874 lines
40 KiB
C++
874 lines
40 KiB
C++
/*
|
|
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
|
|
|
Author: Leonardo de Moura
|
|
*/
|
|
#include "kernel/instantiate.h"
|
|
#include "kernel/inductive/inductive.h"
|
|
#include "library/trace.h"
|
|
#include "library/constants.h"
|
|
#include "library/locals.h"
|
|
#include "library/util.h"
|
|
#include "library/app_builder.h"
|
|
#include "library/replace_visitor_with_tc.h"
|
|
#include "library/equations_compiler/equations.h"
|
|
#include "library/equations_compiler/util.h"
|
|
#include "library/equations_compiler/structural_rec.h"
|
|
#include "library/equations_compiler/elim_match.h"
|
|
|
|
namespace lean {
|
|
#define trace_struct(Code) lean_trace(name({"eqn_compiler", "structural_rec"}), type_context ctx = mk_type_context(); scope_trace_env _scope1(m_env, ctx); Code)
|
|
#define trace_struct_aux(Code) lean_trace(name({"eqn_compiler", "structural_rec"}), scope_trace_env _scope1(m_ctx.env(), m_ctx); Code)
|
|
#define trace_debug_struct(Code) lean_trace(name({"debug", "eqn_compiler", "structural_rec"}), type_context ctx = mk_type_context(); scope_trace_env _scope1(m_env, ctx); Code)
|
|
#define trace_debug_struct_aux(Code) lean_trace(name({"debug", "eqn_compiler", "structural_rec"}), scope_trace_env _scope1(m_ctx.env(), m_ctx); Code)
|
|
|
|
struct structural_rec_fn {
|
|
environment m_env;
|
|
options m_opts;
|
|
metavar_context m_mctx;
|
|
local_context m_lctx;
|
|
|
|
expr m_ref;
|
|
equations_header m_header;
|
|
expr m_fn_type;
|
|
unsigned m_arity;
|
|
unsigned m_arg_pos;
|
|
bool m_reflexive;
|
|
bool m_use_ibelow;
|
|
buffer<unsigned> m_indices_pos;
|
|
expr m_motive_type;
|
|
|
|
structural_rec_fn(environment const & env, options const & opts,
|
|
metavar_context const & mctx, local_context const & lctx):
|
|
m_env(env), m_opts(opts), m_mctx(mctx), m_lctx(lctx) {
|
|
}
|
|
|
|
[[ noreturn ]] void throw_error(char const * msg) {
|
|
throw generic_exception(m_ref, msg);
|
|
}
|
|
|
|
[[ noreturn ]] void throw_error(sstream const & strm) {
|
|
throw generic_exception(m_ref, strm);
|
|
}
|
|
|
|
type_context mk_type_context() {
|
|
return type_context(m_env, m_opts, m_mctx, m_lctx, transparency_mode::Semireducible);
|
|
}
|
|
|
|
environment const & env() const { return m_env; }
|
|
metavar_context const & mctx() const { return m_mctx; }
|
|
|
|
/** \brief Auxiliary object for checking whether recursive application are
|
|
structurally smaller or not */
|
|
struct check_rhs_fn {
|
|
type_context & m_ctx;
|
|
expr m_lhs;
|
|
expr m_fn;
|
|
expr m_pattern;
|
|
unsigned m_arg_idx;
|
|
|
|
check_rhs_fn(type_context & ctx, expr const & lhs, expr const & fn, expr const & pattern, unsigned arg_idx):
|
|
m_ctx(ctx), m_lhs(lhs), m_fn(fn), m_pattern(pattern), m_arg_idx(arg_idx) {}
|
|
|
|
bool is_constructor(expr const & e) const {
|
|
return is_constant(e) && inductive::is_intro_rule(m_ctx.env(), const_name(e));
|
|
}
|
|
|
|
/** \brief Return true iff \c s is structurally smaller than \c t OR equal to \c t */
|
|
bool is_le(expr const & s, expr const & t) {
|
|
return m_ctx.is_def_eq(s, t) || is_lt(s, t);
|
|
}
|
|
|
|
/** Return true iff \c s is structurally smaller than \c t */
|
|
bool is_lt(expr s, expr t) {
|
|
s = m_ctx.whnf(s);
|
|
t = m_ctx.whnf(t);
|
|
if (is_app(s)) {
|
|
expr const & s_fn = get_app_fn(s);
|
|
if (!is_constructor(s_fn))
|
|
return is_lt(s_fn, t); // f < t ==> s := f a_1 ... a_n < t
|
|
}
|
|
buffer<expr> t_args;
|
|
expr const & t_fn = get_app_args(t, t_args);
|
|
if (!is_constructor(t_fn))
|
|
return false;
|
|
return std::any_of(t_args.begin(), t_args.end(),
