diff --git a/hott/algebra/category/functor.hlean b/hott/algebra/category/functor.hlean
index 8546dcc07..ad9696a09 100644
--- a/hott/algebra/category/functor.hlean
+++ b/hott/algebra/category/functor.hlean
@@ -39,7 +39,7 @@ namespace functor
       G (F (g ∘ f)) = G (F g ∘ F f)     : by rewrite respect_comp
                 ... = G (F g) ∘ G (F f) : by rewrite respect_comp)
 
-  infixr `∘f`:60 := compose
+  infixr `∘f`:60 := functor.compose
 
   protected definition id [reducible] {C : Precategory} : functor C C :=
   mk (λa, a) (λ a b f, f) (λ a, idp) (λ a b c f g, idp)
@@ -88,7 +88,7 @@ namespace functor
       apply (respect_id F) ),
   end
 
-  attribute preserve_iso [instance]
+  attribute functor.preserve_iso [instance]
 
   protected definition respect_inv (F : C ⇒ D) {a b : C} (f : hom a b)
     [H : is_iso f] [H' : is_iso (F f)] :
@@ -107,10 +107,10 @@ namespace functor
       H ∘f (G ∘f F) = (H ∘f G) ∘f F :=
   !functor_mk_eq_constant (λa b f, idp)
 
-  protected definition id_left  (F : C ⇒ D) : id ∘f F = F :=
+  protected definition id_left  (F : C ⇒ D) : functor.id ∘f F = F :=
   functor.rec_on F (λF1 F2 F3 F4, !functor_mk_eq_constant (λa b f, idp))
 
-  protected definition id_right (F : C ⇒ D) : F ∘f id = F :=
+  protected definition id_right (F : C ⇒ D) : F ∘f functor.id = F :=
   functor.rec_on F (λF1 F2 F3 F4, !functor_mk_eq_constant (λa b f, idp))
 
   protected definition comp_id_eq_id_comp (F : C ⇒ D) : F ∘f functor.id = functor.id ∘f F :=
@@ -182,7 +182,7 @@ namespace functor
       : ap010 to_fun_ob (apd01111 functor.mk pF pH pid pcomp) c = ap10 pF c :=
   by cases pF; cases pH; cases pid; cases pcomp; apply idp
 
-definition ap010_functor_eq {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ ∼ to_fun_ob F₂)
+  definition ap010_functor_eq {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ ∼ to_fun_ob F₂)
     (q : (λ(a b : C) (f : hom a b), hom_of_eq (p b) ∘ F₁ f ∘ inv_of_eq (p a)) ∼3 to_fun_hom F₂) (c : C) :
     ap010 to_fun_ob (functor_eq p q) c = p c :=
   begin
@@ -198,7 +198,7 @@ definition ap010_functor_eq {F₁ F₂ : C ⇒ D} (p : to_fun_ob F₁ ∼ to_fun
   !ap010_functor_eq
 
   definition ap010_assoc (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B) (a : A) :
-    ap010 to_fun_ob (assoc H G F) a = idp :=
+    ap010 to_fun_ob (functor.assoc H G F) a = idp :=
   by apply ap010_functor_mk_eq_constant
 
   definition compose_pentagon (K : D ⇒ E) (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B) :
diff --git a/hott/algebra/category/nat_trans.hlean b/hott/algebra/category/nat_trans.hlean
index 4416968b9..5c27b9aef 100644
--- a/hott/algebra/category/nat_trans.hlean
+++ b/hott/algebra/category/nat_trans.hlean
@@ -30,13 +30,13 @@ namespace nat_trans
                       ... = η b ∘ (θ b ∘ F f) : by rewrite naturality
                       ... = (η b ∘ θ b) ∘ F f : by rewrite assoc)
 
-  infixr `∘n`:60 := compose
+  infixr `∘n`:60 := nat_trans.compose
 
   protected definition id [reducible] {C D : Precategory} {F : functor C D} : nat_trans F F :=
   mk (λa, id) (λa b f, !id_right ⬝ !id_left⁻¹)
 
   protected definition ID [reducible] {C D : Precategory} (F : functor C D) : nat_trans F F :=
-  (@id C D F)
+  (@nat_trans.id C D F)
 
   definition nat_trans_mk_eq {η₁ η₂ : Π (a : C), hom (F a) (G a)}
     (nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
@@ -52,10 +52,10 @@ namespace nat_trans
       η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
   nat_trans_eq (λa, !assoc)
 
-  protected definition id_left (η : F ⟹ G) : id ∘n η = η :=
+  protected definition id_left (η : F ⟹ G) : nat_trans.id ∘n η = η :=
   nat_trans_eq (λa, !id_left)
 
-  protected definition id_right (η : F ⟹ G) : η ∘n id = η :=
+  protected definition id_right (η : F ⟹ G) : η ∘n nat_trans.id = η :=
   nat_trans_eq (λa, !id_right)
 
   protected definition sigma_char (F G : C ⇒ D) :
@@ -78,7 +78,7 @@ namespace nat_trans
   end
 
   definition is_hset_nat_trans [instance] : is_hset (F ⟹ G) :=
-  by apply is_trunc_is_equiv_closed; apply (equiv.to_is_equiv !sigma_char)
+  by apply is_trunc_is_equiv_closed; apply (equiv.to_is_equiv !nat_trans.sigma_char)
 
   definition nat_trans_functor_compose [reducible] (η : G ⟹ H) (F : E ⇒ C) : G ∘f F ⟹ H ∘f F :=
   nat_trans.mk
diff --git a/hott/hit/circle.hlean b/hott/hit/circle.hlean
index 1a1bcce88..cb410d007 100644
--- a/hott/hit/circle.hlean
+++ b/hott/hit/circle.hlean
@@ -99,7 +99,7 @@ namespace circle
 
   protected definition rec_on [reducible] {P : circle → Type} (x : circle) (Pbase : P base)
     (Ploop : loop ▸ Pbase = Pbase) : P x :=
-  rec Pbase Ploop x
+  circle.rec Pbase Ploop x
 
   theorem rec_loop_helper {A : Type} (P : A → Type)
     {x y : A} {p : x = y} {u : P x} {v : P y} (q : u = p⁻¹ ▸ v) :
@@ -110,9 +110,9 @@ namespace circle
   eq.rec_on p idp
 
   theorem rec_loop {P : circle → Type} (Pbase : P base) (Ploop : loop ▸ Pbase = Pbase) :
-    apd (rec Pbase Ploop) loop = Ploop :=
+    apd (circle.rec Pbase Ploop) loop = Ploop :=
   begin
-    rewrite [↑loop,apd_con,↑rec,↑rec2_on,↑base,rec2_seg1,apd_inv,rec2_seg2,↑ap], --con_idp should work here
+    rewrite [↑loop,apd_con,↑circle.rec,↑circle.rec2_on,↑base,rec2_seg1,apd_inv,rec2_seg2,↑ap], --con_idp should work here
     apply concat, apply (ap (λx, x ⬝ _)), apply con_idp, esimp,
     rewrite [rec_loop_helper,inv_con_inv_left],
     apply con_inv_cancel_left
@@ -120,33 +120,33 @@ namespace circle
 
   protected definition elim {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
     (x : circle) : P :=
-  rec Pbase (tr_constant loop Pbase ⬝ Ploop) x
+  circle.rec Pbase (tr_constant loop Pbase ⬝ Ploop) x
 
   protected definition elim_on [reducible] {P : Type} (x : circle) (Pbase : P)
     (Ploop : Pbase = Pbase) : P :=
-  elim Pbase Ploop x
+  circle.elim Pbase Ploop x
 
   theorem elim_loop {P : Type} (Pbase : P) (Ploop : Pbase = Pbase) :
-    ap (elim Pbase Ploop) loop = Ploop :=
+    ap (circle.elim Pbase Ploop) loop = Ploop :=
   begin
-    apply (@cancel_left _ _ _ _ (tr_constant loop (elim Pbase Ploop base))),
-    rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_loop],
+    apply (@cancel_left _ _ _ _ (tr_constant loop (circle.elim Pbase Ploop base))),
+    rewrite [-apd_eq_tr_constant_con_ap,↑circle.elim,rec_loop],
   end
 
   protected definition elim_type (Pbase : Type) (Ploop : Pbase ≃ Pbase)
     (x : circle) : Type :=
-  elim Pbase (ua Ploop) x
+  circle.elim Pbase (ua Ploop) x
 
   protected definition elim_type_on [reducible] (x : circle) (Pbase : Type)
     (Ploop : Pbase ≃ Pbase) : Type :=
-  elim_type Pbase Ploop x
+  circle.elim_type Pbase Ploop x
 
   theorem elim_type_loop (Pbase : Type) (Ploop : Pbase ≃ Pbase) :
-    transport (elim_type Pbase Ploop) loop = Ploop :=
-  by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_loop];apply cast_ua_fn
+    transport (circle.elim_type Pbase Ploop) loop = Ploop :=
+  by rewrite [tr_eq_cast_ap_fn,↑circle.elim_type,circle.elim_loop];apply cast_ua_fn
 
   theorem elim_type_loop_inv (Pbase : Type) (Ploop : Pbase ≃ Pbase) :
-    transport (elim_type Pbase Ploop) loop⁻¹ = to_inv Ploop :=
+    transport (circle.elim_type Pbase Ploop) loop⁻¹ = to_inv Ploop :=
   by rewrite [tr_inv_fn,↑to_inv]; apply inv_eq_inv; apply elim_type_loop
 end circle
 
