circuit-cas/src/poly/tests.rs

use super::buchberger::{groebner_basis, is_groebner_basis};
use super::flat::{Mono, Poly, lex_cmp};
use super::ideal::{Generators, GroebnerBasis, Ideal};
use super::var::StaticVar;

#[test]
fn test_mono_contains() {
    let a: Mono<StaticVar> = [("x", 2), ("y", 1)].into();

    // Lower exponent of same variable is contained
    assert!(a.contains(&Mono::from([("x", 1)])));

    // Higher exponent of same variable is not contained
    assert!(!a.contains(&Mono::from([("x", 3)])));

    // Identical monomial is contained
    assert!(a.contains(&Mono::from([("x", 2), ("y", 1)])));

    // Variable absent from self is not contained
    assert!(!a.contains(&Mono::from([("x", 2), ("z", 1)])));

    // Subset of variables with lower exponents is contained
    assert!(a.contains(&Mono::from([("x", 1), ("y", 1)])));

    // Single variable with exact exponent is contained
    assert!(a.contains(&Mono::from([("x", 2)])));

    // Insufficient exponent in self means not contained
    assert!(!Mono::<StaticVar>::from([("x", 1)]).contains(&Mono::from([("x", 2)])));

    // Missing variable in self means not contained
    assert!(!Mono::<StaticVar>::from([("x", 1), ("y", 1)]).contains(&Mono::from([("x", 2)])));
    assert!(!Mono::<StaticVar>::from([("x", 1)]).contains(&Mono::from([("x", 1), ("y", 1)])));
}

#[test]
fn test_mono_mul() {
    // Same variable: exponents add
    let a: Mono<StaticVar> = [("x", 2)].into();
    let b: Mono<StaticVar> = [("x", 3)].into();
    assert_eq!(a * b, Mono::from([("x", 5)]));

    // Disjoint variables: both appear in result
    let a: Mono<StaticVar> = [("x", 2)].into();
    let b: Mono<StaticVar> = [("y", 3)].into();
    assert_eq!(a * b, Mono::from([("x", 2), ("y", 3)]));

    // Mixed: shared and disjoint variables
    let a: Mono<StaticVar> = [("x", 1), ("y", 2)].into();
    let b: Mono<StaticVar> = [("y", 1), ("z", 3)].into();
    assert_eq!(a * b, Mono::from([("x", 1), ("y", 3), ("z", 3)]));

    // Commutativity
    let a: Mono<StaticVar> = [("x", 2), ("z", 1)].into();
    let b: Mono<StaticVar> = [("y", 3)].into();
    assert_eq!(a.clone() * b.clone(), b * a);

    // Multiply by constant monomial (empty term vec = 1)
    let a: Mono<StaticVar> = [("x", 4)].into();
    let one: Mono<StaticVar> = Mono { term: vec![] };
    assert_eq!(a.clone() * one, a);
}

#[test]
fn test_poly_add() {
    // Distinct monomials are collected as separate terms
    let a: Poly<StaticVar> = [(1, [("x", 2)]), (2, [("y", 1)])].into();
    let b: Poly<StaticVar> = [(3, [("z", 1)])].into();
    let expected: Poly<StaticVar> = [(1, [("x", 2)]), (2, [("y", 1)]), (3, [("z", 1)])].into();
    assert_eq!(a + b, expected);

    // Coefficients of matching monomials are summed
    let a: Poly<StaticVar> = [(2, [("x", 1)])].into();
    let b: Poly<StaticVar> = [(3, [("x", 1)])].into();
    let expected: Poly<StaticVar> = [(5, [("x", 1)])].into();
    assert_eq!(a + b, expected);

    // Terms that cancel sum to zero are dropped
    let a: Poly<StaticVar> = [(1, [("x", 1)])].into();
    let b: Poly<StaticVar> = [(-1, [("x", 1)])].into();
    let expected: Poly<StaticVar> = Poly::default();
    assert_eq!(a + b, expected);
}

#[test]
fn test_poly_sub() {
    // Distinct monomials are collected as separate terms with negated rhs coefficients
    let a: Poly<StaticVar> = [(3, [("x", 2)])].into();
    let b: Poly<StaticVar> = [(1, [("y", 1)])].into();
    let expected: Poly<StaticVar> = [(3, [("x", 2)]), (-1, [("y", 1)])].into();
    assert_eq!(a - b, expected);

    // Coefficients of matching monomials are subtracted
    let a: Poly<StaticVar> = [(5, [("x", 1)])].into();
    let b: Poly<StaticVar> = [(3, [("x", 1)])].into();
    let expected: Poly<StaticVar> = [(2, [("x", 1)])].into();
    assert_eq!(a - b, expected);

