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import OAI.Probability.LogConcave.Sampling.ProbabilityNormalizedTerm
namespace OAI
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory Quadrature
open scoped Classical BigOperators NNReal
lemma circuitD_dimension {d : ℕ} (F : Point d → ℝ) (x : Point d) :
(d:ℝ)≤(circuitD F x)^2 := by
have h := pow_le_pow_left₀ (Real.sqrt_nonneg (d:ℝ)) (sqrt_le_circuitD F x) 2
simpa only [Real.sq_sqrt (Nat.cast_nonneg d)] using h
lemma harmonicNormalizedTerm_mono (n : ℕ) {R z Z : ℝ} (hR : 0≤R) (hz : 0≤z) (hzZ : z≤Z) :
harmonicNormalizedTerm n R z≤harmonicNormalizedTerm n R Z := by
unfold harmonicNormalizedTerm
rw [abs_of_nonneg hz,abs_of_nonneg (hz.trans hzZ)]
gcongr
lemma kernelNormalizedTerm_mono (d k n N : ℕ) (lam Ap Ah Lp : ℝ≥0)
(r R h : ℝ) {z Z : ℝ} (hR : 0≤R) (hz : 0≤z) (hzZ : z≤Z) :
kernelNormalizedTerm d k n N lam Ap Ah Lp r R h z≤
kernelNormalizedTerm d k n N lam Ap Ah Lp r R h Z := by
dsimp only [kernelNormalizedTerm]
gcongr
exact harmonicNormalizedTerm_mono n hR hz hzZ
lemma weighted_uniform_upper {ι : Type*} [Fintype ι] (a f : ι → ℝ) {B : ℝ}
(ha : ∀i,0≤a i) (hf : ∀i,f i≤B) :
(∑i,a i*f i)≤(∑i,a i)*B := by
rw [Finset.sum_mul]
exact Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hf i) (ha i))
def kernelActionNormalizedBudget (d k n m N : ℕ) (lam Ap Ah Lp : ℝ≥0)
(r R h ψ C J : ℝ) : ℝ :=
let S := (∑i,|derivativeWeight (angleNodes m) i|)/ψ
let V := (numericalContractionCoefficient Ap Ah Lp+1)*numericalErrorCoefficient C J
let B := kernelNormalizedTerm d k n N lam Ap Ah Lp r R h (ψ*m)
600*(V*S*B)^2+2*S^2*((ψ*m)^(m+1)/(m.factorial:ℝ))^2*
(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)
lemma kernelActionNormalizedBudget_nonneg (d k n m N : ℕ) (lam Ap Ah Lp : ℝ≥0)
(r R h ψ C J : ℝ) (hR : 0≤R) : 0≤kernelActionNormalizedBudget d k n m N lam Ap Ah Lp r R h ψ C J := by
unfold kernelActionNormalizedBudget
positivity [harmonicCorrectionBudget_nonneg (m+1)]
theorem kernelActionRmsBudget_normalized {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) (Ap Ah Lp : ℝ≥0) {r R ρ h ψ C J : ℝ}
(hr : 0≤r) (hr1 : r≤1) (hR : 0<R) (hh : 0≤h) (hψ : 0<ψ) (hC : 0≤C) (hJ : 0≤J)
(hl : (lam:ℝ)*r^2≤1/2) (hL : (lam:ℝ)*r≤1) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (k n m N : ℕ) :
kernelActionRmsBudget F x lam Ap Ah Lp r R ρ h ψ C J k n m N ≤
(((lam:ℝ)*r^2)*circuitD F x)^2*
kernelActionNormalizedBudget d k n m N lam Ap Ah Lp r R h ψ C J := by
let l := (lam:ℝ)*r^2
let D := circuitD F x
let S := (∑i,|derivativeWeight (angleNodes m) i|)/ψ
let V := (numericalContractionCoefficient Ap Ah Lp+1)*numericalErrorCoefficient C J
let B := kernelNormalizedTerm d k n N lam Ap Ah Lp r R h (ψ*m)
have hl0 : 0≤l := by positivity
have hD : 0≤D := circuitD_nonneg F x
have hV : 0≤V := by dsimp [V]; positivity [numericalContractionCoefficient_pos Ap Ah Lp,
numericalErrorCoefficient_pos C J]
have hS : 0≤S := by dsimp [S]; positivity
have hw : (∑i,|derivativeWeight (angleNodes m) i/ψ|)=S := by
simp only [abs_div,abs_of_pos hψ,←Finset.sum_div,S]
have hb (i : Fin (m+1)) : kernelValueRmsBudget F x lam Ap Ah Lp r R ρ h
(ψ*angleNodes m i) C J k n N≤300*(V*l*B*D)^2 := by
have hi0 : 0≤angleNodes m i := by dsimp [angleNodes]; positivity
have hi1 : angleNodes m i≤(m:ℝ) := by dsimp [angleNodes]; exact_mod_cast Nat.le_of_lt_succ i.2
have hz : 0≤ψ*angleNodes m i := mul_nonneg hψ.le hi0
apply (kernelValueRmsBudget_normalized hF x Ap Ah Lp hr hr1 hR hh hz hC hJ hl hL hρ0 hρ1 k n N).trans
have hmono := kernelNormalizedTerm_mono d k n N lam Ap Ah Lp r R h hR.le hz
(mul_le_mul_of_nonneg_left hi1 hψ.le)
have hbase : 0≤kernelNormalizedTerm d k n N lam Ap Ah Lp r R h (ψ*angleNodes m i) := by
unfold kernelNormalizedTerm
positivity [probabilityNormalizedTerm_nonneg d k n hh,
harmonicNormalizedTerm_nonneg (z:=ψ*angleNodes m i) n hR.le,
numericalContractionCoefficient_pos Ap Ah Lp]
change 300*(V*l*_*D)^2≤_
gcongr
have hsum := weighted_uniform_upper (fun i => |derivativeWeight (angleNodes m) i/ψ|)
(fun i => kernelValueRmsBudget F x lam Ap Ah Lp r R ρ h (ψ*angleNodes m i) C J k n N)
(fun i => abs_nonneg _) hb
rw [hw] at hsum
have hnum := mul_le_mul_of_nonneg_left hsum (show 0≤2*S by positivity)
have hdim := circuitD_dimension F x
have hc : 0≤harmonicCorrectionBudget (m+1) := harmonicCorrectionBudget_nonneg (m+1)
have hang : 2*(S^2*(((ψ*m)^(m+1)/(m.factorial:ℝ))^2*
((d*l^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1))))≤
2*(S^2*(((ψ*m)^(m+1)/(m.factorial:ℝ))^2*
((D^2*l^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)))) := by
gcongr
unfold kernelActionRmsBudget
rw [hw]
calc
_ ≤ 2*S*(S*(300*(V*l*B*D)^2))+2*(S^2*(((ψ*m)^(m+1)/(m.factorial:ℝ))^2*
((D^2*l^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)))) := by
apply add_le_add
· simpa only [mul_assoc] using hnum
· exact hang
_ = _ := by dsimp [kernelActionNormalizedBudget,V,B,S,l,D]; ring
end LogConcaveSampling
end
end
end OAI
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