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refactor: simp golfing

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jstoobysmith 2024-08-30 10:11:55 -04:00
parent d39c6c3e28
commit 167145acef
8 changed files with 35 additions and 75 deletions
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22
HepLean/Mathematics/SO3/Basic.lean
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@ -47,13 +47,8 @@ lemma coe_inv (A : SO3) : (A⁻¹).1 = A.1⁻¹:=
/-- The inclusion of `SO(3)` into `GL (Fin 3) ℝ`. -/
def toGL : SO(3) →* GL (Fin 3) ℝ where
toFun A := ⟨A.1, (A⁻¹).1, A.2.2, mul_eq_one_comm.mpr A.2.2⟩
map_one' := by
simp
rfl
map_mul' x y := by
simp only [_root_.mul_inv_rev, coe_inv]
ext
rfl
map_one' := (GeneralLinearGroup.ext_iff _ 1).mpr fun _=> congrFun rfl
map_mul' _ _ := (GeneralLinearGroup.ext_iff _ _).mpr fun _ => congrFun rfl
lemma subtype_val_eq_toGL : (Subtype.val : SO3 → Matrix (Fin 3) (Fin 3) ℝ) =
Units.val ∘ toGL.toFun :=
@ -70,14 +65,7 @@ lemma toGL_injective : Function.Injective toGL := by
def toProd : SO(3) →* (Matrix (Fin 3) (Fin 3) ℝ) × (Matrix (Fin 3) (Fin 3) ℝ)ᵐᵒᵖ :=
MonoidHom.comp (Units.embedProduct _) toGL
lemma toProd_eq_transpose : toProd A = (A.1, ⟨A.1ᵀ⟩) := by
simp only [toProd, Units.embedProduct, coe_units_inv, MulOpposite.op_inv, toGL, coe_inv,
MonoidHom.coe_comp, MonoidHom.coe_mk, OneHom.coe_mk, Function.comp_apply, Prod.mk.injEq,
true_and]
refine MulOpposite.unop_inj.mp ?_
simp only [MulOpposite.unop_inv, MulOpposite.unop_op]
rw [← coe_inv]
rfl
lemma toProd_eq_transpose : toProd A = (A.1, ⟨A.1ᵀ⟩) := rfl
lemma toProd_injective : Function.Injective toProd := by
intro A B h
@ -131,7 +119,7 @@ lemma det_minus_id (A : SO(3)) : det (A.1 - 1) = 0 := by
_ = det A.1 * det (1 - A.1ᵀ) := by rw [← det_mul, mul_sub, mul_one]
_ = det (1 - A.1ᵀ) := by simp [A.2.1]
_ = det (1 - A.1ᵀ)ᵀ := by rw [det_transpose]
_ = det (1 - A.1) := by simp
_ = det (1 - A.1) := rfl
_ = det (- (A.1 - 1)) := by simp
_ = (- 1) ^ 3 * det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
_ = - det (A.1 - 1) := by simp [pow_three]
@ -145,7 +133,7 @@ lemma det_id_minus (A : SO(3)) : det (1 - A.1) = 0 := by
_ = (- 1) ^ 3 * det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
_ = - det (A.1 - 1) := by simp [pow_three]
rw [h1, det_minus_id]
simp only [neg_zero]
exact neg_zero
@[simp]
lemma one_in_spectrum (A : SO(3)) : 1 ∈ spectrum ℝ (A.1) := by