| import OAI.Combinatorics.MatrixRemoval.Host |
| import OAI.Combinatorics.MatrixRemoval.BodyRoles |
|
|
| namespace OAI |
|
|
| /-! |
| # Body-role extraction for the canonical host |
|
|
| Anchor traces determine body roles in the actual `Construction.host` table. |
| -/ |
|
|
| namespace Problem348.Construction |
|
|
| theorem rowRole_of_anchor_trace {h : ℕ} (t : Mode h) (r : Position h) |
| (z : Fin 64 → Fin (2 ^ h)) (b : Fin 2) |
| (hsig : ∀ v : Fin 64, |
| host r (Sum.inl (t, v, z v)) = decide (v.val = b.val)) : |
| rowRole t b r := by |
| rcases r with ⟨t', u, a⟩ | r |
| · by_cases ht : t' = t |
| · subst t' |
| have heq : (fun v => anchor64 u v) = |
| (fun v : Fin 64 => decide (v.val = b.val)) := by |
| funext v |
| simpa [host] using hsig v |
| exact False.elim (BodySignatures52.anchor64_row_ne_body_signature u b heq) |
| · have heq : (fun _ : Fin 64 => false) = |
| (fun v : Fin 64 => decide (v.val = b.val)) := by |
| funext v |
| simpa [host, ht] using hsig v |
| exact False.elim (BodySignatures52.zero_ne_body_signature b heq) |
| · fin_cases b |
| · have hv := hsig (0 : Fin 64) |
| simpa [host] using hv |
| · have hv := hsig (1 : Fin 64) |
| simpa [host] using hv |
|
|
| theorem colRole_of_anchor_trace {h : ℕ} (t : Mode h) (c : Position h) |
| (z : Fin 64 → Fin (2 ^ h)) (b : Fin 2) |
| (hsig : ∀ u : Fin 64, |
| host (Sum.inl (t, u, z u)) c = decide (u.val = b.val)) : |
| colRole t b c := by |
| rcases c with ⟨t', v, a⟩ | c |
| · by_cases ht : t = t' |
| · subst t' |
| have heq : (fun u => anchor64 u v) = |
| (fun u : Fin 64 => decide (u.val = b.val)) := by |
| funext u |
| simpa [host] using hsig u |
| exact False.elim (BodySignatures52.anchor64_col_ne_body_signature v b heq) |
| · have heq : (fun _ : Fin 64 => false) = |
| (fun u : Fin 64 => decide (u.val = b.val)) := by |
| funext u |
| simpa [host, ht] using hsig u |
| exact False.elim (BodySignatures52.zero_ne_body_signature b heq) |
| · fin_cases b |
| · have hu := hsig (0 : Fin 64) |
| simpa [host] using hu |
| · have hu := hsig (1 : Fin 64) |
| simpa [host] using hu |
|
|
| /-- Exact converse, useful also when constructing a guarded anchor frame. -/ |
| theorem anchor_trace_of_rowRole {h : ℕ} (t : Mode h) (r : Position h) |
| (z : Fin 64 → Fin (2 ^ h)) (b : Fin 2) (hr : rowRole t b r) : |
| ∀ v : Fin 64, host r (Sum.inl (t, v, z v)) = decide (v.val = b.val) := by |
| have hd := rowRole_disjoint t r |
| rcases r with a | r |
| · simp [rowRole] at hr |
| · fin_cases b |
| · have hn : ¬ rowRole t 1 (Sum.inr r) := fun h1 => hd ⟨hr, h1⟩ |
| intro v |
| simp [host, hn] |
| exact fun _ => hr |
| · have hn : ¬ rowRole t 0 (Sum.inr r) := fun h0 => hd ⟨h0, hr⟩ |
| intro v |
| simp [host, hn] |
| exact fun _ => hr |
|
|
| theorem anchor_trace_of_colRole {h : ℕ} (t : Mode h) (c : Position h) |
| (z : Fin 64 → Fin (2 ^ h)) (b : Fin 2) (hc : colRole t b c) : |
| ∀ u : Fin 64, host (Sum.inl (t, u, z u)) c = decide (u.val = b.val) := by |
| have hd := colRole_disjoint t c |
