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135052a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import Mathlib
import OAI.Probability.SKBarriers.Calculus.ParameterGrowth
namespace OAI
section
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]
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 _ _)
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
end
end
end
end OAI
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