538 lines
15 KiB
Plaintext
538 lines
15 KiB
Plaintext
import Init.Classical
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import Mathlib.Analysis.Analytic.Meromorphic
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import Mathlib.Topology.ContinuousOn
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import Mathlib.Analysis.Analytic.IsolatedZeros
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import Nevanlinna.holomorphic
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noncomputable def zeroDivisor
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(f : ℂ → ℂ) :
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ℂ → ℕ := by
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intro z
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by_cases hf : AnalyticAt ℂ f z
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· exact hf.order.toNat
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· exact 0
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theorem analyticAtZeroDivisorSupport
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : z ∈ Function.support (zeroDivisor f)) :
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AnalyticAt ℂ f z := by
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by_contra h₁f
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simp at h
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dsimp [zeroDivisor] at h
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simp [h₁f] at h
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theorem zeroDivisor_eq_ord_AtZeroDivisorSupport
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : z ∈ Function.support (zeroDivisor f)) :
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zeroDivisor f z = (analyticAtZeroDivisorSupport h).order.toNat := by
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unfold zeroDivisor
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simp [analyticAtZeroDivisorSupport h]
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theorem zeroDivisor_eq_ord_AtZeroDivisorSupport'
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{f : ℂ → ℂ}
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{z : ℂ}
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(h : z ∈ Function.support (zeroDivisor f)) :
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zeroDivisor f z = (analyticAtZeroDivisorSupport h).order := by
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unfold zeroDivisor
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simp [analyticAtZeroDivisorSupport h]
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sorry
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lemma toNatEqSelf_iff {n : ℕ∞} : n.toNat = n ↔ ∃ m : ℕ, m = n := by
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constructor
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· intro H₁
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rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
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by_contra H₂
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rw [H₂] at H₁
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simp at H₁
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· intro H
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obtain ⟨m, hm⟩ := H
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rw [← hm]
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simp
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lemma natural_if_toNatNeZero {n : ℕ∞} : n.toNat ≠ 0 → ∃ m : ℕ, m = n := by
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rw [← ENat.some_eq_coe, ← WithTop.ne_top_iff_exists]
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contrapose; simp; tauto
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theorem zeroDivisor_localDescription
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(h : z₀ ∈ Function.support (zeroDivisor f)) :
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∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ (zeroDivisor f z₀) • g z := by
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have A : zeroDivisor f ↑z₀ ≠ 0 := by exact h
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let B := zeroDivisor_eq_ord_AtZeroDivisorSupport h
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rw [B] at A
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have C := natural_if_toNatNeZero A
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obtain ⟨m, hm⟩ := C
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have h₂m : m ≠ 0 := by
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rw [← hm] at A
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simp at A
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assumption
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rw [eq_comm] at hm
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let E := AnalyticAt.order_eq_nat_iff (analyticAtZeroDivisorSupport h) m
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let F := hm
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rw [E] at F
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have : m = zeroDivisor f z₀ := by
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rw [B, hm]
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simp
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rwa [this] at F
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theorem zeroDivisor_zeroSet
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(h : z₀ ∈ Function.support (zeroDivisor f)) :
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f z₀ = 0 := by
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obtain ⟨g, _, _, h₃⟩ := zeroDivisor_localDescription h
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rw [Filter.Eventually.self_of_nhds h₃]
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simp
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left
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exact h
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theorem zeroDivisor_support_iff
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{f : ℂ → ℂ}
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{z₀ : ℂ} :
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z₀ ∈ Function.support (zeroDivisor f) ↔
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f z₀ = 0 ∧
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AnalyticAt ℂ f z₀ ∧
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∃ (g : ℂ → ℂ), AnalyticAt ℂ g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ (zeroDivisor f z₀) • g z := by
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constructor
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· intro hz
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constructor
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· exact zeroDivisor_zeroSet hz
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· constructor
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· exact analyticAtZeroDivisorSupport hz
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· exact zeroDivisor_localDescription hz
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· intro ⟨h₁, h₂, h₃⟩
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have : zeroDivisor f z₀ = (h₂.order).toNat := by
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unfold zeroDivisor
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simp [h₂]
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simp [this]
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simp [(h₂.order_eq_nat_iff (zeroDivisor f z₀)).2 h₃]
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := h₃
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rw [Filter.Eventually.self_of_nhds h₃g] at h₁
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simp [h₂g] at h₁
