| import OAI.Combinatorics.MatrixRemoval.HostBodyModes |
| import OAI.Combinatorics.MatrixRemoval.HostTreeOrder |
|
|
| namespace OAI |
|
|
| /-! |
| # From increasing body positions to the four node-order inequalities |
|
|
| These lemmas use the actual host roles and tree keys. Each independent |
| coordinate map reflects the corresponding tree-key order on variable |
| positions; no mode-specific ordering is assumed. |
| -/ |
|
|
| universe uAlpha uR uC |
|
|
| namespace Problem348.Construction |
|
|
| theorem vertical_plus_node_order {h : ℕ} {α : Type uAlpha} [Preorder α] |
| (rowIndex : Position h → α) |
| (hkey : ∀ v w : VariablePosition h, |
| rowIndex (Sum.inr (Sum.inl v)) < rowIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.rowKey h v < TreePositions.rowKey h w) |
| {t : Mode h} (hk : kind t = 0) {r₀ r₁ : Position h} |
| (hr₀ : rowRole t 0 r₀) (hr₁ : rowRole t 1 r₁) |
| (hrow : rowIndex r₀ < rowIndex r₁) : |
| positionNode r₀ / 2 ≤ positionNode r₁ := by |
| have hkt : kind t < 2 := by omega |
| obtain ⟨x, rfl⟩ := rowRole_vertical_variable hkt hr₀ |
| obtain ⟨y, rfl⟩ := rowRole_vertical_variable hkt hr₁ |
| 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 hl : level t - 1 + 1 = level t := by have := level_pos t; omega |
| exact (row_plus_lt_iff (hl ▸ hx) hy).mp (hkey x y hrow) |
|
|
| theorem vertical_minus_node_order {h : ℕ} {α : Type uAlpha} [Preorder α] |
| (rowIndex : Position h → α) |
| (hkey : ∀ v w : VariablePosition h, |
| rowIndex (Sum.inr (Sum.inl v)) < rowIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.rowKey h v < TreePositions.rowKey h w) |
| {t : Mode h} (hk : kind t = 1) {r₀ r₁ : Position h} |
| (hr₀ : rowRole t 0 r₀) (hr₁ : rowRole t 1 r₁) |
| (hrow : rowIndex r₀ < rowIndex r₁) : |
| positionNode r₀ ≤ positionNode r₁ / 2 := by |
| have hkt : kind t < 2 := by omega |
| obtain ⟨x, rfl⟩ := rowRole_vertical_variable hkt hr₀ |
| obtain ⟨y, rfl⟩ := rowRole_vertical_variable hkt hr₁ |
| 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 hl : level t - 1 + 1 = level t := by have := level_pos t; omega |
| exact (row_minus_lt_iff (hl ▸ hy) hx).mp (hkey x y hrow) |
|
|
| theorem horizontal_plus_node_order {h : ℕ} {α : Type uAlpha} [Preorder α] |
| (colIndex : Position h → α) |
| (hkey : ∀ v w : VariablePosition h, |
| colIndex (Sum.inr (Sum.inl v)) < colIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.columnKey h v < TreePositions.columnKey h w) |
| {t : Mode h} (hklo : 2 ≤ kind t) (hkhi : kind t < 4) |
| {c₀ c₁ : Position h} (hc₀ : colRole t 0 c₀) (hc₁ : colRole t 1 c₁) |
| (hcol : colIndex c₀ < colIndex c₁) : |
| positionNode c₀ ≤ positionNode c₁ / 2 := by |
| obtain ⟨x, rfl⟩ := colRole_horizontal_variable hklo hc₀ |
| obtain ⟨y, rfl⟩ := colRole_horizontal_variable hklo hc₁ |
| have hkn : ¬ kind t < 2 := by omega |
| have hx : AtLevel (level t - 1) true x := by |
| simpa [colRole, hkn, hkhi] using hc₀ |
| have hy : AtLevel (level t) true y := by |
| have hy' : AtLevel (level t) true y ∧ node y % 2 = kind t % 2 := by |
| simpa [colRole, hkn, hkhi] using hc₁ |
| exact hy'.1 |
| have hl : level t - 1 + 1 = level t := by have := level_pos t; omega |
| exact (column_plus_lt_iff (hl ▸ hy) hx).mp (hkey x y hcol) |
|
|
| theorem horizontal_minus_node_order {h : ℕ} {α : Type uAlpha} [Preorder α] |
| (colIndex : Position h → α) |
| (hkey : ∀ v w : VariablePosition h, |