|
|
[&](expr const & t_arg) { return is_le(s, t_arg); });
|
|
}
|
|
|
|
/** \brief Return true iff all recursive applications in \c e are structurally smaller than \c m_pattern. */
|
|
bool check_rhs(expr const & e) {
|
|
switch (e.kind()) {
|
|
case expr_kind::Var: case expr_kind::Meta:
|
|
case expr_kind::Local: case expr_kind::Constant:
|
|
case expr_kind::Sort:
|
|
return true;
|
|
case expr_kind::Macro:
|
|
for (unsigned i = 0; i < macro_num_args(e); i++)
|
|
if (!check_rhs(macro_arg(e, i)))
|
|
return false;
|
|
return true;
|
|
case expr_kind::App: {
|
|
buffer<expr> args;
|
|
expr const & fn = get_app_args(e, args);
|
|
if (!check_rhs(fn))
|
|
return false;
|
|
for (unsigned i = 0; i < args.size(); i++)
|
|
if (!check_rhs(args[i]))
|
|
return false;
|
|
if (is_local(fn) && mlocal_name(fn) == mlocal_name(m_fn)) {
|
|
/* recusive application */
|
|
if (m_arg_idx < args.size()) {
|
|
expr const & arg = args[m_arg_idx];
|
|
/* arg must be structurally smaller than m_pattern */
|
|
if (!is_lt(arg, m_pattern)) {
|
|
trace_struct_aux(tout() << "structural recursion on argument #" << (m_arg_idx+1)
|
|
<< " was not used "
|
|
<< "for '" << m_fn << "'\nargument #" << (m_arg_idx+1)
|
|
<< " in the application\n "
|
|
<< e << "\nis not structurally smaller than the one occurring in "
|
|
<< "the equation left-hand-side\n "
|
|
<< m_lhs << "\n";);
|
|
return false;
|
|
}
|
|
} else {
|
|
/* function is not fully applied */
|
|
trace_struct_aux(tout() << "structural recursion on argument #" << (m_arg_idx+1) << " was not used "
|
|
<< "for '" << m_fn << "' because of the partial application\n "
|
|
<< e << "\n";);
|
|
return false;
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
case expr_kind::Let:
|
|
if (!check_rhs(let_value(e))) {
|
|
return false;
|
|
} else {
|
|
type_context::tmp_locals locals(m_ctx);
|
|
return check_rhs(instantiate(let_body(e), locals.push_local_from_let(e)));
|
|
}
|
|
case expr_kind::Lambda:
|
|
case expr_kind::Pi:
|
|
if (!check_rhs(binding_domain(e))) {
|
|
return false;
|
|
} else {
|
|
type_context::tmp_locals locals(m_ctx);
|
|
return check_rhs(instantiate(binding_body(e), locals.push_local_from_binding(e)));
|
|
}
|
|
}
|
|
lean_unreachable();
|
|
}
|
|
|
|
bool operator()(expr const & e) {
|
|
return check_rhs(e);
|
|
}
|
|
};
|
|
|
|
bool check_rhs(type_context & ctx, expr const & lhs, expr const & fn, expr pattern, unsigned arg_idx, expr const & rhs) {
|
|
pattern = ctx.whnf(pattern);
|
|
return check_rhs_fn(ctx, lhs, fn, pattern, arg_idx)(rhs);
|
|
}
|
|
|
|
bool check_eq(type_context & ctx, expr const & eqn, unsigned arg_idx) {
|
|
unpack_eqn ue(ctx, eqn);
|
|
buffer<expr> args;
|
|
expr const & fn = get_app_args(ue.lhs(), args);
|
|
return check_rhs(ctx, ue.lhs(), fn, args[arg_idx], arg_idx, ue.rhs());
|
|
}
|
|
|
|
static bool depends_on_locals(expr const & e, type_context::tmp_locals const & locals) {
|
|
return depends_on_any(e, locals.as_buffer().size(), locals.as_buffer().data());
|
|
}
|
|
|
|
/* Return true iff argument arg_idx is a candidate for structural recursion.
|
|
If the argument type is an indexed family, we store the position of the
|
|
indices (in the function being defined) at m_indices_pos.
|
|
This method also updates m_reflexive (true iff the inductive datatype is reflexive). */
|
|
bool check_arg_type(type_context & ctx, unpack_eqns const & ues, unsigned arg_idx) {
|
|
m_indices_pos.clear();
|
|
type_context::tmp_locals locals(ctx);
|
|
/* We can only use structural recursion on arg_idx IF
|
|
1- Type is an inductive datatype with support for the brec_on construction.
