| import OAI.Combinatorics.MatrixRemoval.CanonicalModeEntries |
| import OAI.Combinatorics.MatrixRemoval.ModeLocalization |
|
|
| namespace OAI |
|
|
| /-! Horizontal body localization for the canonical host entry table. -/ |
| namespace Problem348.Construction |
| open ModeLocalization |
|
|
| theorem horizontal_plus_entries_impossible {h i : ℕ} {r c₀ c₁ : VariablePosition h} |
| (hr : AtLevel (i + 1) true r) (hc₀ : AtLevel i true c₀) |
| (hc₁ : AtLevel (i + 1) true c₁) |
| (horder : node c₀ ≤ node c₁ / 2) (hparity : node r % 2 = node c₁ % 2) |
| (h₀ : variableEntry r c₀ = true) (h₁ : variableEntry r c₁ = false) : False := by |
| apply wPlus_not_isP (node r) (node c₁) (node c₀) horder hparity |
| exact ⟨by simpa only [variableEntry_child_parent_plus hr hc₀] using h₀, |
| by simpa only [variableEntry_same_plus hr hc₁] using h₁, rfl, rfl⟩ |
|
|
| /-- A W- body forces the shared leaf diagonal, as literal variable positions. -/ |
| theorem horizontal_minus_entries_localize {h i : ℕ} {r c₀ c₁ : VariablePosition h} |
| (hr : AtLevel (i + 1) false r) (hc₀ : AtLevel (i + 1) false c₀) |
| (hc₁ : AtLevel i false c₁) |
| (horder : node c₀ / 2 ≤ node c₁) (hparity : node r % 2 = node c₀ % 2) |
| (h₀ : variableEntry r c₀ = true) (h₁ : variableEntry r c₁ = false) : |
| i + 1 = h ∧ ∃ z : Fin (2 ^ h), |
| r = (Fin.last (2 * h), z) ∧ c₀ = (Fin.last (2 * h), z) := by |
| have hb : IsP |
| (if i + 1 = h then decide (node r ≤ node c₀) else decide (node r < node c₀)) |
| (decide (node r / 2 < node c₁)) true true := by |
| exact ⟨by simpa only [variableEntry_same_minus hr hc₀] using h₀, |
| by simpa only [variableEntry_child_parent_minus hr hc₁] using h₁, rfl, rfl⟩ |
| obtain ⟨hi, heq⟩ := wMinus_isP_implies_leaf_diagonal h (i + 1) (node r) (node c₀) (node c₁) horder hparity hb |
| have hrd : depth r = h := hr.1.trans hi |
| have hcd : depth c₀ = h := hc₀.1.trans hi |
| have hz : r.2 = c₀.2 := by |
| apply Fin.ext |
| simpa only [node_eq_index_of_depth_eq hrd, node_eq_index_of_depth_eq hcd] using heq |
| exact ⟨hi, r.2, eq_leaf_of_depth_eq hrd, by simpa only [hz] using eq_leaf_of_depth_eq hcd⟩ |
| |
| end Problem348.Construction |
| |
| end OAI |
| |
