File size: 3,772 Bytes
a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Interpolation.ArrayGeometry
import OAI.Probability.Perceptron.Variational.TangentConsistency
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set TopologicalSpace
open scoped NNReal BoundedContinuousFunction ContDiff
namespace SphericalPerceptronFreeEnergy
lemma compact_other_edge_law {K : Type*} [TopologicalSpace K] [MeasurableSpace K]
[BorelSpace K] [MetrizableSpace K] [SecondCountableTopology K]
(μ : ProbabilityMeasure (CompactArray K))
(hGG : ∀ (F : CompactBlock K 2→ᵇℝ) (g : K→ᵇℝ),compactGGDefect μ 2 0 F g=0)
(he : MeasurePreserving (compactRelabel (K:=K) (Equiv.swap 0 1)) (μ : Measure _) (μ : Measure _))
(hsy : ∀ᵐ Q ∂(μ : Measure (CompactArray K)),Q 1 0=Q 0 1) :
(μ : Measure (CompactArray K)).map (fun Q=>(Q 0 1,Q 1 2))=
(1/2:ℝ≥0) • (((μ : Measure (CompactArray K)).map (fun Q=>Q 0 1)).prod
((μ : Measure (CompactArray K)).map (fun Q=>Q 0 1)))+
(1/2:ℝ≥0) • ((μ : Measure (CompactArray K)).map (fun Q=>(Q 0 1,Q 0 1))) := by
have h := congrArg (Measure.map (fun Q : CompactArray K=>(Q 0 1,Q 0 2))) he.map_eq
rw [Measure.map_map (by fun_prop) he.measurable] at h
have h' : (μ : Measure (CompactArray K)).map (fun Q=>(Q 1 0,Q 1 2))=
(μ : Measure (CompactArray K)).map (fun Q=>(Q 0 1,Q 0 2)) := by
simpa [Function.comp_def,compactRelabel,Equiv.swap_apply_def] using h
have hm : (μ : Measure (CompactArray K)).map (fun Q=>(Q 1 0,Q 1 2))=
(μ : Measure (CompactArray K)).map (fun Q=>(Q 0 1,Q 1 2)) :=
Measure.map_congr (hsy.mono (fun Q hQ=>by dsimp only; rw [hQ]))
rw [←hm,h',compact_two_edge_law μ hGG]
lemma array_tangent_self_consistency (μ : ProbabilityMeasure (CompactArray Time))
(hGG : ∀ (F : CompactBlock Time 2→ᵇℝ) (g : Time→ᵇℝ),compactGGDefect μ 2 0 F g=0)
(he : MeasurePreserving (compactRelabel (K:=Time) (Equiv.swap 0 1)) (μ : Measure _) (μ : Measure _))
(hsy : ∀ᵐ Q ∂(μ : Measure (CompactArray Time)),∀ i j,Q i j=Q j i)
(hul : ∀ᵐ Q ∂(μ : Measure (CompactArray Time)),∀ i j k,min (Q i j:ℝ) (Q i k:ℝ)≤(Q j k:ℝ))
(a : Time→ℝ) (ha : Measurable a) {A : ℝ} (hA : 0≤A) (hab : ∀ x,0≤a x ∧ a x≤A)
(hT : ∀ G : ℝ→ᵇℝ,ContDiff ℝ 1 (G : ℝ→ℝ)→
(∫ Q : CompactArray Time,G (Q 0 1)*((Q 0 1:ℝ)-a (Q 0 1)*(1-(Q 0 1:ℝ)^2)+
2*a (Q 0 2)*((Q 1 2:ℝ)-(Q 0 1:ℝ)*(Q 0 2:ℝ))) ∂(μ : Measure _))=0) :
(∀ᵐ x : Time ∂((μ : Measure (CompactArray Time)).map (fun Q=>Q 0 1)),
a x=∫ y in 0..(x:ℝ),(realTail ((μ : Measure (CompactArray Time)).map (fun Q=>Q 0 1)) y)⁻¹^2) ∧
∀ᵐ x : Time ∂((μ : Measure (CompactArray Time)).map (fun Q=>Q 0 1)),(x:ℝ)≤A/(1+A) := by
let ν := (μ : Measure (CompactArray Time)).map (fun Q=>Q 0 1)
have : IsProbabilityMeasure ν := by dsimp [ν]; infer_instance
have hd : ν.map (fun x=>(x,x))=(μ : Measure (CompactArray Time)).map (fun Q=>(Q 0 1,Q 0 1)) := by
change ((μ : Measure (CompactArray Time)).map (fun Q=>Q 0 1)).map (fun x=>(x,x))=_
rw [Measure.map_map (by fun_prop) (by fun_prop)]
rfl
apply tangent_sphere_self_consistency (μ : Measure (CompactArray Time)) ν (fun Q=>Q 0 1) (fun Q=>Q 0 2) (fun Q=>Q 1 2)
(by fun_prop) (by fun_prop) (by fun_prop) rfl
(by rw [hd]; exact compact_two_edge_law μ hGG)
(by rw [hd]; exact compact_other_edge_law μ hGG he (hsy.mono (fun Q hQ=>hQ 1 0)))
_ a ha hA hab hT
filter_upwards [hsy,hul] with Q hS hU
refine ⟨hU 0 1 2,?_,?_⟩
· have h := hU 1 0 2
simpa only [hS 1 0] using h
· have h := hU 2 0 1
simpa only [hS 2 0,hS 2 1] using h
end SphericalPerceptronFreeEnergy
end
end OAI
|