syntax.sig

@@ -1,15 +1,19 @@
 PTm(VarPTm) : Type
 PTag : Type
-Ty : Type
+BTag : Type
 
-Fun : Ty -> Ty -> Ty
+nat : Type
-Prod : Ty -> Ty -> Ty
-Void : Ty
 
 PL : PTag
 PR : PTag
 
+PPi : BTag
+PSig : BTag
+
 PAbs : (bind PTm in PTm) -> PTm
 PApp : PTm -> PTm -> PTm
 PPair : PTm -> PTm -> PTm
 PProj : PTag -> PTm -> PTm
+PBind : BTag -> PTm -> (bind PTm in PTm) -> PTm
+PUniv : nat -> PTm
+PBot : PTm

theories/Autosubst2/syntax.v

@@ -5,6 +5,20 @@ Require Import Setoid Morphisms Relation_Definitions.
 
 Module Core.
 
+Inductive BTag : Type :=
+  | PPi : BTag
+  | PSig : BTag.
+
+Lemma congr_PPi : PPi = PPi.
+Proof.
+exact (eq_refl).
+Qed.
+
+Lemma congr_PSig : PSig = PSig.
+Proof.
+exact (eq_refl).
+Qed.
+
 Inductive PTag : Type :=
   | PL : PTag
   | PR : PTag.
@@ -24,7 +38,10 @@ Inductive PTm (n_PTm : nat) : Type :=
   | 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.
+  | PProj : PTag -> PTm n_PTm -> PTm n_PTm
+  | PBind : BTag -> PTm n_PTm -> PTm (S n_PTm) -> PTm n_PTm
+  | PUniv : nat -> PTm n_PTm
+  | PBot : PTm n_PTm.
 
 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.
@@ -56,6 +73,28 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj m_PTm x s1) H0))
          (ap (fun x => PProj m_PTm t0 x) H1)).
 Qed.
 
+Lemma congr_PBind {m_PTm : nat} {s0 : BTag} {s1 : PTm m_PTm}
+  {s2 : PTm (S m_PTm)} {t0 : BTag} {t1 : PTm m_PTm} {t2 : PTm (S m_PTm)}
+  (H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) :
+  PBind m_PTm s0 s1 s2 = PBind m_PTm t0 t1 t2.
+Proof.
+exact (eq_trans
+         (eq_trans (eq_trans eq_refl (ap (fun x => PBind m_PTm x s1 s2) H0))
+            (ap (fun x => PBind m_PTm t0 x s2) H1))
+         (ap (fun x => PBind m_PTm t0 t1 x) H2)).
+Qed.
+
+Lemma congr_PUniv {m_PTm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
+  PUniv m_PTm s0 = PUniv m_PTm t0.
+Proof.
+exact (eq_trans eq_refl (ap (fun x => PUniv m_PTm x) H0)).
+Qed.
+
+Lemma congr_PBot {m_PTm : nat} : PBot m_PTm = PBot m_PTm.
+Proof.
+exact (eq_refl).
+Qed.
+
 Lemma upRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n) :
   fin (S m) -> fin (S n).
 Proof.
@@ -76,6 +115,10 @@ Fixpoint ren_PTm {m_PTm : nat} {n_PTm : nat}
   | 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)
+  | PBind _ s0 s1 s2 =>
+      PBind n_PTm s0 (ren_PTm xi_PTm s1) (ren_PTm (upRen_PTm_PTm xi_PTm) s2)
+  | PUniv _ s0 => PUniv n_PTm s0
+  | PBot _ => PBot n_PTm
   end.
 
 Lemma up_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm) :
@@ -102,6 +145,11 @@ Fixpoint subst_PTm {m_PTm : nat} {n_PTm : nat}
   | PPair _ s0 s1 =>
       PPair n_PTm (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
   | PProj _ s0 s1 => PProj n_PTm s0 (subst_PTm sigma_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      PBind n_PTm s0 (subst_PTm sigma_PTm s1)
+        (subst_PTm (up_PTm_PTm sigma_PTm) s2)
+  | PUniv _ s0 => PUniv n_PTm s0
+  | PBot _ => PBot n_PTm
   end.
 
 Lemma upId_PTm_PTm {m_PTm : nat} (sigma : fin m_PTm -> PTm m_PTm)
@@ -140,6 +188,11 @@ Fixpoint idSubst_PTm {m_PTm : nat} (sigma_PTm : fin m_PTm -> PTm m_PTm)
         (idSubst_PTm sigma_PTm Eq_PTm s1)
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
+        (idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma upExtRen_PTm_PTm {m : nat} {n : nat} (xi : fin m -> fin n)
@@ -180,6 +233,12 @@ ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
         (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
+        (extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
+           (upExtRen_PTm_PTm _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma upExt_PTm_PTm {m : nat} {n_PTm : nat} (sigma : fin m -> PTm n_PTm)
@@ -221,6 +280,12 @@ subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
         (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
+        (ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
+           (upExt_PTm_PTm _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma up_ren_ren_PTm_PTm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@@ -262,6 +327,13 @@ Fixpoint compRenRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0)
         (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0)
+        (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
+        (compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
+           (upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma up_ren_subst_PTm_PTm {k : nat} {l : nat} {m_PTm : nat}
@@ -312,6 +384,13 @@ Fixpoint compRenSubst_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0)
         (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0)
+        (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
+        (compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
+           (up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma up_subst_ren_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
@@ -382,6 +461,13 @@ ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0)
         (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0)
+        (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
+        (compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
+           (up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma up_subst_subst_PTm_PTm {k : nat} {l_PTm : nat} {m_PTm : nat}
@@ -454,6 +540,13 @@ subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0)
         (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0)
+        (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
+        (compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
+           (up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma renRen_PTm {k_PTm : nat} {l_PTm : nat} {m_PTm : nat}
@@ -566,6 +659,12 @@ Fixpoint rinst_inst_PTm {m_PTm : nat} {n_PTm : nat}
         (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
   | PProj _ s0 s1 =>
       congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
+  | PBind _ s0 s1 s2 =>
+      congr_PBind (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
+        (rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
+           (rinstInst_up_PTm_PTm _ _ Eq_PTm) s2)
+  | PUniv _ s0 => congr_PUniv (eq_refl s0)
+  | PBot _ => congr_PBot
   end.
 
