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ea76aac | 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 | import Mathlib
import OAI.Computability.MinUncut.Analysis.GaussianDensityError
namespace OAI
section
open scoped BigOperators
namespace MinUncut.FiniteGaussian
attribute [local instance] Classical.propDecidable
lemma finite_product_test_bound_general {ι α : Type*} [Fintype ι] [Fintype α] (q g : α → ℝ)
(hg : ∀ a, 0 ≤ g a) (hqg : ∀ a, g a ≤ q a)
(hG : 1 ≤ ∑ a, g a) (F : (ι → α) → ℝ)
(hF : ∀ x, 0 ≤ F x ∧ F x ≤ 1) :
|(∑ x, (∏ i, q (x i))*F x)/(∑ a, q a)^(Fintype.card ι)-
(∑ x, (∏ i, g (x i))*F x)/(∑ a, g a)^(Fintype.card ι)| ≤
(Fintype.card ι:ℝ)*((∑ a, q a)-(∑ a, g a)) := by
classical
have hg0 : 0 < ∑ a, g a := by linarith
have hq0 (a : α) : 0 ≤ q a := (hg a).trans (hqg a)
have hgprod (x : ι → α) : 0 ≤ ∏ i, g (x i) :=
Finset.prod_nonneg (fun i _ => hg (x i))
have hprod (x : ι → α) : (∏ i, g (x i)) ≤ ∏ i, q (x i) :=
Finset.prod_le_prod₀ (fun index _ => hg (x index)) (fun index _ => hqg (x index))
have hsum (w : α → ℝ) : (∑ x : ι → α, ∏ i, w (x i))=(∑ a, w a)^(Fintype.card ι) :=
by simpa only [Finset.prod_const,Finset.card_univ] using (Fintype.prod_sum (fun _ : ι => w)).symm
have htotal : 0 < ∑ x : ι → α, ∏ i, g (x i) := by
rw [hsum]; exact pow_pos hg0 (Fintype.card ι)
have hh := finite_normalized_test_bound (fun x : ι → α => ∏ i, q (x i))
(fun x => ∏ i, g (x i)) F hgprod hprod htotal hF
rw [hsum,hsum] at hh
exact hh.trans (product_residual_bound hG (Finset.sum_le_sum (fun a _ => hqg a)) (Fintype.card ι))
end MinUncut.FiniteGaussian
end
section
open scoped BigOperators
namespace MinUncut.FiniteGaussian
attribute [local instance] Classical.propDecidable
lemma grid_product_bound_general {ι : Type*} [Fintype ι] {T : ℚ} (hT : 1 ≤ T) {L : ℕ} (hL : 0 < L)
(r : ℕ) (F : (ι → Fin L) → ℝ) (hF : ∀ x, 0 ≤ F x ∧ F x ≤ 1) :
|(∑ x, (∏ i, qCell r T L (x i))*F x)/(qIntegral r (-T) T:ℝ)^(Fintype.card ι)-
(∑ x, (∏ i, gCell T L (x i))*F x)/(gIntegral (-T) T)^(Fintype.card ι)| ≤
2*(Fintype.card ι)*(T:ℝ)*error r ((T:ℝ)^2/2) := by
have hT0 : 0 < T := by linarith
have hT' : (1:ℝ) ≤ T := by exact_mod_cast hT
have hb := finite_product_test_bound_general (qCell r T L) (gCell T L)
(gCell_nonneg hT0 hL) (gCell_le_qCell hT0 hL r)
(by rw [sum_gCell T hL]; exact (gIntegral_large hT').le) F hF
rw [sum_qCell r T hL,sum_gCell T hL] at hb
exact hb.trans (by
have hh := mul_le_mul_of_nonneg_left (residual_bounds hT0.le r).2 (Nat.cast_nonneg (α := ℝ) (Fintype.card ι))
nlinarith)
end MinUncut.FiniteGaussian
end
end OAI
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