diff --git a/syntax.sig b/syntax.sig
index d052897..e2bafac 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -9,7 +9,7 @@ Void : Ty
 PL : PTag
 PR : PTag
 
-PAbs : Ty -> (bind PTm in PTm) -> PTm
+PAbs : (bind PTm in PTm) -> PTm
 PApp : PTm -> PTm -> PTm
 PPair : PTm -> PTm -> PTm
 PProj : PTag -> PTm -> PTm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index b37b721..b5cbc66 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -19,43 +19,17 @@ Proof.
 exact (eq_refl).
 Qed.
 
-Inductive Ty : Type :=
-  | Fun : Ty -> Ty -> Ty
-  | Prod : Ty -> Ty -> Ty
-  | Void : Ty.
-
-Lemma congr_Fun {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
-  (H1 : s1 = t1) : Fun s0 s1 = Fun t0 t1.
-Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => Fun x s1) H0))
-         (ap (fun x => Fun t0 x) H1)).
-Qed.
-
-Lemma congr_Prod {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
-  (H1 : s1 = t1) : Prod s0 s1 = Prod t0 t1.
-Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => Prod x s1) H0))
-         (ap (fun x => Prod t0 x) H1)).
-Qed.
-
-Lemma congr_Void : Void = Void.
-Proof.
-exact (eq_refl).
-Qed.
-
 Inductive PTm (n_PTm : nat) : Type :=
   | VarPTm : fin n_PTm -> PTm n_PTm
-  | PAbs : Ty -> PTm (S n_PTm) -> PTm n_PTm
+  | PAbs : PTm (S n_PTm) -> PTm n_PTm
   | PApp : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
   | PPair : PTm n_PTm -> PTm n_PTm -> PTm n_PTm
   | PProj : PTag -> PTm n_PTm -> PTm n_PTm.
 
-Lemma congr_PAbs {m_PTm : nat} {s0 : Ty} {s1 : PTm (S m_PTm)} {t0 : Ty}
-  {t1 : PTm (S m_PTm)} (H0 : s0 = t0) (H1 : s1 = t1) :
-  PAbs m_PTm s0 s1 = PAbs m_PTm t0 t1.
+Lemma congr_PAbs {m_PTm : nat} {s0 : PTm (S m_PTm)} {t0 : PTm (S m_PTm)}
+  (H0 : s0 = t0) : PAbs m_PTm s0 = PAbs m_PTm t0.
 Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PAbs m_PTm x s1) H0))
-         (ap (fun x => PAbs m_PTm t0 x) H1)).
+exact (eq_trans eq_refl (ap (fun x => PAbs m_PTm x) H0)).
 Qed.
 
 Lemma congr_PApp {m_PTm : nat} {s0 : PTm m_PTm} {s1 : PTm m_PTm}
@@ -98,7 +72,7 @@ Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
 (xi_PTm : fin m_PTm -> fin n_PTm) (s : PTm m_PTm) {struct s} : PTm n_PTm :=
   match s with
   | VarPTm _ s0 => VarPTm n_PTm (xi_PTm s0)
-  | PAbs _ s0 s1 => PAbs n_PTm s0 (ren_PTm (upRen_PTm_PTm xi_PTm) s1)
+  | PAbs _ s0 => PAbs n_PTm (ren_PTm (upRen_PTm_PTm xi_PTm) s0)
   | PApp _ s0 s1 => PApp n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
   | PPair _ s0 s1 => PPair n_PTm (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
   | PProj _ s0 s1 => PProj n_PTm s0 (ren_PTm xi_PTm s1)
@@ -122,7 +96,7 @@ Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
 :=
   match s with
   | VarPTm _ s0 => sigma_PTm s0
-  | PAbs _ s0 s1 => PAbs n_PTm s0 (subst_PTm (up_PTm_PTm sigma_PTm) s1)
+  | PAbs _ s0 => PAbs n_PTm (subst_PTm (up_PTm_PTm sigma_PTm) s0)
   | PApp _ s0 s1 =>
       PApp n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
   | PPair _ s0 s1 =>
@@ -155,9 +129,9 @@ Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
    : subst_PTm sigma_PTm s = s :=
   match s with
   | VarPTm _ s0 => Eq_PTm s0
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
-        (idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s1)
+  | PAbs _ s0 =>
+      congr_PAbs
+        (idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (idSubst_PTm sigma_PTm Eq_PTm s0)
         (idSubst_PTm sigma_PTm Eq_PTm s1)
@@ -194,10 +168,10 @@ Fixpoint extRen_PTm {m_PTm : nat} {n_PTm : nat}
 ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
   match s with
   | VarPTm _ s0 => ap (VarPTm n_PTm) (Eq_PTm s0)
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
-           (upExtRen_PTm_PTm _ _ Eq_PTm) s1)
+           (upExtRen_PTm_PTm _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
         (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
@@ -235,10 +209,10 @@ Fixpoint ext_PTm {m_PTm : nat} {n_PTm : nat}
 subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
   match s with
   | VarPTm _ s0 => Eq_PTm s0
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
-           (upExt_PTm_PTm _ _ Eq_PTm) s1)
+           (upExt_PTm_PTm _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
         (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
@@ -275,10 +249,10 @@ Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
 {struct s} : ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm rho_PTm s :=
   match s with
   | VarPTm _ s0 => ap (VarPTm l_PTm) (Eq_PTm s0)
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
-           (upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s1)
+           (upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
         (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
@@ -325,10 +299,10 @@ Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
 {struct s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
   match s with
   | VarPTm _ s0 => Eq_PTm s0
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
-           (up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s1)
+           (up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
         (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
@@ -395,10 +369,10 @@ Fixpoint compSubstRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
 ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
   match s with
   | VarPTm _ s0 => Eq_PTm s0
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
-           (up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s1)
+           (up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
         (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
@@ -467,10 +441,10 @@ Fixpoint compSubstSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
 subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
   match s with
   | VarPTm _ s0 => Eq_PTm s0
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
-           (up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s1)
+           (up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
         (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
@@ -580,10 +554,10 @@ Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
 (s : PTm m_PTm) {struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
   match s with
   | VarPTm _ s0 => Eq_PTm s0
-  | PAbs _ s0 s1 =>
-      congr_PAbs (eq_refl s0)
+  | PAbs _ s0 =>
+      congr_PAbs
         (rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
-           (rinstInst_up_PTm_PTm _ _ Eq_PTm) s1)
+           (rinstInst_up_PTm_PTm _ _ Eq_PTm) s0)
   | PApp _ s0 s1 =>
       congr_PApp (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
         (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
@@ -663,6 +637,30 @@ Proof.
 exact (fun x => eq_refl).
 Qed.
 
