| import Mathlib |
| import OAI.Probability.SKBarriers.Calculus.ParameterGrowth |
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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 |
| section ParameterComposition |
| variable {P E X Y : Type} [NormedAddCommGroup P] [NormedAddCommGroup E] |
| [NormedAddCommGroup X] [NormedSpace ℝ X] [NormedAddCommGroup Y] [NormedSpace ℝ Y] |
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| theorem param_fderiv_compCLM_growth (f : Y → ℝ) (L : X →L[ℝ] Y) (ι : P × E → X) |
| (hf : Differentiable ℝ f) |
| (h : ParamExpGrowth (fun z => fderiv ℝ f (L (ι z)))) : |
| ParamExpGrowth (fun z => fderiv ℝ (fun x => f (L x)) (ι z)) := by |
| apply h.of_norm_le (norm_nonneg L) |
| intro z |
| have he : fderiv ℝ (fun x => f (L x)) (ι z) = (fderiv ℝ f (L (ι z))).comp L := |
| ((hf (L (ι z))).hasFDerivAt.comp (ι z) L.hasFDerivAt).fderiv |
| rw [he] |
| exact (ContinuousLinearMap.opNorm_comp_le _ _).trans_eq (mul_comm _ _) |
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| theorem param_second_compCLM_growth (f : Y → ℝ) (L : X →L[ℝ] Y) (ι : P × E → X) |
| (hf : ContDiff ℝ 2 f) |
| (h : ParamExpGrowth (fun z => fderiv ℝ (fderiv ℝ f) (L (ι z)))) : |
| ParamExpGrowth (fun z => fderiv ℝ (fderiv ℝ (fun x => f (L x))) (ι z)) := by |
| let A : (Y →L[ℝ] ℝ) →L[ℝ] X →L[ℝ] ℝ := (ContinuousLinearMap.compL ℝ X Y ℝ).flip L |
| have hA (B : Y →L[ℝ] ℝ) : A B = B.comp L := rfl |
| have hAn : ‖A‖ ≤ ‖L‖ := by |
| apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _) |
| intro B |
| rw [hA] |
| exact (ContinuousLinearMap.opNorm_comp_le _ _).trans_eq (mul_comm _ _) |
| have hd := hf.differentiable (by norm_num) |
| have hdd := (hf.fderiv_right (m := 1) (by norm_num)).differentiable (by norm_num) |
| have heq : fderiv ℝ (fun x => f (L x)) = fun x => A (fderiv ℝ f (L x)) := by |
| funext x |
| exact ((hd (L x)).hasFDerivAt.comp x L.hasFDerivAt).fderiv |
| apply h.of_norm_le (mul_nonneg (norm_nonneg L) (norm_nonneg L)) |
| intro z |
| have hsec := A.hasFDerivAt.comp (ι z) |
| ((hdd (L (ι z))).hasFDerivAt.comp (ι z) L.hasFDerivAt) |
| change HasFDerivAt (fun x => A (fderiv ℝ f (L x))) |
| (A.comp ((fderiv ℝ (fderiv ℝ f) (L (ι z))).comp L)) (ι z) at hsec |
| rw [heq,hsec.fderiv] |
| calc |
| _ ≤ ‖A‖*‖(fderiv ℝ (fderiv ℝ f) (L (ι z))).comp L‖ := ContinuousLinearMap.opNorm_comp_le _ _ |
| _ ≤ ‖L‖*(‖fderiv ℝ (fderiv ℝ f) (L (ι z))‖*‖L‖) := by |
| gcongr |
| exact ContinuousLinearMap.opNorm_comp_le _ _ |
| _ = _ := by ring |
| end ParameterComposition |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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