    // Subtracting equal polynomials yields zero
    let a: Poly<StaticVar> = [(4, [("x", 2)]), (1, [("y", 1)])].into();
    let b: Poly<StaticVar> = [(4, [("x", 2)]), (1, [("y", 1)])].into();
    assert_eq!(a - b, Poly::default());
}

#[test]
fn test_lex_cmp() {
    use std::cmp::Ordering;

    let x2: Mono<StaticVar> = [("x", 2)].into();
    let xy: Mono<StaticVar> = [("x", 1), ("y", 1)].into();
    let y2: Mono<StaticVar> = [("y", 2)].into();
    let x: Mono<StaticVar> = [("x", 1)].into();
    let one: Mono<StaticVar> = Mono { term: vec![] };

    // x² > xy (x exponent 2 vs 1)
    assert_eq!(lex_cmp(&x2, &xy), Ordering::Greater);
    // x > y² (x has higher priority)
    assert_eq!(lex_cmp(&x, &y2), Ordering::Greater);
    // xy > y² (x present in xy but not y²)
    assert_eq!(lex_cmp(&xy, &y2), Ordering::Greater);
    // 1 < x
    assert_eq!(lex_cmp(&one, &x), Ordering::Less);
    // reflexive
    assert_eq!(lex_cmp(&x2, &x2), Ordering::Equal);
}

#[test]
fn test_mono_div() {
    // x² / x = x
    let a: Mono<StaticVar> = [("x", 2)].into();
    let b: Mono<StaticVar> = [("x", 1)].into();
    assert_eq!(a.div(&b), Mono::from([("x", 1)]));

    // x²y / xy = x
    let a: Mono<StaticVar> = [("x", 2), ("y", 1)].into();
    let b: Mono<StaticVar> = [("x", 1), ("y", 1)].into();
    assert_eq!(a.div(&b), Mono::from([("x", 1)]));

    // x²y / y = x²
    let a: Mono<StaticVar> = [("x", 2), ("y", 1)].into();
    let b: Mono<StaticVar> = [("y", 1)].into();
    assert_eq!(a.div(&b), Mono::from([("x", 2)]));

    // x / x = 1
    let a: Mono<StaticVar> = [("x", 1)].into();
    let b: Mono<StaticVar> = [("x", 1)].into();
    assert_eq!(a.div(&b), Mono { term: vec![] });
}

#[test]
fn test_mono_lcm() {
    // lcm(x², xy) = x²y
    let a: Mono<StaticVar> = [("x", 2)].into();
    let b: Mono<StaticVar> = [("x", 1), ("y", 1)].into();
    assert_eq!(a.lcm(&b), Mono::from([("x", 2), ("y", 1)]));

    // lcm(x, y) = xy  (disjoint)
    let a: Mono<StaticVar> = [("x", 1)].into();
    let b: Mono<StaticVar> = [("y", 1)].into();
    assert_eq!(a.lcm(&b), Mono::from([("x", 1), ("y", 1)]));

    // lcm(x², x²) = x²  (identical)
    let a: Mono<StaticVar> = [("x", 2)].into();
    assert_eq!(a.lcm(&a.clone()), a);
}

#[test]
fn test_s_poly() {
    // S(x², xy) = 0: S-poly of two monomials always vanishes
    let f: Poly<StaticVar> = [(1, [("x", 2)])].into();
    let g: Poly<StaticVar> = [(1, [("x", 1), ("y", 1)])].into();
    assert!(f.s_poly(&g).is_zero());

    // f = x² + y, g = xy + z  (lex: LM(f)=x², LM(g)=xy)
    // L = x²y, t_f = y, t_g = x, d = gcd(1,1) = 1
    // S = y*(x²+y) - x*(xy+z) = x²y + y² - x²y - xz = y² - xz
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32)])),
        (1i32, Mono::from([("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 1u32), ("y", 1u32)])),
        (1i32, Mono::from([("z", 1u32)])),
    ]
    .into_iter()
    .collect();
    let expected: Poly<StaticVar> = [
        (1i32, Mono::from([("y", 2u32)])),
        (-1i32, Mono::from([("x", 1u32), ("z", 1u32)])),
    ]
    .into_iter()
    .collect();
    assert_eq!(f.s_poly(&g), expected);

    // f = 2x + y, g = 3x + z  (same LM=x, d=gcd(2,3)=1)
    // S = (3/1)*f - (2/1)*g = 3(2x+y) - 2(3x+z) = 3y - 2z
    let f: Poly<StaticVar> = [(2, [("x", 1)]), (1, [("y", 1)])].into();
    let g: Poly<StaticVar> = [(3, [("x", 1)]), (1, [("z", 1)])].into();
    let expected: Poly<StaticVar> = [(3, [("y", 1)]), (-2, [("z", 1)])].into();
    assert_eq!(f.s_poly(&g), expected);

