From f3718707f2b4668ee61ec48531a6576dbf29b123 Mon Sep 17 00:00:00 2001
From: Yiyun Liu <liuyiyun@seas.upenn.edu>
Date: Mon, 20 Jan 2025 13:56:40 -0500
Subject: [PATCH] Add constants to the reduction semantics

---
 syntax.sig                   |   6 +-
 theories/Autosubst2/syntax.v |  29 +++---
 theories/fp_red.v            | 169 ++++++++++++-----------------------
 3 files changed, 74 insertions(+), 130 deletions(-)

diff --git a/syntax.sig b/syntax.sig
index 4549f7f..d268994 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -3,14 +3,14 @@ Tm(VarTm) : Type
 PTag : Type
 TTag : Type
 
-TPi : TTag
-TSig : TTag
 PL : PTag
 PR : PTag
+TPi : TTag
+TSig : TTag
 Abs : (bind Tm in Tm) -> Tm
 App : Tm -> Tm -> Tm
 Pair : Tm -> Tm -> Tm
 Proj : PTag -> Tm -> Tm
 TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
-Bot : Tm
+Const : TTag -> Tm
 Univ : nat -> Tm
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index b972476..05fb668 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -40,7 +40,7 @@ Inductive Tm (n_Tm : nat) : Type :=
   | Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
   | Proj : PTag -> Tm n_Tm -> Tm n_Tm
   | TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
-  | Bot : Tm n_Tm
+  | Const : TTag -> Tm n_Tm
   | Univ : nat -> Tm n_Tm.
 
 Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
@@ -83,9 +83,10 @@ exact (eq_trans
          (ap (fun x => TBind m_Tm t0 t1 x) H2)).
 Qed.
 
-Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
+Lemma congr_Const {m_Tm : nat} {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
+  Const m_Tm s0 = Const m_Tm t0.
 Proof.
-exact (eq_refl).
+exact (eq_trans eq_refl (ap (fun x => Const m_Tm x) H0)).
 Qed.
 
 Lemma congr_Univ {m_Tm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
@@ -116,7 +117,7 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
   | Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
   | TBind _ s0 s1 s2 =>
       TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
-  | Bot _ => Bot n_Tm
+  | Const _ s0 => Const n_Tm s0
   | Univ _ s0 => Univ n_Tm s0
   end.
 
@@ -143,7 +144,7 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
   | Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
   | TBind _ s0 s1 s2 =>
       TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
-  | Bot _ => Bot n_Tm
+  | Const _ s0 => Const n_Tm s0
   | Univ _ s0 => Univ n_Tm s0
   end.
 
@@ -183,7 +184,7 @@ subst_Tm sigma_Tm s = s :=
   | TBind _ s0 s1 s2 =>
       congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
         (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -227,7 +228,7 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
       congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
         (extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
            (upExtRen_Tm_Tm _ _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -272,7 +273,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
       congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
         (ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
            s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -317,7 +318,7 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
       congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
         (compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
            (upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -373,7 +374,7 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
         (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
         (compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
            (up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -450,7 +451,7 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
         (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
         (compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
            (up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -528,7 +529,7 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
         (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
         (compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
            (up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -644,7 +645,7 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
       congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
         (rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
            (rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
-  | Bot _ => congr_Bot
+  | Const _ s0 => congr_Const (eq_refl s0)
   | Univ _ s0 => congr_Univ (eq_refl s0)
   end.
 
@@ -846,7 +847,7 @@ Arguments VarTm {n_Tm}.
 
 Arguments Univ {n_Tm}.
 
-Arguments Bot {n_Tm}.
+Arguments Const {n_Tm}.
 
 Arguments TBind {n_Tm}.
 
diff --git a/theories/fp_red.v b/theories/fp_red.v
index d2fbdf6..30f3354 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -69,9 +69,8 @@ Module Par.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  (* Bot is useful for making the prov function computable  *)
-  | BotCong :
-    R Bot Bot
+  | ConstCong k :
+    R (Const k) (Const k)
   | UnivCong i :
     R (Univ i) (Univ i).
 
