feat(frontends/lean): add inductive_cmd

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>

src/emacs/lean-mode.el

@@ -24,7 +24,7 @@
 (define-generic-mode
     'lean-mode     ;; name of the mode to create
   '("--")          ;; comments start with
-  '("import" "abbreviation" "definition" "parameter" "parameters" "conjecture" "hypothesis" "lemma" "corollary" "variable" "variables" "print" "theorem" "axiom" "universe" "alias" "help" "environment" "options" "precedence" "postfix" "calc_trans" "calc_subst" "calc_refl" "infix" "infixl" "infixr" "notation" "eval" "check" "exit" "coercion" "end" "using" "namespace" "builtin" "section" "set_option" "add_rewrite") ;; some keywords
+  '("import" "abbreviation" "definition" "parameter" "parameters" "conjecture" "hypothesis" "lemma" "corollary" "variable" "variables" "print" "theorem" "axiom" "inductive" "with" "universe" "alias" "help" "environment" "options" "precedence" "postfix" "calc_trans" "calc_subst" "calc_refl" "infix" "infixl" "infixr" "notation" "eval" "check" "exit" "coercion" "end" "using" "namespace" "builtin" "section" "set_option" "add_rewrite") ;; some keywords
   '(("\\_<\\(Bool\\|Int\\|Nat\\|Real\\|Type\\|TypeU\\|ℕ\\|ℤ\\)\\_>" . 'font-lock-type-face)
     ("\\_<\\(calc\\|have\\|show\\|by\\|in\\|let\\|forall\\|fun\\|exists\\|if\\|then\\|else\\|assume\\|take\\|obtain\\|from\\)\\_>" . font-lock-keyword-face)
     ("\"[^\"]*\"" . 'font-lock-string-face)

src/frontends/lean/CMakeLists.txt

@@ -2,6 +2,6 @@ add_library(lean_frontend register_module.cpp token_table.cpp
 scanner.cpp parse_table.cpp parser_config.cpp parser.cpp
 parser_pos_provider.cpp builtin_cmds.cpp builtin_tactic_cmds.cpp
 builtin_exprs.cpp interactive.cpp notation_cmd.cpp calc.cpp
-decl_cmds.cpp util.cpp)
+decl_cmds.cpp util.cpp inductive_cmd.cpp)
 
 target_link_libraries(lean_frontend ${LEAN_LIBS})

src/frontends/lean/builtin_cmds.cpp

@@ -16,6 +16,7 @@ Author: Leonardo de Moura
 #include "frontends/lean/parser.h"
 #include "frontends/lean/calc.h"
 #include "frontends/lean/notation_cmd.h"
+#include "frontends/lean/inductive_cmd.h"
 #include "frontends/lean/decl_cmds.h"
 
 namespace lean {
@@ -217,6 +218,7 @@ cmd_table init_cmd_table() {
     add_cmd(r, cmd_info("check",        "type check given expression, and display its type", check_cmd));
     add_cmd(r, cmd_info("#setline",     "modify the current line number, it only affects error/report messages", set_line_cmd));
     register_decl_cmds(r);
+    register_inductive_cmd(r);
     register_notation_cmds(r);
     register_calc_cmds(r);
     return r;

src/frontends/lean/decl_cmds.cpp

@@ -38,7 +38,7 @@ environment universe_cmd(parser & p) {
     return env;
 }
 
-void parse_univ_params(parser & p, buffer<name> & ps) {
+bool parse_univ_params(parser & p, buffer<name> & ps) {
     if (p.curr_is_token(g_llevel_curly)) {
         p.next();
         while (!p.curr_is_token(g_rcurly)) {
@@ -47,10 +47,13 @@ void parse_univ_params(parser & p, buffer<name> & ps) {
             ps.push_back(l);
         }
         p.next();
+        return true;
+    } else{
+        return false;
     }
 }
 
