diff --git a/syntax.sig b/syntax.sig
index 3101cab..4549f7f 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -3,15 +3,14 @@ Tm(VarTm) : Type
 PTag : Type
 TTag : Type
 
-PL : PTag
-PR : PTag
 TPi : TTag
 TSig : TTag
+PL : PTag
+PR : PTag
 Abs : (bind Tm in Tm) -> Tm
 App : Tm -> Tm -> Tm
 Pair : Tm -> Tm -> Tm
 Proj : PTag -> Tm -> Tm
 TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
-Const : TTag -> Tm
-Univ : nat -> Tm
 Bot : Tm
+Univ : nat -> Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index a5cb002..b972476 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -40,9 +40,8 @@ Inductive Tm (n_Tm : nat) : Type :=
   | Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
   | Proj : PTag -> Tm n_Tm -> Tm n_Tm
   | 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
+  | Univ : nat -> 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.
@@ -84,10 +83,9 @@ exact (eq_trans
          (ap (fun x => TBind m_Tm t0 t1 x) H2)).
 Qed.
 
-Lemma congr_Const {m_Tm : nat} {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
-  Const m_Tm s0 = Const m_Tm t0.
+Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
 Proof.
-exact (eq_trans eq_refl (ap (fun x => Const m_Tm x) H0)).
+exact (eq_refl).
 Qed.
 
 Lemma congr_Univ {m_Tm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
@@ -96,11 +94,6 @@ Proof.
 exact (eq_trans eq_refl (ap (fun x => Univ m_Tm x) H0)).
 Qed.
 
-Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
-Proof.
-exact (eq_refl).
-Qed.
-
 Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
   fin (S m) -> fin (S n).
 Proof.
@@ -123,9 +116,8 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
   | Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
   | TBind _ s0 s1 s2 =>
       TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
-  | Const _ s0 => Const n_Tm s0
-  | Univ _ s0 => Univ n_Tm s0
   | Bot _ => Bot n_Tm
+  | Univ _ s0 => Univ n_Tm s0
   end.
 
 Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@@ -151,9 +143,8 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
   | Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
   | TBind _ s0 s1 s2 =>
       TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
-  | Const _ s0 => Const n_Tm s0
-  | Univ _ s0 => Univ n_Tm s0
   | Bot _ => Bot n_Tm
+  | Univ _ s0 => Univ n_Tm s0
   end.
 
 Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@@ -192,9 +183,8 @@ subst_Tm sigma_Tm s = s :=
   | TBind _ s0 s1 s2 =>
       congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
         (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@@ -237,9 +227,8 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
       congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
         (extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
            (upExtRen_Tm_Tm _ _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@@ -283,9 +272,8 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
       congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
         (ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
            s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@@ -329,9 +317,8 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
       congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
         (compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
            (upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@@ -386,9 +373,8 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
         (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
         (compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
            (up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@@ -464,9 +450,8 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
         (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
         (compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
            (up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@@ -543,9 +528,8 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
         (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
         (compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
            (up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@@ -660,9 +644,8 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
       congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
         (rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
            (rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
-  | Const _ s0 => congr_Const (eq_refl s0)
-  | Univ _ s0 => congr_Univ (eq_refl s0)
   | Bot _ => congr_Bot
+  | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
 Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@@ -861,11 +844,9 @@ Core.
 
 Arguments VarTm {n_Tm}.
 
-Arguments Bot {n_Tm}.
-
 Arguments Univ {n_Tm}.
 
-Arguments Const {n_Tm}.
+Arguments Bot {n_Tm}.
 
 Arguments TBind {n_Tm}.
 
