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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 | import OAI.Dynamics.StandardMap.BridgeGrid
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped ENNReal Topology Classical
namespace StandardMapEntropy
lemma aligned_intervals_eventually {R : ℕ} (s t : Fin R → DyadicTime)
(hst : ∀ j,(s j:ℝ)<(t j:ℝ)) :
∀ᶠ p : ℕ in atTop, ∃ (a : Fin R → ℤ) (m : Fin R → ℕ),
(∀ j,0< m j) ∧ (∀ j,((2^p:ℕ):ℝ)*(s j:ℝ)=(a j:ℝ)) ∧
(∀ j,((2^p:ℕ):ℝ)*(t j:ℝ)=(a j:ℝ)+(m j:ℝ)) := by
have he (j : Fin R) : ∀ᶠ p : ℕ in atTop,∃ a : ℤ,∃ m : ℕ,0< m ∧
((2^p:ℕ):ℝ)*(s j:ℝ)=(a:ℝ) ∧ ((2^p:ℕ):ℝ)*(t j:ℝ)=(a:ℝ)+(m:ℝ) := by
obtain ⟨p₀,hp₀⟩ := dyadic_interval_aligned_eventually (s j) (t j) (hst j)
filter_upwards [eventually_ge_atTop p₀] with p hp
obtain ⟨a,m,hm,ha,hb⟩ := hp₀ p hp
exact ⟨a,2*m,by omega,ha,by simpa only [Nat.cast_mul,Nat.cast_ofNat] using hb⟩
filter_upwards [Filter.eventually_all.mpr he] with p hp
choose a m hm ha hb using hp
exact ⟨a,m,hm,ha,hb⟩
lemma exists_bridgeGrid {Rw Rh : ℕ} (u v : DyadicTime) (huv : (u:ℝ)<(v:ℝ))
(sw tw : Fin Rw → DyadicTime) (sh th : Fin Rh → DyadicTime)
(hw : ∀j,(sw j:ℝ)<(tw j:ℝ)) (hh : ∀j,(sh j:ℝ)<(th j:ℝ))
(q : ℕ) (hq : (1:ℝ)/(2:ℝ)^q≤((v:ℝ)-(u:ℝ))/1000)
(hwl : ∀j,-((1:ℝ)/(2:ℝ)^q)≤(sw j:ℝ)-(u:ℝ))
(hwr : ∀j,(tw j:ℝ)-(u:ℝ)≤(1:ℝ)/(2:ℝ)^q)
(hhl : ∀j,-((1:ℝ)/(2:ℝ)^q)≤(sh j:ℝ)-(v:ℝ))
(hhr : ∀j,(th j:ℝ)-(v:ℝ)≤(1:ℝ)/(2:ℝ)^q) :
∃ B : BridgeGrid Rw Rh,B.u=u ∧ B.v=v ∧ B.sw=sw ∧ B.tw=tw ∧ B.sh=sh ∧ B.th=th := by
have hlarge : ∀ᶠ p : ℕ in atTop,(100:ℝ)≤((2^p:ℕ):ℝ)*((v:ℝ)-(u:ℝ)) := by
have ht := (tendsto_pow_atTop_atTop_of_one_lt (by norm_num : (1:ℝ)<2)).atTop_mul_const (sub_pos.mpr huv)
simpa only [Nat.cast_pow,Nat.cast_ofNat] using ht.eventually_ge_atTop 100
have huv' := aligned_intervals_eventually (fun _ : Fin 1 => u) (fun _ : Fin 1 => v) (fun _ => huv)
obtain ⟨p,hpq,hpN,⟨aa,mm,hmm,haa,hmm'⟩,⟨aw,nw,hnw,haw,htw⟩,⟨ah,nh,hnh,hah,hth⟩⟩ :=
(eventually_ge_atTop q |>.and (hlarge.and (huv'.and ((aligned_intervals_eventually sw tw hw).and
(aligned_intervals_eventually sh th hh))))).exists
let b := aa 0
let N := mm 0
let r := 2^(p-q)
have hN : (N:ℝ)=((2^p:ℕ):ℝ)*((v:ℝ)-(u:ℝ)) := by
have h1 := haa 0
have h2 := hmm' 0
dsimp [b,N] at *
linarith
have hr : (r:ℝ)=((2^p:ℕ):ℝ)*((1:ℝ)/(2:ℝ)^q) := by
dsimp [r]
push_cast
rw [show p=q+(p-q) from (Nat.add_sub_of_le hpq).symm,pow_add]
simp only [Nat.add_sub_cancel_left]
field_simp
have hpow : (0:ℝ)≤((2^p:ℕ):ℝ) := by positivity
refine ⟨{
p := p
u := u
v := v
b := b
N := N
r := r
large := ?_
narrow := ?_
hu := haa 0
hv := hmm' 0,
sw := sw
tw := tw
sh := sh
th := th
aw := fun j => aw j-b
nw := nw
ah := fun j => ah j-b
nh := nh,
pw := hnw
ph := hnh
hwleft := ?_
hwright := ?_
hhleft := ?_
hhright := ?_,
hsw := ?_
htw := ?_
hsh := ?_
hth := ?_},rfl,rfl,rfl,rfl,rfl,rfl⟩
· exact_mod_cast (by rw [hN]; exact hpN : (100:ℝ)≤N)
· rw [hr,hN]
nlinarith [mul_le_mul_of_nonneg_left hq hpow]
· intro j
have hb := haa 0
have hj := haw j
have h := mul_le_mul_of_nonneg_left (hwl j) hpow
rw [mul_neg,←hr] at h
exact_mod_cast (show -(r:ℝ)≤(aw j:ℝ)-(b:ℝ) by dsimp [b]; nlinarith)
· intro j
have hb := haa 0
have hj := htw j
have h := mul_le_mul_of_nonneg_left (hwr j) hpow
rw [←hr] at h
exact_mod_cast (show (aw j:ℝ)-(b:ℝ)+(nw j:ℝ)≤r by dsimp [b]; nlinarith)
· intro j
have hb := hmm' 0
have hj := hah j
have h := mul_le_mul_of_nonneg_left (hhl j) hpow
rw [mul_neg,←hr] at h
exact_mod_cast (show -(r:ℝ)≤(ah j:ℝ)-(b:ℝ)-(N:ℝ) by dsimp [b,N]; nlinarith)
· intro j
have hb := hmm' 0
have hj := hth j
have h := mul_le_mul_of_nonneg_left (hhr j) hpow
rw [←hr] at h
exact_mod_cast (show (ah j:ℝ)-(b:ℝ)+(nh j:ℝ)-(N:ℝ)≤r by dsimp [b,N]; nlinarith)
· intro j; simpa only [sub_add_cancel] using haw j
· intro j; simpa only [sub_add_cancel] using htw j
· intro j; simpa only [sub_add_cancel] using hah j
· intro j; simpa only [sub_add_cancel] using hth j
end StandardMapEntropy
end OAI
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