|
|
2- Type parameters do not depend on other arguments of the function being defined. */
|
|
expr fn = ues.get_fn(0);
|
|
expr fn_type = ctx.infer(fn);
|
|
for (unsigned i = 0; i < arg_idx; i++) {
|
|
fn_type = ctx.whnf(fn_type);
|
|
if (!is_pi(fn_type)) throw_ill_formed_eqns();
|
|
fn_type = instantiate(binding_body(fn_type), locals.push_local_from_binding(fn_type));
|
|
}
|
|
if (!is_pi(fn_type)) throw_ill_formed_eqns();
|
|
expr arg_type = ctx.relaxed_whnf(binding_domain(fn_type));
|
|
buffer<expr> I_args;
|
|
expr I = get_app_args(arg_type, I_args);
|
|
if (is_constant(I) && !inductive::is_inductive_decl(m_env, const_name(I))) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because type is not inductive\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
name I_name = const_name(I);
|
|
m_reflexive = is_reflexive_datatype(ctx, I_name);
|
|
if (!m_env.find(name(I_name, "brec_on"))) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because the inductive type '" << I << "' does have brec_on recursor\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
if (m_reflexive && !m_env.find(name(I_name, "binduction_on"))) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because the reflexive inductive type '" << I << "' does "
|
|
<< "have binduction_on recursor\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
unsigned nindices = *inductive::get_num_indices(m_env, I_name);
|
|
if (nindices > 0) {
|
|
lean_assert(I_args.size() >= nindices);
|
|
for (unsigned i = I_args.size() - nindices; i < I_args.size(); i++) {
|
|
expr const & idx = I_args[i];
|
|
if (!is_local(idx)) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because the inductive type '" << I << "' is an indexed family, "
|
|
<< "and index #" << (i+1) << " is not a variable\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
/* Index must be an argument of the function being defined */
|
|
unsigned idx_pos = 0;
|
|
buffer<expr> const & xs = locals.as_buffer();
|
|
for (; idx_pos < xs.size(); idx_pos++) {
|
|
expr const & x = xs[idx_pos];
|
|
if (mlocal_name(x) == mlocal_name(idx)) {
|
|
break;
|
|
}
|
|
}
|
|
if (idx_pos == xs.size()) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because the inductive type '" << I << "' is an indexed family, "
|
|
<< "and index #" << (i+1) << " is not an argument of the function being defined\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
/* Index can only depend on other indices in the function being defined. */
|
|
expr idx_type = ctx.infer(idx);
|
|
for (unsigned j = 0; j < idx_pos; j++) {
|
|
bool j_is_not_index =
|
|
std::find(m_indices_pos.begin(), m_indices_pos.end(), j) == m_indices_pos.end();
|
|
if (j_is_not_index && depends_on(idx_type, xs[j])) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because the inductive type '" << I << "' is an indexed family, "
|
|
<< "and index #" << (i+1) << " depends on argument #" << (j+1) << " of '" << fn << "' "
|
|
<< "which is not an index of the inductive datatype\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
}
|
|
m_indices_pos.push_back(idx_pos);
|
|
/* Each index can only occur once */
|
|
for (unsigned j = 0; j < i; j++) {
|
|
expr const & prev_idx = I_args[j];
|
|
if (mlocal_name(prev_idx) == mlocal_name(idx)) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because the inductive type '" << I << "' is an indexed family, "
|
|
<< "and index #" << (i+1) << " and #" << (j+1) << " must be different variables\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
}
|
|
}
|
|
}
|
|
for (unsigned i = 0; i < I_args.size() - nindices; i++) {
|
|
if (depends_on_locals(I_args[i], locals)) {
|
|
trace_struct(tout() << "structural recursion on argument #" << (arg_idx+1) << " was not used "
|
|
<< "for '" << fn << "' because type parameter depends on previous arguments\n "
|
|
<< arg_type << "\n";);
|
|
return false;
|
|
}
|
|
}
|
|
return true;
|
|
}
|
|
|
|
/* Return true iff structural recursion can be performed on one of the arguments.
|
|
If the result is true, then m_arg_pos will contain the position of the argument,
|
|
and m_indices_pos the position of its indices (when the type of the
|
|
argument is an indexed family). */
|
|
bool find_rec_arg(type_context & ctx, unpack_eqns const & ues) {
|
|
buffer<expr> const & eqns = ues.get_eqns_of(0);
|
|
unsigned arity = ues.get_arity_of(0);
|
|
for (unsigned i = 0; i < arity; i++) {
|
|
if (check_arg_type(ctx, ues, i)) {
|
|
bool ok = true;
|
|
for (expr const & eqn : eqns) {
|
|
if (!check_eq(ctx, eqn, i)) {
|
|
ok = false;
|
|
break;
|
|
}
|
|
}
|
|
if (ok) {
|
|
m_arg_pos = i;
|
|
return true;
|
|
}
|
|
}
|
|
}
|
|
return false;
|
|
}
|
|
|
|
/* Return the type of the new function.
|
|
It also sets the m_motive_type field. */
|
|
expr mk_new_fn_motive_types(type_context & ctx, unpack_eqns const & ues) {
|
|
type_context::tmp_locals locals(ctx);
|
|
expr fn = ues.get_fn(0);
|
|
expr fn_type = ctx.infer(fn);
|
|
unsigned arity = ues.get_arity_of(0);
|
|
expr rec_arg;
|
|
buffer<expr> other_args;
|
|
buffer<expr> idx_args;
|
|
for (unsigned i = 0; i < arity; i++) {
|
|
fn_type = ctx.whnf(fn_type);
|
|
if (!is_pi(fn_type)) throw_ill_formed_eqns();
|
|
expr arg = locals.push_local_from_binding(fn_type);
|
|
if (i == m_arg_pos) {
|
|
rec_arg = arg;
|
|
} else if (std::find(m_indices_pos.begin(), m_indices_pos.end(), i) != m_indices_pos.end()) {
|
|
idx_args.push_back(arg);
|
|
} else {
|
|
other_args.push_back(arg);
|
|
}
|
|
fn_type = instantiate(binding_body(fn_type), arg);
|
|
}
|
|
buffer<expr> I_params;
|
|
expr I = get_app_args(ctx.infer(rec_arg), I_params);
|
|
unsigned nindices = m_indices_pos.size();
|
|
I_params.shrink(I_params.size() - nindices);
|
|
expr motive = ctx.mk_pi(other_args, fn_type);
|
|
level u = get_level(ctx, motive);
|
|
if (m_reflexive) {
|
|
if (!is_zero(u) && !is_not_zero(u))
|
|
throw_error(sstream() << "invalid equations, "
|
|
<< "when trying to recurse over reflexive inductive datatype "
|
|
<< "'" << const_name(I) << "' "
|
|
<< "(argument #" << (m_arg_pos+1) << ") "
|
|
<< "the universe level of the resultant universe must be zero OR "
|
|
<< "not zero for every level assignment "
|
|
<< "(possible solutions: provide universe levels explicitly, "
|
|
<< "or force well_founded recursion by using `using_well_founded` keyword)");
|
|
if (!is_zero(u)) {
|
|
// For reflexive type, the type of brec_on and ibelow perform a +1 on the motive universe.
