cranelift-codegen 0.24.0

"""
Type Inference
"""
from .typevar import TypeVar
from .ast import Def, Var
from copy import copy
from itertools import product

try:
    from typing import Dict, TYPE_CHECKING, Union, Tuple, Optional, Set # noqa
    from typing import Iterable, List, Any, TypeVar as MTypeVar # noqa
    from typing import cast
    from .xform import Rtl, XForm # noqa
    from .ast import Expr # noqa
    from .typevar import TypeSet # noqa
    if TYPE_CHECKING:
        T = MTypeVar('T')
        TypeMap = Dict[TypeVar, TypeVar]
        VarTyping = Dict[Var, TypeVar]
except ImportError:
    TYPE_CHECKING = False
    pass


class TypeConstraint(object):
    """
    Base class for all runtime-emittable type constraints.
    """

    def __init__(self, tv, tc):
        # type: (TypeVar, Union[TypeVar, TypeSet]) -> None
        """
        Abstract "constructor" for linters
        """
        assert False, "Abstract"

    def translate(self, m):
        # type: (Union[TypeEnv, TypeMap]) -> TypeConstraint
        """
        Translate any TypeVars in the constraint according to the map or
        TypeEnv m
        """
        def translate_one(a):
            # type: (Any) -> Any
            if (isinstance(a, TypeVar)):
                return m[a] if isinstance(m, TypeEnv) else subst(a, m)
            return a

        res = None  # type: TypeConstraint
        res = self.__class__(*tuple(map(translate_one, self._args())))
        return res

    def __eq__(self, other):
        # type: (object) -> bool
        if (not isinstance(other, self.__class__)):
            return False

        assert isinstance(other, TypeConstraint)  # help MyPy figure out other
        return self._args() == other._args()

    def is_concrete(self):
        # type: () -> bool
        """
        Return true iff all typevars in the constraint are singletons.
        """
        return [] == list(filter(lambda x:  x.singleton_type() is None,
                                 self.tvs()))

    def __hash__(self):
        # type: () -> int
        return hash(self._args())

    def _args(self):
        # type: () -> Tuple[Any,...]
        """
        Return a tuple with the exact arguments passed to __init__ to create
        this object.
        """
        assert False, "Abstract"

    def tvs(self):
        # type: () -> Iterable[TypeVar]
        """
        Return the typevars contained in this constraint.
        """
        return list(filter(lambda x: isinstance(x, TypeVar), self._args()))

    def is_trivial(self):
        # type: () -> bool
        """
        Return true if this constrain is statically decidable.
        """
        assert False, "Abstract"

    def eval(self):
        # type: () -> bool
        """
        Evaluate this constraint. Should only be called when the constraint has
        been translated to concrete types.
        """
        assert False, "Abstract"

    def __repr__(self):
        # type: () -> str
        return (self.__class__.__name__ + '(' +
                ', '.join(map(str, self._args())) + ')')


class TypesEqual(TypeConstraint):
    """
    Constraint specifying that two derived type vars must have the same runtime
    type.
    """
    def __init__(self, tv1, tv2):
        # type: (TypeVar, TypeVar) -> None
        (self.tv1, self.tv2) = sorted([tv1, tv2], key=repr)

    def _args(self):
        # type: () -> Tuple[Any,...]
        """ See TypeConstraint._args() """
        return (self.tv1, self.tv2)

    def is_trivial(self):
        # type: () -> bool
        """ See TypeConstraint.is_trivial() """
        return self.tv1 == self.tv2 or self.is_concrete()

    def eval(self):
        # type: () -> bool
        """ See TypeConstraint.eval() """
        assert self.is_concrete()
        return self.tv1.singleton_type() == self.tv2.singleton_type()


class InTypeset(TypeConstraint):
    """
    Constraint specifying that a type var must belong to some typeset.
    """
    def __init__(self, tv, ts):
        # type: (TypeVar, TypeSet) -> None
        assert not tv.is_derived and tv.name.startswith("typeof_")
        self.tv = tv
        self.ts = ts

    def _args(self):
        # type: () -> Tuple[Any,...]
        """ See TypeConstraint._args() """
        return (self.tv, self.ts)

    def is_trivial(self):
        # type: () -> bool
        """ See TypeConstraint.is_trivial() """
        tv_ts = self.tv.get_typeset().copy()