@@ -188,17 +188,17 @@ namespace circle
   protected definition code (x : circle) : Type₀ :=
   circle.elim_type_on x ℤ equiv_succ
 
-  definition transport_code_loop (a : ℤ) : transport code loop a = succ a :=
+  definition transport_code_loop (a : ℤ) : transport circle.code loop a = succ a :=
   ap10 !elim_type_loop a
 
   definition transport_code_loop_inv (a : ℤ)
-    : transport code loop⁻¹ a = pred a :=
+    : transport circle.code loop⁻¹ a = pred a :=
   ap10 !elim_type_loop_inv a
 
-  protected definition encode {x : circle} (p : base = x) : code x :=
-  transport code p (of_num 0) -- why is the explicit coercion needed here?
+  protected definition encode {x : circle} (p : base = x) : circle.code x :=
+  transport circle.code p (of_num 0) -- why is the explicit coercion needed here?
 
-  protected definition decode {x : circle} : code x → base = x :=
+  protected definition decode {x : circle} : circle.code x → base = x :=
   begin
     refine circle.rec_on x _ _,
     { exact power loop},
@@ -208,30 +208,30 @@ namespace circle
   end
 
   --remove this theorem after #484
-  theorem encode_decode {x : circle} : Π(a : code x), encode (decode a) = a :=
+  theorem encode_decode {x : circle} : Π(a : circle.code x), circle.encode (circle.decode a) = a :=
   begin
-    unfold decode, refine circle.rec_on x _ _,
+    unfold circle.decode, refine circle.rec_on x _ _,
     { intro a, esimp [base,base1], --simplify after #587
       apply rec_nat_on a,
       { exact idp},
       { intros n p,
-        apply transport (λ(y : base = base), transport code y _ = _), apply power_con,
-        rewrite [▸*,con_tr, transport_code_loop, ↑[encode,code] at p, p]},
+        apply transport (λ(y : base = base), transport circle.code y _ = _), apply power_con,
+        rewrite [▸*,con_tr, transport_code_loop, ↑[circle.encode,circle.code] at p, p]},
       { intros n p,
-        apply transport (λ(y : base = base), transport code y _ = _),
+        apply transport (λ(y : base = base), transport circle.code y _ = _),
         { exact !power_con_inv ⬝ ap (power loop) !neg_succ⁻¹},
-        rewrite [▸*,@con_tr _ code,transport_code_loop_inv, ↑[encode] at p, p, -neg_succ]}},
-    { apply eq_of_homotopy, intro a, apply @is_hset.elim, esimp [code,base,base1], exact _}
+        rewrite [▸*,@con_tr _ circle.code,transport_code_loop_inv, ↑[circle.encode] at p, p, -neg_succ]}},
+    { apply eq_of_homotopy, intro a, apply @is_hset.elim, esimp [circle.code,base,base1], exact _}
         --simplify after #587
   end
 
 
-  definition circle_eq_equiv (x : circle) : (base = x) ≃ code x :=
+  definition circle_eq_equiv (x : circle) : (base = x) ≃ circle.code x :=
   begin
     fapply equiv.MK,
-    { exact encode},
-    { exact decode},
-    { exact encode_decode},
+    { exact circle.encode},
+    { exact circle.decode},
+    { exact circle.encode_decode},
     { intro p, cases p, exact idp},
   end
 
@@ -242,7 +242,7 @@ namespace circle
     base_eq_base_equiv⁻¹ a ⬝ base_eq_base_equiv⁻¹ b = base_eq_base_equiv⁻¹ (a + b) :=
   !power_con_power
 
-  definition encode_con (p q : base = base) : encode (p ⬝ q) = encode p + encode q :=
+  definition encode_con (p q : base = base) : circle.encode (p ⬝ q) = circle.encode p + circle.encode q :=
   preserve_binary_of_inv_preserve base_eq_base_equiv concat add decode_add p q
 
   --the carrier of π₁(S¹) is the set-truncation of base = base.
diff --git a/hott/hit/suspension.hlean b/hott/hit/suspension.hlean
index 62b823cbc..65dbe88e2 100644
--- a/hott/hit/suspension.hlean
+++ b/hott/hit/suspension.hlean
@@ -37,39 +37,39 @@ namespace suspension
 
   protected definition rec_on [reducible] {P : suspension A → Type} (y : suspension A)
     (PN : P !north) (PS : P !south) (Pm : Π(a : A), merid a ▸ PN = PS) : P y :=
-  rec PN PS Pm y
+  suspension.rec PN PS Pm y
 
   theorem rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south)
     (Pm : Π(a : A), merid a ▸ PN = PS) (a : A)
-      : apd (rec PN PS Pm) (merid a) = Pm a :=
+      : apd (suspension.rec PN PS Pm) (merid a) = Pm a :=
   !rec_glue
 
   protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
     (x : suspension A) : P :=
-  rec PN PS (λa, !tr_constant ⬝ Pm a) x
+  suspension.rec PN PS (λa, !tr_constant ⬝ Pm a) x
 
   protected definition elim_on [reducible] {P : Type} (x : suspension A)
     (PN : P) (PS : P)  (Pm : A → PN = PS) : P :=
-  elim PN PS Pm x
+  suspension.elim PN PS Pm x
 
   theorem elim_merid {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS) (a : A)
-    : ap (elim PN PS Pm) (merid a) = Pm a :=
+    : ap (suspension.elim PN PS Pm) (merid a) = Pm a :=
   begin
-    apply (@cancel_left _ _ _ _ (tr_constant (merid a) (elim PN PS Pm !north))),
-    rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_merid],
+    apply (@cancel_left _ _ _ _ (tr_constant (merid a) (suspension.elim PN PS Pm !north))),
+    rewrite [-apd_eq_tr_constant_con_ap,↑suspension.elim,rec_merid],
   end
 
   protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
     (x : suspension A) : Type :=
-  elim PN PS (λa, ua (Pm a)) x
+  suspension.elim PN PS (λa, ua (Pm a)) x
 
   protected definition elim_type_on [reducible] (x : suspension A)
     (PN : Type) (PS : Type)  (Pm : A → PN ≃ PS) : Type :=
-  elim_type PN PS Pm x
+  suspension.elim_type PN PS Pm x
 
   theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
-    (x : suspension A) (a : A) : transport (elim_type PN PS Pm) (merid a) = Pm a :=
-  by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_merid];apply cast_ua_fn
+    (x : suspension A) (a : A) : transport (suspension.elim_type PN PS Pm) (merid a) = Pm a :=
+  by rewrite [tr_eq_cast_ap_fn,↑suspension.elim_type,elim_merid];apply cast_ua_fn
 
 end suspension
 
diff --git a/hott/hit/trunc.hlean b/hott/hit/trunc.hlean
index 5caa67659..778582e61 100644
--- a/hott/hit/trunc.hlean
+++ b/hott/hit/trunc.hlean
@@ -24,7 +24,7 @@ namespace trunc
 
   protected definition elim_on {n : trunc_index} {A : Type} {P : Type} (aa : trunc n A)
     [Pt : is_trunc n P] (H : A → P) : P :=
-  elim H aa
+  trunc.elim H aa
 
