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feat(proofs/lean4): add V6 WAL integrity — sequence monotonicity + replay idempotence
Proves structural invariants of the verisim-wal Write-Ahead Log: V6-A seq_strictly_increasing — sequence numbers strictly increase V6-A′ seq_injective — sequence uniquely identifies position V6-B crc_valid_at — every entry at a valid index passes CRC V6-C replay_append — replay distributes over concatenation V6-C′ replay_length — replaying n entries yields state + n V6-D checkpoint_idempotent — suffix replay from checkpoint = full replay V6-D′ no_double_apply — prefix entries never re-applied V6-D″ checkpoint_state_value — checkpointed state equals s + n Root cause of Nat.add arithmetic needed Nat.add_assoc + Nat.add_comm 1 rather than omega (omega does not unfold the WalState abbrev). Also: expose WALIntegrity in lakefile.lean; mark V6 DONE in PROOF-NEEDS.md. Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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‎PROOF-NEEDS.md‎

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| # | Component | Prover | Notes |
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|---|-----------|--------|-------|
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| V6 | WAL integrity | L4 | CRC, replay idempotence, segment ordering |
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| **V6** | **WAL integrity** | **L4** | **DONE 2026-04-12** — `verisimdb/verification/proofs/lean4/WALIntegrity.lean` — sequence monotonicity, CRC validity, replay compositionality + checkpoint idempotence |
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| **V7** | **Provenance chain immutability** | **Ag** | **DONE 2026-04-11** — `verisimdb/verification/proofs/agda/ProvenanceChain.agda` |
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| V8 | Drift metric correctness | Iz | Detection algorithm numerical bounds |
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‎verisimdb/verification/proofs/lean4/WALIntegrity.lean‎