diff --git a/theories/compile.v b/theories/compile.v
deleted file mode 100644
index 05bc9a8..0000000
--- a/theories/compile.v
+++ /dev/null
@@ -1,181 +0,0 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red.
-Require Import ssreflect ssrbool.
-From Hammer Require Import Tactics.
-From stdpp Require Import relations (rtc(..)).
-
-Module Compile.
-  Fixpoint F {n} (a : Tm n) : Tm n :=
-    match a with
-    | TBind p A B => Pair (Pair (Const p) (F A)) (Abs (F B))
-    | Const k => Const k
-    | Univ i => Univ i
-    | Abs a => Abs (F a)
-    | App a b => App (F a) (F b)
-    | VarTm i => VarTm i
-    | Pair a b => Pair (F a) (F b)
-    | Proj t a => Proj t (F a)
-    | Bot => Bot
-    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.
-
-  #[local]Hint Rewrite Compile.renaming : compile.
-
-  Lemma morphing n m (a : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
-    (forall i, ρ0 i = F (ρ1 i)) ->
-    subst_Tm ρ0 (F a) = F (subst_Tm ρ1 a).
-  Proof.
-    move : m ρ0 ρ1. elim : n / a => n//=.
-    - hauto lq:on inv:option rew:db:compile unfold:funcomp.
-    - hauto lq:on rew:off.
-    - hauto lq:on rew:off.
-    - hauto lq:on.
-    - hauto lq:on inv:option rew:db:compile unfold:funcomp.
-  Qed.
-
-  Lemma substing n b (a : Tm (S n)) :
-    subst_Tm (scons (F b) VarTm) (F a) = F (subst_Tm (scons b VarTm) a).
-  Proof.
-    apply morphing.
-    case => //=.
-  Qed.
-
-End Compile.
-
-#[export] Hint Rewrite Compile.renaming Compile.morphing : compile.
-
-
-Module Join.
-  Definition R {n} (a b : Tm n) := join (Compile.F a) (Compile.F b).
-
-  Lemma BindInj n p0 p1 (A0 A1 : Tm n) B0 B1  :
-    R (TBind p0 A0 B0) (TBind p1 A1 B1) -> p0 = p1 /\ R A0 A1 /\ R B0 B1.
-  Proof.
-    rewrite /R /= !join_pair_inj.
-    move => [[/join_const_inj h0 h1] h2].
-    apply abs_eq in h2.
-    evar (t : Tm (S n)).
-    have : join (App (ren_Tm shift (Abs (Compile.F B1))) (VarTm var_zero)) t by
-      apply Join.FromPar; apply Par.AppAbs; auto using Par.refl.
-    subst t. rewrite -/ren_Tm.
-    move : h2. move /join_transitive => /[apply]. asimpl => h2.
-    tauto.
-  Qed.
-
-  Lemma UnivInj n i j : R (Univ i : Tm n) (Univ j) -> i = j.
-  Proof. hauto l:on use:join_univ_inj. Qed.
-
-  Lemma transitive n (a b c : Tm n) :
-    R a b -> R b c -> R a c.
-  Proof. hauto l:on use:join_transitive unfold:R. Qed.
-
-  Lemma symmetric n (a b : Tm n) :
-    R a b -> R b a.
-  Proof. hauto l:on use:join_symmetric. Qed.
-
-  Lemma reflexive n (a : Tm n) :
-    R a a.
-  Proof. hauto l:on use:join_refl. Qed.
-
-  Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
-    R a b -> R (subst_Tm ρ a) (subst_Tm ρ b).
-  Proof.
-    rewrite /R.
-    rewrite /join.
-    move => [C [h0 h1]].
-    repeat (rewrite <- Compile.morphing with (ρ0 := funcomp Compile.F ρ); last by reflexivity).
-    hauto lq:on use:join_substing.
-  Qed.
-
-End Join.
-
-Module Equiv.
-  Inductive R {n} : Tm n -> Tm n ->  Prop :=
-  (***************** Beta ***********************)
-  | AppAbs a b :
-    R (App (Abs a) b)  (subst_Tm (scons b VarTm) a)
-  | ProjPair p a b :
-    R (Proj p (Pair a b)) (if p is PL then a else b)
-
-  (****************** Eta ***********************)
-  | AppEta a :
-    R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
-  | PairEta a :
-    R a (Pair (Proj PL a) (Proj PR a))
-
-  (*************** Congruence ********************)
-  | Var i : R (VarTm i) (VarTm i)
-  | AbsCong a b :
-    R a b ->
-    R (Abs a) (Abs b)
-  | AppCong a0 a1 b0 b1 :
-    R a0 a1 ->
-    R b0 b1 ->
-    R (App a0 b0) (App a1 b1)
-  | PairCong a0 a1 b0 b1 :
-    R a0 a1 ->
-    R b0 b1 ->
-    R (Pair a0 b0) (Pair a1 b1)
-  | ProjCong p a0 a1 :
-    R a0 a1 ->
-    R (Proj p a0) (Proj p a1)
-  | BindCong p A0 A1 B0 B1:
-    R A0 A1 ->
-    R B0 B1 ->
-    R (TBind p A0 B0) (TBind p A1 B1)
-  | UnivCong i :
-    R (Univ i) (Univ i).
-End Equiv.
-
-Module EquivJoin.
-  Lemma FromEquiv n (a b : Tm n) : Equiv.R a b -> Join.R a b.
-  Proof.
-    move => h. elim : n a b /h => n.
-    - move => a b.
-      rewrite /Join.R /join /=.
-      eexists. split. apply relations.rtc_once.
-      apply Par.AppAbs; auto using Par.refl.
-      rewrite Compile.substing.
-      apply relations.rtc_refl.
-    - move => p a b.
-      apply Join.FromPar.
-      simpl. apply : Par.ProjPair'; auto using Par.refl.
-      case : p => //=.
-    - move => a. apply Join.FromPar => /=.
-      apply : Par.AppEta'; auto using Par.refl.
-      by autorewrite with compile.
-    - move => a. apply Join.FromPar => /=.
-      apply : Par.PairEta; auto using Par.refl.
-    - hauto l:on use:Join.FromPar, Par.Var.
-    - hauto lq:on use:Join.AbsCong.
-    - qauto l:on use:Join.AppCong.
-    - qauto l:on use:Join.PairCong.
-    - qauto use:Join.ProjCong.
-    - rewrite /Join.R => p A0 A1 B0 B1 _ hA _ hB /=.
-      sfirstorder use:Join.PairCong,Join.AbsCong,Join.FromPar,Par.ConstCong.
-    - hauto l:on.
-  Qed.
-End EquivJoin.
-
-Lemma compile_rpar n (a b : Tm n) : RPar'.R a b -> RPar'.R (Compile.F a) (Compile.F b).
-Proof.
-  move => h. elim : n a b /h.
-  - move => n  a0 a1 b0 b1 ha iha hb ihb /=.
-    rewrite -Compile.substing.
-    apply RPar'.AppAbs => //.
-  - hauto q:on use:RPar'.ProjPair'.
-  - qauto ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-  - hauto lq:on ctrs:RPar'.R.
-Qed.
-
-Lemma compile_rpars n (a b : Tm n) : rtc RPar'.R a b -> rtc RPar'.R (Compile.F a) (Compile.F b).
-Proof. induction 1; hauto lq:on ctrs:rtc use:compile_rpar. Qed.
diff --git a/theories/fp_red.v b/theories/fp_red.v
index e9b5591..d2fbdf6 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -69,12 +69,11 @@ Module Par.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | ConstCong k :
-    R (Const k) (Const k)
-  | UnivCong i :
-    R (Univ i) (Univ i)
+  (* Bot is useful for making the prov function computable  *)
   | BotCong :
-    R Bot Bot.
+    R Bot Bot
+  | UnivCong i :
+    R (Univ i) (Univ i).
 