@@ -212,7 +211,7 @@ Module Par.
       move : ihB => [c0 [? ?]]. subst.
       eexists. split. by apply BindCong; eauto.
       by asimpl.
-    - move => n n0 ξ []//=. hauto l:on.
+    - move => n k m ξ []//=. hauto l:on.
     - move => n i n0 ξ []//=. hauto l:on.
   Qed.
 
@@ -275,10 +274,10 @@ Module Pars.
 
 End Pars.
 
-Definition var_or_bot {n} (a : Tm n) :=
+Definition var_or_const {n} (a : Tm n) :=
   match a with
   | VarTm _ => true
-  | Bot => true
+  | Const _ => true
   | _ => false
   end.
 
@@ -324,8 +323,8 @@ Module RPar.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | BotCong :
-    R Bot Bot
+  | ConstCong k :
+    R (Const k) (Const k)
   | UnivCong i :
     R (Univ i) (Univ i).
 
@@ -571,8 +570,8 @@ Module RPar'.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | BotCong :
-    R Bot Bot
+  | ConstCong k :
+    R (Const k) (Const k)
   | UnivCong i :
     R (Univ i) (Univ i).
 
@@ -656,16 +655,16 @@ Module RPar'.
     qauto l:on ctrs:R inv:option.
   Qed.
 
-  Lemma var_or_bot_imp {n} (a b : Tm n) :
-    var_or_bot a ->
-    a = b -> ~~ var_or_bot b -> False.
+  Lemma var_or_const_imp {n} (a b : Tm n) :
+    var_or_const a ->
+    a = b -> ~~ var_or_const b -> False.
   Proof.
     hauto lq:on inv:Tm.
   Qed.
 
-  Lemma var_or_bot_up n m (ρ : fin n -> Tm m) :
-    (forall i, var_or_bot (ρ i)) ->
-    (forall i, var_or_bot (up_Tm_Tm ρ i)).
+  Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
+    (forall i, var_or_const (ρ i)) ->
+    (forall i, var_or_const (up_Tm_Tm ρ i)).
   Proof.
     move => h /= [i|].
     - asimpl.
@@ -676,10 +675,10 @@ Module RPar'.
     - sfirstorder.
   Qed.
 
-  Local Ltac antiimp := qauto l:on use:var_or_bot_imp.
+  Local Ltac antiimp := qauto l:on use:var_or_const_imp.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_bot (ρ i)) ->
+    (forall i, var_or_const (ρ i)) ->
     R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E : (subst_Tm ρ a) => u hρ h.
@@ -690,7 +689,7 @@ Module RPar'.
       case : c => //=; first by antiimp.
       move => c [?]. subst.
       spec_refl.
-      have /var_or_bot_up hρ' := hρ.
+      have /var_or_const_up hρ' := hρ.
       move : iha hρ' => /[apply] iha.
       move : ihb hρ => /[apply] ihb.
       spec_refl.
@@ -714,7 +713,7 @@ Module RPar'.
       hauto l:on.
     - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
       move => t [*]. subst.
-      have /var_or_bot_up {}/iha := hρ => iha.
+      have /var_or_const_up {}/iha := hρ => iha.
       spec_refl.
       move  :iha => [b0 [? ?]]. subst.
       eexists. split. by apply AbsCong; eauto.
@@ -750,7 +749,7 @@ Module RPar'.
                first by antiimp.
       move => ? t t0 [*]. subst.
       have {}/iha := (hρ) => iha.
-      have /var_or_bot_up {}/ihB := (hρ) => ihB.
+      have /var_or_const_up {}/ihB := (hρ) => ihB.
       spec_refl.
       move : iha => [b0 [? ?]].
       move : ihB => [c0 [? ?]]. subst.
@@ -894,8 +893,8 @@ Module EPar.
     R A0 A1 ->
     R B0 B1 ->
     R (TBind p A0 B0) (TBind p A1 B1)
-  | BotCong :
-    R Bot Bot
+  | ConstCong k :
+    R (Const k) (Const k)
   | UnivCong i :
     R (Univ i) (Univ i).
 
