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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | 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
/-- The actual independent enumerations discharge all four numerical mode orders. -/
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
/-- Every correctly assigned ordered body in the canonical host has its
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
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