38 lines
1.1 KiB
Plaintext
38 lines
1.1 KiB
Plaintext
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import Mathlib.Data.Fin.Basic
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import Mathlib.Tactic.NormNum
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import Bookshelf.Real.Basic
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import OneVariableCalculus.Apostol.Real.Set.Partition
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namespace Real.Function
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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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unfold List.pairwise at hI
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have ⟨ys, hys⟩ : ∃ ys, List.tail? p.xs = some ys := sorry
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conv at hI => arg 2; rw [hys]; simp only
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sorry
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/--
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A `Step` function is a function `f` along with a proof of the existence of some
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partition `P` such that `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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def set_def (f : Step) : Set ℝ² := sorry
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-- TODO: Fill out
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end Real.Function.Step
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