/*
Copyright (c) 2018 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Author: Leonardo de Moura
*/
#include <algorithm>
#include <unordered_set>
#include <unordered_map>
#include "runtime/flet.h"
#include "kernel/type_checker.h"
#include "kernel/for_each_fn.h"
#include "kernel/find_fn.h"
#include "kernel/abstract.h"
#include "kernel/instantiate.h"
#include "kernel/inductive.h"
#include "kernel/kernel_exception.h"
#include "kernel/trace.h"
#include "library/util.h"
#include "library/constants.h"
#include "library/class.h"
#include "library/expr_pair_maps.h"
#include "library/compiler/util.h"
#include "library/compiler/cse.h"
#include "library/compiler/elim_dead_let.h"
#include "library/compiler/csimp.h"
#include "library/compiler/extract_closed.h"
#include "library/compiler/reduce_arity.h"
#include "library/compiler/init_attribute.h"

namespace lean {
csimp_cfg::csimp_cfg(options const &):
    csimp_cfg() {
}

csimp_cfg::csimp_cfg() {
    m_inline                          = true;
    m_inline_threshold                = 1;
    m_float_cases_threshold           = 20;
    m_inline_jp_threshold             = 2;
}

/*
@[export lean_fold_un_op]
def fold_un_op (before_erasure : bool) (f : expr) (a : expr) : option expr :=
*/
extern "C" object * lean_fold_un_op(uint8 before_erasure, object * f, object * a);

optional<expr> fold_un_op(bool before_erasure, expr const & f, expr const & a) {
    inc(f.raw()); inc(a.raw());
    return to_optional_expr(lean_fold_un_op(before_erasure, f.raw(), a.raw()));
}

/*
@[export lean_fold_bin_op]
def fold_bin_op (before_erasure : bool) (f : expr) (a : expr) (b : expr) : option expr :=
*/
extern "C" object * lean_fold_bin_op(uint8 before_erasure, object * f, object * a, object * b);

optional<expr> fold_bin_op(bool before_erasure, expr const & f, expr const & a, expr const & b) {
    inc(f.raw()); inc(a.raw()); inc(b.raw());
    return to_optional_expr(lean_fold_bin_op(before_erasure, f.raw(), a.raw(), b.raw()));
}

class csimp_fn {
    typedef expr_pair_struct_map<expr> jp_cache;
    type_checker::state      m_st;
    local_ctx                m_lctx;
    bool                     m_before_erasure;
    csimp_cfg                m_cfg;
    buffer<expr>             m_fvars;
    name                     m_x;
    name                     m_j;
    unsigned                 m_next_idx{1};
    unsigned                 m_next_jp_idx{1};
    expr_set                 m_simplified;
    /* Cache for the method `mk_new_join_point`. It maps the pair `(jp, lambda(x, e))` to the new joint point. */
    jp_cache                 m_jp_cache;
    /* Maps a free variables to a list of joint points that must be inserted after it. */
    expr_map<exprs>          m_fvar2jps;
    /* Maps a new join point to the free variable it must be defined after.
       It is the "inverse" of m_fvar2jps. It maps to `none` if the joint point is in `m_closed_jps` */
    expr_map<optional<expr>> m_jp2fvar;
    /* Join points that do not depend on any free variable. */
    exprs                    m_closed_jps;
    /* Mapping from `casesOn` scrutinee to constructor it is bound to.
       We update the mapping when visiting a `cases_on` branch.
       For example, given
       ```
       List.cases_on x
         <nil_case>
         (fun h t, <cons_case h t>)
       ```
       We can assume `x` is bound to `h::t` when visiting `<cons_case h t>`.
       We use this information to reduce nested cases_on applications and projections. */
    typedef rb_expr_map<expr> expr2ctor;
    expr2ctor                m_expr2ctor;

    environment const & env() const { return m_st.env(); }

    name_generator & ngen() { return m_st.ngen(); }

    unsigned get_fvar_idx(expr const & x) {
        lean_assert(is_fvar(x));
        return m_lctx.get_local_decl(x).get_idx();
    }

    optional<expr> find_max_fvar(expr const & e) {
        if (!has_fvar(e)) return none_expr();
        unsigned max_idx = 0;
        optional<expr> r;
        for_each(e, [&](expr const & x, unsigned) {
                if (!has_fvar(x)) return false;
                if (is_fvar(x)) {
                    auto it = m_jp2fvar.find(x);
                    expr y;
                    if (it != m_jp2fvar.end()) {
                        if (!it->second) {
                            /* `x` is a join point in `m_closed_jps`. */
                            return false;
                        }
                        y = *it->second;
                    } else {
                        y = x;
                    }
                    unsigned curr_idx = get_fvar_idx(y);
                    if (!r || curr_idx > max_idx) {
                        r = y;
                        max_idx = curr_idx;
                    }
                }
                return true;
            });
        return r;
    }

    void register_new_jp(expr const & jp) {
        local_decl jp_decl   = m_lctx.get_local_decl(jp);
        expr jp_val          = *jp_decl.get_value();
        if (optional<expr> max_var = find_max_fvar(jp_val)) {
            m_jp2fvar.insert(mk_pair(jp, some_expr(*max_var)));
            auto it = m_fvar2jps.find(*max_var);
            if (it == m_fvar2jps.end()) {
                m_fvar2jps.insert(mk_pair(*max_var, exprs(jp)));
            } else {
                it->second = exprs(jp, it->second);
            }
        } else {
            m_jp2fvar.insert(mk_pair(jp, none_expr()));
            m_closed_jps = exprs(jp, m_closed_jps);
        }
    }

    void check(expr const & e) {
        if (m_before_erasure) {
            try {
                type_checker(m_st, m_lctx).check(e);
            } catch (exception &) {
                lean_unreachable();
            }
        }
    }

    void mark_simplified(expr const & e) {
        m_simplified.insert(e);
    }

    bool already_simplified(expr const & e) const {
        return m_simplified.find(e) != m_simplified.end();
    }

    bool is_join_point_app(expr const & e) const {
        if (!is_app(e)) return false;
        expr const & fn = get_app_fn(e);
        return
            is_fvar(fn) &&
            is_join_point_name(m_lctx.get_local_decl(fn).get_user_name());
    }

    bool is_small_join_point(expr const & e) const {
        return get_lcnf_size(env(), e) <= m_cfg.m_inline_jp_threshold;
    }

    expr find(expr const & e, bool skip_mdata = true, bool use_expr2ctor = false) const {
        if (use_expr2ctor) {
            if (expr const * ctor = m_expr2ctor.find(e)) {
                return *ctor;
            }
        }
        if (is_fvar(e)) {
            if (optional<local_decl> decl = m_lctx.find_local_decl(e)) {
                if (optional<expr> v = decl->get_value()) {
                    /* Pseudo "do" joinpoints are used to implement a temporary HACK. See `visit_let` method at `lcnf.cpp`.
                       Remark: the condition `is_lambda(*v)` will be false after we perform lambda lifting. */
                    if (!is_pseudo_do_join_point_name(decl->get_user_name()) || !is_lambda(*v)) {
                        if (!is_join_point_name(decl->get_user_name()))
                            return find(*v, skip_mdata, use_expr2ctor);
                        else if (is_small_join_point(*v))
                            return find(*v, skip_mdata, use_expr2ctor);
                    }
                }
            }
        } else if (is_mdata(e) && skip_mdata) {
            return find(mdata_expr(e), true, use_expr2ctor);
        }
        return e;
    }

    expr find_ctor(expr const & e) const {
        return find(e, true, true);
    }

    type_checker tc() {
        lean_assert(m_before_erasure);
        return type_checker(m_st, m_lctx);
    }

    expr infer_type(expr const & e) {
        if (m_before_erasure)
            return type_checker(m_st, m_lctx).infer(e);
        else
            return mk_enf_object_type();
    }

    expr whnf(expr const & e) {
        lean_assert(m_before_erasure);
        return type_checker(m_st, m_lctx).whnf(e);
    }

    expr whnf_infer_type(expr const & e) {
        lean_assert(m_before_erasure);
        type_checker tc(m_st, m_lctx);
        return tc.whnf(tc.infer(e));
    }

    optional<expr> whnf_infer_type_guarded(expr const & e) {
        try {
            expr r = whnf_infer_type(e);
            return optional<expr>(r);
        } catch (kernel_exception &) {
            return optional<expr>();
        }
    }

    name next_name() {
        /* Remark: we use `m_x.append_after(m_next_idx)` instead of `name(m_x, m_next_idx)`
           because the resulting name is confusing during debugging: it looks like a projection application.
           We should replace it with `name(m_x, m_next_idx)` when the compiler code gets more stable. */
        name r = m_x.append_after(m_next_idx);
        m_next_idx++;
        return r;
    }

    name next_jp_name() {
        name r = m_j.append_after(m_next_jp_idx);
        m_next_jp_idx++;
        return mk_join_point_name(r);
    }

    /* Create a new let-declaration `x : t := e`, add `x` to `m_fvars` and return `x`. */
    expr mk_let_decl(expr const & e) {
        lean_assert(!is_lcnf_atom(e));
        expr type = cheap_beta_reduce(infer_type(e));
        expr fvar = m_lctx.mk_local_decl(ngen(), next_name(), type, e);
        m_fvars.push_back(fvar);
        return fvar;
    }

    /* Return `let _x := e in _x` */
    expr mk_trivial_let(expr const & e) {
        expr type = infer_type(e);
        return ::lean::mk_let("_x", type, e, mk_bvar(0));
    }

    /* Create minor premise in LCNF.
       The minor premise is of the form `fun xs, e`.
       However, if `e` is a lambda, we create `fun xs, let _x := e in _x`.
       Thus, we don't "mix" `xs` variables with
       the variables of the `new_minor` lambda */
    expr mk_minor_lambda(buffer<expr> const & xs, expr e) {
        if (is_lambda(e)) {
            /* We don't want to "mix" `xs` variables with
               the variables of the `new_minor` lambda */
            e = mk_trivial_let(e);
        }
        return m_lctx.mk_lambda(xs, e);
    }

    /* See `mk_minor_lambda`. We want to preserve the arity of join-points. */
    expr mk_join_point_lambda(buffer<expr> const & xs, expr e) {
        return mk_minor_lambda(xs, e);
    }

    expr get_lambda_body(expr e, buffer<expr> & xs) {
        while (is_lambda(e)) {
            expr d = instantiate_rev(binding_domain(e), xs.size(), xs.data());
            expr x = m_lctx.mk_local_decl(ngen(), binding_name(e), d, binding_info(e));
            xs.push_back(x);
            e  = binding_body(e);
        }
        return instantiate_rev(e, xs.size(), xs.data());
    }

    expr get_minor_body(expr e, buffer<expr> & xs) {
        while (is_lambda(e)) {
            expr d = instantiate_rev(binding_domain(e), xs.size(), xs.data());
            expr x = m_lctx.mk_local_decl(ngen(), binding_name(e), d, binding_info(e));
            xs.push_back(x);
            e  = binding_body(e);
        }
        return instantiate_rev(e, xs.size(), xs.data());
    }