-void update_parameters(buffer<name> & ls_buffer, name_set const & found, parser const & p) {
+void update_univ_parameters(buffer<name> & ls_buffer, name_set const & found, parser const & p) {
     unsigned old_sz = ls_buffer.size();
     found.for_each([&](name const & n) {
             if (std::find(ls_buffer.begin(), ls_buffer.begin() + old_sz, n) == ls_buffer.begin() + old_sz)
@@ -107,7 +110,7 @@ environment variable_cmd_core(parser & p, bool is_axiom, binder_info const & bi)
     if (in_section(p.env())) {
         ls = to_level_param_names(collect_univ_params(type));
     } else {
-        update_parameters(ls_buffer, collect_univ_params(type), p);
+        update_univ_parameters(ls_buffer, collect_univ_params(type), p);
         ls = to_list(ls_buffer.begin(), ls_buffer.end());
     }
     type = p.elaborate(type, ls);
@@ -132,10 +135,9 @@ environment cast_variable_cmd(parser & p) {
     return variable_cmd_core(p, false, mk_cast_binder_info());
 }
 
-// Collect local (section) constants occurring in type and value, sort them, and store in section_ps
+// Sort local_names by order of occurrence in the section, and copy the associated parameters to section_ps
-void collect_section_locals(expr const & type, expr const & value, parser const & p, buffer<parameter> & section_ps) {
+void mk_section_params(name_set const & local_names, parser const & p, buffer<parameter> & section_ps) {
-    name_set ls = collect_locals(type, collect_locals(value));
+    local_names.for_each([&](name const & n) {
-    ls.for_each([&](name const & n) {
             section_ps.push_back(*p.get_local(n));
         });
     std::sort(section_ps.begin(), section_ps.end(), [&](parameter const & p1, parameter const & p2) {
@@ -143,6 +145,12 @@ void collect_section_locals(expr const & type, expr const & value, parser const
         });
 }
 
+// Collect local (section) constants occurring in type and value, sort them, and store in section_ps
+void collect_section_locals(expr const & type, expr const & value, parser const & p, buffer<parameter> & section_ps) {
+    name_set ls = collect_locals(type, collect_locals(value));
+    return mk_section_params(ls, p, section_ps);
+}
+
 static void parse_modifiers(parser & p, bool & is_private, bool & is_opaque) {
     while (true) {
         if (p.curr_is_token(g_private)) {
@@ -157,6 +165,18 @@ static void parse_modifiers(parser & p, bool & is_private, bool & is_opaque) {
     }
 }
 
+// Return the levels in \c ls that are defined in the section
+levels collect_section_levels(level_param_names const & ls, parser & p) {
+    buffer<level> section_ls_buffer;
+    for (name const & l : ls) {
+        if (p.get_local_level_index(l))
+            section_ls_buffer.push_back(mk_param_univ(l));
+        else
+            break;
+    }
+    return to_list(section_ls_buffer.begin(), section_ls_buffer.end());
+}
+
 environment definition_cmd_core(parser & p, bool is_theorem, bool is_opaque) {
     bool is_private = false;
     parse_modifiers(p, is_private, is_opaque);
@@ -202,7 +222,7 @@ environment definition_cmd_core(parser & p, bool is_theorem, bool is_opaque) {
             p.check_token_next(g_assign, "invalid declaration, ':=' expected");
             value = p.parse_expr();
         }
-        update_parameters(ls_buffer, collect_univ_params(value, collect_univ_params(type)), p);
+        update_univ_parameters(ls_buffer, collect_univ_params(value, collect_univ_params(type)), p);
         ls = to_list(ls_buffer.begin(), ls_buffer.end());
     }
     if (in_section(env)) {
@@ -210,14 +230,7 @@ environment definition_cmd_core(parser & p, bool is_theorem, bool is_opaque) {
         collect_section_locals(type, value, p, section_ps);
         type = p.pi_abstract(section_ps, type);
         value = p.lambda_abstract(section_ps, value);
-        buffer<level> section_ls_buffer;
+        levels section_ls = collect_section_levels(ls, p);
-        for (name const & l : ls) {
-            if (p.get_local_level_index(l))
-                section_ls_buffer.push_back(mk_param_univ(l));
-            else
-                break;
-        }
-        levels section_ls = to_list(section_ls_buffer.begin(), section_ls_buffer.end());
         buffer<expr> section_args;
         for (auto const & p : section_ps)
             section_args.push_back(p.m_local);

