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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 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | import OAI.Dynamics.StandardMap.DyadicArrays
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open Set Filter MeasureTheory
open scoped ENNReal Topology
namespace StandardMapEntropy
noncomputable def arrayTranslate (r : DyadicTime) (d : DistanceArray) : DistanceArray :=
⟨fun s t => d.val (s+r) (t+r),by
obtain ⟨h0,hs,hsy,ht,hb⟩ := d.property
refine ⟨fun s t => h0 _ _,fun s => hs _,fun s t => hsy _ _,fun s t u => ht _ _ _,?_⟩
intro s t
convert hb (s+r) (t+r) using 1
congr 2
change (t:ℝ)-(s:ℝ)=((t:ℝ)+(r:ℝ))-((s:ℝ)+(r:ℝ))
ring⟩
lemma continuous_arrayTranslate (r : DyadicTime) : Continuous (arrayTranslate r) := by
apply Continuous.subtype_mk
apply continuous_pi; intro s
apply continuous_pi; intro t
exact continuous_arrayEval (s+r) (t+r)
@[simp] lemma arrayTranslate_zero (d : DistanceArray) : arrayTranslate 0 d=d := by
apply Subtype.ext
funext s t
simp [arrayTranslate]
lemma arrayTranslate_add (r q : DyadicTime) (d : DistanceArray) :
arrayTranslate r (arrayTranslate q d)=arrayTranslate (r+q) d := by
apply Subtype.ext
funext s t
simp only [arrayTranslate,add_assoc]
noncomputable def arrayTranslation (r : DyadicTime) : DistanceArray ≃ₜ DistanceArray where
toFun := arrayTranslate r
invFun := arrayTranslate (-r)
left_inv := by intro d; rw [arrayTranslate_add,neg_add_cancel,arrayTranslate_zero]
right_inv := by intro d; rw [arrayTranslate_add,add_neg_cancel,arrayTranslate_zero]
continuous_toFun := continuous_arrayTranslate r
continuous_invFun := continuous_arrayTranslate (-r)
noncomputable def arrayDilate (d : DistanceArray) : DistanceArray :=
⟨fun s t => d.val (s+s) (t+t)/2,by
obtain ⟨h0,hs,hsy,ht,hb⟩ := d.property
refine ⟨fun s t => div_nonneg (h0 _ _) (by norm_num),?_,?_,?_,?_⟩
· intro s; simp only [hs,zero_div]
· intro s t; dsimp only; rw [hsy]
· intro s t u; have := ht (s+s) (t+t) (u+u); dsimp only; linarith
· intro s t
have hh := hb (s+s) (t+t)
change d.val (s+s) (t+t)≤|((t:ℝ)+(t:ℝ))-((s:ℝ)+(s:ℝ))|+2 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
dsimp only; linarith⟩
lemma continuous_arrayDilate : Continuous arrayDilate := by
apply Continuous.subtype_mk
apply continuous_pi; intro s
apply continuous_pi; intro t
exact (continuous_arrayEval (s+s) (t+t)).div_const 2
@[simp] lemma arrayDilate_affine (w : Icc (0:ℝ) 1) : arrayDilate (affineArray w)=affineArray w := by
apply Subtype.ext
funext s t
change (w.val*|((t:ℝ)+(t:ℝ))-((s:ℝ)+(s:ℝ))|)/2=w.val*|(t:ℝ)-(s:ℝ)|
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)]
ring
@[simp] lemma arrayTranslate_affine (r : DyadicTime) (w : Icc (0:ℝ) 1) : arrayTranslate r (affineArray w)=affineArray w := by
apply Subtype.ext
funext s t
change w.val*|((t:ℝ)+(r:ℝ))-((s:ℝ)+(r:ℝ))|=w.val*|(t:ℝ)-(s:ℝ)|
congr 2; ring
lemma arrayDilate_mem_affine_iff (d : DistanceArray) : arrayDilate d∈affineLocus ↔ d∈affineLocus := by
constructor
· rintro ⟨w,hw⟩
refine ⟨w,?_⟩
apply Subtype.ext
funext s t
have he := congrArg (fun d : DistanceArray => d.val (dyadicHalf s) (dyadicHalf t)) hw
