| import OAI.Combinatorics.MatrixRemoval.OrderedHost |
| import OAI.Combinatorics.MatrixRemoval.HostBodyOrder |
|
|
| namespace OAI |
|
|
| /-! |
| # Body localization for the independent host enumerations |
|
|
| The canonical full-axis enumeration gives exhaustive body-mode localization. |
| All comparisons use the actual row and column indices; no order-reflection |
| certificate remains as an assumption. |
| -/ |
|
|
| namespace Problem348.OrderedHost |
|
|
| open Construction |
|
|
| theorem rowIndex_reflects_variable_order {h : ℕ} (v w : VariablePosition h) |
| (hvw : rowIndex h (.inr (.inl v)) < rowIndex h (.inr (.inl w))) : |
| TreePositions.rowKey h v < TreePositions.rowKey h w := |
| (rowIndex_variable_lt_iff v w).mp hvw |
|
|
| theorem columnIndex_reflects_variable_order {h : ℕ} (v w : VariablePosition h) |
| (hvw : columnIndex h (.inr (.inl v)) < columnIndex h (.inr (.inl w))) : |
| TreePositions.columnKey h v < TreePositions.columnKey h w := |
| (columnIndex_variable_lt_iff v w).mp hvw |
|
|
| / |
| theorem bodyNodeOrder_of_indices {h : ℕ} {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 h r₀ < rowIndex h r₁) |
| (hcol : columnIndex h c₀ < columnIndex h c₁) : |
| BodyNodeOrder t r₀ r₁ c₀ c₁ := |
| bodyNodeOrder_of_ordered (rowIndex h) (columnIndex h) |
| rowIndex_reflects_variable_order columnIndex_reflects_variable_order |
| hr₀ hr₁ hc₀ hc₁ hrow hcol |
|
|
| / |
| top-left cell on the shared leaf diagonal. -/ |
| theorem body_leaf_diagonal {h : ℕ} {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 h r₀ < rowIndex h r₁) |
| (hcol : columnIndex h c₀ < columnIndex h 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)) := |
| host_ordered_body_leaf_diagonal (rowIndex h) (columnIndex h) |
| rowIndex_reflects_variable_order columnIndex_reflects_variable_order |
| hr₀ hr₁ hc₀ hc₁ hrow hcol hb |
|
|
| end Problem348.OrderedHost |
|
|
| end OAI |
|
|
