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e320e82 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import OAI.Combinatorics.EditDistance.HashCuts
namespace OAI
/-!
# Hash separation and finite histogram inequalities
-/
namespace EditCutHash
attribute [local instance] Classical.propDecidable
open scoped BigOperators
open Finset
section Histogram
variable {J K : Type*} [Fintype J] [Fintype K]
noncomputable def hit (a b : K) : ℝ := if a = b then 1 else 0
omit [Fintype K] in
lemma hit_nonneg (a b : K) : 0 ≤ hit a b := by unfold hit; split <;> norm_num
omit [Fintype K] in
lemma hit_le_one (a b : K) : hit a b ≤ 1 := by unfold hit; split <;> norm_num
omit [Fintype K] in
@[simp] lemma hit_self (a : K) : hit a a = 1 := by simp [hit]
noncomputable def count (f : J → K) (z : K) : ℝ := ∑ j, hit (f j) z
noncomputable def histogramDistance (f g : J → K) : ℝ := ∑ z, |count f z-count g z|
omit [Fintype K] in
lemma count_nonneg (f : J → K) (z : K) : 0 ≤ count f z := sum_nonneg (fun _ _ => hit_nonneg _ _)
lemma histogramDistance_nonneg (f g : J → K) : 0 ≤ histogramDistance f g :=
sum_nonneg (fun _ _ => abs_nonneg _)
lemma hit_sum (a : K) : ∑ z, hit a z = 1 := by
classical
simp [hit]
lemma count_sum (f : J → K) : ∑ z, count f z = Fintype.card J := by
unfold count
rw [sum_comm]
simp only [hit_sum,sum_const,card_univ,nsmul_eq_mul,mul_one]
lemma hit_distance (a b : K) : ∑ z, |hit a z-hit b z| = 2*sep a b := by
classical
by_cases hab : a = b
· subst b
simp [sep]
· have he (z : K) : |hit a z-hit b z| = hit a z+hit b z := by
by_cases haz : a = z
· subst z
simp [hit,Ne.symm hab]
· by_cases hbz : b = z
· subst z
simp [hit,hab]
· simp [hit,haz,hbz]
simp_rw [he]
rw [sum_add_distrib,hit_sum,hit_sum,sep_eq_one hab]
norm_num
lemma histogramDistance_le (f g : J → K) : histogramDistance f g ≤ 2*∑ j, sep (f j) (g j) := by
unfold histogramDistance count
simp_rw [←sum_sub_distrib]
calc
_ ≤ ∑ z : K, ∑ j : J, |hit (f j) z-hit (g j) z| :=
sum_le_sum (fun _ _ => abs_sum_le_sum_abs _ _)
_ = 2*∑ j, sep (f j) (g j) := by
rw [sum_comm]
simp_rw [hit_distance]
rw [mul_sum]
lemma histogramDistance_disjoint (f g : J → K) (h : ∀ i j, f i ≠ g j) :
histogramDistance f g = 2*(Fintype.card J : ℝ) := by
have hzero (z : K) : count f z = 0 ∨ count g z = 0 := by
by_cases hf : ∃ i, f i = z
· obtain ⟨i,rfl⟩ := hf
right
unfold count
apply sum_eq_zero
intro j _
exact ite_eq_right (Ne.symm (h i j))
· left
unfold count
apply sum_eq_zero
intro i _
exact ite_eq_right (fun he => hf ⟨i,he⟩)
have he (z : K) : |count f z-count g z| = count f z+count g z := by
rcases hzero z with hz|hz
· rw [hz,zero_sub,abs_neg,abs_of_nonneg (count_nonneg g z),zero_add]
· rw [hz,sub_zero,abs_of_nonneg (count_nonneg f z),add_zero]
unfold histogramDistance
simp_rw [he]
rw [sum_add_distrib,count_sum,count_sum]
ring
lemma histogramDistance_lower (f g : J → K) :
2*(Fintype.card J : ℝ)*(1-∑ i : J, ∑ j : J, hit (f i) (g j)) ≤
histogramDistance f g := by
classical
by_cases h : ∀ i j, f i ≠ g j
· rw [histogramDistance_disjoint f g h]
have hz : (∑ i : J, ∑ j : J, hit (f i) (g j)) = 0 := by
apply sum_eq_zero
intro i _
apply sum_eq_zero
intro j _
exact ite_eq_right (h i j)
rw [hz]
ring_nf
rfl
· push Not at h
obtain ⟨i,j,hij⟩ := h
have h1 : 1 ≤ ∑ k : J, hit (f i) (g k) := by
have hh := single_le_sum (fun k _ => hit_nonneg (f i) (g k)) (mem_univ j)
simpa only [hij,hit_self] using hh
have h2 : (∑ k : J, hit (f i) (g k)) ≤ ∑ l : J, ∑ k : J, hit (f l) (g k) :=
single_le_sum (fun l _ => sum_nonneg (fun k _ => hit_nonneg (f l) (g k))) (mem_univ i)
have hh : 1-∑ l : J, ∑ k : J, hit (f l) (g k) ≤ 0 := by linarith
exact (mul_nonpos_of_nonneg_of_nonpos (by positivity) hh).trans
(histogramDistance_nonneg f g)
end Histogram
end EditCutHash
end OAI
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