs)) [0]
bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs)) [0]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

< => lt

(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:

bsort(S(x'), Cons(x, xs)) → bsort(x', bubble(x, xs)) [1]
len(Cons(x, xs)) → plus(S(0), len(xs)) [1]
bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1]
len(Nil) → 0 [1]
bubble(x, Nil) → Cons(x, Nil) [1]
bsort(0, xs) → xs [1]
bubblesort(xs) → bsort(len(xs), xs) [1]
plus(x, S(0)) → S(x) [0]
plus(S(0), y) → S(y) [0]
lt(S(x), S(y)) → lt(x, y) [0]
lt(0, S(y)) → True [0]
lt(x, 0) → False [0]
bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs)) [0]
bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs)) [0]

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:

bsort(S(x'), Cons(x, xs)) → bsort(x', bubble(x, xs)) [1]
len(Cons(x, xs)) → plus(S(0), len(xs)) [1]
bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1]
len(Nil) → 0 [1]
bubble(x, Nil) → Cons(x, Nil) [1]
bsort(0, xs) → xs [1]
bubblesort(xs) → bsort(len(xs), xs) [1]
plus(x, S(0)) → S(x) [0]
plus(S(0), y) → S(y) [0]
lt(S(x), S(y)) → lt(x, y) [0]
lt(0, S(y)) → True [0]
lt(x, 0) → False [0]
bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs)) [0]
bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs)) [0]

The TRS has the following type information:

bsort :: S:0 → Cons:Nil → Cons:Nil S :: S:0 → S:0 Cons :: S:0 → Cons:Nil → Cons:Nil bubble :: S:0 → Cons:Nil → Cons:Nil len :: Cons:Nil → S:0 plus :: S:0 → S:0 → S:0 0 :: S:0 bubble[Ite][False][Ite] :: True:False → S:0 → Cons:Nil → Cons:Nil lt :: S:0 → S:0 → True:False Nil :: Cons:Nil bubblesort :: Cons:Nil → Cons:Nil True :: True:False False :: True:False

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:

bsort
bubblesort

(c) The following functions are completely defined:

len
bubble
plus
lt
bubble[Ite][False][Ite]

Due to the following rules being added:

plus(v0, v1) → 0 [0]
lt(v0, v1) → null_lt [0]
bubble[Ite][False][Ite](v0, v1, v2) → Nil [0]

And the following fresh constants:

