refactor: Spellings

HepLean/Lorentz/ComplexTensor/PauliMatrices/Basic.lean

@@ -79,7 +79,7 @@ lemma tensorNode_pauliContrDown : {pauliContrDown | μ α β}ᵀ.tensor =
 -/
 
 set_option maxRecDepth 5000 in
-/-- A rearanging of `pauliCoDown` to place the pauli matrices on the right. -/
+/-- A rearranging of `pauliCoDown` to place the pauli matrices on the right. -/
 lemma pauliCoDown_eq_metric_mul_pauliCo :
     {pauliCoDown | μ α' β' = εL' | α α' ⊗ εR' | β β' ⊗ pauliCo | μ α β}ᵀ := by
   conv =>
@@ -128,7 +128,7 @@ lemma pauliCoDown_eq_metric_mul_pauliCo :
   decide
 
 set_option maxRecDepth 5000 in
-/-- A rearanging of `pauliContrDown` to place the pauli matrices on the right. -/
+/-- A rearranging of `pauliContrDown` to place the pauli matrices on the right. -/
 lemma pauliContrDown_eq_metric_mul_pauliContr :
     {pauliContrDown | μ α' β' = εL' | α α' ⊗
     εR' | β β' ⊗ pauliContr | μ α β}ᵀ := by

HepLean/Lorentz/ComplexVector/Unit.lean

@@ -38,7 +38,7 @@ lemma contrCoUnitVal_expand_tmul : contrCoUnitVal =
   rfl
 
 /-- The contra-co unit for complex lorentz vectors as a morphism
-  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo`, manifesting the invaraince under
+  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo`, manifesting the invariance under
   the `SL(2, ℂ)` action. -/
 def contrCoUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexContr ⊗ complexCo where
   hom := ModuleCat.ofHom {
@@ -88,7 +88,7 @@ lemma coContrUnitVal_expand_tmul : coContrUnitVal =
   rfl
 
 /-- The co-contra unit for complex lorentz vectors as a morphism
-  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr`, manifesting the invaraince under
+  `𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr`, manifesting the invariance under
   the `SL(2, ℂ)` action. -/
 def coContrUnit : 𝟙_ (Rep ℂ SL(2,ℂ)) ⟶ complexCo ⊗ complexContr where
   hom := ModuleCat.ofHom {

HepLean/Lorentz/Group/Orthochronous.lean

@@ -29,7 +29,7 @@ open Lorentz.Contr
 /-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
 def IsOrthochronous : Prop := 0 ≤ Λ.1 (Sum.inl 0) (Sum.inl 0)
 
-/-- A Lorentz transformation is `orthochronous` if and only if its fist column is
+/-- A Lorentz transformation is `orthochronous` if and only if its first column is
   future pointing. -/
 lemma IsOrthochronous_iff_futurePointing :
     IsOrthochronous Λ ↔ toNormOne Λ ∈ NormOne.FuturePointing d := by
@@ -66,7 +66,7 @@ def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => Λ.1 (Sum.inl 0) (Sum.
 
 /-- An auxiliary function used in the definition of `orthchroMapReal`.
   This function takes all elements of `ℝ` less then `-1` to `-1`,
-  all elements of `R` geater then `1` to `1` and peserves all other elements. -/
+  all elements of `R` greater then `1` to `1` and preserves all other elements. -/
 def stepFunction : ℝ → ℝ := fun t =>
   if t ≤ -1 then -1 else
     if 1 ≤ t then 1 else t
@@ -152,7 +152,7 @@ lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h :
   exact NormOne.FuturePointing.metric_reflect_not_mem_not_mem h h'
 
 /-- The product of an orthochronous Lorentz transformations with a
-  non-orthochronous Loentz transformation is not orthochronous. -/
+  non-orthochronous Lorentz transformation is not orthochronous. -/
 lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
     (h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
   rw [not_orthochronous_iff_le_zero, LorentzGroup.inl_inl_mul]
@@ -161,7 +161,7 @@ lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : I
   exact NormOne.FuturePointing.metric_reflect_mem_not_mem h h'
 