   /-
     there are no theorems to eliminate to the universe here,
diff --git a/hott/hit/type_quotient.hlean b/hott/hit/type_quotient.hlean
index 57d4f914c..e685467df 100644
--- a/hott/hit/type_quotient.hlean
+++ b/hott/hit/type_quotient.hlean
@@ -22,34 +22,32 @@ namespace type_quotient
 
   protected definition elim_on [reducible] {P : Type} (x : type_quotient R)
     (Pc : A → P) (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') : P :=
-  elim Pc Pp x
+  type_quotient.elim Pc Pp x
 
   theorem elim_eq_of_rel {P : Type} (Pc : A → P)
     (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a = Pc a') {a a' : A} (H : R a a')
-    : ap (elim Pc Pp) (eq_of_rel R H) = Pp H :=
+    : ap (type_quotient.elim Pc Pp) (eq_of_rel R H) = Pp H :=
   begin
-    apply (@cancel_left _ _ _ _ (tr_constant (eq_of_rel R H) (elim Pc Pp (class_of R a)))),
-    rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_eq_of_rel],
+    apply (@cancel_left _ _ _ _ (tr_constant (eq_of_rel R H) (type_quotient.elim Pc Pp (class_of R a)))),
+    rewrite [-apd_eq_tr_constant_con_ap,↑type_quotient.elim,rec_eq_of_rel],
   end
 
   protected definition elim_type (Pc : A → Type)
     (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') : type_quotient R → Type :=
-  elim Pc (λa a' H, ua (Pp H))
+  type_quotient.elim Pc (λa a' H, ua (Pp H))
 
   protected definition elim_type_on [reducible] (x : type_quotient R) (Pc : A → Type)
     (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') : Type :=
-  elim_type Pc Pp x
+  type_quotient.elim_type Pc Pp x
 
   theorem elim_type_eq_of_rel (Pc : A → Type)
     (Pp : Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a') {a a' : A} (H : R a a')
-    : transport (elim_type Pc Pp) (eq_of_rel R H) = to_fun (Pp H) :=
-  by rewrite [tr_eq_cast_ap_fn, ↑elim_type, elim_eq_of_rel];apply cast_ua_fn
+    : transport (type_quotient.elim_type Pc Pp) (eq_of_rel R H) = to_fun (Pp H) :=
+  by rewrite [tr_eq_cast_ap_fn, ↑type_quotient.elim_type, elim_eq_of_rel];apply cast_ua_fn
 
   definition elim_type_uncurried (H : Σ(Pc : A → Type),  Π⦃a a' : A⦄ (H : R a a'), Pc a ≃ Pc a')
     : type_quotient R → Type :=
-  elim_type H.1 H.2
-
-
+  type_quotient.elim_type H.1 H.2
 end type_quotient
 
 attribute type_quotient.elim [unfold-c 6]
diff --git a/hott/init/hit.hlean b/hott/init/hit.hlean
index 1301c7777..c9ffd5f44 100644
--- a/hott/init/hit.hlean
+++ b/hott/init/hit.hlean
@@ -62,7 +62,7 @@ namespace type_quotient
   protected definition rec_on [reducible] {A : Type} {R : A → A → Type} {P : type_quotient R → Type}
     (x : type_quotient R) (Pc : Π(a : A), P (class_of R a))
     (Pp : Π⦃a a' : A⦄ (H : R a a'), eq_of_rel R H ▸ Pc a = Pc a') : P x :=
-  rec Pc Pp x
+  type_quotient.rec Pc Pp x
 
 end type_quotient
 
diff --git a/hott/init/logic.hlean b/hott/init/logic.hlean
index 555a99bc4..0a6da07c9 100644
--- a/hott/init/logic.hlean
+++ b/hott/init/logic.hlean
@@ -200,13 +200,13 @@ protected definition destruct {A : Type} {B : Type} (H1 : inhabited A) (H2 : A 
 inhabited.rec H2 H1
 
 definition inhabited_fun [instance] (A : Type) {B : Type} [H : inhabited B] : inhabited (A → B) :=
-destruct H (λb, mk (λa, b))
+inhabited.destruct H (λb, mk (λa, b))
 
 definition inhabited_Pi [instance] (A : Type) {B : A → Type} [H : Πx, inhabited (B x)] :
   inhabited (Πx, B x) :=
-mk (λa, destruct (H a) (λb, b))
+mk (λa, inhabited.destruct (H a) (λb, b))
 
-definition default (A : Type) [H : inhabited A] : A := destruct H (take a, a)
+definition default (A : Type) [H : inhabited A] : A := inhabited.destruct H (take a, a)
 
 end inhabited
 
diff --git a/hott/types/W.hlean b/hott/types/W.hlean
index 05c73e728..fc7b6e65d 100644
--- a/hott/types/W.hlean
+++ b/hott/types/W.hlean
@@ -24,7 +24,7 @@ namespace Wtype
   protected definition pr1 [unfold-c 3] (w : W(a : A), B a) : A :=
   by cases w with a f; exact a
 
-  protected definition pr2 [unfold-c 3] (w : W(a : A), B a) : B (pr1 w) → W(a : A), B a :=
+  protected definition pr2 [unfold-c 3] (w : W(a : A), B a) : B (Wtype.pr1 w) → W(a : A), B a :=
   by cases w with a f; exact f
 
   namespace ops
@@ -40,13 +40,13 @@ namespace Wtype
   definition sup_eq_sup (p : a = a') (q : p ▸ f = f') : ⟨a, f⟩ = ⟨a', f'⟩ :=
   by cases p; cases q; exact idp
 
-  protected definition Wtype_eq (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2) : w = w' :=
+  definition Wtype_eq (p : w.1 = w'.1) (q : p ▸ w.2 = w'.2) : w = w' :=
   by cases w; cases w';exact (sup_eq_sup p q)
 
-  protected definition Wtype_eq_pr1 (p : w = w') : w.1 = w'.1 :=
+  definition Wtype_eq_pr1 (p : w = w') : w.1 = w'.1 :=
   by cases p;exact idp
 
-  protected definition Wtype_eq_pr2 (p : w = w') : Wtype_eq_pr1 p ▸ w.2 = w'.2 :=
+  definition Wtype_eq_pr2 (p : w = w') : Wtype_eq_pr1 p ▸ w.2 = w'.2 :=
   by cases p;exact idp
 
   namespace ops
diff --git a/hott/types/equiv.hlean b/hott/types/equiv.hlean
index ee2ea3954..00d85f4ba 100644
--- a/hott/types/equiv.hlean
+++ b/hott/types/equiv.hlean
@@ -76,7 +76,7 @@ namespace is_equiv
   local attribute is_contr_right_coherence [instance] [priority 1600]
 
   theorem is_hprop_is_equiv [instance] : is_hprop (is_equiv f) :=
-  is_hprop_of_imp_is_contr (λ(H : is_equiv f), is_trunc_equiv_closed -2 (equiv.symm !sigma_char'))
+  is_hprop_of_imp_is_contr (λ(H : is_equiv f), is_trunc_equiv_closed -2 (equiv.symm !is_equiv.sigma_char'))
 
   definition inv_eq_inv {A B : Type} {f f' : A → B} {Hf : is_equiv f} {Hf' : is_equiv f'}
     (p : f = f') : f⁻¹ = f'⁻¹ :=
diff --git a/hott/types/int/basic.hlean b/hott/types/int/basic.hlean
index aa056ac36..994d7c3a5 100644
--- a/hott/types/int/basic.hlean
+++ b/hott/types/int/basic.hlean
@@ -171,10 +171,10 @@ have H3 : pr1 p + pr2 r + pr2 q = pr2 p + pr1 r + pr2 q, from
     ... = pr2 p + pr1 r + pr2 q                  : by exact sorry,
 show pr1 p + pr2 r = pr2 p + pr1 r, from add.cancel_right H3
 
-protected definition int_equiv_int_equiv : is_equivalence int_equiv :=
+definition int_equiv_int_equiv : is_equivalence int_equiv :=
 is_equivalence.mk @int_equiv.refl @int_equiv.symm @int_equiv.trans
 
-protected definition int_equiv_cases {p q : ℕ × ℕ} (H : int_equiv p q) :
+definition int_equiv_cases {p q : ℕ × ℕ} (H : int_equiv p q) :
     (pr1 p ≥ pr2 p × pr1 q ≥ pr2 q) ⊎ (pr1 p < pr2 p × pr1 q < pr2 q) :=
 sum.rec_on (@le_or_gt (pr2 p) (pr1 p))
   (assume H1: pr1 p ≥ pr2 p,
@@ -184,7 +184,7 @@ sum.rec_on (@le_or_gt (pr2 p) (pr1 p))
     have H2 : pr2 p + pr1 q < pr2 p + pr2 q, from H ▸ add_lt_add_right H1 (pr2 q),
     sum.inr (pair H1 (lt_of_add_lt_add_left H2)))
 
-protected definition int_equiv_of_eq {p q : ℕ × ℕ} (H : p = q) : p ≡ q := H ▸ int_equiv.refl
+definition int_equiv_of_eq {p q : ℕ × ℕ} (H : p = q) : p ≡ q := H ▸ int_equiv.refl
 