@@ -1070,7 +1069,7 @@ Module RPars.
   Proof. hauto lq:on use:morphing inv:option. Qed.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_bot (ρ i)) ->
+    (forall i, var_or_const (ρ i)) ->
     rtc RPar.R (subst_Tm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E  :(subst_Tm ρ a) => u hρ h.
@@ -1157,7 +1156,7 @@ Module RPars'.
   Proof. hauto lq:on use:morphing inv:option. Qed.
 
   Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
-    (forall i, var_or_bot (ρ i)) ->
+    (forall i, var_or_const (ρ i)) ->
     rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b.
   Proof.
     move E  :(subst_Tm ρ a) => u hρ h.
@@ -1444,15 +1443,15 @@ Proof.
   - hauto l:on ctrs:OExp.R.
 Qed.
 
-Lemma Bot_EPar' n (u : Tm n) :
-  EPar.R Bot u ->
-  rtc OExp.R Bot u.
-  move E : Bot => t h.
-  move : E. elim : n t u /h => //=.
-  - move => n a0 a1 h ih ?. subst.
+Lemma Const_EPar' n k (u : Tm n) :
+  EPar.R (Const k) u ->
+  rtc OExp.R (Const k) u.
+  move E : (Const k) => t h.
+  move : k E. elim : n t u /h => //=.
+  - move => n a0 a1 h ih k ?. subst.
     specialize ih with (1 := eq_refl).
     hauto lq:on ctrs:OExp.R use:rtc_r.
-  - move => n a0 a1 h ih ?. subst.
+  - move => n a0 a1 h ih k ?. subst.
     specialize ih with (1 := eq_refl).
     hauto lq:on ctrs:OExp.R use:rtc_r.
   - hauto l:on ctrs:OExp.R.
@@ -1519,7 +1518,7 @@ Proof.
     move : OExp.commutativity0 h2; repeat move/[apply].
     move => [d h].
     exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
-  - qauto use:Bot_EPar', EPar.refl.
+  - qauto use:Const_EPar', EPar.refl.
   - qauto use:Univ_EPar', EPar.refl.
 Qed.
 
@@ -1536,7 +1535,7 @@ Function tstar {n} (a : Tm n) :=
   | Proj p (Abs a) => (Abs (Proj p (tstar a)))
   | Proj p a => Proj p (tstar a)
   | TBind p a b => TBind p (tstar a) (tstar b)
-  | Bot => Bot
+  | Const k => Const k
   | Univ i => Univ i
   end.
 
@@ -1568,7 +1567,7 @@ Function tstar' {n} (a : Tm n) :=
   | Proj p (Pair a b) => if p is PL then (tstar' a) else (tstar' b)
   | Proj p a => Proj p (tstar' a)
   | TBind p a b => TBind p (tstar' a) (tstar' b)
-  | Bot => Bot
+  | Const k => Const k
   | Univ i => Univ i
   end.
 
@@ -1616,63 +1615,6 @@ Proof.
   sfirstorder use:relations.diamond_confluent, EPar_diamond.
 Qed.
 
