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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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Combinatorics.EditApproximation.Binary.BinaryGatheredValues
namespace OAI
/-! The default-zero finite buffers preserve the nonnegative bounds of
paired initial/history coordinates when the initial buffer covers them. -/
namespace EditApproximation
theorem finiteAnswerTable_eq_zero_of_not_mem {κ : Type*} [DecidableEq κ]
(keys : List κ) (values : List ℚ) (query : κ) (hq : query ∉ keys) :
finiteAnswerTable keys values query = 0 := by
induction keys generalizing values with
| nil => rfl
| cons key keys ih =>
cases values with
| nil => rfl
| cons value values =>
have hk : query ≠ key := fun he => hq (List.mem_cons.mpr (Or.inl he))
have ht : query ∉ keys := fun he => hq (List.mem_cons_of_mem key he)
simpa only [finiteAnswerTable, ite_eq_right hk] using ih values ht
theorem finiteAnswerTable_nonneg {κ : Type*} [DecidableEq κ]
(keys : List κ) (values : List ℚ) (query : κ) (hv : ∀ value ∈ values, 0 ≤ value) :
0 ≤ finiteAnswerTable keys values query := by
induction keys generalizing values with
| nil => exact le_rfl
| cons key keys ih =>
cases values with
| nil => exact le_rfl
| cons value values =>
unfold finiteAnswerTable
split_ifs
· exact hv value List.mem_cons_self
· exact ih values (fun v h => hv v (List.mem_cons_of_mem _ h))
theorem finiteAnswerTable_map_le_of_subset {κ : Type*} [DecidableEq κ]
(keys initialKeys : List κ) (history initial : κ → ℚ)
(hkeys : ∀ key ∈ keys, key ∈ initialKeys)
(hi : ∀ key, 0 ≤ initial key)
(hle : ∀ key, history key ≤ initial key) (query : κ) :
finiteAnswerTable keys (keys.map history) query ≤
finiteAnswerTable initialKeys (initialKeys.map initial) query := by
by_cases hq : query ∈ keys
· rw [finiteAnswerTable_map keys history query hq,
finiteAnswerTable_map initialKeys initial query (hkeys query hq)]
exact hle query
· rw [finiteAnswerTable_eq_zero_of_not_mem keys _ query hq]
apply finiteAnswerTable_nonneg
intro value hv
obtain ⟨key, _, rfl⟩ := List.mem_map.mp hv
exact hi key
end EditApproximation
end OAI
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