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92416a2 | 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 | 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
/-- Actual one-dimensional field directions, with proportional nonzero
vectors identified. -/
abbrev FieldLine (F : Type*) [Field F] := Projectivization F (Vec F)
variable {F : Type*} [Field F]
/-- A chosen nonzero generator, only used with generator-independent data. -/
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]
/-- The denominator of the block erasure probability is exactly `q²+q+1`. -/
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
/-- The same count on one-dimensional submodules, rather than representatives. -/
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 _
/-- Written over rationals, as needed for the finite rational noise law. -/
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
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