From 55a9ad9d9396063bd3e6b497df19247b76865343 Mon Sep 17 00:00:00 2001
From: Junyoung/Clare Jang <jjc9310@gmail.com>
Date: Wed, 4 Sep 2024 18:20:40 -0400
Subject: [PATCH] Use consistent naming

---
 theories/Algorithmic/Subtyping/Definitions.v  |  25 +-
 theories/Algorithmic/Subtyping/Lemmas.v       |  64 ++--
 theories/Algorithmic/Typing/Lemmas.v          |  33 +-
 theories/Core/Completeness/ContextCases.v     |   4 +-
 theories/Core/Completeness/FunctionCases.v    |  36 +-
 .../LogicalRelation/Definitions.v             |  20 +-
 .../Completeness/LogicalRelation/Lemmas.v     |  16 +-
 theories/Core/Completeness/NatCases.v         | 222 +++++++-----
 .../Core/Completeness/SubstitutionCases.v     |   4 +-
 theories/Core/Completeness/SubtypingCases.v   |  23 +-
 .../Core/Completeness/TermStructureCases.v    |  24 +-
 theories/Core/Completeness/UniverseCases.v    |   8 +-
 theories/Core/Semantic/Domain.v               |  20 +-
 .../Core/Semantic/Evaluation/Definitions.v    |  93 ++---
 theories/Core/Semantic/Evaluation/Lemmas.v    | 102 +++---
 theories/Core/Semantic/Evaluation/Tactics.v   |   6 +-
 theories/Core/Semantic/NbE.v                  |  30 +-
 theories/Core/Semantic/PER/Definitions.v      |  61 ++--
 theories/Core/Semantic/PER/Lemmas.v           | 225 ++++++------
 theories/Core/Semantic/Readback/Definitions.v |  22 +-
 theories/Core/Semantic/Readback/Lemmas.v      |  70 ++--
 theories/Core/Semantic/Realizability.v        |  26 +-
 theories/Core/Soundness/FunctionCases.v       |  47 +--
 theories/Core/Soundness/LogicalRelation.v     |   2 +-
 .../Core/Soundness/LogicalRelation/Core.v     |   2 +-
 .../Soundness/LogicalRelation/CoreLemmas.v    | 325 +++++++++---------
 .../Soundness/LogicalRelation/CoreTactics.v   |  13 +
 .../Soundness/LogicalRelation/Definitions.v   | 255 +++++++-------
 .../Core/Soundness/LogicalRelation/Lemmas.v   | 213 ++++++------
 theories/Core/Soundness/NatCases.v            |  89 +++--
 theories/Core/Soundness/Realizability.v       | 159 ++++-----
 theories/Core/Soundness/SubstitutionCases.v   |   7 +-
 theories/Core/Soundness/TermStructureCases.v  |   3 -
 .../Core/Soundness/Weakening/Definition.v     |   2 +-
 theories/Core/Soundness/Weakening/Lemmas.v    |   6 +-
 theories/Extraction/TypeCheck.v               |  38 +-
 36 files changed, 1191 insertions(+), 1104 deletions(-)

diff --git a/theories/Algorithmic/Subtyping/Definitions.v b/theories/Algorithmic/Subtyping/Definitions.v
index 3013865f..78262e0b 100644
--- a/theories/Algorithmic/Subtyping/Definitions.v
+++ b/theories/Algorithmic/Subtyping/Definitions.v
@@ -13,10 +13,10 @@ Definition not_univ_pi (A : nf) : Prop :=
   end.
 
 Inductive alg_subtyping_nf : nf -> nf -> Prop :=
-| asnf_refl : forall M N,
-    not_univ_pi M ->
-    M = N ->
-    {{ ⊢anf M ⊆ N }}
+| asnf_refl : forall A A',
+    not_univ_pi A ->
+    A = A' ->
+    {{ ⊢anf A ⊆ A' }}
 | asnf_univ : forall i j,
     i <= j ->
     {{ ⊢anf Type@i ⊆ Type@j }}
@@ -24,16 +24,15 @@ Inductive alg_subtyping_nf : nf -> nf -> Prop :=
     A = A' ->
     {{ ⊢anf B ⊆ B' }} ->
     {{ ⊢anf Π A B ⊆ Π A' B' }}
-where "⊢anf M ⊆ N" := (alg_subtyping_nf M N) (in custom judg) : type_scope.
-
+where "⊢anf A ⊆ A'" := (alg_subtyping_nf A A') (in custom judg) : type_scope.
 
