openAI-math-dataset / lean /OAI /Probability /LogConcave /Complexity /VelocityNormalizedBudgetCalibrated.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Sampling.VelocityJointLip | |
| import OAI.Probability.LogConcave.Sampling.NumericalLayerCount | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Filter Set MeasureTheory Quadrature | |
| open scoped Topology Classical BigOperators NNReal | |
| theorem velocityNormalizedBudget_calibrated (k n N : ℕ) (Ap Ah Lp : ℝ≥0) (C J₀ D : ℝ) | |
| {lam : ℕ → ℝ≥0} {r T : ℕ → ℝ} {κ J b t w : ℝ} | |
| (hκ : 0<κ) (hJ : 1≤J) (hb : 1/2≤b) (ht : 0<t) (htsmall : t<1/100) | |
| (hw : 0<w) (hwt : w<t) | |
| (hnw : J+10<w*((n:ℝ)-3)) (hnt : J+10<t*((n:ℝ)-3)) | |
| (hnn : 4*(J+10)<(n:ℝ)) (hN : 10*(J+2)≤(N:ℝ)) | |
| (hl : LogPowerRate (fun d => (lam d:ℝ)*(r d)^2) (κ*b)) | |
| (hT0 : ∀ᶠ d : ℕ in atTop,0<T d) (hT1 : ∀ᶠ d : ℕ in atTop,T d<1) | |
| (hRT : ∀ᶠ d : ℕ in atTop,((d:ℝ)^(-(κ*t)))^2≤1-(T d)^2) | |
| (hsmall : ∀ᶠ d : ℕ in atTop,(d:ℝ)^(-(κ*w))≤Real.log 2) | |
| (hlarge : ∀ᶠ d : ℕ in atTop,(d:ℝ)^(-(κ*w))≤logMeshLength (T d)) : | |
| LogPowerRate (fun d => velocityNormalizedBudget d k n n N (lam d) Ap Ah Lp | |
| (r d) ((d:ℝ)^(-(κ*t))) (T d) ((d:ℝ)^(-(κ*w))) ((d:ℝ)^(-(κ*(4*t)))) C J₀ D) | |
| (2*κ*(J+8)) := by | |
| obtain ⟨hp,hh,hiter,ha₁,ha₂,hv₁,hv₂,hlip,hbeta,hcq,hniter,hmq⟩ := | |
| numerical_exponent_margins hJ hb ht htsmall hw hwt hnw hnt hnn hN | |
| have hB := kernelNormalizedTerm_rate k (n+1) N Ap Ah Lp (a:=κ*t) (b:=κ*w) (c:=κ*(4*t)) (e:=κ*(J+10)) (n:ℝ) hl | |
| (show κ*(J+10)≤((n+1:ℕ)+1:ℝ)*(κ*w) by | |
| have he := mul_le_mul_of_nonneg_left hp hκ.le | |
| push_cast; nlinarith) | |
| (show κ*(J+10)≤((n+1:ℕ)+1:ℝ)*(κ*(4*t)-2*(κ*t)) by | |
| have he := mul_le_mul_of_nonneg_left hh hκ.le | |
| push_cast; nlinarith) | |
| (show κ*(J+10)≤N*(κ*b) by | |
| have he := mul_le_mul_of_nonneg_left hiter hκ.le | |
| nlinarith) | |
| have hA := kernelActionNormalizedBudget_rate k (n+1) n N Ap Ah Lp C J₀ hB | |
| (show 2*κ*(J+9)≤2*(κ*(J+10)-κ*(4*t)) by | |
| have he := mul_le_mul_of_nonneg_left ha₁ hκ.le | |
| nlinarith) | |
| (show 2*κ*(J+9)≤2*(κ*(4*t))*n-4*(κ*t)*((n:ℝ)+1) by | |
| have he := mul_le_mul_of_nonneg_left ha₂ hκ.le | |
| nlinarith) | |
| obtain ⟨Dq,hDq,hDqu⟩ := exists_basisDerivative_budget (probabilityNodes (n+1)) | |
