perga is a basic proof assistant based on a dependently typed lambda calculus (calculus of constructions). This implementation is based on the exposition in Nederpelt and Geuvers’ Type Theory and Formal Proof. Right now it is a perfectly capable higher order logic proof checker, though there is lots of room for improved ergonomics and usability, which I intend to work on. At the moment, perga is comparable to Automath in terms of power and ease of use, being slightly more powerful than Automath, and a touch less ergonomic.

Syntax

The syntax is fairly flexible and should work as you expect. Identifiers can be Unicode as long as megaparsec calls them alphanumeric. λ and Π abstractions can be written in the usual ways that should be clear from the examples below. Additionally, arrows can be used as an abbreviation for a Π type where the parameter doesn’t appear in the body as usual.

All of the following example terms correctly parse, and should look familiar if you are used to standard lambda calculus notation or Coq syntax.

λ (α : *) ⇒ λ (β : *) ⇒ λ (x : α) ⇒ λ (y : β) ⇒ x
fun (A B C : *) (g : → C) (f : A → B) (x : A) ⇒ g (f x)
fun (S : *) (P Q : S -> *) (H : Π (x : S) , P x -> Q x) (HP : forall (x : S), P x) => fun (x : S) => H x (HP x)

To be perfectly clear, λ abstractions can be written with either “λ” or “fun”, and are separated from their bodies by either “=>” or “⇒”. Binders with the same type can be grouped together, and multiple binders can occur between the “λ” and the “⇒”.

Π types can be written with either “Π”, “∀”, or “forall”, and are separated from their bodies with a “,”. Arrow types can be written “->” or “→”. Like with λ abstractions, binders with the same type can be grouped, and multiple binders can occur between the “Π” and the “,”.

Let expressions have syntax shown below.

let ( (<ident> := <expr>) )+ in <expr> end

Below is a more concrete example.

let (a := (and_elim_l A (and B C) h))
    (bc := (and_elim_r A (and B C) h))
    (b := (and_elim_l B C bc))
    (c := (and_elim_r B C bc))
in
    and_intro (and A B) C (and_intro A B a b) c
end

You can also directly bind functions. Here’s an example.

let (f (A : *) (x : A) := x) in
    f x
end

The syntax for binding functions is just like with definitions.

Definitions have abstract syntax as shown below.

<ident> (<ident> : <type>)* : <type>? := <term> | axiom;

(The distinction between <type> and <term> is for emphasis; they are the exact same syntactic category.) Here’s a couple definitions of the const function from above showing the options with the syntax, and a more complex example declaring functional extensionality as an axiom (assuming equality has been previously defined having type eq : Π (A : *) → A → A → *). Duplicate definitions are not normally allowed and will result in an error.

const := λ (α : *) ⇒ λ (β : *) ⇒ λ (x : α) => λ (y : β) => x;
const : ∀ (α β : *), α → β → α := fun (α β : *) (x : α) (y : β) ⇒ x;
const (α β : *) (x : α) (y : β) : α := x;

funext (A B : *) (f g : A → B) : (∀ (x : A), eq B (f x) (g x)) → eq (A → B) f g := axiom;

Type ascriptions are optional. If included, perga will check to make sure your definition matches the ascription, and, if so, will remember the way your wrote the type when printing inferred types, which is particularly handy when using abbreviations for complex types. perga has no problem inferring the types of top-level definitions, as they are completely determined by the term, but I recommend including ascriptions most of the time, as they serve as a nice piece of documentation, help guide the implementation process, and make sure you are implementing the type you think you are.

If the RHS of a definition is axiom, then perga will assume that the identifier is an inhabitant of the type ascribed to it (as such when using axioms, a type ascription is required). This allows you to use axioms.

Line comments are -- like in Haskell, and block comments are [* *] somewhat like ML (and nest properly). There is no significant whitespace, so you are free to format code as you wish.

There isn’t a proper module system (yet), but you can include other files in a dumb, C preprocessor way by using @include <filepath> (NOTE: this unfortunately messes up line numbers in error messages). Filepaths are relative to the current file.

Usage

Running perga without any arguments drops you into a basic repl. From here, you can type in definitions which perga will typecheck. Previous definitions are accessible in future definitions. The usual readline keybindings are available, including navigating history, which is saved between sessions (in ~/.cache/perga/history). In the repl, you can enter “:q”, press C-c, or press C-d to quit. Entering “:e” shows everything that has been defined along with their types. Entering “:t ” prints the type of a particular identifier, while “:v ” prints its value. Entering “:n ” will fully normalize (including unfolding definitions) an expression, while “:w ” will reduce it to weak head normal form. Finally “:l ” loads a file.

Here’s an example session showing the capabilities of the repl.

