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5d5048e | 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 | import OAI.Combinatorics.LowSpaceSubsetSum.Separators.BudgetedSeparatorCells
namespace OAI
/-! # Sequencing with the returned object retained by the continuation -/
namespace LowSpaceSubsetSum.LiveSequence
open BudgetedSeparatorCells
variable {α β : Type*}
def run (keep : α → ℕ) (first : Execution α) (next : α → ℕ → Execution β) : Execution β :=
match first.1.1 with
| none => ((none, first.1.2), first.2)
| some (a, left) =>
let out := next a left
((out.1.1, first.1.2 + out.1.2), max first.2 (keep a + out.2))
theorem value (keep : α → ℕ) (first : Execution α) (next : α → ℕ → Execution β) :
(run keep first next).1 = BudgetedSeparatorFlow.bind first.1 (fun a b => (next a b).1) := by
simp only [run, BudgetedSeparatorFlow.bind]
split <;> simp_all only
theorem cells (keep : α → ℕ) (first : Execution α) (next : α → ℕ → Execution β)
(L R K : ℕ) (hfirst : first.2 ≤ L)
(hkeep : ∀ a b, first.1.1 = some (a, b) → keep a ≤ R)
(hnext : ∀ a b, first.1.1 = some (a, b) → (next a b).2 ≤ K) :
(run keep first next).2 ≤ max L (R + K) := by
unfold run
split
· exact hfirst.trans (le_max_left _ _)
next a b h =>
exact max_le_max hfirst (Nat.add_le_add (hkeep a b h) (hnext a b h))
end LowSpaceSubsetSum.LiveSequence
end OAI
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