PHASE_7_PROOF_TOOLS.ipynb

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    "# PHASE 7: Proof Tool Integration\n",
    "\n",
    "**Implements SEC-006: Replace proof tool stubs with real execution**\n",
    "\n",
    "## Objective\n",
    "\n",
    "Integrate actual proof verifiers (Agda, Ada/SPARK, Lean 4, Z3) instead of stub predicates.\n",
    "\n",
    "| Tool | Language | Purpose | Status |\n",
    "|------|----------|---------|--------|\n",
    "| **Agda** | Proof Assistant | Type checking, invariant proofs | PHASE 7.1 |\n",
    "| **Ada/SPARK** | Formal Methods | Loop & invariant verification | PHASE 7.2 |\n",
    "| **Lean 4** | Proof Language | Curry-Howard isomorphism | PHASE 7.3 |\n",
    "| **Z3** | SMT Solver | Symbolic execution | PHASE 7.4 |\n"
   ]
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   "source": [
    "## PHASE 7.1: Agda Integration\n",
    "\n",
    "### Task: Invoke Agda for type checking\n",
    "\n",
    "**File:** `crates/proof-validator/src/adapters/agda_adapter.rs`\n",
    "\n",
    "```rust\n",
    "pub struct AgdaAdapter {\n",
    "    agda_bin: PathBuf,\n",
    "    library_path: Vec<PathBuf>,\n",
    "}\n",
    "\n",
    "impl AgdaAdapter {\n",
    "    pub async fn verify_type(&self, source: &str, expected_type: &str) -> Result<ProofStatus> {\n",
    "        // 1. Write source to temp file\n",
    "        // 2. Invoke: agda --check source.agda\n",
    "        // 3. Parse output (OK / ERROR)\n",
    "        // 4. Return ProofStatus::Proved or ProofStatus::Disproved\n",
    "    }\n",
    "}\n",
    "```\n",
    "\n",
    "**Evidence:**\n",
    "- Creates temp Agda files\n",
    "- Spawns agda process\n",
    "- Parses exit code + stderr\n",
    "- Returns deterministic result"
   ]
  },
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   "source": [
    "## PHASE 7.2: Ada/SPARK Integration\n",
    "\n",
    "### Task: SPARK verifier for loop invariants\n",
    "\n",
    "**File:** `crates/proof-validator/src/adapters/spark_adapter.rs`\n",
    "\n",
    "```rust\n",
    "pub struct SparkAdapter {\n",
    "    spark_bin: PathBuf,\n",
    "    timeout_secs: u64,\n",
    "}\n",
    "\n",
    "impl SparkAdapter {\n",
    "    pub async fn verify_loop_invariant(&self, ada_source: &str, invariant: &str) -> Result<ProofStatus> {\n",
    "        // 1. Extract loop from source\n",
    "        // 2. Annotate with --# loop_invariant pragma\n",
    "        // 3. Run: gnatprove -P project.gpr\n",
    "        // 4. Check proof results\n",
    "        // 5. Return Proved / Disproved / Timeout\n",
    "    }\n",
    "}\n",
    "```\n",
    "\n",
    "**Evidence:**\n",
    "- Parses Ada loop syntax\n",
    "- Instruments with SPARK pragmas\n",
    "- Invokes gnatprove\n",
    "- Deterministic timeout handling"
   ]
  },
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   "source": [
    "## PHASE 7.3: Lean 4 Integration\n",
    "\n",
    "### Task: Curry-Howard type checking\n",
    "\n",
    "**File:** `crates/proof-validator/src/adapters/lean4_adapter.rs`\n",
    "\n",
    "```rust\n",
    "pub struct Lean4Adapter {\n",
    "    lean_bin: PathBuf,\n",
    "    stdlib_path: PathBuf,\n",
    "}\n",
    "\n",
    "impl Lean4Adapter {\n",
    "    pub async fn verify_curry_howard(&self, lean_proof: &str, theorem: &str) -> Result<ProofStatus> {\n",
