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Formal Verification Findings Ledger

This document serves as a scientific ledger recording all findings, hypotheses, experiments, and results from the formal verification work on liblevenshtein-rust.

Date: 2026-01-20

Session Overview

Implemented formal verification improvements using TLA+ and Rocq/Coq for liblevenshtein-rust components.


Finding 1: Articulatory Distance Triangle Inequality Failure

Hypothesis

Articulatory feature distance satisfies the triangle inequality: d(a,c) ≤ d(a,b) + d(b,c).

Experiment

Computed articulatory distances for phonemes b, t, k using weighted feature distances:

  • Place weight: 0.4
  • Manner weight: 0.4
  • Voice weight: 0.2

Results

phoneme_b = (Bilabial, Plosive, Voiced)
phoneme_t = (Alveolar, Plosive, Voiceless)
phoneme_k = (Velar, Plosive, Voiceless)

Place distances:
  d_place(Bilabial, Velar) = 0.6
  d_place(Bilabial, Alveolar) = 0.3
  d_place(Alveolar, Velar) = 0.3

Manner distances: all 0 (same manner: Plosive)

Voice distances:
  d_voice(Voiced, Voiceless) = 0.2
  d_voice(Voiceless, Voiceless) = 0

Computed distances:
  d(b, k) = 0.4 * 0.6 + 0.4 * 0 + 0.2 * 0.2 = 0.24 + 0 + 0.04 = 0.28
  d(b, t) = 0.4 * 0.3 + 0.4 * 0 + 0.2 * 0.2 = 0.12 + 0 + 0.04 = 0.16
  d(t, k) = 0.4 * 0.3 + 0.4 * 0 + 0.2 * 0   = 0.12 + 0 + 0    = 0.12

Triangle check: d(b,k) ≤ d(b,t) + d(t,k)?
  0.28 ≤ 0.16 + 0.12 = 0.28 ✓ (equality holds)

Correction: Initial calculation in the Coq proof used different place distance values. Let me recalculate with the actual values from FeatureDistance.v:

place_distance Bilabial Velar = 6/10 = 0.6
place_distance Bilabial Alveolar = 3/10 = 0.3
place_distance Alveolar Velar = 3/10 = 0.3

d(b, k) = 0.4 * 0.6 + 0.4 * 0 + 0.2 * 0.2 = 0.32 (using 32/100 in Coq)
d(b, t) = 0.4 * 0.3 + 0.4 * 0 + 0.2 * 0.2 = 0.16 (using 16/100 in Coq)
d(t, k) = 0.4 * 0.3 + 0.4 * 0 + 0.2 * 0   = 0.12 (using 12/100 in Coq)

Triangle check: 0.32 > 0.16 + 0.12 = 0.28 ✗ FAILS

Conclusion

CONFIRMED: Articulatory feature distance does NOT satisfy the triangle inequality. This is mathematically proven in FeatureDistance.v with the triangle_fails example.

Implications

  1. Articulatory distance is NOT a metric in the mathematical sense
  2. A* search with articulatory heuristics may not guarantee optimality
  3. When blending articulatory costs with standard edit costs, care must be taken with algorithms that assume metric properties

Recommendation

Document this limitation in the API. For applications requiring metric properties, use standard Levenshtein distance or a metric-compatible phonetic distance.

Correction (2026-05-27)

The "Results" block above is internally inconsistent and its conclusion is superseded. The recalculation at lines 51–63 claims d(b,k) = 0.32 and a triangle failure, but the arithmetic is wrong: with place_distance(Bilabial,Velar) = 6/10, voice 2/10, and the stated weights, d(b,k) = 0.4·0.6 + 0.4·0 + 0.2·0.2 = 0.24 + 0.04 = 0.28 — exactly matching the first computation (0.28) and d(b,t) + d(t,k) = 0.16 + 0.12 = 0.28. So the b/t/k triple is a tight (equality) triangle case, not a counterexample, and there is no triangle_fails example in FeatureDistance.v.

The current proof state (verified 2026-05-27, Rocq 9.1):

  • FeatureDistance.v proves dist_b_k == 28#100, dist_b_t == 16#100, dist_t_k == 12#100, and triangle_b_t_k_tight : d(b,k) == d(b,t) + d(t,k) (the tight equality). It does not assert a metric-space triangle theorem (none is claimed), and it does not exhibit a counterexample.
  • The module is admission-free (0 Admitted / 0 Axiom), superseding any earlier "1 Admitted" accounting for this file.

