Add new syntax for booleans

theories/compile.v

@@ -15,11 +15,14 @@ Module Compile.
     | Pair a b => Pair (F a) (F b)
     | Proj t a => Proj t (F a)
     | Bot => Bot
+    | If a b c => App (App (F a) (F b)) (F c)
+    | BVal b => if b then (Abs (Abs (VarTm (shift var_zero)))) else (Abs (Abs (VarTm var_zero)))
+    | Bool => Bool
     end.
 
   Lemma renaming n m (a : Tm n) (ξ : fin n -> fin m) :
     F (ren_Tm ξ a)= ren_Tm ξ (F a).
-  Proof. move : m ξ. elim : n  / a => //=; scongruence. Qed.
+  Proof. move : m ξ. elim : n  / a => //=; hauto lq:on. Qed.
 
   #[local]Hint Rewrite Compile.renaming : compile.
 
@@ -33,6 +36,8 @@ Module Compile.
     - hauto lq:on rew:off.
     - hauto lq:on.
     - hauto lq:on inv:option rew:db:compile unfold:funcomp.
+    - hauto lq:on rew:off.
+    - hauto lq:on rew:off.
   Qed.
 
   Lemma substing n b (a : Tm (S n)) :