| import OAI.Combinatorics.EditApproximation.Binary.BinaryBellmanKernel |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryReadComposition |
| import OAI.Combinatorics.EditApproximation.Definitions |
|
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|
|
| namespace OAI |
|
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| /-! Bellman action evaluation with literal finite-table reader calls. |
| Child words are gathered once, then Boolean arithmetic |
| operates on that stored list. Reader scans are part of the returned cost. -/ |
|
|
| namespace EditApproximation |
| namespace BinaryFraction |
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| theorem bellmanActionReadWithWork_eq {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (L : BinaryFraction) |
| (read : Child → BinaryFraction × ℕ) (query : Parent) |
| (action : BinaryBellmanAction Parent Child M) : |
| (bellmanActionReadWithWork distance L read query action).1 = |
| (bellmanActionWithWork distance L (fun s => (read s).1) query action).1 := by |
| simp only [bellmanActionReadWithWork, arithmeticMapWithWork_value, |
| bellmanActionWithWork, List.ofFn_eq_map] |
|
|
| theorem bellmanActionReadWithWork_cost {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (L : BinaryFraction) |
| (read : Child → BinaryFraction × ℕ) (query : Parent) |
| (action : BinaryBellmanAction Parent Child M) (R : ℕ) |
| (hread : ∀ i, (read (action.childState i)).2 ≤ R) : |
| (bellmanActionReadWithWork distance L read query action).2 ≤ |
| (bellmanActionWithWork distance L (fun s => (read s).1) query action).2 + M * (R + 3) := by |
| have h := arithmeticMapWithWork_cost (fun i => read (action.childState i)) |
| (List.finRange M) R (fun i _ => hread i) |
| simp only [List.length_finRange] at h |
| simp only [bellmanActionReadWithWork, arithmeticMapWithWork_value, |
| bellmanActionWithWork, List.ofFn_eq_map] |
| omega |
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| theorem bellmanReadWithWork_eq {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (L fallback : BinaryFraction) |
| (read : Child → BinaryFraction × ℕ) (query : Parent) |
| (actions : List (BinaryBellmanAction Parent Child M)) : |
| (bellmanReadWithWork distance L fallback read query actions).1 = |
| (bellmanWithWork distance L fallback (fun s => (read s).1) query actions).1 := by |
| simp only [bellmanReadWithWork, bellmanWithWork, arithmeticMapWithWork_value, |
| bellmanActionReadWithWork_eq] |
|
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| theorem bellmanReadWithWork_cost {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (L fallback : BinaryFraction) |
| (read : Child → BinaryFraction × ℕ) (query : Parent) |
| (actions : List (BinaryBellmanAction Parent Child M)) (R : ℕ) |
| (hread : ∀ action ∈ actions, ∀ i, (read (action.childState i)).2 ≤ R) : |
| (bellmanReadWithWork distance L fallback read query actions).2 ≤ |
| (bellmanWithWork distance L fallback (fun s => (read s).1) query actions).2 + |
| actions.length * (M * (R + 3)) := by |
| have h := arithmeticMapWithWork_extra |
| (bellmanActionReadWithWork distance L read query) |
| (bellmanActionWithWork distance L (fun s => (read s).1) query) |
| (fun _ => M * (R + 3)) actions |
| (fun action _ => bellmanActionReadWithWork_eq distance L read query action) |
| (fun action ha => bellmanActionReadWithWork_cost distance L read query action R (hread action ha)) |
| have hs : (actions.map (fun _ => M * (R + 3))).sum = actions.length * (M * (R + 3)) := by |
| simp only [List.map_const', List.sum_replicate, smul_eq_mul] |
| rw [hs] at h |
| unfold bellmanReadWithWork bellmanWithWork |
| dsimp only |
| rw [h.1] |
| omega |
| |
| theorem bellmanWithWork_action_count {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (L fallback : BinaryFraction) |
| (child : Child → BinaryFraction) (query : Parent) |
| (actions : List (BinaryBellmanAction Parent Child M)) : |
| actions.length ≤ (bellmanWithWork distance L fallback child query actions).2 := by |
| have h := arithmeticMapWithWork_length_le_cost |
| (bellmanActionWithWork distance L child query) actions |
| change actions.length ≤ _ + _ + 1 |
| omega |
| |
| end BinaryFraction |
| end EditApproximation |
| |
| end OAI |
| |
