jltypes.c

/*
  Types
  . type predicates (subtype) and type matching
  . type union and intersection
  . builtin type definitions
*/
#include <stdlib.h>
#include <stdio.h>
#include <string.h>
#include <stdarg.h>
#include <assert.h>
#include <sys/types.h>
#include <limits.h>
#include <errno.h>
#include <math.h>
#ifdef BOEHM_GC
#include <gc.h>
#endif
#include "llt.h"
#include "julia.h"
#include "newobj_internal.h"

jl_tag_type_t *jl_any_type;
jl_tag_type_t *jl_type_type;
jl_struct_type_t *jl_typename_type;
jl_struct_type_t *jl_sym_type;
jl_struct_type_t *jl_symbol_type;
jl_tuple_t *jl_tuple_type;
jl_typename_t *jl_tuple_typename;
jl_tag_type_t *jl_ntuple_type;
jl_typename_t *jl_ntuple_typename;
jl_struct_type_t *jl_tvar_type;

jl_struct_type_t *jl_func_kind;
jl_struct_type_t *jl_union_kind;
jl_struct_type_t *jl_tag_kind;
jl_struct_type_t *jl_tag_type_type;
jl_struct_type_t *jl_struct_kind;
jl_struct_type_t *jl_bits_kind;

jl_type_t *jl_bottom_type;
jl_tag_type_t *jl_seq_type;
jl_tag_type_t *jl_tensor_type;

jl_bits_type_t *jl_bool_type;
jl_bits_type_t *jl_char_type;
jl_bits_type_t *jl_int8_type;
jl_bits_type_t *jl_uint8_type;
jl_bits_type_t *jl_int16_type;
jl_bits_type_t *jl_uint16_type;
jl_bits_type_t *jl_int32_type;
jl_bits_type_t *jl_uint32_type;
jl_bits_type_t *jl_int64_type;
jl_bits_type_t *jl_uint64_type;
jl_bits_type_t *jl_float32_type;
jl_bits_type_t *jl_float64_type;

jl_tuple_t *jl_null;

jl_func_type_t *jl_any_func;
jl_function_t *jl_bottom_func;

void jl_add_constructors(jl_struct_type_t *t);

// --- type properties and predicates ---

int jl_is_type(jl_value_t *v)
{
    if (jl_is_tuple(v)) {
        jl_tuple_t *t = (jl_tuple_t*)v;
        size_t i;
        for(i=0; i < t->length; i++) {
            jl_value_t *vv = jl_tupleref(t, i);
            if (!jl_is_typevar(vv) && !jl_is_type(vv) &&
                !jl_is_typector(vv))
                return 0;
        }
        return 1;
    }
    return (jl_typeis(v, jl_union_kind) ||
            jl_typeis(v, jl_struct_kind) ||
            jl_typeis(v, jl_func_kind) ||
            jl_typeis(v, jl_tag_kind) ||
            jl_typeis(v, jl_bits_kind));
}

int jl_has_typevars_(jl_value_t *v, int incl_wildcard);

DLLEXPORT int jl_is_leaf_type(jl_value_t *v)
{
    if (jl_is_bits_type(v))
        return 1;
    if (jl_is_tuple(v)) {
        jl_tuple_t *t = (jl_tuple_t*)v;
        size_t i;
        for(i=0; i < t->length; i++) {
            if (!jl_is_leaf_type(jl_tupleref(t, i)))
                return 0;
        }
        return 1;
    }
    if (!jl_is_struct_type(v))
        return 0;
    return !jl_has_typevars_(v,1);
}

int jl_has_typevars_(jl_value_t *v, int incl_wildcard)
{
    size_t i;
    if (jl_typeis(v, jl_tvar_type)) {
        if (!((jl_tvar_t*)v)->bound)
            return incl_wildcard;
        return 1;
    }
    if (jl_is_func_type(v))
        return jl_has_typevars_((jl_value_t*)((jl_func_type_t*)v)->from,
                                incl_wildcard) ||
            jl_has_typevars_((jl_value_t*)((jl_func_type_t*)v)->to,
                             incl_wildcard);

    jl_tuple_t *t;
    if (jl_is_union_type(v))
        t = ((jl_uniontype_t*)v)->types;
    else if (jl_is_some_tag_type(v))
        t = ((jl_tag_type_t*)v)->parameters;
    else if (jl_is_tuple(v))
        t = (jl_tuple_t*)v;
    else
        t = jl_null;
    for(i=0; i < t->length; i++) {
        jl_value_t *elt = jl_tupleref(t, i);
        if (elt != v) {
            if (jl_has_typevars_(elt, incl_wildcard))
                return 1;
        }
    }
    // probably not necessary; no reason to use match() instead of subtype()
    // on the unconstrained version of a type
    //if (jl_is_typector(v))
    //    return (((jl_typector_t*)v)->parameters->length > 0);
    return 0;
}

int jl_has_typevars(jl_value_t *v)
{
    return jl_has_typevars_(v, 0);
}

// construct the full type of a value, possibly making a tuple type
jl_value_t *jl_full_type(jl_value_t *v)
{
    if (!jl_is_tuple(v))
        return (jl_value_t*)jl_typeof(v);
    jl_tuple_t *in = (jl_tuple_t*)v;
    jl_tuple_t *out = jl_alloc_tuple(in->length);
    JL_GC_PUSH(&out);
    size_t i;
    for(i=0; i < in->length; i++) {
        jl_tupleset(out, i, jl_full_type(jl_tupleref(in, i)));
    }
    JL_GC_POP();
    return (jl_value_t*)out;
}

typedef struct _jl_value_pair_t {
    jl_value_t *a;
    jl_value_t *b;
    struct _jl_value_pair_t *next;
} jl_value_pair_t;

static int type_eqv_(jl_value_t *a, jl_value_t *b, jl_value_pair_t *stack);

// --- union and intersection ---

static int count_union_components(jl_tuple_t *types)
{
    size_t i, c=0;
    for(i=0; i < types->length; i++) {
        jl_value_t *e = jl_tupleref(types,i);
        if (jl_is_union_type(e)) {
            c += count_union_components(((jl_uniontype_t*)e)->types);
        }
        else {
            c++;
        }
    }
    return c;
}

static void flatten_type_union(jl_tuple_t *types, jl_value_t **out, size_t *idx)
{
    size_t i;
    for(i=0; i < types->length; i++) {
        jl_value_t *e = jl_tupleref(types,i);
        if (jl_is_union_type(e)) {
            flatten_type_union(((jl_uniontype_t*)e)->types, out, idx);
        }
        else {
            out[*idx] = e;
            (*idx)++;
        }
    }
}

DLLEXPORT
jl_tuple_t *jl_compute_type_union(jl_tuple_t *types)
{
    size_t n = count_union_components(types);
    jl_value_t **temp = alloca(n * sizeof(jl_value_t*));
    size_t idx=0;
    flatten_type_union(types, temp, &idx);
    JL_GC_PUSHARGS(temp, n);
    assert(idx == n);
    size_t i, j, ndel=0;
    for(i=0; i < n; i++) {
        for(j=0; j < n; j++) {
            if (j != i && temp[i] && temp[j]) {
                if (temp[i] == temp[j] ||
                    type_eqv_(temp[i], temp[j], NULL) ||
                    (jl_subtype(temp[i], temp[j], 0) /*&&
                     (temp[i] == (jl_value_t*)jl_bottom_type ||
                     !jl_has_typevars(temp[j]))*/)) {
                    temp[i] = NULL;
                    ndel++;
                }
                else if (jl_is_typevar(temp[i]) && jl_is_typevar(temp[j])) {
                    jl_tvar_t *ti = (jl_tvar_t*)temp[i];
                    jl_tvar_t *tj = (jl_tvar_t*)temp[j];
                    if (jl_subtype(ti->lb,tj->lb,0) &&
                        jl_subtype(ti->lb,tj->ub,0) &&
                        jl_subtype(ti->ub,tj->ub,0)) {
                        temp[j] =
                            (jl_value_t*)
                            jl_new_typevar(tj->name, ti->lb, tj->ub);
                        temp[i] = NULL;
                    }
                }
            }
        }
    }
    jl_tuple_t *result = jl_alloc_tuple_uninit(n - ndel);
    j=0;
    for(i=0; i < n; i++) {
        if (temp[i] != NULL) {
            jl_tupleset(result, j, temp[i]);
            j++;
        }
    }
    assert(j == n-ndel);
    JL_GC_POP();
    return result;
}

jl_value_t *jl_type_union(jl_tuple_t *types)
{
    types = jl_compute_type_union(types);
    if (types->length == 1)
        return jl_tupleref(types, 0);
    if (types->length == 0)
        return (jl_value_t*)jl_bottom_type;
    JL_GC_PUSH(&types);
    jl_value_t *tu = (jl_value_t*)jl_new_uniontype(types);
    JL_GC_POP();
    return tu;
}

static jl_value_t *unsafe_type_union(jl_tuple_t *types)
{
    types = jl_compute_type_union(types);
    if (types->length == 1)
        return jl_tupleref(types, 0);
    if (types->length == 0)
        return (jl_value_t*)jl_bottom_type;
    JL_GC_PUSH(&types);
    jl_uniontype_t *t = (jl_uniontype_t*)newobj((jl_type_t*)jl_union_kind, 1);
    t->types = types;
    JL_GC_POP();
    return (jl_value_t*)t;
}

static jl_value_t *jl_type_intersect(jl_value_t *a, jl_value_t *b,
                                     jl_tuple_t **penv);

static jl_value_t *intersect_union(jl_uniontype_t *a, jl_value_t *b,
                                   jl_tuple_t **penv)
{
    jl_tuple_t *t = jl_alloc_tuple(a->types->length);
    JL_GC_PUSH(&t);
    size_t i;
    for(i=0; i < t->length; i++) {
        jl_tupleset(t, i, jl_type_intersect(jl_tupleref(a->types,i),b,penv));
    }
    // problem: an intermediate union type we make here might be too
    // complex, even though the final type after typevars are replaced
    // might be ok.
    jl_value_t *tu = unsafe_type_union(t);
    JL_GC_POP();
    return tu;
}

