Formal Verification AI

# Formal Verification AI

**Category:** Phase 3 Core - Correctness Guarantees
**Status:** Skeleton Implementation
**Dependencies:** `categorical-composition` (correctness as functoriality)

## Overview

Integrates formal verification methods with AI systems: theorem proving for correctness guarantees, interval arithmetic for certified bounds, and categorical proofs for compositional correctness.

## Capabilities

- **Theorem Proving**: Automated verification of AI properties
- **Interval Arithmetic**: Certified bounds on network outputs
- **Categorical Correctness**: Functorial preservation guarantees
- **Adversarial Robustness**: Verified defense certificates

## Core Components

1. **Theorem Prover Interface** (`theorem_proving.jl`)
   - Integration with Z3, Lean, or Coq
   - Encode neural networks as logical formulas
   - Automated proof search

2. **Interval Arithmetic** (`interval_arithmetic.jl`)
   - Interval propagation through networks
   - Certified bounds on outputs
   - Robustness verification

3. **Categorical Proofs** (`categorical_correctness.jl`)
   - Verify functor laws for compositional networks
   - Natural transformation diagrams
   - Commutativity checking

4. **Verification Examples** (`verification_examples.jl`)
   - Adversarial robustness proofs
   - Fairness guarantees
   - Safety-critical system verification

## Integration Points

- **Input from**: All Phase 3 skills (provides verification layer)
- **Output to**: `categorical-composition` (verified transformations)
- **Coordinates with**: `oriented-simplicial-networks` (topological invariants)

## Usage

```julia
using FormalVerificationAI

# Define neural network
network = SimpleNN([Dense(10, 20, relu), Dense(20, 2)])

# Verify robustness using interval arithmetic
input_interval = Interval([0.0, 0.0], [1.0, 1.0])
output_bounds = propagate_intervals(network, input_interval)

# Prove categorical correctness
F = network_to_functor(network)
@assert verify_functor_laws(F)

# Automated theorem proving
property = "∀x. ||x - x'|| < ε ⟹ ||f(x) - f(x')|| < δ"
proof = prove_property(network, property, timeout=60)
```

## References

- Katz et al. "Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks" (2017)
- Singh et al. "An Abstract Domain for Certifying Neural Networks" (POPL 2019)
- Fong & Spivak "Hypergraph Categories" (2019)

## Implementation Status

- [x] Basic interval arithmetic
- [x] Z3 interface skeleton
- [ ] Full neural network encoding
- [ ] Categorical correctness verification
- [ ] Benchmark on standard verification tasks