 Lemma rinstInst'_PTm {m_PTm : nat} {n_PTm : nat}
@@ -637,30 +736,6 @@ 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.
 
@@ -796,6 +871,12 @@ Core.
 
 Arguments VarPTm {n_PTm}.
 
+Arguments PBot {n_PTm}.
+
+Arguments PUniv {n_PTm}.
+
+Arguments PBind {n_PTm}.
+
 Arguments PProj {n_PTm}.
 
 Arguments PPair {n_PTm}.

theories/logrel.v

@@ -0,0 +1,752 @@
+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.
+Require Import Logic.PropExtensionality (propositional_extensionality).
+From stdpp Require Import relations (rtc(..), rtc_subrel).
+Import Psatz.
+Require Import Cdcl.Itauto.
+
+Definition ProdSpace {n} (PA : PTm n -> Prop)
+  (PF : PTm n -> (PTm n -> Prop) -> Prop) b : Prop :=
+  forall a PB, PA a -> PF a PB -> PB (PApp b a).
+
+Definition SumSpace {n} (PA : PTm n -> Prop)
+  (PF : PTm n -> (PTm n -> Prop) -> Prop) t : Prop :=
+  (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.
+
+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 ↘ (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) ->
+  (forall a PB, PF a PB -> ⟦ subst_PTm (scons a VarPTm) B ⟧ i ;; I ↘ PB) ->
+  ⟦ PBind p A B ⟧ i ;; I ↘ BindSpace p PA PF
+
+| InterpExt_Univ j :
+  j < i ->
+  ⟦ PUniv j ⟧ i ;; I ↘ (I j)
+
+| InterpExt_Step A A0 PA :
+  TRedSN A A0 ->
+  ⟦ A0 ⟧ i ;; I ↘ PA ->
+  ⟦ A ⟧ i ;; I ↘ PA
+where "⟦ A ⟧ i ;; I ↘ S" := (InterpExt i I A S).
+
+
+Lemma InterpExt_Univ' n i  I j (PF : PTm n -> Prop) :
+  PF = I j ->
+  j < i ->
+  ⟦ PUniv j ⟧ i ;; I ↘ PF.
+Proof. hauto lq:on ctrs:InterpExt. Qed.
+
+Infix "<?" := Compare_dec.lt_dec (at level 60).
+
+Equations InterpUnivN n (i : nat) : PTm n -> (PTm n -> Prop) -> Prop by wf i lt :=
+  InterpUnivN n i := @InterpExt n i
+                     (fun j A =>
+                        match j <? i  with
+                        | left _ => exists PA, InterpUnivN n j A PA
+                        | right _ => False
+                        end).
+Arguments InterpUnivN {n}.
+
+Lemma InterpExt_lt_impl n i I I' A (PA : PTm n -> Prop) :
+  (forall j, j < i -> I j = I' j) ->
+  ⟦ A ⟧ i ;; I ↘ PA ->
+  ⟦ A ⟧ i ;; I' ↘ PA.
+Proof.
+  move => hI h.
+  elim : A PA /h.
+  - hauto q:on ctrs:InterpExt.
+  - hauto lq:on rew:off ctrs:InterpExt.
+  - hauto q:on ctrs:InterpExt.
+  - hauto lq:on ctrs:InterpExt.
+Qed.
+
+Lemma InterpExt_lt_eq n i I I' A (PA : PTm n -> Prop) :
+  (forall j, j < i -> I j = I' j) ->
+  ⟦ A ⟧ i ;; I ↘ PA =
+  ⟦ A ⟧ i ;; I' ↘ PA.
+Proof.
+  move => hI. apply propositional_extensionality.
+  have : forall j, j < i -> I' j = I j by sfirstorder.
+  firstorder using InterpExt_lt_impl.
+Qed.
+
+Notation "⟦ A ⟧ i ↘ S" := (InterpUnivN i A S) (at level 70).
+
+Lemma InterpUnivN_nolt n i :
+  @InterpUnivN n i = @InterpExt n i (fun j (A : PTm n) => exists PA, ⟦ A ⟧ j ↘ PA).
+Proof.
+  simp InterpUnivN.
+  extensionality A. extensionality PA.
+  set I0 := (fun _ => _).
+  set I1 := (fun _ => _).
+  apply InterpExt_lt_eq.
+  hauto q:on.
+Qed.
+
+#[export]Hint Rewrite @InterpUnivN_nolt : InterpUniv.
+
+Lemma InterpUniv_ind
+  : forall n (P : nat -> PTm n -> (PTm n -> Prop) -> Prop),
+       (forall i (A : PTm n), SNe A -> P i A (fun a : PTm n => exists v : PTm n, rtc TRedSN a v /\ SNe v)) ->
+       (forall i (p : BTag) (A : PTm n) (B : PTm (S n)) (PA : PTm n -> Prop)
+          (PF : PTm n -> (PTm n -> Prop) -> Prop),
+        ⟦ A ⟧ i ↘ PA ->
+        P i 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 i (subst_PTm (scons a VarPTm) B) PB) ->
+        P i (PBind p A B) (BindSpace p PA PF)) ->
