openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /BudgetedSeparatorTerminal.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Separators.BudgetedSeparatorCall | |
| import OAI.Combinatorics.LowSpaceSubsetSum.Storage.StoredEntryTerminal | |
| namespace OAI | |
| /-! # A prepaid terminal sort and weight match | |
| The input lengths are cached by the preceding filter. The budget test uses | |
| only those scalar fields; both sorts and the matching loop are evaluated | |
| only after their full polynomial allowance has been reserved. | |
| -/ | |
| namespace LowSpaceSubsetSum.BudgetedSeparatorTerminal | |
| variable {n : ℕ} | |
| def charge (n nx ny : ℕ) : ℕ := 64 * (n + 1) * (nx + ny + 1) + 1 | |
| def run (target : ℤ) (xs ys : List (StoredEntry n)) (nx ny budget : ℕ) : | |
| BudgetedSeparatorCall.Outcome Bool := | |
| BudgetedSeparatorCall.run (charge n nx ny) | |
| (fun _ => StoredEntryTerminal.run target xs ys) budget | |
| theorem work (target : ℤ) (xs ys : List (StoredEntry n)) (nx ny budget : ℕ) | |
| (hx : xs.length = nx) (hy : ys.length = ny) : | |
| (run target xs ys nx ny budget).2 ≤ budget + 1 := by | |
| apply BudgetedSeparatorCall.work | |
| have h := StoredEntryTerminal.run_work target xs ys | |
| rw [hx, hy] at h | |
| exact Nat.add_le_add_right h 1 | |
| theorem value (target : ℤ) (xs ys : List (StoredEntry n)) (nx ny budget : ℕ) | |
| (hx : xs.length ≤ 2 ^ n) (hy : ys.length ≤ 2 ^ n) | |
| (result : Bool) (left : ℕ) | |
| (h : (run target xs ys nx ny budget).1 = some (result, left)) : | |
| result = (SeparatorTerminal.run StoredEntry.value StoredEntry.value target xs ys).accepted := by | |
| exact (BudgetedSeparatorCall.value _ _ _ _ _ h).1.trans | |
| (StoredEntryTerminal.run_value target xs ys hx hy) | |
| theorem complete (target : ℤ) (xs ys : List (StoredEntry n)) (nx ny budget : ℕ) | |
| (h : charge n nx ny ≤ budget) : | |
| (run target xs ys nx ny budget).1 = | |
| some ((StoredEntryTerminal.run target xs ys).1, budget - charge n nx ny) := by | |
| rw [run, BudgetedSeparatorCall.complete _ _ _ h] | |
| theorem spent (target : ℤ) (xs ys : List (StoredEntry n)) (nx ny budget : ℕ) | |
| (hx : xs.length = nx) (hy : ys.length = ny) | |
| (result : Bool) (left : ℕ) | |
| (h : (run target xs ys nx ny budget).1 = some (result, left)) : | |
| left ≤ budget ∧ (run target xs ys nx ny budget).2 ≤ budget - left := by | |
| apply BudgetedSeparatorCall.spent _ _ _ ?_ _ _ h | |
| have hw := StoredEntryTerminal.run_work target xs ys | |
| rw [hx, hy] at hw | |
| exact Nat.add_le_add_right hw 1 | |
| end LowSpaceSubsetSum.BudgetedSeparatorTerminal | |
| end OAI | |