diff --git a/library/data/examples/vector.lean b/library/data/examples/vector.lean
index b49b163c1..d3d72b5df 100644
--- a/library/data/examples/vector.lean
+++ b/library/data/examples/vector.lean
@@ -261,13 +261,13 @@ namespace vector
     assert e₂ : x = y, begin unfold to_list at h₁, injection h₁, assumption end,
     have   to_list xs = to_list ys, begin unfold to_list at h₁, injection h₁, assumption end,
     assert xs == ys, from heq_of_list_eq this e₁,
-    assert y :: xs == y :: ys, begin clear heq_of_list_eq h₁ h₂, revert xs ys this, induction e₁, intro xs ys h, rewrite [heq.to_eq h] end,
+    assert y :: xs == y :: ys, begin clear heq_of_list_eq h₁ h₂, revert xs ys this, induction e₁, intro xs ys h, rewrite [eq_of_heq h] end,
     show x :: xs == y :: ys, by rewrite e₂; exact this
 
   theorem list_eq_of_heq {n m} {v₁ : vector A n} {v₂ : vector A m} : v₁ == v₂ → n = m → to_list v₁ = to_list v₂ :=
   begin
     intro h₁ h₂, revert v₁ v₂ h₁,
-    subst n, intro v₁ v₂ h₁, rewrite [heq.to_eq h₁]
+    subst n, intro v₁ v₂ h₁, rewrite [eq_of_heq h₁]
   end
 
   theorem of_list_to_list {n : nat} (v : vector A n) : of_list (to_list v) == v :=
diff --git a/library/data/sigma.lean b/library/data/sigma.lean
index a28979563..2a287c6c7 100644
--- a/library/data/sigma.lean
+++ b/library/data/sigma.lean
@@ -18,7 +18,7 @@ namespace sigma
 
   theorem dpair_heq {a : A} {a' : A'} {b : B a} {b' : B' a'}
       (HB : B == B') (Ha : a == a') (Hb : b == b') : ⟨a, b⟩ == ⟨a', b'⟩ :=
-  hcongr_arg4 @mk (heq.type_eq Ha) HB Ha Hb
+  hcongr_arg4 @mk (type_eq_of_heq Ha) HB Ha Hb
 
   protected theorem heq {p : Σa : A, B a} {p' : Σa' : A', B' a'} (HB : B == B') :
     ∀(H₁ : p.1 == p'.1) (H₂ : p.2 == p'.2), p == p' :=
@@ -51,7 +51,7 @@ namespace sigma
   theorem ndtrip_eq {A B : Type} {C : A → B → Type} {a₁ a₂ : A} {b₁ b₂ : B}
       {c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (H₁ : a₁ = a₂) (H₂ : b₁ = b₂)
       (H₃ : cast (congr_arg2 C H₁ H₂) c₁ = c₂) : ⟨a₁, b₁, c₁⟩ = ⟨a₂, b₂, c₂⟩ :=
-  hdcongr_arg3 dtrip H₁ (heq.of_eq H₂) H₃
+  hdcongr_arg3 dtrip H₁ (heq_of_eq H₂) H₃
 
   theorem ndtrip_equal {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
       ∀(H₁ : pr1 p₁ = pr1 p₂) (H₂ : pr2' p₁ = pr2' p₂)
diff --git a/library/data/tuple.lean b/library/data/tuple.lean
index 196ba8ac1..471c20b11 100644
--- a/library/data/tuple.lean
+++ b/library/data/tuple.lean
@@ -91,7 +91,7 @@ namespace tuple
   theorem list_eq_of_heq {n m} {v₁ : tuple A n} {v₂ : tuple A m} : v₁ == v₂ → n = m → to_list v₁ = to_list v₂ :=
   begin
     intro h₁ h₂, revert v₁ v₂ h₁,
-    subst n, intro v₁ v₂ h₁, rewrite [heq.to_eq h₁]
+    subst n, intro v₁ v₂ h₁, rewrite [eq_of_heq h₁]
   end
 
   theorem of_list_to_list {n : nat} (v : tuple A n) : of_list (to_list v) == v :=
diff --git a/library/init/logic.lean b/library/init/logic.lean
index 5476c1559..2a59fb859 100644
--- a/library/init/logic.lean
+++ b/library/init/logic.lean
@@ -168,39 +168,39 @@ end
 
 infixl ` == `:50 := heq
 
-namespace heq
-  universe variable u
-  variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
+section
+universe variable u
+variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
 
