| import OAI.Dynamics.StandardMap.ArrayLimitSupport |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| 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 |
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| end OAI |
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