| 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 |
|
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| end OAI |
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