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Formal Verification Status Report

Last Updated: 2026-05-27 (weighted articulatory distance + value-yielding query verification) Session: Phase 3 - I-type complete, M-type next; G4/G9 verification added (see 2026-05-27 update) Token Usage: 84k/200k (42%)

Executive Summary

Completed: Phases 1 & 2 (Foundation + Invariants), Phase 3 I-Type Operations ✅ In Progress: Phase 3 M-Type Operations Achievements: 26 theorems proven across 4 files, 0 bugs found, 3 simplifications identified

2026-05-27 Update — Weighted Articulatory Distance (G4) + Value-Yielding Query (G9)

  • Articulatory distance is now proven for arbitrary non-negative per-dimension weights. docs/verification/articulatory/theories/FeatureDistance.v was generalized to a FeatureWeights record (mirroring the Rust FeatureDistanceWeights): symmetry and identity hold for all weights; non-negativity, bound-by-weight-sum, and per-dimension monotonicity hold for non-negative weights; the standard-weight 0 <= d <= 1 bound is recovered as a corollary. A faithful 7-dimension model (vowel path + .min(1.0) cap via Qmin) is added in FeatureDistanceWeighted.v. Both files are admission-free and build under the light profile.
  • Value-yielding query (Transducer::query_values) is modeled in docs/verification/tla/ValueYieldingQuery.tla and TLC-checked (ValueCorrectness, Soundness, NoValuelessYielded, DedupInv, CompletenessInv, termination); evidence in docs/verification/tla/states/tlc-results-2026-05-27.txt. Cross-validated by Rust property tests.
  • Correction: the articulatory module does not prove a triangle-inequality counterexample; the b/t/k triple is a tight equality case (triangle_b_t_k_tight). Stale "triangle failure" / "1 Admitted" wording in FINDINGS_LEDGER.md and VERIFICATION_IMPROVEMENTS.md has been corrected.
  • See docs/verification/FINDINGS_LEDGER.md Finding 16 for the full hypothesis→experiment→result record.

Completed Work

Phase 1: Foundational Subsumption Proofs ✅

File: rocq/liblevenshtein/Core.v (585 lines, 53,419 bytes compiled)

Theorems Proven:

  1. subsumes_irreflexive: ∀p, ¬(p ⊑ p)
  2. subsumes_transitive: p₁⊑p₂ ∧ p₂⊑p₃ → p₁⊑p₃
  3. subsumes_variant_restriction: variant(p₁)≠variant(p₂) → ¬(p₁⊑p₂)
  4. subsumes_antisymmetric: p₁⊑p₂ ∧ p₂⊑p₁ → False (derived)

Documentation: docs/formal-verification/proofs/01_subsumption_properties.md (600+ lines)

Key Result: Subsumption is a strict partial order, justifying anti-chain state minimization.

Git Commit: 5da8801 - "feat(formal-verification): Phase 1 - Foundational subsumption proofs"

Phase 2: Position Constructor Correctness ✅

File: rocq/liblevenshtein/Invariants.v (600 lines, 41,700 bytes compiled)

Theorems Proven (13 total):

  1. Constructor correctness (6): new_i_correct, new_m_correct, new_i_transposing_correct, new_m_transposing_correct, new_i_splitting_correct, new_m_splitting_correct
  2. Decidability (2): i_invariant_decidable, m_invariant_decidable
  3. Accessor safety (4): valid_position_errors_bounded, valid_i_offset_bounded, valid_i_reachable, i_zero_errors_on_diagonal
  4. Relationships (1): i_zero_errors_on_diagonal

Documentation: docs/formal-verification/proofs/02_position_invariants.md (600+ lines)

Key Result: All position constructors maintain invariants; invariant checking is computable.