| rcases c with a | c |
| · simp [colRole] at hc |
| · fin_cases b |
| · have hn : ¬ colRole t 1 (Sum.inr c) := fun h1 => hd ⟨hc, h1⟩ |
| intro u |
| simp [host, hn] |
| exact fun _ => hc |
| · have hn : ¬ colRole t 0 (Sum.inr c) := fun h0 => hd ⟨h0, hc⟩ |
| intro u |
| simp [host, hn] |
| exact fun _ => hc |
|
|
| /-- Raw matching version, independent of any enumeration or order on the axes. -/ |
| theorem host_body_roles_of_matching {h : ℕ} |
| (r c : Fin 66 → Position h) |
| (hmatch : ∀ i j, host (r i) (c j) = fixedH i j) |
| (t : Mode h) (zr zc : Fin 64 → Fin (2 ^ h)) |
| (hrows : ∀ u, r (BodySignatures52.anchorIndex u) = Sum.inl (t, u, zr u)) |
| (hcols : ∀ v, c (BodySignatures52.anchorIndex v) = Sum.inl (t, v, zc v)) |
| (b : Fin 2) : |
| rowRole t b (r (BodySignatures52.bodyIndex b)) ∧ |
| colRole t b (c (BodySignatures52.bodyIndex b)) := by |
| constructor |
| · apply rowRole_of_anchor_trace t _ zc b |
| intro v |
| rw [← hcols v] |
| exact (hmatch _ _).trans (BodySignatures52.fixedH_body_anchor b v) |
| · apply colRole_of_anchor_trace t _ zr b |
| intro u |
| rw [← hrows u] |
| exact (hmatch _ _).trans (BodySignatures52.fixedH_anchor_body u b) |
|
|
| /-- The lower-right four cells of any raw matching copy form exactly P. -/ |
| theorem host_body_entries_of_matching {h : ℕ} |
| (r c : Fin 66 → Position h) |
| (hmatch : ∀ i j, host (r i) (c j) = fixedH i j) : |
| host (r (BodySignatures52.bodyIndex 0)) (c (BodySignatures52.bodyIndex 0)) = true ∧ |
| host (r (BodySignatures52.bodyIndex 0)) (c (BodySignatures52.bodyIndex 1)) = false ∧ |
| host (r (BodySignatures52.bodyIndex 1)) (c (BodySignatures52.bodyIndex 0)) = true ∧ |
| host (r (BodySignatures52.bodyIndex 1)) (c (BodySignatures52.bodyIndex 1)) = true := by |
| simp only [hmatch, BodySignatures52.fixedH_body_body] |
| decide |
|
|
| /-- An actual ordered copy in a transported canonical host has the |
| two prescribed body roles, once anchor rigidity pins its prefix. -/ |
| theorem host_body_roles_of_copy {h n : ℕ} {A : BinaryMatrix n} |
| (row col : Fin n → Position h) |
| (hA : ∀ i j, A i j = host (row i) (col j)) |
| {rc : IncreasingMap 66 n × IncreasingMap 66 n} |
| (hcopy : rc ∈ orderedCopies fixedH A) |
| (t : Mode h) (zr zc : Fin 64 → Fin (2 ^ h)) |
| (hrows : ∀ u, row (rc.1.val (BodySignatures52.anchorIndex u)) = |
| Sum.inl (t, u, zr u)) |
| (hcols : ∀ v, col (rc.2.val (BodySignatures52.anchorIndex v)) = |
| Sum.inl (t, v, zc v)) |
| (b : Fin 2) : |
| rowRole t b (row (rc.1.val (BodySignatures52.bodyIndex b))) ∧ |
| colRole t b (col (rc.2.val (BodySignatures52.bodyIndex b))) := by |
| constructor |
| · apply rowRole_of_anchor_trace t _ zc b |
| intro v |
| rw [← hcols v, ← hA] |
| exact BodySignatures52.copy_body_row_signature hcopy b v |
| · apply colRole_of_anchor_trace t _ zr b |
| intro u |
| rw [← hrows u, ← hA] |
| exact BodySignatures52.copy_body_col_signature hcopy b u |
|
|
| end Problem348.Construction |
|
|
| end OAI |
|
|
