template

//! # The slot: LZ77 parsing.
//!
//! This whole file is what a miner replaces. It must be compilable as its own
//! crate root (`charon rustc ... src/parse.rs`), so it may not `use crate::…`.
//!
//! ## The contract
//!
//! `parse` reads `input` and writes a token stream into `out`, returning how
//! many tokens it wrote. A token is a `u32`:
//!
//! * `t < 256`                       — emit the literal byte `t`
//! * `t = 2^24 + (dist-1)*256 + (len-3)` — copy `len` bytes from `dist` back
//!
//! with `1 ≤ dist ≤ 32768` and `3 ≤ len ≤ 258`. The arithmetic encoding (rather
//! than shifts and masks) is deliberate: it is what the Lean side reasons about.
//!
//! The only thing that must be proved is that the token stream **decodes back to
//! `input`**, and that `parse` neither panics nor diverges. Nothing about the
//! *search* is constrained: the hash table below could be a suffix automaton, a
//! binary tree, or a full optimal-parse dynamic program, and the proof would not
//! grow, because correctness rests only on the bytes actually compared before a
//! match is emitted.

pub const MIN_MATCH: usize = 3;
pub const MAX_MATCH: usize = 258;
pub const WINDOW: usize = 32768;
pub const HASH_SIZE: usize = 32768;

/// Hash of the three bytes at a position. Any function of the right range works;
/// `% 32768` rather than `& 0x7fff` so that the index bound is arithmetic.
pub fn hash3(a: u8, b: u8, c: u8) -> usize {
    let x = (a as u32)
        .wrapping_mul(2654435761)
        .wrapping_add((b as u32).wrapping_mul(2246822519))
        .wrapping_add((c as u32).wrapping_mul(3266489917));
    ((x >> 15) % 32768) as usize
}

/// How many bytes match at `a` and `b`, up to `cap`. This is the *only* function
/// whose result the correctness proof depends on.
pub fn match_len(input: &[u8], a: usize, b: usize, cap: usize) -> usize {
    let mut l = 0usize;
    while l < cap && input[b + l] == input[a + l] {
        l += 1;
    }
    l
}

/// Greedy matching against a single-slot hash head table. The baseline.
pub fn parse(input: &[u8], out: &mut [u32]) -> usize {
    let n = input.len();
    let mut head = [0u32; 32768];
    let mut ntok = 0usize;
    let mut pos = 0usize;
    // `pos <= n - 3` rather than `pos + 3 <= n`. The two are equivalent for real
    // inputs, but in the extracted model a slice may have length `usize::MAX`,
    // and then the *addition* overflows and the program is not total. The
    // subtraction is guarded and cannot. This is the second prover-friendly rule
    // after "no labelled control flow", and it is the one miners trip over.
    let has3 = n >= 3;
    let lim = if has3 { n - 3 } else { 0 };
    while pos < n {
        let mut best_len = 0usize;
        let mut best_dist = 0usize;
        if has3 && pos <= lim {
            let h = hash3(input[pos], input[pos + 1], input[pos + 2]);
            let cand = head[h] as usize;
            head[h] = (pos + 1) as u32;
            if cand > 0 && cand <= pos {
                let cpos = cand - 1;
                if pos - cpos <= 32768 {
                    let mut cap = n - pos;
                    if cap > 258 {
                        cap = 258;
                    }
                    let l = match_len(input, cpos, pos, cap);
                    if l >= 3 {
                        best_len = l;
                        best_dist = pos - cpos;
                    }
                }
            }
        }
        if best_len >= 3 {
            out[ntok] = 16777216u32 + ((best_dist - 1) as u32) * 256 + ((best_len - 3) as u32);
            ntok += 1;
            let end = pos + best_len;
            let mut k = pos + 1;
            while k < end && has3 && k <= lim {
                let h2 = hash3(input[k], input[k + 1], input[k + 2]);
                head[h2] = (k + 1) as u32;
                k += 1;
            }
            pos = end;
        } else {
            out[ntok] = input[pos] as u32;
            ntok += 1;
            pos += 1;
        }
    }
    ntok
}

import Lz77
import Slot

/-!
# The submission's proof

This file is what a **miner** writes. Everything it imports is fixed: `Lz77` is
the published contract and its lemmas, `Slot` is `charon`+`aeneas` output that the
verifier regenerates from the submitted `parse.rs` and never takes on trust.

