A Calculus of Constructions Type Checker in Java

TypeSystem.java

/* Copyright (C) Freddy A Cubas "superstar64"
Boost Software License - Version 1.0 - August 17th, 2003

Permission is hereby granted, free of charge, to any person or organization
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// https://en.wikipedia.org/wiki/Calculus_of_constructions
// https://www.staff.city.ac.uk/~ross/papers/debruijn.html
import java.util.Optional;
import java.util.function.Function;

interface Void {
    <A> A absurd();
}


abstract class Term<A> {
    <B> Term<B> inhabited() {
        throw new RuntimeException("Term not inhabited: " + this.toString());
    }
    abstract void unify(Term<A> right);
    void unifyRight(Type<A> left) {
        mismatch(left, this);
    }
    void unifyRight(Kind<A> left) {
        mismatch(left, this);
    }
    void unifyRight(Variable<A> left) {
        mismatch(left, this);
    }
    void unifyRight(Call<A> left) {
        mismatch(left, this);
    }
    void unifyRight(Lambda<A> left) {
        mismatch(left, this);
    }
    void unifyRight(Forall<A> left) {
        mismatch(left, this);
    }
    static <A> void mismatch(Term<A> left, Term<A> right)
    {
        throw new RuntimeException("Type mismatch between \"" + left.toString() + "\" and \"" + right.toString());
    }
    abstract <B> Term<B> substitute(Function<A, Term<B>> f);
    static <A, B> Function<Optional<A>, Term<Optional<B>>> traverse(Function<A, Term<B>> f)
    {
        return o -> o.isPresent() ? f.apply(o.get()).substitute(x -> new Variable<>(Optional.of(x))) : new Variable<>(Optional.empty());
    }
    abstract Term<A> reduce();
    Term<A> reduceCall(Term<A> argument) {
        return new Call<A>(this, argument);
    }
    Term<A> typeCheckCall(Term<A> argumentType, Term<A> argument) {
        throw new RuntimeException("Not a function type: " + this.toString());
    }
    abstract Term<A> typeCheck(Function<A, Term<A>> enviroment);
}

class Type<A> extends Term<A> {
    <B> Term<B> inhabited() {
        return new Type<B>();
    }
    void unify(Term<A> right) {
        right.unifyRight(this);
    }
    void unifyRight(Type<A> left) {}
    <B> Term<B> substitute(Function<A, Term<B>> f) {
        return new Type<B>();
    }
    Term<A> reduce() {
        return this;
    }
    Term<A> typeCheck(Function<A, Term<A>> enviroment) {
        return new Kind<>();
    }
    public String toString() {
        return "P";
    }
}

class Kind<A> extends Term<A> {
    <B> Term<B> inhabited() {
        return new Kind<B>();
    }
    void unify(Term<A> right) {
        right.unifyRight(this);
    }
    void unifyRight(Kind<A> left) {}
    <B> Term<B> substitute(Function<A, Term<B>> f) {
        return new Kind<B>();
    }
    Term<A> reduce() {
        return this;
    }
    Term<A> typeCheck(Function<A, Term<A>> enviroment) {
        throw new RuntimeException("Type of T");
    }
    public String toString() {
        return "T";
    }
}

class Variable<A> extends Term<A> {
    final A name;
    Variable(A name) {
        this.name = name;
    }
    void unify(Term<A> right) {
        right.unifyRight(this);
    }
    void unifyRight(Variable<A> left) {
        if(!left.name.equals(this.name)) {
            mismatch(left, this);
        }
    }
    <B> Term<B> substitute(Function<A, Term<B>> f) {
        return f.apply(name);
    }
    Term<A> reduce() {
        return this;
    }
    Term<A> typeCheck(Function<A, Term<A>> enviroment) {
        return enviroment.apply(name);
    }
    public String toString() {
        return "V(" + name.toString() + ")";
    }
}

