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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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.Combinatorics.EditApproximation.Binary.BinaryCutReads
import OAI.Combinatorics.EditApproximation.Binary.BinaryScoreReads
import OAI.Combinatorics.EditApproximation.Binary.BinaryPreparedRepresentative
import OAI.Combinatorics.EditApproximation.Definitions
namespace OAI
/-! Representative bands with charged child-table reads in every score
and prefix predicate over the materialized wide grid. -/
namespace EditApproximation
namespace BinaryFraction
theorem bandFromActionsReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (b : ℕ) (tau theta velocity : BinaryFraction)
(read : Fin M → TargetInterval ny → BinaryFraction × ℕ)
(actions : List (Fin M → TargetInterval ny)) (hne : actions ≠ []) (R : ℕ)
(hr : ∀ action ∈ actions, ∀ i, (read i (action i)).2 ≤ R) :
(bandFromActionsReadWithWork parent r b tau theta velocity read actions hne).1 =
(bandFromActionsWithWork parent r b tau theta velocity (fun i q => (read i q).1) actions hne).1 ∧
(bandFromActionsReadWithWork parent r b tau theta velocity read actions hne).2 ≤
(bandFromActionsWithWork parent r b tau theta velocity (fun i q => (read i q).1) actions hne).2 +
actions.length * (M * ((actions.length + 1) * (M * (R + 4)))) := by
have costBound (a a' m g A : ℕ) (ha : a ≤ a' + A) :
m + g + a + 2 ≤ m + g + a' + 2 + A := by
omega
have h := filterWithWork_extra
(bandPredicateReadWithWork parent r b tau theta velocity actions hne read)
(bandPredicateWithWork parent r b tau theta velocity actions hne (fun i q => (read i q).1))
(fun _ => M * ((actions.length + 1) * (M * (R + 4)))) actions
(fun states _ => (bandPredicateReadWithWork_eq_cost parent r b tau theta velocity actions hne read states R hr).1)
(fun states _ => (bandPredicateReadWithWork_eq_cost parent r b tau theta velocity actions hne read states R hr).2)
simp only [List.map_const', List.sum_replicate, smul_eq_mul] at h
unfold bandFromActionsReadWithWork bandFromActionsWithWork
dsimp only
split_ifs
· exact ⟨h.1, costBound _ _ _ _ _ h.2⟩
· exact ⟨rfl, Nat.le_add_right _ _⟩
def representativeReadOverhead (M vertices R : ℕ) : ℕ :=
(vertices + 1) * (M * (R + 4)) + vertices * (M * ((vertices + 1) * (M * (R + 4))))
theorem representativeFromActionsReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx)
(r : TargetInterval ny) (sourceLength b : ℕ) (tau theta : BinaryFraction)
(read : Fin M → TargetInterval ny → BinaryFraction × ℕ)
(actions : List (Fin M → TargetInterval ny)) (R : ℕ)
(hr : ∀ action ∈ actions, ∀ i, (read i (action i)).2 ≤ R) :
(representativeFromActionsReadWithWork parent r sourceLength b tau theta read actions).1 =
(representativeFromActionsWithWork parent r sourceLength b tau theta (fun i q => (read i q).1) actions).1 ∧
(representativeFromActionsReadWithWork parent r sourceLength b tau theta read actions).2 ≤
(representativeFromActionsWithWork parent r sourceLength b tau theta (fun i q => (read i q).1) actions).2 +
representativeReadOverhead M actions.length R := by
have costBound (a a' c c' x y z A B : ℕ)
(ha : a ≤ a' + A) (hc : c ≤ c' + B) :
a + x + y + z + c + 3 ≤ a' + x + y + z + c' + 3 + (A + B) := by
omega
have shortCostBound (a a' x A B : ℕ) (ha : a ≤ a' + A) :
a + x + 2 ≤ a' + x + 2 + (A + B) := by
omega
unfold representativeFromActionsReadWithWork representativeFromActionsWithWork
by_cases hne : actions = []
· simp only [dite_eq_left hne]
exact ⟨trivial, Nat.le_add_right _ _⟩
· simp only [dite_eq_right hne]
have hz := wideMinimumReadWithWork_cost r read actions hne R hr
have hmono : (actions.length + 1) * (M * (R + 3)) ≤ (actions.length + 1) * (M * (R + 4)) := by
exact Nat.mul_le_mul_left _ (Nat.mul_le_mul_left _
(Nat.add_le_add_left (by decide : 3 ≤ 4) R))
have hz' := hz.trans (Nat.add_le_add_left hmono _)
rw [wideMinimumReadWithWork_eq]
split_ifs
· have hb := bandFromActionsReadWithWork_eq_cost parent r b tau theta
(divWithWork (stateGapWithWork sourceLength r).1
(wideMinimumWithWork r (fun i q => (read i q).1) actions hne).1).1 read actions hne R hr
refine ⟨hb.1, ?_⟩
exact costBound _ _ _ _ _ _ _ _ _ hz' hb.2
· exact ⟨rfl, shortCostBound _ _ _ _ _ hz'⟩
theorem preparedRepresentativeReadWithWork_eq_cost {nx ny M : ℕ} (parent : TargetInterval nx)
(center : TargetInterval ny) (sourceLength b P : ℕ) (tau theta : BinaryFraction)
(read : Fin M → TargetInterval ny → BinaryFraction × ℕ) (R : ℕ)
(hr : ∀ i q, (read i q).2 ≤ R) :
(preparedRepresentativeReadWithWork parent center sourceLength b P tau theta read).1 =
(preparedRepresentativeWithWork parent center sourceLength b P tau theta (fun i q => (read i q).1)).1 ∧
(preparedRepresentativeReadWithWork parent center sourceLength b P tau theta read).2 ≤
(preparedRepresentativeWithWork parent center sourceLength b P tau theta (fun i q => (read i q).1)).2 +
representativeReadOverhead M (wideActionGridWithWork M parent center b P).1.length R := by
have h := representativeFromActionsReadWithWork_eq_cost parent center sourceLength b tau theta read
((wideActionGridWithWork M parent center b P).1.map Vector.get) R (fun _ _ i => hr i _)
simp only [List.length_map] at h
unfold preparedRepresentativeReadWithWork preparedRepresentativeWithWork
dsimp only
exact ⟨h.1, by omega⟩
end BinaryFraction
end EditApproximation
end OAI
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