    /* Move let-decl `fvar` to the minor premise at position `minor_idx` of cases_on-application `c`. */
    expr move_let_to_minor(expr const & c, unsigned minor_idx, expr const & fvar) {
        lean_assert(is_cases_on_app(env(), c));
        buffer<expr> args;
        expr const & c_fn = get_app_args(c, args);
        expr minor = args[minor_idx];
        buffer<expr> xs;
        minor = get_lambda_body(minor, xs);
        if (minor == fvar) {
            /* `let x := v in x` ==> `v` */
            minor = *m_lctx.get_local_decl(fvar).get_value();
        } else {
            xs.push_back(fvar);
        }
        args[minor_idx] = mk_minor_lambda(xs, minor);
        return mk_app(c_fn, args);
    }

    /* Collect information for deciding whether `float_cases_on` is useful or not, and control
       code blowup. */
    struct cases_info_result {
        /* The number of branches takes into account join-points too. That is,
           it is not just the number of minor premises. */
        unsigned m_num_branches{0};
        /* The number of branches that return a constructor application. */
        unsigned m_num_cnstr_results{0};
        name_hash_set m_visited_jps;
    };

    void collect_cases_info(expr e, cases_info_result & result) {
        while (true) {
            if (is_lambda(e))
                e = binding_body(e);
            else if (is_let(e))
                e = let_body(e);
            else
                break;
        }
        if (is_constructor_app(env(), e)) {
            result.m_num_branches++;
            result.m_num_cnstr_results++;
        } else if (is_cases_on_app(env(), e)) {
            buffer<expr> args;
            expr const & fn = get_app_args(e, args);
            unsigned begin_minors; unsigned end_minors;
            std::tie(begin_minors, end_minors) = get_cases_on_minors_range(env(), const_name(fn), m_before_erasure);
            for (unsigned i = begin_minors; i < end_minors; i++) {
                collect_cases_info(args[i], result);
            }
        } else if (is_join_point_app(e)) {
            expr const & fn = get_app_fn(e);
            lean_assert(is_fvar(fn));
            if (result.m_visited_jps.find(fvar_name(fn)) != result.m_visited_jps.end())
                return;
            result.m_visited_jps.insert(fvar_name(fn));
            local_decl decl = m_lctx.get_local_decl(fn);
            collect_cases_info(*decl.get_value(), result);
        } else {
            result.m_num_branches++;
        }
    }

    /* The `float_cases_on` transformation may produce code duplication.
       The term `e` is "copied" in each branch of the the `cases_on` expression `c`.
       This method creates one (or more) join-point(s) for `e` (if needed).
       Return `none` if the code size increase is above the threshold.
       Remark: it may produce type incorrect terms. */
    expr mk_join_point_float_cases_on(expr const & fvar, expr const & e, expr const & c) {
        lean_assert(is_cases_on_app(env(), c));
        unsigned e_size = get_lcnf_size(env(), e);
        if (e_size == 1) {
            return e;
        }
        cases_info_result c_info;
        collect_cases_info(c, c_info);
        unsigned code_increase = e_size*(c_info.m_num_branches - 1);
        if (code_increase <= m_cfg.m_float_cases_threshold) {
            return e;
        }
        local_decl fvar_decl = m_lctx.get_local_decl(fvar);
        if (is_cases_on_app(env(), e)) {
            buffer<expr> args;
            expr const & fn = get_app_args(e, args);
            inductive_val e_I_val = get_cases_on_inductive_val(env(), fn);
            /* We can control the code blowup by creating join points for each branch.
               In the worst case, each branch becomes a join point jump, and the
               "compressed size" is equal to the number of branches + 1 for the cases_on application. */
            unsigned e_compressed_size = e_I_val.get_ncnstrs() + 1;
            /* We can ignore the cost of branches that return constructors since they will in the worst case become
               join point jumps. */
            unsigned new_code_increase = e_compressed_size*(c_info.m_num_branches - c_info.m_num_cnstr_results);
            if (new_code_increase <= m_cfg.m_float_cases_threshold) {
                unsigned branch_threshold = m_cfg.m_float_cases_threshold / (c_info.m_num_branches - 1);
                unsigned begin_minors; unsigned end_minors;
                std::tie(begin_minors, end_minors) = get_cases_on_minors_range(env(), const_name(fn), m_before_erasure);
                for (unsigned minor_idx = begin_minors; minor_idx < end_minors; minor_idx++) {
                    expr minor = args[minor_idx];
                    if (get_lcnf_size(env(), minor) > branch_threshold) {
                        buffer<bool> used_zs; /* used_zs[i] iff `minor` uses `zs[i]` */
                        bool         used_fvar = false; /* true iff `minor` uses `fvar` */
                        bool         used_unit = false; /* true if we needed to add `unit ->` to joint point */
                        expr         jp_val;
                        /* Create join-point value: `jp-val` */
                        {
                            buffer<expr> zs;
                            minor                = get_lambda_body(minor, zs);
                            mark_used_fvars(minor, zs, used_zs);
                            lean_assert(zs.size() == used_zs.size());
                            used_fvar            = false;
                            jp_val               = minor;
                            buffer<expr> jp_args;
                            if (has_fvar(minor, fvar)) {
                                /* `fvar` is a let-decl variable, we need to convert into a lambda variable.
                                   Remark: we need to use `replace_fvar_with` because replacing the let-decl variable `fvar` with
                                   the lambda variable `new_fvar` may produce a type incorrect term. */
                                used_fvar      = true;
                                expr new_fvar  = m_lctx.mk_local_decl(ngen(), fvar_decl.get_user_name(), fvar_decl.get_type());
                                jp_args.push_back(new_fvar);
                                jp_val = replace_fvar(jp_val, fvar, new_fvar);
                            }
                            for (unsigned i = 0; i < used_zs.size(); i++) {
                                if (used_zs[i])
                                    jp_args.push_back(zs[i]);
                            }
                            if (jp_args.empty()) {
                                jp_args.push_back(m_lctx.mk_local_decl(ngen(), "_", mk_unit()));
                                used_unit = true;
                            }
                            jp_val = mk_join_point_lambda(jp_args, jp_val);
                        }
                        /* Create new jp */
                        expr jp_type = cheap_beta_reduce(infer_type(jp_val));
                        mark_simplified(jp_val);
                        expr jp_var  = m_lctx.mk_local_decl(ngen(), next_jp_name(), jp_type, jp_val);
                        register_new_jp(jp_var);
                        /* Replace minor with new jp */
                        {
                            buffer<expr> zs;
                            minor = args[minor_idx];
                            minor = get_lambda_body(minor, zs);
                            lean_assert(zs.size() == used_zs.size());
                            expr new_minor = jp_var;
                            if (used_unit)
                                new_minor = mk_app(new_minor, mk_unit_mk());
                            if (used_fvar)
                                new_minor = mk_app(new_minor, fvar);
                            for (unsigned i = 0; i < used_zs.size(); i++) {
                                if (used_zs[i])
                                    new_minor = mk_app(new_minor, zs[i]);
                            }
                            new_minor       = mk_minor_lambda(zs, new_minor);
                            args[minor_idx] = new_minor;
                        }
                    }
                }
                lean_trace(name({"compiler", "simp_float_cases"}),
                           tout() << "mk_join " << fvar << "\n" << c << "\n---\n"
                           << e << "\n======>\n" << mk_app(fn, args) << "\n";);
                return mk_app(fn, args);
            }
        }
        /* Create simple join point */
        expr jp_val = e;
        if (is_lambda(e))
            jp_val = mk_trivial_let(jp_val);
        jp_val       = ::lean::mk_lambda(fvar_decl.get_user_name(), fvar_decl.get_type(), abstract(jp_val, fvar));
        expr jp_type = cheap_beta_reduce(infer_type(jp_val));
        mark_simplified(jp_val);
        expr jp_var  = m_lctx.mk_local_decl(ngen(), next_jp_name(), jp_type, jp_val);
        register_new_jp(jp_var);
        return mk_app(jp_var, fvar);
    }

    /* Given `e[x]`, create a let-decl `y := v`, and return `e[y]`
       Note that, this transformation may produce type incorrect terms.

       Remove: if `v` is an atom, we do not create `y`. */
    expr apply_at(expr const & x, expr const & e, expr const & v) {
        if (is_lcnf_atom(v)) {
            expr e_v = replace_fvar(e, x, v);
            return visit(e_v, false);
        } else {
            local_decl x_decl      = m_lctx.get_local_decl(x);
            expr y                 = m_lctx.mk_local_decl(ngen(), x_decl.get_user_name(), x_decl.get_type(), v);
            expr e_y               = replace_fvar(e, x, y);
            m_fvars.push_back(y);
            return visit(e_y, false);
        }
    }

    expr_pair mk_jp_cache_key(expr const & x, expr const & e, expr const & jp) {
        expr x_type = m_lctx.get_local_decl(x).get_type();
        expr abst_e = ::lean::mk_lambda("_x", x_type, abstract(e, x));
        return mk_pair(abst_e, jp);
    }

    /*
      Given `e[x]`
      ```
      let jp := fun z, let .... in e'
      ```
      ==>
      ```
      let jp' := fun z, let ... y := e' in e[y]
      ```
      If `e'` is a `cases_on` application, we use `float_cases_on_core`. That is,
      ```
      let jp := fun z, let ... in
                cases_on m
                 (fun y_1, let ... in e_1)
                 ...
                 (fun y_n, let ... in e_n)
      ```
      ==>
      ```
      let jp := fun z, let ... in
                cases_on m
                 (fun y_1, let ... y := e_1 in e[y])
                 ...
                 (fun y_n, let ... y := e_n in e[y])
      ```

      Remark: this method may produce type incorrect terms because of dependent types. */
    expr mk_new_join_point(expr const & x, expr const & e, expr const & jp) {
        expr_pair key = mk_jp_cache_key(x, e, jp);
        auto it = m_jp_cache.find(key);
        if (it != m_jp_cache.end())
            return it->second;
        local_decl jp_decl = m_lctx.get_local_decl(jp);
        lean_assert(is_join_point_name(jp_decl.get_user_name()));
        expr jp_val = *jp_decl.get_value();
        buffer<expr> zs;
        unsigned saved_fvars_size = m_fvars.size();
        jp_val = visit(get_lambda_body(jp_val, zs), false);
        expr e_y;
        if (is_join_point_app(jp_val)) {
            buffer<expr> jp2_args;
            expr const & jp2  = get_app_args(jp_val, jp2_args);
            expr new_jp2      = mk_new_join_point(x, e, jp2);
            e_y = mk_app(new_jp2, jp2_args);
        } else if (is_cases_on_app(env(), jp_val)) {
            e_y = float_cases_on_core(x, e, jp_val);
        } else {
            e_y = apply_at(x, e, jp_val);
        }
        expr new_jp_val  = e_y;
        new_jp_val = mk_let(zs, saved_fvars_size, new_jp_val, false);
        new_jp_val = mk_join_point_lambda(zs, new_jp_val);
        mark_simplified(new_jp_val);
        expr new_jp_type = cheap_beta_reduce(infer_type(new_jp_val));
        expr new_jp_var  = m_lctx.mk_local_decl(ngen(), next_jp_name(), new_jp_type, new_jp_val);
        register_new_jp(new_jp_var);
        m_jp_cache.insert(mk_pair(key, new_jp_var));
        return new_jp_var;
    }

    /* Add entry `x := cidx fields` to m_expr2ctor */
    void update_expr2ctor(expr const & x, expr const & c_fn, buffer<expr> const & c_args, unsigned cidx, buffer<expr> const & fields) {
        inductive_val I_val = get_cases_on_inductive_val(env(), c_fn);
        name ctor_name      = get_ith(I_val.get_cnstrs(), cidx);
        levels ctor_lvls;
        buffer<expr> ctor_args;
        if (m_before_erasure) {
            ctor_lvls = tail(const_levels(c_fn));
            ctor_args.append(I_val.get_nparams(), c_args.data());
        } else {
            for (unsigned i = 0; i < I_val.get_nparams(); i++)
                ctor_args.push_back(mk_enf_neutral());
        }
        ctor_args.append(fields);
        expr ctor = mk_app(mk_constant(ctor_name, ctor_lvls), ctor_args);
        m_expr2ctor.insert(x, ctor);
    }

    /* Given `e[x]`
      ```
      cases_on m
              (fun zs, let ... in e_1)
              ...
              (fun zs, let ... in e_n)
      ```
      ==>
      ```
      cases_on m
        (fun zs, let ... y := e_1 in e[y])
        ...
        (fun y_n, let ... y := e_n in e[y])
      ``` */
    expr float_cases_on_core(expr const & x, expr const & e, expr const & c) {
        lean_assert(is_cases_on_app(env(), c));
        local_decl x_decl     = m_lctx.get_local_decl(x);
        buffer<expr> c_args;
        expr c_fn             = get_app_args(c, c_args);
        inductive_val I_val   = get_cases_on_inductive_val(env(), c_fn);
        unsigned major_idx;
        /* Update motive and get major_idx */
        if (m_before_erasure) {
            unsigned motive_idx      = I_val.get_nparams();
            unsigned first_index     = motive_idx + 1;
            unsigned nindices        = I_val.get_nindices();
            major_idx                = first_index + nindices;
            buffer<expr> zs;
            expr result_type         = whnf_infer_type(e);
            expr motive              = c_args[motive_idx];
            expr motive_type         = whnf_infer_type(motive);
            for (unsigned i = 0; i < nindices + 1; i++) {
                lean_assert(is_pi(motive_type));
                expr z = m_lctx.mk_local_decl(ngen(), binding_name(motive_type), binding_domain(motive_type), binding_info(motive_type));
                zs.push_back(z);
                motive_type = whnf(instantiate(binding_body(motive_type), z));
            }
            level result_lvl       = sort_level(tc().ensure_type(result_type));
            if (has_fvar(result_type, x)) {
                /* `x` will be deleted after the float_cases_on transformation.
                   So, if the result type depends on it, we must replace it with its value. */
                result_type = replace_fvar(result_type, x, *x_decl.get_value());
            }
            expr new_motive        = m_lctx.mk_lambda(zs, result_type);
            c_args[motive_idx] = new_motive;
            /* We need to update the resultant universe. */
            levels new_cases_lvls  = levels(result_lvl, tail(const_levels(c_fn)));
            c_fn = update_constant(c_fn, new_cases_lvls);
        } else {
            /* After erasure, we keep only major and minor premises. */
            major_idx = 0;
        }
        /* Update minor premises */
        expr const & major = c_args[major_idx];
        unsigned first_minor_idx = major_idx + 1;
        unsigned nminors         = I_val.get_ncnstrs();
        for (unsigned i = 0; i < nminors; i++) {
            unsigned minor_idx        = first_minor_idx + i;
            expr minor                = c_args[minor_idx];
            buffer<expr> zs;
            unsigned saved_fvars_size = m_fvars.size();
            expr minor_val            = get_minor_body(minor, zs);
            {
                flet<expr2ctor> save_expr2ctor(m_expr2ctor, m_expr2ctor);
                update_expr2ctor(major, c_fn, c_args, i, zs);
                minor_val             = visit(minor_val, false);
            }
            expr new_minor;
            if (is_join_point_app(minor_val)) {
                buffer<expr> jp_args;
                expr const & jp = get_app_args(minor_val, jp_args);
                expr new_jp     = mk_new_join_point(x, e, jp);
                new_minor       = visit(mk_app(new_jp, jp_args), false);
            } else {
                new_minor       = apply_at(x, e, minor_val);
            }
            new_minor                 = mk_let(zs, saved_fvars_size, new_minor, false);
            new_minor                 = mk_minor_lambda(zs, new_minor);
            c_args[minor_idx]         = new_minor;
        }
        lean_trace(name({"compiler", "simp_float_cases"}),
                   tout() << "float_cases_on [" << get_lcnf_size(env(), e) << "]\n" << c << "\n----\n" << e << "\n=====>\n"
                   << mk_app(c_fn, c_args) << "\n";);
        return mk_app(c_fn, c_args);
    }

    /* Float cases transformation (see: `float_cases_on_core`).
       This version may create join points if `e` is big, or "good" join-points could not be created. */
    expr float_cases_on(expr const & x, expr const & e, expr const & c) {
        expr new_e = mk_join_point_float_cases_on(x, e, c);
        return float_cases_on_core(x, new_e, c);
    }

    /* Given the buffer `entries`: `[(x_1, w_1), ..., (x_n, w_n)]`, and `e`.
       Create the let-expression
       ```
       let x_n := w_n
           ...
           x_1 := w_1
       in e
       ```
       The values `w_i` are the "simplified values" for the let-declaration `x_i`. */
    expr mk_let_core(buffer<pair<expr, expr>> const & entries, expr e) {
        buffer<expr> fvars;
        buffer<name> user_names;
        buffer<expr> types;
        buffer<expr> vals;
        unsigned i = entries.size();
        while (i > 0) {
            --i;
            expr const & fvar = entries[i].first;
            fvars.push_back(fvar);
            expr const & val  = entries[i].second;
            vals.push_back(val);
            local_decl fvar_decl = m_lctx.get_local_decl(fvar);
            user_names.push_back(fvar_decl.get_user_name());
            types.push_back(fvar_decl.get_type());
        }
        e = abstract(e, fvars.size(), fvars.data());
        i = fvars.size();
        while (i > 0) {
            --i;
            expr new_value = abstract(vals[i], i, fvars.data());
            expr new_type  = abstract(types[i], i, fvars.data());
            e = ::lean::mk_let(user_names[i], new_type, new_value, e);
        }
        return e;
    }

    /* Split `entries` into two groups: `entries_dep_x` and `entries_ndep_x`.
       The first group contains the entries that depend on `x` and the second the ones that doesn't.
       This auxiliary method is used to float cases_on over expressions.

       `entries` is of the form `[(x_1, w_1), ..., (x_n, w_n)]`, where `x_i`s are
       let-decl free variables, and `w_i`s their new values. We use `entries`
       and an expression `e` to create a `let` expression:
       ```
       let x_n := w_n
           ...
           x_1 := w_1
       in e
       ``` */
    void split_entries(buffer<pair<expr, expr>> const & entries,
                       expr const & x,
                       buffer<pair<expr, expr>> & entries_dep_x,
                       buffer<pair<expr, expr>> & entries_ndep_x) {
        if (entries.empty())
            return;
        name_hash_set deps;
        deps.insert(fvar_name(x));
        /* Recall that `entries` are in reverse order. That is, pos 0 is the inner most variable. */
        unsigned i = entries.size();
        while (i > 0) {
            --i;
            expr const & fvar = entries[i].first;
            expr fvar_type    = m_lctx.get_type(fvar);
            expr fvar_new_val = entries[i].second;
            if (depends_on(fvar_type, deps) ||
                depends_on(fvar_new_val, deps)) {
                deps.insert(fvar_name(fvar));
                entries_dep_x.push_back(entries[i]);
            } else {
                entries_ndep_x.push_back(entries[i]);
            }
        }
        std::reverse(entries_dep_x.begin(), entries_dep_x.end());
        std::reverse(entries_ndep_x.begin(), entries_ndep_x.end());
    }

    bool push_dep_jps(expr const & fvar) {
        lean_assert(is_fvar(fvar));
        auto it = m_fvar2jps.find(fvar);
        if (it == m_fvar2jps.end())
            return false;
        buffer<expr> tmp;
        to_buffer(it->second, tmp);
        m_fvar2jps.erase(fvar);
        std::reverse(tmp.begin(), tmp.end());
        m_fvars.append(tmp);
        return true;
    }

    bool push_dep_jps(buffer<expr> const & zs, bool top) {
        buffer<expr> tmp;
        if (top) {
            to_buffer(m_closed_jps, tmp);
            m_closed_jps = exprs();
        }
        for (expr const & z : zs) {
            auto it = m_fvar2jps.find(z);
            if (it != m_fvar2jps.end()) {
                to_buffer(it->second, tmp);
                m_fvar2jps.erase(z);
            }
        }
        if (tmp.empty())
            return false;
        sort_fvars(m_lctx, tmp);
        m_fvars.append(tmp);
        return true;
    }

    void sort_entries(buffer<expr_pair> & entries) {
        std::sort(entries.begin(), entries.end(), [&](expr_pair const & p1, expr_pair const & p2) {
                /* We use `>` because entries in `entries` are in reverse dependency order */
                return m_lctx.get_local_decl(p1.first).get_idx() > m_lctx.get_local_decl(p2.first).get_idx();
            });
    }

