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f61e547 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | import OAI.Probability.FKPressure.ComplexSeries
namespace OAI
noncomputable section
open Finset Set
open scoped Classical Topology
namespace BelowOne.PressureSeries
variable {V : Type*} [Fintype V]
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
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)))
end BelowOne.PressureSeries
end
end OAI
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