openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /BudgetedSeparatorCall.lean
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! # Reserving a known bound before a checker subroutine | |
| The action is a thunk: its body is evaluated only inside the successful | |
| budget branch. The returned counter includes the guard comparison. | |
| -/ | |
| namespace LowSpaceSubsetSum.BudgetedSeparatorCall | |
| variable {α : Type*} | |
| abbrev Outcome (α : Type*) := Option (α × ℕ) × ℕ | |
| def run (charge : ℕ) (action : Unit → α × ℕ) (budget : ℕ) : Outcome α := | |
| if charge ≤ budget then | |
| let out := action () | |
| (some (out.1, budget - charge), out.2 + 1) | |
| else (none, 1) | |
| theorem insufficient (charge : ℕ) (action : Unit → α × ℕ) (budget : ℕ) | |
| (h : budget < charge) : run charge action budget = (none, 1) := by | |
| simp only [run, ite_eq_right (by omega : ¬charge ≤ budget)] | |
| theorem complete (charge : ℕ) (action : Unit → α × ℕ) (budget : ℕ) | |
| (h : charge ≤ budget) : | |
| run charge action budget = (some ((action ()).1, budget - charge), (action ()).2 + 1) := by | |
| simp only [run, ite_eq_left h] | |
| theorem value (charge : ℕ) (action : Unit → α × ℕ) (budget : ℕ) | |
| (a : α) (left : ℕ) (h : (run charge action budget).1 = some (a, left)) : | |
| a = (action ()).1 ∧ left = budget - charge ∧ charge ≤ budget := by | |
| unfold run at h | |
| split at h | |
| · cases h | |
| exact ⟨rfl, rfl, by assumption⟩ | |
| · contradiction | |
| theorem work (charge : ℕ) (action : Unit → α × ℕ) (budget : ℕ) | |
| (hcharge : (action ()).2 + 1 ≤ charge) : | |
| (run charge action budget).2 ≤ budget + 1 := by | |
| unfold run | |
| split <;> dsimp only <;> omega | |
| theorem spent (charge : ℕ) (action : Unit → α × ℕ) (budget : ℕ) | |
| (hcharge : (action ()).2 + 1 ≤ charge) | |
| (a : α) (left : ℕ) (h : (run charge action budget).1 = some (a, left)) : | |
| left ≤ budget ∧ (run charge action budget).2 ≤ budget - left := by | |
| obtain ⟨_, rfl, hb⟩ := value charge action budget a left h | |
| rw [complete charge action budget hb] | |
| constructor <;> omega | |
| end LowSpaceSubsetSum.BudgetedSeparatorCall | |
| end OAI | |