    /* Copy `src_entries` and the new joint points that depend on them to `entries`, and update `entries_fvars`.
       This method is used after we perform a `float_cases_on`. */
    void move_to_entries(buffer<expr_pair> const & src_entries, buffer<expr_pair> & entries, name_hash_set & entries_fvars) {
        buffer<expr_pair> todo;
        for (unsigned i = 0; i < src_entries.size(); i++) {
            expr_pair const & entry = src_entries[i];
            /* New join points may have been attached to `ndep_entry` */
            todo.push_back(entry);
            while (!todo.empty()) {
                expr_pair const & curr = todo.back();
                auto it = m_fvar2jps.find(curr.first);
                if (it != m_fvar2jps.end()) {
                    buffer<expr> tmp;
                    to_buffer(it->second, tmp);
                    for (expr const & jp : tmp) {
                        /* Recall that new join points have already been simplified.
                           So, it is ok to move them to `entries`. */
                        todo.emplace_back(jp, *m_lctx.get_local_decl(jp).get_value());
                    }
                    m_fvar2jps.erase(curr.first);
                } else {
                    entries.push_back(curr);
                    collect_used(curr.second, entries_fvars);
                    todo.pop_back();
                }
            }
        }
        /* The following sorting operation is necessary because of non trivial dependencies between entries.
           For example, consider the following scenario. When starting a `float_cases_on` operation, we determine
           that the already processed entries `[_j_1._join, _x_1]` do not depend on the operation.
           Moreover, `_j_1._join` is a new join-point that depends on `_x_1`. Recall that entries are in reverse
           dependency order, and this is why `_j_1._join` occurs before `_x_1`.
           Then, during the actual execution of the `float_cases_on` operation, we create a new joint point `_j_2._join` that depends on `_j_1._join`,
           and is consequently attached to `_x_1`, that is, `m_fvar2jps[_x_1]` contains `_j_2._join`.
           After executing this procedure, `entries` will contain `[_j_1._join, _j_2._join, _x_1]` which is incorrect
           since `_j_2._join` depends on `_j_1._join`. */
        sort_entries(entries);
    }

    /* Given a casesOn application `c`, return `some idx` iff `c` has more than one branch, `fvar` only occurs
       in the argument `idx`, this argument is a minor premise.

       Recall this method is used to implement the float `let` inwards transformation.
       Thus, it doesn't really help to move `let` inwards if there is only one branch.

       Moreover, it may negatively impact performance because we use `casesOn` applications to
       guide the insertion of reset/reuse IR instructions.

       Here is a problematic example:
       ```
       let p := Array.index a i in                -- Get pair `p` at `a[i]`
       let a := Array.update a i (default _) in   -- "Reset" `a[i]` to make sure `p` is now the owner
       casesOn p (fun fst snd, Array.update a i (fst+1, snd))
       ```
       Before this commit the compiler would move
       ```
       a := Array.update a i (default _)
       ```
       into the `casesOn` branch, and we would get
       ```
       let p := Array.index a i in                -- Get pair `p` at `a[i]`
       casesOn p (fun fst snd,
         let a := Array.update a i (default _) in -- "Reset" `a[i]` to make sure `p` is now the owner
         Array.update a i (fst+1, snd))
       ```
       Then, we would get
       ```
       let p := Array.index a i in                -- Get pair `p` at `a[i]`
       casesOn p (fun fst snd,
         let p := reset p in
         let a := Array.update a i (default _) in -- "Reset" `a[i]` to make sure `p` is now the owner
         let p := reuse p (fst+1, snd) in
         Array.update a i p)
       ```
       But, this `reset p` will always fail since the `Array` still contains a
       reference to `p` when we execute `reset p`.
    */
    optional<unsigned> used_in_one_minor(expr const & c, expr const & fvar) {
        lean_assert(is_cases_on_app(env(), c));
        lean_assert(is_fvar(fvar));
        buffer<expr> args;
        expr const & c_fn = get_app_args(c, args);
        unsigned minors_begin; unsigned minors_end;
        std::tie(minors_begin, minors_end) = get_cases_on_minors_range(env(), const_name(c_fn), m_before_erasure);
        if (minors_end <= minors_begin + 1) {
            /* casesOn has only one branch */
            return optional<unsigned>();
        }
        unsigned i = 0;
        for (; i < minors_begin; i++) {
            if (has_fvar(args[i], fvar)) {
                /* Free variable occurs in a term that is a not a minor premise. */
                return optional<unsigned>();
            }
        }
        lean_assert(i == minors_begin);
        /* The following #pragma is to disable a bogus g++ 4.9 warning at `optional<unsigned> r` */
        #if defined(__GNUC__) && !defined(__CLANG__)
        #pragma GCC diagnostic ignored "-Wmaybe-uninitialized"
        #endif
        optional<unsigned> r;
        for (; i < minors_end; i++) {
            expr minor = args[i];
            while (is_lambda(minor)) {
                if (has_fvar(binding_domain(minor), fvar)) {
                    /* Free variable occurs in the type of a field */
                    return optional<unsigned>();
                }
                minor = binding_body(minor);
            }
            if (has_fvar(minor, fvar)) {
                if (r) {
                    /* Free variable occur in more than one minor premise. */
                    return optional<unsigned>();
                }
                r = i;
            }
        }
        return r;
    }

    /*
      Given `x := val`, the entries `y_1 := w_1; ...; y_n := w_n`, and the set `S` of all free variables
      in `entries`. Return true if we may move `x := val` after these entries.

      This method is used to implement the float `let` inwards transformation. */
    bool may_move_after(expr const & x, expr const & /* val */, buffer<expr_pair> const & entries, name_hash_set const & S) {
        lean_assert(is_fvar(x));
        if (S.find(fvar_name(x)) != S.end()) {
            /* If `x` is used in the entries `y_1 := w_1; ...; y_n := w_n`,
               then we must *not* move `x` after them since it would produce
               an ill-formed expression. */
            return false;
        }
        /* The condition above is sufficient to make sure the resulting expression is well-formed.
           However, moving `x := val` after `entries` may affect perform by preventing destructive
           updates from happening and memory from being reused. Consider the following example
           ```
           let x := z.1 in
           let y := f z in
           C
           ```
           If we move `x := z.1` after `y := f z` obtaining the expression:
           ```
           let y := f z in
           let x := z.1 in
           C
           ```
           Then, `RC(z)` will be greater than 1 when we invoke `f z` because we would need to include
           an `inc z` instruction before `y := f z`. The `inc z` is needed because `z` would still be
           alive after `f z`.

           In the example above, `val` contains a variable (`z`) used in `entries`.
           However, this test is not sufficient. Here is a more intricate example:
           ```
           let w := z.1 in
           let x := Array.size w in
           let y := f z in
           C
           ```
           If we move `x := Array.size w` after `y := f z`, we get
           ```
           let w := z.1 in
           let y := f z in
           let x := Array.size w in
           C
           ```
           `f z` and `Array.size w` do not share any free variable, but `w` is an reference to a field of `z`.
           In the example above, `w` is an array, and `f z` will not be able to update the array nested there if
           we have `let x := Array.size w` after it.

           The example above suggests that a sufficient condition for preventing this issue is:
           - Any memory cell reachable from `val` is not reachable from `entries`.

           A simpler sufficient condition for preventing the issue is:
           - `entries` code does not perform destructive updates or tries to reuse memory cells.
           Here we use an even simpler check: `entries` contains only projection operations.
        */
        for (expr_pair const & p : entries) {
            expr const & w = p.second;
            if (!is_proj(w))
                return false;
        }
        return true;
    }

    /* Create a let-expression with body `e`, and
       all "used" let-declarations `m_fvars[i]` for `i in [saved_fvars_size, m_fvars.size)`.
       We also include all join points that depends on these free variables,
       nad join points that depends on `zs`. The buffer `zs` (when non empty) contains
       the free variables for a lambda expression that will be created around the let-expression.

       BTW, we also visit the lambda expressions in used let-declarations of the form
       `x : t := fun ...`


       Note that, we don't visit them when we have visit let-expressions. */
    expr mk_let(buffer<expr> const & zs, unsigned saved_fvars_size, expr e, bool top) {
        if (saved_fvars_size == m_fvars.size()) {
            if (!push_dep_jps(zs, top))
                return e;
        }
        /* `entries` contains pairs (let-decl fvar, new value) for building the resultant let-declaration.
           We simplify the value of some let-declarations in this method, but we don't want to create
           a new temporary declaration just for this. */
        buffer<expr_pair> entries;
        name_hash_set e_fvars; /* Set of free variables names used in `e` */
        name_hash_set entries_fvars; /* Set of free variable names used in `entries` */
        collect_used(e, e_fvars);
        bool e_is_cases      = is_cases_on_app(env(), e);
        /*
          Recall that all free variables in `m_fvars` are let-declarations.
          In the following loop, we have the following "order" for the let-declarations:
          ```
             m_fvars[saved_fvars_size]
             ...
             m_fvars[m_fvars.size() - 1]

             entries[entries.size() - 1]
             ...
             entries[0]
          ```
          The "body" of the let-declaration is `e`.
          The mapping `m_fvar2jps` maps a free variable `x to join points that must be inserted after `x`.
        */
        while (true) {
            if (m_fvars.size() == saved_fvars_size) {
                if (!push_dep_jps(zs, top))
                    break;
            }
            lean_assert(m_fvars.size() > saved_fvars_size);
            expr x               = m_fvars.back();
            if (push_dep_jps(x)) {
                /* We must process the join points that depend on `x` before we process `x`. */
                continue;
            }
            m_fvars.pop_back();
            bool used_in_e       = (e_fvars.find(fvar_name(x)) != e_fvars.end());
            bool used_in_entries = (entries_fvars.find(fvar_name(x)) != entries_fvars.end());
            if (!used_in_e && !used_in_entries) {
                /* Skip unused variables */
                continue;
            }
            local_decl x_decl = m_lctx.get_local_decl(x);
            expr type       = x_decl.get_type();
            expr val        = *x_decl.get_value();
            bool is_jp      = false;
            bool modified_val = false;
            if (is_lambda(val)) {
                /* We don't simplify lambdas when we visit `let`-expressions. */
                DEBUG_CODE(unsigned saved_fvars_size = m_fvars.size(););
                is_jp        = is_join_point_name(x_decl.get_user_name());
                val          = visit_lambda(val, is_jp, false);
                modified_val = true;
                lean_assert(m_fvars.size() == saved_fvars_size);
            }

            if (is_lc_unreachable_app(val)) {
                /* `let x := lc_unreachable in e` => `lc_unreachable` */
                e = val;
                e_is_cases = false;
                e_fvars.clear(); entries_fvars.clear();
                collect_used(e, e_fvars);
                entries.clear();
                continue;
            }

            if (entries.empty() && e == x) {
                /* `let x := v in x` ==> `v` */
                e = val;
                collect_used(val, e_fvars);
                e_is_cases = is_cases_on_app(env(), e);
                continue;
            }

