File size: 4,262 Bytes
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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | 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
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