LSTS is a proof assistant and maybe a programming language.
Large Scale Type Systems is a library for performance-sensitive type system operations, specifically designed for cases where type logic may greatly outscale AST logic. The LSTS code works only with backreferences to AST code thereby permitting lazy generation of AST nodes.
LSTS implements three methods of type unification which form a basis for logical operations. Type expressions are made up of some combination of logical statements. When two type expressions are merged somehow, this is called unification. Depending on which direction information can propogate (narrow), we end up with three possible operators.
| A x B | A can narrow | A cannot narrow | | --------------- | -------------------- | ------------------- | | B can narrow | Most General Unifier | B => A | | B cannot narrow | A => B | Structural Equality |
The three binary operations on types are
let v: Kilo<Meter>/Second = 123.456;
let s: Minute = 78.9;
let d: Mile = (v as Mile/Minute) * s;
Unit is a Kind separate from Term. Unit typing adds a check for dimension analysis. Inference rules are programmable on a Kind by Kind basis. The three builtin Kinds are Nil, Term, and Constant.
LSTS does not ensure against all forms of logical errors, however it does complain about some famous ones.
/* The square root of 2 is irrational */
let $"/"(x:X, y:Y): X/Y;
let $"*"(x:X, y:Y): X*Y;
let square(x:X): X*X;
type Pt; let p:Pt;
type Qt; let q:Qt;
let sqrt_of_two: Pt/Qt;
square(sqrt_of_two) * square(q): Pt*Pt; //2 * q*q = p*p
square(p) / square(sqrt_of_two): Qt*Qt; //p*p / 2 = q*q
p / square(sqrt_of_two) : ?/(); //2 is a factor of p
/* Infinitude of primes */
import "number_theory.tlc";
let primes:Prime[]; //assume there are a finite number of primes
let p = primes.product() + 1; //let p be the product of all primes + 1
forall d:primes. p%d == 1; //p mod d, forall d in primes list, is 1
LSTS natively supports dependent types, inhabiting the "Constant" Kind. Constant Types are just untyped Terms minus abstraction. This middle ground Term language allows us to describe many desirable scenarios while still keeping the type system Strongly Normalizing.
let x: [1 + 2 * 3]; x: [7];
//constant folding is programmable
let f(x: [x], y: [y]): [2*x+y];
//algebraic substitution is allowed in dependent contexts
[if a%2==0 then 1 else 2]\[a%2|1] : [if 1==0 then 1 else 2] : [2]
For further information there is a tutorial and reference documentation.