+Inductive Ty : Type :=
+  | Fun : Ty -> Ty -> Ty
+  | Prod : Ty -> Ty -> Ty
+  | Void : Ty.
+
+Lemma congr_Fun {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
+  (H1 : s1 = t1) : Fun s0 s1 = Fun t0 t1.
+Proof.
+exact (eq_trans (eq_trans eq_refl (ap (fun x => Fun x s1) H0))
+         (ap (fun x => Fun t0 x) H1)).
+Qed.
+
+Lemma congr_Prod {s0 : Ty} {s1 : Ty} {t0 : Ty} {t1 : Ty} (H0 : s0 = t0)
+  (H1 : s1 = t1) : Prod s0 s1 = Prod t0 t1.
+Proof.
+exact (eq_trans (eq_trans eq_refl (ap (fun x => Prod x s1) H0))
+         (ap (fun x => Prod t0 x) H1)).
+Qed.
+
+Lemma congr_Void : Void = Void.
+Proof.
+exact (eq_refl).
+Qed.
+
 Class Up_PTm X Y :=
     up_PTm : X -> Y.
 
diff --git a/theories/fp_red.v b/theories/fp_red.v
index babb3c8..a53a692 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -1,5 +1,6 @@
 From Ltac2 Require Ltac2.
 Import Ltac2.Notations.
+
 Import Ltac2.Control.
 Require Import ssreflect ssrbool.
 Require Import FunInd.
@@ -22,16 +23,16 @@ Ltac spec_refl := ltac2:(spec_refl ()).
 Module ERed.
   Inductive R {n} : PTm n -> PTm n ->  Prop :=
   (****************** Eta ***********************)
-  | AppEta A a0 a1 :
+  | AppEta a0 a1 :
     R a0 a1 ->
-    R (PAbs A (PApp (ren_PTm shift a0) (VarPTm var_zero))) a1
+    R (PAbs (PApp (ren_PTm shift a0) (VarPTm var_zero))) a1
   | PairEta a0 a1 :
     R a0 a1 ->
     R (PPair (PProj PL a0) (PProj PR a0)) a1
   (*************** Congruence ********************)
-  | AbsCong A a0 a1 :
+  | AbsCong a0 a1 :
     R a0 a1 ->
-    R (PAbs A a0) (PAbs A a1)
+    R (PAbs a0) (PAbs a1)
   | AppCong a0 a1 b0 b1 :
     R a0 a1 ->
     R b0 b1 ->
@@ -53,8 +54,8 @@ Module ERed.
 
   Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
 
-  Lemma AppEta' n A a0 a1 (u : PTm n) :
-    u = (PAbs A (PApp (ren_PTm shift a0) (VarPTm var_zero))) ->
+  Lemma AppEta' n a0 a1 (u : PTm n) :
+    u = (PAbs (PApp (ren_PTm shift a0) (VarPTm var_zero))) ->
     R a0 a1 ->
     R u a1.
   Proof. move => ->. apply AppEta. Qed.
@@ -65,8 +66,8 @@ Module ERed.
     move => h. move : m ξ.
     elim : n a b /h.
 
-    move => n A a0 a1 ha iha m ξ /=.
-    eapply AppEta' with (A := A); eauto. by asimpl.
+    move => n a0 a1 ha iha m ξ /=.
+    eapply AppEta'; eauto. by asimpl.
     all : qauto ctrs:R.
   Qed.
 
@@ -91,18 +92,11 @@ Module ERed.
     R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
   Proof.
     move => + h. move : m ρ0 ρ1. elim : n a b / h => n.
-    move => A a0 a1 ha iha m ρ0 ρ1 hρ /=.
-    eapply AppEta' with (A := A); eauto. by asimpl.
+    move => a0 a1 ha iha m ρ0 ρ1 hρ /=.
+    eapply AppEta'; eauto. by asimpl.
     all : hauto lq:on ctrs:R use:morphing_up.
   Qed.
 
-  Lemma substing n m (a : PTm n) b (ρ : fin n -> PTm m) :
-    R a b ->
-    R (subst_PTm ρ a) (subst_PTm ρ b).
-  Proof.
-    hauto l:on use:morphing, refl.
-  Qed.
-
 End ERed.
 
 Inductive SNe {n} : PTm n -> Prop :=
@@ -120,9 +114,9 @@ with SN  {n} : PTm n -> Prop :=
   SN a ->
   SN b ->
   SN (PPair a b)
-| N_Abs A a :
+| N_Abs a :
   SN a ->
-  SN (PAbs A a)
+  SN (PAbs a)
 | N_SNe a :
   SNe a ->
   SN a
@@ -131,10 +125,11 @@ with SN  {n} : PTm n -> Prop :=
   SN b ->
   SN a
 with TRedSN {n} : PTm n -> PTm n -> Prop :=
-| N_β A a b :
+| N_β a b :
   SN b ->
-  TRedSN (PApp (PAbs A a) b) (subst_PTm (scons b VarPTm) a)
+  TRedSN (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
 | N_AppL a0 a1 b :
+  SN b ->
   TRedSN a0 a1 ->
   TRedSN (PApp a0 b) (PApp a1 b)
 | N_ProjPairL a b  :
@@ -147,12 +142,64 @@ with TRedSN {n} : PTm n -> PTm n -> Prop :=
   TRedSN a b ->
   TRedSN (PProj p a) (PProj p b).
 
+Derive Dependent Inversion tred_inv with (forall n (a b : PTm n), TRedSN a b) Sort Prop.
+
+Inductive SNe' {n} : PTm n -> Prop :=
+| N_Var' i :
+  SNe' (VarPTm i)
+| N_App' a b :
+  SNe a ->
+  SNe' (PApp a b)
+| N_Proj' p a :
+  SNe a ->
+  SNe' (PProj p a).
+
+Lemma PProjAbs_imp n p (a : PTm (S n)) :
+  ~ SN (PProj p (PAbs a)).
+Proof.
+  move E : (PProj p (PAbs a)) => u hu.
+  move : p a E.
+  elim : n u / hu=>//=.
+  hauto lq:on inv:SNe.
+  hauto lq:on inv:TRedSN.
+Qed.
+
+Lemma PProjPair_imp n (a b0 b1 : PTm n ) :
+  ~ SN (PApp (PPair b0 b1) a).
+Proof.
+  move E : (PApp (PPair b0 b1) a) => u hu.
+  move : a b0 b1 E.
+  elim : n u / hu=>//=.
+  hauto lq:on inv:SNe.
+  hauto lq:on inv:TRedSN.
+Qed.
+
 Scheme sne_ind := Induction for SNe Sort Prop
   with sn_ind := Induction for SN Sort Prop
   with sred_ind := Induction for TRedSN Sort Prop.
 
 Combined Scheme sn_mutual from sne_ind, sn_ind, sred_ind.
 
