File size: 4,232 Bytes
92416a2 | 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 | import OAI.Dynamics.StandardMap.CriticalLimits
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open Set Filter MeasureTheory Topology
open scoped Topology CompactlySupported Classical
namespace StandardMapEntropy
lemma arrayTranslate_mem_affine_iff (r : DyadicTime) (d : DistanceArray) :
arrayTranslate r d ∈ affineLocus ↔ d ∈ affineLocus := by
constructor
· rintro ⟨w,hw⟩
have hh := congrArg (arrayTranslate (-r)) hw
rw [arrayTranslate_affine,arrayTranslate_add,neg_add_cancel,arrayTranslate_zero] at hh
exact ⟨w,hh⟩
· rintro ⟨w,rfl⟩; exact ⟨w,(arrayTranslate_affine r w).symm⟩
noncomputable def nonaffineTranslation (r : DyadicTime) : NonAffineArray ≃ₜ NonAffineArray where
toFun d := ⟨arrayTranslate r d.val,by simpa only [arrayTranslate_mem_affine_iff] using d.property⟩
invFun d := ⟨arrayTranslate (-r) d.val,by simpa only [arrayTranslate_mem_affine_iff] using d.property⟩
left_inv d := by apply Subtype.ext; exact (arrayTranslation r).left_inv d.val
right_inv d := by apply Subtype.ext; exact (arrayTranslation r).right_inv d.val
continuous_toFun := (continuous_arrayTranslate r).comp continuous_subtype_val |>.subtype_mk _
continuous_invFun := (continuous_arrayTranslate (-r)).comp continuous_subtype_val |>.subtype_mk _
noncomputable def nonaffineDilate (d : NonAffineArray) : NonAffineArray :=
⟨arrayDilate d.val,by simpa only [arrayDilate_mem_affine_iff] using d.property⟩
lemma continuous_nonaffineDilate : Continuous nonaffineDilate :=
(continuous_arrayDilate.comp continuous_subtype_val).subtype_mk _
lemma proper_nonaffineDilate : IsProperMap nonaffineDilate := by
apply isProperMap_iff_isCompact_preimage.mpr
refine ⟨continuous_nonaffineDilate,?_⟩
intro K hK
have hA : IsCompact (Subtype.val '' K : Set DistanceArray) := hK.image continuous_subtype_val
have hB : IsCompact (arrayDilate ⁻¹' (Subtype.val '' K)) :=
(hA.isClosed.preimage continuous_arrayDilate).isCompact
have hsub : arrayDilate ⁻¹' (Subtype.val '' K) ⊆ affineLocusᶜ := by
rintro d ⟨u,hu,he⟩
exact fun hd => u.property (he ▸ (arrayDilate_mem_affine_iff d).mpr hd)
have hsub' : arrayDilate ⁻¹' (Subtype.val '' K) ⊆ Set.range ((↑) : NonAffineArray → DistanceArray) :=
fun d hd => ⟨⟨d,hsub hd⟩,rfl⟩
have hh := IsEmbedding.subtypeVal.isInducing.isCompact_preimage' hB hsub'
convert! hh using 1
ext d
change nonaffineDilate d ∈ K ↔ ∃ u ∈ K,u.val=arrayDilate d.val
exact ⟨fun hd => ⟨nonaffineDilate d,hd,rfl⟩,fun ⟨u,hu,he⟩ => (Subtype.ext he : u=nonaffineDilate d) ▸ hu⟩
noncomputable def nonaffineDilation : CocompactMap NonAffineArray NonAffineArray where
toFun := nonaffineDilate
continuous_toFun := continuous_nonaffineDilate
cocompact_tendsto' := (isProperMap_iff_tendsto_cocompact.mp proper_nonaffineDilate).2
noncomputable def arrayTestExtend (g : C_c(NonAffineArray,ℝ)) : C_c(DistanceArray,ℝ) := {
toFun := Subtype.val.extend g 0
continuous_toFun := HasCompactSupport.continuous_extend_zero isClosed_affineLocus.isOpen_compl g.continuous g.hasCompactSupport
hasCompactSupport' := g.hasCompactSupport.extend_zero continuous_subtype_val }
@[simp] lemma arrayTestExtend_apply (g : C_c(NonAffineArray,ℝ)) (d : NonAffineArray) :
arrayTestExtend g d.val=g d := Subtype.val_injective.extend_apply _ _ _
lemma arrayTestExtend_affine (g : C_c(NonAffineArray,ℝ)) (d : DistanceArray) (hd : d∈affineLocus) :
arrayTestExtend g d=0 := by
apply Function.extend_apply' _ _ _
rintro ⟨u,rfl⟩
exact u.property hd
lemma integral_arrayTestExtend (μ : Measure DistanceArray) [IsFiniteMeasure μ]
(g : C_c(NonAffineArray,ℝ)) : (∫ d,arrayTestExtend g d ∂μ)=∫ d,g d ∂nonaffinePart μ := by
have he : (∫ d,g d ∂nonaffinePart μ)=∫ d in affineLocusᶜ,arrayTestExtend g d ∂μ := by
convert! integral_subtype_comap isClosed_affineLocus.isOpen_compl.measurableSet (arrayTestExtend g) using 1
simp only [arrayTestExtend_apply]
rfl
rw [he]
symm
apply setIntegral_eq_integral_of_forall_compl_eq_zero
intro d hd
exact arrayTestExtend_affine g d (not_not.mp hd)
end StandardMapEntropy
end OAI
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