 Inductive alg_subtyping : ctx -> typ -> typ -> Prop :=
-| alg_subtyp_run : forall Γ M N A B,
-    nbe_ty Γ M A ->
-    nbe_ty Γ N B ->
-    {{ ⊢anf A ⊆ B }} ->
-    {{ Γ ⊢a M ⊆ N }}
-where "Γ ⊢a M ⊆ N" := (alg_subtyping Γ M N) (in custom judg) : type_scope.
+| alg_subtyp_run : forall Γ A B A' B',
+    nbe_ty Γ A A' ->
+    nbe_ty Γ B B' ->
+    {{ ⊢anf A' ⊆ B' }} ->
+    {{ Γ ⊢a A ⊆ B }}
+where "Γ ⊢a A ⊆ B" := (alg_subtyping Γ A B) (in custom judg) : type_scope.
 
 #[export]
-  Hint Constructors alg_subtyping_nf alg_subtyping: mcltt.
+Hint Constructors alg_subtyping_nf alg_subtyping: mcltt.
diff --git a/theories/Algorithmic/Subtyping/Lemmas.v b/theories/Algorithmic/Subtyping/Lemmas.v
index 421a4d65..fb223de8 100644
--- a/theories/Algorithmic/Subtyping/Lemmas.v
+++ b/theories/Algorithmic/Subtyping/Lemmas.v
@@ -8,17 +8,17 @@ Import Syntax_Notations.
 #[local]
 Ltac apply_subtyping :=
   repeat match goal with
-    | H : {{ ~?Γ ⊢ ~?A : ~?M }},
-        H1 : {{ ~?Γ ⊢ ~?M ⊆ ~?N }} |- _ =>
-        assert {{ Γ ⊢ A : N }} by mauto; clear H
+    | H : {{ ~?Γ ⊢ ~?M : ~?A }},
+        H1 : {{ ~?Γ ⊢ ~?A ⊆ ~?B }} |- _ =>
+        assert {{ Γ ⊢ M : B }} by mauto; clear H
     end.
 
-Lemma alg_subtyping_nf_sound : forall M N,
-    {{ ⊢anf M ⊆ N }} ->
+Lemma alg_subtyping_nf_sound : forall A B,
+    {{ ⊢anf A ⊆ B }} ->
     forall Γ i,
-      {{ Γ ⊢ M : Type@i }} ->
-      {{ Γ ⊢ N : Type@i }} ->
-      {{ Γ ⊢ M ⊆ N }}.
+      {{ Γ ⊢ A : Type@i }} ->
+      {{ Γ ⊢ B : Type@i }} ->
+      {{ Γ ⊢ A ⊆ B }}.
 Proof.
   induction 1; intros; subst; simpl in *.
   - eapply wf_subtyp_refl'; mauto.
@@ -27,7 +27,7 @@ Proof.
     destruct_all.
     gen_presups.
     repeat match goal with
-           | H : {{ ~?Γ ⊢ ~?M ⊆ ~?N }}, H1: {{ ⊢ ~?Γ , ~_ }} |- _ =>
+           | H : {{ ~?Γ ⊢ ~?A ⊆ ~?B }}, H1: {{ ⊢ ~?Γ , ~_ }} |- _ =>
                pose proof (wf_subtyp_univ_weaken _ _ _ _ H H1);
                fail_if_dup
            end.
@@ -38,12 +38,12 @@ Proof.
     mauto 3.
 Qed.
 
-Lemma alg_subtyping_nf_trans : forall M1 M0 M2,
-    {{ ⊢anf M0 ⊆ M1 }} ->
-    {{ ⊢anf M1 ⊆ M2 }} ->
-    {{ ⊢anf M0 ⊆ M2 }}.
+Lemma alg_subtyping_nf_trans : forall A0 A1 A2,
+    {{ ⊢anf A0 ⊆ A1 }} ->
+    {{ ⊢anf A1 ⊆ A2 }} ->
+    {{ ⊢anf A0 ⊆ A2 }}.
 Proof.
-  intros * H1; gen M2.
+  intros * H1; gen A2.
   induction H1; subst; intros ? H2;
     dependent destruction H2;
     simpl in *;
@@ -53,20 +53,20 @@ Proof.
   constructor; lia.
 Qed.
 
