feat(hott): replace assert by have and merge namespace equiv.ops into equiv

The coercion A ≃ B -> (A -> B) is now in namespace equiv. The notation ⁻¹ for symmetry of equivalences is not supported anymore. Use ⁻¹ᵉ

hott/algebra/binary.hlean

@@ -104,7 +104,7 @@ namespace is_equiv
   end
 end is_equiv
 namespace equiv
-  open is_equiv equiv.ops
+  open is_equiv
   definition inv_preserve_binary {A B : Type} (f : A ≃ B)
     (mA : A → A → A) (mB : B → B → B) (H : Π(a a' : A), mB (f a) (f a') = f (mA a a'))
     (b b' : B) : f⁻¹ (mB b b') = mA (f⁻¹ b) (f⁻¹ b') :=

hott/algebra/category/category.hlean

@@ -6,7 +6,7 @@ Author: Jakob von Raumer
 
 import .iso
 
-open iso is_equiv equiv eq is_trunc sigma equiv.ops
+open iso is_equiv equiv eq is_trunc sigma
 
 /-
   A category is a precategory extended by a witness

hott/algebra/category/functor/yoneda.hlean

@@ -8,10 +8,12 @@ Yoneda embedding and Yoneda lemma
 
 import .examples .attributes
 
-open category eq functor prod.ops is_trunc iso is_equiv equiv category.set nat_trans lift
+open category eq functor prod.ops is_trunc iso is_equiv category.set nat_trans lift
 
 namespace yoneda
 
+  universe variables u v
+  variable {C : Precategory.{u v}}
   /-
     These attributes make sure that the fields of the category "set" reduce to the right things
     However, we don't want to have them globally, because that will unfold the composition g ∘ f
@@ -28,7 +30,7 @@ namespace yoneda
     we use (change_fun) to make sure that (to_fun_ob (yoneda_embedding C) c) will reduce to
     (hom_functor_left c) instead of (functor_curry_rev_ob (hom_functor C) c)
   -/
-  definition yoneda_embedding [constructor] (C : Precategory) : C ⇒ cset ^c Cᵒᵖ :=
+  definition yoneda_embedding [constructor] (C : Precategory.{u v}) : C ⇒ cset ^c Cᵒᵖ :=
 --(functor_curry_rev !hom_functor)
   change_fun
     (functor_curry_rev !hom_functor)
@@ -39,17 +41,30 @@ namespace yoneda
 
   notation `ɏ` := yoneda_embedding _
 
-  definition yoneda_lemma_hom [constructor] {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ cset)
+  definition yoneda_lemma_hom_fun [unfold_full] (c : C) (F : Cᵒᵖ ⇒ cset)
+    (x : trunctype.carrier (F c)) (c' : Cᵒᵖ) : to_fun_ob (ɏ c) c' ⟶ F c' :=
+  begin
+    esimp [yoneda_embedding], intro f, exact F f x
+  end
+
+  definition yoneda_lemma_hom_nat (c : C) (F : Cᵒᵖ ⇒ cset)
+    (x : trunctype.carrier (F c)) {c₁ c₂ : Cᵒᵖ} (f : c₁ ⟶ c₂)
+      : F f ∘ yoneda_lemma_hom_fun c F x c₁ = yoneda_lemma_hom_fun c F x c₂ ∘ to_fun_hom (ɏ c) f :=
+  begin
+    esimp [yoneda_embedding], apply eq_of_homotopy, intro f',
+    refine _ ⬝ ap (λy, to_fun_hom F y x) !(@id_left _ C)⁻¹,
+    exact ap10 !(@respect_comp Cᵒᵖ cset)⁻¹ x
+  end
+
+  definition yoneda_lemma_hom [constructor] (c : C) (F : Cᵒᵖ ⇒ cset)
     (x : trunctype.carrier (F c)) : ɏ c ⟹ F :=
   begin
     fapply nat_trans.mk,
-    { intro c', esimp [yoneda_embedding], intro f, exact F f x},
+    { exact yoneda_lemma_hom_fun c F x},
-    { intro c' c'' f, esimp [yoneda_embedding], apply eq_of_homotopy, intro f',
+    { intro c₁ c₂ f, exact yoneda_lemma_hom_nat c F x f}
-      refine _ ⬝ ap (λy, to_fun_hom F y x) !(@id_left _ C)⁻¹,
-      exact ap10 !(@respect_comp Cᵒᵖ cset)⁻¹ x}
   end
 
-  definition yoneda_lemma_equiv [constructor] {C : Precategory} (c : C)
+  definition yoneda_lemma_equiv [constructor] (c : C)
     (F : Cᵒᵖ ⇒ cset) : hom (ɏ c) F ≃ lift (trunctype.carrier (to_fun_ob F c)) :=
   begin
     fapply equiv.MK,
@@ -64,13 +79,13 @@ namespace yoneda
       rewrite naturality, esimp [yoneda_embedding], rewrite [id_left], apply ap _ !id_left end end},
   end
 
