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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 86 87 88 89 90 91 92 93 94 95 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.NormalizedLaplace
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
namespace LogConcaveSampling.JetCalculus
open scoped Topology
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {ι κ : Type*}
lemma mixedJet_add {f : E → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f)
(v : ι → E) (w : κ → E) (l : List ι) (k : List κ) :
mixedJet v w l k (fun x => f (x.1+x.2)) =
jet (Sum.elim v w) (l.map Sum.inl++k.map Sum.inr) f 0 := by
let A : E × E →L[ℝ] E := ContinuousLinearMap.fst ℝ E E+ContinuousLinearMap.snd ℝ E E
have hh := congrFun (jet_comp_affine hf A 0
(Sum.elim (fun i => (v i,(0:E))) (fun j => ((0:E),w j)))
(l.map Sum.inl++k.map Sum.inr)) 0
have hdir : (fun i => A (Sum.elim (fun i => (v i,(0:E)))
(fun j => ((0:E),w j)) i))=Sum.elim v w := by
funext i
cases i <;> simp [A]
have hs : ContDiff ℝ (⊤ : ℕ∞) (fun x : E × E => f (x.1+x.2)) :=
hf.comp (contDiff_fst.add contDiff_snd)
rw [mixedJet_eq_joint hs]
rw [hdir] at hh
simpa [A] using hh
end LogConcaveSampling.JetCalculus
namespace LogConcaveSampling.JetCalculus
open scoped Topology
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {ι : Type*}
lemma dir_dir {f : E → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (v w : E) :
dir v (dir w f)=dir w (dir v f) := by
funext x
have hs : ContDiff ℝ (⊤ : ℕ∞) (fderiv ℝ f) := hf.fderiv_right (by simp)
have hd : Differentiable ℝ (fderiv ℝ f) := hs.differentiable (by simp)
change (fderiv ℝ (fun y => fderiv ℝ f y w) x) v =
(fderiv ℝ (fun y => fderiv ℝ f y v) x) w
rw [fderiv_clm_apply (hd x) (differentiableAt_const _),
fderiv_clm_apply (hd x) (differentiableAt_const _)]
simpa only [add_apply,ContinuousLinearMap.comp_apply,
zero_apply,ContinuousLinearMap.map_zero,zero_add,
ContinuousLinearMap.flip_apply,fderiv_const_apply] using
((hf.contDiffAt (x := x)).isSymmSndFDerivAt (by
simp only [minSmoothness_of_isRCLikeNormedField]
exact WithTop.coe_le_coe.mpr (show (2:ℕ∞)≤⊤ from le_top))).eq v w
lemma jet_perm {f : E → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f)
(v : ι → E) {l k : List ι} (hp : l.Perm k) : jet v l f=jet v k f := by
induction hp with
| nil => rfl
| cons i hp ih => simp only [jet,ih]
| swap i j l => exact dir_dir (smooth_jet hf v l) (v j) (v i)
| trans h1 h2 ih1 ih2 => exact ih1.trans ih2
lemma dir_smul (v : E) (c : ℝ) (f : E → ℝ) :
dir (c • v) f=fun x => c*dir v f x := by
funext x
simp only [dir,map_smul,smul_eq_mul]
lemma jet_smul {f : E → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f)
(v : ι → E) (l : List ι) (c : ℝ) :
jet (fun i => c • v i) l f=fun x => c^l.length*jet v l f x := by
induction l with
| nil => simp [jet]
| cons i l ih =>
simp only [jet,ih,List.length_cons]
rw [dir_const_mul ((smooth_jet hf v l).differentiable (by simp)),dir_smul]
funext x
simp only [pow_succ,dir]
ring
end LogConcaveSampling.JetCalculus
end UpperProof
end
end
end
end OAI
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