openAI-math-dataset / lean /OAI /Combinatorics /EditApproximation /Binary /BinaryContinuationReadCosts.lean
| import OAI.Combinatorics.EditApproximation.Binary.BinaryReadCostBounds | |
| import OAI.Combinatorics.EditApproximation.Binary.BinaryContinuationReads | |
| namespace OAI | |
| /-! Continuation cost transfer includes both Bellman scans and the | |
| literal online score array, with each child lookup charged. -/ | |
| namespace EditApproximation | |
| namespace BinaryFraction | |
| theorem bellmanActionReadWithWork_factor {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) | |
| (hr : ∀ 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 * tableReadFactor R := by | |
| apply tableReadFactor_absorb _ _ M R | |
| (bellmanActionReadWithWork_cost distance L read query action R hr) | |
| have hs : M ≤ (sumListWithWork (List.ofFn fun i => (read (action.childState i)).1)).2 := by | |
| simp only [sumListWithWork, List.length_ofFn] | |
| omega | |
| unfold bellmanActionWithWork | |
| dsimp only | |
| omega | |
| theorem bellmanReadWithWork_factor {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) | |
| (hr : ∀ 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 * tableReadFactor R := by | |
| have hs := arithmeticMapWithWork_factor | |
| (bellmanActionReadWithWork distance L read query) | |
| (bellmanActionWithWork distance L (fun s => (read s).1) query) | |
| actions (tableReadFactor R) (tableReadFactor_pos R) | |
| (fun action _ => bellmanActionReadWithWork_eq distance L read query action) | |
| (fun action ha => bellmanActionReadWithWork_factor distance L read query action R (hr action ha)) | |
| unfold bellmanReadWithWork bellmanWithWork | |
| dsimp only | |
| rw [hs.1] | |
| apply cost_replace_factor _ _ _ _ _ hs.2 (tableReadFactor_pos R) <;> omega | |
| theorem localContinuationReadWithWork_factor {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).2 ≤ | |
| (localContinuationWithWork distance L delta fallback (fun s => (old s).1) | |
| (fun s => (current s).1) query wide groups online).2 * tableReadFactor R := by | |
| have hw := bellmanReadWithWork_factor distance L fallback old query wide R (fun _ _ _ => ho _) | |
| have hwq := bellmanReadWithWork_eq distance L fallback old query wide | |
| have hg := bellmanReadWithWork_factor distance L | |
| (bellmanWithWork distance L fallback (fun s => (old s).1) query wide).1 current query groups R | |
| (fun _ _ _ => hn _) | |
| have hgq := bellmanReadWithWork_eq distance L | |
| (bellmanWithWork distance L fallback (fun s => (old s).1) query wide).1 current query groups | |
| have hs := arithmeticMapWithWork_factor | |
| (bellmanActionReadWithWork distance L current query) | |
| (bellmanActionWithWork distance L (fun s => (current s).1) query) | |
| online (tableReadFactor R) (tableReadFactor_pos R) | |
| (fun action _ => bellmanActionReadWithWork_eq distance L current query action) | |
| (fun action _ => bellmanActionReadWithWork_factor distance L current query action R (fun _ => hn _)) | |
| have h := Nat.add_le_add (Nat.add_le_add hw hg) hs.2 | |
| simp only [← Nat.add_mul] at h | |
| unfold localContinuationReadWithWork localContinuationWithWork | |
| dsimp only | |
| rw [hwq, hgq, hs.1] | |
| apply cost_replace_factor _ _ _ _ _ h (tableReadFactor_pos R) <;> omega | |
| theorem physicalContinuationReadWithWork_factor {nx n M : ℕ} (parent : TargetInterval nx) | |
| (representatives : List (ℕ × TargetInterval n)) (P F : ℕ) | |
| (delta fallback : BinaryFraction) (old current : Fin M × TargetInterval n → BinaryFraction × ℕ) | |
| (groups online : List (BinaryBellmanAction (TargetInterval n) (Fin M × TargetInterval n) M)) | |
| (query : TargetInterval n) (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).2 ≤ | |
| (physicalContinuationWithWork parent representatives P F delta fallback | |
| (fun s => (old s).1) (fun s => (current s).1) groups online query).2 * tableReadFactor R := by | |
| have h := localContinuationReadWithWork_factor 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 | |
| apply cost_replace_factor _ _ _ _ _ h (tableReadFactor_pos R) <;> omega | |
| end BinaryFraction | |
| end EditApproximation | |
| end OAI | |