| import Mathlib |
| 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 |
| |
