diff --git a/src/kernel/arith/arith.h b/src/kernel/arith/arith.h
index 1fc3b7c26..a30159c84 100644
--- a/src/kernel/arith/arith.h
+++ b/src/kernel/arith/arith.h
@@ -13,7 +13,6 @@ Author: Leonardo de Moura
 namespace lean {
 expr mk_int_type();
 #define Int mk_int_type()
-bool is_int_type(expr const & e);
 
 expr mk_int_value(mpz const & v);
 inline expr mk_int_value(int v) { return mk_int_value(mpz(v)); }
@@ -22,35 +21,27 @@ bool is_int_value(expr const & e);
 mpz const & int_value_numeral(expr const & e);
 
 expr mk_int_add_fn();
-bool is_int_add_fn(expr const & e);
 inline expr iAdd(expr const & e1, expr const & e2) { return mk_app(mk_int_add_fn(), e1, e2); }
 
 expr mk_int_sub_fn();
-bool is_int_sub_fn(expr const & e);
 inline expr iSub(expr const & e1, expr const & e2) { return mk_app(mk_int_sub_fn(), e1, e2); }
 
 expr mk_int_mul_fn();
-bool is_int_mul_fn(expr const & e);
 inline expr iMul(expr const & e1, expr const & e2) { return mk_app(mk_int_mul_fn(), e1, e2); }
 
 expr mk_int_div_fn();
-bool is_int_div_fn(expr const & e);
 inline expr iDiv(expr const & e1, expr const & e2) { return mk_app(mk_int_div_fn(), e1, e2); }
 
 expr mk_int_le_fn();
-bool is_int_le_fn(expr const & e);
 inline expr iLe(expr const & e1, expr const & e2) { return mk_app(mk_int_le_fn(), e1, e2); }
 
 expr mk_int_ge_fn();
-bool is_int_ge_fn(expr const & e);
 inline expr iGe(expr const & e1, expr const & e2) { return mk_app(mk_int_ge_fn(), e1, e2); }
 
 expr mk_int_lt_fn();
-bool is_int_lt_fn(expr const & e);
 inline expr iLt(expr const & e1, expr const & e2) { return mk_app(mk_int_lt_fn(), e1, e2); }
 
 expr mk_int_gt_fn();
-bool is_int_gt_fn(expr const & e);
 inline expr iGt(expr const & e1, expr const & e2) { return mk_app(mk_int_gt_fn(), e1, e2); }
 
 inline expr iIf(expr const & c, expr const & t, expr const & e) { return mk_if(Int, c, t, e); }
diff --git a/src/kernel/builtin.h b/src/kernel/builtin.h
index 4667e9a8f..82c5fd0c3 100644
--- a/src/kernel/builtin.h
+++ b/src/kernel/builtin.h
@@ -26,8 +26,6 @@ expr mk_type_u();
 /** \brief Return the Lean Boolean type. */
 expr mk_bool_type();
 #define Bool mk_bool_type()
-/** \brief Return true iff \c e is the Lean Boolean type. */
-bool is_bool_type(expr const & e);
 
 /** \brief Create a Lean Boolean value (true/false) */
 expr mk_bool_value(bool v);
@@ -47,9 +45,6 @@ bool is_false(expr const & e);
 
 /** \brief Return the Lean If-Then-Else operator. It has type: pi (A : Type), bool -> A -> A -> A */
 expr mk_if_fn();
-/** \brief Return true iff \c e is the Lean If-Then-Else operator */
-bool is_if_fn(expr const & e);
-
 /** \brief Return the term (if A c t e) */
 inline expr mk_if(expr const & A, expr const & c, expr const & t, expr const & e) { return mk_app(mk_if_fn(), A, c, t, e); }
 inline expr If(expr const & A, expr const & c, expr const & t, expr const & e) { return mk_if(A, c, t, e); }
@@ -59,17 +54,12 @@ inline expr bIf(expr const & c, expr const & t, expr const & e) { return mk_bool
 
 /** \brief Return the Lean Implies operator */
 expr mk_implies_fn();
-/** \brief Return true iff \c e is the Lean implies operator */
-bool is_implies_fn(expr const & e);
 /** \brief Return the term (e1 => e2) */
 inline expr mk_implies(expr const & e1, expr const & e2) { return mk_app(mk_implies_fn(), e1, e2); }
 inline expr Implies(expr const & e1, expr const & e2) { return mk_implies(e1, e2); }
 