 /-- The product of a non-orthochronous Lorentz transformations with an
-  orthochronous Loentz transformation is not orthochronous. -/
+  orthochronous Lorentz transformation is not orthochronous. -/
 lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
     (h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
   rw [not_orthochronous_iff_le_zero, LorentzGroup.inl_inl_mul]

HepLean/Lorentz/MinkowskiMatrix.lean

@@ -29,7 +29,7 @@ variable {d : ℕ}
 /-- Notation for `minkowskiMatrix`. -/
 scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
 
-/-- The Minkowski matrix is self-inveting. -/
+/-- The Minkowski matrix is self-inverting. -/
 @[simp]
 lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
   simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]
@@ -55,8 +55,8 @@ lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
 lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
   simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
 
-/-- The deteminant of the Minkowski matrix is equal to `-1` to the power
-  of the number of spactial dimensions. -/
+/-- The determinant of the Minkowski matrix is equal to `-1` to the power
+  of the number of spatial dimensions. -/
 @[simp]
 lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
   simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@@ -68,7 +68,7 @@ lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ =
   | Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
   | Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
 
-/-- The Minkowski matix as a block matrix. -/
+/-- The Minkowski matrix as a block matrix. -/
 lemma as_block : @minkowskiMatrix d =
     Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ) := by
   rw [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, ← fromBlocks_diagonal]

HepLean/Lorentz/PauliMatrices/SelfAdjoint.lean

@@ -301,7 +301,7 @@ lemma σSAL_span : ⊤ ≤ Submodule.span ℝ (Set.range σSAL') := by
 def σSAL : Basis (Fin 1 ⊕ Fin 3) ℝ (selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :=
   Basis.mk σSAL_linearly_independent σSAL_span
 
-/-- The decomposition of a self-adjint matrix into the Pauli matrices (where `σi` are negated). -/
+/-- The decomposition of a self-adjoint matrix into the Pauli matrices (where `σi` are negated). -/
 lemma σSAL_decomp (M : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
     M = (1/2 * (Matrix.trace (σ0 * M.1)).re) • σSAL (Sum.inl 0)
     + (-1/2 * (Matrix.trace (σ1 * M.1)).re) • σSAL (Sum.inr 0)

HepLean/Lorentz/SL2C/SelfAdjoint.lean

@@ -175,7 +175,7 @@ lemma toSelfAdjointMap_det_one' {M : ℂ²ˣ²} (hM : M.IsUpperTriangular) (detM
     _ = 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`, to a linear equivalence by recognizing 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

HepLean/Lorentz/Weyl/Contraction.lean

@@ -174,11 +174,11 @@ lemma altLeftContraction_basis (i j : Fin 2) :
   exact Eq.propIntro (fun a => id (Eq.symm a)) fun a => id (Eq.symm a)
 
 /--
-The linear map from rightHandedWeyl ⊗ altRightHandedWeyl to ℂ given by
-  summing over components of rightHandedWeyl and altRightHandedWeyl in the
+The linear map from `rightHandedWeyl ⊗ altRightHandedWeyl` to `ℂ` given by
+  summing over components of `rightHandedWeyl` and `altRightHandedWeyl` in the
   standard basis (i.e. the dot product).
   The contraction of a right-handed Weyl fermion with a left-handed Weyl fermion.
-  In index notation this is ψ^{dot a} φ_{dot a}.
+  In index notation this is `ψ^{dot a} φ_{dot a}`.
 -/
 def rightAltContraction : rightHanded ⊗ altRightHanded ⟶ 𝟙_ (Rep ℂ SL(2,ℂ)) where
   hom := ModuleCat.ofHom <| TensorProduct.lift rightAltBi