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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 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.Combinatorics.HadwigerCounterexample.CellPositivityScales
namespace OAI
/-! # Conditional Signature Positivity -/
noncomputable section
namespace HadwigerCounterexample
open Filter Finset Matrix Function
open scoped BigOperators Topology
variable {p:Shape} {n N K r d : ℕ}
namespace StatusAnalysis
open Template
namespace Appearance
variable (x : Appearance p) (Y : Silhouette p N r)
theorem smudges (o : UnitType p n N) (m:Mixer p n) (v : Ax p n d) (i:Nodes) (j : Knobs p.g n) :
(x.profile i o m v j).2.1=((scan p x.l x.f j).diagO,stains x.trot m v j) := by
apply Prod.ext
· exact role_scan p _ _ _ _
funext i; exact T_swap m _ _
def nearly (p: Shape) (z: Digest p.g) (u: Par d) (x: Magazine p d) :=
x.1=z∧ x.2.1=u
set_option maxHeartbeats 800000 in
theorem demonstrates (o : UnitType p n N) {m : Mixer p n}
(v : Ax p n d) (K:ℕ) (i:Nodes) (j:Fin r) (δ : ℝ) (hd:0<δ)
(z : Digest p.g) (u:Cell d)
(he : z.feasible x.l x.f) (hf : z.sumBit=x.q) (ht : 1≤p.r) (hn : 2*p.r≤n)
(hh : δ≤ x.reference Y j)
(hs : smoothSilk p δ δ Y m K (onSide o i)) (hK : d≤K^2)
(hV : Injective v) (hv : ∀ x,totalRank (v x) ≤ K) :
basebeta p K/2 ≤ pr (x.reflected Y i o m v j) (nearly p z (z.diagO,u)) := by
classical
have ha : x.apt := ⟨z,he,hf⟩
let μ:= x.reference Y j
let f := onSide o i
obtain ⟨hM,hB⟩:= hs x
specialize hM ha
specialize hB z j he hf hh
let U (y:AtomKey (p:=p) (x.trot.t ()) N) := Y x.l x.q y=j
have hm : basebeta p K*μ ≤
pr (x.wit n) (fun y=> (scan p x.l x.f y=z∧Y x.l x.q (x.face f y)=j) ∧
stains x.trot m v y=u) := by
let h := patrol_beta x.trot U m K z he ht hn f
dsimp only at h
rw [hf] at h
exact h hB d hK v hV hv u
let w := nearly p z (z.diagO,u)
have hq (y : Knobs p.g n) :
(scan p x.l x.f y=z ∧ Y x.l x.q (x.face f y)=j) ∧ stains x.trot m v y=u →
x.showTag Y (onSide o i) y=j∧ w (x.profile i o m v y) := by
rintro ⟨⟨h,h1⟩,h2⟩
refine ⟨h1,h,?_⟩; rw [smudges,h,h2]
have hW := witLaw ha ht hn
have heq := (hm.trans (pr_mono hW.nz hq)).trans_eq (x.show_pr Y i o m v hW j w)
let ν:= x.empirical Y f j
have h2 : ν ≤ 2*μ := by
have hz : |μ-ν|≤δ := le_trans
(Finset.single_le_sum (fun k _ => abs_nonneg (x.reference Y k-x.empirical Y f k))
(Finset.mem_univ j)) hM
change δ≤ μ at hh
linarith [(abs_le.mp hz).1]
let Q := pr (x.reflected Y i o m v j) w
have h3 : basebeta p K*μ≤ ν*Q := heq
have ht := mul_le_mul_of_nonneg_right h2
(show 0≤ Q from pr_nonneg (x.reflectedLaw Y hW i o m v j).nz)
apply le_of_mul_le_mul_right _ (hd.trans_le hh)
change _*μ ≤ Q*μ
linarith only [h3,ht]
def fitsink (x : Appearance p) (w : Magazine p d) :=
w.1.feasible x.l x.f ∧ w.1.sumBit=x.q ∧ w.1.diagO=w.2.1.1
theorem rawink (x : Appearance p) (m : Mixer p n) (i : Nodes) (o : UnitType p n N)
(v : Ax p n d) :
carried (x.wit n) (x.fitsink ∘ x.profile i o m v) := by
let w:=x.wit n
have hC : carried w (fun y=>dialed p x.l x.f y=x.q) :=
normW_carried (subW_carried (spins p n) _)
intro y hi
refine hC y (fun h=>?_); apply hi
refine ⟨?_,h, (role_scan p _ _ _ _).symm⟩
intro j hj
unfold profile; change scan p x.l x.f y j=0
unfold scan
simp [hj]
theorem refInk (i : Nodes) (o : UnitType p n N) (m:Mixer p n) (v : Ax p n d)
(j:Fin r) : carried (x.reflected Y i o m v j) x.fitsink :=
reflection_exact x Y i o m v j _ (rawink x m i o v)
end Appearance
theorem haze_small (w : Wgt (Fin r)) (δ : ℝ) (hδ : 0≤δ) :
pr w (fun x=> w x < δ) ≤ r*δ := by
classical
unfold pr avg
refine (Finset.sum_le_sum (s:=Finset.univ) (g:=fun _=>δ) (fun i _=>?_)).trans_eq (by simp)
dsimp only
by_cases h:w i<δ
· rw [ind_one h,mul_one]; exact h.le
rw [ind_zero h,mul_zero]; exact hδ
end StatusAnalysis
end HadwigerCounterexample
end
end OAI
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