File size: 2,643 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 | import Mathlib
import OAI.Probability.Perceptron.Variational.FieldDerivative
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology ENNReal NNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
def sourceFieldPolynomial {k : ℕ} (a : Fin (k+1)→ℝ) : Polynomial ℝ :=
Lagrange.interpolate Finset.univ (fun l : Fin (k+1) => (l.val:ℝ)/(k+1:ℕ)) a
lemma sourceFieldPolynomial_eval {k : ℕ} (a : Fin (k+1)→ℝ) (l : Fin (k+1)) :
(sourceFieldPolynomial a).eval ((l.val:ℝ)/(k+1:ℕ))=a l := by
unfold sourceFieldPolynomial
apply Lagrange.eval_interpolate_at_node (s := Finset.univ)
(v := fun l : Fin (k+1) => (l.val:ℝ)/(k+1:ℕ)) (i := l) a
· intro i hi j hj he
apply Fin.ext
have hv : (i.val:ℝ)=(j.val:ℝ) := (div_left_inj' (by positivity : ((k+1:ℕ):ℝ)≠0)).mp he
exact_mod_cast hv
· exact Finset.mem_univ l
lemma sourceFieldPolynomial_continuous {k : ℕ} (a : Fin (k+1)→ℝ) :
Continuous (fun t : unitInterval => (sourceFieldPolynomial a).eval t.val) :=
(sourceFieldPolynomial a).continuous.comp continuous_subtype_val
def sourceFieldArrayTest {k : ℕ} (a : Fin (k+1)→ℝ) : CompactArray CompactJointOverlap→ᵇ ℝ :=
BoundedContinuousFunction.mkOfCompact ⟨fun Q => (sourceFieldPolynomial a).eval (Q 1 0).2.val*(Q 1 0).1.val,
by fun_prop⟩
lemma sourceGibbsArray_field_integral (n k : ℕ) (f : ℝ →ᵇ ℝ)
(p d : Fin (n+1)→ℕ) (h : Fin (k+1)→ℝ) (hh0 : ∀ l, 0≤h l) (hh : Monotone h)
(u : Fin (n+1)→ℝ) (z : Fin k→ℝ) (t : ℝ≥0) (a : Fin (k+1)→ℝ) :
(∫ Q, sourceFieldArrayTest a Q ∂sourceGibbsArrayLaw n k f p d h u z t)=
∫ b, gibbsReplicaMean (sourceSpinLeafKernel n k b.1)
(sourceCouplingHamiltonian n k f p d h u b) 2 (sourceFieldObservable a)
∂((sourceBaseDataLaw n k z t).prod countableGaussianLaw) := by
let F : CompactBlock CompactJointOverlap 2→ᵇ ℝ := BoundedContinuousFunction.mkOfCompact
⟨fun Q => (sourceFieldPolynomial a).eval (Q 1 0).2.val*(Q 1 0).1.val,by fun_prop⟩
have he := sourceGibbsArray_block_integral n k f p d h hh0 hh u z t 2 F.measurable
(fun Q => F.norm_coe_le_norm Q)
change (∫ Q, sourceFieldArrayTest a Q ∂sourceGibbsArrayLaw n k f p d h u z t)=_ at he
rw [he]
apply integral_congr_ae
exact ae_of_all _ fun b => by
change gibbsReplicaMean (sourceSpinLeafKernel n k b.1) _ 2 _ = _
congr 1
funext x
dsimp [F,sourceJointOverlap,sourceFieldObservable]
rw [sourceFieldPolynomial_eval]
end SphericalPerceptronFreeEnergy
end
end OAI
|