   Lemma refl n (a : Tm n) : R a a.
     elim : n /a; hauto ctrs:R.
@@ -136,7 +135,6 @@ Module Par.
     - hauto l:on inv:option ctrs:R use:renaming.
     - sfirstorder.
     - sfirstorder.
-    - sfirstorder.
   Qed.
 
   Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -214,9 +212,8 @@ Module Par.
       move : ihB => [c0 [? ?]]. subst.
       eexists. split. by apply BindCong; eauto.
       by asimpl.
-    - move => n k m ξ []//=. hauto l:on.
+    - move => n n0 ξ []//=. hauto l:on.
     - move => n i n0 ξ []//=. hauto l:on.
-    - hauto q:on inv:Tm ctrs:R.
   Qed.
 
 End Par.
@@ -278,7 +275,7 @@ Module Pars.
 
 End Pars.
 
-Definition var_or_const {n} (a : Tm n) :=
+Definition var_or_bot {n} (a : Tm n) :=
   match a with
   | VarTm _ => true
   | Bot => true
@@ -327,12 +324,10 @@ Module RPar.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | ConstCong k :
-    R (Const k) (Const k)
-  | UnivCong i :
-    R (Univ i) (Univ i)
   | BotCong :
-    R Bot Bot.
+    R Bot Bot
+  | UnivCong i :
+    R (Univ i) (Univ i).
 
   Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
 
@@ -400,7 +395,6 @@ Module RPar.
     - hauto lq:on ctrs:R use:morphing_up.
     - hauto lq:on ctrs:R.
     - hauto lq:on ctrs:R.
-    - hauto lq:on ctrs:R.
   Qed.
 
   Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -417,16 +411,16 @@ Module RPar.
     qauto l:on ctrs:R inv:option.
   Qed.
 
-  Lemma var_or_const_imp {n} (a b : Tm n) :
-    var_or_const a ->
-    a = b -> ~~ var_or_const b -> False.
+  Lemma var_or_bot_imp {n} (a b : Tm n) :
+    var_or_bot a ->
+    a = b -> ~~ var_or_bot b -> False.
   Proof.
     hauto lq:on inv:Tm.
   Qed.
 
-  Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
-    (forall i, var_or_const (ρ i)) ->
-    (forall i, var_or_const (up_Tm_Tm ρ i)).
+  Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
+    (forall i, var_or_bot (ρ i)) ->
+    (forall i, var_or_bot (up_Tm_Tm ρ i)).
   Proof.
     move => h /= [i|].
     - asimpl.
@@ -437,10 +431,10 @@ Module RPar.
     - sfirstorder.
   Qed.
 
-  Local Ltac antiimp := qauto l:on use:var_or_const_imp.
+  Local Ltac antiimp := qauto l:on use:var_or_bot_imp.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_const (ρ i)) ->
+    (forall i, var_or_bot (ρ i)) ->
     R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E : (subst_Tm ρ a) => u hρ h.
@@ -451,7 +445,7 @@ Module RPar.
       case : c => //=; first by antiimp.
       move => c [?]. subst.
       spec_refl.
-      have /var_or_const_up hρ' := hρ.
+      have /var_or_bot_up hρ' := hρ.
       move : iha hρ' => /[apply] iha.
       move : ihb hρ => /[apply] ihb.
       spec_refl.
@@ -477,7 +471,7 @@ Module RPar.
     - move => n p a0 a1 ha iha m ρ hρ []//=;
                first by antiimp.
       move => p0 []//= t [*]; first by antiimp. subst.
-      have  /var_or_const_up {}/iha  := hρ => iha.
+      have  /var_or_bot_up {}/iha  := hρ => iha.
       spec_refl. move : iha => [b0 [? ?]]. subst.
       eexists. split. apply ProjAbs; eauto. by asimpl.
     - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
@@ -495,7 +489,7 @@ Module RPar.
       hauto l:on.
     - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
       move => t [*]. subst.
-      have /var_or_const_up {}/iha := hρ => iha.
+      have /var_or_bot_up {}/iha := hρ => iha.
       spec_refl.
       move  :iha => [b0 [? ?]]. subst.
       eexists. split. by apply AbsCong; eauto.
@@ -531,7 +525,7 @@ Module RPar.
                first by antiimp.
       move => ? t t0 [*]. subst.
       have {}/iha := (hρ) => iha.
-      have /var_or_const_up {}/ihB := (hρ) => ihB.
+      have /var_or_bot_up {}/ihB := (hρ) => ihB.
       spec_refl.
       move : iha => [b0 [? ?]].
       move : ihB => [c0 [? ?]]. subst.
@@ -540,7 +534,6 @@ Module RPar.
     - hauto q:on ctrs:R inv:Tm.
     - move => n i n0 ρ hρ []//=; first by antiimp.
       hauto l:on.
-    - move => n m ρ hρ []//=; hauto lq:on ctrs:R.
   Qed.
 End RPar.
 
@@ -578,12 +571,10 @@ Module RPar'.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | ConstCong k :
-    R (Const k) (Const k)
-  | UnivCong i :
-    R (Univ i) (Univ i)
   | BotCong :
-    R Bot Bot.
+    R Bot Bot
+  | UnivCong i :
+    R (Univ i) (Univ i).
 
   Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
 
@@ -649,7 +640,6 @@ Module RPar'.
     - hauto lq:on ctrs:R use:morphing_up.
     - hauto lq:on ctrs:R.
     - hauto lq:on ctrs:R.
-    - hauto lq:on ctrs:R.
   Qed.
 
   Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
@@ -666,16 +656,16 @@ Module RPar'.
     qauto l:on ctrs:R inv:option.
   Qed.
 