+       (forall i j : nat, j < i ->  (forall A PA, ⟦ A ⟧ j ↘ PA -> P j A PA) -> P i (PUniv j) (fun A => exists PA, ⟦ A ⟧ j ↘ PA)) ->
+       (forall i (A A0 : PTm n) (PA : PTm n -> Prop), TRedSN A A0 -> ⟦ A0 ⟧ i ↘ PA -> P i A0 PA -> P i A PA) ->
+       forall i (p : PTm n) (P0 : PTm n -> Prop), ⟦ p ⟧ i ↘ P0 -> P i p P0.
+Proof.
+  move => n P  hSN hBind hUniv hRed.
+  elim /Wf_nat.lt_wf_ind => i ih . simp InterpUniv.
+  move => A PA. move => h. set I := fun _ => _ in h.
+  elim : A PA / h; rewrite -?InterpUnivN_nolt; eauto.
+Qed.
+
+Derive Dependent Inversion iinv with (forall n i I (A : PTm n) PA, InterpExt i I A PA) Sort Prop.
+
+Lemma InterpUniv_Ne n i (A : PTm n) :
+  SNe A ->
+  ⟦ A ⟧ i ↘ (fun a => exists v, rtc TRedSN a v /\ SNe v).
+Proof. simp InterpUniv. apply InterpExt_Ne. Qed.
+
+Lemma InterpUniv_Bind n i p A B PA PF :
+  ⟦ A : PTm n ⟧ i ↘ PA ->
+  (forall a, PA a -> exists PB, PF a PB) ->
+  (forall a PB, PF a PB -> ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB) ->
+  ⟦ PBind p A B ⟧ i ↘ BindSpace p PA PF.
+Proof. simp InterpUniv. apply InterpExt_Bind. Qed.
+
+Lemma InterpUniv_Univ n i j :
+  j < i -> ⟦ PUniv j : PTm n ⟧ i ↘ (fun A => exists PA, ⟦ A ⟧ j ↘ PA).
+Proof.
+  simp InterpUniv. simpl.
+  apply InterpExt_Univ'. by simp InterpUniv.
+Qed.
+
+Lemma InterpUniv_Step i n A A0 PA :
+  TRedSN A A0 ->
+  ⟦ A0 : PTm n ⟧ i ↘ PA ->
+  ⟦ A ⟧ i ↘ PA.
+Proof. simp InterpUniv. apply InterpExt_Step. Qed.
+
+
+#[export]Hint Resolve InterpUniv_Bind InterpUniv_Step InterpUniv_Ne InterpUniv_Univ : InterpUniv.
+
+Lemma InterpExt_cumulative n i j I (A : PTm n) PA :
+  i <= j ->
+   ⟦ A ⟧ i ;; I ↘ PA ->
+   ⟦ A ⟧ j ;; I ↘ PA.
+Proof.
+  move => h h0.
+  elim : A PA /h0;
+    hauto l:on ctrs:InterpExt solve+:(by lia).
+Qed.
+
+Lemma InterpUniv_cumulative n i (A : PTm n) PA :
+   ⟦ A ⟧ i ↘ PA -> forall j, i <= j ->
+   ⟦ A ⟧ j ↘ PA.
+Proof.
+  hauto l:on rew:db:InterpUniv use:InterpExt_cumulative.
+Qed.
+
+Definition CR {n} (P : PTm n -> Prop) :=
+  (forall a, P a -> SN a) /\
+    (forall a, SNe a -> P a).
+
+Lemma N_Exps n (a b : PTm n) :
+  rtc TRedSN a b ->
+  SN b ->
+  SN a.
+Proof.
+  induction 1; eauto using N_Exp.
+Qed.
+
+Lemma adequacy : forall i n A PA,
+   ⟦ A : PTm n ⟧ i ↘ PA ->
+  CR PA /\ SN A.
+Proof.
+  move => + n. apply : InterpUniv_ind.
+  - hauto l:on use:N_Exps ctrs:SN,SNe.
+  - move => i 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.
+    rewrite /CR.
+    repeat split.
+    + case : p =>//=.
+      * rewrite /ProdSpace.
+        qauto use:SN_AppInv unfold:CR.
+      * 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.
+      * sfirstorder.
+    + apply N_Bind=>//=.
+      have : SN (PApp (PAbs B) PBot).
+      apply : N_Exp; eauto using N_β.
+      hauto lq:on.
+      qauto l:on use:SN_AppInv, SN_NoForbid.P_AbsInv.
+  - hauto l:on ctrs:InterpExt rew:db:InterpUniv.
+  - hauto l:on ctrs:SN unfold:CR.
+Qed.
+
+Lemma InterpUniv_Steps i n A A0 PA :
+  rtc TRedSN A A0 ->
+  ⟦ A0 : PTm n ⟧ i ↘ PA ->
+  ⟦ A ⟧ i ↘ PA.
+Proof. induction 1; hauto l:on use:InterpUniv_Step. 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.
+  move : i A PA . apply : InterpUniv_ind; eauto.
+  - hauto q:on ctrs:rtc.
+  - move => i 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.
+    + hauto lq:on ctrs:rtc unfold:SumSpace.
+  - hauto l:on use:InterpUniv_Step.
+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 InterpUniv_case n i (A : PTm n) PA :
+  ⟦ A ⟧ i ↘ PA ->
+  exists H, rtc TRedSN A H /\  ⟦ H ⟧ i ↘ PA /\ (SNe H \/ isbind H \/ isuniv H).
+Proof.
+  move : i A PA. apply InterpUniv_ind => //=.
+  hauto lq:on ctrs:rtc use:InterpUniv_Ne.