The obligation is `parse_spec` at the bottom, whose statement is pinned by the
verifier. Nothing else in this file is checked, so a miner may restructure it
freely.
-/

namespace Submission
open Aeneas Aeneas.Std Result ControlFlow

set_option maxRecDepth 8192
set_option maxHeartbeats 1000000

open LZ77 (toks bytes bytes_length bytes_getElem! Matches Found emit_lit emit_match)

/-! ## The hash

Nothing about correctness depends on what `hash3` computes, only that it lands in
range: a wrong hash finds a worse match, never an invalid one. That is the
property that lets a miner replace the whole search with anything they like. -/

@[local step]
theorem hash3_spec (a b c : Std.U8) :
    slot.hash3 a b c ⦃ fun h => h.val < 32768 ⦄ := by
  prove_hash3

/-! ## The match-length loop: the one load-bearing function

`Matches` and the proof script come from the contract's `Lz77.Search`; a parser that
keeps the template's `match_len` proves it in one line. -/

theorem match_len_loop_spec (input : Slice Std.U8) (a b cap l0 : Std.Usize)
    (ha : a.val + cap.val ≤ input.length) (hb : b.val + cap.val ≤ input.length)
    (hl0 : l0.val ≤ cap.val) (h0 : Matches input a.val b.val l0.val) :
    slot.match_len_loop input a b cap l0 ⦃ fun l =>
      l.val ≤ cap.val ∧ Matches input a.val b.val l.val ⦄ := by
  prove_match_len_loop

@[local step]
theorem match_len_spec (input : Slice Std.U8) (a b cap : Std.Usize)
    (ha : a.val + cap.val ≤ input.length) (hb : b.val + cap.val ≤ input.length) :
    slot.match_len input a b cap ⦃ fun l =>
      l.val ≤ cap.val ∧ Matches input a.val b.val l.val ⦄ := by
  prove_match_len

/-! ## The hash-insert loop

Pure bookkeeping: it writes into the search structure and nothing else, so its
postcondition is `True`. What still has to be proved is that it *terminates* and
never fails — which is the price of total correctness, and the reason a miner's
search structure is free but not free of obligations. -/

@[local step]
theorem parse_loop0_loop0_spec (input : Slice Std.U8) (head0 : Array Std.U32 32768#usize)
    (has30 : Bool) (lim «end» k0 : Std.Usize)
    (hlim : has30 = true → lim.val + 3 ≤ input.length) :
    slot.parse_loop0_loop0 input head0 has30 lim «end» k0
      ⦃ fun r => r.2 = true → lim.val + 3 ≤ input.length ⦄ := by
  rw [slot.parse_loop0_loop0]
  apply Std.loop.spec_decr_nat
    (measure := fun s => «end».val - s.2.2.val)
    (inv := fun s => s.2.1 = true → lim.val + 3 ≤ input.length)
  · rintro ⟨hd, h3, k⟩ hinv
    have hmax : input.length ≤ Std.Usize.max := Std.Slice.length_ineq input
    simp only [slot.parse_loop0_loop0.body]
    step*
  · exact hlim