-  definition yoneda_lemma {C : Precategory} (c : C) (F : Cᵒᵖ ⇒ cset) :
+  definition yoneda_lemma (c : C) (F : Cᵒᵖ ⇒ cset) :
     homset (ɏ c) F ≅ functor_lift (F c) :=
   begin
     apply iso_of_equiv, esimp, apply yoneda_lemma_equiv,
   end
 
-  theorem yoneda_lemma_natural_ob {C : Precategory} (F : Cᵒᵖ ⇒ cset) {c c' : C} (f : c' ⟶ c)
+  theorem yoneda_lemma_natural_ob (F : Cᵒᵖ ⇒ cset) {c c' : C} (f : c' ⟶ c)
     (η : ɏ c ⟹ F) :
      to_fun_hom (functor_lift ∘f F) f (to_hom (yoneda_lemma c F) η) =
      to_hom (yoneda_lemma c' F) (η ∘n to_fun_hom ɏ f) :=
@@ -89,7 +104,7 @@ namespace yoneda
   -- attribute yoneda_lemma functor_lift Precategory_Set precategory_Set homset
   --   yoneda_embedding nat_trans.compose functor_nat_trans_compose [reducible]
   -- attribute tlift functor.compose [reducible]
-  theorem yoneda_lemma_natural_functor.{u v} {C : Precategory.{u v}} (c : C) (F F' : Cᵒᵖ ⇒ cset)
+  theorem yoneda_lemma_natural_functor (c : C) (F F' : Cᵒᵖ ⇒ cset)
     (θ : F ⟹ F') (η : to_fun_ob ɏ c ⟹ F) :
      (functor_lift.{v u} ∘fn θ) c (to_hom (yoneda_lemma c F) η) =
      proof to_hom (yoneda_lemma c F') (θ ∘n η) qed :=
@@ -107,20 +122,21 @@ namespace yoneda
   --    proof _ qed :=
   -- by reflexivity
 
-  definition fully_faithful_yoneda_embedding [instance] (C : Precategory) :
+  open equiv
+  definition fully_faithful_yoneda_embedding [instance] (C : Precategory.{u v}) :
     fully_faithful (ɏ : C ⇒ cset ^c Cᵒᵖ) :=
   begin
     intro c c',
     fapply is_equiv_of_equiv_of_homotopy,
     { symmetry, transitivity _, apply @equiv_of_iso (homset _ _),
-      rexact yoneda_lemma c (ɏ c'), esimp [yoneda_embedding], exact !equiv_lift⁻¹ᵉ},
+      exact @yoneda_lemma C c (ɏ c'), esimp [yoneda_embedding], exact !equiv_lift⁻¹ᵉ},
     { intro f, apply nat_trans_eq, intro c, apply eq_of_homotopy, intro f',
       esimp [equiv.symm,equiv.trans],
       esimp [yoneda_lemma,yoneda_embedding,Opposite],
       rewrite [id_left,id_right]}
   end
 
-  definition is_embedding_yoneda_embedding (C : Category) :
+  definition is_embedding_yoneda_embedding (C : Category.{u v}) :
     is_embedding (ɏ : C → Cᵒᵖ ⇒ cset) :=
   begin
     intro c c', fapply is_equiv_of_equiv_of_homotopy,
@@ -134,15 +150,18 @@ namespace yoneda
       rewrite [▸*, category.category.id_left], apply id_right}
   end
 
-  definition is_representable {C : Precategory} (F : Cᵒᵖ ⇒ cset) := Σ(c : C), ɏ c ≅ F
+  definition is_representable (F : Cᵒᵖ ⇒ cset) := Σ(c : C), ɏ c ≅ F
 
   section
     set_option apply.class_instance false
-    definition is_prop_representable {C : Category} (F : Cᵒᵖ ⇒ cset)
+    open functor.ops
+    definition is_prop_representable {C : Category.{u v}} (F : Cᵒᵖ ⇒ cset)
       : is_prop (is_representable F) :=
     begin
       fapply is_trunc_equiv_closed,
-      { exact proof fiber.sigma_char ɏ F qed ⬝e sigma.sigma_equiv_sigma_right (λc, !eq_equiv_iso)},
+      { unfold [is_representable],
+        rexact fiber.sigma_char ɏ F ⬝e sigma.sigma_equiv_sigma_right
+                 (λc, @eq_equiv_iso (cset ^c2 Cᵒᵖ) _ (hom_functor_left c) F)},
       { apply function.is_prop_fiber_of_is_embedding, apply is_embedding_yoneda_embedding}
     end
   end

hott/algebra/category/limits/functor_preserve.hlean

@@ -52,7 +52,8 @@ namespace category
 
   local attribute category.to_precategory [constructor]
 