 /- the representation and abstraction functions -/
 
diff --git a/hott/types/nat/order.hlean b/hott/types/nat/order.hlean
index bfba715fd..0852c90b4 100644
--- a/hott/types/nat/order.hlean
+++ b/hott/types/nat/order.hlean
@@ -328,10 +328,10 @@ H1 !lt_succ_self
 
 protected theorem case_strong_induction_on {P : nat → Type} (a : nat) (H0 : P 0)
   (Hind : Π(n : nat), (Πm, m ≤ n → P m) → P (succ n)) : P a :=
-strong_induction_on a (
-  take n,
-  show (Πm, m < n → P m) → P n, from
-    nat.cases_on n
+nat.strong_induction_on a
+  (take n,
+   show (Π m, m < n → P m) → P n, from
+     nat.cases_on n
        (assume H : (Πm, m < 0 → P m), show P 0, from H0)
        (take n,
          assume H : (Πm, m < succ n → P m),
diff --git a/hott/types/prod.hlean b/hott/types/prod.hlean
index 549fdbbab..2bf48ac71 100644
--- a/hott/types/prod.hlean
+++ b/hott/types/prod.hlean
@@ -27,8 +27,8 @@ namespace prod
   begin
     cases u with a₁ b₁,
     cases v with a₂ b₂,
-    apply transport _ (eta (a₁, b₁)),
-    apply transport _ (eta (a₂, b₂)),
+    apply transport _ (prod.eta (a₁, b₁)),
+    apply transport _ (prod.eta (a₂, b₂)),
     apply pair_eq H₁ H₂,
   end
 
diff --git a/hott/types/sigma.hlean b/hott/types/sigma.hlean
index 23a290040..47704b091 100644
--- a/hott/types/sigma.hlean
+++ b/hott/types/sigma.hlean
@@ -273,7 +273,7 @@ namespace sigma
   definition sigma_assoc_equiv (C : (Σa, B a) → Type) : (Σa b, C ⟨a, b⟩) ≃ (Σu, C u) :=
   equiv.mk _ (adjointify
     (λav, ⟨⟨av.1, av.2.1⟩, av.2.2⟩)
-    (λuc, ⟨uc.1.1, uc.1.2, !eta⁻¹ ▸ uc.2⟩)
+    (λuc, ⟨uc.1.1, uc.1.2, !sigma.eta⁻¹ ▸ uc.2⟩)
     begin intro uc, cases uc with u c, cases u, reflexivity end
     begin intro av, cases av with a v, cases v, reflexivity end)
 
@@ -331,18 +331,18 @@ namespace sigma
   ⟨fg.1 a, fg.2 a⟩
 
   protected definition coind (f : Π a, B a) (g : Π a, C a (f a)) (a : A) : Σ(b : B a), C a b :=
-  coind_uncurried ⟨f, g⟩ a
+  sigma.coind_uncurried ⟨f, g⟩ a
 
   --is the instance below dangerous?
   --in Coq this can be done without function extensionality
   definition is_equiv_coind [instance] (C : Πa, B a → Type)
-    : is_equiv (@coind_uncurried _ _ C) :=
+    : is_equiv (@sigma.coind_uncurried _ _ C) :=
   adjointify _ (λ h, ⟨λa, (h a).1, λa, (h a).2⟩)
-               (λ h, proof eq_of_homotopy (λu, !eta) qed)
+               (λ h, proof eq_of_homotopy (λu, !sigma.eta) qed)
                (λfg, destruct fg (λ(f : Π (a : A), B a) (g : Π (x : A), C x (f x)), proof idp qed))
 
   definition sigma_pi_equiv_pi_sigma : (Σ(f : Πa, B a), Πa, C a (f a)) ≃ (Πa, Σb, C a b) :=
-  equiv.mk coind_uncurried _
+  equiv.mk sigma.coind_uncurried _
   end
 
   /- ** Subtypes (sigma types whose second components are hprops) -/
diff --git a/hott/types/trunc.hlean b/hott/types/trunc.hlean
index d32424ad8..99cdc78d2 100644
--- a/hott/types/trunc.hlean
+++ b/hott/types/trunc.hlean
@@ -148,27 +148,27 @@ namespace trunc
   protected definition code (n : trunc_index) (aa aa' : trunc n.+1 A) : n-Type :=
   trunc.rec_on aa (λa, trunc.rec_on aa' (λa', trunctype.mk' n (trunc n (a = a'))))
 
-  protected definition encode (n : trunc_index) (aa aa' : trunc n.+1 A) : aa = aa' → code n aa aa' :=
+  protected definition encode (n : trunc_index) (aa aa' : trunc n.+1 A) : aa = aa' → trunc.code n aa aa' :=
   begin
     intro p, cases p, apply (trunc.rec_on aa),
-    intro a, esimp [code,trunc.rec_on], exact (tr idp)
+    intro a, esimp [trunc.code,trunc.rec_on], exact (tr idp)
   end
 
-  protected definition decode (n : trunc_index) (aa aa' : trunc n.+1 A) : code n aa aa' → aa = aa' :=
+  protected definition decode (n : trunc_index) (aa aa' : trunc n.+1 A) : trunc.code n aa aa' → aa = aa' :=
   begin
     eapply (trunc.rec_on aa'), eapply (trunc.rec_on aa),
-    intro a a' x, esimp [code, trunc.rec_on] at x,
+    intro a a' x, esimp [trunc.code, trunc.rec_on] at x,
     apply (trunc.rec_on x), intro p, exact (ap tr p)
   end
 
   definition trunc_eq_equiv (n : trunc_index) (aa aa' : trunc n.+1 A)
-    : aa = aa' ≃ code n aa aa' :=
+    : aa = aa' ≃ trunc.code n aa aa' :=
   begin
     fapply equiv.MK,
-    { apply encode},
-    { apply decode},
+    { apply trunc.encode},
+    { apply trunc.decode},
     { eapply (trunc.rec_on aa'), eapply (trunc.rec_on aa),
-      intro a a' x, esimp [code, trunc.rec_on] at x,
+      intro a a' x, esimp [trunc.code, trunc.rec_on] at x,
       apply (trunc.rec_on x), intro p, cases p, exact idp},
     { intro p, cases p, apply (trunc.rec_on aa), intro a, exact idp},
   end
diff --git a/library/algebra/category/constructions.lean b/library/algebra/category/constructions.lean
index ed0b72010..aad183b3b 100644
--- a/library/algebra/category/constructions.lean
+++ b/library/algebra/category/constructions.lean
@@ -176,32 +176,32 @@ namespace category
   open sigma function
   variables {ob : Type} {C : category ob} {c : ob}
   protected definition slice_obs (C : category ob) (c : ob) := Σ(b : ob), hom b c
-  variables {a b : slice_obs C c}
-  protected definition to_ob  (a : slice_obs C c) : ob              := sigma.pr1 a
-  protected definition to_ob_def (a : slice_obs C c) : to_ob a = sigma.pr1 a := rfl
-  protected definition ob_hom (a : slice_obs C c) : hom (to_ob a) c := sigma.pr2 a
+  variables {a b : slice.slice_obs C c}
+  protected definition to_ob  (a : slice.slice_obs C c) : ob              := sigma.pr1 a
+  protected definition to_ob_def (a : slice.slice_obs C c) : slice.to_ob a = sigma.pr1 a := rfl
+  protected definition ob_hom (a : slice.slice_obs C c) : hom (slice.to_ob a) c := sigma.pr2 a
   -- protected theorem slice_obs_equal (H₁ : to_ob a = to_ob b)
   --     (H₂ : eq.drec_on H₁ (ob_hom a) = ob_hom b) : a = b :=
   -- sigma.equal H₁ H₂
 
 
-  protected definition slice_hom (a b : slice_obs C c) : Type :=
-  Σ(g : hom (to_ob a) (to_ob b)), ob_hom b ∘ g = ob_hom a
+  protected definition slice_hom (a b : slice.slice_obs C c) : Type :=
+  Σ(g : hom (slice.to_ob a) (slice.to_ob b)), slice.ob_hom b ∘ g = slice.ob_hom a
 
-  protected definition hom_hom  (f : slice_hom a b) : hom (to_ob a) (to_ob b)          := sigma.pr1 f
-  protected definition commute (f : slice_hom a b) : ob_hom b ∘ (hom_hom f) = ob_hom a := sigma.pr2 f
+  protected definition hom_hom  (f : slice.slice_hom a b) : hom (slice.to_ob a) (slice.to_ob b)                := sigma.pr1 f
+  protected definition commute (f : slice.slice_hom a b) : slice.ob_hom b ∘ (slice.hom_hom f) = slice.ob_hom a := sigma.pr2 f
   -- protected theorem slice_hom_equal (f g : slice_hom a b) (H : hom_hom f = hom_hom g) : f = g :=
   -- sigma.equal H !proof_irrel
 