+Fixpoint ne {n} (a : PTm n) :=
+  match a with
+  | VarPTm i => true
+  | PApp a b => ne a && nf b
+  | PAbs a => false
+  | PPair _ _ => false
+  | PProj _ a => ne a
+  end
+with nf {n} (a : PTm n) :=
+  match a with
+  | VarPTm i => true
+  | PApp a b => ne a && nf b
+  | PAbs a => nf a
+  | PPair a b => nf a && nf b
+  | PProj _ a => ne a
+end.
+
+Lemma ne_nf n a : @ne n a -> nf a.
+Proof. elim : a => //=. Qed.
+
 Inductive TRedSN' {n} (a : PTm n) : PTm n -> Prop :=
 | T_Refl :
   TRedSN' a a
@@ -168,6 +215,124 @@ Proof.
   elim : n u / h => n //=; sauto.
 Qed.
 
+Lemma N_β' n a (b : PTm n) u :
+  u = (subst_PTm (scons b VarPTm) a) ->
+  SN b ->
+  TRedSN (PApp (PAbs a) b) u.
+Proof. move => ->. apply N_β. Qed.
+
+Lemma sn_renaming n :
+  (forall (a : PTm n) (s : SNe a), forall m (ξ : fin n -> fin m), SNe (ren_PTm ξ a)) /\
+  (forall (a : PTm n) (s : SN a), forall m (ξ : fin n -> fin m), SN (ren_PTm ξ a)) /\
+  (forall (a b : PTm n) (_ : TRedSN a b), forall m (ξ : fin n -> fin m), TRedSN (ren_PTm ξ a) (ren_PTm ξ b)).
+Proof.
+  move : n. apply sn_mutual => n; try qauto ctrs:SN, SNe, TRedSN depth:1.
+  move => a b ha iha m ξ /=.
+  apply N_β'. by asimpl. eauto.
+Qed.
+
+#[export]Hint Constructors SN SNe TRedSN : sn.
+
+Ltac2 rec solve_anti_ren () :=
+  let x := Fresh.in_goal (Option.get (Ident.of_string "x")) in
+  intro $x;
+  lazy_match! Constr.type (Control.hyp x) with
+  | fin ?x -> _ ?y => (ltac1:(case;qauto depth:2 db:sn))
+  | _ => solve_anti_ren ()
+  end.
+
+Ltac solve_anti_ren := ltac2:(Control.enter solve_anti_ren).
+
+Lemma sn_antirenaming n :
+  (forall (a : PTm n) (s : SNe a), forall m (ξ : fin m -> fin n) b, a = ren_PTm ξ b -> SNe b) /\
+  (forall (a : PTm n) (s : SN a), forall m (ξ : fin m -> fin n) b, a = ren_PTm ξ b -> SN b) /\
+  (forall (a b : PTm n) (_ : TRedSN a b), forall m (ξ : fin m -> fin n) a0,
+      a = ren_PTm ξ a0 -> exists b0, TRedSN a0 b0 /\ b = ren_PTm ξ b0).
+Proof.
+  move : n. apply sn_mutual => n; try solve_anti_ren.
+
+  move => a b ha iha m ξ []//= u u0 [+ ?]. subst.
+  case : u => //= => u [*]. subst.
+  spec_refl. eexists. split. apply N_β=>//. by asimpl.
+
+  move => a b hb ihb m ξ[]//= p p0 [? +]. subst.
+  case : p0 => //= p p0 [*]. subst.
+  spec_refl. by eauto with sn.
+
+  move => a b ha iha m ξ[]//= u u0 [? ]. subst.
+  case : u0 => //=. move => p p0 [*]. subst.
+  spec_refl. by eauto with sn.
+Qed.
+
+Lemma sn_unmorphing n :
+  (forall (a : PTm n) (s : SNe a), forall m (ρ : fin m -> PTm n) b, a = subst_PTm ρ b -> SNe b) /\
+  (forall (a : PTm n) (s : SN a), forall m (ρ : fin m -> PTm n) b, a = subst_PTm ρ b -> SN b) /\
+  (forall (a b : PTm n) (_ : TRedSN a b), forall m (ρ : fin m -> PTm n) a0,
+      a = subst_PTm ρ a0 -> (exists b0, b = subst_PTm ρ b0 /\ TRedSN a0 b0) \/ SNe a0).
+Proof.
+  move : n. apply sn_mutual => n; try solve_anti_ren.
+  - move => a b ha iha  m ξ b0.
+    case : b0 => //=.
+    + hauto lq:on rew:off db:sn.
+    + move => p p0 [+ ?]. subst.
+      case : p => //=.
+      hauto lq:on db:sn.
+      move => p [?]. subst.
+      asimpl. left.
+      spec_refl.
+      eexists. split; last by eauto using N_β.
+      by asimpl.
+  - move => a0 a1 b hb ihb ha iha m ρ []//=.
+    + hauto lq:on rew:off db:sn.
+    + move => t0  t1 [*]. subst.
+      spec_refl.
+      case : iha.
+      * move => [u [? hu]]. subst.
+        left. eexists. split; eauto using N_AppL.
+        reflexivity.
+      * move => h.
+        right.
+        apply N_App => //.
+  - move => a b hb ihb m ρ []//=.
+    + hauto l:on ctrs:TRedSN.
+    + move => p p0 [?]. subst.
+      case : p0 => //=.
+      * hauto lq:on rew:off db:sn.
+      * move => p p0 [*]. subst.
+        hauto lq:on db:sn.
+  - move => a b ha iha m ρ []//=; first by hauto l:on db:sn.
+    hauto q:on inv:PTm db:sn.
+  - move => p a b ha iha m ρ []//=; first by hauto l:on db:sn.
+    move => t0 t1 [*]. subst.
+    spec_refl.
+    case : iha.
+    + move => [b0 [? h]]. subst.
+      left. eexists. split; last by eauto with sn.
+      reflexivity.
+    + hauto lq:on db:sn.
+Qed.
+
+Lemma SN_AppInv : forall n (a b : PTm n), SN (PApp a b) -> SN a /\ SN b.
+Proof.
+  move => n a b. move E : (PApp a b) => u hu. move : a b E.
+  elim : n u /hu=>//=.
+  hauto lq:on rew:off inv:SNe db:sn.
+  move => n a b ha hb ihb a0 b0 ?. subst.
+  inversion ha; subst.
+  move {ihb}.
+  hecrush use:sn_unmorphing.
+  hauto lq:on db:sn.
+Qed.
+
+Lemma SN_ProjInv : forall n p (a : PTm n), SN (PProj p a) -> SN a.
+Proof.
+  move => n p a. move E : (PProj p a) => u hu.
+  move : p a E.
+  elim : n u / hu => //=.
+  hauto lq:on rew:off inv:SNe db:sn.
+  hauto lq:on rew:off inv:TRedSN db:sn.
+Qed.
+
 Lemma ered_sn_preservation n :
   (forall (a : PTm n) (s : SNe a), forall b, ERed.R a b -> SNe b) /\
   (forall (a : PTm n) (s : SN a), forall b, ERed.R a b -> SN b) /\
@@ -182,17 +347,18 @@ Proof.
     + have /iha : (ERed.R (PProj PL a0) (PProj PL b0)) by sauto lq:on.
       sfirstorder use:SN_Proj.
     + sauto lq:on.
-  - move => A a ha iha b.
+  - move => a ha iha b.
     inversion 1; subst.
     + have : ERed.R (PApp (ren_PTm shift a0) (VarPTm var_zero))  (PApp (ren_PTm shift b) (VarPTm var_zero)).
       apply ERed.AppCong; eauto using ERed.refl.
       sfirstorder use:ERed.renaming.
       move /iha.
-      admit.
+      move /SN_AppInv => [+ _].
+      hauto l:on use:sn_antirenaming.
     + sauto lq:on.
   - sauto lq:on.
   - sauto lq:on.
-  - move => A a b ha iha c h0.
+  - move => a b ha iha c h0.
     inversion h0; subst.
     inversion H1; subst.
     + exists (PApp a1 b1). split. sfirstorder.
@@ -208,7 +374,7 @@ Proof.
     elim /ERed.inv => //= _.
     move => p a0 a1 ha [*]. subst.
     elim  /ERed.inv :  ha => //= _.
-    + move => a0 a2 ha [*]. subst.
+    + move => a0 a2 ha' [*]. subst.
       exists (PProj PL a1).
       split. sauto.
       sauto lq:on.
@@ -217,27 +383,27 @@ Proof.
     elim /ERed.inv => //=_.
     move => p a0 a1 + [*]. subst.
     elim /ERed.inv => //=_.
-    + move => a0 a2 h [*]. subst.
+    + move => a0  a2 h1 [*]. subst.
       exists (PProj PR a1).
       split. sauto.
       sauto lq:on.
     + sauto lq:on.
   - sauto.
-Admitted.
+Qed.
 