> :l examples/computation.pg
loading: examples/computation.pg
> :e
eight : nat
eq : ∏ (A : *) . A -> A -> *
eq_cong : ∏ (A B : *) (x y : A) (f : A -> B) . eq A x y -> eq B (f x) (f y)
eq_refl : ∏ (A : *) (x : A) . eq A x x
eq_sym : ∏ (A : *) (x y : A) . eq A x y -> eq A y x
eq_trans : ∏ (A : *) (x y z : A) . eq A x y -> eq A y z -> eq A x z
five : nat
four : nat
nat : *
nine : nat
one : nat
one_plus_one_is_two : eq nat (plus one one) two
plus : nat -> nat -> nat
seven : nat
six : nat
suc : nat -> nat
ten : nat
three : nat
times : nat -> nat -> nat
two : nat
two_plus_two_is_four : eq nat (plus two two) four
two_times_five_is_ten : eq nat (times two five) ten
zero : nat
> :n plus one one
λ (A : *) (f : A -> A) (x : A) . f (f x)
> :n two
λ (A : *) (f : A -> A) (x : A) . f (f x)
> :w plus one one
λ (A : *) (f : A -> A) (x : A) . one A f (one A f x)
> :w two
λ (A : *) (f : A -> A) (x : A) . f (one A f x)

You can also give perga a filename as an argument, in which case it will typecheck every definition in the file. If you give perga multiple filenames, it will process each one in turn, sharing an environment between them. Upon finishing, which should be nearly instantaneous, it will print out all files it processed, and “success!” if it successfully typechecked, and the first error it encountered otherwise.

Simple Example

There are many very well commented examples in the <./examples/> folder. These include

• <./examples/logic.pg>, which defines the standard logical operators and proves standard results about them,
• <./examples/computation.pg>, which demonstrates using perga for computational purposes,
• <./examples/algebra.pg>, which defines standard algebraic structures and proves results for them, and
• <./examples/peano.pg>, which proves standard arithmetic results from the Peano axioms.

I intend to extend these examples further.

Here is an example file defining Leibniz equality and proving that it is reflexive, symmetric, and transitive.

-- file: equality.pg

-- Defining Leibniz equality
-- Note that we can leave the ascription off
eq (A : *) (x y : A) := forall (P : A -> *), P x -> P y;

-- Equality is reflexive, which is easy to prove
-- Here we give an ascription so that when `perga` reports the type,
-- it references `eq` rather than inferring the type.
eq_refl (A : *) (x : A) : eq A x x := fun (P : A -> *) (Hx : P x) => Hx;

-- Equality is symmetric. This one's a little harder to prove.
eq_sym (A : *) (x y : A) (Hxy : eq A x y) : eq A y x :=
    fun (P : A -> *) (Hy : P y) =>
        Hxy (fun (z : A) => P z -> P x) (fun (Hx : P x) => Hx) Hy;

-- Equality is transitive.
eq_trans (A : *) (x y z : A) (Hxy : eq A x y) (Hyz : eq A y z) : eq A x z :=
    fun (P : A -> *) (Hx : P x) => Hyz P (Hxy P Hx);

Running perga equality.pg yields the following output.

eq : ∏ (A : *) . A -> A -> *
eq_refl : ∏ (A : *) (x : A) . eq A x x
eq_sym : ∏ (A : *) (x y : A) . eq A x y -> eq A y x
eq_trans : ∏ (A : *) (x y z : A) . eq A x y -> eq A y z -> eq A x z

This means our proofs were accepted. Furthermore, as a sanity check, we can see that the types correspond exactly to what we wanted to prove.

Future Goals

Substantive

DONE Let-expressions

I decided to bake let expressions into the formal language, rather than being a layer on top of the syntax. This means we can treat typing let expressions different from functions. The only substantial difference between

let (x := value) in body end

and

(fun (x : type) => body) value

is that the latter must have a proper function type while the former doesn’t need to. So, for instance,

let (x := *) in ...

is possible, while you could not write a function whose argument is type □. We are justified in doing this because we always know what the argument is, and can just immediately substitute it, both into the value of the body, as well as the type of the body.

TODO Sections

Coq-style sections would be very handy, and probably relatively easy to implement (compared to everything else on this todo list). Upon parsing a definition inside a section, will somehow need to look ahead to see what variables are used to see how I need to modify binders, or just make every definition require every section variable as an argument.

TODO Inference

Not decidable, but I might be able to implement some basic unification algorithm, or switch to bidirectional type checking. This isn’t super necessary though, I find leaving off the types of arguments to generally be a bad idea, but in some cases it can be handy, especially not at the top level.

TODO Implicits

Much, much more useful than inference, implicit arguments would be amazing. It also seems a lot more complicated, but any system for dealing with implicit arguments is far better than none.

