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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 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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
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