-  Lemma var_or_const_imp {n} (a b : Tm n) :
-    var_or_const a ->
-    a = b -> ~~ var_or_const b -> False.
+  Lemma var_or_bot_imp {n} (a b : Tm n) :
+    var_or_bot a ->
+    a = b -> ~~ var_or_bot b -> False.
   Proof.
     hauto lq:on inv:Tm.
   Qed.
 
-  Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
-    (forall i, var_or_const (ρ i)) ->
-    (forall i, var_or_const (up_Tm_Tm ρ i)).
+  Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
+    (forall i, var_or_bot (ρ i)) ->
+    (forall i, var_or_bot (up_Tm_Tm ρ i)).
   Proof.
     move => h /= [i|].
     - asimpl.
@@ -686,10 +676,10 @@ Module RPar'.
     - sfirstorder.
   Qed.
 
-  Local Ltac antiimp := qauto l:on use:var_or_const_imp.
+  Local Ltac antiimp := qauto l:on use:var_or_bot_imp.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_const (ρ i)) ->
+    (forall i, var_or_bot (ρ i)) ->
     R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E : (subst_Tm ρ a) => u hρ h.
@@ -700,7 +690,7 @@ Module RPar'.
       case : c => //=; first by antiimp.
       move => c [?]. subst.
       spec_refl.
-      have /var_or_const_up hρ' := hρ.
+      have /var_or_bot_up hρ' := hρ.
       move : iha hρ' => /[apply] iha.
       move : ihb hρ => /[apply] ihb.
       spec_refl.
@@ -724,7 +714,7 @@ Module RPar'.
       hauto l:on.
     - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
       move => t [*]. subst.
-      have /var_or_const_up {}/iha := hρ => iha.
+      have /var_or_bot_up {}/iha := hρ => iha.
       spec_refl.
       move  :iha => [b0 [? ?]]. subst.
       eexists. split. by apply AbsCong; eauto.
@@ -760,7 +750,7 @@ Module RPar'.
                first by antiimp.
       move => ? t t0 [*]. subst.
       have {}/iha := (hρ) => iha.
-      have /var_or_const_up {}/ihB := (hρ) => ihB.
+      have /var_or_bot_up {}/ihB := (hρ) => ihB.
       spec_refl.
       move : iha => [b0 [? ?]].
       move : ihB => [c0 [? ?]]. subst.
@@ -769,7 +759,6 @@ Module RPar'.
     - hauto q:on ctrs:R inv:Tm.
     - move => n i n0 ρ hρ []//=; first by antiimp.
       hauto l:on.
-    - hauto q:on inv:Tm ctrs:R.
   Qed.
 End RPar'.
 
@@ -905,12 +894,10 @@ Module EPar.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | ConstCong k :
-    R (Const k) (Const k)
-  | UnivCong i :
-    R (Univ i) (Univ i)
   | BotCong :
-    R Bot Bot.
+    R Bot Bot
+  | UnivCong i :
+    R (Univ i) (Univ i).
 
   Lemma refl n (a : Tm n) : EPar.R a a.
   Proof.
@@ -955,7 +942,6 @@ Module EPar.
     - hauto l:on ctrs:R use:renaming inv:option.
     - hauto lq:on ctrs:R.
     - hauto lq:on ctrs:R.
-    - hauto lq:on ctrs:R.
   Qed.
 
   Lemma substing n a0 a1 (b0 b1 : Tm n) :
@@ -1084,7 +1070,7 @@ Module RPars.
   Proof. hauto lq:on use:morphing inv:option. Qed.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_const (ρ i)) ->
+    (forall i, var_or_bot (ρ i)) ->
     rtc RPar.R (subst_Tm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E  :(subst_Tm ρ a) => u hρ h.
@@ -1171,7 +1157,7 @@ Module RPars'.
   Proof. hauto lq:on use:morphing inv:option. Qed.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_const (ρ i)) ->
+    (forall i, var_or_bot (ρ i)) ->
     rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E  :(subst_Tm ρ a) => u hρ h.
@@ -1359,7 +1345,6 @@ Proof.
   - hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
   - hauto l:on ctrs:EPar.R inv:RPar.R.
   - hauto l:on ctrs:EPar.R inv:RPar.R.
-  - hauto l:on ctrs:EPar.R inv:RPar.R.
 Qed.
 
 Lemma commutativity1 n (a b0 b1 : Tm n) :
@@ -1459,24 +1444,10 @@ Proof.
   - hauto l:on ctrs:OExp.R.
 Qed.
 
-Lemma Const_EPar' n k (u : Tm n) :
-  EPar.R (Const k) u ->
-  rtc OExp.R (Const k) u.
-  move E : (Const k) => t h.
-  move : k E. elim : n t u /h => //=.
-  - move => n a0 a1 h ih k ?. subst.
-    specialize ih with (1 := eq_refl).
-    hauto lq:on ctrs:OExp.R use:rtc_r.
-  - move => n a0 a1 h ih k ?. subst.
-    specialize ih with (1 := eq_refl).
-    hauto lq:on ctrs:OExp.R use:rtc_r.
-  - hauto l:on ctrs:OExp.R.
-Qed.
-
 Lemma Bot_EPar' n (u : Tm n) :
-  EPar.R (Bot) u ->
-  rtc OExp.R (Bot) u.
-  move E : (Bot) => t h.
+  EPar.R Bot u ->
+  rtc OExp.R Bot u.
+  move E : Bot => t h.
   move : E. elim : n t u /h => //=.
   - move => n a0 a1 h ih ?. subst.
     specialize ih with (1 := eq_refl).
@@ -1548,9 +1519,8 @@ Proof.
     move : OExp.commutativity0 h2; repeat move/[apply].
     move => [d h].
     exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
-  - qauto use:Const_EPar', EPar.refl.
-  - qauto use:Univ_EPar', EPar.refl.
   - qauto use:Bot_EPar', EPar.refl.
+  - qauto use:Univ_EPar', EPar.refl.
 Qed.
 
