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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | 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
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