| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.VelocityWorkCoefficient |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Filter Quadrature MeanTree |
| open scoped Topology |
|
|
| lemma logMeshCount_rate {T : ℕ → ℝ} {b : ℝ} (hb : 0≤b) |
| (hT0 : ∀ᶠ d : ℕ in atTop,0<T d) (hT1 : ∀ᶠ d : ℕ in atTop,T d<1) |
| (hL : LogPowerRate (fun d => logMeshLength (T d)) 0) : |
| LogPowerRate (fun d => (logMeshCount (T d) ((d:ℝ)^(-b)):ℝ)) (-b) := by |
| have hr := hL.mul (LogPowerRate.inv_power b) |
| simp only [zero_add] at hr |
| apply LogPowerRate.of_le (hr.add ((LogPowerRate.const 1).mono (by linarith))) |
| · exact Eventually.of_forall (fun _ => Nat.cast_nonneg _) |
| · filter_upwards [hT0,hT1,eventually_ge_atTop (1:ℕ)] with d hd0 hd1 hd |
| have hp : (0:ℝ)<d := by exact_mod_cast hd |
| simpa only [div_eq_mul_inv] using (logMeshCount_bound hd0 hd1 (Real.rpow_pos_of_pos hp (-b))).le |
|
|
| lemma probabilityNode_card_rate {T : ℕ → ℝ} {b : ℝ} (hb : 0≤b) (n : ℕ) |
| (hT0 : ∀ᶠ d : ℕ in atTop,0<T d) (hT1 : ∀ᶠ d : ℕ in atTop,T d<1) |
| (hL : LogPowerRate (fun d => logMeshLength (T d)) 0) : |
| LogPowerRate (fun d => (Fintype.card (ProbabilityNode (T d) ((d:ℝ)^(-b)) n):ℝ)) (-b) := by |
| simpa only [probabilityNode_card,Nat.cast_mul,Nat.cast_add,Nat.cast_one] using |
| (logMeshCount_rate hb hT0 hT1 hL).mul_const ((n:ℝ)+1) |
|
|
| theorem meanWork_rate {p : ℕ → ℕ} {b : ℝ} (hb : 0≤b) |
| (hp : LogPowerRate (fun d => (p d:ℝ)) (-b)) (q m N nc Nc : ℕ) : |
| LogPowerRate (fun d => (meanWork (p d) q m N nc Nc 0 0 0:ℝ)) (-((Nc*(2*N+2):ℕ):ℝ)*b) := by |
| have hr := (hp.add ((LogPowerRate.const 1).mono (by linarith))).pow (Nc*(2*N+2)) |
| have he := hr.const_mul (((nc+1)*((nc+1)*velocityWorkCoefficient q m N)^Nc:ℕ):ℝ) |
| simp only [mul_neg,neg_mul] at he ⊢ |
| apply LogPowerRate.of_le he |
| · exact Eventually.of_forall (fun _ => Nat.cast_nonneg _) |
| · apply Eventually.of_forall |
| intro d |
| exact_mod_cast meanWork_polynomial (p d) q m N nc Nc |
|
|
| lemma sampleWork_rate {p : ℕ → ℕ} {b : ℝ} (hb : 0≤b) |
| (hp : LogPowerRate (fun d => (p d:ℝ)) (-b)) (N : ℕ) : |
| LogPowerRate (fun d => (flowWork (p d) N 0 0:ℝ)) (-(N:ℝ)*b) := by |
| have he := (hp.add ((LogPowerRate.const 1).mono (by linarith))).pow N |
| simp only [mul_neg,neg_mul] at he ⊢ |
| apply LogPowerRate.of_le he |
| · exact Eventually.of_forall (fun _ => Nat.cast_nonneg _) |
| · apply Eventually.of_forall |
| intro d |
| have hh := flowWork_bound (p d) N 0 |
| simp only [zero_add,one_mul] at hh |
| have : flowWork (p d) N 0 0≤(p d+1)^N := by omega |
| exact_mod_cast this |
|
|
| lemma routineBudget_rate {c : ℕ → ℕ} {a : ℝ} (ha : 0≤a) |
| (hc : LogPowerRate (fun d => (c d:ℝ)) (-a)) (rank : ℕ) : |
| LogPowerRate (fun d => ((2*(40*c d+40)^rank:ℕ):ℝ)) (-(rank:ℝ)*a) := by |
| have he := (((hc.const_mul 40).add ((LogPowerRate.const 40).mono (by linarith))).pow rank).const_mul 2 |
| simp only [mul_neg,neg_mul] at he ⊢ |