+  hauto l:on use:InterpUniv_Bind.
+  hauto l:on use:InterpUniv_Univ.
+  hauto lq:on ctrs:rtc.
+Qed.
+
+Lemma redsn_preservation_mutual n :
+  (forall (a : PTm n) (s : SNe a), forall b, TRedSN a b -> False) /\
+    (forall (a : PTm n) (s : SN a), forall b, TRedSN a b -> SN b) /\
+  (forall (a b : PTm n) (_ : TRedSN a b), forall c, TRedSN a c -> b = c).
+Proof.
+  move : n. apply sn_mutual; sauto lq:on rew:off.
+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.
+
+#[export]Hint Resolve DJoin.sne_bind_noconf DJoin.sne_univ_noconf DJoin.bind_univ_noconf : noconf.
+
+Lemma InterpUniv_SNe_inv n i (A : PTm n) PA :
+  SNe A ->
+  ⟦ A ⟧ i ↘ PA ->
+  PA = (fun a => exists v, rtc TRedSN a v /\ SNe v).
+Proof.
+  simp InterpUniv.
+  hauto lq:on rew:off inv:InterpExt,SNe use:redsn_preservation_mutual.
+Qed.
+
+Lemma InterpUniv_Bind_inv n i p A B S :
+  ⟦ PBind p A B ⟧ i ↘ S -> exists PA PF,
+  ⟦ A : PTm n ⟧ i ↘ PA /\
+  (forall a, PA a -> exists PB, PF a PB) /\
+  (forall a PB, PF a PB -> ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB) /\
+  S = BindSpace p PA PF.
+Proof. simp InterpUniv.
+       inversion 1; try hauto inv:SNe q:on use:redsn_preservation_mutual.
+       rewrite -!InterpUnivN_nolt.
+       sauto lq:on.
+Qed.
+
+Lemma InterpUniv_Univ_inv n i j S :
+  ⟦ PUniv j : PTm n ⟧ i ↘ S ->
+  S = (fun A => exists PA, ⟦ A ⟧ j ↘ PA) /\ j < i.
+Proof.
+  simp InterpUniv. inversion 1;
+    try hauto inv:SNe use:redsn_preservation_mutual.
+  rewrite -!InterpUnivN_nolt. sfirstorder.
+  subst. hauto lq:on inv:TRedSN.
+Qed.
+
+Lemma bindspace_iff n p (PA : PTm n -> Prop) PF PF0 b  :
+  (forall (a : PTm n) (PB PB0 : PTm n -> Prop), PA a -> PF a PB -> PF0 a PB0 -> PB = PB0) ->
+  (forall a, PA a -> exists PB, PF a PB) ->
+  (forall a, PA a -> exists PB0, PF0 a PB0) ->
+  (BindSpace p PA PF b <-> BindSpace p PA PF0 b).
+Proof.
+  rewrite /BindSpace => h hPF hPF0.
+  case : p => /=.
+  - rewrite /ProdSpace.
+    split.
+    move => h1 a PB ha hPF'.
+    specialize hPF with (1 := ha).
+    specialize hPF0 with (1 := ha).
+    sblast.
+    move => ? a PB ha.
+    specialize hPF with (1 := ha).
+    specialize hPF0 with (1 := ha).
+    sblast.
+  - rewrite /SumSpace.
+    hauto lq:on rew:off.
+Qed.
+
+Lemma InterpUniv_Join n i (A B : PTm n) PA PB :
+  ⟦ A ⟧ i ↘ PA ->
+  ⟦ B ⟧ i ↘ PB ->
+  DJoin.R A B ->
+  PA = PB.
+Proof.
+  move => hA.
+  move : i A PA hA B PB.
+  apply : InterpUniv_ind.
+  - 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 [/DJoin.FromRedSNs h [h1 h0]]].
+    have {hAB} {}h : DJoin.R A H by eauto using DJoin.transitive.
+    have {}h0 : SNe H.
+    suff : ~ isbind H /\ ~ isuniv H by itauto.
+    move : h hA. clear. hauto lq:on db:noconf.
+    hauto lq:on use:InterpUniv_SNe_inv.
+  - move => i p A B PA PF hPA ihPA hTot hRes ihPF U PU hU.
+    have hU' : SN U by hauto l:on use:adequacy.
+    move /InterpUniv_case : hU => [H [/DJoin.FromRedSNs h [h1 h0]]] hU.
+    have {hU} {}h : DJoin.R (PBind p A B) H by eauto using DJoin.transitive.
+    have{h0} : isbind H.
+    suff : ~ SNe H /\ ~ isuniv H by itauto.
+    have : isbind (PBind p A B) by scongruence.
+    hauto l:on use: DJoin.sne_bind_noconf, DJoin.bind_univ_noconf, DJoin.symmetric.
+    case : H h1 h => //=.
+    move => p0 A0 B0 h0 /DJoin.bind_inj.
+    move => [? [hA hB]] _. subst.
+    move /InterpUniv_Bind_inv : h0.
+    move => [PA0][PF0][hPA0][hTot0][hRes0 ?]. subst.
+    have ? : PA0 = PA by qauto l:on. subst.
+    have : forall a PB PB', PA a -> PF a PB -> PF0 a PB' -> PB = PB'.
