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9655094 | 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 | import OAI.Analysis.CircleDomains.Selection.HitConfigurations
import Mathlib.Topology.MetricSpace.Infsep
namespace OAI
/-! Testing all sizes and reciprocal separation scales is sufficient to control
the full square norm of the point-hit probabilities. -/
open Set Function MeasureTheory
open scoped Topology ENNReal
noncomputable section
namespace CircleDomainRigidity
theorem exists_reciprocal_separation {X I : Type*} [MetricSpace X] [Finite I]
(x : I → X) (hx : Injective x) :
∃ m : ℕ, ∀ i j, i ≠ j → 1 / ((m : ℝ) + 1) ≤ dist (x i) (x j) := by
obtain ⟨c, hc, hsep⟩ := (Set.finite_range x).relatively_discrete
obtain ⟨a, ha, hac⟩ := ENNReal.lt_iff_exists_nnreal_btwn.mp hc
have ha0 : 0 < (a : ℝ) := by exact_mod_cast ha
obtain ⟨m, hm⟩ := exists_nat_one_div_lt ha0
refine ⟨m, fun i j hij => hm.le.trans ?_⟩
have hd := hac.le.trans (hsep (x i) (mem_range_self i) (x j) (mem_range_self j)
(fun h => hij (hx h)))
rw [edist_dist] at hd
have hdreal := ENNReal.toReal_mono ENNReal.ofReal_ne_top hd
simpa only [ENNReal.coe_toReal, ENNReal.toReal_ofReal dist_nonneg] using hdreal
variable {X : Type*} [MetricSpace X]
[MeasurableSpace C(PathTime, X)]
(μ : ProbabilityMeasure C(PathTime, X)) {B : ℝ}
/-- This hypothesis runs over a countable list of compact configuration spaces. -/
def HasConfigurationBounds (B : ℝ) : Prop :=
∀ n m : ℕ, ∀ p : HitConfiguration X n (1 / ((m : ℝ) + 1)),
weightedConfigurationHits (fun _ : Unit => μ) () p ≤ B
theorem configuration_bound_of_injective
(h : HasConfigurationBounds μ B) {n : ℕ} (x : Fin n → X) (hx : Injective x)
(w : Fin n → Icc (0 : ℝ) 1) (hw : (∑ i, (w i : ℝ) ^ 2) ≤ 1) :
∑ i, (w i : ℝ) * pathHitReal μ (x i) ≤ B := by
obtain ⟨m, hm⟩ := exists_reciprocal_separation x hx
exact h n m ⟨(x, w), hw, hm⟩
theorem configuration_square_sum_le
(h : HasConfigurationBounds μ B) (hB : 0 ≤ B)
{n : ℕ} (x : Fin n → X) (hx : Injective x) :
∑ i, pathHitReal μ (x i) ^ 2 ≤ B ^ 2 := by
let v : Fin n → ℝ := fun i => pathHitReal μ (x i)
let s : ℝ := Real.sqrt (∑ i, v i ^ 2)
have hv (i) : 0 ≤ v i := pathHitReal_nonneg _ _
have hsum : 0 ≤ ∑ i, v i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
have hs : s ^ 2 = ∑ i, v i ^ 2 := Real.sq_sqrt hsum
have hs0 : 0 ≤ s := Real.sqrt_nonneg _
by_cases hsz : s = 0
· have hz : (∑ i, v i ^ 2) = 0 := by nlinarith
change (∑ i, v i ^ 2) ≤ B ^ 2
rw [hz]
positivity
have hsp : 0 < s := lt_of_le_of_ne hs0 (Ne.symm hsz)
have hvi (i) : v i ≤ s := by
have hi : v i ^ 2 ≤ ∑ j, v j ^ 2 :=
Finset.single_le_sum (fun _ _ => sq_nonneg _) (Finset.mem_univ i)
nlinarith
let w : Fin n → Icc (0 : ℝ) 1 := fun i =>
⟨v i / s, div_nonneg (hv i) hs0, (div_le_one hsp).mpr (hvi i)⟩
have hw : (∑ i, (w i : ℝ) ^ 2) ≤ 1 := by
change (∑ i, (v i / s) ^ 2) ≤ 1
simp_rw [div_pow]
rw [← Finset.sum_div, ← hs, div_self (pow_ne_zero 2 hsz)]
have hb := configuration_bound_of_injective μ h x hx w hw
have heq : (∑ i, (w i : ℝ) * pathHitReal μ (x i)) = s := by
change (∑ i, (v i / s) * v i) = s
calc
_ = (∑ i, v i ^ 2) / s := by
rw [Finset.sum_div]
apply Finset.sum_congr rfl
intro i _
ring
_ = s := by rw [← hs]; field_simp
rw [heq] at hb
change (∑ i, v i ^ 2) ≤ B ^ 2
nlinarith
theorem hasHitNormBound_of_configuration_bounds
(h : HasConfigurationBounds μ B) (hB : 0 ≤ B) : HasHitNormBound μ B := by
classical
apply (hasHitNormBound_iff_sum_sq_le μ hB).mpr
intro F
let e : Fin (Fintype.card F) ≃ F := (Fintype.equivFin F).symm
have he : Injective (fun i : Fin (Fintype.card F) => (e i).val) :=
Subtype.val_injective.comp e.injective
have hb := configuration_square_sum_le μ h hB (fun i => (e i).val) he
rw [e.sum_comp (fun a : F => pathHitReal μ a.val ^ 2)] at hb
rw [Finset.sum_coe_sort F (fun x : X => pathHitReal μ x ^ 2)] at hb
exact hb
end CircleDomainRigidity
end
end OAI
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