src/frontends/lean/decl_cmds.h

@@ -7,11 +7,29 @@ Author: Leonardo de Moura
 #pragma once
 #include "util/buffer.h"
 #include "kernel/expr.h"
+#include "frontends/lean/parameter.h"
 #include "frontends/lean/cmd_table.h"
 namespace lean {
 class parser;
-void parse_univ_params(parser & p, buffer<name> & ps);
+/**
-void update_parameters(buffer<name> & ls_buffer, name_set const & found, parser const & p);
+    \brief Parse (optional) universe parameters <tt>'.{' l_1 ... l_k '}'</tt>
-void collect_section_locals(expr const & type, expr const & value, parser const & p, buffer<parameter> & section_ps);
+    Store the result in \c ps.
+    Return true when levels were provided.
+*/
+bool parse_univ_params(parser & p, buffer<name> & ps);
+/**
+   \brief Add universe levels from \c found_ls to \c ls_buffer (only the levels that do not already occur in \c ls_buffer are added).
+   Then sort \c ls_buffer (using the order in which the universe levels were declared).
+*/
+void update_univ_parameters(buffer<name> & ls_buffer, name_set const & found_ls, parser const & p);
+/**
+   \brief Copy the parameters associated with the local names in \c local_names to \c section_ps.
+   Then sort \c section_ps (using the order in which they were declared).
+*/
+void mk_section_params(name_set const & local_names, parser const & p, buffer<parameter> & section_ps);
+/**
+   \brief Return the levels in \c ls that are defined in the section.
+*/
+levels collect_section_levels(level_param_names const & ls, parser & p);
 void register_decl_cmds(cmd_table & r);
 }

