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dd14c26 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import Mathlib
import OAI.Probability.LatticeSurface.Convexity.USCProduct
namespace OAI
open Set Filter
open scoped Topology
noncomputable section
namespace Overlap
end Overlap
namespace Overlap
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E]
def kernel (p : E → ℝ) (x χ : E) : ℝ := Real.sqrt (p (x+χ)*p (x-χ))
def shifts (l : E →L[ℝ] ℝ) (t R : ℝ) : Set E :=
Metric.closedBall 0 R ∩ l ⁻¹' {t}
def overlap (p : E → ℝ) (l : E →L[ℝ] ℝ) (t R : ℝ) (x : E) : ℝ :=
max 0 (sSup (kernel p x '' shifts l t R))
lemma compact_shifts (l : E →L[ℝ] ℝ) (t R : ℝ) : IsCompact (shifts l t R) :=
(isCompact_closedBall 0 R).inter_right (isClosed_singleton.preimage l.continuous)
omit [NormedSpace ℝ E] [ProperSpace E] in
lemma usc_kernel {p : E → ℝ} (hp : UpperSemicontinuous p) (hp0 : ∀ x,0≤p x) :
UpperSemicontinuous (fun z : E × E => kernel p z.1 z.2) := by
exact usc_sqrt (usc_mul (hp.comp (continuous_fst.add continuous_snd))
(hp.comp (continuous_fst.sub continuous_snd)) (fun z => hp0 _) (fun z => hp0 _))
lemma usc_overlap {p : E → ℝ} (hp : UpperSemicontinuous p) (hp0 : ∀ x,0≤p x)
(l : E →L[ℝ] ℝ) (t R : ℝ) : UpperSemicontinuous (overlap p l t R) :=
upperSemicontinuous_const.sup (usc_sup_compact (compact_shifts l t R) (usc_kernel hp hp0))
omit [ProperSpace E] in
lemma overlap_nonneg (p : E → ℝ) (l : E →L[ℝ] ℝ) (t R : ℝ) (x : E) :
0≤overlap p l t R x := le_max_left _ _
omit [ProperSpace E] in
lemma overlap_le {p : E → ℝ} {l : E →L[ℝ] ℝ} {t R : ℝ} {x : E} {b : ℝ}
(hb : 0≤b) (h : ∀ χ∈shifts l t R,kernel p x χ≤b) : overlap p l t R x≤b := by
apply max_le hb
by_cases hne : (shifts l t R).Nonempty
· exact csSup_le (hne.image _) (by rintro _ ⟨χ,hχ,rfl⟩;exact h χ hχ)
· rw [Set.not_nonempty_iff_eq_empty.mp hne,Set.image_empty,Real.sSup_empty]
exact hb
lemma kernel_le_overlap {p : E → ℝ} (hp : UpperSemicontinuous p) (hp0 : ∀ x,0≤p x)
(l : E →L[ℝ] ℝ) (t R : ℝ) (x : E) {χ : E} (hχ : χ∈shifts l t R) :
kernel p x χ≤overlap p l t R x := by
apply le_trans _ (le_max_right _ _)
exact le_csSup (((usc_kernel hp hp0).comp (continuous_const.prodMk continuous_id)).upperSemicontinuousOn _
|>.bddAbove_of_isCompact (compact_shifts l t R)) (mem_image_of_mem _ hχ)
omit [ProperSpace E] in
lemma midpoint_norms {x χ : E} {R : ℝ} (hplus : ‖x+χ‖≤R) (hminus : ‖x-χ‖≤R) :
‖x‖≤R ∧ ‖χ‖≤R := by
have hxa : (x+χ)+(x-χ)=(2:ℝ) • x := by module
have hχa : (x+χ)-(x-χ)=(2:ℝ) • χ := by module
have hx := norm_add_le (x+χ) (x-χ)
have hχ := norm_sub_le (x+χ) (x-χ)
rw [hxa,norm_smul,Real.norm_of_nonneg (by norm_num : (0:ℝ)≤2)] at hx
rw [hχa,norm_smul,Real.norm_of_nonneg (by norm_num : (0:ℝ)≤2)] at hχ
constructor <;> linarith
omit [ProperSpace E] in
lemma kernel_support {p : E → ℝ} {R : ℝ} (hpK : ∀ z,R<‖z‖ → p z=0)
{x χ : E} (h : kernel p x χ≠0) : ‖x‖≤R ∧ ‖χ‖≤R := by
have hp : p (x+χ)≠0 ∧ p (x-χ)≠0 := by
constructor
· intro hz;apply h;simp [kernel,hz]
· intro hz;apply h;simp [kernel,hz]
apply midpoint_norms
· exact le_of_not_gt (fun hx => hp.1 (hpK _ hx))
· exact le_of_not_gt (fun hx => hp.2 (hpK _ hx))
omit [ProperSpace E] in
lemma overlap_support {p : E → ℝ} {l : E →L[ℝ] ℝ} {t R : ℝ}
(hpK : ∀ z,R<‖z‖ → p z=0) {x : E} (hx : R<‖x‖) : overlap p l t R x=0 := by
apply le_antisymm _ (overlap_nonneg _ _ _ _ _)
apply overlap_le le_rfl
intro χ hχ
have hz : kernel p x χ=0 := by
by_contra hn
exact (not_lt_of_ge (kernel_support hpK hn).1) hx
exact hz.le
lemma kernel_le_overlap_all {p : E → ℝ} (hp : UpperSemicontinuous p) (hp0 : ∀ x,0≤p x)
{l : E →L[ℝ] ℝ} {t R : ℝ} (hpK : ∀ z,R<‖z‖ → p z=0)
(x χ : E) (hχ : l χ=t) : kernel p x χ≤overlap p l t R x := by
by_cases hz : kernel p x χ=0
· rw [hz];exact overlap_nonneg _ _ _ _ _
· apply kernel_le_overlap hp hp0 l t R x
exact ⟨by simpa [Metric.mem_closedBall,dist_zero_right] using (kernel_support hpK hz).2,hχ⟩
lemma overlap_eq_sup {p : E → ℝ} (hp : UpperSemicontinuous p) (hp0 : ∀ x,0≤p x)
{l : E →L[ℝ] ℝ} {t R : ℝ} (hpK : ∀ z,R<‖z‖ → p z=0)
(hne : (l ⁻¹' {t}).Nonempty) (x : E) :
overlap p l t R x=sSup (kernel p x '' (l ⁻¹' {t})) := by
have hbd : BddAbove (kernel p x '' (l ⁻¹' {t})) := ⟨overlap p l t R x,by
rintro _ ⟨χ,hχ,rfl⟩;exact kernel_le_overlap_all hp hp0 hpK x χ hχ⟩
apply le_antisymm
· apply overlap_le
· obtain ⟨χ,hχ⟩ := hne
exact (Real.sqrt_nonneg _).trans (le_csSup hbd (mem_image_of_mem _ hχ))
· intro χ hχ
exact le_csSup hbd (mem_image_of_mem _ hχ.2)
· apply csSup_le (hne.image _)
rintro _ ⟨χ,hχ,rfl⟩
exact kernel_le_overlap_all hp hp0 hpK x χ hχ
end Overlap
end
end OAI
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