 Function tstar {n} (a : Tm n) :=
@@ -1566,9 +1536,8 @@ Function tstar {n} (a : Tm n) :=
   | Proj p (Abs a) => (Abs (Proj p (tstar a)))
   | Proj p a => Proj p (tstar a)
   | TBind p a b => TBind p (tstar a) (tstar b)
-  | Const k => Const k
-  | Univ i => Univ i
   | Bot => Bot
+  | Univ i => Univ i
   end.
 
 Lemma RPar_triangle n (a : Tm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
@@ -1587,7 +1556,6 @@ Proof.
   - hauto lq:on inv:RPar.R ctrs:RPar.R.
   - hauto lq:on inv:RPar.R ctrs:RPar.R.
   - hauto lq:on inv:RPar.R ctrs:RPar.R.
-  - hauto lq:on inv:RPar.R ctrs:RPar.R.
 Qed.
 
 Function tstar' {n} (a : Tm n) :=
@@ -1600,9 +1568,8 @@ Function tstar' {n} (a : Tm n) :=
   | Proj p (Pair a b) => if p is PL then (tstar' a) else (tstar' b)
   | Proj p a => Proj p (tstar' a)
   | TBind p a b => TBind p (tstar' a) (tstar' b)
-  | Const k => Const k
-  | Univ i => Univ i
   | Bot => Bot
+  | Univ i => Univ i
   end.
 
 Lemma RPar'_triangle n (a : Tm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a).
@@ -1619,7 +1586,6 @@ Proof.
   - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
   - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
   - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
-  - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
 Qed.
 
 Lemma RPar_diamond n (c a1 b1 : Tm n) :
@@ -1650,6 +1616,63 @@ Proof.
   sfirstorder use:relations.diamond_confluent, EPar_diamond.
 Qed.
 
+Fixpoint depth_tm {n} (a : Tm n) :=
+  match a with
+  | VarTm _ => 1
+  | TBind _ A B => 1 + max (depth_tm A) (depth_tm B)
+  | Abs a => 1 + depth_tm a
+  | App a b => 1 + max (depth_tm a) (depth_tm b)
+  | Proj p a => 1 + depth_tm a
+  | Pair a b => 1 + max (depth_tm a) (depth_tm b)
+  | Bot => 1
+  | Univ i => 1
+  end.
+
+Lemma depth_ren n m (ξ: fin n -> fin m) a :
+  depth_tm a = depth_tm (ren_Tm ξ a).
+Proof.
+  move : m ξ. elim : n / a; scongruence.
+Qed.
+
+Lemma depth_subst n m (ρ : fin n -> Tm m) a :
+  (forall i, depth_tm (ρ i) = 1) ->
+  depth_tm a = depth_tm (subst_Tm ρ a).
+Proof.
+  move : m ρ. elim : n / a.
+  - sfirstorder.
+  - move => n a iha m ρ hρ.
+    simpl.
+    f_equal. apply iha.
+    destruct i as [i|].
+    + simpl.
+      by rewrite -depth_ren.
+    + by simpl.
+  - hauto lq:on rew:off.
+  - hauto lq:on rew:off.
+  - hauto lq:on rew:off.
+  - move => n p a iha b ihb m ρ hρ.
+    simpl. f_equal.
+    f_equal.
+    by apply iha.
+    apply ihb.
+    destruct i as [i|].
+    + simpl.
+      by rewrite -depth_ren.
+    + by simpl.
+  - sfirstorder.
+  - sfirstorder.
+Qed.
+
+Lemma depth_subst_bool n (a : Tm (S n)) :
+  depth_tm a = depth_tm (subst_Tm (scons Bot VarTm) a).
+Proof.
+  apply depth_subst.
+  destruct i as [i|] => //=.
+Qed.
+
+Local Ltac prov_tac := sfirstorder use:depth_ren.
+Local Ltac extract_tac := rewrite -?depth_subst_bool;hauto use:depth_subst_bool.
+
 Definition prov_bind {n} p0 A0 B0 (a : Tm n)  :=
   match a with
   | TBind p A B => p = p0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
@@ -1681,14 +1704,12 @@ Inductive prov {n} : Tm n -> Tm n -> Prop :=
 | P_Proj h p a :
   prov h a ->
   prov h (Proj p a)
-| P_Const k  :
-  prov (Const k) (Const k)
+| P_Bot  :
+  prov Bot Bot
 | P_Var i :
   prov (VarTm i) (VarTm i)
 | P_Univ i :
-  prov (Univ i) (Univ i)
-| P_Bot :
-  prov Bot Bot.
+  prov (Univ i) (Univ i).
 
 Lemma ERed_EPar n (a b : Tm n) : ERed.R a b -> EPar.R a b.
 Proof.
@@ -1708,7 +1729,6 @@ Proof.
   - eauto using EReds.BindCong.
   - auto using rtc_refl.
   - auto using rtc_refl.
-  - auto using rtc_refl.
 Qed.
 
 Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
@@ -1761,7 +1781,6 @@ Proof.
   - hauto lq:on ctrs:prov inv:RPar.R.
   - hauto l:on ctrs:RPar.R inv:RPar.R.
   - hauto l:on ctrs:RPar.R inv:RPar.R.
-  - hauto l:on ctrs:RPar.R inv:RPar.R.
 Qed.
 
 
@@ -1770,7 +1789,7 @@ Proof.
   split.
   move => h. constructor. move => b. asimpl. by constructor.
   inversion 1; subst.
-  specialize H2 with (b := Const TPi).
+  specialize H2 with (b := Bot).
   move : H2. asimpl. inversion 1; subst. done.
 Qed.
 
