feat(hott/types): add characterization of lift, prove that Type.{u} is not an hset

hott/init/equiv.hlean

@@ -45,18 +45,19 @@ namespace is_equiv
   is_equiv.mk id id (λa, idp) (λa, idp) (λa, idp)
 
   -- The composition of two equivalences is, again, an equivalence.
-  definition is_equiv_compose [Hf : is_equiv f] [Hg : is_equiv g] : is_equiv (g ∘ f) :=
+  definition is_equiv_compose [constructor] [Hf : is_equiv f] [Hg : is_equiv g]
-    is_equiv.mk (g ∘ f) (f⁻¹ ∘ g⁻¹)
+    : is_equiv (g ∘ f) :=
-               (λc, ap g (right_inv f (g⁻¹ c)) ⬝ right_inv g c)
+  is_equiv.mk (g ∘ f) (f⁻¹ ∘ g⁻¹)
-               (λa, ap (inv f) (left_inv g (f a)) ⬝ left_inv f a)
+             (λc, ap g (right_inv f (g⁻¹ c)) ⬝ right_inv g c)
-               (λa, (whisker_left _ (adj g (f a))) ⬝
+             (λa, ap (inv f) (left_inv g (f a)) ⬝ left_inv f a)
-                    (ap_con g _ _)⁻¹ ⬝
+             (λa, (whisker_left _ (adj g (f a))) ⬝
-                    ap02 g ((ap_con_eq_con (right_inv f) (left_inv g (f a)))⁻¹ ⬝
+                  (ap_con g _ _)⁻¹ ⬝
-                            (ap_compose f (inv f) _ ◾  adj f a) ⬝
+                  ap02 g ((ap_con_eq_con (right_inv f) (left_inv g (f a)))⁻¹ ⬝
-                            (ap_con f _ _)⁻¹
+                          (ap_compose f (inv f) _ ◾  adj f a) ⬝
-                           ) ⬝
+                          (ap_con f _ _)⁻¹
-                    (ap_compose g f _)⁻¹
+                         ) ⬝
-               )
+                  (ap_compose g f _)⁻¹
+             )
 
   -- Any function equal to an equivalence is an equivlance as well.
   variable {f}
@@ -106,7 +107,7 @@ namespace is_equiv
   end
 
   -- Any function pointwise equal to an equivalence is an equivalence as well.
-  definition homotopy_closed [constructor] {A B : Type} {f f' : A → B} [Hf : is_equiv f]
+  definition homotopy_closed [constructor] {A B : Type} (f : A → B) {f' : A → B} [Hf : is_equiv f]
     (Hty : f ~ f') : is_equiv f' :=
   adjointify f'
              (inv f)
@@ -120,7 +121,7 @@ namespace is_equiv
              (λ b, ap f !Hty⁻¹ ⬝ right_inv f b)
              (λ a, !Hty⁻¹ ⬝ left_inv f a)
 
-  definition is_equiv_up [instance] (A : Type) : is_equiv (up : A → lift A) :=
+  definition is_equiv_up [instance] [constructor] (A : Type) : is_equiv (up : A → lift A) :=
   adjointify up down (λa, by induction a;reflexivity) (λa, idp)
 
   section
@@ -128,7 +129,7 @@ namespace is_equiv
   include Hf
 
   --The inverse of an equivalence is, again, an equivalence.
-  definition is_equiv_inv [instance] : is_equiv f⁻¹ :=
+  definition is_equiv_inv [instance] [constructor] : is_equiv f⁻¹ :=
   adjointify f⁻¹ f (left_inv f) (right_inv f)
 
   definition cancel_right (g : B → C) [Hgf : is_equiv (g ∘ f)] : (is_equiv g) :=
@@ -201,8 +202,9 @@ namespace is_equiv
   end
 