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assumption
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theorem topOnPreconnected
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) :
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∀ (hz : z ∈ U), (h₁f z hz).order ≠ ⊤ := by
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by_contra H
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push_neg at H
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obtain ⟨z', hz'⟩ := H
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rw [AnalyticAt.order_eq_top_iff] at hz'
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rw [← AnalyticAt.frequently_zero_iff_eventually_zero (h₁f z z')] at hz'
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have A := AnalyticOn.eqOn_zero_of_preconnected_of_frequently_eq_zero h₁f hU z' hz'
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tauto
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theorem supportZeroSet
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) :
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U ∩ Function.support (zeroDivisor f) = U ∩ f⁻¹' {0} := by
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ext x
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constructor
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· intro hx
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constructor
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· exact hx.1
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exact zeroDivisor_zeroSet hx.2
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· simp
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intro h₁x h₂x
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constructor
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· exact h₁x
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· let A := zeroDivisor_support_iff (f := f) (z₀ := x)
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simp at A
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rw [A]
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constructor
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· exact h₂x
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· constructor
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· exact h₁f x h₁x
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· have B := AnalyticAt.order_eq_nat_iff (h₁f x h₁x) (zeroDivisor f x)
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simp at B
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rw [← B]
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dsimp [zeroDivisor]
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simp [h₁f x h₁x]
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refine Eq.symm (ENat.coe_toNat ?h.mpr.right.right.right.a)
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exact topOnPreconnected hU h₁f h₂f h₁x
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theorem discreteZeros
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{f : ℂ → ℂ} :
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DiscreteTopology (Function.support (zeroDivisor f)) := by
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simp_rw [← singletons_open_iff_discrete, Metric.isOpen_singleton_iff]
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intro z
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have A : zeroDivisor f ↑z ≠ 0 := by exact z.2
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let B := zeroDivisor_eq_ord_AtZeroDivisorSupport z.2
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rw [B] at A
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have C := natural_if_toNatNeZero A
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obtain ⟨m, hm⟩ := C
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have h₂m : m ≠ 0 := by
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rw [← hm] at A
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simp at A
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assumption
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rw [eq_comm] at hm
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let E := AnalyticAt.order_eq_nat_iff (analyticAtZeroDivisorSupport z.2) m
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rw [E] at hm
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := hm
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rw [Metric.eventually_nhds_iff_ball] at h₃g
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have : ∃ ε > 0, ∀ y ∈ Metric.ball (↑z) ε, g y ≠ 0 := by
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have h₄g : ContinuousAt g z := AnalyticAt.continuousAt h₁g
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have : {0}ᶜ ∈ nhds (g z) := by
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exact compl_singleton_mem_nhds_iff.mpr h₂g
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let F := h₄g.preimage_mem_nhds this
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rw [Metric.mem_nhds_iff] at F
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obtain ⟨ε, h₁ε, h₂ε⟩ := F
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use ε
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constructor; exact h₁ε
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intro y hy
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let G := h₂ε hy
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simp at G
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exact G
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obtain ⟨ε₁, h₁ε₁⟩ := this
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obtain ⟨ε₂, h₁ε₂, h₂ε₂⟩ := h₃g
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use min ε₁ ε₂
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constructor
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· have : 0 < min ε₁ ε₂ := by
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rw [lt_min_iff]
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exact And.imp_right (fun _ => h₁ε₂) h₁ε₁
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exact this
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intro y
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intro h₁y
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have h₂y : ↑y ∈ Metric.ball (↑z) ε₂ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₂ := by exact min_le_right ε₁ ε₂
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have h₃y : ↑y ∈ Metric.ball (↑z) ε₁ := by
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simp
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calc dist y z
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_ < min ε₁ ε₂ := by assumption
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_ ≤ ε₁ := by exact min_le_left ε₁ ε₂
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let F := h₂ε₂ y.1 h₂y
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rw [zeroDivisor_zeroSet y.2] at F
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simp at F
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simp [h₂m] at F
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have : g y.1 ≠ 0 := by
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exact h₁ε₁.2 y h₃y
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simp [this] at F
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ext
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rwa [sub_eq_zero] at F
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theorem zeroDivisor_finiteOnCompact
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hU : IsPreconnected U)
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(h₁f : AnalyticOn ℂ f U)
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(h₂f : ∃ z ∈ U, f z ≠ 0) -- not needed!
