bool is_lt_no_level_params(level const & a, level const & b) {
if (is_eqp(a, b)) return false;
if (kind(a) != kind(b)) {
if (kind(a) == level_kind::Param || kind(b) == level_kind::Param)
return false;
return kind(a) < kind(b);
}
switch (kind(a)) {
case level_kind::Zero:
lean_unreachable(); // LCOV_EXCL_LINE
case level_kind::Param:
return false;
case level_kind::Global:
return global_id(a) < global_id(b);
case level_kind::Meta:
return meta_id(a) < meta_id(b);
case level_kind::Max:
if (is_lt_no_level_params(max_lhs(a), max_lhs(b)))
return true;
else if (is_lt_no_level_params(max_lhs(b), max_lhs(a)))
return false;
else
return is_lt_no_level_params(max_rhs(a), max_rhs(b));
case level_kind::IMax:
if (is_lt_no_level_params(imax_lhs(a), imax_lhs(b)))
return true;
else if (is_lt_no_level_params(imax_lhs(b), imax_lhs(a)))
return false;
else
return is_lt_no_level_params(imax_rhs(a), imax_rhs(b));
case level_kind::Succ:
return is_lt_no_level_params(succ_of(a), succ_of(b));
}
lean_unreachable();
}
unsigned level_cases_on(vm_obj const & o, buffer<vm_obj> & data) {
level const & l = to_level(o);
switch (l.kind()) {
case level_kind::Zero:
break;
case level_kind::Succ:
data.push_back(to_obj(succ_of(l)));
break;
case level_kind::Max:
data.push_back(to_obj(max_lhs(l)));
data.push_back(to_obj(max_rhs(l)));
break;
case level_kind::IMax:
data.push_back(to_obj(imax_lhs(l)));
data.push_back(to_obj(imax_rhs(l)));
break;
case level_kind::Param:
data.push_back(to_obj(param_id(l)));
break;
case level_kind::Global:
data.push_back(to_obj(global_id(l)));
break;
case level_kind::Meta:
data.push_back(to_obj(meta_id(l)));
break;
}
return static_cast<unsigned>(l.kind());
}
void push_max_args(level const & l, buffer<level> & r) {
if (is_max(l)) {
push_max_args(max_lhs(l), r);
push_max_args(max_rhs(l), r);
} else {
r.push_back(l);
}
}
bool is_geq_core(level l1, level l2) {
if (l1 == l2 || is_zero(l2))
return true;
if (is_max(l2))
return is_geq(l1, max_lhs(l2)) && is_geq(l1, max_rhs(l2));
if (is_max(l1) && (is_geq(max_lhs(l1), l2) || is_geq(max_rhs(l1), l2)))
return true;
if (is_imax(l2))
return is_geq(l1, imax_lhs(l2)) && is_geq(l1, imax_rhs(l2));
if (is_imax(l1))
return is_geq(imax_rhs(l1), l2);
auto p1 = to_offset(l1);
auto p2 = to_offset(l2);
if (p1.first == p2.first || is_zero(p2.first))
return p1.second >= p2.second;
if (p1.second == p2.second && p1.second > 0)
return is_geq(p1.first, p2.first);
return false;
}
bool is_not_zero(level const & l) {
switch (kind(l)) {
case level_kind::Zero: case level_kind::Param: case level_kind::Global: case level_kind::Meta:
return false;
case level_kind::Succ:
return true;
case level_kind::Max:
return is_not_zero(max_lhs(l)) || is_not_zero(max_rhs(l));
case level_kind::IMax:
return is_not_zero(imax_rhs(l));
}
lean_unreachable(); // LCOV_EXCL_LINE
}
level mk_max(level const & l1, level const & l2) {
if (is_explicit(l1) && is_explicit(l2)) {
return get_depth(l1) >= get_depth(l2) ? l1 : l2;
} else if (l1 == l2) {
return l1;
} else if (is_zero(l1)) {
return l2;
} else if (is_zero(l2)) {
return l1;
} else if (is_max(l2) && (max_lhs(l2) == l1 || max_rhs(l2) == l1)) {
return l2; // if l2 == (max l1 l'), then max l1 l2 == l2
} else if (is_max(l1) && (max_lhs(l1) == l2 || max_rhs(l1) == l2)) {
return l1; // if l1 == (max l2 l'), then max l1 l2 == l1
} else {
auto p1 = to_offset(l1);
auto p2 = to_offset(l2);
if (p1.first == p2.first) {
lean_assert(p1.second != p2.second);
return p1.second > p2.second ? l1 : l2;
} else {
return cache(level(new level_max_core(false, l1, l2)));
}
}
}
level apply(level const & l) {
auto r = m_lvl_cache.find(l);
if (r != m_lvl_cache.end())
return *r;
level res;
switch (l.kind()) {
case level_kind::Zero: case level_kind::Param:
case level_kind::Global: case level_kind::Meta:
res = l;
break;
case level_kind::Succ:
res = update_succ(l, apply(succ_of(l)));
break;
case level_kind::Max:
res = update_max(l, apply(max_lhs(l)), apply(max_rhs(l)));
break;
case level_kind::IMax:
res = update_max(l, apply(imax_lhs(l)), apply(imax_rhs(l)));
break;
}
m_lvl_cache.insert(res);
return res;
}
static environment mk_below(environment const & env, name const & n, bool ibelow) {
if (!is_recursive_datatype(env, n))
return env;
if (is_inductive_predicate(env, n))
return env;
inductive::inductive_decls decls = *inductive::is_inductive_decl(env, n);
type_checker tc(env);
name_generator ngen;
unsigned nparams = std::get<1>(decls);
declaration ind_decl = env.get(n);
declaration rec_decl = env.get(inductive::get_elim_name(n));
unsigned nindices = *inductive::get_num_indices(env, n);
unsigned nminors = *inductive::get_num_minor_premises(env, n);
unsigned ntypeformers = length(std::get<2>(decls));
level_param_names lps = rec_decl.get_univ_params();
bool is_reflexive = is_reflexive_datatype(tc, n);
level lvl = mk_param_univ(head(lps));
levels lvls = param_names_to_levels(tail(lps));
level_param_names blvls; // universe level parameters of ibelow/below
level rlvl; // universe level of the resultant type
// The arguments of below (ibelow) are the ones in the recursor - minor premises.
// The universe we map to is also different (l+1 for below of reflexive types) and (0 fo ibelow).
expr ref_type;
expr Type_result;
if (ibelow) {
// we are eliminating to Prop
blvls = tail(lps);
rlvl = mk_level_zero();
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_level_zero());
} else if (is_reflexive) {
blvls = lps;
rlvl = get_datatype_level(ind_decl.get_type());
// if rlvl is of the form (max 1 l), then rlvl <- l
if (is_max(rlvl) && is_one(max_lhs(rlvl)))
rlvl = max_rhs(rlvl);
rlvl = mk_max(mk_succ(lvl), rlvl);
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_succ(lvl));
} else {
// we can simplify the universe levels for non-reflexive datatypes
blvls = lps;
rlvl = mk_max(mk_level_one(), lvl);
ref_type = rec_decl.get_type();
}
Type_result = mk_sort(rlvl);
buffer<expr> ref_args;
to_telescope(ngen, ref_type, ref_args);
if (ref_args.size() != nparams + ntypeformers + nminors + nindices + 1)
throw_corrupted(n);
// args contains the below/ibelow arguments
buffer<expr> args;
buffer<name> typeformer_names;
// add parameters and typeformers
for (unsigned i = 0; i < nparams; i++)
args.push_back(ref_args[i]);
for (unsigned i = nparams; i < nparams + ntypeformers; i++) {
args.push_back(ref_args[i]);
typeformer_names.push_back(mlocal_name(ref_args[i]));
}
// we ignore minor premises in below/ibelow
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
args.push_back(ref_args[i]);
// We define below/ibelow using the recursor for this type
levels rec_lvls = cons(mk_succ(rlvl), lvls);
expr rec = mk_constant(rec_decl.get_name(), rec_lvls);
for (unsigned i = 0; i < nparams; i++)
rec = mk_app(rec, args[i]);
// add type formers
for (unsigned i = nparams; i < nparams + ntypeformers; i++) {
buffer<expr> targs;
to_telescope(ngen, mlocal_type(args[i]), targs);
rec = mk_app(rec, Fun(targs, Type_result));
}
// add minor premises
for (unsigned i = nparams + ntypeformers; i < nparams + ntypeformers + nminors; i++) {
expr minor = ref_args[i];
expr minor_type = mlocal_type(minor);
buffer<expr> minor_args;
minor_type = to_telescope(ngen, minor_type, minor_args);
buffer<expr> prod_pairs;
for (expr & minor_arg : minor_args) {
buffer<expr> minor_arg_args;
expr minor_arg_type = to_telescope(tc, mlocal_type(minor_arg), minor_arg_args);
if (is_typeformer_app(typeformer_names, minor_arg_type)) {
expr fst = mlocal_type(minor_arg);
minor_arg = update_mlocal(minor_arg, Pi(minor_arg_args, Type_result));
expr snd = Pi(minor_arg_args, mk_app(minor_arg, minor_arg_args));
prod_pairs.push_back(mk_prod(tc, fst, snd, ibelow));
