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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 | import OAI.Combinatorics.MatrixRemoval.CanonicalHorizontal
import OAI.Combinatorics.MatrixRemoval.HostVerticalLocalization
namespace OAI
/-!
# Exhaustive localization of bodies in canonical host modes
All entry and role hypotheses refer to the actual canonical host. The numerical
order premise is exactly the four parent comparisons supplied by the two tree
orders.
-/
namespace Problem348.Construction
open ModeLocalization
/-- The necessary parent comparison for an ordered body in each of the six modes. -/
def BodyNodeOrder {h : ℕ} (t : Mode h) (r₀ r₁ c₀ c₁ : Position h) : Prop :=
if kind t = 0 then positionNode r₀ / 2 ≤ positionNode r₁
else if kind t = 1 then positionNode r₀ ≤ positionNode r₁ / 2
else if kind t < 4 then positionNode c₀ ≤ positionNode c₁ / 2
else positionNode c₀ / 2 ≤ positionNode c₁
theorem rowRole_first_variable {h : ℕ} {t : Mode h} {x : Position h}
(hx : rowRole t 0 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
theorem colRole_horizontal_variable {h : ℕ} {t : Mode h} {j : Fin 2}
{x : Position h} (hk : 2 ≤ kind t) (hx : colRole t j 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
omega
/-- The horizontal plus roles cannot realize the body's top row. -/
theorem host_horizontal_plus_not_body {h : ℕ} {t : Mode h}
(hklo : 2 ≤ kind t) (hkhi : kind t < 4) {r₀ r₁ c₀ c₁ : Position h}
(hr₀ : rowRole t 0 r₀) (hc₀ : colRole t 0 c₀) (hc₁ : colRole t 1 c₁)
(horder : positionNode c₀ ≤ positionNode c₁ / 2) :
¬ IsP (host r₀ c₀) (host r₀ c₁) (host r₁ c₀) (host r₁ c₁) := by
obtain ⟨r, rfl⟩ := rowRole_first_variable hr₀
obtain ⟨c, rfl⟩ := colRole_horizontal_variable hklo hc₀
obtain ⟨d, rfl⟩ := colRole_horizontal_variable hklo hc₁
have hk0 : kind t ≠ 0 := by omega
have hk1 : kind t ≠ 1 := by omega
have hk2 : ¬ kind t < 2 := by omega
have hr : AtLevel (level t) true r ∧ node r % 2 = kind t % 2 := by
simpa [rowRole, hk0, hk1, hkhi] using hr₀
have hc : AtLevel (level t - 1) true c := by
simpa [colRole, hk2, hkhi] using hc₀
have hd : AtLevel (level t) true d ∧ node d % 2 = kind t % 2 := by
simpa [colRole, hk2, hkhi] using hc₁
have hl : level t - 1 + 1 = level t := by have := level_pos t; omega
intro hb
exact horizontal_plus_entries_impossible (hl ▸ hr.1) hc (hl ▸ hd.1)
horder (hr.2.trans hd.2.symm) hb.1 hb.2.1
/-- The horizontal minus roles force the body's top-left entry onto a shared leaf. -/
theorem host_horizontal_minus_leaf_diagonal {h : ℕ} {t : Mode h}
(hklo : 4 ≤ kind t) {r₀ r₁ c₀ c₁ : Position h}
(hr₀ : rowRole t 0 r₀) (hc₀ : colRole t 0 c₀) (hc₁ : colRole t 1 c₁)
(horder : positionNode c₀ / 2 ≤ positionNode c₁)
(hb : 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
obtain ⟨r, rfl⟩ := rowRole_first_variable hr₀
obtain ⟨c, rfl⟩ := colRole_horizontal_variable (by omega) hc₀
obtain ⟨d, rfl⟩ := colRole_horizontal_variable (by omega) hc₁
have hk0 : kind t ≠ 0 := by omega
have hk1 : kind t ≠ 1 := by omega
have hk2 : ¬ kind t < 2 := by omega
have hk4 : ¬ kind t < 4 := by omega
have hr : AtLevel (level t) false r ∧ node r % 2 = kind t % 2 := by
simpa [rowRole, hk0, hk1, hk4] using hr₀
have hc : AtLevel (level t) false c ∧ node c % 2 = kind t % 2 := by
simpa [colRole, hk2, hk4] using hc₀
have hd : AtLevel (level t - 1) false d := by
simpa [colRole, hk2, hk4] using hc₁
have hl : level t - 1 + 1 = level t := by have := level_pos t; omega
obtain ⟨_, z, hz, hcz⟩ := horizontal_minus_entries_localize
(hl ▸ hr.1) (hl ▸ hc.1) hd horder (hr.2.trans hc.2.symm) hb.1 hb.2.1
exact ⟨z, congrArg (fun v => Sum.inr (Sum.inl v)) hz,
congrArg (fun v => Sum.inr (Sum.inl v)) hcz⟩
/-- Under `BodyNodeOrder`, every ordered body in any canonical mode has its
top-left cell on the shared leaf diagonal. -/
theorem host_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)
(horder : BodyNodeOrder t r₀ r₁ c₀ c₁)
(hb : 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
by_cases hk0 : kind t = 0
· exact False.elim (host_vertical_plus_not_body hk0 hr₀ hr₁ hc₀ hc₁
(by simpa [BodyNodeOrder, hk0] using horder) hb)
by_cases hk1 : kind t = 1
· exact False.elim (host_vertical_minus_not_body hk1 hr₀ hr₁ hc₀ hc₁
(by simpa [BodyNodeOrder, hk0, hk1] using horder) hb)
by_cases hk4 : kind t < 4
· exact False.elim (host_horizontal_plus_not_body (by omega) hk4 hr₀ hc₀ hc₁
(by simpa [BodyNodeOrder, hk0, hk1, hk4] using horder) hb)
· exact host_horizontal_minus_leaf_diagonal (by omega) hr₀ hc₀ hc₁
(by simpa [BodyNodeOrder, hk0, hk1, hk4] using horder) hb
end Problem348.Construction
end OAI
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