 Module RRed.
   Inductive R {n} : PTm n -> PTm n ->  Prop :=
   (****************** Eta ***********************)
-  | AppAbs A a b :
-    R (PApp (PAbs A a) b)  (subst_PTm (scons b VarPTm) a)
+  | AppAbs a b :
+    R (PApp (PAbs a) b)  (subst_PTm (scons b VarPTm) a)
 
   | ProjPair p a b :
     R (PProj p (PPair a b)) (if p is PL then a else b)
 
   (*************** Congruence ********************)
-  | AbsCong A a0 a1 :
+  | AbsCong a0 a1 :
     R a0 a1 ->
-    R (PAbs A a0) (PAbs A a1)
+    R (PAbs a0) (PAbs a1)
   | AppCong0 a0 a1 b :
     R a0 a1 ->
     R (PApp a0 b) (PApp a1 b)
@@ -255,4 +421,693 @@ Module RRed.
     R (PProj p a0) (PProj p a1).
 
   Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
+
+  Lemma AppAbs' n a (b : PTm n) u :
+    u = (subst_PTm (scons b VarPTm) a) ->
+    R (PApp (PAbs a) b)  u.
+  Proof.
+    move => ->. by apply AppAbs. Qed.
+
+  Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+    R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+  Proof.
+    move => h. move : m ξ.
+    elim : n a b /h.
+
+    move => n a b m ξ /=.
+    apply AppAbs'. by asimpl.
+    all : qauto ctrs:R.
+  Qed.
+
+
+  Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) :
+    R (ren_PTm ξ a) b -> exists b0, R a b0 /\ ren_PTm ξ b0 = b.
+  Proof.
+    move E : (ren_PTm ξ a) => u h.
+    move : n ξ a E. elim : m u b/h.
+    - move => n a b m ξ []//=. move => []//= t t0 [*]. subst.
+      eexists. split. apply AppAbs. by asimpl.
+    - move => n p a b m ξ []//=.
+      move => p0 []//=. hauto q:on ctrs:R.
+    - move => n a0 a1 ha iha m ξ []//=.
+      move => p [*]. subst.
+      spec_refl.
+      move : iha => [t [h0 h1]]. subst.
+      eexists. split. eauto using AbsCong.
+      by asimpl.
+    - move => n a0 a1 b ha iha m ξ []//=.
+      hauto lq:on ctrs:R.
+    - move => n a b0 b1 h ih m ξ []//=.
+      hauto lq:on ctrs:R.
+    - move => n a0 a1 b ha iha m ξ []//=.
+      hauto lq:on ctrs:R.
+    - move => n a b0 b1 h ih m ξ []//=.
+      hauto lq:on ctrs:R.
+    - move => n p a0 a1 ha iha m ξ []//=.
+      hauto lq:on ctrs:R.
+  Qed.
+
 End RRed.
+
+Module RPar.
+  Inductive R {n} : PTm n -> PTm n ->  Prop :=
+  (****************** Beta ***********************)
+  | AppAbs a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (PApp (PAbs a0) b0)  (subst_PTm (scons b1 VarPTm) a1)
+
+  | ProjPair p a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
+
+  (*************** Congruence ********************)
+  | AbsCong a0 a1 :
+    R a0 a1 ->
+    R (PAbs a0) (PAbs a1)
+  | AppCong a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (PApp a0 b0) (PApp a1 b1)
+  | PairCong a0 a1 b0 b1 :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (PPair a0 b0) (PPair a1 b1)
+  | ProjCong p a0 a1 :
+    R a0 a1 ->
+    R (PProj p a0) (PProj p a1)
+  | VarTm i :
+    R (VarPTm i) (VarPTm i).
+
+  Lemma refl n (a : PTm n) : R a a.
+  Proof.
+    elim : n / a; hauto lq:on ctrs:R.
+  Qed.
+
+  Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
+
+  Lemma AppAbs' n a0 a1 (b0 b1 : PTm n) u :
+    u = (subst_PTm (scons b1 VarPTm) a1) ->
+    R a0 a1 ->
+    R b0 b1 ->
+    R (PApp (PAbs a0) b0)  u.
+  Proof. move => ->. apply AppAbs. Qed.
+
+  Lemma ProjPair' n p u (a0 a1 b0 b1 : PTm n) :
+    u = (if p is PL then a1 else b1) ->
+    R a0 a1 ->
+    R b0 b1 ->
+    R (PProj p (PPair a0 b0)) u.
+  Proof. move => ->. apply ProjPair. Qed.
+
+  Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+    R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+  Proof.
+    move => h. move : m ξ.
+    elim : n a b /h.
+
+    move => n a0 a1 b0 b1 ha iha hb ihb m ξ /=.
+    eapply AppAbs'; eauto. by asimpl.
+    all : qauto ctrs:R use:ProjPair'.
+  Qed.
+
+  Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
+    (forall i, R (ρ0 i) (ρ1 i)) ->
+    (forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
+  Proof. eauto using renaming. Qed.
+
+  Lemma morphing_ext n m (ρ0 ρ1 : fin n -> PTm m) a b  :
+    R a b ->
+    (forall i, R (ρ0 i) (ρ1 i)) ->
+    (forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
+  Proof. hauto q:on inv:option. Qed.
+
+  Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
+    (forall i, R (ρ0 i) (ρ1 i)) ->
+    (forall i, R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
+  Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_PTm_PTm. Qed.
+
+  Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
+    (forall i, R (ρ0 i) (ρ1 i)) ->
+    R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
+  Proof.
