# circuit-cas

A computer algebra system in Rust for algebraic circuit verification over polynomial rings with integer coefficients.

## Overview

`circuit-cas` provides two main layers:

- **`poly`** — multivariate polynomials over ℤ, Gröbner bases, and ideals
- **`circuit`** — arithmetic circuits (DAGs of add/mul nodes) quotiented by an ideal

The intended use case is verifying arithmetic circuits modulo an ideal: two circuits computing the same polynomial in the quotient ring are equivalent.

## Modules

### `poly`

#### Variables — `poly::var`

Variables are any type implementing the `Var` trait. The built-in `StaticVar` carries a static string name and optional integer indices, with a `var!` macro for construction:

```rust
use circuit_cas::var;

let x   = var!("x");          // x
let x0  = var!("x", 0);       // x₀
let x01 = var!("x", 0, 1);    // x₀,₁
```

#### Polynomials — `poly::flat`

`Poly<V>` is a multivariate polynomial over ℤ backed by a `HashMap<Mono<V>, i32>`. Arithmetic operators (`+`, `-`, `*`) and scalar multiplication are provided. Monomial ordering uses pure lexicographic order (lex) with variable names sorted alphabetically.

```rust
// x² - 2xy  (using array-of-pairs syntax)
let f: Poly<StaticVar> = [
    (1i32,  Mono::from([("x", 2u32)])),
    (-2i32, Mono::from([("x", 1u32), ("y", 1u32)])),
].into_iter().collect();
```

Key operations:

| Method | Description |
|--------|-------------|
| `Poly::is_zero` | whether the polynomial is the zero polynomial |
| `Poly::leading_term_lex` | leading `(monomial, coefficient)` under lex |
| `Poly::s_poly(&other)` | S-polynomial of two polynomials |
| `Poly::div_rem(divisor)` | pseudo-division: returns `(d, q, r)` with `lc(g)^d·f = q·g + r` |

#### Ideals — `poly::ideal`

`Ideal<V, S>` uses the typestate pattern to track at the type level whether its generators form a Gröbner basis:

```rust
use circuit_cas::poly::ideal::{Ideal, Generators, GroebnerBasis};

// Arbitrary generators
let ideal: Ideal<StaticVar, Generators> = Ideal::new(vec![f1, f2]);

// Compute the reduced Gröbner basis — consumes the original ideal
let gb: Ideal<StaticVar, GroebnerBasis> = ideal.groebner_basis();

// Ideal membership test (only available on GroebnerBasis state)
assert!(gb.contains(&some_poly));
```

`Ideal` implements `Display` as `<g1, g2, ...>` and `FromIterator<Poly<V>>` for construction from an iterator.

#### Buchberger's algorithm — `poly::buchberger`

```rust
use circuit_cas::poly::buchberger::{groebner_basis, is_groebner_basis};

let basis = groebner_basis(vec![f1, f2]);
assert!(is_groebner_basis(&basis));
```

`groebner_basis` returns the **reduced** Gröbner basis: after running Buchberger's algorithm it minimizes (removes redundant generators) and interreduces (fully reduces each generator modulo the others).

### `circuit`

#### DAG — `circuit::dag`

`Circuit<V>` is a DAG of `Node<V>` values, with structural sharing via interning. Nodes are `Leaf(V)`, `Scale(id, i32)`, `Sum(id, id)`, or `Prod(id, id)`.

#### Quotient ring — `circuit::quotient`

`Quotient<V>` pairs an arithmetic circuit with a `Ideal<V, GroebnerBasis>`, representing a circuit in the quotient ring `ℤ[vars] / I`.

```rust
use circuit_cas::circuit::quotient::Quotient;

// Build from a list of generators — Gröbner basis is computed automatically
let quotient: Quotient<StaticVar> = vec![f1, f2, f3].into_iter().collect();

// Or from a pre-computed Gröbner basis
let quotient = Quotient::from(gb);

println!("{quotient}"); // C/<g1, g2>
```

## Example

```rust
use circuit_cas::{var, circuit::quotient::Quotient, poly::var::StaticVar};

fn main() {
    let x  = var!("x");
    let xb = var!("x\u{0304}");  // x̄

    // Idempotency relations: x² = x, x̄² = x̄, x·x̄ = x
    let idem = vec![
        1 * (&x  ^ 2) - 1 * (&x  ^ 1),
        1 * (&xb ^ 2) - 1 * (&xb ^ 1),
        1 * ((&x ^ 1) * (&xb ^ 1)) - 1 * (&x ^ 1),
    ];

    let quotient: Quotient<StaticVar> = idem.into_iter().collect();
    println!("{quotient}");
}
```

Run with:

```
cargo run --example quotient
```

## Development

```
cargo test     # run all tests
```