| import OAI.Probability.ClassicalON.AmplitudeLaw |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped InnerProductSpace |
| namespace ClassicalON |
|
|
| theorem planarCos_sq_add_sin_sq (θ : PlanarAngle) : planarCos θ^2+planarSin θ^2=1 := by |
| have h := Circle.normSq_coe (AddCircle.toCircle θ) |
| simpa [Complex.normSq_apply,planarCos,planarSin,planarCharacter,pow_two] using h |
|
|
| theorem planarCos_add (θ φ : PlanarAngle) : |
| planarCos (θ+φ)=planarCos θ*planarCos φ-planarSin θ*planarSin φ := by |
| simp [planarCos,planarSin,planarCharacter_add,Complex.mul_re] |
| |
| theorem planarSin_add (θ φ : PlanarAngle) : |
| planarSin (θ+φ)=planarSin θ*planarCos φ+planarCos θ*planarSin φ := by |
| simp [planarCos,planarSin,planarCharacter_add,Complex.mul_im,add_comm] |
| |
| theorem signValue_sq (τ : Bool) : signValue τ^2=1 := by cases τ < |
|
|
| theorem transverseAmplitude_sq (r : Amplitude) : transverseAmplitude r^2=1-(r:ℝ)^2 := by |
| apply Real.sq_sqrt |
| have h0:=r.property.1 |
| have h1:=r.property.2 |
| nlinarith |
|
|
| theorem normSq_three (v : EuclideanSpace ℝ (Fin 3)) : ‖v‖^2=v 0^2+v 1^2+v 2^2 := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp [Fin.sum_univ_succ,add_assoc] |
| |
| def cylindricalSpin (r : Amplitude) (τ : Bool) (θ : PlanarAngle) : Spin 3 := |
| ⟨WithLp.toLp 2 ![(r:ℝ)*signValue τ,transverseAmplitude r*planarCos θ, |
| transverseAmplitude r*planarSin θ],by |
| rw [mem_sphere_zero_iff_norm] |
| have h := planarCos_sq_add_sin_sq θ |
| have hq := transverseAmplitude_sq r |
| have ht := signValue_sq τ |
| apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0:ℝ)≤1)).mp |
| rw [normSq_three] |
| change ((r:ℝ)*signValue τ)^2+(transverseAmplitude r*planarCos θ)^2+ |
| (transverseAmplitude r*planarSin θ)^2=1^2 |
| calc |
| _ = (r:ℝ)^2*signValue τ^2+transverseAmplitude r^2*(planarCos θ^2+planarSin θ^2) := by ring |
| _ = 1^2 := by rw [ht,h,hq] |
|
|
| theorem continuous_cylindricalSpin : |
| Continuous (fun p : Amplitude×Bool×PlanarAngle => cylindricalSpin p.1 p.2.1 p.2.2) := by |
| apply Continuous.subtype_mk |
| apply (PiLp.continuous_toLp 2 _).comp |
| apply continuous_pi |
| intro i |
| fin_cases i |
| · change Continuous (fun p : Amplitude×Bool×PlanarAngle => (p.1:ℝ)*signValue p.2.1) |
| exact continuous_fst.subtype_val.mul ((show Continuous signValue from continuous_of_discreteTopology).comp (continuous_fst.comp continuous_snd)) |
| · exact (continuous_transverseAmplitude.comp continuous_fst).mul |
| (continuous_planarCos.comp (continuous_snd.comp continuous_snd)) |
| · exact (continuous_transverseAmplitude.comp continuous_fst).mul |
| (continuous_planarSin.comp (continuous_snd.comp continuous_snd)) |
|
|
| def cylinderLinear (τ : Bool) (θ : PlanarAngle) : |
| EuclideanSpace ℝ (Fin 3) →ₗ[ℝ] EuclideanSpace ℝ (Fin 3) where |
| toFun v := WithLp.toLp 2 ![signValue τ*v 0, |
| planarCos θ*v 1-planarSin θ*v 2,planarSin θ*v 1+planarCos θ*v 2] |
| map_add' v w := by ext i; fin_cases i <;> simp <;> ring |
| map_smul' a v := by ext i |
|
|
| theorem cylinderLinear_norm (τ : Bool) (θ : PlanarAngle) (v : EuclideanSpace ℝ (Fin 3)) : |
| ‖cylinderLinear τ θ v‖=‖v‖ := by |
| apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp |
| rw [normSq_three,normSq_three] |
| have h := planarCos_sq_add_sin_sq θ |
| have ht := signValue_sq τ |
| change (signValue τ*v 0)^2+(planarCos θ*v 1-planarSin θ*v 2)^2+ |
| (planarSin θ*v 1+planarCos θ*v 2)^2=_ |
| nlinarith [congrArg (fun a : ℝ => a*(v 1^2+v 2^2)) h, |
| congrArg (fun a : ℝ => a*v 0^2) ht] |
|
|
| def cylinderRotation (τ : Bool) (θ : PlanarAngle) : |
| EuclideanSpace ℝ (Fin 3) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 3) := |
| LinearIsometryEquiv.ofSurjective |
| {cylinderLinear τ θ with norm_map' := cylinderLinear_norm τ θ} |
| (LinearMap.surjective_of_injective |
| (show Function.Injective (cylinderLinear τ θ) from |
| (show EuclideanSpace ℝ (Fin 3) →ₗᵢ[ℝ] EuclideanSpace ℝ (Fin 3) from |
| {cylinderLinear τ θ with norm_map' := cylinderLinear_norm τ θ}).injective)) |
|
|
| @[simp] theorem cylinderRotation_apply (τ : Bool) (θ : PlanarAngle) (v : EuclideanSpace ℝ (Fin 3)) : |
| cylinderRotation τ θ v=cylinderLinear τ θ v := rfl |
|
|
| end ClassicalON |
|
|
| end |
|
|
| end OAI |
|
|