|
|
// Example: for a reflexive formula type, we have:
|
|
// formula.below.{l_1} : Π {C : formula → Type.{l_1+1}}, formula → Type.{max (l_1+1) 1}
|
|
if (auto dlvl = dec_level(u)) {
|
|
u = *dlvl;
|
|
} else {
|
|
throw_error(sstream() << "invalid equations, "
|
|
<< "when trying to recurse over reflexive inductive datatype "
|
|
<< "'" << const_name(I) << "' "
|
|
<< "(argument #" << (m_arg_pos+1) << ") "
|
|
<< "the universe level of the resultant universe must be zero OR "
|
|
<< "not zero for every level assignment. "
|
|
<< "The compiler managed to establish that the resultant "
|
|
<< "universe level u := (" << u << ") is never zero, but failed to compute "
|
|
<< "the new resulting level (u - 1) "
|
|
<< "(possible solutions: provide universe levels explicitly, "
|
|
<< "or force well_founded recursion by using `using_well_founded` keyword)");
|
|
}
|
|
}
|
|
}
|
|
m_use_ibelow = m_reflexive && is_zero(u);
|
|
motive = ctx.mk_lambda(idx_args, ctx.mk_lambda(rec_arg, motive));
|
|
lean_assert(is_constant(I));
|
|
buffer<level> below_lvls;
|
|
if (!m_use_ibelow)
|
|
below_lvls.push_back(u);
|
|
for (level const & v : const_levels(I))
|
|
below_lvls.push_back(v);
|
|
name below_name = name(const_name(I), m_use_ibelow ? "ibelow" : "below");
|
|
expr below = mk_app(mk_constant(below_name, to_list(below_lvls)), I_params);
|
|
m_motive_type = binding_domain(ctx.relaxed_whnf(ctx.infer(below)));
|
|
below = mk_app(mk_app(mk_app(below, motive), idx_args), rec_arg);
|
|
locals.push_local("_F", below);
|
|
return locals.mk_pi(fn_type);
|
|
}
|
|
|
|
struct elim_rec_apps_failed {};
|
|
|
|
struct elim_rec_apps_fn : public replace_visitor_with_tc {
|
|
expr m_fn;
|
|
unsigned m_arg_pos;
|
|
buffer<unsigned> const & m_indices_pos;
|
|
expr m_F;
|
|
expr m_C;
|
|
|
|
elim_rec_apps_fn(type_context & ctx, expr const & fn,
|
|
unsigned arg_pos, buffer<unsigned> const & indices_pos, expr const & F, expr const & C):
|
|
replace_visitor_with_tc(ctx),
|
|
m_fn(fn), m_arg_pos(arg_pos), m_indices_pos(indices_pos), m_F(F), m_C(C) {}
|
|
|
|
/** \brief Retrieve result for \c a from the below dictionary \c d. \c d is a term made of products,
|
|
and m_C (the abstract local). */
|
|
optional<expr> to_below(expr const & d, expr const & a, expr const & F) {
|
|
expr const & fn = get_app_fn(d);
|
|
if (is_constant(fn, get_prod_name())) {
|
|
expr d_arg1 = m_ctx.whnf(app_arg(app_fn(d)));
|
|
expr d_arg2 = m_ctx.whnf(app_arg(d));
|
|
if (auto r = to_below(d_arg1, a, mk_pr1(m_ctx, F)))
|
|
return r;
|
|
else if (auto r = to_below(d_arg2, a, mk_pr2(m_ctx, F)))
|
|
return r;
|
|
else
|
|
return none_expr();
|
|
} else if (is_local(fn)) {
|
|
if (mlocal_name(m_C) == mlocal_name(fn) && m_ctx.is_def_eq(app_arg(d), a))
|
|
return some_expr(F);
|
|
return none_expr();
|
|
} else if (is_pi(d)) {
|
|
if (is_app(a)) {
|
|
expr new_d = m_ctx.whnf(instantiate(binding_body(d), app_arg(a)));
|
|
return to_below(new_d, a, mk_app(F, app_arg(a)));
|
|
} else {
|
|
return none_expr();
|
|
}
|
|
} else {
|
|
return none_expr();
|
|
}
|
|
}
|
|
|
|
bool is_index_pos(unsigned idx) const {
|
|
return std::find(m_indices_pos.begin(), m_indices_pos.end(), idx) != m_indices_pos.end();
|
|
}
|
|
|
|
expr elim(buffer<expr> const & args, tag g) {
|
|
/* Replace motives with abstract one m_C.