-  definition slice_category (C : category ob) (c : ob) : category (slice_obs C c) :=
-  mk (λa b, slice_hom a b)
-     (λ a b c g f, sigma.mk (hom_hom g ∘ hom_hom f)
-       (show ob_hom c ∘ (hom_hom g ∘ hom_hom f) = ob_hom a,
+  definition slice_category (C : category ob) (c : ob) : category (slice.slice_obs C c) :=
+  mk (λa b, slice.slice_hom a b)
+     (λ a b c g f, sigma.mk (slice.hom_hom g ∘ slice.hom_hom f)
+       (show slice.ob_hom c ∘ (slice.hom_hom g ∘ slice.hom_hom f) = slice.ob_hom a,
          proof
          calc
-           ob_hom c ∘ (hom_hom g ∘ hom_hom f) = (ob_hom c ∘ hom_hom g) ∘ hom_hom f : !assoc
-             ... = ob_hom b ∘ hom_hom f : {commute g}
-             ... = ob_hom a : {commute f}
+           slice.ob_hom c ∘ (slice.hom_hom g ∘ slice.hom_hom f) = (slice.ob_hom c ∘ slice.hom_hom g) ∘ slice.hom_hom f : !assoc
+             ... = slice.ob_hom b ∘ slice.hom_hom f : {slice.commute g}
+             ... = slice.ob_hom a : {slice.commute f}
          qed))
      (λ a, sigma.mk id !id_right)
      (λ a b c d h g f, dpair_eq    !assoc    !proof_irrel)
@@ -231,20 +231,20 @@ namespace category
   attribute slice_category [instance]
   variables {D : Category}
   definition forgetful (x : D) : (Slice_category D x) ⇒ D :=
-  functor.mk (λ a, to_ob a)
-             (λ a b f, hom_hom f)
+  functor.mk (λ a, slice.to_ob a)
+             (λ a b f, slice.hom_hom f)
              (λ a, rfl)
              (λ a b c g f, rfl)
 
   definition postcomposition_functor {x y : D} (h : x ⟶ y)
       : Slice_category D x ⇒ Slice_category D y :=
   functor.mk
-       (λ a, sigma.mk (to_ob a) (h ∘ ob_hom a))
+       (λ a, sigma.mk (slice.to_ob a) (h ∘ slice.ob_hom a))
        (λ a b f,
-         ⟨hom_hom f,
+         ⟨slice.hom_hom f,
           calc
-            (h ∘ ob_hom b) ∘ hom_hom f = h ∘ (ob_hom b ∘ hom_hom f) : by rewrite assoc
-                                   ... = h ∘ ob_hom a               : by rewrite commute⟩)
+            (h ∘ slice.ob_hom b) ∘ slice.hom_hom f = h ∘ (slice.ob_hom b ∘ slice.hom_hom f) : by rewrite assoc
+                                   ... = h ∘ slice.ob_hom a               : by rewrite slice.commute⟩)
        (λ a, rfl)
        (λ a b c g f, dpair_eq rfl !proof_irrel)
 
@@ -330,30 +330,32 @@ namespace category
   -- make these definitions private?
   variables {ob : Type} {C : category ob}
   protected definition arrow_obs (ob : Type) (C : category ob) := Σ(a b : ob), hom a b
-  variables {a b : arrow_obs ob C}
-  protected definition src    (a : arrow_obs ob C) : ob                  := sigma.pr1 a
-  protected definition dst    (a : arrow_obs ob C) : ob                  := sigma.pr2' a
-  protected definition to_hom (a : arrow_obs ob C) : hom (src a) (dst a) := sigma.pr3 a
+  variables {a b : category.arrow_obs ob C}
+  protected definition src    (a : category.arrow_obs ob C) : ob                  := sigma.pr1 a
+  protected definition dst    (a : category.arrow_obs ob C) : ob                  := sigma.pr2' a
+  protected definition to_hom (a : category.arrow_obs ob C) : hom (category.src a) (category.dst a) := sigma.pr3 a
 
-  protected definition arrow_hom (a b : arrow_obs ob C) : Type :=
-  Σ (g : hom (src a) (src b)) (h : hom (dst a) (dst b)), to_hom b ∘ g = h ∘ to_hom a
+  protected definition arrow_hom (a b : category.arrow_obs ob C) : Type :=
+  Σ (g : hom (category.src a) (category.src b)) (h : hom (category.dst a) (category.dst b)),
+    category.to_hom b ∘ g = h ∘ category.to_hom a
 
-  protected definition hom_src (m : arrow_hom a b) : hom (src a) (src b) := sigma.pr1 m
-  protected definition hom_dst (m : arrow_hom a b) : hom (dst a) (dst b) := sigma.pr2' m
-  protected definition commute (m : arrow_hom a b) : to_hom b ∘ (hom_src m) = (hom_dst m) ∘ to_hom a
+  protected definition hom_src (m : category.arrow_hom a b) : hom (category.src a) (category.src b) := sigma.pr1 m
+  protected definition hom_dst (m : category.arrow_hom a b) : hom (category.dst a) (category.dst b) := sigma.pr2' m
+  protected definition commute (m : category.arrow_hom a b) :
+     category.to_hom b ∘ (category.hom_src m) = (category.hom_dst m) ∘ category.to_hom a
   := sigma.pr3 m
 
-  definition arrow (ob : Type) (C : category ob) : category (arrow_obs ob C) :=
-  mk (λa b, arrow_hom a b)
-     (λ a b c g f, sigma.mk (hom_src g ∘ hom_src f) (sigma.mk (hom_dst g ∘ hom_dst f)
-        (show to_hom c ∘ (hom_src g ∘ hom_src f) = (hom_dst g ∘ hom_dst f) ∘ to_hom a,
+  definition arrow (ob : Type) (C : category ob) : category (category.arrow_obs ob C) :=
+  mk (λa b, category.arrow_hom a b)
+     (λ a b c g f, sigma.mk (category.hom_src g ∘ category.hom_src f) (sigma.mk (category.hom_dst g ∘ category.hom_dst f)
+        (show category.to_hom c ∘ (category.hom_src g ∘ category.hom_src f) = (category.hom_dst g ∘ category.hom_dst f) ∘ category.to_hom a,
          proof
          calc
-         to_hom c ∘ (hom_src g ∘ hom_src f) = (to_hom c ∘ hom_src g) ∘ hom_src f : by rewrite assoc
-           ... = (hom_dst g ∘ to_hom b) ∘ hom_src f                              : by rewrite commute
-           ... = hom_dst g ∘ (to_hom b ∘ hom_src f)                              : by rewrite assoc
-           ... = hom_dst g ∘ (hom_dst f ∘ to_hom a)                              : by rewrite commute
-           ... = (hom_dst g ∘ hom_dst f) ∘ to_hom a                              : by rewrite assoc
+         category.to_hom c ∘ (category.hom_src g ∘ category.hom_src f) = (category.to_hom c ∘ category.hom_src g) ∘ category.hom_src f : by rewrite assoc
+           ... = (category.hom_dst g ∘ category.to_hom b) ∘ category.hom_src f                     : by rewrite category.commute
+           ... = category.hom_dst g ∘ (category.to_hom b ∘ category.hom_src f)                     : by rewrite assoc
+           ... = category.hom_dst g ∘ (category.hom_dst f ∘ category.to_hom a)                     : by rewrite category.commute
+           ... = (category.hom_dst g ∘ category.hom_dst f) ∘ category.to_hom a                     : by rewrite assoc
          qed)
        ))
      (λ a, sigma.mk id (sigma.mk id (!id_right ⬝ (symm !id_left))))
diff --git a/library/algebra/category/functor.lean b/library/algebra/category/functor.lean
index 8d53e2c40..97dc6bc01 100644
--- a/library/algebra/category/functor.lean
+++ b/library/algebra/category/functor.lean
@@ -39,7 +39,7 @@ namespace functor
       G (F (g ∘ f)) = G (F g ∘ F f)     : respect_comp F g f
                 ... = G (F g) ∘ G (F f) : respect_comp G (F g) (F f) qed)
 
-  infixr `∘f`:60 := compose
+  infixr `∘f`:60 := functor.compose
 
   protected theorem assoc (H : functor C D) (G : functor B C) (F : functor A B) :
       H ∘f (G ∘f F) = (H ∘f G) ∘f F :=
@@ -47,11 +47,11 @@ namespace functor
 
   protected definition id [reducible] {C : Category} : functor C C :=
   mk (λa, a) (λ a b f, f) (λ a, rfl) (λ a b c f g, rfl)
-  protected definition ID [reducible] (C : Category) : functor C C := @id C
+  protected definition ID [reducible] (C : Category) : functor C C := @functor.id C
 
-  protected theorem id_left  (F : functor C D) : (@id D) ∘f F = F :=
+  protected theorem id_left  (F : functor C D) : (@functor.id D) ∘f F = F :=
   functor.rec (λ obF homF idF compF, dcongr_arg4 mk rfl rfl !proof_irrel !proof_irrel) F
-  protected theorem id_right (F : functor C D) : F ∘f (@id C) = F :=
+  protected theorem id_right (F : functor C D) : F ∘f (@functor.id C) = F :=
   functor.rec (λ obF homF idF compF, dcongr_arg4 mk rfl rfl !proof_irrel !proof_irrel) F
 
 end functor
diff --git a/library/algebra/category/natural_transformation.lean b/library/algebra/category/natural_transformation.lean
index e47c65b2f..184e4732b 100644
--- a/library/algebra/category/natural_transformation.lean
+++ b/library/algebra/category/natural_transformation.lean
@@ -36,7 +36,7 @@ namespace natural_transformation
                       ... = (η b ∘ θ b) ∘ F f : assoc)
 --congr_arg (λx, η b ∘ x) (naturality θ f) -- this needed to be explicit for some reason (on Oct 24)
 