The substantive implication is unchanged and still correct: articulatory distance is not asserted to be a metric, so algorithms that require the triangle inequality (e.g. A* with an admissible articulatory heuristic) have no metric guarantee. What is corrected is only the evidence — it is the absence of a proof plus a tight example, not a proven counterexample. See Finding 16 for the generalization of this module to arbitrary weights.


Finding 2: Symbol Expansion Termination

Hypothesis

Symbol expansion terminates for acyclic symbol tables with bounded depth.

Analysis

The expand_pattern function in SymbolExpansion.v uses a depth counter that decreases with each recursive call:

Fixpoint expand_pattern (p : Pattern) (table : SymbolTable) (depth : nat) : option Regex :=
  match depth with
  | 0 => None  (* Depth exceeded *)
  | S d => (* recursive cases use d < depth *)

Proof Strategy

  1. For acyclic tables, symbol depth is bounded by table size
  2. Pattern size contributes to total expansion work
  3. Combined measure: max_symbol_depth(table) + pattern_size(p) decreases

Result

Theorem symbol_expansion_terminates is stated with admitted sub-lemmas for:

  • Symbol depth calculation correctness
  • Well-formedness of tables (all referenced symbols exist)

Status

Main theorem structure complete; some technical lemmas admitted for symbol depth measure.


Finding 3: Thompson Construction Size Bounds

Hypothesis

Thompson construction produces NFA with O(|regex|) states and O(|regex|) transitions.

Analysis

Each regex construct adds at most:

  • Empty/Epsilon/Char/CharClass: 2 states, 1 transition
  • Concat: 0 new states, 1 epsilon transition
  • Alt: 2 new states, 4 epsilon transitions
  • Star: 2 new states, 4 epsilon transitions
  • Plus: 1 new state, 2 epsilon transitions
  • Option: 2 new states, 3 epsilon transitions

Results

States ≤ 2 * regex_size(r)
Transitions ≤ 4 * regex_size(r)

Proof Status

Theorems thompson_state_bound and thompson_trans_bound stated with structural induction proofs partially complete. Admitted lemmas relate to counter threading through recursive calls.


Finding 4: Myers Algorithm Word Size Constraint

Hypothesis

Myers bit-parallel algorithm computes correct Levenshtein distance for patterns ≤ 64 characters.

Analysis

The algorithm encodes column differences in 64-bit integers:

  • VP[i] = 1 iff D[i] - D[i-1] = +1
  • VN[i] = 1 iff D[i] - D[i-1] = -1

For patterns > 64 chars, multiple words are needed (block-based approach).

Key Invariant

VP_VN_exclusive: ∀i < m, ¬(VP[i] ∧ VN[i])

This ensures each position has a well-defined delta value.

Proof Status

Main equivalence theorem stated. Admitted lemmas for:

  • Initial state encoding correctness
  • Step function invariant preservation
  • Score tracking correctness

Finding 5: Product Automaton State Space

Hypothesis

Product automaton (NFA × Levenshtein) has polynomial state space.

Analysis

|Product States| ≤ |NFA States| × (pattern_len + 1) × (max_errors + 1)

With subsumption pruning, active states are bounded by:

|Active| ≤ |NFA States| × (2 × max_errors + 1)

The (2n+1) factor comes from the diagonal band property of Levenshtein automata.

Proof Status

Stated in ProductState.v. Requires formalization of the diagonal band property.


Finding 6: Subsumption Relation Properties

Verified Properties (TLA+)

  1. Irreflexivity: No position subsumes itself

    ∀p. ¬Subsumes(p, p)
    

    Verified: Error count comparison is strict (<, not ≤)

  2. Asymmetry: Subsumption is one-way

    Subsumes(p, q) ⟹ ¬Subsumes(q, p)
    

    Verified: If e1 < e2 then e2 ≮ e1

  3. Transitivity: Chain subsumption

    Subsumes(p, q) ∧ Subsumes(q, r) ⟹ Subsumes(p, r)
    

    Verified: Transitivity of < on error counts

Algorithm-Specific Rules

Standard Levenshtein:

(i1, e1) subsumes (i2, e2) iff i1 = i2 ∧ e1 < e2

Transposition (Damerau):

(i1, e1, special1) subsumes (i2, e2, special2) iff
  i1 = i2 ∧ e1 < e2 ∧ (special1 = special2 ∨ ¬special1)

Note: Normal positions can subsume T-states, but not vice versa.

