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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Probability.ClassicalON.OrthogonalAverage
namespace OAI
noncomputable section
open MeasureTheory
open scoped InnerProductSpace Classical
namespace ClassicalON
abbrev ThreeBlock (k : ℕ) := WithLp 2 (EuclideanSpace ℝ (Fin 3) × EuclideanSpace ℝ (Fin k))
def splitThree (k : ℕ) : EuclideanSpace ℝ (Fin (3+k)) ≃ₗᵢ[ℝ] ThreeBlock k :=
(LinearIsometryEquiv.piLpCongrLeft 2 ℝ ℝ (finSumFinEquiv.symm : Fin (3+k) ≃ Fin 3 ⊕ Fin k)).trans
(PiLp.sumPiLpEquivProdLpPiLp 2 (fun _ : Fin 3 ⊕ Fin k => ℝ))
def blockHead (k : ℕ) (s : Spin (3+k)) : EuclideanSpace ℝ (Fin 3) := (splitThree k s.val).fst
def blockTail (k : ℕ) (s : Spin (3+k)) : EuclideanSpace ℝ (Fin k) := (splitThree k s.val).snd
def blockRadius (k : ℕ) (s : Spin (3+k)) : ℝ := ‖blockHead k s‖
theorem blockRadius_nonneg (k : ℕ) (s : Spin (3+k)) : 0≤blockRadius k s := norm_nonneg _
theorem blockRadius_le_one (k : ℕ) (s : Spin (3+k)) : blockRadius k s≤1 := by
have h := WithLp.norm_fst_le _ (splitThree k s.val)
simpa only [blockRadius,blockHead,LinearIsometryEquiv.norm_map,spin_norm] using h
theorem block_norm_sq (k : ℕ) (s : Spin (3+k)) : blockRadius k s^2+‖blockTail k s‖^2=1 := by
have h := WithLp.prod_norm_sq_eq_of_L2 (splitThree k s.val)
simpa only [blockRadius,blockHead,blockTail,LinearIsometryEquiv.norm_map,spin_norm,one_pow] using h.symm
theorem block_inner (k : ℕ) (s t : Spin (3+k)) :
⟪s.val,t.val⟫_ℝ=⟪blockHead k s,blockHead k t⟫_ℝ+⟪blockTail k s,blockTail k t⟫_ℝ := by
rw [← (splitThree k).inner_map_map]
rfl
theorem blockHead_first (k : ℕ) (s : Spin (3+k)) : (blockHead k s) 0=s.val ⟨0,by omega⟩ := rfl
theorem continuous_blockHead (k : ℕ) : Continuous (blockHead k) :=
(WithLp.continuous_fst _ _ _).comp ((splitThree k).continuous.comp continuous_subtype_val)
theorem continuous_blockTail (k : ℕ) : Continuous (blockTail k) :=
(WithLp.continuous_snd _ _ _).comp ((splitThree k).continuous.comp continuous_subtype_val)
theorem continuous_blockRadius (k : ℕ) : Continuous (blockRadius k) := (continuous_blockHead k).norm
def blockDirection (k : ℕ) (s : Spin (3+k)) : Spin 3 :=
if h : blockHead k s=0 then cylindricalSpin ⊤ true 0 else
⟨(blockRadius k s)⁻¹ • blockHead k s,by
rw [Metric.mem_sphere,dist_zero_right,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (blockRadius_nonneg k s))]
exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr h)⟩
theorem block_head_direction (k : ℕ) (s : Spin (3+k)) :
blockRadius k s • (blockDirection k s).val=blockHead k s := by
unfold blockDirection
split_ifs with h
· simp only [blockRadius,h,norm_zero,zero_smul]
· have hn : blockRadius k s ≠ 0 := norm_ne_zero_iff.mpr h
simp only [smul_smul,mul_inv_cancel₀ hn,one_smul]
def blockFiber (k : ℕ) (s : Spin (3+k)) (t : Spin 3) : Spin (3+k) :=
⟨(splitThree k).symm (WithLp.toLp 2 (blockRadius k s • t.val,blockTail k s)),by
rw [Metric.mem_sphere,dist_zero_right,LinearIsometryEquiv.norm_map]
