openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Counting /CountedReplayPotential.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Counting.CountedReplay | |
| namespace OAI | |
| /-! # Telescoping callback costs through a stopped replay | |
| The callback may itself contain a replay. A potential on its returned state | |
| charges that actual inner work; a uniform worst-case inner fuel bound is not | |
| multiplied by the outer stream length. | |
| -/ | |
| namespace LowSpaceSubsetSum.CountedReplay | |
| universe u v w | |
| variable {α : Type u} {σ : Type v} {τ : Type w} | |
| theorem runAcc_work_potential | |
| (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) | |
| (report : α → τ → (τ × Bool) × ℕ) (potential : τ → ℕ) (R L C : ℕ) | |
| (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) | |
| (hc : ∀ x a, (report x a).2 + potential a ≤ potential (report x a).1.1 + C) | |
| (fuel : ℕ) (s : σ) (a : τ) (q r : ℕ) : | |
| (runAcc read key K report fuel s a q r).2 + (R + L + C + 8) * q + potential a ≤ | |
| (R + L + C + 8) * (runAcc read key K report fuel s a q r).1.queries + | |
| potential (runAcc read key K report fuel s a q r).1.state + 1 := by | |
| induction fuel generalizing s a q r with | |
| | zero => simp only [runAcc]; omega | |
| | succ fuel ih => | |
| have hread := hr s | |
| cases hv : (read s).1 with | |
| | none => simp only [runAcc, hv, Nat.mul_add, Nat.mul_one]; omega | |
| | some z => | |
| rcases z with ⟨x, s'⟩ | |
| have hkey := hk x | |
| simp only [runAcc, hv] | |
| split | |
| · have hcallback := hc x a | |
| split | |
| · simp only [Nat.mul_add, Nat.mul_one]; omega | |
| · have hrest := ih s' (report x a).1.1 (q + 1) (r + 1) | |
| simp only [Nat.mul_add, Nat.mul_one] at hrest | |
| dsimp only | |
| omega | |
| · simp only [Nat.mul_add, Nat.mul_one]; omega | |
| theorem run_work_potential | |
| (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (K : ℤ) | |
| (report : α → τ → (τ × Bool) × ℕ) (potential : τ → ℕ) (R L C : ℕ) | |
| (hr : ∀ s, (read s).2 ≤ R) (hk : ∀ x, (key x).2 ≤ L) | |
| (hc : ∀ x a, (report x a).2 + potential a ≤ potential (report x a).1.1 + C) | |
| (fuel : ℕ) (s : σ) (a : τ) : | |
| (run read key K report fuel s a).2 + potential a ≤ | |
| (R + L + C + 8) * (run read key K report fuel s a).1.queries + | |
| potential (run read key K report fuel s a).1.state + 1 := by | |
| simpa only [run, Nat.mul_zero, Nat.add_zero, Nat.zero_add] using | |
| runAcc_work_potential read key K report potential R L C hr hk hc fuel s a 0 0 | |
| end LowSpaceSubsetSum.CountedReplay | |
| end OAI | |