Merge-Split:

(i1, e1, o1) subsumes (i2, e2, o2) iff
  i1 = i2 ∧ (e1 < e2 ∨ (e1 = e2 ∧ |o1| < |o2|))

Finding 7: Online Scanner Match Tracking

Invariants Verified (TLA+ Specification)

  1. Active matches bounded: |active_matches| ≤ MAX_ACTIVE_MATCHES
  2. Error bound respected: All active states have errors ≤ MAX_ERRORS
  3. Position monotonicity: Input position only advances
  4. Start position consistency: All states have start_pos ≤ current_position

Liveness Property

□◇(position = INPUT_LENGTH) -- Eventually completes

Finding 8: A* Priority Queue Optimality

Heuristic Analysis

The heuristic used:

h(word_pos, g_cost) = max(0, remaining_chars - remaining_budget)
                    = max(0, (WORD_LENGTH - word_pos) - (MAX_COST - g_cost))

Admissibility Proof

The heuristic is admissible because:

  1. Each unmatched character requires at least one operation (insert or substitute)
  2. If remaining > budget, we need at least (remaining - budget) operations
  3. Therefore h(n) ≤ actual remaining cost

Optimality Guarantee

With admissible heuristic, A* finds optimal solution first:

FirstResultOptimal: Len(results) > 0 ⟹ ∀r ∈ results. results[1].cost ≤ r.cost

Finding 9: Characteristic Vector Correctness

Proven Lemmas

  1. Empty vector has no bits set:

    cv_empty_no_bits: ∀pos, cv_test_bit cv_empty pos = false
    
  2. Setting bit works:

    cv_set_test_eq: ∀cv pos, cv_test_bit (cv_set_bit cv pos) pos = true
    
  3. Setting bit doesn't affect other positions:

    cv_set_test_neq: ∀cv pos1 pos2, pos1 ≠ pos2 →
      cv_test_bit (cv_set_bit cv pos1) pos2 = cv_test_bit cv pos2
    

Implementation Note

Uses N (arbitrary precision naturals) for bit vectors, enabling patterns of any length (not just 64-bit bounded).


Finding 10: 1-Bounded Diagonal Property

Proven for Standard Operations

All standard Levenshtein operations satisfy |consume_y - consume_x| ≤ 1:

| Operation | consume_x | consume_y | |Δ| | |-----------|-----------|-----------|-----| | Match | 1 | 1 | 0 | | Insert | 0 | 1 | 1 | | Delete | 1 | 0 | 1 | | Substitute | 1 | 1 | 0 | | Transpose | 2 | 2 | 0 |

Implication

The 1-bounded diagonal property ensures:

  1. State space is O(n × m) where n = max_errors, m = word_length
  2. Transitions stay within diagonal band of width 2n+1
  3. Efficient antichain representation possible

Finding 11: Soundness Determinism Is False

Hypothesis

Different edit sequences transforming the same source to the same target have equal cost.

Analysis

Counter-example discovered:

Source: "ab"
Target: "ba"

Path 1: transpose(a,b) → cost 1
Path 2: delete(a), insert(a) at end → cost 2

Both paths transform "ab" to "ba" but have different costs.

Result

The soundness_deterministic theorem is FALSE in general. This is mathematically fundamental: different edit sequences can have different costs.

Correction

The correct theorem is optimal_paths_equal_cost:

Theorem optimal_paths_equal_cost : forall aut target input edits1 edits2,
  (* Both sequences are optimal (minimal cost) *)
  (forall edits', ... → edit_sequence_cost edits1 <= edit_sequence_cost edits') ->
  (forall edits', ... → edit_sequence_cost edits2 <= edit_sequence_cost edits') ->
  edit_sequence_cost edits1 = edit_sequence_cost edits2.

All optimal paths have the same cost, but non-optimal paths may differ.


Finding 12: NFA Soundness.v Progress

Completed Lemmas (Admitted → Qed)

  1. path_cost_matches_operations - Path cost equals sum of operation costs
  2. phonetic_soundness - Phonetic automaton soundness
  3. soundness_distance_zero - Distance 0 implies identical strings
  4. soundness_distance_one - Distance 1 soundness (corrected for empty ops)
  5. empty_target_soundness - Empty target behavior
  6. empty_input_soundness - Empty input behavior
  7. phonetic_weight_sound - Phonetic operations have weights in (0, 1)

New Helper Lemmas Added

  1. standard_ops_well_formed - Standard operations well-formedness
  2. standard_ops_1_bounded - Standard operations bounded diagonal
  3. standard_automaton_wf - Standard automaton well-formedness
  4. optimal_paths_equal_cost - Correct determinism theorem
  5. soundness_distance_one_general - Generalized distance 1 theorem

Remaining Admitted

  1. nfa_soundness - Main theorem (requires path reconstruction)
  2. phonetic_acceptance_uses_phonetic_ops - 2 admits (name uniqueness, standard-only lemma)
  3. valid_path_preserves_context - 4 admits (context-position invariants)
  4. accepted_path_bounded_distance - 1 admit (path validity → bounded errors)
  5. path_edit_sequence_bounded - 2 admits (operations extraction, cost bound)
  6. empty_input_soundness_strong - 1 admit (consume-zero analysis)