-  definition preserves_existing_limits_yoneda_embedding_lemma [constructor] (y : cone_obj F)
+  definition preserves_existing_limits_yoneda_embedding_lemma [constructor]
+    (y : cone_obj F)
     [H : is_terminal y] {G : Cᵒᵖ ⇒ cset} (η : constant_functor I G ⟹ ɏ ∘f F) :
     G ⟹ hom_functor_left (cone_to_obj y) :=
   begin
@@ -70,15 +71,14 @@ namespace category
       refine _ ⬝ ap10 (naturality (η i) f) x, rewrite [▸*, id_left]}
       -- abstracting here fails
   end
---  print preserves_existing_limits_yoneda_embedding_lemma_11
 
   theorem preserves_existing_limits_yoneda_embedding (C : Precategory)
     : preserves_existing_limits (yoneda_embedding C) :=
   begin
     intro I F H Gη, induction H with y H, induction Gη with G η, esimp at *,
-    have lem : Π (i : carrier I),
+    have lem : Π(i : carrier I),
     nat_trans_hom_functor_left (natural_map (cone_to_nat y) i)
-      ∘n preserves_existing_limits_yoneda_embedding_lemma y η = natural_map η i,
+      ∘n @preserves_existing_limits_yoneda_embedding_lemma I C F y H G η = natural_map η i,
     begin
         intro i, apply nat_trans_eq, intro c, apply eq_of_homotopy, intro x,
         esimp, refine !assoc ⬝ !id_right ⬝ !to_hom_limit_commute

hott/algebra/category/precategory.hlean

@@ -5,7 +5,7 @@ Authors: Floris van Doorn
 -/
 import types.trunc types.pi arity
 
-open eq is_trunc pi equiv equiv.ops
+open eq is_trunc pi equiv
 
 namespace category
 

hott/algebra/hott.hlean

@@ -8,7 +8,7 @@ Theorems about algebra specific to HoTT
 
 import .group arity types.pi prop_trunc types.unit .bundled
 
-open equiv eq equiv.ops is_trunc unit
+open equiv eq is_trunc unit
 
 namespace algebra
 

hott/cubical/cube.hlean

@@ -8,7 +8,7 @@ Cubes
 
 import .square
 
-open equiv equiv.ops is_equiv sigma sigma.ops
+open equiv is_equiv sigma sigma.ops
 
 namespace eq
 

hott/cubical/squareover.hlean

@@ -8,7 +8,7 @@ Squareovers
 
 import .square
 
-open eq equiv is_equiv equiv.ops
+open eq equiv is_equiv
 
 namespace eq
 

hott/hit/coeq.hlean

@@ -8,7 +8,7 @@ Declaration of the coequalizer
 
 import .quotient_functor types.equiv
 
-open quotient eq equiv equiv.ops is_trunc sigma sigma.ops
+open quotient eq equiv is_trunc sigma sigma.ops
 
 namespace coeq
 section

hott/hit/colimit.hlean

@@ -7,7 +7,7 @@ Definition of general colimits and sequential colimits.
 -/
 
 /- definition of a general colimit -/
-open eq nat quotient sigma equiv equiv.ops is_trunc
+open eq nat quotient sigma equiv is_trunc
 
 namespace colimit
 section

hott/hit/pushout.hlean

@@ -8,7 +8,7 @@ Declaration of the pushout
 
 import .quotient cubical.square types.sigma
 
-open quotient eq sum equiv equiv.ops is_trunc
+open quotient eq sum equiv is_trunc
 
 namespace pushout
 section

hott/hit/quotient.hlean

@@ -16,7 +16,7 @@ See also .set_quotient
 
 import arity cubical.squareover types.arrow cubical.pathover2 types.pointed
 
-open eq equiv sigma sigma.ops equiv.ops pi is_trunc pointed
+open eq equiv sigma sigma.ops pi is_trunc pointed
 
 namespace quotient
 

hott/hit/quotient_functor.hlean

@@ -104,7 +104,6 @@ section
 end
 
 section
-  open equiv.ops
   variables {A : Type} (R : A → A → Type)
              {B : Type} (Q : B → B → Type)
              (f : A ≃ B) (k : Πa a' : A, R a a' ≃ Q (f a) (f a'))

hott/hit/trunc.hlean

@@ -82,7 +82,6 @@ namespace trunc
   λxx, trunc.rec_on xx (λx, ap tr (p x))
 
   section
-    open equiv.ops
     definition trunc_equiv_trunc [constructor] (f : X ≃ Y) : trunc n X ≃ trunc n Y :=
     equiv.mk _ (is_equiv_trunc_functor n f)
   end

hott/homotopy/circle.hlean

@@ -10,7 +10,7 @@ import .sphere
 import types.bool types.int.hott types.equiv
 import algebra.homotopy_group algebra.hott .connectedness
 
-open eq susp bool sphere_index is_equiv equiv equiv.ops is_trunc pi algebra homotopy
+open eq susp bool sphere_index is_equiv equiv is_trunc pi algebra homotopy
 
 definition circle : Type₀ := sphere 1
 
@@ -235,7 +235,7 @@ namespace circle
   preserve_binary_of_inv_preserve base_eq_base_equiv concat (@add ℤ _) decode_add p q
 
