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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 | import OAI.Combinatorics.MatrixRemoval.Host
import OAI.Combinatorics.MatrixRemoval.TreePositions
namespace OAI
/-!
Comparison lemmas for the concrete depth-first tree orders used in
mode localization.
-/
namespace Problem348.Construction
open TreePositions
theorem eq_plusPos_of_atLevel {h i : ℕ} {v : VariablePosition h}
(hv : AtLevel i true v) :
v = TreePositions.plusPos i (hv.1 ▸ depth_le v) v.2 := by
apply Prod.ext
· apply Fin.ext
have hc := v.1.isLt
obtain ⟨hd, hs⟩ := hv
simp only [depth] at hd
change v.1.val = 2 * i
rcases hs with hi | hs
· omega
· simp only [plus, decide_eq_true_eq] at hs
omega
· rfl
theorem eq_minusPos_of_atLevel {h i : ℕ} {v : VariablePosition h}
(hv : AtLevel i false v) :
v = TreePositions.minusPos i (hv.1 ▸ depth_le v) v.2 := by
apply Prod.ext
· apply Fin.ext
have hc := v.1.isLt
obtain ⟨hd, hs⟩ := hv
simp only [depth] at hd
change v.1.val = if i = h then 2 * h else 2 * i + 1
split_ifs with hi
· omega
· rcases hs with hs | hs
· exact (hi hs).elim
· simp only [plus, decide_eq_false_iff_not] at hs
omega
· rfl
theorem node_eq_level_node {h i : ℕ} {v : VariablePosition h}
(hv : depth v = i) : node v = TreePositions.node i v.2 := by
simp [node, TreePositions.node, hv]
/-- Row plus child precedes row plus parent exactly for the stated parent-node inequality. -/
theorem row_plus_lt_iff {h i : ℕ} {child parent : VariablePosition h}
(hc : AtLevel (i + 1) true child) (hp : AtLevel i true parent) :
TreePositions.rowKey h child < TreePositions.rowKey h parent ↔
node child / 2 ≤ node parent := by
have hi : i < h := by have := hc.1 ▸ depth_le child; omega
have hn₁ := node_eq_level_node hc.1
have hn₂ := node_eq_level_node hp.1
rw [hn₁, hn₂, eq_plusPos_of_atLevel hc, eq_plusPos_of_atLevel hp]
exact TreePositions.row_plus_child_lt_parent hi child.2 parent.2
/-- Row minus parent precedes row minus child exactly for the parent-node inequality. -/
theorem row_minus_lt_iff {h i : ℕ} {child parent : VariablePosition h}
(hc : AtLevel (i + 1) false child) (hp : AtLevel i false parent) :
TreePositions.rowKey h parent < TreePositions.rowKey h child ↔
node parent ≤ node child / 2 := by
have hi : i < h := by have := hc.1 ▸ depth_le child; omega
have hn₁ := node_eq_level_node hc.1
have hn₂ := node_eq_level_node hp.1
rw [hn₁, hn₂, eq_minusPos_of_atLevel hc, eq_minusPos_of_atLevel hp]
exact TreePositions.row_minus_parent_lt_child hi child.2 parent.2
/-- Column plus parent precedes column plus child exactly for the parent-node inequality. -/
theorem column_plus_lt_iff {h i : ℕ} {child parent : VariablePosition h}
(hc : AtLevel (i + 1) true child) (hp : AtLevel i true parent) :
TreePositions.columnKey h parent < TreePositions.columnKey h child ↔
node parent ≤ node child / 2 := by
have hi : i < h := by have := hc.1 ▸ depth_le child; omega
have hn₁ := node_eq_level_node hc.1
have hn₂ := node_eq_level_node hp.1
rw [hn₁, hn₂, eq_plusPos_of_atLevel hc, eq_plusPos_of_atLevel hp]
exact TreePositions.column_plus_parent_lt_child hi child.2 parent.2
/-- Column minus child precedes column minus parent exactly for the parent-node inequality. -/
theorem column_minus_lt_iff {h i : ℕ} {child parent : VariablePosition h}
(hc : AtLevel (i + 1) false child) (hp : AtLevel i false parent) :
TreePositions.columnKey h child < TreePositions.columnKey h parent ↔
node child / 2 ≤ node parent := by
have hi : i < h := by have := hc.1 ▸ depth_le child; omega
have hn₁ := node_eq_level_node hc.1
have hn₂ := node_eq_level_node hp.1
rw [hn₁, hn₂, eq_minusPos_of_atLevel hc, eq_minusPos_of_atLevel hp]
exact TreePositions.column_minus_child_lt_parent hi child.2 parent.2
end Problem348.Construction
end OAI
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