diff --git a/theories/admissible.v b/theories/admissible.v
index 392e3cf..830ceec 100644
--- a/theories/admissible.v
+++ b/theories/admissible.v
@@ -1,45 +1,24 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax common typing structural.
+Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax common typing structural.
 From Hammer Require Import Tactics.
 Require Import ssreflect.
 Require Import Psatz.
 Require Import Coq.Logic.FunctionalExtensionality.
 
-Derive Dependent Inversion wff_inv with (forall n (Γ : fin n -> PTm n), ⊢ Γ) Sort Prop.
+Derive Inversion wff_inv with (forall Γ, ⊢ Γ) Sort Prop.
 
-Lemma Wff_Cons_Inv n Γ (A : PTm n) :
-  ⊢ funcomp (ren_PTm shift) (scons A Γ) ->
-  ⊢ Γ /\ exists i, Γ ⊢ A ∈ PUniv i.
-Proof.
-  elim /wff_inv => //= _.
-  move => n0 Γ0 A0 i0 hΓ0 hA0 [? _]. subst.
-  Equality.simplify_dep_elim.
-  have h : forall i, (funcomp (ren_PTm shift) (scons A0 Γ0)) i = (funcomp (ren_PTm shift) (scons A Γ)) i by scongruence.
-  move /(_ var_zero) : (h).
-  rewrite /funcomp. asimpl.
-  move /ren_injective. move /(_ ltac:(hauto lq:on rew:off inv:option)).
-  move => ?. subst.
-  have : Γ0 = Γ. extensionality i.
-  move /(_ (Some i)) : h.
-  rewrite /funcomp. asimpl.
-  move /ren_injective. move /(_ ltac:(hauto lq:on rew:off inv:option)).
-  done.
-  move => ?. subst. sfirstorder.
-Qed.
-
-Lemma T_Abs n Γ (a : PTm (S n)) A B :
-  funcomp (ren_PTm shift) (scons A Γ) ⊢ a ∈ B ->
+Lemma T_Abs Γ (a : PTm ) A B :
+  (cons A Γ) ⊢ a ∈ B ->
   Γ ⊢ PAbs a ∈ PBind PPi A B.
 Proof.
   move => ha.
-  have [i hB] : exists i, funcomp (ren_PTm shift) (scons A Γ) ⊢ B ∈ PUniv i by sfirstorder use:regularity.
-  have hΓ : ⊢ funcomp (ren_PTm shift) (scons A Γ) by sfirstorder use:wff_mutual.
-  move /Wff_Cons_Inv : hΓ => [hΓ][j]hA.
-  hauto lq:on rew:off use:T_Bind', typing.T_Abs.
+  have [i hB] : exists i, (cons A Γ) ⊢ B ∈ PUniv i by sfirstorder use:regularity.
+  have hΓ : ⊢  (cons A Γ) by sfirstorder use:wff_mutual.
+  hauto lq:on rew:off inv:Wff use:T_Bind', typing.T_Abs.
 Qed.
 
-Lemma E_Bind n Γ i p (A0 A1 : PTm n) B0 B1 :
+Lemma E_Bind Γ i p (A0 A1 : PTm) B0 B1 :
   Γ ⊢ A0 ≡ A1 ∈ PUniv i ->
-  funcomp (ren_PTm shift) (scons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i ->
+  (cons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i ->
   Γ ⊢ PBind p A0 B0 ≡ PBind p A1 B1 ∈ PUniv i.
 Proof.
   move => h0 h1.
@@ -49,7 +28,7 @@ Proof.
   firstorder.
 Qed.
 
-Lemma E_App n Γ (b0 b1 a0 a1 : PTm n) A B :
+Lemma E_App Γ (b0 b1 a0 a1 : PTm ) A B :
   Γ ⊢ b0 ≡ b1 ∈ PBind PPi A B ->
   Γ ⊢ a0 ≡ a1 ∈ A ->
   Γ ⊢ PApp b0 a0 ≡ PApp b1 a1 ∈ subst_PTm (scons a0 VarPTm) B.
@@ -59,7 +38,7 @@ Proof.
   move : E_App h. by repeat move/[apply].
 Qed.
 
