| import OAI.Combinatorics.MatrixRemoval.Host | |
| import OAI.Combinatorics.MatrixRemoval.TreePositions | |
| import OAI.Combinatorics.MatrixRemoval.AnchorFrames | |
| namespace OAI | |
| /-! | |
| # The canonical host with both full-axis orders | |
| The raw host, variable depth-first orders, and anchor/dummy order extension | |
| give two orders. They are not global instances on the same carrier. Their | |
| increasing enumerations are independent equivalences to the matrix size. | |
| -/ | |
| noncomputable section | |
| namespace Problem348.OrderedHost | |
| open Construction | |
| abbrev size (h : ℕ) := (386 * h + 2) * 2 ^ h | |
| /-- Rows: anchors first, then the minus/subtree/plus tree order, then dummy. -/ | |
| abbrev rowOrder (h : ℕ) : LinearOrder (Position h) := by | |
| exact @AnchorFrames.rowOrder (Mode h) (VariablePosition h) (Fin (2 ^ h)) | |
| inferInstance (TreePositions.positionOrder false h) inferInstance | |
| /-- Columns: anchors first, then dummy, then the plus/subtree/minus tree order. -/ | |
| abbrev columnOrder (h : ℕ) : LinearOrder (Position h) := by | |
| exact @AnchorFrames.colOrder (Mode h) (VariablePosition h) (Fin (2 ^ h)) | |
| inferInstance (TreePositions.positionOrder true h) inferInstance | |
| /-- The increasing row enumeration, from raw positions to actual indices. -/ | |
| def rowIndex (h : ℕ) : Position h ≃ Fin (size h) := by | |
| letI := rowOrder h | |
| exact (monoEquivOfFin (Position h) (card_position h)).toEquiv.symm | |
| /-- The increasing column enumeration, independent of the row enumeration. -/ | |
| def columnIndex (h : ℕ) : Position h ≃ Fin (size h) := by | |
| letI := columnOrder h | |
| exact (monoEquivOfFin (Position h) (card_position h)).toEquiv.symm | |
| theorem rowIndex_lt_iff (h : ℕ) (r s : Position h) : | |
| rowIndex h r < rowIndex h s ↔ @LT.lt _ (rowOrder h).toLT r s := by | |
| let := rowOrder h | |
| exact @OrderIso.lt_iff_lt _ _ (rowOrder h).toPreorder inferInstance | |
| (@OrderIso.symm _ _ inferInstance (rowOrder h).toLE | |
| (monoEquivOfFin (Position h) (card_position h))) r s | |
| theorem columnIndex_lt_iff (h : ℕ) (r s : Position h) : | |
| columnIndex h r < columnIndex h s ↔ @LT.lt _ (columnOrder h).toLT r s := by | |
| let := columnOrder h | |
| exact @OrderIso.lt_iff_lt _ _ (columnOrder h).toPreorder inferInstance | |
| (@OrderIso.symm _ _ inferInstance (columnOrder h).toLE | |
| (monoEquivOfFin (Position h) (card_position h))) r s | |
| /-- On variable positions the full row enumeration has exactly the tree-key order. -/ | |
| theorem rowIndex_variable_lt_iff {h : ℕ} (v w : VariablePosition h) : | |
| rowIndex h (Sum.inr (Sum.inl v)) < rowIndex h (Sum.inr (Sum.inl w)) ↔ | |
| TreePositions.rowKey h v < TreePositions.rowKey h w := by | |
| rw [rowIndex_lt_iff] | |
| exact @AnchorFrames.row_variable_lt_variable (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder false h) inferInstance v w | |
| /-- On variable positions the full column enumeration has exactly the tree-key order. -/ | |
| theorem columnIndex_variable_lt_iff {h : ℕ} (v w : VariablePosition h) : | |
| columnIndex h (Sum.inr (Sum.inl v)) < columnIndex h (Sum.inr (Sum.inl w)) ↔ | |
| TreePositions.columnKey h v < TreePositions.columnKey h w := by | |
| rw [columnIndex_lt_iff] | |
| exact @AnchorFrames.col_variable_lt_variable (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder true h) inferInstance v w | |
| /-- Any representatives of the 64 successive anchor groups are increasing rows. -/ | |
| theorem row_anchor_strictMono {h : ℕ} (t : Mode h) | |
| (a : Fin 64 → Fin (2 ^ h)) : | |
| StrictMono (fun u => rowIndex h (Sum.inl (t, u, a u))) := by | |
| intro u v huv | |
| rw [rowIndex_lt_iff] | |
| exact @AnchorFrames.row_anchor_strictMono (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder false h) inferInstance | |
| t a u v huv | |
| /-- The analogous anchor representatives are increasing columns. -/ | |
| theorem column_anchor_strictMono {h : ℕ} (t : Mode h) | |
| (a : Fin 64 → Fin (2 ^ h)) : | |
| StrictMono (fun u => columnIndex h (Sum.inl (t, u, a u))) := by | |
| intro u v huv | |
| rw [columnIndex_lt_iff] | |
| exact @AnchorFrames.col_anchor_strictMono (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder true h) inferInstance | |
