# Makkai

A simple type theory.

## Type Theory

### Judgements
- `C ctx` means `C` is a context.
- `C ⊢ X set` means `X` is a set in context `C`.
- `C ⊢ m : X` means `m` is a match of set `X`.
- `C ⊢ S signature` means `S` is a signature in context `C`.
- `C ⊢ I :: S` means `I` is an instance of signature `S`.

Contexts are built by extension with either kind of binding:
- `· ctx`.
- `C, x : X` ctx, when `C ⊢ X set`.
- `C, i :: S` ctx, when `C ⊢ S signature`.

Everywhere below, `...` ranges over a finite (possibly zero) index, and the labels `xi`, `si`, `vi` are assumed distinct within any single list.

### The set layer

#### Built-Ins
For each `B ∈ { Nat, Int, Float, Str }`:
- **Formation.** `C ⊢ B set`.
- **Intro.** Literals of the appropriate form match the corresponding built-in (e.g. `C ⊢ 7 : Nat`).

#### Variables
- `C ⊢ x : X` when `x : X` is in `C`.

#### Records
- **Formation.** `C ⊢ record { x0 : X0, ..., xn : Xn } set` when `C ⊢ X0 set`, `C, x0 : X0 ⊢ X1 set`, ..., `C, x0 : X0, ..., xn-1 : Xn-1 ⊢ Xn set`.
- **Intro.** `C ⊢ { x0 = m0, ..., xn = mn } : record { x0 : X0, ..., xn : Xn }` when `C ⊢ m0 : X0`, `C ⊢ m1 : X1[m0/x0]`, ..., `C ⊢ mn : Xn[m0/x0, ..., mn-1/xn-1]`.
- **Elim.** `C ⊢ m.xi : Xi[(m .x0)/x0, ..., (m .xi-1)/xi-1]` when `C ⊢ m : record { x0 : X0, ..., xn : Xn }`.
- **β.** `{ ..., xi = mi, ... } .xi=mi`.
- **η.** `m=record { x0=m .x0, ..., xn=m .xn }` when `m : record { ... }`.

#### Variants
- **Formation.** `C ⊢ variant { v0 : X0 | ... | vn : Xn } set` when each `C ⊢ Xi set`.
- **Intro.** `C ⊢ vi. m : variant { v0 : X0 | ... | vn : Xn }` when `C ⊢ m : Xi`.
- **Elim.** `C ⊢ case m of { v0. x0 => n0 | ... | vn. xn => nn } : X` when `C ⊢ m : variant { v0 : X0 | ... | vn : Xn }` and, for each `i`, `C, xi : Xi ⊢ ni : X`.
- **β.** `case vi. m of { ... | vi. xi => ni | ... }=ni[m/xi]`.
- **η.** `m=case m of { v0. x0 => v0. x0 | ... | vn. xn => vn. xn }` when `m : variant { ... }`.

### The signature layer

#### Theory
- **Formation.** `C ⊢ theory { s0 :: S0, ..., sn :: Sn } signature` when `C ⊢ S0 signature`, `C, s0 :: S0 ⊢ S1 signature`, ..., `C, s0 :: S0, ..., sn-1 :: Sn-1 ⊢ Sn signature`.
- **Intro.** `C ⊢ { s0 = I0, ..., sn = In } :: theory { s0 :: S0, ..., sn :: Sn }` when `C ⊢ I0 :: S0`, `C ⊢ I1 :: S1[I0/s0]`, ..., `C ⊢ In :: Sn[I0/s0, ..., In-1/sn-1]`.
- **Elim.** `C ⊢ I .si :: Si[(I .s0)/s0, ..., (I .si-1)/si-1]` when `C ⊢ I :: theory { s0 :: S0, ..., sn :: Sn }`.
- **β.** `{ ..., si=Ii, ... } .si=Ii`.
- **η.** `I={ s0=I .s0, ..., sn=I .sn }` when `I :: theory { ... }`.

#### Variables
- `C ⊢ i :: S` when `i :: S` is in `C`.

### The interface between signatures and sets

#### The signature `Set`
- **Formation.** `C ⊢ Set signature`.
- **Intro.** `C ⊢ X :: Set` when `C ⊢ X set`.
- **Elim.** `C ⊢ set-of(I) set` when `C ⊢ I :: Set`.
- **β.** `set-of(X)=X` when `X :: Set` arises from `C ⊢ X set`.
- **η.** `I=set-of(I)` viewed as an instance, when `I :: Set`.

#### Extension signatures
- **Formation.** `C ⊢ (x: X) -> S signature` when `C ⊢ X set` and `C, x : X ⊢ S signature`.
- **Intro.** `C ⊢ for (y: X). I :: (x: X) -> S` when `C, y : X ⊢ I :: S[y/x]`.
- **β.** `(for (x: X). I)(y)=I[y/x]`.
- **η.** `I=for (x: X). I(x)` when x is not free in I.

#### Instances via case analysis
A construction admitting an instance of any signature, given an element of a variant set and a branch for every tag.
- **Intro.** `C ⊢ case m of { v0. x0 => I0 | ... | vn. xn => In } :: S` when `C ⊢ S signature`, `C ⊢ m : variant { v0. : X0 | ... | vn. : Xn }`, and, for each `i`, `C, xi : Xi ⊢ Ii :: S`.
- **β.** `case vi. m of { ... | vi. xi => Ii | ... }=Ii[m/xi]`.


## Rust implementation

Usage `makkai [--debug] file1.makkai ... fileN.makkai`

See the [grammar](grammar.txt) for details, and the examples in `examples/`.

Note: the checker does not presently support η-equivalence in all cases.

# License

Copyright tslil clingman 2026, this programme is free software and is made available under the terms of the GPL v3 or later. See LICENSE for details.