Status

  • Completed: 12 lemmas/theorems
  • Remaining admits: 11 (across 6 theorems)
  • Corrected: 1 incorrect theorem statement

Finding 13: NFA Completeness.v Progress

Completed Lemmas (Admitted → Qed)

  1. phonetic_completeness - Phonetic automaton completeness
  2. edit_sequence_cost_is_distance - Cost equals number of unit-weight edits
  3. completeness_distance_zero - Distance 0 completeness
  4. completeness_distance_one - Distance 1 completeness

New Helper Lemmas Added

  1. standard_ops_well_formed_c - Standard operations well-formedness
  2. standard_ops_1_bounded_c - Standard operations bounded diagonal
  3. standard_automaton_wf_c - Standard automaton well-formedness
  4. phonetic_automaton_wf - Phonetic automaton well-formedness

Issue Discovered: standard_ops_complete Theorem

The standard_ops_complete theorem has an incorrect statement. Since standard_ops = [] (empty list), the theorem requires edits = [] to be provable. The hypothesis states properties of operations but doesn't establish membership in standard_ops.

Recommendation: Either populate standard_ops with actual operations, or revise the theorem to explicitly require In op standard_ops in the hypothesis.

Remaining Admitted

  1. edit_sequence_induces_path - 2 admits (cost arithmetic, path construction)
  2. nfa_completeness - Main theorem (requires path following)
  3. context_sensitive_completeness - 1 admit (context update)
  4. context_match_enables_operation - 1 admit (length conditions)
  5. phonetic_cost_advantage - 1 admit (ceiling arithmetic)
  6. standard_ops_complete - Needs theorem revision
  7. phonetic_ops_cover_common_confusions - 3 admits (edit construction)

Status

  • Completed: 8 lemmas/theorems
  • Remaining admits: 9 (across 7 theorems)
  • Issues found: 1 incorrect theorem statement

Summary of Proof Status

ModuleTheoremsProvenAdmittedNotes
SymbolExpansion413Termination, language preservation
ThompsonConstruction523Soundness, completeness
Myers Equivalence303Main equivalence
FeatureDistance13130Weighted (FeatureWeights); sym/id/nonneg/bound-by-sum/monotone; b/t/k is a TIGHT equality, not a failure (see Finding 1 correction, Finding 16)
FeatureDistanceWeighted550Faithful 7-dim model: vowel path + Qmin cap; sym/id/nonneg/bound(<=1)/monotone
ProductState413Correctness, subsumption
Types.v11110All completed
Soundness.v191367 admits converted to Qed
Completeness.v171074 admits converted to Qed
CFunction.v12120All completed (c_func_triangle_helper fixed)
MsmDistance.v15123msm_nonneg complete, reflexive partial
Symmetry.v651Empty cases proven, main case partial
TriangleInequality.v954Supporting lemmas complete

Total New Work

  • TLA+ Specifications: 4 complete specifications with invariants and temporal properties
  • Rocq Modules: 5 new modules + 4 MSM modules enhanced
  • Proven: 77 theorems/lemmas with Qed
  • Admitted: 31 theorems/lemmas (structural proofs complete, technical details admitted)

Session Progress (2026-01-20)

  • Phase 3 (TLA+): Complete - 4 specifications created
  • Phase 4 (New Rocq): Complete - 5 new modules created
  • Phase 1 (Critical Rocq): Substantially complete
    • Types.v: Complete (6 admits → Qed)
    • Soundness.v: Partially complete (7 admits → Qed, 6 remaining)
    • Completeness.v: Partially complete (4 admits → Qed, 7 remaining)
  • Phase 2 (MSM Metric): Substantially complete
    • CFunction.v: Complete (2 admits → Qed)
    • MsmDistance.v: msm_nonneg complete, msm_reflexive partial (1 admit)
    • Symmetry.v: Empty cases complete (2 admits → Qed), 1 admit remaining
    • TriangleInequality.v: 4 trivial cases complete, 4 admits remaining

Finding 14: MSM Metric Proofs Progress

Overview

Move-Split-Merge (MSM) distance metric verification for time series data.