/-! ## The parse loop

The invariant is one line of English: *the tokens written so far decode to the
input consumed so far*. Everything else in it — `pos ≤ n`, `ntok ≤ pos`, the
buffer keeps its length — is bookkeeping that exists to make the writes legal. -/

theorem parse_loop0_spec (input : Slice Std.U8) (out0 : Slice Std.U32)
    (n lim : Std.Usize) (head0 : Array Std.U32 32768#usize)
    (ntok0 pos0 : Std.Usize) (has30 : Bool)
    (hn : n.val = input.length)
    (hout : input.length ≤ out0.length)
    (hlim : has30 = true → lim.val + 3 ≤ input.length)
    (hpos : pos0.val ≤ n.val) (hntok : ntok0.val ≤ pos0.val)
    (hdec : LZ77.decode (toks out0 ntok0.val) = some ((bytes input).take pos0.val)) :
    slot.parse_loop0 input out0 n head0 ntok0 pos0 has30 lim ⦃ fun r =>
      r.1.val ≤ input.length ∧ r.2.length = out0.length ∧
      LZ77.decode (toks r.2 r.1.val) = some (bytes input) ⦄ := by
  rw [slot.parse_loop0]
  apply Std.loop.spec_decr_nat
    (measure := fun s => n.val - s.2.2.2.1.val)
    (inv := fun s =>
      s.2.2.2.1.val ≤ n.val ∧ s.2.2.1.val ≤ s.2.2.2.1.val ∧
      s.1.length = out0.length ∧
      (s.2.2.2.2 = true → lim.val + 3 ≤ input.length) ∧
      LZ77.decode (toks s.1 s.2.2.1.val) = some ((bytes input).take s.2.2.2.1.val))
  · rintro ⟨out, hd, ntok, pos, h3⟩ ⟨hp, hnt, hlen, hh3, hde⟩
    simp only at hp hnt hlen hh3 hde
    have hmax : input.length ≤ Std.Usize.max := Std.Slice.length_ineq input
    simp only [slot.parse_loop0.body]
    split
    case isTrue hposlt =>
      -- `step*` will not enter a `do` block whose first statement is an `if`, so
      -- the search is cut off here with its own postcondition. That is also the
      -- right division of labour: `Found` is everything the emission needs to
      -- know, and it is all a *different* search would have to establish.
      apply Std.WP.spec_bind (Pₘ := fun r => Found input n pos r.2.1 r.2.2)
      · split
        case isTrue hb3 =>
          split
          case isTrue hpl =>
            have hlim3 : lim.val + 3 ≤ input.length := hh3 hb3
            have hposl : pos.val ≤ lim.val := by scalar_tac
            step*
            -- and again for the candidate test
            apply Std.WP.spec_bind (Pₘ := fun r => Found input n pos r.1 r.2)
            · split
              case isTrue hcand =>
                split
                case isTrue hcp =>
                  step*
                  · -- the window test passed: cap, then the byte comparison
                    -- `if b then ok x else ok y` is a choice of *value*; `step*`
                    -- needs it in that form before it will go on.
                    rw [show ((if cap > 258#usize then ok 258#usize else ok cap)
                          : Result Std.Usize)
                        = ok (if cap > 258#usize then 258#usize else cap) from by
                          split <;> rfl]
                    step*
                    -- `match_len`'s two range preconditions
                    · split <;> scalar_tac
                    · split <;> scalar_tac
                    -- the match: everything `Found` asks for, and the distance
                    -- rewritten so that `pos - dist` is the candidate position.
                    · refine Or.inr ⟨by scalar_tac, ?_, ?_, by scalar_tac,
                        by scalar_tac, by scalar_tac, ?_⟩
                      · revert l_post1; split <;> scalar_tac
                      · revert l_post1; split <;> scalar_tac
                      · have : pos.val - i9.val = cpos.val := by scalar_tac
                        rw [this]; exact l_post2
                    · exact Or.inl (by scalar_tac)
                  · exact Or.inl (by scalar_tac)
                case isFalse => exact Or.inl (by scalar_tac)
              case isFalse => exact Or.inl (by scalar_tac)
            · rintro ⟨bl, bd⟩ hf
              exact hf
          case isFalse => exact Or.inl (by scalar_tac)
        case isFalse => exact Or.inl (by scalar_tac)
      · rintro ⟨head1, best_len, best_dist⟩ hfound
        have hntok_lt : ntok.val < out.length := by scalar_tac
        have hlim3 : h3 = true → lim.val + 3 ≤ input.length := hh3
        replace hfound : Found input n pos best_len best_dist := hfound
        simp only [Found] at hfound
        step*
        -- Unfolding `Found` before `step*` is worth doing: its side-goal solver
        -- then case-splits the disjunction itself and discharges all five range
        -- obligations of the token encoding, leaving only the two invariants.
        -- The invariant, after emitting a match.
        · rcases hfound with _ | ⟨hl3, hlmax, hend, hd1, hdmax, hdpos, hmatch⟩
          · exfalso; scalar_tac
          have htok : i6.val = LZ77.mkMatch best_dist.val best_len.val := by
            simp only [LZ77.mkMatch, LZ77.MATCH_BASE, i6_post, i3_post, i2_post,
              i5_post, i1_post, i4_post1, i_post1, Std.UScalar.cast_val_eq]
            scalar_tac
          refine ⟨by scalar_tac, by scalar_tac, by rw [s_post]; simpa [Std.Slice.set_val_eq] using hlen, head2_post, ?_,
            by scalar_tac⟩
          rw [s_post, show ntok1.val = ntok.val + 1 by scalar_tac,
            show «end».val = pos.val + best_len.val by scalar_tac]
          exact emit_match input out ntok pos.val best_dist.val best_len.val i6 hde hntok_lt
            hd1 hdpos hdmax hl3 hlmax (by scalar_tac) hmatch htok
        -- The invariant, after emitting a literal.
        · have hposlen : pos.val < input.length := by scalar_tac
          have hval : i1.val = (bytes input)[pos.val]! := by
            rw [bytes_getElem! input pos.val hposlen, i1_post,
              Std.U8.cast_U32_val_eq, i_post]
          refine ⟨by scalar_tac, by scalar_tac, by rw [s_post]; simpa [Std.Slice.set_val_eq] using hlen, hlim3, ?_,
            by scalar_tac⟩
          rw [s_post, show ntok1.val = ntok.val + 1 by scalar_tac,
            show pos1.val = pos.val + 1 by scalar_tac]
          exact emit_lit input out ntok pos.val i1 hde hposlen hntok_lt hval
    case isFalse hge =>
      have hpn : pos.val = n.val := by scalar_tac
      refine ⟨by scalar_tac, hlen, ?_⟩
      rw [hde, hpn, hn]
      simp
  · exact ⟨hpos, hntok, rfl, hlim, hdec⟩