        # Trivially True
        if (tv_ts.issubset(self.ts)):
            return True

        # Trivially false
        tv_ts &= self.ts
        if (tv_ts.size() == 0):
            return True

        return self.is_concrete()

    def eval(self):
        # type: () -> bool
        """ See TypeConstraint.eval() """
        assert self.is_concrete()
        return self.tv.get_typeset().issubset(self.ts)


class WiderOrEq(TypeConstraint):
    """
    Constraint specifying that a type var tv1 must be wider than or equal to
    type var tv2 at runtime. This requires that:
        1) They have the same number of lanes
        2) In a lane tv1 has at least as many bits as tv2.
    """
    def __init__(self, tv1, tv2):
        # type: (TypeVar, TypeVar) -> None
        self.tv1 = tv1
        self.tv2 = tv2

    def _args(self):
        # type: () -> Tuple[Any,...]
        """ See TypeConstraint._args() """
        return (self.tv1, self.tv2)

    def is_trivial(self):
        # type: () -> bool
        """ See TypeConstraint.is_trivial() """
        # Trivially true
        if (self.tv1 == self.tv2):
            return True

        ts1 = self.tv1.get_typeset()
        ts2 = self.tv2.get_typeset()

        def set_wider_or_equal(s1, s2):
            # type: (Set[int], Set[int]) -> bool
            return len(s1) > 0 and len(s2) > 0 and min(s1) >= max(s2)

        # Trivially True
        if set_wider_or_equal(ts1.ints, ts2.ints) and\
           set_wider_or_equal(ts1.floats, ts2.floats) and\
           set_wider_or_equal(ts1.bools, ts2.bools):
            return True

        def set_narrower(s1, s2):
            # type: (Set[int], Set[int]) -> bool
            return len(s1) > 0 and len(s2) > 0 and min(s1) < max(s2)

        # Trivially False
        if set_narrower(ts1.ints, ts2.ints) and\
           set_narrower(ts1.floats, ts2.floats) and\
           set_narrower(ts1.bools, ts2.bools):
            return True

        # Trivially False
        if len(ts1.lanes.intersection(ts2.lanes)) == 0:
            return True

        return self.is_concrete()

    def eval(self):
        # type: () -> bool
        """ See TypeConstraint.eval() """
        assert self.is_concrete()
        typ1 = self.tv1.singleton_type()
        typ2 = self.tv2.singleton_type()

        return typ1.wider_or_equal(typ2)


class SameWidth(TypeConstraint):
    """
    Constraint specifying that two types have the same width. E.g. i32x2 has
    the same width as i64x1, i16x4, f32x2, f64, b1x64 etc.
    """
    def __init__(self, tv1, tv2):
        # type: (TypeVar, TypeVar) -> None
        self.tv1 = tv1
        self.tv2 = tv2

    def _args(self):
        # type: () -> Tuple[Any,...]
        """ See TypeConstraint._args() """
        return (self.tv1, self.tv2)

    def is_trivial(self):
        # type: () -> bool
        """ See TypeConstraint.is_trivial() """
        # Trivially true
        if (self.tv1 == self.tv2):
            return True

        ts1 = self.tv1.get_typeset()
        ts2 = self.tv2.get_typeset()

        # Trivially False
        if len(ts1.widths().intersection(ts2.widths())) == 0:
            return True

        return self.is_concrete()

    def eval(self):
        # type: () -> bool
        """ See TypeConstraint.eval() """
        assert self.is_concrete()
        typ1 = self.tv1.singleton_type()
        typ2 = self.tv2.singleton_type()

        return (typ1.width() == typ2.width())


class TypeEnv(object):
    """
    Class encapsulating the neccessary book keeping for type inference.
        :attribute type_map: dict holding the equivalence relations between tvs
        :attribute constraints: a list of accumulated constraints - tuples
                            (tv1, tv2)) where tv1 and tv2 are equal
        :attribute ranks: dictionary recording the (optional) ranks for tvs.
                          'rank' is a partial ordering on TVs based on their
                          origin. See comments in rank() and register().
        :attribute vars: a set containing all known Vars
        :attribute idx: counter used to get fresh ids
    """