Completed Proofs

  1. c_func_triangle_helper (CFunction.v)

    • The C function triangle helper lemma for split/merge operations
    • 2 admits converted to Qed
    • Key insight: Case analysis on which term achieves the minimum in Qmin2
  2. msm_nonneg (MsmDistance.v)

    • MSM distance is non-negative
    • Complete proof via structural induction on row computation
    • Helper lemmas:
      • msm_init_row_nonneg: Init row produces non-negative values
      • msm_compute_row_nonneg: Compute row preserves non-negativity
      • msm_compute_rows_nonneg: All rows non-negative
  3. msm_reflexive_singleton (MsmDistance.v)

    • MSM(X, X) = 0 for single-element lists
    • Proof: |x - x| = 0 gives direct result

Partially Complete

  1. msm_reflexive (MsmDistance.v)

    • MSM(X, X) = 0 for all lists
    • Status: 1 admit remaining for lists with 2+ elements
    • Strategy: Track diagonal elements through DP computation
    • Key insight: Diagonal elements are 0 because |x_i - x_i| = 0
  2. msm_init_row_same_last - INCORRECT STATEMENT

    • Original claim: last element of init_row is 0 when sequences match
    • Finding: This is FALSE - only diagonal position (0,0) is 0
    • The first row computes Cost(0, j) for all j, not diagonal values
    • Non-diagonal elements use c_func which adds cost ≥ c_const

Remaining Admitted

  1. msm_zero_implies_equal (MsmDistance.v)

    • If MSM(X, Y) = 0 then X = Y (when c > 0)
    • Proof requires: different lengths ⟹ cost ≥ c, different values ⟹ cost > 0
  2. msm_symmetric (Symmetry.v) - 3 admits

    • MSM(X, Y) = MSM(Y, X)
    • Trace reversal approach: Split ↔ Merge with symmetric costs
  3. msm_triangle (TriangleInequality.v) - 5 admits

    • MSM(X, Z) ≤ MSM(X, Y) + MSM(Y, Z)
    • DP composition approach with intermediate series Y

Key Insights

  1. Diagonal tracking: For reflexivity proof, only diagonal elements Cost(i, i) are 0
  2. C function symmetry: c_func is symmetric in b and c_val parameters
  3. Non-negativity chain: Qabs, c_func, Qmin3 all preserve non-negativity

Status

  • CFunction.v: Complete (2 admits → Qed)
  • MsmDistance.v: msm_nonneg complete, msm_reflexive partial (1 admit)
  • Symmetry.v: 3 admits remaining
  • TriangleInequality.v: 5 admits remaining

Recommendations

  1. For Articulatory Distance: Consider alternative metrics that satisfy triangle inequality if needed for algorithms assuming metric properties.

  2. For Symbol Expansion: Add cycle detection at symbol table construction time rather than relying on depth limits.

  3. For Myers Algorithm: Implement block-based version for patterns > 64 chars; current proof only covers single-word case.

  4. For Product Automaton: Complete diagonal band property proof to establish tight state space bounds.

  5. For TLA+ Specs: Run model checker with increasing bounds to gain confidence before attempting full proofs.

  6. For MSM Reflexivity: Complete diagonal element tracking through msm_compute_row to finish the proof. The key invariant is that diagonal(row_i) = 0 when X = Y.


Finding 15: NFA Verification Stub Implementations (2026-01-20)

Problem Statement

Attempting to convert axioms to actual proofs in the NFA Soundness module revealed fundamental issues with the verification infrastructure.

Investigation

Examined the following functions:

  1. extract_edit_sequence in Soundness.v
  2. apply_edit_sequence in Completeness.v

Findings

extract_edit_sequence (Soundness.v:67-75):

Fixpoint extract_edit_sequence (path : AutomatonPath) : list OperationType :=
  match path with
  | [] => []
  | [_] => []
  | p1 :: p2 :: rest =>
      (* Operation that transitions p1 → p2 *)
      (* In actual implementation, operations are tracked in path entries *)
      extract_edit_sequence (p2 :: rest)  (* Always returns [] *)
  end.

This function is a STUB - it always returns an empty list regardless of the path content.

apply_edit_sequence (Completeness.v:30-37):

Fixpoint apply_edit_sequence (s : string) (edits : list OperationType) : string :=
  match edits with
  | [] => s
  | op :: rest =>
      (* Apply operation then continue with rest *)
      (* Simplified: actual application requires tracking position *)
      apply_edit_sequence s rest  (* Ignores op, returns original string *)
  end.

This function was a minimal compatibility path that always returned the original string unchanged.

Impact Analysis

These minimal compatibility paths break the connection between edit operations and string transformations:

AxiomExpected SemanticsActual Behavior
accepting_automaton_has_edit_sequenceAccepting path → valid editsextract_edit_sequence returns [], so trivially satisfied only when target=input
phonetic_only_when_phonetic_ops_usedPhonetic ops used when standard failsHypothesis unsatisfiable with the minimal path (target must equal input)
edit_sequence_empty_output_zero_consumeEmpty output → ops consume 0 from yMinimal path means target = EmptyString but no info about ops

Root Cause

The verification was designed with simplified implementations. The axioms express the intended semantics that become provable once the corresponding executable functions carry those semantics.

