refactor: Change case of type and props

HepLean/SpaceTime/LorentzGroup/Orthochronous.lean

@@ -20,19 +20,19 @@ matrices.
 
 noncomputable section
 
-namespace spaceTime
+namespace SpaceTime
 
 open Manifold
 open Matrix
 open Complex
 open ComplexConjugate
 
-namespace lorentzGroup
+namespace LorentzGroup
 open PreFourVelocity
 
 /-- The first column of a lorentz matrix as a `PreFourVelocity`. -/
 @[simp]
-def fstCol (Λ : lorentzGroup) : PreFourVelocity := ⟨Λ.1 *ᵥ stdBasis 0, by
+def fstCol (Λ : LorentzGroup) : PreFourVelocity := ⟨Λ.1 *ᵥ stdBasis 0, by
   rw [mem_PreFourVelocity_iff, ηLin_expand]
   simp only [Fin.isValue, stdBasis_mulVec]
   have h00 := congrFun (congrFun ((PreservesηLin.iff_matrix Λ.1).mp  Λ.2) 0) 0
@@ -46,18 +46,18 @@ def fstCol (Λ : lorentzGroup) : PreFourVelocity := ⟨Λ.1 *ᵥ stdBasis 0, by
   exact h00⟩
 
 /-- A Lorentz transformation is  `orthochronous` if its `0 0` element is non-negative. -/
-def IsOrthochronous (Λ : lorentzGroup) : Prop := 0 ≤ Λ.1 0 0
+def IsOrthochronous (Λ : LorentzGroup) : Prop := 0 ≤ Λ.1 0 0
 
-lemma IsOrthochronous_iff_transpose (Λ : lorentzGroup) :
+lemma IsOrthochronous_iff_transpose (Λ : LorentzGroup) :
     IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
 
-lemma IsOrthochronous_iff_fstCol_IsFourVelocity (Λ : lorentzGroup) :
+lemma IsOrthochronous_iff_fstCol_IsFourVelocity (Λ : LorentzGroup) :
     IsOrthochronous Λ ↔ IsFourVelocity (fstCol Λ) := by
   simp [IsOrthochronous, IsFourVelocity]
   rw [stdBasis_mulVec]
 
 /-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
-def mapZeroZeroComp : C(lorentzGroup, ℝ) := ⟨fun Λ => Λ.1 0 0,
+def mapZeroZeroComp : C(LorentzGroup, ℝ) := ⟨fun Λ => Λ.1 0 0,
    Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) 0 0⟩
 
 /-- An auxillary function used in the definition of `orthchroMapReal`. -/
@@ -78,10 +78,10 @@ lemma stepFunction_continuous : Continuous stepFunction := by
 
 /-- The continuous map from `lorentzGroup` to `ℝ` wh
 taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
-def orthchroMapReal : C(lorentzGroup, ℝ) := ContinuousMap.comp
+def orthchroMapReal : C(LorentzGroup, ℝ) := ContinuousMap.comp
   ⟨stepFunction, stepFunction_continuous⟩ mapZeroZeroComp
 
-lemma orthchroMapReal_on_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
+lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup} (h : IsOrthochronous Λ) :
     orthchroMapReal Λ = 1 := by
   rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
   simp only [IsFourVelocity] at h
@@ -92,7 +92,7 @@ lemma orthchroMapReal_on_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronou
   rw [stepFunction, if_neg h1, if_pos h]
 
 
-lemma orthchroMapReal_on_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
+lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup} (h : ¬ IsOrthochronous Λ) :
     orthchroMapReal Λ = - 1 := by
   rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h
   rw [not_IsFourVelocity_iff, zero_nonpos_iff] at h
@@ -100,7 +100,7 @@ lemma orthchroMapReal_on_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrtho
   change stepFunction (Λ.1 0 0) = - 1
   rw [stepFunction, if_pos h]
 
-lemma orthchroMapReal_minus_one_or_one (Λ : lorentzGroup) :
+lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup) :
     orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
   by_cases h : IsOrthochronous Λ
   apply Or.inr $ orthchroMapReal_on_IsOrthochronous h
@@ -109,21 +109,21 @@ lemma orthchroMapReal_minus_one_or_one (Λ : lorentzGroup) :
 local notation  "ℤ₂" => Multiplicative (ZMod 2)
 
