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:

selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs))
select(Cons(x, xs)) → selects(x, Nil, xs)
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest))
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil)
select(Nil) → Nil
revapp(Nil, rest) → rest

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:

selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1]
select(Cons(x, xs)) → selects(x, Nil, xs) [1]
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest)) [1]
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1]
select(Nil) → Nil [1]
revapp(Nil, rest) → rest [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:

selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1]
select(Cons(x, xs)) → selects(x, Nil, xs) [1]
revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest)) [1]
selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1]
select(Nil) → Nil [1]
revapp(Nil, rest) → rest [1]

The TRS has the following type information:

selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil Cons :: Cons:Nil → Cons:Nil → Cons:Nil revapp :: Cons:Nil → Cons:Nil → Cons:Nil select :: Cons:Nil → Cons:Nil Nil :: Cons:Nil

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:

selects
select
revapp

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

(6) Obligation:

Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1] select(Cons(x, xs)) → selects(x, Nil, xs) [1] revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest)) [1] selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1] select(Nil) → Nil [1] revapp(Nil, rest) → rest [1] The TRS has the following type information: selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil Cons :: Cons:Nil → Cons:Nil → Cons:Nil revapp :: Cons:Nil → Cons:Nil → Cons:Nil select :: Cons:Nil → Cons:Nil Nil :: Cons:Nil 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: selects(x', revprefix, Cons(x, xs)) → Cons(Cons(x', revapp(revprefix, Cons(x, xs))), selects(x, Cons(x', revprefix), xs)) [1] select(Cons(x, xs)) → selects(x, Nil, xs) [1] revapp(Cons(x, xs), rest) → revapp(xs, Cons(x, rest)) [1] selects(x, revprefix, Nil) → Cons(Cons(x, revapp(revprefix, Nil)), Nil) [1] select(Nil) → Nil [1] revapp(Nil, rest) → rest [1] The TRS has the following type information: selects :: Cons:Nil → Cons:Nil → Cons:Nil → Cons:Nil Cons :: Cons:Nil → Cons:Nil → Cons:Nil revapp :: Cons:Nil → Cons:Nil → Cons:Nil select :: Cons:Nil → Cons:Nil Nil :: Cons:Nil 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:

Nil => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 1 }→ rest :|: z' = rest, rest >= 0, z = 0
revapp(z, z') -{ 1 }→ revapp(xs, 1 + x + rest) :|: z = 1 + x + xs, xs >= 0, z' = rest, x >= 0, rest >= 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + x + revapp(revprefix, 0)) + 0 :|: z'' = 0, z' = revprefix, x >= 0, revprefix >= 0, z = x
selects(z, z', z'') -{ 1 }→ 1 + (1 + x' + revapp(revprefix, 1 + x + xs)) + selects(x, 1 + x' + revprefix, xs) :|: xs >= 0, z' = revprefix, x' >= 0, x >= 0, z'' = 1 + x + xs, revprefix >= 0, z = x'

(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:

revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
revapp(z, z') -{ 1 }→ revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ revapp } { selects } { select }

(14) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
revapp(z, z') -{ 1 }→ revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

Function symbols to be analyzed: {revapp}, {selects}, {select}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
revapp(z, z') -{ 1 }→ revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

Function symbols to be analyzed: {revapp}, {selects}, {select}
Previous analysis results are:

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

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
revapp(z, z') -{ 1 }→ revapp(xs, 1 + x + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 0)) + 0 :|: z'' = 0, z >= 0, z' >= 0
selects(z, z', z'') -{ 1 }→ 1 + (1 + z + revapp(z', 1 + x + xs)) + selects(x, 1 + z + z', xs) :|: xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0

Function symbols to be analyzed: {selects}, {select}
Previous analysis results are:

revapp: runtime: O(n1) [1 + z], size: O(n1) [z + 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:

revapp(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s) + selects(x, 1 + z + z', xs) :|: s >= 0, s <= 1 * z' + 1 * (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= 1 * z' + 1 * 0, z'' = 0, z >= 0, z' >= 0

Function symbols to be analyzed: {selects}, {select}
Previous analysis results are:

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

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

Computed SIZE bound using CoFloCo for: selects
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + z + z·z'' + z' + z'·z'' + 3·z'' + z''²

(22) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s) + selects(x, 1 + z + z', xs) :|: s >= 0, s <= 1 * z' + 1 * (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= 1 * z' + 1 * 0, z'' = 0, z >= 0, z' >= 0

Function symbols to be analyzed: {selects}, {select}
Previous analysis results are:

revapp: runtime: O(n1) [1 + z], size: O(n1) [z + z'] selects: runtime: ?, size: O(n2) [2 + z + z·z'' + z' + z'·z'' + 3·z'' + z''2]