-  theorem to_eq (H : a == a') : a = a' :=
-  have H₁ : ∀ (Ht : A = A), eq.rec a Ht = a, from
-    λ Ht, eq.refl a,
-  heq.rec H₁ H (eq.refl A)
+theorem eq_of_heq (H : a == a') : a = a' :=
+have H₁ : ∀ (Ht : A = A), eq.rec a Ht = a, from
+  λ Ht, eq.refl a,
+heq.rec H₁ H (eq.refl A)
 
-  theorem elim {A : Type} {a : A} {P : A → Type} {b : A} (H₁ : a == b)
-    : P a → P b := eq.rec_on (to_eq H₁)
+theorem heq.elim {A : Type} {a : A} {P : A → Type} {b : A} (H₁ : a == b)
+: P a → P b := eq.rec_on (eq_of_heq H₁)
 
-  theorem subst {P : ∀T : Type, T → Prop} : a == b → P A a → P B b :=
-  heq.rec_on
+theorem heq.subst {P : ∀T : Type, T → Prop} : a == b → P A a → P B b :=
+heq.rec_on
 
-  theorem symm (H : a == b) : b == a :=
-  heq.rec_on H (refl a)
+theorem heq.symm (H : a == b) : b == a :=
+heq.rec_on H (heq.refl a)
 
-  theorem of_eq (H : a = a') : a == a' :=
-  eq.subst H (refl a)
+theorem heq_of_eq (H : a = a') : a == a' :=
+eq.subst H (heq.refl a)
 
-  theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
-  subst H₂ H₁
+theorem heq.trans (H₁ : a == b) (H₂ : b == c) : a == c :=
+heq.subst H₂ H₁
 
-  theorem of_heq_of_eq (H₁ : a == b) (H₂ : b = b') : a == b' :=
-  trans H₁ (of_eq H₂)
+theorem heq_of_heq_of_eq (H₁ : a == b) (H₂ : b = b') : a == b' :=
+heq.trans H₁ (heq_of_eq H₂)
 
-  theorem of_eq_of_heq (H₁ : a = a') (H₂ : a' == b) : a == b :=
-  trans (of_eq H₁) H₂
+theorem heq_of_eq_of_heq (H₁ : a = a') (H₂ : a' == b) : a == b :=
+heq.trans (heq_of_eq H₁) H₂
 
-  definition type_eq (H : a == b) : A = B :=
-  heq.rec_on H (eq.refl A)
-end heq
+definition type_eq_of_heq (H : a == b) : A = B :=
+heq.rec_on H (eq.refl A)
+end
 
 open eq.ops
 theorem eq_rec_heq {A : Type} {P : A → Type} {a a' : A} (H : a = a') (p : P a) : H ▹ p == p :=
@@ -213,7 +213,7 @@ theorem heq_of_eq_rec_right {A : Type} {P : A → Type} : ∀ {a a' : A} {p₁ :
 | a a p₁ p₂ (eq.refl a) h := eq.rec_on h !heq.refl
 
 theorem of_heq_true {a : Prop} (H : a == true) : a :=
-of_eq_true (heq.to_eq H)
+of_eq_true (eq_of_heq H)
 
 theorem eq_rec_compose : ∀ {A B C : Type} (p₁ : B = C) (p₂ : A = B) (a : A), p₁ ▹ (p₂ ▹ a : B) = (p₂ ⬝ p₁) ▹ a
 | A A A (eq.refl A) (eq.refl A) a := calc
@@ -223,16 +223,16 @@ theorem eq_rec_compose : ∀ {A B C : Type} (p₁ : B = C) (p₂ : A = B) (a : A
 theorem eq_rec_eq_eq_rec {A₁ A₂ : Type} {p : A₁ = A₂} : ∀ {a₁ : A₁} {a₂ : A₂}, p ▹ a₁ = a₂ → a₁ = p⁻¹ ▹ a₂ :=
 eq.drec_on p (λ a₁ a₂ h, eq.drec_on h rfl)
 
