openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Streams /CountedDictionaryCallback.lean
| import OAI.Combinatorics.LowSpaceSubsetSum.Streams.CountedRecordedDictionary | |
| namespace OAI | |
| /-! # Counted dictionary callbacks with an intrinsic trie invariant | |
| The executable state carries the proved width/count invariant. Encoding is | |
| computed once per emitted record, and its work plus record copying is included. | |
| -/ | |
| namespace LowSpaceSubsetSum.CountedDictionaryCallback | |
| universe u v w | |
| variable {α : Type u} {β : Type v} {γ : Type w} | |
| def SafeState (C width : ℕ) (encode : α → List Bool × ℕ) := | |
| {r : RecordedDictionary.State α // | |
| RecordedDictionary.Invariant C width (fun x => (encode x).1) r} | |
| def empty (C width : ℕ) (encode : α → List Bool × ℕ) : SafeState C width encode := | |
| ⟨RecordedDictionary.empty, RecordedDictionary.empty_invariant _ _ _⟩ | |
| def feed (C width : ℕ) (encode : α → List Bool × ℕ) | |
| (hwidth : ∀ x, (encode x).1.length = width) (cells : α → ℕ) | |
| (x : α) (r : SafeState C width encode) : SafeState C width encode × ℕ := | |
| let code := encode x | |
| let out := CountedTrie.recordedFeed C (fun _ => code.1) x r.val | |
| have hv : out.1 = RecordedDictionary.feed C (fun x => (encode x).1) x r.val := by | |
| change (CountedTrie.recordedFeed C (fun x => (encode x).1) x r.val).1 = _ | |
| exact CountedTrie.recordedFeed_value x r.val r.property (hwidth x) | |
| (⟨out.1, hv.symm ▸ RecordedDictionary.feed_invariant r.property x (hwidth x)⟩, | |
| code.2 + out.2 + cells x + 3) | |
| theorem feed_value (C width : ℕ) (encode : α → List Bool × ℕ) | |
| (hwidth : ∀ x, (encode x).1.length = width) (cells : α → ℕ) | |
| (x : α) (r : SafeState C width encode) : | |
| (feed C width encode hwidth cells x r).1.val = | |
| RecordedDictionary.feed C (fun x => (encode x).1) x r.val := | |
| CountedTrie.recordedFeed_value x r.val r.property (hwidth x) | |
| theorem feed_work (C width : ℕ) (encode : α → List Bool × ℕ) | |
| (hwidth : ∀ x, (encode x).1.length = width) (cells : α → ℕ) | |
| (K S : ℕ) (hk : ∀ x, (encode x).2 ≤ K) (hs : ∀ x, cells x ≤ S) | |
| (x : α) (r : SafeState C width encode) : | |
| (feed C width encode hwidth cells x r).2 ≤ K + S + 2 * width + 14 := by | |
| have h := CountedTrie.recordedFeed_work C (fun _ : α => (encode x).1) x r.val | |
| rw [hwidth x] at h | |
| have he := hk x | |
| have hc := hs x | |
| change (encode x).2 + (CountedTrie.recordedFeed C (fun _ : α => (encode x).1) x r.val).2 + | |
| cells x + 3 ≤ _ | |
| omega | |
| def callback (C width : ℕ) (encode : α → List Bool × ℕ) | |
| (hwidth : ∀ x, (encode x).1.length = width) (cells : α → ℕ) | |
| (emit : β → γ → Option α × ℕ) (x : β) (y : γ) (r : SafeState C width encode) : | |
| (SafeState C width encode × Bool) × ℕ := | |
| let found := emit x y | |
| match found.1 with | |
| | none => ((r, false), found.2 + 4) | |
| | some z => | |
| let out := feed C width encode hwidth cells z r | |
| ((out.1, false), found.2 + out.2 + 5) | |
| theorem callback_value (C width : ℕ) (encode : α → List Bool × ℕ) | |
| (hwidth : ∀ x, (encode x).1.length = width) (cells : α → ℕ) | |
| (emit : β → γ → Option α × ℕ) (x : β) (y : γ) (r : SafeState C width encode) : | |
| ((callback C width encode hwidth cells emit x y r).1.1.val, | |
| (callback C width encode hwidth cells emit x y r).1.2) = | |
| RecordedDictionary.callback C (fun x => (encode x).1) (fun x y => (emit x y).1) x y r.val := by | |
| cases he : (emit x y).1 with | |
| | none => simp only [callback, he, RecordedDictionary.callback] | |
| | some z => simp only [callback, he, feed_value, RecordedDictionary.callback] | |
| theorem callback_work (C width : ℕ) (encode : α → List Bool × ℕ) | |
| (hwidth : ∀ x, (encode x).1.length = width) (cells : α → ℕ) | |
| (emit : β → γ → Option α × ℕ) (K S E : ℕ) | |
| (hk : ∀ x, (encode x).2 ≤ K) (hs : ∀ x, cells x ≤ S) | |
| (he : ∀ x y, (emit x y).2 ≤ E) | |
| (x : β) (y : γ) (r : SafeState C width encode) : | |
| (callback C width encode hwidth cells emit x y r).2 ≤ E + K + S + 2 * width + 19 := by | |
| have hemit := he x y | |
| cases hv : (emit x y).1 with | |
| | none => simp only [callback, hv]; omega | |
| | some z => | |
| have hf := feed_work C width encode hwidth cells K S hk hs z r | |
| simp only [callback, hv] | |
| omega | |
| end LowSpaceSubsetSum.CountedDictionaryCallback | |
| end OAI | |