   --Transporting is an equivalence
-  definition is_equiv_tr [instance] {A : Type} (P : A → Type) {x y : A} (p : x = y) : (is_equiv (transport P p)) :=
+  definition is_equiv_tr [instance] [constructor] {A : Type} (P : A → Type) {x y : A} (p : x = y)
-    is_equiv.mk _ (transport P p⁻¹) (tr_inv_tr p) (inv_tr_tr p) (tr_inv_tr_lemma p)
+    : (is_equiv (transport P p)) :=
+  is_equiv.mk _ (transport P p⁻¹) (tr_inv_tr p) (inv_tr_tr p) (tr_inv_tr_lemma p)
 
 end is_equiv
 open is_equiv
@@ -241,10 +243,10 @@ namespace equiv
   protected definition refl [refl] [constructor] : A ≃ A :=
   equiv.mk id !is_equiv_id
 
-  protected definition symm [symm] [unfold 3] (f : A ≃ B) : B ≃ A :=
+  protected definition symm [symm] (f : A ≃ B) : B ≃ A :=
   equiv.mk f⁻¹ !is_equiv_inv
 
-  protected definition trans [trans] (f : A ≃ B) (g: B ≃ C) : A ≃ C :=
+  protected definition trans [trans] (f : A ≃ B) (g : B ≃ C) : A ≃ C :=
   equiv.mk (g ∘ f) !is_equiv_compose
 
   infixl `⬝e`:75 := equiv.trans
@@ -252,8 +254,12 @@ namespace equiv
     -- notation for inverse which is not overloaded
   abbreviation erfl [constructor] := @equiv.refl
 
+  definition to_inv_trans [reducible] [unfold-full] (f : A ≃ B) (g : B ≃ C)
+    : to_inv (f ⬝e g) = to_fun (g⁻¹ᵉ ⬝e f⁻¹ᵉ) :=
+  idp
+
   definition equiv_change_fun [constructor] (f : A ≃ B) {f' : A → B} (Heq : f ~ f') : A ≃ B :=
-  equiv.mk f' (is_equiv.homotopy_closed Heq)
+  equiv.mk f' (is_equiv.homotopy_closed f Heq)
 
   definition equiv_change_inv [constructor] (f : A ≃ B) {f' : B → A} (Heq : f⁻¹ ~ f')
     : A ≃ B :=

hott/init/ua.hlean

@@ -15,11 +15,16 @@ section
   universe variable l
   variables {A B : Type.{l}}
 
-  definition is_equiv_cast_of_eq (H : A = B) : is_equiv (cast H) :=
+  definition is_equiv_cast_of_eq [constructor] (H : A = B) : is_equiv (cast H) :=
-    (@is_equiv_tr Type (λX, X) A B H)
+  is_equiv_tr (λX, X) H
+
+  definition equiv_of_eq [constructor] (H : A = B) : A ≃ B :=
+  equiv.mk _ (is_equiv_cast_of_eq H)
+
+  definition equiv_of_eq_refl [reducible] [unfold-full] (A : Type)
+    : equiv_of_eq (refl A) = equiv.refl :=
+  idp
 
-  definition equiv_of_eq (H : A = B) : A ≃ B :=
-    equiv.mk _ (is_equiv_cast_of_eq H)
 
 end
 
@@ -50,6 +55,9 @@ definition eq_of_equiv_lift {A B : Type} (f : A ≃ B) : A = lift B :=
 ua (f ⬝e !equiv_lift)
 
 namespace equiv
+  definition ua_refl (A : Type) : ua erfl = idpath A :=
+  eq_of_fn_eq_fn !eq_equiv_equiv (right_inv !eq_equiv_equiv erfl)
+
   -- One consequence of UA is that we can transport along equivalencies of types
   -- We can use this for calculation evironments
   protected definition transport_of_equiv [subst] (P : Type → Type) {A B : Type} (H : A ≃ B)

hott/types/bool.hlean

@@ -29,9 +29,6 @@ namespace bool
   assert H2 : bnot = id, from !cast_ua_fn⁻¹ ⬝ ap cast H,
   absurd (ap10 H2 tt) ff_ne_tt
 