-theorem eq_rec_of_heq_left : ∀ {A₁ A₂ : Type} {a₁ : A₁} {a₂ : A₂} (h : a₁ == a₂), heq.type_eq h ▹ a₁ = a₂
+theorem eq_rec_of_heq_left : ∀ {A₁ A₂ : Type} {a₁ : A₁} {a₂ : A₂} (h : a₁ == a₂), type_eq_of_heq h ▹ a₁ = a₂
 | A A a a (heq.refl a) := rfl
 
-theorem eq_rec_of_heq_right {A₁ A₂ : Type} {a₁ : A₁} {a₂ : A₂} (h : a₁ == a₂) : a₁ = (heq.type_eq h)⁻¹ ▹ a₂ :=
+theorem eq_rec_of_heq_right {A₁ A₂ : Type} {a₁ : A₁} {a₂ : A₂} (h : a₁ == a₂) : a₁ = (type_eq_of_heq h)⁻¹ ▹ a₂ :=
 eq_rec_eq_eq_rec (eq_rec_of_heq_left h)
 
 attribute heq.refl [refl]
 attribute heq.trans [trans]
-attribute heq.of_heq_of_eq [trans]
-attribute heq.of_eq_of_heq [trans]
+attribute heq_of_heq_of_eq [trans]
+attribute heq_of_eq_of_heq [trans]
 attribute heq.symm [symm]
 
 /- and -/
diff --git a/library/init/quot.lean b/library/init/quot.lean
index f353844f8..27ac8b459 100644
--- a/library/init/quot.lean
+++ b/library/init/quot.lean
@@ -78,7 +78,7 @@ namespace quot
   protected definition hrec_on [reducible]
      (q : quot s) (f : Π a, B ⟦a⟧) (c : ∀ (a b : A) (p : a ≈ b), f a == f b) : B q :=
   quot.rec_on q f
-    (λ a b p, heq.to_eq (calc
+    (λ a b p, eq_of_heq (calc
       eq.rec (f a) (sound p) == f a : eq_rec_heq
                          ... == f b : c a b p))
   end
diff --git a/library/logic/cast.lean b/library/logic/cast.lean
index 98f86e7d0..d32d13b78 100644
--- a/library/logic/cast.lean
+++ b/library/logic/cast.lean
@@ -33,7 +33,7 @@ namespace heq
     C b H₁ :=
   heq.rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
 
-  theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
+  theorem to_cast_eq (H : a == b) : cast (type_eq_of_heq H) a = b :=
   drec_on H !cast_eq
 end heq
 
@@ -96,7 +96,7 @@ section
 
   theorem rec_on_pull (H : P = P') (f : Π x, P x) (a : A) :
     eq.rec_on H f a = eq.rec_on (congr_fun H a) (f a) :=
-  heq.to_eq (calc
+  eq_of_heq (calc
     eq.rec_on H f a == f a                   : rec_on_app H f a
       ... == eq.rec_on (congr_fun H a) (f a) : heq.symm (eq_rec_heq (congr_fun H a) (f a)))
 
@@ -107,7 +107,7 @@ end
 -- function extensionality wrt heterogeneous equality
 theorem hfunext {A : Type} {B : A → Type} {B' : A → Type} {f : Π x, B x} {g : Π x, B' x}
                 (H : ∀ a, f a == g a) : f == g :=
-cast_to_heq (funext (λ a, heq.to_eq (heq.trans (cast_app (funext (λ x, heq.type_eq (H x))) f a) (H a))))
+cast_to_heq (funext (λ a, eq_of_heq (heq.trans (cast_app (funext (λ x, type_eq_of_heq (H x))) f a) (H a))))
 
 section
   variables {A : Type} {B : A → Type} {C : Πa, B a → Type} {D : Πa b, C a b → Type}
@@ -123,34 +123,34 @@ section
 
   theorem dcongr_arg2 (f : Πa, B a → F) (Ha : a = a') (Hb : eq.rec_on Ha b = b')
       : f a b = f a' b' :=
-  heq.to_eq (hcongr_arg2 f Ha (eq_rec_to_heq Hb))
+  eq_of_heq (hcongr_arg2 f Ha (eq_rec_to_heq Hb))
 