    "        // 1. Write proof to .lean file\n",
    "        // 2. Run: lean --check proof.lean\n",
    "        // 3. Parse Lean diagnostics\n",
    "        // 4. Match theorem signature\n",
    "        // 5. Return Proved or error message\n",
    "    }\n",
    "}\n",
    "```\n",
    "\n",
    "**Evidence:**\n",
    "- Validates proof term structure\n",
    "- Type-checks against theorem statement\n",
    "- Extracts proof witness\n",
    "- Returns serializable proof object"
   ]
  },
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   "source": [
    "## PHASE 7.4: Z3 SMT Solver Integration\n",
    "\n",
    "### Task: Symbolic execution for invariant extraction\n",
    "\n",
    "**File:** `crates/proof-validator/src/adapters/z3_adapter.rs`\n",
    "\n",
    "```rust\n",
    "pub struct Z3Adapter {\n",
    "    z3_ctx: z3::Context,\n",
    "}\n",
    "\n",
    "impl Z3Adapter {\n",
    "    pub fn verify_invariant(&self, formula: &str, invariant: &str) -> Result<ProofStatus> {\n",
    "        // 1. Parse formula to Z3 expression\n",
    "        // 2. Assert (NOT invariant)\n",
    "        // 3. Check satisfiability\n",
    "        // 4. If UNSAT: invariant proven\n",
    "        // 5. If SAT: return counterexample\n",
    "    }\n",
    "}\n",
    "```\n",
    "\n",
    "**Evidence:**\n",
    "- Uses z3-rs bindings\n",
    "- SMT-LIB syntax support\n",
    "- Counterexample extraction\n",
    "- Proof object construction"
   ]
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   "source": [
    "## Implementation Checklist\n",
    "\n",
    "### PHASE 7.1: Agda\n",
    "- [ ] Create `agda_adapter.rs`\n",
    "- [ ] Implement `verify_type()` method\n",
    "- [ ] Add temp file creation\n",
    "- [ ] Parse agda exit codes\n",
    "- [ ] Unit tests (3 test cases)\n",
    "- [ ] Integration test with sample proof\n",
    "\n",
    "### PHASE 7.2: Ada/SPARK\n",
    "- [ ] Create `spark_adapter.rs`\n",
    "- [ ] Implement `verify_loop_invariant()`\n",
    "- [ ] Pragma injection\n",
    "- [ ] gnatprove subprocess handling\n",
    "- [ ] Timeout + cleanup\n",
    "- [ ] Unit tests (2 test cases)\n",
    "\n",
    "### PHASE 7.3: Lean 4\n",
    "- [ ] Create `lean4_adapter.rs`\n",
    "- [ ] Implement `verify_curry_howard()`\n",
    "- [ ] .lean file writing\n",
    "- [ ] Lean diagnostics parsing\n",
    "- [ ] Proof term extraction\n",
    "- [ ] Unit tests (3 test cases)\n",
    "\n",
    "### PHASE 7.4: Z3\n",
    "- [ ] Create `z3_adapter.rs`\n",
    "- [ ] Implement `verify_invariant()`\n",
    "- [ ] SMT-LIB formula construction\n",
    "- [ ] Satisfiability checking\n",
    "- [ ] Counterexample extraction\n",
    "- [ ] Unit tests (4 test cases)"
   ]
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   "metadata": {},
   "source": [
    "## Proof Obligations\n",
    "\n",
    "### InvariantPreservation\n",
    "```\n",
    "∀ state: State.\n",
    "  invariant(state) ∧ transition(state, state') →\n",
    "  invariant(state')\n",
    "```\n",
    "\n",
    "**Verified by:** Z3 SMT solver (symbolic execution)\n",
    "\n",
    "### SemanticPreservation\n",
    "```\n",
    "∀ source: String.\n",
    "  semantics(compile(source)) = semantics(source)\n",
    "```\n",
    "\n",
    "**Verified by:** Lean 4 (Curry-Howard proof)\n",
    "\n",
    "### LoopInvariantMaintenance\n",
    "```\n",
    "∀ i: Nat.\n",
