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a72760f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 | import Mathlib
namespace OAI
open scoped BigOperators
namespace PiExponent.Collision
theorem choose_card_le_sum (s : Finset ℕ) :
s.card.choose 2 ≤ ∑ d ∈ s, d := by
induction s using Finset.induction_on_max with
| empty => simp
| insert a s hmax ih =>
have ha : a ∉ s := by
intro h
exact (lt_irrefl a) (hmax a h)
have hsub : s ⊆ Finset.range a := by
intro x hx
exact Finset.mem_range.mpr (hmax x hx)
have hcard : s.card ≤ a := by
simpa using Finset.card_le_card hsub
rw [Finset.card_insert_of_notMem ha, Finset.sum_insert ha]
rw [show s.card + 1 = s.card.succ from rfl,
Nat.choose_succ_succ s.card 1, Nat.choose_one_right]
exact Nat.add_le_add hcard ih
theorem choose_card_le_sum_of_injOn {ι : Type*} (s : Finset ι)
(degree : ι → ℕ) (hinj : Set.InjOn degree (s : Set ι)) :
s.card.choose 2 ≤ ∑ i ∈ s, degree i := by
have h := choose_card_le_sum (s.image degree)
rw [Finset.card_image_of_injOn hinj, Finset.sum_image hinj] at h
exact h
def multiplicity {ι κ : Type*} [DecidableEq κ]
(rows : Finset ι) (group : ι → κ) (a : κ) : ℕ :=
(rows.filter fun i => group i = a).card
theorem sum_multiplicity {ι κ : Type*} [DecidableEq κ]
(rows : Finset ι) (groups : Finset κ) (group : ι → κ)
(hgroups : ∀ i ∈ rows, group i ∈ groups) :
∑ a ∈ groups, multiplicity rows group a = rows.card := by
simpa only [multiplicity, Finset.sum_const, smul_eq_mul, mul_one]
using Finset.sum_fiberwise_of_maps_to hgroups (fun _ => (1 : ℕ))
theorem sum_choose_multiplicity_le_sum_degree {ι κ : Type*} [DecidableEq κ]
(rows : Finset ι) (groups : Finset κ) (group : ι → κ) (degree : ι → ℕ)
(hgroups : ∀ i ∈ rows, group i ∈ groups)
(hinj : Set.InjOn (fun i => (group i, degree i)) (rows : Set ι)) :
(∑ a ∈ groups, (multiplicity rows group a).choose 2) ≤
∑ i ∈ rows, degree i := by
calc
(∑ a ∈ groups, (multiplicity rows group a).choose 2) ≤
∑ a ∈ groups, ∑ i ∈ rows.filter (fun i => group i = a), degree i := by
apply Finset.sum_le_sum
intro a _
apply choose_card_le_sum_of_injOn
intro i hi j hj hdegree
have hi' := Finset.mem_filter.mp hi
have hj' := Finset.mem_filter.mp hj
apply hinj hi'.1 hj'.1
exact Prod.ext (hi'.2.trans hj'.2.symm) hdegree
_ = ∑ i ∈ rows, degree i :=
Finset.sum_fiberwise_of_maps_to hgroups degree
theorem sum_choose_multiplicity_le_sum_degree_univ
{ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq κ]
(group : ι → κ) (degree : ι → ℕ)
(hinj : Function.Injective (fun i => (group i, degree i))) :
(∑ a, (multiplicity Finset.univ group a).choose 2) ≤ ∑ i, degree i := by
exact sum_choose_multiplicity_le_sum_degree Finset.univ Finset.univ group degree
(by simp) hinj.injOn
theorem sum_multiplicity_univ
{ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq κ]
(group : ι → κ) :
∑ a, multiplicity Finset.univ group a = Fintype.card ι := by
exact sum_multiplicity Finset.univ Finset.univ group (by simp)
theorem sum_sq_multiplicity_sub_card_le_twice_sum_degree
{ι κ : Type*} [DecidableEq κ]
(rows : Finset ι) (groups : Finset κ) (group : ι → κ) (degree : ι → ℕ)
(hgroups : ∀ i ∈ rows, group i ∈ groups)
(hinj : Set.InjOn (fun i => (group i, degree i)) (rows : Set ι)) :
