LaTTe

http://latte-central.github.io/LaTTe/

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LaTTe : a Laboratory for Type Theory experiments (in clojure)

What?

LaTTe is a proof assistant library based on type theory (a variant of λD as described in the book Type theory and formal proof: an introduction).

• Hot! Watch Latte live at: https://www.youtube.com/watch?v=5YTCY7wm0Nw

• Sizzling! A paper about LaTTe at the European Lisp Symposium, 2017: https://github.com/latte-central/latte-ELS-2017 [PDF]

• Blistering LaTTe was in the Hacker news!

The specific feature of LaTTe is its design as a library (unlike most proof assistant, generally designed as tools) tightly integrated with the Clojure language. It is of course fully implemented in Clojure, but most importantly all the definitional aspects of the assistant (definitions, theorem and axioms) are handled using Clojure namespaces, definitions and macros.

For example, the fact that logical implication is reflexive can be stated directly as a Clojure top-level form:

(defthm impl-refl
  "Implication is reflexive."
  [[A :type]]
  (==> A A))
;; => [:declared :theorem impl-refl]

in plain text:

assuming a type A, then A implies A.

The proof of the theorem can be also constructed as a Clojure form:

• either giving a lambda-term as a direct proof (exploiting the proposition-as-type, proof-as-term correspondance) :

(proof 'impl-refl
  (qed (lambda [x A] x)))
;; => [:qed impl-refl]

(i.e. the identity function is a proof of reflexivity for implication)

• or using a declarative proof script:

(proof 'impl-refl
   (assume [x A]
     (have concl A :by x))
   (qed concl))
;; => [:qed impl-refl]

... which, with some training, can be read as a "standard" mathematical proof:

assuming A holds, as an hypothesis named x we can deduce A by x hence A implies A as stated (QED).

Of course, all the proofs are checked for correctness. Without the introduction of an inconsistent axiom (and assuming the correctness of the implementation of the LaTTe kernel), no mathematical inconsistency can be introduced by the proof form.

Yes, but what?

LaTTe helps you formalize mathematic concepts and construct formal proofs of theorems (propositions) about such concepts. Given the tight integration with the Clojure language, existing Clojure development environments (e.g. Cider, Cursive) can be used as (quite effective) interactive proof assistants.

How?

• There will be a tutorial at some point ...

• The reference documentation is at: http://latte-central.github.io/LaTTe/

Standard library :

• The prelude library is at: https://github.com/latte-central/latte-prelude

• The (typed) sets library is at: https://github.com/latte-central/latte-sets

• The integer arithmetic library is at: https://github.com/latte-central/latte-integers

(obviously more to come ...)

Who?

LaTTe may be of some interest for you:

• obviously if you are interested in type theory and the way it can be implemented on a Computer. LaTTe has been implemented with readability and simplicity in mind (more so than efficiency or correctness),
• probably if you are interested in the "mechanical" formalization of mathematics, intuitionistic logic, etc. (although you might not learn much, you may be interested in contributing definitions, theorems and proofs),
• maybe if you are curious about the lambda-calculus (the underlying theory of your favorite programming language) and dependent types (a current trend) and what you can do with these besides programming.

When?

LaTTe is, at least for now, an experiment more than a finalized product, but it is already usable.

A few non-trivial formalizations have been conducted using LaTTe:

• e.g. a (gorilla REPL) document about Knaster-Tarski fixed point theorem(s): https://github.com/latte-central/fixed-points

Contributions such as mathematical content or enhancement/correction of the underlying machinery are very much welcomed.

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Copyright (C) 2015-2018 Frederic Peschanski (MIT license, cf. LICENSE)