| import OAI.LinearAlgebra.MatrixMultiplication.Recovery.OrbitCounting |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Tactic.Linarith |
|
|
| namespace OAI |
|
|
| /-! Finite orbit symmetries, masks and exact recovery operations. -/ |
|
|
| namespace MatrixMultiplication.OrbitLoss |
|
|
| variable {X : Type*} [DecidableEq X] |
|
|
| theorem missing_card_partition (s : Finset X) (pass assigned : X → Prop) |
| [DecidablePred pass] [DecidablePred assigned] : |
| (s.filter fun x => ¬(pass x ∧ assigned x)).card = |
| (s.filter fun x => ¬pass x).card + |
| (s.filter fun x => pass x ∧ ¬assigned x).card := by |
| classical |
| have heq : (s.filter fun x => ¬(pass x ∧ assigned x)) = |
| (s.filter fun x => ¬pass x) ∪ (s.filter fun x => pass x ∧ ¬assigned x) := by |
| ext x |
| by_cases hp : pass x <;> simp [hp] |
| have hd : Disjoint (s.filter fun x => ¬pass x) |
| (s.filter fun x => pass x ∧ ¬assigned x) := by |
| apply Finset.disjoint_left.mpr |
| intro x hx hy |
| exact (Finset.mem_filter.mp hx).2 (Finset.mem_filter.mp hy).2.1 |
| rw [heq, Finset.card_union_of_disjoint hd] |
|
|
| theorem missing_fraction_partition (s : Finset X) (pass assigned : X → Prop) |
| [DecidablePred pass] [DecidablePred assigned] : |
| ((s.filter fun x => ¬(pass x ∧ assigned x)).card : ℝ) / s.card = |
| ((s.filter fun x => ¬pass x).card : ℝ) / s.card + |
| ((s.filter fun x => pass x ∧ ¬assigned x).card : ℝ) / s.card := by |
| rw [missing_card_partition, Nat.cast_add, add_div] |
|
|
| theorem combined_inverse_linear (s : Finset X) (pass assigned : X → Prop) |
| [DecidablePred pass] [DecidablePred assigned] (C : ℝ) (N : ℕ) |
| (hmask : ((s.filter fun x => ¬pass x).card : ℝ) / s.card ≤ C / N) |
| (hcollision : ((s.filter fun x => pass x ∧ ¬assigned x).card : ℝ) / s.card ≤ 1 / N) : |
| ((s.filter fun x => ¬(pass x ∧ assigned x)).card : ℝ) / s.card ≤ (C + 1) / N := by |
| rw [missing_fraction_partition, add_div] |
| exact add_le_add hmask hcollision |
|
|
| end MatrixMultiplication.OrbitLoss |
|
|
| end OAI |
|
|