-Lemma alg_subtyping_nf_refl : forall M,
-    {{ ⊢anf M ⊆ M }}.
+Lemma alg_subtyping_nf_refl : forall A,
+    {{ ⊢anf A ⊆ A }}.
 Proof.
-  intro M; induction M;
+  induction A;
     solve [constructor; simpl; trivial].
 Qed.
 
 #[local]
 Hint Resolve alg_subtyping_nf_trans alg_subtyping_nf_refl : mcltt.
 
-Lemma alg_subtyping_trans : forall Γ M0 M1 M2,
-    {{ Γ ⊢a M0 ⊆ M1 }} ->
-    {{ Γ ⊢a M1 ⊆ M2 }} ->
-    {{ Γ ⊢a M0 ⊆ M2 }}.
+Lemma alg_subtyping_trans : forall Γ A0 A1 A2,
+    {{ Γ ⊢a A0 ⊆ A1 }} ->
+    {{ Γ ⊢a A1 ⊆ A2 }} ->
+    {{ Γ ⊢a A0 ⊆ A2 }}.
 Proof.
   intros. progressive_inversion.
   functional_nbe_rewrite_clear.
@@ -76,9 +76,9 @@ Qed.
 #[local]
 Hint Resolve alg_subtyping_trans : mcltt.
 
-Lemma alg_subtyping_complete : forall Γ M N,
-    {{ Γ ⊢ M ⊆ N }} ->
-    {{ Γ ⊢a M ⊆ N }}.
+Lemma alg_subtyping_complete : forall Γ A B,
+    {{ Γ ⊢ A ⊆ B }} ->
+    {{ Γ ⊢a A ⊆ B }}.
 Proof.
   induction 1; mauto.
   - apply completeness in H0 as [W [? ?]].
@@ -106,11 +106,11 @@ Proof.
     mauto 2.
 Qed.
 
