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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 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | import OAI.Dynamics.StandardMap.ArrayLimitSupport
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open Set Filter MeasureTheory Topology
open scoped Topology Classical
namespace StandardMapEntropy
noncomputable def arrayHalf (d : DistanceArray) (hu : UnitArray d) : DistanceArray :=
⟨fun s t => 2*d.val (dyadicHalf s) (dyadicHalf t),by
obtain ⟨h0,hs,hsy,ht,hb⟩ := d.property
refine ⟨fun s t => mul_nonneg (by norm_num) (h0 _ _),?_,?_,?_,?_⟩
· intro s; simp only [hs,mul_zero]
· intro s t; dsimp only; rw [hsy]
· intro s t u; have := ht (dyadicHalf s) (dyadicHalf t) (dyadicHalf u); dsimp only; linarith
· intro s t
have hh := hu (dyadicHalf s) (dyadicHalf t)
change d.val (dyadicHalf s) (dyadicHalf t)≤|(t:ℝ)/2-(s:ℝ)/2| at hh
rw [←sub_div,abs_div,abs_of_pos (by norm_num : (0:ℝ)<2)] at hh
dsimp only; linarith⟩
lemma arrayHalf_unit (d : DistanceArray) (hu : UnitArray d) : UnitArray (arrayHalf d hu) := by
intro s t
have hh := hu (dyadicHalf s) (dyadicHalf t)
change d.val (dyadicHalf s) (dyadicHalf t)≤|(t:ℝ)/2-(s:ℝ)/2| at hh
rw [←sub_div,abs_div,abs_of_pos (by norm_num : (0:ℝ)<2)] at hh
change 2*d.val (dyadicHalf s) (dyadicHalf t)≤|(t:ℝ)-(s:ℝ)|
linarith
lemma arrayDilate_unit (d : DistanceArray) (hu : UnitArray d) : UnitArray (arrayDilate d) := by
intro s t
have hh := hu (s+s) (t+t)
change d.val (s+s) (t+t)≤|((t:ℝ)+(t:ℝ))-((s:ℝ)+(s:ℝ))| at hh
rw [show ((t:ℝ)+(t:ℝ))-((s:ℝ)+(s:ℝ))=2*((t:ℝ)-(s:ℝ)) by ring,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] at hh
change d.val (s+s) (t+t)/2≤|(t:ℝ)-(s:ℝ)|
linarith
lemma arrayDilate_half (d : DistanceArray) (hu : UnitArray d) : arrayDilate (arrayHalf d hu)=d := by
apply Subtype.ext; funext s t
change (2*d.val (dyadicHalf (s+s)) (dyadicHalf (t+t)))/2=d.val s t
rw [dyadicHalf_double,dyadicHalf_double]; ring
lemma arrayHalf_dilate (d : DistanceArray) (hu : UnitArray d) : arrayHalf (arrayDilate d) (arrayDilate_unit d hu)=d := by
apply Subtype.ext; funext s t
change 2*(d.val (dyadicHalf s+dyadicHalf s) (dyadicHalf t+dyadicHalf t)/2)=d.val s t
rw [dyadicHalf_add_self,dyadicHalf_add_self]; ring
noncomputable def nonaffineHalfOn (d : {d : NonAffineArray // UnitArray d.val}) : NonAffineArray :=
⟨arrayHalf d.val.val d.property,by
intro ha
have hh := (arrayDilate_mem_affine_iff _).mpr ha
rw [arrayDilate_half] at hh
exact d.val.property hh⟩
lemma continuous_nonaffineHalfOn : Continuous nonaffineHalfOn := by
apply Continuous.subtype_mk
apply Continuous.subtype_mk
apply continuous_pi; intro s
apply continuous_pi; intro t
exact continuous_const.mul ((continuous_arrayEval (dyadicHalf s) (dyadicHalf t)).comp (continuous_subtype_val.comp continuous_subtype_val))
noncomputable def nonaffineHalf (d : NonAffineArray) : NonAffineArray :=
if h : UnitArray d.val then nonaffineHalfOn ⟨d,h⟩ else d
lemma measurable_nonaffineHalf : Measurable nonaffineHalf := by
exact continuous_nonaffineHalfOn.measurable.dite measurable_subtype_coe
(isClosed_unitArray.preimage continuous_subtype_val).measurableSet
lemma nonaffineHalf_unit (d : NonAffineArray) (hu : UnitArray d.val) : UnitArray (nonaffineHalf d).val := by
rw [nonaffineHalf,dite_eq_left hu]
exact arrayHalf_unit d.val hu
lemma nonaffineDilate_half (d : NonAffineArray) (hu : UnitArray d.val) : nonaffineDilate (nonaffineHalf d)=d := by
apply Subtype.ext
rw [nonaffineHalf,dite_eq_left hu]
exact arrayDilate_half d.val hu
lemma nonaffineHalf_dilate (d : NonAffineArray) (hu : UnitArray d.val) : nonaffineHalf (nonaffineDilate d)=d := by
have hh : UnitArray (nonaffineDilate d).val := arrayDilate_unit d.val hu
rw [nonaffineHalf,dite_eq_left hh]
apply Subtype.ext
exact arrayHalf_dilate d.val hu
lemma map_half_dilate (μ : Measure NonAffineArray) (hu : ∀ᵐ d ∂μ,UnitArray d.val) :
(μ.map nonaffineDilate).map nonaffineHalf=μ := by
rw [Measure.map_map measurable_nonaffineHalf continuous_nonaffineDilate.measurable]
calc
μ.map (nonaffineHalf ∘ nonaffineDilate)=μ.map id := Measure.map_congr (hu.mono (fun d hd => nonaffineHalf_dilate d hd))
_=μ := Measure.map_id
lemma map_dilate_half (μ : Measure NonAffineArray) (hu : ∀ᵐ d ∂μ,UnitArray d.val) :
(μ.map nonaffineHalf).map nonaffineDilate=μ := by
rw [Measure.map_map continuous_nonaffineDilate.measurable measurable_nonaffineHalf]
calc
μ.map (nonaffineDilate ∘ nonaffineHalf)=μ.map id := Measure.map_congr (hu.mono (fun d hd => nonaffineDilate_half d hd))
_=μ := Measure.map_id
namespace CriticalScaleSequence
variable (S : CriticalScaleSequence) (L : S.LimitLaws)
lemma half_balance : L.multi=L.multi.map nonaffineHalf+L.terminal.map nonaffineHalf := by
have hh := congrArg (fun μ : Measure NonAffineArray => μ.map nonaffineHalf) (S.dilation_balance L)
rw [map_half_dilate L.multi (S.unit_aemulti L),Measure.map_add] at hh <;> try exact measurable_nonaffineHalf
exact hh
end CriticalScaleSequence
end StandardMapEntropy
end OAI
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