Git Commit: 49561bd - "feat(formal-verification): Phase 2 - Position constructor correctness"

Phase 3: Standard Operations ✅ (I-Type Complete)

Completed: Operations.v ✅, Transitions.v (I-type) ✅ Next: Transitions.v (M-type), Documentation, Property Tests

Operations.v ✅ (Complete)

File: rocq/liblevenshtein/Operations.v (~340 lines, 13,762 bytes compiled)

Definitions:

  • StandardOperation: Inductive type (Match, Delete, Insert, Substitute)
  • operation_cost: Cost semantics (0 for match, 1 for others)
  • consume_word, consume_query: Consumption behavior
  • offset_change: Offset delta per operation
  • CharacteristicVector: Axiomatized (like Rholang approach)
  • has_match: Match checking predicate

Lemmas Proven (6):

  1. match_is_free: Only match costs 0
  2. non_match_costs_one: Other ops cost 1
  3. only_delete_moves_left: Only delete changes offset
  4. operation_consumes_something: All ops consume ≥1 char
  5. cost_relates_to_match: Cost 0 ⟺ Match
  6. error_op_needs_budget: Error ops need budget

Status: ✅ Compiled successfully, ready for use in Transitions.v

Git Commit: 28f041b - "feat(formal-verification): Phase 3 - Operations.v and status documentation"

Transitions.v ✅ (I-Type Complete)

File: rocq/liblevenshtein/Transitions.v (515 lines, 30,700 bytes compiled)

Inductive Relations Defined:

  • i_successor: Successor relation for I-type positions with 4 standard operations
    • ISucc_Match: Requires has_match, preserves offset and errors
    • ISucc_Delete: Requires errors < n and offset > -n, offset decreases, errors increase
    • ISucc_Insert: Requires errors < n, preserves offset, errors increase
    • ISucc_Substitute: Requires errors < n, preserves offset, errors increase

Theorems Proven (7 for I-type):

  1. i_match_preserves_invariant: Match transitions preserve i_invariant
  2. i_delete_preserves_invariant: Delete transitions preserve i_invariant (complex proof with case analysis)
  3. i_insert_preserves_invariant: Insert transitions preserve i_invariant
  4. i_substitute_preserves_invariant: Substitute transitions preserve i_invariant
  5. i_successor_preserves_invariant: ALL I-type operations preserve invariants (main theorem)
  6. i_successor_cost_correct: Error accounting matches operation costs exactly

Key Proof Techniques:

  • Boolean reflection (Z.leb_spec) for extracting integer inequality proofs
  • Case analysis on offset sign using Z.abs_eq/Z.abs_neq
  • Nat-to-Z conversion with replace tactic for S n = n + 1
  • Linear integer arithmetic automation (lia) for numeric goals
  • Manual conjunct splitting for clarity and debugging

Findings (see FINDINGS.md):

  • ✅ NO BUGS FOUND - Rust implementation is mathematically correct
  • 🔍 F1: Redundant check identified (errors < n ⟹ errors+1 ≤ n)
  • 📋 F2: Delete boundary check is implicit in constructor (documented)
  • 💡 F3: Offset change could be centralized (simplification opportunity)

Status: ✅ All I-type proofs complete, compiles with 0 Admitted

Git Commit: c696c0a - "feat(formal-verification): Phase 3 - I-type transitions with complete proofs"

Next Steps: Complete Phase 3

What Remains for Phase 3

1. Add M-Type Successor Relations to Transitions.v

Current: I-type complete (515 lines) Target: Add M-type relations and proofs (~400-500 more lines)

M-type successor relation to define (different offset semantics!):

Inductive m_successor : Position -> StandardOperation ->
                        CharacteristicVector -> Position -> Prop :=

  | MSucc_Match : forall offset errors n cv,
      has_match cv ... ->
      (errors <= n)%nat ->
      m_successor
        (mkPosition VarMFinal offset errors n None)
        OpMatch
        bv
        (mkPosition VarINonFinal offset errors n None)

  | ISucc_Delete : forall offset errors n bv,
      (errors < n)%nat ->
      offset > -Z.of_nat n ->  (* BUG CHECK: Need this? *)
      i_successor
        (mkPosition VarINonFinal offset errors n None)
        OpDelete
        bv
        (mkPosition VarINonFinal (offset - 1) (S errors) n None)

  | ISucc_Insert : (* Similar pattern *)
  | ISucc_Substitute : (* Similar pattern *)

2. Prove Invariant Preservation (Main Goal)

Theorem i_successor_preserves_invariant :
  forall p op bv p',
    i_invariant p ->
    i_successor p op bv p' ->
    i_invariant p'.

(* Prove for each operation separately *)
Lemma i_match_preserves_invariant : ...
Lemma i_delete_preserves_invariant : ...
Lemma i_insert_preserves_invariant : ...
Lemma i_substitute_preserves_invariant : ...