-Fixpoint depth_tm {n} (a : Tm n) :=
-  match a with
-  | VarTm _ => 1
-  | TBind _ A B => 1 + max (depth_tm A) (depth_tm B)
-  | Abs a => 1 + depth_tm a
-  | App a b => 1 + max (depth_tm a) (depth_tm b)
-  | Proj p a => 1 + depth_tm a
-  | Pair a b => 1 + max (depth_tm a) (depth_tm b)
-  | Bot => 1
-  | Univ i => 1
-  end.
-
-Lemma depth_ren n m (ξ: fin n -> fin m) a :
-  depth_tm a = depth_tm (ren_Tm ξ a).
-Proof.
-  move : m ξ. elim : n / a; scongruence.
-Qed.
-
-Lemma depth_subst n m (ρ : fin n -> Tm m) a :
-  (forall i, depth_tm (ρ i) = 1) ->
-  depth_tm a = depth_tm (subst_Tm ρ a).
-Proof.
-  move : m ρ. elim : n / a.
-  - sfirstorder.
-  - move => n a iha m ρ hρ.
-    simpl.
-    f_equal. apply iha.
-    destruct i as [i|].
-    + simpl.
-      by rewrite -depth_ren.
-    + by simpl.
-  - hauto lq:on rew:off.
-  - hauto lq:on rew:off.
-  - hauto lq:on rew:off.
-  - move => n p a iha b ihb m ρ hρ.
-    simpl. f_equal.
-    f_equal.
-    by apply iha.
-    apply ihb.
-    destruct i as [i|].
-    + simpl.
-      by rewrite -depth_ren.
-    + by simpl.
-  - sfirstorder.
-  - sfirstorder.
-Qed.
-
-Lemma depth_subst_bool n (a : Tm (S n)) :
-  depth_tm a = depth_tm (subst_Tm (scons Bot VarTm) a).
-Proof.
-  apply depth_subst.
-  destruct i as [i|] => //=.
-Qed.
-
-Local Ltac prov_tac := sfirstorder use:depth_ren.
-Local Ltac extract_tac := rewrite -?depth_subst_bool;hauto use:depth_subst_bool.
-
 Definition prov_bind {n} p0 A0 B0 (a : Tm n)  :=
   match a with
   | TBind p A B => p = p0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
@@ -1704,8 +1646,8 @@ Inductive prov {n} : Tm n -> Tm n -> Prop :=
 | P_Proj h p a :
   prov h a ->
   prov h (Proj p a)
-| P_Bot  :
-  prov Bot Bot
+| P_Const k  :
+  prov (Const k) (Const k)
 | P_Var i :
   prov (VarTm i) (VarTm i)
 | P_Univ i :
@@ -1789,7 +1731,7 @@ Proof.
   split.
   move => h. constructor. move => b. asimpl. by constructor.
   inversion 1; subst.
-  specialize H2 with (b := Bot).
+  specialize H2 with (b := Const TPi).
   move : H2. asimpl. inversion 1; subst. done.
 Qed.
 
@@ -1845,11 +1787,11 @@ Qed.
 Fixpoint extract {n} (a : Tm n) : Tm n :=
   match a with
   | TBind p A B => TBind p A B
-  | Abs a => subst_Tm (scons Bot VarTm) (extract a)
+  | Abs a => subst_Tm (scons (Const TPi) VarTm) (extract a)
   | App a b => extract a
   | Pair a b => extract a
   | Proj p a => extract a
-  | Bot => Bot
+  | Const k => Const k
   | VarTm i => VarTm i
   | Univ i => Univ i
   end.
@@ -1886,7 +1828,7 @@ Proof.
 Qed.
 
 Lemma ren_subst_bot n (a : Tm (S n)) :
-  extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
+  extract (subst_Tm (scons (Const TPi) VarTm) a) = subst_Tm (scons (Const TPi) VarTm) (extract a).
 Proof.
   apply ren_morphing. destruct i as [i|] => //=.
 Qed.
@@ -1896,7 +1838,7 @@ Definition prov_extract_spec {n} u (a : Tm n) :=
   | TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
   | Univ i => extract a = Univ i
   | VarTm i => extract a = VarTm i
-  | Bot => extract a = Bot
+  | (Const TPi) => extract a = (Const TPi)
   | _ => True
   end.
 
