https://github.com/forked-from-1kasper/anders

Anders: Cubical Type Checker
https://github.com/forked-from-1kasper/anders

ctt cubical-type-theory dependent-type-theory dependent-types homotopy-type-theory hott mltt proof-assistant theorem-prover type-checker type-system

Last synced: 29 days ago
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Anders: Cubical Type Checker

• Host: GitHub
• URL: https://github.com/forked-from-1kasper/anders
• Owner: forked-from-1kasper
• License: isc
• Created: 2022-02-28T12:25:34.000Z (about 3 years ago)
• Default Branch: master
• Last Pushed: 2023-10-23T16:26:40.000Z (over 1 year ago)
• Last Synced: 2025-04-02T21:51:25.005Z (about 2 months ago)
• Topics: ctt, cubical-type-theory, dependent-type-theory, dependent-types, homotopy-type-theory, hott, mltt, proof-assistant, theorem-prover, type-checker, type-system
• Language: OCaml
• Homepage:
• Size: 4.02 MB
• Stars: 24
• Watchers: 1
• Forks: 1
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • Changelog: CHANGELOG.md
  • License: LICENSE

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README

        
Anders

======

```OCaml

type exp =

  | EPre of Z.t | EKan of Z.t                                                          (* cosmos *)

  | EVar of ident | EHole                                                           (* variables *)

  | EPi of exp * (ident * exp) | ELam of exp * (ident * exp) | EApp of exp * exp           (* pi *)

  | ESig of exp * (ident * exp) | EPair of tag * exp * exp                              (* sigma *)

  | EFst of exp | ESnd of exp | EField of exp * string                    (* simga elims/records *)

  | EId of exp | ERef of exp | EJ of exp                                      (* strict equality *)

  | EPathP of exp | EPLam of exp | EAppFormula of exp * exp                     (* path equality *)

  | EI | EDir of dir | EAnd of exp * exp | EOr of exp * exp | ENeg of exp       (* CCHM interval *)

  | ETransp of exp * exp | EHComp of exp * exp * exp * exp                     (* Kan operations *)

  | EPartial of exp | EPartialP of exp * exp | ESystem of exp System.t      (* partial functions *)

  | ESub of exp * exp * exp | EInc of exp * exp | EOuc of exp                (* cubical subtypes *)

  | EGlue of exp | EGlueElem of exp * exp * exp | EUnglue of exp * exp * exp          (* glueing *)

  | EEmpty | EIndEmpty of exp                                                               (* 𝟎 *)

  | EUnit | EStar | EIndUnit of exp                                                         (* 𝟏 *)

  | EBool | EFalse | ETrue | EIndBool of exp                                                (* 𝟐 *)

  | EW of exp * (ident * exp) | ESup of exp * exp | EIndW of exp * exp * exp                (* W *)

  | EIm of exp | EInf of exp | EIndIm of exp * exp | EJoin of exp      (* Infinitesimal Modality *)

```

Anders is a HoTT proof assistant based on [CCHM](https://arxiv.org/pdf/1611.02108.pdf)

in flavour of [Cubical Agda](https://agda.readthedocs.io/en/v2.6.2.1/language/cubical.html)

plus strict equality for 2LTT and ℑ modality for synthetic differential geometry.

Features

--------

* 𝟎, 𝟏, 𝟐, W.

* Pretypes & strict equality.

* Generalized Transport and Homogeneous Composition as primitive Kan operations.

* Cubical subtypes.

* Glue types.

* Coequalizer.

* ℑ modality.

* UTF-8 support including universe levels (i.e. `U₁₂₃`).

* Lean syntax for ΠΣW.

* Poor man’s records as Σ with named accessors to telescope variables.

* 1D syntax with top-level declarations.

Setup

-----

```shell

$ make

$ dune exec anders help

```

Samples

-------

You can find some examples in [`library/`](https://github.com/forked-from-1kasper/anders/tree/master/library).

```Lean

def inv′ (A : U) (a b : A) (p : Path A a b) : Path A b a :=

 hcomp A (∂ i) (λ (j : I), [(i = 0) → p @ j, (i = 1) → a]) a


def kan (A : U) (a b c d : A) (p : Path A a c) (q : Path A b d) (r : Path A a b) : Path A c d :=

 hcomp A (∂ i) (λ (j : I), [(i = 0) → p @ j, (i = 1) → q @ j]) (r @ i)


def comp (A : I → U) (r : I) (u : Π (i : I), Partial (A i) r) (u₀ : (A 0)[r ↦ u 0]) : A 1 :=

hcomp (A 1) r (λ (i : I), [(r = 1) → transp ( A (i ∨ j)) i (u i 1=1)]) (transp ( A i) 0 (ouc u₀))


def ghcomp (A : U) (r : I) (u : I → Partial A r) (u₀ : A[r ↦ u 0]) : A :=

hcomp A (∂ r) (λ (j : I), [(r = 1) → u j 1=1, (r = 0) → ouc u₀]) (ouc u₀)

```

```shell

$ anders check library/everything.anders

```

## Related publications

### MLTT

Type Checker is based on classical MLTT-80 with 0, 1, 2 and W-types.

* Intuitionistic Type Theory [Martin-Löf]

### CCHM

* CTT: 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]

* Cubical Methods in HoTT/UF [Mörtberg]

### HTS

* A simple type system with two identity types [Voevodsky]

* Two-level type theory and applications [Annenkov, Capriotti, Kraus, Sattler]

* Syntax for two-level type theory [Bonacina, Ahrens]

### Modalities

Infinitesimal Modality was added for direct support of Synthetic Differential Geometry.

* Differential cohomology in a cohesive ∞-topos [Schreiber]

* Cartan Geometry in Modal Homotopy Type Theory [Cherubini]

* Differential Cohesive Type Theory [Gross, Licata, New, Paykin, Riley, Shulman, Cherubini]

* Brouwer's fixed-point theorem in real-cohesive homotopy type theory [Shulman]

Acknowledgements

----------------

* Univalent People

Authors

-------

* @siegment

* @tonpaguru