+    move => a PB PB' ha hPB hPB'. apply : ihPF; eauto.
+    have hj0 : DJoin.R (PAbs B) (PAbs B0) by eauto using DJoin.AbsCong.
+    have {}hj0 : DJoin.R (PApp (PAbs B) a) (PApp (PAbs B0) a) by eauto using DJoin.AppCong, DJoin.refl.
+    have [? ?] : SN (PApp (PAbs B) a) /\ SN (PApp (PAbs B0) a) by
+      hauto lq:on rew:off use:N_Exp, N_β, adequacy.
+    have [? ?] : DJoin.R (PApp (PAbs B0) a) (subst_PTm (scons a VarPTm) B0) /\
+           DJoin.R (subst_PTm (scons a VarPTm) B) (PApp (PAbs B) a)
+      by hauto lq:on ctrs:RRed.R use:DJoin.FromRRed0, DJoin.FromRRed1.
+    eauto using DJoin.transitive.
+    move => h. extensionality b. apply propositional_extensionality.
+    hauto l:on use:bindspace_iff.
+  - move => i j jlti ih B PB hPB.
+    have ? : SN B by hauto l:on use:adequacy.
+    move /InterpUniv_case : hPB => [H [/DJoin.FromRedSNs h [h1 h0]]].
+    move => hj.
+    have {hj}{}h : DJoin.R (PUniv j) H by eauto using DJoin.transitive.
+    have {h0} : isuniv H.
+    suff : ~ SNe H /\ ~ isbind H by tauto.
+    hauto l:on use: DJoin.sne_univ_noconf, DJoin.bind_univ_noconf, DJoin.symmetric.
+    case : H h1 h => //=.
+    move => j' hPB h _.
+    have {}h : j' = j by hauto lq:on use: DJoin.univ_inj. subst.
+    hauto lq:on use:InterpUniv_Univ_inv.
+  - move => i A A0 PA hr hPA ihPA B PB hPB hAB.
+    have /DJoin.symmetric ? : DJoin.R A A0 by hauto lq:on rew:off ctrs:rtc use:DJoin.FromRedSNs.
+    have ? : SN A0 by hauto l:on use:adequacy.
+    have ? : SN A by eauto using N_Exp.
+    have : DJoin.R A0 B by eauto using DJoin.transitive.
+    eauto.
+Qed.
+
+Lemma InterpUniv_Functional n i  (A : PTm n) PA PB :
+  ⟦ A ⟧ i ↘ PA ->
+  ⟦ A ⟧ i ↘ PB ->
+  PA = PB.
+Proof. hauto use:InterpUniv_Join, DJoin.refl. Qed.
+
+Lemma InterpUniv_Join' n i j (A B : PTm n) PA PB :
+  ⟦ A ⟧ i ↘ PA ->
+  ⟦ B ⟧ j ↘ PB ->
+  DJoin.R A B ->
+  PA = PB.
+Proof.
+  have [? ?] : i <= max i j /\ j <= max i j by lia.
+  move => hPA hPB.
+  have : ⟦ A ⟧ (max i j) ↘ PA by eauto using InterpUniv_cumulative.
+  have : ⟦ B ⟧ (max i j) ↘ PB by eauto using InterpUniv_cumulative.
+  eauto using InterpUniv_Join.
+Qed.
+
+Lemma InterpUniv_Functional' n i j A PA PB :
+  ⟦ A : PTm n ⟧ i ↘ PA ->
+  ⟦ A ⟧ j ↘ PB ->
+  PA = PB.
+Proof.
+  hauto l:on use:InterpUniv_Join', DJoin.refl.
+Qed.
+
+Lemma InterpUniv_Bind_inv_nopf n i p A B P (h :  ⟦PBind p A B ⟧ i ↘ P) :
+  exists (PA : PTm n -> Prop),
+     ⟦ A ⟧ i ↘ PA /\
+    (forall a, PA a -> exists PB, ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB) /\
+      P = BindSpace p PA (fun a PB => ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB).
+Proof.
+  move /InterpUniv_Bind_inv : h.
+  move => [PA][PF][hPA][hPF][hPF']?. subst.
+  exists PA. repeat split => //.
+  - sfirstorder.
+  - extensionality b.
+    case : p => /=.
+    + extensionality a.
+      extensionality PB.
+      extensionality ha.
+      apply propositional_extensionality.
+      split.
+      * move => h hPB.  apply h.
+        have {}/hPF := ha.
+        move => [PB0 hPB0].
+        have {}/hPF' := hPB0 => ?.
+        have : PB = PB0 by hauto l:on use:InterpUniv_Functional.
+        congruence.
+      * sfirstorder.
+    + rewrite /SumSpace. apply propositional_extensionality.
+      split; hauto q:on use:InterpUniv_Functional.
+Qed.
+
+Definition ρ_ok {n} (Γ : fin n -> PTm n) (ρ : fin n -> PTm 0) := forall i k PA,
+    ⟦ subst_PTm ρ (Γ i) ⟧ k ↘ PA -> PA (ρ i).
+
+Definition SemWt {n} Γ (a A : PTm n) := forall ρ, ρ_ok Γ ρ -> exists k PA, ⟦ subst_PTm ρ A ⟧ k ↘ PA /\ PA (subst_PTm ρ a).
+Notation "Γ ⊨ a ∈ A" := (SemWt Γ a A) (at level 70).
