| import OAI.Combinatorics.MatrixRemoval.Host |
| import OAI.Combinatorics.MatrixRemoval.TreePositions |
|
|
| namespace OAI |
|
|
| /-! |
| Comparison lemmas for the concrete depth-first tree orders used in |
| mode localization. |
| -/ |
|
|
| namespace Problem348.Construction |
|
|
| open TreePositions |
|
|
| theorem eq_plusPos_of_atLevel {h i : ℕ} {v : VariablePosition h} |
| (hv : AtLevel i true v) : |
| v = TreePositions.plusPos i (hv.1 ▸ depth_le v) v.2 := by |
| apply Prod.ext |
| · apply Fin.ext |
| have hc := v.1.isLt |
| obtain ⟨hd, hs⟩ := hv |
| simp only [depth] at hd |
| change v.1.val = 2 * i |
| rcases hs with hi | hs |
| · omega |
| · simp only [plus, decide_eq_true_eq] at hs |
| omega |
| · rfl |
|
|
| theorem eq_minusPos_of_atLevel {h i : ℕ} {v : VariablePosition h} |
| (hv : AtLevel i false v) : |
| v = TreePositions.minusPos i (hv.1 ▸ depth_le v) v.2 := by |
| apply Prod.ext |
| · apply Fin.ext |
| have hc := v.1.isLt |
| obtain ⟨hd, hs⟩ := hv |
| simp only [depth] at hd |
| change v.1.val = if i = h then 2 * h else 2 * i + 1 |
| split_ifs with hi |
| · omega |
| · rcases hs with hs | hs |
| · exact (hi hs).elim |
| · simp only [plus, decide_eq_false_iff_not] at hs |
| omega |
| · rfl |
|
|
| theorem node_eq_level_node {h i : ℕ} {v : VariablePosition h} |
| (hv : depth v = i) : node v = TreePositions.node i v.2 := by |
| simp [node, TreePositions.node, hv] |
|
|
| / |
| theorem row_plus_lt_iff {h i : ℕ} {child parent : VariablePosition h} |
| (hc : AtLevel (i + 1) true child) (hp : AtLevel i true parent) : |
| TreePositions.rowKey h child < TreePositions.rowKey h parent ↔ |
| node child / 2 ≤ node parent := by |
| have hi : i < h := by have := hc.1 ▸ depth_le child; omega |
| have hn₁ := node_eq_level_node hc.1 |
| have hn₂ := node_eq_level_node hp.1 |
| rw [hn₁, hn₂, eq_plusPos_of_atLevel hc, eq_plusPos_of_atLevel hp] |
| exact TreePositions.row_plus_child_lt_parent hi child.2 parent.2 |
|
|
| / |
| theorem row_minus_lt_iff {h i : ℕ} {child parent : VariablePosition h} |
| (hc : AtLevel (i + 1) false child) (hp : AtLevel i false parent) : |
| TreePositions.rowKey h parent < TreePositions.rowKey h child ↔ |
| node parent ≤ node child / 2 := by |
| have hi : i < h := by have := hc.1 ▸ depth_le child; omega |
| have hn₁ := node_eq_level_node hc.1 |
| have hn₂ := node_eq_level_node hp.1 |
| rw [hn₁, hn₂, eq_minusPos_of_atLevel hc, eq_minusPos_of_atLevel hp] |
| exact TreePositions.row_minus_parent_lt_child hi child.2 parent.2 |
|
|
| / |
| theorem column_plus_lt_iff {h i : ℕ} {child parent : VariablePosition h} |
| (hc : AtLevel (i + 1) true child) (hp : AtLevel i true parent) : |
| TreePositions.columnKey h parent < TreePositions.columnKey h child ↔ |
| node parent ≤ node child / 2 := by |
| have hi : i < h := by have := hc.1 ▸ depth_le child; omega |
| have hn₁ := node_eq_level_node hc.1 |
| have hn₂ := node_eq_level_node hp.1 |
| rw [hn₁, hn₂, eq_plusPos_of_atLevel hc, eq_plusPos_of_atLevel hp] |
| exact TreePositions.column_plus_parent_lt_child hi child.2 parent.2 |
|
|
| / |
| theorem column_minus_lt_iff {h i : ℕ} {child parent : VariablePosition h} |
| (hc : AtLevel (i + 1) false child) (hp : AtLevel i false parent) : |
| TreePositions.columnKey h child < TreePositions.columnKey h parent ↔ |
| node child / 2 ≤ node parent := by |
| have hi : i < h := by have := hc.1 ▸ depth_le child; omega |
| have hn₁ := node_eq_level_node hc.1 |
| have hn₂ := node_eq_level_node hp.1 |
| rw [hn₁, hn₂, eq_minusPos_of_atLevel hc, eq_minusPos_of_atLevel hp] |
| exact TreePositions.column_minus_child_lt_parent hi child.2 parent.2 |
|
|
| end Problem348.Construction |
|
|
| end OAI |
|
|