// if returns with *bot!=0, then intersection is None
static size_t tuple_intersect_size(jl_tuple_t *a, jl_tuple_t *b, int *bot)
{
    size_t al = a->length;
    size_t bl = b->length;
    *bot = 0;
    if (al == bl) return al;
    if (al > bl) return tuple_intersect_size(b, a, bot);
    assert(al < bl);
    if (jl_is_seq_type(jl_tupleref(b,bl-1))) {
        if (al > 0 && jl_is_seq_type(jl_tupleref(a,al-1))) {
            return bl;
        }
        else {
            if (bl == al+1)
                return al;
            *bot=1;
            return 0;
        }
    }
    if (al > 0 && jl_is_seq_type(jl_tupleref(a,al-1))) {
        return bl;
    }
    *bot=1;
    return 0;
}

static jl_value_t *intersect_tuple(jl_tuple_t *a, jl_tuple_t *b,
                                   jl_tuple_t **penv)
{
    size_t al = a->length;
    size_t bl = b->length;
    int bot=0;
    size_t n = tuple_intersect_size(a, b, &bot);
    if (bot)
        return (jl_value_t*)jl_bottom_type;
    if (n == 0) return (jl_value_t*)jl_null;
    jl_tuple_t *tc = jl_alloc_tuple(n);
    jl_value_t *result = (jl_value_t*)tc;
    jl_value_t *ce = NULL;
    JL_GC_PUSH(&tc, &ce);
    size_t ai=0, bi=0, ci;
    jl_value_t *ae=NULL, *be=NULL;
    int aseq=0, bseq=0;
    for(ci=0; ci < n; ci++) {
        if (ai < al) {
            ae = jl_tupleref(a,ai);
            if (jl_is_seq_type(ae)) {
                aseq=1;
                ae = jl_tparam0(ae);
            }
            ai++;
        }
        if (bi < bl) {
            be = jl_tupleref(b,bi);
            if (jl_is_seq_type(be)) {
                bseq=1;
                be = jl_tparam0(be);
            }
            bi++;
        }
        assert(ae!=NULL && be!=NULL);
        ce = jl_type_intersect(ae,be,penv);
        if (ce == (jl_value_t*)jl_bottom_type) {
            if (aseq && bseq) {
                // (X∩Y)==∅ → (X...)∩(Y...) == ()
                if (n == 1) {
                    result = (jl_value_t*)jl_null;
                    goto done_intersect_tuple;
                }
                tc->length--;
                goto done_intersect_tuple;
            }
            result = (jl_value_t*)jl_bottom_type;
            goto done_intersect_tuple;
        }
        if (aseq && bseq) {
            ce = (jl_value_t*)jl_tuple1(ce);
            ce = (jl_value_t*)jl_apply_type((jl_value_t*)jl_seq_type,
                                            (jl_tuple_t*)ce);
        }
        jl_tupleset(tc, ci, ce);
    }
 done_intersect_tuple:
    JL_GC_POP();
    return result;
}

static jl_value_t *intersect_tag(jl_tag_type_t *a, jl_tag_type_t *b,
                                 jl_tuple_t **penv)
{
    assert(a->name == b->name);
    assert(a->parameters->length == b->parameters->length);
    jl_tuple_t *p = jl_alloc_tuple(a->parameters->length);
    JL_GC_PUSH(&p);
    size_t i;
    for(i=0; i < p->length; i++) {
        jl_value_t *ap = jl_tupleref(a->parameters,i);
        jl_value_t *bp = jl_tupleref(b->parameters,i);
        jl_value_t *ti;
        if (a->name == jl_ntuple_typename || jl_is_tag_type(a) ||
            jl_has_typevars_(ap,1) || jl_has_typevars_(bp,1)) {
            ti = jl_type_intersect(ap,bp,penv);
        }
        else if (type_eqv_(ap,bp,NULL)) {
            ti = ap;
        }
        else {
            JL_GC_POP();
            return (jl_value_t*)jl_bottom_type;
        }
        if (ti == (jl_value_t*)jl_bottom_type) {
            JL_GC_POP();
            return (jl_value_t*)jl_bottom_type;
        }
        jl_tupleset(p, i, ti);
    }
    if (a->name->primary != NULL) {
        jl_value_t *res = (jl_value_t*)jl_apply_type(a->name->primary, p);
        JL_GC_POP();
        return res;
    }
    assert(0 && "not yet implemented");
    return NULL;
}

static void tvar_union(jl_tuple_t **penv, jl_tvar_t *a, jl_value_t *b)
{
    // when we add X->Y, change all Z->X to Z->Y
    if ((jl_value_t*)a == b) return;
    jl_tuple_t *p = *penv;
    while (p != jl_null) {
        if (jl_t1(p) == (jl_value_t*)a) {
            jl_t1(p) = b;
            tvar_union(penv, (jl_tvar_t*)jl_t0(p), b);
        }
        p = (jl_tuple_t*)jl_nextpair(p);
    }
}

static jl_value_t *tvar_find(jl_tuple_t **penv, jl_value_t *b)
{
    jl_tuple_t *p = *penv;
    while (p != jl_null) {
        if (jl_t0(p) == b)
            return jl_t1(p);
        p = (jl_tuple_t*)jl_nextpair(p);
    }
    return b;
}

static jl_value_t *meet_tvars(jl_tvar_t *a, jl_tvar_t *b)
{
    jl_value_t *lb=NULL, *ub=NULL;
    if (type_eqv_((jl_value_t*)a->lb, (jl_value_t*)b->lb, NULL) &&
        type_eqv_((jl_value_t*)a->ub, (jl_value_t*)b->ub, NULL))
        return (jl_value_t*)b;
    ub = jl_type_intersection((jl_value_t*)a->ub, (jl_value_t*)b->ub);
    if (ub == (jl_value_t*)jl_bottom_type)
        return ub;
    JL_GC_PUSH(&lb, &ub);
    lb = jl_type_union(jl_tuple2(a->lb, b->lb));
    if (!jl_subtype(lb, ub, 0)) {
        JL_GC_POP();
        return (jl_value_t*)jl_bottom_type;
    }
    // TODO: might not want to collapse tvar to non-tvar in all cases
    if (jl_is_leaf_type(ub)) {
        JL_GC_POP();
        return ub;
    }
    jl_value_t *res = (jl_value_t*)jl_new_typevar(jl_symbol("_"), lb, ub);
    JL_GC_POP();
    return res;
}

static jl_value_t *meet_tvar(jl_tvar_t *tv, jl_value_t *ty)
{
    if (jl_is_typevar(ty))
        return (jl_value_t*)meet_tvars(tv, (jl_tvar_t*)ty);
    //if (jl_types_equal((jl_value_t*)tv->ub, ty))
    //    return ty;
    if (jl_subtype((jl_value_t*)tv->ub, ty, 0))
        return (jl_value_t*)tv;
    // TODO: should we check type_intersection(tv->ub, ty) instead?
    if (!jl_subtype(ty, (jl_value_t*)tv->ub, 0))
        return (jl_value_t*)jl_bottom_type;
    //if (jl_types_equal((jl_value_t*)tv->lb, ty))
    //    return ty;
    if (jl_subtype((jl_value_t*)tv->lb, ty, 0)) {
        if (jl_is_leaf_type(ty) || jl_is_int32(ty))
            return ty;
        assert(ty != (jl_value_t*)jl_bottom_type);
        return (jl_value_t*)jl_new_typevar(jl_symbol("_"), tv->lb, ty);
    }
    return (jl_value_t*)jl_bottom_type;
}

// use, essentially, union-find on type variables to group them
// into equivalence classes.
static jl_value_t *intersect_typevar(jl_tvar_t *a, jl_value_t *b,
                                     jl_tuple_t **penv)
{
    if (jl_subtype(b, (jl_value_t*)a, 0)) {
        //if (!a->bound) return b;
    }
    else if (jl_subtype((jl_value_t*)a, b, 0)) {
        if (!a->bound) return (jl_value_t*)a;
    }
    else {
        return (jl_value_t*)jl_bottom_type;
    }
    jl_tuple_t *p = *penv;
    while (p != jl_null) {
        if (jl_t0(p) == (jl_value_t*)a) {
            assert(jl_t1(p) != (jl_value_t*)a);
            jl_value_t *ti = jl_type_intersect(jl_t1(p), b, penv);
            if (ti == (jl_value_t*)jl_bottom_type)
                return (jl_value_t*)jl_bottom_type;
            if (jl_is_typevar(ti))
                ti = tvar_find(penv, ti);
            jl_t1(p) = ti;
            tvar_union(penv, a, ti);
            return (jl_value_t*)a;
        }
        p = (jl_tuple_t*)jl_nextpair(p);
    }
    if (jl_is_typevar(b))
        b = tvar_find(penv, b);
    if ((jl_value_t*)a != b) {
        jl_tuple_t *np = jl_tuple3((jl_value_t*)a, b, (jl_value_t*)*penv);
        *penv = np;
        tvar_union(penv, a, b);
    }
    return (jl_value_t*)a;
}