   theorem dcongr_arg3 (f : Πa b, C a b → F) (Ha : a = a') (Hb : eq.rec_on Ha b = b')
       (Hc : cast (dcongr_arg2 C Ha Hb) c = c') : f a b c = f a' b' c' :=
-  heq.to_eq (hcongr_arg3 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc))
+  eq_of_heq (hcongr_arg3 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc))
 
   theorem dcongr_arg4 (f : Πa b c, D a b c → F) (Ha : a = a') (Hb : eq.rec_on Ha b = b')
       (Hc : cast (dcongr_arg2 C Ha Hb) c = c')
       (Hd : cast (dcongr_arg3 D Ha Hb Hc) d = d') : f a b c d = f a' b' c' d' :=
-  heq.to_eq (hcongr_arg4 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
+  eq_of_heq (hcongr_arg4 f Ha (eq_rec_to_heq Hb) (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
 
   -- mixed versions (we want them for example if C a' b' is a subsingleton, like a proposition.
   -- Then proving eq is easier than proving heq)
   theorem hdcongr_arg3 (f : Πa b, C a b → F) (Ha : a = a') (Hb : b == b')
-      (Hc : cast (heq.to_eq (hcongr_arg2 C Ha Hb)) c = c')
+      (Hc : cast (eq_of_heq (hcongr_arg2 C Ha Hb)) c = c')
         : f a b c = f a' b' c' :=
-  heq.to_eq (hcongr_arg3 f Ha Hb (eq_rec_to_heq Hc))
+  eq_of_heq (hcongr_arg3 f Ha Hb (eq_rec_to_heq Hc))
 
   theorem hhdcongr_arg4 (f : Πa b c, D a b c → F) (Ha : a = a') (Hb : b == b')
       (Hc : c == c')
       (Hd : cast (dcongr_arg3 D Ha (!eq.rec_on_irrel_arg ⬝ heq.to_cast_eq Hb)
                                    (!eq.rec_on_irrel_arg ⬝ heq.to_cast_eq Hc)) d = d')
         : f a b c d = f a' b' c' d' :=
-  heq.to_eq (hcongr_arg4 f Ha Hb Hc (eq_rec_to_heq Hd))
+  eq_of_heq (hcongr_arg4 f Ha Hb Hc (eq_rec_to_heq Hd))
 
