File size: 3,082 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 | import OAI.Combinatorics.MatrixRemoval.AnchorPatternBridge
import OAI.Combinatorics.MatrixRemoval.AnchorProperties
namespace OAI
/-!
# Structural anchor bridges for the fixed pattern
The generic bridges are specialized to the explicit `anchor64` definition.
-/
universe uR uC
namespace Problem348.AnchorBridge
/-- The exact anchor realizes every five-bit word on the specified suffix. -/
theorem anchor64_word_block_complete (w : Fin 5 → Bool) :
∃ a : Fin 32, ∀ b : Fin 5,
anchor64 (wordRow a) (wordCol b) = w b := by
obtain ⟨a, ha⟩ := anchorWord_bijective.2 w
refine ⟨a, fun b => ?_⟩
have hb := congrFun ha b
simpa only [anchorWord, wordRow, wordCol, Nat.add_comm] using hb
/-- An exact anchor copy has at least 33 anchors on one of its two axes,
provided anchors form prefixes and non-anchor rows do not shatter five
non-anchor columns. -/
theorem anchor64_anchor_majority {R : Type uR} {C : Type uC}
(A : R → C → Bool) (row : Fin 64 → R) (col : Fin 64 → C)
(anchorR : R → Prop) (anchorC : C → Prop)
[DecidablePred anchorR] [DecidablePred anchorC]
(hcopy : ∀ i j, A (row i) (col j) = anchor64 i j)
(hcol : Function.Injective col)
(hprefixR : ∀ i j, i ≤ j → anchorR (row j) → anchorR (row i))
(hprefixC : ∀ i j, i ≤ j → anchorC (col j) → anchorC (col i))
(hnoshatter : ∀ cs : Fin 5 → C, Function.Injective cs →
(∀ b, ¬ anchorC (cs b)) → ∃ w : Fin 5 → Bool,
∀ r, ¬ anchorR r → ∃ b, A r (cs b) ≠ w b) :
33 ≤ (Finset.univ.filter (fun i => anchorR (row i))).card ∨
33 ≤ (Finset.univ.filter (fun j => anchorC (col j))).card := by
rcases anchor_at_32_or_anchor_at_32 anchor64 A row col anchorR anchorC
hcopy hcol hprefixR hprefixC anchor64_word_block_complete hnoshatter with hr | hc
· exact Or.inl (prefix_card_lower_bound _ ⟨32, by decide⟩ hprefixR hr)
· exact Or.inr (prefix_card_lower_bound _ ⟨32, by decide⟩ hprefixC hc)
/-- A row class of equal host rows occurs at most once in an anchor copy. -/
theorem anchor64_row_class_single_use {R : Type uR} {C : Type uC}
(A : R → C → Bool) (row : Fin 64 → R) (col : Fin 64 → C)
(P : R → Prop) [DecidablePred P]
(hcopy : ∀ i j, A (row i) (col j) = anchor64 i j)
(hclass : ∀ r s, P r → P s → A r = A s) :
(Finset.univ.filter (fun i => P (row i))).card ≤ 1 :=
card_selected_row_class_le_one anchor64 A row col P hcopy
anchor64_rows_injective hclass
/-- A column class of equal host columns occurs at most once in an anchor copy. -/
theorem anchor64_col_class_single_use {R : Type uR} {C : Type uC}
(A : R → C → Bool) (row : Fin 64 → R) (col : Fin 64 → C)
(P : C → Prop) [DecidablePred P]
(hcopy : ∀ i j, A (row i) (col j) = anchor64 i j)
(hclass : ∀ c d, P c → P d → ∀ r, A r c = A r d) :
(Finset.univ.filter (fun j => P (col j))).card ≤ 1 :=
card_selected_col_class_le_one anchor64 A row col P hcopy
anchor64_columns_injective hclass
end Problem348.AnchorBridge
end OAI
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