            if (is_cases_on_app(env(), val)) {
                /* We first create a let-declaration with all entries that depends on the current
                   `x` which is a cases_on application. */
                buffer<pair<expr, expr>> entries_dep_curr;
                buffer<pair<expr, expr>> entries_ndep_curr;
                split_entries(entries, x, entries_dep_curr, entries_ndep_curr);
                expr new_e = mk_let_core(entries_dep_curr, e);
                e = float_cases_on(x, new_e, val);
                lean_assert(is_cases_on_app(env(), e));
                e_is_cases = true;
                /* Reset `e_fvars` and `entries_fvars`, we need to reconstruct them. */
                e_fvars.clear(); entries_fvars.clear();
                collect_used(e, e_fvars);
                entries.clear();
                /* Copy `entries_ndep_curr` to `entries` */
                move_to_entries(entries_ndep_curr, entries, entries_fvars);
                continue;
            }

            if (!is_jp && e_is_cases && used_in_e) {
                optional<unsigned> minor_idx = used_in_one_minor(e, x);
                if (minor_idx && may_move_after(x, val, entries, entries_fvars)) {
                    /* If x is only used in only one minor declaration,
                       and it passed the may_move_after test. */
                    if (modified_val) {
                        /* We need to create a new free variable since the new
                           simplified value `val` */
                        expr new_x = m_lctx.mk_local_decl(ngen(), x_decl.get_user_name(), type, val);
                        e = replace_fvar(e, x, new_x);
                        x = new_x;
                    }
                    collect_used(type, e_fvars);
                    collect_used(val, e_fvars);
                    e = move_let_to_minor(e, *minor_idx, x);
                    continue;
                }
            }

            collect_used(type, entries_fvars);
            collect_used(val,  entries_fvars);
            entries.emplace_back(x, val);
        }
        return mk_let_core(entries, e);
    }

    name mk_let_name(name const & n) {
        if (is_internal_name(n)) {
            if (is_join_point_name(n))
                return next_jp_name();
            else if (is_pseudo_do_join_point_name(n))
                return n;
            else
                return next_name();
        } else {
            return n;
        }
    }

    expr visit_let(expr e) {
        buffer<expr> let_fvars;
        while (is_let(e)) {
            expr new_type = instantiate_rev(let_type(e), let_fvars.size(), let_fvars.data());
            expr new_val  = visit(instantiate_rev(let_value(e), let_fvars.size(), let_fvars.data()), true);
            if (!is_pseudo_do_join_point_name(let_name(e)) && is_lcnf_atom(new_val)) {
                let_fvars.push_back(new_val);
            } else {
                name n = mk_let_name(let_name(e));
                expr new_fvar = m_lctx.mk_local_decl(ngen(), n, new_type, new_val);
                let_fvars.push_back(new_fvar);
                m_fvars.push_back(new_fvar);
            }
            e = let_body(e);
        }
        return visit(instantiate_rev(e, let_fvars.size(), let_fvars.data()), false);
    }

    /* - `is_join_point_def` is true if the lambda is the value of a join point.
       - `root` is true if the lambda is the value of a definition. */
    expr visit_lambda(expr e, bool is_join_point_def, bool top) {
        lean_assert(is_lambda(e));
        lean_assert(!top || m_fvars.size() == 0);
        if (already_simplified(e))
            return e;
        // Hack to avoid eta-expansion of implicit lambdas
        // Example: `fun {a} => ReaderT.pure`
        if (!is_join_point_def && !top) {
            expr new_e = eta_reduce(e);
           // Remark: we should only keep result if it is not a term must be in eta-expanded form.
            if (is_app(new_e) && !must_be_eta_expanded(env(), new_e))
                return visit(new_e, true);
        }
        buffer<expr> binding_fvars;
        while (is_lambda(e)) {
            /* Types are ignored in compilation steps. So, we do not invoke visit for d. */
            expr new_d    = instantiate_rev(binding_domain(e), binding_fvars.size(), binding_fvars.data());
            expr new_fvar = m_lctx.mk_local_decl(ngen(), binding_name(e), new_d, binding_info(e));
            binding_fvars.push_back(new_fvar);
            e = binding_body(e);
        }
        e = instantiate_rev(e, binding_fvars.size(), binding_fvars.data());
        /* When we simplify before erasure, we eta-expand all lambdas which are not join points. */
        buffer<expr> eta_args;
        if (m_before_erasure && !is_join_point_def) {
            // HACK: infer_type may fail
            if (auto e_type_opt = whnf_infer_type_guarded(e)) {
                expr e_type = *e_type_opt;
                while (is_pi(e_type)) {
                    expr arg = m_lctx.mk_local_decl(ngen(), binding_name(e_type), binding_domain(e_type), binding_info(e_type));
                    eta_args.push_back(arg);
                    e_type = whnf(instantiate(binding_body(e_type), arg));
                }
            }
        }
        unsigned saved_fvars_size = m_fvars.size();
        expr new_body             = visit(e, false);
        if (!eta_args.empty()) {
            if (is_join_point_app(new_body)) {
                /* Remark: we cannot simply set
                   ```
                   new_body = mk_app(new_body, eta_args);
                   ```
                   when `new_body` is a join-point, because the result will not be a valid LCNF term.
                   We could expand the join-point, but it this will create a copy.
                   So, for now, we simply avoid eta-expansion.
                */
                eta_args.clear();
            } else {
                if (is_lcnf_atom(new_body)) {
                    new_body = mk_app(new_body, eta_args);
                } else if (is_app(new_body) && !is_cases_on_app(env(), new_body)) {
                    new_body = mk_app(new_body, eta_args);
                } else {
                    expr f   = mk_let_decl(new_body);
                    new_body = mk_app(f, eta_args);
                }
                new_body = visit(new_body, false);
            }
            binding_fvars.append(eta_args);
        }
        new_body      = mk_let(binding_fvars, saved_fvars_size, new_body, top);
        expr r;
        if (is_join_point_def) {
            lean_assert(eta_args.empty());
            r = mk_join_point_lambda(binding_fvars, new_body);
        } else {
            r = m_lctx.mk_lambda(binding_fvars, new_body);
        }
        mark_simplified(r);
        return r;
    }

    /* Auxiliary method for `beta_reduce` and `beta_reduce_if_not_cases` */
    expr beta_reduce_cont(expr r, unsigned i, unsigned nargs, expr const * args, bool is_let_val) {
        r = visit(r, false);
        if (i == nargs)
            return r;
        lean_assert(i < nargs);
        if (is_join_point_app(r)) {
            /* Expand join-point */
            lean_assert(!is_let_val);
            buffer<expr> new_args;
            expr const & jp = get_app_args(r, new_args);
            lean_assert(is_fvar(jp));
            for (; i < nargs; i++)
                new_args.push_back(args[i]);
            expr jp_val     = *m_lctx.get_local_decl(jp).get_value();
            lean_assert(is_lambda(jp_val));
            return beta_reduce(jp_val, new_args.size(), new_args.data(), false);
        } else {
            if (!is_lcnf_atom(r))
                r = mk_let_decl(r);
            return visit(mk_app(r, nargs - i, args + i), is_let_val);
        }
    }

    expr beta_reduce(expr fn, unsigned nargs, expr const * args, bool is_let_val) {
        unsigned i = 0;
        while (is_lambda(fn) && i < nargs) {
            i++;
            fn = binding_body(fn);
        }
        expr r = instantiate_rev(fn, i, args);
        if (is_lambda(r)) {
            lean_assert(i == nargs);
            return visit(r, is_let_val);
        } else {
            return beta_reduce_cont(r, i, nargs, args, is_let_val);
        }
    }

    /* Remark: if `fn` is not a lambda expression, then this function
       will simply create the application `fn args_of(e)` */
    expr beta_reduce(expr fn, expr const & e, bool is_let_val) {
        buffer<expr> args;
        get_app_args(e, args);
        return beta_reduce(fn, args.size(), args.data(), is_let_val);
    }

    bool should_inline_instance(name const & n) const {
        if (is_instance(env(), n))
            return !has_noinline_attribute(env(), n) && !has_init_attribute(env(), n);
        else
            return false;
    }

    expr proj_constructor(expr const & k_app, unsigned proj_idx) {
        lean_assert(is_constructor_app(env(), k_app));
        buffer<expr> args;
        expr const & k        = get_app_args(k_app, args);
        constructor_val k_val = env().get(const_name(k)).to_constructor_val();
        lean_assert(k_val.get_nparams() + proj_idx < args.size());
        return args[k_val.get_nparams() + proj_idx];
    }

    optional<expr> try_inline_proj_instance_aux(expr s) {
        lean_assert(m_before_erasure);
        s = find(s);
        if (is_constructor_app(env(), s)) {
            return some_expr(s);
        } else if (is_proj(s)) {
            if (optional<expr> new_nested_s = try_inline_proj_instance_aux(proj_expr(s))) {
                lean_assert(is_constructor_app(env(), *new_nested_s));
                expr r = proj_constructor(*new_nested_s, proj_idx(s).get_small_value());
                return try_inline_proj_instance_aux(r);
            }
        } else {
            expr const & s_fn = get_app_fn(s);
            if (!is_constant(s_fn) || !should_inline_instance(const_name(s_fn)))
                return none_expr();
            optional<constant_info> info = env().find(mk_cstage1_name(const_name(s_fn)));
            if (!info || !info->is_definition()) return none_expr();
            if (get_app_num_args(s) < get_num_nested_lambdas(info->get_value())) return none_expr();
            expr new_s_fn = instantiate_value_lparams(*info, const_levels(s_fn));
            expr r        = find(beta_reduce(new_s_fn, s, false));
            if (is_constructor_app(env(), r)) {
                return some_expr(r);
            } else if (optional<expr> new_r = try_inline_proj_instance_aux(r)) {
                return new_r;
            }
        }
        return none_expr();
    }

    bool is_type_class(expr type) {
        type = cheap_beta_reduce(type);
        expr const & fn = get_app_fn(type);
        if (!is_constant(fn)) return false;
        return is_class(env(), const_name(fn));
    }

    /* Auxiliary function for projecting "type class dictionary access".
       That is, we are trying to extract one of the type class instance elements.

