| import OAI.Combinatorics.MatrixRemoval.AnchorPatternBridge |
| import OAI.Combinatorics.MatrixRemoval.AnchorProperties |
|
|
| namespace OAI |
|
|
| /-! |
| |
|
|
| 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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|
