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135052a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.FiniteLaw
namespace OAI
section
noncomputable section
open scoped BigOperators
open Classical MeasureTheory Set
namespace SK.Analytic
namespace FiniteLaw
def uniformFin (n : ℕ) (hn : 0 < n) : FiniteLaw (Fin n) where
weight _ := 1/(n:ℝ)
nonneg _ := by positivity
sum_one := by
simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
field_simp
theorem uniformFin_prob (n : ℕ) (hn : 0 < n) (E : Fin n → Prop) :
(uniformFin n hn).prob E=(Finset.univ.filter E).card/(n:ℝ) := by
simp only [prob,expect,uniformFin,← Finset.mul_sum]
rw [Finset.sum_boole]
ring
theorem markov {n : ℕ} (P : FiniteLaw (Fin n)) (f : Fin n → ℝ) (hf : ∀ x, 0 ≤ f x)
(c : ℝ) (_hc : 0 < c) : c*P.prob (fun x => c < f x) ≤ P.expect f := by
rw [prob,← expect_const_mul]
apply expect_mono
intro x
split_ifs with h
· simpa only [mul_one] using h.le
· simpa only [mul_zero] using hf x
end FiniteLaw
def thresholdGrid (n : ℕ) (a ρ : ℝ) (i : Fin n) : ℝ := a+ρ*(i:ℕ)/(n:ℝ)
theorem thresholdGrid_mem {n : ℕ} (hn : 0 < n) {ρ : ℝ} (hρ : 0 < ρ)
(a : ℝ) (i : Fin n) : thresholdGrid n a ρ i ∈ Icc a (a+ρ) := by
have hnR : (0:ℝ)<n := by exact_mod_cast hn
have hi : (i:ℝ)<n := by exact_mod_cast i.isLt
unfold thresholdGrid
constructor
· have : 0 ≤ ρ*(i:ℕ)/(n:ℝ) := by positivity
linarith only [this]
· have : ρ*(i:ℕ)/(n:ℝ) ≤ ρ := (div_le_iff₀ hnR).mpr (by gcongr)
linarith only [this]
theorem thresholdGrid_count {n : ℕ} (hn : 0 < n) {ρ w : ℝ} (hρ : 0 < ρ) (hw : 0 ≤ w)
(a x : ℝ) :
((Finset.univ.filter (fun i : Fin n => |thresholdGrid n a ρ i-x| ≤ w)).card:ℝ) ≤
2*w*(n:ℝ)/ρ+1 := by
let S := Finset.univ.filter (fun i : Fin n => |thresholdGrid n a ρ i-x| ≤ w)
have hnR : (0:ℝ)<n := by exact_mod_cast hn
by_cases hs : S.Nonempty
· let i := S.min' hs
let j := S.max' hs
have hi : i∈S := Finset.min'_mem S hs
have hj : j∈S := Finset.max'_mem S hs
have hij : i≤j := Finset.min'_le S j hj
have hcard : S.card ≤ (j:ℕ)+1-(i:ℕ) := by
rw [← Fin.card_Icc]
apply Finset.card_le_card
intro z hz
exact Finset.mem_Icc.mpr ⟨Finset.min'_le S z hz,Finset.le_max' S z hz⟩
have hcard' : (S.card:ℝ) ≤ (j:ℝ)-(i:ℝ)+1 := by
have hnat : (i:ℕ)≤(j:ℕ)+1 := by omega
have hc : (S.card:ℝ) ≤ (((j:ℕ)+1-(i:ℕ):ℕ):ℝ) := by exact_mod_cast hcard
rw [Nat.cast_sub hnat,Nat.cast_add,Nat.cast_one] at hc
linarith only [hc]
have hdi := abs_le.mp ((Finset.mem_filter.mp hi).2)
have hdj := abs_le.mp ((Finset.mem_filter.mp hj).2)
have hgap : ρ*((j:ℝ)-(i:ℝ)) ≤ 2*w*(n:ℝ) := by
have H : thresholdGrid n a ρ j-thresholdGrid n a ρ i ≤ 2*w := by linarith only [hdi.1,hdj.2]