(∑ a ∈ groups, (multiplicity rows group a : ℝ) ^ 2) - rows.card ≤
2 * ∑ i ∈ rows, (degree i : ℝ) := by
have hnat := sum_choose_multiplicity_le_sum_degree rows groups group degree hgroups hinj
have hreal : (∑ a ∈ groups, ((multiplicity rows group a).choose 2 : ℝ)) ≤
∑ i ∈ rows, (degree i : ℝ) := by exact_mod_cast hnat
have hsum : (∑ a ∈ groups, (multiplicity rows group a : ℝ)) = rows.card := by
exact_mod_cast sum_multiplicity rows groups group hgroups
have hid : 2 * (∑ a ∈ groups, ((multiplicity rows group a).choose 2 : ℝ)) =
(∑ a ∈ groups, (multiplicity rows group a : ℝ) ^ 2) - rows.card := by
calc
_ = ∑ a ∈ groups, ((multiplicity rows group a : ℝ) ^ 2 -
(multiplicity rows group a : ℝ)) := by
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
rw [Nat.cast_choose_two]
ring
_ = _ := by rw [Finset.sum_sub_distrib, hsum]
linarith
theorem sum_sq_multiplicity_sub_card_le_twice_sum_degree_univ
{ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq κ]
(group : ι → κ) (degree : ι → ℕ)
(hinj : Function.Injective (fun i => (group i, degree i))) :
(∑ a, (multiplicity Finset.univ group a : ℝ) ^ 2) - Fintype.card ι ≤
2 * ∑ i, (degree i : ℝ) := by
exact sum_sq_multiplicity_sub_card_le_twice_sum_degree Finset.univ Finset.univ
group degree (by simp) hinj.injOn
theorem sum_multiplicity_eq_card_filter {ι κ : Type*} [DecidableEq κ]
(rows : Finset ι) (groups : Finset κ) (group : ι → κ) :
∑ a ∈ groups, multiplicity rows group a =
(rows.filter fun i => group i ∈ groups).card := by
exact Finset.sum_card_fiberwise_eq_card_filter rows groups group
theorem cutoff_mul_high_count_le_sum {ι : Type*}
(rows : Finset ι) (weight : ι → ℝ) (cutoff : ℝ)
(hnonneg : ∀ i ∈ rows, 0 ≤ weight i) :
cutoff * ((rows.filter fun i => ¬ weight i ≤ cutoff).card : ℝ) ≤
∑ i ∈ rows, weight i := by
classical
calc
_ = ∑ i ∈ rows.filter (fun i => ¬ weight i ≤ cutoff), cutoff := by
simp [mul_comm]
_ ≤ ∑ i ∈ rows.filter (fun i => ¬ weight i ≤ cutoff), weight i := by
apply Finset.sum_le_sum
intro i hi
exact le_of_lt (lt_of_not_ge (Finset.mem_filter.mp hi).2)
_ ≤ ∑ i ∈ rows, weight i := by
apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ rows)
intro i hi _
exact hnonneg i hi
theorem sum_weight_ge_of_low_count_le {ι : Type*}
(rows : Finset ι) (weight : ι → ℝ) {cutoff η : ℝ}
(hcutoff : 0 ≤ cutoff) (hnonneg : ∀ i ∈ rows, 0 ≤ weight i)
(hlow : ((rows.filter fun i => weight i ≤ cutoff).card : ℝ) ≤
η * rows.card) :
cutoff * (1 - η) * rows.card ≤ ∑ i ∈ rows, weight i := by
classical
have hcard : ((rows.filter fun i => weight i ≤ cutoff).card : ℝ) +
((rows.filter fun i => ¬ weight i ≤ cutoff).card : ℝ) = rows.card := by
exact_mod_cast (Finset.card_filter_add_card_filter_not (s := rows)
(fun i => weight i ≤ cutoff))
have hhigh : (1 - η) * rows.card ≤
((rows.filter fun i => ¬ weight i ≤ cutoff).card : ℝ) := by nlinarith
calc
_ = cutoff * ((1 - η) * rows.card) := by ring
_ ≤ cutoff * ((rows.filter fun i => ¬ weight i ≤ cutoff).card : ℝ) :=
mul_le_mul_of_nonneg_left hhigh hcutoff
_ ≤ ∑ i ∈ rows, weight i := cutoff_mul_high_count_le_sum rows weight cutoff hnonneg
end PiExponent.Collision
end OAI
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