| have hW := terminalQuadratureWeight_rate n Dq (by linarith) hDqu (κ*t) (κ*w) | |
| hT0 hT1 hRT hsmall hlarge | |
| exact velocityNormalizedBudget_rate k n n N Ap Ah Lp C J₀ D hW hA | |
| (logMeshLength_rate (κ*t) hT0 hT1 hRT) | |
| (by have he := mul_le_mul_of_nonneg_left hv₁ hκ.le; nlinarith) | |
| (by have he := mul_le_mul_of_nonneg_left hv₂ hκ.le; nlinarith) | |
| lemma velocityJointLip_rate (n m : ℕ) (A : ℝ≥0) | |
| {lam : ℕ → ℝ≥0} {r s T : ℕ → ℝ} {b c u : ℝ} | |
| (hbc : 0≤b+c) (hr : ∀ᶠ d : ℕ in atTop,0<r d) (hs : ∀ᶠ d : ℕ in atTop,0<s d) | |
| (hT0 : ∀ᶠ d : ℕ in atTop,0<T d) (hT1 : ∀ᶠ d : ℕ in atTop,T d<1) | |
| (hW : LogPowerRate (fun d => ∑v,|terminalQuadratureWeight (T d) ((d:ℝ)^(-b)) n v|) (-b)) | |
| (hα : LogPowerRate (fun d => (lam d:ℝ)*r d/s d) u) : | |
| LogPowerRate (fun d => (velocityJointLip n m A (lam d) (r d) (T d) | |
| ((d:ℝ)^(-b)) ((d:ℝ)^(-c)) (s d):ℝ)) (u-b-c) := by | |
| have hS := (LogPowerRate.inv_power c).const_mul (∑i,|derivativeWeight (angleNodes m) i|) | |
| have hQS := (hW.const_mul (16*(A:ℝ))).mul hS | |
| have h1 := (LogPowerRate.const 1).mono (show -b+-c≤0 by linarith) | |
| have he := (hα.const_mul (Real.pi^2)).mul (hQS.add h1) | |
| have he' : LogPowerRate (fun d => Real.pi^2*((lam d:ℝ)*r d/s d)* | |
| (16*(A:ℝ)*(∑v,|terminalQuadratureWeight (T d) ((d:ℝ)^(-b)) n v|)* | |
| ((∑i,|derivativeWeight (angleNodes m) i|)/(d:ℝ)^(-c))+1)) (u-b-c) := by | |
| convert! he using 1 | |
| first | (simp only [div_eq_mul_inv]) | ring | |
| apply LogPowerRate.of_le he' | |
| · exact Eventually.of_forall (fun d => NNReal.coe_nonneg _) | |
| · filter_upwards [hr,hs,hT0,hT1,eventually_ge_atTop (1:ℕ)] with d hdr hds hdT0 hdT1 hd | |
| have hd0 : (0:ℝ)<d := by exact_mod_cast hd | |
| exact velocityJointLip_coefficient hdr hds (Real.rpow_pos_of_pos hd0 (-c)) hdT0 hdT1 (lam d) A n m | |
| namespace LogPowerRate | |
| lemma eventually_lt_const {f : ℕ → ℝ} {a C : ℝ} (hf : LogPowerRate f a) | |
| (ha : 0<a) (hC : 0<C) : ∀ᶠ d : ℕ in atTop, |f d|<C := by | |
| have he := hf.eventually_small (show a/2<a by linarith) | |
| have ht := (tendsto_rpow_neg_atTop (show 0<a/2 by linarith)).comp | |
| (tendsto_natCast_atTop_atTop (R:=ℝ)) | |
| have hb := ht.eventually (gt_mem_nhds hC) | |
| filter_upwards [he,hb] with d hd hb | |
| exact hd.trans_lt hb | |
| end LogPowerRate | |
| end LogConcaveSampling | |
| end | |
| end | |
| end OAI | |