|
|
We use the abstract motive m_C as "marker". */
|
|
buffer<expr> below_args;
|
|
expr const & below_cnst = get_app_args(m_ctx.infer(m_F), below_args);
|
|
unsigned nindices = m_indices_pos.size();
|
|
below_args[below_args.size() - 1 - 1 /* major */ - nindices] = m_C;
|
|
expr abst_below = mk_app(below_cnst, below_args);
|
|
expr below_dict = m_ctx.whnf(abst_below);
|
|
expr rec_arg = m_ctx.whnf(args[m_arg_pos]);
|
|
if (optional<expr> b = to_below(below_dict, rec_arg, m_F)) {
|
|
expr r = *b;
|
|
for (unsigned i = 0; i < args.size(); i++) {
|
|
if (i != m_arg_pos && !is_index_pos(i))
|
|
r = mk_app(r, args[i], g);
|
|
}
|
|
return r;
|
|
} else {
|
|
throw elim_rec_apps_failed();
|
|
}
|
|
}
|
|
|
|
virtual expr visit_local(expr const & e) {
|
|
if (mlocal_name(e) == mlocal_name(m_fn)) {
|
|
/* unexpected occurrence of recursive function */
|
|
throw elim_rec_apps_failed();
|
|
}
|
|
return e;
|
|
}
|
|
|
|
virtual expr visit_app(expr const & e) {
|
|
expr const & fn = get_app_fn(e);
|
|
if (is_local(fn) && mlocal_name(fn) == mlocal_name(m_fn)) {
|
|
buffer<expr> args;
|
|
get_app_args(e, args);
|
|
if (m_arg_pos >= args.size()) throw elim_rec_apps_failed();
|
|
buffer<expr> new_args;
|
|
for (expr const & arg : args)
|
|
new_args.push_back(visit(arg));
|
|
return elim(new_args, e.get_tag());
|
|
} else {
|
|
return replace_visitor_with_tc::visit_app(e);
|
|
}
|
|
}
|
|
};
|
|
|
|
void update_eqs(type_context & ctx, unpack_eqns & ues, expr const & fn, expr const & new_fn) {
|
|
/* C is a temporary "abstract" motive, we use it to access the "brec_on dictionary".
|
|
The "brec_on dictionary is an element of type below, and it is the last argument of the new function. */
|
|
expr C = mk_local(mk_fresh_name(), "_C", m_motive_type, binder_info());
|
|
buffer<expr> & eqns = ues.get_eqns_of(0);
|
|
for (expr & eqn : eqns) {
|
|
unpack_eqn ue(ctx, eqn);
|
|
expr lhs = ue.lhs();
|
|
expr rhs = ue.rhs();
|
|
buffer<expr> lhs_args;
|
|
get_app_args(lhs, lhs_args);
|
|
expr new_lhs = mk_app(new_fn, lhs_args);
|
|
expr type = ctx.whnf(ctx.infer(new_lhs));
|
|
lean_assert(is_pi(type));
|
|
expr F = ue.add_var(binding_name(type), binding_domain(type));
|
|
new_lhs = mk_app(new_lhs, F);
|
|
ue.lhs() = new_lhs;
|
|
ue.rhs() = elim_rec_apps_fn(ctx, fn, m_arg_pos, m_indices_pos, F, C)(rhs);
|
|
eqn = ue.repack();
|
|
}
|
|
}
|
|
|
|
optional<expr> elim_recursion(expr const & e) {
|
|
type_context ctx = mk_type_context();
|
|
unpack_eqns ues(ctx, e);
|
|
if (ues.get_num_fns() != 1) {
|
|
trace_struct(tout() << "structural recursion is not supported for mutually recursive functions:";
|
|
for (unsigned i = 0; i < ues.get_num_fns(); i++)
|
|
tout() << " " << ues.get_fn(i);
|
|
tout() << "\n";);
|
|
return none_expr();
|
|
}
|
|
m_fn_type = ctx.infer(ues.get_fn(0));
|
|
m_arity = ues.get_arity_of(0);
|
|
if (!find_rec_arg(ctx, ues)) return none_expr();
|
|
expr fn = ues.get_fn(0);
|
|
trace_struct(tout() << "using structural recursion on argument #" << (m_arg_pos+1) <<
|
|
" for '" << fn << "'\n";);
|
|
expr new_fn_type = mk_new_fn_motive_types(ctx, ues);
|
|
trace_struct(
|
|
tout() << "\n";
|
|
tout() << "new function type: " << new_fn_type << "\n";
|
|
tout() << "motive type: " << m_motive_type << "\n";);
|
|
expr new_fn = ues.update_fn_type(0, new_fn_type);
|
|
try {
|
|
update_eqs(ctx, ues, fn, new_fn);
|
|
} catch (elim_rec_apps_failed &) {
|
|
trace_struct(tout() << "failed to compile equations/match using structural recursion, "
|
|
<< "when creating new set of equations\n";);
|
|
return none_expr();
|
|
}
|
|
expr new_eqns = ues.repack();
|
|
lean_trace("eqn_compiler", tout() << "using structural recursion:\n" << new_eqns << "\n";);
|
|
m_mctx = ctx.mctx();
|
|
return some_expr(new_eqns);
|
|
}
|
|
|
|
expr whnf_upto_below(type_context & ctx, name const & I_name, expr const & below_type) {
|
|
name below_name(I_name, "below");
|
|
return ctx.whnf_pred(below_type, [&](expr const & e) {