-Lemma alg_subtyping_sound : forall Γ M N i,
-    {{ Γ ⊢a M ⊆ N }} ->
-    {{ Γ ⊢ M : Type@i }} ->
-    {{ Γ ⊢ N : Type@i }} ->
-    {{ Γ ⊢ M ⊆ N }}.
+Lemma alg_subtyping_sound : forall Γ A B i,
+    {{ Γ ⊢a A ⊆ B }} ->
+    {{ Γ ⊢ A : Type@i }} ->
+    {{ Γ ⊢ B : Type@i }} ->
+    {{ Γ ⊢ A ⊆ B }}.
 Proof.
   intros. destruct H.
   on_all_hyp: fun H => apply soundness in H.
@@ -118,8 +118,8 @@ Proof.
   on_all_hyp: fun H => apply nbe_type_to_nbe_ty in H.
   functional_nbe_rewrite_clear.
   gen_presups.
-  assert {{ Γ ⊢ A ⊆ B }} by mauto 3 using alg_subtyping_nf_sound.
-  transitivity A; [mauto |].
-  transitivity B; [eassumption |].
+  assert {{ Γ ⊢ A' ⊆ B' }} by mauto 3 using alg_subtyping_nf_sound.
+  transitivity A'; [mauto |].
+  transitivity B'; [eassumption |].
   mauto.
 Qed.
diff --git a/theories/Algorithmic/Typing/Lemmas.v b/theories/Algorithmic/Typing/Lemmas.v
index d96e4a68..14cd716f 100644
--- a/theories/Algorithmic/Typing/Lemmas.v
+++ b/theories/Algorithmic/Typing/Lemmas.v
@@ -125,7 +125,7 @@ Proof with (f_equiv; mautosolve 4).
     simplify_evals.
     dir_inversion_by_head read_typ; subst.
     functional_initial_env_rewrite_clear.
-    assert (initial_env {{{ Γ, ~(C : exp) }}} d{{{ p ↦ ⇑! a (length Γ) }}}) by mauto 3.
+    assert (initial_env {{{ Γ, ~(C : exp) }}} d{{{ ρ ↦ ⇑! a (length Γ) }}}) by mauto 3.
     assert (nbe_ty Γ A C) by mauto 3.
     assert (nbe_ty Γ C A0) by mauto 3.
     replace A0 with C by mauto 3.
@@ -148,9 +148,9 @@ Lemma alg_type_check_typ_implies_alg_type_infer_typ : forall {Γ A i},
     exists j, {{ Γ ⊢a A ⟹ Type@j }} /\ j <= i.
 Proof.
   intros * ? Hcheck.
-  inversion Hcheck as [? ? ? ? Hinfer Hsub]; subst.
-  inversion Hsub as [? ? ? ? ? Hnbe1 Hnbe2 Hnfsub]; subst.
-  replace A0 with A1 in * by (symmetry; mauto 3).
+  inversion Hcheck as [? A' ? ? Hinfer Hsub]; subst.
+  inversion Hsub as [? ? ? A'' ? Hnbe1 Hnbe2 Hnfsub]; subst.
+  replace A' with A'' in * by (symmetry; mauto 3).
   inversion Hnbe2; subst.
   dir_inversion_by_head eval_exp; subst.
   dir_inversion_by_head read_typ; subst.
@@ -169,9 +169,9 @@ Lemma alg_type_check_pi_implies_alg_type_infer_pi : forall {Γ M A B i},
 Proof.
   intros * ? ? Hcheck.
   assert ({{ Γ ⊢ A : Type@i }} /\ {{ Γ, A ⊢ B : Type@i }}) as [] by mauto 3.
-  inversion Hcheck as [? ? ? ? Hinfer Hsub]; subst.
-  inversion Hsub as [? ? ? ? C Hnbe1 Hnbe2 Hnfsub]; subst.
-  replace A0 with A1 in * by (symmetry; mauto 3).
+  inversion Hcheck as [? A' ? ? Hinfer Hsub]; subst.
+  inversion Hsub as [? ? ? A'' C Hnbe1 Hnbe2 Hnfsub]; subst.
+  replace A' with A'' in * by (symmetry; mauto 3).
   inversion Hnbe2; subst.
   dir_inversion_by_head eval_exp; subst.
   dir_inversion_by_head read_typ; subst.
@@ -181,18 +181,18 @@ Proof.
   dir_inversion_by_head eval_exp; subst.
   dir_inversion_by_head read_typ; subst.
   simpl in *.
-  assert {{ Γ ⊢ M : ~n{{{ Π A2 B0 }}} }} by mauto 3 using alg_type_infer_sound.
-  assert (exists j, {{ Γ ⊢ Π A2 B0 : Type@j }}) as [j] by (gen_presups; eauto 2).
-  assert ({{ Γ ⊢ A2 : Type@j }} /\ {{ Γ, ~(A2 : exp) ⊢ B0 : Type@j }}) as [] by mauto 3.
+  assert {{ Γ ⊢ M : ~n{{{ Π A0 B0 }}} }} by mauto 3 using alg_type_infer_sound.
+  assert (exists j, {{ Γ ⊢ Π A0 B0 : Type@j }}) as [j] by (gen_presups; eauto 2).
+  assert ({{ Γ ⊢ A0 : Type@j }} /\ {{ Γ, ~(A0 : exp) ⊢ B0 : Type@j }}) as [] by mauto 3.
   do 2 eexists.
-  assert (nbe_ty Γ A A2) by mauto 3.
-  assert {{ Γ ⊢ A ≈ A2 : Type@i }} by mauto 3 using soundness_ty'.
+  assert (nbe_ty Γ A A0) by mauto 3.
+  assert {{ Γ ⊢ A ≈ A0 : Type@i }} by mauto 3 using soundness_ty'.
   repeat split; mauto 3.
-  assert (initial_env {{{ Γ, A }}} d{{{ p ↦ ⇑! a (length Γ) }}}) by mauto 3.
+  assert (initial_env {{{ Γ, A }}} d{{{ ρ ↦ ⇑! a (length Γ) }}}) by mauto 3.
   assert (nbe_ty {{{ Γ, A }}} B B') by mauto 3.
-  assert (initial_env {{{ Γ, ~(A2 : exp) }}} d{{{ p ↦ ⇑! a0 (length Γ) }}}) by mauto 3.
-  assert (nbe_ty {{{ Γ, ~(A2 : exp) }}} B0 B0) by mauto 3.
-  assert {{ ⊢ Γ, A ≈ Γ, ~(A2 : exp) }} by mauto 3.
+  assert (initial_env {{{ Γ, ~(A0 : exp) }}} d{{{ ρ ↦ ⇑! a0 (length Γ) }}}) by mauto 3.
+  assert (nbe_ty {{{ Γ, ~(A0 : exp) }}} B0 B0) by mauto 3.
+  assert {{ ⊢ Γ, A ≈ Γ, ~(A0 : exp) }} by mauto 3.
   assert (nbe_ty {{{ Γ, A }}} B0 B0) by mauto 4 using ctxeq_nbe_ty_eq'.
   mauto 3.
 Qed.
@@ -312,4 +312,3 @@ Qed.
 