    RANK_SINGLETON = 5
    RANK_INPUT = 4
    RANK_INTERMEDIATE = 3
    RANK_OUTPUT = 2
    RANK_TEMP = 1
    RANK_INTERNAL = 0

    def __init__(self, arg=None):
        # type: (Optional[Tuple[TypeMap, List[TypeConstraint]]]) -> None
        self.ranks = {}  # type: Dict[TypeVar, int]
        self.vars = set()  # type: Set[Var]

        if arg is None:
            self.type_map = {}  # type: TypeMap
            self.constraints = []  # type: List[TypeConstraint]
        else:
            self.type_map, self.constraints = arg

        self.idx = 0

    def __getitem__(self, arg):
        # type: (Union[TypeVar, Var]) -> TypeVar
        """
        Lookup the canonical representative for a Var/TypeVar.
        """
        if (isinstance(arg, Var)):
            assert arg in self.vars
            tv = arg.get_typevar()
        else:
            assert (isinstance(arg, TypeVar))
            tv = arg

        while tv in self.type_map:
            tv = self.type_map[tv]

        if tv.is_derived:
            tv = TypeVar.derived(self[tv.base], tv.derived_func)
        return tv

    def equivalent(self, tv1, tv2):
        # type: (TypeVar, TypeVar) -> None
        """
        Record a that the free tv1 is part of the same equivalence class as
        tv2. The canonical representative of the merged class is tv2's
        cannonical representative.
        """
        assert not tv1.is_derived
        assert self[tv1] == tv1

        # Make sure we don't create cycles
        if tv2.is_derived:
            assert self[tv2.base] != tv1

        self.type_map[tv1] = tv2

    def add_constraint(self, constr):
        # type: (TypeConstraint) -> None
        """
        Add a new constraint
        """
        if (constr in self.constraints):
            return

        # InTypeset constraints can be expressed by constraining the typeset of
        # a variable. No need to add them to self.constraints
        if (isinstance(constr, InTypeset)):
            self[constr.tv].constrain_types_by_ts(constr.ts)
            return

        self.constraints.append(constr)

    def get_uid(self):
        # type: () -> str
        r = str(self.idx)
        self.idx += 1
        return r

    def __repr__(self):
        # type: () -> str
        return self.dot()

    def rank(self, tv):
        # type: (TypeVar) -> int
        """
        Get the rank of tv in the partial order. TVs directly associated with a
        Var get their rank from the Var (see register()). Internally generated
        non-derived TVs implicitly get the lowest rank (0). Derived variables
        get their rank from their free typevar. Singletons have the highest
        rank. TVs associated with vars in a source pattern have a higher rank
        than TVs associted with temporary vars.
        """
        default_rank = TypeEnv.RANK_INTERNAL if tv.singleton_type() is None \
            else TypeEnv.RANK_SINGLETON

        if tv.is_derived:
            tv = tv.free_typevar()

        return self.ranks.get(tv, default_rank)

    def register(self, v):
        # type: (Var) -> None
        """
        Register a new Var v. This computes a rank for the associated TypeVar
        for v, which is used to impose a partial order on type variables.
        """
        self.vars.add(v)

        if v.is_input():
            r = TypeEnv.RANK_INPUT
        elif v.is_intermediate():
            r = TypeEnv.RANK_INTERMEDIATE
        elif v.is_output():
            r = TypeEnv.RANK_OUTPUT
        else:
            assert(v.is_temp())
            r = TypeEnv.RANK_TEMP

        self.ranks[v.get_typevar()] = r

    def free_typevars(self):
        # type: () -> List[TypeVar]
        """
        Get the free typevars in the current type env.
        """
        tvs = set([self[tv].free_typevar() for tv in self.type_map.keys()])
        tvs = tvs.union(set([self[v].free_typevar() for v in self.vars]))
        # Filter out None here due to singleton type vars
        return sorted(filter(lambda x: x is not None, tvs),
                      key=lambda x:   x.name)

    def normalize(self):
        # type: () -> None
        """
        Normalize by:
            - collapsing any roots that don't correspond to a concrete TV AND
              have a single TV derived from them or equivalent to them

