openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Streams /MeasuredDistinctCount.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.DistinctStreamCount | |
| namespace OAI | |
| /-! # Tail-recursive distinct counting with measured reader calls | |
| Only the current reader state, previous key, count, remaining fuel and work | |
| counter survive each iteration. No list of emitted keys is constructed. | |
| -/ | |
| namespace LowSpaceSubsetSum.MeasuredDistinctCount | |
| variable {α σ : Type*} | |
| def loop (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) : | |
| ℕ → σ → Option ℤ → ℕ → ℕ → ℕ × ℕ | |
| | 0, _, _, count, work => (count, work + 1) | |
| | fuel + 1, state, previous, count, work => | |
| let out := read state | |
| match out.1 with | |
| | none => (count, work + out.2 + 1) | |
| | some (x, next) => | |
| let k := key x | |
| let fresh := match previous with | none => true | some p => decide (p ≠ k.1) | |
| loop read key fuel next (some k.1) (count + if fresh then 1 else 0) | |
| (work + out.2 + k.2 + 3) | |
| theorem loop_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) | |
| (fuel : ℕ) (s : σ) (previous : Option ℤ) (count work : ℕ) : | |
| (loop read key fuel s previous count work).1 = count + | |
| match previous with | |
| | none => (DistinctStreamCount.run (fun s => (read s).1) (fun x => (key x).1) fuel s).1 | |
| | some p => (DistinctStreamCount.tail (fun s => (read s).1) (fun x => (key x).1) fuel s p).1 := by | |
| induction fuel generalizing s previous count work with | |
| | zero => cases previous <;> simp only [loop, DistinctStreamCount.run, DistinctStreamCount.tail, Nat.add_zero] | |
| | succ fuel ih => | |
| cases previous with | |
| | none => | |
| cases hr : (read s).1 with | |
| | none => simp only [loop, hr, DistinctStreamCount.run, Nat.add_zero] | |
| | some out => | |
| rcases out with ⟨x, next⟩ | |
| simp only [loop, hr, ↓reduceIte, ih, DistinctStreamCount.run] | |
| omega | |
| | some p => | |
| cases hr : (read s).1 with | |
| | none => simp only [loop, hr, DistinctStreamCount.tail, Nat.add_zero] | |
| | some out => | |
| rcases out with ⟨x, next⟩ | |
| by_cases he : p = (key x).1 | |
| · subst p | |
| simp only [loop, hr, ne_eq, not_true_eq_false, decide_false, Bool.false_eq_true, | |
| ↓reduceIte, Nat.add_zero, ih, DistinctStreamCount.tail] | |
| · simp only [loop, hr, ne_eq, he, not_false_eq_true, decide_true, ↓reduceIte, | |
| ih, DistinctStreamCount.tail] | |
| omega | |
| theorem loop_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) | |
| (R K : ℕ) (hread : ∀ s, (read s).2 ≤ R) (hkey : ∀ x, (key x).2 ≤ K) | |
| (fuel : ℕ) (s : σ) (previous : Option ℤ) (count work : ℕ) : | |
| (loop read key fuel s previous count work).2 ≤ work + (R + K + 4) * fuel + 1 := by | |
| induction fuel generalizing s previous count work with | |
| | zero => simp only [loop, Nat.mul_zero, Nat.add_zero, le_refl] | |
| | succ fuel ih => | |
| have hr := hread s | |
| simp only [loop] | |
| split | |
| · dsimp only | |
| nlinarith only [hr, Nat.zero_le ((R + K + 4) * fuel)] | |
| next x next _ => | |
| have hk := hkey x | |
| have ht := ih next (some (key x).1) | |
| (count + if (match previous with | none => true | some p => decide (p ≠ (key x).1)) | |
| then 1 else 0) (work + (read s).2 + (key x).2 + 3) | |
| nlinarith only [hr, hk, ht] | |
| def run (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (fuel : ℕ) (s : σ) : ℕ × ℕ := | |
| loop read key fuel s none 0 0 | |
| theorem run_value (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) (fuel : ℕ) (s : σ) : | |
| (run read key fuel s).1 = | |
| (DistinctStreamCount.run (fun s => (read s).1) (fun x => (key x).1) fuel s).1 := by | |
| simpa only [run, Nat.zero_add] using loop_value read key fuel s none 0 0 | |
| theorem run_work (read : σ → Option (α × σ) × ℕ) (key : α → ℤ × ℕ) | |
| (R K : ℕ) (hread : ∀ s, (read s).2 ≤ R) (hkey : ∀ x, (key x).2 ≤ K) | |
| (fuel : ℕ) (s : σ) : | |
| (run read key fuel s).2 ≤ (R + K + 4) * fuel + 1 := by | |
| simpa only [run, Nat.zero_add] using loop_work read key R K hread hkey fuel s none 0 0 | |
| end LowSpaceSubsetSum.MeasuredDistinctCount | |
| end OAI | |