| import OAI.Combinatorics.EditApproximation.Binary.BinaryBellmanReads |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryGeometrySpace |
| import OAI.Combinatorics.EditApproximation.Definitions |
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|
| namespace OAI |
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| namespace EditApproximation.BinaryFraction |
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| /-- Actual temporary cells allocated while scoring one action; child values |
| are retained pointers and their lookup scratch is counted by `footprint`. -/ |
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| theorem bellmanActionReadAllocation_le_work {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (distanceAllocation : Parent → Parent → ℕ) |
| (L : BinaryFraction) (read : Child → BinaryFraction × ℕ) (footprint : Child → ℕ) |
| (query : Parent) (action : BinaryBellmanAction Parent Child M) |
| (hd : distanceAllocation query action.center ≤ (distance query action.center).2) |
| (hr : ∀ i, footprint (action.childState i) ≤ (read (action.childState i)).2) : |
| bellmanActionReadAllocation distance distanceAllocation L read footprint query action ≤ |
| (bellmanActionReadWithWork distance L read query action).2 := by |
| have hw := arithmeticMapAllocation_le_work (fun i => read (action.childState i)) |
| (fun i => footprint (action.childState i)) (List.finRange M) (fun i _ => hr i) |
| have hm := canonicalMulAllocation_le_work L (distance query action.center).1 |
| have ha := canonicalAddAllocation_le_work (canonicalMulWithWork L (distance query action.center).1).1 action.connection |
| let words := (arithmeticMapWithWork (fun i => read (action.childState i)) (List.finRange M)).1 |
| have hs := sumListAllocation_le_work words |
| have hf := canonicalAddAllocation_le_work |
| (canonicalAddWithWork (canonicalMulWithWork L (distance query action.center).1).1 action.connection).1 |
| (sumListWithWork words).1 |
| unfold bellmanActionReadAllocation bellmanActionReadWithWork |
| dsimp only [words] at hs hf ⊢ |
| omega |
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| theorem bellmanReadAllocation_le_work {Parent Child : Type*} {M : ℕ} |
| (distance : Parent → Parent → BinaryFraction × ℕ) (distanceAllocation : Parent → Parent → ℕ) |
| (L fallback : BinaryFraction) (read : Child → BinaryFraction × ℕ) (footprint : Child → ℕ) |
| (query : Parent) (actions : List (BinaryBellmanAction Parent Child M)) |
| (hd : ∀ action ∈ actions, distanceAllocation query action.center ≤ (distance query action.center).2) |
| (hr : ∀ action ∈ actions, ∀ i, footprint (action.childState i) ≤ (read (action.childState i)).2) : |
| bellmanReadAllocation distance distanceAllocation L fallback read footprint query actions ≤ |
| (bellmanReadWithWork distance L fallback read query actions).2 := by |
| have hs := arithmeticMapAllocation_le_work (bellmanActionReadWithWork distance L read query) |
| (bellmanActionReadAllocation distance distanceAllocation L read footprint query) actions |
| (fun action ha => bellmanActionReadAllocation_le_work distance distanceAllocation L read footprint |
| query action (hd action ha) (hr action ha)) |
| have hm := minimumAllocation_le_work fallback |
| (arithmeticMapWithWork (bellmanActionReadWithWork distance L read query) actions).1 |
| unfold bellmanReadAllocation bellmanReadWithWork |
| dsimp only |
| omega |
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| end EditApproximation.BinaryFraction |
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| end OAI |
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