diff --git a/syntax.sig b/syntax.sig
index 3101cab..9e04431 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -2,6 +2,7 @@ nat  : Type
 Tm(VarTm) : Type
 PTag : Type
 TTag : Type
+bool : Type
 
 PL : PTag
 PR : PTag
@@ -15,3 +16,6 @@ TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
 Const : TTag -> Tm
 Univ : nat -> Tm
 Bot : Tm
+BVal : bool -> Tm
+Bool : Tm
+If : Tm -> Tm -> Tm -> Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index a5cb002..5b06281 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -42,7 +42,10 @@ Inductive Tm (n_Tm : nat) : Type :=
   | TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
   | Const : TTag -> Tm n_Tm
   | Univ : nat -> Tm n_Tm
-  | Bot : Tm n_Tm.
+  | Bot : Tm n_Tm
+  | BVal : bool -> Tm n_Tm
+  | Bool : Tm n_Tm
+  | If : Tm n_Tm -> Tm n_Tm -> Tm n_Tm -> Tm n_Tm.
 
 Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
   (H0 : s0 = t0) : Abs m_Tm s0 = Abs m_Tm t0.
@@ -101,6 +104,27 @@ Proof.
 exact (eq_refl).
 Qed.
 
+Lemma congr_BVal {m_Tm : nat} {s0 : bool} {t0 : bool} (H0 : s0 = t0) :
+  BVal m_Tm s0 = BVal m_Tm t0.
+Proof.
+exact (eq_trans eq_refl (ap (fun x => BVal m_Tm x) H0)).
+Qed.
+
+Lemma congr_Bool {m_Tm : nat} : Bool m_Tm = Bool m_Tm.
+Proof.
+exact (eq_refl).
+Qed.
+
+Lemma congr_If {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm m_Tm} {s2 : Tm m_Tm}
+  {t0 : Tm m_Tm} {t1 : Tm m_Tm} {t2 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1)
+  (H2 : s2 = t2) : If m_Tm s0 s1 s2 = If m_Tm t0 t1 t2.
+Proof.
+exact (eq_trans
+         (eq_trans (eq_trans eq_refl (ap (fun x => If m_Tm x s1 s2) H0))
+            (ap (fun x => If m_Tm t0 x s2) H1))
+         (ap (fun x => If m_Tm t0 t1 x) H2)).
+Qed.
+
 Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
   fin (S m) -> fin (S n).
 Proof.
@@ -126,6 +150,10 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
   | Const _ s0 => Const n_Tm s0
   | Univ _ s0 => Univ n_Tm s0
   | Bot _ => Bot n_Tm
+  | BVal _ s0 => BVal n_Tm s0
+  | Bool _ => Bool n_Tm
+  | If _ s0 s1 s2 =>
+      If n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1) (ren_Tm xi_Tm s2)
   end.
 
 Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@@ -154,6 +182,11 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
   | Const _ s0 => Const n_Tm s0
   | Univ _ s0 => Univ n_Tm s0
   | Bot _ => Bot n_Tm
+  | BVal _ s0 => BVal n_Tm s0
+  | Bool _ => Bool n_Tm
+  | If _ s0 s1 s2 =>
+      If n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
+        (subst_Tm sigma_Tm s2)
   end.
 
 Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@@ -195,6 +228,11 @@ subst_Tm sigma_Tm s = s :=
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
+        (idSubst_Tm sigma_Tm Eq_Tm s2)
   end.
 
 Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@@ -240,6 +278,11 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
+        (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s2)
   end.
 
 Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@@ -286,6 +329,11 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
+        (ext_Tm sigma_Tm tau_Tm Eq_Tm s1) (ext_Tm sigma_Tm tau_Tm Eq_Tm s2)
   end.
 
 Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@@ -332,6 +380,12 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
+        (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
+        (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s2)
   end.
 
 Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@@ -389,6 +443,12 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
+        (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
+        (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s2)
   end.
 
 Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@@ -467,6 +527,12 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
+        (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
+        (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s2)
   end.
 
 Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@@ -546,6 +612,12 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
+        (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
+        (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s2)
   end.
 
 Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@@ -663,6 +735,12 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
   | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | BVal _ s0 => congr_BVal (eq_refl s0)
+  | Bool _ => congr_Bool
+  | If _ s0 s1 s2 =>
+      congr_If (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
+        (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
+        (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s2)
   end.
 
 Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@@ -861,6 +939,12 @@ Core.
 
 Arguments VarTm {n_Tm}.
 
+Arguments If {n_Tm}.
+
+Arguments Bool {n_Tm}.
+
+Arguments BVal {n_Tm}.
+
 Arguments Bot {n_Tm}.
 
 Arguments Univ {n_Tm}.
diff --git a/theories/compile.v b/theories/compile.v
index 05bc9a8..e481d71 100644
--- a/theories/compile.v
+++ b/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)) :