| import Mathlib |
| import OAI.Probability.Perceptron.Variational.GlobalContact |
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|
| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| lemma source_global_contact_density (n k : ℕ) (P : Measure BrownianPath) (f : ℝ →ᵇ ℝ) |
| (p d : Fin (n+1)→ℕ) (w : Fin (k+1)→ℝ) (q : Fin (k+1)→Time) |
| (hw : ∀ i, 0<w i) (hw1 : ∑ i, w i=1) (δ : ℝ) (α : ℝ≥0) (H : ℝ) |
| {a : SourceContactParameter n k} (ha : a∈sourceContactCompactDomain n k α H) |
| (hmin : IsMinOn (sourceStepContactObjective n k P f p d w q (fun i => (hw i).le) hw1 δ) |
| (sourceContactCompactDomain n k α H) a) (ht : 0<a.1) : |
| controlValue P f (weightedStepTrial w q (fun i => (hw i).le) hw1)+δ ≤ |
| sourceDensityIncrement n k f p d a.2.1 (stepCumulative w) a.2.2 a.1 := by |
| let V := controlValue P f (weightedStepTrial w q (fun i => (hw i).le) hw1)+δ |
| let F : ℝ → ℝ := fun s => -sourceExpectedPressure n k f p d a.2.1 (stepCumulative w) |
| s.toNNReal a.2.2+s*V |
| have hm : IsMinOn F (Icc 0 (a.1:ℝ)) a.1 := by |
| intro s hs |
| have hs' : s.toNNReal ≤ α := (Real.toNNReal_le_toNNReal hs.2).trans (by simpa using ha.1.2) |
| have he := hmin (show (s.toNNReal,a.2.1,a.2.2)∈sourceContactCompactDomain n k α H from |
| ⟨⟨zero_le,hs'⟩,ha.2⟩) |
| simp only [mem_ofPred_eq,sourceStepContactObjective,sourceJointPressure] at he |
| rw [Real.coe_toNNReal _ hs.1] at he |
| change F (a.1:ℝ) ≤ F s |
| dsimp [F,V] |
| simp only [Real.toNNReal_coe] |
| linarith |
| have hd : HasDerivAt F (-sourceDensityIncrement n k f p d a.2.1 (stepCumulative w) a.2.2 a.1+V) a.1 := by |
| convert |
| (sourceExpectedPressure_density_derivative n k f p d a.2.1 (stepCumulative w) a.2.2 |
| (stepCumulative_strictMono w hw) (stepCumulative_pos w hw) (stepCumulative_lt_one w hw hw1) |
| a.1 ht).neg.add ((hasDerivAt_id (a.1:ℝ)).mul_const V) using 1 <;> first | rfl | simp |
| have hseg : segment ℝ (a.1:ℝ) 0 ⊆ Icc 0 (a.1:ℝ) := by |
| rw [segment_symm,segment_eq_Icc a.1.coe_nonneg] |
| have hn := hm.isLocalMinOn.hasFDerivWithinAt_nonneg hd.hasFDerivAt.hasFDerivWithinAt |
| (sub_mem_posTangentConeAt_of_segment_subset hseg) |
| change 0 ≤ (0-(a.1:ℝ))*(-sourceDensityIncrement n k f p d a.2.1 (stepCumulative w) a.2.2 a.1+V) at hn |
| change V ≤ _ |
| nlinarith [show (0:ℝ)<(a.1:ℝ) from ht] |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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