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9b6e014 | 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 | import OAI.Probability.ContinuumTransition.ActiveLimits
namespace OAI
section
open scoped Topology
open Set Filter Metric
noncomputable section
namespace ContinuumTemperature
lemma active_uniform_near {Q : ℕ → Parameter → ℝ} {q : Parameter → ℝ}
{P : ℕ → Parameter → ℝ} {p : Parameter → ℝ} {a : ℝ}
(hQ : TendstoLocallyUniformlyOn Q q atTop pressureDomain)
(hconv : ∀ n, ConvexOn ℝ pressureDomain (Q n)) (hq : ContinuousOn q pressureDomain)
(hP : ∀ n, IsMeanField (Q n) a (P n)) (hp : IsMeanField q a p)
{x : Parameter} (hx : x ∈ pressureDomain) {r : ℝ} (hr : 0 < r)
(hball : ball x r ⊆ pressureDomain) {D : ℝ}
(hD : ∀ n, ∀ y ∈ ball x r, ∀ w ∈ activeLifts (Q n) a (P n) y, w.2 ∈ Icc 0 D)
{U : Set Parameter} (hU : IsOpen U) (hAU : activeLifts q a p x ⊆ U) :
∃ δ > 0, δ < r ∧ ∃ N : ℕ, ∀ n ≥ N, ∀ y ∈ ball x δ, activeLifts (Q n) a (P n) y ⊆ U := by
by_contra hn
push Not at hn
have hex : ∀ m : ℕ, ∃ n ≥ m, ∃ y ∈ ball x (min (r/2) ((1/2 : ℝ)^m)),
∃ w ∈ activeLifts (Q n) a (P n) y, w ∉ U := by
intro m
have hd : 0 < min (r/2) ((1/2 : ℝ)^m) := lt_min (by positivity) (pow_pos (by norm_num) _)
have hdr : min (r/2) ((1/2 : ℝ)^m) < r := (min_le_left _ _).trans_lt (by linarith)
obtain ⟨n,hn,y,hy,hbad⟩ := hn _ hd hdr m
obtain ⟨w,hw,hwu⟩ := Set.not_subset.mp hbad
exact ⟨n,hn,y,hy,w,hw,hwu⟩
choose k hk X hX W hW hbad using hex
have hk' : Tendsto k atTop atTop := tendsto_atTop_mono hk tendsto_id
have hXr : ∀ n, X n ∈ ball x r := fun n => (hX n).trans_le ((min_le_left _ _).trans (show r/2 ≤ r by linarith))
have hXt : Tendsto X atTop (𝓝 x) := by
apply tendsto_iff_dist_tendsto_zero.mpr
exact squeeze_zero (fun n => dist_nonneg) (fun n => (hX n).le.trans (min_le_right _ _))
(tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num))
obtain ⟨φ,_,w,hw,hwt⟩ := active_subseq_of_density_bound (locallyUniform_reindex hQ hk')
(fun n => hconv (k n)) hq tendsto_const_nhds (fun n => hP (k n)) hp hx hXt
(fun n => hball (hXr n)) hW (fun n => hD (k n) _ (hXr n) _ (hW n))
obtain ⟨n,hn⟩ := (hwt.eventually (hU.mem_nhds (hAU hw))).exists
exact hbad (φ n) hn
end ContinuumTemperature
end
end
end OAI
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