-Lemma E_Proj2 n Γ (a b : PTm n) A B :
+Lemma E_Proj2 Γ (a b : PTm) A B :
   Γ ⊢ a ≡ b ∈ PBind PSig A B ->
   Γ ⊢ PProj PR a ≡ PProj PR b ∈ subst_PTm (scons (PProj PL a) VarPTm) B.
 Proof.
diff --git a/theories/logrel.v b/theories/logrel.v
index ae4169c..a479f81 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -1583,7 +1583,7 @@ Lemma SSu_Pi Γ (A0 A1 : PTm ) B0 B1 :
   cons A0 Γ ⊨ B0 ≲ B1 ->
   Γ ⊨ PBind PPi A0 B0 ≲ PBind PPi A1 B1.
 Proof.
-  move => hΓ hA hB.
+  move =>  hA hB.
   have ? : SN A0 /\ SN A1 /\ SN B0 /\ SN B1
     by hauto l:on use:SemLEq_SN_Sub.
   apply SemLEq_SemWt in hA, hB.
@@ -1592,7 +1592,6 @@ Proof.
   apply : SemWt_SemLEq; last by hauto l:on use:Sub.PiCong.
   hauto l:on use:ST_Bind'.
   apply ST_Bind'; eauto.
-  have hΓ' : ⊨ (cons A1 Γ) by hauto l:on use:SemWff_cons.
   move => ρ hρ.
   suff : ρ_ok (cons A0 Γ) ρ by hauto l:on.
   move : hρ. suff : Γ_sub' (A0 :: Γ) (A1 :: Γ)
@@ -1614,8 +1613,6 @@ Proof.
   apply : SemWt_SemLEq; last by hauto l:on use:Sub.SigCong.
   2 : { hauto l:on use:ST_Bind'. }
   apply ST_Bind'; eauto.
-  have hΓ' : ⊨ cons A1 Γ by hauto l:on use:SemWff_cons.
-  have hΓ'' : ⊨ cons A0 Γ by hauto l:on use:SemWff_cons.
   move => ρ hρ.
   suff : ρ_ok (cons A1 Γ) ρ by hauto l:on.
   move : hρ. suff : Γ_sub' (A1 :: Γ) (A0 :: Γ) by hauto l:on.
diff --git a/theories/preservation.v b/theories/preservation.v
index fe0a920..c8a2106 100644
--- a/theories/preservation.v
+++ b/theories/preservation.v
@@ -30,12 +30,12 @@ Proof.
   - hauto lq:on rew:off ctrs:LEq.
 Qed.
 
-Lemma Proj1_Inv n Γ (a : PTm n) U :
+Lemma Proj1_Inv Γ (a : PTm ) U :
   Γ ⊢ PProj PL a ∈ U ->
   exists A B, Γ ⊢ a ∈ PBind PSig A B /\ Γ ⊢ A ≲ U.
 Proof.
   move E : (PProj PL a) => u hu.
-  move :a E. elim : n Γ u U / hu => n //=.
+  move :a E. elim : Γ u U / hu => //=.
   - move => Γ a A B ha _ a0 [*]. subst.
     exists A, B. split => //=.
     eapply regularity in ha.
@@ -45,12 +45,12 @@ Proof.
   - hauto lq:on rew:off ctrs:LEq.
 Qed.
 
-Lemma Proj2_Inv n Γ (a : PTm n) U :
+Lemma Proj2_Inv Γ (a : PTm) U :
   Γ ⊢ PProj PR a ∈ U ->
   exists A B, Γ ⊢ a ∈ PBind PSig A B /\ Γ ⊢ subst_PTm (scons (PProj PL a) VarPTm) B ≲ U.
 Proof.
   move E : (PProj PR a) => u hu.
-  move :a E. elim : n Γ u U / hu => n //=.
+  move :a E. elim : Γ u U / hu => //=.
   - move => Γ a A B ha _ a0 [*]. subst.
     exists A, B. split => //=.
     have ha' := ha.
@@ -62,30 +62,30 @@ Proof.
   - hauto lq:on rew:off ctrs:LEq.
 Qed.
 
-Lemma Pair_Inv n Γ (a b : PTm n) U :
+Lemma Pair_Inv Γ (a b : PTm ) U :
   Γ ⊢ PPair a b ∈ U ->
   exists A B, Γ ⊢ a ∈ A /\
          Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B /\
          Γ ⊢ PBind PSig A B ≲ U.
 Proof.
   move E : (PPair a b) => u hu.
-  move : a b E. elim : n Γ u U / hu => n //=.
+  move : a b E. elim : Γ u U / hu => //=.
   - move => Γ a b A B i hS _ ha _ hb _ a0 b0 [*]. subst.
     exists A,B. repeat split => //=.
     move /E_Refl /Su_Eq : hS. apply.
   - hauto lq:on rew:off ctrs:LEq.
 Qed.
 
