Update holomorphic_zero.lean

This commit is contained in:
Stefan Kebekus 2024-08-16 10:43:57 +02:00
parent 83b3e0da1e
commit 9410087ddf
1 changed files with 75 additions and 0 deletions

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@ -12,9 +12,84 @@ noncomputable def zeroDivisor
exact 0
theorem analyticAtZeroDivisorSupport
{f : }
{z : }
(h : z ∈ Function.support (zeroDivisor f)) :
AnalyticAt f z := by
by_contra h₁f
simp at h
dsimp [zeroDivisor] at h
simp [h₁f] at h
lemma toNatEqSelf_iff {n : ℕ∞} : n.toNat = n ↔ ∃ m : , m = n := by
constructor
· intro H₁
rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
by_contra H₂
rw [H₂] at H₁
simp at H₁
· intro H
obtain ⟨m, hm⟩ := H
rw [← hm]
simp
theorem discreteZeros
{f : } :
DiscreteTopology (Function.support (zeroDivisor f)) := by
apply singletons_open_iff_discrete.mp
intro z
let A := analyticAtZeroDivisorSupport z.2
let c : WithTop := A.order
let B := AnalyticAt.order_eq_nat_iff A
let n := zeroDivisor f z.1
have : ∃ a : , a = A.order := by
rw [← ENat.some_eq_coe]
rw [← WithTop.ne_top_iff_exists]
by_contra H
rw [AnalyticAt.order_eq_top_iff] at H
dsimp [n, zeroDivisor]
simp [A]
sorry
let C := (B n).1 this
apply Metric.isOpen_singleton_iff.mpr
/-
Try this: refine Metric.isOpen_singleton_iff.mpr ?_
Remaining subgoals:
⊢ ∃ ε > 0, ∀ (y : ↑(Function.support (zeroDivisor f))), dist y z < ε → y = z
Suggestions
Try this: refine isClosed_compl_iff.mp ?_
Remaining subgoals:
⊢ IsClosed {z}ᶜ
Suggestions
Try this: refine disjoint_frontier_iff_isOpen.mp ?_
Remaining subgoals:
⊢ Disjoint (frontier {z}) {z}
Suggestions
Try this: refine isOpen_iff_forall_mem_open.mpr ?_
Remaining subgoals:
⊢ ∀ x ∈ {z}, ∃ t ⊆ {z}, IsOpen t ∧ x ∈ t
-/
sorry