-  infixr `∘n`:60 := compose
+  infixr `∘n`:60 := natural_transformation.compose
   protected theorem assoc (η₃ : H ⟹ I) (η₂ : G ⟹ H) (η₁ : F ⟹ G) :
       η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
   dcongr_arg2 mk (funext (take x, !assoc)) !proof_irrel
diff --git a/library/data/empty.lean b/library/data/empty.lean
index 20d50aaa6..7f5eb70fc 100644
--- a/library/data/empty.lean
+++ b/library/data/empty.lean
@@ -12,7 +12,7 @@ namespace empty
   empty.rec (λe, A) H
 
   protected theorem subsingleton [instance] : subsingleton empty :=
-  subsingleton.intro (λ a b, !elim a)
+  subsingleton.intro (λ a b, !empty.elim a)
 end empty
 
 protected definition empty.has_decidable_eq [instance] : decidable_eq empty :=
diff --git a/library/data/finset/basic.lean b/library/data/finset/basic.lean
index f7bb4d11f..6003d273a 100644
--- a/library/data/finset/basic.lean
+++ b/library/data/finset/basic.lean
@@ -223,7 +223,7 @@ protected theorem induction_on {P : finset A → Prop} (s : finset A)
     (H1 : P empty)
     (H2 : ∀⦃s : finset A⦄, ∀{a : A}, a ∉ s → P s → P (insert a s)) :
   P s :=
-induction H1 H2 s
+finset.induction H1 H2 s
 end insert
 
 /- erase -/
diff --git a/library/data/int/basic.lean b/library/data/int/basic.lean
index d6d1bcf2c..25463b3b5 100644
--- a/library/data/int/basic.lean
+++ b/library/data/int/basic.lean
@@ -181,10 +181,10 @@ have H3 : pr1 p + pr2 r + pr2 q = pr2 p + pr1 r + pr2 q, from
     ... = pr2 p + pr1 r + pr2 q                  : by simp,
 show pr1 p + pr2 r = pr2 p + pr1 r, from add.cancel_right H3
 
-protected theorem equiv_equiv : is_equivalence equiv :=
+protected theorem equiv_equiv : is_equivalence int.equiv :=
 is_equivalence.mk @equiv.refl @equiv.symm @equiv.trans
 
-protected theorem equiv_cases {p q : ℕ × ℕ} (H : equiv p q) :
+protected theorem equiv_cases {p q : ℕ × ℕ} (H : int.equiv p q) :
     (pr1 p ≥ pr2 p ∧ pr1 q ≥ pr2 q) ∨ (pr1 p < pr2 p ∧ pr1 q < pr2 q) :=
 or.elim (@le_or_gt (pr2 p) (pr1 p))
   (assume H1: pr1 p ≥ pr2 p,
@@ -232,7 +232,7 @@ theorem repr_abstr (p : ℕ × ℕ) : repr (abstr p) ≡ p :=
 !prod.eta ▸ !repr_sub_nat_nat
 
 theorem abstr_eq {p q : ℕ × ℕ} (Hequiv : p ≡ q) : abstr p = abstr q :=
-or.elim (equiv_cases Hequiv)
+or.elim (int.equiv_cases Hequiv)
   (assume H2,
     have H3 : pr1 p ≥ pr2 p, from and.elim_left H2,
     have H4 : pr1 q ≥ pr2 q, from and.elim_right H2,
@@ -264,9 +264,9 @@ or.elim (equiv_cases Hequiv)
 
 theorem equiv_iff (p q : ℕ × ℕ) : (p ≡ q) ↔ ((p ≡ p) ∧ (q ≡ q) ∧ (abstr p = abstr q)) :=
 iff.intro
-  (assume H : equiv p q,
+  (assume H : int.equiv p q,
     and.intro !equiv.refl (and.intro !equiv.refl (abstr_eq H)))
-  (assume H : equiv p p ∧ equiv q q ∧ abstr p = abstr q,
+  (assume H : int.equiv p p ∧ int.equiv q q ∧ abstr p = abstr q,
     have H1 : abstr p = abstr q, from and.elim_right (and.elim_right H),
     equiv.trans (H1 ▸ equiv.symm (repr_abstr p)) (repr_abstr q))
 
@@ -335,7 +335,7 @@ int.cases_on a
     int.cases_on b
       (take n, !equiv.refl)
       (take n',
-        have H1 : equiv (repr (add (of_nat m) (neg_succ_of_nat n'))) (m, succ n'),
+        have H1 : int.equiv (repr (add (of_nat m) (neg_succ_of_nat n'))) (m, succ n'),
           from !repr_sub_nat_nat,
         have H2 : padd (repr (of_nat m)) (repr (neg_succ_of_nat n')) = (m, 0 + succ n'),
           from rfl,
@@ -343,7 +343,7 @@ int.cases_on a
   (take m',
     int.cases_on b
       (take n,
-        have H1 : equiv (repr (add (neg_succ_of_nat m') (of_nat n))) (n, succ m'),
+        have H1 : int.equiv (repr (add (neg_succ_of_nat m') (of_nat n))) (n, succ m'),
           from !repr_sub_nat_nat,
         have H2 : padd (repr (neg_succ_of_nat m')) (repr (of_nat n)) = (0 + n, succ m'),
           from rfl,
@@ -566,7 +566,7 @@ eq_of_repr_equiv_repr
       ... = repr (a * (b * c)) : repr_mul) ▸ !equiv.refl)
 
 theorem mul_one (a : ℤ) : a * 1 = a :=
-eq_of_repr_equiv_repr (equiv_of_eq
+eq_of_repr_equiv_repr (int.equiv_of_eq
   ((calc
     repr (a * 1) = pmul (repr a) (repr 1) : repr_mul
       ... = (pr1 (repr a), pr2 (repr a)) : by simp
diff --git a/library/data/nat/basic.lean b/library/data/nat/basic.lean
index 6c1b35e1f..79959b6dd 100644
--- a/library/data/nat/basic.lean
+++ b/library/data/nat/basic.lean
@@ -307,7 +307,7 @@ section migrate_algebra
     mul_zero       := mul_zero,
     mul_comm       := mul.comm⦄
 
-  local attribute comm_semiring [instance]
+  local attribute nat.comm_semiring [instance]
   definition dvd (a b : ℕ) : Prop := algebra.dvd a b
   notation a ∣ b := dvd a b
 
diff --git a/library/data/nat/choose.lean b/library/data/nat/choose.lean
index 6244b0117..b8d9a4952 100644
--- a/library/data/nat/choose.lean
+++ b/library/data/nat/choose.lean
@@ -104,6 +104,6 @@ end find_x
 protected definition choose {p : nat → Prop} [d : decidable_pred p] : (∃ x, p x) → nat :=
 assume h, elt_of (find_x h)
 
-protected theorem choose_spec {p : nat → Prop} [d : decidable_pred p] (ex : ∃ x, p x) : p (choose ex) :=
+protected theorem choose_spec {p : nat → Prop} [d : decidable_pred p] (ex : ∃ x, p x) : p (nat.choose ex) :=
 has_property (find_x ex)
 end nat
diff --git a/library/data/nat/order.lean b/library/data/nat/order.lean
index 0b1a25632..47f689d32 100644
--- a/library/data/nat/order.lean
+++ b/library/data/nat/order.lean
@@ -326,10 +326,10 @@ H1 !lt_succ_self
 
 protected theorem case_strong_induction_on {P : nat → Prop} (a : nat) (H0 : P 0)
   (Hind : ∀(n : nat), (∀m, m ≤ n → P m) → P (succ n)) : P a :=
-strong_induction_on a (
-  take n,
-  show (∀m, m < n → P m) → P n, from
-    nat.cases_on n
+nat.strong_induction_on a
+  (take n,
+   show (∀ m, m < n → P m) → P n, from
+     nat.cases_on n
        (assume H : (∀m, m < 0 → P m), show P 0, from H0)
        (take n,
          assume H : (∀m, m < succ n → P m),
diff --git a/library/data/set/classical_inverse.lean b/library/data/set/classical_inverse.lean
index 47502f182..ac1d6b4db 100644
--- a/library/data/set/classical_inverse.lean
+++ b/library/data/set/classical_inverse.lean
@@ -71,15 +71,15 @@ protected definition inverse (f : map a b) {dflt : X} (dflta : dflt ∈ a) :=
 map.mk (inv_fun f a dflt) (@maps_to_inv_fun _ _ _ _ b _ dflta)
 
 theorem left_inverse_inverse {f : map a b} {dflt : X} (dflta : dflt ∈ a) (H : map.injective f) :
-  map.left_inverse (inverse f dflta) f :=
+  map.left_inverse (map.inverse f dflta) f :=
 left_inv_on_inv_fun_of_inj_on dflt H
 
 theorem right_inverse_inverse {f : map a b} {dflt : X} (dflta : dflt ∈ a) (H : map.surjective f) :
-  map.right_inverse (inverse f dflta) f :=
+  map.right_inverse (map.inverse f dflta) f :=
 right_inv_on_inv_fun_of_surj_on dflt H
 
 theorem is_inverse_inverse {f : map a b} {dflt : X} (dflta : dflt ∈ a) (H : map.bijective f) :
-map.is_inverse (inverse f dflta) f :=
+map.is_inverse (map.inverse f dflta) f :=
 and.intro
   (left_inverse_inverse dflta (and.left H))
   (right_inverse_inverse dflta (and.right H))
diff --git a/library/data/set/map.lean b/library/data/set/map.lean
index 29ffd5e79..2a1878004 100644
--- a/library/data/set/map.lean
+++ b/library/data/set/map.lean
@@ -38,57 +38,57 @@ assume H₁ : f₁ ~ f₂, assume H₂ : f₂ ~ f₃,
 take x, assume Ha : x ∈ a, eq.trans (H₁ Ha) (H₂ Ha)
 
 protected theorem equiv.is_equivalence {X Y : Type} (a : set X) (b : set Y) :
-  equivalence (@equiv X Y a b) :=
-mk_equivalence (@equiv X Y a b) (@equiv.refl X Y a b) (@equiv.symm X Y a b) (@equiv.trans X Y a b)
+  equivalence (@map.equiv X Y a b) :=
+mk_equivalence (@map.equiv X Y a b) (@equiv.refl X Y a b) (@equiv.symm X Y a b) (@equiv.trans X Y a b)
 