@@ -1909,22 +1851,22 @@ Proof.
   - move => h a ha ih.
     case : h ha ih => //=.
     + move => i ha ih.
-      move /(_ Bot) in ih.
+      move /(_ (Const TPi)) in ih.
       rewrite -ih.
       by rewrite ren_subst_bot.
     + move => p A B h ih.
-      move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
+      move /(_ (Const TPi)) : ih => [A0][B0][h0][h1]h2.
       rewrite ren_subst_bot in h0.
       rewrite h0.
       eauto.
-    + move => _ /(_ Bot).
+    + move => p _ /(_ (Const TPi)).
       by rewrite ren_subst_bot.
-    + move => i h /(_ Bot).
+    + move => i h /(_ (Const TPi)).
       by rewrite ren_subst_bot => ->.
   - hauto lq:on.
   - hauto lq:on.
   - hauto lq:on.
-  - sfirstorder.
+  - case => //=.
   - sfirstorder.
   - sfirstorder.
 Qed.
@@ -2400,23 +2342,24 @@ Fixpoint ne {n} (a : Tm n) :=
   match a with
   | VarTm i => true
   | TBind _ A B => false
-  | Bot => true
   | App a b => ne a && nf b
   | Abs a => false
   | Univ _ => false
   | Proj _ a => ne a
   | Pair _ _ => false
+  | Const _ => false
   end
 with nf {n} (a : Tm n) :=
   match a with
   | VarTm i => true
   | TBind _ A B => nf A && nf B
-  | Bot => true
+  | (Const TPi) => true
   | App a b => ne a && nf b
   | Abs a => nf a
   | Univ _ => true
   | Proj _ a => ne a
   | Pair a b => nf a && nf b
+  | Const _ => true
 end.
 
 Lemma ne_nf n a : @ne n a -> nf a.
@@ -2482,15 +2425,15 @@ Create HintDb nfne.
 #[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne.
 
 Lemma ne_nf_antiren n m (a : Tm n) (ρ : fin n -> Tm m) :
-  (forall i, var_or_bot (ρ i)) ->
+  (forall i, var_or_const (ρ i)) ->
   (ne (subst_Tm ρ a) -> ne a) /\ (nf (subst_Tm ρ a) -> nf a).
 Proof.
   move : m ρ. elim : n / a => //;
-   hauto b:on drew:off use:RPar.var_or_bot_up.
+   hauto b:on drew:off use:RPar.var_or_const_up.
 Qed.
 
 Lemma wn_antirenaming n m a (ρ : fin n -> Tm m) :
-  (forall i, var_or_bot (ρ i)) ->
+  (forall i, var_or_const (ρ i)) ->
   wn (subst_Tm ρ a) -> wn a.
 Proof.
   rewrite /wn => hρ.
@@ -2503,10 +2446,10 @@ Proof.
 Qed.
 
 Lemma ext_wn n (a : Tm n) :
-    wn (App a Bot) ->
+    wn (App a (Const TPi)) ->
     wn a.
 Proof.
-  move E : (App a Bot) => a0 [v [hr hv]].
+  move E : (App a (Const TPi)) => a0 [v [hr hv]].
   move : a E.
   move : hv.
   elim : a0 v / hr.
@@ -2517,9 +2460,9 @@ Proof.
     case : a0 hr0=>// => b0 b1.
     elim /RPar'.inv=>// _.
     + move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
-      have ? : b3 = Bot by hauto lq:on inv:RPar'.R. subst.
+      have ? : b3 = (Const TPi) by hauto lq:on inv:RPar'.R. subst.
       suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
-      have : wn (subst_Tm (scons Bot VarTm) a3) by sfirstorder.
+      have : wn (subst_Tm (scons (Const TPi) VarTm) a3) by sfirstorder.
       move => h. apply wn_abs.
       move : h. apply wn_antirenaming.
       hauto lq:on rew:off inv:option.