+
+(* Semantic context wellformedness *)
+Definition SemWff {n} Γ := forall (i : fin n), exists j, Γ ⊨ Γ i ∈ PUniv j.
+Notation "⊨ Γ" := (SemWff Γ) (at level 70).
+
+Lemma ρ_ok_bot n (Γ : fin n -> PTm n)  :
+  ρ_ok Γ (fun _ => PBot).
+Proof.
+  rewrite /ρ_ok.
+  hauto q:on use:adequacy ctrs:SNe.
+Qed.
+
+Lemma ρ_ok_cons n i (Γ : fin n -> PTm n) ρ a PA A :
+  ⟦ subst_PTm ρ A ⟧ i ↘ PA -> PA a ->
+  ρ_ok Γ ρ ->
+  ρ_ok (funcomp (ren_PTm shift) (scons A Γ)) (scons a ρ).
+Proof.
+  move => h0 h1 h2.
+  rewrite /ρ_ok.
+  move => j.
+  destruct j as [j|].
+  - move => m PA0. asimpl => ?.
+    asimpl.
+    firstorder.
+  - move => m PA0. asimpl => h3.
+    have ? : PA0 = PA by eauto using InterpUniv_Functional'.
+    by subst.
+Qed.
+
+Definition renaming_ok {n m} (Γ : fin n -> PTm n) (Δ : fin m -> PTm m) (ξ : fin m -> fin n) :=
+  forall (i : fin m), ren_PTm ξ (Δ i) = Γ (ξ i).
+
+Lemma ρ_ok_renaming n m (Γ : fin n -> PTm n) ρ :
+  forall (Δ : fin m -> PTm m) ξ,
+    renaming_ok Γ Δ ξ ->
+    ρ_ok Γ ρ ->
+    ρ_ok Δ (funcomp ρ ξ).
+Proof.
+  move => Δ ξ hξ hρ.
+  rewrite /ρ_ok => i m' PA.
+  rewrite /renaming_ok in hξ.
+  rewrite /ρ_ok in hρ.
+  move => h.
+  rewrite /funcomp.
+  apply hρ with (k := m').
+  move : h. rewrite -hξ.
+  by asimpl.
+Qed.
+
+Lemma renaming_SemWt {n} Γ a A :
+  Γ ⊨ a ∈ A ->
+  forall {m} Δ (ξ : fin n -> fin m),
+    renaming_ok Δ Γ ξ ->
+    Δ ⊨ ren_PTm ξ a ∈ ren_PTm ξ A.
+Proof.
+  rewrite /SemWt => h m Δ ξ hξ ρ hρ.
+  have /h hρ' : (ρ_ok Γ (funcomp ρ ξ)) by eauto using ρ_ok_renaming.
+  hauto q:on solve+:(by asimpl).
+Qed.
+
+Lemma weakening_Sem n Γ (a : PTm n) A B i
+  (h0 : Γ ⊨ B ∈ PUniv i)
+  (h1 : Γ ⊨ a ∈ A) :
+   funcomp (ren_PTm shift) (scons B Γ) ⊨ ren_PTm shift a ∈ ren_PTm shift A.
+Proof.
+  apply : renaming_SemWt; eauto.
+  hauto lq:on inv:option unfold:renaming_ok.
+Qed.
+
+Lemma SemWt_SN n Γ (a : PTm n) A :
+  Γ ⊨ a ∈ A ->
+  SN a /\ SN A.
+Proof.
+  move => h.
+  have {}/h := ρ_ok_bot _ Γ => h.
+  have h0 : SN (subst_PTm (fun _ : fin n => (PBot : PTm 0)) A) by hauto l:on use:adequacy.
+  have h1 : SN (subst_PTm (fun _ : fin n => (PBot : PTm 0)) a)by hauto l:on use:adequacy.
+  move {h}. hauto l:on use:sn_unmorphing.
+Qed.
+
+Lemma SemWt_Univ n Γ (A : PTm n) i  :
+  Γ ⊨ A ∈ PUniv i <->
+  forall ρ, ρ_ok Γ ρ -> exists S, ⟦ subst_PTm ρ A ⟧ i ↘ S.
+Proof.
+  rewrite /SemWt.
+  split.
+  - hauto lq:on rew:off use:InterpUniv_Univ_inv.
+  - move => /[swap] ρ /[apply].
+    move => [PA hPA].
+    exists (S i). eexists.
+    split.
+    + simp InterpUniv. apply InterpExt_Univ. lia.
+    + simpl. eauto.
+Qed.
+
+(* Structural laws for Semantic context wellformedness *)
+Lemma SemWff_nil : SemWff null.
+Proof. case. Qed.
+
+Lemma SemWff_cons n Γ (A : PTm n) i :
+    ⊨ Γ ->
+    Γ ⊨ A ∈ PUniv i ->
+    (* -------------- *)
+    ⊨ funcomp (ren_PTm shift) (scons A Γ).
+Proof.
+  move => h h0.
+  move => j. destruct j as [j|].
+  - move /(_ j) : h => [k hk].
+    exists k. change (PUniv k) with (ren_PTm shift (PUniv k : PTm n)).
+    eauto using weakening_Sem.
+  - hauto q:on use:weakening_Sem.
+Qed.
+
+(* Semantic typing rules *)
+Lemma ST_Var n Γ (i : fin n) :
+  ⊨ Γ ->
+  Γ ⊨ VarPTm i ∈ Γ i.
+Proof.
+  move /(_ i) => [j /SemWt_Univ h].
+  rewrite /SemWt => ρ /[dup] hρ {}/h [S hS].
+  exists j, S.
+  asimpl. firstorder.
+Qed.
+
+Lemma InterpUniv_Bind_nopf n p i (A : PTm n) B PA :
+  ⟦ A ⟧ i ↘ PA ->
+  (forall a, PA a -> exists PB, ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB) ->
+  ⟦ PBind p A B ⟧ i ↘ (BindSpace p PA (fun a PB => ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB)).