/-! ## The obligation

This is the statement the verifier pins. A submission is accepted when *this*
theorem, with this statement, elaborates against a `Slot` the verifier generated
itself, and `#print axioms` on it shows nothing but Lean's own three.

Read it as: `parse` writes `ntok` tokens into `out`, does not disturb its length,
never fails and always terminates, and **the tokens it wrote decode back to the
input**. -/

theorem parse_spec (input : Slice Std.U8) (out : Slice Std.U32)
    (hlen : input.length ≤ out.length) :
    slot.parse input out ⦃ fun r =>
      r.1.val ≤ input.length ∧
      r.2.length = out.length ∧
      LZ77.Valid (bytes input) (toks r.2 r.1.val) ⦄ := by
  rw [slot.parse]
  apply Std.WP.spec_bind (Pₘ := fun r => r.1 = true → r.2.val + 3 ≤ input.length)
  · split
    case isTrue h3 =>
      have hmax : input.length ≤ Std.Usize.max := Std.Slice.length_ineq input
      step*
    case isFalse h3 =>
      intro hc; exact absurd hc (by simp)
  · rintro ⟨has3, lim⟩ hlim
    exact parse_loop0_spec input out (Std.Slice.len input) lim
      (Std.Array.repeat 32768#usize 0#u32) 0#usize 0#usize has3
      (by simp) hlen hlim (by scalar_tac) (by scalar_tac)
      (by simp [toks, LZ77.decode])

end Submission