        E.g. if we have a root of the tree that looks like:

          typeof_a   typeof_b
                 \\  /
              typeof_x
                  |
                half_width(1)
                  |
                  1

        we want to collapse the linear path between 1 and typeof_x. The
        resulting graph is:

          typeof_a   typeof_b
                 \\  /
              typeof_x
        """
        source_tvs = set([v.get_typevar() for v in self.vars])
        children = {}  # type: Dict[TypeVar, Set[TypeVar]]
        for v in self.type_map.values():
            if not v.is_derived:
                continue

            t = v.free_typevar()
            s = children.get(t, set())
            s.add(v)
            children[t] = s

        for (a, b) in self.type_map.items():
            s = children.get(b, set())
            s.add(a)
            children[b] = s

        for r in self.free_typevars():
            while (r not in source_tvs and r in children and
                   len(children[r]) == 1):
                child = list(children[r])[0]
                if child in self.type_map:
                    assert self.type_map[child] == r
                    del self.type_map[child]

                r = child

    def extract(self):
        # type: () -> TypeEnv
        """
        Extract a clean type environment from self, that only mentions
        TVs associated with real variables
        """
        vars_tvs = set([v.get_typevar() for v in self.vars])
        new_type_map = {tv: self[tv] for tv in vars_tvs if tv != self[tv]}

        new_constraints = []  # type: List[TypeConstraint]
        for constr in self.constraints:
            constr = constr.translate(self)

            if constr.is_trivial() or constr in new_constraints:
                continue

            # Sanity: translated constraints should refer to only real vars
            for arg in constr._args():
                if (not isinstance(arg, TypeVar)):
                    continue

                arg_free_tv = arg.free_typevar()
                assert arg_free_tv is None or arg_free_tv in vars_tvs

            new_constraints.append(constr)

        # Sanity: translated typemap should refer to only real vars
        for (k, v) in new_type_map.items():
            assert k in vars_tvs
            assert v.free_typevar() is None or v.free_typevar() in vars_tvs

        t = TypeEnv()
        t.type_map = new_type_map
        t.constraints = new_constraints
        # ranks and vars contain only TVs associated with real vars
        t.ranks = copy(self.ranks)
        t.vars = copy(self.vars)
        return t

    def concrete_typings(self):
        # type: () -> Iterable[VarTyping]
        """
        Return an iterable over all possible concrete typings permitted by this
        TypeEnv.
        """
        free_tvs = self.free_typevars()
        free_tv_iters = [tv.get_typeset().concrete_types() for tv in free_tvs]
        for concrete_types in product(*free_tv_iters):
            # Build type substitutions for all free vars
            m = {tv: TypeVar.singleton(typ)
                 for (tv, typ) in zip(free_tvs, concrete_types)}

            concrete_var_map = {v: subst(self[v.get_typevar()], m)
                                for v in self.vars}

            # Check if constraints are satisfied for this typing
            failed = None
            for constr in self.constraints:
                concrete_constr = constr.translate(m)
                if not concrete_constr.eval():
                    failed = concrete_constr
                    break

            if (failed is not None):
                continue

            yield concrete_var_map

    def permits(self, concrete_typing):
        # type: (VarTyping) -> bool
        """
        Return true iff this TypeEnv permits the (possibly partial) concrete
        variable type mapping concrete_typing.
        """
        # Each variable has a concrete type, that is a subset of its inferred
        # typeset.
        for (v, typ) in concrete_typing.items():
            assert typ.singleton_type() is not None
            if not typ.get_typeset().issubset(self[v].get_typeset()):
                return False

