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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r))
isort(Nil, r) → Nil
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r)
inssort(xs) → isort(xs, Nil)

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

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

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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r)) [1]
isort(Nil, r) → Nil [1]
insert(S(x), r) → insert[Ite](<(S(x), x), S(x), r) [1]
inssort(xs) → isort(xs, Nil) [1]
<(S(x), S(y)) → <(x, y) [0]
<(0, S(y)) → True [0]
<(x, 0) → False [0]
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs)) [0]
insert[Ite](True, x, r) → Cons(x, r) [0]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

< => 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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r)) [1]
isort(Nil, r) → Nil [1]
insert(S(x), r) → insert[Ite](lt(S(x), x), S(x), r) [1]
inssort(xs) → isort(xs, Nil) [1]
lt(S(x), S(y)) → lt(x, y) [0]
lt(0, S(y)) → True [0]
lt(x, 0) → False [0]
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs)) [0]
insert[Ite](True, x, r) → Cons(x, r) [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:

isort(Cons(x, xs), r) → isort(xs, insert(x, r)) [1]
isort(Nil, r) → Nil [1]
insert(S(x), r) → insert[Ite](lt(S(x), x), S(x), r) [1]
inssort(xs) → isort(xs, Nil) [1]
lt(S(x), S(y)) → lt(x, y) [0]
lt(0, S(y)) → True [0]
lt(x, 0) → False [0]
insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs)) [0]
insert[Ite](True, x, r) → Cons(x, r) [0]

The TRS has the following type information:

isort :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: S:0 → Cons:Nil → Cons:Nil insert :: S:0 → Cons:Nil → Cons:Nil Nil :: Cons:Nil S :: S:0 → S:0 insert[Ite] :: True:False → S:0 → Cons:Nil → Cons:Nil lt :: S:0 → S:0 → True:False inssort :: Cons:Nil → Cons:Nil 0 :: S:0 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:

isort
inssort

(c) The following functions are completely defined:

insert
lt
insert[Ite]

Due to the following rules being added:

lt(v0, v1) → null_lt [0]
insert[Ite](v0, v1, v2) → Nil [0]
insert(v0, v1) → 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: isort(Cons(x, xs), r) → isort(xs, insert(x, r)) [1] isort(Nil, r) → Nil [1] insert(S(x), r) → insert[Ite](lt(S(x), x), S(x), r) [1] inssort(xs) → isort(xs, Nil) [1] lt(S(x), S(y)) → lt(x, y) [0] lt(0, S(y)) → True [0] lt(x, 0) → False [0] insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs)) [0] insert[Ite](True, x, r) → Cons(x, r) [0] lt(v0, v1) → null_lt [0] insert[Ite](v0, v1, v2) → Nil [0] insert(v0, v1) → Nil [0] The TRS has the following type information: isort :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: S:0 → Cons:Nil → Cons:Nil insert :: S:0 → Cons:Nil → Cons:Nil Nil :: Cons:Nil S :: S:0 → S:0 insert[Ite] :: True:False:null_lt → S:0 → Cons:Nil → Cons:Nil lt :: S:0 → S:0 → True:False:null_lt inssort :: Cons:Nil → Cons:Nil 0 :: S:0 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: isort(Cons(S(x''), xs), r) → isort(xs, insert[Ite](lt(S(x''), x''), S(x''), r)) [2] isort(Cons(x, xs), r) → isort(xs, Nil) [1] isort(Nil, r) → Nil [1] insert(S(S(y')), r) → insert[Ite](lt(S(y'), y'), S(S(y')), r) [1] insert(S(0), r) → insert[Ite](False, S(0), r) [1] insert(S(x), r) → insert[Ite](null_lt, S(x), r) [1] inssort(xs) → isort(xs, Nil) [1] lt(S(x), S(y)) → lt(x, y) [0] lt(0, S(y)) → True [0] lt(x, 0) → False [0] insert[Ite](False, x', Cons(x, xs)) → Cons(x, insert(x', xs)) [0] insert[Ite](True, x, r) → Cons(x, r) [0] lt(v0, v1) → null_lt [0] insert[Ite](v0, v1, v2) → Nil [0] insert(v0, v1) → Nil [0] The TRS has the following type information: isort :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: S:0 → Cons:Nil → Cons:Nil insert :: S:0 → Cons:Nil → Cons:Nil Nil :: Cons:Nil S :: S:0 → S:0 insert[Ite] :: True:False:null_lt → S:0 → Cons:Nil → Cons:Nil lt :: S:0 → S:0 → True:False:null_lt inssort :: Cons:Nil → Cons:Nil 0 :: S:0 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:

Nil => 0
0 => 0
True => 2
False => 1
null_lt => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](lt(1 + y', y'), 1 + (1 + y'), r) :|: r >= 0, z = 1 + (1 + y'), y' >= 0, z' = r
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, r) :|: r >= 0, z = 1 + 0, z' = r
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + x, r) :|: r >= 0, x >= 0, z = 1 + x, z' = r
insert(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + r :|: z = 2, z'' = r, r >= 0, z' = x, x >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(x', xs) :|: z' = x', xs >= 0, z = 1, x' >= 0, x >= 0, z'' = 1 + x + xs
inssort(z) -{ 1 }→ isort(xs, 0) :|: xs >= 0, z = xs
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', r)) :|: xs >= 0, r >= 0, z = 1 + (1 + x'') + xs, x'' >= 0, z' = r
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, r >= 0, x >= 0, z' = r
isort(z, z') -{ 1 }→ 0 :|: r >= 0, z = 0, z' = r
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

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

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

Found the following analysis order by SCC decomposition:

{ lt } { insert[Ite], insert } { isort } { inssort }

(16) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort}

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {lt}, {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:

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

(19) 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

(20) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](lt(1 + (z - 2), z - 2), 1 + (1 + (z - 2)), z') :|: z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](lt(1 + x'', x''), 1 + x'', z')) :|: xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:

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

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {insert[Ite],insert}, {isort}, {inssort}
Previous analysis results are:

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

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

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 }→ insert[Ite](s', 1 + (1 + (z - 2)), z') :|: s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 }→ insert[Ite](1, 1 + 0, z') :|: z' >= 0, z = 1 + 0
insert(z, z') -{ 1 }→ insert[Ite](0, 1 + (z - 1), z') :|: z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 1 + x + insert(z', xs) :|: xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 }→ isort(xs, insert[Ite](s, 1 + x'', z')) :|: s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {isort}, {inssort}
Previous analysis results are:

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

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + (z - 2))) + 1 * z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + 0) + 1 * z' + 1, z' >= 0, z = 1 + 0
insert(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * z' + 1, z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 1 + xs }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 + z' }→ isort(xs, s1) :|: s1 >= 0, s1 <= 1 * (1 + x'') + 1 * z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {isort}, {inssort}
Previous analysis results are:

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

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + (z - 2))) + 1 * z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + 0) + 1 * z' + 1, z' >= 0, z = 1 + 0
insert(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * z' + 1, z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 1 + xs }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 + z' }→ isort(xs, s1) :|: s1 >= 0, s1 <= 1 * (1 + x'') + 1 * z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed: {isort}, {inssort}
Previous analysis results are:

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

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

Computed RUNTIME bound using CoFloCo for: isort
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + z + z·z' + z²

(32) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + (z - 2))) + 1 * z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + 0) + 1 * z' + 1, z' >= 0, z = 1 + 0
insert(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * z' + 1, z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 1 + xs }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 1 }→ isort(z, 0) :|: z >= 0
isort(z, z') -{ 2 + z' }→ isort(xs, s1) :|: s1 >= 0, s1 <= 1 * (1 + x'') + 1 * z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 1 }→ isort(xs, 0) :|: z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