+Proof.
+  move => h0 h1. apply InterpUniv_Bind => //=.
+Qed.
+
+
+Lemma ST_Bind n Γ i j p (A : PTm n) (B : PTm (S n)) :
+  Γ ⊨ A ∈ PUniv i ->
+  funcomp (ren_PTm shift) (scons A Γ) ⊨ B ∈ PUniv j ->
+  Γ ⊨ PBind p A B ∈ PUniv (max i j).
+Proof.
+  move => /SemWt_Univ h0 /SemWt_Univ h1.
+  apply SemWt_Univ => ρ hρ.
+  move /h0 : (hρ){h0} => [S hS].
+  eexists => /=.
+  have ? : i <= Nat.max i j by lia.
+  apply InterpUniv_Bind_nopf; eauto.
+  - eauto using InterpUniv_cumulative.
+  - move => *. asimpl. hauto l:on use:InterpUniv_cumulative, ρ_ok_cons.
+Qed.
+
+Lemma ST_Abs n Γ (a : PTm (S n)) A B i :
+  Γ ⊨ PBind PPi A B ∈ (PUniv i) ->
+  funcomp (ren_PTm shift) (scons A Γ) ⊨ a ∈ B ->
+  Γ ⊨ PAbs a ∈ PBind PPi A B.
+Proof.
+  rename a into b.
+  move /SemWt_Univ => + hb ρ hρ.
+  move /(_ _ hρ) => [PPi hPPi].
+  exists i, PPi. split => //.
+  simpl in hPPi.
+  move /InterpUniv_Bind_inv_nopf : hPPi.
+  move => [PA [hPA [hTot ?]]]. subst=>/=.
+  move => a PB ha. asimpl => hPB.
+  move : ρ_ok_cons (hPA) (hρ) (ha). repeat move/[apply].
+  move /hb.
+  intros (m & PB0 & hPB0 & hPB0').
+  replace PB0 with PB in * by hauto l:on use:InterpUniv_Functional'.
+  apply : InterpUniv_back_clos; eauto.
+  apply N_β'. by asimpl.
+  move : ha hPA. clear. hauto q:on use:adequacy.
+Qed.
+
+Lemma ST_App n Γ (b a : PTm n) A B :
+  Γ ⊨ b ∈ PBind PPi A B ->
+  Γ ⊨ a ∈ A ->
+  Γ ⊨ PApp b a ∈ subst_PTm (scons a VarPTm) B.
+Proof.
+  move => hf hb ρ hρ.
+  move /(_ ρ hρ) : hf; intros (i & PPi & hPi & hf).
+  move /(_ ρ hρ) : hb; intros (j & PA & hPA & hb).
+  simpl in hPi.
+  move /InterpUniv_Bind_inv_nopf : hPi. intros (PA0 & hPA0 & hTot & ?). subst.
+  have ? : PA0 = PA by eauto using InterpUniv_Functional'. subst.
+  move  : hf (hb). move/[apply].
+  move : hTot hb. move/[apply].
+  asimpl. hauto lq:on.
+Qed.
+
+Lemma ST_Pair n Γ (a b : PTm n) A B i :
+  Γ ⊨ PBind PSig A B ∈ (PUniv i) ->
+  Γ ⊨ a ∈ A ->
+  Γ ⊨ b ∈ subst_PTm (scons a VarPTm) B ->
+  Γ ⊨ PPair a b ∈ PBind PSig A B.
+Proof.
+  move /SemWt_Univ => + ha hb ρ hρ.
+  move /(_ _ hρ) => [PPi hPPi].
+  exists i, PPi. split => //.
+  simpl in hPPi.
+  move /InterpUniv_Bind_inv_nopf : hPPi.
+  move => [PA [hPA [hTot ?]]]. subst=>/=.
+  rewrite /SumSpace. right.
+  exists (subst_PTm ρ a), (subst_PTm ρ b).
+  split.
+  - apply rtc_refl.
+  - move /ha : (hρ){ha}.
+    move => [m][PA0][h0]h1.
+    move /hb : (hρ){hb}.
+    move => [k][PB][h2]h3.
+    have ? : PA0 = PA by eauto using InterpUniv_Functional'. subst.
+    split => // PB0.
+    move : h2. asimpl => *.
+    have ? : PB0 = PB by eauto using InterpUniv_Functional'. by subst.
+Qed.
+
+Lemma N_Projs n p (a b : PTm n) :
+  rtc TRedSN a b ->
+  rtc TRedSN (PProj p a) (PProj p b).
+Proof. induction 1; hauto lq:on ctrs:rtc, TRedSN. Qed.
+
+Lemma ST_Proj1 n Γ (a : PTm n) A B :
+  Γ ⊨ a ∈ PBind PSig A B ->
+  Γ ⊨ PProj PL a ∈ A.
+Proof.
+  move => h ρ /[dup]hρ {}/h [m][PA][/= /InterpUniv_Bind_inv_nopf h0]h1.
+  move : h0 => [S][h2][h3]?. subst.
+  move : h1 => /=.
+  rewrite /SumSpace.
+  case.
+  - move => [v [h0 h1]].
+    have {}h0 : rtc TRedSN (PProj PL (subst_PTm ρ a)) (PProj PL v) by hauto lq:on use:N_Projs.
+    have {}h1 : SNe (PProj PL v) by hauto lq:on ctrs:SNe.
+    hauto q:on use:InterpUniv_back_closs,adequacy.
+  - move => [a0 [b0 [h4 [h5 h6]]]].
+    exists m, S. split => //=.
+    have {}h4 : rtc TRedSN (PProj PL (subst_PTm ρ a)) (PProj PL (PPair a0 b0)) by eauto using N_Projs.
+    have ? : rtc TRedSN (PProj PL (PPair a0 b0)) a0 by hauto q:on ctrs:rtc, TRedSN use:adequacy.
