| import OAI.Combinatorics.MatrixRemoval.Model |
|
|
| namespace OAI |
|
|
| /-! |
| The raw finite host; row and column orders are supplied separately. |
| The second coordinate in a variable class is uniform on `Fin (2^h)`; |
| its quotient by the block size is the node index. |
| -/ |
| noncomputable section |
| namespace Problem348.Construction |
|
|
| abbrev VariablePosition (h : ℕ) := Fin (2 * h + 1) × Fin (2 ^ h) |
| abbrev AnchorPosition (h : ℕ) := Fin (6 * h) × Fin 64 × Fin (2 ^ h) |
| abbrev Position (h : ℕ) := AnchorPosition h ⊕ (VariablePosition h ⊕ Fin (2 ^ h)) |
| abbrev Mode (h : ℕ) := Fin (6 * h) |
|
|
| def depth {h : ℕ} (x : VariablePosition h) : ℕ := x.1.val / 2 |
| def node {h : ℕ} (x : VariablePosition h) : ℕ := x.2.val / 2 ^ (h - depth x) |
| def plus {h : ℕ} (x : VariablePosition h) : Bool := decide (x.1.val % 2 = 0) |
| def level {h : ℕ} (t : Mode h) : ℕ := t.val / 6 + 1 |
| def kind {h : ℕ} (t : Mode h) : ℕ := t.val % 6 |
|
|
| def AtLevel {h : ℕ} (i : ℕ) (s : Bool) (x : VariablePosition h) : Prop := |
| depth x = i ∧ (i = h ∨ plus x = s) |
|
|
| def rowRole {h : ℕ} (t : Mode h) (j : Fin 2) (x : Position h) : Prop := |
| match x with |
| | Sum.inl _ => False |
| | Sum.inr (Sum.inr _) => 2 ≤ kind t ∧ j.val = 1 |
| | Sum.inr (Sum.inl v) => |
| if kind t = 0 then |
| AtLevel (if j.val = 0 then level t else level t - 1) true v |
| else if kind t = 1 then |
| AtLevel (if j.val = 0 then level t - 1 else level t) false v |
| else |
| j.val = 0 ∧ AtLevel (level t) (decide (kind t < 4)) v ∧ |
| node v % 2 = (kind t % 2) |
|
|
| def colRole {h : ℕ} (t : Mode h) (j : Fin 2) (x : Position h) : Prop := |
| match x with |
| | Sum.inl _ => False |
| | Sum.inr (Sum.inr _) => kind t < 2 ∧ j.val = 0 |
| | Sum.inr (Sum.inl v) => |
| if kind t < 2 then |
| j.val = 1 ∧ AtLevel (level t - 1) (decide (kind t = 0)) v |
| else if kind t < 4 then |
| if j.val = 0 then AtLevel (level t - 1) true v |
| else AtLevel (level t) true v ∧ node v % 2 = kind t % 2 |
| else |
| if j.val = 0 then AtLevel (level t) false v ∧ node v % 2 = kind t % 2 |
| else AtLevel (level t - 1) false v |
|
|
| instance {h : ℕ} (i : ℕ) (s : Bool) (v : VariablePosition h) : |
| Decidable (AtLevel i s v) := inferInstanceAs (Decidable (_ ∧ _)) |
| instance {h : ℕ} (t : Mode h) (j : Fin 2) (x : Position h) : |
| Decidable (rowRole t j x) := by unfold rowRole; split <;> infer_instance |
| instance {h : ℕ} (t : Mode h) (j : Fin 2) (x : Position h) : |
| Decidable (colRole t j x) := by unfold colRole; split <;> infer_instance |
|
|
| def variableEntry {h : ℕ} (r c : VariablePosition h) : Bool := |
| if depth r = depth c then |
| if depth r = h then decide (node r ≤ node c) |
| else if plus r = plus c then |
| if plus r then decide (node r ≤ node c) else decide (node r < node c) |
| else false |
| else if depth r = depth c + 1 ∧ depth c < h ∧ |
| (depth r = h ∨ plus r = plus c) then |
| if plus c then decide (node r / 2 ≤ node c) else decide (node r / 2 < node c) |
