bookshelf/Common/Geometry/StepFunction.lean

79 lines
1.9 KiB
Plaintext
Raw Blame History

This file contains ambiguous Unicode characters!

This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.

import Common.Geometry.Rectangle.Orthogonal
import Common.List.Basic
import Common.List.NonEmpty
/-! # Common.Geometry.StepFunction
Characterization of step functions.
-/
namespace Geometry
/--
An interval defines a range of values, characterized by a "left" value and a
"right" value. We require these values to be distinct; we do not support the
notion of an empty interval.
-/
structure Interval (α : Type _) [LT α] where
left : α
right : α
h : left < right
namespace Interval
/--
Computes the size of the interval.
-/
def size [LT α] [Sub α] (i : Interval α) : α := i.right - i.left
/--
Computes the midpoint of the interval.
-/
def midpoint [LT α] [Add α] [HDiv α α] (i : Interval α) : α :=
(i.left + i.right) / (2 : )
/--
Convert an `Interval` into a `Set.Ico`.
-/
def toIco [Preorder α] (i : Interval α) : Set α := Set.Ico i.left i.right
/--
Convert an `Interval` into a `Set.Ioc`.
-/
def toIoc [Preorder α] (i : Interval α) : Set α := Set.Ioc i.left i.right
/--
Convert an `Interval` into a `Set.Icc`.
-/
def toIcc [Preorder α] (i : Interval α) : Set α := Set.Icc i.left i.right
/--
Convert an `Interval` into a `Set.Ioo`.
-/
def toIoo [Preorder α] (i : Interval α) : Set α := Set.Ioo i.left i.right
end Interval
/--
A function `f`, whose domain is a closed interval `[a, b]`, is a `StepFunction`
if there exists a partition `P = {x₀, x₁, …, xₙ}` of `[a, b]` such that `f` is
constant on each open subinterval of `P`.
Instead of maintaining a function from `[a, b]` to ``, we instead maintain a
function that maps each partition index to some constant value.
-/
structure StepFunction where
ivls : List.NonEmpty (Interval )
connected : ∀ I ∈ ivls.toList.pairwise (·.right = ·.left), I
toFun : Fin ivls.length →
namespace StepFunction
/--
The ordinate set of the `StepFunction`.
-/
def toSet (sf : StepFunction) : Set Point := sorry
end StepFunction
end Geometry