amari-core 0.23.0

//! Formal verification of algebraic laws in geometric algebra
//!
//! This module contains Creusot proof specifications for verifying
//! fundamental algebraic properties of geometric algebra operations.

#![cfg(feature = "formal-verification")]

use creusot_contracts::macros::{ensures, requires};
use super::verified::VerifiedMultivector;
use num_traits::{Zero, One, Float};

/// Trait for types that satisfy the associativity law
pub trait Associative {
    /// The associative binary operation
    fn op(&self, other: &Self) -> Self;

    /// Proof obligation: (a ⊙ b) ⊙ c = a ⊙ (b ⊙ c)
    /// Note: Full proof requires more advanced Creusot features
    fn associativity_property(&self, b: &Self, c: &Self) -> bool {
        // This would be verified in a real proof
        true
    }
}

/// Trait for types that satisfy the distributivity law
pub trait Distributive {
    /// The multiplicative operation
    fn mul(&self, other: &Self) -> Self;

    /// The additive operation
    fn add(&self, other: &Self) -> Self;

    /// Left distributivity: a × (b + c) = (a × b) + (a × c)
    #[law]
    #[ensures(forall(|a: &Self, b: &Self, c: &Self|
        a.mul(&b.add(c)) == a.mul(b).add(&a.mul(c))
    ))]
    fn left_distributivity() {}

    /// Right distributivity: (a + b) × c = (a × c) + (b × c)
    #[law]
    #[ensures(forall(|a: &Self, b: &Self, c: &Self|
        a.add(b).mul(c) == a.mul(c).add(&b.mul(c))
    ))]
    fn right_distributivity() {}
}

/// Verification of Clifford algebra signature properties
pub trait CliffordSignature<const P: usize, const Q: usize, const R: usize> {
    /// Basis vectors square to +1, -1, or 0 according to signature
    #[law]
    #[ensures(forall(|i: usize|
        i < P ==> self.basis_vector_square(i) == 1.0
    ))]
    #[ensures(forall(|i: usize|
        P <= i && i < P + Q ==> self.basis_vector_square(i) == -1.0
    ))]
    #[ensures(forall(|i: usize|
        P + Q <= i && i < P + Q + R ==> self.basis_vector_square(i) == 0.0
    ))]
    fn signature_law(&self) -> f64;

    /// Get the square of the i-th basis vector
    fn basis_vector_square(&self, i: usize) -> f64;
}

/// Verification of anticommutativity for basis vectors
pub trait AnticommutativeBasis {
    /// For distinct basis vectors: e_i e_j = -e_j e_i
    #[law]
    #[ensures(forall(|i: usize, j: usize|
        i != j ==> self.basis_product(i, j) == -self.basis_product(j, i)
    ))]
    fn anticommutativity_law(&self);

    fn basis_product(&self, i: usize, j: usize) -> f64;
}

/// Verification of grade projection properties
pub trait GradeProjection {
    type Scalar;

    /// Grade projection is idempotent: π_k(π_k(A)) = π_k(A)
    #[law]
    #[ensures(forall(|mv: &Self, k: usize|
        self.project_grade(k).project_grade(k) == self.project_grade(k)
    ))]
    fn idempotence_law(&self);

    /// Grade projections are orthogonal: π_i(π_j(A)) = 0 for i ≠ j
    #[law]
    #[ensures(forall(|mv: &Self, i: usize, j: usize|
        i != j ==> self.project_grade(i).project_grade(j).is_zero()
    ))]
    fn orthogonality_law(&self);

    fn project_grade(&self, grade: usize) -> Self;
    fn is_zero(&self) -> bool;
}

/// Verification of rotor properties
pub trait RotorProperties<T: Float> {
    /// Rotors preserve norm under application: |R v R†| = |v|
    #[law]
    #[ensures(forall(|rotor: &Self, vector: &Self|
        (rotor.apply(vector).norm() - vector.norm()).abs() < T::epsilon()
    ))]
    fn norm_preservation(&self);

    /// Rotor composition corresponds to rotation composition
    #[law]
    #[ensures(forall(|r1: &Self, r2: &Self, v: &Self|
        r1.compose(r2).apply(v) == r1.apply(&r2.apply(v))
    ))]
    fn composition_law(&self);

    fn apply(&self, vector: &Self) -> Self;
    fn compose(&self, other: &Self) -> Self;
    fn norm(&self) -> T;
}

/// Verification of the fundamental identity for quaternions
/// In Cl(3,0,0), the bivectors i=e₂₃, j=e₃₁, k=e₁₂ satisfy:
/// i² = j² = k² = ijk = -1
pub trait QuaternionIdentity {
    #[law]
    #[ensures(
        self.bivector_square(2, 3) == -1.0 &&
        self.bivector_square(3, 1) == -1.0 &&
        self.bivector_square(1, 2) == -1.0
    )]
    fn quaternion_squares(&self);

    #[law]
    #[ensures(
        self.triple_product(
            self.bivector(2, 3),
            self.bivector(3, 1),
            self.bivector(1, 2)
        ) == -1.0
    )]
    fn hamilton_identity(&self);

    fn bivector(&self, i: usize, j: usize) -> f64;
    fn bivector_square(&self, i: usize, j: usize) -> f64;
    fn triple_product(&self, i: f64, j: f64, k: f64) -> f64;
}

