docs: Some doc strings for instances
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jstoobysmith
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9 changed files with 40 additions and 36 deletions
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@ -112,8 +112,11 @@ instance (i : Fin 2) : Module ℝ (EdgeVertexMomentaMap F i) :=
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/-- The direct sum of `EdgeMomenta` and `VertexMomenta`. -/
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def EdgeVertexMomenta : Type := DirectSum (Fin 2) (EdgeVertexMomentaMap F)
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/-- The structure of a addative commutative group on `EdgeVertexMomenta` for a
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Feynman diagram `F`. -/
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instance : AddCommGroup F.EdgeVertexMomenta := DirectSum.instAddCommGroup _
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/-- The structure of a module over `ℝ` on `EdgeVertexMomenta` for a Feynman diagram `F`. -/
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instance : Module ℝ F.EdgeVertexMomenta := DirectSum.instModule
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/-!
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@ -37,6 +37,7 @@ inductive Color
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| up : Color
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| down : Color
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/-- Color for complex Lorentz tensors is decidable. -/
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instance : DecidableEq Color := fun x y =>
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match x, y with
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| Color.upL, Color.upL => isTrue rfl
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@ -93,6 +93,7 @@ end LorentzGroup
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-/
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/-- The instance of a group on `LorentzGroup d`. -/
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@[simps! mul_coe one_coe inv div]
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instance lorentzGroupIsGroup : Group (LorentzGroup d) where
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mul A B := ⟨A.1 * B.1, LorentzGroup.mem_mul A.2 B.2⟩
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@ -114,6 +115,7 @@ variable {Λ Λ' : LorentzGroup d}
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lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= (inv_eq_left_inv (mem_iff_dual_mul_self.mp Λ.2)).symm
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/-- The underlying matrix of a Lorentz transformation is invertible. -/
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instance (M : LorentzGroup d) : Invertible M.1 where
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invOf := M⁻¹
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invOf_mul_self := by
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@ -256,6 +258,8 @@ lemma toGL_embedding : IsEmbedding (@toGL d).toFun where
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isOpen_induced_iff]
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exact exists_exists_and_eq_and
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/-- The embedding of the Lorentz group into `GL(n, ℝ)` gives `LorentzGroup d` an instance
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of a topological group. -/
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instance : TopologicalGroup (LorentzGroup d) :=
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IsInducing.topologicalGroup toGL toGL_embedding.toIsInducing
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@ -281,6 +285,7 @@ def toComplex : LorentzGroup d →* Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d)
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simp only [← Matrix.map_mul, RingHom.map_matrix_mul]
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rfl
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/-- The image of a Lorentz transformation under `toComplex` is invertible. -/
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instance (M : LorentzGroup d) : Invertible (toComplex M) where
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invOf := toComplex M⁻¹
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invOf_mul_self := by
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@ -324,42 +329,6 @@ lemma toComplex_mulVec_ofReal (v : Fin 1 ⊕ Fin d → ℝ) (Λ : LorentzGroup d
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funext i
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rw [← RingHom.map_mulVec]
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rfl
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/-!
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/-!
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# To a norm one Lorentz vector
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-/
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/-- The first column of a Lorentz matrix as a `NormOneLorentzVector`. -/
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@[simps!]
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def toNormOneLorentzVector (Λ : LorentzGroup d) : NormOneLorentzVector d :=
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⟨Λ.1 *ᵥ timeVec, by rw [NormOneLorentzVector.mem_iff, Λ.2, minkowskiMetric.on_timeVec]⟩
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/-!
