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a2ee19a | 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 | import OAI.Probability.RadialTransition.StarIntegral
namespace OAI
noncomputable section
open scoped BigOperators ENNReal NNReal
open MeasureTheory Set Filter
namespace RadialTransition
lemma cluster_rate_le {h α : ℝ} (h1 : h ≤ α) (h4 : 4*h-9 ≤ α)
{n : ℕ} (hn : n ≤ 3) : h*(n+1)-3*n ≤ α := by
have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg n
have hn3 : (n:ℝ) ≤ 3 := by exact_mod_cast hn
by_cases hh : h ≤ 3
· nlinarith [mul_nonneg hn0 (sub_nonneg.mpr hh)]
· nlinarith [mul_nonneg (sub_nonneg.mpr hn3) (sub_nonneg.mpr (le_of_not_ge hh))]
lemma core_cluster_coefficient_bound {t h α : ℝ} (ht : 0 ≤ t)
(h1 : h ≤ α) (h4 : 4*h-9 ≤ α) {n : ℕ} (hn : n ≤ 3) :
Real.exp (t*h*(n+1))/(n+1).factorial * Real.exp (-t*α) *
((n+1)*((2*((1+(1:ℝ)/1000)*microLength t))^3)^n) ≤ 3000 := by
have hl := (microLength_pos t).le
have hfac : (1:ℝ) ≤ (n+1).factorial := by exact_mod_cast (Nat.succ_le_of_lt (Nat.factorial_pos (n+1)))
have he : Real.exp (t*h*(n+1))*Real.exp (-t*α)*
(((2*((1+(1:ℝ)/1000)*microLength t))^3)^n) =
Real.exp (t*(h*(n+1)-3*n-α))*((2*(1+(1:ℝ)/1000))^3)^n := by
have hbase : 2*((1+(1:ℝ)/1000)*microLength t) = (2*(1+(1:ℝ)/1000))*microLength t := by ring
rw [hbase, mul_pow, mul_pow]
have hexp : (microLength t^3)^n = Real.exp ((n:ℝ)*(3*(-t))) := by
simp only [microLength, ← Real.exp_nat_mul, Nat.cast_ofNat]
rw [hexp]
calc
_ = (Real.exp (t*h*(n+1))*Real.exp (-t*α)*Real.exp ((n:ℝ)*(3*(-t))))*
((2*(1+(1:ℝ)/1000))^3)^n := by ring
_ = _ := by
rw [← Real.exp_add, ← Real.exp_add]
congr 2
ring
have hrate : Real.exp (t*(h*(n+1)-3*n-α)) ≤ 1 := by
rw [Real.exp_le_one_iff]
exact mul_nonpos_of_nonneg_of_nonpos ht (sub_nonpos.mpr (cluster_rate_le h1 h4 hn))
have hpow : (n+1:ℝ)*((2*(1+(1:ℝ)/1000))^3)^n ≤ 3000 := by
interval_cases n <;> norm_num
calc
_ ≤ Real.exp (t*h*(n+1))*Real.exp (-t*α)*
(((2*((1+(1:ℝ)/1000)*microLength t))^3)^n)*(n+1) := by
have hc : Real.exp (t*h*(n+1))/(n+1).factorial ≤ Real.exp (t*h*(n+1)) :=
div_le_self (Real.exp_pos _).le hfac
nlinarith [mul_le_mul_of_nonneg_right hc
(mul_nonneg (Real.exp_pos (-t*α)).le
(mul_nonneg (show (0:ℝ) ≤ n+1 by positivity) (by positivity : 0 ≤ ((2*((1+(1:ℝ)/1000)*microLength t))^3)^n)))]
_ = Real.exp (t*(h*(n+1)-3*n-α))*(((2*(1+(1:ℝ)/1000))^3)^n)*(n+1) := by rw [he]
_ ≤ _ := by
nlinarith [mul_le_mul_of_nonneg_right hrate
(mul_nonneg (by positivity : 0 ≤ ((2*(1+(1:ℝ)/1000))^3)^n) (show (0:ℝ) ≤ n+1 by positivity))]
lemma lintegral_cellClusterWeight_sector {t α r : ℝ} (hr : 0 ≤ r) (k : Cell)
{n : ℕ} (hn : n ≤ 3) :
(∫⁻ x : Fin (n+1) → Position, cellClusterWeight t α r k (configurationMultiset x)
∂Measure.pi (fun _ => volume)) =
ENNReal.ofReal (Real.exp (-t*α))*((n+1 : ℕ)*(volume (gridRegion k)*(ENNReal.ofReal (2*r)^3)^n)) := by
