oxilean-std

Standard Library for the OxiLean Theorem Prover

The OxiLean standard library provides essential inductive types, type classes, and theorems that form the foundation for formal mathematics in OxiLean. It is logically a layer above the kernel: everything defined here is ultimately verified by oxilean-kernel, so bugs in the standard library can cause incorrect proofs but never unsoundness.

The library is organized into primitive types (Nat, Bool, List, Option, Result), algebraic type classes (Eq, Ord, Functor, Monad, Monoid), and core theorems covering equality, logic, arithmetic, and ordering.

416,133 SLOC -- the largest crate in the OxiLean workspace, providing comprehensive standard library coverage (1,105 source files, 7,934 tests passing).

Part of the OxiLean project -- a Lean-compatible theorem prover implemented in pure Rust.

Overview

oxilean-std covers three broad areas:

Primitive Types

Inductive definitions of the core data types used throughout OxiLean proofs:

Nat : Type 0          -- Peano natural numbers
  | zero : Nat
  | succ : Nat -> Nat

Bool : Prop
  | true  : Bool
  | false : Bool

List alpha : Type 0
  | nil  : List alpha
  | cons : alpha -> List alpha -> List alpha

Option alpha : Type 0
  | none : Option alpha
  | some : alpha -> Option alpha

Each inductive type is accompanied by its recursor (for primitive recursion) and induction principle (for structural inductive proofs).

Type Classes

Ad-hoc polymorphism via type classes mirrors the Lean 4 Std hierarchy:

Class	Description
`Eq`	Setoid equality with `refl`, `symm`, `trans`
`Ord`	Total ordering via `compare : alpha -> alpha -> Ordering`
`Functor`	Covariant mapping over a type constructor
`Monad`	Sequential computation with `pure` and `bind`
`Monoid`	Associative binary operation with an identity element
`Semigroup`	Associative binary operation
`Decidable`	Computationally decidable propositions

Core Theorems & Lemmas

Reusable lemmas grouped by subject:

eq.rs -- Eq.refl, Eq.symm, Eq.trans, congruence
logic.rs / prop.rs -- And, Or, Not, Iff, classical axioms
nat.rs -- arithmetic (add_comm, mul_assoc, zero_add, ...)
int.rs -- integer arithmetic and order
ord.rs / order.rs -- ordering properties and monotonicity lemmas

Usage

Add to your Cargo.toml:

[dependencies]
oxilean-std = "0.1.3"

Example OxiLean proof using standard library theorems:

-- Uses Nat from the standard library
theorem two_plus_three : 2 + 3 = 5 := by
  norm_num

-- Uses List from the standard library
def my_list : List Nat := [1, 2, 3]

-- Uses Eq type class instance for Nat
#check (inferInstance : Eq Nat)

Dependencies

oxilean-kernel -- expression types (Expr, Level, Name) and environment

Testing

cargo test -p oxilean-std
cargo test -p oxilean-std -- --nocapture

oxilean-std 0.1.3