diff --git a/src/kernel/inductive/inductive.cpp b/src/kernel/inductive/inductive.cpp
index 8be087814..5bee4298a 100644
--- a/src/kernel/inductive/inductive.cpp
+++ b/src/kernel/inductive/inductive.cpp
@@ -47,8 +47,7 @@ struct add_inductive_fn {
         expr             m_major_premise;  // major premise for each inductive decl
         buffer<expr>     m_minor_premises; // minor premise for each introduction rule
     };
-
-    buffer<elim_info> m_elim_info; // for each datatype being declared
+    buffer<elim_info>    m_elim_info; // for each datatype being declared
 
     add_inductive_fn(environment                  env,
                      level_param_names const &    level_params,
@@ -63,13 +62,13 @@ struct add_inductive_fn {
     /** \brief Return the number of inductive datatypes being defined. */
     unsigned get_num_its() const { return m_decls_sz; }
 
-    /** \brief Make sure the latest environment is being used by m_tc */
+    /** \brief Make sure the latest environment is being used by m_tc. */
     void updt_type_checker() {
         type_checker tc(m_env);
         m_tc.swap(tc);
     }
 
-    /** \brief Return a fresh name */
+    /** \brief Return a fresh name. */
     name mk_fresh_name() { return m_ngen.next(); }
 
     /** \brief Create a local constant for the given binding. */
@@ -176,7 +175,7 @@ struct add_inductive_fn {
             std::any_of(m_it_consts.begin(), m_it_consts.end(), [&](expr const & c) { return const_name(e) == const_name(c); });
     }
 
-    /** \brief Return true if \c t does not contain any occurrence of a datatype being declared */
+    /** \brief Return true if \c t does not contain any occurrence of a datatype being declared. */
     bool has_it_occ(expr const & t) {
         return (bool)find(t, [&](expr const & e, unsigned) { return is_it_occ(e); }); // NOLINT
     }
@@ -275,7 +274,7 @@ struct add_inductive_fn {
         }
     }
 
-    /** \brief Add all introduction rules (aka constructors) to environment */
+    /** \brief Add all introduction rules (aka constructors) to environment. */
     void declare_intro_rules() {
         for (auto d : m_decls) {
             for (auto ir : inductive_decl_intros(d))
@@ -285,7 +284,7 @@ struct add_inductive_fn {
     }
 
 
-    /** \brief Initialize m_elim_level */
+    /** \brief Initialize m_elim_level. */
     void mk_elim_level() {
         if (m_env.impredicative() && is_zero(m_it_levels[0]) && (get_num_its() > 1 || length(inductive_decl_intros(head(m_decls))) != 1)) {
             // environment is impredicative, datatype maps to Bool/Prop, we have more than one introduction rule.
@@ -301,7 +300,7 @@ struct add_inductive_fn {
         }
     }
 
-    /** \brief Initialize m_dep_elim flag */
+    /** \brief Initialize m_dep_elim flag. */
     void set_dep_elim() {
         if (m_env.impredicative() && is_zero(m_it_levels[0]))
             m_dep_elim = false;
@@ -324,7 +323,7 @@ struct add_inductive_fn {
         return *r;
     }
 
-    /** \brief Populate m_elim_info */
+    /** \brief Populate m_elim_info. */
     void mk_elim_info() {
         unsigned d_idx = 0;
         // First, populate the fields, m_C, m_indices, m_major_premise
@@ -368,7 +367,7 @@ struct add_inductive_fn {
                     if (i < m_num_params) {
                         t = instantiate(binding_body(t), m_param_consts[i]);
                     } else {
-                        expr l = mk_local(mk_fresh_name(), binding_name(t), binding_domain(t));
+                        expr l = mk_local_for(t);
                         if (!is_rec_argument(binding_domain(t)))
                             b.push_back(l);
                         else
@@ -390,7 +389,7 @@ struct add_inductive_fn {
                     expr u_i_ty = m_tc.whnf(mlocal_type(u_i));
                     buffer<expr> xs;
                     while (is_pi(u_i_ty)) {
-                        expr x = mk_local(mk_fresh_name(), binding_name(u_i_ty), binding_domain(u_i_ty));
+                        expr x = mk_local_for(u_i_ty);
                         xs.push_back(x);
                         u_i_ty = m_tc.whnf(instantiate(binding_body(u_i_ty), x));
                     }
@@ -414,7 +413,28 @@ struct add_inductive_fn {
         }
     }
 
