| import Mathlib |
| import OAI.Probability.LogConcave.Complexity.Envelope |
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|
| namespace OAI |
|
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| section |
| noncomputable section |
| namespace LogConcaveSampling.OracleCompiler |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Classical NNReal Topology |
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| theorem exists_primitive_producer {κ P : ℝ} (hκ : 0<κ) (hP : 1≤P) : |
| ∃t : ℝ,0<t ∧ t<1/100 ∧ 40*(P+2)*t<1/4 ∧ |
| ∃K : ℝ,0≤K ∧ ∃e : ℕ → ℝ,LogPowerRate e (κ*P) ∧ |
| (∀ᶠ d : ℕ in atTop,0<e d) ∧ |
| ∀ᶠ d : ℕ in atTop,∃S : ℝ → ℝ → ReservedProgram d,∃B : ℕ, |
| (B:ℝ)≤(d:ℝ)^(κ*t) ∧ |
| (∀r : ℝ,SampleBudget B r 1 (S r 1)) ∧ |
| ∀{F : Point d → ℝ} {lam : ℝ≥0},Primitive F lam → 0<lam → (lam:ℝ)≤1 → |
| SampleCorrect F lam ((d:ℝ)^(-κ)) K (e d) ((d:ℝ)^(-(κ*t))) S := by |
| let t : ℝ := 1/(1000*(P+2)) |
| have ht : 0<t := by dsimp [t]; positivity |
| have heq : t*(1000*(P+2))=1 := by dsimp [t]; exact one_div_mul_cancel (by positivity) |
| have hsmall : 40*(P+2)*t<1/4 := by nlinarith only [heq] |
| have hts : t<1/100 := by nlinarith [mul_nonneg ht.le (by linarith : 0≤P)] |
| let J := P+1 |
| have hJ : 1≤J := by dsimp [J]; linarith |
| let N := numericalLayerCount J |
| obtain ⟨w,hw,hwt,hbudget⟩ := exists_uniform_routine_mesh (K:=P) hκ ht (by linarith) N |
| obtain ⟨n,hnw,hnt,hnn,hnum⟩ := exists_source_numerical_ready hκ hJ ht hts hw hwt |
| have hR : ∀ᶠ d : ℕ in atTop,(d:ℝ)^(-(κ*t))≤1 := by |
| filter_upwards [eventually_ge_atTop (1:ℕ)] with d hd |
| exact Real.rpow_le_one_of_one_le_of_nonpos (by exact_mod_cast hd) (by nlinarith) |
| have hb : ∀ᶠ d : ℕ in atTop,∀C : CircuitParameters,SourceShape κ t w n N d C → |
| ∃B : ℕ,(B:ℝ)≤C.D ∧ ∀L : RoutineLabel,L.Admissible t P → |
| RoutineBudget B (Real.sqrt (1-C.T^2)/C.T) 1 (constructRoutine d C ht (by linarith) L) := by |
| filter_upwards [hbudget n,hR] with d hd hRd |
| intro C hC |
| obtain ⟨B,hB,hbud⟩ := hd C hC |
| exact ⟨B,hB,fun L hL => hbud L hL 1 hRd le_rfl⟩ |
| have ha := uniform_routine_accuracy hκ ht hts hw hwt (show P≤J by dsimp [J]; linarith) |
| (show N=numericalLayerCount J from rfl) hb hnum |
| ⟨.sample,1,P⟩ (topLabel_admissible (by linarith) ht.le hsmall) |
| have hq : LogPowerRate (fun d : ℕ => (d:ℝ)^(-κ)) (κ*1) := by |
| simpa only [mul_one] using LogPowerRate.power κ |
| have hqp : ∀ᶠ d : ℕ in atTop,0<(d:ℝ)^(-κ) := by |
| filter_upwards [eventually_ge_atTop (1:ℕ)] with d hd |
| exact Real.rpow_pos_of_pos (by exact_mod_cast hd) _ |
| obtain ⟨K,hK,e,her,hep,he⟩ := ha _ hq hqp |
| refine ⟨t,ht,hts,hsmall,K,hK,e,her,hep,?_⟩ |
| filter_upwards [he,hb,exists_source_shape hκ ht hw n N] with d hd hbd hshape |
| obtain ⟨C,hC⟩ := hshape |
| obtain ⟨B,hB,hbud⟩ := hbd C hC |
| refine ⟨constructRoutine d C ht (by linarith) ⟨.sample,1,P⟩,B,?_,?_,?_⟩ |
| · simpa only [hC.D_eq] using hB |
| · intro r |
| exact hbud _ (topLabel_admissible (by linarith) ht.le hsmall) r 1 (Or.inr rfl) |
| · intro F lam hF hlam hlam1 |
| exact hd C hC hF hlam hlam1 |
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| end LogConcaveSampling.OracleCompiler |
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| end |
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| end |
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| end OAI |
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