-  definition not_is_hset_type : ¬is_hset Type₀ :=
-  assume H : is_hset Type₀,
-  absurd !is_hset.elim eq_bnot_ne_idp
 
   definition bool_equiv_option_unit : bool ≃ option unit :=
   begin

hott/types/equiv.hlean

@@ -109,7 +109,7 @@ end is_equiv
 
 namespace equiv
   open is_equiv
-  variables {A B : Type}
+  variables {A B C : Type}
 
   definition equiv_mk_eq {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f')
       : equiv.mk f H = equiv.mk f' H' :=
@@ -118,6 +118,15 @@ namespace equiv
   definition equiv_eq {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' :=
   by (cases f; cases f'; apply (equiv_mk_eq p))
 
+  definition equiv_eq' {f f' : A ≃ B} (p : to_fun f ~ to_fun f') : f = f' :=
+  by apply equiv_eq;apply eq_of_homotopy p
+
+  definition trans_symm (f : A ≃ B) (g : B ≃ C) : (f ⬝e g)⁻¹ᵉ = g⁻¹ᵉ ⬝e f⁻¹ᵉ :> (C ≃ A) :=
+  equiv_eq idp
+
+  definition symm_symm (f : A ≃ B) : f⁻¹ᵉ⁻¹ᵉ = f :> (A ≃ B) :=
+  equiv_eq idp
+
   protected definition equiv.sigma_char [constructor]
     (A B : Type) : (A ≃ B) ≃ Σ(f : A → B), is_equiv f :=
   begin