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# The time like element
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-/
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/-- The time like element of a Lorentz matrix. -/
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@[simp]
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def timeComp (Λ : LorentzGroup d) : ℝ := Λ.1 (Sum.inl 0) (Sum.inl 0)
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lemma timeComp_eq_toNormOneLorentzVector : timeComp Λ = (toNormOneLorentzVector Λ).1.time := by
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simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue, timeComp]
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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rfl
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lemma timeComp_mul (Λ Λ' : LorentzGroup d) : timeComp (Λ * Λ') =
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⟪toNormOneLorentzVector (transpose Λ), (toNormOneLorentzVector Λ').1.spaceReflection⟫ₘ := by
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simp only [timeComp, Fin.isValue, lorentzGroupIsGroup_mul_coe, mul_apply, Fintype.sum_sum_type,
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Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton, toNormOneLorentzVector,
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transpose, timeVec, right_spaceReflection, time, space, PiLp.inner_apply, Function.comp_apply,
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RCLike.inner_apply, conj_trivial]
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erw [Pi.basisFun_apply, Matrix.mulVec_single_one]
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simp
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-/
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/-- The parity transformation. -/
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def parity : LorentzGroup d := ⟨minkowskiMatrix, by
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@ -30,13 +30,17 @@ lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
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refine mul_self_eq_one_iff.mp ?_
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simpa only [det_mul, det_dual, det_one] using congrArg det ((mem_iff_self_mul_dual).mp Λ.2)
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/-- The group `ℤ₂`. -/
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local notation "ℤ₂" => Multiplicative (ZMod 2)
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/-- The instance of a topological space on `ℤ₂` corresponding to the discrete topology. -/
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instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
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/-- The topological space defined by `ℤ₂` is discrete. -/
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instance : DiscreteTopology ℤ₂ := by
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exact forall_open_iff_discrete.mp fun _ => trivial
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/-- The instance of a topological group on `ℤ₂` defined via the discrete topology. -/
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instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
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/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
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@ -138,8 +142,12 @@ def detRep : 𝓛 d →* ℤ₂ where
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apply (detContinuous_eq_one _).mpr
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simp only [lorentzGroupIsGroup_mul_coe, det_mul, h1, h2, mul_neg, mul_one, neg_neg]
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/-- The representation of the Lorentz group defined by taking the determinant `detRep` is
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continuous. -/
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lemma detRep_continuous : Continuous (@detRep d) := detContinuous.2
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/-- Two Lorentz transformations which are in the same connected component have the same
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determinant. -/
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lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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Λ.1.det = Λ'.1.det := by
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obtain ⟨s, hs, hΛ'⟩ := h
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@ -149,12 +157,15 @@ lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connecte
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(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
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(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
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/-- Two Lorentz transformations which are in the same connected component have the same
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image under `detRep`, the determinant representation. -/
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lemma detRep_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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detRep Λ = detRep Λ' := by
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simp only [detRep_apply, detContinuous, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
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coeForℤ₂_apply, Subtype.mk.injEq]
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rw [det_on_connected_component h]
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/-- Two Lorentz transformations which are joined by a path have the same determinant. -/
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lemma det_of_joined {Λ Λ' : LorentzGroup d} (h : Joined Λ Λ') : Λ.1.det = Λ'.1.det :=
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det_on_connected_component $ pathComponent_subset_component _ h
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@ -162,18 +173,24 @@ lemma det_of_joined {Λ Λ' : LorentzGroup d} (h : Joined Λ Λ') : Λ.1.det =
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@[simp]
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def IsProper (Λ : LorentzGroup d) : Prop := Λ.1.det = 1
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/-- The predicate `IsProper` is decidable. -/
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instance : DecidablePred (@IsProper d) := by
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intro Λ
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apply Real.decidableEq
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/-- A Lorentz transformation is proper if it's image under the det-representation
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`detRep` is `1`. -/
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lemma IsProper_iff (Λ : LorentzGroup d) : IsProper Λ ↔ detRep Λ = 1 := by
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rw [show 1 = detRep 1 from Eq.symm (MonoidHom.map_one detRep), detRep_apply, detRep_apply,
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detContinuous_eq_iff_det_eq]
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simp only [IsProper, lorentzGroupIsGroup_one_coe, det_one]
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/-- The identity Lorentz transformation is proper. -/
