File size: 2,300 Bytes
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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | import OAI.Combinatorics.EditApproximation.Queries.QueryDependencyExpansion
import OAI.Combinatorics.EditApproximation.Geometry.RankedMemoSimulation
namespace OAI
/-! Weighted read counts are the actual same-index and earlier-index
lengths used by the dependency expansion, after identifying its values
with the local mathematical recurrence. -/
namespace EditApproximation
theorem list_filter_length_eq_sum {κ : Type*} (keys : List κ) (predicate : κ → Bool) :
(keys.filter predicate).length = (keys.map fun key => if predicate key then 1 else 0).sum := by
induction keys with
| nil => rfl
| cons key keys ih =>
cases hp : predicate key <;> simp [hp, ih, Nat.add_comm]
section Queries
variable {κ : Type*} (depth index : κ → ℕ)
(program : (key : κ) → FiniteQuery {child : κ // depth child + index child < depth key + index key})
(expected : κ → ℚ)
(hstep : ∀ key, (program key).value (fun child => expected child.val) = expected key)
include hstep
theorem queryRequestList_expected (key : κ) :
queryRequestList depth index program key =
((program key).keys (fun child => expected child.val)).map Subtype.val := by
unfold queryRequestList
simp only [rankedProgramValue_eq (fun key => depth key + index key) program expected hstep]
theorem querySameRequests_length_eq_cost (key : κ) :
(querySameRequests depth index program key).length =
(program key).cost (fun child => expected child.val)
(fun child => if ¬ index child.val < index key then 1 else 0) := by
unfold querySameRequests
rw [queryRequestList_expected depth index program expected hstep,
list_filter_length_eq_sum, FiniteQuery.cost_eq_sum, List.map_map]
simp only [Function.comp_def, Bool.not_eq_true', decide_eq_false_iff_not]
theorem queryEarlierRequests_length_eq_cost (key : κ) :
(queryEarlierRequests depth index program key).length =
(program key).cost (fun child => expected child.val)
(fun child => if index child.val < index key then 1 else 0) := by
unfold queryEarlierRequests
rw [queryRequestList_expected depth index program expected hstep,
list_filter_length_eq_sum, FiniteQuery.cost_eq_sum, List.map_map]
simp only [Function.comp_def, decide_eq_true_eq]
end Queries
end EditApproximation
end OAI
|