| colIndex (Sum.inr (Sum.inl v)) < colIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.columnKey h v < TreePositions.columnKey h w) |
| {t : Mode h} (hk : 4 ≤ kind t) |
| {c₀ c₁ : Position h} (hc₀ : colRole t 0 c₀) (hc₁ : colRole t 1 c₁) |
| (hcol : colIndex c₀ < colIndex c₁) : |
| positionNode c₀ / 2 ≤ positionNode c₁ := by |
| have hklo : 2 ≤ kind t := by omega |
| obtain ⟨x, rfl⟩ := colRole_horizontal_variable hklo hc₀ |
| obtain ⟨y, rfl⟩ := colRole_horizontal_variable hklo hc₁ |
| have hkn : ¬ kind t < 2 := by omega |
| have hkhi : ¬ kind t < 4 := by omega |
| have hx : AtLevel (level t) false x := by |
| have hx' : AtLevel (level t) false x ∧ node x % 2 = kind t % 2 := by |
| simpa [colRole, hkn, hkhi] using hc₀ |
| exact hx'.1 |
| have hy : AtLevel (level t - 1) false y := by |
| simpa [colRole, hkn, hkhi] using hc₁ |
| have hl : level t - 1 + 1 = level t := by have := level_pos t; omega |
| exact (column_minus_lt_iff (hl ▸ hx) hy).mp (hkey x y hcol) |
|
|
| / |
| node-order premise of the exhaustive six-mode localization theorem. -/ |
| theorem bodyNodeOrder_of_ordered {h : ℕ} {R : Type uR} {C : Type uC} [Preorder R] [Preorder C] |
| (rowIndex : Position h → R) (colIndex : Position h → C) |
| (hrowKey : ∀ v w : VariablePosition h, |
| rowIndex (Sum.inr (Sum.inl v)) < rowIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.rowKey h v < TreePositions.rowKey h w) |
| (hcolKey : ∀ v w : VariablePosition h, |
| colIndex (Sum.inr (Sum.inl v)) < colIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.columnKey h v < TreePositions.columnKey h w) |
| {t : Mode h} {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) |
| (hrow : rowIndex r₀ < rowIndex r₁) (hcol : colIndex c₀ < colIndex c₁) : |
| BodyNodeOrder t r₀ r₁ c₀ c₁ := by |
| unfold BodyNodeOrder |
| split_ifs with hk₀ hk₁ hk₄ |
| · exact vertical_plus_node_order rowIndex hrowKey hk₀ hr₀ hr₁ hrow |
| · exact vertical_minus_node_order rowIndex hrowKey hk₁ hr₀ hr₁ hrow |
| · exact horizontal_plus_node_order colIndex hcolKey (by omega) hk₄ hc₀ hc₁ hcol |
| · exact horizontal_minus_node_order colIndex hcolKey (by omega) hc₀ hc₁ hcol |
|
|
| / |
| certificate discharged by the constructed key orders and increasing coordinates. -/ |
| theorem host_ordered_body_leaf_diagonal {h : ℕ} {R : Type uR} {C : Type uC} |
| [Preorder R] [Preorder C] |
| (rowIndex : Position h → R) (colIndex : Position h → C) |
| (hrowKey : ∀ v w : VariablePosition h, |
| rowIndex (Sum.inr (Sum.inl v)) < rowIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.rowKey h v < TreePositions.rowKey h w) |
| (hcolKey : ∀ v w : VariablePosition h, |
| colIndex (Sum.inr (Sum.inl v)) < colIndex (Sum.inr (Sum.inl w)) → |
| TreePositions.columnKey h v < TreePositions.columnKey h w) |
| {t : Mode h} {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) |
| (hrow : rowIndex r₀ < rowIndex r₁) (hcol : colIndex c₀ < colIndex c₁) |
| (hb : ModeLocalization.IsP (host r₀ c₀) (host r₀ c₁) |
| (host r₁ c₀) (host r₁ c₁)) : |
| ∃ z : Fin (2 ^ h), |
| r₀ = Sum.inr (Sum.inl (Fin.last (2 * h), z)) ∧ |
| c₀ = Sum.inr (Sum.inl (Fin.last (2 * h), z)) := by |
| exact host_body_leaf_diagonal hr₀ hr₁ hc₀ hc₁ |
| (bodyNodeOrder_of_ordered rowIndex colIndex hrowKey hcolKey hr₀ hr₁ hc₀ hc₁ hrow hcol) hb |
|
|
| end Problem348.Construction |
|
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| end OAI |
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|