hott/types/lift.hlean

@@ -0,0 +1,141 @@
+/-
+Copyright (c) 2015 Floris van Doorn. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Floris van Doorn
+
+Theorems about lift
+-/
+
+import ..function
+open eq equiv equiv.ops is_equiv is_trunc
+
+namespace lift
+
+  universe variables u v
+  variables {A : Type.{u}} (z z' : lift.{u v} A)
+
+  protected definition eta : up (down z) = z :=
+  by induction z; reflexivity
+
+  protected definition code [unfold 2 3] : lift A → lift A → Type
+  | code (up a) (up a') := a = a'
+
+  protected definition decode [unfold 2 3] : Π(z z' : lift A), lift.code z z' → z = z'
+  | decode (up a) (up a') := λc, ap up c
+
+  variables {z z'}
+  protected definition encode [unfold 3 4 5] (p : z = z') : lift.code z z' :=
+  by induction p; induction z; esimp
+
+  variables (z z')
+  definition lift_eq_equiv : (z = z') ≃ lift.code z z' :=
+  equiv.MK lift.encode
+           !lift.decode
+           abstract begin
+             intro c, induction z with a, induction z' with a', esimp at *, induction c,
+             reflexivity
+           end end
+           abstract begin
+             intro p, induction p, induction z, reflexivity
+           end end
+
+
+  section
+  variables {a a' : A}
+  definition eq_of_up_eq_up [unfold 4] (p : up a = up a') : a = a' :=
+  lift.encode p
+
+  definition lift_transport {P : A → Type} (p : a = a') (z : lift (P a))
+    : p ▸ z = up (p ▸ down z) :=
+  by induction p; induction z; reflexivity
+  end
+
+  variables {A' : Type} (f : A → A') (g : lift A → lift A')
+  definition lift_functor [unfold 4] : lift A → lift A'
+  | lift_functor (up a) := up (f a)
+
+  definition is_equiv_lift_functor [constructor] [Hf : is_equiv f] : is_equiv (lift_functor f) :=
+  adjointify (lift_functor f)
+             (lift_functor f⁻¹)
+             abstract begin
+               intro z', induction z' with a',
+               esimp, exact ap up !right_inv
+             end end
+             abstract begin
+               intro z, induction z with a,
+               esimp, exact ap up !left_inv
+             end end
+
+  definition lift_equiv_lift_of_is_equiv [constructor] [Hf : is_equiv f] : lift A ≃ lift A' :=
+  equiv.mk _ (is_equiv_lift_functor f)
+
+  definition lift_equiv_lift [constructor] (f : A ≃ A') : lift A ≃ lift A' :=
+  equiv.mk _ (is_equiv_lift_functor f)
+
+  definition lift_equiv_lift_refl (A : Type) : lift_equiv_lift (erfl : A ≃ A) = erfl :=
+  by apply equiv_eq'; intro z; induction z with a; reflexivity
+
+  definition lift_inv_functor [unfold-full] (a : A) : A' :=
+  down (g (up a))
+
+  definition is_equiv_lift_inv_functor [constructor] [Hf : is_equiv g]
+    : is_equiv (lift_inv_functor g) :=
+  adjointify (lift_inv_functor g)
+             (lift_inv_functor g⁻¹)
+             abstract begin
+               intro z', rewrite [▸*,lift.eta,right_inv g],
+             end end
+             abstract begin
+               intro z', rewrite [▸*,lift.eta,left_inv g],
+             end end
+
+  definition equiv_of_lift_equiv_lift [constructor] (g : lift A ≃ lift A') : A ≃ A' :=
+  equiv.mk _ (is_equiv_lift_inv_functor g)
+
+  definition lift_functor_left_inv  : lift_inv_functor (lift_functor f) = f :=
+  eq_of_homotopy (λa, idp)
+
+  definition lift_functor_right_inv : lift_functor (lift_inv_functor g) = g :=
+  begin
+    apply eq_of_homotopy, intro z, induction z with a, esimp, apply lift.eta
+  end
+
+  variables (A A')
+  definition is_equiv_lift_functor_fn [constructor]
+    : is_equiv (lift_functor : (A → A') → (lift A → lift A')) :=
+  adjointify lift_functor
+             lift_inv_functor
+             lift_functor_right_inv
+             lift_functor_left_inv
+
+  definition lift_imp_lift_equiv [constructor] : (lift A → lift A') ≃ (A → A') :=
+  (equiv.mk _ (is_equiv_lift_functor_fn A A'))⁻¹ᵉ
+
+  -- can we deduce this from lift_imp_lift_equiv?
+  definition lift_equiv_lift_equiv [constructor] : (lift A ≃ lift A') ≃ (A ≃ A') :=
+  equiv.MK equiv_of_lift_equiv_lift
+           lift_equiv_lift
+           abstract begin
+             intro f, apply equiv_eq, reflexivity
+           end end
+           abstract begin
+             intro g, apply equiv_eq, esimp, apply eq_of_homotopy, intro z,
+             induction z with a, esimp, apply lift.eta
+           end end
+
+  definition lift_eq_lift_equiv.{u1 u2} (A A' : Type.{u1})
+    : (lift.{u1 u2} A = lift.{u1 u2} A') ≃ (A = A') :=
+  !eq_equiv_equiv ⬝e !lift_equiv_lift_equiv ⬝e !eq_equiv_equiv⁻¹ᵉ
+
+  definition is_embedding_lift [instance] : is_embedding lift :=
+  begin
+    apply is_embedding.mk, intro A A', fapply is_equiv.homotopy_closed,
+      exact to_inv !lift_eq_lift_equiv,
+      exact _,
+    { intro p, induction p,
+      esimp [lift_eq_lift_equiv,equiv.trans,equiv.symm,eq_equiv_equiv],
+      rewrite [equiv_of_eq_refl,lift_equiv_lift_refl],
+      apply ua_refl}
+  end
+
+end lift

hott/types/prod.hlean

@@ -68,14 +68,14 @@ namespace prod
   definition prod_eq_unc_eta (p : u = v) : prod_eq_unc (p..1, p..2) = p :=
   prod_eq_eta p
 
-  definition is_equiv_prod_eq [instance] (u v : A × B)
+  definition is_equiv_prod_eq [instance] [constructor] (u v : A × B)
     : is_equiv (prod_eq_unc : u.1 = v.1 × u.2 = v.2 → u = v) :=
   adjointify prod_eq_unc
              (λp, (p..1, p..2))
              prod_eq_unc_eta
              pair_prod_eq_unc
 
-  definition prod_eq_equiv (u v : A × B) : (u = v) ≃ (u.1 = v.1 × u.2 = v.2) :=
+  definition prod_eq_equiv [constructor] (u v : A × B) : (u = v) ≃ (u.1 = v.1 × u.2 = v.2) :=
   (equiv.mk prod_eq_unc _)⁻¹ᵉ
 