    "  loop_invariant(i) ∧ loop_condition(i) →\n",
    "  loop_invariant(i+1)\n",
    "```\n",
    "\n",
    "**Verified by:** Ada/SPARK (gnatprove)\n",
    "\n",
    "### ReceiptChainIntegrity\n",
    "```\n",
    "∀ r1, r2: Receipt.\n",
    "  r1.sequence < r2.sequence ∧\n",
    "  r2.previous_hash = r1.hash →\n",
    "  chain_valid(r1, r2)\n",
    "```\n",
    "\n",
    "**Verified by:** Agda (type checking)"
   ]
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   "source": [
    "## Release Gate Integration\n",
    "\n",
    "### Prolog Gate (logic/rules/release_ready.pl)\n",
    "\n",
    "```prolog\n",
    "% Before Phase 7: stubs always return true\n",
    "proof_verified(_, stub, true) :- !.\n",
    "\n",
    "% After Phase 7: real verifiers required\n",
    "proof_verified(Obligation, agda, true) :-\n",
    "    agda_adapter:verify_type(Obligation, _).\n",
    "\n",
    "proof_verified(Obligation, spark, true) :-\n",
    "    spark_adapter:verify_loop_invariant(Obligation, _).\n",
    "\n",
    "proof_verified(Obligation, lean4, true) :-\n",
    "    lean4_adapter:verify_curry_howard(Obligation, _).\n",
    "\n",
    "proof_verified(Obligation, z3, true) :-\n",
    "    z3_adapter:verify_invariant(Obligation, _).\n",
    "```\n",
    "\n",
    "### Release Readiness Check\n",
    "\n",
    "```prolog\n",
    "release_ready(proof_obligations_discharged) :-\n",
    "    forall(\n",
    "        proof_obligation(Obligation, Tool, _),\n",
    "        proof_verified(Obligation, Tool, true)\n",
    "    ).\n",
    "```"
   ]
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   "source": [
    "## Execution Flow\n",
    "\n",
    "### Cell Execution → Proof Validation → Receipt\n",
    "\n",
    "```\n",
    "1. Notebook cell executes (user code)\n",
    "   ↓\n",
    "2. Invariant extractor infers proof obligations\n",
    "   ↓\n",
    "3. Dispatch to appropriate verifier:\n",
    "   - Agda for type safety\n",
    "   - Ada/SPARK for loop invariants\n",
    "   - Lean 4 for semantic preservation\n",
    "   - Z3 for symbolic execution\n",
    "   ↓\n",
    "4. Verifier returns ProofStatus (Proved/Disproved/Manual/Error)\n",
    "   ↓\n",
    "5. If all proofs Proved:\n",
    "   - Create WORM-sealed receipt\n",
    "   - Link to receipt chain\n",
    "   - Mark cell as VERIFIED\n",
    "   ↓\n",
    "6. If any proof fails:\n",
    "   - Rollback cell\n",
    "   - Return error to notebook\n",
    "   - Operator must fix and retry\n",
    "```"
   ]
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   "metadata": {},
   "source": [
    "## Success Criteria\n",
    "\n",
    "✅ **PHASE 7 Complete When:**\n",
    "\n",
    "1. All 4 adapters implemented (Agda, SPARK, Lean 4, Z3)\n",
    "2. Each adapter has 2-4 unit tests (12+ total)\n",
    "3. Each adapter can be invoked from release gate\n",
    "4. Prolog predicates updated to call real verifiers\n",
    "5. End-to-end test: execute cell → verify proof → seal receipt\n",
    "6. No stubs remaining (all ProofStatus calls reach actual tools)\n",
    "7. Timeout + error handling for each tool\n",
    "8. 37+ existing tests still passing\n",
    "9. All commits pushed to GitHub\n",
    "10. Notebook artifact committed (`.ipynb` file)\n",
    "\n",
    "**Status: READY FOR IMPLEMENTATION**"
   ]
  }
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