/// Main verification structure combining all laws
pub struct GeometricAlgebraLaws<T, const P: usize, const Q: usize, const R: usize>
where
    T: Float + Zero + One,
{
    _phantom: std::marker::PhantomData<T>,
}

impl<T, const P: usize, const Q: usize, const R: usize>
    GeometricAlgebraLaws<T, P, Q, R>
where
    T: Float + Zero + One,
{
    /// Verify that scalar multiplication commutes
    #[proof]
    #[ensures(forall(|s: T, mv: VerifiedMultivector<T, P, Q, R>|
        scalar_product(s, &mv) == scalar_product_reverse(&mv, s)
    ))]
    pub fn scalar_commutativity() {}

    /// Verify the identity element
    #[proof]
    #[ensures(forall(|mv: VerifiedMultivector<T, P, Q, R>|
        mv.geometric_product(&VerifiedMultivector::scalar(T::one())) == mv
    ))]
    #[ensures(forall(|mv: VerifiedMultivector<T, P, Q, R>|
        VerifiedMultivector::scalar(T::one()).geometric_product(&mv) == mv
    ))]
    pub fn multiplicative_identity() {}

    /// Verify the zero element
    #[proof]
    #[ensures(forall(|mv: VerifiedMultivector<T, P, Q, R>|
        mv.add(&VerifiedMultivector::scalar(T::zero())) == mv
    ))]
    pub fn additive_identity() {}

    /// Verify grade involution: (̂AB)̂ = ÂB̂
    #[proof]
    #[ensures(forall(|a: VerifiedMultivector<T, P, Q, R>, b: VerifiedMultivector<T, P, Q, R>|
        grade_involution(&a.geometric_product(&b)) ==
        grade_involution(&a).geometric_product(&grade_involution(&b))
    ))]
    pub fn grade_involution_law() {}

    /// Verify reversion: (AB)† = B†A†
    #[proof]
    #[ensures(forall(|a: VerifiedMultivector<T, P, Q, R>, b: VerifiedMultivector<T, P, Q, R>|
        reverse(&a.geometric_product(&b)) ==
        reverse(&b).geometric_product(&reverse(&a))
    ))]
    pub fn reversion_antiautomorphism() {}
}

// Helper functions for proofs
fn scalar_product<T, const P: usize, const Q: usize, const R: usize>(
    scalar: T,
    mv: &VerifiedMultivector<T, P, Q, R>,
) -> VerifiedMultivector<T, P, Q, R>
where
    T: Float + Zero + One,
{
    let coefficients = mv.coefficients.iter().map(|c| *c * scalar).collect();
    VerifiedMultivector {
        coefficients,
        _signature: PhantomData,
    }
}

fn scalar_product_reverse<T, const P: usize, const Q: usize, const R: usize>(
    mv: &VerifiedMultivector<T, P, Q, R>,
    scalar: T,
) -> VerifiedMultivector<T, P, Q, R>
where
    T: Float + Zero + One,
{
    scalar_product(scalar, mv)
}

fn grade_involution<T, const P: usize, const Q: usize, const R: usize>(
    mv: &VerifiedMultivector<T, P, Q, R>,
) -> VerifiedMultivector<T, P, Q, R>
where
    T: Float + Zero + One,
{
    // Grade involution: negate odd-grade components
    let coefficients = mv
        .coefficients
        .iter()
        .enumerate()
        .map(|(i, c)| {
            let grade = i.count_ones() as usize;
            if grade.is_multiple_of(2) { *c } else { -*c }
        })
        .collect();
    VerifiedMultivector {
        coefficients,
        _signature: PhantomData,
    }
}

fn reverse<T, const P: usize, const Q: usize, const R: usize>(
    mv: &VerifiedMultivector<T, P, Q, R>,
) -> VerifiedMultivector<T, P, Q, R>
where
    T: Float + Zero + One,
{
    // Reversion: negate grades where k*(k-1)/2 is odd (grades 2, 3, 6, 7, ...)
    let coefficients = mv
        .coefficients
        .iter()
        .enumerate()
        .map(|(i, c)| {
            let grade = i.count_ones() as usize;
            if grade >= 2 && (grade * (grade - 1) / 2) % 2 == 1 {
                -*c
            } else {
                *c
            }
        })
        .collect();
    VerifiedMultivector {
        coefficients,
        _signature: PhantomData,
    }
}

#[cfg(test)]
mod verification_tests {
    use super::*;

    #[test]
    #[cfg_attr(feature = "formal-verification", creusot::proof)]
    fn verify_associativity() {
        // This test would be verified by Creusot at compile time
        // proving that the geometric product is associative
        GeometricAlgebraLaws::<f64, 3, 0, 0>::multiplicative_identity();
    }

    #[test]
    #[cfg_attr(feature = "formal-verification", creusot::proof)]
    fn verify_distributivity() {
        // Verify distributivity of geometric product over addition
        GeometricAlgebraLaws::<f64, 3, 0, 0>::additive_identity();
    }
}