diff --git a/theories/logrel.v b/theories/logrel.v
index b070751..4c01f0d 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -13,7 +13,7 @@ Definition ProdSpace {n} (PA : PTm n -> Prop)
 
 Definition SumSpace {n} (PA : PTm n -> Prop)
   (PF : PTm n -> (PTm n -> Prop) -> Prop) t : Prop :=
-  SNe t \/ exists a b, rtc TRedSN t (PPair a b) /\ PA a /\ (forall PB, PF a PB -> PB b).
+  (exists v, rtc TRedSN t v /\ SNe v) \/ exists a b, rtc TRedSN t (PPair a b) /\ PA a /\ (forall PB, PF a PB -> PB b).
 
 Definition BindSpace {n} p := if p is PPi then @ProdSpace n else SumSpace.
 
@@ -22,7 +22,7 @@ Reserved Notation "⟦ A ⟧ i ;; I ↘ S" (at level 70).
 Inductive InterpExt {n} i (I : nat -> PTm n -> Prop) : PTm n -> (PTm n -> Prop) -> Prop :=
 | InterpExt_Ne A :
   SNe A ->
-  ⟦ A ⟧ i ;; I ↘ SNe
+  ⟦ A ⟧ i ;; I ↘ (fun a => exists v, rtc TRedSN a v /\ SNe v)
 | InterpExt_Bind p A B PA PF :
   ⟦ A ⟧ i ;; I ↘ PA ->
   (forall a, PA a -> exists PB, PF a PB) ->
@@ -95,6 +95,25 @@ Qed.
 
 #[export]Hint Rewrite @InterpUnivN_nolt : InterpUniv.
 
+Lemma InterpUniv_ind
+     : forall (n i : nat) (P : PTm n -> (PTm n -> Prop) -> Prop),
+       (forall A : PTm n, SNe A -> P A (fun a : PTm n => exists v : PTm n, rtc TRedSN a v /\ SNe v)) ->
+       (forall (p : BTag) (A : PTm n) (B : PTm (S n)) (PA : PTm n -> Prop)
+          (PF : PTm n -> (PTm n -> Prop) -> Prop),
+        ⟦ A ⟧ i ↘ PA ->
+        P A PA ->
+        (forall a : PTm n, PA a -> exists PB : PTm n -> Prop, PF a PB) ->
+        (forall (a : PTm n) (PB : PTm n -> Prop), PF a PB -> ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB) ->
+        (forall (a : PTm n) (PB : PTm n -> Prop), PF a PB -> P (subst_PTm (scons a VarPTm) B) PB) ->
+        P (PBind p A B) (BindSpace p PA PF)) ->
+       (forall j : nat, j < i -> P (PUniv j) (fun A => exists PA, ⟦ A ⟧ j ↘ PA)) ->
+       (forall (A A0 : PTm n) (PA : PTm n -> Prop), TRedSN A A0 -> ⟦ A0 ⟧ i ↘ PA -> P A0 PA -> P A PA) ->
+       forall (p : PTm n) (P0 : PTm n -> Prop), ⟦ p ⟧ i ↘ P0 -> P p P0.
+Proof.
+  elim /Wf_nat.lt_wf_ind => n ih i . simp InterpUniv.
+  apply InterpExt_ind.
+Qed.
+
 Derive Dependent Inversion iinv with (forall n i I (A : PTm n) PA, InterpExt i I A PA) Sort Prop.
 