 /-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
-def orthchroMap : C(lorentzGroup, ℤ₂) :=
+def orthchroMap : C(LorentzGroup, ℤ₂) :=
   ContinuousMap.comp coeForℤ₂ {
     toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
     continuous_toFun :=  Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
 
-lemma orthchroMap_IsOrthochronous {Λ : lorentzGroup} (h : IsOrthochronous Λ) :
+lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup} (h : IsOrthochronous Λ) :
     orthchroMap Λ = 1 := by
   simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
 
-lemma orthchroMap_not_IsOrthochronous {Λ : lorentzGroup} (h : ¬ IsOrthochronous Λ) :
+lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup} (h : ¬ IsOrthochronous Λ) :
     orthchroMap Λ =  Additive.toMul (1 : ZMod 2) := by
   simp [orthchroMap, orthchroMapReal_on_not_IsOrthochronous h]
   rfl
 
-lemma zero_zero_mul (Λ Λ' : lorentzGroup) :
+lemma zero_zero_mul (Λ Λ' : LorentzGroup) :
     (Λ * Λ').1 0 0 =  (fstCol (transpose Λ)).1 0 * (fstCol Λ').1 0 +
     ⟪(fstCol (transpose Λ)).1.space, (fstCol Λ').1.space⟫_ℝ := by
   simp only [Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fin.sum_univ_four, fstCol,
@@ -132,21 +132,21 @@ lemma zero_zero_mul (Λ Λ' : lorentzGroup) :
     cons_val_one, head_cons, cons_val_two, tail_cons]
   ring
 
-lemma mul_othchron_of_othchron_othchron {Λ Λ' : lorentzGroup} (h : IsOrthochronous Λ)
+lemma mul_othchron_of_othchron_othchron {Λ Λ' : LorentzGroup} (h : IsOrthochronous Λ)
     (h' : IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
   rw [IsOrthochronous_iff_transpose] at h
   rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
   rw [IsOrthochronous, zero_zero_mul]
   exact euclid_norm_IsFourVelocity_IsFourVelocity h h'
 
-lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : lorentzGroup} (h : ¬  IsOrthochronous Λ)
+lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup} (h : ¬  IsOrthochronous Λ)
     (h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') := by
   rw [IsOrthochronous_iff_transpose] at h
   rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
   rw [IsOrthochronous, zero_zero_mul]
   exact euclid_norm_not_IsFourVelocity_not_IsFourVelocity h h'
 
-lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : lorentzGroup} (h :  IsOrthochronous Λ)
+lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup} (h :  IsOrthochronous Λ)
     (h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
   rw [IsOrthochronous_iff_transpose] at h
   rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
@@ -156,7 +156,7 @@ lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : lorentzGroup} (h :  Is
   rw [zero_zero_mul]
   exact euclid_norm_IsFourVelocity_not_IsFourVelocity h h'
 
-lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : lorentzGroup} (h : ¬ IsOrthochronous Λ)
+lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup} (h : ¬ IsOrthochronous Λ)
     (h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
   rw [IsOrthochronous_iff_transpose] at h
   rw [IsOrthochronous_iff_fstCol_IsFourVelocity] at h h'
@@ -167,7 +167,7 @@ lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : lorentzGroup} (h : ¬
   exact euclid_norm_not_IsFourVelocity_IsFourVelocity h h'
 
 /-- The homomorphism from `lorentzGroup` to `ℤ₂` whose kernel are the Orthochronous elements. -/
-def orthchroRep : lorentzGroup →* ℤ₂ where
+def orthchroRep : LorentzGroup →* ℤ₂ where
   toFun := orthchroMap
   map_one' := orthchroMap_IsOrthochronous (by simp [IsOrthochronous])
   map_mul' Λ Λ' := by
@@ -187,7 +187,7 @@ def orthchroRep : lorentzGroup →* ℤ₂ where
       orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
     rfl
 
-end lorentzGroup
+end LorentzGroup
 
-end spaceTime
+end SpaceTime
 end