 /** \brief Return the Lean And operator */
 expr mk_and_fn();
-/** \brief Return true iff \c e is the Lean and operator. */
-bool is_and_fn(expr const & e);
-
 /** \brief Return (e1 and e2) */
 inline expr mk_and(expr const & e1, expr const & e2) { return mk_app(mk_and_fn(), e1, e2); }
 inline expr mk_and(unsigned num_args, expr const * args) { return mk_bin_op(mk_and_fn(), True, num_args, args); }
@@ -78,8 +68,6 @@ inline expr And(std::initializer_list<expr> const & l) { return mk_and(l.size(),
 
 /** \brief Return the Lean Or operator */
 expr mk_or_fn();
-bool is_or_fn(expr const & e);
-
 /** \brief Return (e1 Or e2) */
 inline expr mk_or(expr const & e1, expr const & e2) { return mk_app(mk_or_fn(), e1, e2); }
 inline expr mk_or(unsigned num_args, expr const * args) { return mk_bin_op(mk_or_fn(), False, num_args, args); }
@@ -88,68 +76,55 @@ inline expr Or(std::initializer_list<expr> const & l) { return mk_or(l.size(), l
 
 /** \brief Return the Lean not operator */
 expr mk_not_fn();
-bool is_not_fn(expr const & e);
-
 /** \brief Return (Not e) */
 inline expr mk_not(expr const & e) { return mk_app(mk_not_fn(), e); }
 inline expr Not(expr const & e) { return mk_not(e); }
 
 /** \brief Return the Lean forall operator. It has type: <tt>Pi (A : Type), (A -> bool) -> Bool<\tt> */
 expr mk_forall_fn();
-/** \brief Return true iff \c e is the Lean forall operator */
-bool is_forall_fn(expr const & e);
 /** \brief Return the term (Forall A P), where A is a type and P : A -> bool */
 inline expr mk_forall(expr const & A, expr const & P) { return mk_app(mk_forall_fn(), A, P); }
 inline expr Forall(expr const & A, expr const & P) { return mk_forall(A, P); }
 
 /** \brief Return the Lean exists operator. It has type: <tt>Pi (A : Type), (A -> Bool) -> Bool<\tt> */
 expr mk_exists_fn();
-/** \brief Return true iff \c e is the Lean exists operator */
-bool is_exists_fn(expr const & e);
 /** \brief Return the term (exists A P), where A is a type and P : A -> bool */
 inline expr mk_exists(expr const & A, expr const & P) { return mk_app(mk_exists_fn(), A, P); }
 inline expr Exists(expr const & A, expr const & P) { return mk_exists(A, P); }
 
 /** \brief Modus Ponens axiom */
 expr mk_mp_fn();
-bool is_mp_fn(const expr & e);
 /** \brief (Axiom) a : Bool, b : Bool, H1 : a => b, H2 : a |- MP(a, b, H1, H2) : b */
 inline expr MP(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_mp_fn(), a, b, H1, H2); }
 
 /** \brief Discharge axiom */
 expr mk_discharge_fn();
-bool is_discharge_fn(const expr & e);
 /** \brief (Axiom) a : Bool, b : Bool, H : a -> b |- Discharge(a, b, H) : a => b */
 inline expr Discharge(expr const & a, expr const & b, expr const & H) { return mk_app(mk_discharge_fn(), a, b, H); }
 
 /** \brief Reflexivity axiom */
 expr mk_refl_fn();
-bool is_refl_fn(expr const & e);
 /** \brief (Axiom) A : Type u, a : A |- Refl(A, a) : a = a */
 inline expr Refl(expr const & A, expr const & a) { return mk_app(mk_refl_fn(), A, a); }
 #define Trivial Refl(Bool, True)
 
 /** \brief Case analysis axiom */
 expr mk_case_fn();
-bool is_case_fn(expr const & e);
 /** \brief (Axiom) P : Bool -> Bool, H1 : P True, H2 : P False, a : Bool |- Case(P, H1, H2, a) : P a */
 inline expr Case(expr const & P, expr const & H1, expr const & H2, expr const & a) { return mk_app(mk_case_fn(), P, H1, H2, a); }
 