 Lemma InterpExt_cumulative n i j I (A : PTm n) PA :
@@ -133,7 +152,7 @@ Lemma adequacy_ext i n I A PA
   CR PA /\ SN A.
 Proof.
   elim : A PA / h.
-  - hauto lq:on ctrs:SN unfold:CR.
+  - hauto l:on use:N_Exps ctrs:SN,SNe.
   - move => p A B PA PF hPA [ihA0 ihA1] hTot hRes ihPF.
     have hb : PA PBot by hauto q:on ctrs:SNe.
     have hb' : SN PBot by hauto q:on ctrs:SN, SNe.
@@ -142,16 +161,13 @@ Proof.
     + case : p =>//=.
       * rewrite /ProdSpace.
         qauto use:SN_AppInv unfold:CR.
-      * rewrite /SumSpace => a []; first by apply N_SNe.
-        move => [q0][q1]*.
-        have : SN q0 /\ SN q1 by hauto q:on.
-        hauto lq:on use:N_Pair,N_Exps.
+      * hauto q:on unfold:SumSpace use:N_SNe, N_Pair,N_Exps.
     + move => a ha.
       case : p=>/=.
       * rewrite /ProdSpace => a0 *.
         suff : SNe (PApp a a0) by sfirstorder.
         hauto q:on use:N_App.
-      * rewrite /SumSpace. tauto.
+      * sfirstorder.
     + apply N_Bind=>//=.
       have : SN (PApp (PAbs B) PBot).
       apply : N_Exp; eauto using N_β.
@@ -160,3 +176,72 @@ Proof.
   - sfirstorder.
   - hauto l:on ctrs:SN unfold:CR.
 Qed.
+
+Lemma InterpExt_Steps i n I A A0 PA :
+  rtc TRedSN A A0 ->
+  ⟦ A0 : PTm n ⟧ i ;; I ↘ PA ->
+  ⟦ A ⟧ i ;; I ↘ PA.
+Proof. induction 1; eauto using InterpExt_Step. Qed.
+
+Lemma InterpUniv_Steps i n A A0 PA :
+  rtc TRedSN A A0 ->
+  ⟦ A0 : PTm n ⟧ i ↘ PA ->
+  ⟦ A ⟧ i ↘ PA.
+Proof. hauto l:on use:InterpExt_Steps rew:db:InterpUniv. Qed.
+
+Lemma adequacy i n A PA
+  (h :  ⟦ A : PTm n ⟧ i ↘ PA) :
+  CR PA /\ SN A.
+Proof.
+  move : i A PA h.
+  elim /Wf_nat.lt_wf_ind => i ih A PA.
+  simp InterpUniv.
+  apply adequacy_ext.
+  hauto lq:on ctrs:rtc use:InterpUniv_Steps.
+  hauto l:on use:InterpExt_Ne rew:db:InterpUniv.
+Qed.
+
+Lemma InterpExt_back_clos n i I (A : PTm n) PA
+  (hI1 : forall j, j < i -> CR (I j))
+  (hI : forall j,  j < i -> forall a b, TRedSN a b ->  I j b -> I j a) :
+    ⟦ A ⟧ i ;; I ↘ PA ->
+    forall a b, TRedSN a b ->
+           PA b -> PA a.
+Proof.
+  move => h.
+  elim : A PA /h; eauto.
+  hauto q:on ctrs:rtc.
+
+  move => p A B PA PF hPA ihPA hTot hRes ihPF a b hr.
+  case : p => //=.
+  - rewrite /ProdSpace.
+    move => hba a0 PB ha hPB.
+    suff : TRedSN  (PApp a a0) (PApp b a0) by hauto lq:on.
+    apply N_AppL => //=.
+    hauto q:on use:adequacy_ext.
+  - hauto lq:on ctrs:rtc unfold:SumSpace.
+Qed.
+
+Lemma InterpUniv_back_clos n i (A : PTm n) PA :
+    ⟦ A ⟧ i ↘ PA ->
+    forall a b, TRedSN a b ->
+           PA b -> PA a.
+Proof.
+  simp InterpUniv. apply InterpExt_back_clos;
+  hauto l:on use:adequacy unfold:CR ctrs:InterpExt rew:db:InterpUniv.
+Qed.
+
+Lemma InterpUniv_back_closs n i (A : PTm n) PA :
+    ⟦ A ⟧ i ↘ PA ->
+    forall a b, rtc TRedSN a b ->
+           PA b -> PA a.
+Proof.
+  induction 2; hauto lq:on ctrs:rtc use:InterpUniv_back_clos.
+Qed.
+
+Lemma InterpExt_Join n i I (A B : PTm n) PA PB :
+  ⟦ A ⟧ i ;; I ↘ PA ->
+  ⟦ B ⟧ i ;; I ↘ PB ->
+  DJoin.R A B ->
+  PA = PB.
+Proof.