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refactor: Last batch of multi-goal proofs

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jstoobysmith 2024-08-21 06:40:58 -04:00
parent b9479c904d
commit c0499483a8
43 changed files with 910 additions and 955 deletions
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50
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
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@ -56,10 +56,10 @@ lemma not_orthochronous_iff_le_neg_one :
lemma not_orthochronous_iff_le_zero :
¬ IsOrthochronous Λ ↔ timeComp Λ ≤ 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
rw [not_orthochronous_iff_le_neg_one] at h
linarith
rw [IsOrthochronous_iff_ge_one]
linarith
· rw [not_orthochronous_iff_le_neg_one] at h
linarith
· rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
@ -73,13 +73,13 @@ def stepFunction : ℝ → ℝ := fun t =>
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
· rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
· rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
exact id (Eq.symm ha)
/-- The continuous map from `lorentzGroup` to `ℝ` wh
taking Orthochronous elements to `1` and non-orthochronous to `-1`. -/
@ -102,8 +102,8 @@ lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrt
lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
orthchroMapReal Λ = -1 ∨ orthchroMapReal Λ = 1 := by
by_cases h : IsOrthochronous Λ
apply Or.inr $ orthchroMapReal_on_IsOrthochronous h
apply Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
· exact Or.inr $ orthchroMapReal_on_IsOrthochronous h
· exact Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
local notation "ℤ₂" => Multiplicative (ZMod 2)
@ -158,18 +158,18 @@ def orthchroRep : LorentzGroup d →* ℤ₂ where
simp only
by_cases h : IsOrthochronous Λ
<;> by_cases h' : IsOrthochronous Λ'
rw [orthchroMap_IsOrthochronous h, orthchroMap_IsOrthochronous h',
orthchroMap_IsOrthochronous (mul_othchron_of_othchron_othchron h h')]
rfl
rw [orthchroMap_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_othchron_not_othchron h h')]
rfl
rw [orthchroMap_not_IsOrthochronous h, orthchroMap_IsOrthochronous h',
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_not_othchron_othchron h h')]
rfl
rw [orthchroMap_not_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
rfl
· rw [orthchroMap_IsOrthochronous h, orthchroMap_IsOrthochronous h',
orthchroMap_IsOrthochronous (mul_othchron_of_othchron_othchron h h')]
rfl
· rw [orthchroMap_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_othchron_not_othchron h h')]
rfl
· rw [orthchroMap_not_IsOrthochronous h, orthchroMap_IsOrthochronous h',
orthchroMap_not_IsOrthochronous (mul_not_othchron_of_not_othchron_othchron h h')]
rfl
· rw [orthchroMap_not_IsOrthochronous h, orthchroMap_not_IsOrthochronous h',
orthchroMap_IsOrthochronous (mul_othchron_of_not_othchron_not_othchron h h')]
rfl
end LorentzGroup