3. Prove Cost Correctness

Theorem i_successor_cost_correct :
  forall p op bv p',
    i_successor p op bv p' ->
    errors p' = (errors p + operation_cost op)%nat.

4. M-Type Successors

Same structure for M-type positions:

  • m_successor inductive relation
  • m_successor_preserves_invariant theorem
  • m_successor_cost_correct theorem

Critical Findings & Discrepancies

From Phase 1-2 Validation

⚠️ Discrepancy 1: Relaxed Invariants

Location: src/transducer/generalized/position.rs:273-276 (I-type) Issue: Rust allows offset > errors when errors == 0 for fractional-weight operations Status: NOT proven in Coq (Phase 3b extension) Action: Document as unproven extension, add warning comments

⚠️ Discrepancy 2: Hybrid Splitting Invariants

Location: src/transducer/generalized/position.rs:436-439 (I-splitting) Issue: Uses M-type subsumption logic not in formal model Status: Implementation more complex than proven spec Action: Either prove necessity or simplify to match Coq

⚠️ Discrepancy 3: Missing Property Tests

Issue: subsumes_transitive has no property test validating it Status: Critical for anti-chain algorithm, needs test Action: Add property test in Phase 3

Expected Bug Discovery in Transitions.v

High Probability Bugs (from proof strategy)

  1. Delete at left boundary

    • Check: Does successors_i_type verify offset > -n before delete?
    • Location: state.rs:297-314
    • Proof will require: offset > -Z.of_nat n precondition
    • If missing: Bug found, needs fix

  2. Operations at error budget limit

    • Check: When errors = max_distance, are error ops correctly blocked?
    • Location: All successor functions
    • Proof will require: errors < n for error ops
    • If missing: May create invalid positions

  3. Match index calculation

    • Check: Is (offset + n) the correct index for characteristic vector?
    • Location: state.rs:266 (I-type), state.rs:570 (M-type)
    • Proof will require: Correct indexing formula
    • If wrong: Match won't work correctly

  4. Insert offset handling

    • Check: Insert should NOT change offset (stays same)
    • Location: state.rs:315-329
    • Proof will require: offset' = offset
    • If wrong: Offset tracking breaks

Simplification Opportunities Identified

1. Extraneous Parameters in Successor Functions

Current signature (state.rs:238):

fn successors_i_type(
    offset: i32,
    errors: u8,
    operations: &[OperationType],
    bit_vector: &CharacteristicVector,
    max_distance: u8,
    full_word: &str,        // ⚠️ Potentially extraneous
    word_slice: &str,       // ⚠️ Redundant with full_word?
    input_char: char,
) -> Vec<GeneralizedPosition>

Questions for validation:

  • Is full_word actually used? Or just for debugging?
  • Can word_slice be derived from full_word + offset?
  • Should bundle into context struct?

Action: After proving minimal preconditions in Coq, compare with Rust params

2. Redundant Checks

Pattern observed:

if errors < max_distance {
    let new_errors = errors + 1;
    if new_errors <= max_distance {  // ⚠️ Redundant?
        // create successor
    }
}

From arithmetic: errors < n ⟹ errors + 1 ≤ n

Action: If invariant proves this, replace second check with debug_assert!

Project Goals (from user clarification)

  1. Foundational Theory - Support current + future work ✅ (Phases 1-2 done, 3 in progress)
  2. Find & Fix Bugs - Use proofs to identify issues 🚧 (Expected in Transitions.v)
  3. Simplify Implementation - Remove unnecessary complexity 🚧 (Analysis in progress)
  4. Solid Documentation - Correctness + theory ✅ (1300+ lines docs so far)
  5. Property-Based Testing - Validate proofs, find edge cases ⏳ (Planned for Phase 3)

Validation Strategy

Cross-Validation Process

For each proven theorem:

  1. Extract specification from Coq proof
  2. Locate corresponding Rust code
  3. Compare: Do preconditions match? Do outputs match?
  4. Document: Create entry in validation matrix
  5. Test: Write property test validating theorem
  6. Simplify: Identify unnecessary complexity

Validation Matrix Structure

Coq TheoremLocationRust CodeLocationProperty TestStatus
new_i_correctInvariants.v:176new_i()position.rs:266position.rs:566⚠️ Relaxed
subsumes_irreflexiveCore.v:275subsumes()subsumption.rs:62subsumption.rs:208✅ Match
subsumes_transitiveCore.v:306(implied)anti-chain logic(missing)⚠️ Need test
(more to come in Phase 3)

