| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| open MeasureTheory ProbabilityTheory Filter Real | |
| namespace SKRatio.Bins | |
| variable {ι α : Type*} [Fintype ι] [Fintype α] [DecidableEq α] | |
| attribute [local instance] Classical.propDecidable | |
| def sum (σ : ι → α) (f : ι → ℝ) (a : α) : ℝ := | |
| ∑ i with σ i = a, f i | |
| def overlap (σ : ι → α) (u z : ι → ℝ) (a : α) : ℝ := | |
| sum σ (fun i => u i*z i) a | |
| def ZeroSum (σ : ι → α) (u : ι → ℝ) : Prop := ∀ a, sum σ u a = 0 | |
| lemma sum_fibers (σ : ι → α) (f : ι → ℝ) : ∑ a, sum σ f a = ∑ i, f i := by | |
| classical | |
| simpa only [sum,Finset.mem_univ,true_and] using | |
| (Finset.sum_fiberwise_of_maps_to (s := Finset.univ) (t := Finset.univ) (g := σ) | |
| (fun _ _ => Finset.mem_univ _) f) | |
| lemma sum_weighted (σ : ι → α) (f : ι → ℝ) (c : α → ℝ) : | |
| ∑ i, f i*c (σ i) = ∑ a, sum σ f a*c a := by | |
| rw [←sum_fibers σ (fun i => f i*c (σ i))] | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| unfold sum | |
| rw [Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| change f i*c (σ i)=f i*c a | |
| rw [(Finset.mem_filter.mp hi).2] | |
| lemma zero_sum_weighted {σ : ι → α} {u : ι → ℝ} (hu : ZeroSum σ u) (c : α → ℝ) : | |
| (∑ i, u i*c (σ i)) = 0 := by | |
| rw [sum_weighted] | |
| simp only [show ∀ a, sum σ u a=0 from hu,zero_mul,Finset.sum_const_zero] | |
| lemma zero_sum_bilinear {σ : ι → α} {u : ι → ℝ} (hu : ZeroSum σ u) | |
| (z : ι → ℝ) (K : α → α → ℝ) : | |
| (∑ i, ∑ j, u i*z j*K (σ i) (σ j)) = 0 := by | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_eq_zero | |
| intro j _ | |
| simpa only [mul_assoc] using zero_sum_weighted hu (fun a => z j*K a (σ j)) | |
| omit [Fintype α] in | |
| lemma overlap_symm (σ : ι → α) (u z : ι → ℝ) : overlap σ u z = overlap σ z u := by | |
| funext a | |
| simp only [overlap,mul_comm] | |
| omit [Fintype α] in | |
| lemma overlap_self_nonneg (σ : ι → α) (u : ι → ℝ) (a : α) : 0 ≤ overlap σ u u a := by | |
| exact Finset.sum_nonneg (fun i _ => mul_self_nonneg (u i)) | |
| omit [Fintype α] in | |
| lemma overlap_le {σ : ι → α} {u z : ι → ℝ} {s : α → ℝ} | |
| (hu : overlap σ u u = s) (hz : overlap σ z z = s) (a : α) : | |
| overlap σ u z a ≤ s a := by | |
| have h : 0 ≤ ∑ i with σ i=a, (u i-z i)^2 := Finset.sum_nonneg (fun _ _ => sq_nonneg _) | |
| have he : (∑ i with σ i=a, (u i-z i)^2) = | |
| overlap σ u u a+overlap σ z z a-2*overlap σ u z a := by | |
| unfold overlap sum | |
| simp only [Finset.mul_sum] | |
| rw [←Finset.sum_add_distrib,←Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| ring | |
| rw [he,hu,hz] at h | |
| linarith | |
| omit [Fintype α] in | |
| lemma overlap_abs_le {σ : ι → α} {u z : ι → ℝ} {s : α → ℝ} | |
| (hu : overlap σ u u = s) (hz : overlap σ z z = s) (a : α) : | |
| |overlap σ u z a| ≤ s a := by | |
| have hz' : overlap σ (fun i => -z i) (fun i => -z i) = s := by | |
| calc | |
| _ = overlap σ z z := by ext a; simp only [overlap,neg_mul_neg] | |
| _ = s := hz | |
| have hh := overlap_le hu hz' a | |
| have he : overlap σ u (fun i => -z i) a = -overlap σ u z a := by | |
| simp [overlap,sum] | |
| rw [he] at hh | |
| exact abs_le.mpr ⟨by linarith,overlap_le hu hz a⟩ | |
| omit [Fintype α] [DecidableEq α] in | |
| lemma overlap_product_bound {s q : α → ℝ} (hq : ∀ a, q a ≤ s a) (a d : α) : | |
| s a*s d-q a*q d ≤ s a*(s d-q d)+s d*(s a-q a) := by | |
| nlinarith only [mul_nonneg (sub_nonneg.mpr (hq a)) (sub_nonneg.mpr (hq d))] | |
| lemma double_weighted (σ : ι → α) (f g : ι → ℝ) (K : α → α → ℝ) : | |
| (∑ i, ∑ j, f i*g j*K (σ i) (σ j)) = | |
| ∑ a, ∑ d, sum σ f a*sum σ g d*K a d := by | |
| simp_rw [mul_assoc,←Finset.mul_sum,sum_weighted σ g] | |
| rw [sum_weighted σ f (fun a => ∑ d, sum σ g d*K a d)] | |
| lemma sum_overlap (σ : ι → α) (u z : ι → ℝ) : | |
| (∑ a, overlap σ u z a) = ∑ i, u i*z i := sum_fibers _ _ | |
| end SKRatio.Bins | |
| end | |
| end OAI | |