| import OAI.Combinatorics.MatrixRemoval.Host |
| import OAI.Combinatorics.MatrixRemoval.SampledPath |
|
|
| namespace OAI |
|
|
| /-! |
| Concrete sampled-path role membership for the canonical host. |
| The input is one common leaf seed and arbitrary within-class offset seeds. |
| -/ |
| namespace Problem348.HostSampledPath |
|
|
| open SampledPath |
|
|
| / |
| def position (h i : ℕ) (hi : i ≤ h) (plus : Bool) |
| (z u : Fin (2 ^ h)) : Construction.Position h := |
| Sum.inr (Sum.inl (sample h i hi plus z u)) |
|
|
| / |
| def dummy (h : ℕ) (u : Fin (2 ^ h)) : Construction.Position h := |
| Sum.inr (Sum.inr u) |
|
|
| @[simp] theorem sample_atLevel (h i : ℕ) (hi : i ≤ h) (s : Bool) |
| (z u : Fin (2 ^ h)) : Construction.AtLevel i s (sample h i hi s z u) := by |
| constructor |
| · exact sample_depth h i hi s z u |
| · by_cases heq : i = h |
| · exact Or.inl heq |
| · right |
| cases s <;> |
| simp [Construction.plus, sample, classAt, heq] |
|
|
| @[simp] theorem sample_hostNode (h i : ℕ) (hi : i ≤ h) (s : Bool) |
| (z u : Fin (2 ^ h)) : |
| Construction.node (sample h i hi s z u) = z.val / blockSize h i := |
| sample_node h i hi s z u |
|
|
| @[simp] theorem position_shared_leaf (h : ℕ) (s t : Bool) |
| (z u v : Fin (2 ^ h)) : |
| position h h le_rfl s z u = position h h le_rfl t z v := by |
| exact congrArg (fun p : Construction.VariablePosition h => |
| (Sum.inr (Sum.inl p) : Construction.Position h)) |
| (sample_shared_leaf h s t z u v) |
|
|
| / |
| def mode {h : ℕ} (i : Fin h) (k : Fin 6) : Construction.Mode h := |
| ⟨6 * i.val + k.val, by have := i.isLt; have := k.isLt; omega⟩ |
|
|
| @[simp] theorem mode_level {h : ℕ} (i : Fin h) (k : Fin 6) : |
| Construction.level (mode i k) = i.val + 1 := by |
| have := k.isLt |
| unfold Construction.level mode |
| dsimp |
| omega |
|
|
| @[simp] theorem mode_kind {h : ℕ} (i : Fin h) (k : Fin 6) : |
| Construction.kind (mode i k) = k.val := by |
| have := k.isLt |
| unfold Construction.kind mode |
| dsimp |
| omega |
|
|
| def verticalKind : Bool → Fin 6 |
| | true => 0 |
| | false => 1 |
|
|
| def horizontalKind : Bool → Bool → Fin 6 |
| | true, false => 2 |
| | true, true => 3 |
| | false, false => 4 |
| | false, true => 5 |
|
|
| def vertical {h : ℕ} (i : Fin h) (s : Bool) : Construction.Mode h := |
| mode i (verticalKind s) |
| def horizontal {h : ℕ} (i : Fin h) (s d : Bool) : Construction.Mode h := |
| mode i (horizontalKind s d) |
|
|
| def parity (n : ℕ) : Bool := decide (n % 2 = 1) |
|
|
| @[simp] theorem parity_eq_false (n : ℕ) : parity n = false ↔ n % 2 = 0 := by |
| simp only [parity, decide_eq_false_iff_not] |
| omega |
|
|
| @[simp] theorem parity_eq_true (n : ℕ) : parity n = true ↔ n % 2 = 1 := by |
| simp [parity] |
|
|
| theorem vertical_plus_row_first {h : ℕ} (i : Fin h) (z u : Fin (2 ^ h)) : |
| Construction.rowRole (vertical i true) 0 |
| (position h (i.val + 1) (by omega) true z u) := by |
| simp [vertical, verticalKind, Construction.rowRole, position] |
|
|
| theorem vertical_plus_row_second {h : ℕ} (i : Fin h) (z u : Fin (2 ^ h)) : |
| Construction.rowRole (vertical i true) 1 |
| (position h i.val (by omega) true z u) := by |
| simp [vertical, verticalKind, Construction.rowRole, position] |