       Remark: We do not consider parent instances to be elements.
       For example, suppose `e` is `_x_4.1`, and we have
       ```
       _x_2 : Monad (ReaderT Bool (ExceptT String Id)) := @ReaderT.Monad Bool (ExceptT String Id) _x_1,
       _x_3 : Applicative (ReaderT Bool (ExceptT String Id)) := _x_2.1
       _x_4 : Functor (ReaderT Bool (ExceptT String Id)) := _x_3.1
       ```
       Then, we will expand `_x_4.1` since it corresponds to the `Functor` `map` element,
       and its type is not a type class, but is of the form
       ```
       (Π {α β : Type u}, (α → β) → ...)
       ```
       In the example above, the compiler should not expand `_x_3.1` or `_x_2.1` since their
       types type class applications: `Functor` and `Applicative` respectively.
       By eagerly expanding them, we may produce inefficient and bloated code.
       For example, we may be using `_x_3.1` to invoke a function that expects a `Functor` instance.
       By expanding `_x_3.1` we will be just expanding the code that creates this instance.
    */
    optional<expr> try_inline_proj_instance(expr const & e, bool is_let_val) {
        lean_assert(is_proj(e));
        if (!m_before_erasure) return none_expr();
        try {
            expr e_type = infer_type(e);
            if (is_type_class(e_type)) {
                /* If `typeof(e)` is a type class, then we should not instantiate it.
                   See comment above. */
                return none_expr();
            }

            unsigned saved_fvars_size = m_fvars.size();
            if (optional<expr> new_s = try_inline_proj_instance_aux(proj_expr(e))) {
                lean_assert(is_constructor_app(env(), *new_s));
                expr r = proj_constructor(*new_s, proj_idx(e).get_small_value());
                return some_expr(visit(r, is_let_val));
            }
            m_fvars.resize(saved_fvars_size);
            return none_expr();
        } catch (kernel_exception &) {
            return none_expr();
        }
    }

    /* Return true iff `e` is of the form `fun (xs), let ys := ts in (ctor ...)`.
       This auxiliary method is used at try_inline_proj_instance_aux.
       It is a "quick" filter. */
    bool inline_proj_app_candidate(expr e) {
        while (is_lambda(e))
            e = binding_body(e);
        while (is_let(e))
            e = let_body(e);
        return static_cast<bool>(is_constructor_app(env(), e));
    }

    /*
      Given `let x := f as in ... x.i`, where where `f` is defined as
      ```
      def f (xs) :=
      ...
      let y_i := t[xs] in
      ...
      ctor ... y_i ...
      ```
      reduce `x.i` into `t[as]`.
      `y_i` may depend on other let-declarations, but we only inline if the number
      of let-decl dependencies is less than `m_inline_threshold`.

      Remark: this transformation is only applied before erasure.
      Remark: this transformation complements eager lambda lifting,
      and has been designed to optimize code such as:
      ```
      def f (x : nat) : Pro (Nat -> Nat) (Nat -> Bool) :=
      ((fun y, <code1 using x y>), (fun z, <code2 using x z>))
      ```
      That is, `f` is "packing" functions in a structure and returning it.
      Now, consider the following application:
      ```
      (f a).1 b
      ```
      With eager lambda lifting, we transform `f` into
      ```
      def f._elambda_1 (x y) : Nat :=
      <code1 using x y>
      def f._elambda_2 (x z) : Bool :=
      <code2 using x z>
      def f (x : nat) : Pro (Nat -> Nat) (Nat -> Bool) :=
      (f._elambda_1 x, f._elambda_2 x)
      ```
      Then, with this transformation, we transform `(f a).1` into
      `f._elambda_1 a`, and then with application merge, we transform
      `(f a).1 b` into `f._elambda_1 a b`

      See additional comments at `eager_lambda_lifting.cpp` */
    optional<expr> try_inline_proj_app(expr const & e, bool is_let_val) {
        lean_assert(is_proj(e));
        if (!m_before_erasure) return none_expr();
        if (!proj_idx(e).is_small()) return none_expr();
        unsigned idx = proj_idx(e).get_small_value();
        expr s = find(proj_expr(e));
        buffer<expr> s_args;
        expr const & s_fn = get_app_rev_args(s, s_args);
        if (!is_constant(s_fn)) return none_expr();
        if (has_init_attribute(env(), const_name(s_fn))) return none_expr();
        if (has_noinline_attribute(env(), const_name(s_fn))) return none_expr();
        optional<constant_info> info = env().find(mk_cstage1_name(const_name(s_fn)));
        if (!info || !info->is_definition()) return none_expr();
        if (s_args.size()  < get_num_nested_lambdas(info->get_value())) return none_expr();
        if (!inline_proj_app_candidate(info->get_value())) return none_expr();
        expr s_val = instantiate_value_lparams(*info, const_levels(s_fn));
        s_val = apply_beta(s_val, s_args.size(), s_args.data());
        buffer<expr> fvars;
        while (is_let(s_val)) {
            name n        = mk_let_name(let_name(s_val));
            expr new_type = instantiate_rev(let_type(s_val), fvars.size(), fvars.data());
            expr new_val  = instantiate_rev(let_value(s_val), fvars.size(), fvars.data());
            expr new_fvar = m_lctx.mk_local_decl(ngen(), n, new_type, new_val);
            fvars.push_back(new_fvar);
            s_val = let_body(s_val);
        }
        s_val = instantiate_rev(s_val, fvars.size(), fvars.data());
        lean_assert(is_constructor_app(env(), s_val));
        buffer<expr> k_args;
        expr const & k = get_app_args(s_val, k_args);
        constructor_val k_val = env().get(const_name(k)).to_constructor_val();
        lean_assert(k_val.get_nparams() + idx < k_args.size());
        expr val = k_args[k_val.get_nparams() + idx];
        buffer<expr> fvars_to_keep;
        name_hash_set used_fvars; /* Set of free variables names used */
        collect_used(val, used_fvars);
        unsigned i = fvars.size();
        while (i > 0) {
            i--;
            expr x = fvars[i];
            if (used_fvars.find(fvar_name(x)) != used_fvars.end()) {
                local_decl x_decl = m_lctx.get_local_decl(x);
                expr x_type       = x_decl.get_type();
                expr x_val        = *x_decl.get_value();
                collect_used(x_type, used_fvars);
                collect_used(x_val,  used_fvars);
                fvars_to_keep.push_back(x);
                if (fvars_to_keep.size() > m_cfg.m_inline_threshold) return none_expr();
            }
        }
        std::reverse(fvars_to_keep.begin(), fvars_to_keep.end());
        val = m_lctx.mk_lambda(fvars_to_keep, val);
        return some_expr(visit(val, is_let_val));
    }

    expr visit_proj(expr const & e, bool is_let_val) {
        expr s = find_ctor(proj_expr(e));

        if (is_constructor_app(env(), s)) {
            return proj_constructor(s, proj_idx(e).get_small_value());
        }

        if (optional<expr> r = try_inline_proj_instance(e, is_let_val)) {
            return *r;
        }

        if (optional<expr> r = try_inline_proj_app(e, is_let_val)) {
            return *r;
        }

        expr new_arg = visit_arg(proj_expr(e));
        if (is_eqp(proj_expr(e), new_arg))
            return e;
        else
            return update_proj(e, new_arg);
    }

    expr reduce_cases_cnstr(buffer<expr> const & args, inductive_val const & I_val, expr const & major, bool is_let_val) {
        lean_assert(is_constructor_app(env(), major));
        unsigned nparams = I_val.get_nparams();
        buffer<expr> k_args;
        expr const & k   = get_app_args(major, k_args);
        lean_assert(is_constant(k));
        lean_assert(nparams <= k_args.size());
        unsigned first_minor_idx = m_before_erasure ? (nparams + 1 /* typeformer/motive */ + I_val.get_nindices() + 1 /* major */) : 1;
        constructor_val k_val = env().get(const_name(k)).to_constructor_val();
        expr const & minor    = args[first_minor_idx + k_val.get_cidx()];
        return beta_reduce(minor, k_args.size() - nparams, k_args.data() + nparams, is_let_val);
    }

    /* Just simplify minor premises. */
    expr visit_cases_default(expr const & e) {
        if (already_simplified(e))
            return e;
        lean_assert(is_cases_on_app(env(), e));
        buffer<expr> args;
        expr const & c = get_app_args(e, args);
        /* simplify minor premises */
        bool all_equal_opt = true;
        optional<expr> a_minor;
        unsigned minor_idx; unsigned minors_end;
        std::tie(minor_idx, minors_end) = get_cases_on_minors_range(env(), const_name(c), m_before_erasure);
        expr const & major = args[minor_idx-1];
        for (unsigned cidx = 0; minor_idx < minors_end; minor_idx++, cidx++) {
            expr minor                = args[minor_idx];
            unsigned saved_fvars_size = m_fvars.size();
            buffer<expr> zs;
            minor = get_minor_body(minor, zs);
            expr new_minor;
            {
                flet<expr2ctor> save_expr2ctor(m_expr2ctor, m_expr2ctor);
                update_expr2ctor(major, c, args, cidx, zs);
                new_minor = visit(minor, false);
            }
            try {
              new_minor = mk_let(zs, saved_fvars_size, new_minor, false);
              expr result_minor = mk_minor_lambda(zs, new_minor);
              if (all_equal_opt) {
                  expr result_minor_body = result_minor;
                  for (unsigned i = 0; i < zs.size(); i++) {
                      result_minor_body = binding_body(result_minor_body);
                      if (has_loose_bvars(result_minor_body)) {
                          /* Minor premise depends on constructor fields. */
                          all_equal_opt = false;
                          break;
                      }
                  }
              }
              if (all_equal_opt) {
                  if (!a_minor) {
                      a_minor = new_minor;
                  } else if (new_minor != *a_minor) {
                      all_equal_opt = false;
                  }
              }
              args[minor_idx] = result_minor;
            } catch (kernel_exception &) {
                // HACK: the code above depends on `infer_type`, and it may fail.
                // The compiler performs transformations that do not preserve typability.
                // When we rewrite the compiler in Lean, we must write a custom `infer_type` that returns an
                // `Any` type in this kind of situation.
            }
        }
        if (all_equal_opt && a_minor && !is_join_point_app(*a_minor)) {
            /*
               Remark: we must make sure `a_minor` is not a joint-point. Otherwise, we would break our joint point application invariant.
               In the current implementation, this may seen as a hack or temporary workaround.
               Since the joint point inside of a non-terminal casesOn should not be
               allowed in the first place. When we reimplement this module in Lean,
               we should make sure this kind of term is not created by previous steps.
            */
            return *a_minor;
        }
        expr r = mk_app(c, args);
        mark_simplified(r);
        return r;
    }

