approx_of_notMem_U — Mathlib

English

If x lies in C, then the nth approximation at x is 0.

Русский

Если x ∈ C, то для любой n аппроксимации c.approx n x = 0.

LaTeX

$$$approx(n,c,x) = 0 \\quad\\text{when } x \\in c.C$$$

Lean4

theorem approx_of_notMem_U (c : CU P) (n : ℕ) {x : X} (hx : x ∉ c.U) : c.approx n x = 1 := by
  induction n generalizing c with
  | zero =>
    rw [← mem_compl_iff] at hx
    exact indicator_of_mem hx _
  | succ n ihn =>
    simp only [approx]
    rw [ihn, ihn, midpoint_self]
    exacts [hx, fun hU => hx <| c.left_U_subset hU]

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