|
|
| theorem vertical_minus_row_first {h : ℕ} (i : Fin h) (z u : Fin (2 ^ h)) : |
| Construction.rowRole (vertical i false) 0 |
| (position h i.val (by omega) false z u) := by |
| simp [vertical, verticalKind, Construction.rowRole, position] |
|
|
| theorem vertical_minus_row_second {h : ℕ} (i : Fin h) (z u : Fin (2 ^ h)) : |
| Construction.rowRole (vertical i false) 1 |
| (position h (i.val + 1) (by omega) false z u) := by |
| simp [vertical, verticalKind, Construction.rowRole, position] |
|
|
| theorem vertical_col_first {h : ℕ} (i : Fin h) (s : Bool) (u : Fin (2 ^ h)) : |
| Construction.colRole (vertical i s) 0 (dummy h u) := by |
| cases s <;> simp [vertical, verticalKind, Construction.colRole, dummy] |
|
|
| theorem vertical_col_second {h : ℕ} (i : Fin h) (s : Bool) (z u : Fin (2 ^ h)) : |
| Construction.colRole (vertical i s) 1 (position h i.val (by omega) s z u) := by |
| cases s <;> simp [vertical, verticalKind, Construction.colRole, position] |
|
|
| theorem horizontal_row_second {h : ℕ} (i : Fin h) (s d : Bool) |
| (u : Fin (2 ^ h)) : Construction.rowRole (horizontal i s d) 1 (dummy h u) := by |
| cases s <;> cases d <;> simp [horizontal, horizontalKind, Construction.rowRole, dummy] |
|
|
| theorem horizontal_row_first {h : ℕ} (i : Fin h) (s : Bool) |
| (z u : Fin (2 ^ h)) : |
| Construction.rowRole (horizontal i s (parity (z.val / blockSize h (i.val + 1)))) 0 |
| (position h (i.val + 1) (by omega) s z u) := by |
| have hp := Nat.mod_lt (z.val / blockSize h (i.val + 1)) (by decide : 0 < 2) |
| cases s <;> by_cases hd : (z.val / blockSize h (i.val + 1)) % 2 = 1 <;> |
| simp_all [horizontal, horizontalKind, Construction.rowRole, position, parity] |
|
|
| theorem horizontal_plus_col_first {h : ℕ} (i : Fin h) (d : Bool) |
| (z u : Fin (2 ^ h)) : |
| Construction.colRole (horizontal i true d) 0 (position h i.val (by omega) true z u) := by |
| cases d <;> simp [horizontal, horizontalKind, Construction.colRole, position] |
|
|
| theorem horizontal_plus_col_second {h : ℕ} (i : Fin h) (z u : Fin (2 ^ h)) : |
| Construction.colRole (horizontal i true (parity (z.val / blockSize h (i.val + 1)))) 1 |
| (position h (i.val + 1) (by omega) true z u) := by |
| have hp := Nat.mod_lt (z.val / blockSize h (i.val + 1)) (by decide : 0 < 2) |
| by_cases hd : (z.val / blockSize h (i.val + 1)) % 2 = 1 <;> |
| simp_all [horizontal, horizontalKind, Construction.colRole, position, parity] |
|
|
| theorem horizontal_minus_col_first {h : ℕ} (i : Fin h) (z u : Fin (2 ^ h)) : |
| Construction.colRole (horizontal i false (parity (z.val / blockSize h (i.val + 1)))) 0 |
| (position h (i.val + 1) (by omega) false z u) := by |
| have hp := Nat.mod_lt (z.val / blockSize h (i.val + 1)) (by decide : 0 < 2) |
| by_cases hd : (z.val / blockSize h (i.val + 1)) % 2 = 1 <;> |
| simp_all [horizontal, horizontalKind, Construction.colRole, position, parity] |
|
|
| theorem horizontal_minus_col_second {h : ℕ} (i : Fin h) (d : Bool) |
| (z u : Fin (2 ^ h)) : |
| Construction.colRole (horizontal i false d) 1 (position h i.val (by omega) false z u) := by |
| cases d <;> simp [horizontal, horizontalKind, Construction.colRole, position] |
|
|
| end Problem348.HostSampledPath |
|
|
| end OAI |
|
|
