| import OAI.NumberTheory.TwoPoint.Walks.ColumnRunEncoding |
|
|
| namespace OAI |
|
|
| /-! Run bits may force a split at a perfect-block boundary without changing its label. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| variable {α : Type*} [DecidableEq α] |
|
|
| def columnFlagsWithCuts : List (Option α × Bool) → Option α → List (Bool × Bool) |
| | [], _ => [] |
| | (none, _) :: rest, _ => (true, false) :: columnFlagsWithCuts rest none |
| | (some a, cut) :: rest, previous => |
| (false, cut || decide (previous ≠ some a)) :: columnFlagsWithCuts rest (some a) |
|
|
| def columnRunHeadsWithCuts : List (Option α × Bool) → Option α → List α |
| | [], _ => [] |
| | (none, _) :: rest, _ => columnRunHeadsWithCuts rest none |
| | (some a, cut) :: rest, previous => |
| if cut = false ∧ previous = some a then columnRunHeadsWithCuts rest (some a) |
| else a :: columnRunHeadsWithCuts rest (some a) |
|
|
| @[simp] theorem columnFlagsWithCuts_length (entries : List (Option α × Bool)) (previous : Option α) : |
| (columnFlagsWithCuts entries previous).length = entries.length := by |
| induction entries generalizing previous with |
| | nil => rfl |
| | cons entry rest ih => rcases entry with ⟨a, cut⟩; cases a <;> simp [columnFlagsWithCuts, ih] |
|
|
| /-- A forced boundary consumes a run label even when the adjacent blocks |
| end and start with the same label. Arbitrary unused labels remain harmless. -/ |
| theorem expandColumnRuns_cuts_recover (entries : List (Option CanonicalColumnLabel × Bool)) |
| (previous : Option CanonicalColumnLabel) (tail : List CanonicalColumnLabel) : |
| expandColumnRuns (columnFlagsWithCuts entries previous) |
| (columnRunHeadsWithCuts entries previous ++ tail) previous = entries.map Prod.fst := by |
| induction entries generalizing previous with |
| | nil => rfl |
| | cons entry rest ih => |
| rcases entry with ⟨a, cut⟩ |
| cases a with |
| | none => simpa [columnFlagsWithCuts, columnRunHeadsWithCuts, expandColumnRuns] using |
| congrArg (List.cons none) (ih none) |
| | some a => |
| cases cut with |
| | true => |
| simpa [columnFlagsWithCuts, columnRunHeadsWithCuts, expandColumnRuns] using |
| congrArg (List.cons (some a)) (ih (some a)) |
| | false => |
| by_cases h : previous = some a |
| · subst previous |
| simpa [columnFlagsWithCuts, columnRunHeadsWithCuts, expandColumnRuns] using |
| congrArg (List.cons (some a)) (ih (some a)) |
| · simpa [columnFlagsWithCuts, columnRunHeadsWithCuts, h, expandColumnRuns] using |
| congrArg (List.cons (some a)) (ih (some a)) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