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lemma id_IsProper : @IsProper d 1 := by
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simp [IsProper]
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/-- If two Lorentz transformations are in the same connected componenet, and one is proper then
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the other is also proper. -/
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lemma isProper_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
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IsProper Λ ↔ IsProper Λ' := by
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simp only [IsProper]
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@ -278,6 +278,8 @@ lemma toSelfAdjoint_symm_basis (i : Fin 1 ⊕ Fin 3) :
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## Topology
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-/
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/-- The type `ContrMod d` carries an instance of a topological group, induced by
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it's equivalence to `Fin 1 ⊕ Fin d → ℝ`. -/
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instance : TopologicalSpace (ContrMod d) := TopologicalSpace.induced
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ContrMod.toFin1dℝEquiv (Pi.topologicalSpace)
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@ -36,6 +36,7 @@ lemma mem_mulAction (g : LorentzGroup d) (v : Contr d) :
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v ∈ NormOne d ↔ (Contr d).ρ g v ∈ NormOne d := by
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rw [mem_iff, mem_iff, contrContrContractField.action_tmul]
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/-- The instance of a topological space on `NormOne d` defined as the subspace topology. -/
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instance : TopologicalSpace (NormOne d) := instTopologicalSpaceSubtype
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variable (v w : NormOne d)
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@ -24,6 +24,7 @@ namespace HomogeneousQuadratic
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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/-- A homogenous quadratic equation can be treated as a function from `V` to `ℚ`. -/
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instance instFun : FunLike (HomogeneousQuadratic V) V ℚ where
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coe f := f.toFun
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coe_injective' f g h := by
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@ -49,6 +50,7 @@ open BigOperators
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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/-- A symmetric bilinear form can be treated as a function from `V` to `V →ₗ[ℚ] ℚ`. -/
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instance instFun (V : Type) [AddCommMonoid V] [Module ℚ V] :
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FunLike (BiLinearSymm V) V (V →ₗ[ℚ] ℚ) where
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coe f := f.toFun
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@ -146,6 +148,7 @@ namespace HomogeneousCubic
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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/-- A homogenous cubic equation can be treated as a function from `V` to `ℚ`. -/
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instance instFun : FunLike (HomogeneousCubic V) V ℚ where
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coe f := f.toFun
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coe_injective' f g h := by
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@ -168,6 +171,7 @@ namespace TriLinearSymm
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open BigOperators
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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/-- A symmetric trilinear form can be treated as a function from `V` to `V →ₗ[ℚ] V →ₗ[ℚ] ℚ`. -/
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instance instFun : FunLike (TriLinearSymm V) V (V →ₗ[ℚ] V →ₗ[ℚ] ℚ) where
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coe f := f.toFun
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coe_injective' f g h := by
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@ -19,6 +19,7 @@ open Matrix
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/-- The group of `3×3` real matrices with determinant 1 and `A * Aᵀ = 1`. -/
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def SO3 : Type := {A : Matrix (Fin 3) (Fin 3) ℝ // A.det = 1 ∧ A * Aᵀ = 1}
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/-- The instance of a group on `SO3`. -/
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@[simps! mul_coe one_coe inv div]
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instance SO3Group : Group SO3 where
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mul A B := ⟨A.1 * B.1,
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@ -110,6 +111,8 @@ lemma toGL_embedding : IsEmbedding toGL.toFun where
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apply And.intro (isOpen_induced hU1)
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exact hU2
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/-- The instance of a topological group on `SO(3)`, defined through the embedding of `SO(3)`
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into `GL(n)`. -/
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instance : TopologicalGroup SO(3) :=
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IsInducing.topologicalGroup toGL toGL_embedding.toIsInducing
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@ -24,9 +24,12 @@ instance : AddCommMonoid SpaceTime := Pi.addCommMonoid
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/-- Give spacetime the structure of a module over the reals. -/
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instance : Module ℝ SpaceTime := Pi.module _ _ _
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/-- The instance of a normed group on spacetime defined via the Euclidean norm. -/
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instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normedAddCommGroup
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/-- The Euclidean norm-structure on space time. This is used to give it a smooth structure. -/
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instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
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namespace SpaceTime
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open Manifold
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@ -41,6 +44,7 @@ def space (x : SpaceTime) : EuclideanSpace ℝ (Fin 3) := ![x 1, x 2, x 3]
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/-- The structure of a smooth manifold on spacetime. -/
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def asSmoothManifold : ModelWithCorners ℝ SpaceTime SpaceTime := 𝓘(ℝ, SpaceTime)
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/-- The instance of a `ChartedSpace` on `SpaceTime`. -/
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instance : ChartedSpace SpaceTime SpaceTime := chartedSpaceSelf SpaceTime
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/-- The standard basis for spacetime. -/
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