null_lt

(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: bsort(S(x'), Cons(x, xs)) → bsort(x', bubble(x, xs)) [1] len(Cons(x, xs)) → plus(S(0), len(xs)) [1] bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](lt(x', x), x', Cons(x, xs)) [1] len(Nil) → 0 [1] bubble(x, Nil) → Cons(x, Nil) [1] bsort(0, xs) → xs [1] bubblesort(xs) → bsort(len(xs), xs) [1] plus(x, S(0)) → S(x) [0] plus(S(0), y) → S(y) [0] lt(S(x), S(y)) → lt(x, y) [0] lt(0, S(y)) → True [0] lt(x, 0) → False [0] bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs)) [0] bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs)) [0] plus(v0, v1) → 0 [0] lt(v0, v1) → null_lt [0] bubble[Ite][False][Ite](v0, v1, v2) → Nil [0] The TRS has the following type information: bsort :: S:0 → Cons:Nil → Cons:Nil S :: S:0 → S:0 Cons :: S:0 → Cons:Nil → Cons:Nil bubble :: S:0 → Cons:Nil → Cons:Nil len :: Cons:Nil → S:0 plus :: S:0 → S:0 → S:0 0 :: S:0 bubble[Ite][False][Ite] :: True:False:null_lt → S:0 → Cons:Nil → Cons:Nil lt :: S:0 → S:0 → True:False:null_lt Nil :: Cons:Nil bubblesort :: Cons:Nil → Cons:Nil True :: True:False:null_lt False :: True:False:null_lt null_lt :: True:False:null_lt 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: bsort(S(x'), Cons(x, Cons(x1, xs'))) → bsort(x', bubble[Ite][False][Ite](lt(x, x1), x, Cons(x1, xs'))) [2] bsort(S(x'), Cons(x, Nil)) → bsort(x', Cons(x, Nil)) [2] len(Cons(x, Cons(x'', xs''))) → plus(S(0), plus(S(0), len(xs''))) [2] len(Cons(x, Nil)) → plus(S(0), 0) [2] bubble(S(x2), Cons(S(y'), xs)) → bubble[Ite][False][Ite](lt(x2, y'), S(x2), Cons(S(y'), xs)) [1] bubble(0, Cons(S(y''), xs)) → bubble[Ite][False][Ite](True, 0, Cons(S(y''), xs)) [1] bubble(x', Cons(0, xs)) → bubble[Ite][False][Ite](False, x', Cons(0, xs)) [1] bubble(x', Cons(x, xs)) → bubble[Ite][False][Ite](null_lt, x', Cons(x, xs)) [1] len(Nil) → 0 [1] bubble(x, Nil) → Cons(x, Nil) [1] bsort(0, xs) → xs [1] bubblesort(Cons(x3, xs1)) → bsort(plus(S(0), len(xs1)), Cons(x3, xs1)) [2] bubblesort(Nil) → bsort(0, Nil) [2] plus(x, S(0)) → S(x) [0] plus(S(0), y) → S(y) [0] lt(S(x), S(y)) → lt(x, y) [0] lt(0, S(y)) → True [0] lt(x, 0) → False [0] bubble[Ite][False][Ite](False, x', Cons(x, xs)) → Cons(x, bubble(x', xs)) [0] bubble[Ite][False][Ite](True, x', Cons(x, xs)) → Cons(x', bubble(x, xs)) [0] plus(v0, v1) → 0 [0] lt(v0, v1) → null_lt [0] bubble[Ite][False][Ite](v0, v1, v2) → Nil [0] The TRS has the following type information: bsort :: S:0 → Cons:Nil → Cons:Nil S :: S:0 → S:0 Cons :: S:0 → Cons:Nil → Cons:Nil bubble :: S:0 → Cons:Nil → Cons:Nil len :: Cons:Nil → S:0 plus :: S:0 → S:0 → S:0 0 :: S:0 bubble[Ite][False][Ite] :: True:False:null_lt → S:0 → Cons:Nil → Cons:Nil lt :: S:0 → S:0 → True:False:null_lt Nil :: Cons:Nil bubblesort :: Cons:Nil → Cons:Nil True :: True:False:null_lt False :: True:False:null_lt null_lt :: True:False:null_lt 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
Nil => 0
True => 2
False => 1
null_lt => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ xs :|: xs >= 0, z' = xs, z = 0
bsort(z, z') -{ 2 }→ bsort(x', bubble[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x', x1 >= 0, x' >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(x', 1 + x + 0) :|: z = 1 + x', x' >= 0, x >= 0, z' = 1 + x + 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](lt(x2, y'), 1 + x2, 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, z = 1 + x2, y' >= 0, x2 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, x', 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, x' >= 0, z = x'
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
bubble(z, z') -{ 1 }→ 1 + x + 0 :|: x >= 0, z = x, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x' + bubble(x, xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 2 }→ plus(1 + 0, 0) :|: x >= 0, z = 1 + x + 0
len(z) -{ 1 }→ 0 :|: z = 0
lt(z, z') -{ 0 }→ lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
lt(z, z') -{ 0 }→ 2 :|: z' = 1 + y, y >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: x >= 0, z = x, z' = 0
lt(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 0 }→ 1 + x :|: x >= 0, z' = 1 + 0, z = x
plus(z, z') -{ 0 }→ 1 + y :|: z = 1 + 0, y >= 0, z' = y

(13) InliningProof (UPPER BOUND(ID) transformation)

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

plus(z, z') -{ 0 }→ 1 + x :|: x >= 0, z' = 1 + 0, z = x
plus(z, z') -{ 0 }→ 1 + y :|: z = 1 + 0, y >= 0, z' = y
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