 /** \brief Substitution axiom */
 expr mk_subst_fn();
-bool is_subst_fn(expr const & e);
 /** \brief (Axiom) A : Type u, P : A -> Bool, a b : A, H1 : P a, H2 : a = b |- Subst(A, P, a, b, H1, H2) : P b */
 inline expr Subst(expr const & A, expr const & P, expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app({mk_subst_fn(), A, P, a, b, H1, H2}); }
 
 /** \brief Eta conversion axiom */
 expr mk_eta_fn();
-bool is_eta_fn(expr const & e);
 /** \brief (Axiom) A : Type u, B : A -> Type u, f : (Pi x : A, B x) |- Eta(A, B, f) : ((Fun x : A => f x) = f) */
 inline expr Eta(expr const & A, expr const & B, expr const & f) { return mk_app(mk_eta_fn(), A, B, f); }
 
 /** \brief Implies Anti-symmetry */
 expr mk_imp_antisym_fn();
-bool is_imp_antisym_fn(expr const & e);
 /** \brief (Axiom) a : Bool, b : Bool, H1 : a => b, H2 : b => a |- ImpAntisym(a, b, H1, H2) : a = b */
 inline expr ImpAntisym(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_imp_antisym_fn(), a, b, H1, H2); }
 
@@ -177,8 +152,5 @@ static name Name ## _name = NameObj;                            \
 expr mk_##Name() {                                              \
     static thread_local expr r = mk_constant(Name ## _name);    \
     return r ;                                                  \
-}                                                               \
-bool is_##Name(expr const & e) {                                \
-    return is_constant(e) && const_name(e) == Name ## _name;    \
 }
 }
diff --git a/src/kernel/type_check.cpp b/src/kernel/type_check.cpp
index f402479a8..97baff3f4 100644
--- a/src/kernel/type_check.cpp
+++ b/src/kernel/type_check.cpp
@@ -57,7 +57,7 @@ struct infer_type_fn {
         expr u = normalize(infer_type(t, ctx), m_env, ctx);
         if (is_type(u))
             return ty_level(u);
-        if (is_bool_type(u))
+        if (u == Bool)
             return level();
         std::ostringstream buffer;
         buffer << "type expected, ";
diff --git a/src/tests/kernel/arith.cpp b/src/tests/kernel/arith.cpp
index 55db67431..10981b7da 100644
--- a/src/tests/kernel/arith.cpp
+++ b/src/tests/kernel/arith.cpp
@@ -31,8 +31,6 @@ static void tst2() {
     std::cout << infer_type(mk_int_add_fn(), env) << "\n";
     lean_assert(infer_type(e, env) == Int);
     lean_assert(infer_type(mk_app(mk_int_add_fn(), iVal(10)), env) == (Int >> Int));
-    lean_assert(is_int_add_fn(mk_int_add_fn()));
-    lean_assert(!is_int_add_fn(mk_int_mul_fn()));
     lean_assert(is_int_value(normalize(e, env)));
     expr e2 = Fun("a", Int, iAdd(Const("a"), iAdd(iVal(10), iVal(30))));
     std::cout << e2 << " --> " << normalize(e2, env) << "\n";
@@ -49,8 +47,6 @@ static void tst3() {
     std::cout << infer_type(mk_int_mul_fn(), env) << "\n";
     lean_assert(infer_type(e, env) == Int);
     lean_assert(infer_type(mk_app(mk_int_mul_fn(), iVal(10)), env) == arrow(Int, Int));
-    lean_assert(is_int_mul_fn(mk_int_mul_fn()));
-    lean_assert(!is_int_mul_fn(mk_int_add_fn()));
     lean_assert(is_int_value(normalize(e, env)));
     expr e2 = Fun("a", Int, iMul(Const("a"), iMul(iVal(10), iVal(30))));
     std::cout << e2 << " --> " << normalize(e2, env) << "\n";
@@ -67,8 +63,6 @@ static void tst4() {
     std::cout << infer_type(mk_int_sub_fn(), env) << "\n";
     lean_assert(infer_type(e, env) == Int);
     lean_assert(infer_type(mk_app(mk_int_sub_fn(), iVal(10)), env) == arrow(Int, Int));
-    lean_assert(is_int_sub_fn(mk_int_sub_fn()));
-    lean_assert(!is_int_sub_fn(mk_int_add_fn()));
     lean_assert(is_int_value(normalize(e, env)));
     expr e2 = Fun("a", Int, iSub(Const("a"), iSub(iVal(10), iVal(30))));
     std::cout << e2 << " --> " << normalize(e2, env) << "\n";