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fcab081 | 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 | import OAI.Combinatorics.HadwigerCounterexample.SimultaneousProjection
namespace OAI
/-! # Conditional Large Set Hits -/
noncomputable section
namespace HadwigerCounterexample
open Filter Finset Matrix LinearMap Function
open scoped BigOperators Topology
open Template
variable {p:Shape} {M n N K r s0:ℕ}
namespace StatusAnalysis
namespace Quill
variable {q:Format} {d:Rates} {b:Bool} (w:Quill p q d b) (t : Tagging p (Messenger q))
(z:Messenger q→F) (m:Mixer p n)
def prelim (K:ℕ) (s:Charts p n d) (i:Messenger q) :=
d (parcel b i)≤ K^2 ∧ Injective (s (parcel b i)) ∧
∀ x,totalRank (s (parcel b i) x)≤ K
def sameInk (i:Messenger q) (v: Magazine p (d (parcel b i))) :=
Appearance.nearly p (w.amp i).1 (w.amp i).2.1 v
structure steady (K:ℕ) (s:Charts p n d) (o: UnitType p n N) : Prop where
frame : PreparationParameters.frameSide p m o
steadyAxes (i:Messenger q) : prelim (b:=b) K s i
enough i : (w.amp i).1.diagO=(w.amp i).2.1.1
sufficesP i :
let j:= parcel b i; let x:=w.allure t z i
carried (w.policies t z n i) (fun y=> w.sameInk i (x.profile j o m (s j) y) →
x.profile j o m (s j) y=w.amp i)
set_option maxHeartbeats 800000 in
theorem thinGlaze (Y : Silhouette p N r) (o : UnitType p n N) (i : Messenger q)
(s : Charts p n d) (k:AtomKey (p:=p) (t i) N)
{δ : ℝ} (hd:0<δ)
(hw:w.moral t z) (hp : w.steady t z m K s o)
(he : gauze p δ δ Y m K o) (ht : 1≤p.r) (hn : 2*p.r≤n) :
let x:=w.allure t z i
δ ≤ x.reference Y (Y (t i) (z i) k) →
basebeta p K/2 ≤ w.seraph t z o m s Y i k := by
classical
intro x h
let j:=parcel b i
let z':= w.amp i; let a:=Y (t i) (z i) k
let μ:=x.reflected Y j o m (s j) a
have hx : x.apt := w.allure_apt t z hw i
have hm : IsLaw μ:= x.reflectedLaw Y (Appearance.witLaw hx ht hn) _ _ _ _ _
obtain ⟨h1,h2,h3⟩:=hp.steadyAxes i
have hh := x.demonstrates Y o (m:=m) (s j) K j a δ hd z'.1 z'.2.1.2 (hw.2.2 i)
(hw.2.1 i) ht hn h (he _) h1 h2 h3
have hq (y) (h:μ y≠0) (hz:w.sameInk i y): y=w.amp i := by
let P := fun x=>w.sameInk i x → x=w.amp i
have hb : carried μ P:= x.reflection_exact Y j o m (s j) a P (hp.sufficesP i)
exact (show P y from not_not.mp fun hi=>h (hb y hi)) hz
have h4 : (z'.1.diagO,z'.2.1.2)=z'.2.1 := Prod.ext (hp.enough i) rfl
rw [h4] at hh
exact hh.trans ((pr_on_mono hm.nz hq).trans_eq (pr_point' ..))
theorem tinyCell_loss (Y:Silhouette p N r) {δ:ℝ} (hd:0≤δ) (i : Messenger q)
(hN: 2*num (Messenger q×Toast p)-(N:ℝ)≤ -1)
(hh:2*num (Toast p)-(N:ℝ)≤ -1) :
let x:= w.allure t z i
let μ:=lawNu p N t z
pr μ (fun y=> x.reference Y (Y (t i) (z i) (y i)) < δ) ≤
r*δ+2*netFact (trayPort p t)*bp (-(N:ℝ)/2) := by
classical
intro a μ
let I:=Messenger q
let Ω:=fun i=> lawOmega p N (t i) (z i)
let S:=fun y=>a.reference Y (Y (t i) (z i) y)<δ
let f : ∀ i:Messenger q,Wgt (AtomKey (p:=p) (t i) N) := fun j=>
if h:j=i then h ▸ (fun x : AtomKey (p:=p) (t i) N=> ind (S x)) else fun _=>1
have hi : f i=fun x=>ind (S x) := by unfold f; simp
have hf j (h:j≠i) : f j=fun _ => 1 := by unfold f; simp [h]
have hg j x : |f j x| ≤ 1 := by
by_cases h:j=i
· subst j; rw [hi]; exact (abs_of_nonneg (ind_nonneg _)).le.trans (ind_le_one _)
rw [hf j h]; simp
have he x : prodW f x= ind (S (x i)) := by
unfold prodW; rw [Finset.prod_eq_single i _ (by simp),hi]
intro b _ hb; rw [hf _ hb]
have hR :
∏ i,avg (Ω i) (f i) = pr (Ω i) S := by
rw [Finset.prod_eq_single i _ (by simp),hi]; rfl
intro b _ hb; rw [hf _ hb,(omegaLaw _ _ _ hh).avg_const]
let L:=2*netFact (trayPort p t)*bp (-(N:ℝ)/2)
have hM : pr μ (fun x=> S (x i)) ≤ pr (Ω i) S+L := by
have hm := reference_product t z f hg hN
change |avg μ (prodW f)-∏ i,avg (Ω i) (f i)|≤L at hm
rw [show prodW f=(fun x=>ind (S (x i))) from funext he,hR] at hm
exact sub_le_iff_le_add'.mp ((abs_le.mp hm).2)
have hZ : pr (Ω i) S ≤ r*δ := by
let Q:=fun x=> a.reference Y x < δ
rw [show pr (Ω i) S= pr (a.reference Y) Q from (mapW_pr (Ω i) (Y (t i) (z i)) Q).symm]
exact haze_small _ δ hd
exact hM.trans (add_le_add_left hZ _)
end Quill
end StatusAnalysis
end HadwigerCounterexample
end
end OAI
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