-    /** \brief Declare elimination rule */
+    /** \brief Return the name of the eliminator/recursor for \c d. */
+    name get_elim_name(inductive_decl const & d) { return inductive_decl_name(d).append_after("_rec"); }
+
+    name get_elim_name(unsigned d_idx) { return get_elim_name(get_ith(m_decls, d_idx)); }
+
+    /** \brief Return the level parameter names for the eliminator. */
+    level_param_names get_elim_level_param_names() {
+        if (is_param(m_elim_level))
+            return level_param_names(param_id(m_elim_level), m_level_names);
+        else
+            return m_level_names;
+    }
+
+    /** \brief Return the levels for the eliminator application. */
+    levels get_elim_level_params() {
+        if (is_param(m_elim_level))
+            return levels(m_elim_level, m_levels);
+        else
+            return m_levels;
+    }
+
+    /** \brief Declare elimination rule. */
     void declare_elim_rule(inductive_decl const & d, unsigned d_idx) {
         elim_info const & info = m_elim_info[d_idx];
         expr C_app   = mk_app(info.m_C, info.m_indices);
@@ -438,16 +458,10 @@ struct add_inductive_fn {
             elim_ty = Pi(m_elim_info[i].m_C, elim_ty);
         }
         elim_ty = Pi(m_param_consts, elim_ty);
-        level_param_names ls;
-        if (is_param(m_elim_level))
-            ls = level_param_names(param_id(m_elim_level), m_level_names);
-        else
-            ls = m_level_names;
-        name elim_name = inductive_decl_name(d).append_after("_rec");
-        m_env = m_env.add(check(m_env, mk_var_decl(elim_name, ls, elim_ty)));
+        m_env = m_env.add(check(m_env, mk_var_decl(get_elim_name(d), get_elim_level_param_names(), elim_ty)));
     }
 
-    /** \brief Declare the eliminator/recursor for each datatype */
+    /** \brief Declare the eliminator/recursor for each datatype. */
     void declare_elim_rules() {
         set_dep_elim();
         mk_elim_level();
@@ -457,6 +471,73 @@ struct add_inductive_fn {
             declare_elim_rule(d, i);
             i++;
         }
+        updt_type_checker();
+    }
+
+    /** \brief Store all set formers in \c Cs */
+    void collect_Cs(buffer<expr> & Cs) {
+        for (unsigned i = 0; i < get_num_its(); i++)
+            Cs.push_back(m_elim_info[i].m_C);
+    }
+
+    /** \brief Store all minor premises in \c es. */
+    void collect_minor_premises(buffer<expr> & es) {
+        for (unsigned i = 0; i < get_num_its(); i++)
+            es.append(m_elim_info[i].m_minor_premises);
+    }
+
+    /** \brief Create computional rules RHS. They are used by the normalizer extension. */
+    void mk_comp_rules_rhs() {
+        unsigned d_idx  = 0;
+        unsigned minor_idx = 0;
+        buffer<expr> C; collect_Cs(C);
+        buffer<expr> e; collect_minor_premises(e);
+        levels ls = get_elim_level_params();
+        for (auto d : m_decls) {
+            for (auto ir : inductive_decl_intros(d)) {
+                buffer<expr> b;
+                buffer<expr> u;
+                expr t = intro_rule_type(ir);
+                unsigned i = 0;
+                while (is_pi(t)) {
+                    if (i < m_num_params) {
+                        t = instantiate(binding_body(t), m_param_consts[i]);
+                    } else {
+                        expr l = mk_local_for(t);
+                        if (!is_rec_argument(binding_domain(t)))
+                            b.push_back(l);
+                        else
+                            u.push_back(l);
+                        t = instantiate(binding_body(t), l);
+                    }
+                    i++;
+                }
+                buffer<expr> v;
+                if (m_dep_elim) {
+                    for (unsigned i = 0; i < u.size(); i++) {
+                        expr u_i    = u[i];
+                        expr u_i_ty = m_tc.whnf(mlocal_type(u_i));
+                        buffer<expr> xs;
+                        while (is_pi(u_i_ty)) {
+                            expr x = mk_local_for(u_i_ty);
+                            xs.push_back(x);
+                            u_i_ty = m_tc.whnf(instantiate(binding_body(u_i_ty), x));
+                        }
+                        buffer<expr> it_indices;
+                        unsigned it_idx = get_I_indices(u_i_ty, it_indices);
+                        expr elim_app = mk_constant(get_elim_name(it_idx), ls);
+                        elim_app = mk_app(mk_app(mk_app(mk_app(mk_app(elim_app, m_param_consts), C), e), it_indices), mk_app(u_i, xs));
+                        v.push_back(Fun(xs, elim_app));
+                    }
+                }
+                expr e_app = mk_app(mk_app(mk_app(e[minor_idx], b), u), v);
+                expr comp_rhs   = Fun(m_param_consts, Fun(C, Fun(e, Fun(b, Fun(u, e_app)))));
+                m_tc.check(comp_rhs, get_elim_level_param_names());
+                // TODO(Leo): store computational rule RHS
+                minor_idx++;
+            }
+            d_idx++;
+        }
     }
 