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

Computed RUNTIME bound using CoFloCo for: selects
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + z + z·z'' + z' + z'·z'' + 2·z'' + z''²

(24) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
select(z) -{ 1 }→ selects(x, 0, xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s) + selects(x, 1 + z + z', xs) :|: s >= 0, s <= 1 * z' + 1 * (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= 1 * z' + 1 * 0, z'' = 0, z >= 0, z' >= 0

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

revapp: runtime: O(n1) [1 + z], size: O(n1) [z + z'] selects: runtime: O(n2) [2 + z + z·z'' + z' + z'·z'' + 2·z'' + z''2], size: O(n2) [2 + z + z·z'' + z' + z'·z'' + 3·z'' + z''2]

(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:

revapp(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
select(z) -{ 3 + x + x·xs + 2·xs + xs² }→ s2 :|: s2 >= 0, s2 <= 1 * x + 1 * 0 + 2 + 3 * xs + 1 * (x * xs) + 1 * (0 * xs) + 1 * (xs * xs), z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 5 + x + x·xs + 3·xs + xs·z + xs·z' + xs² + z + 2·z' }→ 1 + (1 + z + s) + s1 :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + z + z') + 2 + 3 * xs + 1 * (x * xs) + 1 * ((1 + z + z') * xs) + 1 * (xs * xs), s >= 0, s <= 1 * z' + 1 * (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= 1 * z' + 1 * 0, z'' = 0, z >= 0, z' >= 0

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

revapp: runtime: O(n1) [1 + z], size: O(n1) [z + z'] selects: runtime: O(n2) [2 + z + z·z'' + z' + z'·z'' + 2·z'' + z''2], size: O(n2) [2 + z + z·z'' + z' + z'·z'' + 3·z'' + z''2]

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

Computed SIZE bound using CoFloCo for: select
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(28) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
select(z) -{ 3 + x + x·xs + 2·xs + xs² }→ s2 :|: s2 >= 0, s2 <= 1 * x + 1 * 0 + 2 + 3 * xs + 1 * (x * xs) + 1 * (0 * xs) + 1 * (xs * xs), z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 5 + x + x·xs + 3·xs + xs·z + xs·z' + xs² + z + 2·z' }→ 1 + (1 + z + s) + s1 :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + z + z') + 2 + 3 * xs + 1 * (x * xs) + 1 * ((1 + z + z') * xs) + 1 * (xs * xs), s >= 0, s <= 1 * z' + 1 * (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= 1 * z' + 1 * 0, z'' = 0, z >= 0, z' >= 0

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

revapp: runtime: O(n1) [1 + z], size: O(n1) [z + z'] selects: runtime: O(n2) [2 + z + z·z'' + z' + z'·z'' + 2·z'' + z''2], size: O(n2) [2 + z + z·z'' + z' + z'·z'' + 3·z'' + z''2] select: runtime: ?, size: INF

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

Computed RUNTIME bound using KoAT for: select
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 4 + 3·z + 2·z²

(30) Obligation:

Complexity RNTS consisting of the following rules:

revapp(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + x + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
revapp(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
select(z) -{ 3 + x + x·xs + 2·xs + xs² }→ s2 :|: s2 >= 0, s2 <= 1 * x + 1 * 0 + 2 + 3 * xs + 1 * (x * xs) + 1 * (0 * xs) + 1 * (xs * xs), z = 1 + x + xs, xs >= 0, x >= 0
select(z) -{ 1 }→ 0 :|: z = 0
selects(z, z', z'') -{ 5 + x + x·xs + 3·xs + xs·z + xs·z' + xs² + z + 2·z' }→ 1 + (1 + z + s) + s1 :|: s1 >= 0, s1 <= 1 * x + 1 * (1 + z + z') + 2 + 3 * xs + 1 * (x * xs) + 1 * ((1 + z + z') * xs) + 1 * (xs * xs), s >= 0, s <= 1 * z' + 1 * (1 + x + xs), xs >= 0, z >= 0, x >= 0, z'' = 1 + x + xs, z' >= 0
selects(z, z', z'') -{ 2 + z' }→ 1 + (1 + z + s'') + 0 :|: s'' >= 0, s'' <= 1 * z' + 1 * 0, z'' = 0, z >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

revapp: runtime: O(n1) [1 + z], size: O(n1) [z + z'] selects: runtime: O(n2) [2 + z + z·z'' + z' + z'·z'' + 2·z'' + z''2], size: O(n2) [2 + z + z·z'' + z' + z'·z'' + 3·z'' + z''2] select: runtime: O(n2) [4 + 3·z + 2·z2], size: INF

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^2)