| 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 |
|
|