|
|
expr const & fn = get_app_fn(e);
|
|
return !is_constant(fn) || const_name(fn) != below_name;
|
|
});
|
|
}
|
|
|
|
bool is_index_pos(unsigned idx) const {
|
|
return std::find(m_indices_pos.begin(), m_indices_pos.end(), idx) != m_indices_pos.end();
|
|
}
|
|
|
|
expr mk_function(expr const & aux_fn) {
|
|
type_context ctx = mk_type_context();
|
|
type_context::tmp_locals locals(ctx);
|
|
buffer<expr> fn_args;
|
|
expr aux_fn_type = ctx.infer(aux_fn);
|
|
for (unsigned i = 0; i < m_arity + 1 /* below argument */; i++) {
|
|
aux_fn_type = ctx.relaxed_whnf(aux_fn_type);
|
|
lean_assert(is_pi(aux_fn_type));
|
|
expr arg = locals.push_local_from_binding(aux_fn_type);
|
|
if (i < m_arity) fn_args.push_back(arg);
|
|
aux_fn_type = instantiate(binding_body(aux_fn_type), arg);
|
|
}
|
|
buffer<expr> const & aux_fn_args = locals.as_buffer();
|
|
unsigned nindices = m_indices_pos.size();
|
|
expr rec_arg = aux_fn_args[m_arg_pos];
|
|
expr rec_arg_type = ctx.relaxed_whnf(ctx.infer(rec_arg));
|
|
buffer<expr> I_args;
|
|
expr const & I = get_app_args(rec_arg_type, I_args);
|
|
name I_name = const_name(I);
|
|
unsigned nparams = I_args.size() - nindices;
|
|
expr below_arg = aux_fn_args.back();
|
|
expr below_type = whnf_upto_below(ctx, I_name, ctx.infer(below_arg));
|
|
buffer<expr> below_args;
|
|
expr below = get_app_args(below_type, below_args);
|
|
expr motive = below_args[nparams];
|
|
name brec_on_name = name(I_name, m_use_ibelow ? "binduction_on" : "brec_on");
|
|
expr brec_on_fn = mk_constant(brec_on_name, const_levels(below));
|
|
buffer<expr> brec_on_args;
|
|
buffer<expr> F_domain; /* domain for F argument for brec_on */
|
|
brec_on_args.append(nparams, I_args.data());
|
|
brec_on_args.push_back(motive);
|
|
for (unsigned idx_pos : m_indices_pos) {
|
|
brec_on_args.push_back(aux_fn_args[idx_pos]);
|
|
F_domain.push_back(aux_fn_args[idx_pos]);
|
|
}
|
|
brec_on_args.push_back(rec_arg);
|
|
F_domain.push_back(rec_arg);
|
|
F_domain.push_back(below_arg);
|
|
buffer<expr> extra_args;
|
|
for (unsigned i = 0; i < fn_args.size(); i++) {
|
|
if (i != m_arg_pos && !is_index_pos(i)) {
|
|
F_domain.push_back(aux_fn_args[i]);
|
|
extra_args.push_back(aux_fn_args[i]);
|
|
}
|
|
}
|
|
expr F = ctx.mk_lambda(F_domain, mk_app(aux_fn, aux_fn_args));
|
|
brec_on_args.push_back(F);
|
|
expr new_fn = ctx.mk_lambda(fn_args, mk_app(mk_app(brec_on_fn, brec_on_args), extra_args));
|
|
lean_trace("eqn_compiler", tout() << "result:\n" << new_fn << "\ntype:\n" << ctx.infer(new_fn) << "\n";);
|
|
if (m_header.m_is_meta) {
|
|
/* We don't create auxiliary definitions for meta-definitions because we don't create lemmas
|
|
for them. */
|
|
return new_fn;
|
|
} else {
|
|
expr r;
|
|
std::tie(m_env, r) = mk_aux_definition(m_env, m_mctx, m_lctx, m_header.m_is_private, head(m_header.m_fn_names),
|
|
m_fn_type, new_fn);
|
|
return r;
|
|
}
|
|
}
|
|
|
|
struct mk_lemma_rhs_fn : public replace_visitor_with_tc {
|
|
expr m_fn;
|
|
expr m_F;
|
|
expr m_lhs_rec_arg;
|
|
unsigned m_arg_pos;
|
|
buffer<unsigned> const & m_indices_pos;
|
|
public:
|
|
mk_lemma_rhs_fn(type_context & ctx, expr const & fn, expr const & F, expr const & rec_arg,
|
|
unsigned arg_pos, buffer<unsigned> const & indices_pos):
|
|
replace_visitor_with_tc(ctx), m_fn(fn), m_F(F), m_lhs_rec_arg(rec_arg),
|
|
m_arg_pos(arg_pos), m_indices_pos(indices_pos) {}
|
|
|
|
environment const & env() const { return m_ctx.env(); }
|
|
|
|
/* Auxiliary method for detecting terms representing "recursive calls". Examples:
|
|
F.1.1
|
|
F.2.2.1.2.1.1
|
|
|
|
It stores the path (.e.g., [2,2,1,2,1,1]) in the output parameter `path`.
|
|
|
|
\remark This is encoding is defined in the automatically generated `below` functions.
|
|
Example: consider the inductive datatype
|
|
|
|
inductive tree (A : Type)
|
|
| leaf : A → tree
|
|
| node : tree → tree → tree → tree
|
|
|
|
Then, we can use the follow command to "visualize" how the information is encoded.