    // f = 4x, g = 6x  (d = gcd(4,6) = 2)
    // S = (6/2)*x*4x - (4/2)*x*6x = 3*4x - 2*6x = 12x - 12x = 0
    let f: Poly<StaticVar> = [(4, [("x", 1)])].into();
    let g: Poly<StaticVar> = [(6, [("x", 1)])].into();
    assert!(f.s_poly(&g).is_zero());
}

fn make_const_poly(c: i32) -> Poly<StaticVar> {
    Poly {
        mono: [(Mono { term: vec![] }, c)].into_iter().collect(),
    }
}

fn verify_div_rem(
    f: Poly<StaticVar>,
    g: &Poly<StaticVar>,
    d: u32,
    q: Poly<StaticVar>,
    r: Poly<StaticVar>,
) {
    // lc(g)^d * f == q * g + r
    let (_, lc_g) = g.leading_term_lex().unwrap();
    let lhs = lc_g.pow(d) * f;
    let rhs = q * g.clone() + r;
    assert_eq!(lhs, rhs);
}

#[test]
fn test_div_rem() {
    // x³ / x² = x, r=0, d=0
    let f: Poly<StaticVar> = [(1, [("x", 3)])].into();
    let g: Poly<StaticVar> = [(1, [("x", 2)])].into();
    let expected_q: Poly<StaticVar> = [(1, [("x", 1)])].into();
    let (d, q, r) = f.clone().div_rem(&g);
    assert_eq!(q, expected_q);
    assert!(r.is_zero());
    verify_div_rem(f, &g, d, q, r);

    // (x³ + x²y) / x² = x + y, r=0
    // f = x²(x + y), g = x²
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 3u32)])),
        (1i32, Mono::from([("x", 2u32), ("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [(1, [("x", 2)])].into();
    let expected_q: Poly<StaticVar> = [(1, [("x", 1)]), (1, [("y", 1)])].into();
    let (d, q, r) = f.clone().div_rem(&g);
    assert_eq!(q, expected_q);
    assert!(r.is_zero());
    verify_div_rem(f, &g, d, q, r);

    // (x³ + y) / x² = x, r=y
    // LT(x³) divisible by x², LT(y) is not
    let f: Poly<StaticVar> = [(1, [("x", 3)]), (1, [("y", 1)])].into();
    let g: Poly<StaticVar> = [(1, [("x", 2)])].into();
    let expected_q: Poly<StaticVar> = [(1, [("x", 1)])].into();
    let expected_r: Poly<StaticVar> = [(1, [("y", 1)])].into();
    let (d, q, r) = f.clone().div_rem(&g);
    assert_eq!(q, expected_q);
    assert_eq!(r, expected_r);
    verify_div_rem(f, &g, d, q, r);

    // 3x² / (2x): 2 ∤ 3, needs pseudo-division
    // 2¹ · 3x² = 3x · 2x  =>  d=1, q=3x, r=0
    let f: Poly<StaticVar> = [(3, [("x", 2)])].into();
    let g: Poly<StaticVar> = [(2, [("x", 1)])].into();
    let expected_q: Poly<StaticVar> = [(3, [("x", 1)])].into();
    let (d, q, r) = f.clone().div_rem(&g);
    assert_eq!(d, 1);
    assert_eq!(q, expected_q);
    assert!(r.is_zero());
    verify_div_rem(f, &g, d, q, r);

    // 6xy / 2 = 3xy, r=0, d=0
    let f: Poly<StaticVar> = [(6, [("x", 1), ("y", 1)])].into();
    let g = make_const_poly(2);
    let expected_q: Poly<StaticVar> = [(3, [("x", 1), ("y", 1)])].into();
    let (d, q, r) = f.clone().div_rem(&g);
    assert_eq!(d, 0);
    assert_eq!(q, expected_q);
    assert!(r.is_zero());
    verify_div_rem(f, &g, d, q, r);

    // (x² + xy) / (x + y) = x, r=0
    // f = x(x + y), g = x + y
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32)])),
        (1i32, Mono::from([("x", 1u32), ("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [(1, [("x", 1)]), (1, [("y", 1)])].into();
    let expected_q: Poly<StaticVar> = [(1, [("x", 1)])].into();
    let (d, q, r) = f.clone().div_rem(&g);
    assert_eq!(q, expected_q);
    assert!(r.is_zero());
    verify_div_rem(f, &g, d, q, r);
}

#[test]
fn test_groebner() {
    // Ideal (x²) — already a Gröbner basis
    let f: Poly<StaticVar> = [(1, [("x", 2)])].into();
    let _basis = groebner_basis(vec![f]);

    // Ideal (x³ - x², x² - x): gcd is x² - x
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 3u32)])),
        (-1i32, Mono::from([("x", 2u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32)])),
        (-1i32, Mono::from([("x", 1u32)])),
    ]
    .into_iter()
    .collect();
    let basis = groebner_basis(vec![f, g]);
    assert!(is_groebner_basis(&basis));

