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2c7d975 | 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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | import Mathlib.LinearAlgebra.Prod
import Mathlib.Tactic.Module
import Mathlib.Algebra.CharP.Two
namespace OAI
namespace ArExplicit
variable {k V : Type*} [Field k] [AddCommGroup V] [Module k V]
def twoActionMap (a b : k) : (V × V) →ₗ[k] (V × V) :=
(a • LinearMap.fst k V V + LinearMap.snd k V V).prod
(b • LinearMap.fst k V V + LinearMap.snd k V V)
@[simp] theorem twoActionMap_apply (a b : k) (x y : V) :
twoActionMap a b (x, y) = (a • x + y, b • x + y) := rfl
private def twoActionInverse (a b : k) (p : V × V) : V × V :=
let g := (a - b)⁻¹ • (p.1 - p.2)
(g, p.1 - a • g)
private theorem twoActionInverse_left (a b : k) (hab : a ≠ b) (p : V × V) :
twoActionInverse a b (twoActionMap a b p) = p := by
rcases p with ⟨x, y⟩
have hd : a - b ≠ 0 := sub_ne_zero.mpr hab
have hsub : (a • x + y) - (b • x + y) = (a - b) • x := by module
have hx : (a - b)⁻¹ • ((a • x + y) - (b • x + y)) = x := by
rw [hsub, smul_smul, inv_mul_cancel₀ hd, one_smul]
change (_, _) = (x, y)
simp only [twoActionMap_apply, hx]
congr 1
module
private theorem twoActionInverse_right (a b : k) (hab : a ≠ b) (p : V × V) :
twoActionMap a b (twoActionInverse a b p) = p := by
rcases p with ⟨u, v⟩
have hd : a - b ≠ 0 := sub_ne_zero.mpr hab
let g : V := (a - b)⁻¹ • (u - v)
have hg : (a - b) • g = u - v := by
dsimp [g]
rw [smul_smul, mul_inv_cancel₀ hd, one_smul]
change (a • g + (u - a • g), b • g + (u - a • g)) = (u, v)
apply Prod.ext
· module
· calc
b • g + (u - a • g) = u - (a - b) • g := by module
_ = v := by rw [hg]; module
def twoActionEquiv (a b : k) (hab : a ≠ b) : (V × V) ≃ₗ[k] (V × V) where
toLinearMap := twoActionMap a b
invFun := twoActionInverse a b
left_inv := twoActionInverse_left a b hab
right_inv := twoActionInverse_right a b hab
theorem twoActionMap_bijective (a b : k) (hab : a ≠ b) :
Function.Bijective (twoActionMap a b : (V × V) → V × V) :=
(twoActionEquiv a b hab).bijective
def degreeZeroMap : (V × V) →ₗ[k] V :=
LinearMap.fst k V V + LinearMap.snd k V V
theorem degreeZeroMap_surjective :
Function.Surjective (degreeZeroMap (k := k) : (V × V) → V) := by
intro v
exact ⟨(v, 0), add_zero v⟩
theorem degreeZeroMap_kernel [CharP k 2] (x y : V) :
degreeZeroMap (k := k) (x, y) = 0 ↔ x = y := by
change x + y = 0 ↔ x = y
rw [add_eq_zero_iff_eq_neg]
have hy : -y = y := by
have htwo : (2 : k) = 0 := CharP.cast_eq_zero k 2
have : y + y = 0 := by rw [← two_smul k y, htwo, zero_smul]
exact neg_eq_iff_add_eq_zero.mpr this
rw [hy]
theorem twoActionMap_same_not_injective [Nontrivial V] (a : k) :
¬ Function.Injective (twoActionMap a a : (V × V) → V × V) := by
obtain ⟨x, hx⟩ := exists_ne (0 : V)
intro hinj
have h : twoActionMap a a (x, -(a • x)) = twoActionMap a a (0, 0) := by
simp
have he := congrArg Prod.fst (hinj h)
exact hx he
end ArExplicit
end OAI
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