    /* Applies `Bool.casesOn x false true` ==> `x`

       This transformation is often applicable to code that goes back and forth between `Decidable` and `Bool`.
       After `erase_irrelevant` both are `Bool`. */
    optional<expr> is_identity_bool_cases_on (inductive_val const & I_val, buffer<expr> const & args) {
        if (m_before_erasure) return none_expr();
        if (args.size() == 3 && I_val.to_constant_val().get_name() == get_bool_name() &&
            args[1] == mk_bool_false() && args[2] == mk_bool_true()) {
            return some_expr(args[0]);
        }
        return none_expr();
    }

    expr visit_cases(expr const & e, bool is_let_val) {
        buffer<expr> args;
        expr const & c = get_app_args(e, args);
        lean_assert(is_constant(c));
        inductive_val I_val      = get_cases_on_inductive_val(env(), c);
        unsigned major_idx       = get_cases_on_major_idx(env(), const_name(c), m_before_erasure);
        lean_assert(major_idx < args.size());
        expr major               = find_ctor(args[major_idx]);

        if (is_nat_lit(major)) {
            major = nat_lit_to_constructor(major);
        }

        if (optional<expr> r = is_identity_bool_cases_on(I_val, args)) {
            return *r;
        }

        if (is_constructor_app(env(), major)) {
            return reduce_cases_cnstr(args, I_val, major, is_let_val);
        } else if (!is_let_val) {
            return visit_cases_default(e);
        } else {
            return e;
        }
    }

    expr merge_app_app(expr const & fn, expr const & e, bool is_let_val) {
        lean_assert(is_app(fn));
        lean_assert(is_eqp(find(get_app_fn(e)), fn));
        lean_assert(!is_join_point_app(fn));
        if (!is_cases_on_app(env(), fn)) {
            buffer<expr> args;
            get_app_args(e, args);
            return visit_app(mk_app(fn, args), is_let_val);
        } else {
            return e;
        }
    }

    struct is_recursive_fn {
        environment const & m_env;
        csimp_cfg const &   m_cfg;
        bool                m_before_erasure;
        name                m_target;

        is_recursive_fn(environment const & env, csimp_cfg const & cfg, bool before_erasure):
            m_env(env), m_cfg(cfg), m_before_erasure(before_erasure) {
        }

        optional<constant_info> is_inline_candidate(name const  & f) {
            name c = m_before_erasure ? mk_cstage1_name(f) : mk_cstage2_name(f);
            optional<constant_info> info = m_env.find(c);
            if (!info || !info->is_definition()) {
                return optional<constant_info>();
            } else if (has_inline_attribute(m_env, f)) {
                return info;
            } else if (get_lcnf_size(m_env, info->get_value()) <= m_cfg.m_inline_threshold) {
                return info;
            } else {
                return optional<constant_info>();
            }
        }

        bool visit(name const & f, name_set visited) {
            if (optional<constant_info> info = is_inline_candidate(f)) {
                if (visited.contains(f))
                    return true;
                visited.insert(f);
                return static_cast<bool>(::lean::find(info->get_value(), [&](expr const & e, unsigned) {
                            return is_constant(e) && (const_name(e) == m_target || visit(const_name(e), visited));
                        }));
            } else {
                return false;
            }
        }

        bool operator()(name const & f) {
            m_target = f;
            return visit(f, name_set());
        }
    };

    /* We don't inline recursive functions. */
    bool is_recursive(name const & c) {
        return is_recursive_fn(env(), m_cfg, m_before_erasure)(c);
    }

    bool uses_unsafe_inductive(name const & c) {
        constant_info info = env().get(c);
        return static_cast<bool>(::lean::find(info.get_value(), [&](expr const & e, unsigned) {
                    if (!is_constant(e) || !is_cases_on_recursor(env(), const_name(e))) return false;
                    name const & I = const_name(e).get_prefix();
                    constant_info I_cinfo = env().get(I);
                    return I_cinfo.is_unsafe();
                }));
    }

    bool is_stuck_at_cases(expr e) {
        type_checker tc(m_st, m_lctx);
        while (true) {
            expr e1 = tc.whnf_core_cheap(e);
            expr const & fn = get_app_fn(e1);
            if (!is_constant(fn)) return false;
            if (is_recursor(env(), const_name(fn))) return true;
            if (!is_cases_on_recursor(env(), const_name(fn))) return false;
            auto next_e = tc.unfold_definition(e1);
            if (!next_e) return true;
            e = *next_e;
        }
    }

    optional<expr> beta_reduce_if_not_cases(expr fn, unsigned nargs, expr const * args, bool is_let_val) {
        unsigned i = 0;
        while (is_lambda(fn) && i < nargs) {
            i++;
            fn = binding_body(fn);
        }
        expr r = instantiate_rev(fn, i, args);
        if (is_lambda(r) || is_stuck_at_cases(r)) return none_expr();
        return some_expr(beta_reduce_cont(r, i, nargs, args, is_let_val));
    }

    /* Auxiliary method used to inline functions marked with `[inline_if_reduce]`. It is similar to `beta_reduce`
       but it fails if the head is a `cases_on` application after `whnf_core`. */
    optional<expr> beta_reduce_if_not_cases(expr fn, expr const & e, bool is_let_val) {
        buffer<expr> args;
        get_app_args(e, args);
        return beta_reduce_if_not_cases(fn, args.size(), args.data(), is_let_val);
    }

    bool check_noinline_attribute(name const & n) {
        if (!has_noinline_attribute(env(), n)) return false;
        /* Even if the function has `@[noinline]` attribute, we must still inline if its arguments
           were reduced by `reduce_arity`. This should only be checked after erasure. */
        if (m_before_erasure) return true;
        name c = mk_cstage2_name(n);
        optional<constant_info> info = env().find(c);
        if (!info || !info->is_definition()) return true;
        return !arity_was_reduced(comp_decl(n, info->get_value()));
    }

    optional<expr> try_inline(expr const & fn, expr const & e, bool is_let_val) {
        lean_assert(is_constant(fn));
        lean_assert(is_constant(e) || is_eqp(find(get_app_fn(e)), fn));
        if (!m_cfg.m_inline) return none_expr();
        if (has_init_attribute(env(), const_name(fn))) return none_expr();
        if (check_noinline_attribute(const_name(fn))) return none_expr();
        if (m_before_erasure) {
            if (already_simplified(e)) return none_expr();
            name c = mk_cstage1_name(const_name(fn));
            optional<constant_info> info = env().find(c);
            if (!info || !info->is_definition()) return none_expr();
            if (get_app_num_args(e) < get_num_nested_lambdas(info->get_value())) return none_expr();
            bool inline_attr           = has_inline_attribute(env(), const_name(fn));
            bool inline_if_reduce_attr = has_inline_if_reduce_attribute(env(), const_name(fn));
            if (!inline_attr && !inline_if_reduce_attr &&
                (get_lcnf_size(env(), info->get_value()) > m_cfg.m_inline_threshold ||
                 is_constant(e))) { /* We only inline constants if they are marked with the `[inline]` or `[inline_if_reduce]` attrs */
                return none_expr();
            }
            if (!inline_if_reduce_attr && is_recursive(const_name(fn))) return none_expr();
            if (!is_matcher(env(), const_name(fn))) {
                // Hack for test `inliner_loop`. We don't generate code for auxiliary matcher applications.
                // However, they are safe to be inline even when they use unsafe inductive types.
                // REMARK: the to be implemented `[strong_inline]` attribute should not be used in unsafe code.
                if (uses_unsafe_inductive(c)) return none_expr();
            }
            lean_trace(name({"compiler", "inline"}), tout() << const_name(fn) << "\n";);
            expr new_fn = instantiate_value_lparams(*info, const_levels(fn));
            if (inline_if_reduce_attr && !inline_attr) {
                return beta_reduce_if_not_cases(new_fn, e, is_let_val);
            } else {
                return some_expr(beta_reduce(new_fn, e, is_let_val));
            }
        } else {
            /* We should not inline closed constants we have extracted. */
            if (is_extract_closed_aux_fn(const_name(fn))) return none_expr();
            name c = mk_cstage2_name(const_name(fn));
            optional<constant_info> info = env().find(c);
            if (!info || !info->is_definition()) return none_expr();
            unsigned arity = get_num_nested_lambdas(info->get_value());
            if (get_app_num_args(e) < arity || arity == 0) return none_expr();
            if (get_lcnf_size(env(), info->get_value()) > m_cfg.m_inline_threshold) return none_expr();
            if (is_recursive(const_name(fn))) return none_expr();
            if (uses_unsafe_inductive(c)) return none_expr();
            return some_expr(beta_reduce(info->get_value(), e, is_let_val));
        }
    }

    expr visit_inline_app(expr const & e, bool is_let_val) {
        buffer<expr> args;
        get_app_args(e, args);
        lean_assert(!args.empty());
        if (args.size() < 2)
            return visit_app_default(e);
        buffer<expr> new_args;
        expr fn = get_app_args(find(args[1]), new_args);
        new_args.append(args.size() - 2, args.data() + 2);
        expr r  = mk_app(fn, new_args);
        if (!m_cfg.m_inline || !is_constant(fn))
            return visit(r, is_let_val);
        name main  = const_name(fn);
        bool first = true;
        while (true) {
            name c = mk_cstage1_name(const_name(fn));
            optional<constant_info> info = env().find(c);
            if (!info || !info->is_definition())
                return first ? visit(r, is_let_val) : r;
            expr new_fn = instantiate_value_lparams(*info, const_levels(fn));
            r = beta_reduce(new_fn, new_args.size(), new_args.data(), is_let_val);
            if (!is_app(r)) return r;
            fn = get_app_fn(r);
            /* If `r` is an application of the form `g ...` where
               `g` is an internal name and `g` prefix of the main function, we unfold this
               application too. */
            if (!is_constant(fn) || !is_internal_name(const_name(fn)) ||
                const_name(fn).get_prefix() != main)
                return r;
            new_args.clear();
            get_app_args(r, new_args);
            first = false;
        }
    }

    expr visit_app_default(expr const & e) {
        if (already_simplified(e)) return e;
        buffer<expr> args;
        bool modified   = true;
        expr const & fn = get_app_args(e, args);
        for (expr & arg : args) {
            expr new_arg = visit_arg(arg);
            if (!is_eqp(arg, new_arg))
                modified = true;
            arg = new_arg;
        }
        expr new_e = modified ? mk_app(fn, args) : e;
        mark_simplified(new_e);
        return new_e;
    }

    expr visit_nat_succ(expr const & e) {
        expr arg = visit(app_arg(e), false);
        return mk_app(mk_constant(get_nat_add_name()), arg, mk_lit(literal(nat(1))));
    }

    expr visit_thunk_get(expr const & e, bool is_let_val) {
        buffer<expr> args;
        expr fn = get_app_args(e, args);
        lean_assert(is_constant(fn, get_thunk_get_name()));
        if (args.size() != 2) return visit_app_default(e);
        expr mk = find(args[1]);
        if (!is_app_of(mk, get_thunk_mk_name(), 2)) return visit_app_default(e);
        // @Thunk.get _ (@Thunk.mk _ g) => g ()
        expr g = app_arg(mk);
        return visit(mk_app(g, mk_unit_mk()), is_let_val);
    }