   theorem hddcongr_arg4 (f : Πa b c, D a b c → F) (Ha : a = a') (Hb : b == b')
-      (Hc : cast (heq.to_eq (hcongr_arg2 C Ha Hb)) c = c')
+      (Hc : cast (eq_of_heq (hcongr_arg2 C Ha Hb)) c = c')
       (Hd : cast (hdcongr_arg3 D Ha Hb Hc) d = d')
         : f a b c d = f a' b' c' d' :=
-  heq.to_eq (hcongr_arg4 f Ha Hb (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
+  eq_of_heq (hcongr_arg4 f Ha Hb (eq_rec_to_heq Hc) (eq_rec_to_heq Hd))
 end
diff --git a/src/library/constants.cpp b/src/library/constants.cpp
index 1377abade..61c3c430c 100644
--- a/src/library/constants.cpp
+++ b/src/library/constants.cpp
@@ -44,6 +44,7 @@ name const * g_eq_refl = nullptr;
 name const * g_eq_subst = nullptr;
 name const * g_eq_symm = nullptr;
 name const * g_eq_trans = nullptr;
+name const * g_eq_of_heq = nullptr;
 name const * g_exists_elim = nullptr;
 name const * g_false = nullptr;
 name const * g_false_of_true_iff_false = nullptr;
@@ -61,7 +62,9 @@ name const * g_has_zero = nullptr;
 name const * g_has_zero_zero = nullptr;
 name const * g_heq = nullptr;
 name const * g_heq_refl = nullptr;
-name const * g_heq_to_eq = nullptr;
+name const * g_heq_symm = nullptr;
+name const * g_heq_trans = nullptr;
+name const * g_heq_of_eq = nullptr;
 name const * g_iff = nullptr;
 name const * g_iff_elim_left = nullptr;
 name const * g_iff_elim_right = nullptr;
@@ -304,6 +307,7 @@ void initialize_constants() {
     g_eq_subst = new name{"eq", "subst"};
     g_eq_symm = new name{"eq", "symm"};
     g_eq_trans = new name{"eq", "trans"};
+    g_eq_of_heq = new name{"eq_of_heq"};
     g_exists_elim = new name{"exists", "elim"};
     g_false = new name{"false"};
     g_false_of_true_iff_false = new name{"false_of_true_iff_false"};
@@ -321,7 +325,9 @@ void initialize_constants() {
     g_has_zero_zero = new name{"has_zero", "zero"};
     g_heq = new name{"heq"};
     g_heq_refl = new name{"heq", "refl"};
-    g_heq_to_eq = new name{"heq", "to_eq"};
+    g_heq_symm = new name{"heq", "symm"};
+    g_heq_trans = new name{"heq", "trans"};
+    g_heq_of_eq = new name{"heq_of_eq"};
     g_iff = new name{"iff"};
     g_iff_elim_left = new name{"iff", "elim_left"};
     g_iff_elim_right = new name{"iff", "elim_right"};
@@ -565,6 +571,7 @@ void finalize_constants() {
     delete g_eq_subst;
     delete g_eq_symm;
     delete g_eq_trans;
+    delete g_eq_of_heq;
     delete g_exists_elim;
     delete g_false;
     delete g_false_of_true_iff_false;
@@ -582,7 +589,9 @@ void finalize_constants() {
     delete g_has_zero_zero;
     delete g_heq;
     delete g_heq_refl;
-    delete g_heq_to_eq;
+    delete g_heq_symm;
+    delete g_heq_trans;
+    delete g_heq_of_eq;
     delete g_iff;
     delete g_iff_elim_left;
     delete g_iff_elim_right;
@@ -825,6 +834,7 @@ name const & get_eq_refl_name() { return *g_eq_refl; }
 name const & get_eq_subst_name() { return *g_eq_subst; }
 name const & get_eq_symm_name() { return *g_eq_symm; }
 name const & get_eq_trans_name() { return *g_eq_trans; }
+name const & get_eq_of_heq_name() { return *g_eq_of_heq; }
 name const & get_exists_elim_name() { return *g_exists_elim; }
 name const & get_false_name() { return *g_false; }
 name const & get_false_of_true_iff_false_name() { return *g_false_of_true_iff_false; }
@@ -842,7 +852,9 @@ name const & get_has_zero_name() { return *g_has_zero; }
 name const & get_has_zero_zero_name() { return *g_has_zero_zero; }
 name const & get_heq_name() { return *g_heq; }
 name const & get_heq_refl_name() { return *g_heq_refl; }
-name const & get_heq_to_eq_name() { return *g_heq_to_eq; }
+name const & get_heq_symm_name() { return *g_heq_symm; }
+name const & get_heq_trans_name() { return *g_heq_trans; }
+name const & get_heq_of_eq_name() { return *g_heq_of_eq; }
 name const & get_iff_name() { return *g_iff; }
 name const & get_iff_elim_left_name() { return *g_iff_elim_left; }
 name const & get_iff_elim_right_name() { return *g_iff_elim_right; }
diff --git a/src/library/constants.h b/src/library/constants.h
index f201f4306..fb64c1f94 100644
--- a/src/library/constants.h
+++ b/src/library/constants.h
@@ -46,6 +46,7 @@ name const & get_eq_refl_name();
 name const & get_eq_subst_name();
 name const & get_eq_symm_name();
 name const & get_eq_trans_name();
+name const & get_eq_of_heq_name();
 name const & get_exists_elim_name();
 name const & get_false_name();
 name const & get_false_of_true_iff_false_name();
@@ -63,7 +64,9 @@ name const & get_has_zero_name();
 name const & get_has_zero_zero_name();
 name const & get_heq_name();
 name const & get_heq_refl_name();
-name const & get_heq_to_eq_name();
+name const & get_heq_symm_name();
+name const & get_heq_trans_name();
+name const & get_heq_of_eq_name();
 name const & get_iff_name();
 name const & get_iff_elim_left_name();
 name const & get_iff_elim_right_name();
diff --git a/src/library/constants.txt b/src/library/constants.txt
index 411a73e6d..93cfbd659 100644
--- a/src/library/constants.txt
+++ b/src/library/constants.txt
@@ -39,6 +39,7 @@ eq.refl
 eq.subst
 eq.symm
 eq.trans
+eq_of_heq
 exists.elim
 false
 false_of_true_iff_false
@@ -56,7 +57,9 @@ has_zero
 has_zero.zero
 heq
 heq.refl
-heq.to_eq
+heq.symm
+heq.trans
+heq_of_eq
 iff
 iff.elim_left
 iff.elim_right
diff --git a/src/library/tactic/injection_tactic.cpp b/src/library/tactic/injection_tactic.cpp
index 763040003..a7e7c5c00 100644
--- a/src/library/tactic/injection_tactic.cpp
+++ b/src/library/tactic/injection_tactic.cpp
@@ -123,7 +123,7 @@ tactic intros_num_tactic(unsigned num, list<name> _ns) {
                 goal new_goal(new_meta, new_type);
                 expr H         = mk_local(ngen.next(), binding_domain(t));
                 levels heq_lvl = const_levels(get_app_fn(Htype));
-                expr arg       = mk_app(mk_constant(get_heq_to_eq_name(), heq_lvl), A, lhs, rhs, H);
+                expr arg       = mk_app(mk_constant(get_eq_of_heq_name(), heq_lvl), A, lhs, rhs, H);
                 expr V         = Fun(H, mk_app(new_meta, arg));
                 substitution new_subst = s.get_subst();
                 assign(new_subst, g, V);
diff --git a/src/library/tactic/inversion_tactic.cpp b/src/library/tactic/inversion_tactic.cpp
index d607de435..def28e2bb 100644
--- a/src/library/tactic/inversion_tactic.cpp
+++ b/src/library/tactic/inversion_tactic.cpp
@@ -605,7 +605,7 @@ class inversion_tac {
             goal new_g(new_meta, new_type);
             hyps.pop_back();
             expr H        = mk_local(m_ngen.next(), g.get_unused_name(binding_name(type)), binding_domain(type), binder_info());
-            expr to_eq    = mk_app(mk_constant(get_heq_to_eq_name(), const_levels(heq_fn)), args[0], args[1], args[3], H);
+            expr to_eq    = mk_app(mk_constant(get_eq_of_heq_name(), const_levels(heq_fn)), args[0], args[1], args[3], H);
             expr val      = Fun(H, mk_app(mk_app(new_mvar, hyps), to_eq));
             assign(g, val);
             return new_g;
diff --git a/tests/lean/run/cases_bug.lean b/tests/lean/run/cases_bug.lean
index 98ab7452a..ddacd3fb8 100644
--- a/tests/lean/run/cases_bug.lean
+++ b/tests/lean/run/cases_bug.lean
@@ -1,4 +1,4 @@
 import logic.cast
 
