| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.ParameterAlgebra |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| namespace SK.Analytic |
| theorem ParamExpGrowth.weakSmul {P E F : Type} [NormedAddCommGroup P] |
| [NormedAddCommGroup E] [Norm F] [SMul ℝ F] {f : P × E → ℝ} {g : P × E → F} |
| (hf : ParamExpGrowth f) (hg : ParamExpGrowth g) |
| (hn : ∀ z, 0 ≤ ‖g z‖) (hm : ∀ z, ‖f z • g z‖ ≤ ‖f z‖*‖g z‖) : |
| ParamExpGrowth (fun z => f z • g z) := by |
| apply (hf.norm.mul (hg.weakNorm hn)).of_norm_le zero_le_one |
| intro z |
| rw [one_mul,Real.norm_eq_abs,abs_mul,abs_of_nonneg (norm_nonneg (f z)),abs_of_nonneg (hn z)] |
| exact hm z |
|
|
| section ParameterProduct |
| variable {P E : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| theorem ParamSmooth.mul {f g : P × E → ℝ} (hf : ParamSmooth f) (hg : ParamSmooth g) : |
| ParamSmooth (fun z => f z*g z) := by |
| have hd := hf.1.differentiable (by norm_num) |
| have he := hg.1.differentiable (by norm_num) |
| have hdd := (hf.1.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) |
| have hee := (hg.1.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) |
| have h₁ : fderiv ℝ (fun z => f z*g z) = fun z => |
| f z • fderiv ℝ g z+g z • fderiv ℝ f z := by |
| funext z |
| exact ((hd z).hasFDerivAt.fun_mul (he z).hasFDerivAt).fderiv |
| have h₂ : fderiv ℝ (fderiv ℝ (fun z => f z*g z)) = fun z => |
| (f z • fderiv ℝ (fderiv ℝ g) z+(fderiv ℝ f z).smulRight (fderiv ℝ g z))+ |
| (g z • fderiv ℝ (fderiv ℝ f) z+(fderiv ℝ g z).smulRight (fderiv ℝ f z)) := by |
| rw [h₁] |
| funext z |
| exact (((hd z).hasFDerivAt.fun_smul (hee z).hasFDerivAt).fun_add |
| ((he z).hasFDerivAt.fun_smul (hdd z).hasFDerivAt)).fderiv |
| refine ⟨hf.1.mul hg.1,hf.2.1.mul hg.2.1,?_,?_⟩ |
| · rw [h₁] |
| exact (hf.2.1.smul hg.2.2.1).add (hg.2.1.smul hf.2.2.1) |
| · rw [h₂] |
| exact ((hf.2.1.weakSmul hg.2.2.2 (fun _ => ContinuousLinearMap.opNorm_nonneg _) |
| (fun _ => ContinuousLinearMap.opNorm_smul_le _ _)).weakAdd (hf.2.2.1.smulRight hg.2.2.1) |
| (fun _ => ContinuousLinearMap.opNorm_add_le _ _)).weakAdd |
| ((hg.2.1.weakSmul hf.2.2.2 (fun _ => ContinuousLinearMap.opNorm_nonneg _) |
| (fun _ => ContinuousLinearMap.opNorm_smul_le _ _)).weakAdd (hg.2.2.1.smulRight hf.2.2.1) |
| (fun _ => ContinuousLinearMap.opNorm_add_le _ _)) |
| (fun _ => ContinuousLinearMap.opNorm_add_le _ _) |
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| theorem ParamSmooth.const_mul {f : P × E → ℝ} (hf : ParamSmooth f) (c : ℝ) : |
| ParamSmooth (fun z => c*f z) := (paramSmooth_const c).mul hf |
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| theorem ParamSmooth.gaussian_integral {f : P × (E × ℝ) → ℝ} (hf : ParamSmooth f) : |
| ParamSmooth (fun z : P × E => ∫ y, f (z.1,(z.2,y)) ∂gaussianReal 0 1) := by |
| have h := param_gaussian_smooth f hf.1 hf.2.1 hf.2.2.1 hf.2.2.2 |
| exact ⟨h.1,hf.2.1.gaussian_integral,h.2⟩ |
| end ParameterProduct |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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