| t a u v huv | |
| theorem row_anchor_lt_nonanchor {h : ℕ} (t : Mode h) (u : Fin 64) | |
| (a : Fin (2 ^ h)) (x : VariablePosition h ⊕ Fin (2 ^ h)) : | |
| rowIndex h (Sum.inl (t, u, a)) < rowIndex h (Sum.inr x) := by | |
| rw [rowIndex_lt_iff] | |
| exact @AnchorFrames.row_anchor_lt_nonanchor (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder false h) inferInstance | |
| t u a x | |
| theorem column_anchor_lt_nonanchor {h : ℕ} (t : Mode h) (u : Fin 64) | |
| (a : Fin (2 ^ h)) (x : VariablePosition h ⊕ Fin (2 ^ h)) : | |
| columnIndex h (Sum.inl (t, u, a)) < columnIndex h (Sum.inr x) := by | |
| rw [columnIndex_lt_iff] | |
| exact @AnchorFrames.col_anchor_lt_nonanchor (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder true h) inferInstance | |
| t u a x | |
| theorem row_variable_lt_dummy {h : ℕ} (v : VariablePosition h) (x : Fin (2 ^ h)) : | |
| rowIndex h (Sum.inr (Sum.inl v)) < rowIndex h (Sum.inr (Sum.inr x)) := by | |
| rw [rowIndex_lt_iff] | |
| exact @AnchorFrames.row_variable_lt_dummy (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder false h) inferInstance v x | |
| theorem column_dummy_lt_variable {h : ℕ} (v : VariablePosition h) (x : Fin (2 ^ h)) : | |
| columnIndex h (Sum.inr (Sum.inr x)) < columnIndex h (Sum.inr (Sum.inl v)) := by | |
| rw [columnIndex_lt_iff] | |
| exact @AnchorFrames.col_dummy_lt_variable (Mode h) (VariablePosition h) | |
| (Fin (2 ^ h)) inferInstance (TreePositions.positionOrder true h) inferInstance v x | |
| /-- Initial-segment property for anchors, stated without an extra host predicate. -/ | |
| theorem row_anchor_initial {h : ℕ} {r s : Position h} | |
| (hrs : rowIndex h r < rowIndex h s) (hs : ∃ a, s = Sum.inl a) : | |
| ∃ a, r = Sum.inl a := by | |
| obtain ⟨⟨t, u, x⟩, rfl⟩ := hs | |
| rcases r with a | v | |
| · exact ⟨a, rfl⟩ | |
| · exact False.elim (lt_asymm hrs (row_anchor_lt_nonanchor t u x v)) | |
| theorem column_anchor_initial {h : ℕ} {r s : Position h} | |
| (hrs : columnIndex h r < columnIndex h s) (hs : ∃ a, s = Sum.inl a) : | |
| ∃ a, r = Sum.inl a := by | |
| obtain ⟨⟨t, u, x⟩, rfl⟩ := hs | |
| rcases r with a | v | |
| · exact ⟨a, rfl⟩ | |
| · exact False.elim (lt_asymm hrs (column_anchor_lt_nonanchor t u x v)) | |
| /-- First tree-order equivalence, now for the actual row indices. -/ | |
| theorem row_plus_child_lt_parent {h i : ℕ} (hi : i < h) (x y : Fin (2 ^ h)) : | |
| rowIndex h (.inr (.inl (TreePositions.plusPos (i + 1) (by omega) x))) < | |
| rowIndex h (.inr (.inl (TreePositions.plusPos i (by omega) y))) ↔ | |
| TreePositions.node (i + 1) x / 2 ≤ TreePositions.node i y := by | |
| rw [rowIndex_variable_lt_iff] | |
| exact TreePositions.row_plus_child_lt_parent hi x y | |
| /-- Second tree-order equivalence, now for the actual row indices. -/ | |
| theorem row_minus_parent_lt_child {h i : ℕ} (hi : i < h) (x y : Fin (2 ^ h)) : | |
| rowIndex h (.inr (.inl (TreePositions.minusPos i (by omega) y))) < | |
| rowIndex h (.inr (.inl (TreePositions.minusPos (i + 1) (by omega) x))) ↔ | |
| TreePositions.node i y ≤ TreePositions.node (i + 1) x / 2 := by | |
| rw [rowIndex_variable_lt_iff] | |
| exact TreePositions.row_minus_parent_lt_child hi x y | |
| /-- Third tree-order equivalence, now for the actual column indices. -/ | |
| theorem column_plus_parent_lt_child {h i : ℕ} (hi : i < h) (x y : Fin (2 ^ h)) : | |
| columnIndex h (.inr (.inl (TreePositions.plusPos i (by omega) y))) < | |
| columnIndex h (.inr (.inl (TreePositions.plusPos (i + 1) (by omega) x))) ↔ | |
| TreePositions.node i y ≤ TreePositions.node (i + 1) x / 2 := by | |
| rw [columnIndex_variable_lt_iff] | |
| exact TreePositions.column_plus_parent_lt_child hi x y | |
| /-- Fourth tree-order equivalence, now for the actual column indices. -/ | |
| theorem column_minus_child_lt_parent {h i : ℕ} (hi : i < h) (x y : Fin (2 ^ h)) : | |
| columnIndex h (.inr (.inl (TreePositions.minusPos (i + 1) (by omega) x))) < | |
| columnIndex h (.inr (.inl (TreePositions.minusPos i (by omega) y))) ↔ | |
| TreePositions.node (i + 1) x / 2 ≤ TreePositions.node i y := by | |
| rw [columnIndex_variable_lt_iff] | |
| exact TreePositions.column_minus_child_lt_parent hi x y | |
| end Problem348.OrderedHost | |
| end | |
| end OAI | |