// TODO: handle NTuple
static jl_value_t *jl_type_intersect(jl_value_t *a, jl_value_t *b,
                                     jl_tuple_t **penv)
{
    if (jl_is_typector(a))
        a = (jl_value_t*)((jl_typector_t*)a)->body;
    if (jl_is_typector(b))
        b = (jl_value_t*)((jl_typector_t*)b)->body;
    if (a == b) return a;
    if (a == (jl_value_t*)jl_bottom_type || b == (jl_value_t*)jl_bottom_type)
        return (jl_value_t*)jl_bottom_type;
    if (jl_is_typevar(a))
        return intersect_typevar((jl_tvar_t*)a, b, penv);
    if (jl_is_typevar(b))
        return intersect_typevar((jl_tvar_t*)b, a, penv);
    if (!jl_has_typevars(a) && !jl_has_typevars(b)) {
        if (jl_subtype(a, b, 0))
            return a;
        if (jl_subtype(b, a, 0))
            return b;
    }
    // union
    if (jl_is_union_type(a))
        return intersect_union((jl_uniontype_t*)a, b, penv);
    if (jl_is_union_type(b))
        return intersect_union((jl_uniontype_t*)b, a, penv);
    if (a == (jl_value_t*)jl_undef_type) return (jl_value_t*)jl_bottom_type;
    if (b == (jl_value_t*)jl_undef_type) return (jl_value_t*)jl_bottom_type;
    if (a == (jl_value_t*)jl_any_type) return b;
    if (b == (jl_value_t*)jl_any_type) return a;
    // tuple
    if (jl_is_tuple(a)) {
        if (!jl_is_tuple(b))
            return (jl_value_t*)jl_bottom_type;
        return intersect_tuple((jl_tuple_t*)a, (jl_tuple_t*)b, penv);
    }
    if (jl_is_tuple(b))
        return (jl_value_t*)jl_bottom_type;
    // function
    if (jl_is_func_type(a)) {
        if (!jl_is_func_type(b))
            return (jl_value_t*)jl_bottom_type;
        jl_value_t *t1=NULL, *t2=NULL;
        JL_GC_PUSH(&t1, &t2);
        t1 = (jl_value_t*)jl_tuple2(((jl_func_type_t*)a)->from,
                                    ((jl_func_type_t*)b)->from);
        t1 = jl_type_union((jl_tuple_t*)t1);
        t2 = jl_type_intersect((jl_value_t*)((jl_func_type_t*)a)->to,
                               (jl_value_t*)((jl_func_type_t*)b)->to, penv);
        jl_value_t *result =
            (jl_value_t*)jl_new_functype((jl_type_t*)t1, (jl_type_t*)t2);
        JL_GC_POP();
        return result;
    }
    if (jl_is_func_type(b))
        return (jl_value_t*)jl_bottom_type;
    if (jl_is_int32(a) || jl_is_int32(b))
        return (jl_value_t*)jl_bottom_type;
    // tag
    assert(jl_is_some_tag_type(a));
    assert(jl_is_some_tag_type(b));
    jl_tag_type_t *tta = (jl_tag_type_t*)a;
    jl_tag_type_t *ttb = (jl_tag_type_t*)b;
    if (tta->name == ttb->name)
        return (jl_value_t*)intersect_tag(tta, ttb, penv);
    jl_tag_type_t *super = NULL;
    jl_tag_type_t *sub = NULL;
    jl_value_t *ti = NULL;
    jl_value_t *env = NULL;
    jl_tuple_t *p = NULL;
    JL_GC_PUSH(&super, &sub, &env, &p);
    while (tta != jl_any_type) {
        if (tta->name == ttb->name) {
            ti = intersect_tag(tta, ttb, penv);
            if (ti == (jl_value_t*)jl_bottom_type) {
                JL_GC_POP();
                return ti;
            }
            super = (jl_tag_type_t*)ti;
            sub = (jl_tag_type_t*)a;
            break;
        }
        tta = tta->super;
    }
    if (super == NULL) {
        tta = (jl_tag_type_t*)a;
        while (ttb != jl_any_type) {
            if (tta->name == ttb->name) {
                ti = intersect_tag(tta, ttb, penv);
                if (ti == (jl_value_t*)jl_bottom_type) {
                    JL_GC_POP();
                    return ti;
                }
                super = (jl_tag_type_t*)ti;
                sub = (jl_tag_type_t*)b;
                break;
            }
            ttb = ttb->super;
        }
    }
    if (super == NULL) {
        JL_GC_POP();
        return (jl_value_t*)jl_bottom_type;
    }

    size_t n = sub->parameters->length;
    if (n == 0) {
        JL_GC_POP();
        return (jl_value_t*)sub;
    }

    assert(sub->name->primary != NULL);
    jl_value_t *tc = sub->name->primary;
    // compute what constraints the supertype imposes on the subtype
    jl_tuple_t *sup_params =
        ((jl_tag_type_t*)((jl_tag_type_t*)tc)->super)->parameters;
    // match the intersected supertype against the pattern this subtype
    // uses to instantiate its supertype. this tells us what subtype parameter
    // values are implied by the intersected supertype, or that the
    // intersected supertype cannot come from this subtype (in which case
    // our final answer is None).
    size_t i;
    // hack: we need type_match to find assignments for these
    // typevars even though they normally aren't constrained. temporarily
    // switch them.
    jl_tuple_t *tc_params = ((jl_tag_type_t*)tc)->parameters;
    for(i=0; i < tc_params->length; i++) {
        jl_tvar_t *tv = (jl_tvar_t*)jl_tupleref(tc_params,i);
        assert(jl_is_typevar(tv));
        assert(!tv->bound);
        tv->bound = 1;
    }
    env = jl_type_match((jl_value_t*)super->parameters,
                        (jl_value_t*)sup_params);
    for(i=0; i < tc_params->length; i++) {
        jl_tvar_t *tv = (jl_tvar_t*)jl_tupleref(tc_params,i);
        tv->bound = 0;
    }
    if (env == jl_false) {
        JL_GC_POP();
        return (jl_value_t*)jl_bottom_type;
    }

    p = jl_alloc_tuple(n);
    for(i=0; i < n; i++) {
        jl_value_t *tp = jl_tupleref(((jl_tag_type_t*)tc)->parameters, i);
        jl_value_t *elt = jl_tupleref(sub->parameters, i);
        jl_value_t *e = env;
        while (e != (jl_value_t*)jl_null) {
            if (jl_t0(e) == tp) {
                elt = jl_type_intersect(elt, jl_t1(e), penv);
                assert(elt != (jl_value_t*)jl_bottom_type);
                break;
            }
            e = jl_nextpair(e);
        }
        jl_tupleset(p, i, elt);
    }
    jl_value_t *result = (jl_value_t*)jl_apply_type(tc, p);
    JL_GC_POP();
    return result;
}

jl_value_t *jl_type_intersection(jl_value_t *a, jl_value_t *b)
{
    jl_tuple_t *env = jl_null;
    JL_GC_PUSH(&env);
    jl_value_t *ti = jl_type_intersection_matching(a, b, &env);
    JL_GC_POP();
    return ti;
}

jl_value_t *jl_type_intersection_matching(jl_value_t *a, jl_value_t *b,
                                          jl_tuple_t **penv)
{
    jl_value_t *ti = jl_type_intersect(a, b, penv);
    if (ti == (jl_value_t*)jl_bottom_type)
        return ti;
    jl_value_t *m=NULL;
    jl_tuple_t *t=NULL;
    JL_GC_PUSH(&ti, &m, &t);
    if (*penv != jl_null) {
        //jl_show(*penv); ios_printf(ios_stdout,"\n");
        jl_tuple_t *p = *penv;
        while (p != jl_null) {
            jl_tvar_t *tv = (jl_tvar_t*)jl_t0(p);
            jl_value_t *pv = jl_t1(p);
            m = meet_tvar(tv, pv);
            if (m == (jl_value_t*)jl_bottom_type) {
                JL_GC_POP();
                return m;
            }
            if (m != pv) {
                if (jl_is_typevar(pv))
                    tvar_union(penv, (jl_tvar_t*)pv, m);
                jl_t1(p) = m;
                tvar_union(penv, tv, m);
            }
            p = (jl_tuple_t*)jl_nextpair(p);
        }
        t = jl_flatten_pairs(*penv);
        ti = (jl_value_t*)
            jl_instantiate_type_with((jl_type_t*)ti,
                                     &jl_tupleref(t, 0), t->length/2);
    }
    JL_GC_POP();
    return ti;
}

// --- type instantiation and cache ---

static int extensionally_same_type(jl_value_t *a, jl_value_t *b)
{
    return (jl_subtype(a, b, 0) && jl_subtype(b, a, 0));
}

static int type_eqv_(jl_value_t *a, jl_value_t *b, jl_value_pair_t *stack)
{
    jl_value_pair_t top;
    if (a == b) return 1;
    if (jl_is_typector(a)) a = (jl_value_t*)((jl_typector_t*)a)->body;
    if (jl_is_typector(b)) b = (jl_value_t*)((jl_typector_t*)b)->body;
    if (jl_is_typevar(a)) return 0;
    if (jl_is_int32(a)) {
        if (jl_is_int32(b))
            return (jl_unbox_int32(a) == jl_unbox_int32(b));
        return 0;
    }
    if (jl_is_tuple(a)) {
        if (jl_is_tuple(b)) {
            return extensionally_same_type(a, b);
        }
        return 0;
    }
    if (jl_is_union_type(a)) {
        if (jl_is_union_type(b)) {
            return extensionally_same_type(a, b);
        }
        return 0;
    }
    jl_value_pair_t *p = stack;
    while (p != NULL) {
        if (p->a == a && p->b == b)
            return 1;
        p = p->next;
    }
    top.a = a;
    top.b = b;
    top.next = stack;
    stack = &top;
    if (jl_is_func_type(a)) {
        if (jl_is_func_type(b)) {
            return (type_eqv_((jl_value_t*)((jl_func_type_t*)a)->from,
                              (jl_value_t*)((jl_func_type_t*)b)->from, stack) &&
                    type_eqv_((jl_value_t*)((jl_func_type_t*)a)->to,
                              (jl_value_t*)((jl_func_type_t*)b)->to, stack));
        }
        return 0;
    }
    assert(jl_is_some_tag_type(a));
    if (!jl_is_some_tag_type(b)) return 0;
    jl_tag_type_t *tta = (jl_tag_type_t*)a;
    jl_tag_type_t *ttb = (jl_tag_type_t*)b;
    if (tta->name != ttb->name) return 0;
    jl_tuple_t *ap = tta->parameters;
    jl_tuple_t *bp = ttb->parameters;
    assert(ap->length == bp->length);
    size_t i;
    for(i=0; i < ap->length; i++) {
        jl_value_t *api = jl_tupleref(ap,i);
        jl_value_t *bpi = jl_tupleref(bp,i);
        if (api == bpi) continue;
        if (!type_eqv_(api, bpi, stack))
            return 0;
    }
    return 1;
}