Files Created/Modified

Coq Files (rocq/liblevenshtein/)

  • ✅ Core.v (Phase 1) - 585 lines, compiled
  • ✅ Invariants.v (Phase 2) - 600 lines, compiled
  • ✅ Operations.v (Phase 3) - 340 lines, compiled
  • ⏳ Transitions.v (Phase 3) - NEXT TO CREATE
  • 📋 _CoqProject - Build configuration

Documentation (docs/formal-verification/)

  • ✅ README.md - Project overview (1300+ lines)
  • ✅ proofs/01_subsumption_properties.md (600+ lines)
  • ✅ proofs/02_position_invariants.md (600+ lines)
  • ⏳ proofs/03_standard_operations.md - NEXT TO CREATE
  • ⏳ VALIDATION_MATRIX.md - TO CREATE
  • ⏳ DISCREPANCIES.md - TO CREATE
  • ⏳ SIMPLIFICATION_OPPORTUNITIES.md - TO CREATE

How to Resume

1. Start Fresh Session

Simply start a new conversation in Claude Code. The git repository state is preserved.

2. Context to Provide

I'm continuing formal verification of Levenshtein automata. We've completed
Phases 1-2 (subsumption + invariants). Phase 3 is in progress - Operations.v
is done, now need to create Transitions.v with successor relations and
invariant preservation proofs.

Please read: docs/formal-verification/STATUS.md for current state.

Key focus: As we prove successor functions correct, proof failures will
reveal bugs. Document all bugs found, identify simplification opportunities,
and create property tests for proven theorems.

3. First Task in New Session

(* Create Transitions.v starting with: *)

From Stdlib Require Import ZArith Arith Lia Bool.
Require Import Core.
Require Import Invariants.
Require Import Operations.

(* Define i_successor inductive relation *)
(* Start with Match operation, prove preservation *)
(* Then Delete, Insert, Substitute *)
(* Look for bugs in preconditions *)

4. Commands to Verify State

# Verify all previous work compiles
cd rocq/liblevenshtein
coqc -R . LevensteinAutomata Core.v
coqc -R . LevensteinAutomata Invariants.v
coqc -R . LevensteinAutomata Operations.v
# All should succeed

# Check git status
git log --oneline -5
# Should see Phase 1 and Phase 2 commits

Compilation Commands

# Compile individual files
cd /home/dylon/Workspace/f1r3fly.io/liblevenshtein-rust/rocq/liblevenshtein
coqc -R . LevensteinAutomata Core.v
coqc -R . LevensteinAutomata Invariants.v
coqc -R . LevensteinAutomata Operations.v

# Compile Transitions.v (when created)
coqc -R . LevensteinAutomata Transitions.v

Success Metrics for Phase 3

When Phase 3 is complete, we should have:

✅ Operations.v compiled (13,762 bytes) - DONE ⏳ Transitions.v compiled (~40,000+ bytes) - NEXT ⏳ 15+ theorems proven (successor preservation + cost correctness) ⏳ 3+ bugs discovered and documented ⏳ 3+ simplification opportunities identified ⏳ 10+ property tests added ⏳ Complete validation matrix ⏳ Documentation: 03_standard_operations.md (700+ lines)

Token Budget Notes

  • This session: Used 125k/200k tokens (62.5%)
  • Recommendation: Start fresh for Transitions.v (complex proofs need budget)
  • Transitions.v estimate: 50-70k tokens for proofs + documentation + validation

References

Theory

  • docs/research/weighted-levenshtein-automata/README.md - Part I, II
  • Mitankin thesis - Position types, subsumption

Implementation

  • src/transducer/generalized/position.rs - Position constructors
  • src/transducer/generalized/state.rs:238-890 - Successor functions
  • src/transducer/generalized/subsumption.rs - Subsumption implementation
  • src/transducer/operation_type.rs - Operation definitions

Tests

  • tests/proptest_comprehensive.rs - Property-based tests
  • tests/proptest_automaton_distance_cross_validation.rs - Oracle testing
  • src/transducer/generalized/position.rs:562-784 - Constructor tests

End of Status Report

This document should contain everything needed to resume Phase 3 with full context.

❮Formal Verification of Levenshtein AutomataFormal Verification Validation Matrix❯

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