 /- compose -/
 
 protected definition compose (g : map b c) (f : map a b) : map a c :=
 map.mk (#function g ∘ f) (maps_to_compose (mapsto g) (mapsto f))
 
-notation g ∘ f := compose g f
+notation g ∘ f := map.compose g f
 
 /- range -/
 
 protected definition range (f : map a b) : set Y := image f a
 
-theorem range_eq_range_of_equiv {f1 f2 : map a b} (H : f1 ~ f2) : range f1 = range f2 :=
+theorem range_eq_range_of_equiv {f1 f2 : map a b} (H : f1 ~ f2) : map.range f1 = map.range f2 :=
 image_eq_image_of_eq_on H
 
 /- injective -/
 
 protected definition injective (f : map a b) : Prop := inj_on f a
 
-theorem injective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : injective f1) :
-  injective f2 :=
+theorem injective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : map.injective f1) :
+  map.injective f2 :=
 inj_on_of_eq_on H1 H2
 
-theorem injective_compose {g : map b c} {f : map a b} (Hg : injective g) (Hf: injective f) :
-  injective (g ∘ f) :=
+theorem injective_compose {g : map b c} {f : map a b} (Hg : map.injective g) (Hf: map.injective f) :
+  map.injective (g ∘ f) :=
 inj_on_compose (mapsto f) Hg Hf
 
 /- surjective -/
 
 protected definition surjective (f : map a b) : Prop := surj_on f a b
 
-theorem surjective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : surjective f1) :
-  surjective f2 :=
+theorem surjective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : map.surjective f1) :
+  map.surjective f2 :=
 surj_on_of_eq_on H1 H2
 
-theorem surjective_compose {g : map b c} {f : map a b} (Hg : surjective g) (Hf: surjective f) :
-  surjective (g ∘ f) :=
+theorem surjective_compose {g : map b c} {f : map a b} (Hg : map.surjective g) (Hf: map.surjective f) :
+  map.surjective (g ∘ f) :=
 surj_on_compose Hg Hf
 
 /- bijective -/
 
-protected definition bijective (f : map a b) : Prop := injective f ∧ surjective f
+protected definition bijective (f : map a b) : Prop := map.injective f ∧ map.surjective f
 
-theorem bijective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : bijective f1) :
-  bijective f2 :=
+theorem bijective_of_equiv {f1 f2 : map a b} (H1 : f1 ~ f2) (H2 : map.bijective f1) :
+  map.bijective f2 :=
 and.intro (injective_of_equiv H1 (and.left H2)) (surjective_of_equiv H1 (and.right H2))
 
-theorem bijective_compose {g : map b c} {f : map a b} (Hg : bijective g) (Hf: bijective f) :
-  bijective (g ∘ f) :=
+theorem bijective_compose {g : map b c} {f : map a b} (Hg : map.bijective g) (Hf: map.bijective f) :
+  map.bijective (g ∘ f) :=
 obtain Hg₁ Hg₂, from Hg,
 obtain Hf₁ Hf₂, from Hf,
 and.intro (injective_compose Hg₁ Hf₁) (surjective_compose Hg₂ Hf₂)
@@ -99,53 +99,53 @@ and.intro (injective_compose Hg₁ Hf₁) (surjective_compose Hg₂ Hf₂)
 protected definition left_inverse (g : map b a) (f : map a b) : Prop := left_inv_on g f a
 
 theorem left_inverse_of_equiv_left {g1 g2 : map b a} {f : map a b} (eqg : g1 ~ g2)
-  (H : left_inverse g1 f) : left_inverse g2 f :=
+  (H : map.left_inverse g1 f) : map.left_inverse g2 f :=
 left_inv_on_of_eq_on_left (mapsto f) eqg H
 
 theorem left_inverse_of_equiv_right {g : map b a} {f1 f2 : map a b} (eqf : f1 ~ f2)
-  (H : left_inverse g f1) : left_inverse g f2 :=
+  (H : map.left_inverse g f1) : map.left_inverse g f2 :=
 left_inv_on_of_eq_on_right eqf H
 
-theorem injective_of_left_inverse {g : map b a} {f : map a b} (H : left_inverse g f) :
-  injective f :=
+theorem injective_of_left_inverse {g : map b a} {f : map a b} (H : map.left_inverse g f) :
+  map.injective f :=
 inj_on_of_left_inv_on H
 
 theorem left_inverse_compose {f' : map b a} {g' : map c b} {g : map b c} {f : map a b}
-    (Hf : left_inverse f' f) (Hg : left_inverse g' g) : left_inverse (f' ∘ g') (g ∘ f) :=
+    (Hf : map.left_inverse f' f) (Hg : map.left_inverse g' g) : map.left_inverse (f' ∘ g') (g ∘ f) :=
 left_inv_on_compose (mapsto f) Hf Hg
 
 /- right inverse -/
 
 -- g is a right inverse to f
-protected definition right_inverse (g : map b a) (f : map a b) : Prop := left_inverse f g
+protected definition right_inverse (g : map b a) (f : map a b) : Prop := map.left_inverse f g
 
 theorem right_inverse_of_equiv_left {g1 g2 : map b a} {f : map a b} (eqg : g1 ~ g2)
-  (H : right_inverse g1 f) : right_inverse g2 f :=
-left_inverse_of_equiv_right eqg H
+  (H : map.right_inverse g1 f) : map.right_inverse g2 f :=
+map.left_inverse_of_equiv_right eqg H
 
 theorem right_inverse_of_equiv_right {g : map b a} {f1 f2 : map a b} (eqf : f1 ~ f2)
-  (H : right_inverse g f1) : right_inverse g f2 :=
-left_inverse_of_equiv_left eqf H
+  (H : map.right_inverse g f1) : map.right_inverse g f2 :=
+map.left_inverse_of_equiv_left eqf H
 
-theorem surjective_of_right_inverse {g : map b a} {f : map a b} (H : right_inverse g f) :
-  surjective f :=
+theorem surjective_of_right_inverse {g : map b a} {f : map a b} (H : map.right_inverse g f) :
+  map.surjective f :=
 surj_on_of_right_inv_on (mapsto g) H
 
 theorem right_inverse_compose {f' : map b a} {g' : map c b} {g : map b c} {f : map a b}
-    (Hf : right_inverse f' f) (Hg : right_inverse g' g) : right_inverse (f' ∘ g') (g ∘ f) :=
-left_inverse_compose Hg Hf
+    (Hf : map.right_inverse f' f) (Hg : map.right_inverse g' g) : map.right_inverse (f' ∘ g') (g ∘ f) :=
+map.left_inverse_compose Hg Hf
 
-theorem equiv_of_left_inverse_of_right_inverse {g1 g2 : map b a} {f : map a b}
-  (H1 : left_inverse g1 f) (H2 : right_inverse g2 f) : g1 ~ g2 :=
+theorem equiv_of_map.left_inverse_of_right_inverse {g1 g2 : map b a} {f : map a b}
+  (H1 : map.left_inverse g1 f) (H2 : map.right_inverse g2 f) : g1 ~ g2 :=
 eq_on_of_left_inv_of_right_inv (mapsto g2) H1 H2
 
 /- inverse -/
 
 -- g is an inverse to f
 protected definition is_inverse (g : map b a) (f : map a b) : Prop :=
-left_inverse g f ∧ right_inverse g f
+map.left_inverse g f ∧ map.right_inverse g f
 