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(h₂U : IsCompact U) :
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Set.Finite (U ∩ Function.support (zeroDivisor f)) := by
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have hinter : IsCompact (U ∩ Function.support (zeroDivisor f)) := by
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apply IsCompact.of_isClosed_subset h₂U
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rw [supportZeroSet]
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apply h₁f.continuousOn.preimage_isClosed_of_isClosed
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exact IsCompact.isClosed h₂U
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exact isClosed_singleton
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assumption
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assumption
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assumption
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exact Set.inter_subset_left
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apply hinter.finite
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apply DiscreteTopology.of_subset (s := Function.support (zeroDivisor f))
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exact discreteZeros (f := f)
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exact Set.inter_subset_right
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theorem AnalyticOn.order_eq_nat_iff
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{z₀ : ℂ}
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(hf : AnalyticOn ℂ f U)
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(hz₀ : z₀ ∈ U)
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(n : ℕ) :
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(hf z₀ hz₀).order = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
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constructor
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-- Direction →
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intro hn
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obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
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-- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
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-- removable singularity removed
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let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
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-- Describe g near z₀
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have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
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rw [eventually_nhds_iff]
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
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use t
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constructor
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· intro y h₁y
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by_cases h₂y : y = z₀
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· dsimp [g]; simp [h₂y]
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· dsimp [g]; simp [h₂y]
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rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
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exact h₁t y h₁y
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norm_num
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rw [sub_eq_zero]
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tauto
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· constructor
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· assumption
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· assumption
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-- Describe g near points z₁ that are different from z₀
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have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
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intro hz₁
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rw [eventually_nhds_iff]
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use {z₀}ᶜ
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constructor
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· intro y hy
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simp at hy
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simp [g, hy]
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· exact ⟨isOpen_compl_singleton, hz₁⟩
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-- Use g and show that it has all required properties
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use g
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constructor
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· -- AnalyticOn ℂ g U
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intro z h₁z
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by_cases h₂z : z = z₀
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· rw [h₂z]
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apply AnalyticAt.congr h₁gloc
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exact Filter.EventuallyEq.symm g_near_z₀
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· simp_rw [eq_comm] at g_near_z₁
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apply AnalyticAt.congr _ (g_near_z₁ h₂z)
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apply AnalyticAt.div
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exact hf z h₁z
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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simp
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rw [sub_eq_zero]
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tauto
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· constructor
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· simp [g]; tauto
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· intro z
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by_cases h₂z : z = z₀
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· rw [h₂z, g_near_z₀.self_of_nhds]
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exact h₃gloc.self_of_nhds
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· rw [(g_near_z₁ h₂z).self_of_nhds]
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simp [h₂z]
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rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel]
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simp; norm_num
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rw [sub_eq_zero]
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tauto
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-- direction ←
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intro h
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obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
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rw [AnalyticAt.order_eq_nat_iff]
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use g
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exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.eventually_of_forall h₃g⟩⟩
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theorem AnalyticOn.order_eq_nat_iff'
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{f : ℂ → ℂ}
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{U : Set ℂ}
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{A : Finset U}
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(hf : AnalyticOn ℂ f U)
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(n : A → ℕ) :
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∀ a : A, (hf a (Subtype.coe_prop a.val)).order = n a → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a, g a ≠ 0) ∧ ∀ z, f z = (∏ a, (z - a) ^ (n a)) • g z := by
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apply Finset.induction
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let a : A := by sorry
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let b : ℂ := by sorry
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let u : U := by sorry
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let X := n a
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have : a = (3 : ℂ) := by sorry
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have : b ∈ ↑A := by sorry
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have : ↑a ∈ U := by exact Subtype.coe_prop a.val
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let Y := ∀ a : A, (hf a (Subtype.coe_prop a.val)).order = n a
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--∀ a : A, (hf (ha a)).order = ↑(n a) →
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intro hn
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obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
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-- Define a candidate function
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let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
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-- Describe g near z₀
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have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
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rw [eventually_nhds_iff]
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obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
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use t
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constructor
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· intro y h₁y
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by_cases h₂y : y = z₀
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· dsimp [g]; simp [h₂y]
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· dsimp [g]; simp [h₂y]
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rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
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exact h₁t y h₁y
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norm_num
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rw [sub_eq_zero]
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tauto
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· constructor
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· assumption
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· assumption
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-- Describe g near points z₁ different from z₀
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have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
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intro hz₁
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rw [eventually_nhds_iff]
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use {z₀}ᶜ
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constructor
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· intro y hy
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simp at hy
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simp [g, hy]
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· exact ⟨isOpen_compl_singleton, hz₁⟩
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-- Use g and show that it has all required properties
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use g
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constructor
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· -- AnalyticOn ℂ g U
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intro z h₁z
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by_cases h₂z : z = z₀
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· rw [h₂z]
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apply AnalyticAt.congr h₁gloc
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exact Filter.EventuallyEq.symm g_near_z₀
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· simp_rw [eq_comm] at g_near_z₁
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apply AnalyticAt.congr _ (g_near_z₁ h₂z)
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apply AnalyticAt.div
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exact hf z h₁z
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id
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apply analyticAt_const
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simp
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rw [sub_eq_zero]
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tauto
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· constructor
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· simp [g]; tauto
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· intro z
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by_cases h₂z : z = z₀
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· rw [h₂z, g_near_z₀.self_of_nhds]
|
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exact h₃gloc.self_of_nhds
|
||
· rw [(g_near_z₁ h₂z).self_of_nhds]
|
||
simp [h₂z]
|
||
rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel]
|
||
simp; norm_num
|
||
rw [sub_eq_zero]
|
||
tauto
|
||
|
||
|
||
noncomputable def zeroDivisorDegree
|
||
{f : ℂ → ℂ}
|
||
{U : Set ℂ}
|
||
(h₁U : IsPreconnected U) -- not needed!
