# Formal Verification Strategy for Amari Mathematical Library
## Overview
This document outlines the systematic approach to formally verify the mathematical correctness of the Amari library using:
1. **Phantom Types** - Encode mathematical invariants at the type level
2. **Creusot** - Deductive verification framework for Rust
3. **Refinement Types** - Express precise mathematical properties
## Goals
- **Prove mathematical invariants** at compile time
- **Prevent invalid operations** through type-level constraints
- **Verify algebraic laws** (associativity, distributivity, etc.)
- **Ensure dimensional consistency** in geometric operations
- **Guarantee numerical stability** properties
## Verification Roadmap
### Phase 1: Foundation (amari-core)
- [ ] Phantom types for Clifford algebra signatures (p,q,r)
- [ ] Invariants for multivector grade consistency
- [ ] Geometric product laws verification
- [ ] Basis blade orthogonality constraints
### Phase 2: Algebraic Structures (amari-tropical, amari-dual)
- [ ] Semiring laws for tropical arithmetic
- [ ] Dual number algebraic properties
- [ ] Automatic differentiation correctness
### Phase 3: Geometric Invariants (amari-enumerative)
- [ ] Dimension bounds for varieties
- [ ] Intersection theory constraints
- [ ] Degree calculations correctness
- [ ] Moduli space stability conditions
### Phase 4: Information Geometry (amari-info-geom)
- [ ] Manifold tangent space properties
- [ ] Metric tensor positive definiteness
- [ ] Connection compatibility conditions
## Phantom Type Design Patterns
### 1. Dimension Tracking
```rust
use std::marker::PhantomData;
// Phantom types for compile-time dimension checking
struct Dim<const N: usize>;
pub struct Vector<T, const D: usize> {
data: Vec<T>,
_phantom: PhantomData<Dim<D>>,
}
impl<T, const D: usize> Vector<T, D> {
// Only allows addition of same-dimension vectors
pub fn add(self, other: Vector<T, D>) -> Vector<T, D> {
// Implementation
}
}
```
### 2. Algebraic Signature Encoding
```rust
// Encode Clifford algebra signature at type level
pub struct Signature<const P: usize, const Q: usize, const R: usize>;
pub struct Multivector<T, const P: usize, const Q: usize, const R: usize> {
coefficients: Vec<T>,
_signature: PhantomData<Signature<P, Q, R>>,
}
```
### 3. Grade Constraints
```rust
// Phantom type for multivector grade
pub struct Grade<const G: usize>;
pub struct KVector<T, const G: usize, Sig> {
data: Vec<T>,
_grade: PhantomData<Grade<G>>,
_sig: PhantomData<Sig>,
}
```
## Creusot Annotations
### Pre/Post Conditions
```rust
use creusot_contracts::*;
#[requires(v1.len() == v2.len())]
#[ensures(result.len() == v1.len())]
pub fn vector_add(v1: &Vec<f64>, v2: &Vec<f64>) -> Vec<f64> {
// Implementation
}
```
### Loop Invariants
```rust
#[invariant(i <= n)]
#[invariant(result == factorial(i))]
for i in 1..=n {
result *= i;
}
```
### Mathematical Properties
```rust
// Verify associativity
#[ensures(
geometric_product(a, geometric_product(b, c)) ==
geometric_product(geometric_product(a, b), c)
)]
pub fn geometric_product(a: &Multivector, b: &Multivector) -> Multivector {
// Implementation
}
```
## Verification Targets
### amari-core
1. **Clifford Algebra Laws**
- Anticommutativity of basis vectors
- Geometric product associativity
- Distributivity over addition
2. **Rotor Properties**
- Unit norm preservation
- Composition as rotation composition
- Inverse existence
3. **Grade Projection**
- Grade preservation under projection
- Orthogonality of different grades
### amari-tropical
1. **Tropical Semiring**
- Associativity of ⊕ (min/max)
- Associativity of ⊗ (addition)
- Distributivity of ⊗ over ⊕
2. **Tropical Polynomial Properties**
- Newton polygon correspondence
- Tropical root preservation
### amari-enumerative
1. **Intersection Theory**
- Bézout's theorem bounds
- Dimension formula correctness
- Degree calculations
2. **Moduli Spaces**
- Stability conditions (2g-2+n > 0)
- Dimension formulas
- Compactification properties
## Implementation Strategy
### Step 1: Core Type Definitions
Start with amari-core, adding phantom types to existing structures without breaking the API.
