diff --git a/theories/fp_red.v b/theories/fp_red.v
index 5048bab..4fd2ed4 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -170,6 +170,9 @@ Definition ishf {n} (a : PTm n) :=
   | PBind _ _ _ => true
   | _ => false
   end.
+Definition isbind {n} (a : PTm n) := if a is PBind _ _ _ then true else false.
+
+Definition isuniv {n} (a : PTm n) := if a is PUniv _ then true else false.
 
 Definition ispair {n} (a : PTm n) :=
   match a with
@@ -195,7 +198,6 @@ Definition ispair_ren n m (a : PTm n)  (ξ : fin n -> fin m) :
   ispair (ren_PTm ξ a) = ispair a.
 Proof. case : a => //=. Qed.
 
-
 Lemma PProj_imp n p a :
   @ishf n a ->
   ~~ ispair a ->
@@ -848,6 +850,11 @@ Module RReds.
     rtc RRed.R a b -> nf a -> a = b.
   Proof. induction 1; sfirstorder use:RRed.nf_imp. Qed.
 
+  Lemma FromRedSNs n (a b : PTm n) :
+    rtc TRedSN a b ->
+    rtc RRed.R a b.
+  Proof. induction 1; hauto lq:on ctrs:rtc use:RRed.FromRedSN. Qed.
+
 End RReds.
 
 
@@ -1534,6 +1541,11 @@ Module EReds.
       apply ERed.PairEta.
   Qed.
 
+  Lemma FromEPars n (a b : PTm n) :
+    rtc EPar.R a b ->
+    rtc ERed.R a b.
+  Proof. induction 1; hauto l:on use:FromEPar, @relations.rtc_transitive. Qed.
+
 End EReds.
 
 #[export]Hint Constructors ERed.R RRed.R EPar.R : red.
@@ -1606,6 +1618,20 @@ Module RERed.
     SN b.
   Proof. hauto q:on use:ToBetaEtaPar, epar_sn_preservation, red_sn_preservation, RPar.FromRRed. Qed.
 
+  Lemma bind_preservation n (a b : PTm n) :
+    R a b -> isbind a -> isbind b.
+  Proof. hauto q:on inv:R. Qed.
+
+  Lemma univ_preservation n (a b : PTm n) :
+    R a b -> isuniv a -> isuniv b.
+  Proof. hauto q:on inv:R. Qed.
+
+  Lemma sne_preservation n (a b : PTm n) :
+    R a b -> SNe a -> SNe b.
+  Proof.
+    hauto q:on use:ToBetaEtaPar, RPar.FromRRed use:red_sn_preservation, epar_sn_preservation.
+  Qed.
+
 End RERed.
 
 Module REReds.
@@ -1660,6 +1686,18 @@ Module REReds.
     rtc RERed.R (PBind p A0 B0) (PBind p A1 B1).
   Proof. solve_s. Qed.
 
+  Lemma bind_preservation n (a b : PTm n) :
+    rtc RERed.R a b -> isbind a -> isbind b.
+  Proof. induction 1; qauto l:on ctrs:rtc use:RERed.bind_preservation. Qed.
+
+  Lemma univ_preservation n (a b : PTm n) :
+    rtc RERed.R a b -> isuniv a -> isuniv b.
+  Proof. induction 1; qauto l:on ctrs:rtc use:RERed.univ_preservation. Qed.
+
+  Lemma sne_preservation n (a b : PTm n) :
+    rtc RERed.R a b -> SNe a -> SNe b.
+  Proof. induction 1; qauto l:on ctrs:rtc use:RERed.sne_preservation. Qed.
+
 End REReds.
 
 Module LoRed.
@@ -2051,4 +2089,20 @@ Module DJoin.
     R (PProj p a0) (PProj p a1).
   Proof. hauto q:on use:REReds.ProjCong. Qed.
 
+  Lemma FromRedSNs n (a b : PTm n) :
+    rtc TRedSN a b ->
+    R a b.
+  Proof.
+    move /RReds.FromRedSNs /REReds.FromRReds.
+    sfirstorder use:@rtc_refl unfold:R.
+  Qed.
+
+  Lemma sne_bind_imp n (a b : PTm n) :
+    R a b -> SNe a -> isbind b -> False.
+  Proof.
+    move => [c [? ?]] *.
+    have : SNe c /\ isbind c by sfirstorder use:REReds.sne_preservation, REReds.bind_preservation.
+    qauto l:on inv:SNe.
+  Qed.
+
 End DJoin.
diff --git a/theories/logrel.v b/theories/logrel.v
index 76e7746..424351b 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -215,9 +215,6 @@ Proof.
   induction 2; hauto lq:on ctrs:rtc use:InterpUniv_back_clos.
 Qed.
 
-Definition isbind {n} (a : PTm n) := if a is PBind _ _ _ then true else false.
-
-Definition isuniv {n} (a : PTm n) := if a is PUniv _ then true else false.
 
 Lemma InterpUniv_case n i (A : PTm n) PA :
   ⟦ A ⟧ i ↘ PA ->
@@ -237,6 +234,11 @@ Qed.
 Lemma redsns_preservation : forall n a b, @SN n a -> rtc TRedSN a b -> SN b.
 Proof. induction 2; sfirstorder use:redsn_preservation_mutual ctrs:rtc. Qed.
 
+Lemma sne_bind_noconf n (a b : PTm n) :
+  SNe a -> isbind b -> DJoin.R a b -> False.
+Proof.
+
+
 Lemma InterpUniv_Join n i (A B : PTm n) PA PB :
   ⟦ A ⟧ i ↘ PA ->
   ⟦ B ⟧ i ↘ PB ->
@@ -249,5 +251,5 @@ Proof.
   - move => i A hA B PB hPB hAB.
     have [*] : SN B /\ SN A by hauto l:on use:adequacy.
     move /InterpUniv_case : hPB.
-    move => [H [h ?]].
-    (* have ? : SN H by sfirstorder use:redsns_preservation. *)
+    move => [H [/DJoin.FromRedSNs h ?]].
+    have ? : DJoin.R A H by eauto using DJoin.transitive.