Features

• Fibrant MLTT-style ΠΣ primitives with Leibniz equality in 500 LOC
• Cofibrant CHM-style I primitives with pretypes hierarchy Vₙ in 500 LOC
• Parser in 50 LOC
• Lexer in 50 LOC
• Full Agda-style UTF-8 support including universe levels
• Lean comma-syntax for ΠΣ
• 1D syntax with top-level declarations
• Groupoid Infinity CCHM base library: https://groupoid.space/math
• Best suited for academic papers
• Homepage: https://groupoid.space/homotopy

Prerequisites

Here is example for Ubuntu:

$ apt install ocaml ocamlbuild menhir

Samples

You can find some examples in the experiments directory. For instance reality checking by internalizing MLTT can be performed by the following usage:

def MLTT (A: U) : U := Σ
    (Π-form  : Π (B : A → U), U) -- Pi Type
    (Π-ctor₁ : Π (B : A → U), Pi A B → Pi A B)
    (Π-elim₁ : Π (B : A → U), Pi A B → Pi A B)
    (Π-comp₁ : Π (B : A → U) (a : A) (f : Pi A B), Equ (B a) (Π-elim₁ B (Π-ctor₁ B f) a) (f a))
    (Π-comp₂ : Π (B : A → U) (a : A) (f : Pi A B), Equ (Pi A B) f (λ (x : A), f x))
    (Σ-form  : Π (B : A → U), U) -- Sigma Type
    (Σ-ctor₁ : Π (B : A → U) (a : A) (b : B a), Sigma A B)
    (Σ-elim₁ : Π (B : A → U) (_ : Sigma A B), A)
    (Σ-elim₂ : Π (B : A → U) (x : Sigma A B), B (pr₁ A B x))
    (Σ-comp₁ : Π (B : A → U) (a : A) (b: B a), Equ A a (Σ-elim₁ B (Σ-ctor₁ B a b)))
    (Σ-comp₂ : Π (B : A → U) (a : A) (b: B a), Equ (B a) b (Σ-elim₂ B (a, b)))
    (Σ-comp₃ : Π (B : A → U) (p : Sigma A B), Equ (Sigma A B) p (pr₁ A B p, pr₂ A B p))
    (=-form  : Π (a : A), A → U) -- Identity Type
    (=-ctor₁ : Π (a : A), Equ A a a)
    (=-elim₁ : Π (a : A) (C: D A) (d: C a a (=-ctor₁ a)) (y: A) (p: Equ A a y), C a y p)
    (=-comp₁ : Π (a : A) (C: D A) (d: C a a (=-ctor₁ a)),
       Equ (C a a (=-ctor₁ a)) d (=-elim₁ a C d a (=-ctor₁ a))),
    U

def instance (A : U) : MLTT A :=
    (Pi A, lambda A, app A, comp₁ A, comp₂ A,
     Sigma A, pair A, pr₁ A, pr₂ A, comp₃ A, comp₄ A, comp₅ A,
     Equ A, refl A, J A, comp₆ A, A)

$ ocamlbuild -use-menhir -yaccflag "--explain" anders.native
$ ./anders.native girard check experiments/mltt.anders

Papers

Anders was built by strictly following these publications:

• A simple type-theoretic language: Mini-TT [Coquand, Kinoshita, Nordström, Takeyama]
• Cubical Type Theory: a constructive interpretation of the univalence axiom [Cohen, Coquand, Huber, Mörtberg]
• On Higher Inductive Types in Cubical Type Theory [Coquand, Huber, Mörtberg]
• Canonicity for Cubical Type Theory [Huber]
• Canonicity and homotopy canonicity for cubical type theory [Coquand, Huber, Sattler]
• Cubical Synthetic Homotopy Theory [Mörtberg, Pujet]
• Unifying Cubical Models of Univalent Type Theory [Cavallo, Mörtberg, Swan]
• Cubical Agda: A Dependently Typed PL with Univalence and HITs [Vezzosi, Mörtberg, Abel]
• A Cubical Type Theory for Higher Inductive Types [Huber]
• Gluing for type theory [Kaposi, Huber, Sattler]

Acknowledgements

• Univalent People

Authors

• Siegmentation Fault
• 5HT