| import OAI.Probability.ClassicalON.CylindricalGeometry |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory |
| namespace ClassicalON |
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| def firstAmplitude (s : Spin 3) : Amplitude := ⟨|s.val 0|,abs_nonneg _,by |
| simpa only [Real.norm_eq_abs,spin_norm] using PiLp.norm_apply_le s.val 0⟩ |
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| theorem continuous_firstAmplitude : Continuous firstAmplitude := by |
| apply Continuous.subtype_mk |
| exact ((PiLp.continuous_apply 2 _ 0).comp continuous_subtype_val).abs |
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| @[simp] theorem firstAmplitude_cylindricalSpin (r : Amplitude) (τ : Bool) (θ : PlanarAngle) : |
| firstAmplitude (cylindricalSpin r τ θ)=r := by |
| apply Subtype.ext |
| change |(r:ℝ)*signValue τ|=(r:ℝ) |
| cases τ < |
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| theorem planarAngle_exists (a b : ℝ) (hab : a^2+b^2=1) : |
| ∃ θ : PlanarAngle,planarCos θ=a ∧ planarSin θ=b := by |
| let z : Circle := ⟨⟨a,b⟩,by |
| change (⟨a,b⟩ : ℂ) ∈ Metric.sphere 0 1 |
| rw [mem_sphere_zero_iff_norm] |
| apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num : (0:ℝ)≤1)).mp |
| rw [← Complex.normSq_eq_norm_sq] |
| simpa [Complex.normSq_apply,pow_two] using hab⟩ |
| obtain ⟨θ,hθ⟩ := (AddCircle.homeomorphCircle (T := (1:ℝ)) (by norm_num)).surjective z |
| rw [AddCircle.homeomorphCircle_apply] at hθ |
| exact ⟨θ,congrArg (fun z : Circle => (z:ℂ).re) hθ,congrArg (fun z : Circle => (z:ℂ).im) hθ⟩ |
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| theorem cylindricalSpin_surjective_fiber (s : Spin 3) : |
| ∃ τ θ,cylindricalSpin (firstAmplitude s) τ θ=s := by |
| let r := firstAmplitude s |
| let q := transverseAmplitude r |
| have hq : q^2=s.val 1^2+s.val 2^2 := by |
| have hh := normSq_three s.val |
| rw [spin_norm] at hh |
| have hr : (r:ℝ)^2=s.val 0^2 := sq_abs _ |
| dsimp only [q] |
| rw [transverseAmplitude_sq,hr] |
| nlinarith |
| obtain ⟨τ,hτ⟩ : ∃ τ,(r:ℝ)*signValue τ=s.val 0 := by |
| by_cases hs : 0 ≤ s.val 0 |
| · exact ⟨true,by simp [r,firstAmplitude,signValue,abs_of_nonneg hs]⟩ |
| · exact ⟨false,by simp [r,firstAmplitude,signValue,abs_of_neg (lt_of_not_ge hs)]⟩ |
| by_cases hzero : q=0 |
| · refine ⟨τ,0,Subtype.ext ?_⟩ |
| ext i |
| have h1 : s.val 1=0 := by nlinarith [sq_nonneg (s.val 2)] |
| have h2 : s.val 2=0 := by nlinarith [sq_nonneg (s.val 1)] |
| fin_cases i |
| · exact hτ |
| · change q*planarCos 0=s.val 1 |
| rw [hzero,h1,zero_mul] |
| · change q*planarSin 0=s.val 2 |
| rw [hzero,h2,zero_mul] |
| · have hab : (s.val 1/q)^2+(s.val 2/q)^2=1 := by |
| field_simp |
| nlinarith |
| obtain ⟨θ,hc,hs⟩ := planarAngle_exists _ _ hab |
| refine ⟨τ,θ,Subtype.ext ?_⟩ |
| ext i |
| fin_cases i |
| · exact hτ |
| · change q*planarCos θ=s.val 1 |
| rw [hc,mul_div_cancel₀ _ hzero] |
| · change q*planarSin θ=s.val 2 |
| rw [hs,mul_div_cancel₀ _ hzero] |
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| theorem cylinderRotation_cylindricalSpin (τ υ : Bool) (θ φ : PlanarAngle) (r : Amplitude) : |
| spinRotation (cylinderRotation τ θ) (cylindricalSpin r υ φ)= |
| cylindricalSpin r (τ==υ) (θ+φ) := by |
| apply Subtype.ext |
| ext i |
| fin_cases i |
| · change signValue τ*((r:ℝ)*signValue υ)=(r:ℝ)*signValue (τ==υ) |
| cases τ < |
| · change planarCos θ*(transverseAmplitude r*planarCos φ)- |
| planarSin θ*(transverseAmplitude r*planarSin φ)=transverseAmplitude r*planarCos (θ+φ) |
| rw [planarCos_add] |
| · change planarSin θ*(transverseAmplitude r*planarCos φ)+ |
| planarCos θ*(transverseAmplitude r*planarSin φ)=transverseAmplitude r*planarSin (θ+φ) |
| rw [planarSin_add] |
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| theorem continuous_cylinderSpinRotation : |
| Continuous (fun p : (Bool×PlanarAngle)×Spin 3 => |
| spinRotation (cylinderRotation p.1.1 p.1.2) p.2) := by |
| apply Continuous.subtype_mk |
| change Continuous (fun p : (Bool×PlanarAngle)×Spin 3 => cylinderLinear p.1.1 p.1.2 p.2.val) |
| unfold cylinderLinear |
| apply (PiLp.continuous_toLp 2 _).comp |
| apply continuous_pi |
| intro i |
| have hc : Continuous (fun p : (Bool×PlanarAngle)×Spin 3 => planarCos p.1.2) := continuous_planarCos.comp (continuous_snd.comp continuous_fst) |
| have hs : Continuous (fun p : (Bool×PlanarAngle)×Spin 3 => planarSin p.1.2) := continuous_planarSin.comp (continuous_snd.comp continuous_fst) |
| have ht : Continuous (fun p : (Bool×PlanarAngle)×Spin 3 => signValue p.1.1) := (show Continuous signValue from continuous_of_discreteTopology).comp (continuous_fst.comp continuous_fst) |
| have hv (j : Fin 3) : Continuous (fun p : (Bool×PlanarAngle)×Spin 3 => p.2.val j) := (PiLp.continuous_apply 2 _ j).comp (continuous_subtype_val.comp continuous_snd) |
| fin_cases i |
| · exact ht.mul (hv 0) |
| · exact (hc.mul (hv 1)).sub (hs.mul (hv 2)) |
| · exact (hs.mul (hv 1)).add (hc.mul (hv 2)) |
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| end ClassicalON |
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| end |
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| end OAI |
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