+    move => + h. move : m ρ0 ρ1. elim : n a b / h => n.
+    move => a0 a1 b0 b1 ha iha hb ihb m ρ0 ρ1 hρ /=.
+    eapply AppAbs'; eauto; cycle 1. sfirstorder use:morphing_up.
+    by asimpl.
+    all : hauto lq:on ctrs:R use:morphing_up, ProjPair'.
+  Qed.
+
+  Lemma substing n m (a : PTm n) b (ρ : fin n -> PTm m) :
+    R a b ->
+    R (subst_PTm ρ a) (subst_PTm ρ b).
+  Proof.
+    hauto l:on use:morphing, refl.
+  Qed.
+
+
+  Lemma cong n (a0 a1 : PTm (S n)) b0 b1  :
+    R a0 a1 ->
+    R b0 b1 ->
+    R (subst_PTm (scons b0 VarPTm) a0) (subst_PTm (scons b1 VarPTm) a1).
+  Proof.
+    move => h0 h1. apply morphing=>//.
+    hauto q:on inv:option ctrs:R.
+  Qed.
+
+  Lemma FromRRed n (a b : PTm n) :
+    RRed.R a b -> RPar.R a b.
+  Proof.
+    induction 1; qauto l:on use:RPar.refl ctrs:RPar.R.
+  Qed.
+
+End RPar.
+
+Lemma red_sn_preservation n :
+  (forall (a : PTm n) (s : SNe a), forall b, RPar.R a b -> SNe b) /\
+  (forall (a : PTm n) (s : SN a), forall b, RPar.R a b -> SN b) /\
+  (forall (a b : PTm n) (_ : TRedSN a b), forall c, RPar.R a c -> exists d, TRedSN' c d /\ RPar.R b d).
+Proof.
+  move : n. apply sn_mutual => n.
+  - hauto l:on inv:RPar.R.
+  - qauto l:on inv:RPar.R,SNe,SN ctrs:SNe.
+  - hauto lq:on inv:RPar.R, SNe ctrs:SNe.
+  - qauto l:on ctrs:SN inv:RPar.R.
+  - hauto lq:on ctrs:SN inv:RPar.R.
+  - hauto lq:on ctrs:SN.
+  - hauto q:on ctrs:SN inv:SN, TRedSN'.
+  - move => a b ha iha hb ihb.
+    elim /RPar.inv : ihb => //=_.
+    + move => a0 a1 b0 b1 ha0 hb0 [*]. subst.
+      eauto using RPar.cong, T_Refl.
+    + move => a0 a1 b0 b1 h0 h1 [*]. subst.
+      elim /RPar.inv : h0 => //=_.
+      move => a0 a2 h [*]. subst.
+      eexists. split. apply T_Once. hauto lq:on ctrs:TRedSN.
+      eauto using RPar.cong.
+  - move => a0 a1 b hb ihb ha iha c.
+    elim /RPar.inv => //=_.
+    + qauto l:on inv:TRedSN.
+    + move => a2 a3 b0 b1 h0 h1 [*]. subst.
+      have {}/iha := h0.
+      move => [d [iha0 iha1]].
+      hauto lq:on rew:off inv:TRedSN' ctrs:TRedSN, RPar.R, TRedSN'.
+  - hauto lq:on inv:RPar.R ctrs:RPar.R, TRedSN', TRedSN.
+  - hauto lq:on inv:RPar.R ctrs:RPar.R, TRedSN', TRedSN.
+  - sauto.
+Qed.
+
+Function tstar {n} (a : PTm n) :=
+  match a with
+  | VarPTm i => a
+  | PAbs a => PAbs (tstar a)
+  | PApp (PAbs a) b => subst_PTm (scons (tstar b) VarPTm) (tstar a)
+  | PApp a b => PApp (tstar a) (tstar b)
+  | PPair a b => PPair (tstar a) (tstar b)
+  | PProj p (PPair a b) => if p is PL then (tstar a) else (tstar b)
+  | PProj p a => PProj p (tstar a)
+  end.
+
+Module TStar.
+  Lemma renaming n m (ξ : fin n -> fin m) (a : PTm n) :
+    tstar (ren_PTm ξ a) = ren_PTm ξ (tstar a).
+  Proof.
+    move : m ξ.
+    apply tstar_ind => {}n {}a => //=.
+    - hauto lq:on.
+    - scongruence.
+    - move => a0 b ? h ih m ξ.
+      rewrite ih.
+      asimpl; congruence.
+    - qauto l:on.
+    - scongruence.
+    - hauto q:on.
+    - qauto l:on.
+  Qed.
+
+  Lemma pair n (a b : PTm n) :
+    tstar (PPair a b) = PPair (tstar a) (tstar b).
+    reflexivity. Qed.
+End TStar.
+
+Definition isPair {n} (a : PTm n) := if a is PPair _ _ then true else false.
+
+Lemma tstar_proj n (a : PTm n) :
+  ((~~ isPair a) /\ forall p, tstar (PProj p a) = PProj p (tstar a)) \/
+    exists a0 b0, a = PPair a0 b0 /\ forall p, tstar (PProj p a) = (if p is PL then (tstar a0) else (tstar b0)).
+Proof. sauto lq:on. Qed.
+
+Module ERed'.
+  Inductive R {n} : PTm n -> PTm n ->  Prop :=
+  (****************** Eta ***********************)
+  | AppEta a :
+    R (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) a
+  | PairEta a :
+    R (PPair (PProj PL a) (PProj PR a)) a
+  (*************** Congruence ********************)
+  | AbsCong a0 a1 :
+    R a0 a1 ->
+    R (PAbs a0) (PAbs a1)
+  | AppCong0 a0 a1 b :
+    R a0 a1 ->
+    R (PApp a0 b) (PApp a1 b)
+  | AppCong1 a b0 b1 :
+    R b0 b1 ->
+    R (PApp a b0) (PApp a b1)
+  | PairCong0 a0 a1 b :
+    R a0 a1 ->
+    R (PPair a0 b) (PPair a1 b)
+  | PairCong1 a b0 b1 :
+    R b0 b1 ->
+    R (PPair a b0) (PPair a b1)
+  | ProjCong p a0 a1 :
+    R a0 a1 ->
+    R (PProj p a0) (PProj p a1).
+
+  Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
+
+  Lemma AppEta' n a (u : PTm n) :
+    u = (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) ->
+    R u a.
+  Proof. move => ->. apply AppEta. Qed.
+
+  Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+    R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+  Proof.
+    move => h. move : m ξ.
+    elim : n a b /h.
+
+    move => n a m ξ /=.
+    eapply AppEta'; eauto. by asimpl.
