| import Mathlib |
| import OAI.Probability.LatticeSurface.Convexity.USCProduct |
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| namespace OAI |
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| open Set Filter |
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| open scoped Topology |
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| noncomputable section |
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| namespace Overlap |
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| end Overlap |
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| namespace Overlap |
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| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] |
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| def kernel (p : E → ℝ) (x χ : E) : ℝ := Real.sqrt (p (x+χ)*p (x-χ)) |
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| def shifts (l : E →L[ℝ] ℝ) (t R : ℝ) : Set E := |
| Metric.closedBall 0 R ∩ l ⁻¹' {t} |
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| 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)) |
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| 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) |
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| 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 _)) |
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| 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)) |
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| 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 _ _ |
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| 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 |
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| 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) |
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| 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 |
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| 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)) |
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| 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 |
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| 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⟩ |
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| 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 |
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| end Overlap |
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| end |
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| end OAI |
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