File size: 4,155 Bytes
e63eb7f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 | 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
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