+    have : rtc TRedSN (PProj PL (subst_PTm ρ a)) a0 by hauto q:on ctrs:rtc use:@relations.rtc_r.
+    move => h.
+    apply : InterpUniv_back_closs; eauto.
+Qed.
+
+Lemma ST_Proj2 n Γ (a : PTm n) A B :
+  Γ ⊨ a ∈ PBind PSig A B ->
+  Γ ⊨ PProj PR a ∈ subst_PTm (scons (PProj PL a) VarPTm) B.
+Proof.
+  move => h ρ hρ.
+  move : (hρ) => {}/h [m][PA][/= /InterpUniv_Bind_inv_nopf h0]h1.
+  move : h0 => [S][h2][h3]?. subst.
+  move : h1 => /=.
+  rewrite /SumSpace.
+  case.
+  - move => h.
+    move : h => [v [h0 h1]].
+    have hp : forall p, SNe (PProj p v) by hauto lq:on ctrs:SNe.
+    have hp' : forall p, rtc TRedSN (PProj p(subst_PTm ρ a)) (PProj p v) by eauto using N_Projs.
+    have hp0 := hp PL. have hp1 := hp PR => {hp}.
+    have hp0' := hp' PL. have hp1' := hp' PR => {hp'}.
+    have : S (PProj PL (subst_PTm ρ a)). apply : InterpUniv_back_closs; eauto. hauto q:on use:adequacy.
+    move /h3 => [PB]. asimpl => hPB.
+    do 2 eexists. split; eauto.
+    apply : InterpUniv_back_closs; eauto. hauto q:on use:adequacy.
+  - move => [a0 [b0 [h4 [h5 h6]]]].
+    have h3_dup := h3.
+    specialize h3 with (1 := h5).
+    move : h3 => [PB hPB].
+    have hr : forall p, rtc TRedSN (PProj p (subst_PTm ρ a)) (PProj p (PPair a0 b0)) by hauto l:on use: N_Projs.
+    have hSN : SN a0 by move : h5 h2; clear; hauto q:on use:adequacy.
+    have hSN' : SN b0 by hauto q:on use:adequacy.
+    have hrl : TRedSN (PProj PL (PPair a0 b0)) a0 by hauto lq:on ctrs:TRedSN.
+    have hrr : TRedSN (PProj PR (PPair a0 b0)) b0 by hauto lq:on ctrs:TRedSN.
+    exists m, PB.
+    asimpl. split.
+    + have hr' : rtc TRedSN (PProj PL (subst_PTm ρ a)) a0 by hauto l:on use:@relations.rtc_r.
+      have : S (PProj PL (subst_PTm ρ a)) by hauto lq:on use:InterpUniv_back_closs.
+      move => {}/h3_dup.
+      move => [PB0]. asimpl => hPB0.
+      suff : PB = PB0 by congruence.
+      move : hPB. asimpl => hPB.
+      suff : DJoin.R (subst_PTm (scons (PProj PL (subst_PTm ρ a)) ρ) B) (subst_PTm (scons a0 ρ) B).
+      move : InterpUniv_Join hPB0 hPB; repeat move/[apply]. done.
+      suff : BJoin.R (subst_PTm (scons (PProj PL (subst_PTm ρ a)) ρ) B) (subst_PTm (scons a0 ρ) B)
+        by hauto q:on use:DJoin.FromBJoin.
+      have : BJoin.R (PApp (PAbs (subst_PTm (up_PTm_PTm ρ) B)) (PProj PL (subst_PTm ρ a)))
+               (subst_PTm (scons (PProj PL (subst_PTm ρ a)) ρ) B).
+      eexists. split. apply relations.rtc_once. apply RRed.AppAbs.
+      asimpl. apply rtc_refl.
+      have /BJoin.symmetric : BJoin.R (PApp (PAbs (subst_PTm (up_PTm_PTm ρ)B)) a0)
+                                (subst_PTm (scons a0 ρ) B).
+      eexists. split. apply relations.rtc_once. apply RRed.AppAbs.
+      asimpl. apply rtc_refl.
+      suff : BJoin.R (PApp (PAbs (subst_PTm (up_PTm_PTm ρ) B)) (PProj PL (subst_PTm ρ a)))
+               (PApp (PAbs (subst_PTm (up_PTm_PTm ρ)B)) a0) by eauto using BJoin.transitive, BJoin.symmetric.
+      apply BJoin.AppCong. apply BJoin.refl.
+      move /RReds.FromRedSNs : hr'.
+      hauto lq:on ctrs:rtc unfold:BJoin.R.
+    + hauto lq:on use:@relations.rtc_r, InterpUniv_back_closs.
+Qed.
+
+Lemma ST_Conv n Γ (a : PTm n) A B i :
+  Γ ⊨ a ∈ A ->
+  Γ ⊨ B ∈ PUniv i ->
+  DJoin.R A B ->
+  Γ ⊨ a ∈ B.
+Proof.
+  move => ha /SemWt_Univ h h0.
+  move => ρ hρ.
+  have {}h0 : DJoin.R (subst_PTm ρ A) (subst_PTm ρ B) by eauto using DJoin.substing.
+  move /ha : (hρ){ha} => [m [PA [h1 h2]]].
+  move /h : (hρ){h} => [S hS].
+  have ? : PA = S by eauto using InterpUniv_Join'. subst.
+  eauto.
+Qed.