class Call<A> extends Term<A> {
    final Term<A> function;
    final Term<A> argument;
    Call(Term<A> function, Term<A> argument) {
        this.function = function;
        this.argument = argument;
    }
    void unify(Term<A> right) {
        right.unifyRight(this);
    }
    void unifyRight(Call<A> left) {
        left.function.unify(this.function);
        left.argument.unify(this.argument);
    }
    <B> Term<B> substitute(Function<A, Term<B>> f) {
        return new Call<B>(function.substitute(f), argument.substitute(f));
    }
    Term<A> reduce() {
        return function.reduce().reduceCall(argument.reduce());
    }
    Term<A> typeCheck(Function<A, Term<A>> enviroment) {
        return function.typeCheck(enviroment).typeCheckCall(argument.typeCheck(enviroment), argument.reduce());
    }
    public String toString() {
        return "(" + function.toString() + ")" + "(" + argument.toString() + ")";
    }
}

class Lambda<A> extends Term<A> {
    final Term<A> argumentType;
    final Term<Optional<A>> bound;
    Lambda(Term<A> argumentType, Term<Optional<A>> bound) {
        this.argumentType = argumentType;
        this.bound = bound;
    }
    void unify(Term<A> right) {
        right.unifyRight(this);
    }
    void unifyRight(Lambda<A> left) {
        left.argumentType.unify(this.argumentType);
        left.bound.unify(this.bound);
    }
    <B> Term<B> substitute(Function<A, Term<B>> f) {
        return new Lambda<B>(argumentType.substitute(f), bound.substitute(traverse(f)) );
    }
    Term<A> reduce() {
        return new Lambda<A>(argumentType.reduce(), bound.reduce());
    }
    Term<A> reduceCall(Term<A> argument) {
        return bound.substitute(o -> o.isPresent() ? new Variable<>(o.get()) : argument).reduce();
    }
    Term<A> typeCheck(Function<A, Term<A>> enviroment) {
        argumentType.typeCheck(enviroment).inhabited();
        Term<A> argumentType = this.argumentType.reduce();
        return new Forall<>(argumentType, bound.typeCheck(o ->
                            (o.isPresent() ? enviroment.apply(o.get()) : argumentType).substitute(x -> new Variable<>(Optional.of(x)))));
    }
    public String toString() {
        return "λ _ : (" + argumentType.toString() + "). " + bound.toString();
    }
}

class Forall<A> extends Term<A> {
    final Term<A> argumentType;
    final Term<Optional<A>> resultType;
    Forall(Term<A> argumentType, Term<Optional<A>> resultType) {
        this.argumentType = argumentType;
        this.resultType = resultType;
    }
    void unify(Term<A> right) {
        right.unifyRight(this);
    }
    void unifyRight(Forall<A> left) {
        left.argumentType.unify(this.argumentType);
        left.resultType.unify(this.resultType);
    }
    <B> Term<B> substitute(Function<A, Term<B>> f) {
        return new Forall<B>(argumentType.substitute(f), resultType.substitute(traverse(f)) );
    }
    Term<A> reduce() {
        return new Forall<A>(argumentType.reduce(), resultType.reduce());
    }
    Term<A> typeCheckCall(Term<A> argumentType, Term<A> argument)
    {
        this.argumentType.unify(argumentType);
        return resultType.substitute(o -> o.isPresent() ? new Variable<>(o.get()) : argument).reduce();
    }
    Term<A> typeCheck(Function<A, Term<A>> enviroment) {
        argumentType.typeCheck(enviroment).inhabited();
        Term<A> argumentType = this.argumentType.reduce();
        return resultType.typeCheck(o -> (o.isPresent() ? enviroment.apply(o.get()) : argumentType).substitute(x -> new Variable<>(Optional.of(x)))).inhabited();
    }
    public String toString() {
        return "∀ _ : (" + argumentType.toString() + "). " + resultType.toString();
    }
}


public class TypeSystem {
    public static void main(String[] $) {
        Term<Void> id = new Lambda<>(new Type<>(), new Lambda<>(new Variable<>(Optional.empty()), new Variable<>(Optional.empty())) );
        System.out.println(id);
        Term<Void> endo = id.typeCheck(v -> v.absurd());
        System.out.println(endo);
        Term<Void> p = endo.typeCheck(v -> v.absurd());
        System.out.println(p);
        Term<Void> idUniversal = new Call<>(id, endo);
        System.out.println(idUniversal.typeCheck(v -> v.absurd()));
        System.out.println(idUniversal);
        System.out.println(idUniversal.reduce());
    }
}