Type checking for the lambda-Pi calculus modulo rewriting.

This is the library underlying the Kontroli proof checker.

Usage

Users communicate with Kontroli using commands. A command either introduces of a new name (by declaration, definition, or theorem), or adds a rewrite rule. The state of a Kontroli typechecking session consists of a [Symbols] table, keeping track of all previously introduced names, and a global context [GCtx], recording types and rewrite rules attached to symbols.

How is a user command processed? A command is parsed from a string to yield a [parse::Command]. Once we have a command, we distinguish whether it introduces a name or adds a rewrite rule: In case of a rewrite rule, we add the rewrite rule to the global context. In case of a name introduction, we first update the [Symbols] table with the newly introduced name and verify that the given types and terms are valid, yielding a [Typing]. In both cases, share refines terms in a command to shared terms, verifying whether the names referenced in the terms, have been previously declared in the [Symbols] table. Once we have a typing, we add it to the global context.

The following example parses a few commands and executes them on a global context. (By the way, this example, just as all other code examples in this library, can be executed by running cargo test.)

# use kontroli::{Command, Error, Share, Symbols};
# use kontroli::kernel::{self, GCtx};
# use colosseum::unsync::Arena;
let cmds = r#"
(; declarations ;)
prop : Type.
imp : prop -> prop -> prop.

(; definition with a rewrite rule ;)
def proof : prop -> Type.
[x: prop, y: prop] proof (imp x y) --> proof x -> proof y.

(; theorem ;)
thm imp_refl (x : prop) : proof (imp x x) := p : proof x => p.
"#;

let arena = Arena::new();
let mut syms = Symbols::new();
let mut gc = GCtx::new();

for cmd in kontroli::parse::Strict::new(cmds) {
// match constants in the command to previously introduced constants
let cmd = cmd.unwrap().share(&syms)?;
match cmd {
// introduction of a new name
Command::Intro(id, it) => {
let owned = kontroli::symbol::Owned::new(id.clone());
// add symbol to symbol table and fail if it is not new
let sym = syms.insert(id, arena.alloc(owned))?;

// typecheck and insert into global context
let rewritable = it.rewritable();
let (typing, check) = kernel::intro(it, &gc)?;
if let Some(check) = check {
check.check(&gc)?
}
gc.insert(sym, typing, rewritable)?
}
// addition of rewrite rules (without typechecking them!)
Command::Rules(rules) => rules.into_iter().try_for_each(|r| gc.add_rule(r))?,
}
}
# Ok::<_, Error>(())

Organisation

This library is divided into several modules:

• The [parse] module contains unshared, reference-free data structures,
• the [share] module contains data structures with references, and
• the [kernel] module contains data structures with references and shared pointers.

For many data structures, we have counterparts in the [parse], [share], and [kernel] modules. We call types from the [parse] and [share] modules "parsed structures" and "shared structures", respectively. For example, we distinguish parsed terms, shared terms, and (kernel) terms (the latter being defined in the [kernel] module). Parsed structures are constructed by the parser and refined into their corresponding shared structures by the sharer. Parsed and shared structures also implement the Send and Sync traits, meaning that they can be transferred and shared between threads. This allows parsing and checking to be performed in parallel.