| else false |
|
|
| def host {h : ℕ} (r c : Position h) : Bool := |
| match r, c with |
| | Sum.inl (t, u, _), Sum.inl (t', v, _) => |
| if t = t' then Problem348.anchor64 u v else false |
| | Sum.inl (t, u, _), Sum.inr _ => |
| decide ((u.val = 0 ∧ colRole t 0 c) ∨ (u.val = 1 ∧ colRole t 1 c)) |
| | Sum.inr _, Sum.inl (t, v, _) => |
| decide ((v.val = 0 ∧ rowRole t 0 r) ∨ (v.val = 1 ∧ rowRole t 1 r)) |
| | Sum.inr (Sum.inl v), Sum.inr (Sum.inl w) => variableEntry v w |
| | Sum.inr _, Sum.inr _ => true |
|
|
| theorem depth_le {h : ℕ} (x : VariablePosition h) : depth x ≤ h := by |
| have hx := x.1.isLt |
| unfold depth |
| omega |
|
|
| theorem level_pos {h : ℕ} (t : Mode h) : 1 ≤ level t := by |
| unfold level |
| omega |
|
|
| theorem level_le {h : ℕ} (t : Mode h) : level t ≤ h := by |
| have ht := t.isLt |
| unfold level |
| omega |
|
|
| theorem kind_lt {h : ℕ} (t : Mode h) : kind t < 6 := by |
| exact Nat.mod_lt _ (by decide) |
|
|
| theorem card_variablePosition (h : ℕ) : |
| Fintype.card (VariablePosition h) = (2 * h + 1) * 2 ^ h := by |
| simp [VariablePosition] |
|
|
| theorem card_position (h : ℕ) : |
| Fintype.card (Position h) = (386 * h + 2) * 2 ^ h := by |
| simp only [Position, AnchorPosition, VariablePosition, Fintype.card_sum, |
| Fintype.card_prod, Fintype.card_fin] |
| ring |
|
|
| @[simp] theorem host_anchor_same {h : ℕ} (t : Mode h) (u v : Fin 64) |
| (x y : Fin (2 ^ h)) : |
| host (Sum.inl (t, u, x)) (Sum.inl (t, v, y)) = anchor64 u v := by |
| simp [host] |
|
|
| @[simp] theorem host_anchor_different {h : ℕ} (t t' : Mode h) (ht : t ≠ t') |
| (u v : Fin 64) (x y : Fin (2 ^ h)) : |
| host (Sum.inl (t, u, x)) (Sum.inl (t', v, y)) = false := by |
| simp [host, ht] |
|
|
| @[simp] theorem host_variable_variable {h : ℕ} (v w : VariablePosition h) : |
| host (Sum.inr (Sum.inl v)) (Sum.inr (Sum.inl w)) = variableEntry v w := rfl |
|
|
| @[simp] theorem host_dummy_variable {h : ℕ} (x : Fin (2 ^ h)) |
| (v : VariablePosition h) : |
| host (Sum.inr (Sum.inr x)) (Sum.inr (Sum.inl v)) = true := rfl |
|
|
| @[simp] theorem host_variable_dummy {h : ℕ} (x : Fin (2 ^ h)) |
| (v : VariablePosition h) : |
| host (Sum.inr (Sum.inl v)) (Sum.inr (Sum.inr x)) = true := rfl |
|
|
| theorem rowRole_disjoint {h : ℕ} (t : Mode h) (x : Position h) : |
| ¬ (rowRole t 0 x ∧ rowRole t 1 x) := by |
| have hp := level_pos t |
| have hk := kind_lt t |
| rcases x with a | (v | d) |
| · simp [rowRole] |
| · interval_cases e : kind t <;> |
| simp [rowRole, e, AtLevel] <;> omega |
| · simp [rowRole] |
|
|
| theorem colRole_disjoint {h : ℕ} (t : Mode h) (x : Position h) : |
| ¬ (colRole t 0 x ∧ colRole t 1 x) := by |
| have hp := level_pos t |
| have hk := kind_lt t |
| rcases x with a | (v | d) |
| · simp [colRole] |
| · interval_cases e : kind t <;> |
| simp [colRole, e, AtLevel] <;> omega |
| · simp [colRole] |
|
|
| end Problem348.Construction |
|
|
| end |
|
|
| end OAI |
|
|