-theorem bijective_of_is_inverse {g : map b a} {f : map a b} (H : is_inverse g f) : bijective f :=
+theorem bijective_of_is_inverse {g : map b a} {f : map a b} (H : map.is_inverse g f) : map.bijective f :=
 and.intro
   (injective_of_left_inverse (and.left H))
   (surjective_of_right_inverse (and.right H))
diff --git a/library/data/unit.lean b/library/data/unit.lean
index dfeb3f253..b48be9e2e 100644
--- a/library/data/unit.lean
+++ b/library/data/unit.lean
@@ -17,11 +17,11 @@ namespace unit
   unit.equal a star
 
   protected theorem subsingleton [instance] : subsingleton unit :=
-  subsingleton.intro (λ a b, equal a b)
+  subsingleton.intro (λ a b, unit.equal a b)
 
   protected definition is_inhabited [instance] : inhabited unit :=
   inhabited.mk unit.star
 
   protected definition has_decidable_eq [instance] : decidable_eq unit :=
-  take (a b : unit), decidable.inl (equal a b)
+  take (a b : unit), decidable.inl (unit.equal a b)
 end unit
diff --git a/library/init/funext.lean b/library/init/funext.lean
index ea2c4e1c7..73dc630e8 100644
--- a/library/init/funext.lean
+++ b/library/init/funext.lean
@@ -28,8 +28,8 @@ namespace function
   protected theorem equiv.trans {f₁ f₂ f₃ : Πx: A, B x} : f₁ ~ f₂ → f₂ ~ f₃ → f₁ ~ f₃ :=
   λH₁ H₂ x, eq.trans (H₁ x) (H₂ x)
 
-  protected theorem equiv.is_equivalence (A : Type) (B : A → Type) : equivalence (@equiv A B) :=
-  mk_equivalence (@equiv A B) (@equiv.refl A B) (@equiv.symm A B) (@equiv.trans A B)
+  protected theorem equiv.is_equivalence (A : Type) (B : A → Type) : equivalence (@function.equiv A B) :=
+  mk_equivalence (@function.equiv A B) (@equiv.refl A B) (@equiv.symm A B) (@equiv.trans A B)
 end function
 
 section
diff --git a/library/init/quot.lean b/library/init/quot.lean
index 66e1ab305..dd5c4c937 100644
--- a/library/init/quot.lean
+++ b/library/init/quot.lean
@@ -49,19 +49,19 @@ namespace quot
 
   protected lemma indep_coherent (f : Π a, B ⟦a⟧)
                        (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b)
-                       : ∀ a b, a ≈ b → indep f a = indep f b  :=
+                       : ∀ a b, a ≈ b → quot.indep f a = quot.indep f b  :=
   λa b e, sigma.equal (sound e) (H a b e)
 
   protected lemma lift_indep_pr1
     (f : Π a, B ⟦a⟧) (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b)
-    (q : quot s) : (lift (indep f) (indep_coherent f H) q).1 = q  :=
+    (q : quot s) : (lift (quot.indep f) (quot.indep_coherent f H) q).1 = q  :=
   quot.ind (λ a, by esimp) q
 
   protected definition rec [reducible]
      (f : Π a, B ⟦a⟧) (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b)
      (q : quot s) : B q :=
-  let p := lift (indep f) (indep_coherent f H) q in
-  eq.rec_on (lift_indep_pr1 f H q) (p.2)
+  let p := lift (quot.indep f) (quot.indep_coherent f H) q in
+  eq.rec_on (quot.lift_indep_pr1 f H q) (p.2)
 
   protected definition rec_on [reducible]
      (q : quot s) (f : Π a, B ⟦a⟧) (H : ∀ (a b : A) (p : a ≈ b), eq.rec (f a) (sound p) = f b) : B q :=
@@ -126,8 +126,8 @@ namespace quot
   protected definition hrec_on₂ [reducible]
      {C : quot s₁ → quot s₂ → Type₁} (q₁ : quot s₁) (q₂ : quot s₂)
      (f : Π a b, C ⟦a⟧ ⟦b⟧) (c : ∀ a₁ a₂ b₁ b₂, a₁ ≈ b₁ → a₂ ≈ b₂ → f a₁ a₂ == f b₁ b₂) : C q₁ q₂:=
-  hrec_on q₁
-    (λ a, hrec_on q₂ (λ b, f a b) (λ b₁ b₂ p, c _ _ _ _ !setoid.refl p))
+  quot.hrec_on q₁
+    (λ a, quot.hrec_on q₂ (λ b, f a b) (λ b₁ b₂ p, c _ _ _ _ !setoid.refl p))
     (λ a₁ a₂ p, quot.induction_on q₂
       (λ b,
         have aux : f a₁ b == f a₂ b, from c _ _ _ _ p !setoid.refl,
diff --git a/src/frontends/lean/decl_cmds.cpp b/src/frontends/lean/decl_cmds.cpp
index 4f2150d5e..e0e86f610 100644
--- a/src/frontends/lean/decl_cmds.cpp
+++ b/src/frontends/lean/decl_cmds.cpp
@@ -152,8 +152,12 @@ static environment declare_var(parser & p, environment env,
             env = module::add(env, check(env, mk_constant_assumption(full_n, ls, new_type)));
             p.add_decl_index(full_n, pos, get_variable_tk(), new_type);
         }
-        if (!ns.is_anonymous())
-            env = add_expr_alias(env, n, full_n);
+        if (!ns.is_anonymous()) {
+            if (is_protected)
+                env = add_expr_alias(env, get_protected_shortest_name(full_n), full_n);
+            else
+                env = add_expr_alias(env, n, full_n);
+        }
         if (is_protected)
             env = add_protected(env, full_n);
         return env;
@@ -1101,8 +1105,12 @@ class definition_cmd_fn {
                 m_p.add_decl_index(real_n, m_pos, m_p.get_cmd_token(), type);
             if (m_is_protected)
                 m_env = add_protected(m_env, real_n);
-            if (n != real_n)
-                m_env = add_expr_alias_rec(m_env, n, real_n);
+            if (n != real_n) {
+                if (m_is_protected)
+                    m_env = add_expr_alias_rec(m_env, get_protected_shortest_name(real_n), real_n);
+                else
+                    m_env = add_expr_alias_rec(m_env, n, real_n);
+            }
             if (m_kind == Abbreviation || m_kind == LocalAbbreviation) {
                 bool persistent = m_kind == Abbreviation;
                 m_env = add_abbreviation(m_env, real_n, m_attributes.m_is_parsing_only, persistent);
diff --git a/src/library/protected.cpp b/src/library/protected.cpp
index 91057b9bc..1cfb79d50 100644
--- a/src/library/protected.cpp
+++ b/src/library/protected.cpp
@@ -53,6 +53,15 @@ bool is_protected(environment const & env, name const & n) {
     return get_extension(env).m_protected.contains(n);
 }
 
+name get_protected_shortest_name(name const & n) {
+    if (n.is_atomic() || n.get_prefix().is_atomic()) {
+        return n;
+    } else {
+        name new_prefix = n.get_prefix().replace_prefix(n.get_prefix().get_prefix(), name());
+        return n.replace_prefix(n.get_prefix(), new_prefix);
+    }
+}
+
 void initialize_protected() {
     g_ext     = new protected_ext_reg();
     g_prt_key = new std::string("prt");
diff --git a/src/library/protected.h b/src/library/protected.h
index af427ce3f..2e049ff7d 100644
--- a/src/library/protected.h
+++ b/src/library/protected.h
@@ -12,6 +12,8 @@ namespace lean {
 environment add_protected(environment const & env, name const & n);
 /** \brief Return true iff \c n was marked as protected in the environment \c n. */
 bool is_protected(environment const & env, name const & n);
+/** \brief Return the shortest name that can be used to reference the given name */
+name get_protected_shortest_name(name const & n);
 
 void initialize_protected();
 void finalize_protected();
diff --git a/tests/lean/hott/433.hlean b/tests/lean/hott/433.hlean
index 7667c9616..16ad87e56 100644
--- a/tests/lean/hott/433.hlean
+++ b/tests/lean/hott/433.hlean
@@ -95,7 +95,7 @@ namespace pi
   definition transport_V [reducible] (P : A → Type) {x y : A} (p : x = y) (u : P y) : P x :=
   p⁻¹ ▸ u
 
-  protected definition functor_pi : (Π(a:A), B a) → (Π(a':A'), B' a') := (λg a', f1 a' (g (f0 a')))
+  definition functor_pi : (Π(a:A), B a) → (Π(a':A'), B' a') := (λg a', f1 a' (g (f0 a')))
   /- Equivalences -/
   definition isequiv_functor_pi [instance] (f0 : A' → A) (f1 : Π(a':A'), B (f0 a') → B' a')
     [H0 : is_equiv f0] [H1 : Πa', @is_equiv (B (f0 a')) (B' a') (f1 a')]
diff --git a/tests/lean/protected_consts.lean b/tests/lean/protected_consts.lean
index b5e15624c..61d8f0043 100644
--- a/tests/lean/protected_consts.lean
+++ b/tests/lean/protected_consts.lean
@@ -1,10 +1,10 @@
 namespace foo
   protected axiom A : Prop
   axiom B : Prop
-  protected constant a : A
+  protected constant a : foo.A
   constant b : B
   protected axioms (A₁ A₂ : Prop)
-  protected constants (a₁ a₂ : A)
+  protected constants (a₁ a₂ : foo.A)
   axioms (B₁ B₂ : Prop)
   constants (b₁ b₂ : B)
 end foo