@@ -1816,7 +1835,6 @@ Proof.
   - hauto lq:on inv:ERed.R, prov ctrs:prov.
   - hauto lq:on inv:ERed.R, prov ctrs:prov.
   - hauto lq:on inv:ERed.R, prov ctrs:prov.
-  - hauto lq:on inv:ERed.R, prov ctrs:prov.
 Qed.
 
 Lemma prov_ereds n (u : Tm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
@@ -1831,10 +1849,9 @@ Fixpoint extract {n} (a : Tm n) : Tm n :=
   | App a b => extract a
   | Pair a b => extract a
   | Proj p a => extract a
-  | Const k => Const k
+  | Bot => Bot
   | VarTm i => VarTm i
   | Univ i => Univ i
-  | Bot => Bot
   end.
 
 Lemma ren_extract n m (a : Tm n) (ξ : fin n -> fin m) :
@@ -1851,7 +1868,6 @@ Proof.
   - hauto q:on.
   - sfirstorder.
   - sfirstorder.
-  - sfirstorder.
 Qed.
 
 Lemma ren_morphing n m (a : Tm n) (ρ : fin n -> Tm m) :
@@ -1880,7 +1896,6 @@ Definition prov_extract_spec {n} u (a : Tm n) :=
   | TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
   | Univ i => extract a = Univ i
   | VarTm i => extract a = VarTm i
-  | (Const i) => extract a = (Const i)
   | Bot => extract a = Bot
   | _ => True
   end.
@@ -1902,17 +1917,13 @@ Proof.
       rewrite ren_subst_bot in h0.
       rewrite h0.
       eauto.
-    + move => p _ /(_ Bot).
+    + move => _ /(_ Bot).
       by rewrite ren_subst_bot.
     + move => i h /(_ Bot).
       by rewrite ren_subst_bot => ->.
-    + move /(_ Bot).
-      move => h /(_ Bot).
-      by rewrite ren_subst_bot.
   - hauto lq:on.
   - hauto lq:on.
   - hauto lq:on.
-  - case => //=.
   - sfirstorder.
   - sfirstorder.
   - sfirstorder.
@@ -2119,7 +2130,6 @@ Proof.
   - sfirstorder use:ERPars.BindCong.
   - sfirstorder.
   - sfirstorder.
-  - sfirstorder.
 Qed.
 
 Lemma Pars_ERPar n (a b : Tm n) : rtc Par.R a b -> rtc ERPar.R a b.
@@ -2298,15 +2308,6 @@ Proof.
   apply prov_extract.
 Qed.
 
-Lemma pars_const_inv n i (c : Tm n) :
-  rtc Par.R (Const i) c ->
-  extract c = Const i.
-Proof.
-  have : prov (Const i) (Const i : Tm n) by sfirstorder.
-  move : prov_pars. repeat move/[apply].
-  apply prov_extract.
-Qed.
-
 Lemma pars_pi_inv n p (A : Tm n) B C :
   rtc Par.R (TBind p A B) C ->
   exists A0 B0, extract C = TBind p A0 B0 /\
@@ -2334,14 +2335,6 @@ Proof.
   sauto l:on use:pars_univ_inv.
 Qed.
 
-Lemma pars_const_inj n i j (C : Tm n) :
-  rtc Par.R (Const i) C ->
-  rtc Par.R (Const j) C ->
-  i = j.
-Proof.
-  sauto l:on use:pars_const_inv.
-Qed.
-
 Lemma pars_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 C :
   rtc Par.R (TBind p0 A0 B0) C ->
   rtc Par.R (TBind p1 A1 B1) C ->
@@ -2379,12 +2372,6 @@ Proof.
   sfirstorder use:pars_univ_inj.
 Qed.
 
-Lemma join_const_inj n i j :
-  join (Const i : Tm n) (Const j) -> i = j.
-Proof.
-  sfirstorder use:pars_const_inj.
-Qed.
-
 Lemma join_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 :
   join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
   p0 = p1 /\ join A0 A1 /\ join B0 B1.
@@ -2394,6 +2381,16 @@ Proof.
   sfirstorder unfold:join.
 Qed.
 
+Lemma join_univ_pi_contra n p (A : Tm n) B i :
+  join (TBind p A B) (Univ i) -> False.
+Proof.
+  rewrite /join.
+  move => [c [h0 h1]].
+  move /pars_univ_inv : h1.
+  move /pars_pi_inv : h0.
+  hauto l:on.
+Qed.
+
 Lemma join_substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
   join a b ->
   join (subst_Tm ρ a) (subst_Tm ρ b).
@@ -2403,25 +2400,23 @@ Fixpoint ne {n} (a : Tm n) :=
   match a with
   | VarTm i => true
   | TBind _ A B => false
+  | Bot => true
   | App a b => ne a && nf b
   | Abs a => false
   | Univ _ => false
   | Proj _ a => ne a
   | Pair _ _ => false
-  | Const _ => false
-  | Bot => true
   end
 with nf {n} (a : Tm n) :=
   match a with
   | VarTm i => true
   | TBind _ A B => nf A && nf B
+  | Bot => true
   | App a b => ne a && nf b
   | Abs a => nf a
   | Univ _ => true
   | Proj _ a => ne a
   | Pair a b => nf a && nf b
-  | Const _ => true
-  | Bot => true
 end.
 
 Lemma ne_nf n a : @ne n a -> nf a.
@@ -2487,15 +2482,15 @@ Create HintDb nfne.
 #[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne.
 
