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e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Coarse.CoarseArraySimulation
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
namespace EditApproximation
namespace BinaryFraction
/-- Read one array cell; absent arrays represent exact zero. -/
theorem coarseBinaryRead_value (array : CoarseBinaryArray) (i : ℕ) :
(coarseBinaryRead array i).value = ((coarseBinaryContents array).getD [])[i]?.getD 0 := by
cases array with
| none => simp only [coarseBinaryRead, nat_value, Nat.cast_zero, coarseBinaryContents,
Option.map_none, Option.getD_none, List.getElem?_nil]
| some cells =>
simp only [coarseBinaryRead, coarseBinaryContents, Option.map_some, Option.getD_some,
List.getElem?_map]
cases cells[i]? <;> simp only [ Option.getD_none, Option.map_none, nat_value,
Nat.cast_zero, Option.getD_some, Option.map_some]
theorem coarseBinaryRead_rep {n : ℕ} (array : CoarseBinaryArray) (spec : CoarseStoredArray n)
(h : CoarseArrayRep array spec) (i : Fin n) :
(coarseBinaryRead array i.val).value = coarseStoredRead spec i := by
rw [coarseBinaryRead_value, h]
cases spec with
| none => rfl
| some cells =>
have hi : i.val < cells.1.length := by rw [cells.2]; exact i.isLt
simp only [coarseRationalContents, Option.map_some, Option.getD_some, List.getElem?_eq_getElem hi,
Option.getD_some, coarseStoredRead]
end BinaryFraction
end EditApproximation
end OAI
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