| import OAI.Probability.ClassicalON.CoarseGeometry |
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|
| namespace OAI |
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|
| noncomputable section |
| open scoped BigOperators Classical |
| namespace ClassicalON |
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|
| abbrev CoarseDirection := Fin 3 × Fin 3 |
| abbrev CoarseCode (L : ℕ) := Fin L → CoarseDirection |
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|
| def directionVector (d : CoarseDirection) : Site := ((d.1.val:ℤ)-1,(d.2.val:ℤ)-1) |
| def codePosition {L : ℕ} (z : Site) (c : CoarseCode L) (i : Fin (L+1)) : Site := |
| z+Fin.partialSum (directionVector ∘ c) i |
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| @[simp] theorem codePosition_zero {L : ℕ} (z : Site) (c : CoarseCode L) : codePosition z c 0=z := by |
| simp [codePosition] |
|
|
| theorem codePosition_succ {L : ℕ} (z : Site) (c : CoarseCode L) (i : Fin L) : |
| codePosition z c i.succ=codePosition z c i.castSucc+directionVector (c i) := by |
| simp only [codePosition,Fin.partialSum_succ,Function.comp_apply,add_assoc] |
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|
| theorem directionVector_surj {z w : Site} (h : siteRadius z w≤1) : |
| ∃ d,directionVector d=w-z := by |
| have hh := (siteRadius_le_iff z w 1).mp h |
| have h1 := abs_le.mp hh.1 |
| have h2 := abs_le.mp hh.2 |
| refine ⟨(⟨(w.1-z.1+1).toNat,by omega⟩,⟨(w.2-z.2+1).toNat,by omega⟩),?_⟩ |
| apply Prod.ext <;> dsimp [directionVector] <;> omega |
|
|
| theorem codePosition_exists {L : ℕ} (p : Fin (L+1) → Site) |
| (h : ∀ i : Fin L,siteRadius (p i.castSucc) (p i.succ)≤1) : |
| ∃ c : CoarseCode L,codePosition (p 0) c=p := by |
| choose c hc using fun i => directionVector_surj (h i) |
| refine ⟨c,funext fun i => ?_⟩ |
| induction i using Fin.induction with |
| | zero => exact codePosition_zero _ _ |
| | succ i ih => rw [codePosition_succ,ih,hc]; abel |
|
|
| abbrev SimpleCoarseCode (z : Site) (L : ℕ) := {c : CoarseCode L // Function.Injective (codePosition z c)} |
|
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| theorem card_simpleCoarseCode (z : Site) (L : ℕ) : Fintype.card (SimpleCoarseCode z L)≤9^L := by |
| calc |
| _≤Fintype.card (CoarseCode L) := Fintype.card_subtype_le _ |
| _=9^L := by simp [CoarseCode,CoarseDirection] |
|
|
| theorem codeColor_exists (t : ℕ) (z : Site) (c : CoarseCode (25*t)) : |
| ∃ q : SiteColor,t≤Fintype.card {i : Fin (25*t+1) // siteColor (codePosition z c i)=q} := by |
| obtain ⟨q,hq⟩ := Fintype.exists_le_card_fiber_of_mul_le_card |
| (fun i => siteColor (codePosition z c i)) (n := t) (by simp [SiteColor]) |
| exact ⟨q,by simpa only [Fintype.card_subtype] using hq⟩ |
|
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| def codeColor (t : ℕ) (z : Site) (c : CoarseCode (25*t)) : SiteColor := |
| (codeColor_exists t z c).choose |
|
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| theorem codeColor_card (t : ℕ) (z : Site) (c : CoarseCode (25*t)) : |
| t≤Fintype.card {i : Fin (25*t+1) // siteColor (codePosition z c i)=codeColor t z c} := |
| (codeColor_exists t z c).choose_spec |
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| end ClassicalON |
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| end |
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| end OAI |
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