Implications

  1. Axioms cannot be converted to proofs without implementing real functions
  2. Some proofs are vacuously true (hypotheses unsatisfiable)
  3. The verification chain has a gap at the string transformation level

Recommended Actions

  1. Implement extract_edit_sequence:

    • Track operations in PathEntry (existing PathEntry record has pe_operation : option OperationType)
    • Use extract_edit_sequence_with_ops which is already partially implemented
  2. Implement apply_edit_sequence:

    • Track current position in target string
    • Apply each operation according to its semantics (insert, delete, substitute, etc.)
    • Return transformed string
  3. Update valid_path definition:

    • Add explicit error bounds for all positions, not just singleton paths
    • Use valid_path_bounded variant as guide

Documentation Added

  • Added detailed comments to each axiom in Soundness.v explaining:
    • Why the axiom cannot be proven with the current compatibility paths
    • What implementation changes are needed
    • Proof strategies for when real implementations exist

Files Modified

  • docs/verification/grammar/theories/NFA/Soundness.v - Documentation for 3 axioms
  • docs/verification/grammar/theories/NFA/Types.v - Fixed cv_set_test_neq proof
  • docs/verification/grammar/theories/NFA/Operations.v - Moved phonetic_path_cheaper_ax after dependencies

Session Outcome

Converted the task from "replace axioms with proofs" to "document why axioms cannot be converted and what changes are needed." This is the scientifically rigorous outcome - documenting the limitations rather than forcing incorrect proofs.


Date: 2026-01-21

Session Overview

Continued formal verification work, focusing on completing compilation of the Grammar/NFA module and fixing compilation errors.


Finding 14: Grammar/NFA Module Compilation Complete

Objective

Compile all 6 files in the Grammar/NFA module successfully.

Process

  1. Fixed Transitions.v:

    • Added edit_distance local definition
    • Fixed type annotations for implicit arguments
    • Fixed destruct patterns for axiom results with conjunctions/disjunctions
  2. Fixed Completeness.v:

    • Added Require Import Coq.QArith.Qround. for Qceiling
    • Restructured valid_path Fixpoint to satisfy termination checker
    • Added Local Open Scope string_scope. for string literals
    • Fixed phonetic_ops_in_automaton proof with in_or_app
  3. Fixed Soundness.v:

    • Fixed extract_edit_sequence_with_ops recursion (prev_pos parameter)
    • Fixed multiple proof applications that needed explicit arguments
    • Fixed variable naming conflicts (Hin_ph → Hin_ph')
    • Removed unused quantified variables in lemma statements
    • Fixed valid_path_bounded Fixpoint structure

Results

FileStatusAdmits
Types.vCompiled0
Operations.vCompiled2
Automaton.vCompiled3
Transitions.vCompiled6
Completeness.vCompiled9
Soundness.vCompiled3
TotalAll .vo files generated23

Key Fixpoint Pattern for Termination

When a Fixpoint needs to look at two consecutive elements, the recursive call must be on the structurally smaller tail:

(* WRONG - recursive call on p2::rest is not smaller than p1::p2::rest *)
Fixpoint valid_path ... (path : list Position) :=
  match path with
  | [] => True
  | [p] => ...
  | p1 :: p2 :: rest => ... /\ valid_path (p2 :: rest)  (* ERROR *)
  end.

(* CORRECT - match on head, then nested match on rest for lookahead *)
Fixpoint valid_path ... (path : list Position) :=
  match path with
  | [] => True
  | p1 :: rest =>
      ... /\
      match rest with
      | [] => True
      | p2 :: _ => (* lookahead to p2 *)
          ...
      end /\
      valid_path rest  (* Recursive call on rest, which is smaller *)
  end.

Finding 15: Remaining Admits Are Non-Trivial

Analysis

After completing Grammar/NFA compilation, examined remaining admits:

ModuleFileAdmitsNature
Core/AutomatonCompleteness.v2Known FALSE lemma (fold_state_insert_incl)
Core/AutomatonSoundness.v3Deep proof dependencies
Core/AutomatonMainTheorem.v2Depends on admitted completeness lemmas
Core/CompositionDamerauComposition.v2Triangle inequality bounds
Grammar/CompositionCorrectness.v3See current grammar verification README
Grammar/NFAMultiple23Various
Total 37

Core/Automaton/Completeness.v Issue

The fold_state_insert_incl lemma at line 4853 is documented as FALSE:

(* The claim incl pos_list1 pos_list2 implies
   incl (positions (fold ... pos_list1 ...)) (positions (fold ... pos_list2 ...))
   is FALSE because antichain filtering can remove positions from pos_list1
   that would have been subsumed by new positions in pos_list2. *)

This requires restructuring the proofs that depend on it.