|
||
(h₂U : IsCompact U)
|
||
(h₁f : AnalyticOn ℂ f U)
|
||
(h₂f : ∃ z ∈ U, f z ≠ 0) : -- not needed!
|
||
ℕ := (zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U).toFinset.card
|
||
|
||
|
||
lemma zeroDivisorDegreeZero
|
||
{f : ℂ → ℂ}
|
||
{U : Set ℂ}
|
||
(h₁U : IsPreconnected U) -- not needed!
|
||
(h₂U : IsCompact U)
|
||
(h₁f : AnalyticOn ℂ f U)
|
||
(h₂f : ∃ z ∈ U, f z ≠ 0) : -- not needed!
|
||
0 = zeroDivisorDegree h₁U h₂U h₁f h₂f ↔ U ∩ (zeroDivisor f).support = ∅ := by
|
||
sorry
|
||
|
||
|
||
lemma eliminatingZeros₀
|
||
{U : Set ℂ}
|
||
(h₁U : IsPreconnected U)
|
||
(h₂U : IsCompact U) :
|
||
∀ n : ℕ, ∀ f : ℂ → ℂ, (h₁f : AnalyticOn ℂ f U) → (h₂f : ∃ z ∈ U, f z ≠ 0) →
|
||
(n = zeroDivisorDegree h₁U h₂U h₁f h₂f) →
|
||
∃ F : ℂ → ℂ, (AnalyticOn ℂ F U) ∧ (f = F * ∏ᶠ a ∈ (U ∩ (zeroDivisor f).support), fun z ↦ (z - a) ^ (zeroDivisor f a)) := by
|
||
|
||
intro n
|
||
induction' n with n ih
|
||
-- case zero
|
||
intro f h₁f h₂f h₃f
|
||
use f
|
||
rw [zeroDivisorDegreeZero] at h₃f
|
||
rw [h₃f]
|
||
simpa
|
||
|
||
-- case succ
|
||
intro f h₁f h₂f h₃f
|
||
|
||
let Supp := (zeroDivisor_finiteOnCompact h₁U h₁f h₂f h₂U).toFinset
|
||
|
||
have : Supp.Nonempty := by
|
||
rw [← Finset.one_le_card]
|
||
calc 1
|
||
_ ≤ n + 1 := by exact Nat.le_add_left 1 n
|
||
_ = zeroDivisorDegree h₁U h₂U h₁f h₂f := by exact h₃f
|
||
_ = Supp.card := by rfl
|
||
obtain ⟨z₀, hz₀⟩ := this
|
||
dsimp [Supp] at hz₀
|
||
simp only [Set.Finite.mem_toFinset, Set.mem_inter_iff] at hz₀
|
||
|
||
|
||
|
||
|
||
let A := AnalyticOn.order_eq_nat_iff h₁f hz₀.1 (zeroDivisor f z₀)
|
||
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport hz₀.2
|
||
let B := zeroDivisor_eq_ord_AtZeroDivisorSupport' hz₀.2
|
||
rw [eq_comm] at B
|
||
let C := A B
|
||
obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := C
|
||
|
||
have h₄g₀ : ∃ z ∈ U, g₀ z ≠ 0 := by sorry
|
||
have h₅g₀ : n = zeroDivisorDegree h₁U h₂U h₁g₀ h₄g₀ := by sorry
|
||
|
||
obtain ⟨F, h₁F, h₂F⟩ := ih g₀ h₁g₀ h₄g₀ h₅g₀
|
||
use F
|
||
constructor
|
||
· assumption
|
||
·
|
||
sorry
|