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9b6e014 | 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 | import OAI.Probability.ContinuumTransition.FreeFamily
namespace OAI
section
open scoped BigOperators
open MeasureTheory Set
namespace ContinuumTemperature
open Classical
lemma freeFamily_integral (t d L : ℝ) (ι : Type) [Fintype ι] :
familyIntegral (freeFamily t d) (cube L) ι =
volume.real ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ {x | FreeConfig t x})*
(Real.exp d)^Fintype.card ι := by
have he : freeFamily t d ι = {x : ι → Space | FreeConfig t x}.indicator
(fun _ => Real.exp (d*Fintype.card ι)) := by
funext x
simp only [freeFamily, indicator_apply, mem_ofPred_eq]
rw [familyIntegral, he, setIntegral_indicator (measurableSet_freeConfig t), setIntegral_const,
smul_eq_mul, mul_comm d, Real.exp_nat_mul]
lemma power_trunc_bound {w : ℝ} (hw : 0 ≤ w) {j k : ℕ} (hjk : j ≤ 3*k) :
w^j ≤ (max 1 w)^ (3*k) := by
exact (pow_le_pow_left₀ hw (le_max_right 1 w) j).trans
(pow_le_pow_right₀ (le_max_left 1 w) hjk)
theorem free_grand_upper {t : ℝ} (ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (d L : ℝ) :
grandIntegral (freeFamily t d) (cube L) ≤
(∑ k ∈ Finset.range (packingNumber L+1),
((volume.real (cube L))*Real.exp d*
(max 1 (volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t)))*Real.exp d))^3)^k /
(k.factorial : ℝ))*Real.exp (packingNumber L) := by
let V := volume.real (cube L)
let z := Real.exp d
let w := volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t)))
let M := packingNumber L
let D := max 1 (w*z)
let a : ℕ → ℝ := fun k => if k ≤ M then (V*z*D^3)^k/(k.factorial : ℝ) else 0
let b : ℕ → ℝ := fun j => (M : ℝ)^j/(j.factorial : ℝ)
have hV : 0 ≤ V := ENNReal.toReal_nonneg
have hz : 0 ≤ z := (Real.exp_pos _).le
have hw : 0 ≤ w := ENNReal.toReal_nonneg
have hD : 1 ≤ D := le_max_left _ _
have ha : Summable a := summable_of_ne_finset_zero (s := Finset.range (M+1)) (by
intro k hk
have hh : ¬k ≤ M := by simpa only [Finset.mem_range, Nat.lt_add_one_iff] using hk
simp only [a, ite_eq_right hh])
have hb : HasSum b (Real.exp (M : ℝ)) := by
simpa only [b, Real.exp_eq_exp_ℝ] using NormedSpace.expSeries_div_hasSum_exp (M : ℝ)
have hea : (∑' k, a k) = ∑ k ∈ Finset.range (M+1), (V*z*D^3)^k/(k.factorial : ℝ) := by
rw [tsum_eq_sum (s := Finset.range (M+1)) (by
intro k hk
have hh : ¬k ≤ M := by simpa only [Finset.mem_range, Nat.lt_add_one_iff] using hk
simp only [a, ite_eq_right hh])]
apply Finset.sum_congr rfl
intro k hk
exact ite_eq_left (by have := Finset.mem_range.mp hk; omega)
rw [show (∑ k ∈ Finset.range (packingNumber L+1), _) = ∑' k, a k by exact hea.symm,
← hb.tsum_eq]
apply tsum_le_convolution (freeFamily_summable t d L) ha hb.summable
intro n
have hv := freeConfig_volume_le (ι := Fin n) ht L
rw [Fintype.card_fin] at hv
have hh := mul_le_mul_of_nonneg_right hv (pow_nonneg hz n)
have hh' := div_le_div_of_nonneg_right hh (Nat.cast_nonneg n.factorial : (0 : ℝ) ≤ n.factorial)
rw [familyTerm, freeFamily_integral, Fintype.card_fin]
apply hh'.trans
rw [Finset.sum_mul, Finset.sum_div]
apply Finset.sum_le_sum
intro k hk
have hkn : k ≤ n := by have := Finset.mem_range.mp hk; omega
by_cases hc : k ≤ M ∧ n ≤ 4*k
· simp only [show packingNumber L = M from rfl, ite_eq_left hc, a, ite_eq_left hc.1, b]
rw [Nat.cast_choose ℝ hkn]
have hf : (n.factorial : ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero n
have he : (((n.factorial : ℝ)/((k.factorial : ℝ)*((n-k).factorial : ℝ)))*V^k*
((k : ℝ)*w)^(n-k)*z^n)/(n.factorial : ℝ) =
(V*z)^k/(k.factorial : ℝ)*((k : ℝ)^(n-k)*(w*z)^(n-k)/((n-k).factorial : ℝ)) := by
have hzpow : z^n = z^k*z^(n-k) := by rw [← pow_add, Nat.add_sub_of_le hkn]
rw [hzpow, mul_pow]
field_simp
ring
change (((n.factorial : ℝ)/((k.factorial : ℝ)*((n-k).factorial : ℝ)))*V^k*
((k : ℝ)*w)^(n-k)*z^n)/(n.factorial : ℝ) ≤ _
rw [he]
have hkM : (k : ℝ) ≤ M := by exact_mod_cast hc.1
have hp := power_trunc_bound (mul_nonneg hw hz) (show n-k ≤ 3*k by omega)
have hpow : (k : ℝ)^(n-k)*(w*z)^(n-k) ≤ (M : ℝ)^(n-k)*D^(3*k) :=
mul_le_mul (pow_le_pow_left₀ (Nat.cast_nonneg k) hkM _) hp
(pow_nonneg (mul_nonneg hw hz) _) (pow_nonneg (Nat.cast_nonneg M) _)
have hm := mul_le_mul_of_nonneg_left
(div_le_div_of_nonneg_right hpow (Nat.cast_nonneg (n-k).factorial))
(div_nonneg (pow_nonneg (mul_nonneg hV hz) k) (Nat.cast_nonneg k.factorial))
apply hm.trans_eq
rw [mul_pow, pow_mul]
ring
· simp only [show packingNumber L = M from rfl, ite_eq_right hc, zero_mul, zero_div]
apply mul_nonneg
· dsimp only [a]
split_ifs <;> positivity
· dsimp only [b]
positivity
end ContinuumTemperature
end
end OAI
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