    // Classic example: I = (x²y - x, xy² - y)
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32), ("y", 1u32)])),
        (-1i32, Mono::from([("x", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 1u32), ("y", 2u32)])),
        (-1i32, Mono::from([("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let basis = groebner_basis(vec![f, g]);
    assert!(is_groebner_basis(&basis));
}

#[test]
fn test_is_groebner_basis() {
    // {x} is a GB: only one element, no pairs to check.
    let f: Poly<StaticVar> = [(1, [("x", 1)])].into();
    assert!(is_groebner_basis(&[f]));

    // {x, y} is a GB: S(x, y) = y*x - x*y = 0.
    let x: Poly<StaticVar> = [(1, [("x", 1)])].into();
    let y: Poly<StaticVar> = [(1, [("y", 1)])].into();
    assert!(is_groebner_basis(&[x, y]));

    // {x² + y, xy} is NOT a GB:
    // S(x²+y, xy) = y*(x²+y) - x*(xy) = y² ≠ 0 mod the set.
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32)])),
        (1i32, Mono::from([("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [(1i32, Mono::from([("x", 1u32), ("y", 1u32)]))]
        .into_iter()
        .collect();
    assert!(!is_groebner_basis(&[f, g]));

    // After running groebner_basis, the result must always pass.
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32)])),
        (1i32, Mono::from([("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [(1i32, Mono::from([("x", 1u32), ("y", 1u32)]))]
        .into_iter()
        .collect();
    let basis = groebner_basis(vec![f, g]);
    assert!(is_groebner_basis(&basis));
}

#[test]
fn test_ideal() {
    // Construction from Vec and iterator
    let f: Poly<StaticVar> = [(1, [("x", 2)])].into();
    let g: Poly<StaticVar> = [(1, [("y", 1)])].into();
    let ideal = Ideal::new(vec![f.clone(), g.clone()]);
    assert_eq!(ideal.generators().len(), 2);
    let ideal: Ideal<StaticVar, Generators> = [f.clone(), g.clone()].into_iter().collect();
    assert_eq!(ideal.generators().len(), 2);

    // Construction via From / .into()
    let ideal: Ideal<StaticVar, Generators> = vec![f.clone(), g.clone()].into();
    assert_eq!(ideal.generators().len(), 2);

    // Display: <x², y>
    let ideal = Ideal::new(vec![f, g]);
    assert_eq!(ideal.to_string(), "<x\u{00B2}, y>");

    // groebner_basis transitions state and result satisfies the criterion
    let f: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32), ("y", 1u32)])),
        (-1i32, Mono::from([("x", 1u32)])),
    ]
    .into_iter()
    .collect();
    let g: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 1u32), ("y", 2u32)])),
        (-1i32, Mono::from([("y", 1u32)])),
    ]
    .into_iter()
    .collect();
    let ideal: Ideal<StaticVar, Generators> = [f, g].into_iter().collect();
    let gb: Ideal<StaticVar, GroebnerBasis> = ideal.groebner_basis();
    assert!(is_groebner_basis(gb.generators()));
}

#[test]
fn test_groebner_sagemath() {
    // I = (x³ - 2xy, x²y - 2y² + x) ⊆ k[x, y]
    // grobner basis: {4y³, x - 2y²}

    // f1 = x³ - 2xy
    let f1: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 3u32)])),
        (-2i32, Mono::from([("x", 1u32), ("y", 1u32)])),
    ]
    .into_iter()
    .collect();

    // f2 = x²y - 2y² + x
    let f2: Poly<StaticVar> = [
        (1i32, Mono::from([("x", 2u32), ("y", 1u32)])),
        (-2i32, Mono::from([("y", 2u32)])),
        (1i32, Mono::from([("x", 1u32)])),
    ]
    .into_iter()
    .collect();

    let gb = Ideal::new(vec![f1.clone(), f2.clone()]).groebner_basis();

    assert!(is_groebner_basis(gb.generators()));
    assert_eq!(gb.generators().len(), 2);

    // -x + 2y²
    let neg_x_plus_2y2: Poly<StaticVar> = [
        (-1i32, Mono::from([("x", 1u32)])),
        (2i32, Mono::from([("y", 2u32)])),
    ]
    .into_iter()
    .collect();

    // -4y³
    let neg_4y3: Poly<StaticVar> = [(-4i32, Mono::from([("y", 3u32)]))].into_iter().collect();

    let expected = [neg_x_plus_2y2, neg_4y3];
    for e in &expected {
        assert!(
            gb.generators()
                .iter()
                .any(|a| *a == *e || *a == -1i32 * e.clone())
        );
    }
}