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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Probability.LogConcave.JetEstimates.SplitBundle
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open Set Function MeasureTheory
open scoped Classical BigOperators Matrix.Norms.L2Operator
namespace JetCalculus
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
lemma jet_compact_integral {f : E × ℝ → ℝ} (hf : ContDiff ℝ (⊤:ℕ∞) f)
{s : Set ℝ} (hs : IsCompact s) {ι : Type} (v : ι → E) (l : List ι) (y : E) :
jet v l (fun y => ∫t in s,f (y,t)) y=
∫t in s,jet (fun i => (v i,0)) l f (y,t) := by
induction l generalizing y with
| nil => rfl
| cons k l ih =>
let J := jet (fun i => (v i,0)) l f
have hJ : ContDiff ℝ (⊤:ℕ∞) J := smooth_jet hf _ _
have he : jet v l (fun y => ∫t in s,f (y,t))=fun y => ∫t in s,J (y,t) := funext ih
change dir (v k) (jet v l (fun y => ∫t in s,f (y,t))) y=_
rw [he]
have hd := CompactIntegral.parametric_hasFDerivAt (hJ.of_le (by simp)) hs y
unfold dir
rw [hd.fderiv]
have hI : IntegrableOn (fun t => (fderiv ℝ J (y,t)).comp (ContinuousLinearMap.inl ℝ E ℝ)) s :=
(((hJ.continuous_fderiv (by simp)).comp (continuous_const.prodMk continuous_id)).clm_comp
continuous_const).continuousOn.integrableOn_compact hs
rw [ContinuousLinearMap.integral_apply hI]
rfl
end JetCalculus
namespace TensorEnergy
lemma AllSplitBound.compact_integral {S : Type} [Fintype S] [DecidableEq S] {d : ℕ}
{A : ℝ → (S → Fin d) → ℝ} {s : Set ℝ} (hs : IsCompact s)
(hA : ∀c,Continuous (fun t => A t c)) {B : ℝ} (hB : 0≤B)
(hb : ∀t∈s,AllSplitBound (A t) B) :
AllSplitBound (fun c => ∫t in s,A t c) (volume.real s*B) := by
have hc : Continuous A := continuous_pi hA
have hi : IntegrableOn A s := hc.continuousOn.integrableOn_compact hs
have he : (fun c => ∫t in s,A t c)=∫t in s,A t := by
funext c
exact ((ContinuousLinearMap.proj c : ((S → Fin d) → ℝ) →L[ℝ] ℝ).integral_comp_comm hi)
rw [he]
have hbi := (splitBundle (S:=S) (d:=d)).integral_comp_comm hi
have hbound := norm_setIntegral_le_of_norm_le_const (μ:=volume) (s:=s)
(f:=fun t => splitBundle (A t)) hs.measure_lt_top
(fun t ht => bundle_norm_le hB (hb t ht))
rw [hbi] at hbound
exact (allSplitBound_bundle _).mono (norm_nonneg _) (by simpa only [mul_comm] using hbound)
lemma array_iterated_eq_jet {d n : ℕ} {S : Type} [Fintype S] [DecidableEq S]
{g : Point d → (S → Fin d) → ℝ} (hg : ContDiff ℝ (⊤:ℕ∞) g)
(y : Point d) (c : S ⊕ Fin n → Fin d) :
arrayTensor (iteratedFDeriv ℝ n g y) c=
JetCalculus.jet (fun i => EuclideanSpace.basisFun (Fin d) ℝ (c (Sum.inr i)))
(List.finRange n) (fun z => g z (fun s => c (Sum.inl s))) y := by
let L : ((S → Fin d) → ℝ) →L[ℝ] ℝ := ContinuousLinearMap.proj (fun s => c (Sum.inl s))
change _=JetCalculus.jet _ _ (L ∘ g) y
rw [JetCalculus.jet_finRange_eq_iteratedFDeriv (L.contDiff.comp hg),