src/frontends/lean/inductive_cmd.cpp

@@ -0,0 +1,343 @@
+/*
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Author: Leonardo de Moura
+*/
+#include "util/sstream.h"
+#include "util/name_map.h"
+#include "kernel/replace_fn.h"
+#include "kernel/type_checker.h"
+#include "kernel/instantiate.h"
+#include "kernel/inductive/inductive.h"
+#include "library/scoped_ext.h"
+#include "library/locals.h"
+#include "library/placeholder.h"
+#include "library/aliases.h"
+#include "frontends/lean/decl_cmds.h"
+#include "frontends/lean/util.h"
+#include "frontends/lean/parser.h"
+
+namespace lean {
+static name g_assign(":=");
+static name g_with("with");
+static name g_colon(":");
+static name g_bar("|");
+
+using inductive::intro_rule;
+using inductive::inductive_decl;
+using inductive::inductive_decl_name;
+using inductive::inductive_decl_type;
+using inductive::inductive_decl_intros;
+using inductive::intro_rule_name;
+using inductive::intro_rule_type;
+
+// Mark all parameters as implicit
+static void make_implicit(buffer<parameter> & ps) {
+    for (parameter & p : ps)
+        p.m_bi = mk_implicit_binder_info();
+}
+
+// Make sure that every inductive datatype (in decls) occurring in \c type has
+// the universe levels \c lvl_params and section parameters \c section_params
+static expr fix_inductive_occs(expr const & type, buffer<inductive_decl> const & decls,
+                               buffer<name> const & lvl_params, buffer<parameter> const & section_params) {
+    return replace(type, [&](expr const & e, unsigned) {
+            if (!is_constant(e))
+                return none_expr();
+            if (!std::any_of(decls.begin(), decls.end(),
+                             [&](inductive_decl const & d) { return const_name(e) == inductive_decl_name(d); }))
+                return none_expr();
+            // found target
+            levels  ls = const_levels(e);
+            unsigned n = length(ls);
+            if (n < lvl_params.size()) {
+                unsigned i = lvl_params.size() - n;
+                while (i > 0) {
+                    --i;
+                    ls = cons(mk_param_univ(lvl_params[i]), ls);
+                }
+            }
+            expr r = update_constant(e, ls);
+            for (unsigned i = 0; i < section_params.size(); i++)
+                r = mk_app(r, section_params[i].m_local);
+            return some_expr(r);
+        });
+}
+
+static level mk_result_level(bool impredicative, buffer<level> const & ls) {
+    if (ls.empty()) {
+        return impredicative ? mk_level_one() : mk_level_zero();
+    } else {
+        level r = ls[0];
+        for (unsigned i = 1; i < ls.size(); i++)
+            r = mk_max(r, ls[i]);
+        return impredicative ? mk_max(r, mk_level_one()) : r;
+    }
+}
+
+static expr update_result_sort(expr const & t, level const & l) {
+    if (is_pi(t)) {
+        return update_binding(t, binding_domain(t), update_result_sort(binding_body(t), l));
+    } else if (is_sort(t)) {
+        return update_sort(t, l);
+    } else {
+        lean_unreachable();
+    }
+}
+
+static name g_tmp_prefix = name::mk_internal_unique_name();
+static void set_result_universes(buffer<inductive_decl> & decls, level_param_names const & lvls, unsigned num_params, parser & p) {
+    if (std::all_of(decls.begin(), decls.end(), [](inductive_decl const & d) {
+                return !has_placeholder(inductive_decl_type(d));
+            }))
+        return; // nothing to be done
+    // We can't infer the type of intro rule arguments because we did declare the inductive datatypes.
+    // So, we use the following trick, we create a "draft" environment where the inductive datatypes
+    // are asserted as variable declarations, and keep doing that until we reach a "fix" point.
+    unsigned num_rounds = 0;
+    while (true) {
+        if (num_rounds > 2*decls.size() + 1) {
+            // TODO(Leo): this is test is a hack to avoid non-termination.
+            // We should use a better termination condition
+            throw exception("failed to compute resultant universe level for inductive datatypes, "
+                            "provide explicit universe levels");
+        }
+        num_rounds++;
+        bool progress = false;
+        environment env    = p.env();
+        bool impredicative = env.impredicative();
+        // first assert inductive types that do not have placeholders
+        for (auto const & d : decls) {
+            expr type = inductive_decl_type(d);
+            if (!has_placeholder(type))
+                env = env.add(check(env, mk_var_decl(inductive_decl_name(d), lvls, inductive_decl_type(d))));
+        }
+        type_checker   tc(env);
+        name_generator ngen(g_tmp_prefix);
+        // try to update resultant universe levels
+        for (auto & d : decls) {
+            expr d_t = inductive_decl_type(d);
+            while (is_pi(d_t)) {
+                d_t = binding_body(d_t);
+            }
+            if (!is_sort(d_t))
+                throw exception(sstream() << "invalid inductive datatype '" << inductive_decl_name(d) << "', "
+                                "resultant type is not a sort");