int jl_types_equal(jl_value_t *a, jl_value_t *b)
{
    return type_eqv_(a, b, NULL);
}

static int type_le_generic(jl_value_t *a, jl_value_t *b)
{
    jl_value_t *env = jl_type_match(a, b);
    if (env == jl_false) return 0;
    jl_value_t *vp = env;
    jl_value_t *x;
    // make sure all typevars correspond to other unique typevars
    while (vp != (jl_value_t*)jl_null) {
        if (!jl_is_typevar(jl_t1(vp)))
            return 0;
        x = env;
        while (x != (jl_value_t*)jl_null) {
            if (x != vp) {
                if (jl_t1(x) == jl_t1(vp))
                    return 0;
            }
            x = jl_nextpair(x);
        }
        vp = jl_nextpair(vp);
    }
    return 1;
}

int jl_types_equal_generic(jl_value_t *a, jl_value_t *b)
{
    return type_le_generic(a, b) && type_le_generic(b, a);
}

static jl_value_t *apply_type_(jl_value_t *tc, jl_value_t **params, size_t n)
{
    if (n == 0) {
        if (jl_is_typector(tc))
            return (jl_value_t*)((jl_typector_t*)tc)->body;
        return tc;
    }
    size_t i;
    char *tname;
    jl_tuple_t *tp;
    if (jl_is_typector(tc)) {
        tp = ((jl_typector_t*)tc)->parameters;
        tname = "alias";
    }
    else {
        assert(jl_is_some_tag_type(tc));
        tp = ((jl_tag_type_t*)tc)->parameters;
        tname = ((jl_tag_type_t*)tc)->name->name->name;
    }
    for(i=0; i < n; i++) {
        jl_value_t *pi = params[i];
        if (!jl_is_typector(pi) && !jl_is_type(pi) && !jl_is_int32(pi) &&
            !jl_is_typevar(pi))
            jl_errorf("invalid parameter %s for type %s",
                      jl_show_to_string(pi), tname);
    }
    if (tc == (jl_value_t*)jl_ntuple_type && (n==1||n==2) &&
        jl_is_int32(params[0])) {
        size_t nt = jl_unbox_int32(params[0]);
        if (nt == 0) return (jl_value_t*)jl_null;
        if (n==2) {
            jl_tuple_t *tup = jl_alloc_tuple_uninit(nt);
            jl_value_t *eltype = params[1];
            for(i=0; i < nt; i++) {
                jl_tupleset(tup, i, eltype);
            }
            return (jl_value_t*)tup;
        }
    }
    if (n > tp->length)
        jl_errorf("too many parameters for type %s", tname);
    jl_value_t **env = alloca(2 * tp->length * sizeof(jl_value_t*));
    for(i=0; i < tp->length; i++) {
        jl_tvar_t *tv = (jl_tvar_t*)jl_tupleref(tp,i);
        if (!jl_is_typevar(tv))
            continue;
        env[i*2+0] = (jl_value_t*)tv;
        if (i >= n) {
            env[i*2+1] = (jl_value_t*)tv;
        }
        else {
            if (!jl_is_typevar(params[i]) &&
                // TODO: Undef should not be special here; fix.
                // maybe introduce Top == Union(Any,Undef), and make this
                // the default upper bound.
                params[i] != (jl_value_t*)jl_undef_type &&
                !jl_subtype(params[i], (jl_value_t*)tv, 0))
                jl_errorf("%s: %s does not match type parameter %s",
                          tname, jl_show_to_string(params[i]),
                          jl_show_to_string((jl_value_t*)tv));
            if (jl_is_typector(params[i]))
                env[i*2+1] = (jl_value_t*)((jl_typector_t*)params[i])->body;
            else
                env[i*2+1] = params[i];
        }
    }
    if (jl_is_typector(tc)) tc = (jl_value_t*)((jl_typector_t*)tc)->body;
    return (jl_value_t*)jl_instantiate_type_with((jl_type_t*)tc, env, i);
}

jl_value_t *jl_apply_type(jl_value_t *tc, jl_tuple_t *params)
{
    return apply_type_(tc, &jl_tupleref(params,0), params->length);
}

typedef struct _typekey_stack_t {
    jl_typename_t *tn;
    jl_value_t **key;
    size_t n;  // key length
    jl_type_t *type;
    struct _typekey_stack_t *next;
} typekey_stack_t;

static jl_type_t *lookup_type(typekey_stack_t *table,
                              jl_typename_t *tn, jl_value_t **key, size_t n)
{
    assert(n > 0);
    while (table != NULL) {
        assert(table->n > 0);
        if (table->n == n && table->tn == tn) {
            size_t i;
            for(i=0; i < n; i++) {
                if (!type_eqv_(table->key[i], key[i], NULL))
                    break;
            }
            if (i==n) return table->type;
        }
        table = table->next;
    }
    return NULL;
}

static int t_uid_ctr = 1;  // TODO: lock

int jl_assign_type_uid()
{
    return t_uid_ctr++;
}

// TODO: synchronize
static void cache_type_(jl_value_t **key, size_t n, jl_type_t *type)
{
    // only cache concrete types
    if (jl_has_typevars((jl_value_t*)type))
        return;
    // assign uid
    if (jl_is_struct_type(type))
        ((jl_struct_type_t*)type)->uid = t_uid_ctr++;
    else if (jl_is_bits_type(type)) 
        ((jl_bits_type_t*)type)->uid = t_uid_ctr++;
    typekey_stack_t *tc =
        (typekey_stack_t*)((jl_tag_type_t*)type)->name->cache;
    typekey_stack_t *tk = (typekey_stack_t*)allocb(sizeof(typekey_stack_t));
    tk->tn = ((jl_tag_type_t*)type)->name;
    tk->type = type;
    tk->next = tc;
    ((jl_tag_type_t*)type)->name->cache = tk;
    tk->n = 0;
    tk->key = NULL;

    tk->key = (jl_value_t**)allocb(n * sizeof(void*));
    size_t i;
    for(i=0; i < n; i++) tk->key[i] = key[i];
    tk->n = n;
}

#ifdef JL_GC_MARKSWEEP
void jl_mark_type_cache(void *tc)
{
    typekey_stack_t *tk = (typekey_stack_t*)tc;
    while (tk != NULL) {
        jl_gc_setmark(tk);
        if (tk->key) jl_gc_setmark(tk->key);
        size_t i;
        for(i=0; i < tk->n; i++)
            jl_gc_markval(tk->key[i]);
        jl_gc_markval((jl_value_t*)tk->type);
        tk = tk->next;
    }
}
#endif

JL_CALLABLE(jl_f_tuple);
JL_CALLABLE(jl_constructor_factory_trampoline);

static jl_type_t *inst_type_w_(jl_value_t *t, jl_value_t **env, size_t n,
                               typekey_stack_t *stack)
{
    typekey_stack_t top;
    size_t i;
    if (n == 0) return (jl_type_t*)t;
    if (jl_is_typevar(t)) {
        for(i=0; i < n; i++) {
            if (env[i*2] == t)
                return (jl_type_t*)env[i*2+1];
        }
        return (jl_type_t*)t;
    }
    if (jl_is_tuple(t)) {
        jl_tuple_t *p = (jl_tuple_t*)t;
        jl_tuple_t *nt = jl_alloc_tuple(p->length);
        JL_GC_PUSH(&nt);
        for(i=0; i < p->length; i++) {
            jl_tupleset(nt, i, (jl_value_t*)inst_type_w_(jl_tupleref(p,i), env, n, stack));
        }
        JL_GC_POP();
        return (jl_type_t*)nt;
    }
    if (jl_is_union_type(t)) {
        jl_tuple_t *tw = (jl_tuple_t*)inst_type_w_((jl_value_t*)((jl_uniontype_t*)t)->types,
                                                   env, n, stack);
        JL_GC_PUSH(&tw);
        jl_type_t *res = (jl_type_t*)jl_new_uniontype(tw);
        JL_GC_POP();
        return res;
    }
    if (jl_is_func_type(t)) {
        jl_func_type_t *ft = (jl_func_type_t*)t;
        jl_type_t *fr=NULL, *to=NULL;
        JL_GC_PUSH(&fr, &to);
        fr = inst_type_w_((jl_value_t*)ft->from, env, n, stack);
        to = inst_type_w_((jl_value_t*)ft->to  , env, n, stack);
        jl_type_t *res = (jl_type_t*)jl_new_functype(fr, to);
        JL_GC_POP();
        return res;
    }
    if (jl_is_some_tag_type(t)) {
        jl_tag_type_t *tt = (jl_tag_type_t*)t;
        jl_tuple_t *tp = tt->parameters;
        if (jl_is_null(tp))
            return (jl_type_t*)t;
        jl_type_t *result;
        size_t ntp = tp->length;
        jl_value_t **iparams = (jl_value_t**)alloca((ntp+2) * sizeof(void*));
        for(i=0; i < ntp+2; i++) iparams[i] = NULL;
        jl_value_t **rt1 = &iparams[ntp+0];  // some extra gc roots
        jl_value_t **rt2 = &iparams[ntp+1];
        JL_GC_PUSHARGS(iparams, ntp+2);
        for(i=0; i < ntp; i++) {
            jl_value_t *elt = jl_tupleref(tp, i);
            if (elt == t)
                iparams[i] = t;
            else
                iparams[i] = (jl_value_t*)inst_type_w_(elt, env, n, stack);
        }
        jl_typename_t *tn = tt->name;

        // if an identical instantiation is already in process somewhere
        // up the stack, return it. this computes a fixed point for
        // recursive types.
        jl_type_t *lkup = lookup_type(stack, tn, iparams, ntp);
        if (lkup != NULL) { result = lkup; goto done_inst_tt; }

