diff --git a/src/kernel/instantiate.cpp b/src/kernel/instantiate.cpp
index dd54af0a3..f064d2c8d 100644
--- a/src/kernel/instantiate.cpp
+++ b/src/kernel/instantiate.cpp
@@ -9,7 +9,7 @@ Author: Leonardo de Moura
 #include "replace.h"
 
 namespace lean {
-expr instantiate(unsigned n, expr const * s, expr const & e) {
+expr instantiate_with_closed(unsigned n, expr const * s, expr const & e) {
     lean_assert(std::all_of(s, s+n, closed));
 
     auto f = [=](expr const & e, unsigned offset) -> expr {
@@ -24,7 +24,21 @@ expr instantiate(unsigned n, expr const * s, expr const & e) {
         }
         return e;
     };
-
+    return replace_fn<decltype(f)>(f)(e);
+}
+expr instantiate(unsigned n, expr const * s, expr const & e) {
+    auto f = [=](expr const & e, unsigned offset) -> expr {
+        if (is_var(e)) {
+            unsigned vidx = var_idx(e);
+            if (vidx >= offset) {
+                if (vidx < offset + n)
+                    return lift_free_vars(s[n - (vidx - offset) - 1], offset);
+                else
+                    return var(vidx - n);
+            }
+        }
+        return e;
+    };
     return replace_fn<decltype(f)>(f)(e);
 }
 }
diff --git a/src/kernel/instantiate.h b/src/kernel/instantiate.h
index 98d606bc4..e21a4108f 100644
--- a/src/kernel/instantiate.h
+++ b/src/kernel/instantiate.h
@@ -13,6 +13,13 @@ namespace lean {
 
    \pre s[0], ..., s[n-1] must be closed expressions (i.e., no free variables).
 */
+expr instantiate_with_closed(unsigned n, expr const * s, expr const & e);
+inline expr instantiate_with_closed(expr const & s, expr const & e) { return instantiate_with_closed(1, &s, e); }
+
+/**
+   \brief Replace the free variables with indices 0,...,n-1 with s[n-1],...,s[0] in e.
+*/
 expr instantiate(unsigned n, expr const * s, expr const & e);
 inline expr instantiate(expr const & s, expr const & e) { return instantiate(1, &s, e); }
+
 }
diff --git a/src/kernel/type_check.cpp b/src/kernel/type_check.cpp
index 5e2fbb498..68e082c9f 100644
--- a/src/kernel/type_check.cpp
+++ b/src/kernel/type_check.cpp
@@ -7,6 +7,7 @@ Author: Leonardo de Moura
 #include <sstream>
 #include "type_check.h"
 #include "normalize.h"
+#include "instantiate.h"
 #include "free_vars.h"
 #include "exception.h"
 #include "trace.h"
@@ -83,23 +84,19 @@ class infer_type_fn {
         case expr_kind::Var:      return lookup(ctx, var_idx(e));
         case expr_kind::Type:     return type(ty_level(e) + 1);
         case expr_kind::App: {
-            expr f_t     = infer_pi(arg(e, 0), ctx);
+            expr f       = arg(e, 0);
             unsigned i   = 1;
             unsigned num = num_args(e);
             lean_assert(num >= 2);
             while (true) {
                 expr const & c = arg(e, i);
+                expr f_t       = infer_pi(f, ctx);
                 expr c_t       = infer_type(c, ctx);
                 check_type(e, i, abst_domain(f_t), c_t, ctx);
-                // Remark: if f_t is an arrow, we don't have to call normalize and
-                // lower_free_vars
-                f_t = normalize(abst_body(f_t), ctx, c);
-                f_t = lower_free_vars(f_t, 1);
+                f = instantiate(abst_body(f_t), c);
                 i++;
                 if (i == num)
-                    return f_t;
-                if (!is_pi(f_t))
-                    throw exception("function expected");
+                    return f;
             }
         }
         case expr_kind::Lambda: {
diff --git a/src/tests/kernel/replace.cpp b/src/tests/kernel/replace.cpp
index 7e9d70074..b3f9b845d 100644
--- a/src/tests/kernel/replace.cpp
+++ b/src/tests/kernel/replace.cpp
@@ -30,13 +30,15 @@ static void tst1() {
 
 static void tst2() {
     expr r = lambda("x", type(level()), app(var(0), var(1), var(2)));
-    std::cout << instantiate(constant("a"), r) << std::endl;
-    lean_assert(instantiate(constant("a"), r) == lambda("x", type(level()), app(var(0), constant("a"), var(1))));
-    lean_assert(instantiate(constant("b"), instantiate(constant("a"), r)) ==
+    std::cout << instantiate_with_closed(constant("a"), r) << std::endl;
+    lean_assert(instantiate_with_closed(constant("a"), r) == lambda("x", type(level()), app(var(0), constant("a"), var(1))));
+    lean_assert(instantiate_with_closed(constant("b"), instantiate_with_closed(constant("a"), r)) ==
                 lambda("x", type(level()), app(var(0), constant("a"), constant("b"))));
-    std::cout << instantiate(constant("a"), abst_body(r)) << std::endl;
-    lean_assert(instantiate(constant("a"), abst_body(r)) == app(constant("a"), var(0), var(1)));
-}
+    std::cout << instantiate_with_closed(constant("a"), abst_body(r)) << std::endl;
+    lean_assert(instantiate_with_closed(constant("a"), abst_body(r)) == app(constant("a"), var(0), var(1)));
+    std::cout << instantiate(var(10), r) << std::endl;
+    lean_assert(instantiate(var(10), r) == lambda("x", type(level()), app(var(0), var(11), var(1))));
+ }
 
 int main() {
     continue_on_violation(true);