     environment operator()() {
@@ -467,6 +548,7 @@ struct add_inductive_fn {
         check_intro_rules();
         declare_intro_rules();
         declare_elim_rules();
+        mk_comp_rules_rhs();
         return m_env;
     }
 };
diff --git a/tests/lua/ind1.lua b/tests/lua/ind1.lua
index 742a439f0..53dea0666 100644
--- a/tests/lua/ind1.lua
+++ b/tests/lua/ind1.lua
@@ -33,8 +33,7 @@ env = add_inductive(env,
                     "nil", Pi({{A, U_l, true}}, list_l(A)),
                     "cons", Pi({{A, U_l, true}}, mk_arrow(A, list_l(A), list_l(A))))
 env = add_inductive(env,
-                    "vec", {l}, 1,
-                    mk_arrow(U_l, Nat, U_l1),
+                    "vec", {l}, 1, Pi({{A, U_l}, {n, Nat}}, U_l1),
                     "vnil",  Pi({{A, U_l, true}}, vec_l(A, zero)),
                     "vcons", Pi({{A, U_l, true}, {n, Nat, true}}, mk_arrow(A, vec_l(A, n), vec_l(A, succ(n)))))
 
@@ -87,9 +86,9 @@ env = add_inductive(env, {},
 
 local flist_l = Const("flist", {l})
 env = add_inductive(env,
-                    "flist", {l}, 1, mk_arrow(U_l, U_l1),
+                    "flist", {l}, 1, Pi(A, U_l, U_l1),
                     "fnil", Pi({{A, U_l, true}}, flist_l(A)),
-                    "fcons", Pi({{A, U_l, true}}, mk_arrow(A, mk_arrow(Nat, flist_l(A)), flist_l(A))))
+                    "fcons", Pi({{A, U_l, true}}, mk_arrow(mk_arrow(Nat, A), mk_arrow(Nat, Bool, flist_l(A)), flist_l(A))))
 
 local eq_l = Const("eq", {l})
 env = add_inductive(env,
@@ -100,3 +99,6 @@ display_type(env, Const("exists_rec", {v, u}))
 display_type(env, Const("list_rec", {v, u}))
 display_type(env, Const("Even_rec"))
 display_type(env, Const("Odd_rec"))
+display_type(env, Const("and_rec", {v}))
+display_type(env, Const("vec_rec", {v, u}))
+display_type(env, Const("flist_rec", {v, u}))