   /- Transport -/
@@ -96,7 +96,7 @@ namespace prod
 
   /- Equivalences -/
 
-  definition is_equiv_prod_functor [instance] [H : is_equiv f] [H : is_equiv g]
+  definition is_equiv_prod_functor [instance] [constructor] [H : is_equiv f] [H : is_equiv g]
     : is_equiv (prod_functor f g) :=
   begin
     apply adjointify _ (prod_functor f⁻¹ g⁻¹),
@@ -104,33 +104,34 @@ namespace prod
       intro u, induction u, rewrite [▸*,left_inv f,left_inv g],
   end
 
-  definition prod_equiv_prod_of_is_equiv [H : is_equiv f] [H : is_equiv g]
+  definition prod_equiv_prod_of_is_equiv [constructor] [H : is_equiv f] [H : is_equiv g]
     : A × B ≃ A' × B' :=
   equiv.mk (prod_functor f g) _
 
-  definition prod_equiv_prod (f : A ≃ A') (g : B ≃ B') : A × B ≃ A' × B' :=
+  definition prod_equiv_prod [constructor] (f : A ≃ A') (g : B ≃ B') : A × B ≃ A' × B' :=
   equiv.mk (prod_functor f g) _
 
-  definition prod_equiv_prod_left (g : B ≃ B') : A × B ≃ A × B' :=
+  definition prod_equiv_prod_left [constructor] (g : B ≃ B') : A × B ≃ A × B' :=
   prod_equiv_prod equiv.refl g
 
-  definition prod_equiv_prod_right (f : A ≃ A') : A × B ≃ A' × B :=
+  definition prod_equiv_prod_right [constructor] (f : A ≃ A') : A × B ≃ A' × B :=
   prod_equiv_prod f equiv.refl
 
   /- Symmetry -/
 
-  definition is_equiv_flip [instance] (A B : Type) : is_equiv (@flip A B) :=
+  definition is_equiv_flip [instance] [constructor] (A B : Type)
+    : is_equiv (flip : A × B → B × A) :=
   adjointify flip
              flip
              (λu, destruct u (λb a, idp))
              (λu, destruct u (λa b, idp))
 
-  definition prod_comm_equiv (A B : Type) : A × B ≃ B × A :=
+  definition prod_comm_equiv [constructor] (A B : Type) : A × B ≃ B × A :=
   equiv.mk flip _
 
   /- Associativity -/
 
-  definition prod_assoc_equiv (A B C : Type) : A × (B × C) ≃ (A × B) × C :=
+  definition prod_assoc_equiv [constructor] (A B C : Type) : A × (B × C) ≃ (A × B) × C :=
   begin
     fapply equiv.MK,
     { intro z, induction z with a z, induction z with b c, exact (a, b, c)},
@@ -139,24 +140,24 @@ namespace prod
     { intro z, induction z with a z, induction z with b c, reflexivity},
   end
 
-  definition prod_contr_equiv (A B : Type) [H : is_contr B] : A × B ≃ A :=
+  definition prod_contr_equiv [constructor] (A B : Type) [H : is_contr B] : A × B ≃ A :=
   equiv.MK pr1
            (λx, (x, !center))
            (λx, idp)
            (λx, by cases x with a b; exact pair_eq idp !center_eq)
 
-  definition prod_unit_equiv (A : Type) : A × unit ≃ A :=
+  definition prod_unit_equiv [constructor] (A : Type) : A × unit ≃ A :=
   !prod_contr_equiv
 
   /- Universal mapping properties -/
-  definition is_equiv_prod_rec [instance] (P : A × B → Type)
+  definition is_equiv_prod_rec [instance] [constructor] (P : A × B → Type)
     : is_equiv (prod.rec : (Πa b, P (a, b)) → Πu, P u) :=
   adjointify _
              (λg a b, g (a, b))
              (λg, eq_of_homotopy (λu, by induction u;reflexivity))
              (λf, idp)
 
