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f18e7de | 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 | 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
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