|
|
|
|
eval ∀ (A : Type) (n₁ n₂ n₃ n₄ n₅ : tree A) (C : tree A → Type), @tree.below A C (node (node n₁ n₂ n₃) n₄ n₅)
|
|
*/
|
|
bool is_F_instance(expr const & e, buffer<unsigned> & path) const {
|
|
if (e == m_F) return true;
|
|
buffer<expr> args;
|
|
expr const & fn = get_app_args(e, args);
|
|
if (args.size() == 3) {
|
|
if (is_constant(fn, get_prod_pr1_name())) {
|
|
bool r = is_F_instance(args[2], path);
|
|
path.push_back(1);
|
|
return r;
|
|
} else if (is_constant(fn, get_prod_pr2_name())) {
|
|
bool r = is_F_instance(args[2], path);
|
|
path.push_back(2);
|
|
return r;
|
|
}
|
|
}
|
|
return false;
|
|
}
|
|
|
|
/* Return the ith recursive argument of \c e */
|
|
pair<expr, unsigned> get_ith_rec_arg(expr e, unsigned ith) {
|
|
e = m_ctx.relaxed_whnf(e);
|
|
buffer<pair<expr, unsigned>> rec_args;
|
|
if (get_constructor_rec_args(m_ctx.env(), e, rec_args)) {
|
|
lean_assert(ith < rec_args.size());
|
|
return rec_args[ith];
|
|
}
|
|
lean_unreachable();
|
|
}
|
|
|
|
optional<name> is_constructor(name const & n) const { return inductive::is_intro_rule(m_ctx.env(), n); }
|
|
optional<name> is_constructor(expr const & e) const {
|
|
if (!is_constant(e)) return optional<name>();
|
|
return is_constructor(const_name(e));
|
|
}
|
|
optional<name> is_constructor_app(expr const & e) const { return is_constructor(get_app_fn(e)); }
|
|
|
|
/* Move to different module? */
|
|
expr normalize_constructor_apps(expr const & e) {
|
|
expr new_e = m_ctx.relaxed_whnf(e);
|
|
if (optional<name> I_name = is_constructor_app(new_e)) {
|
|
buffer<expr> args;
|
|
expr const & c_fn = get_app_args(new_e, args);
|
|
unsigned nparams = *inductive::get_num_params(m_ctx.env(), *I_name);
|
|
for (unsigned i = nparams; i < args.size(); i++) {
|
|
args[i] = normalize_constructor_apps(args[i]);
|
|
}
|
|
return mk_app(c_fn, args);
|
|
} else if (is_local(new_e)) {
|
|
return new_e;
|
|
} else {
|
|
return e;
|
|
}
|
|
}
|
|
|
|
/* Auxiliary method for decode_rec_arg */
|
|
pair<expr, unsigned> decode_rec_arg_core(expr const & e, unsigned i, buffer<unsigned> const & path) {
|
|
unsigned rec_arg_idx = 0;
|
|
while (true) {
|
|
lean_assert(i < path.size());
|
|
if (path[i] == 1)
|
|
break;
|
|
rec_arg_idx++;
|
|
i++;
|
|
}
|
|
i++;
|
|
auto result = get_ith_rec_arg(e, rec_arg_idx);
|
|
if (path[i] == 1) {
|
|
return mk_pair(normalize_constructor_apps(result.first), result.second);
|
|
} else {
|
|
i++;
|
|
return decode_rec_arg_core(result.first, i, path);
|
|
}
|
|
}
|
|
|
|
/* Decode the argument for the recursive call.
|
|
|
|
Example: consider the inductive datatype
|
|
|
|
inductive tree (A : Type)
|
|
| leaf : A → tree
|
|
| node : tree → tree → tree → tree
|
|
|
|
and the term (F.1.2.2.2.1.1). We extract the path
|
|
[1,2,2,2,1,1] for this term.
|
|
Now, assume the recursive argument in the left-hand-side is
|
|
|
|
(node (node n₁ n₂ n₃) n₄ n₅).
|
|
|
|
Then, the result of this method is the pair (n₃, 0).
|
|
|
|
The unsigned value is only relevant for reflexive inductive types.
|
|
|
|
Example: consider the reflexive inductive datatype
|
|
|
|
inductive inftree (A : Type)
|
|
| leaf : A → inftree
|
|
| node : (nat → inftree) → inftree
|
|
|
|
and the term (F.1.1). We extract the path [1,1] for this term.
|
|
Now, assume the recursive argument in the left-hand-side is
|
|
|
|
(node f).
|
|
|
|
Then, the result of this method is the pair (f, 1).
