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Verification Coverage Report

Generated: $(date -u +"%Y-%m-%d %H:%M:%S UTC") Version: see CHANGELOG.md (release history) and impl/rust-cli/Cargo.toml [package].version (semver pin) Phase: M14 Complete - Conditionals & Logical Operators

Summary

This report tracks the correspondence between Lean 4 formal proofs and Rust implementation, along with property-based test coverage for each proven theorem.

Legend

  • ✅ Proven: Formal proof exists in Lean 4 (and cross-validated in other systems)
  • 🔗 Implemented: Rust implementation exists
  • 🧪 Tested: Property-based tests validate the implementation
  • ⚠️ Partial: Incomplete or pending work
  • ❌ Missing: Not implemented or tested

Core Filesystem Operations

mkdir Operation

  • ✅ Proven: mkdir_rmdir_reversible (Lean 4, Coq, Agda, Isabelle, Mizar)
  • 🔗 Implemented: impl/rust-cli/src/commands.rs:mkdir()
  • 🧪 Tested: prop_mkdir_rmdir_reversible (1000+ iterations)
  • Status: ✅ COMPLETE

Correspondence:

-- Lean 4: proofs/lean4/FilesystemModel.lean:158
theorem mkdir_rmdir_reversible (fs : Filesystem) (p : Path)
  (h_pre : MkdirPrecondition p fs) :
  rmdir p (mkdir p fs) = fs
// Rust: impl/rust-cli/src/commands.rs:14-55
pub fn mkdir(state: &mut ShellState, path: &str, quiet: bool) -> Result<()> {
    // Preconditions: path must not exist, parent must exist
    // ...
}

rmdir Operation

  • ✅ Proven: mkdir_rmdir_reversible (Lean 4, Coq, Agda, Isabelle, Mizar)
  • 🔗 Implemented: impl/rust-cli/src/commands.rs:rmdir()
  • 🧪 Tested: prop_mkdir_rmdir_reversible (1000+ iterations)
  • Status: ✅ COMPLETE

touch (create_file) Operation

  • ✅ Proven: create_delete_file_reversible (Lean 4, Coq, Agda, Isabelle)
  • 🔗 Implemented: impl/rust-cli/src/commands.rs:touch()
  • 🧪 Tested: prop_create_delete_file_reversible (1000+ iterations)
  • Status: ✅ COMPLETE

rm (delete_file) Operation

  • ✅ Proven: create_delete_file_reversible (Lean 4, Coq, Agda, Isabelle)
  • 🔗 Implemented: impl/rust-cli/src/commands.rs:rm()
  • 🧪 Tested: prop_create_delete_file_reversible (1000+ iterations)
  • Status: ✅ COMPLETE

Composition & Equivalence

Operation Sequences

  • ✅ Proven: operation_sequence_reversible (Lean 4, Coq, Agda, Isabelle)
  • 🔗 Implemented: impl/rust-cli/src/state.rs (undo/redo stack)
  • 🧪 Tested: prop_sequence_reversible, prop_composition_correctness
  • Status: ✅ COMPLETE

Filesystem Equivalence

  • ✅ Proven: fs_equiv_refl/sym/trans (Lean 4, Coq, Agda, Isabelle)
  • 🔗 Implemented: impl/rust-cli/src/state.rs (equivalence via snapshots)
  • 🧪 Tested: prop_equivalence_reflexive
  • Status: ✅ COMPLETE

CNO (Certified Null Operations)

  • ✅ Proven: cno_identity_element (Lean 4)
  • 🔗 Implemented: impl/rust-cli/src/state.rs (identity detection)
  • 🧪 Tested: prop_cno_identity
  • Status: ✅ COMPLETE

File Content Operations (Phase 6 M2)

File Truncation (> redirection)

  • ⚠️ Proven: Pending in Lean 4 (implementation exists, proof TODO)
  • 🔗 Implemented: impl/rust-cli/src/redirection.rs
  • 🧪 Tested: prop_truncate_restore_reversible (1000+ iterations)
  • Status: ⚠️ PARTIAL - Implementation complete, proof pending

File Append (>> redirection)

  • ⚠️ Proven: Pending in Lean 4 (implementation exists, proof TODO)
  • 🔗 Implemented: impl/rust-cli/src/redirection.rs
  • 🧪 Tested: prop_append_truncate_reversible (1000+ iterations)
  • Status: ⚠️ PARTIAL - Implementation complete, proof pending

Conditional Operations (Phase 6 M14)

test -f (file test)

  • ❌ Proven: Not yet proven in Lean 4 (POSIX semantics)
  • 🔗 Implemented: impl/rust-cli/src/test_command.rs:FileIsRegular
  • 🧪 Tested: prop_test_f_file_detection (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

test -d (directory test)

  • ❌ Proven: Not yet proven in Lean 4 (POSIX semantics)
  • 🔗 Implemented: impl/rust-cli/src/test_command.rs:FileIsDirectory
  • 🧪 Tested: prop_test_d_directory_detection (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

test -e (existence test)

  • ❌ Proven: Not yet proven in Lean 4 (POSIX semantics)
  • 🔗 Implemented: impl/rust-cli/src/test_command.rs:FileExists
  • 🧪 Tested: prop_test_e_existence_check (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

test string comparison (=, !=)

  • ❌ Proven: Not yet proven in Lean 4 (POSIX semantics)
  • 🔗 Implemented: impl/rust-cli/src/test_command.rs:StringEqual/NotEqual
  • 🧪 Tested: prop_test_string_equality (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

test integer comparison (-eq, -ne, -lt, etc.)

