| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
|
|
| / |
| 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⟩ |
|
|
| / |
| 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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