(14) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ xs :|: xs >= 0, z' = xs, z = 0
bsort(z, z') -{ 2 }→ bsort(x', bubble[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: z = 1 + x', x1 >= 0, x' >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(x', 1 + x + 0) :|: z = 1 + x', x' >= 0, x >= 0, z' = 1 + x + 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](lt(x2, y'), 1 + x2, 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, z = 1 + x2, y' >= 0, x2 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, x', 1 + 0 + xs) :|: xs >= 0, z' = 1 + 0 + xs, x' >= 0, z = x'
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, x', 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
bubble(z, z') -{ 1 }→ 1 + x + 0 :|: x >= 0, z = x, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x' + bubble(x, xs) :|: z = 2, z' = x', xs >= 0, x' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: x >= 0, z = 1 + x + 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: x >= 0, z = 1 + x + 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
lt(z, z') -{ 0 }→ 2 :|: z' = 1 + y, y >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: x >= 0, z = x, z' = 0
lt(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
plus(z, z') -{ 0 }→ 1 + x :|: x >= 0, z' = 1 + 0, z = x
plus(z, z') -{ 0 }→ 1 + y :|: z = 1 + 0, y >= 0, z' = y

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ lt } { plus } { bubble[Ite][False][Ite], bubble } { len } { bsort } { bubblesort }

(18) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {lt}, {plus}, {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}

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

Computed SIZE bound using CoFloCo for: lt
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

(20) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {lt}, {plus}, {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: ?, size: O(1) [2]

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](lt(x, x1), x, 1 + x1 + xs')) :|: x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](lt(z - 1, y'), 1 + (z - 1), 1 + (1 + y') + xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2]

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2]

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {plus}, {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: ?, size: O(n1) [z + z']

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

Computed RUNTIME bound using CoFloCo for: plus
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:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z']

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z']

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

Computed SIZE bound using CoFloCo for: bubble[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z' + z''

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {bubble[Ite][False][Ite],bubble}, {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: ?, size: O(n1) [1 + z' + z''] bubble: runtime: ?, size: O(n1) [1 + z + z']

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: bubble[Ite][False][Ite]
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2 + z''

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 2 }→ bsort(z - 1, bubble[Ite][False][Ite](s, x, 1 + x1 + xs')) :|: s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](s', 1 + (z - 1), 1 + (1 + y') + xs) :|: s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](2, 0, 1 + (1 + y'') + xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](1, z, 1 + 0 + (z' - 1)) :|: z' - 1 >= 0, z >= 0
bubble(z, z') -{ 1 }→ bubble[Ite][False][Ite](0, z, 1 + x + xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + x + bubble(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 1 + z' + bubble(x, xs) :|: z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z']

(35) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 5 + x1 + xs' }→ bsort(z - 1, s1) :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z']

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

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 5 + x1 + xs' }→ bsort(z - 1, s1) :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {len}, {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: ?, size: O(n1) [z]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(40) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 5 + x1 + xs' }→ bsort(z - 1, s1) :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 2 }→ bsort(plus(1 + 0, len(xs1)), 1 + x3 + xs1) :|: z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 2 }→ plus(1 + 0, plus(1 + 0, len(xs''))) :|: z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z]

(41) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(42) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 5 + x1 + xs' }→ bsort(z - 1, s1) :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 4 + 2·xs1 }→ bsort(s12, 1 + x3 + xs1) :|: s11 >= 0, s11 <= 1 * xs1, s12 >= 0, s12 <= 1 * (1 + 0) + 1 * s11, z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 4 + 2·xs'' }→ s10 :|: s8 >= 0, s8 <= 1 * xs'', s9 >= 0, s9 <= 1 * (1 + 0) + 1 * s8, s10 >= 0, s10 <= 1 * (1 + 0) + 1 * s9, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(44) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 5 + x1 + xs' }→ bsort(z - 1, s1) :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 4 + 2·xs1 }→ bsort(s12, 1 + x3 + xs1) :|: s11 >= 0, s11 <= 1 * xs1, s12 >= 0, s12 <= 1 * (1 + 0) + 1 * s11, z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 4 + 2·xs'' }→ s10 :|: s8 >= 0, s8 <= 1 * xs'', s9 >= 0, s9 <= 1 * (1 + 0) + 1 * s8, s10 >= 0, s10 <= 1 * (1 + 0) + 1 * s9, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed: {bsort}, {bubblesort}
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z] bsort: runtime: ?, size: O(n1) [z']

