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+# 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
+```