+    all : qauto ctrs:R.
+  Qed.
+
+  Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
+    (forall i, R (ρ0 i) (ρ1 i)) ->
+    (forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
+  Proof. eauto using renaming. Qed.
+
+End ERed'.
+
+Module EReds.
+
+  #[local]Ltac solve_s_rec :=
+  move => *; eapply rtc_l; eauto;
+         hauto lq:on ctrs:ERed'.R.
+
+  #[local]Ltac solve_s :=
+    repeat (induction 1; last by solve_s_rec); apply rtc_refl.
+
+  Lemma AbsCong n (a b : PTm (S n)) :
+    rtc ERed'.R a b ->
+    rtc ERed'.R (PAbs a) (PAbs b).
+  Proof. solve_s. Qed.
+
+  Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
+    rtc ERed'.R a0 a1 ->
+    rtc ERed'.R b0 b1 ->
+    rtc ERed'.R (PApp a0 b0) (PApp a1 b1).
+  Proof. solve_s. Qed.
+
+  Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
+    rtc ERed'.R a0 a1 ->
+    rtc ERed'.R b0 b1 ->
+    rtc ERed'.R (PPair a0 b0) (PPair a1 b1).
+  Proof. solve_s. Qed.
+
+  Lemma ProjCong n p (a0 a1  : PTm n) :
+    rtc ERed'.R a0 a1 ->
+    rtc ERed'.R (PProj p a0) (PProj p a1).
+  Proof. solve_s. Qed.
+End EReds.
+
+Module RReds.
+
+  #[local]Ltac solve_s_rec :=
+  move => *; eapply rtc_l; eauto;
+         hauto lq:on ctrs:RRed.R.
+
+  #[local]Ltac solve_s :=
+    repeat (induction 1; last by solve_s_rec); apply rtc_refl.
+
+  Lemma AbsCong n (a b : PTm (S n)) :
+    rtc RRed.R a b ->
+    rtc RRed.R (PAbs a) (PAbs b).
+  Proof. solve_s. Qed.
+
+  Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
+    rtc RRed.R a0 a1 ->
+    rtc RRed.R b0 b1 ->
+    rtc RRed.R (PApp a0 b0) (PApp a1 b1).
+  Proof. solve_s. Qed.
+
+  Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
+    rtc RRed.R a0 a1 ->
+    rtc RRed.R b0 b1 ->
+    rtc RRed.R (PPair a0 b0) (PPair a1 b1).
+  Proof. solve_s. Qed.
+
+  Lemma ProjCong n p (a0 a1  : PTm n) :
+    rtc RRed.R a0 a1 ->
+    rtc RRed.R (PProj p a0) (PProj p a1).
+  Proof. solve_s. Qed.
+
+  Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
+    rtc RRed.R a b -> rtc RRed.R (ren_PTm ξ a) (ren_PTm ξ b).
+  Proof.
+    move => h. move : m ξ. elim : a b /h; hauto lq:on ctrs:rtc use:RRed.renaming.
+  Qed.
+
+End RReds.
+
+
+Lemma ne_nf_ren n m (a : PTm n) (ξ : fin n -> fin m) :
+  (ne a <-> ne (ren_PTm ξ a)) /\ (nf a <-> nf (ren_PTm ξ a)).
+Proof.
+  move : m ξ. elim : n / a => //=; solve [hauto b:on].
+Qed.
+
+Lemma ne_ered n (a b : PTm n) (h : ERed'.R a b ) :
+  (ne a -> ne b) /\ (nf a -> nf b).
+Proof.
+  elim : n a b /h=>//=; hauto qb:on use:ne_nf_ren, ne_nf.
+Qed.
+
+Definition ishf {n} (a : PTm n) :=
+  match a with
+  | PPair _ _ => true
+  | PAbs _ => true
+  | _ => false
+  end.
+
+Module NeERed.
+  Inductive R_nonelim {n} : PTm n -> PTm n ->  Prop :=
+  (****************** Eta ***********************)
+  | AppEta a0 a1 :
+    ~~ ishf a0 ->
+    R_elim a0 a1 ->
+    R_nonelim (PAbs (PApp (ren_PTm shift a0) (VarPTm var_zero))) a1
+  | PairEta a0 a1 :
+    ~~ ishf a0 ->
+    R_elim a0 a1 ->
+    R_nonelim (PPair (PProj PL a0) (PProj PR a0)) a1
+  (*************** Congruence ********************)
+  | AbsCong a0 a1 :
+    R_nonelim a0 a1 ->
+    R_nonelim (PAbs a0) (PAbs a1)
+  | AppCong a0 a1 b0 b1 :
+    R_elim a0 a1 ->
+    R_nonelim b0 b1 ->
+    R_nonelim (PApp a0 b0) (PApp a1 b1)
+  | PairCong a0 a1 b0 b1 :
+    R_nonelim a0 a1 ->
+    R_nonelim b0 b1 ->
+    R_nonelim (PPair a0 b0) (PPair a1 b1)
+  | ProjCong p a0 a1 :
+    R_elim a0 a1 ->
+    R_nonelim (PProj p a0) (PProj p a1)
+  | VarTm i :
+    R_nonelim (VarPTm i) (VarPTm i)
+  with R_elim {n} : PTm n -> PTm n -> Prop :=
+  | NAbsCong a0 a1 :
+    R_nonelim a0 a1 ->
+    R_elim (PAbs a0) (PAbs a1)
+  | NAppCong a0 a1 b0 b1 :
+    R_elim a0 a1 ->
+    R_nonelim b0 b1 ->
+    R_elim (PApp a0 b0) (PApp a1 b1)
+  | NPairCong a0 a1 b0 b1 :
+    R_nonelim a0 a1 ->
+    R_nonelim b0 b1 ->
+    R_elim (PPair a0 b0) (PPair a1 b1)
+  | NProjCong p a0 a1 :
+    R_elim a0 a1 ->
+    R_elim (PProj p a0) (PProj p a1)
+  | NVarTm i :
+    R_elim (VarPTm i) (VarPTm i).
+
+  Scheme ered_elim_ind := Induction for R_elim Sort Prop
+      with ered_nonelim_ind := Induction for R_nonelim Sort Prop.
+
+  Combined Scheme ered_mutual from ered_elim_ind, ered_nonelim_ind.
+
+  Lemma R_elim_nf n :
+    (forall (a b : PTm n), R_elim a b -> nf b -> nf a) /\
+      (forall (a b : PTm n), R_nonelim a b -> nf b -> nf a).
+  Proof.
+    move : n. apply ered_mutual => n //=.
+    - move => a0 a1 b0 b1 h ih h' ih' /andP [h0 h1].
+      have hb0 : nf b0 by eauto.
+      suff : ne a0 by qauto b:on.
+      qauto l:on inv:R_elim.