+            level r_lvl = sort_level(d_t);
+            if (impredicative && is_zero(r_lvl))
+                continue;
+            buffer<level>  lvls;
+            for (intro_rule const & ir : inductive_decl_intros(d)) {
+                expr t     = intro_rule_type(ir);
+                unsigned i = 0;
+                while (is_pi(t)) {
+                    if (i >= num_params) {
+                        try {
+                            expr s = tc.ensure_sort(tc.infer(binding_domain(t)));
+                            level lvl = sort_level(s);
+                            if (std::find(lvls.begin(), lvls.end(), lvl) == lvls.end())
+                                lvls.push_back(lvl);
+                        } catch (...) {
+                        }
+                    }
+                    t = instantiate(binding_body(t), mk_local(ngen.next(), binding_name(t), binding_domain(t)));
+                    i++;
+                }
+            }
+            level m_lvl = normalize(mk_result_level(impredicative, lvls));
+            if (is_placeholder(r_lvl) || !(is_geq(r_lvl, m_lvl))) {
+                progress = true;
+                // update result level
+                expr new_type = update_result_sort(inductive_decl_type(d), m_lvl);
+                d = inductive_decl(inductive_decl_name(d), new_type, inductive_decl_intros(d));
+            }
+        }
+        if (!progress)
+            break;
+    }
+}
+
+static environment create_alias(environment const & env, name const & full_id, name const & id, levels const & section_leves,
+                                buffer<parameter> const & section_params, parser & p) {
+    if (in_section(env)) {
+        expr r = mark_explicit(mk_constant(full_id, section_leves));
+        for (unsigned i = 0; i < section_params.size(); i++)
+            r = mk_app(r, section_params[i].m_local);
+        p.add_local_expr(id, r);
+        return env;
+    } else if (full_id != id) {
+        return add_alias(env, id, mk_constant(full_id));
+    } else {
+        return env;
+    }
+}
+
+environment inductive_cmd(parser & p) {
+    parser::no_undef_id_error_scope err_scope(p);
+    environment            env = p.env();
+    name const &           ns = get_namespace(env);
+    bool                   first = true;
+    buffer<name>           ls_buffer;
+    name_map<name>         id_to_short_id;
+    unsigned               num_params = 0;
+    bool                   explicit_levels = false;
+    buffer<inductive_decl> decls;
+    while (true) {
+        parser::local_scope l_scope(p);
+        auto id_pos  = p.pos();
+        name id      = p.check_id_next("invalid inductive declaration, identifier expected");
+        check_atomic(id);
+        name full_id = ns + id;
+        id_to_short_id.insert(full_id, id);
+        buffer<name> curr_ls_buffer;
+        expr type;
+        optional<parser::param_universe_scope> pu_scope;
+        if (parse_univ_params(p, curr_ls_buffer)) {
+            if (first) {
+                explicit_levels = true;
+                ls_buffer.append(curr_ls_buffer);
+            } else if (!explicit_levels) {
+                throw parser_error("invalid mutually recursive declaration, "
+                                   "explicit universe levels were not provided for previous inductive types in this declaration",
+                                   id_pos);
+            } else if (curr_ls_buffer.size() != ls_buffer.size()) {
+                throw parser_error("invalid mutually recursive declaration, "
+                                   "all inductive types must have the same number of universe paramaters", id_pos);
+            } else {
+                for (unsigned i = 0; i < ls_buffer.size(); i++) {
+                    if (curr_ls_buffer[i] != ls_buffer[i])
+                        throw parser_error("invalid mutually recursive inductive declaration, "
+                                           "all inductive types must have the same universe paramaters", id_pos);
+                }
+            }
+        } else {
+            if (first) {
+                explicit_levels = false;
+            } else if (explicit_levels)  {
+                throw parser_error("invalid mutually recursive declaration, "
+                                   "explicit universe levels were provided for previous inductive types in this declaration",
+                                   id_pos);
+            }
+            // initialize param_universe_scope, we are using implicit universe levels
+            pu_scope.emplace(p);
+        }
+        buffer<parameter> ps;
+        local_environment lenv = env;
+        auto params_pos = p.pos();
+        if (!p.curr_is_token(g_colon)) {
+            lenv = p.parse_binders(ps);
+            p.check_token_next(g_colon, "invalid inductive declaration, ':' expected");
+            {
+                parser::placeholder_universe_scope place_scope(p);
+                type = p.parse_scoped_expr(ps, lenv);
+            }
+            type = p.pi_abstract(ps, type);
+        } else {
+            p.next();
+            parser::placeholder_universe_scope place_scope(p);
