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