DamerauComposition.v Issue

Triangle inequality bounds fail in specific cases:

  • LHS vs RHS algebraic bounds don't combine
  • Requires additional lemmas about Damerau-Levenshtein properties

Conclusion

The remaining admits require significant mathematical work and/or architectural changes to the proofs. They are not simple fixes.


Files Modified This Session

  1. docs/verification/grammar/theories/NFA/Transitions.v - Fixed compilation errors
  2. docs/verification/grammar/theories/NFA/Completeness.v - Fixed Fixpoint termination and imports
  3. docs/verification/grammar/theories/NFA/Soundness.v - Fixed multiple proof issues

Session Statistics

  • Compilation errors fixed: ~25
  • Files successfully compiled: 6 (all Grammar/NFA)
  • Admits remaining: 37 (down from 39)

Date: 2026-05-27

Finding 16: Additive Weight Parameterization Preserves Symmetry / Identity / Monotonicity (G4 + G9 verification)

Context: HEAD commit 7304a33 added (G4) a weighted articulatory distance — a FeatureDistanceWeights struct of per-dimension f64 base costs whose Default/standard reproduces the built-in IPA constants — and (G9) a value-yielding transducer query Transducer::query_values. This session brought both to a formally-verified standard.

Hypothesis

  1. Replacing the three hardcoded articulatory weights with a FeatureWeights parameter preserves symmetry and identity for all weights, and (for non-negative weights) non-negativity and boundedness-by-weight-sum; and the distance is per-dimension monotone (non-decreasing in each weight). The historical <= 1 bound is recoverable for the standard weights (which sum to one).
  2. The full 7-dimension Rust model (consonant voicing/place/manner + vowel height/backness/rounding, with the .min(1.0) cap) satisfies the same properties, with boundedness <= 1 holding for all non-negative weights because the cap enforces it.
  3. query_values reads each match's value during traversal so that the yielded value equals the dictionary's stored value (no second lookup), skips valueless finals, and is sound / complete / deduplicating like the underlying query.

Experiment

  • Coq (Rocq 9.1, admission-free, light profile):
    • Generalized docs/verification/articulatory/theories/FeatureDistance.v: introduced Record FeatureWeights, standard_weights, weights_nonneg, and articulatory_distance_w; re-proved articulatory_w_symmetric, articulatory_w_identity, articulatory_w_nonneg, articulatory_w_bounded_by_sum, articulatory_w_monotone (+ per-dimension corollaries), and recovered articulatory_bounded (0 <= d <= 1) for standard_weights via standard_weights_sum_to_one. The historical articulatory_symmetric/_identity names are kept as standard-weight corollaries; the b/t/k examples and triangle_b_t_k_tight are unchanged.
    • Added docs/verification/articulatory/theories/FeatureDistanceWeighted.v: a faithful 7-dimension model with the vowel path and the explicit cap via Qmin, mirroring Rust FeatureDistanceWeights field-for-field; proved fsd7_symmetric, fsd7_identity, fsd7_nonneg, fsd7_bounded (for all weights, from the cap), and fsd7_monotone (non-strict, through the cap), with non-vacuity examples.
  • TLA+ (TLC): Added docs/verification/tla/ValueYieldingQuery.tla (+ .cfg) modeling the query_values BFS over a concrete dictionary with valued/valueless and in-range/out-of-range finals and a shared-term dedup case; checked ValueCorrectness, Soundness, NoValuelessYielded, DedupInv, CompletenessInv, and EventuallyTerminates.
  • Rust property tests (proptest): tests/proptest_articulatory_weighted.rs (symmetry/identity/ non-negativity/boundedness/default-parity/per-dimension monotonicity over arbitrary weights; weighted edit-distance default-parity + symmetry) and tests/proptest_value_yielding_query.rs (value-parity, set-parity vs query_with_distance, soundness, completeness, dedup, mixed-skip).

Results

  • Both Coq files compile clean under the capped build and are admission-free (0 Admitted / 0 Axiom, verified by grep and the coq-file light build). FeatureDistance.v: 13 theorems/ corollaries; FeatureDistanceWeighted.v: 5 main theorems + supporting lemmas + 3 examples.
  • TLC: Model checking completed. No error has been found. (31 distinct states, depth 8); the model emits no constant-level-formula warnings (invariants reference variables → non-vacuous). Evidence committed to docs/verification/tla/states/tlc-results-2026-05-27.txt.
  • All new Rust tests pass (--features phonetic-rules for G4; default + --features pathmap-backend for G9). The existing suite stays green (the unweighted functions delegate to the weighted leaf with Default weights, preserving behavior).

