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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.ParameterAlgebra
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section ParameterFieldMap
variable {P E D : Type} [NormedAddCommGroup P] [NormedSpace ℝ P]
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup D] [NormedSpace ℝ D]
omit [NormedSpace ℝ P] in
theorem ParamExpGrowth.fieldComp {F : Type} [Norm F] {f : P × E → F}
(hf : ParamExpGrowth f) (L : D →L[ℝ] E) :
ParamExpGrowth (fun z : P × D => f (z.1,L z.2)) := by
intro R
obtain ⟨C,M,hC,hM,hf⟩ := hf R
refine ⟨C,M*‖L‖,hC,mul_nonneg hM (norm_nonneg L),?_⟩
intro p e hp
apply (hf p (L e) hp).trans
apply mul_le_mul_of_nonneg_left _ hC
apply Real.exp_le_exp.2
exact (mul_le_mul_of_nonneg_left (L.le_opNorm e) hM).trans_eq (mul_assoc _ _ _).symm
omit [NormedSpace ℝ P] in
theorem ParamLinearGrowth.fieldComp {f : P × E → ℝ}
(hf : ParamLinearGrowth f) (L : D →L[ℝ] E) :
ParamLinearGrowth (fun z : P × D => f (z.1,L z.2)) := by
intro R
obtain ⟨C,hC,hf⟩ := hf R
refine ⟨C*(1+‖L‖),mul_nonneg hC (by positivity),?_⟩
intro p e hp
calc
_ ≤ C*(1+‖L e‖) := hf p (L e) hp
_ ≤ C*((1+‖L‖)*(1+‖e‖)) := by
apply mul_le_mul_of_nonneg_left _ hC
nlinarith [L.le_opNorm e,norm_nonneg L,norm_nonneg e]
_ = _ := by ring
theorem ParamSmooth.fieldComp {f : P × E → ℝ} (hf : ParamSmooth f) (L : D →L[ℝ] E) :
ParamSmooth (fun z : P × D => f (z.1,L z.2)) := by
let A : P × D →L[ℝ] P × E := (ContinuousLinearMap.id ℝ P).prodMap L
exact ⟨hf.1.comp A.contDiff,hf.2.1.fieldComp L,
param_fderiv_compCLM_growth f A id (hf.1.differentiable (by norm_num))
(hf.2.2.1.fieldComp L),
param_second_compCLM_growth f A id hf.1 (hf.2.2.2.fieldComp L)⟩
theorem ParamRegular.fieldComp {f : P × E → ℝ} (hf : ParamRegular f) (L : D →L[ℝ] E) :
ParamRegular (fun z : P × D => f (z.1,L z.2)) := by
have h := hf.toSmooth.fieldComp L
exact ⟨h.1,hf.2.1.fieldComp L,h.2.2⟩
end ParameterFieldMap
end SK.Analytic
end
end
end
end OAI
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