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import OAI.Probability.SKBarriers.Replicas.TripleFiniteExpansion
namespace OAI
section
noncomputable section
open scoped NNReal Topology
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
def narrowExpansionDenominator (β : ℝ) : ℝ := 1+24*β^2+288*β^4
def narrowExpansionErrorConstant (β : ℝ) : ℝ :=
(12*(susceptibilityLipschitzConstant:ℝ)+64)*(narrowExpansionDenominator β)^2
theorem narrowExpansionDenominator_ge_one (β : ℝ) : 1≤ narrowExpansionDenominator β := by
unfold narrowExpansionDenominator; nlinarith [sq_nonneg β,sq_nonneg (β^2)]
theorem narrowExpansionErrorConstant_pos (β : ℝ) : 0< narrowExpansionErrorConstant β := by
have hd : 0< narrowExpansionDenominator β := lt_of_lt_of_le zero_lt_one (narrowExpansionDenominator_ge_one β)
unfold narrowExpansionErrorConstant
positivity
theorem narrowExpansion_smallness (β : ℝ) {h a θ : ℝ} (hh : 0< h) (hh1 : h≤1)
(ha : 0≤ a) (hθ : 0≤θ) (haθ : a+θ=1) :
(h/narrowExpansionDenominator β)*(4*β^2*((2*a^2+θ^2)/h)+(2*β^2*(5*a+θ))^2)≤1/2 := by
have ha1 : a≤1 := by linarith
have hθ1 : θ≤1 := by linarith
have hK : 2*a^2+θ^2≤3 := by nlinarith [sq_nonneg (1-a),sq_nonneg (1-θ)]
have hscore : (5*a+θ)^2≤36 := by nlinarith [sq_nonneg (6-(5*a+θ))]
have hd : 0< narrowExpansionDenominator β := lt_of_lt_of_le zero_lt_one (narrowExpansionDenominator_ge_one β)
have HP : 4*β^2*(2*a^2+θ^2)+h*(2*β^2*(5*a+θ))^2≤12*β^2+144*β^4 := by
have HK := mul_le_mul_of_nonneg_left hK (show 0≤4*β^2 by positivity)
have HS := mul_le_mul_of_nonneg_left hscore (show 0≤4*β^4 by positivity)
have HH := mul_le_mul_of_nonneg_left hh1 (show 0≤(2*β^2*(5*a+θ))^2 by positivity)
nlinarith only [HK,HS,HH]
have HE : (h/narrowExpansionDenominator β)*(4*β^2*((2*a^2+θ^2)/h)+(2*β^2*(5*a+θ))^2)=
(4*β^2*(2*a^2+θ^2)+h*(2*β^2*(5*a+θ))^2)/narrowExpansionDenominator β := by field_simp
rw [HE,div_le_iff₀ hd]
dsimp only [narrowExpansionDenominator]
linarith
theorem tripleExpansionConstant_scaled_bound (β : ℝ) {h : ℝ} (hh : 0< h) (hh1 : h≤1) :
tripleExpansionConstant (h/narrowExpansionDenominator β)≤ narrowExpansionErrorConstant β/h^2 := by
have hd1 := narrowExpansionDenominator_ge_one β
have hd : 0< narrowExpansionDenominator β := lt_of_lt_of_le zero_lt_one hd1
have hL : 0≤(susceptibilityLipschitzConstant:ℝ) := NNReal.coe_nonneg _
have hsq : h^2≤(narrowExpansionDenominator β)^2 := by nlinarith
have HE : tripleExpansionConstant (h/narrowExpansionDenominator β)*h^2=
4*(susceptibilityLipschitzConstant:ℝ)*h^2+(8*(susceptibilityLipschitzConstant:ℝ)+64)*(narrowExpansionDenominator β)^2 := by
unfold tripleExpansionConstant
field_simp
ring
rw [le_div_iff₀ (sq_pos_of_pos hh),HE]
unfold narrowExpansionErrorConstant
nlinarith [mul_le_mul_of_nonneg_left hsq (show 0≤4*(susceptibilityLipschitzConstant:ℝ) by positivity)]
end SK.Analytic
end
end
end OAI
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