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import OAI.Probability.SKBarriers.Calculus.HessianAlgebra
import OAI.Probability.SKBarriers.Calculus.ParameterProducts
namespace OAI
section
section
noncomputable section
open scoped BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
noncomputable section
open MeasureTheory Set Filter
open scoped Topology Interval
noncomputable section
open MeasureTheory Set
open scoped Interval
namespace SK.Analytic
theorem cube_linear_variance_le_aux (n : ℕ) (V : ParameterSpace n → ℝ)
(hV : ContDiff ℝ 2 V) (hV₀ : ∀ z, parameterDerivative n V z = 0)
(L : ParameterSpace n →L[ℝ] ℝ) (hL₀ : L (parameterAxis n) = 0)
{c ε A : ℝ} (hε : 0 < ε) (hεc : ε < c) (hA : 0 ≤ A)
(hL : ∀ u, (L u)^2 ≤ A*coordinateSquare n u)
(hH : ∀ z u, c*coordinateSquare n u ≤ Hessian V z u u)
{a b : ℝ} (hab : a < b) :
let w := fun z => Real.exp (-V z)
let Z := cubeIntegral n w a b 0
cubeIntegral n (fun z => (L z)^2*w z) a b 0/Z -
(cubeIntegral n (fun z => L z*w z) a b 0/Z)^2 ≤ A/(c-ε) := by
let K := A/(c-ε)
let T := tiltedPotential n V L K
let w : ParameterSpace n → ℝ := fun z => Real.exp (-V z)
let W : ParameterSpace n → ℝ := fun z => Real.exp (-T z)
let Z := cubeIntegral n w a b 0
let I₁ := cubeIntegral n (fun z => L z*w z) a b 0
let I₂ := cubeIntegral n (fun z => (L z)^2*w z) a b 0
have hT : ContDiff ℝ 2 T := contDiff_tiltedPotential n V hV L K
have hessianT (z u : ParameterSpace n) : ε*coordinateSquare n u ≤ Hessian T z u u := by
exact hessian_tilt_lower hV (parameter n) L (coordinateSquare n) hεc hA
(coordinateSquare_nonneg n) hL hH z u
have hpos := (cubePotential_hessian_lower n T hT hε hessianT hab).2 0 1
have hformula := cubePotential_hessian_formula n T hT hab 0
have hW (z : ParameterSpace n) (hz : parameter n z = 0) : W z = w z := by
simp only [W, T, tiltedPotential, hz, zero_mul, sub_zero, mul_zero, add_zero, w]
have hZ : 0 < Z := cubeIntegral_pos n w hV.neg.exp.continuous
(fun _ => Real.exp_pos _) hab 0
have hZeq : cubeIntegral n W a b 0 = Z :=
cubeIntegral_congr_parameter n W w a b 0 hW
have hZ₁eq : cubeIntegral n (parameterDerivative n W) a b 0 = I₁ := by
apply cubeIntegral_congr_parameter
intro z hz
rw [parameterDerivative_exp_neg n T (hT.differentiable (by norm_num)),
parameterDerivative_tiltedPotential n V (hV.differentiable (by norm_num)) hV₀ L hL₀]
change -(K*parameter n z-L z)*W z = L z*w z
rw [hz, mul_zero, zero_sub, neg_neg, hW z hz]
have hZ₂eq : cubeIntegral n (parameterDerivative n (parameterDerivative n W)) a b 0 = I₂-K*Z := by
have he := cubeIntegral_congr_parameter n
(parameterDerivative n (parameterDerivative n W))
(fun z => (L z)^2*w z-K*w z) a b 0 (by
intro z hz
rw [parameterDerivative2_exp_neg n T hT,
parameterDerivative_tiltedPotential n V (hV.differentiable (by norm_num)) hV₀ L hL₀,
parameterDerivative2_tiltedPotential n V (hV.differentiable (by norm_num)) hV₀ L hL₀]
change ((K*parameter n z-L z)^2-K)*W z = _
rw [hz, hW z hz]
ring)
have hcF : Continuous (fun z => (L z)^2*w z) := (L.continuous.pow 2).mul hV.neg.exp.continuous
have hcG : Continuous (fun z => K*w z) := hV.neg.exp.continuous.const_mul K
rw [he, cubeIntegral_sub n _ _ hcF hcG, cubeIntegral_const_mul]
change Hessian (cubePotential n T a b) 0 1 1 =
(cubeIntegral n (parameterDerivative n W) a b 0/cubeIntegral n W a b 0)^2 -
cubeIntegral n (parameterDerivative n (parameterDerivative n W)) a b 0/cubeIntegral n W a b 0
at hformula
rw [hZeq, hZ₁eq, hZ₂eq] at hformula
change I₂/Z-(I₁/Z)^2 ≤ K
have he : (I₁/Z)^2-(I₂-K*Z)/Z = K-(I₂/Z-(I₁/Z)^2) := by
field_simp
ring
rw [he] at hformula
linarith
theorem cube_linear_variance_le (n : ℕ) (V : ParameterSpace n → ℝ)
(hV : ContDiff ℝ 2 V) (hV₀ : ∀ z, parameterDerivative n V z = 0)
(L : ParameterSpace n →L[ℝ] ℝ) (hL₀ : L (parameterAxis n) = 0)
{c A : ℝ} (hc : 0 < c) (hA : 0 ≤ A)
(hL : ∀ u, (L u)^2 ≤ A*coordinateSquare n u)
(hH : ∀ z u, c*coordinateSquare n u ≤ Hessian V z u u)
{a b : ℝ} (hab : a < b) :
let w := fun z => Real.exp (-V z)
let Z := cubeIntegral n w a b 0
cubeIntegral n (fun z => (L z)^2*w z) a b 0/Z -
(cubeIntegral n (fun z => L z*w z) a b 0/Z)^2 ≤ A/c := by
let w : ParameterSpace n → ℝ := fun z => Real.exp (-V z)
let Z := cubeIntegral n w a b 0
let v := cubeIntegral n (fun z => (L z)^2*w z) a b 0/Z -
(cubeIntegral n (fun z => L z*w z) a b 0/Z)^2
change v ≤ A/c
by_contra hh
have hv : A/c < v := lt_of_not_ge hh
have hvpos := lt_of_le_of_lt (div_nonneg hA hc.le) hv
have hAv : A < v*c := (div_lt_iff₀ hc).mp hv
have hAc : A/v < c := (div_lt_iff₀ hvpos).mpr (by nlinarith)
let ε := (c-A/v)/2
have hε : 0 < ε := by dsimp [ε]; linarith
have hεc : ε < c := by dsimp [ε]; have := div_nonneg hA hvpos.le; linarith
have hbound := cube_linear_variance_le_aux n V hV hV₀ L hL₀ hε hεc hA hL hH hab
change v ≤ A/(c-ε) at hbound
have hstrict : A/(c-ε) < v := by
apply (div_lt_iff₀ (sub_pos.mpr hεc)).mpr
have he : A/v*v = A := div_mul_cancel₀ A (ne_of_gt hvpos)
dsimp only [ε]
nlinarith
exact (lt_irrefl v) (hbound.trans_lt hstrict)
end SK.Analytic
end
end
end
end
end
end OAI
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