File size: 3,491 Bytes
9b6e014 | 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 | import OAI.Probability.ClassicalON.AnnulusLabels
namespace OAI
universe uι
noncomputable section
open scoped Classical
namespace ClassicalON
def coarseSite (N : ℤ) (v : Site) : Site := (v.1/N,v.2/N)
def coarseCenter (N : ℤ) (v : Site) : Site := (N*v.1,N*v.2)
theorem siteRadius_scale (N : ℤ) (hN : 0≤N) (x y : Site) :
siteRadius (coarseCenter N x) (coarseCenter N y)=N*siteRadius x y := by
simp only [siteRadius,coarseCenter,← mul_sub,abs_mul,abs_of_nonneg hN]
exact (mul_max_of_nonneg _ _ hN).symm
theorem coarseSite_close (N : ℤ) (hN : 0<N) (v : Site) :
siteRadius (coarseCenter N (coarseSite N v)) v<N := by
change max |v.1-N*(v.1/N)| |v.2-N*(v.2/N)|<N
rw [max_lt_iff]
constructor
· have he := Int.emod_add_ediv_mul v.1 N
rw [mul_comm (v.1/N) N] at he
have hn := Int.emod_nonneg v.1 (ne_of_gt hN)
have hl := Int.emod_lt_of_pos v.1 hN
rw [show v.1-N*(v.1/N)=v.1%N by omega,abs_of_nonneg hn]
exact hl
· have he := Int.emod_add_ediv_mul v.2 N
rw [mul_comm (v.2/N) N] at he
have hn := Int.emod_nonneg v.2 (ne_of_gt hN)
have hl := Int.emod_lt_of_pos v.2 hN
rw [show v.2-N*(v.2/N)=v.2%N by omega,abs_of_nonneg hn]
exact hl
theorem coarse_quotient_step (N : ℤ) (hN : 0<N) (a b : ℤ) (h : |b-a|≤1) :
|b/N-a/N|≤1 := by
have hab := abs_le.mp h
have h1 : b≤a+N := by omega
have h2 : a≤b+N := by omega
have hh1 := Int.ediv_le_ediv hN h1
have hh2 := Int.ediv_le_ediv hN h2
rw [Int.add_ediv_of_dvd_right (dvd_refl N),Int.ediv_self (ne_of_gt hN)] at hh1 hh2
exact abs_le.mpr ⟨by omega,by omega⟩
theorem coarseSite_step (N : ℤ) (hN : 0<N) {x y : Site} (h : siteRadius x y≤1) :
siteRadius (coarseSite N x) (coarseSite N y)≤1 := by
have hh := siteRadius_le_iff x y 1 |>.mp h
exact max_le (coarse_quotient_step N hN _ _ hh.1) (coarse_quotient_step N hN _ _ hh.2)
theorem coarseSite_distance (N : ℤ) (hN : 0<N) (x y : Site) :
siteRadius x y≤N*(siteRadius (coarseSite N x) (coarseSite N y)+2) := by
have h1 := coarseSite_close N hN x
have h2 := coarseSite_close N hN y
have h3 := siteRadius_triangle x (coarseCenter N (coarseSite N x)) y
have h4 := siteRadius_triangle (coarseCenter N (coarseSite N x)) (coarseCenter N (coarseSite N y)) y
rw [siteRadius_scale N (le_of_lt hN)] at h4
rw [siteRadius_symm x (coarseCenter N (coarseSite N x))] at h3
nlinarith
abbrev SiteColor := Fin 5 × Fin 5
def siteColor (v : Site) : SiteColor :=
(⟨(v.1%5).toNat,by omega⟩,⟨(v.2%5).toNat,by omega⟩)
theorem sameColor_far {x y : Site} (hxy : x≠y) (h : siteColor x=siteColor y) :
4<siteRadius x y := by
have h1 := congrArg (fun c : SiteColor => c.1.val) h
have h2 := congrArg (fun c : SiteColor => c.2.val) h
change (x.1%5).toNat=(y.1%5).toNat at h1
change (x.2%5).toNat=(y.2%5).toNat at h2
by_contra hn
have hr : |y.1-x.1|≤4 ∧ |y.2-x.2|≤4 := (siteRadius_le_iff x y 4).mp (by omega)
have ha := abs_le.mp hr.1
have hb := abs_le.mp hr.2
apply hxy
apply Prod.ext <;> omega
theorem sameColor_separated {ι : Type uι} (N : ℤ) (hN : 0<N) (v : ι → Site)
(hv : Function.Injective v) (c : SiteColor) :
SeparatedCenters N (fun i : {i // siteColor (v i)=c} => coarseCenter N (v i.val)) := by
intro i j hij
rw [siteRadius_scale N (le_of_lt hN)]
have hd : v i.val≠v j.val := fun h => hij (Subtype.ext (hv h))
have hh := sameColor_far hd (i.property.trans j.property.symm)
nlinarith
end ClassicalON
end
end OAI
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