        // check type cache
        lkup = lookup_type(tn->cache, tn, iparams, ntp);
        if (lkup != NULL) { result = lkup; goto done_inst_tt; }

        // always use original type constructor (?)
        // only necessary for special cases like NTuple
        jl_value_t *tc = tn->primary;
        if (tc == (jl_value_t*)jl_ntuple_type && tc != t) {
            //(tc != NULL && tc != t)
            result = (jl_type_t*)apply_type_(tc, iparams, ntp);
            goto done_inst_tt;
        }

        // move array of instantiated parameters to heap; we need to keep it
        jl_tuple_t *iparams_tuple = jl_alloc_tuple_uninit(ntp);
        for(i=0; i < ntp; i++)
            jl_tupleset(iparams_tuple, i, iparams[i]);
        *rt1 = (jl_value_t*)iparams_tuple;
        if (jl_is_tag_type(t)) {
            jl_tag_type_t *tagt = (jl_tag_type_t*)t;
            jl_tag_type_t *ntt =
                (jl_tag_type_t*)newobj((jl_type_t*)jl_tag_kind, TAG_TYPE_NW);
            *rt2 = (jl_value_t*)ntt;
            top.tn = tn;
            top.key = iparams;
            top.n = ntp;
            top.type = (jl_type_t*)ntt;
            top.next = stack;
            stack = &top;
            ntt->name = tn;
            // temporarily initialize all fields so object is valid during
            // allocation of other objects (possible GC)
            ntt->fptr = NULL;
            ntt->env = NULL;
            ntt->linfo = NULL;
            ntt->super = jl_any_type;
            ntt->parameters = iparams_tuple;
            ntt->super = (jl_tag_type_t*)inst_type_w_((jl_value_t*)tagt->super,env,n,stack);
            cache_type_(iparams, ntp, (jl_type_t*)ntt);
            result = (jl_type_t*)ntt;
        }
        else if (jl_is_bits_type(t)) {
            jl_bits_type_t *bitst = (jl_bits_type_t*)t;
            jl_bits_type_t *nbt =
                (jl_bits_type_t*)newobj((jl_type_t*)jl_bits_kind, BITS_TYPE_NW);
            *rt2 = (jl_value_t*)nbt;
            top.tn = tn;
            top.key = iparams;
            top.n = ntp;
            top.type = (jl_type_t*)nbt;
            top.next = stack;
            stack = &top;
            nbt->name = tn;
            nbt->fptr = NULL;
            nbt->env = NULL;
            nbt->linfo = NULL;
            nbt->super = jl_any_type;
            nbt->parameters = iparams_tuple;
            nbt->nbits = bitst->nbits;
            nbt->bnbits = bitst->bnbits;
            nbt->super = (jl_tag_type_t*)inst_type_w_((jl_value_t*)bitst->super, env, n, stack);
            cache_type_(iparams, ntp, (jl_type_t*)nbt);
            result = (jl_type_t*)nbt;
        }
        else {
            assert(jl_is_struct_type(t));
            jl_struct_type_t *st = (jl_struct_type_t*)t;
            // create and initialize new struct type
            jl_struct_type_t *nst =
                (jl_struct_type_t*)newobj((jl_type_t*)jl_struct_kind,
                                          STRUCT_TYPE_NW);
            *rt2 = (jl_value_t*)nst;
            // associate these parameters with the new struct type on
            // the stack, in case one of its field types references it.
            top.tn = tn;
            top.key = iparams;
            top.n = ntp;
            top.type = (jl_type_t*)nst;
            top.next = stack;
            stack = &top;
            nst->name = tn;
            nst->super = jl_any_type;
            nst->parameters = iparams_tuple;
            nst->names = st->names;
            nst->types = jl_null; // to be filled in below
            nst->fptr = jl_constructor_factory_trampoline;
            nst->env = (jl_value_t*)nst;
            nst->linfo = NULL;
            nst->ctor_factory = st->ctor_factory;
            nst->instance = NULL;
            nst->uid = 0;
            nst->types = jl_null;
            nst->super = (jl_tag_type_t*)inst_type_w_((jl_value_t*)st->super, env,n,stack);
            jl_tuple_t *ftypes = st->types;
            if (ftypes != NULL) {
                // recursively instantiate the types of the fields
                jl_tuple_t *nftypes = jl_alloc_tuple(ftypes->length);
                nst->types = nftypes;
                for(i=0; i < ftypes->length; i++) {
                    jl_tupleset(nftypes, i,
                                (jl_value_t*)inst_type_w_(jl_tupleref(ftypes,i),
                                                          env,n,stack));
                }
            }
            cache_type_(iparams, ntp, (jl_type_t*)nst);
            if (!jl_has_typevars_((jl_value_t*)nst,1)) {
                jl_add_constructors(nst);
            }
            result = (jl_type_t*)nst;
        }
    done_inst_tt:
        JL_GC_POP();
        return result;
    }
    return (jl_type_t*)t;
}

jl_type_t *jl_instantiate_type_with(jl_type_t *t, jl_value_t **env, size_t n)
{
    return inst_type_w_((jl_value_t*)t, env, n, NULL);
}

int jl_subtype_le(jl_value_t *a, jl_value_t *b, int ta, int morespecific);

int jl_tuple_subtype(jl_value_t **child, size_t cl,
                     jl_value_t **parent, size_t pl, int ta, int morespecific)
{
    size_t ci=0, pi=0;
    while(1) {
        int cseq = !ta && (ci<cl) && jl_is_seq_type(child[ci]);
        int pseq = (pi<pl) && jl_is_seq_type(parent[pi]);
        if (ci >= cl)
            return (pi>=pl || pseq);
        if (cseq && !pseq)
            return 0;
        if (pi >= pl)
            return 0;
        jl_value_t *ce = child[ci];
        jl_value_t *pe = parent[pi];
        if (cseq) ce = jl_tparam0(ce);
        if (pseq) pe = jl_tparam0(pe);

        if (!jl_subtype_le(ce, pe, ta, morespecific))
            return 0;

        if (cseq && pseq) return 1;
        if (!cseq) ci++;
        if (!pseq) pi++;
    }
    return 0;
}

static int tuple_all_subtype(jl_tuple_t *t, jl_value_t *super,
                             int ta, int morespecific)
{
    size_t ci;
    for(ci=0; ci < t->length; ci++) {
        jl_value_t *ce = jl_tupleref(t,ci);
        if (!ta && jl_is_seq_type(ce))
            ce = jl_tparam0(ce);
        if (!jl_subtype_le(ce, super, ta, morespecific))
            return 0;
    }
    return 1;
}

/*
  ta specifies whether typeof() should be implicitly applied to a.
  this is used for tuple types to avoid allocating them explicitly.
  morespecific means we only care whether a is more specific than b,
  not necessarily a strict subtype
*/
int jl_subtype_le(jl_value_t *a, jl_value_t *b, int ta, int morespecific)
{
    if (!ta&&jl_is_typector(a)) a = (jl_value_t*)((jl_typector_t*)a)->body;
    if (jl_is_typector(b)) b = (jl_value_t*)((jl_typector_t*)b)->body;
    if (ta) {
        if (jl_is_tag_type(b) &&
            ((jl_tag_type_t*)b)->name == jl_type_type->name) {
            if (jl_is_type(a) || jl_is_typector(a))
                return jl_subtype_le(a, jl_tparam0(b), 0, morespecific);
        }
    }
    size_t i;
    if (jl_is_tuple(a)) {
        if ((jl_tuple_t*)b == jl_tuple_type) return 1;
        if (jl_is_tag_type(b) &&
            ((jl_tag_type_t*)b)->name == jl_ntuple_typename) {
            jl_tuple_t *tp = ((jl_tag_type_t*)b)->parameters;
            return tuple_all_subtype((jl_tuple_t*)a,
                                     jl_tupleref(tp,1), ta, morespecific);
        }
        if (jl_is_tuple(b)) {
            return jl_tuple_subtype(&jl_tupleref(a,0), ((jl_tuple_t*)a)->length,
                                    &jl_tupleref(b,0), ((jl_tuple_t*)b)->length,
                                    ta, morespecific);
        }
    }

    if (!ta && jl_is_union_type(a)) {
        jl_tuple_t *ap = ((jl_uniontype_t*)a)->types;
        for(i=0; i < ap->length; i++) {
            if (!jl_subtype_le(jl_tupleref(ap,i), b, 0, morespecific))
                return 0;
        }
        return 1;
    }

    if (jl_is_union_type(b)) {
        // None <: None
        if (!ta && a == b && b==(jl_value_t*)jl_bottom_type) return 1;
        jl_tuple_t *bp = ((jl_uniontype_t*)b)->types;
        for(i=0; i < bp->length; i++) {
            if (jl_subtype_le(a, jl_tupleref(bp,i), ta, morespecific))
                return 1;
        }
        return 0;
    }

    if (ta) a = (jl_value_t*)jl_typeof(a);

    if (a == b) return 1;
    if (a==(jl_value_t*)jl_undef_type || b==(jl_value_t*)jl_undef_type)
        return 0;
    if ((jl_tag_type_t*)b == jl_any_type) return 1;
    if (jl_is_typevar(a)) {
        if (jl_is_typevar(b)) {
            return
                jl_subtype_le((jl_value_t*)((jl_tvar_t*)a)->ub,
                              (jl_value_t*)((jl_tvar_t*)b)->ub, 0, 0) &&
                jl_subtype_le((jl_value_t*)((jl_tvar_t*)b)->lb,
                              (jl_value_t*)((jl_tvar_t*)a)->lb, 0, 0);
        }
        return jl_subtype_le((jl_value_t*)((jl_tvar_t*)a)->ub, b, 0, 0);
    }
    if (jl_is_typevar(b)) {
        return jl_subtype_le(a, (jl_value_t*)((jl_tvar_t*)b)->ub, 0, 0) &&
            jl_subtype_le((jl_value_t*)((jl_tvar_t*)b)->lb, a, 0, 0);
    }
    if ((jl_tag_type_t*)a == jl_any_type) return 0;
    if (jl_is_tuple(b)) {
        if (jl_is_tag_type(a) &&
            ((jl_tag_type_t*)a)->name == jl_ntuple_typename) {
            // only ((T>:S)...,) can be a supertype of NTuple[N,S]
            jl_tuple_t *tp = (jl_tuple_t*)b;
            jl_value_t *ntp = jl_tupleref(((jl_tag_type_t*)a)->parameters, 1);
            if (tp->length == 1 && jl_is_seq_type(jl_tupleref(tp,0))) {
                return jl_subtype_le(ntp, jl_tparam0(jl_tupleref(tp,0)),
                                     0, morespecific);
            }
        }
        return 0;
    }
    if (jl_is_tuple(a)) return 0;

