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dd14c26 | 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 | 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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