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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
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