}
}
expr new_arg = foldr([&](expr const & a, expr const & b) { return mk_prod(tc, a, b, ibelow); },
[&]() { return mk_unit(rlvl, ibelow); },
prod_pairs.size(), prod_pairs.data());
rec = mk_app(rec, Fun(minor_args, new_arg));
}
// add indices and major premise
for (unsigned i = nparams + ntypeformers; i < args.size(); i++) {
rec = mk_app(rec, args[i]);
}
name below_name = ibelow ? name{n, "ibelow"} : name{n, "below"};
expr below_type = Pi(args, Type_result);
expr below_value = Fun(args, rec);
bool use_conv_opt = true;
declaration new_d = mk_definition(env, below_name, blvls, below_type, below_value,
use_conv_opt);
environment new_env = module::add(env, check(env, new_d));
new_env = set_reducible(new_env, below_name, reducible_status::Reducible);
if (!ibelow)
new_env = add_unfold_hint(new_env, below_name, nparams + nindices + ntypeformers);
return add_protected(new_env, below_name);
}
static environment mk_brec_on(environment const & env, name const & n, bool ind) {
if (!is_recursive_datatype(env, n))
return env;
if (is_inductive_predicate(env, n))
return env;
inductive::inductive_decls decls = *inductive::is_inductive_decl(env, n);
type_checker tc(env);
name_generator ngen;
unsigned nparams = std::get<1>(decls);
declaration ind_decl = env.get(n);
declaration rec_decl = env.get(inductive::get_elim_name(n));
// declaration below_decl = env.get(name(n, ind ? "ibelow" : "below"));
unsigned nindices = *inductive::get_num_indices(env, n);
unsigned nminors = *inductive::get_num_minor_premises(env, n);
unsigned ntypeformers = length(std::get<2>(decls));
level_param_names lps = rec_decl.get_univ_params();
bool is_reflexive = is_reflexive_datatype(tc, n);
level lvl = mk_param_univ(head(lps));
levels lvls = param_names_to_levels(tail(lps));
level rlvl;
level_param_names blps;
levels blvls; // universe level parameters of brec_on/binduction_on
// The arguments of brec_on (binduction_on) are the ones in the recursor - minor premises.
// The universe we map to is also different (l+1 for below of reflexive types) and (0 fo ibelow).
expr ref_type;
if (ind) {
// we are eliminating to Prop
blps = tail(lps);
blvls = lvls;
rlvl = mk_level_zero();
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_level_zero());
} else if (is_reflexive) {
blps = lps;
blvls = cons(lvl, lvls);
rlvl = get_datatype_level(ind_decl.get_type());
// if rlvl is of the form (max 1 l), then rlvl <- l
if (is_max(rlvl) && is_one(max_lhs(rlvl)))
rlvl = max_rhs(rlvl);
rlvl = mk_max(mk_succ(lvl), rlvl);
// inner_prod, inner_prod_intro, pr1, pr2 do not use the same universe levels for
// reflective datatypes.
ref_type = instantiate_univ_param(rec_decl.get_type(), param_id(lvl), mk_succ(lvl));
} else {
// we can simplify the universe levels for non-reflexive datatypes
blps = lps;
blvls = cons(lvl, lvls);
rlvl = mk_max(mk_level_one(), lvl);
ref_type = rec_decl.get_type();
}
buffer<expr> ref_args;
to_telescope(ngen, ref_type, ref_args);
if (ref_args.size() != nparams + ntypeformers + nminors + nindices + 1)
throw_corrupted(n);
// args contains the brec_on/binduction_on arguments
buffer<expr> args;
buffer<name> typeformer_names;
// add parameters and typeformers
for (unsigned i = 0; i < nparams; i++)
args.push_back(ref_args[i]);
for (unsigned i = nparams; i < nparams + ntypeformers; i++) {
args.push_back(ref_args[i]);
typeformer_names.push_back(mlocal_name(ref_args[i]));
}
// add indices and major premise
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
args.push_back(ref_args[i]);
// create below terms (one per datatype)
// (below.{lvls} params type-formers)
// Remark: it also creates the result type
buffer<expr> belows;
expr result_type;
unsigned k = 0;
for (auto const & decl : std::get<2>(decls)) {
name const & n1 = inductive::inductive_decl_name(decl);
if (n1 == n) {
