| import Mathlib.Algebra.Field.ZMod |
| import Mathlib.Data.Fintype.Prod |
| import Mathlib.Basic.Real.Basic |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.LinearCombination |
|
|
| namespace OAI |
|
|
| /-! # Exact line counts in the ideal giant residues -/ |
|
|
| namespace Ostmann |
|
|
| open scoped BigOperators |
|
|
| / |
| def unitLineEquiv {G : Type*} [CommGroup G] (a b : G) : |
| {xy : G × G // a * xy.1 = b * xy.2} ≃ G where |
| toFun xy := xy.1.2 |
| invFun y := ⟨(a⁻¹ * b * y, y), by simp [mul_assoc]⟩ |
| left_inv xy := by |
| apply Subtype.ext |
| apply Prod.ext |
| · change a⁻¹ * b * xy.1.2 = xy.1.1 |
| rw [mul_assoc, ← xy.property] |
| simp |
| · rfl |
| right_inv _ := rfl |
|
|
| theorem card_unitLine {G : Type*} [CommGroup G] [Fintype G] [DecidableEq G] (a b : G) : |
| Fintype.card {xy : G × G // a * xy.1 = b * xy.2} = Fintype.card G := |
| Fintype.card_congr (unitLineEquiv a b) |
|
|
| / |
| noncomputable def externalLineEquiv {p : ℕ} [Fact p.Prime] |
| (a : (ZMod p)ˣ) (b : ZMod p) : |
| {xy : ZMod p × (ZMod p)ˣ // (a : ZMod p) * xy.1 = b * xy.2} ≃ (ZMod p)ˣ where |
| toFun xy := xy.1.2 |
| invFun y := ⟨(b / (a : ZMod p) * y, y), by field_simp⟩ |
| left_inv xy := by |
| apply Subtype.ext |
| apply Prod.ext |
| · change b / (a : ZMod p) * xy.1.2 = xy.1.1 |
| apply (mul_left_cancel₀ (Units.ne_zero a)) |
| calc |
| (a : ZMod p) * (b / (a : ZMod p) * xy.1.2) = b * xy.1.2 := by field_simp |
| _ = _ := xy.property.symm |
| · rfl |
| right_inv _ := rfl |
|
|
| theorem card_externalLine {p : ℕ} [Fact p.Prime] |
| (a : (ZMod p)ˣ) (b : ZMod p) : |
| Fintype.card {xy : ZMod p × (ZMod p)ˣ // (a : ZMod p) * xy.1 = b * xy.2} = |
| Fintype.card (ZMod p)ˣ := |
| Fintype.card_congr (externalLineEquiv a b) |
|
|
| theorem unitLine_probability {p : ℕ} [Fact p.Prime] (a b : (ZMod p)ˣ) : |
| (Fintype.card {xy : (ZMod p)ˣ × (ZMod p)ˣ // a * xy.1 = b * xy.2} : ℝ) / |
| (Fintype.card ((ZMod p)ˣ × (ZMod p)ˣ) : ℝ) = |
| (Fintype.card (ZMod p)ˣ : ℝ)⁻¹ := by |
| rw [card_unitLine, Fintype.card_prod, Nat.cast_mul] |
| have hU : (Fintype.card (ZMod p)ˣ : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| field_simp |
|
|
| theorem externalLine_probability {p : ℕ} [Fact p.Prime] |
| (a : (ZMod p)ˣ) (b : ZMod p) : |
| (Fintype.card {xy : ZMod p × (ZMod p)ˣ // (a : ZMod p) * xy.1 = b * xy.2} : ℝ) / |
| (Fintype.card (ZMod p × (ZMod p)ˣ) : ℝ) = (p : ℝ)⁻¹ := by |
| rw [card_externalLine, Fintype.card_prod, ZMod.card, Nat.cast_mul] |
| have hU : (Fintype.card (ZMod p)ˣ : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero |
| field_simp |
|
|
| theorem independent_lines_no_unit_solution {K : Type*} [Field K] |
| (a b c d x y : K) (hdet : a * d - b * c ≠ 0) (hy : y ≠ 0) |
| (h₁ : a * x + b * y = 0) (h₂ : c * x + d * y = 0) : False := by |
| have he : (a * d - b * c) * y = 0 := by |
| linear_combination a * h₂ - c * h₁ |
| exact hy ((mul_eq_zero.mp he).resolve_left hdet) |
|
|
| theorem line_zero_first_coefficient {K : Type*} [Field K] |
| (b x y : K) (hb : b ≠ 0) (hy : y ≠ 0) : ¬ (0 : K) * x + b * y = 0 := by |
| simpa only [zero_mul, zero_add] using mul_ne_zero hb hy |
|
|
| / |
| theorem dependent_lines_iff {K I : Type*} [Field K] |
| (a b : I → K) (i : I) (ha : a i ≠ 0) |
| (hdet : ∀ j, a i * b j - b i * a j = 0) (x y : K) : |
| (∀ j, a j * x + b j * y = 0) ↔ a i * x + b i * y = 0 := by |
| constructor |
| · exact fun h => h i |
| · intro h j |
| have he : a i * (a j * x + b j * y) = 0 := by |
| linear_combination a j * h + y * hdet j |
| exact (mul_eq_zero.mp he).resolve_left ha |
|
|
| end Ostmann |
|
|
| end OAI |
|
|
