| 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 | |