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110f4f3 | 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 94 95 96 97 98 99 | import Mathlib
import OAI.Analysis.BiholderTransport.Contact.CompactMinty
namespace OAI
section
section
noncomputable section
open Set Filter
open scoped Topology ContDiff
namespace WeakMTWTransport
section
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
lemma contDiffAt_parametric_quadratic_upper {f : E×F → ℝ} {x : E} {a : F}
(hf : ContDiffAt ℝ ∞ f (x,a)) :
∃ δ>0, ∃ K>0, ∀ z∈Metric.ball x δ, ∀ w∈Metric.ball x δ,
∀ b∈Metric.ball a δ,
f (w,b)-f (z,b)-fderiv ℝ f (z,b) (w-z,0) ≤ K*‖w-z‖^2 := by
let K := ‖fderiv ℝ (fderiv ℝ f) (x,a)‖+1
have hK : 0<K := by dsimp [K]; positivity
have hc : ContinuousAt (fun q => ‖fderiv ℝ (fderiv ℝ f) q‖) (x,a) :=
((hf.fderiv_right (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)).continuousAt_fderiv (by norm_num)).norm
have hf2 : ContDiffAt ℝ 2 f (x,a) := hf.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)
have hk : ‖fderiv ℝ (fderiv ℝ f) (x,a)‖<K := by dsimp [K]; linarith
have he : ∀ᶠ q in 𝓝 (x,a), ContDiffAt ℝ 2 f q ∧
‖fderiv ℝ (fderiv ℝ f) q‖<K :=
(hf2.eventually (by norm_num)).and (hc.preimage_mem_nhds (Iio_mem_nhds hk))
obtain ⟨δ,hδ,hbound⟩ := Metric.mem_nhds_iff.mp he
refine ⟨δ,hδ,K,hK,?_⟩
intro z hz w hw b hb
have hzb : (z,b)∈Metric.ball (x,a) δ := by
simpa only [Metric.mem_ball,Prod.dist_eq,max_lt_iff] using And.intro hz hb
have hwb : (w,b)∈Metric.ball (x,a) δ := by
simpa only [Metric.mem_ball,Prod.dist_eq,max_lt_iff] using And.intro hw hb
have H := c2_quadratic_remainder_on_convex hK.le (convex_ball (x,a) δ)
(fun q hq => (hbound hq).1) (fun q hq => (hbound hq).2.le) hzb hwb
have H' := (abs_le.mp (show |f (w,b)-f (z,b)-fderiv ℝ f (z,b) ((w,b)-(z,b))|
≤ K*‖(w,b)-(z,b)‖^2 from H)).2
simpa only [Prod.mk_sub_mk,sub_self,Prod.norm_def,norm_zero,max_eq_left (norm_nonneg _)] using H'
end
end WeakMTWTransport
end
end
section
noncomputable section
open Set Filter
open scoped Topology
namespace WeakMTWTransport
lemma compact_eventually_uniform_radius_bound {A T : Type*}
[TopologicalSpace A] [TopologicalSpace T] {K : Set A} (hK : IsCompact K)
{t : T} {P : ℝ → ℝ → T → A → Prop}
(hmono : ∀ {r R b B : ℝ}, 0<r → r≤R → b≤B → ∀ s a, P R b s a → P r B s a)
(hlocal : ∀ a∈K, ∃ r>0, ∃ b>0, ∀ᶠ q : T×A in 𝓝 (t,a), P r b q.1 q.2) :
∃ r>0, ∃ b>0, ∀ᶠ s in 𝓝 t, ∀ a∈K, P r b s a := by
apply hK.induction_on
(p := fun J => ∃ r>0, ∃ b>0, ∀ᶠ s in 𝓝 t, ∀ a∈J, P r b s a)
· exact ⟨1,by norm_num,1,by norm_num,Filter.Eventually.of_forall (by simp)⟩
· intro J L hJL hL
obtain ⟨r,hr,b,hb,H⟩ := hL
refine ⟨r,hr,b,hb,?_⟩
filter_upwards [H] with s hs
exact fun a ha => hs a (hJL ha)
· intro J L hJ hL
obtain ⟨r,hr,b,hb,H⟩ := hJ
obtain ⟨R,hR,B,hB,G⟩ := hL
refine ⟨min r R,lt_min hr hR,max b B,lt_max_of_lt_left hb,?_⟩
filter_upwards [H,G] with s hs gs
intro a ha
rcases ha with ha|ha
· exact hmono (lt_min hr hR) (min_le_left _ _) (le_max_left _ _) s a (hs a ha)
· exact hmono (lt_min hr hR) (min_le_right _ _) (le_max_right _ _) s a (gs a ha)
· intro a ha
obtain ⟨r,hr,b,hb,H⟩ := hlocal a ha
obtain ⟨U,hU,V,hV,hUV⟩ := mem_nhds_prod_iff.mp H
refine ⟨V,mem_nhdsWithin_of_mem_nhds hV,r,hr,b,hb,?_⟩
filter_upwards [hU] with s hs
intro a ha
exact hUV (show (s,a)∈U×ˢV from ⟨hs,ha⟩)
end WeakMTWTransport
end
end
end
end OAI
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