File size: 2,642 Bytes
180bc03 | 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 | 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
|