L.iteratedFDeriv_comp_left hg.contDiffAt (by exact_mod_cast (le_top : (n:ℕ∞)≤⊤))]
rfl
lemma array_iterated_joint_jet {d n : ℕ} {S : Type} [Fintype S] [DecidableEq S]
{g : Point d × ℝ → (S → Fin d) → ℝ} (hg : ContDiff ℝ (⊤:ℕ∞) g)
(y : Point d) (t : ℝ) (c : S ⊕ Fin n → Fin d) :
arrayTensor (iteratedFDeriv ℝ n (fun z => g (z,t)) y) c=
JetCalculus.jet (fun i => (EuclideanSpace.basisFun (Fin d) ℝ (c (Sum.inr i)),(0:ℝ)))
(List.finRange n) (fun z => g z (fun s => c (Sum.inl s))) (y,t) := by
have hz : ContDiff ℝ (⊤:ℕ∞) (fun z => g (z,t)) := hg.comp (contDiff_id.prodMk contDiff_const)
rw [array_iterated_eq_jet hz]
exact JetCalculus.jet_left_slice
((ContinuousLinearMap.proj _ : ((S → Fin d) → ℝ) →L[ℝ] ℝ).contDiff.comp hg) _ _ _ _
lemma arrayTensor_compact_integral {d n : ℕ} {S : Type} [Fintype S] [DecidableEq S]
{g : Point d × ℝ → (S → Fin d) → ℝ} (hg : ContDiff ℝ (⊤:ℕ∞) g)
{s : Set ℝ} (hs : IsCompact s) (y : Point d) (c : S ⊕ Fin n → Fin d) :
arrayTensor (iteratedFDeriv ℝ n (fun z => ∫t in s,g (z,t)) y) c=
∫t in s,arrayTensor (iteratedFDeriv ℝ n (fun z => g (z,t)) y) c := by
rw [array_iterated_eq_jet (CompactIntegral.smooth_parametric hg hs)]
have he : (fun z => (∫t in s,g (z,t)) (fun s => c (Sum.inl s)))=
fun z => ∫t in s,g (z,t) (fun s => c (Sum.inl s)) := by
funext z
exact ((ContinuousLinearMap.proj (fun s => c (Sum.inl s)) : ((S → Fin d) → ℝ) →L[ℝ] ℝ).integral_comp_comm
((hg.continuous.comp (continuous_const.prodMk continuous_id)).continuousOn.integrableOn_compact hs)).symm
have hc : ContDiff ℝ (⊤:ℕ∞) (fun z => g z (fun s => c (Sum.inl s))) :=
(ContinuousLinearMap.proj _ : ((S → Fin d) → ℝ) →L[ℝ] ℝ).contDiff.comp hg
rw [he,JetCalculus.jet_compact_integral hc hs]
apply integral_congr_ae
filter_upwards [] with t
exact (array_iterated_joint_jet hg y t c).symm
lemma compactIntegral_iterated_split {d n : ℕ} {S : Type} [Fintype S] [DecidableEq S]
{g : Point d × ℝ → (S → Fin d) → ℝ} (hg : ContDiff ℝ (⊤:ℕ∞) g)
{s : Set ℝ} (hs : IsCompact s) (y : Point d) {B : ℝ} (hB : 0≤B)
(hb : ∀t∈s,AllSplitBound (arrayTensor (iteratedFDeriv ℝ n (fun z => g (z,t)) y)) B) :
AllSplitBound (arrayTensor (iteratedFDeriv ℝ n (fun z => ∫t in s,g (z,t)) y)) (volume.real s*B) := by
have hc (c : S ⊕ Fin n → Fin d) : Continuous
(fun t => arrayTensor (iteratedFDeriv ℝ n (fun z => g (z,t)) y) c) := by
simp_rw [array_iterated_joint_jet hg]
exact (JetCalculus.smooth_jet
((ContinuousLinearMap.proj _ : ((S → Fin d) → ℝ) →L[ℝ] ℝ).contDiff.comp hg) _ _).continuous.comp
(continuous_const.prodMk continuous_id)
have he : arrayTensor (iteratedFDeriv ℝ n (fun z => ∫t in s,g (z,t)) y)=
fun c => ∫t in s,arrayTensor (iteratedFDeriv ℝ n (fun z => g (z,t)) y) c :=
funext (arrayTensor_compact_integral hg hs y)
rw [he]
exact AllSplitBound.compact_integral hs hc hB hb
end TensorEnergy
end LogConcaveSampling
end
end
end
end OAI
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