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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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | 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)
/-- Increasing positions in the actual two tree orders satisfy precisely the
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
/-- The concrete host localization conclusion, with the numerical mode-order
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
end OAI
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