diff --git a/coq/typing.v b/coq/typing.v
index 98a4481..9606c9d 100644
--- a/coq/typing.v
+++ b/coq/typing.v
@@ -29,25 +29,24 @@ Inductive expr_type : context -> expr_term -> type_term -> Prop :=
 
   | T_Let : forall Γ s σ t τ x,
     (Γ |- s \is σ) ->
-    (Γ |- t \is τ) ->
-    (Γ |- (expr_let x σ s t) \is τ)
+    ((ctx_assign x σ Γ) |- t \is τ) ->
+    (Γ |- (expr_let x s t) \is τ)
 
   | T_TypeAbs : forall Γ e τ α,
     Γ |- e \is τ ->
     Γ |- (expr_ty_abs α e) \is (type_univ α τ)
 
-  | T_TypeApp : forall Γ α e σ τ,
+  | T_TypeApp : forall Γ α e σ τ τ',
     Γ |- e \is (type_univ α τ) ->
-    Γ |- (expr_ty_app e σ) \is (type_subst α σ τ)
+    (type_subst1 α σ τ τ') ->
+    Γ |- (expr_ty_app e σ) \is τ'
 
   | T_Abs : forall Γ x σ t τ,
-    (context_contains Γ x σ) ->
-    Γ |- t \is τ ->
-    Γ |- (expr_abs x σ t) \is (type_fun σ τ)
+    ((ctx_assign x σ Γ) |- t \is τ) ->
+    (Γ |- (expr_abs x σ t) \is (type_fun σ τ))
 
   | T_MorphAbs : forall Γ x σ e τ,
-    (context_contains Γ x σ) ->
-    Γ |- e \is τ ->
+    ((ctx_assign x σ Γ) |- e \is τ) ->
     Γ |- (expr_morph x σ e) \is (type_morph σ τ)
 
   | T_App : forall Γ f a σ' σ τ,
@@ -80,31 +79,24 @@ Definition is_well_typed (e:expr_term) : Prop :=
 (* Examples *)
 
 Example typing1 :
-  forall Γ,
-  (context_contains Γ "x" [< %"T"% >]) ->
-  Γ |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
-   (* Γ |- [ ΛT ↦ λx:T ↦ x ] : ∀T.(T->T) *)
+  ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"T", %"T"% -> %"T"% >].
 Proof.
   intros.
   apply T_TypeAbs.
   apply T_Abs.
-  apply H.
   apply T_Var.
-  apply H.
+  apply C_take.
 Qed.
 
 Example typing2 :
-  forall Γ,
-  (context_contains Γ "x" [< %"T"% >]) ->
-  Γ |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
+  ctx_empty |- [{ Λ"T" ↦ λ "x" %"T"% ↦ %"x"% }] \is [< ∀"U", %"U"% -> %"U"% >].
 Proof.
   intros.
   apply T_Descend with (τ:=[< ∀"T",(%"T"% -> %"T"%) >]).
   apply T_TypeAbs.
   apply T_Abs.
-  apply H.
   apply T_Var.
-  apply H.
+  apply C_take.
 
   apply TSubRepr_Refl.
   apply TEq_Alpha.
@@ -115,10 +107,7 @@ Proof.
 Qed.
 
 Example typing3 :
-  forall Γ,
-  (context_contains Γ "x" [< %"T"% >]) ->
-  (context_contains Γ "y" [< %"U"% >]) ->
-  Γ |- [{
+  ctx_empty |- [{
     Λ"T" ↦ Λ"U" ↦ λ"x" %"T"% ↦ λ"y" %"U"% ↦ %"y"%
   }] \is [<
        ∀"S",∀"T",(%"S"%->%"T"%->%"T"%)
@@ -127,10 +116,9 @@ Proof.
   intros.
   apply T_Descend with (τ:=[< ∀"T",∀"U",(%"T"%->%"U"%->%"U"%) >]) (τ':=[< ∀"S",∀"T",(%"S"%->%"T"%->%"T"%) >]).
   apply T_TypeAbs, T_TypeAbs, T_Abs.
-  apply H.
   apply T_Abs.
-  apply H0.
-  apply T_Var, H0.
+  apply T_Var.
+  apply C_take.
 
   apply TSubRepr_Refl.
   apply TEq_Trans with (y:= [< ∀"S",∀"U",(%"S"%->%"U"%->%"U"%) >] ).
@@ -159,13 +147,11 @@ Example typing4 : (is_well_typed
 ).
 Proof.
   unfold is_well_typed.
-  exists (ctx_assign "x" [< %"T"% >]
-            (ctx_assign "y" [< %"U"% >] ctx_empty)).
+  exists ctx_empty.
   exists [< ∀"T",∀"U",%"T"%->%"U"%->%"T"% >].
   apply T_TypeAbs.
   apply T_TypeAbs.
   apply T_Abs.
-  apply C_take.
   apply T_Abs.
   apply C_shuffle. apply C_take.
   apply T_Var.