bookshelf/OneVariableCalculus/Real/Function/Step.lean

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import Bookshelf.Real.Basic
import OneVariableCalculus.Real.Set.Partition
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 => i(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 => i(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