bookshelf/Common/Real/Function/Step.lean

67 lines
2.0 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.Real.Basic
import Common.Real.Set.Partition
/-! # Common.Real.Function.Step
A characterization of step functions.
-/
namespace Real.Function
open Partition
/--
Any member of a subinterval of a partition `P` must also be a member of `P`.
-/
lemma mem_open_subinterval_imp_mem_partition {p : Partition}
(hI : I ∈ p.xs.pairwise (fun x₁ x₂ => Set.Ioo x₁ x₂))
(hy : y ∈ I) : y ∈ p := by
cases h : p.xs with
| nil =>
-- By definition, a partition must always have at least two points in the
-- interval. Discharge the empty case.
rw [h] at hI
cases hI
| cons x ys =>
have ⟨i, x₁, ⟨x₂, ⟨hx₁, ⟨hx₂, hI'⟩⟩⟩⟩ :=
List.mem_pairwise_imp_exists_adjacent hI
have hx₁ : x₁ ∈ p.xs := by
rw [hx₁]
let j : Fin (List.length p.xs) := ⟨i.1, Nat.lt_of_lt_pred i.2⟩
exact List.mem_iff_exists_get.mpr ⟨j, rfl⟩
have hx₂ : x₂ ∈ p.xs := by
rw [hx₂]
let j : Fin (List.length p.xs) := ⟨i.1 + 1, lt_tsub_iff_right.mp i.2⟩
exact List.mem_iff_exists_get.mpr ⟨j, rfl⟩
rw [hI'] at hy
apply And.intro
· calc p.left
_ ≤ x₁ := (subdivision_point_mem_partition hx₁).left
_ ≤ y := le_of_lt hy.left
· calc y
_ ≤ x₂ := le_of_lt hy.right
_ ≤ p.right := (subdivision_point_mem_partition hx₂).right
/--
A function `f` is a `Step` function if there exists a `Partition` `p` such that
`f` is constant on every open subinterval of `p`.
-/
structure Step where
p : Partition
f : ∀ x ∈ p,
const_open_subintervals :
∀ (hI : I ∈ p.xs.pairwise (fun x₁ x₂ => Set.Ioo x₁ x₂)),
∃ c : , ∀ (hy : y ∈ I),
f y (mem_open_subinterval_imp_mem_partition hI hy) = c
namespace Step
/--
The set definition of a `Step` function is the region between the constant
values of the function's subintervals and the real axis.
-/
def set_def (f : Step) : Set ℝ² := sorry
end Step
end Real.Function