    if (jl_is_int32(a)) {
        if (jl_is_int32(b))
            return (jl_unbox_int32(a)==jl_unbox_int32(b));
        return 0;
    }
    if (jl_is_int32(b)) return 0;

    if (jl_is_func_type(a)) {
        if (jl_is_func_type(b)) {
            jl_func_type_t *fa = (jl_func_type_t*)a;
            jl_func_type_t *fb = (jl_func_type_t*)b;
            return ( (jl_is_typevar(fb->from) ||
                      jl_subtype_le((jl_value_t*)fb->from,
                                    (jl_value_t*)fa->from, 0, morespecific)) &&
                      jl_subtype_le((jl_value_t*)fa->to,
                                    (jl_value_t*)fb->to,   0, morespecific) );
        }
        return 0;
    }
    else if (jl_is_func_type(b)) {
        return 0;
    }

    assert(jl_is_some_tag_type(a));
    assert(jl_is_some_tag_type(b));
    jl_tag_type_t *tta = (jl_tag_type_t*)a;
    jl_tag_type_t *ttb = (jl_tag_type_t*)b;
    int super=0;
    while (tta != (jl_tag_type_t*)jl_any_type) {
        if (tta->name == ttb->name) {
            if (super && morespecific) {
                if (tta->name != jl_type_type->name)
                    return 1;
            }
            if (tta->name == jl_ntuple_typename) {
                // NTuple must be covariant
                return jl_subtype_le(jl_tupleref(tta->parameters,1),
                                     jl_tupleref(ttb->parameters,1),
                                     0, morespecific);
            }
            assert(tta->parameters->length == ttb->parameters->length);
            for(i=0; i < tta->parameters->length; i++) {
                jl_value_t *apara = jl_tupleref(tta->parameters,i);
                jl_value_t *bpara = jl_tupleref(ttb->parameters,i);
                if (jl_is_typevar(bpara)) {
                    if (!jl_subtype_le(apara, bpara, 0, morespecific))
                        goto check_Type;
                }
                else {
                    if (!type_eqv_(apara, bpara, NULL))
                        goto check_Type;
                }
            }
            return 1;
        }
        tta = tta->super; super = 1;
    }
 check_Type:
    if (((jl_tag_type_t*)a)->name == jl_type_type->name) {
        // Type{T} matches either Type{>:T} or >:typeof(T)
        return jl_subtype_le(jl_tparam0(a), b, 1, morespecific);
    }

    return 0;
}

int jl_subtype(jl_value_t *a, jl_value_t *b, int ta)
{
    return jl_subtype_le(a, b, ta, 0);
}

int jl_type_morespecific(jl_value_t *a, jl_value_t *b, int ta)
{
    return jl_subtype_le(a, b, ta, 1);
}

static jl_value_t *type_match_(jl_value_t *child, jl_value_t *parent,
                               jl_tuple_t **env, int morespecific);

static jl_value_t *tuple_match(jl_tuple_t *child, jl_tuple_t *parent,
                               jl_tuple_t **env, int morespecific)
{
    size_t ci=0, pi=0;
    size_t cl = child->length;
    size_t pl = parent->length;
    while(1) {
        int cseq = (ci<cl) && jl_is_seq_type(jl_tupleref(child,ci));
        int pseq = (pi<pl) && jl_is_seq_type(jl_tupleref(parent,pi));
        if (ci >= cl)
            return (pi>=pl || pseq) ? (jl_value_t*)*env : jl_false;
        if (cseq && !pseq)
            return jl_false;
        if (pi >= pl)
            return jl_false;
        jl_value_t *ce = jl_tupleref(child,ci);
        jl_value_t *pe = jl_tupleref(parent,pi);
        if (cseq) ce = jl_tparam0(ce);
        if (pseq) pe = jl_tparam0(pe);

        *env = (jl_tuple_t*)type_match_(ce, pe, env, morespecific);
        if ((jl_value_t*)*env == jl_false) return (jl_value_t*)*env;

        if (cseq && pseq) return (jl_value_t*)*env;
        if (!cseq) ci++;
        if (!pseq) pi++;
    }
    return (jl_value_t*)*env;
}

jl_tag_type_t *jl_wrap_Type(jl_value_t *t)
{
    jl_value_t *env[2];
    env[0] = jl_tparam0(jl_type_type);
    env[1] = t;
    return (jl_tag_type_t*)
        jl_instantiate_type_with((jl_type_t*)jl_type_type, env, 1);
}

static jl_value_t *type_match_(jl_value_t *child, jl_value_t *parent,
                               jl_tuple_t **env, int morespecific)
{
    if (jl_is_typector(child))
        child = (jl_value_t*)((jl_typector_t*)child)->body;
    if (jl_is_typector(parent))
        parent = (jl_value_t*)((jl_typector_t*)parent)->body;
    size_t i;
    if (jl_is_typevar(parent)) {
        // make sure type is within this typevar's bounds
        if (!jl_subtype_le(child, parent, 0, 0))
            return jl_false;
        if (!((jl_tvar_t*)parent)->bound) return (jl_value_t*)*env;
        jl_tuple_t *p = *env;
        while (p != jl_null) {
            if (jl_t0(p) == (jl_value_t*)parent) {
                jl_value_t *pv = jl_t1(p);
                if (jl_is_typevar(pv) && jl_is_typevar(child)) {
                    if (pv == (jl_value_t*)child)
                        return (jl_value_t*)*env;
                    return jl_false;
                }
                if (morespecific) {
                    if (jl_subtype(child, pv, 0)) {
                        return (jl_value_t*)*env;
                    }
                    else if (jl_subtype(pv, child, 0)) {
                        jl_t1(p) = (jl_value_t*)child;
                        return (jl_value_t*)*env;
                    }
                }
                else {
                    if (type_eqv_(child, pv, NULL))
                        return (jl_value_t*)*env;
                }
                return jl_false;
            }
            p = (jl_tuple_t*)jl_nextpair(p);
        }
        jl_tuple_t *np = jl_tuple3(parent, child, (jl_value_t*)*env);
        return (jl_value_t*)np;
    }

    if (child == parent) return (jl_value_t*)*env;

    if (jl_is_typevar(child)) {
        if (jl_subtype_le(child, parent, 0, morespecific))
            return (jl_value_t*)*env;
        return jl_false;
    }
    if (jl_is_int32(child)) {
        if (jl_is_int32(parent)) {
            if (jl_unbox_int32((jl_value_t*)child) ==
                jl_unbox_int32((jl_value_t*)parent))
                return (jl_value_t*)*env;
        }
        return jl_false;
    }
    if (jl_is_int32(parent))
        return jl_false;
    if (parent == (jl_value_t*)jl_any_type) return (jl_value_t*)*env;
    if (child  == (jl_value_t*)jl_any_type) return jl_false;

    if (jl_is_union_type(child)) {
        jl_tuple_t *t = ((jl_uniontype_t*)child)->types;
        for(i=0; i < t->length; i++) {
            *env = (jl_tuple_t*)type_match_(jl_tupleref(t,i), parent, env,
                                            morespecific);
            if ((jl_value_t*)*env == jl_false) return (jl_value_t*)*env;
        }
        return (jl_value_t*)*env;
    }
    if (jl_is_union_type(parent)) {
        jl_tuple_t *t = ((jl_uniontype_t*)parent)->types;
        for(i=0; i < t->length; i++) {
            jl_value_t *p = type_match_(child, jl_tupleref(t,i), env,
                                        morespecific);
            if (p != jl_false) return p;
        }
        return jl_false;
    }

    if (jl_is_func_type(parent)) {
        if (jl_is_func_type(child)) {
            *env = (jl_tuple_t*)
                type_match_((jl_value_t*)((jl_func_type_t*)child)->from,
                            (jl_value_t*)((jl_func_type_t*)parent)->from, env,
                            morespecific);
            if ((jl_value_t*)*env == jl_false) return (jl_value_t*)*env;
            return type_match_((jl_value_t*)((jl_func_type_t*)child)->to,
                               (jl_value_t*)((jl_func_type_t*)parent)->to, env,
                               morespecific);
        }
        return jl_false;
    }
    else if (jl_is_func_type(child)) {
        return jl_false;
    }

    if (jl_is_tuple(child)) {
        if (jl_is_tag_type(parent) &&
            ((jl_tag_type_t*)parent)->name == jl_ntuple_typename) {
            jl_tuple_t *tp = ((jl_tag_type_t*)parent)->parameters;
            size_t alen = ((jl_tuple_t*)child)->length;
            // if child has a sequence type, there exists no N such that
            // NTuple[N,Any] could be its supertype.
            if (alen>0 && jl_is_seq_type(jl_tupleref(child,alen-1)))
                return jl_false;
            jl_value_t *nt_len = jl_tupleref(tp,0);
            jl_value_t *childlen = jl_box_int32(((jl_tuple_t*)child)->length);
            if (jl_is_typevar(nt_len)) {
                *env = (jl_tuple_t*)type_match_(childlen, nt_len, env,
                                                morespecific);
                if ((jl_value_t*)*env == jl_false) return (jl_value_t*)*env;
            }
            else {
                return jl_false;
            }
            jl_tuple_t *p_seq =
                jl_tuple1(jl_apply_type((jl_value_t*)jl_seq_type,
                                        jl_tuple1(jl_tupleref(tp,1))));
            return tuple_match((jl_tuple_t*)child, p_seq,
                               env, morespecific);
        }