   --the carrier of π₁(S¹) is the set-truncation of base = base.
-  open algebra trunc equiv.ops
+  open algebra trunc
 
   definition fg_carrier_equiv_int : π[1](S¹.) ≃ ℤ :=
   trunc_equiv_trunc 0 base_eq_base_equiv ⬝e @(trunc_equiv ℤ _) proof _ qed

hott/homotopy/connectedness.hlean

@@ -275,10 +275,11 @@ namespace homotopy
   end
 
   open trunc_index
+  -- Corollary 8.2.2
   theorem is_conn_sphere [instance] (n : ℕ₋₁) : is_conn (n..-1) (sphere n) :=
   begin
     induction n with n IH,
-    { apply is_trunc_trunc},
+    { apply minus_two_conn},
     { rewrite [succ_sub_one, sphere.sphere_succ], apply is_conn_susp}
   end
 

hott/homotopy/cylinder.hlean

@@ -8,7 +8,7 @@ Declaration of mapping cylinders
 
 import hit.quotient
 
-open quotient eq sum equiv equiv.ops
+open quotient eq sum equiv
 
 namespace cylinder
 section

hott/homotopy/interval.hlean

@@ -7,7 +7,7 @@ Declaration of the interval
 -/
 
 import .susp types.eq types.prod cubical.square
-open eq susp unit equiv equiv.ops is_trunc nat prod
+open eq susp unit equiv is_trunc nat prod
 
 definition interval : Type₀ := susp unit
 

hott/homotopy/susp.hlean

@@ -8,7 +8,7 @@ Declaration of suspension
 
 import hit.pushout types.pointed cubical.square
 
-open pushout unit eq equiv equiv.ops
+open pushout unit eq equiv
 
 definition susp (A : Type) : Type := pushout (λ(a : A), star) (λ(a : A), star)
 
@@ -220,7 +220,7 @@ namespace susp
   end
 
   definition susp_adjoint_loop_nat_left (f : Y →* Ω Z) (g : X →* Y)
-    : (susp_adjoint_loop X Z)⁻¹ (f ∘* g) ~* (susp_adjoint_loop Y Z)⁻¹ f ∘* psusp_functor g :=
+    : (susp_adjoint_loop X Z)⁻¹ᵉ (f ∘* g) ~* (susp_adjoint_loop Y Z)⁻¹ᵉ f ∘* psusp_functor g :=
   begin
     esimp [susp_adjoint_loop],
     refine _ ⬝* !passoc⁻¹*,

hott/init/equiv.hlean

@@ -269,14 +269,11 @@ namespace eq
   definition cast_inv {A B : Type} (p : A = B) (b : B) : cast p⁻¹ b = (cast p)⁻¹ b := idp
 end eq
 
-namespace equiv
+infix ` ≃ `:25 := equiv
-  namespace ops
+attribute equiv.to_is_equiv [instance]
-    attribute to_fun [coercion]
-  end ops
-  open equiv.ops
-  attribute to_is_equiv [instance]
 
-  infix ` ≃ `:25 := equiv
+namespace equiv
+  attribute to_fun [coercion]
 
   section
   variables {A B C : Type}
@@ -395,17 +392,12 @@ namespace equiv
 
   end
 
-
-  namespace ops
-    postfix ⁻¹ := equiv.symm -- overloaded notation for inverse
-  end ops
-
   infixl ` ⬝pe `:75 := equiv_of_equiv_of_eq
   infixl ` ⬝ep `:75 := equiv_of_eq_of_equiv
 
 end equiv
 
-open equiv equiv.ops
+open equiv
 namespace is_equiv
 
   definition is_equiv_of_equiv_of_homotopy [constructor] {A B : Type} (f : A ≃ B)

hott/init/pathover.hlean

@@ -9,7 +9,7 @@ Basic theorems about pathovers
 prelude
 import .path .equiv
 
-open equiv is_equiv equiv.ops function
+open equiv is_equiv function
 
 variables {A A' : Type} {B B' : A → Type} {B'' : A' → Type} {C : Π⦃a⦄, B a → Type}
           {a a₂ a₃ a₄ : A} {p p' : a = a₂} {p₂ : a₂ = a₃} {p₃ : a₃ = a₄} {p₁₃ : a = a₃}

hott/init/trunc.hlean

@@ -278,11 +278,12 @@ namespace is_trunc
 
   theorem is_trunc_is_equiv_closed (n : ℕ₋₂) (f : A → B) [H : is_equiv f]
     [HA : is_trunc n A] : is_trunc n B :=
-  trunc_index.rec_on n
+  begin
-    (λA (HA : is_contr A) B f (H : is_equiv f), is_contr_is_equiv_closed f)
+    revert A HA B f H, induction n with n IH: intros,
-    (λn IH A (HA : is_trunc n.+1 A) B f (H : is_equiv f), @is_trunc_succ_intro _ _ (λ x y : B,
+    { exact is_contr_is_equiv_closed f},
-      IH (f⁻¹ x = f⁻¹ y) _ (x = y) (ap f⁻¹)⁻¹ !is_equiv_inv))
+    { apply is_trunc_succ_intro, intro x y,
-    A HA B f H
+      exact IH (f⁻¹ x = f⁻¹ y) _ (x = y) (ap f⁻¹)⁻¹ !is_equiv_inv}
+  end
 
   definition is_trunc_is_equiv_closed_rev (n : ℕ₋₂) (f : A → B) [H : is_equiv f]
     [HA : is_trunc n B] : is_trunc n A :=

hott/init/ua.hlean

@@ -8,7 +8,7 @@ Ported from Coq HoTT
 
 prelude
 import .equiv
-open eq equiv is_equiv equiv.ops
+open eq equiv is_equiv
 
 --Ensure that the types compared are in the same universe
 section

hott/types/arrow.hlean

@@ -9,7 +9,7 @@ Theorems about arrow types (function spaces)
 
 import types.pi
 
-open eq equiv is_equiv funext pi equiv.ops is_trunc unit
+open eq equiv is_equiv funext pi is_trunc unit
 
 namespace pi
 

hott/types/eq.hlean

@@ -8,7 +8,7 @@ Theorems about path types (identity types)
 -/
 
 import types.sigma
-open eq sigma sigma.ops equiv is_equiv equiv.ops is_trunc
+open eq sigma sigma.ops equiv is_equiv is_trunc
 
 -- TODO: Rename transport_eq_... and pathover_eq_... to eq_transport_... and eq_pathover_...
 

hott/types/equiv.hlean

@@ -9,7 +9,7 @@ Theorems about the types equiv and is_equiv
 
 import .fiber .arrow arity ..prop_trunc
 
-open eq is_trunc sigma sigma.ops pi fiber function equiv equiv.ops
+open eq is_trunc sigma sigma.ops pi fiber function equiv
 
 namespace is_equiv
   variables {A B : Type} (f : A → B) [H : is_equiv f]

hott/types/int/hott.hlean

@@ -7,7 +7,7 @@ Theorems about the integers specific to HoTT
 -/
 
 import .basic types.eq arity algebra.bundled
-open core eq is_equiv equiv equiv.ops algebra is_trunc
+open core eq is_equiv equiv algebra is_trunc
 open nat (hiding pred)
 
 namespace int

hott/types/lift.hlean

@@ -7,7 +7,7 @@ Theorems about lift
 -/
 
 import ..function
-open eq equiv equiv.ops is_equiv is_trunc pointed
+open eq equiv is_equiv is_trunc pointed
 
 namespace lift
 

hott/types/nat/div.hlean

@@ -168,15 +168,15 @@ begin
     intro x IH,
     show succ x = succ x / y * y + succ x % y, from
       if H1 : succ x < y then
-         assert H2 : succ x / y = 0, from div_eq_zero_of_lt H1,
+         have H2 : succ x / y = 0, from div_eq_zero_of_lt H1,
-         assert H3 : succ x % y = succ x, from mod_eq_of_lt H1,
+         have H3 : succ x % y = succ x, from mod_eq_of_lt H1,
          begin rewrite [H2, H3, zero_mul, zero_add] end
       else
          have H2 : y ≤ succ x, from le_of_not_gt H1,
-         assert H3 : succ x / y = succ ((succ x - y) / y), from div_eq_succ_sub_div H H2,
+         have H3 : succ x / y = succ ((succ x - y) / y), from div_eq_succ_sub_div H H2,
-         assert H4 : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H H2,
+         have H4 : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H H2,
          have H5 : succ x - y < succ x, from sub_lt !succ_pos H,
-         assert H6 : succ x - y ≤ x, from le_of_lt_succ H5,
+         have H6 : succ x - y ≤ x, from le_of_lt_succ H5,
          (calc
              succ x / y * y + succ x % y =
                   succ ((succ x - y) / y) * y + succ x % y      : by rewrite H3
@@ -213,7 +213,7 @@ by_cases_zero_pos n
     assume npos : n > 0,
     assume H1 : (m + i) % n = (k + i) % n,
     have H2 : (m + i % n) % n = (k + i % n) % n, by rewrite [*add_mod_mod, H1],
-    assert H3 : (m + i % n + (n - i % n)) % n = (k + i % n + (n - i % n)) % n,
+    have H3 : (m + i % n + (n - i % n)) % n = (k + i % n + (n - i % n)) % n,
       from add_mod_eq_add_mod_right _ H2,
     begin
       revert H3,
@@ -301,11 +301,11 @@ theorem mul_mod_eq_mul_mod_mod (m n k : nat) : (m * n) % k = (m * (n % k)) % k :
 !mul.comm ▸ !mul.comm ▸ !mul_mod_eq_mod_mul_mod
 
 protected theorem div_one (n : ℕ) : n / 1 = n :=
-assert n / 1 * 1 + n % 1 = n, from !eq_div_mul_add_mod⁻¹,
+have n / 1 * 1 + n % 1 = n, from !eq_div_mul_add_mod⁻¹,
 begin rewrite [-this at {2}, mul_one, mod_one] end
 
 protected theorem div_self {n : ℕ} (H : n > 0) : n / n = 1 :=
-assert (n * 1) / (n * 1) = 1 / 1, from !nat.mul_div_mul_left H,
+have (n * 1) / (n * 1) = 1 / 1, from !nat.mul_div_mul_left H,
 by rewrite [nat.div_one at this, -this, *mul_one]
 
 theorem div_mul_cancel_of_mod_eq_zero {m n : ℕ} (H : m % n = 0) : m / n * n = m :=
@@ -451,11 +451,11 @@ end
 
 lemma le_of_dvd {m n : nat} : n > 0 → m ∣ n → m ≤ n :=
 assume (h₁ : n > 0) (h₂ : m ∣ n),
-assert h₃ : n % m = 0, from mod_eq_zero_of_dvd h₂,
+have h₃ : n % m = 0, from mod_eq_zero_of_dvd h₂,
 by_contradiction
  (λ nle : ¬ m ≤ n,
    have   h₄ : m > n, from lt_of_not_ge nle,
-   assert h₅ : n % m = n, from mod_eq_of_lt h₄,
+   have h₅ : n % m = n, from mod_eq_of_lt h₄,
    begin
      rewrite h₃ at h₅, subst n,
      exact absurd h₁ (lt.irrefl 0)
@@ -516,7 +516,7 @@ lt_of_mul_lt_mul_right (calc
     ... < n * k                 : H)
 
 protected theorem lt_mul_of_div_lt {m n k : ℕ} (H1 : k > 0) (H2 : m / k < n) : m < n * k :=
-assert H3 : succ (m / k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2),
+have H3 : succ (m / k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2),
 have H4 : m / k * k + k ≤ n * k, by rewrite [succ_mul at H3]; apply H3,
 calc
   m     = m / k * k + m % k : eq_div_mul_add_mod
@@ -567,28 +567,28 @@ nat.strong_rec_on a
 (λ a ih,
   let k₁ := a / (b*c) in
   let k₂ := a %(b*c) in
-  assert bc_pos : b*c > 0, from mul_pos `b > 0` `c > 0`,
+  have bc_pos : b*c > 0, from mul_pos `b > 0` `c > 0`,
-  assert k₂ < b * c,       from mod_lt _ bc_pos,
+  have k₂ < b * c,       from mod_lt _ bc_pos,
-  assert k₂ ≤ a,           from !mod_le,
+  have k₂ ≤ a,           from !mod_le,
   sum.elim (eq_sum_lt_of_le this)
     (suppose k₂ = a,
-     assert i₁ : a < b * c,   by rewrite -this; assumption,
+     have i₁ : a < b * c,   by rewrite -this; assumption,
-     assert k₁ = 0,           from div_eq_zero_of_lt i₁,
+     have k₁ = 0,           from div_eq_zero_of_lt i₁,
-     assert a / b < c,      by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁,
+     have a / b < c,      by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁,
      begin
        rewrite [`k₁ = 0`],
        show (a / b) / c = 0, from div_eq_zero_of_lt `a / b < c`
      end)
     (suppose k₂ < a,
-     assert a = k₁*(b*c) + k₂,         from eq_div_mul_add_mod a (b*c),
+     have a = k₁*(b*c) + k₂,         from eq_div_mul_add_mod a (b*c),
-     assert a / b = k₁*c + k₂ / b, by
+     have a / b = k₁*c + k₂ / b, by
        rewrite [this at {1}, mul.comm b c at {2}, -mul.assoc,
                 add.comm, add_mul_div_self `b > 0`, add.comm],
-     assert e₁ : (a / b) / c = k₁ + (k₂ / b) / c, by
+     have e₁ : (a / b) / c = k₁ + (k₂ / b) / c, by
        rewrite [this, add.comm, add_mul_div_self `c > 0`, add.comm],
-     assert e₂ : (k₂ / b) / c = k₂ / (b * c), from ih k₂ `k₂ < a`,
+     have e₂ : (k₂ / b) / c = k₂ / (b * c), from ih k₂ `k₂ < a`,
-     assert e₃ : k₂ / (b * c)   = 0,              from div_eq_zero_of_lt `k₂ < b * c`,
+     have e₃ : k₂ / (b * c)   = 0,              from div_eq_zero_of_lt `k₂ < b * c`,
-     assert (k₂ / b) / c      = 0,              by rewrite [e₂, e₃],
+     have (k₂ / b) / c      = 0,              by rewrite [e₂, e₃],
      show (a / b) / c = k₁,                     by rewrite [e₁, this]))
 
 protected lemma div_div_eq_div_mul (a b c : nat) : (a / b) / c = a / (b * c) :=

hott/types/pointed.hlean

@@ -7,7 +7,7 @@ Ported from Coq HoTT
 -/
 
 import arity .eq .bool .unit .sigma .nat.basic prop_trunc
-open is_trunc eq prod sigma nat equiv option is_equiv bool unit algebra equiv.ops sigma.ops
+open is_trunc eq prod sigma nat equiv option is_equiv bool unit algebra sigma.ops
 
 structure pointed [class] (A : Type) :=
   (point : A)
@@ -97,7 +97,6 @@ namespace pointed
   definition iterated_loop_space [unfold 3] (A : Type) [H : pointed A] (n : ℕ) : Type :=
   Ω[n] (pointed.mk' A)
 
-  open equiv.ops
   definition pType_eq {A B : Type*} (f : A ≃ B) (p : f pt = pt) : A = B :=
   begin
     cases A with A a, cases B with B b, esimp at *,

hott/types/pointed2.hlean

@@ -9,7 +9,7 @@ Ported from Coq HoTT
 
 import .equiv cubical.square
 
-open eq is_equiv equiv equiv.ops pointed is_trunc
+open eq is_equiv equiv pointed is_trunc
 
 -- structure pequiv (A B : Type*) :=
 --   (to_pmap : A →* B)

hott/types/prod.hlean

@@ -7,7 +7,7 @@ Ported from Coq HoTT
 Theorems about products
 -/
 
-open eq equiv is_equiv is_trunc prod prod.ops unit equiv.ops
+open eq equiv is_equiv is_trunc prod prod.ops unit
 
 variables {A A' B B' C D : Type} {P Q : A → Type}
           {a a' a'' : A} {b b₁ b₂ b' b'' : B} {u v w : A × B}

hott/types/pullback.hlean

@@ -7,7 +7,7 @@ Theorems about pullbacks
 -/
 
 import cubical.square
-open eq equiv is_equiv function equiv.ops prod unit is_trunc sigma
+open eq equiv is_equiv function prod unit is_trunc sigma
 
 variables {A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ : Type}
           (f₁₀ : A₀₀ → A₂₀) (f₃₀ : A₂₀ → A₄₀)

hott/types/sum.hlean

@@ -8,7 +8,7 @@ Theorems about sums/coproducts/disjoint unions
 
 import .pi .equiv logic
 
-open lift eq is_equiv equiv equiv.ops prod prod.ops is_trunc sigma bool
+open lift eq is_equiv equiv prod prod.ops is_trunc sigma bool
 
 namespace sum
   universe variables u v u' v'
@@ -226,9 +226,8 @@ namespace sum
   end
 
   definition unit_sum_equiv_cancel_inv (b : B) :
-    unit_sum_equiv_cancel_map H (unit_sum_equiv_cancel_map H⁻¹ b) = b :=
+    unit_sum_equiv_cancel_map H (unit_sum_equiv_cancel_map H⁻¹ᵉ b) = b :=
   begin
-    have HH : to_fun H⁻¹ = (to_fun H)⁻¹, begin cases H, reflexivity end,
     esimp[unit_sum_equiv_cancel_map], apply sum.rec,
     { intro x, cases x with u Hu, esimp, apply sum.rec,
       { intro x, exfalso, cases x with u' Hu', apply empty_of_inl_eq_inr,
@@ -238,28 +237,28 @@ namespace sum
                ... = H⁻¹ (H (inr _)) : {Hu⁻¹}
                ... = inr _ : to_left_inv H },
       { intro x, cases x with b' Hb', esimp, cases sum.mem_cases (H⁻¹ (inr b)) with x x,
-        { cases x with u' Hu', cases u', apply eq_of_inr_eq_inr, rewrite -HH at Hu',
+        { cases x with u' Hu', cases u', apply eq_of_inr_eq_inr,
-          calc inr b' = H (inl ⋆) : Hb'⁻¹
+          calc inr b' = H (inl ⋆)       : Hb'⁻¹
-                  ... = H (H⁻¹ (inr b)) : {(ap (to_fun H) Hu')⁻¹}
+                  ... = H (H⁻¹ (inr b)) : (ap (to_fun H) Hu')⁻¹
-                  ... = inr b : to_right_inv H (inr b) },
+                  ... = inr b           : to_right_inv H (inr b)},
-        { exfalso, cases x with a Ha, rewrite -HH at Ha, apply empty_of_inl_eq_inr,
+        { exfalso, cases x with a Ha, apply empty_of_inl_eq_inr,
           cases u, apply concat, apply Hu⁻¹, apply concat, rotate 1, apply !(to_right_inv H),
-          apply ap (to_fun H), krewrite -HH,
+          apply ap (to_fun H),
           apply concat, rotate 1, apply Ha⁻¹, apply ap inr, esimp,
           apply sum.rec, intro x, exfalso, apply empty_of_inl_eq_inr,
           apply concat, exact x.2⁻¹, apply Ha,
           intro x, cases x with a' Ha', esimp, apply eq_of_inr_eq_inr, apply Ha'⁻¹ ⬝ Ha } } },
     { intro x, cases x with b' Hb', esimp, apply eq_of_inr_eq_inr, refine Hb'⁻¹ ⬝ _,
-      cases sum.mem_cases (to_fun H⁻¹ (inr b)) with x x,
+      cases sum.mem_cases (to_fun H⁻¹ᵉ (inr b)) with x x,
-      { cases x with u Hu, esimp, cases sum.mem_cases (to_fun H⁻¹ (inl ⋆)) with x x,
+      { cases x with u Hu, esimp, cases sum.mem_cases (to_fun H⁻¹ᵉ (inl ⋆)) with x x,
         { cases x with u' Hu', exfalso, apply empty_of_inl_eq_inr,
-          calc inl ⋆ = H (H⁻¹ (inl ⋆)) : (to_right_inv H (inl ⋆))⁻¹
+          calc inl ⋆ = H (H⁻¹ (inl ⋆))  : (to_right_inv H (inl ⋆))⁻¹
-                 ... = H (inl u') : {ap H Hu'}
+                 ... = H (inl u')       : ap H Hu'
-                 ... = H (inl u) : {!is_prop.elim}
+                 ... = H (inl u)        : by rewrite [is_prop.elim u' u]
-                 ... = H (H⁻¹ (inr b)) : {ap H Hu⁻¹}
+                 ... = H (H⁻¹ᵉ (inr b)) : ap H Hu⁻¹
-                 ... = inr b : to_right_inv H (inr b) },
+                 ... = inr b            : to_right_inv H (inr b) },
       { cases x with a Ha, exfalso, apply empty_of_inl_eq_inr,
-        apply concat, rotate 1, exact Hb', krewrite HH at Ha,
+        apply concat, rotate 1, exact Hb',
         have Ha' : inl ⋆ = H (inr a), by apply !(to_right_inv H)⁻¹ ⬝ ap H Ha,
         apply concat Ha', apply ap H, apply ap inr, apply sum.rec,
           intro x, cases x with u' Hu', esimp, apply sum.rec,
@@ -269,15 +268,15 @@ namespace sum
           intro x, exfalso, cases x with a'' Ha'', apply empty_of_inl_eq_inr,
           apply Hu⁻¹ ⬝ Ha'', } },
     { cases x with a' Ha', esimp, refine _ ⬝ !(to_right_inv H), apply ap H,
-      rewrite -HH, apply Ha'⁻¹ } }
+      apply Ha'⁻¹ } }
   end
 
   definition unit_sum_equiv_cancel : A ≃ B :=
   begin
     fapply equiv.MK, apply unit_sum_equiv_cancel_map H,
-    apply unit_sum_equiv_cancel_map H⁻¹,
+    apply unit_sum_equiv_cancel_map H⁻¹ᵉ,
     intro b, apply unit_sum_equiv_cancel_inv,
-    { intro a, have H = (H⁻¹)⁻¹, from !equiv.symm_symm⁻¹, rewrite this at {2},
+    { intro a, have H = (H⁻¹ᵉ)⁻¹ᵉ, from !equiv.symm_symm⁻¹, rewrite this at {2},
       apply unit_sum_equiv_cancel_inv }
   end
 

hott/types/trunc.hlean

@@ -428,7 +428,6 @@ namespace trunc
         { exact (IH _ _ _)}}}
   end
 
-  open equiv.ops
   definition unique_choice {P : A → Type} [H : Πa, is_prop (P a)] (f : Πa, ∥ P a ∥) (a : A)
     : P a :=
   !trunc_equiv (f a)

hott/types/univ.hlean

@@ -10,7 +10,7 @@ Theorems about the universe
 
 import .bool .trunc .lift .pullback
 
-open is_trunc bool lift unit eq pi equiv equiv.ops sum sigma fiber prod pullback is_equiv sigma.ops
+open is_trunc bool lift unit eq pi equiv sum sigma fiber prod pullback is_equiv sigma.ops
      pointed
 namespace univ
 
@@ -117,9 +117,9 @@ namespace univ
                                                    (λb, sigma_equiv_sigma_right
                                                    (λX, !comm_equiv_nondep))
       ... ≃ Σb (v : ΣX, X), v.1 = fiber f b    : sigma_equiv_sigma_right
-                                                   (λb, !sigma_assoc_equiv⁻¹)
+                                                   (λb, !sigma_assoc_equiv⁻¹ᵉ)
       ... ≃ Σb (Y : Type*), Y = fiber f b      : sigma_equiv_sigma_right
-                                     (λb, sigma_equiv_sigma (pType.sigma_char)⁻¹
+                                     (λb, sigma_equiv_sigma (pType.sigma_char)⁻¹ᵉ
                                                             (λv, sigma.rec_on v (λx y, equiv.refl)))
       ... ≃ Σ(Y : Type*) b, Y = fiber f b      : sigma_comm_equiv
       ... ≃ pullback pType.carrier (fiber f) : !pullback.sigma_char⁻¹ᵉ

tests/lean/extra/597a.hlean

@@ -1,4 +1,3 @@
 open equiv
-open equiv.ops
 constants (A B : Type₀) (f : A ≃ B)
 definition foo : A → B := f

tests/lean/extra/597b.hlean

@@ -1,4 +1,3 @@
-open equiv
+-- open equiv
--- open equiv.ops
 constants (A B : Type₀) (f : A ≃ B)
 definition foo : A → B := f -- should fail