        m = {self[v]: typ for (v, typ) in concrete_typing.items()}

        # Constraints involving vars in concrete_typing are satisfied
        for constr in self.constraints:
            try:
                # If the constraint includes only vars in concrete_typing, we
                # can translate it using m. Otherwise we encounter a KeyError
                # and ignore it
                constr = constr.translate(m)
                if not constr.eval():
                    return False
            except KeyError:
                pass

        return True

    def dot(self):
        # type: () -> str
        """
        Return a representation of self as a graph in dot format.
            Nodes correspond to TypeVariables.
            Dotted edges correspond to equivalences between TVS
            Solid edges correspond to derivation relations between TVs.
            Dashed edges correspond to equivalence constraints.
        """
        def label(s):
            # type: (TypeVar) -> str
            return "\"" + str(s) + "\""

        # Add all registered TVs (as some of them may be singleton nodes not
        # appearing in the graph
        nodes = set()  # type: Set[TypeVar]
        edges = set()  # type: Set[Tuple[TypeVar, TypeVar, str, str, Optional[str]]] # noqa

        def add_nodes(*args):
            # type: (*TypeVar) -> None
            for tv in args:
                nodes.add(tv)
                while (tv.is_derived):
                    nodes.add(tv.base)
                    edges.add((tv, tv.base, "solid", "forward",
                               tv.derived_func))
                    tv = tv.base

        for v in self.vars:
            add_nodes(v.get_typevar())

        for (tv1, tv2) in self.type_map.items():
            # Add all intermediate TVs appearing in edges
            add_nodes(tv1, tv2)
            edges.add((tv1, tv2, "dotted", "forward", None))

        for constr in self.constraints:
            if isinstance(constr, TypesEqual):
                add_nodes(constr.tv1, constr.tv2)
                edges.add((constr.tv1, constr.tv2, "dashed", "none", "equal"))
            elif isinstance(constr, WiderOrEq):
                add_nodes(constr.tv1, constr.tv2)
                edges.add((constr.tv1, constr.tv2, "dashed", "forward", ">="))
            elif isinstance(constr, SameWidth):
                add_nodes(constr.tv1, constr.tv2)
                edges.add((constr.tv1, constr.tv2, "dashed", "none",
                           "same_width"))
            else:
                assert False, "Can't display constraint {}".format(constr)

        root_nodes = set([x for x in nodes
                          if x not in self.type_map and not x.is_derived])

        r = "digraph {\n"
        for n in nodes:
            r += label(n)
            if n in root_nodes:
                r += "[xlabel=\"{}\"]".format(self[n].get_typeset())
            r += ";\n"

        for (n1, n2, style, direction, elabel) in edges:
            e = label(n1) + "->" + label(n2)
            e += "[style={},dir={}".format(style, direction)

            if elabel is not None:
                e += ",label=\"{}\"".format(elabel)
            e += "];\n"

            r += e
        r += "}"

        return r


if TYPE_CHECKING:
    TypingError = str
    TypingOrError = Union[TypeEnv, TypingError]


def get_error(typing_or_err):
    # type: (TypingOrError) -> Optional[TypingError]
    """
    Helper function to appease mypy when checking the result of typing.
    """
    if isinstance(typing_or_err, str):
        if (TYPE_CHECKING):
            return cast(TypingError, typing_or_err)
        else:
            return typing_or_err
    else:
        return None


def get_type_env(typing_or_err):
    # type: (TypingOrError) -> TypeEnv
    """
    Helper function to appease mypy when checking the result of typing.
    """
    assert isinstance(typing_or_err, TypeEnv), \
        "Unexpected error: {}".format(typing_or_err)

    if (TYPE_CHECKING):
        return cast(TypeEnv, typing_or_err)
    else:
        return typing_or_err


def subst(tv, tv_map):
    # type: (TypeVar, TypeMap) -> TypeVar
    """
    Perform substition on the input tv using the TypeMap tv_map.
    """
    if tv in tv_map:
        return tv_map[tv]

    if tv.is_derived:
        return TypeVar.derived(subst(tv.base, tv_map), tv.derived_func)

    return tv


def normalize_tv(tv):
    # type: (TypeVar) -> TypeVar
    """
    Normalize a (potentially derived) TV using the following rules:
        - vector and width derived functions commute
        {HALF,DOUBLE}VECTOR({HALF,DOUBLE}WIDTH(base)) ->
            {HALF,DOUBLE}WIDTH({HALF,DOUBLE}VECTOR(base))