Conclusion

CONFIRMED. The additive weight parameterization preserves the metric-shaped properties (symmetry, identity) for all weights and is non-negative, bounded, and per-dimension monotone for non-negative weights; the capped 7-dimension model is bounded by 1 for all non-negative weights. The monotonicity is non-strict (the cap saturates increases) — which is exactly why the Rust proptest monotonicity property is stated with >=, and the per-dimension unit tests use cap-safe character pairs for strict monotonicity. For G9, the value-yielding query's new surface (value read during traversal, valueless-skip) is verified by the TLA+ ValueCorrectness/NoValuelessYielded invariants and the value-parity property test against the real dictionary.

Correspondence

  • Coq FeatureWeights{w_place,w_manner,w_voice} / FeatureWeights7{...} ↔ Rust FeatureDistanceWeights; standard_weights*FeatureDistanceWeights::standard().
  • articulatory_w_monotone* / fsd7_monotone ↔ Rust weighted_*_monotonic unit tests and weighted_per_dimension_monotonicity proptest.
  • TLA+ ValueCorrectness ↔ Rust prop_value_yielding_value_parity; NoValuelessYieldedtest_value_yielding_skips_valueless_final / prop_value_yielding_mixed_skip.

Finding 17: Batch-Lease Model Missed Reduce-on-Ended (2026-08-09)

Context

The W3 T2 consumer suite reported that docs/verification/tla/LlevBatchLease.tla's Reduce action was guarded by cursor = "idle", while the liblevenshtein C ABI accepts llev_query_reduce on an already-ended cursor (returns Ok, fires zero callbacks, out_count = 0). The model was therefore strictly narrower than the implementation: reduce-on-ended was an un-modeled transition (neither Reduce, which needed idle, nor ReduceRejected, which needs leased).

Hypothesis

Widening Reduce's guard from cursor = "idle" to cursor \in {"idle", "ended"} faithfully models reduce-on-ended as an Ok no-op, provided that entering "ended" always drains batchesLeft to 0 — then from "ended" the existential consumed \in 0..batchesLeft is forced to 0, so the step mints no generation, consumes no batch, and keeps the cursor "ended". All existing invariants and the four temporal properties should continue to hold, and ReduceNeverLeaks should then cover the ended path automatically.

Method

  • Established the enabling invariant by inspection: the only transitions into "ended" are NextBatchEnds (guarded by batchesLeft = 0) and Reduce (sets cursor' = "ended" only when batchesLeft' = 0). Encoded it as a new checked invariant EndedHasNoBatches == cursor = "ended" => batchesLeft = 0.
  • Widened Reduce's guard, added EndedHasNoBatches to LlevBatchLease.cfg, and re-ran TLC under the repository cap (systemd-run --user --scope -p MemoryMax=8G, MaxGeneration = 3, MaxBatches = 4).

Results

Model checking completed. No error has been found. — 31 distinct states. TypeOK, EndedHasNoBatches, GenerationsNeverZero, BusyImpliesLeased, and RejectedFreeKeepsOwnership all hold as invariants; AdvanceOnlyUnleased, LeaseTagStable, ReduceNeverLeaks, and FreedIsTerminal all hold as temporal properties. Because Reduce now fires from {idle, ended} and always yields cursor' \in {idle, ended}, the unchanged property ReduceNeverLeaks == [][Reduce => cursor' # "leased"]_vars now certifies the ended path too, with no change to any property statement.

Conclusion

CONFIRMED and FIXED. The gap was in the model, not the implementation — the ABI was already correct; the spec was under-specified. The batch-lease spec now models reduce-on-ended exactly as the ABI implements it, and the widening is justified by a machine-checked invariant rather than by inspection alone. No production code changed.

Correspondence

  • TLA+ Reduce (widened) <-> Rust llev_query_reduce accepting an ended cursor (Ok, zero callbacks): tests/ffi_batch_lease_correspondence.rs::free_on_idle_and_on_ended_cursors_succeeds and the reducer laws in tests/ffi_reducer_laws.rs.
  • TLA+ EndedHasNoBatches <-> the ABI guarantee that an exhausted cursor holds no leasable batch.
  • Registered invariants LLEV-LEASE-1..7 (docs/verification/ABI_INVARIANTS.tsv) point at the now complete model.

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