        if (jl_is_tuple(parent)) {
            return tuple_match((jl_tuple_t*)child, (jl_tuple_t*)parent, env,
                               morespecific);
        }
        return jl_false;
    }
    if (jl_is_tuple(parent)) {
        if (jl_is_tag_type(child) &&
            ((jl_tag_type_t*)child)->name == jl_ntuple_typename) {
            // only ((T>:S)...,) can be a supertype of NTuple[N,S]
            jl_tuple_t *tp = (jl_tuple_t*)parent;
            jl_value_t *ntp = jl_tupleref(((jl_tag_type_t*)child)->parameters,
                                          1);
            if (tp->length == 1 && jl_is_seq_type(jl_tupleref(tp,0))) {
                return type_match_(ntp, jl_tparam0(jl_tupleref(tp,0)),
                                   env, morespecific);
            }
        }
        return jl_false;
    }

    assert(jl_is_some_tag_type(child));
    assert(jl_is_some_tag_type(parent));
    jl_tag_type_t *tta = (jl_tag_type_t*)child;
    jl_tag_type_t *ttb = (jl_tag_type_t*)parent;
    jl_tuple_t *env0 = *env;
    int super = 0;
    while (tta != (jl_tag_type_t*)jl_any_type) {
        if (tta->name == ttb->name) {
            if (super && morespecific)
                return (jl_value_t*)*env;
            assert(tta->parameters->length == ttb->parameters->length);
            for(i=0; i < tta->parameters->length; i++) {
                *env = (jl_tuple_t*)type_match_(jl_tupleref(tta->parameters,i),
                                                jl_tupleref(ttb->parameters,i),
                                                env, morespecific);
                if ((jl_value_t*)*env == jl_false) goto check_Type;
            }
            return (jl_value_t*)*env;
        }
        tta = tta->super; super = 1;
    }
 check_Type:
    if (((jl_tag_type_t*)child)->name == jl_type_type->name &&
        ttb->name != jl_type_type->name) {
        // Type{T} matches either Type{>:T} or >:typeof(T)
        *env = env0;
        return type_match_(jl_full_type(jl_tparam0(child)),
                           parent, env, morespecific);
    }

    return jl_false;
}

/*
  typically a is a concrete type and b is a type containing typevars.
  this function tries to find a typevar assignment such that "a" is a subtype
  of "b".
  returns a linked list of (typevar,type) pairs.
  used to infer static parameter values in generic method definitions.
*/
jl_value_t *jl_type_match(jl_value_t *a, jl_value_t *b)
{
    jl_tuple_t *env = jl_null;
    JL_GC_PUSH(&env);
    jl_value_t *m = type_match_(a, b, &env, 0);
    JL_GC_POP();
    return m;
}

jl_value_t *jl_type_match_morespecific(jl_value_t *a, jl_value_t *b)
{
    jl_tuple_t *env = jl_null;
    JL_GC_PUSH(&env);
    jl_value_t *m = type_match_(a, b, &env, 1);
    JL_GC_POP();
    return m;
}

// given a (possibly-generic) function type and some argument types,
// determine the result type. this is using a function type A-->B as a
// transfer function.
DLLEXPORT
jl_value_t *jl_func_type_tfunc(jl_func_type_t *ft, jl_tuple_t *argtypes)
{
    if (!jl_has_typevars((jl_value_t*)ft->from)) {
        return (jl_value_t*)ft->to;
    }
    jl_value_t *env=jl_type_match((jl_value_t*)argtypes,(jl_value_t*)ft->from);
    jl_tuple_t *te = (env == jl_false) ? jl_null : (jl_tuple_t*)env;
    JL_GC_PUSH(&te);
    te = jl_flatten_pairs(te);
    jl_value_t *ty =
        (jl_value_t*)jl_instantiate_type_with(ft->to, &jl_t0(te), te->length/2);
    JL_GC_POP();
    return ty;
}

// initialization -------------------------------------------------------------

jl_tvar_t *jl_new_typevar(jl_sym_t *name, jl_value_t *lb, jl_value_t *ub)
{
    jl_tvar_t *tv = (jl_tvar_t*)newobj((jl_type_t*)jl_tvar_type, 4);
    tv->name = name;
    tv->lb = lb;
    tv->ub = ub;
    tv->bound = 1;
    return tv;
}

static jl_tvar_t *tvar(const char *name)
{
    jl_tvar_t *tv =
        jl_new_typevar(jl_symbol(name), (jl_value_t*)jl_bottom_type,
                       (jl_value_t*)jl_any_type);
    tv->bound=0;
    return tv;
}

static jl_tuple_t *jl_typevars(size_t n, ...)
{
#ifdef JL_GC_MARKSWEEP
    assert(!jl_gc_is_enabled());
#endif
    va_list args;
    va_start(args, n);
    jl_tuple_t *t = jl_alloc_tuple(n);
    size_t i;
    for(i=0; i < n; i++) {
        jl_tupleset(t, i, (jl_value_t*)tvar(va_arg(args, char*)));
    }
    va_end(args);
    return t;
}

extern void jl_init_int32_cache();
extern JL_CALLABLE(jl_generic_array_ctor);

void jl_init_types()
{
    // create base objects
    jl_struct_kind = (jl_struct_type_t*)newobj(NULL, STRUCT_TYPE_NW);
    jl_struct_kind->type = (jl_type_t*)jl_struct_kind;
    jl_tag_kind = (jl_struct_type_t*)newobj((jl_type_t*)jl_struct_kind, STRUCT_TYPE_NW);
    jl_tag_type_type = jl_tag_kind;
    jl_func_kind = (jl_struct_type_t*)newobj((jl_type_t*)jl_struct_kind, STRUCT_TYPE_NW);

    jl_typename_type = (jl_struct_type_t*)newobj((jl_type_t*)jl_struct_kind, STRUCT_TYPE_NW);
    jl_sym_type = (jl_struct_type_t*)newobj((jl_type_t*)jl_struct_kind, STRUCT_TYPE_NW);
    jl_symbol_type = jl_sym_type;

    jl_any_type = (jl_tag_type_t*)newobj((jl_type_t*)jl_tag_kind, TAG_TYPE_NW);
    jl_type_type = (jl_tag_type_t*)newobj((jl_type_t*)jl_tag_kind, TAG_TYPE_NW);
    jl_tuple_type = jl_alloc_tuple(1);
    jl_tuple_type->type = (jl_type_t*)jl_tuple_type;

    jl_null = (jl_tuple_t*)newobj((jl_type_t*)jl_tuple_type, 1);
    jl_null->length = 0;

    // initialize them. lots of cycles.
    jl_struct_kind->name = jl_new_typename(jl_symbol("StructKind"));
    jl_struct_kind->super = (jl_tag_type_t*)jl_tag_kind;
    jl_struct_kind->parameters = jl_null;
    jl_struct_kind->names = jl_tuple(5, jl_symbol("name"), jl_symbol("super"),
                                     jl_symbol("parameters"),
                                     jl_symbol("names"), jl_symbol("types"));
    jl_struct_kind->types = jl_tuple(5, jl_typename_type, jl_type_type,
                                     jl_tuple_type, jl_tuple_type,
                                     jl_tuple_type);
    jl_struct_kind->fptr = jl_f_no_function;
    jl_struct_kind->env = NULL;
    jl_struct_kind->linfo = NULL;
    jl_struct_kind->ctor_factory = NULL;
    jl_struct_kind->instance = NULL;
    jl_struct_kind->uid = t_uid_ctr++;

    jl_tag_kind->name = jl_new_typename(jl_symbol("TagKind"));
    jl_tag_kind->super = jl_type_type;
    jl_tag_kind->parameters = jl_null;
    jl_tag_kind->names = jl_tuple(3, jl_symbol("name"), jl_symbol("super"),
                                  jl_symbol("parameters"));
    jl_tag_kind->types = jl_tuple(3, jl_typename_type, jl_type_type,
                                  jl_tuple_type);
    jl_tag_kind->fptr = jl_f_no_function;
    jl_tag_kind->env = NULL;
    jl_tag_kind->linfo = NULL;
    jl_tag_kind->ctor_factory = NULL;
    jl_tag_kind->instance = NULL;
    jl_tag_kind->uid = t_uid_ctr++;

    jl_func_kind->name = jl_new_typename(jl_symbol("FuncKind"));
    jl_func_kind->super = jl_type_type;
    jl_func_kind->parameters = jl_null;
    jl_func_kind->names = jl_tuple(2, jl_symbol("from"), jl_symbol("to"));
    jl_func_kind->types = jl_tuple(2, jl_type_type, jl_type_type);
    jl_func_kind->fptr = jl_f_no_function;
    jl_func_kind->env = NULL;
    jl_func_kind->linfo = NULL;
    jl_func_kind->ctor_factory = NULL;
    jl_func_kind->instance = NULL;
    jl_func_kind->uid = t_uid_ctr++;

    jl_typename_type->name = jl_new_typename(jl_symbol("TypeName"));
    jl_typename_type->super = jl_any_type;
    jl_typename_type->parameters = jl_null;
    jl_typename_type->names = jl_tuple(1, jl_symbol("name"));
    jl_typename_type->types = jl_tuple(1, jl_sym_type);
    jl_typename_type->uid = t_uid_ctr++;
    jl_typename_type->fptr = jl_f_no_function;
    jl_typename_type->env = NULL;
    jl_typename_type->linfo = NULL;
    jl_typename_type->ctor_factory = NULL;
    jl_typename_type->instance = NULL;

    jl_sym_type->name = jl_new_typename(jl_symbol("Symbol"));
    jl_sym_type->super = jl_any_type;
    jl_sym_type->parameters = jl_null;
    jl_sym_type->names = jl_null;
    jl_sym_type->types = jl_null;
    jl_sym_type->fptr = jl_f_no_function;
    jl_sym_type->env = NULL;
    jl_sym_type->linfo = NULL;
    jl_sym_type->ctor_factory = NULL;
    jl_sym_type->instance = NULL;
    jl_sym_type->uid = t_uid_ctr++;

    jl_any_type->name = jl_new_typename(jl_symbol("Any"));
    jl_any_type->super = jl_any_type;
    jl_any_type->parameters = jl_null;
    jl_any_type->fptr = NULL;
    jl_any_type->env = NULL;
    jl_any_type->linfo = NULL;

    jl_type_type->name = jl_new_typename(jl_symbol("Type"));
    jl_type_type->name->primary = (jl_value_t*)jl_type_type;
    jl_type_type->super = jl_any_type;
    jl_type_type->fptr = NULL;
    jl_type_type->env = NULL;
    jl_type_type->linfo = NULL;

    jl_tuple_typename = jl_new_typename(jl_symbol("Tuple"));