(45) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: bsort
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + 3·z + z·z'

(46) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bsort(z, z') -{ 5 + x1 + xs' }→ bsort(z - 1, s1) :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 2 }→ bsort(z - 1, 1 + (z' - 1) + 0) :|: z - 1 >= 0, z' - 1 >= 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 4 + 2·xs1 }→ bsort(s12, 1 + x3 + xs1) :|: s11 >= 0, s11 <= 1 * xs1, s12 >= 0, s12 <= 1 * (1 + 0) + 1 * s11, z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
bubblesort(z) -{ 2 }→ bsort(0, 0) :|: z = 0
len(z) -{ 4 + 2·xs'' }→ s10 :|: s8 >= 0, s8 <= 1 * xs'', s9 >= 0, s9 <= 1 * (1 + 0) + 1 * s8, s10 >= 0, s10 <= 1 * (1 + 0) + 1 * s9, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

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

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z] bsort: runtime: O(n2) [1 + 3·z + z·z'], size: O(n1) [z']

(47) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(48) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 3 + -1·s1 + s1·z + x1 + xs' + 3·z }→ s13 :|: s13 >= 0, s13 <= 1 * s1, s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 3·z + z·z' + -1·z' }→ s14 :|: s14 >= 0, s14 <= 1 * (1 + (z' - 1) + 0), z - 1 >= 0, z' - 1 >= 0
bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 3 }→ s15 :|: s15 >= 0, s15 <= 1 * 0, z = 0
bubblesort(z) -{ 5 + 4·s12 + s12·x3 + s12·xs1 + 2·xs1 }→ s16 :|: s16 >= 0, s16 <= 1 * (1 + x3 + xs1), s11 >= 0, s11 <= 1 * xs1, s12 >= 0, s12 <= 1 * (1 + 0) + 1 * s11, z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
len(z) -{ 4 + 2·xs'' }→ s10 :|: s8 >= 0, s8 <= 1 * xs'', s9 >= 0, s9 <= 1 * (1 + 0) + 1 * s8, s10 >= 0, s10 <= 1 * (1 + 0) + 1 * s9, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

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

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z] bsort: runtime: O(n2) [1 + 3·z + z·z'], size: O(n1) [z']

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(50) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 3 + -1·s1 + s1·z + x1 + xs' + 3·z }→ s13 :|: s13 >= 0, s13 <= 1 * s1, s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 3·z + z·z' + -1·z' }→ s14 :|: s14 >= 0, s14 <= 1 * (1 + (z' - 1) + 0), z - 1 >= 0, z' - 1 >= 0
bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 3 }→ s15 :|: s15 >= 0, s15 <= 1 * 0, z = 0
bubblesort(z) -{ 5 + 4·s12 + s12·x3 + s12·xs1 + 2·xs1 }→ s16 :|: s16 >= 0, s16 <= 1 * (1 + x3 + xs1), s11 >= 0, s11 <= 1 * xs1, s12 >= 0, s12 <= 1 * (1 + 0) + 1 * s11, z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
len(z) -{ 4 + 2·xs'' }→ s10 :|: s8 >= 0, s8 <= 1 * xs'', s9 >= 0, s9 <= 1 * (1 + 0) + 1 * s8, s10 >= 0, s10 <= 1 * (1 + 0) + 1 * s9, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

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

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z] bsort: runtime: O(n2) [1 + 3·z + z·z'], size: O(n1) [z'] bubblesort: runtime: ?, size: O(n1) [z]

(51) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: bubblesort
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 12 + 8·z + 2·z²

(52) Obligation:

Complexity RNTS consisting of the following rules:

bsort(z, z') -{ 3 + -1·s1 + s1·z + x1 + xs' + 3·z }→ s13 :|: s13 >= 0, s13 <= 1 * s1, s1 >= 0, s1 <= 1 * x + 1 * (1 + x1 + xs') + 1, s >= 0, s <= 2, x1 >= 0, z - 1 >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
bsort(z, z') -{ 3·z + z·z' + -1·z' }→ s14 :|: s14 >= 0, s14 <= 1 * (1 + (z' - 1) + 0), z - 1 >= 0, z' - 1 >= 0
bsort(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
bubble(z, z') -{ 5 + xs + y' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (z - 1)) + 1 * (1 + (1 + y') + xs) + 1, s' >= 0, s' <= 2, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z - 1 >= 0
bubble(z, z') -{ 5 + xs + y'' }→ s3 :|: s3 >= 0, s3 <= 1 * 0 + 1 * (1 + (1 + y'') + xs) + 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z = 0
bubble(z, z') -{ 3 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * z + 1 * (1 + 0 + (z' - 1)) + 1, z' - 1 >= 0, z >= 0
bubble(z, z') -{ 4 + x + xs }→ s5 :|: s5 >= 0, s5 <= 1 * z + 1 * (1 + x + xs) + 1, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
bubble(z, z') -{ 1 }→ 1 + z + 0 :|: z >= 0, z' = 0
bubble[Ite][False][Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + x + s6 :|: s6 >= 0, s6 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
bubble[Ite][False][Ite](z, z', z'') -{ 3 + xs }→ 1 + z' + s7 :|: s7 >= 0, s7 <= 1 * x + 1 * xs + 1, z = 2, xs >= 0, z' >= 0, x >= 0, z'' = 1 + x + xs
bubblesort(z) -{ 3 }→ s15 :|: s15 >= 0, s15 <= 1 * 0, z = 0
bubblesort(z) -{ 5 + 4·s12 + s12·x3 + s12·xs1 + 2·xs1 }→ s16 :|: s16 >= 0, s16 <= 1 * (1 + x3 + xs1), s11 >= 0, s11 <= 1 * xs1, s12 >= 0, s12 <= 1 * (1 + 0) + 1 * s11, z = 1 + x3 + xs1, xs1 >= 0, x3 >= 0
len(z) -{ 4 + 2·xs'' }→ s10 :|: s8 >= 0, s8 <= 1 * xs'', s9 >= 0, s9 <= 1 * (1 + 0) + 1 * s8, s10 >= 0, s10 <= 1 * (1 + 0) + 1 * s9, z = 1 + x + (1 + x'' + xs''), x >= 0, xs'' >= 0, x'' >= 0
len(z) -{ 1 }→ 0 :|: z = 0
len(z) -{ 2 }→ 0 :|: z - 1 >= 0, v0 >= 0, v1 >= 0, 1 + 0 = v0, 0 = v1
len(z) -{ 2 }→ 1 + y :|: z - 1 >= 0, 1 + 0 = 1 + 0, y >= 0, 0 = y
lt(z, z') -{ 0 }→ s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
lt(z, z') -{ 0 }→ 2 :|: z' - 1 >= 0, z = 0
lt(z, z') -{ 0 }→ 1 :|: z >= 0, z' = 0
lt(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
plus(z, z') -{ 0 }→ 1 + z :|: z >= 0, z' = 1 + 0
plus(z, z') -{ 0 }→ 1 + z' :|: z = 1 + 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] plus: runtime: O(1) [0], size: O(n1) [z + z'] bubble[Ite][False][Ite]: runtime: O(n1) [2 + z''], size: O(n1) [1 + z' + z''] bubble: runtime: O(n1) [3 + z'], size: O(n1) [1 + z + z'] len: runtime: O(n1) [2 + 2·z], size: O(n1) [z] bsort: runtime: O(n2) [1 + 3·z + z·z'], size: O(n1) [z'] bubblesort: runtime: O(n2) [12 + 8·z + 2·z2], size: O(n1) [z]

(53) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(54) BOUNDS(1, n^2)