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import OAI.Probability.LogConcave.OraclePrograms.FullTape
namespace OAI
section
noncomputable section
namespace LogConcaveSampling.OracleCompiler.CircuitParameters
open MeanTree Coupling MeasureTheory ProbabilityTheory
open scoped Classical NNReal
variable (C : CircuitParameters) {d : ℕ}
theorem sampleFrom_correct {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam)
{q Kf Ki ef ei en ηmin : ℝ} (hd : 1≤d) (hlam : (lam:ℝ)≤1)
(hq : 0≤q) (hql : q≤1/4) (hR : Real.sqrt (1-C.T^2)≤ηmin)
(hD : 1≤C.D) (hA : C.A=C.actualA) (hAf : C.Af=C.actualAf)
(hKf : 0≤Kf) (hKi : 0≤Ki) (hef : 0<ef) (hei : 0≤ei) (hen : 0≤en)
(hsmall : 2*C.A*q*(2*Real.pi+2)≤1) (hcon : 4*C.A*q*Ki≤1)
(Mf Mi : ℝ → ℝ → SeedProgram d) (B : ℕ) (hBD : (B:ℝ)≤C.D^2)
(hbudget : ∀(s : Bool) r τ,MeanBudget B r ((if s then Mf else Mi) r τ))
(hf : MeanCorrect F lam ((1+4*C.Af)*q) Kf ef Mf)
(hi : MeanCorrect F lam (q*(1+16*C.D*C.A/Real.sqrt (1-C.T^2))) Ki ei Mi)
(hnum : C.SampleNumericalReady hF q en) :
SampleCorrect F lam q (C.Af*C.depthBound*Kf)
(C.sampleError C.depthBound Kf q ei ef en) ηmin (C.sampleFrom Mf Mi) := by
have hA0 : 0<C.A := hA ▸ C.actualA_pos
have hAf0 : 0<C.Af := hAf ▸ C.actualAf_pos
have small {r η : ℝ} (hs : SampleSize lam q r η) :=
physical_small_of_scaled hA0.le hq hKi
(show 0≤(lam:ℝ)*C.sampleCenterBudget r by unfold sampleCenterBudget; positivity)
(C.sampleCenter_scaled hs lam.coe_nonneg hA0.le) hsmall hcon
constructor
· intro r η hs hη x y g
have he := C.sampleFrom_lipschitz hs (hR.trans hη) (by linarith) hA hAf hKf hKi
(small hs).2.2 Mf Mi hf hi x y g
apply he.trans
have hdep : (depth (C.sampleTree (d:=d) r η hs.2.2.1 hs.2.2.2.1):ℝ)≤C.depthBound :=
by exact_mod_cast C.sampleTree_depth (d:=d) r η hs.2.2.1 hs.2.2.2.1
have hh := mul_le_mul_of_nonneg_right hdep
(show 0≤(lam:ℝ)*r*C.Af*Kf*‖x-y‖ by positivity [hs.1])
nlinarith only [hh]
· intro r η hs hη x
have hL : (lam:ℝ)*r≤1 := (mul_le_of_le_one_left hs.1.le hlam).trans hs.2.1
have hlr : (lam:ℝ)*r^2≤1/2 := by
have hh := hs.2.2.2.2.trans (mul_le_of_le_one_right hq hs.2.2.2.1)
linarith
have hp := C.sampleFrom_squared hF hd hs hql hL (hR.trans hη) hD hA hAf hKf hKi hef hei
(small hs).1 (small hs).2.1 Mf Mi B hBD hbudget hf hi x hlr
(hnum x r η hs hlr).1 (hnum x r η hs hlr).2
apply hp.mono
let dep := (depth (C.sampleTree (d:=d) r η hs.2.2.1 hs.2.2.2.1):ℝ)
let a := 2*(1+dep)*(C.Af*Kf*q*ei+ef)
have hdep : dep≤C.depthBound := by dsimp only [dep]; exact_mod_cast C.sampleTree_depth (d:=d) r η hs.2.2.1 hs.2.2.2.1
have hdep0 : 0≤dep := by dsimp [dep]; positivity
have ha : 0≤a := by dsimp [a]; positivity
have ha' : a≤C.compilerError C.depthBound Kf q ei ef := by
unfold compilerError
dsimp only [a]
have hh : 0≤C.Af*Kf*q*ei+ef := by positivity
nlinarith [mul_nonneg hh (sub_nonneg.mpr hdep)]
apply (sample_error_envelope hs.2.2.1.le hs.2.2.2.1 ha hen).trans
have he : 20*a+2*en≤C.sampleError C.depthBound Kf q ei ef en := by unfold sampleError; linarith
exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (by positivity : 0≤20*a+2*en) he 2) (sq_nonneg _)
end LogConcaveSampling.OracleCompiler.CircuitParameters
end
end
end OAI
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