    // now they can be used to create the remaining base kinds and types
    jl_methtable_type =
        jl_new_struct_type(jl_symbol("MethodTable"),
                           jl_any_type, jl_null, jl_null, jl_null);
    jl_methtable_type->fptr = jl_f_no_function;

    jl_union_kind = jl_new_struct_type(jl_symbol("UnionKind"),
                                       jl_type_type, jl_null,
                                       jl_tuple(1, jl_symbol("types")),
                                       jl_tuple(1, jl_tuple_type));
    jl_union_kind->fptr = jl_f_no_function;

    jl_bottom_type = (jl_type_t*)jl_new_struct(jl_union_kind, jl_null);

    // the universal (compatible w/ any signature) function type is Any-->None.
    // the most general (any function is compatible with it) function type
    // is None-->Any.
    // generic functions have the type Any-->Any, as they take any arguments
    // but don't satisfy any particular return-type requirements.
    jl_any_func = jl_new_functype((jl_type_t*)jl_any_type, (jl_type_t*)jl_any_type);

    jl_bottom_func = jl_new_closure(jl_f_no_function, NULL);

    jl_bits_kind =
        jl_new_struct_type(jl_symbol("BitsKind"), (jl_tag_type_t*)jl_tag_kind,
                           jl_null,
                           jl_tuple(4, jl_symbol("name"), jl_symbol("super"),
                                    jl_symbol("parameters"),
                                    jl_symbol("nbits")),
                           jl_tuple(4, jl_typename_type, jl_type_type,
                                    jl_tuple_type, jl_any_type));
    // cannot be created with normal constructor due to hidden fields
    jl_bits_kind->fptr = jl_f_no_function;
    
    jl_tvar_type = jl_new_struct_type(jl_symbol("TypeVar"),
                                      jl_any_type, jl_null,
                                      jl_tuple(3, jl_symbol("name"),
                                               jl_symbol("lb"),
                                               jl_symbol("ub")),
                                      jl_tuple(3, jl_sym_type, jl_type_type,
                                               jl_type_type));
    jl_tvar_type->fptr = jl_f_no_function;

    jl_tvar_t *tttvar = tvar("T");
    jl_type_type->parameters = jl_tuple(1, tttvar);

    jl_tuple_t *tv;

    tv = jl_typevars(1, "T");
    jl_seq_type = jl_new_tagtype((jl_value_t*)jl_symbol("..."),
                                 jl_any_type, tv);

    jl_tupleset(jl_tuple_type, 0,
                (jl_value_t*)jl_apply_type((jl_value_t*)jl_seq_type,
                                           jl_tuple(1,jl_any_type)));

    tv = jl_typevars(2, "N", "T");
    jl_ntuple_type = jl_new_tagtype((jl_value_t*)jl_symbol("NTuple"),
                                    jl_any_type, tv);
    jl_ntuple_typename = jl_ntuple_type->name;

    // non-primitive definitions follow
    jl_int32_type = NULL;
    jl_int32_type = jl_new_bitstype((jl_value_t*)jl_symbol("Int32"),
                                    jl_any_type, jl_null, 32);
    jl_init_int32_cache();
    jl_int32_type->bnbits = jl_box_int32(32);
    jl_tupleset(jl_bits_kind->types, 3, (jl_value_t*)jl_int32_type);

    jl_bool_type = NULL;
    jl_bool_type = jl_new_bitstype((jl_value_t*)jl_symbol("Bool"),
                                   jl_any_type, jl_null, 8);
    jl_false = jl_box8(jl_bool_type, 0);
    jl_true  = jl_box8(jl_bool_type, 1);

    tv = jl_typevars(2, "T", "N");
    jl_tensor_type = jl_new_tagtype((jl_value_t*)jl_symbol("Tensor"),
                                    jl_any_type, tv);

    tv = jl_typevars(2, "T", "N");
    jl_array_type = 
        jl_new_struct_type(jl_symbol("Array"),
                           (jl_tag_type_t*)
                           jl_apply_type((jl_value_t*)jl_tensor_type, tv),
                           tv,
                           jl_tuple(1, jl_symbol("dims")),
                           jl_tuple(1, jl_apply_type((jl_value_t*)jl_ntuple_type,
                                                     jl_tuple(2, jl_tupleref(tv,1),
                                                              jl_int32_type))));
    jl_array_typename = jl_array_type->name;
    jl_array_type->fptr = jl_generic_array_ctor;
    jl_array_type->env = NULL;
    jl_array_type->linfo = NULL;

    jl_array_any_type =
        (jl_type_t*)jl_apply_type((jl_value_t*)jl_array_type,
                                  jl_tuple(2, jl_any_type,
                                           jl_box_int32(1)));

    jl_expr_type =
        jl_new_struct_type(jl_symbol("Expr"),
                           jl_any_type, jl_null,
                           jl_tuple(3, jl_symbol("head"), jl_symbol("args"),
                                    jl_symbol("type")),
                           jl_tuple(3, jl_sym_type, jl_array_any_type,
                                    jl_any_type));
    jl_add_constructors(jl_expr_type);

    jl_lambda_info_type =
        jl_new_struct_type(jl_symbol("LambdaStaticData"),
                           jl_any_type, jl_null,
                           jl_tuple(4, jl_symbol("ast"), jl_symbol("sparams"),
                                    jl_symbol("tfunc"), jl_symbol("name")),
                           jl_tuple(4, jl_expr_type, jl_tuple_type,
                                    jl_any_type, jl_sym_type));
    jl_lambda_info_type->fptr = jl_f_no_function;

    tv = jl_typevars(1, "T");
    jl_box_type =
        jl_new_struct_type(jl_symbol("Box"),
                           jl_any_type, tv,
                           jl_tuple(1, jl_symbol("contents")), tv);
    jl_add_constructors(jl_box_type);
    jl_box_typename = jl_box_type->name;
    jl_box_any_type =
        (jl_type_t*)jl_apply_type((jl_value_t*)jl_box_type,
                                  jl_tuple(1, jl_any_type));

    jl_typector_type =
        jl_new_struct_type(jl_symbol("TypeConstructor"),
                           jl_any_type, jl_null,
                           jl_tuple(2, jl_symbol("parameters"),
                                    jl_symbol("body")),
                           jl_tuple(2, jl_tuple_type, jl_any_type));

    tv = jl_typevars(2, "A", "B");
    jl_functype_ctor =
        jl_new_type_ctor(tv,
                         (jl_type_t*)jl_new_functype((jl_type_t*)jl_tupleref(tv,0),
                                                     (jl_type_t*)jl_tupleref(tv,1)));
    jl_function_type = jl_functype_ctor;

    jl_intrinsic_type = jl_new_bitstype((jl_value_t*)jl_symbol("IntrinsicFunction"),
                                        jl_any_type, jl_null, 32);

    tv = jl_typevars(1, "T");
    jl_pointer_type =
        jl_new_bitstype((jl_value_t*)jl_symbol("Ptr"), jl_any_type, tv,
#ifdef BITS64
                        64
#else
                        32
#endif
                        );

    jl_pointer_void_type =
        (jl_bits_type_t*)jl_apply_type((jl_value_t*)jl_pointer_type,
                                       jl_tuple(1, jl_bottom_type));

    jl_undef_type = jl_new_tagtype((jl_value_t*)jl_symbol("Undef"),
                                   jl_any_type, jl_null);

    // Type{Type}
    jl_typetype_type = (jl_tag_type_t*)jl_apply_type((jl_value_t*)jl_type_type,
                                                     jl_tuple(1,jl_type_type));

    call_sym = jl_symbol("call");
    call1_sym = jl_symbol("call1");
    quote_sym = jl_symbol("quote");
    top_sym = jl_symbol("top");
    dots_sym = jl_symbol("...");
    dollar_sym = jl_symbol("$");
    line_sym = jl_symbol("line");
    continue_sym = jl_symbol("continue");
    error_sym = jl_symbol("error");
    goto_sym = jl_symbol("goto");
    goto_ifnot_sym = jl_symbol("gotoifnot");
    label_sym = jl_symbol("label");
    return_sym = jl_symbol("return");
    lambda_sym = jl_symbol("lambda");
    vinf_sym = jl_symbol("vinf");
    macro_sym = jl_symbol("macro");
    unexpanded_sym = jl_symbol("unexpanded");
    assign_sym = jl_symbol("=");
    null_sym = jl_symbol("null");
    unbound_sym = jl_symbol("unbound");
    symbol_sym = jl_symbol("symbol");
    body_sym = jl_symbol("body");
    locals_sym = jl_symbol("locals");
    colons_sym = jl_symbol("::");
    method_sym = jl_symbol("method");
    exc_sym = jl_symbol("the_exception");
    enter_sym = jl_symbol("enter");
    leave_sym = jl_symbol("leave");
    Any_sym = jl_symbol("Any");
    static_typeof_sym = jl_symbol("static_typeof");
}