        - half/double pairs collapse
        {HALF,DOUBLE}WIDTH({DOUBLE,HALF}WIDTH(base)) -> base
        {HALF,DOUBLE}VECTOR({DOUBLE,HALF}VECTOR(base)) -> base
    """
    vector_derives = [TypeVar.HALFVECTOR, TypeVar.DOUBLEVECTOR]
    width_derives = [TypeVar.HALFWIDTH, TypeVar.DOUBLEWIDTH]

    if not tv.is_derived:
        return tv

    df = tv.derived_func

    if (tv.base.is_derived):
        base_df = tv.base.derived_func

        # Reordering: {HALFWIDTH, DOUBLEWIDTH} commute with {HALFVECTOR,
        # DOUBLEVECTOR}. Arbitrarily pick WIDTH < VECTOR
        if df in vector_derives and base_df in width_derives:
            return normalize_tv(
                    TypeVar.derived(
                        TypeVar.derived(tv.base.base, df), base_df))

        # Cancelling: HALFWIDTH, DOUBLEWIDTH and HALFVECTOR, DOUBLEVECTOR
        # cancel each other. Note: This doesn't hide any over/underflows,
        # since we 1) assert the safety of each TV in the chain upon its
        # creation, and 2) the base typeset is only allowed to shrink.

        if (df, base_df) in \
                [(TypeVar.HALFVECTOR, TypeVar.DOUBLEVECTOR),
                 (TypeVar.DOUBLEVECTOR, TypeVar.HALFVECTOR),
                 (TypeVar.HALFWIDTH, TypeVar.DOUBLEWIDTH),
                 (TypeVar.DOUBLEWIDTH, TypeVar.HALFWIDTH)]:
            return normalize_tv(tv.base.base)

    return TypeVar.derived(normalize_tv(tv.base), df)


def constrain_fixpoint(tv1, tv2):
    # type: (TypeVar, TypeVar) -> None
    """
    Given typevars tv1 and tv2 (which could be derived from one another)
    constrain their typesets to be the same. When one is derived from the
    other, repeat the constrain process until fixpoint.
    """
    # Constrain tv2's typeset as long as tv1's typeset is changing.
    while True:
        old_tv1_ts = tv1.get_typeset().copy()
        tv2.constrain_types(tv1)
        if tv1.get_typeset() == old_tv1_ts:
            break

    old_tv2_ts = tv2.get_typeset().copy()
    tv1.constrain_types(tv2)
    assert old_tv2_ts == tv2.get_typeset()


def unify(tv1, tv2, typ):
    # type: (TypeVar, TypeVar, TypeEnv) -> TypingOrError
    """
    Unify tv1 and tv2 in the current type environment typ, and return an
    updated type environment or error.
    """
    tv1 = normalize_tv(typ[tv1])
    tv2 = normalize_tv(typ[tv2])

    # Already unified
    if tv1 == tv2:
        return typ

    if typ.rank(tv2) < typ.rank(tv1):
        return unify(tv2, tv1, typ)

    constrain_fixpoint(tv1, tv2)

    if (tv1.get_typeset().size() == 0 or tv2.get_typeset().size() == 0):
        return "Error: empty type created when unifying {} and {}"\
               .format(tv1, tv2)

    # Free -> Derived(Free)
    if not tv1.is_derived:
        typ.equivalent(tv1, tv2)
        return typ

    if (tv1.is_derived and TypeVar.is_bijection(tv1.derived_func)):
        inv_f = TypeVar.inverse_func(tv1.derived_func)
        return unify(tv1.base, normalize_tv(TypeVar.derived(tv2, inv_f)), typ)

    typ.add_constraint(TypesEqual(tv1, tv2))
    return typ


def move_first(l, i):
    # type: (List[T], int) -> List[T]
    return [l[i]] + l[:i] + l[i+1:]


def ti_def(definition, typ):
    # type: (Def, TypeEnv) -> TypingOrError
    """
    Perform type inference on one Def in the current type environment typ and
    return an updated type environment or error.

