| import OAI.Combinatorics.MatrixRemoval.BodyExtension |
| import OAI.Combinatorics.MatrixRemoval.PathPropagation |
|
|
| namespace OAI |
|
|
| /-! Exact selected-anchor frames and their protected cells. |
| The frame does not constrain any of the four body entries. -/ |
|
|
| noncomputable section |
| namespace Problem348.GuardedPath |
|
|
| structure AnchorFrame (n : ℕ) where |
| rows : IncreasingMap 64 n |
| cols : IncreasingMap 64 n |
|
|
| namespace AnchorFrame |
|
|
| variable {n : ℕ} |
|
|
| /-- All data needed to extend a body, except the body entries themselves. -/ |
| structure Valid (F : AnchorFrame n) (A : BinaryMatrix n) |
| (r₀ r₁ c₀ c₁ : Fin n) : Prop where |
| row_before : ∀ u, F.rows.val u < r₀ |
| row_lt : r₀ < r₁ |
| col_before : ∀ v, F.cols.val v < c₀ |
| col_lt : c₀ < c₁ |
| anchor : ∀ u v, A (F.rows.val u) (F.cols.val v) = anchor64 u v |
| row_zero : ∀ v, A r₀ (F.cols.val v) = decide (v.val = 0) |
| row_one : ∀ v, A r₁ (F.cols.val v) = decide (v.val = 1) |
| col_zero : ∀ u, A (F.rows.val u) c₀ = decide (u.val = 0) |
| col_one : ∀ u, A (F.rows.val u) c₁ = decide (u.val = 1) |
|
|
| /-- Precisely the anchor and signature cells of one selected frame. -/ |
| def Cell (F : AnchorFrame n) (r₀ r₁ c₀ c₁ : Fin n) |
| (rc : Fin n × Fin n) : Prop := |
| (∃ u v, rc = (F.rows.val u, F.cols.val v)) ∨ |
| (∃ v, rc = (r₀, F.cols.val v) ∨ rc = (r₁, F.cols.val v)) ∨ |
| (∃ u, rc = (F.rows.val u, c₀) ∨ rc = (F.rows.val u, c₁)) |
|
|
| /-- Agreement only on frame cells preserves anchor and signature validity. -/ |
| theorem Valid.transfer {F : AnchorFrame n} {A B : BinaryMatrix n} |
| {r₀ r₁ c₀ c₁ : Fin n} (hF : F.Valid A r₀ r₁ c₀ c₁) |
| (hag : ∀ rc, F.Cell r₀ r₁ c₀ c₁ rc → A rc.1 rc.2 = B rc.1 rc.2) : |
| F.Valid B r₀ r₁ c₀ c₁ := by |
| refine ⟨hF.row_before, hF.row_lt, hF.col_before, hF.col_lt, ?_, ?_, ?_, ?_, ?_⟩ |
| · intro u v |
| exact (hag _ (Or.inl ⟨u, v, rfl⟩)).symm.trans (hF.anchor u v) |
| · intro v |
| exact (hag _ (Or.inr (Or.inl ⟨v, Or.inl rfl⟩))).symm.trans (hF.row_zero v) |
| · intro v |
| exact (hag _ (Or.inr (Or.inl ⟨v, Or.inr rfl⟩))).symm.trans (hF.row_one v) |
| · intro u |
| exact (hag _ (Or.inr (Or.inr ⟨u, Or.inl rfl⟩))).symm.trans (hF.col_zero u) |
| · intro u |
| exact (hag _ (Or.inr (Or.inr ⟨u, Or.inr rfl⟩))).symm.trans (hF.col_one u) |
|
|
| /-- A valid frame and the body give a genuine ordered copy of the fixed H. -/ |
| theorem Valid.not_HFree {F : AnchorFrame n} {B : BinaryMatrix n} |
| {r₀ r₁ c₀ c₁ : Fin n} (hF : F.Valid B r₀ r₁ c₀ c₁) |
| (hbody : PathPropagation.Body B r₀ r₁ c₀ c₁) : ¬ HFree fixedH B := by |
| exact not_HFree_of_anchor_body B F.rows F.cols r₀ r₁ c₀ c₁ |
| hF.row_before hF.row_lt hF.col_before hF.col_lt |
| hF.anchor hF.row_zero hF.row_one hF.col_zero hF.col_one |
| hbody.1 hbody.2.1 hbody.2.2.1 hbody.2.2.2 |
|
|
| end AnchorFrame |
| end Problem348.GuardedPath |
|
|
| end |
|
|
| end OAI |
|
|