+            type = p.parse_scoped_expr(ps, lenv);
+        }
+        // check if number of parameters
+        if (first) {
+            num_params = ps.size();
+        } else {
+            // mutually recursive declaration checks
+            if (num_params != ps.size()) {
+                throw parser_error("invalid mutually recursive inductive declaration, "
+                                   "all inductive types must have the same number of arguments",
+                                   params_pos);
+            }
+        }
+        make_implicit(ps); // parameters are implicit for introduction rules
+        // parse introduction rules
+        p.check_token_next(g_assign, "invalid inductive declaration, ':=' expected");
+        buffer<intro_rule> intros;
+        while (p.curr_is_token(g_bar)) {
+            p.next();
+            name intro_id = p.check_id_next("invalid introduction rule, identifier expected");
+            check_atomic(intro_id);
+            name full_intro_id = ns + intro_id;
+            id_to_short_id.insert(full_intro_id, intro_id);
+            p.check_token_next(g_colon, "invalid introduction rule, ':' expected");
+            expr intro_type = p.parse_scoped_expr(ps, lenv);
+            intro_type = p.pi_abstract(ps, intro_type);
+            intros.push_back(intro_rule(full_intro_id, intro_type));
+        }
+        decls.push_back(inductive_decl(full_id, type, to_list(intros.begin(), intros.end())));
+        if (!p.curr_is_token(g_with))
+            break;
+        p.next();
+        first = false;
+    }
+    // Collect (section) locals occurring in inductive_decls, and abstract them
+    // using these additional parameters.
+    name_set used_levels;
+    name_set section_locals;
+    for (inductive_decl const & d : decls) {
+        used_levels    = collect_univ_params(inductive_decl_type(d), used_levels);
+        section_locals = collect_locals(inductive_decl_type(d), section_locals);
+        for (auto const & ir : inductive_decl_intros(d)) {
+            used_levels    = collect_univ_params(intro_rule_type(ir), used_levels);
+            section_locals = collect_locals(intro_rule_type(ir), section_locals);
+        }
+    }
+    update_univ_parameters(ls_buffer, used_levels, p);
+    buffer<parameter> section_params;
+    mk_section_params(section_locals, p, section_params);
+    // First, add section_params to inductive types type.
+    // We don't update the introduction rules in the first pass, because
+    // we will mark all section_params as implicit for them.
+    for (inductive_decl & d : decls) {
+        d = inductive_decl(inductive_decl_name(d),
+                           p.pi_abstract(section_params, inductive_decl_type(d)),
+                           inductive_decl_intros(d));
+    }
+    make_implicit(section_params);
+    // Add section_params to introduction rules type, and also "fix"
+    // occurrences of inductive types.
+    for (inductive_decl & d : decls) {
+        buffer<intro_rule> new_irs;
+        for (auto const & ir : inductive_decl_intros(d)) {
+            expr type = intro_rule_type(ir);
+            type = fix_inductive_occs(type, decls, ls_buffer, section_params);
+            type = p.pi_abstract(section_params, type);
+            new_irs.push_back(intro_rule(intro_rule_name(ir), type));
+        }
+        d = inductive_decl(inductive_decl_name(d),
+                           inductive_decl_type(d),
+                           to_list(new_irs.begin(), new_irs.end()));
+    }
+    num_params += section_params.size();
+    level_param_names ls = to_list(ls_buffer.begin(), ls_buffer.end());
+
+    // Check if introduction rules do not have placeholders
+    for (inductive_decl const & d : decls) {
+        for (auto const & ir : inductive_decl_intros(d)) {
+            if (has_placeholder(intro_rule_type(ir)))
+                throw exception(sstream() << "invalid inductive datatype '" << inductive_decl_name(d) << "', "
+                                << "introduction rule '" << intro_rule_name(ir) << "' has placeholders");
+        }
+    }
+    // "Fix" the inductive type resultant type universe level, if it was not explicitly provided.
+    set_result_universes(decls, ls, num_params, p);
+    env = module::add_inductive(env, ls, num_params, to_list(decls.begin(), decls.end()));
+    // Create aliases/local refs
+    levels section_levels = collect_section_levels(ls, p);
+    for (inductive_decl const & d : decls) {
+        name const & n = inductive_decl_name(d);
+        env = create_alias(env, n, *id_to_short_id.find(n), section_levels, section_params, p);
+        env = create_alias(env, n.append_after("_rec"), id_to_short_id.find(n)->append_after("_rec"), section_levels, section_params, p);
+        for (intro_rule const & ir : inductive_decl_intros(d)) {
+            name const & n = intro_rule_name(ir);
+            env = create_alias(env, n, *id_to_short_id.find(n), section_levels, section_params, p);
+        }
+    }
+    return env;
+}
+void register_inductive_cmd(cmd_table & r) {
+    add_cmd(r, cmd_info("inductive",   "declare an inductive datatype", inductive_cmd));
+}
+}
+
+

