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namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open Set Filter MeasureTheory Topology
open scoped ENNReal Classical
namespace StandardMapEntropy
lemma twoRay_real_properties {d:DistanceArray} {c:ℝ} (hu:UnitArray d) (hc:c<1) (hs:SlowShape (realArray d) c)
(hd:d∈twoRayClass) : ExteriorUnitRays (realArray d) (endpointLeft d) (endpointRight d) ∧
(∀s t:ℝ,s∈Icc (endpointLeft d) (endpointRight d) → t∈Icc (endpointLeft d) (endpointRight d) → realArray d s t≤c*|t-s|) := by
rcases hs with hs|⟨a,hl,hs⟩|⟨b,hr,hs⟩|⟨a,b,hab,⟨hl,hr⟩,hs⟩
· have hh := endpoints_slow (d:=d) hc (fun s t => by simpa only [realArray_coe d hu] using hs s t)
exact (hd.1 hh.1).elim
· have hh := endpoints_left_ray (d:=d) hc
(fun s t hp hq => by simpa only [realArray_coe d hu] using hl s hp t hq)
(fun s t hp hq => by simpa only [realArray_coe d hu] using hs s hp t hq)
exact (hd.2.2.2 hh.2).elim
· have hh := endpoints_right_ray (d:=d) hc
(fun s t hp hq => by simpa only [realArray_coe d hu] using hr s hp t hq)
(fun s t hp hq => by simpa only [realArray_coe d hu] using hs s hp t hq)
exact (hd.1 hh.1).elim
· have hh := endpoints_two_rays (d:=d) hab hc
(fun s t hp hq => by simpa only [realArray_coe d hu] using hl s t hp hq)
(fun s t hp hq => by simpa only [realArray_coe d hu] using hr s t hp hq)
(fun s t hp hq => by simpa only [realArray_coe d hu] using hs s hp t hq)
have ha : endpointLeft d=a := by simp only [endpointLeft,hh.1,EReal.toReal_coe]
have hb : endpointRight d=b := by simp only [endpointRight,hh.2,EReal.toReal_coe]
rw [ha,hb]
exact ⟨⟨hl,hr⟩,fun s t hp hq => hs s hp t hq⟩
lemma band_shortfall_bound (d:NonAffineArray) (hd:d∈lengthBand 0)
(hs:SlowShape (realArray d.val) (999/1000)) (ht:TreeArray d.val)
(hc:NoFastCancellation (realArray d.val) (1/100000000)) (t:DyadicTime) (hp:0<(t:ℝ)) :
arrayShortfall 0 t d.val≤1200000004/(t:ℝ) := by
have hh := twoRay_real_properties hd.1 (by norm_num) hs hd.2.1
have hab := (twoRay_properties hd.1 (by norm_num) hs hd.2.1).1
have hb := (exterior_ray_shortfall (realArray_treeLine d.val hd.1 ht) hab.le (by norm_num) hh.1 hc 0 t hp).2
have hlen : coreLength d.val<2 := by simpa only [lengthBand,zpow_zero,zero_add,zpow_one] using hd.2.2.2
have hreal : realArray d.val 0 (0+(t:ℝ))=d.val.val 0 t := by
simpa only [ZeroMemClass.coe_zero,zero_add] using realArray_coe d.val hd.1 (0:DyadicTime) t
rw [hreal] at hb
change 1-d.val.val 0 t/((t:ℝ)-0)≤_
rw [sub_zero]
apply hb.trans
apply (div_le_div_iff_of_pos_right hp).mpr
unfold coreLength at hlen
norm_num at *
linarith
lemma band_cap_support (α:ℝ) (d:NonAffineArray) (hd:d∈lengthBand 0)
(hs:SlowShape (realArray d.val) (999/1000)) (t:DyadicTime) (hp:0<(t:ℝ))
(he:endpointLeft d.val∉Icc (-2) (t:ℝ)) : entropyCap α (arrayShortfall 0 t d.val)=0 := by
obtain ⟨hab,hl,hr,hslow⟩ := twoRay_properties hd.1 (by norm_num) hs hd.2.1
have hlen : endpointRight d.val-endpointLeft d.val<2 := by simpa only [lengthBand,zpow_zero,zero_add,zpow_one,coreLength] using hd.2.2.2
have hh : d.val.val 0 t=(t:ℝ) := by
by_cases h:endpointLeft d.val<(t:ℝ)
· have ha : endpointLeft d.val< -2 := by by_contra hn; exact he ⟨le_of_not_gt hn,h.le⟩
have hrr := hr 0 t (by change endpointRight d.val≤0; linarith) (by linarith)
simpa only [ZeroMemClass.coe_zero,sub_zero,abs_of_pos hp] using hrr
· have hll := hl 0 t (by change (0:ℝ)≤endpointLeft d.val; linarith) (le_of_not_gt h)
simpa only [ZeroMemClass.coe_zero,sub_zero,abs_of_pos hp] using hll
have heq : arrayShortfall 0 t d.val=0 := by
unfold arrayShortfall; rw [hh,ZeroMemClass.coe_zero,sub_zero,div_self hp.ne']; ring
rw [heq,entropyCap_linear (by norm_num),mul_zero]
end StandardMapEntropy
end OAI
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