import Bookshelf.Real.Basic import Bookshelf.Real.Set.Partition /-! # Bookshelf.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 => 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, ℝ const_open_subintervals : ∀ (hI : I ∈ p.xs.pairwise (fun x₁ x₂ => i(x₁, x₂))), ∃ c : ℝ, ∀ (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