src/frontends/lean/inductive_cmd.h

@@ -0,0 +1,11 @@
+/*
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Author: Leonardo de Moura
+*/
+#pragma once
+#include "frontends/lean/cmd_table.h"
+namespace lean {
+void register_inductive_cmd(cmd_table & r);
+}

src/frontends/lean/parser.cpp

@@ -24,6 +24,7 @@ Author: Leonardo de Moura
 #include "library/placeholder.h"
 #include "library/deep_copy.h"
 #include "library/module.h"
+#include "library/scoped_ext.h"
 #include "library/error_handling/error_handling.h"
 #include "frontends/lean/parser.h"
 #include "frontends/lean/notation_cmd.h"
@@ -58,6 +59,13 @@ parser::param_universe_scope::~param_universe_scope() {
     m_p.m_type_use_placeholder = m_old;
 }
 
+parser::placeholder_universe_scope::placeholder_universe_scope(parser & p):m_p(p), m_old(m_p.m_type_use_placeholder) {
+    m_p.m_type_use_placeholder = true;
+}
+parser::placeholder_universe_scope::~placeholder_universe_scope() {
+    m_p.m_type_use_placeholder = m_old;
+}
+
 parser::no_undef_id_error_scope::no_undef_id_error_scope(parser & p):m_p(p), m_old(m_p.m_no_undef_id_error) {
     m_p.m_no_undef_id_error = true;
 }
@@ -817,7 +825,7 @@ expr parser::parse_id() {
         r = mk_choice(new_as.size(), new_as.data());
     }
     if (m_no_undef_id_error)
-        r = mk_constant(id, ls);
+        r = mk_constant(get_namespace(m_env) + id, ls);
     if (!r)
         throw parser_error(sstream() << "unknown identifier '" << id << "'", p);
     return *r;

src/frontends/lean/parser.h

@@ -18,21 +18,13 @@ Author: Leonardo de Moura
 #include "library/io_state.h"
 #include "library/io_state_stream.h"
 #include "library/kernel_bindings.h"
+#include "frontends/lean/parameter.h"
 #include "frontends/lean/scanner.h"
 #include "frontends/lean/local_decls.h"
 #include "frontends/lean/parser_config.h"
 #include "frontends/lean/parser_pos_provider.h"
 
 namespace lean {
-struct parameter {
-    pos_info    m_pos;
-    expr        m_local;
-    binder_info m_bi;
-    parameter(pos_info const & p, expr const & l, binder_info const & bi):
-        m_pos(p), m_local(l), m_bi(bi) {}
-    parameter():m_pos(0, 0) {}
-};
-
 /** \brief Exception used to track parsing erros, it does not leak outside of this class. */
 struct parser_error : public exception {
     pos_info m_pos;
@@ -248,6 +240,8 @@ public:
         The new parameter is automatically added to \c m_local_level_decls.
     */
     struct param_universe_scope { parser & m_p; bool m_old; param_universe_scope(parser &); ~param_universe_scope(); };
+    /** \brief Switch back to <tt>Type.{_}</tt>, see \c param_universe_scope */
+    struct placeholder_universe_scope { parser & m_p; bool m_old; placeholder_universe_scope(parser &); ~placeholder_universe_scope(); };
     expr mk_Type();
 
