| 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 | |