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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 | import OAI.Probability.ClassicalON.BlockGeometry
namespace OAI
universe uV
noncomputable section
open MeasureTheory
open scoped InnerProductSpace Classical
namespace ClassicalON
theorem orthogonalSpin_eval_preserves (n : ℕ) [NeZero n] (s : Spin n) :
MeasurePreserving (fun Q : SpinOrthogonal n => orthogonalSpin Q s)
(orthogonalLaw n) (sphereProbability n) := by
have hc : Continuous (fun Q : SpinOrthogonal n => orthogonalSpin Q s) :=
(continuous_orthogonalSpin n).comp (continuous_id.prodMk continuous_const)
refine ⟨hc.measurable,?_⟩
apply Measure.ext_of_integral_eq_on_compactlySupported
intro f
rw [integral_map (f := fun x => f x) hc.measurable.aemeasurable f.continuous.aestronglyMeasurable]
exact integral_orthogonalSpin_eq f f.continuous s
variable {V : Type uV} [Fintype V]
def blockFiberConfiguration (k : ℕ) (s : V → Spin (3+k)) (t : V → Spin 3) : V → Spin (3+k) :=
fun v => blockFiber k (s v) (t v)
omit [Fintype V] in
theorem continuous_blockFiberConfiguration (k : ℕ) :
Continuous (Function.uncurry (blockFiberConfiguration (V := V) k)) := by
apply continuous_pi
intro v
exact (continuous_blockFiber k).comp
(f := fun p : (V → Spin (3+k))×(V → Spin 3) => (p.1 v,p.2 v))
(((continuous_apply v).comp continuous_fst).prodMk ((continuous_apply v).comp continuous_snd))
def blockRotateConfiguration (k : ℕ) (Q : V → SpinOrthogonal 3) :
(V → Spin (3+k)) ≃ₜ (V → Spin (3+k)) :=
SpinSystem.rotateConfiguration (fun v => blockRotation k (Q v))
omit [Fintype V] in
theorem blockRotateConfiguration_fiber (k : ℕ) (Q : V → SpinOrthogonal 3) (s : V → Spin (3+k)) :
blockRotateConfiguration k Q s=
blockFiberConfiguration k s (fun v => orthogonalSpin (Q v) (blockDirection k (s v))) := by
funext v
exact blockRotateSpin_fiber k (Q v) (s v)
omit [Fintype V] in
theorem continuous_blockRotateConfiguration (k : ℕ) : Continuous
(fun p : (V → SpinOrthogonal 3)×(V → Spin (3+k)) => blockRotateConfiguration k p.1 p.2) := by
apply continuous_pi
intro v
exact (continuous_blockRotateSpin k).comp
(f := fun p : (V → SpinOrthogonal 3)×(V → Spin (3+k)) => (p.1 v,p.2 v))
(((continuous_apply v).comp continuous_fst).prodMk ((continuous_apply v).comp continuous_snd))
theorem blockRotateConfiguration_preserves (k : ℕ) (Q : V → SpinOrthogonal 3) :
MeasurePreserving (blockRotateConfiguration k Q) (freeSpinReference (3+k)) (freeSpinReference (3+k)) :=
measurePreserving_pi (fun _ : V => sphereProbability (3+k)) (fun _ => sphereProbability (3+k))
(fun v => spinRotation_preserves_probability (blockRotation k (Q v)))
def orthogonalConfigurationLaw : Measure (V → SpinOrthogonal 3) := Measure.pi (fun _ => orthogonalLaw 3)
instance orthogonalConfigurationLaw_isProbability : IsProbabilityMeasure (orthogonalConfigurationLaw (V := V)) := by
unfold orthogonalConfigurationLaw
infer_instance
theorem integral_block_orbit (k : ℕ) (f : (V → Spin (3+k)) → ℝ) (hf : Continuous f)
(s : V → Spin (3+k)) :
(∫ Q,f (blockRotateConfiguration k Q s) ∂orthogonalConfigurationLaw)=
∫ t,f (blockFiberConfiguration k s t) ∂freeSpinReference 3 := by
have hp := measurePreserving_pi (fun _ : V => orthogonalLaw 3) (fun _ => sphereProbability 3)
(fun v => orthogonalSpin_eval_preserves 3 (blockDirection k (s v)))
have hc : Continuous (fun t : V → Spin 3 => f (blockFiberConfiguration k s t)) :=
hf.comp ((continuous_blockFiberConfiguration k).comp
(f := fun t : V → Spin 3 => (s,t)) (continuous_const.prodMk continuous_id))
rw [show freeSpinReference (V := V) 3=
(orthogonalConfigurationLaw (V := V)).map
(fun Q v => orthogonalSpin (Q v) (blockDirection k (s v))) from hp.map_eq.symm,
integral_map hp.measurable.aemeasurable hc.aestronglyMeasurable]
apply integral_congr_ae
filter_upwards with Q
rw [blockRotateConfiguration_fiber]
theorem integral_freeSpinReference_block (k : ℕ) (f : (V → Spin (3+k)) → ℝ) (hf : Continuous f) :
(∫ s,f s ∂freeSpinReference (3+k))=
∫ s,∫ t,f (blockFiberConfiguration k s t) ∂freeSpinReference 3 ∂freeSpinReference (3+k) := by
let : NeZero (3+k) := ⟨by omega⟩
have hc : Continuous (fun p : (V → Spin (3+k))×(V → SpinOrthogonal 3) =>
f (blockRotateConfiguration k p.2 p.1)) :=
hf.comp ((continuous_blockRotateConfiguration k).comp
(f := (Prod.swap : (V → Spin (3+k))×(V → SpinOrthogonal 3) →
(V → SpinOrthogonal 3)×(V → Spin (3+k)))) continuous_swap)
have he := integral_integral_swap (μ := freeSpinReference (3+k)) (ν := orthogonalConfigurationLaw)
(f := fun s Q => f (blockRotateConfiguration k Q s)) (compact_integrable hc)
have hi (Q : V → SpinOrthogonal 3) :
(∫ s,f (blockRotateConfiguration k Q s) ∂freeSpinReference (3+k))=∫ s,f s ∂freeSpinReference (3+k) :=
(blockRotateConfiguration_preserves k Q).integral_comp
(blockRotateConfiguration k Q).measurableEmbedding f
simp_rw [hi] at he
simp only [integral_const,probReal_univ,one_smul] at he
rw [← he]
exact integral_congr_ae (Filter.Eventually.of_forall (integral_block_orbit k f hf))
end ClassicalON
end
end OAI
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