-Lemma Ind_Inv n Γ P (a : PTm n) b c U :
+Lemma Ind_Inv Γ P (a : PTm) b c U :
   Γ ⊢ PInd P a b c ∈ U ->
-  exists i, funcomp (ren_PTm shift) (scons PNat Γ) ⊢ P ∈ PUniv i /\
+  exists i, (cons PNat Γ) ⊢ P ∈ PUniv i /\
        Γ ⊢ a ∈ PNat /\
        Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P /\
-      funcomp (ren_PTm shift)(scons P (funcomp (ren_PTm shift) (scons PNat Γ))) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) /\
+          (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) /\
        Γ ⊢ subst_PTm (scons a VarPTm) P ≲ U.
 Proof.
   move E : (PInd P a b c)=> u hu.
-  move : P a b c E. elim : n Γ u U / hu => n //=.
+  move : P a b c E. elim : Γ u U / hu => //=.
   - move => Γ P a b c i hP _ ha _ hb _ hc _ P0 a0 b0 c0 [*]. subst.
     exists i. repeat split => //=.
     have : Γ ⊢ subst_PTm (scons a VarPTm) P ∈ subst_PTm (scons a VarPTm) (PUniv i) by hauto l:on use:substing_wt.
@@ -93,10 +93,10 @@ Proof.
   - hauto lq:on rew:off ctrs:LEq.
 Qed.
 
-Lemma E_AppAbs : forall n (a : PTm (S n)) (b : PTm n) (Γ : fin n -> PTm n) (A : PTm n),
+Lemma E_AppAbs : forall (a : PTm) (b : PTm) Γ (A : PTm),
   Γ ⊢ PApp (PAbs a) b ∈ A -> Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ A.
 Proof.
-  move => n a b Γ A ha.
+  move => a b Γ A ha.
   move /App_Inv : ha.
   move => [A0][B0][ha][hb]hS.
   move /Abs_Inv : ha => [A1][B1][ha]hS0.
@@ -112,10 +112,10 @@ Proof.
   eauto using T_Conv.
 Qed.
 
-Lemma E_ProjPair1 : forall n (a b : PTm n) (Γ : fin n -> PTm n) (A : PTm n),
+Lemma E_ProjPair1 : forall (a b : PTm) Γ (A : PTm),
     Γ ⊢ PProj PL (PPair a b) ∈ A -> Γ ⊢ PProj PL (PPair a b) ≡ a ∈ A.
 Proof.
-  move => n a b Γ A.
+  move => a b Γ A.
   move /Proj1_Inv. move => [A0][B0][hab]hA0.
   move /Pair_Inv : hab => [A1][B1][ha][hb]hS.
   have [i ?]  : exists i, Γ ⊢ PBind PSig A1 B1 ∈ PUniv i by sfirstorder use:regularity_sub0.
@@ -125,25 +125,25 @@ Proof.
   apply : E_ProjPair1; eauto.
 Qed.
 
-Lemma Suc_Inv n Γ (a : PTm n) A :
+Lemma Suc_Inv Γ (a : PTm) A :
   Γ ⊢ PSuc a ∈ A -> Γ ⊢ a ∈ PNat /\ Γ ⊢ PNat ≲ A.
 Proof.
   move E : (PSuc a) => u hu.
   move : a E.
-  elim : n Γ u A /hu => //=.
-  - move => n Γ a ha iha a0 [*]. subst.
+  elim : Γ u A /hu => //=.
+  - move => Γ a ha iha a0 [*]. subst.
     split => //=. eapply wff_mutual in ha.
     apply : Su_Eq.
     apply E_Refl. by apply T_Nat'.
   - hauto lq:on rew:off ctrs:LEq.
 Qed.
 
-Lemma RRed_Eq n Γ (a b : PTm n) A :
+Lemma RRed_Eq Γ (a b : PTm) A :
   Γ ⊢ a ∈ A ->
   RRed.R a b ->
   Γ ⊢ a ≡ b ∈ A.
 Proof.
-  move => + h. move : Γ A. elim : n a b /h => n.
+  move => + h. move : Γ A. elim : a b /h.
   - apply E_AppAbs.
   - move => p a b Γ A.
     case : p => //=.
@@ -214,7 +214,7 @@ Proof.
   - hauto lq:on use:Suc_Inv, E_Conv, E_SucCong.
 Qed.
 
-Theorem subject_reduction n Γ (a b A : PTm n) :
+Theorem subject_reduction Γ (a b A : PTm) :
   Γ ⊢ a ∈ A ->
   RRed.R a b ->
   Γ ⊢ b ∈ A.
diff --git a/theories/soundness.v b/theories/soundness.v
index b11dded..8bfeedf 100644
--- a/theories/soundness.v
+++ b/theories/soundness.v
@@ -1,4 +1,4 @@
-Require Import Autosubst2.fintype Autosubst2.syntax.
+Require Import Autosubst2.unscoped Autosubst2.syntax.
 Require Import fp_red logrel typing.
 From Hammer Require Import Tactics.