have hn4 : n+1 ≤ 4 := by omega
simp only [cellClusterWeight, emptyWeight, configurationMultiset_card, Fintype.card_fin,
Nat.add_eq_zero_iff, Nat.one_ne_zero, and_false, ite_false, ite_eq_left hn4, zero_add]
rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, lintegral_starWeight _ (measurableSet_gridRegion _) hr]
lemma particleBase_cellClusterWeight_le {t h α : ℝ} (ht : 0 ≤ t)
(h1 : h ≤ α) (h4 : 4*h-9 ≤ α) (k : Cell) :
(∫⁻ x, cellClusterWeight t α ((1+(1:ℝ)/1000)*microLength t) k (particleMultiset x)
∂particleBase volume (t*h)) ≤ 15001 := by
have hl := (microLength_pos t).le
have hm := measurable_particleObservable _ (measurable_cellClusterWeight t α ((1+(1:ℝ)/1000)*microLength t) k) 0
simp only [zero_add] at hm
rw [lintegral_particleBase _ _ hm]
have hz (n : ℕ) (hn : n ∉ Finset.range 5) :
ENNReal.ofReal (Real.exp (t*h*n)/n.factorial)*
(∫⁻ x : Fin n → Position, cellClusterWeight t α ((1+(1:ℝ)/1000)*microLength t) k (particleMultiset ⟨n,x⟩)
∂Measure.pi (fun _ => volume)) = 0 := by
have hn4 : ¬n ≤ 4 := by simpa only [Finset.mem_range, Nat.lt_succ_iff] using hn
have hn0 : n ≠ 0 := by omega
simp [particleMultiset, cellClusterWeight, emptyWeight, configurationMultiset_card, hn4, hn0]
rw [tsum_eq_sum hz]
have hterm (n : ℕ) (hn : n ∈ Finset.range 5) :
ENNReal.ofReal (Real.exp (t*h*n)/n.factorial)*
(∫⁻ x : Fin n → Position, cellClusterWeight t α ((1+(1:ℝ)/1000)*microLength t) k (particleMultiset ⟨n,x⟩)
∂Measure.pi (fun _ => volume)) ≤ 3000 := by
rcases n with _|n
· norm_num [particleMultiset, cellClusterWeight, emptyWeight, starWeight, configurationMultiset, Finset.univ_eq_empty]
have hn3 : n ≤ 3 := by have := Finset.mem_range.mp hn; omega
change ENNReal.ofReal _ * (∫⁻ x : Fin (n+1) → Position,
cellClusterWeight t α ((1+(1:ℝ)/1000)*microLength t) k (configurationMultiset x) ∂Measure.pi (fun _ => volume)) ≤ _
rw [lintegral_cellClusterWeight_sector (by positivity) k hn3]
have hv : volume (gridRegion k) ≤ (1:ℝ≥0∞) :=
(volume_gridRegion_le k).trans (by norm_num)
apply le_trans (mul_le_mul' le_rfl (mul_le_mul' le_rfl (mul_le_mul' le_rfl (mul_le_mul' hv le_rfl)) ))
simp only [one_mul]
rw [← ENNReal.ofReal_pow (by positivity : 0 ≤ 2*((1+(1:ℝ)/1000)*microLength t)),
← ENNReal.ofReal_pow (by positivity : 0 ≤ (2*((1+(1:ℝ)/1000)*microLength t))^3)]
have hnat : ((n+1:ℕ):ℝ≥0∞) = ENNReal.ofReal ((n+1:ℕ):ℝ) := by norm_cast
rw [hnat, ← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity), ← ENNReal.ofReal_mul (by positivity)]
have hb := ENNReal.ofReal_le_ofReal (core_cluster_coefficient_bound ht h1 h4 hn3)
norm_num only [ENNReal.ofReal_ofNat, Nat.cast_add, Nat.cast_one] at hb ⊢
convert hb using 1
congr 1
ring
calc
_ ≤ ∑ _n ∈ Finset.range 5, (3000:ℝ≥0∞) := Finset.sum_le_sum hterm
_ ≤ 15001 := by norm_num
end RadialTransition
end
end OAI
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