 #[export]
 Hint Resolve alg_type_infer_pi_complete : mcltt.
-
diff --git a/theories/Core/Completeness/ContextCases.v b/theories/Core/Completeness/ContextCases.v
index c83daf65..aa2a338c 100644
--- a/theories/Core/Completeness/ContextCases.v
+++ b/theories/Core/Completeness/ContextCases.v
@@ -25,9 +25,9 @@ Proof with intuition.
   handle_per_ctx_env_irrel.
   eexists.
   per_ctx_env_econstructor; eauto.
-  - instantiate (1 := fun p p' (equiv_p_p' : env_relΓ p p') m m' =>
+  - instantiate (1 := fun ρ ρ' (equiv_ρ_ρ' : env_relΓ ρ ρ') m m' =>
                         forall i R,
-                          rel_typ i A p A' p' R ->
+                          rel_typ i A ρ A' ρ' R ->
                           R m m').
     intros.
     (on_all_hyp: destruct_rel_by_assumption env_relΓ).
diff --git a/theories/Core/Completeness/FunctionCases.v b/theories/Core/Completeness/FunctionCases.v
index 3fa0ea4b..757729be 100644
--- a/theories/Core/Completeness/FunctionCases.v
+++ b/theories/Core/Completeness/FunctionCases.v
@@ -7,11 +7,11 @@ Import Domain_Notations.
 Lemma rel_exp_of_pi_inversion : forall {Γ M M' A B},
     {{ Γ ⊨ M ≈ M' : Π A B }} ->
     exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i,
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
     exists in_rel out_rel,
-      rel_typ i A p A p' in_rel /\
-        (forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_typ i B d{{{ p ↦ c }}} B d{{{ p' ↦ c' }}} (out_rel c c' equiv_c_c')) /\
-        rel_exp M p M' p'
+      rel_typ i A ρ A ρ' in_rel /\
+        (forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_typ i B d{{{ ρ ↦ c }}} B d{{{ ρ' ↦ c' }}} (out_rel c c' equiv_c_c')) /\
+        rel_exp M ρ M' ρ'
           (fun f f' : domain => forall (c c' : domain) (equiv_c_c' : in_rel c c'), rel_mod_app f c f' c' (out_rel c c' equiv_c_c')).
 Proof.
   intros * [env_relΓ].
@@ -26,11 +26,11 @@ Qed.
 
 Lemma rel_exp_of_pi : forall {Γ M M' A B},
     (exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i j,
-      forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
+      forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
       exists in_rel out_rel,
-        rel_typ i A p A p' in_rel /\
-          (forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_typ j B d{{{ p ↦ c }}} B d{{{ p' ↦ c' }}} (out_rel c c' equiv_c_c')) /\
-          rel_exp M p M' p'
+        rel_typ i A ρ A ρ' in_rel /\
+          (forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_typ j B d{{{ ρ ↦ c }}} B d{{{ ρ' ↦ c' }}} (out_rel c c' equiv_c_c')) /\
+          rel_exp M ρ M' ρ'
             (fun f f' : domain => forall (c c' : domain) (equiv_c_c' : in_rel c c'), rel_mod_app f c f' c' (out_rel c c' equiv_c_c'))) ->
     {{ Γ ⊨ M ≈ M' : Π A B }}.
 Proof.
@@ -57,9 +57,9 @@ Ltac eexists_rel_exp_of_pi :=
   eexists.
 
