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7d9f027 | 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 | 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
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