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

lt: runtime: O(1) [0], size: O(1) [2] insert[Ite]: runtime: O(n1) [z''], size: O(n1) [1 + z' + z''] insert: runtime: O(n1) [1 + z'], size: O(n1) [1 + z + z'] isort: runtime: O(n2) [1 + z + z·z' + z2], size: O(1) [0]

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

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + (z - 2))) + 1 * z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + 0) + 1 * z' + 1, z' >= 0, z = 1 + 0
insert(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * z' + 1, z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 1 + xs }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 2 + z + z² }→ s8 :|: s8 >= 0, s8 <= 0, z >= 0
isort(z, z') -{ 3 + s1·xs + xs + xs² + z' }→ s6 :|: s6 >= 0, s6 <= 0, s1 >= 0, s1 <= 1 * (1 + x'') + 1 * z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 2 + xs + xs² }→ s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

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

lt: runtime: O(1) [0], size: O(1) [2] insert[Ite]: runtime: O(n1) [z''], size: O(n1) [1 + z' + z''] insert: runtime: O(n1) [1 + z'], size: O(n1) [1 + z + z'] isort: runtime: O(n2) [1 + z + z·z' + z2], size: O(1) [0]

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

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + (z - 2))) + 1 * z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + 0) + 1 * z' + 1, z' >= 0, z = 1 + 0
insert(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * z' + 1, z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 1 + xs }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 2 + z + z² }→ s8 :|: s8 >= 0, s8 <= 0, z >= 0
isort(z, z') -{ 3 + s1·xs + xs + xs² + z' }→ s6 :|: s6 >= 0, s6 <= 0, s1 >= 0, s1 <= 1 * (1 + x'') + 1 * z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 2 + xs + xs² }→ s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

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

lt: runtime: O(1) [0], size: O(1) [2] insert[Ite]: runtime: O(n1) [z''], size: O(n1) [1 + z' + z''] insert: runtime: O(n1) [1 + z'], size: O(n1) [1 + z + z'] isort: runtime: O(n2) [1 + z + z·z' + z2], size: O(1) [0] inssort: runtime: ?, size: O(1) [0]

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

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

insert(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + (1 + (z - 2))) + 1 * z' + 1, s' >= 0, s' <= 2, z' >= 0, z - 2 >= 0
insert(z, z') -{ 1 + z' }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + 0) + 1 * z' + 1, z' >= 0, z = 1 + 0
insert(z, z') -{ 1 + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)) + 1 * z' + 1, z' >= 0, z - 1 >= 0
insert(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert[Ite](z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
insert[Ite](z, z', z'') -{ 1 + xs }→ 1 + x + s5 :|: s5 >= 0, s5 <= 1 * z' + 1 * xs + 1, xs >= 0, z = 1, z' >= 0, x >= 0, z'' = 1 + x + xs
insert[Ite](z, z', z'') -{ 0 }→ 1 + z' + z'' :|: z = 2, z'' >= 0, z' >= 0
inssort(z) -{ 2 + z + z² }→ s8 :|: s8 >= 0, s8 <= 0, z >= 0
isort(z, z') -{ 3 + s1·xs + xs + xs² + z' }→ s6 :|: s6 >= 0, s6 <= 0, s1 >= 0, s1 <= 1 * (1 + x'') + 1 * z' + 1, s >= 0, s <= 2, xs >= 0, z' >= 0, z = 1 + (1 + x'') + xs, x'' >= 0
isort(z, z') -{ 2 + xs + xs² }→ s7 :|: s7 >= 0, s7 <= 0, z = 1 + x + xs, xs >= 0, z' >= 0, x >= 0
isort(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
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

Function symbols to be analyzed:
Previous analysis results are:

lt: runtime: O(1) [0], size: O(1) [2] insert[Ite]: runtime: O(n1) [z''], size: O(n1) [1 + z' + z''] insert: runtime: O(n1) [1 + z'], size: O(n1) [1 + z + z'] isort: runtime: O(n2) [1 + z + z·z' + z2], size: O(1) [0] inssort: runtime: O(n2) [2 + z + z2], size: O(1) [0]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^2)