| import OAI.Probability.MatroidProphet.LayerOnline |
| import OAI.Probability.MatroidProphet.Density.Birth |
| import Mathlib.Data.Finset.Sort |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet |
| open Set Finset |
|
|
| structure MainMasks (n : ℕ) where |
| H : Finset (Fin n) |
| D : Finset (Fin n) |
| C : Finset (Fin n) |
| T : Finset (Fin n) |
| odd : Bool |
|
|
| def mainSeedBits (n : ℕ) : ℕ := 4*n+2 |
|
|
| def seedMask {n : ℕ} (r : Seed (mainSeedBits n)) (j : Fin 4) : Finset (Fin n) := |
| univ.filter fun e => r ⟨4*e.val+j.val, by dsimp [mainSeedBits]; omega⟩ = true |
|
|
| def mainMasks {n : ℕ} (r : Seed (mainSeedBits n)) : MainMasks n where |
| H := seedMask r 0 |
| D := seedMask r 1 |
| C := seedMask r 2 |
| T := seedMask r 3 |
| odd := r ⟨4*n+1, by dsimp [mainSeedBits]; omega⟩ |
|
|
| def mainBranch {n : ℕ} (r : Seed (mainSeedBits n)) : Bool := |
| r ⟨4*n, by dsimp [mainSeedBits]; omega⟩ |
|
|
| namespace MainAlgorithm |
| variable {n : ℕ} |
|
|
| def higher (f : Fin n) (a : Option ℤ) (e : Fin n) (b : Option ℤ) : Prop := |
| match a, b with |
| | some i, some j => j < i ∨ j = i ∧ f < e |
| | some _, none => True |
| | _, _ => False |
|
|
| def candidate (M : Matroid (Fin n)) (d : MainMasks n) (seen : Fin n → Option ℤ) |
| (e : Fin n) (w : Option ℤ) : Prop := |
| w ≠ none ∧ e ∉ d.H ∧ e ∉ M.closure {f | f ∈ d.H ∧ higher f (seen f) e w} |
|
|
| noncomputable def listed (M : Matroid (Fin n)) (d : MainMasks n) |
| (seen : Fin n → Option ℤ) : Finset ℤ := by |
| classical |
| exact d.D.biUnion fun e => if candidate M d seen e (seen e) then (seen e).toFinset else ∅ |
|
|
| noncomputable def groups (M : Matroid (Fin n)) (d : MainMasks n) |
| (seen : Fin n → Option ℤ) : List ℤ := (listed M d seen).sort (· ≤ ·) |
|
|
| def groupMask (M : Matroid (Fin n)) (d : MainMasks n) (seen : Fin n → Option ℤ) |
| (mask : Finset (Fin n)) (h : ℕ) : Set (Fin n) := |
| {e | e ∈ mask ∧ candidate M d seen e (seen e) ∧ seen e = (groups M d seen)[h]?} |
|
|
| noncomputable def finalPath (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (seen : Fin n → Option ℤ) : ℤ → Set (Fin n) := |
| guardedPath M hE (2^100) (groupMask M d seen d.D) (groupMask M d seen d.C) |
| (groups M d seen).length |
|
|
| noncomputable def birth (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (seen : Fin n → Option ℤ) (e : Fin n) (i : ℤ) : ℤ := |
| nominalBirth M hE (2^100) (groupMask M d seen d.D) (groupMask M d seen d.C) |
| ((groups M d seen).idxOf i) e |
|
|
| def eligible (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (seen : Fin n → Option ℤ) (e : Fin n) (i : ℤ) : Prop := |
| e ∉ d.H ∪ d.D ∪ d.C ∧ candidate M d seen e (some i) ∧ i ∈ groups M d seen ∧ |
| e ∈ d.T ∧ activation ((groups M d seen).idxOf i) + 2 ≤ birth M hE d seen e i ∧ |
| (birth M hE d seen e i) % 2 = (if d.odd then 1 else 0) ∧ |
| e ∉ finalPath M hE d seen (birth M hE d seen e i - 2) |
|
|
| noncomputable def assign (M : Matroid (Fin n)) (hE : M.E = Set.univ) |
| (d : MainMasks n) (seen : Fin n → Option ℤ) (e : Fin n) : Option ℤ → Option ℤ := by |
| classical |
| exact fun w => match w with |
| | none => none |
| | some i => if eligible M hE d seen e i then some (birth M hE d seen e i) else none |
|
|
| noncomputable def core (M : Matroid (Fin n)) (hE : M.E = Set.univ) : |
| OnlineRule n (mainSeedBits n) := |
| layerOnlineRule M (roundedLevel weightBase) (measurable_roundedLevel weightBase) |
| (fun r seen b => finalPath M hE (mainMasks r) seen (b-2)) |
| (fun r seen => assign M hE (mainMasks r) seen) |
|
|
| noncomputable def hidden (M : Matroid (Fin n)) (hE : M.E = Set.univ) : |
| HiddenRule n (mainSeedBits n) where |
| mask r := (mainMasks r).H ∪ (mainMasks r).D ∪ (mainMasks r).C |
| core := core M hE |
|
|
| end MainAlgorithm |
| end MatroidProphet |
|
|
| end OAI |
|
|