    At a high level this works by creating fresh copies of each formal type var
    in the Def's instruction's signature, and unifying the formal tv with the
    corresponding actual tv.
    """
    expr = definition.expr
    inst = expr.inst

    # Create a dict m mapping each free typevar in the signature of definition
    # to a fresh copy of itself.
    free_formal_tvs = inst.all_typevars()
    m = {tv: tv.get_fresh_copy(str(typ.get_uid())) for tv in free_formal_tvs}

    # Update m with any explicitly bound type vars
    for (idx, bound_typ) in enumerate(expr.typevars):
        m[free_formal_tvs[idx]] = TypeVar.singleton(bound_typ)

    # Get fresh copies for each typevar in the signature (both free and
    # derived)
    fresh_formal_tvs = \
        [subst(inst.outs[i].typevar, m) for i in inst.value_results] +\
        [subst(inst.ins[i].typevar, m) for i in inst.value_opnums]

    # Get the list of actual Vars
    actual_vars = []  # type: List[Expr]
    actual_vars += [definition.defs[i] for i in inst.value_results]
    actual_vars += [expr.args[i] for i in inst.value_opnums]

    # Get the list of the actual TypeVars
    actual_tvs = []
    for v in actual_vars:
        assert(isinstance(v, Var))
        # Register with TypeEnv that this typevar corresponds ot variable v,
        # and thus has a given rank
        typ.register(v)
        actual_tvs.append(v.get_typevar())

    # Make sure we unify the control typevar first.
    if inst.is_polymorphic:
        idx = fresh_formal_tvs.index(m[inst.ctrl_typevar])
        fresh_formal_tvs = move_first(fresh_formal_tvs, idx)
        actual_tvs = move_first(actual_tvs, idx)

    # Unify each actual typevar with the correpsonding fresh formal tv
    for (actual_tv, formal_tv) in zip(actual_tvs, fresh_formal_tvs):
        typ_or_err = unify(actual_tv, formal_tv, typ)
        err = get_error(typ_or_err)
        if (err):
            return "fail ti on {} <: {}: ".format(actual_tv, formal_tv) + err

        typ = get_type_env(typ_or_err)

    # Add any instruction specific constraints
    for constr in inst.constraints:
        typ.add_constraint(constr.translate(m))

    return typ


def ti_rtl(rtl, typ):
    # type: (Rtl, TypeEnv) -> TypingOrError
    """
    Perform type inference on an Rtl in a starting type env typ. Return an
    updated type environment or error.
    """
    for (i, d) in enumerate(rtl.rtl):
        assert (isinstance(d, Def))
        typ_or_err = ti_def(d, typ)
        err = get_error(typ_or_err)  # type: Optional[TypingError]
        if (err):
            return "On line {}: ".format(i) + err

        typ = get_type_env(typ_or_err)

    return typ


def ti_xform(xform, typ):
    # type: (XForm, TypeEnv) -> TypingOrError
    """
    Perform type inference on an Rtl in a starting type env typ. Return an
    updated type environment or error.
    """
    typ_or_err = ti_rtl(xform.src, typ)
    err = get_error(typ_or_err)  # type: Optional[TypingError]
    if (err):
        return "In src pattern: " + err

    typ = get_type_env(typ_or_err)

    typ_or_err = ti_rtl(xform.dst, typ)
    err = get_error(typ_or_err)
    if (err):
        return "In dst pattern: " + err

    typ = get_type_env(typ_or_err)

    return get_type_env(typ_or_err)