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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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 | import Mathlib
import OAI.Probability.Perceptron.Variational.CompactGGDensity
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
def compactScalarMonomial (p : ℕ) : CompactOverlap →ᵇ ℝ :=
BoundedContinuousFunction.mkOfCompact ⟨fun x : CompactOverlap => (x:ℝ)^p,by fun_prop⟩
@[simp] lemma compactScalarMonomial_apply (p : ℕ) (x : CompactOverlap) :
compactScalarMonomial p x = (x:ℝ)^p := rfl
lemma compactScalarMonomial_add (p q : ℕ) :
compactScalarMonomial (p+q)=compactScalarMonomial p*compactScalarMonomial q := by
ext x
simp only [compactScalarMonomial_apply,BoundedContinuousFunction.mul_apply,pow_add]
lemma compactScalar_monomials_total (L : (CompactOverlap →ᵇ ℝ) →L[ℝ] ℝ)
(hL : ∀ p, L (compactScalarMonomial p)=0) : L=0 := by
let e := ContinuousMap.linearIsometryBoundedOfCompact CompactOverlap ℝ ℝ
let L' := L.comp e.toContinuousLinearEquiv.toContinuousLinearMap
let m (a : ℕ) : C(CompactOverlap,ℝ) := (compactScalarMonomial a).toContinuousMap
have hm0 : m 0=1 := by ext x; simp [m,compactScalarMonomial_apply]
have hmadd (a b : ℕ) : m (a+b)=m a*m b := by
ext x
exact congrArg (fun f : CompactOverlap →ᵇ ℝ => f x) (compactScalarMonomial_add _ _)
let M : Submonoid C(CompactOverlap,ℝ) :=
{ carrier := Set.range m
one_mem' := ⟨0,hm0⟩
mul_mem' := by rintro _ _ ⟨a,rfl⟩ ⟨b,rfl⟩; exact ⟨a+b,hmadd a b⟩ }
let A := Algebra.adjoin ℝ (M : Set C(CompactOverlap,ℝ))
have hsep : A.SeparatesPoints := by
intro x y hxy
have hpd : compactScalarMonomial 1 x ≠ compactScalarMonomial 1 y := by
simpa only [compactScalarMonomial_apply,pow_one] using (Subtype.val_injective.ne hxy)
have hmem : m 1 ∈ A :=
Algebra.subset_adjoin (show m 1 ∈ M from ⟨1,rfl⟩)
exact ⟨m 1,⟨m 1,hmem,rfl⟩,hpd⟩
have hA : (A : Set C(CompactOverlap,ℝ)) ⊆ (L'.ker : Set C(CompactOverlap,ℝ)) := by
have hspan : Submodule.span ℝ (M : Set C(CompactOverlap,ℝ)) ≤ L'.ker := by
apply Submodule.span_le.mpr
rintro _ ⟨p,rfl⟩
change L (e ((compactScalarMonomial p).toContinuousMap)) = 0
have he : e ((compactScalarMonomial p).toContinuousMap) = compactScalarMonomial p := by
ext x; rfl
rw [he]
exact hL p
intro f hf
apply hspan
change f ∈ A.toSubmodule at hf
simpa only [A,Algebra.adjoin_eq_span,Submonoid.closure_eq] using hf
have hclosed := L'.isClosed_ker
have hc := closure_minimal hA hclosed
have htop := ContinuousMap.subalgebra_topologicalClosure_eq_top_of_separatesPoints A hsep
have hall (f : C(CompactOverlap,ℝ)) : L' f=0 := by
apply hc
change f ∈ A.topologicalClosure
rw [htop]
trivial
ext g
have h := hall (e.symm g)
simpa [L'] using h
lemma compactGG_of_scalar_monomials (μ : ProbabilityMeasure (CompactArray CompactOverlap))
(n : ℕ) (i : Fin n) (f : CompactBlock CompactOverlap n →ᵇ ℝ)
(h : ∀ p, compactGGDefect μ n i f (compactScalarMonomial p)=0)
(g : CompactOverlap →ᵇ ℝ) : compactGGDefect μ n i f g=0 := by
have hz := compactScalar_monomials_total (compactGGTestCLM μ n i f)
(fun p => (compactGGTestCLM_apply μ n i f _).trans (h p))
rw [← compactGGTestCLM_apply,hz]
rfl
end SphericalPerceptronFreeEnergy
end
end OAI
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