TODO Module System

A proper module system would be wonderful. To me, ML style modules with structures, signatures, and functors seems like the right way to handle algebraic structures for a relatively simple language, rather than records (or, worse, a bunch of and’s like I currently have; especially painful without implicits) or type classes (probably much harder, but could be nicer), but any way of managing scope, importing files, etc. is a necessity. The F-ing modules paper is probably a good reference.

TODO Universes?

Not super necessary, but occasionally extremely useful. Could be fun, idk.

TODO Inductive Definitions

This is definitely a stretch goal. It would be cool though, and would turn this proof checker into a much more competent programming language. It’s not necessary for the math, but inductive definitions let you leverage computation in proofs, which is amazing. They also make certain definitions way easier, by avoiding needing to manually stipulate elimination rules, including induction principles, and let you keep more math constructive and understandable to the computer.

Cosmetic/usage/technical

TODO Prettier pretty printing

Right now, everything defaults to one line, which can be a problem with how large the proof terms get. Probably want to use prettyprinter to be able to nicely handle indentation and line breaks.

TODO Better repl

The repl is decent, probably the most fully-featured repl I’ve ever made, but implementing something like this would be awesome.

TODO Improve error messages

Error messages are decent, but a little buggy. Syntax error messages are pretty ok, but could have better labeling. The type check error messages are decent, but could do with better location information. Right now, the location defaults to the end of the current definition, which is often good enough, but more detail can’t hurt. The errors are generally very janky and hard to read. Having had quite a bit of practice reading them now, they actually provide very useful information, but could be made a lot more readable.

TODO Document library code

Low priority, as I’m the only one working on this, I’m working on it very actively, and things will continue rapidly changing, but I’ll want to get around to it once things stabilize, before I forget how everything works.

TODO Add versions to perga.cabal and/or nixify

Probably a smart idea.

TODO More incremental parsing/typechecking

Right now, if there’s a failure, everything just stops immediately. More incremental parsing/typechecking could pave the way for more interactivity, e.g. development with holes, an LSP server, etc., not to mention better error messages.

TODO Multiple levels of AST

Right now, the parsing and typechecking kind of happens all at once on a single syntax representation. As I add fancier and fancier elaboration, it might be a good idea to have multiple syntax representations. So we’d have one level of syntax representing what is actually in the file (modulo some easy elaboration like with function definitions), and through a series of transformations transform it into something like the current Expr type with all the information needed for typechecking and all the complex language features removed.

TODO Alternate syntax

I’ve had a bunch of ideas for a more mathematician-friendly syntax bouncing around my head for a while. Implementing one of them would be awesome, but probably quite tricky.

Something like

Theorem basic (S : *) (P : S → *) :
    (∀ (x : S), P x → Q x) → (∀ (x : S), P x) → ∀ (x : S), Q x.
Proof
        1. Suppose ∀ (x : S), P x → Q x
        2. Suppose ∀ (x : S), P x
        3. Let x : S
        4. P x by [2 x]
        5. Q x by [1 x 4]
Qed

I think could be reliably translated into

basic (S : *) (P : S → *) : (Π (x : S), P x → Q x) → (Π (x : S), P x) → Π (x : S), Q x :=
      fun (a1 : Π (x : S), P x → Q x) ⇒
          fun (a2 : Π (x : S), P x) ⇒
              fun (x : S) ⇒
                  a1 x (a2 x);

and is more intuitively understandable to a mathematician not familiar with type theory, while the latter would be utter nonsense.

I’m imagining the parser could be chosen based on the file extension or something. Some way to mix the syntaxes could be nice too.

TODO Infix/misfix operators

Infix/misfix operators would be very nice and make perga look more normal. It’s funny, at the moment it looks a lot like a lisp, even though it’s totally not.

(eq_trans nat (plus n (suc m)) (suc (plus n m)) (plus (suc m) n)
    (plus_s_r n m)
    (eq_trans nat (suc (plus n m)) (suc (plus m n)) (plus (suc m) n)
      (eq_cong nat nat (plus n m) (plus m n) suc IH)
      (eq_sym nat (plus (suc m) n) (suc (plus m n)) (plus_s_l m n))))

DONE treesitter parser and/or emacs mode

There’s a tree-sitter parser and neovim plugin available now, but no emacs-mode.

TODO TUI

This is definitely a stretch goal, and I’m not sure how good of an idea it would be, but I’m imagining a TUI split into two panels. On the left you can see the term you are building with holes in it. On the right you have the focused hole’s type as well as the types of everything in scope (like Coq and Lean show while you’re in the middle of a proof). Then you can interact with the system by entering commands (e.g. intros, apply, etc.) which changes the proof term on the left. You’d also just be able to type in the left window as well, and edit the proof term directly. This way you’d get the benefits of working with tactics, making it way faster to construct proof terms, and the benefits of working with proof terms directly, namely transparency and simplicity. I’ll probably want to look into brick if I want to make this happen.