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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 | import OAI.Combinatorics.MatrixRemoval.Host
import OAI.Combinatorics.MatrixRemoval.ModeLocalization
namespace OAI
/-!
# Vertical-mode localization for the concrete host
The only additional input to the exclusion theorems is the node inequality
provided by the tree order. All entry values and role memberships follow
from the host definitions.
-/
namespace Problem348.Construction
/-- Extend the variable node index to all positions. -/
def positionNode {h : ℕ} : Position h → ℕ
| Sum.inr (Sum.inl v) => node v
| _ => 0
@[simp] theorem positionNode_variable {h : ℕ} (v : VariablePosition h) :
positionNode (Sum.inr (Sum.inl v)) = node v := rfl
theorem variableEntry_same_level {h i : ℕ} {r c : VariablePosition h}
(s : Bool) (hr : AtLevel i s r) (hc : AtLevel i s c) :
variableEntry r c =
if i = h then decide (node r ≤ node c)
else if s then decide (node r ≤ node c) else decide (node r < node c) := by
rcases hr with ⟨hrd, hrs⟩
rcases hc with ⟨hcd, hcs⟩
by_cases hi : i = h
· simp [variableEntry, hrd, hcd, hi]
· have hrp : plus r = s := hrs.resolve_left hi
have hcp : plus c = s := hcs.resolve_left hi
simp [variableEntry, hrd, hcd, hi, hrp, hcp]
theorem variableEntry_child_parent {h i : ℕ} {r c : VariablePosition h}
(s : Bool) (hr : AtLevel (i + 1) s r) (hc : AtLevel i s c) :
variableEntry r c =
if s then decide (node r / 2 ≤ node c) else decide (node r / 2 < node c) := by
rcases hr with ⟨hrd, hrs⟩
rcases hc with ⟨hcd, hcs⟩
have hih : i < h := by have := depth_le r; omega
have hcp : plus c = s := hcs.resolve_left (by omega)
have hsign : i + 1 = h ∨ plus r = s := hrs
simp [variableEntry, hrd, hcd, hih, hcp, hsign]
theorem rowRole_vertical_variable {h : ℕ} {t : Mode h} {j : Fin 2}
{x : Position h} (hk : kind t < 2) (hx : rowRole t j x) :
∃ v : VariablePosition h, x = Sum.inr (Sum.inl v) := by
rcases x with a | (v | d)
· simp [rowRole] at hx
· exact ⟨v, rfl⟩
· simp [rowRole] at hx
omega
theorem colRole_vertical_first_dummy {h : ℕ} {t : Mode h}
{x : Position h} (hk : kind t < 2) (hx : colRole t 0 x) :
∃ d : Fin (2 ^ h), x = Sum.inr (Sum.inr d) := by
rcases x with a | (v | d)
· simp [colRole] at hx
· simp [colRole, hk] at hx
· exact ⟨d, rfl⟩
theorem colRole_vertical_second_variable {h : ℕ} {t : Mode h}
{x : Position h} (_hk : kind t < 2) (hx : colRole t 1 x) :
∃ v : VariablePosition h, x = Sum.inr (Sum.inl v) := by
rcases x with a | (v | d)
· simp [colRole] at hx
· exact ⟨v, rfl⟩
· simp [colRole] at hx
/-- An actual vertical plus mode of the host cannot contain the body. -/
theorem host_vertical_plus_not_body {h : ℕ} {t : Mode h}
(hk : kind t = 0) {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)
(horder : positionNode r₀ / 2 ≤ positionNode r₁) :
¬ ModeLocalization.IsP (host r₀ c₀) (host r₀ c₁)
(host r₁ c₀) (host r₁ c₁) := by
have hkt : kind t < 2 := by omega
obtain ⟨x, rfl⟩ := rowRole_vertical_variable hkt hr₀
obtain ⟨y, rfl⟩ := rowRole_vertical_variable hkt hr₁
obtain ⟨d, rfl⟩ := colRole_vertical_first_dummy hkt hc₀
obtain ⟨z, rfl⟩ := colRole_vertical_second_variable hkt hc₁
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 hz : AtLevel (level t - 1) true z := by simpa [colRole, hk] using hc₁
have hl : level t - 1 + 1 = level t := by have := level_pos t; omega
have hparent := variableEntry_child_parent true (hl ▸ hx) hz
have hsame := variableEntry_same_level true hy hz
have hnotleaf : level t - 1 ≠ h := by
have := level_le t
have := level_pos t
omega
simpa [host, hparent, hsame, hnotleaf, positionNode] using
ModeLocalization.vPlus_not_isP (node x) (node y) (node z) horder
/-- An actual vertical minus mode of the host cannot contain the body. -/
theorem host_vertical_minus_not_body {h : ℕ} {t : Mode h}
(hk : kind t = 1) {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)
(horder : positionNode r₀ ≤ positionNode r₁ / 2) :
¬ ModeLocalization.IsP (host r₀ c₀) (host r₀ c₁)
(host r₁ c₀) (host r₁ c₁) := by
have hkt : kind t < 2 := by omega
obtain ⟨x, rfl⟩ := rowRole_vertical_variable hkt hr₀
obtain ⟨y, rfl⟩ := rowRole_vertical_variable hkt hr₁
obtain ⟨d, rfl⟩ := colRole_vertical_first_dummy hkt hc₀
obtain ⟨z, rfl⟩ := colRole_vertical_second_variable hkt hc₁
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 hz : AtLevel (level t - 1) false z := by simpa [colRole, hk] using hc₁
have hl : level t - 1 + 1 = level t := by have := level_pos t; omega
have hparent := variableEntry_child_parent false (hl ▸ hy) hz
have hsame := variableEntry_same_level false hx hz
have hnotleaf : level t - 1 ≠ h := by
have := level_le t
have := level_pos t
omega
simpa [host, hparent, hsame, hnotleaf, positionNode] using
ModeLocalization.vMinus_not_isP (node y) (node x) (node z) horder
end Problem348.Construction
end OAI
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