| import Mathlib |
| import OAI.Probability.LogConcave.Complexity.KernelAnalyticNormalizedBudget |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Filter |
| open scoped Classical |
|
|
| theorem eventually_forall_of_sequences {X : Type*} [Nonempty X] |
| {P Q : ℕ → X → Prop} |
| (hne : ∀ᶠ d : ℕ in atTop, ∃x,P d x) |
| (hall : ∀f : ℕ → X,(∀ᶠ d : ℕ in atTop,P d (f d)) → |
| ∀ᶠ d : ℕ in atTop,Q d (f d)) : |
| ∀ᶠ d : ℕ in atTop,∀x,P d x → Q d x := by |
| let f : ℕ → X := fun d => if h : ∃x,P d x ∧ ¬Q d x then Classical.choose h |
| else if h' : ∃x,P d x then Classical.choose h' else Classical.choice ‹Nonempty X› |
| have hp : ∀ᶠ d : ℕ in atTop,P d (f d) := by |
| filter_upwards [hne] with d hd |
| dsimp only [f] |
| split_ifs with h |
| · exact (Classical.choose_spec h).1 |
| · exact Classical.choose_spec hd |
| filter_upwards [hall f hp] with d hd x hx |
| by_contra hn |
| have he : ∃x,P d x ∧ ¬Q d x := ⟨x,hx,hn⟩ |
| have hf : f d=Classical.choose he := dite_eq_left he |
| rw [hf] at hd |
| exact (Classical.choose_spec he).2 hd |
|
|
| lemma logPowerEnvelope_rate (C : ℝ) (k : ℕ) (a : ℝ) : |
| LogPowerRate (fun d => C*dimensionLog d^k*(d:ℝ)^(-a)) a := by |
| have he := ((LogPowerRate.log.pow k).const_mul C).mul (LogPowerRate.power a) |
| simpa only [mul_zero,zero_add] using he |
|
|
| lemma logPowerRate_of_envelope {f : ℕ → ℝ} {a C : ℝ} {k : ℕ} |
| (hn : ∀ᶠ d in atTop,0≤f d) |
| (he : ∀ᶠ d in atTop,f d≤C*dimensionLog d^k*(d:ℝ)^(-a)) : |
| LogPowerRate f a := LogPowerRate.of_le (logPowerEnvelope_rate C k a) hn he |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