 Lemma ne_nf_antiren n m (a : Tm n) (ρ : fin n -> Tm m) :
-  (forall i, var_or_const (ρ i)) ->
+  (forall i, var_or_bot (ρ i)) ->
   (ne (subst_Tm ρ a) -> ne a) /\ (nf (subst_Tm ρ a) -> nf a).
 Proof.
   move : m ρ. elim : n / a => //;
-   hauto b:on drew:off use:RPar.var_or_const_up.
+   hauto b:on drew:off use:RPar.var_or_bot_up.
 Qed.
 
 Lemma wn_antirenaming n m a (ρ : fin n -> Tm m) :
-  (forall i, var_or_const (ρ i)) ->
+  (forall i, var_or_bot (ρ i)) ->
   wn (subst_Tm ρ a) -> wn a.
 Proof.
   rewrite /wn => hρ.
@@ -2511,7 +2506,7 @@ Lemma ext_wn n (a : Tm n) :
     wn (App a Bot) ->
     wn a.
 Proof.
-  move E : (App a (Bot)) => a0 [v [hr hv]].
+  move E : (App a Bot) => a0 [v [hr hv]].
   move : a E.
   move : hv.
   elim : a0 v / hr.
@@ -2522,9 +2517,9 @@ Proof.
     case : a0 hr0=>// => b0 b1.
     elim /RPar'.inv=>// _.
     + move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
-      have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
+      have ? : b3 = Bot by hauto lq:on inv:RPar'.R. subst.
       suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
-      have : wn (subst_Tm (scons (Bot) VarTm) a3) by sfirstorder.
+      have : wn (subst_Tm (scons Bot VarTm) a3) by sfirstorder.
       move => h. apply wn_abs.
       move : h. apply wn_antirenaming.
       hauto lq:on rew:off inv:option.
diff --git a/theories/logrel.v b/theories/logrel.v
index b9c854d..a4b15d2 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -1,4 +1,5 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red compile.
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
+Require Import fp_red.
 From Hammer Require Import Tactics.
 From Equations Require Import Equations.
 Require Import ssreflect ssrbool.
@@ -262,12 +263,8 @@ Lemma RPar's_Pars n (A B : Tm n) :
 Proof. hauto lq:on use:RPar'_Par, rtc_subrel. Qed.
 
 Lemma RPar's_join n (A B : Tm n) :
-  rtc RPar'.R A B -> Join.R A B.
-Proof.
-  rewrite /Join.R => h.
-  have {}h : rtc RPar'.R (Compile.F A) (Compile.F B) by eauto using compile_rpars.
-  rewrite /join. eauto using RPar's_Pars, rtc_refl.
-Qed.
+  rtc RPar'.R A B -> join A B.
+Proof. hauto lq:on ctrs:rtc use:RPar's_Pars. Qed.
 
 Lemma bindspace_iff n p (PA : Tm n -> Prop) PF PF0 b  :
   (forall (a : Tm n) (PB PB0 : Tm n -> Prop), PF a PB -> PF0 a PB0 -> PB = PB0) ->
@@ -298,9 +295,9 @@ Proof.
   - hauto lq:on ctrs:prov.
   - hauto lq:on rew:off ctrs:prov b:on.
   - hauto lq:on ctrs:prov.
-  - move => n k.
+  - move => n.
     have : @prov n Bot Bot by auto using P_Bot.
-    eauto.
+    tauto.
 Qed.
 
 Lemma ne_pars_inv n (a b : Tm n) :
@@ -314,59 +311,32 @@ Lemma ne_pars_extract  n (a b : Tm n) :
   ne a -> rtc Par.R a b -> (exists i, extract b = (VarTm i)) \/ extract b = Bot.
 Proof. hauto lq:on rew:off use:ne_pars_inv, prov_extract. Qed.
 
-Lemma const_pars_extract n k b :
-  rtc Par.R (Const k : Tm n) b -> extract b = Const k.
-Proof. hauto l:on use:pars_const_inv, prov_extract. Qed.
-
-Lemma compile_ne n (a : Tm n) :
-  ne a = ne (Compile.F a) /\ nf a = nf (Compile.F a).
-Proof.
-  elim : n / a => //=; sfirstorder b:on.
-Qed.
-
-Lemma join_univ_pi_contra n p (A : Tm n) B i :
-  Join.R (TBind p A B) (Univ i) -> False.
-Proof.
-  rewrite /Join.R /=.
-  rewrite !pair_eq.
-  move => [[h _ ]_ ].
-  move : h => [C [h0 h1]].
-  have ?  : extract C = Const p by hauto l:on use:pars_const_inv.
-  have h : prov (Univ i : Tm n) (Proj PL (Proj PL (Univ i))) by hauto lq:on ctrs:prov.
-  have {h} : prov (Univ i) C by eauto using prov_pars.
-  move /prov_extract=>/=. congruence.
-Qed.
-
 Lemma join_bind_ne_contra n p (A : Tm n) B C :
   ne C ->
-  Join.R (TBind p A B) C -> False.
+  join (TBind p A B) C -> False.
 Proof.
-  rewrite /Join.R. move => hC /=.
-  rewrite !pair_eq.
-  move => [[[D [h0 h1]] _] _].
-  have {}hC : ne (Compile.F C) by hauto lq:on use:compile_ne.
-  have {}hC : ne (Proj PL (Proj PL (Compile.F C))) by scongruence.
+  move => hC [D [h0 h1]].
+  move /pars_pi_inv : h0 => [A0 [B0 [h2 [h3 h4]]]].
   have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
-  have : extract D = Const p by eauto using const_pars_extract.
   sfirstorder.
 Qed.
 
 Lemma join_univ_ne_contra n i C :
   ne C ->
-  Join.R (Univ i : Tm n) C -> False.
+  join (Univ i : Tm n) C -> False.
 Proof.
   move => hC [D [h0 h1]].
   move /pars_univ_inv : h0 => ?.
-  have : (exists i, extract D = (VarTm i)) \/ exists k, extract D =(Const k)  by hauto q:on use:ne_pars_extract, compile_ne.
+  have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
   sfirstorder.
 Qed.
 