change w.val*|((t:ℝ)/2)-((s:ℝ)/2)|=d.val (dyadicHalf s+dyadicHalf s) (dyadicHalf t+dyadicHalf t)/2 at he
rw [dyadicHalf_add_self,dyadicHalf_add_self,← sub_div,abs_div,abs_of_pos (by norm_num : (0:ℝ)<2)] at he
change w.val*|(t:ℝ)-(s:ℝ)|=d.val s t
linarith
· rintro ⟨w,rfl⟩; exact ⟨w,(arrayDilate_affine w).symm⟩
noncomputable def arrayShortfall (s t : DyadicTime) (d : DistanceArray) : ℝ :=
1-d.val s t/((t:ℝ)-(s:ℝ))
noncomputable def arrayJ (s t : DyadicTime) (d : DistanceArray) : ℝ :=
arrayShortfall s t d-(arrayShortfall s (dyadicMid s t) d+arrayShortfall (dyadicMid s t) t d)/2
noncomputable def arrayV (s t : DyadicTime) (d : DistanceArray) : ℝ :=
(arrayShortfall s (dyadicMid s t) d-arrayShortfall (dyadicMid s t) t d)^2
lemma continuous_arrayShortfall (s t : DyadicTime) : Continuous (arrayShortfall s t) :=
continuous_const.sub ((continuous_arrayEval s t).div_const _)
lemma continuous_arrayJ (s t : DyadicTime) : Continuous (arrayJ s t) :=
(continuous_arrayShortfall s t).sub (((continuous_arrayShortfall s (dyadicMid s t)).add
(continuous_arrayShortfall (dyadicMid s t) t)).div_const 2)
lemma continuous_arrayV (s t : DyadicTime) : Continuous (arrayV s t) :=
((continuous_arrayShortfall s (dyadicMid s t)).sub (continuous_arrayShortfall (dyadicMid s t) t)).pow 2
lemma arrayJ_eq (s t : DyadicTime) (hst : (s:ℝ)<(t:ℝ)) (d : DistanceArray) :
arrayJ s t d=(d.val s (dyadicMid s t)+d.val (dyadicMid s t) t-d.val s t)/((t:ℝ)-(s:ℝ)) := by
unfold arrayJ arrayShortfall
rw [dyadicMid_val]
have h1 : (t:ℝ)-(s:ℝ)≠0 := sub_ne_zero.mpr hst.ne'
have h2 : ((s:ℝ)+(t:ℝ))/2-(s:ℝ)≠0 := by linarith
have h3 : (t:ℝ)-((s:ℝ)+(t:ℝ))/2≠0 := by linarith
rw [show ((s:ℝ)+(t:ℝ))/2-(s:ℝ)=((t:ℝ)-(s:ℝ))/2 by ring,
show (t:ℝ)-((s:ℝ)+(t:ℝ))/2=((t:ℝ)-(s:ℝ))/2 by ring]
field_simp [h1]; ring
lemma arrayJ_nonneg (s t : DyadicTime) (hst : (s:ℝ)<(t:ℝ)) (d : DistanceArray) : 0≤arrayJ s t d := by
rw [arrayJ_eq s t hst d]
exact div_nonneg (sub_nonneg.mpr (d.property.2.2.2.1 _ _ _)) (sub_pos.mpr hst).le
lemma arrayV_nonneg (s t : DyadicTime) (d : DistanceArray) : 0≤arrayV s t d := sq_nonneg _
lemma array_zero_test_halves (s t : DyadicTime) (hst : (s:ℝ)<(t:ℝ)) (d : DistanceArray)
(hzero : arrayJ s t d+arrayV s t d=0) :
d.val s (dyadicMid s t)=d.val s t/2 ∧ d.val (dyadicMid s t) t=d.val s t/2 := by
have hJ : arrayJ s t d=0 := by linarith [arrayJ_nonneg s t hst d,arrayV_nonneg s t d]
have hV : arrayV s t d=0 := by linarith
have he : arrayShortfall s (dyadicMid s t) d=arrayShortfall (dyadicMid s t) t d := by
exact sub_eq_zero.mp (sq_eq_zero_iff.mp hV)
rw [arrayJ_eq s t hst d] at hJ
have hsum : d.val s (dyadicMid s t)+d.val (dyadicMid s t) t=d.val s t := by
have := (div_eq_zero_iff.mp hJ).resolve_right (sub_ne_zero.mpr hst.ne')
linarith
unfold arrayShortfall at he
rw [dyadicMid_val] at he
have hm : ((s:ℝ)+(t:ℝ))/2-(s:ℝ)=((t:ℝ)-(s:ℝ))/2 := by ring
have hm' : (t:ℝ)-((s:ℝ)+(t:ℝ))/2=((t:ℝ)-(s:ℝ))/2 := by ring
rw [hm,hm'] at he
have hn : ((t:ℝ)-(s:ℝ))/2≠0 := by positivity
have he' : d.val s (dyadicMid s t)=d.val (dyadicMid s t) t := (div_left_inj' hn).mp (by linarith)
constructor <;> linarith
end StandardMapEntropy
end OAI
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