Kind2/README.md

# Kind

A cute proof and programming language.

Usage
-----

```bash
npm i -g kind-lang                             # installs Kind
git clone https://github.com/uwu-tech/kind     # clones base libs
cd Kind/base                                   # enters base libs
kind Main                                      # checks Main.kind
kind Main --run                                # runs Main
```

*Right now, you must be at `kind/base` to use the language.*

Examples
--------

### A 'Hello, world!'

```c
Main: IO(Unit)
  do IO {
    IO.print("Hello, world!")
  }
```

### Some algorithms

```c
// List sum using recursion
sum(list: List(Nat)): Nat
  case list {
    nil  : 0
    cons : list.head + sum(list.tail)
  }

// List sum using a fold
sum(list: List(Nat)): Nat
  List.fold!(list)!(0, Nat.add)

// List sum using a loop
sum(list: List(Nat)): Nat
  let sum = 0
  for x in list:
    sum = x + sum
  sum
```

### Some types

```c
// A struct
type User {
  new(name: String, birth: Date, avatar: Image)
}

// A simple pair
type Pair <A: Type> <B: Type> {
  new(fst: A, snd: B)
}

// A dependent pair
type Sigma <A: Type> <B: A -> Type> {
  new(fst: A, snd: B(fst))
}

// A list
type List <A: Type> {
  nil
  cons(head: A, tail: List(A))
}

// A list with a statically known size
type Vector <A: Type> ~ (size: Nat) {
  nil                                              ~ (size = 0) 
  cons(size: Nat, head: Nat, tail: Vector(A,size)) ~ (size = 1 + size)
}

// The propositional equality
type Equal <A: Type> <a: A> ~ (b: A) {
  refl ~ (b = a)
}
```

### Some proofs

```
// Proof that `a == a + 0`
Nat.add.zero(a: Nat): a == Nat.add(a, 0)
  case a {
    zero: refl
    succ: apply(Nat.succ, Nat.add.zero(a.pred))
  }!

// Proof that `1 + (a + b) == a + (1 + b)`
Nat.add.succ(a: Nat, b: Nat): Nat.succ(a + b) == (a + Nat.succ(b))
  case a {
    zero: refl
    succ: apply(Nat.succ, Nat.add.succ(a.pred, b))
  }!

// Proof that addition is commutative
Nat.add.comm(a: Nat, b: Nat): (a + b) == (b + a)
  case a {
    zero:
      Nat.add.zero(b)
    succ: 
      let p0 = Nat.add.succ(b, a.pred)
      let p1 = Nat.add.comm(b, a.pred)
      p0 :: rewrite X in Nat.succ(X) == _ with p1
  }!
```

Resources
---------

- Syntax reference: [SYNTAX.md](SYNTAX.md).

- Theorem proving tutorial: [THEOREMS.md](THEOREMS.md).

- Base library: [base](https://github.com/uwu-tech/Kind/tree/master/base).

- [Telegram chat](https://t.me/formality_lang)! 

- Discord server (TODO)

[trusted core]: https://github.com/moonad/FormCoreJS

[FormCore-to-Haskell]: https://github.com/moonad/FormCoreJS/blob/master/FmcToHs.js

[kind.js]: https://github.com/uwu-tech/Kind/blob/master/bin/js/base/kind.js

[Agda]: https://github.com/agda/agda

[Idris]: https://github.com/idris-lang/Idris-dev

[Coq]: https://github.com/coq/coq

[Lean]: https://github.com/leanprover/lean

[Absal]: https://medium.com/@maiavictor/solving-the-mystery-behind-abstract-algorithms-magical-optimizations-144225164b07

[JavaScript compiler]:https://github.com/moonad/FormCoreJS/blob/master/FmcToJs.js