-  definition equiv_prod_rec (P : A × B → Type) : (Πa b, P (a, b)) ≃ (Πu, P u) :=
+  definition equiv_prod_rec [constructor] (P : A × B → Type) : (Πa b, P (a, b)) ≃ (Πu, P u) :=
   equiv.mk prod.rec _
 
   definition imp_imp_equiv_prod_imp (A B C : Type) : (A → B → C) ≃ (A × B → C) :=
@@ -166,17 +167,19 @@ namespace prod
     : P a × Q a :=
   (u.1 a, u.2 a)
 
-  definition is_equiv_prod_corec (P Q : A → Type)
+  definition is_equiv_prod_corec [constructor] (P Q : A → Type)
     : is_equiv (prod_corec_unc : (Πa, P a) × (Πa, Q a) → Πa, P a × Q a) :=
   adjointify _
              (λg, (λa, (g a).1, λa, (g a).2))
              (by intro g; apply eq_of_homotopy; intro a; esimp; induction (g a); reflexivity)
              (by intro h; induction h with f g; reflexivity)
 
-  definition equiv_prod_corec (P Q : A → Type) : ((Πa, P a) × (Πa, Q a)) ≃ (Πa, P a × Q a) :=
+  definition equiv_prod_corec [constructor] (P Q : A → Type)
+    : ((Πa, P a) × (Πa, Q a)) ≃ (Πa, P a × Q a) :=
   equiv.mk _ !is_equiv_prod_corec
 
-  definition imp_prod_imp_equiv_imp_prod (A B C : Type) : (A → B) × (A → C) ≃ (A → (B × C)) :=
+  definition imp_prod_imp_equiv_imp_prod [constructor] (A B C : Type)
+    : (A → B) × (A → C) ≃ (A → (B × C)) :=
   !equiv_prod_corec
 
   definition is_trunc_prod (A B : Type) (n : trunc_index) [HA : is_trunc n A] [HB : is_trunc n B]

hott/types/sum.hlean

@@ -31,7 +31,7 @@ namespace sum
   by induction p; induction z; all_goals exact up idp
 
   variables (z z')
-  definition sum_eq_equiv : (z = z') ≃ sum.code z z' :=
+  definition sum_eq_equiv [constructor] : (z = z') ≃ sum.code z z' :=
   equiv.MK sum.encode
            !sum.decode
            abstract begin
@@ -63,7 +63,8 @@ namespace sum
   | sum_functor (inl a) := inl (f a)
   | sum_functor (inr b) := inr (g b)
 
-  definition is_equiv_sum_functor [Hf : is_equiv f] [Hg : is_equiv g] : is_equiv (sum_functor f g) :=
+  definition is_equiv_sum_functor [constructor] [Hf : is_equiv f] [Hg : is_equiv g]
+    : is_equiv (sum_functor f g) :=
   adjointify (sum_functor f   g)
              (sum_functor f⁻¹ g⁻¹)
              abstract begin
@@ -75,23 +76,24 @@ namespace sum
                all_goals (esimp; (apply ap inl | apply ap inr); apply right_inv)
              end end
 
-  definition sum_equiv_sum_of_is_equiv [Hf : is_equiv f] [Hg : is_equiv g] : A + B ≃ A' + B' :=
+  definition sum_equiv_sum_of_is_equiv [constructor] [Hf : is_equiv f] [Hg : is_equiv g]
+    : A + B ≃ A' + B' :=
   equiv.mk _ (is_equiv_sum_functor f g)
 
-  definition sum_equiv_sum (f : A ≃ A') (g : B ≃ B') : A + B ≃ A' + B' :=
+  definition sum_equiv_sum [constructor] (f : A ≃ A') (g : B ≃ B') : A + B ≃ A' + B' :=
   equiv.mk _ (is_equiv_sum_functor f g)
 
-  definition sum_equiv_sum_left (g : B ≃ B') : A + B ≃ A + B' :=
+  definition sum_equiv_sum_left [constructor] (g : B ≃ B') : A + B ≃ A + B' :=
   sum_equiv_sum equiv.refl g
 
-  definition sum_equiv_sum_right (f : A ≃ A') : A + B ≃ A' + B :=
+  definition sum_equiv_sum_right [constructor] (f : A ≃ A') : A + B ≃ A' + B :=
   sum_equiv_sum f equiv.refl
 