| simpa only [Nat.cast_mul,Nat.cast_pow,Nat.cast_add,Nat.cast_ofNat] using he |
| end LogConcaveSampling |
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|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Filter Quadrature |
| open scoped Topology |
|
|
| lemma sampleCorrelation_antitone {a b : ℝ} (ha : 0≤a) (hab : a≤b) : |
| sampleCorrelation b≤ sampleCorrelation a := by |
| unfold sampleCorrelation |
| apply (inv_le_inv₀ (Real.sqrt_pos.mpr (by positivity)) |
| (Real.sqrt_pos.mpr (by positivity))).mpr |
| exact Real.sqrt_le_sqrt (by nlinarith) |
|
|
| lemma logMeshCount_mono {T U h : ℝ} (hT : T≤U) (hU : U<1) (hh : 0<h) : |
| logMeshCount T h≤logMeshCount U h := by |
| apply Nat.ceil_mono |
| apply div_le_div_of_nonneg_right _ hh.le |
| unfold logMeshLength |
| have he := Real.log_le_log (by linarith : 0<1-U) (by linarith : 1-U≤1-T) |
| linarith |
|
|
| lemma sampleCorrelation_radius_lower {η : ℝ} (hη : 0<η) (hη1 : η≤1) : |
| η/4≤Real.sqrt (1-(sampleCorrelation η)^2) := by |
| have hp := sampleCorrelation_properties hη hη1 |
| have ht : 0<sampleCorrelation η := by linarith |
| have he : Real.sqrt (1-(sampleCorrelation η)^2)=sampleCorrelation η*(η/2) := by |
| have h := congrArg (fun z : ℝ => sampleCorrelation η*z) hp.2.2 |
| simpa only [←mul_assoc,mul_inv_cancel₀ ht.ne',one_mul] using h |
| rw [he] |
| nlinarith |
| |
| lemma logMeshLength_sample_rate {η : ℕ → ℝ} {a : ℝ} |
| (hη : ∀ᶠ d : ℕ in atTop,(d:ℝ)^(-a)≤η d ∧ η d≤1) : |
| LogPowerRate (fun d => logMeshLength (sampleCorrelation (η d))) 0 := by |
| have hr := ((LogPowerRate.const (Real.log 2)).add |
| (((LogPowerRate.log_power a).add (LogPowerRate.const (-Real.log 4))).const_mul (-2))) |
| apply LogPowerRate.of_le hr |
| · filter_upwards [hη,eventually_ge_atTop (1:ℕ)] with d hd hd1 |
| have hp : (0:ℝ)<d := by exact_mod_cast hd1 |
| have hpη := sampleCorrelation_properties (lt_of_lt_of_le (Real.rpow_pos_of_pos hp _) hd.1) hd.2 |
| exact (logMeshLength_pos (by linarith [hpη.1]) hpη.2.1).le |
| · filter_upwards [hη,eventually_ge_atTop (1:ℕ)] with d hd hd1 |
| have hp : (0:ℝ)<d := by exact_mod_cast hd1 |
| have hR : 0<(d:ℝ)^(-a) := Real.rpow_pos_of_pos hp _ |
| have hηpos := lt_of_lt_of_le hR hd.1 |
| have hpη := sampleCorrelation_properties hηpos hd.2 |
| have hh := sampleCorrelation_radius_lower hηpos hd.2 |
| have hu : (d:ℝ)^(-a)/4≤Real.sqrt (1-(sampleCorrelation (η d))^2) := by linarith |
| have hs : 0≤1-(sampleCorrelation (η d))^2 := by nlinarith [hpη.1,hpη.2.1] |
| have hdR : ((d:ℝ)^(-a)/4)^2≤1-(sampleCorrelation (η d))^2 := by |
| have he := pow_le_pow_left₀ (by positivity : 0≤(d:ℝ)^(-a)/4) hu 2 |
| rwa [Real.sq_sqrt hs] at he |
| have he := logMeshLength_le_scale (by linarith [hpη.1]) hpη.2.1 |
| (by positivity : 0<(d:ℝ)^(-a)/4) hdR |
| rw [Real.log_div hR.ne' (by norm_num)] at he |
| linarith |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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