| import OAI.Combinatorics.MatrixRemoval.Host |
| import OAI.Combinatorics.MatrixRemoval.ModeLocalization |
|
|
| namespace OAI |
|
|
| /-! |
| # Vertical-mode localization for the concrete host |
|
|
| The only additional input to the exclusion theorems is the node inequality |
| provided by the tree order. All entry values and role memberships follow |
| from the host definitions. |
| -/ |
|
|
| namespace Problem348.Construction |
|
|
| / |
| def positionNode {h : ℕ} : Position h → ℕ |
| | Sum.inr (Sum.inl v) => node v |
| | _ => 0 |
|
|
| @[simp] theorem positionNode_variable {h : ℕ} (v : VariablePosition h) : |
| positionNode (Sum.inr (Sum.inl v)) = node v := rfl |
|
|
| theorem variableEntry_same_level {h i : ℕ} {r c : VariablePosition h} |
| (s : Bool) (hr : AtLevel i s r) (hc : AtLevel i s c) : |
| variableEntry r c = |
| if i = h then decide (node r ≤ node c) |
| else if s then decide (node r ≤ node c) else decide (node r < node c) := by |
| rcases hr with ⟨hrd, hrs⟩ |
| rcases hc with ⟨hcd, hcs⟩ |
| by_cases hi : i = h |
| · simp [variableEntry, hrd, hcd, hi] |
| · have hrp : plus r = s := hrs.resolve_left hi |
| have hcp : plus c = s := hcs.resolve_left hi |
| simp [variableEntry, hrd, hcd, hi, hrp, hcp] |
|
|
| theorem variableEntry_child_parent {h i : ℕ} {r c : VariablePosition h} |
| (s : Bool) (hr : AtLevel (i + 1) s r) (hc : AtLevel i s c) : |
| variableEntry r c = |
| if s then decide (node r / 2 ≤ node c) else decide (node r / 2 < node c) := by |
| rcases hr with ⟨hrd, hrs⟩ |
| rcases hc with ⟨hcd, hcs⟩ |
| have hih : i < h := by have := depth_le r; omega |
| have hcp : plus c = s := hcs.resolve_left (by omega) |
| have hsign : i + 1 = h ∨ plus r = s := hrs |
| simp [variableEntry, hrd, hcd, hih, hcp, hsign] |
|
|
| theorem rowRole_vertical_variable {h : ℕ} {t : Mode h} {j : Fin 2} |
| {x : Position h} (hk : kind t < 2) (hx : rowRole t j x) : |
| ∃ v : VariablePosition h, x = Sum.inr (Sum.inl v) := by |
| rcases x with a | (v | d) |
| · simp [rowRole] at hx |
| · exact ⟨v, rfl⟩ |
| · simp [rowRole] at hx |
| omega |
|
|
| theorem colRole_vertical_first_dummy {h : ℕ} {t : Mode h} |
| {x : Position h} (hk : kind t < 2) (hx : colRole t 0 x) : |
| ∃ d : Fin (2 ^ h), x = Sum.inr (Sum.inr d) := by |
| rcases x with a | (v | d) |
| · simp [colRole] at hx |
| · simp [colRole, hk] at hx |
| · exact ⟨d, rfl⟩ |
|
|
| theorem colRole_vertical_second_variable {h : ℕ} {t : Mode h} |
| {x : Position h} (_hk : kind t < 2) (hx : colRole t 1 x) : |
| ∃ v : VariablePosition h, x = Sum.inr (Sum.inl v) := by |
| rcases x with a | (v | d) |
| · simp [colRole] at hx |
| · exact ⟨v, rfl⟩ |
| · simp [colRole] at hx |
|
|
| / |
| theorem host_vertical_plus_not_body {h : ℕ} {t : Mode h} |
| (hk : kind t = 0) {r₀ r₁ c₀ c₁ : Position h} |
| (hr₀ : rowRole t 0 r₀) (hr₁ : rowRole t 1 r₁) |
| (hc₀ : colRole t 0 c₀) (hc₁ : colRole t 1 c₁) |
| (horder : positionNode r₀ / 2 ≤ positionNode r₁) : |
| ¬ ModeLocalization.IsP (host r₀ c₀) (host r₀ c₁) |
| (host r₁ c₀) (host r₁ c₁) := by |
| have hkt : kind t < 2 := by omega |
| obtain ⟨x, rfl⟩ := rowRole_vertical_variable hkt hr₀ |
| obtain ⟨y, rfl⟩ := rowRole_vertical_variable hkt hr₁ |
| obtain ⟨d, rfl⟩ := colRole_vertical_first_dummy hkt hc₀ |
| obtain ⟨z, rfl⟩ := colRole_vertical_second_variable hkt hc₁ |
| have hx : AtLevel (level t) true x := by simpa [rowRole, hk] using hr₀ |
| have hy : AtLevel (level t - 1) true y := by simpa [rowRole, hk] using hr₁ |
| have hz : AtLevel (level t - 1) true z := by simpa [colRole, hk] using hc₁ |
| have hl : level t - 1 + 1 = level t := by have := level_pos t; omega |
| have hparent := variableEntry_child_parent true (hl ▸ hx) hz |
| have hsame := variableEntry_same_level true hy hz |
| have hnotleaf : level t - 1 ≠ h := by |
| have := level_le t |
| have := level_pos t |
| omega |
| simpa [host, hparent, hsame, hnotleaf, positionNode] using |
| ModeLocalization.vPlus_not_isP (node x) (node y) (node z) horder |
|
|
| / |
| theorem host_vertical_minus_not_body {h : ℕ} {t : Mode h} |
| (hk : kind t = 1) {r₀ r₁ c₀ c₁ : Position h} |
| (hr₀ : rowRole t 0 r₀) (hr₁ : rowRole t 1 r₁) |
| (hc₀ : colRole t 0 c₀) (hc₁ : colRole t 1 c₁) |
| (horder : positionNode r₀ ≤ positionNode r₁ / 2) : |
| ¬ ModeLocalization.IsP (host r₀ c₀) (host r₀ c₁) |
| (host r₁ c₀) (host r₁ c₁) := by |
| have hkt : kind t < 2 := by omega |
| obtain ⟨x, rfl⟩ := rowRole_vertical_variable hkt hr₀ |
| obtain ⟨y, rfl⟩ := rowRole_vertical_variable hkt hr₁ |
| obtain ⟨d, rfl⟩ := colRole_vertical_first_dummy hkt hc₀ |
| obtain ⟨z, rfl⟩ := colRole_vertical_second_variable hkt hc₁ |
| have hx : AtLevel (level t - 1) false x := by simpa [rowRole, hk] using hr₀ |
| have hy : AtLevel (level t) false y := by simpa [rowRole, hk] using hr₁ |
| have hz : AtLevel (level t - 1) false z := by simpa [colRole, hk] using hc₁ |
| have hl : level t - 1 + 1 = level t := by have := level_pos t; omega |
| have hparent := variableEntry_child_parent false (hl ▸ hy) hz |
| have hsame := variableEntry_same_level false hx hz |
| have hnotleaf : level t - 1 ≠ h := by |
| have := level_le t |
| have := level_pos t |
| omega |
| simpa [host, hparent, hsame, hnotleaf, positionNode] using |
| ModeLocalization.vMinus_not_isP (node y) (node x) (node z) horder |
|
|
| end Problem348.Construction |
|
|
| end OAI |
|
|