have HH : thresholdGrid n a ρ j-thresholdGrid n a ρ i=ρ*((j:ℝ)-(i:ℝ))/(n:ℝ) := by
unfold thresholdGrid
ring
rw [HH,div_le_iff₀ hnR] at H
exact H
have hgap' : (j:ℝ)-(i:ℝ) ≤ 2*w*(n:ℝ)/ρ := (le_div_iff₀ hρ).mpr (by nlinarith only [hgap])
exact hcard'.trans (by linarith only [hgap'])
· have he : S=∅ := Finset.not_nonempty_iff_eq_empty.mp hs
change (S.card:ℝ)≤_
rw [he,Finset.card_empty,Nat.cast_zero]
positivity
theorem thresholdGrid_window {n : ℕ} (hn : 0 < n) {ρ w : ℝ} (hρ : 0 < ρ) (hw : 0 ≤ w)
(a x : ℝ) :
(FiniteLaw.uniformFin n hn).prob (fun i => |thresholdGrid n a ρ i-x| ≤ w) ≤ 2*w/ρ+1/(n:ℝ) := by
rw [FiniteLaw.uniformFin_prob]
have h := div_le_div_of_nonneg_right (thresholdGrid_count hn hρ hw a x) (by positivity : (0:ℝ)≤n)
refine h.trans_eq ?_
have hnR : (n:ℝ)≠0 := by exact_mod_cast hn.ne'
field_simp
theorem thresholdGrid_mass_expect {n : ℕ} (hn : 0 < n) (μ : ProbabilityMeasure ℝ)
{ρ h : ℝ} (hρ : 0 < ρ) (hh : 0 ≤ h) (a : ℝ) :
(FiniteLaw.uniformFin n hn).expect (fun i => (μ:Measure ℝ).real
(Icc (thresholdGrid n a ρ i-h) (thresholdGrid n a ρ i+h))) ≤ 2*h/ρ+1/(n:ℝ) := by
let f (i : Fin n) := (Icc (thresholdGrid n a ρ i-h) (thresholdGrid n a ρ i+h)).indicator (fun _ : ℝ => (1:ℝ))
have hi (i : Fin n) : Integrable (f i) (μ:Measure ℝ) :=
(integrable_const 1).indicator measurableSet_Icc
have he : (FiniteLaw.uniformFin n hn).expect (fun i => (μ:Measure ℝ).real
(Icc (thresholdGrid n a ρ i-h) (thresholdGrid n a ρ i+h))) =
∫ x, (1/(n:ℝ))*∑ i, f i x ∂(μ:Measure ℝ) := by
unfold FiniteLaw.expect FiniteLaw.uniformFin
rw [integral_const_mul,integral_finsetSum _ (fun i _ => hi i),Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
dsimp only [f]
rw [integral_indicator_const (1:ℝ) measurableSet_Icc]
simp only [smul_eq_mul,mul_one]
rw [he]
have hb (x : ℝ) : (1/(n:ℝ))*∑ i, f i x ≤ 2*h/ρ+1/(n:ℝ) := by
have H := thresholdGrid_window hn hρ hh a x
change (1/(n:ℝ))*(∑ i, f i x)≤_
convert H using 1
unfold FiniteLaw.prob FiniteLaw.expect FiniteLaw.uniformFin
rw [← Finset.mul_sum]
congr 1
apply Finset.sum_congr rfl
intro i _
have he : x∈Icc (thresholdGrid n a ρ i-h) (thresholdGrid n a ρ i+h) ↔ |thresholdGrid n a ρ i-x|≤h := by
rw [Set.mem_Icc,abs_le]
constructor <;> intro H <;> constructor <;> linarith only [H.1,H.2]
simp only [f,Set.indicator_apply,he]
exact (integral_mono_of_nonneg (ae_of_all _ (fun x => by
apply mul_nonneg (by positivity)
exact Finset.sum_nonneg (fun i _ => Set.indicator_nonneg (fun _ _ => zero_le_one) x)))
(integrable_const _) (ae_of_all _ hb)).trans_eq (by simp)
end SK.Analytic
end
end
end OAI
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