Create analyticAt.lean

Nevanlinna/analyticAt.lean

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+import Mathlib.Analysis.Analytic.IsolatedZeros
+import Mathlib.Analysis.Complex.Basic
+
+
+theorem AnalyticAt.order_mul
+  {f₁ f₂ : ℂ → ℂ}
+  {z₀ : ℂ}
+  (hf₁ : AnalyticAt ℂ f₁ z₀)
+  (hf₂ : AnalyticAt ℂ f₂ z₀) :
+  (AnalyticAt.mul hf₁ hf₂).order = hf₁.order + hf₂.order := by
+  by_cases h₂f₁ : hf₁.order = ⊤
+  · simp [h₂f₁]
+    rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
+    rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₁
+    obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₁
+    use t
+    constructor
+    · intro y hy
+      rw [h₁t y hy]
+      ring
+    · exact ⟨h₂t, h₃t⟩
+  · by_cases h₂f₂ : hf₂.order = ⊤
+    · simp [h₂f₂]
+      rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff]
+      rw [AnalyticAt.order_eq_top_iff, eventually_nhds_iff] at h₂f₂
+      obtain ⟨t, h₁t, h₂t, h₃t⟩ := h₂f₂
+      use t
+      constructor
+      · intro y hy
+        rw [h₁t y hy]
+        ring
+      · exact ⟨h₂t, h₃t⟩
+    · obtain ⟨g₁, h₁g₁, h₂g₁, h₃g₁⟩ := (AnalyticAt.order_eq_nat_iff hf₁ ↑hf₁.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₁))
+      obtain ⟨g₂, h₁g₂, h₂g₂, h₃g₂⟩ := (AnalyticAt.order_eq_nat_iff hf₂ ↑hf₂.order.toNat).1 (eq_comm.1 (ENat.coe_toNat h₂f₂))
+      rw [← ENat.coe_toNat h₂f₁, ← ENat.coe_toNat h₂f₂, ← ENat.coe_add]
+      rw [AnalyticAt.order_eq_nat_iff (AnalyticAt.mul hf₁ hf₂) ↑(hf₁.order.toNat + hf₂.order.toNat)]
+      use g₁ * g₂
+      constructor
+      · exact AnalyticAt.mul h₁g₁ h₁g₂
+      · constructor
+        · simp; tauto
+        · obtain ⟨t₁, h₁t₁, h₂t₁, h₃t₁⟩ := eventually_nhds_iff.1 h₃g₁
+          obtain ⟨t₂, h₁t₂, h₂t₂, h₃t₂⟩ := eventually_nhds_iff.1 h₃g₂
+          rw [eventually_nhds_iff]
+          use t₁ ∩ t₂
+          constructor
+          · intro y hy
+            rw [h₁t₁ y hy.1, h₁t₂ y hy.2]
+            simp; ring
+          · constructor
+            · exact IsOpen.inter h₂t₁ h₂t₂
+            · exact Set.mem_inter h₃t₁ h₃t₂