openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Binary /BinaryContinuationReads.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryBellmanReads | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryCachedContinuation | |
| import OAI.Combinatorics.EditApproximation.Definitions | |
| namespace OAI | |
| /-! Both Bellman minima and the online score list execute their table | |
| lookups before evaluating the final stored-fraction decrement. -/ | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| theorem localContinuationReadWithWork_eq_cost {Parent Child : Type*} {M : ℕ} | |
| (distance : Parent → Parent → BinaryFraction × ℕ) (L delta fallback : BinaryFraction) | |
| (old current : Child → BinaryFraction × ℕ) (query : Parent) | |
| (wide groups online : List (BinaryBellmanAction Parent Child M)) (R : ℕ) | |
| (ho : ∀ s, (old s).2 ≤ R) (hn : ∀ s, (current s).2 ≤ R) : | |
| (localContinuationReadWithWork distance L delta fallback old current query wide groups online).1 = | |
| (localContinuationWithWork distance L delta fallback (fun s => (old s).1) | |
| (fun s => (current s).1) query wide groups online).1 ∧ | |
| (localContinuationReadWithWork distance L delta fallback old current query wide groups online).2 ≤ | |
| (localContinuationWithWork distance L delta fallback (fun s => (old s).1) | |
| (fun s => (current s).1) query wide groups online).2 + | |
| (wide.length + groups.length + online.length) * (M * (R + 3)) := by | |
| have hwv := bellmanReadWithWork_eq distance L fallback old query wide | |
| have hwc := bellmanReadWithWork_cost distance L fallback old query wide R (fun _ _ _ => ho _) | |
| have hgv := bellmanReadWithWork_eq distance L | |
| (bellmanWithWork distance L fallback (fun s => (old s).1) query wide).1 current query groups | |
| have hgc := bellmanReadWithWork_cost distance L | |
| (bellmanWithWork distance L fallback (fun s => (old s).1) query wide).1 current query groups R | |
| (fun _ _ _ => hn _) | |
| have hs := arithmeticMapWithWork_extra | |
| (bellmanActionReadWithWork distance L current query) | |
| (bellmanActionWithWork distance L (fun s => (current s).1) query) | |
| (fun _ => M * (R + 3)) online | |
| (fun a _ => bellmanActionReadWithWork_eq distance L current query a) | |
| (fun a _ => bellmanActionReadWithWork_cost distance L current query a R (fun _ => hn _)) | |
| have hsum : (online.map (fun _ => M * (R + 3))).sum = online.length * (M * (R + 3)) := by | |
| simp only [List.map_const', List.sum_replicate, smul_eq_mul] | |
| rw [hsum] at hs | |
| unfold localContinuationReadWithWork localContinuationWithWork | |
| dsimp only | |
| rw [hwv, hgv, hs.1] | |
| exact ⟨rfl, by nlinarith only [hwc, hgc, hs.2]⟩ | |
| theorem physicalContinuationReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx) | |
| (representatives : List (ℕ × TargetInterval ny)) (P F : ℕ) | |
| (delta fallback : BinaryFraction) | |
| (old current : Fin M × TargetInterval ny → BinaryFraction × ℕ) | |
| (groups online : List (BinaryBellmanAction (TargetInterval ny) (Fin M × TargetInterval ny) M)) | |
| (query : TargetInterval ny) (R : ℕ) (ho : ∀ s, (old s).2 ≤ R) (hn : ∀ s, (current s).2 ≤ R) : | |
| (physicalContinuationReadWithWork parent representatives P F delta fallback old current groups online query).1 = | |
| (physicalContinuationWithWork parent representatives P F delta fallback | |
| (fun s => (old s).1) (fun s => (current s).1) groups online query).1 ∧ | |
| (physicalContinuationReadWithWork parent representatives P F delta fallback old current groups online query).2 ≤ | |
| (physicalContinuationWithWork parent representatives P F delta fallback | |
| (fun s => (old s).1) (fun s => (current s).1) groups online query).2 + | |
| ((wideActionsFromRepresentativesWithWork M parent representatives P).1.length + | |
| groups.length + online.length) * (M * (R + 3)) := by | |
| have h := localContinuationReadWithWork_eq_cost endpointDistanceWithWork | |
| (canonicalMulWithWork (nat 128) (nat F)).1 delta fallback old current query | |
| (wideActionsFromRepresentativesWithWork M parent representatives P).1 groups online R ho hn | |
| unfold physicalContinuationReadWithWork physicalContinuationWithWork | |
| dsimp only | |
| exact ⟨h.1, by omega⟩ | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |