diff --git a/hott/algebra/binary.hlean b/hott/algebra/binary.hlean
index 2c9f3d93f..9645529e0 100644
--- a/hott/algebra/binary.hlean
+++ b/hott/algebra/binary.hlean
@@ -41,12 +41,12 @@ namespace binary
     definition right_distributive := ∀a b c, (a + b) * c = a * c + b * c
   end
 
-  context
+  section
     variable {A : Type}
     variable {f : A → A → A}
     variable H_comm : commutative f
     variable H_assoc : associative f
-    infixl `*` := f
+    local infixl `*` := f
     theorem left_comm : ∀a b c, a*(b*c) = b*(a*c) :=
     take a b c, calc
       a*(b*c) = (a*b)*c  : H_assoc
@@ -60,11 +60,11 @@ namespace binary
         ...   = (a*c)*b : H_assoc
   end
 
-  context
+  section
     variable {A : Type}
     variable {f : A → A → A}
     variable H_assoc : associative f
-    infixl `*` := f
+    local infixl `*` := f
     theorem assoc4helper (a b c d) : (a*b)*(c*d) = a*((b*c)*d) :=
     calc
       (a*b)*(c*d) = a*(b*(c*d)) : H_assoc
diff --git a/hott/init/axioms/funext_of_ua.hlean b/hott/init/axioms/funext_of_ua.hlean
index ddeeec0ee..ab14fca08 100644
--- a/hott/init/axioms/funext_of_ua.hlean
+++ b/hott/init/axioms/funext_of_ua.hlean
@@ -14,7 +14,7 @@ import .funext_varieties .ua
 
 open eq function prod is_trunc sigma equiv is_equiv unit
 
-context
+section
   universe variables l
 
   private theorem ua_isequiv_postcompose {A B : Type.{l}} {C : Type}
diff --git a/hott/init/equiv.hlean b/hott/init/equiv.hlean
index 3e2911493..0e653471e 100644
--- a/hott/init/equiv.hlean
+++ b/hott/init/equiv.hlean
@@ -95,7 +95,7 @@ namespace is_equiv
   is_equiv.mk (inv f) sect' retr' adj'
   end
 
-  context
+  section
   parameters {A B : Type} (f : A → B) (g : B → A)
             (ret : f ∘ g ∼ id) (sec : g ∘ f ∼ id)
 
diff --git a/hott/init/hedberg.hlean b/hott/init/hedberg.hlean
index f22be3193..661a9b2dd 100644
--- a/hott/init/hedberg.hlean
+++ b/hott/init/hedberg.hlean
@@ -16,7 +16,7 @@ private definition const {A B : Type} (f : A → B) := ∀ x y, f x = f y
 private definition coll (A : Type) := Σ f : A → A, const f
 private definition path_coll (A : Type) := ∀ x y : A, coll (x = y)
 
-context
+section
   parameter  {A : Type}
   hypothesis [h : decidable_eq A]
   variables  {x y : A}
diff --git a/hott/init/types/prod.hlean b/hott/init/types/prod.hlean
index 77885a0ee..be336c4e6 100644
--- a/hott/init/types/prod.hlean
+++ b/hott/init/types/prod.hlean
@@ -56,10 +56,10 @@ namespace prod
   intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
   end
 
-  context
+  section
   parameters {A B : Type}
   parameters {Ra  : A → A → Type} {Rb  : B → B → Type}
-  infix `≺`:50 := lex Ra Rb
+  local infix `≺`:50 := lex Ra Rb
 
   definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
   acc.rec_on aca
diff --git a/hott/init/wf.hlean b/hott/init/wf.hlean
index 79ae373f5..1e41258b6 100644
--- a/hott/init/wf.hlean
+++ b/hott/init/wf.hlean
@@ -26,9 +26,9 @@ namespace well_founded
   definition apply [coercion] {A : Type} {R : A → A → Type} (wf : well_founded R) : ∀a, acc R a :=
   take a, well_founded.rec_on wf (λp, p) a
 
-  context
+  section
   parameters {A : Type} {R : A → A → Type}
-  infix `≺`:50    := R
+  local infix `≺`:50    := R
 
   hypothesis [Hwf : well_founded R]
 
@@ -98,7 +98,7 @@ well_founded.intro (λ (a : A),
 
 -- Subrelation of a well-founded relation is well-founded
 namespace subrelation
-context
+section
   parameters {A : Type} {R Q : A → A → Type}
   parameters (H₁ : subrelation Q R)
   parameters (H₂ : well_founded R)
@@ -115,7 +115,7 @@ end subrelation
 
 -- The inverse image of a well-founded relation is well-founded
 namespace inv_image
-context
+section
   parameters {A B : Type} {R : B → B → Type}
   parameters (f : A → B)
   parameters (H : well_founded R)
@@ -136,9 +136,9 @@ end inv_image
 
 -- The transitive closure of a well-founded relation is well-founded
 namespace tc
-context
+section
   parameters {A : Type} {R : A → A → Type}
-  notation `R⁺` := tc R
+  local notation `R⁺` := tc R
 
   definition accessible {z} (ac: acc R z) : acc R⁺ z :=
   acc.rec_on ac
diff --git a/hott/types/sigma.hlean b/hott/types/sigma.hlean
index f71c5c9fb..0102ed4f5 100644
--- a/hott/types/sigma.hlean
+++ b/hott/types/sigma.hlean
@@ -228,9 +228,9 @@ namespace sigma
     : (Σa, B a) ≃ (Σa', B' a') :=
   equiv.mk (sigma_functor f g) !is_equiv_sigma_functor
 
-  context
-  attribute inv [irreducible]
-  attribute function.compose [irreducible] --this is needed for the following class inference problem
+  section
+  local attribute inv [irreducible]
+  local attribute function.compose [irreducible] --this is needed for the following class inference problem
   definition sigma_equiv_sigma (Hf : A ≃ A') (Hg : Π a, B a ≃ B' (to_fun Hf a)) :
       (Σa, B a) ≃ (Σa', B' a') :=
   sigma_equiv_sigma_of_is_equiv (to_fun Hf) (λ a, to_fun (Hg a))