diff --git a/doc/lua/univ_poly.lua b/doc/lua/univ_poly.lua
new file mode 100644
index 000000000..f2b68a3c0
--- /dev/null
+++ b/doc/lua/univ_poly.lua
@@ -0,0 +1,125 @@
+-- Given a term t that does not contain global universe levels,
+-- This procedure produces a new environment based on env, but
+-- without using universe polymorphism. The optional argument
+-- def_level is used to instantiate global universe levels used
+-- in the definitions t depends on.
+function elim_univ_poly(env, t, def_level)
+   if not def_level then
+      -- If def_level is not provided, set it to universe 1.
+      def_level = level()+1
+   end
+
+   -- Create the new environment
+   local new_env = environment()
+   -- Already processed definitions
+   local map     = rb_map()
+
+   -- TODO(Leo): convert inductive datatypes.
+
+   -- Return a new (based on n) that is not used in new_env
+   function mk_name(n)
+      local i = 1
+      local r = n
+      while new_env:find(r) do
+         r = name(n, i)
+         i = i + 1
+      end
+      return r
+   end
+
+   -- Instantiate the declaration decl from env using the levels ls,
+   -- and add it to new_env. Return the name of this instance in new_env.
+   function process_declaration(decl, ls)
+      local new_name = mk_name(decl:name())
+      local ps       = decl:univ_params()
+      local new_type = process_expr(decl:type(), ps, ls)
+      local new_decl
+      if decl:is_axiom() then
+         new_decl = mk_axiom(new_name, new_type)
+      elseif decl:is_var_decl() then
+         new_decl = mk_var_decl(new_name, new_type)
+      else
+         local new_value = process_expr(decl:type(), ps, ls)
+         if decl:is_theorem() then
+            new_decl = mk_theorem(new_name, new_type, new_value)
+         else
+            -- TODO(Leo): expose update_declaration functions in the
+            -- Lua API.
+            local attrs = {opaque=decl:opaque(),
+                           weight=decl:weight(),
+                           module_idx=decl:module_idx(),
+                           use_conv_opt=decl:use_conv_opt()}
+            new_decl = mk_definition(new_name, new_type, new_value, attrs)
+         end
+      end
+      new_env = add_decl(new_env, new_decl)
+      return new_name
+   end
+
+   -- Convert a constant from env into a constant for new_env
+   function process_constant(e)
+      if e:is_constant() then
+         local n, ls = e:data()
+         local new_e = map:find(e)
+         if new_e then
+            -- constant was already mapped to new_env
+            return new_e
+         else
+            local decl  = env:find(n)
+            if not decl then
+               error("undefined constant '" .. n .. "'")
+            end
+            local new_name = process_declaration(decl, ls)
+            local new_e    = mk_constant(new_name)
+            map:insert(e, new_e)
+            return new_e
+         end
+      end
+   end
+
+   -- Convert the expression e from env to new_env.
+   -- The universe level parameters ps are instantiated with
+   -- the explicit universe levels ls
+   function process_expr(e, ps, ls)
+      local new_e = e:instantiate_levels(ps, ls)
+      -- replace all constants occurring in new_env with the
+      -- corresponding ones from env
+      return new_e:replace(process_constant)
+   end
+
+   -- Convert all constants occurring in t, and return the new environment.
+   local new_t = process_expr(t, {}, {})
+   return new_env, new_t
+end
+
+function display_env(env)
+   env:for_each(function(d)
+                   print(tostring(d:name()) .. " : " .. tostring(d:type()))
+                end)
+end
+
+function example1()
+   local env = environment()
+   local l   = mk_param_univ("l")
+   local v   = mk_param_univ("v")
+   local A   = Local("A", mk_sort(l))
+   local a   = Local("a", A)
+   env = add_decl(env, mk_var_decl("eq", {l}, Pi(A, mk_arrow(A, A, Bool))))
+   env = add_decl(env, mk_var_decl("group", {l}, mk_sort(l+1)))
+   env = add_decl(env, mk_var_decl("prod", {l, v}, mk_arrow(mk_sort(l), mk_sort(v), mk_sort(max_univ(l, v)))))
+   local eq2 = Const("eq", {2})
+   local group1 = Const("group", {1})
+   local group2 = Const("group", {2})
+   local prod1  = Const("prod", {2, 3})(group1, group2)
+   env:type_check(prod1) -- Type check prod1
+   local prod2  = Const("prod", {3, 0})(prod1, eq2(Type, Bool, Bool))
+   env:type_check(prod2) -- Type check prod1
+
+   local new_env, new_prod2 = elim_univ_poly(env, prod2)
+   print("converted term: " .. tostring(new_prod2))
+   print("converted environment: ")
+   display_env(new_env)
+   new_env:type_check(new_prod2)
+end
+
+example1()