Add renaming

theories/typing.v

@@ -8,27 +8,31 @@ Definition get_level a :=
   | Floating ℓ => ℓ
   end.
 
+Inductive lookup {A} : nat -> list A -> A -> Prop :=
+| here A Γ : lookup 0 (cons A Γ) A
+| there i Γ A B :
+  lookup i Γ A ->
+  lookup (S i) (cons B Γ) A.
+
 Definition ctx := list (Level * Ty).
 
-Fixpoint fix_ctx (Γ : ctx) :=
-  match Γ with
-  | nil => nil
-  | (Fixed ℓ, A) :: Γ => (Fixed ℓ, A) :: fix_ctx Γ
-  | (Floating ℓ, A) :: Γ => (Fixed ℓ, A) :: fix_ctx Γ
+Definition squash a :=
+  match a with
+  | Fixed ℓ => false
+  | Floating ℓ => true
   end.
 
 Fixpoint fixed_ctx (Γ : ctx) :=
   match Γ with
   | nil => nil
-  | (Fixed ℓ, A) :: Γ => false :: fixed_ctx Γ
-  | (Floating ℓ, A) :: Γ => true :: fixed_ctx Γ
+  | (a, _) :: Γ => squash a :: fixed_ctx Γ
   end.
 
 (* false <= true *)
 (* A term is liftable if it remains well-typed raising the levels of the floating levels to the current level *)
 Fixpoint liftable Φ (a : Tm) :=
   match a with
-  | VarTm i => nth_default true Φ i = false
+  | VarTm i => lookup i Φ false
   | FlApp a b => liftable Φ a /\ liftable Φ b
   | App a b => liftable Φ a
   | FlAbs a => liftable (false :: Φ) a
@@ -38,7 +42,7 @@ Fixpoint liftable Φ (a : Tm) :=
 Inductive Wt (Γ : ctx) : Tm -> bool -> Ty -> Prop :=
 | T_Var ℓ i ℓ0 A :
   Bool.le (get_level ℓ0) ℓ ->
-  nth_error Γ i = Some (ℓ0, A) ->
+  lookup i Γ (ℓ0, A) ->
   Γ ⊢ VarTm i ;; ℓ ∈ A
 
 | T_FlApp ℓ a b A B :
@@ -53,6 +57,7 @@ Inductive Wt (Γ : ctx) : Tm -> bool -> Ty -> Prop :=
 | T_App ℓ ℓ0 a b A B :
   Γ ⊢ b ;; ℓ ∈ Fun ℓ0 A B ->
   Γ ⊢ a ;; ℓ0 ∈ A ->
+  liftable (fixed_ctx Γ) a ->
   Γ ⊢ App b a ;; ℓ ∈ A
 
 | T_Abs ℓ ℓ0 b A B :