An proof refinement logic for computational type theory based on Brouwer-realizability & the verificationist meaning explanation. Inspired by Nuprl. [For up-to-date development, see JonPRL's successor, Red JonPRL: https://github.com/JonPRL/sml-red-jonprl]
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README.md

JonPRL is a proof refinement logic in the sense of λ-PRL or Nuprl; JonPRL inherits its computational type theory from Constable, Bates, Harper, Allen, Bickford, Howe, Smith and many other names. Computational Type Theory is based on a meaning explanation similar to the ones which Martin-Löf introduced in 1979, to which I have written a self-contained introduction, Type Theory and its Meaning Explanations.

Discussion and Questions

There is an IRC channel on freenode at #jonprl.

Building & Installing JonPRL

JonPRL uses SML/NJ's CM for its build. First make sure you have SML/NJ set up properly. Then, install JonPRL's dependencies:

git submodule update --init --recursive

Then, JonPRL may be built using its Makefile:

make smlnj
make test

This puts a binary in ./bin/jonprl. Optionally, you may install JonPRL globally using:

sudo make install

Running JonPRL

To run JonPRL, simply direct it at your development:

jonprl example/test.jonprl

You may specify as many files as you like in this command; they will be refined in order, in case of any dependencies.

Emacs Mode

Optionally, you may install the JonPRL Mode for Emacs.

screenshot of jonprl-mode

Basic Syntax

JonPRL has a two-level syntax. There is the syntax of terms in the underlying lambda calculus (the object language) and the syntax of tactics and definitions in the metalanguage. Terms from the underlying lambda calculus are embedded into the metalanguage using brackets ([ and ]). When referring to names from the object language in the metalanguage, they are quoted in angle brackets (< and >).

The syntax of the object language represents all binders in a consistent manner. The variables to be bound in a subterm are written before it with a dot. For example, the identity function is written λ(x.x) where the first x indicates the bound name and the second refers back to it, and the first projection of a pair P is written spread(P; x.y.x). The semicolon separates arguments to spread.

Top-level declarations

JonPRL provides four top-level declarations:

  • Operator gives the binding structure of a new operator.
  • =def= defines the meaning of an operator in terms of the existing operators.
  • Theorem declares a theorem, and allows it to be proven.
  • Tactic defines a tactic in terms of the built-in tactics.

Built-in operators

Together with the syntax for binding trees, the built-in operators of JonPRL constitute the core type theory, in combination with the rules for CTT which are built into JonPRL's refiner. In this section, several of the operators is presented together with its arity and a brief informal description. An arity is a list of the valences of an operator's subterms; valence is the number of variables to bind.

abstract form concrete form description
unit() unit The unit type
<>() <> The trivial inhabitant of unit
fun(0;1) (x:A) B, A -> B The dependent function type
lam(1) lam(x.E) The lambda abstraction
ap(0;0) M N Function application
isect(0;1) {x:A} B, A => B The family intersection type
prod(0;1) (x:A) * B, A * B The dependent pair type
pair(0;0) <M,N> The pair operator
spread(0;2) spread(M; x.y.E) Pattern matching for pairs
subset(0;1) `{x:A B}`
plus(0;0) A + B The disjoint union type
inl(0) inl(M) The left union introduction form
inr(0) inr(M) The right union introduction form
decide(0;1;1) decide(M; l.E; r.F) Pattern matching for disjoint unions
void() void The empty type
=(0;0;0) =(M;N;A) The equality type (M,N are equal at type A)
member(0;0) member(M;A) The membership type (M is a member of A)
so_apply(0;0) F[x] Application for second order variables

Second-order variables

Unlike some other implementations of type theory that use the same syntax for function application and filling in second-order variables, JonPRL's second-order variables must be applied using the so_apply(0;0) operator (written using brackets, F[x]).

As an example, unique existence might be defined as follows:

Operator ex_uni : (0;1).

[ex_uni(T;x.P[x])] =def= [(x:T) * P[x] * {y:T} P[y] => =(x;y;T)].

Note that P is applied to x and y using so_apply rather than ap, which is reserved for function application.

Computational Equivalence

The elements of base() are all closed terms. Their equality is ceq(0;0), which denotes Howe's computational equivalence. Two terms are computationally equivalent if they both diverge or if they run to equivalent results. Computational equivalence is a congruence, which means that one can also prove that two terms are computationally equivalent if their subterms are computationally equivalent.