have he := WithLp.prod_norm_sq_eq_of_L2 (WithLp.toLp 2 (blockRadius k s • t.val,blockTail k s))
have hn := block_norm_sq k s
simp only [WithLp.fst,WithLp.snd,norm_smul,Real.norm_of_nonneg (blockRadius_nonneg k s),spin_norm,mul_one] at he
nlinarith [norm_nonneg (WithLp.toLp 2 (blockRadius k s • t.val,blockTail k s))]⟩
@[simp] theorem blockHead_fiber (k : ℕ) (s : Spin (3+k)) (t : Spin 3) :
blockHead k (blockFiber k s t)=blockRadius k s • t.val := by
simp [blockHead,blockFiber]
@[simp] theorem blockTail_fiber (k : ℕ) (s : Spin (3+k)) (t : Spin 3) :
blockTail k (blockFiber k s t)=blockTail k s := by
simp [blockTail,blockFiber]
theorem blockFiber_inner (k : ℕ) (s q : Spin (3+k)) (t u : Spin 3) :
⟪(blockFiber k s t).val,(blockFiber k q u).val⟫_ℝ=
blockRadius k s*blockRadius k q*⟪t.val,u.val⟫_ℝ+⟪blockTail k s,blockTail k q⟫_ℝ := by
rw [block_inner,blockHead_fiber,blockHead_fiber,blockTail_fiber,blockTail_fiber]
simp only [real_inner_smul_left,real_inner_smul_right]
ring
theorem continuous_blockFiber (k : ℕ) : Continuous (Function.uncurry (blockFiber k)) := by
apply Continuous.subtype_mk
apply (splitThree k).symm.continuous.comp
exact (WithLp.prodContinuousLinearEquiv 2 ℝ _ _).symm.continuous.comp
((((continuous_blockRadius k).comp continuous_fst).smul (continuous_subtype_val.comp continuous_snd)).prodMk
((continuous_blockTail k).comp continuous_fst))
def blockRotation (k : ℕ) (Q : SpinOrthogonal 3) :
EuclideanSpace ℝ (Fin (3+k)) ≃ₗᵢ[ℝ] EuclideanSpace ℝ (Fin (3+k)) :=
((splitThree k).trans (LinearIsometryEquiv.withLpProdCongr 2 (Unitary.linearIsometryEquiv Q) (.refl ℝ _))).trans (splitThree k).symm
def blockRotateSpin (k : ℕ) (Q : SpinOrthogonal 3) (s : Spin (3+k)) : Spin (3+k) :=
spinRotation (blockRotation k Q) s
theorem blockRotateSpin_fiber (k : ℕ) (Q : SpinOrthogonal 3) (s : Spin (3+k)) :
blockRotateSpin k Q s=blockFiber k s (orthogonalSpin Q (blockDirection k s)) := by
apply Subtype.ext
apply (splitThree k).injective
change splitThree k ((splitThree k).symm _)=splitThree k ((splitThree k).symm _)
simp only [LinearIsometryEquiv.apply_symm_apply]
apply (WithLp.equiv 2 _).injective
apply Prod.ext
· change (Unitary.linearIsometryEquiv Q) (blockHead k s)=
blockRadius k s • ((Unitary.linearIsometryEquiv Q) (blockDirection k s).val)
rw [← map_smul,block_head_direction]
· rfl
theorem continuous_blockRotateSpin (k : ℕ) : Continuous (Function.uncurry (blockRotateSpin k)) := by
apply Continuous.subtype_mk
change Continuous (fun p : SpinOrthogonal 3×Spin (3+k) =>
(splitThree k).symm (WithLp.toLp 2 (((p.1:SpinOperator 3) (blockHead k p.2)),blockTail k p.2)))
apply (splitThree k).symm.continuous.comp
apply (WithLp.prodContinuousLinearEquiv 2 ℝ _ _).symm.continuous.comp
exact ((continuous_subtype_val.comp continuous_fst).clm_apply ((continuous_blockHead k).comp continuous_snd)).prodMk
((continuous_blockTail k).comp continuous_snd)
end ClassicalON
end
end OAI
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