|
|
The value 1 indicates that f takes one argument. */
|
|
pair<expr, unsigned> decode_rec_arg(buffer<unsigned> const & path) {
|
|
return decode_rec_arg_core(m_lhs_rec_arg, 0, path);
|
|
}
|
|
|
|
virtual expr visit_app(expr const & e) {
|
|
buffer<expr> args;
|
|
expr const & fn = get_app_args(e, args);
|
|
if (is_constant(fn, get_prod_pr1_name()) && args.size() >= 3) {
|
|
buffer<unsigned> path;
|
|
if (is_F_instance(args[2], path)) {
|
|
path.push_back(1);
|
|
unsigned b_next_idx = 3; /* pr1 A B F b_1 ... */
|
|
expr rec_arg; unsigned rec_arg_arity;
|
|
std::tie(rec_arg, rec_arg_arity) = decode_rec_arg(path);
|
|
for (unsigned i = 0; i < rec_arg_arity; i++, b_next_idx++) {
|
|
rec_arg = mk_app(rec_arg, args[b_next_idx]);
|
|
}
|
|
expr rec_arg_type = m_ctx.relaxed_whnf(m_ctx.infer(rec_arg));
|
|
buffer<expr> I_args;
|
|
expr const & I = get_app_args(rec_arg_type, I_args);
|
|
lean_assert(is_constant(I));
|
|
name I_name = const_name(I);
|
|
lean_assert(inductive::is_inductive_decl(env(), I_name));
|
|
unsigned nindices = m_indices_pos.size();
|
|
lean_assert(*inductive::get_num_indices(env(), I_name) == m_indices_pos.size());
|
|
unsigned I_next_idx = I_args.size() - nindices;
|
|
unsigned new_nargs = nindices + 1 + args.size() - b_next_idx;
|
|
buffer<expr> new_args;
|
|
for (unsigned i = 0; i < new_nargs; i++) {
|
|
if (i == m_arg_pos) {
|
|
new_args.push_back(rec_arg);
|
|
} else if (std::find(m_indices_pos.begin(), m_indices_pos.end(), i) != m_indices_pos.end()) {
|
|
new_args.push_back(I_args[I_next_idx]);
|
|
I_next_idx++;
|
|
} else {
|
|
new_args.push_back(args[b_next_idx]);
|
|
b_next_idx++;
|
|
}
|
|
}
|
|
expr new_e = mk_app(m_fn, new_args);
|
|
trace_debug_struct_aux(tout() << "decoded equation rhs term:\n" << e << "\n==>\n" << new_e << "\n";);
|
|
return new_e;
|
|
}
|
|
}
|
|
return replace_visitor_with_tc::visit_app(e);
|
|
}
|
|
};
|
|
|
|
expr mk_lemma_rhs(type_context & ctx, expr const & fn, expr const & F, expr const & rec_arg, expr const & rhs) {
|
|
return mk_lemma_rhs_fn(ctx, fn, F, rec_arg, m_arg_pos, m_indices_pos)(rhs);
|
|
}
|
|
|
|
void mk_lemmas(expr const & fn, list<expr> const & lemmas) {
|
|
name base_name(const_name(get_app_fn(fn)), "equations");
|
|
unsigned eqn_idx = 1;
|
|
type_context ctx = mk_type_context();
|
|
for (expr type : lemmas) {
|
|
type_context::tmp_locals locals(ctx);
|
|
type = ctx.relaxed_whnf(type);
|
|
while (is_pi(type)) {
|
|
expr local = locals.push_local_from_binding(type);
|
|
type = instantiate(binding_body(type), local);
|
|
}
|
|
lean_assert(is_eq(type));
|
|
expr lhs = app_arg(app_fn(type));
|
|
expr rhs = app_arg(type);
|
|
trace_debug_struct(tout() << "elim_match equation rhs:\n" << rhs << "\n";);
|
|
buffer<expr> lhs_args;
|
|
get_app_args(lhs, lhs_args);
|
|
lean_assert(lhs_args.size() == m_arity + 1);
|
|
expr F = lhs_args.back();
|
|
expr new_lhs = mk_app(fn, lhs_args.size() - 1, lhs_args.data());
|
|
expr new_rhs = mk_lemma_rhs(ctx, fn, F, lhs_args[m_arg_pos], rhs);
|
|
buffer<expr> new_locals;
|
|
for (expr const & local : locals.as_buffer()) {
|
|
if (local != F)
|
|
new_locals.push_back(local);
|
|
}
|
|
expr new_type = ctx.mk_pi(new_locals, mk_eq(ctx, new_lhs, new_rhs));
|
|
trace_struct(tout() << "lemma:\n" << new_type << "\n";);
|
|
expr new_proof = prove_eqn_lemma(ctx, new_locals, new_lhs, new_rhs);
|
|
name lemma_name = base_name + name("eqn").append_after(eqn_idx);
|
|
m_env = mk_equation_lemma(m_env, m_opts, m_mctx, m_lctx, m_header.m_is_private,
|
|
lemma_name, new_type, new_proof);
|
|
eqn_idx++;
|
|
}
|
|
}
|
|
|
|
optional<expr> operator()(expr const & eqns) {
|
|
m_ref = eqns;
|
|
m_header = get_equations_header(eqns);
|
|
auto new_eqns = elim_recursion(eqns);
|
|
if (!new_eqns) return none_expr();
|
|
elim_match_result R = elim_match(m_env, m_opts, m_mctx, m_lctx, *new_eqns);
|
|
expr fn = mk_function(R.m_fn);
|
|
if (m_header.m_lemmas) {
|
|
lean_assert(!m_header.m_is_meta);
|
|
mk_lemmas(fn, R.m_lemmas);
|
|
}
|
|
return some_expr(fn);
|
|
}
|
|
};
|
|
|
|
optional<expr> try_structural_rec(environment & env, options const & opts, metavar_context & mctx,
|
|
local_context const & lctx, expr const & eqns) {
|
|
structural_rec_fn F(env, opts, mctx, lctx);
|
|
if (auto r = F(eqns)) {
|
|
env = F.env();
|
|
mctx = F.mctx();
|
|
return r;
|
|
} else {
|
|
return none_expr();
|
|
}
|
|
}
|
|
|
|
void initialize_structural_rec() {
|
|
register_trace_class({"eqn_compiler", "structural_rec"});
|
|
register_trace_class({"debug", "eqn_compiler", "structural_rec"});
|
|
}
|
|
void finalize_structural_rec() {}
|
|
}
|