  • ❌ Proven: Not yet proven in Lean 4 (POSIX semantics)
  • 🔗 Implemented: impl/rust-cli/src/test_command.rs:IntEqual/IntLessThan/etc
  • 🧪 Tested: prop_test_integer_transitivity (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

Logical Operators (Phase 6 M14)

&& (Logical AND)

  • ❌ Proven: Not yet proven in Lean 4 (short-circuit semantics)
  • 🔗 Implemented: impl/rust-cli/src/parser.rs:LogicalOp::And
  • 🧪 Tested: prop_logical_and_short_circuit (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

Semantics Validated:

  • ✅ Short-circuit evaluation (cmd2 doesn't run if cmd1 fails)
  • ✅ Exit code propagation
  • ✅ Proper error handling

|| (Logical OR)

  • ❌ Proven: Not yet proven in Lean 4 (short-circuit semantics)
  • 🔗 Implemented: impl/rust-cli/src/parser.rs:LogicalOp::Or
  • 🧪 Tested: prop_logical_or_short_circuit (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

Semantics Validated:

  • ✅ Short-circuit evaluation (cmd2 doesn't run if cmd1 succeeds)
  • ✅ Exit code propagation
  • ✅ Proper error handling

Quote Processing (Phase 6 M13)

Single Quote Literals

  • ❌ Proven: Not yet proven in Lean 4 (POSIX quote semantics)
  • 🔗 Implemented: impl/rust-cli/src/quotes.rs:parse_quotes
  • 🧪 Tested: prop_quote_prevents_glob (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

Quote-Aware Glob Expansion

  • ❌ Proven: Not yet proven in Lean 4
  • 🔗 Implemented: impl/rust-cli/src/parser.rs:quoted_word_to_string
  • 🧪 Tested: prop_quote_prevents_glob (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

Glob Expansion (Phase 6 M12)

Glob Determinism

  • ❌ Proven: Not yet proven in Lean 4
  • 🔗 Implemented: impl/rust-cli/src/glob.rs:expand_glob
  • 🧪 Tested: prop_glob_deterministic (1000+ iterations)
  • Status: 🔗 Implemented & Tested (proof pending)

Property Test Coverage

Total Property Tests: 30+

By Category:

  • Core filesystem: 6 tests
  • Composition/equivalence: 5 tests
  • File content: 5 tests
  • Conditionals: 5 tests
  • Logical operators: 2 tests
  • Quote processing: 1 test
  • Glob expansion: 1 test
  • Other properties: 5+ tests

Iteration Count: 1000+ per test (configurable)

Test Framework: Rust PropTest (property-based testing)

Verification Gaps

High Priority (For v1.0)

  1. Formal proofs for POSIX operations

    • test/[ commands (file tests, string tests, integer comparisons)
    • Logical operators (&&, ||) with short-circuit semantics
    • Quote processing semantics
    • Glob expansion guarantees

  2. Mechanized correspondence proofs

    • Currently: Manual documentation + property tests (85% confidence)
    • Needed: Mechanized proofs that Rust matches Lean 4 (99%+ confidence)
    • Approach: Echidna integration (planned) or manual Lean ↔ Rust proofs

  3. Complete file content operation proofs

    • Truncation reversibility proof
    • Append reversibility proof

Medium Priority

  1. Pipeline composition proofs

    • Prove that pipelines preserve reversibility
    • Prove stdio plumbing correctness

  2. Process substitution proofs

    • Prove <(cmd) and >(cmd) semantics
    • Prove cleanup guarantees

Confidence Levels

Overall Correspondence Confidence: 85%

Breakdown by Feature:

  • ✅ Core operations (mkdir, rmdir, touch, rm): 95% (proven + tested)
  • ✅ Composition/equivalence: 95% (proven + tested)
  • ⚠️ File content operations: 75% (tested, proof pending)
  • ⚠️ Conditionals: 70% (tested, proof pending)
  • ⚠️ Logical operators: 70% (tested, proof pending)
  • ⚠️ Quote processing: 65% (tested, proof pending)
  • ⚠️ Glob expansion: 65% (tested, proof pending)

Methodology:

  • Proven + Implemented + Tested = 95% confidence
  • Implemented + Tested (no proof) = 70% confidence
  • Implemented only = 50% confidence
  • No implementation = 0% confidence

Next Steps for Verification

  1. Week 1-2: Add Lean 4 proofs for test/[ operations
  2. Week 3-4: Add Lean 4 proofs for logical operators (&&, ||)
  3. Week 5-6: Add Lean 4 proofs for quote processing
  4. Week 7-8: Mechanized correspondence proofs (Echidna or manual)

Report Generated: $(date -u +"%Y-%m-%d %H:%M:%S UTC") Valence Shell Version: see CHANGELOG.md (release history) and impl/rust-cli/Cargo.toml [package].version (semver pin) Phase: M14 Complete