    /*
      Replace `fixCore<n> f a_1 ... a_m`
      with `fixCore<m> f a_1 ... a_m` whenever `n < m`.
      This optimization is for writing reusable/generic code. For
      example, we cannot write an efficient `rec_t` monad transformer
      without it because we don't know the arity of `m A` when we write `rec_t`.
      Remark: the runtime provides a small set of `fixCore<i>` implementations (`i in [1, 6]`).
      This methods does nothing if `m > 6`. */
    expr visit_fix_core(expr const & e, unsigned n) {
        if (m_before_erasure) return visit_app_default(e);
        buffer<expr> args;
        expr fn = get_app_args(e, args);
        lean_assert(is_constant(fn) && is_fix_core(const_name(fn)));
        unsigned arity =
            n + /* α_1 ... α_n Type arguments */
            1 + /* β : Type */
            1 + /* (base : α_1 → ... → α_n → β) */
            1 + /* (rec : (α_1 → ... → α_n → β) → α_1 → ... → α_n → β) */
            n; /* α_1 → ... → α_n */
        if (args.size() <= arity) return visit_app_default(e);
        /* This `fixCore<n>` application is an overapplication.
           The `fixCore<n>` is implemented by the runtime, and the result
           is a closure. This is bad for performance. We should
           replace it with `fixCore<m>` (if the runtime contains one) */
        unsigned num_extra = args.size() - arity;
        unsigned m = n + num_extra;
        optional<expr> fix_core_m = mk_enf_fix_core(m);
        if (!fix_core_m) return visit_app_default(e);
        buffer<expr> new_args;
        /* Add α_1 ... α_n and β */
        for (unsigned i = 0; i < m+1; i++) {
            new_args.push_back(mk_enf_neutral());
        }
        /* `(base : α_1 → ... → α_n → β)` is not used in the runtime primitive.
           So, we replace it with a neutral value :) */
        new_args.push_back(mk_enf_neutral());
        new_args.append(args.size() - n - 2, args.data() + n + 2);
        return mk_app(*fix_core_m, new_args);
    }

    expr visit_app(expr const & e, bool is_let_val) {
        if (is_cases_on_app(env(), e)) {
            return visit_cases(e, is_let_val);
        } else if (is_app_of(e, get_inline_name())) {
            return visit_inline_app(e, is_let_val);
        }
        expr fn = find(get_app_fn(e));
        if (is_lambda(fn)) {
            return beta_reduce(fn, e, is_let_val);
        } else if (is_cases_on_app(env(), fn)) {
            expr new_e = float_cases_on_core(get_app_fn(e), e, fn);
            mark_simplified(new_e);
            return new_e;
        } else if (is_lc_unreachable_app(fn)) {
            lean_assert(m_before_erasure);
            expr type = infer_type(e);
            return mk_lc_unreachable(m_st, m_lctx, type);
        } else if (is_app(fn)) {
            return merge_app_app(fn, e, is_let_val);
        } else if (is_constant(fn)) {
            unsigned nargs = get_app_num_args(e);
            if (nargs == 1) {
                expr a1 = find(visit_arg(app_arg(e)));
                if (optional<expr> r = fold_un_op(m_before_erasure, fn, a1)) {
                    return *r;
                }
            } else if (nargs == 2) {
                expr a1 = find(visit_arg(app_arg(app_fn(e))));
                expr a2 = find(visit_arg(app_arg(e)));
                if (optional<expr> r = fold_bin_op(m_before_erasure, fn, a1, a2)) {
                    return *r;
                }
            }
            name const & n = const_name(fn);
            if (n == get_nat_succ_name()) {
                return visit_nat_succ(e);
            } else if (n == get_nat_zero_name()) {
                return mk_lit(literal(nat(0)));
            } else if (n == get_thunk_get_name()) {
                return visit_thunk_get(e, is_let_val);
            } else if (optional<expr> r = try_inline(fn, e, is_let_val)) {
                return *r;
            } else if (optional<unsigned> i = is_fix_core(n)) {
                return visit_fix_core(e, *i);
            } else {
                return visit_app_default(e);
            }
        } else {
            return visit_app_default(e);
        }
    }

    expr visit_constant(expr const & e, bool is_let_val) {
        if (optional<expr> r = try_inline(e, e, is_let_val))
            return *r;
        else
            return e;
    }

    expr visit_arg(expr const & e) {
        if (!is_lcnf_atom(e)) {
            /* non-atomic arguments are irrelevant in LCNF */
            return e;
        }
        expr new_e = visit(e, false);
        if (is_lcnf_atom(new_e))
            return new_e;
        else
            return mk_let_decl(new_e);
    }

    expr visit(expr const & e, bool is_let_val) {
        switch (e.kind()) {
        case expr_kind::Lambda: return is_let_val ? e : visit_lambda(e, false, false);
        case expr_kind::Let:    return visit_let(e);
        case expr_kind::Proj:   return visit_proj(e, is_let_val);
        case expr_kind::App:    return visit_app(e, is_let_val);
        case expr_kind::Const:  return visit_constant(e, is_let_val);
        /* We erase MData for now. We should revisit this decision when we rewrite the compiler in Lean, since we
           are probably going to store debugging information in this kind of node.
           We erase it here because `ir.cpp` does not support it, and produced `unreachable` code at issue #616 */
        case expr_kind::MData:  return visit(mdata_expr(e), is_let_val);
        default:                return e;
        }
    }

public:
    csimp_fn(environment const & env, local_ctx const & lctx, bool before_erasure, csimp_cfg const & cfg):
        m_st(env), m_lctx(lctx), m_before_erasure(before_erasure), m_cfg(cfg), m_x("_x"), m_j("j") {}

    expr operator()(expr const & e) {
        if (is_lambda(e)) {
            return visit_lambda(e, false, true);
        } else {
            buffer<expr> empty_xs;
            expr r = visit(e, false);
            return mk_let(empty_xs, 0, r, true);
        }
    }
};

extern "C" uint8 lean_at_most_once(obj_arg e, obj_arg x);

bool at_most_once(expr const & e, name const & x) {
    inc_ref(e.raw()); inc_ref(x.raw());
    return lean_at_most_once(e.raw(), x.raw());
}

/* Eliminate join-points that are used only once */
class elim_jp1_fn {
    environment const & m_env;
    local_ctx           m_lctx;
    bool                m_before_erasure;
    name_generator      m_ngen;
    name_set            m_to_expand;
    bool                m_expanded{false};

    void mark_to_expand(expr const & e) {
        m_to_expand.insert(fvar_name(e));
    }

    bool is_to_expand_jp_app(expr const & e) {
        expr const & f = get_app_fn(e);
        return is_fvar(f) && m_to_expand.contains(fvar_name(f));
    }

    expr visit_lambda(expr e) {
        buffer<expr> fvars;
        while (is_lambda(e)) {
            expr domain     = visit(instantiate_rev(binding_domain(e), fvars.size(), fvars.data()));
            expr fvar       = m_lctx.mk_local_decl(m_ngen, binding_name(e), domain, binding_info(e));
            fvars.push_back(fvar);
            e = binding_body(e);
        }
        e = visit(instantiate_rev(e, fvars.size(), fvars.data()));
        return m_lctx.mk_lambda(fvars, e);
    }

    expr visit_cases(expr const & e) {
        lean_assert(is_cases_on_app(m_env, e));
        buffer<expr> args;
        expr const & c = get_app_args(e, args);
        /* simplify minor premises */
        unsigned minor_idx; unsigned minors_end;
        std::tie(minor_idx, minors_end) = get_cases_on_minors_range(m_env, const_name(c), m_before_erasure);
        for (; minor_idx < minors_end; minor_idx++) {
            args[minor_idx] = visit(args[minor_idx]);
        }
        return mk_app(c, args);
    }

    expr visit_app(expr const & e) {
        lean_assert(is_app(e));
        if (is_cases_on_app(m_env, e)) {
            return visit_cases(e);
        } else if (is_to_expand_jp_app(e)) {
            buffer<expr> args;
            expr const & jp    = get_app_rev_args(e, args);
            local_decl jp_decl = m_lctx.get_local_decl(jp);
            lean_assert(is_join_point_name(jp_decl.get_user_name()));
            lean_assert(jp_decl.get_value());
            lean_assert(is_lambda(*jp_decl.get_value()));
            return apply_beta(*jp_decl.get_value(), args.size(), args.data());
        } else {
            return e;
        }
    }

    bool at_most_once(expr const & e, expr const & jp) {
        lean_assert(is_fvar(jp));
        return lean::at_most_once(e, fvar_name(jp));
    }

    expr visit_let(expr e) {
        buffer<expr> fvars;
        buffer<expr> all_fvars;
        while (is_let(e)) {
            expr new_type = visit(instantiate_rev(let_type(e), fvars.size(), fvars.data()));
            expr new_val  = visit(instantiate_rev(let_value(e), fvars.size(), fvars.data()));
            expr fvar     = m_lctx.mk_local_decl(m_ngen, let_name(e), new_type, new_val);
            fvars.push_back(fvar);
            if (is_join_point_name(let_name(e))) {
                e = instantiate_rev(let_body(e), fvars.size(), fvars.data());
                fvars.clear();
                if (at_most_once(e, fvar)) {
                    m_expanded = true;
                    mark_to_expand(fvar);
                } else {
                    /* Keep join point */
                    all_fvars.push_back(fvar);
                }
            } else {
                all_fvars.push_back(fvar);
                e = let_body(e);
            }
        }
        e = instantiate_rev(e, fvars.size(), fvars.data());
        e = visit(e);
        return m_lctx.mk_lambda(all_fvars, e);
    }

    expr visit(expr const & e) {
        switch (e.kind()) {
        case expr_kind::Lambda: return visit_lambda(e);
        case expr_kind::Let:    return visit_let(e);
        case expr_kind::App:    return visit_app(e);
        default:                return e;
        }
    }

public:
    elim_jp1_fn(environment const & env, local_ctx const & lctx, bool before_erasure):
        m_env(env), m_lctx(lctx), m_before_erasure(before_erasure) {}
    expr operator()(expr const & e) {
        m_expanded = false;
        return visit(e);
    }

    bool expanded() const { return m_expanded; }
};

expr csimp_core(environment const & env, local_ctx const & lctx, expr const & e0, bool before_erasure, csimp_cfg const & cfg) {
    csimp_fn simp(env, lctx, before_erasure, cfg);
    elim_jp1_fn elim_jp1(env, lctx, before_erasure);
    expr e = e0;
    while (true) {
        e = simp(e);
        bool modified = false;
        e = elim_jp1(e);
        if (elim_jp1.expanded())
            modified = true;
        expr new_e = cse_core(env, e, before_erasure);
        new_e = elim_dead_let(new_e);
        if (e != new_e)
            modified = true;
        if (!modified)
            return e;
        e = new_e;
    }
}
}
