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refactor: Spelling

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jstoobysmith 2025-02-07 15:43:59 +00:00
parent b82791d671
commit f8f1e1757f
25 changed files with 80 additions and 80 deletions
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4
HepLean/Lorentz/Group/Boosts.lean
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@ -31,7 +31,7 @@ open TensorProduct
variable {d : ℕ}
/-- An auxillary linear map used in the definition of a generalised boost. -/
/-- An auxiliary linear map used in the definition of a generalised boost. -/
def genBoostAux₁ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d where
toFun x := (2 * ⟪x, u.val.val⟫ₘ) • v.1.1
map_add' x y := by
@ -43,7 +43,7 @@ def genBoostAux₁ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d
smul_tmul, tmul_smul, map_smul, smul_eq_mul, RingHom.id_apply]
rw [← mul_assoc, mul_comm 2 c, mul_assoc, mul_smul]
/-- An auxillary linear map used in the definition of a genearlised boost. -/
/-- An auxiliary linear map used in the definition of a genearlised boost. -/
def genBoostAux₂ (u v : FuturePointing d) : ContrMod d →ₗ[ℝ] ContrMod d where
toFun x := - (⟪x, u.1.1 + v.1.1⟫ₘ / (1 + ⟪u.1.1, v.1.1⟫ₘ)) • (u.1.1 + v.1.1)
map_add' x y := by

14
HepLean/Lorentz/Group/Orthochronous.lean
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@ -64,7 +64,7 @@ lemma not_orthochronous_iff_le_zero : ¬ IsOrthochronous Λ ↔ Λ.1 (Sum.inl 0)
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => Λ.1 (Sum.inl 0) (Sum.inl 0),
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`.
/-- 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. -/
def stepFunction : ℝ → ℝ := fun t =>
@ -135,7 +135,7 @@ lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochron
· rfl
· linarith
/-- The product of two orthochronous Lorentz transfomations is orthochronous. -/
/-- The product of two orthochronous Lorentz transformations is orthochronous. -/
lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
@ -143,7 +143,7 @@ lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthoch
rw [IsOrthochronous, LorentzGroup.inl_inl_mul]
exact NormOne.FuturePointing.metric_reflect_mem_mem h h'
/-- The product of two non-orthochronous Lorentz transfomations is orthochronous. -/
/-- The product of two non-orthochronous Lorentz transformations is orthochronous. -/
lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
rw [IsOrthochronous_iff_transpose] at h
@ -151,8 +151,8 @@ lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h :
rw [IsOrthochronous, LorentzGroup.inl_inl_mul]
exact NormOne.FuturePointing.metric_reflect_not_mem_not_mem h h'
/-- The product of an orthochronous Lorentz transfomations with a
non-orthchronous Loentz transformation is not orthochronous. -/
/-- The product of an orthochronous Lorentz transformations with a
non-orthochronous Loentz 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]
@ -160,8 +160,8 @@ lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : I
rw [IsOrthochronous_iff_futurePointing] at h h'
exact NormOne.FuturePointing.metric_reflect_mem_not_mem h h'
/-- The product of a non-orthochronous Lorentz transfomations with an
orthchronous Loentz transformation is not orthochronous. -/
/-- The product of a non-orthochronous Lorentz transformations with an
orthochronous Loentz 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]

2
HepLean/Lorentz/MinkowskiMatrix.lean
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@ -88,7 +88,7 @@ lemma off_diag_zero {μ ν : Fin 1 ⊕ Fin d} (h : μ ≠ ν) : η μ ν = 0 :=
lemma inl_0_inl_0 : @minkowskiMatrix d (Sum.inl 0) (Sum.inl 0) = 1 := by
rfl
/-- The space diagonal components of the Minkowski matriz are `-1`. -/
/-- The space diagonal components of the Minkowski matrix are `-1`. -/
lemma inr_i_inr_i (i : Fin d) : @minkowskiMatrix d (Sum.inr i) (Sum.inr i) = -1 := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
simp_all only [diagonal_apply_eq, Sum.elim_inr]

16
HepLean/Lorentz/PauliMatrices/SelfAdjoint.lean
View file

@ -12,7 +12,7 @@ import HepLean.Lorentz.PauliMatrices.Basic
namespace PauliMatrix
open Matrix
/-- The trace of `σ0` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ0` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ0_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ0 * A.1)).re = Matrix.trace (σ0 * A.1) := by
rw [eta_fin_two A.1]
@ -35,7 +35,7 @@ lemma selfAdjoint_trace_σ0_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
cons_val_fin_one, head_fin_const] at h11
exact Complex.conj_eq_iff_re.mp h11
/-- The trace of `σ1` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ1` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ1_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ1 * A.1)).re = Matrix.trace (σ1 * A.1) := by
rw [eta_fin_two A.1]
@ -55,7 +55,7 @@ lemma selfAdjoint_trace_σ1_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
simp only [Fin.isValue, Complex.ofReal_mul, Complex.ofReal_ofNat]
ring
/-- The trace of `σ2` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ2` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ2_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ2 * A.1)).re = Matrix.trace (σ2 * A.1) := by
rw [eta_fin_two A.1]
@ -79,7 +79,7 @@ lemma selfAdjoint_trace_σ2_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
simp
· ring
/-- The trace of `σ3` multiplied by a self-adjiont `2×2` matrix is real. -/
/-- The trace of `σ3` multiplied by a self-adjoint `2×2` matrix is real. -/
lemma selfAdjoint_trace_σ3_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :
(Matrix.trace (σ3 * A.1)).re = Matrix.trace (σ3 * A.1) := by
rw [eta_fin_two A.1]
@ -100,7 +100,7 @@ lemma selfAdjoint_trace_σ3_real (A : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ))
open Complex
/-- Two `2×2` self-adjiont matrices are equal if the (complex) traces of each matrix multiplied by
/-- Two `2×2` self-adjoint matrices are equal if the (complex) traces of each matrix multiplied by
each of the Pauli-matrices are equal. -/
lemma selfAdjoint_ext_complex {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
(h0 : Matrix.trace (PauliMatrix.σ0 * A.1) = Matrix.trace (PauliMatrix.σ0 * B.1))
@ -133,7 +133,7 @@ lemma selfAdjoint_ext_complex {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
| 1, 1 =>
linear_combination (norm := ring_nf) (h0 - h3) / 2
/-- Two `2×2` self-adjiont matrices are equal if the real traces of each matrix multiplied by
/-- Two `2×2` self-adjoint matrices are equal if the real traces of each matrix multiplied by
each of the Pauli-matrices are equal. -/
lemma selfAdjoint_ext {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
(h0 : ((Matrix.trace (PauliMatrix.σ0 * A.1))).re = ((Matrix.trace (PauliMatrix.σ0 * B.1))).re)
@ -154,7 +154,7 @@ lemma selfAdjoint_ext {A B : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)}
noncomputable section
/-- An auxillary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
/-- An auxiliary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
Pauli-matrix as a self-adjoint matrix. -/
def σSA' (i : Fin 1 ⊕ Fin 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
match i with
@ -225,7 +225,7 @@ lemma σSA_span : ⊤ ≤ Submodule.span ℝ (Set.range σSA') := by
def σSA : Basis (Fin 1 ⊕ Fin 3) ℝ (selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)) :=
Basis.mk σSA_linearly_independent σSA_span
/-- An auxillary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
/-- An auxiliary function which on `i : Fin 1 ⊕ Fin 3` returns the corresponding
Pauli-matrix as a self-adjoint matrix with a minus sign for `Sum.inr _`. -/
def σSAL' (i : Fin 1 ⊕ Fin 3) : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) :=
match i with