| import Mathlib.LinearAlgebra.Projectivization.Cardinality |
| import OAI.Computability.UniqueGames.Quadratic.Block |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! Exact field-line count for the quadratic block's erasure probability. -/ |
|
|
| namespace UniqueGamesTheorem.Quadratic |
|
|
| / |
| vectors identified. -/ |
| abbrev FieldLine (F : Type*) [Field F] := Projectivization F (Vec F) |
|
|
| variable {F : Type*} [Field F] |
|
|
| / |
| noncomputable def lineGenerator (A : FieldLine F) : Vec F := A.rep |
|
|
| theorem lineGenerator_ne_zero (A : FieldLine F) : lineGenerator A ≠ 0 := |
| A.rep_nonzero |
|
|
| theorem line_lineGenerator (A : FieldLine F) : line (lineGenerator A) = A.submodule := |
| A.submodule_eq.symm |
|
|
| theorem fieldLine_finrank (A : FieldLine F) : Module.finrank F A.submodule = 1 := |
| A.finrank_submodule |
|
|
| variable [Finite F] |
|
|
| / |
| theorem card_FieldLine : Nat.card (FieldLine F) = Nat.card F ^ 2 + Nat.card F + 1 := by |
| have h := Projectivization.card_of_finrank F (Vec F) (n := 3) |
| (Module.finrank_fin_fun (R := F)) |
| simpa only [Finset.sum_range_succ, Finset.sum_range_zero, zero_add, |
| pow_zero, pow_one, add_comm, add_left_comm, add_assoc] using h |
|
|
| / |
| theorem card_fieldLine_submodules : |
| Nat.card {A : Submodule F (Vec F) // Module.finrank F A = 1} = |
| Nat.card F ^ 2 + Nat.card F + 1 := by |
| rw [← Nat.card_congr (Projectivization.equivSubmodule F (Vec F))] |
| exact card_FieldLine |
|
|
| theorem card_FieldLine_pos : 0 < Nat.card (FieldLine F) := by |
| rw [card_FieldLine] |
| exact Nat.succ_pos _ |
|
|
| / |
| theorem reciprocal_card_FieldLine : |
| (Nat.card (FieldLine F) : ℚ)⁻¹ = |
| ((Nat.card F : ℚ) ^ 2 + Nat.card F + 1)⁻¹ := by |
| rw [card_FieldLine] |
| simp only [Nat.cast_add, Nat.cast_pow, Nat.cast_one] |
|
|
| end UniqueGamesTheorem.Quadratic |
|
|
| end |
|
|
| end OAI |
|
|