 #[local]
-Ltac extract_output_info_with p c p' c' env_rel :=
+Ltac extract_output_info_with ρ c ρ' c' env_rel :=
   let Hequiv := fresh "equiv" in
-  (assert (Hequiv : {{ Dom p ↦ c ≈ p' ↦ c' ∈ env_rel }}) by (apply_relation_equivalence; mauto 4);
+  (assert (Hequiv : {{ Dom ρ ↦ c ≈ ρ' ↦ c' ∈ env_rel }}) by (apply_relation_equivalence; mauto 4);
    apply_relation_equivalence;
    (on_all_hyp: fun H => destruct (H _ _ Hequiv));
    destruct_conjs;
@@ -130,7 +130,7 @@ Proof with mautosolve.
   eexists_rel_exp_of_typ.
   intros.
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
-  assert {{ Dom o' ≈ o' ∈ env_relΔ }} by (etransitivity; [symmetry |]; eassumption).
+  assert {{ Dom ρ'σ' ≈ ρ'σ' ∈ env_relΔ }} by (etransitivity; [symmetry |]; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΔ).
   destruct_by_head per_univ.
   handle_per_univ_elem_irrel.
@@ -139,7 +139,7 @@ Proof with mautosolve.
   per_univ_elem_econstructor; eauto.
   - eapply rel_exp_pi_core; eauto; try reflexivity.
     intros.
-    extract_output_info_with o c o' c' env_relΔA...
+    extract_output_info_with ρσ c ρ'σ' c' env_relΔA...
   - (* `reflexivity` does not work as (simple) unification fails for some unknown reason. *)
     apply Equivalence_Reflexive.
 Qed.
@@ -166,11 +166,11 @@ Proof with mautosolve.
   repeat split; [econstructor | | econstructor]; mauto.
   - eapply rel_exp_pi_core; eauto; try reflexivity.
     intros.
-    extract_output_info_with p c p' c' env_relΓA.
+    extract_output_info_with ρ c ρ' c' env_relΓA.
     econstructor; eauto.
     eexists...
   - intros.
-    extract_output_info_with p c p' c' env_relΓA.
+    extract_output_info_with ρ c ρ' c' env_relΓA.
     econstructor; mauto.
     intros.
     destruct_by_head rel_typ.
@@ -202,12 +202,12 @@ Proof with mautosolve.
     clear dependent c.
     clear dependent c'.
     intros.
-    extract_output_info_with o c o' c' env_relΔA.
+    extract_output_info_with ρσ c ρ'σ' c' env_relΔA.
     econstructor; eauto.
     eexists.
     eapply per_univ_elem_cumu_max_left...
   - intros ? **.
-    extract_output_info_with o c o' c' env_relΔA.
+    extract_output_info_with ρσ c ρ'σ' c' env_relΔA.
     econstructor; mauto.
     intros.
     destruct_by_head rel_typ.
@@ -228,7 +228,7 @@ Proof with intuition.
   handle_per_ctx_env_irrel.
   eexists_rel_exp.
   intros.
-  assert (equiv_p'_p' : env_relΓ p' p') by (etransitivity; [symmetry |]; eassumption).
+  assert (equiv_p'_p' : env_relΓ ρ' ρ') by (etransitivity; [symmetry |]; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   rename x2 into in_rel.
   destruct_by_head rel_typ.
@@ -290,7 +290,7 @@ Proof with mautosolve.
   destruct_by_head rel_exp.
   rename m into n.
   rename m' into n'.
-  extract_output_info_with p n p' n' env_relΓA.
+  extract_output_info_with ρ n ρ' n' env_relΓA.
   eexists.
   split; econstructor...
 Qed.
diff --git a/theories/Core/Completeness/LogicalRelation/Definitions.v b/theories/Core/Completeness/LogicalRelation/Definitions.v
index cf09a2cb..7ddc4b6e 100644
--- a/theories/Core/Completeness/LogicalRelation/Definitions.v
+++ b/theories/Core/Completeness/LogicalRelation/Definitions.v
@@ -3,8 +3,8 @@ From Mcltt Require Import Base.
 From Mcltt.Core Require Export PER.
 Import Domain_Notations.
 
-Inductive rel_exp M p M' p' (R : relation domain) : Prop :=
-| mk_rel_exp : forall m m', {{ ⟦ M ⟧ p ↘ m }} -> {{ ⟦ M' ⟧ p' ↘ m' }} -> {{ Dom m ≈ m' ∈ R }} -> rel_exp M p M' p' R.
+Inductive rel_exp M ρ M' ρ' (R : relation domain) : Prop :=
+| mk_rel_exp : forall m m', {{ ⟦ M ⟧ ρ ↘ m }} -> {{ ⟦ M' ⟧ ρ' ↘ m' }} -> {{ Dom m ≈ m' ∈ R }} -> rel_exp M ρ M' ρ' R.
 #[global]
 Arguments mk_rel_exp {_ _ _ _ _}.
 #[export]
@@ -12,9 +12,9 @@ Hint Constructors rel_exp : mcltt.
 
