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 [Signature], 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]. The scoping operation then refines the parse command to a [scope::Command], verifying whether the names referenced in the parse command have been previously declared in the Symbols table. Once we have a scope 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 signature. 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]. Once we have a typing, we add it to the signature.

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

# use kontroli::Error;
# use kontroli::scope::{Command, Symbols};
# use kontroli::rc::{Intro, Rule, Signature, Typing};
# use colosseum::unsync::Arena;
let cmds = [
// declarations
"prop : Type.\n",
"imp : prop -> prop -> prop.\n",

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

// theorem
r"thm imp_refl (x : prop) : proof (imp x x) := p : proof x => p.\n",
];

let arena = Arena::new();
let mut syms = Symbols::new();
let mut sig = Signature::new();

for c in cmds.iter() {
// parse and scope command in one go
let cmd: Command<String> = Command::parse(c, &syms)?;
match cmd {
// introduction of a new name
Command::Intro(id, it) => {
let id: &str = arena.alloc(id);
// add symbol to symbol table and fail if it is not new
let sym = syms.insert(id)?;

// typecheck and insert into signature
let typing: Typing = Typing::new(Intro::from(it), &sig)?.check(&sig)?;
sig.insert(sym, typing)?
}
// addition of rewrite rules
Command::Rules(rules) => sig.add_rules(rules.into_iter().map(Rule::from))?
}
}
# Ok::<_, Error>(())

Organisation

This library is divided into several modules:

• The [parse] module contains unshared, reference-free data structures,
• the [scope] module contains data structures with references, and
• the [rc] and [arc] modules contain data structures with references and shared pointers.

The [rc] and [arc] modules expose completely the same API, the difference being that the structures in [rc] cannot be used in multi-threaded scenarios. Due to the performance overhead incurred by the data structures in [arc], it is advisable to use these only in multi-threaded scenarios, and to prefer [rc] whenever possible.

For many data structures, we have counterparts in the [parse], [scope], and [rc]/[arc] modules. We call types from the [parse] and [scope] modules "parse structures" and "scope structures", respectively. For example, we distinguish parse terms, scope terms, and terms (the latter being defined in the [rc]/[arc] modules). Parse structures are constructed by the parser and refined into their corresponding scope structures by the scoper. Parse and scope 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.