result_type = ref_args[nparams + k];
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
result_type = mk_app(result_type, ref_args[i]);
}
k++;
name bname = name(n1, ind ? "ibelow" : "below");
expr below = mk_constant(bname, blvls);
for (unsigned i = 0; i < nparams; i++)
below = mk_app(below, ref_args[i]);
for (unsigned i = nparams; i < nparams + ntypeformers; i++)
below = mk_app(below, ref_args[i]);
belows.push_back(below);
}
// create functionals (one for each type former)
// Pi idxs t, below idxs t -> C idxs t
buffer<expr> Fs;
name F_name("F");
for (unsigned i = nparams, j = 0; i < nparams + ntypeformers; i++, j++) {
expr const & C = ref_args[i];
buffer<expr> F_args;
to_telescope(ngen, mlocal_type(C), F_args);
expr F_result = mk_app(C, F_args);
expr F_below = mk_app(belows[j], F_args);
F_args.push_back(mk_local(ngen.next(), "f", F_below, binder_info()));
expr F_type = Pi(F_args, F_result);
expr F = mk_local(ngen.next(), F_name.append_after(j+1), F_type, binder_info());
Fs.push_back(F);
args.push_back(F);
}
// We define brec_on/binduction_on using the recursor for this type
levels rec_lvls = cons(rlvl, lvls);
expr rec = mk_constant(rec_decl.get_name(), rec_lvls);
// add parameters to rec
for (unsigned i = 0; i < nparams; i++)
rec = mk_app(rec, ref_args[i]);
// add type formers to rec
// Pi indices t, prod (C ... t) (below ... t)
for (unsigned i = nparams, j = 0; i < nparams + ntypeformers; i++, j++) {
expr const & C = ref_args[i];
buffer<expr> C_args;
to_telescope(ngen, mlocal_type(C), C_args);
expr C_t = mk_app(C, C_args);
expr below_t = mk_app(belows[j], C_args);
expr prod = mk_prod(tc, C_t, below_t, ind);
rec = mk_app(rec, Fun(C_args, prod));
}
// add minor premises to rec
for (unsigned i = nparams + ntypeformers, j = 0; i < nparams + ntypeformers + nminors; i++, j++) {
expr minor = ref_args[i];
expr minor_type = mlocal_type(minor);
buffer<expr> minor_args;
minor_type = to_telescope(ngen, minor_type, minor_args);
buffer<expr> pairs;
for (expr & minor_arg : minor_args) {
buffer<expr> minor_arg_args;
expr minor_arg_type = to_telescope(tc, mlocal_type(minor_arg), minor_arg_args);
if (auto k = is_typeformer_app(typeformer_names, minor_arg_type)) {
buffer<expr> C_args;
get_app_args(minor_arg_type, C_args);
expr new_minor_arg_type = mk_prod(tc, minor_arg_type, mk_app(belows[*k], C_args), ind);
minor_arg = update_mlocal(minor_arg, Pi(minor_arg_args, new_minor_arg_type));
if (minor_arg_args.empty()) {
pairs.push_back(minor_arg);
} else {
expr r = mk_app(minor_arg, minor_arg_args);
expr r_1 = Fun(minor_arg_args, mk_pr1(tc, r, ind));
expr r_2 = Fun(minor_arg_args, mk_pr2(tc, r, ind));
pairs.push_back(mk_pair(tc, r_1, r_2, ind));
}
}
}
expr b = foldr([&](expr const & a, expr const & b) { return mk_pair(tc, a, b, ind); },
[&]() { return mk_unit_mk(rlvl, ind); },
pairs.size(), pairs.data());
unsigned F_idx = *is_typeformer_app(typeformer_names, minor_type);
expr F = Fs[F_idx];
buffer<expr> F_args;
get_app_args(minor_type, F_args);
F_args.push_back(b);
expr new_arg = mk_pair(tc, mk_app(F, F_args), b, ind);
rec = mk_app(rec, Fun(minor_args, new_arg));
}
// add indices and major to rec
for (unsigned i = nparams + ntypeformers + nminors; i < ref_args.size(); i++)
rec = mk_app(rec, ref_args[i]);
name brec_on_name = name(n, ind ? "binduction_on" : "brec_on");
expr brec_on_type = Pi(args, result_type);
expr brec_on_value = Fun(args, mk_pr1(tc, rec, ind));
bool use_conv_opt = true;
declaration new_d = mk_definition(env, brec_on_name, blps, brec_on_type, brec_on_value,
use_conv_opt);
environment new_env = module::add(env, check(env, new_d));
new_env = set_reducible(new_env, brec_on_name, reducible_status::Reducible);
if (!ind)
new_env = add_unfold_hint(new_env, brec_on_name, nparams + nindices + ntypeformers);
return add_protected(new_env, brec_on_name);
}