-  definition flip : A + B → B + A
+  definition flip [unfold 3] : A + B → B + A
   | flip (inl a) := inr a
   | flip (inr b) := inl b
 
-  definition sum_comm_equiv (A B : Type) : A + B ≃ B + A :=
+  definition sum_comm_equiv [constructor] (A B : Type) : A + B ≃ B + A :=
   begin
     fapply equiv.MK,
       exact flip,
@@ -107,7 +109,7 @@ namespace sum
   --     all_goals try (repeat (apply inl | apply inr | assumption); now),
   -- end
 
-  definition sum_empty_equiv (A : Type) : A + empty ≃ A :=
+  definition sum_empty_equiv [constructor] (A : Type) : A + empty ≃ A :=
   begin
     fapply equiv.MK,
       intro z, induction z, assumption, contradiction,
@@ -119,7 +121,7 @@ namespace sum
   definition sum_rec_unc {P : A + B → Type} (fg : (Πa, P (inl a)) × (Πb, P (inr b))) : Πz, P z :=
   sum.rec fg.1 fg.2
 
-  definition is_equiv_sum_rec (P : A + B → Type)
+  definition is_equiv_sum_rec [constructor] (P : A + B → Type)
     : is_equiv (sum_rec_unc : (Πa, P (inl a)) × (Πb, P (inr b)) → Πz, P z) :=
   begin
      apply adjointify sum_rec_unc (λf, (λa, f (inl a), λb, f (inr b))),
@@ -127,13 +129,15 @@ namespace sum
        intro h, induction h with f g, reflexivity
   end
 
-  definition equiv_sum_rec (P : A + B → Type) : (Πa, P (inl a)) × (Πb, P (inr b)) ≃ Πz, P z :=
+  definition equiv_sum_rec [constructor] (P : A + B → Type)
+    : (Πa, P (inl a)) × (Πb, P (inr b)) ≃ Πz, P z :=
   equiv.mk _ !is_equiv_sum_rec
 
-  definition imp_prod_imp_equiv_sum_imp (A B C : Type) : (A → C) × (B → C) ≃ (A + B → C) :=
+  definition imp_prod_imp_equiv_sum_imp [constructor] (A B C : Type)
+    : (A → C) × (B → C) ≃ (A + B → C) :=
   !equiv_sum_rec
 
-  definition is_trunc_sum  (n : trunc_index) [HA : is_trunc (n.+2) A]  [HB : is_trunc (n.+2) B]
+  definition is_trunc_sum (n : trunc_index) [HA : is_trunc (n.+2) A]  [HB : is_trunc (n.+2) B]
     : is_trunc (n.+2) (A + B) :=
   begin
     apply is_trunc_succ_intro, intro z z',
@@ -150,7 +154,8 @@ namespace sum
   definition sigma_bool_of_sum {A B : Type} (z : A + B) : Σ(b : bool), bool.rec A B b :=
   by induction z with a b; exact ⟨ff, a⟩; exact ⟨tt, b⟩
 
-  definition sum_equiv_sigma_bool (A B : Type) : A + B ≃ Σ(b : bool), bool.rec A B b :=
+  definition sum_equiv_sigma_bool [constructor] (A B : Type)
+    : A + B ≃ Σ(b : bool), bool.rec A B b :=
   equiv.MK sigma_bool_of_sum
            sum_of_sigma_bool
            begin intro v, induction v with b x, induction b, all_goals reflexivity end

hott/types/univ.hlean

@@ -0,0 +1,26 @@
+/-
+Copyright (c) 2015 Floris van Doorn. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Floris van Doorn
+
+Theorems about the universe
+-/
+
+-- see also init.ua
+
+import .bool .trunc .lift
+
+open is_trunc bool lift unit
+
+namespace univ
+
+  definition not_is_hset_type0 : ¬is_hset Type₀ :=
+  assume H : is_hset Type₀,
+  absurd !is_hset.elim eq_bnot_ne_idp
+
+  definition not_is_hset_type.{u} : ¬is_hset Type.{u} :=
+  assume H : is_hset Type,
+  absurd (is_trunc_is_embedding_closed lift star) not_is_hset_type0
+
+
+end univ