| import OAI.Combinatorics.MatrixRemoval.Host |
|
|
| namespace OAI |
|
|
| /-! Exact level-based entry formulas for the canonical flat-class host. -/ |
| namespace Problem348.Construction |
|
|
| theorem variableEntry_same_plus {h i : ℕ} {r c : VariablePosition h} |
| (hr : AtLevel i true r) (hc : AtLevel i true c) : |
| variableEntry r c = decide (node r ≤ node c) := by |
| by_cases hi : i = h |
| · simp [variableEntry, hr.1, hc.1, hi] |
| · have hpr := hr.2.resolve_left hi |
| have hpc := hc.2.resolve_left hi |
| simp [variableEntry, hr.1, hc.1, hi, hpr, hpc] |
|
|
| theorem variableEntry_same_minus {h i : ℕ} {r c : VariablePosition h} |
| (hr : AtLevel i false r) (hc : AtLevel i false c) : |
| variableEntry r c = |
| if i = h then decide (node r ≤ node c) else decide (node r < node c) := by |
| by_cases hi : i = h |
| · simp [variableEntry, hr.1, hc.1, hi] |
| · have hpr := hr.2.resolve_left hi |
| have hpc := hc.2.resolve_left hi |
| simp [variableEntry, hr.1, hc.1, hi, hpr, hpc] |
|
|
| theorem variableEntry_child_parent_plus {h i : ℕ} {r c : VariablePosition h} |
| (hr : AtLevel (i + 1) true r) (hc : AtLevel i true c) : |
| variableEntry r c = decide (node r / 2 ≤ node c) := by |
| have hi : i < h := by have := depth_le r; have hrd := hr.1; omega |
| have hpc := hc.2.resolve_left (Nat.ne_of_lt hi) |
| simp [variableEntry, hr.1, hc.1, hi, hpc, hr.2] |
|
|
| theorem variableEntry_child_parent_minus {h i : ℕ} {r c : VariablePosition h} |
| (hr : AtLevel (i + 1) false r) (hc : AtLevel i false c) : |
| variableEntry r c = decide (node r / 2 < node c) := by |
| have hi : i < h := by have := depth_le r; have hrd := hr.1; omega |
| have hpc := hc.2.resolve_left (Nat.ne_of_lt hi) |
| simp [variableEntry, hr.1, hc.1, hi, hpc, hr.2] |
|
|
| / |
| theorem class_eq_leaf_of_depth_eq {h : ℕ} {v : VariablePosition h} |
| (hv : depth v = h) : v.1 = Fin.last (2 * h) := by |
| apply Fin.ext |
| have := v.1.isLt |
| simp only [Fin.val_last] |
| unfold depth at hv |
| omega |
|
|
| / |
| theorem node_eq_index_of_depth_eq {h : ℕ} {v : VariablePosition h} |
| (hv : depth v = h) : node v = v.2.val := by |
| simp [node, hv] |
|
|
| theorem eq_leaf_of_depth_eq {h : ℕ} {v : VariablePosition h} |
| (hv : depth v = h) : v = (Fin.last (2 * h), v.2) := by |
| exact Prod.ext (class_eq_leaf_of_depth_eq hv) rfl |
|
|
| end Problem348.Construction |
|
|
| end OAI |
|
|
