openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Binary /BinaryContinuationSpace.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryBellmanSpace | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryContinuationReads | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| namespace EditApproximation.BinaryFraction | |
| theorem onlineMinimumAllocation_le_work (values : List BinaryFraction) : | |
| onlineMinimumAllocation values ≤ (onlineMinimumWithWork values).2 := by | |
| cases values with | |
| | nil => exact Nat.zero_le _ | |
| | cons first rest => exact (minimumAllocation_le_work first rest).trans (Nat.le_add_right _ _) | |
| theorem refinementUpdateAllocation_le_work (delta cone : BinaryFraction) (online : Option BinaryFraction) : | |
| refinementUpdateAllocation delta cone online ≤ (refinementUpdateWithWork delta cone online).2 := by | |
| cases online with | |
| | none => exact Nat.zero_le _ | |
| | some online => | |
| have ha := canonicalAddAllocation_le_work one delta | |
| have hd := divAllocation_le_work cone (canonicalAddWithWork one delta).1 | |
| have hm := maxAllocation_le_work (divWithWork cone (canonicalAddWithWork one delta).1).1 online | |
| have hf := minAllocation_le_work cone | |
| (maxWithWork (divWithWork cone (canonicalAddWithWork one delta).1).1 online).1 | |
| unfold refinementUpdateAllocation refinementUpdateWithWork | |
| dsimp only | |
| omega | |
| /-- The two Bellman passes, materialized online scores and final arithmetic | |
| retain their actual primitive allocation traces. -/ | |
| theorem localContinuationReadAllocation_le_work {Parent Child : Type*} {M : ℕ} | |
| (distance : Parent → Parent → BinaryFraction × ℕ) (distanceAllocation : Parent → Parent → ℕ) | |
| (L delta fallback : BinaryFraction) (old current : Child → BinaryFraction × ℕ) | |
| (oldAllocation currentAllocation : Child → ℕ) (query : Parent) | |
| (wide groups online : List (BinaryBellmanAction Parent Child M)) | |
| (hd : ∀ state, distanceAllocation query state ≤ (distance query state).2) | |
| (ho : ∀ child, oldAllocation child ≤ (old child).2) | |
| (hc : ∀ child, currentAllocation child ≤ (current child).2) : | |
| localContinuationReadAllocation distance distanceAllocation L delta fallback old current | |
| oldAllocation currentAllocation query wide groups online ≤ | |
| (localContinuationReadWithWork distance L delta fallback old current query wide groups online).2 := by | |
| have hw := bellmanReadAllocation_le_work distance distanceAllocation L fallback old oldAllocation query wide | |
| (fun action _ => hd action.center) (fun action _ i => ho (action.childState i)) | |
| have hg := bellmanReadAllocation_le_work distance distanceAllocation L | |
| (bellmanReadWithWork distance L fallback old query wide).1 current currentAllocation query groups | |
| (fun action _ => hd action.center) (fun action _ i => hc (action.childState i)) | |
| have hs := arithmeticMapAllocation_le_work (bellmanActionReadWithWork distance L current query) | |
| (bellmanActionReadAllocation distance distanceAllocation L current currentAllocation query) online | |
| (fun action _ => bellmanActionReadAllocation_le_work distance distanceAllocation L current | |
| currentAllocation query action (hd action.center) (fun i => hc (action.childState i))) | |
| let scores := (arithmeticMapWithWork (bellmanActionReadWithWork distance L current query) online).1 | |
| have hm := onlineMinimumAllocation_le_work scores | |
| have hf := refinementUpdateAllocation_le_work delta | |
| (bellmanReadWithWork distance L (bellmanReadWithWork distance L fallback old query wide).1 current query groups).1 | |
| (onlineMinimumWithWork scores).1 | |
| unfold localContinuationReadAllocation localContinuationReadWithWork | |
| dsimp only [scores] at hm hf ⊢ | |
| omega | |
| end EditApproximation.BinaryFraction | |
| end OAI | |