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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 | import OAI.Combinatorics.EditApproximation.Queries.QueryInputCodes
import OAI.Combinatorics.EditApproximation.Binary.BinaryBellmanKernel
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Once the selected group and online actions have been computed, the
last adaptive gather lists exactly their child coordinates, sorted by
the same injective encoding as the reference query. -/
namespace EditApproximation
theorem query_mem_action_cover {Parent Child : Type*} [DecidableEq Child] {M : ℕ}
(actions : List (RationalBellmanAction Parent Child M)) (input : Child) :
input ∈ actionCurrentInputCover actions ↔
∃ action ∈ actions, ∃ i, action.childState i = input := by
constructor
· intro h
obtain ⟨index, _, hi⟩ := Finset.mem_biUnion.mp h
obtain ⟨i, _, heq⟩ := Finset.mem_image.mp hi
exact ⟨actions.get index, List.get_mem actions index, i, heq⟩
· rintro ⟨action, ha, i, rfl⟩
exact actionCurrentInputCover_mem actions action ha i
theorem queryActionInputsWithWork_value {Parent Child : Type*} [DecidableEq Child] {M : ℕ}
(actions : List (BinaryBellmanAction Parent Child M)) :
(queryActionInputsWithWork actions).1.toFinset =
actionCurrentInputCover (actions.map BinaryBellmanAction.decode) := by
ext input
simp only [queryActionInputsWithWork, arithmeticFlatMapWithWork_value, arithmeticMapWithWork_value,
List.mem_toFinset, List.mem_flatMap, List.mem_map, List.mem_finRange, true_and, query_mem_action_cover]
constructor
· rintro ⟨action, ha, i, hi⟩
exact ⟨action.decode, ⟨action, ha, rfl⟩, i, hi⟩
· rintro ⟨action, ⟨word, hw, rfl⟩, i, hi⟩
exact ⟨word, hw, i, hi⟩
theorem queryNeededInputsWithWork_value {n M : ℕ}
(selected online : List (BinaryBellmanAction (TargetInterval n) (Fin M × TargetInterval n) M)) :
(queryNeededInputsWithWork selected online).1 =
orderedInputs childInputCode (childInputCode_injective M n)
(actionCurrentInputCover (selected.map BinaryBellmanAction.decode) ∪
actionCurrentInputCover (online.map BinaryBellmanAction.decode)) := by
simp only [queryNeededInputsWithWork, queryOrderedChildInputsWithWork_value,
List.toFinset_append, queryActionInputsWithWork_value]
end EditApproximation
end OAI
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