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9b6e014 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 | 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 <;> norm_num [signValue]
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]; ring⟩
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; fin_cases i <;> simp <;> ring
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
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