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dd14c26 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 | import Mathlib
import OAI.Probability.LogConcave.Model
namespace OAI
section
section
noncomputable section
open MeasureTheory Filter
open scoped ENNReal NNReal Topology
section UpperProof
open MeasureTheory ProbabilityTheory Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open Function MeasureTheory Set Filter
open scoped Topology NNReal
namespace LogConcaveSampling
open scoped RealInnerProductSpace
open MeasureTheory
def directional {d : ℕ} (v : Point d) (f : Point d → ℝ) : Point d → ℝ :=
fun x => fderiv ℝ f x v
lemma directional_smooth {d : ℕ} {f : Point d → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (v : Point d) :
ContDiff ℝ (⊤ : ℕ∞) (directional v f) :=
(hf.fderiv_right (by simp)).clm_apply contDiff_const
lemma directional_mul {d : ℕ} {f g : Point d → ℝ} (hf : Differentiable ℝ f)
(hg : Differentiable ℝ g) (v : Point d) :
directional v (fun x => f x*g x)=fun x => directional v f x*g x+f x*directional v g x := by
funext x
unfold directional
change (fderiv ℝ (f*g) x) v=_
rw [((hf x).hasFDerivAt.mul (hg x).hasFDerivAt).fderiv]
simp only [add_apply,smul_apply,smul_eq_mul]
ring
lemma directional_add {d : ℕ} {f g : Point d → ℝ} (hf : Differentiable ℝ f)
(hg : Differentiable ℝ g) (v : Point d) :
directional v (fun x => f x+g x)=fun x => directional v f x+directional v g x := by
funext x
exact congrArg (fun A : Point d →L[ℝ] ℝ => A v) ((hf x).hasFDerivAt.add (hg x).hasFDerivAt).fderiv
lemma directional_neg {d : ℕ} {f : Point d → ℝ} (hf : Differentiable ℝ f) (v : Point d) :
directional v (fun x => -f x)=fun x => -directional v f x := by
funext x
exact congrArg (fun A : Point d →L[ℝ] ℝ => A v) ((hf x).hasFDerivAt.neg).fderiv
lemma directional_sum {d : ℕ} {ι : Type*} (s : Finset ι) {f : ι → Point d → ℝ}
(hf : ∀ i∈s,Differentiable ℝ (f i)) (v : Point d) :
directional v (fun x => ∑ i∈s,f i x)=fun x => ∑ i∈s,directional v (f i) x := by
funext x
have hh := HasFDerivAt.fun_sum (u:=s) (fun i hi => (hf i hi x).hasFDerivAt)
simpa only [directional,sum_apply] using
congrArg (fun A : Point d →L[ℝ] ℝ => A v) hh.fderiv
lemma directional_directional {d : ℕ} {f : Point d → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (u v x : Point d) :
directional u (directional v f) x=(fderiv ℝ (fderiv ℝ f) x u) v := by
have hdf : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp)
unfold directional
rw [fderiv_clm_apply (hdf.differentiable (by norm_num) x) (differentiableAt_const _)]
simp
lemma directional_commute {d : ℕ} {f : Point d → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (u v : Point d) :
directional u (directional v f)=directional v (directional u f) := by
funext x
rw [directional_directional hf,directional_directional hf]
exact ((hf.contDiffAt (x:=x)).isSymmSndFDerivAt (by
simp only [minSmoothness_of_isRCLikeNormedField]
exact WithTop.coe_le_coe.mpr (show (2:ℕ∞)≤⊤ from le_top))).eq u v
lemma directional_gradient {d : ℕ} (H : Point d → ℝ) (v x : Point d) :
directional v H x=inner ℝ (gradient H x) v := by
rw [directional,←toDual_gradient,InnerProductSpace.toDual_apply_apply]
end LogConcaveSampling
end UpperProof
end
end
end
end OAI
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