DibyashanuPati/PhysLean
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor: Lint style

Browse source
This commit is contained in:
kuotsanhsu 2025-01-11 20:09:29 +08:00
parent 1053ccaa3a
commit 6a3bb431bf
4 changed files with 73 additions and 29 deletions
Download patch file Download diff file Expand all files Collapse all files

27
HepLean/Lorentz/SL2C/SelfAdjoint.lean
View file

@ -6,11 +6,17 @@ namespace Lorentz
open scoped Matrix
open scoped ComplexConjugate
notation "ℂ²ˣ²" => Matrix (Fin 2) (Fin 2) ℂ
/-- A notation for the type of complex 2-by-2 matrices. It would have been better to make it an
abbreviation if it wasn't for Lean's inability to recognize `ℂ²ˣ²` as an identifier. -/
scoped notation "ℂ²ˣ²" => Matrix (Fin 2) (Fin 2) ℂ
/-- A convenient abbreviation for the type of self-adjoint complex 2-by-2 matrices. -/
noncomputable abbrev ℍ₂ := selfAdjoint ℂ²ˣ²
namespace SL2C
/-- Definitially equal to `Lorentz.SL2C.toSelfAdjointMap` but dropping the requirement that `M` be
special linear. -/
noncomputable def toSelfAdjointMap' (M : ℂ²ˣ²) : ℍ₂ →ₗ[ℝ] ℍ₂ where
toFun | ⟨A, hA⟩ => ⟨M * A * Mᴴ, hA.conjugate M⟩
map_add' | ⟨A, _⟩, ⟨B, _⟩ => Subtype.ext <|
@ -47,12 +53,12 @@ theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (de
Matrix.ext fun | 0, 0 | 1, 0 | 1, 1 => rfl | 0, 1 => by simp [hM10]
have detA_one : normSq x * normSq y = 1 := congrArg Complex.re <|
calc ↑(normSq x * normSq y)
calc ↑(normSq x * normSq y)
_ = x * conj x * (y * conj y) := by simp [Complex.mul_conj]
_ = x * y * (conj y * conj x) := by ring
_ = x * y * conj (x * y) := congrArg _ (star_mul ..).symm
_ = 1 := suffices x * y = 1 by simp [this]
calc x * y
calc x * y
_ = !![x, _; 0, y].det := by simp
_ = M.det := congrArg _ he.symm
_ = 1 := detM
@ -67,7 +73,7 @@ theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (de
have hD : D = !![z.re, -z.im; z.im, z.re] := Matrix.ext fun
| 0, 0 => congrArg Complex.re k₀ | 1, 0 => congrArg Complex.im k₀
| 0, 1 => congrArg Complex.re k₁ | 1, 1 => congrArg Complex.im k₁
calc D.det
calc D.det
_ = normSq z := by simp [hD, z.normSq_apply]
_ = normSq x * normSq y := by simp [x.normSq_mul]
_ = 1 := detA_one
@ -95,23 +101,26 @@ theorem toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (de
show (M * E₂ * Mᴴ) 1 1 = 0 by rw [he', he] ; simp [E₂]
have hB11 : B 1 1 = 0 := congrArg Complex.re <|
show (M * E₃ * Mᴴ) 1 1 = 0 by rw [he', he] ; simp [E₃]
calc A 1 1 - (B * ⅟D * C) 1 1
calc A 1 1 - (B * ⅟D * C) 1 1
_ = A 1 1 - B 1 ⬝ᵥ ((⅟D * C) · 1) := by noncomm_ring
_ = normSq y := by simp [hB10, hB11, hA11]
calc LinearMap.det (toSelfAdjointMap' M)
calc LinearMap.det (toSelfAdjointMap' M)
_ = F.det := (LinearMap.det_toMatrix ..).symm
_ = D.det * (A - B * ⅟D * C).det := F.fromBlocks_toBlocks ▸ Matrix.det_fromBlocks₂₂ ..
_ = 1 := by rw [hE] ; simp [detD_one, detA_one]
/-- This promotes `Lorentz.SL2C.toSelfAdjointMap M` and its definitional equivalence,
`Lorentz.SL2C.toSelfAdjointMap' M`, to a linear equivalence by recognising the linear inverse to be
`Lorentz.SL2C.toSelfAdjointMap M⁻¹`, i.e., `Lorentz.SL2C.toSelfAdjointMap' M⁻¹`. -/
noncomputable def toSelfAdjointEquiv (M : ℂ²ˣ²) [Invertible M] : ℍ₂ ≃ₗ[ℝ] ℍ₂ where
toLinearMap := toSelfAdjointMap' M
invFun := toSelfAdjointMap' M⁻¹
left_inv | ⟨A, _⟩ => Subtype.ext <|
calc M⁻¹ * (M * A * Mᴴ) * M⁻¹ᴴ
calc M⁻¹ * (M * A * Mᴴ) * M⁻¹ᴴ
_ = M⁻¹ * ↑M * A * (M⁻¹ * M)ᴴ := by noncomm_ring [Matrix.conjTranspose_mul]
_ = A := by simp
right_inv | ⟨A, _⟩ => Subtype.ext <|
calc M * (M⁻¹ * A * M⁻¹ᴴ) * Mᴴ
calc M * (M⁻¹ * A * M⁻¹ᴴ) * Mᴴ
_ = M * M⁻¹ * A * (M * M⁻¹)ᴴ := by noncomm_ring [Matrix.conjTranspose_mul]
_ = A := by simp
@ -120,7 +129,7 @@ theorem toSelfAdjointMap_mul (M N : ℂ²ˣ²)
LinearMap.ext fun A => Subtype.ext <|
show M * N * A * (M * N)ᴴ = M * (N * A * Nᴴ) * Mᴴ by noncomm_ring [Matrix.conjTranspose_mul]
theorem toSelfAdjointMap_similar_det (M N : ℂ²ˣ²) [Invertible M]
theorem toSelfAdjointMap_similar_det (M N : ℂ²ˣ²) [Invertible M]
: LinearMap.det (toSelfAdjointMap' (M * N * M⁻¹)) = LinearMap.det (toSelfAdjointMap' N) :=
let e := toSelfAdjointEquiv M
let f := toSelfAdjointMap' N