 theorem cast_heq : ∀ {A B : Type} (H : A = B) (a : A), cast H a == a
-| A A (eq.refl A) a := heq.of_eq (cast_refl a)
+| A A (eq.refl A) a := heq_of_eq (cast_refl a)
diff --git a/tests/lean/run/pquot.lean b/tests/lean/run/pquot.lean
index 708327380..282d67a64 100644
--- a/tests/lean/run/pquot.lean
+++ b/tests/lean/run/pquot.lean
@@ -43,7 +43,7 @@ pquot.rec f
     have aux₁ : f a == f b, from sound a b H,
     have aux₂ : pquot.abs R a = pquot.abs R b, from pquot.eq R H,
     have aux₃ : ∀ (c₁ c₂ : C (pquot.abs R a)) (e : c₁ == c₂), P (pquot.abs R a) c₁ → P (pquot.abs R a) c₂, from
-      λ c₁ c₂ e H, eq.rec_on (heq.to_eq e) H,
+      λ c₁ c₂ e H, eq.rec_on (eq_of_heq e) H,
     have aux₄ : ∀ (c₁ : C (pquot.abs R a)) (c₂ : C (pquot.abs R b)) (e : c₁ == c₂), P (pquot.abs R a) c₁ → P (pquot.abs R b) c₂, from
       eq.rec_on aux₂ aux₃,
     show P (pquot.abs R b) (f b), from
@@ -55,7 +55,7 @@ definition pquot.lift {A : Type} {R : A → A → Prop} {B : Type}
                       (sound : ∀ (a b : A), R a b → f a = f b)
                       (q : pquot R)
                       : B :=
-pquot.rec_on q f (λ (a b : A) (H : R a b), heq.of_eq (sound a b H))
+pquot.rec_on q f (λ (a b : A) (H : R a b), heq_of_eq (sound a b H))
 
