Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Smooth urysohn support Ioo

smooth_urysohn_support_Ioo

analytic number theory prime numbers asymptotics smooth existence

Plain-language statement

Given $a<b$ and $c<d$ , there is a smooth compactly supported function $\Psi:\mathbb R\to\mathbb R$ with $\mathbf 1_{[b,c]}\le\Psi\le\mathbf 1_{(a,d)}$ and support exactly $(a,d)$ . In the usual ordered case $a<b\le c<d$ , this is a smooth cutoff equal to $1$ on $[b,c]$ , strictly supported in $(a,d)$ , and taking values between $0$ and $1$ .

Exact Lean statement

lemma smooth_urysohn_support_Ioo {a b c d : ℝ} (h1 : a < b) (h3 : c < d) :
    ∃ Ψ : ℝ → ℝ, (ContDiff ℝ ∞ Ψ) ∧ (HasCompactSupport Ψ) ∧
    Set.indicator (Set.Icc b c) 1 ≤ Ψ ∧ Ψ ≤ Set.indicator (Set.Ioo a d) 1 ∧
    (Function.support Ψ = Set.Ioo a d)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma smooth_urysohn_support_Ioo {a b c d : ℝ} (h1 : a < b) (h3 : c < d) :    ∃ Ψ : ℝ → ℝ, (ContDiff ℝ ∞ Ψ) ∧ (HasCompactSupport Ψ) ∧    Set.indicator (Set.Icc b c) 1 ≤ Ψ ∧ Ψ ≤ Set.indicator (Set.Ioo a d) 1 ∧    (Function.support Ψ = Set.Ioo a d) := by  have := exists_contMDiff_zero_iff_one_iff_of_isClosed (n := ⊤)    (modelWithCornersSelf ℝ ℝ) (s := Set.Iic a ∪ Set.Ici d) (t := Set.Icc b c)    (IsClosed.union isClosed_Iic isClosed_Ici) isClosed_Icc    (by      simp_rw [Set.disjoint_union_left, Set.disjoint_iff, Set.subset_def,        Set.mem_inter_iff, Set.mem_Iic, Set.mem_Icc, Set.mem_empty_iff_false,        and_imp, imp_false, not_le, Set.mem_Ici]      constructor <;> intros <;> linarith)  obtain ⟨Ψ, hΨSmooth, hΨrange, hΨ0, hΨ1⟩ := this  simp only [Set.mem_union, Set.mem_Iic, Set.mem_Ici, Set.mem_Icc] at *  use Ψ  simp only [range_subset_iff, mem_Icc] at hΨrange  refine ⟨ContMDiff.contDiff hΨSmooth, ?_, ?_, ?_, ?_⟩  · apply HasCompactSupport.of_support_subset_isCompact (K := Set.Icc a d) isCompact_Icc    simp only [Function.support_subset_iff, ne_eq, mem_Icc, ← hΨ0, not_or]    bound  · apply Set.indicator_le'    · intro x hx      rw [hΨ1 x |>.mp, Pi.one_apply]      simpa using hx    · exact fun x _ ↦ (hΨrange x).1  · intro x    apply Set.le_indicator_apply    · exact fun _ ↦ (hΨrange x).2    · intro hx      rw [← hΨ0 x |>.mp]      simpa [-not_and, mem_Ioo, not_and_or, not_lt] using hx  · ext x    simp only [Function.mem_support, ne_eq, mem_Ioo, ← hΨ0, not_or, not_le]

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/SmoothExistence.lean:12-44

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