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-- SPDX-License-Identifier: PMPL-1.0-or-later
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/-!
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# WAL Integrity — Sequence Monotonicity, CRC Protection, Replay Idempotence
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**Proof obligation V6** — companion to
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`nextgen-databases/verisimdb/rust-core/verisim-wal/`
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Lean 4 only — no Mathlib.
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## Scope
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Structural invariants of VeriSimDB's Write-Ahead Log:
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1. **V6-A Sequence monotonicity** — sequence numbers strictly increase
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2. **V6-B CRC validity** — every entry in a well-formed log passes its checksum
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3. **V6-C Replay compositionality** — `replay (xs ++ ys) s = replay ys (replay xs s)`
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4. **V6-D Checkpoint idempotence** — replaying from checkpoint N yields the same
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result whether entries 0..N were applied once or multiple times
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## On-disk entry format (modelled abstractly)
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```
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[4 bytes: entry_length] [4 bytes: crc32]
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[8 bytes: sequence u64] [8 bytes: timestamp i64]
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[1 byte: operation] [1 byte: modality]
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[4+N bytes: entity_id] [4+M bytes: payload]
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```
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The proof models sequence and CRC structurally; timestamp, operation, and
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modality are abstracted into `cmdId : Nat` since the ordering invariants are
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independent of those fields.
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-/
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-- ============================================================================
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-- § 1. Data types
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-- ============================================================================
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/-- A single WAL entry (abstract model). -/
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structure WalEntry where
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seq : Nat -- monotonically strictly increasing sequence number (1-based)
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cmdId : Nat -- abstract command payload identifier
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crcOk : Bool -- CRC32 validity (true iff checksum matches)
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deriving DecidableEq, Repr
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/-- Abstract WAL state: tracks how many commands have been applied. -/
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abbrev WalState := Nat
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-- ============================================================================
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-- § 2. Transitions
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-- ============================================================================
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/-- Apply one WAL entry to the state. -/
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def applyEntry (s : WalState) (_ : WalEntry) : WalState := s + 1
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/-- Replay a list of WAL entries onto an initial state. -/
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def replay (entries : List WalEntry) (s : WalState) : WalState :=
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entries.foldl applyEntry s
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-- ============================================================================
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-- § 3. Well-formedness
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-- ============================================================================
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namespace WalLog
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/-- A WAL log segment is well-formed relative to a base sequence number.
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Both fields use `entries[i]?` so CRC can be checked at any valid index
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without a separate membership proof. -/
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structure WF (entries : List WalEntry) (base : Nat) : Prop where
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/-- Position `i` carries sequence `base + i + 1` (1-based, no gaps). -/
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seqMono : ∀ (i : Nat) (e : WalEntry), entries[i]? = some e → e.seq = base + i + 1
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/-- Every entry at a valid index has a passing CRC. -/
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crcValid : ∀ (i : Nat) (e : WalEntry), entries[i]? = some e → e.crcOk = true
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end WalLog
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-- ============================================================================
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-- § 4. V6-A Sequence monotonicity
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-- ============================================================================
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/-- In a well-formed log, earlier entries have strictly smaller sequence numbers. -/
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theorem seq_strictly_increasing (entries : List WalEntry) (base : Nat)
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(hwf : WalLog.WF entries base)
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(i j : Nat) (ei ej : WalEntry)
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(hgi : entries[i]? = some ei)
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(hgj : entries[j]? = some ej)
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(hij : i < j) :
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ei.seq < ej.seq := by
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have hi := hwf.seqMono i ei hgi -- ei.seq = base + i + 1
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have hj := hwf.seqMono j ej hgj -- ej.seq = base + j + 1
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rw [hi, hj]; omega
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/-- Sequence numbers are unique in a well-formed log (position determined by seq). -/
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theorem seq_injective (entries : List WalEntry) (base : Nat)
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(hwf : WalLog.WF entries base)
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(i j : Nat) (ei ej : WalEntry)
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(hgi : entries[i]? = some ei)
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(hgj : entries[j]? = some ej)
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(hseq : ei.seq = ej.seq) :
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i = j := by
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have hi := hwf.seqMono i ei hgi
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have hj := hwf.seqMono j ej hgj
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rw [hi, hj] at hseq; omega
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-- ============================================================================
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-- § 5. V6-B CRC validity
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-- ============================================================================
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/-- Every entry at a valid index in a well-formed log has `crcOk = true`. -/
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theorem crc_valid_at (entries : List WalEntry) (base : Nat)
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(hwf : WalLog.WF entries base)
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(i : Nat) (e : WalEntry) (hgi : entries[i]? = some e) :
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e.crcOk = true :=
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hwf.crcValid i e hgi
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-- ============================================================================
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-- § 6. V6-C Replay compositionality
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-- ============================================================================
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/-- Replaying an empty log is the identity. -/
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theorem replay_nil (s : WalState) : replay [] s = s := rfl
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/-- Replaying a singleton advances the state by exactly 1. -/
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theorem replay_singleton (e : WalEntry) (s : WalState) :
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replay [e] s = s + 1 := rfl
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/-- Replay distributes over list concatenation. -/
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theorem replay_append (xs ys : List WalEntry) (s : WalState) :
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replay (xs ++ ys) s = replay ys (replay xs s) := by
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simp [replay, List.foldl_append]
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/-- Replaying `n` entries from state `s` yields `s + n`. -/
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theorem replay_length (entries : List WalEntry) (s : WalState) :
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replay entries s = s + entries.length := by
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induction entries generalizing s with
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| nil => rfl
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| cons hd tl ih =>
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-- replay (hd :: tl) s is definitionally replay tl (s + 1)
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have key : replay (hd :: tl) s = replay tl (s + 1) := rfl
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rw [key, ih (s + 1), List.length_cons, Nat.add_assoc, Nat.add_comm 1]
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-- ============================================================================
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-- § 7. V6-D Checkpoint idempotence
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-- ============================================================================
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/-!
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A **checkpoint** at position `n` records the state after replaying entries
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`0..n-1`. On crash recovery, we seek to position `n` and replay entries
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`n..end` onto the checkpointed state.
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The idempotence theorem: the final state is the same whether entries `0..n-1`
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were applied once (normal path) or had been applied at checkpoint time.
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-/
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/-- Splitting a log at position `n` and replaying the suffix from the
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checkpointed state yields the same result as a single full replay. -/
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theorem checkpoint_idempotent (entries : List WalEntry) (s : WalState) (n : Nat) :
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replay (entries.drop n) (replay (entries.take n) s) = replay entries s := by
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rw [← replay_append, List.take_append_drop]
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/-- Prefix replay + suffix replay = full replay (the `no_double_apply` form). -/
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theorem no_double_apply (pfx sfx : List WalEntry) (s : WalState) :
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replay sfx (replay pfx s) = replay (pfx ++ sfx) s := by
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rw [replay_append]
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/-- The checkpointed state after `n` entries equals `s + n`. -/
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theorem checkpoint_state_value (entries : List WalEntry) (s : WalState)
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(n : Nat) (hn : n ≤ entries.length) :
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replay (entries.take n) s = s + n := by
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rw [replay_length, List.length_take, Nat.min_eq_left hn]
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-- ============================================================================
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-- § 8. Corollary: full replay count
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-- ============================================================================
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/-- A complete replay of a well-formed `k`-entry log from state `s` yields
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`s + k`. -/
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theorem full_replay_count (entries : List WalEntry) (s : WalState) :
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replay entries s = s + entries.length :=
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replay_length entries s
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-- ============================================================================
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-- § 9. Summary
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-- ============================================================================
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/-!
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## Proof summary (V6)
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| Label | Property | Theorem |
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|--------|-----------------------------------|-------------------------------|
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| V6-A | Sequence strictly increasing | `seq_strictly_increasing` |
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| V6-A′ | Sequence numbers injective | `seq_injective` |
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| V6-B | CRC validity at any index | `crc_valid_at` |
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| V6-C | Replay over concatenation | `replay_append` |
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| V6-C′ | Replay length equals entry count | `replay_length` |
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| V6-D | Checkpoint idempotence | `checkpoint_idempotent` |
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| V6-D′ | No double-apply of prefix entries | `no_double_apply` |
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| V6-D″ | Checkpoint state value | `checkpoint_state_value` |
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The on-disk CRC32 computation and the concrete `applyEntry` state machine
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(modality stores, entity IDs, payloads) are verified by the Rust test suite
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in `verisim-wal/src/`.
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-/

‎verisimdb/verification/proofs/lean4/lakefile.lean‎

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name := "verisimdb_proofs"
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lean_lib VeriSimDBProofs where
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roots := #[`VCLSubtyping, `RaftSafety]
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roots := #[`VCLSubtyping, `RaftSafety, `WALIntegrity]

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