File size: 2,899 Bytes
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 | 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
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