### Step 2: Creusot Setup
1. Install Creusot: `cargo install --git https://github.com/xldenis/creusot`
2. Add creusot-contracts dependency
3. Configure verification targets
### Step 3: Incremental Verification
- Begin with simple invariants (dimension checks)
- Progress to algebraic laws
- Finally tackle complex geometric properties
### Step 4: Documentation
- Document all phantom type invariants
- Explain verification conditions
- Provide proof sketches for complex properties
## Example: Verified Multivector
```rust
use std::marker::PhantomData;
use creusot_contracts::*;
/// Phantom type encoding Clifford algebra signature
pub struct CliffordSignature<const P: usize, const Q: usize, const R: usize>;
/// Type-level grade marker
pub struct Grade<const G: usize>;
/// Verified multivector with compile-time signature and runtime contracts
pub struct VerifiedMultivector<T, const P: usize, const Q: usize, const R: usize> {
/// Coefficients in lexicographic basis order
coefficients: Vec<T>,
/// Phantom data for signature
_signature: PhantomData<CliffordSignature<P, Q, R>>,
}
impl<T: Field, const P: usize, const Q: usize, const R: usize>
VerifiedMultivector<T, P, Q, R>
{
const BASIS_SIZE: usize = 1 << (P + Q + R);
#[requires(coefficients.len() == Self::BASIS_SIZE)]
pub fn new(coefficients: Vec<T>) -> Self {
Self {
coefficients,
_signature: PhantomData,
}
}
#[ensures(result.grade() <= P + Q + R)]
pub fn grade(&self) -> usize {
// Compute highest non-zero grade
}
/// Geometric product with verified associativity
#[ensures(
// Associativity property
forall<a: Self, b: Self, c: Self>
a.geometric_product(&b.geometric_product(&c)) ==
a.geometric_product(&b).geometric_product(&c)
)]
pub fn geometric_product(&self, other: &Self) -> Self {
// Implementation with Cayley table
}
/// Grade projection with verified idempotence
#[requires(grade <= P + Q + R)]
#[ensures(result.project_grade(grade).project_grade(grade) == result.project_grade(grade))]
pub fn project_grade(&self, grade: usize) -> Self {
// Extract k-vector component
}
}
/// Type-safe rotor with unit norm guarantee
pub struct VerifiedRotor<T, const P: usize, const Q: usize, const R: usize> {
multivector: VerifiedMultivector<T, P, Q, R>,
_phantom: PhantomData<()>,
}
impl<T: Field, const P: usize, const Q: usize, const R: usize>
VerifiedRotor<T, P, Q, R>
{
#[requires(mv.norm() - 1.0 < EPSILON)]
#[requires(mv.is_even_grade())]
pub fn new(mv: VerifiedMultivector<T, P, Q, R>) -> Result<Self, Error> {
// Verify rotor conditions
}
#[ensures(result.norm() == 1.0)]
pub fn compose(&self, other: &Self) -> Self {
// Rotor composition preserves unit norm
}
}
```
## Testing Strategy
1. **Property-based testing** with quickcheck
2. **Symbolic verification** for small dimensions
3. **Numerical validation** against known results
4. **Counterexample generation** for failed proofs
## Success Metrics
- [ ] 100% of dimension checks verified at compile time
- [ ] Core algebraic laws proven with Creusot
- [ ] Zero runtime panics from mathematical violations
- [ ] API maintains ergonomics while adding safety
## References
- [Creusot: A Foundry for the Deductive Verification of Rust Programs](https://github.com/xldenis/creusot)
- [Phantom Types in Rust](https://doc.rust-lang.org/nomicon/phantom-data.html)
- [Geometric Algebra for Computer Science](https://geometricalgebra.org)
- [Formal Methods for Cryptographic Protocol Analysis](https://www.di.ens.fr/~blanchet/publications/)