| import OAI.Probability.FKPressure.ComplexSeries |
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| namespace OAI |
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| noncomputable section |
| open Finset Set |
| open scoped Classical Topology |
| namespace BelowOne.PressureSeries |
| variable {V : Type*} [Fintype V] |
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| lemma identity_on_disk_of_real_cube {f g : ℂ → ℂ} {r : ℝ} (hr : 1 < r) |
| (hf : AnalyticOnNhd ℂ f (Metric.ball 0 r)) |
| (hg : AnalyticOnNhd ℂ g (Metric.ball 0 r)) |
| (he : ∀ t : ℝ, t ∈ Set.Icc 0 1 → f (t : ℂ) = g (t : ℂ)) : |
| EqOn f g (Metric.ball 0 r) := by |
| apply hf.eqOn_of_preconnected_of_mem_closure hg (convex_ball (0 : ℂ) r).isPreconnected |
| (show (0 : ℂ) ∈ Metric.ball 0 r by simpa using zero_lt_one.trans hr) |
| rw [Metric.mem_closure_iff] |
| intro ε hε |
| let t : ℝ := min ε 1 / 2 |
| have ht : 0 < t := div_pos (lt_min hε zero_lt_one) (by norm_num) |
| have ht1 : t ≤ 1 := by dsimp [t]; linarith [min_le_right ε 1] |
| have htε : t < ε := by dsimp [t]; linarith [min_le_left ε 1] |
| refine ⟨(t : ℂ), ⟨he t ⟨ht.le,ht1⟩, ?_⟩, ?_⟩ |
| · simpa only [mem_singleton_iff, Complex.ofReal_eq_zero] using ne_of_gt ht |
| · simpa only [dist_zero_left, Complex.norm_real, Real.norm_of_nonneg ht.le] using htε |
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| lemma identity_on_polydisk {f g : (V → ℂ) → ℂ} {r : ℝ} (hr : 1 < r) |
| (hf : AnalyticOnNhd ℂ f {x | ∀ i, ‖x i‖ < r}) |
| (hg : AnalyticOnNhd ℂ g {x | ∀ i, ‖x i‖ < r}) |
| (he : ∀ t : V → ℝ, (∀ i, t i ∈ Set.Icc 0 1) → |
| f (fun i => (t i : ℂ)) = g (fun i => (t i : ℂ))) : |
| EqOn f g {x | ∀ i, ‖x i‖ < r} := by |
| have hfinite : ∀ s : Finset V, ∀ x : V → ℂ, |
| (∀ i, ‖x i‖ < r) → (∀ i ∉ s, ∃ t : ℝ, t ∈ Set.Icc 0 1 ∧ x i = (t : ℂ)) → |
| f x = g x := by |
| intro s |
| induction s using Finset.induction with |
| | empty => |
| intro x hx hreal |
| have hreal' : ∀ i, ∃ t : ℝ, t ∈ Set.Icc 0 1 ∧ x i = (t : ℂ) := |
| fun i => hreal i (by simp) |
| choose t ht htx using hreal' |
| have hh : x = fun i => (t i : ℂ) := funext htx |
| rw [hh] |
| exact he t ht |
| | @insert i s his ih => |
| intro x hx hreal |
| let curve : ℂ → V → ℂ := fun z => Function.update x i z |
| have hc (z : ℂ) : AnalyticAt ℂ curve z := by |
| apply AnalyticAt.pi |
| intro j |
| by_cases hji : j = i |
| · subst j |
| simp only [Function.update_self] |
| exact analyticAt_id |
| · simpa only [curve, Function.update_of_ne hji] using |
| (analyticAt_const : AnalyticAt ℂ (fun _ : ℂ => x j) z) |
| have hb {z : ℂ} (hz : z ∈ Metric.ball 0 r) : ∀ j, ‖curve z j‖ < r := by |
| intro j |
| by_cases hji : j = i |
| · subst j |
| simpa only [curve, Function.update_self, Metric.mem_ball, dist_zero_right] using hz |
| · simpa only [curve, Function.update_of_ne hji] using hx j |
| have hfc : AnalyticOnNhd ℂ (fun z => f (curve z)) (Metric.ball 0 r) := |
| fun z hz => (hf (curve z) (hb hz)).comp (hc z) |
| have hgc : AnalyticOnNhd ℂ (fun z => g (curve z)) (Metric.ball 0 r) := |
| fun z hz => (hg (curve z) (hb hz)).comp (hc z) |
| have hec : ∀ t : ℝ, t ∈ Set.Icc 0 1 → f (curve (t : ℂ)) = g (curve (t : ℂ)) := by |
| intro t ht |
| apply ih (curve (t : ℂ)) |
| · apply hb |
| simpa only [Metric.mem_ball, dist_zero_right, Complex.norm_real, Real.norm_of_nonneg ht.1] using ht.2.trans_lt hr |
| · intro j hjs |
| by_cases hji : j = i |
| · subst j |
| exact ⟨t,ht,Function.update_self _ _ _⟩ |
| · obtain ⟨u,hu,hxu⟩ := hreal j (by simp only [Finset.mem_insert]; exact not_or.mpr ⟨hji,hjs⟩) |
| exact ⟨u,hu, by simpa only [curve, Function.update_of_ne hji] using hxu⟩ |
| have hres := identity_on_disk_of_real_cube hr hfc hgc hec |
| simpa only [curve, Function.update_eq_self] using |
| hres (show x i ∈ Metric.ball 0 r by simpa only [Metric.mem_ball, dist_zero_right] using hx i) |
| intro x hx |
| exact hfinite univ x hx (fun i hi => False.elim (hi (mem_univ i))) |
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| end BelowOne.PressureSeries |
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| end |
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| end OAI |
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