     /** \brief Use tactic \c t for "synthesizing" the placeholder \c e. */

src/frontends/lean/token_table.cpp

@@ -56,7 +56,7 @@ token_table init_token_table() {
     std::pair<char const *, unsigned> builtin[] =
         {{"fun", 0}, {"Pi", 0}, {"let", 0}, {"in", 0}, {"have", 0}, {"show", 0}, {"by", 0},
          {"from", 0}, {"(", g_max_prec}, {")", 0}, {"{", g_max_prec}, {"}", 0}, {"_", g_max_prec},
-         {"[", g_max_prec}, {"]", 0}, {".{", 0}, {"Type", g_max_prec},
+         {"[", g_max_prec}, {"]", 0}, {".{", 0}, {"Type", g_max_prec}, {"|", 0}, {"with", 0},
          {"...", 0}, {",", 0}, {".", 0}, {":", 0}, {"calc", 0}, {":=", 0}, {"--", 0},
          {"(*", 0}, {"(--", 0}, {"proof", 0}, {"qed", 0},
          {"+", g_plus_prec}, {g_cup, g_cup_prec}, {"->", g_arrow_prec}, {nullptr, 0}};

tests/lean/run/ind0.lean

@@ -0,0 +1,6 @@
+inductive nat : Type :=
+| zero : nat
+| succ : nat → nat
+
+check nat
+check nat_rec.{1}

tests/lean/run/ind1.lean

@@ -0,0 +1,7 @@
+inductive list (A : Type) : Type :=
+| nil  : list A
+| cons : A → list A → list A
+
+check list.{1}
+check cons.{1}
+check list_rec.{1 1}

tests/lean/run/ind2.lean

@@ -0,0 +1,12 @@
+inductive nat : Type :=
+| zero : nat
+| succ : nat → nat
+
+inductive vector (A : Type) : nat → Type :=
+| vnil  : vector A zero
+| vcons : Π {n : nat}, A → vector A n → vector A (succ n)
+
+check vector.{1}
+check vnil.{1}
+check vcons.{1}
+check vector_rec.{1 1}

tests/lean/run/ind3.lean

@@ -0,0 +1,21 @@
+inductive tree (A : Type) : Type :=
+| node : A → forest A → tree A
+with forest {A : Type} : Type :=
+| nil  : forest A
+| cons : tree A → forest A → forest A
+
+check tree.{1}
+check forest.{1}
+check tree_rec.{1 1}
+check forest_rec.{1 1}
+
+print "==============="
+
+inductive group : Type :=
+| mk_group : Π (carrier : Type) (mul : carrier → carrier → carrier) (one : carrier), group
+
+check group.{1}
+check group.{2}
+check group_rec.{1 1}
+
+

tests/lean/run/ind4.lean

@@ -0,0 +1,23 @@
+section
+  parameter A : Type
+  inductive list : Type :=
+  | nil  : list
+  | cons : A → list → list
+end
+
+check list.{1}
+check cons.{1}
+
+section
+  parameter A : Type
+  inductive tree : Type :=
+  | node : A → forest → tree
+  with forest : Type :=
+  | fnil  : forest
+  | fcons : tree → forest → forest
+  check tree
+  check forest
+end
+
+check tree.{1}
+check forest.{1}

tests/lean/run/ind5.lean

@@ -0,0 +1,9 @@
+definition [inline] Bool : Type.{1} := Type.{0}
+
+inductive or (A B : Bool) : Bool :=
+| or_intro_left  : A → or A B
+| or_intro_right : B → or A B
+
+check or
+check or_intro_left
+check or_rec

tests/lean/run/ind6.lean

@@ -0,0 +1,10 @@
+inductive tree.{u} (A : Type.{u}) : Type.{max u 1} :=
+| node : A → forest.{u} A → tree.{u} A
+with forest.{u} {A : Type.{u}} : Type.{max u 1} :=
+| nil  : forest.{u} A
+| cons : tree.{u} A → forest.{u} A → forest.{u} A
+
+check tree.{1}
+check forest.{1}
+check tree_rec.{1 1}
+check forest_rec.{1 1}

tests/lean/run/ind7.lean

@@ -0,0 +1,11 @@
+namespace list
+  inductive list (A : Type) : Type :=
+  | nil  : list A
+  | cons : A → list A → list A
+
+  check list.{1}
+  check cons.{1}
+  check list_rec.{1 1}
+end
+
+check list.list.{1}