+    - hauto lb:on.
+    - hauto lq:on inv:R_elim.
+    - move => a0 a1 /negP ha' ha ih ha1.
+      have {ih} := ih ha1.
+      move => ha0.
+      suff : ne a0 by hauto lb:on drew:off use:ne_nf_ren.
+      inversion ha; subst => //=.
+    - move => a0 a1 /negP ha' ha ih ha1.
+      have {}ih := ih ha1.
+      have : ne a0 by hauto lq:on inv:PTm.
+      qauto lb:on.
+    - move => a0 a1 b0 b1 ha iha hb ihb /andP [h0 h1].
+      have {}ihb := ihb h1.
+      have {}iha := iha ltac:(eauto using ne_nf).
+      suff : ne a0 by hauto lb:on.
+      move : ha h0. hauto lq:on inv:R_elim.
+    - hauto lb: on drew: off.
+    - hauto lq:on rew:off inv:R_elim.
+  Qed.
+
+  Lemma R_nonelim_nothf n (a b : PTm n) :
+    R_nonelim a b ->
+    ~~ ishf a ->
+    R_elim a b.
+  Proof.
+    move => h. elim : n a b /h=>//=; hauto lq:on ctrs:R_elim.
+  Qed.
+
+  Lemma R_elim_nonelim n (a b : PTm n) :
+    R_elim a b ->
+    R_nonelim a b.
+    move => h. elim : n a b /h=>//=; hauto lq:on ctrs:R_nonelim.
+  Qed.
+
+End NeERed.
+
+Module Type NoForbid.
+  Parameter P : forall n, PTm n -> Prop.
+  Arguments P {n}.
+
+  Axiom P_ERed : forall n (a b : PTm n), ERed.R a b -> P a -> P b.
+  Axiom P_RRed : forall n (a b : PTm n), RRed.R a b -> P a -> P b.
+  Axiom P_AppPair : forall n (a b c : PTm n), ~ P (PApp (PPair a b) c).
+  Axiom P_ProjAbs : forall n p (a : PTm (S n)), ~ P (PProj p (PAbs a)).
+
+  Axiom P_AppInv : forall n (a b : PTm n), P (PApp a b) -> P a /\ P b.
+  Axiom P_AbsInv : forall n (a : PTm (S n)), P (PAbs a) -> P a.
+  Axiom P_PairInv : forall n (a b : PTm n), P (PPair a b) -> P a /\ P b.
+  Axiom P_ProjInv : forall n p (a : PTm n), P (PProj p a) -> P a.
+  Axiom P_renaming : forall n m (ξ : fin n -> fin m) a , P (ren_PTm ξ a) <-> P a.
+
+End NoForbid.
+
+Module Type NoForbid_FactSig (M : NoForbid).
+
+  Axiom P_EReds : forall n (a b : PTm n), rtc ERed.R a b -> M.P a -> M.P b.
+
+  Axiom P_RReds : forall n (a b : PTm n), rtc RRed.R a b -> M.P a -> M.P b.
+
+End NoForbid_FactSig.
+
+Module NoForbid_Fact (M : NoForbid) : NoForbid_FactSig M.
+  Import M.
+
+  Lemma P_EReds : forall n (a b : PTm n), rtc ERed.R a b -> P a -> P b.
+  Proof.
+    induction 1; eauto using P_ERed, rtc_l, rtc_refl.
+  Qed.
+
+  Lemma P_RReds : forall n (a b : PTm n), rtc RRed.R a b -> P a -> P b.
+  Proof.
+    induction 1; eauto using P_RRed, rtc_l, rtc_refl.
+  Qed.
+End NoForbid_Fact.
+
+Module SN_NoForbid : NoForbid.
+  Definition P := @SN.
+  Arguments P {n}.
+
+  Lemma P_ERed : forall n (a b : PTm n), ERed.R a b -> P a -> P b.
+  Proof. sfirstorder use:ered_sn_preservation. Qed.
+
+  Lemma P_RRed : forall n (a b : PTm n), RRed.R a b -> P a -> P b.
+  Proof. hauto q:on use:red_sn_preservation, RPar.FromRRed. Qed.
+
+  Lemma P_AppPair : forall n (a b c : PTm n), ~ P (PApp (PPair a b) c).
+  Proof. sfirstorder use:PProjPair_imp. Qed.
+
+  Lemma P_ProjAbs : forall n p (a : PTm (S n)), ~ P (PProj p (PAbs a)).
+  Proof. sfirstorder use:PProjAbs_imp. Qed.
+
+  Lemma P_AppInv : forall n (a b : PTm n), P (PApp a b) -> P a /\ P b.
+  Proof. sfirstorder use:SN_AppInv. Qed.
+
+  Lemma P_PairInv : forall n (a b : PTm n), P (PPair a b) -> P a /\ P b.
+    move => n a b. move E : (PPair a b) => u h.
+    move : a b E. elim : n u / h; sauto lq:on rew:off. Qed.
+
+  Lemma P_ProjInv : forall n p (a : PTm n), P (PProj p a) -> P a.
+  Proof. sfirstorder use:SN_ProjInv. Qed.
+
+  Lemma P_AbsInv : forall n (a : PTm (S n)), P (PAbs a) -> P a.
+  Proof.
+    move => n a. move E : (PAbs a) => u h.
+    move : E. move : a.
+    induction h; sauto lq:on rew:off.
+  Qed.
+
+  Lemma P_renaming : forall n m (ξ : fin n -> fin m) a , P (ren_PTm ξ a) <-> P  a.
+  Proof. hauto lq:on use:sn_antirenaming, sn_renaming. Qed.
+
+End SN_NoForbid.
+
+Module UniqueNF (M : NoForbid) (MFacts : NoForbid_FactSig M).
+  Import M MFacts.
+  #[local]Hint Resolve P_ERed P_RRed P_AppPair P_ProjAbs : forbid.
+
+  Lemma η_split n (a0 a1 : PTm n) :
+    ERed.R a0 a1 ->
+    P a0 ->
+    exists b, rtc RRed.R a0 b /\ NeERed.R_nonelim b a1.
+  Proof.
+    move => h. elim : n a0 a1 /h .
+    - move => n a0 a1 ha ih /[dup] hP.
+      move /P_AbsInv /P_AppInv => [/P_renaming ha0 _].
+      have {ih} := ih ha0.
+      move => [b [ih0 ih1]].
+      case /orP : (orNb (ishf b)).
+      exists (PAbs (PApp (ren_PTm shift b) (VarPTm var_zero))).
+      split. apply RReds.AbsCong. apply RReds.AppCong; auto using rtc_refl.
+      by eauto using RReds.renaming.
+      apply NeERed.AppEta=>//.
+      sfirstorder use:NeERed.R_nonelim_nothf.
+
+      case : b ih0 ih1 => //=.