 theorem pquot.induction_on {A : Type} {R : A → A → Prop} {P : pquot R → Prop}
                            (q : pquot R)
diff --git a/tests/lean/run/vector.lean b/tests/lean/run/vector.lean
index 6c270907c..5085b635d 100644
--- a/tests/lean/run/vector.lean
+++ b/tests/lean/run/vector.lean
@@ -19,7 +19,7 @@ namespace vector
 
   theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ = v₂ :=
   begin
-     intro h, apply heq.to_eq, apply (no_confusion h), intros, eassumption,
+     intro h, apply eq_of_heq, apply (no_confusion h), intros, eassumption,
   end
 
   set_option pp.universes true
diff --git a/tests/lean/run/vector_subterm_pred.lean b/tests/lean/run/vector_subterm_pred.lean
index 869047e7a..8e9e75d9c 100644
--- a/tests/lean/run/vector_subterm_pred.lean
+++ b/tests/lean/run/vector_subterm_pred.lean
@@ -34,8 +34,8 @@ well_founded.intro (λ (bv : vec A),
                 have gen₂ : ∀ (m : nat) (Heq₁ : n₂ = m) (v : vector A m) (ih : acc (direct_subterm A) (to_vec v))
                               (Heq₂ : @cons A a₂ n₂ v₂ == @cons A a₁ m v), acc (direct_subterm A) (to_vec v₂), from
                   λ m Heq₁, eq.rec_on Heq₁ (λ (v : vector A n₂) (ih : acc (direct_subterm A) (to_vec v)) (Heq₂ : @cons A a₂ n₂ v₂ == @cons A a₁ n₂ v),
-                    vector.no_confusion (heq.to_eq Heq₂) (λ (e₄ : a₂ = a₁) (e₅ : n₂ = n₂) (e₆ : v₂ == v),
-                      eq.rec_on (heq.to_eq (heq.symm e₆)) ih)),
+                    vector.no_confusion (eq_of_heq Heq₂) (λ (e₄ : a₂ = a₁) (e₅ : n₂ = n₂) (e₆ : v₂ == v),
+                      eq.rec_on (eq_of_heq (heq.symm e₆)) ih)),
                 gen₂ n₁ e₃ v₁ ih e₂))),
             gen (to_vec (cons a₁ v₁)) lt₁ rfl))))
 
diff --git a/tests/lean/unifier_bug.lean b/tests/lean/unifier_bug.lean
index af6f1136d..cd28be983 100644
--- a/tests/lean/unifier_bug.lean
+++ b/tests/lean/unifier_bug.lean
@@ -1,7 +1,7 @@
 import logic
 
 theorem test {A B : Type} {a : A} {b : B} (H : a == b) :
-           eq.rec_on (heq.type_eq H) a = b
+           eq.rec_on (type_eq_of_heq H) a = b
 :=
 -- Remark the error message should not occur in the token theorem
 heq.rec_on H rfl
diff --git a/tests/lean/unifier_bug.lean.expected.out b/tests/lean/unifier_bug.lean.expected.out
index 58fd36010..7ffcedcb5 100644
--- a/tests/lean/unifier_bug.lean.expected.out
+++ b/tests/lean/unifier_bug.lean.expected.out
@@ -3,6 +3,6 @@ unifier_bug.lean:7:0: error: type mismatch at application
 term
   rfl
 has type
-  eq.rec_on (heq.type_eq H) a = eq.rec_on (heq.type_eq H) a
+  eq.rec_on (type_eq_of_heq H) a = eq.rec_on (type_eq_of_heq H) a
 but is expected to have type
-  eq.rec_on (heq.type_eq H) a = b
+  eq.rec_on (type_eq_of_heq H) a = b