-#[export]Hint Resolve join_univ_ne_contra join_bind_ne_contra join_univ_pi_contra Join.symmetric Join.transitive : join.
+#[export]Hint Resolve join_univ_ne_contra join_bind_ne_contra join_univ_pi_contra join_symmetric join_transitive : join.
 
 Lemma InterpExt_Join n i I (A B : Tm n) PA PB :
   ⟦ A ⟧ i ;; I ↘ PA ->
   ⟦ B ⟧ i ;; I ↘ PB ->
-  Join.R A B ->
+  join A B ->
   PA = PB.
 Proof.
   move => h. move : B PB. elim : A PA /h.
@@ -387,9 +357,9 @@ Proof.
       move /InterpExt_Bind_inv : hPi.
       move => [PA0][PF0][hPA0][hTot0][hRes0]?. subst.
       move => hjoin.
-      have{}hr : Join.R U (TBind p0 A0 B0) by auto using RPar's_join.
-      have hj : Join.R (TBind p A B) (TBind p0 A0 B0) by eauto using Join.transitive.
-      have {hj} : p0 = p /\ Join.R A A0 /\ Join.R B B0 by hauto l:on use:Join.BindInj.
+      have{}hr : join U (TBind p0 A0 B0) by auto using RPar's_join.
+      have hj : join (TBind p A B) (TBind p0 A0 B0) by eauto using join_transitive.
+      have {hj} : p0 = p /\ join A A0 /\ join B B0 by hauto l:on use:join_pi_inj.
       move => [? [h0 h1]]. subst.
       have ? : PA0 = PA by hauto l:on. subst.
       rewrite /ProdSpace.
@@ -398,15 +368,13 @@ Proof.
       apply bindspace_iff; eauto.
       move => a PB PB0 hPB hPB0.
       apply : ihPF; eauto.
-      rewrite /Join.R.
-      rewrite -!Compile.substing.
-      hauto l:on use:join_substing.
+      by apply join_substing.
     + move => j ?. subst.
       move => [h0 h1] h.
-      have ? : Join.R U (Univ j) by eauto using RPar's_join.
-      have : Join.R (TBind p A B) (Univ j) by eauto using Join.transitive.
+      have ? : join U (Univ j) by eauto using RPar's_join.
+      have : join (TBind p A B) (Univ j) by eauto using join_transitive.
       move => ?. exfalso.
-      hauto l: on use: join_univ_pi_contra.
+      eauto using join_univ_pi_contra.
     + move => A0 ? ? [/RPar's_join ?]. subst.
       move => _ ?. exfalso. eauto with join.
   - move => j ? B PB /InterpExtInv.
@@ -416,19 +384,19 @@ Proof.
       move /RPar's_join => *.
       exfalso. eauto with join.
     + move => m _ [/RPar's_join h0 + h1].
-      have /join_univ_inj {h0 h1} ?  : Join.R (Univ j : Tm n) (Univ m) by eauto using Join.transitive.
+      have /join_univ_inj {h0 h1} ?  : join (Univ j : Tm n) (Univ m) by eauto using join_transitive.
       subst.
       move /InterpExt_Univ_inv. firstorder.
     + move => A ? ? [/RPar's_join] *. subst. exfalso. eauto with join.
   - move => A A0 PA h.
-    have /Join.symmetric {}h : Join.R A A0 by hauto lq:on ctrs:rtc use:RPar's_join, relations.rtc_once.
-    eauto with join.
+    have /join_symmetric {}h : join A A0 by hauto lq:on ctrs:rtc use:RPar'_Par, relations.rtc_once.
+    eauto using join_transitive.
 Qed.
 
 Lemma InterpUniv_Join n i (A B : Tm n) PA PB :
   ⟦ A ⟧ i ↘ PA ->
   ⟦ B ⟧ i ↘ PB ->
-  Join.R A B ->
+  join A B ->
   PA = PB.
 Proof. hauto l:on use:InterpExt_Join rew:db:InterpUniv. Qed.
 
@@ -461,7 +429,7 @@ Proof. hauto use:InterpExt_Functional rew:db:InterpUniv. Qed.
 Lemma InterpUniv_Join' n i j (A B : Tm n) PA PB :
   ⟦ A ⟧ i ↘ PA ->
   ⟦ B ⟧ j ↘ PB ->
-  Join.R A B ->
+  join A B ->
   PA = PB.
 Proof.
   have [? ?] : i <= max i j /\ j <= max i j by lia.
@@ -768,12 +736,12 @@ Qed.
 Lemma ST_Conv n Γ (a : Tm n) A B i :
   Γ ⊨ a ∈ A ->
   Γ ⊨ B ∈ Univ i ->
-  Join.R A B ->
+  join A B ->
   Γ ⊨ a ∈ B.
 Proof.
   move => ha /SemWt_Univ h h0.
   move => ρ hρ.
-  have {}h0 : Join.R (subst_Tm ρ A) (subst_Tm ρ B) by eauto using Join.substing.
+  have {}h0 : join (subst_Tm ρ A) (subst_Tm ρ B) by eauto using join_substing.
   move /ha : (hρ){ha} => [m [PA [h1 h2]]].
   move /h : (hρ){h} => [S hS].
   have ? : PA = S by eauto using InterpUniv_Join'. subst.
@@ -928,7 +896,6 @@ Fixpoint size_tm {n} (a : Tm n) :=
   | Proj p a => 1 + size_tm a
   | Pair a b => 1 + Nat.add (size_tm a) (size_tm b)
   | Bot => 1
-  | Const _ => 1
   | Univ _ => 1
   end.