+      + move => p ih0 ih1 _.
+        set q := PAbs _.
+        suff : rtc RRed.R q (PAbs p) by sfirstorder.
+        subst q.
+        apply : rtc_r.
+        apply RReds.AbsCong. apply RReds.AppCong.
+        by eauto using RReds.renaming.
+        apply rtc_refl.
+        apply : RRed.AbsCong => /=.
+        apply RRed.AppAbs'. by asimpl.
+      (* violates SN *)
+      + move => p p0 h. exfalso.
+        have : P (PApp (ren_PTm shift a0) (VarPTm var_zero))
+          by sfirstorder use:P_AbsInv.
+
+        have : rtc RRed.R (PApp (ren_PTm shift a0) (VarPTm var_zero))
+                 (PApp (ren_PTm shift (PPair p p0)) (VarPTm var_zero))
+          by hauto lq:on use:RReds.AppCong, RReds.renaming, rtc_refl.
+
+        move : P_RReds. repeat move/[apply] => /=.
+        hauto l:on use:P_AppPair.
+    - move => n a0 a1 h ih /[dup] hP.
+      move /P_PairInv => [/P_ProjInv + _].
+      move : ih => /[apply].
+      move => [b [ih0 ih1]].
+      case /orP : (orNb (ishf b)).
+      exists (PPair (PProj PL b) (PProj PR b)).
+      split. sfirstorder use:RReds.PairCong,RReds.ProjCong.
+      hauto lq:on ctrs:NeERed.R_nonelim use:NeERed.R_nonelim_nothf.
+
+      case : b ih0 ih1 => //=.
+      (* violates SN *)
+      + move => p ?. exfalso.
+        have {}hP : P (PProj PL a0) by sfirstorder use:P_PairInv.
+        have : rtc RRed.R (PProj PL a0) (PProj PL (PAbs p))
+          by eauto using RReds.ProjCong.
+        move : P_RReds hP. repeat move/[apply] => /=.
+        sfirstorder use:P_ProjAbs.
+      + move => t0 t1 ih0 h1 _.
+        exists (PPair t0 t1).
+        split => //=.
+        apply RReds.PairCong.
+        apply : rtc_r; eauto using RReds.ProjCong.
+        apply RRed.ProjPair.
+        apply : rtc_r; eauto using RReds.ProjCong.
+        apply RRed.ProjPair.
+    - hauto lq:on ctrs:NeERed.R_nonelim use:RReds.AbsCong, P_AbsInv.
+    - move => n a0 a1 b0 b1 ha iha hb ihb.
+      move => /[dup] hP /P_AppInv [hP0 hP1].
+      have {iha} [a2 [iha0 iha1]]  := iha hP0.
+      have {ihb} [b2 [ihb0 ihb1]]  := ihb hP1.
+      case /orP : (orNb (ishf a2)) => [h|].
+      + exists (PApp a2 b2). split; first by eauto using RReds.AppCong.
+        hauto lq:on ctrs:NeERed.R_nonelim use:NeERed.R_nonelim_nothf.
+      + case : a2 iha0 iha1 => //=.
+        * move => p h0 h1 _.
+          inversion h1; subst.
+          ** exists (PApp a2 b2).
+             split.
+             apply : rtc_r.
+             apply RReds.AppCong; eauto.
+             apply RRed.AppAbs'. by asimpl.
+             hauto lq:on ctrs:NeERed.R_nonelim.
+          ** hauto lq:on ctrs:NeERed.R_nonelim,NeERed.R_elim use:RReds.AppCong.
+        (* Impossible *)
+        * move => u0 u1 h. exfalso.
+          have : rtc RRed.R (PApp a0 b0) (PApp (PPair u0 u1) b0)
+            by hauto lq:on ctrs:rtc use:RReds.AppCong.
+          move : P_RReds hP; repeat move/[apply].
+          sfirstorder use:P_AppPair.
+    - hauto lq:on ctrs:NeERed.R_nonelim use:RReds.PairCong, P_PairInv.
+    - move => n p a0 a1 ha ih /[dup] hP /P_ProjInv.
+      move : ih => /[apply]. move => [a2 [iha0 iha1]].
+      case /orP : (orNb (ishf a2)) => [h|].
+      exists (PProj p a2).
+      split.  eauto using RReds.ProjCong.
+      qauto l:on ctrs:NeERed.R_nonelim, NeERed.R_elim use:NeERed.R_nonelim_nothf.
+
+      case : a2 iha0 iha1 => //=.
+      + move => u iha0. exfalso.
+        have : rtc RRed.R (PProj p a0) (PProj p (PAbs u))
+          by sfirstorder use:RReds.ProjCong ctrs:rtc.
+        move : P_RReds hP. repeat move/[apply].
+        sfirstorder use:P_ProjAbs.
+      + move => u0 u1 iha0 iha1 _.
+        inversion iha1; subst.
+        * exists (PProj p a2). split.
+          apply : rtc_r.
+          apply RReds.ProjCong; eauto.
+          clear. hauto l:on inv:PTag.
+          hauto lq:on ctrs:NeERed.R_nonelim.
+        * hauto lq:on ctrs:NeERed.R_nonelim,NeERed.R_elim use:RReds.ProjCong.
+    - hauto lq:on ctrs:rtc, NeERed.R_nonelim.
+  Qed.
+
+End UniqueNF.
+
+Lemma η_nf_to_ne n (a0 a1 : PTm n) :
+  ERed'.R a0 a1 -> nf a0 -> ~~ ne a0 -> ne a1 ->
+  (a0 = PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) \/
+    (a0 = PPair (PProj PL a1) (PProj PR a1)).
+Proof.
+  move => h.
+  elim : n a0 a1 /h => n /=.
+  - sfirstorder use:ne_ered.
+  - hauto l:on use:ne_ered.
+  - scongruence use:ne_ered.
+  - hauto qb:on use:ne_ered, ne_nf.
+  - move => a b0 b1 h0 ih0 /andP [h1 h2] h3 /andP [h4 h5].
+    have {h3} : ~~ ne a by sfirstorder b:on.
+    by move /negP.
+  - hauto lqb:on.
+  - sfirstorder b:on.
+  - scongruence b:on.
+Qed.