 Definition rel_exp_under_ctx Γ A M M' :=
   exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i,
-  forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
+  forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
   exists (elem_rel : relation domain),
-     rel_typ i A p A p' elem_rel /\ rel_exp M p M' p' elem_rel.
+     rel_typ i A ρ A ρ' elem_rel /\ rel_exp M ρ M' ρ' elem_rel.
 
 Definition valid_exp_under_ctx Γ A M := rel_exp_under_ctx Γ A M M.
 #[global]
@@ -24,8 +24,8 @@ Hint Transparent valid_exp_under_ctx : mcltt.
 #[export]
 Hint Unfold valid_exp_under_ctx : mcltt.
 
-Inductive rel_sub σ p σ' p' (R : relation env) : Prop :=
-| mk_rel_sub : forall o o', {{ ⟦ σ ⟧s p ↘ o }} -> {{ ⟦ σ' ⟧s p' ↘ o' }} -> {{ Dom o ≈ o' ∈ R }} -> rel_sub σ p σ' p' R.
+Inductive rel_sub σ ρ σ' ρ' (R : relation env) : Prop :=
+| mk_rel_sub : forall ρσ ρ'σ', {{ ⟦ σ ⟧s ρ ↘ ρσ }} -> {{ ⟦ σ' ⟧s ρ' ↘ ρ'σ' }} -> {{ Dom ρσ ≈ ρ'σ' ∈ R }} -> rel_sub σ ρ σ' ρ' R.
 #[global]
 Arguments mk_rel_sub {_ _ _ _ _ _}.
 #[export]
@@ -34,8 +34,8 @@ Hint Constructors rel_sub : mcltt.
 Definition rel_sub_under_ctx Γ Δ σ σ' :=
   exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }})
      env_rel' (_ : {{ EF Δ ≈ Δ ∈ per_ctx_env ↘ env_rel' }}),
-  forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-    rel_sub σ p σ' p' env_rel'.
+  forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+    rel_sub σ ρ σ' ρ' env_rel'.
 
 Definition valid_sub_under_ctx Γ Δ σ := rel_sub_under_ctx Γ Δ σ σ.
 #[global]
@@ -47,8 +47,8 @@ Hint Unfold valid_sub_under_ctx : mcltt.
 
 Definition subtyp_under_ctx Γ M M' :=
   exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i,
-  forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-    exists R R', rel_typ i M p M p' R /\ rel_typ i M' p M' p' R' /\ rel_exp M p M' p' (per_subtyp i).
+  forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+    exists R R', rel_typ i M ρ M ρ' R /\ rel_typ i M' ρ M' ρ' R' /\ rel_exp M ρ M' ρ' (per_subtyp i).
 
 Notation "⊨ Γ ≈ Γ'" := (per_ctx Γ Γ')  (in custom judg at level 80, Γ custom exp, Γ' custom exp).
 Notation "⊨ Γ" := (valid_ctx Γ) (in custom judg at level 80, Γ custom exp).
diff --git a/theories/Core/Completeness/LogicalRelation/Lemmas.v b/theories/Core/Completeness/LogicalRelation/Lemmas.v
index 4e3d5bca..d35c2f69 100644
--- a/theories/Core/Completeness/LogicalRelation/Lemmas.v
+++ b/theories/Core/Completeness/LogicalRelation/Lemmas.v
@@ -3,23 +3,23 @@ From Mcltt Require Import Base LibTactics.
 From Mcltt.Core.Completeness Require Import LogicalRelation.Definitions LogicalRelation.Tactics.
 Import Domain_Notations.
 
-Add Parametric Morphism M p M' p' : (rel_exp M p M' p')
+Add Parametric Morphism M ρ M' ρ' : (rel_exp M ρ M' ρ')
     with signature (@relation_equivalence domain) ==> iff as rel_exp_morphism.
 Proof.
   intros R R' HRR'.
   split; intros []; econstructor; intuition.
 Qed.
 
-Add Parametric Morphism σ p σ' p' : (rel_sub σ p σ' p')
+Add Parametric Morphism σ ρ σ' ρ' : (rel_sub σ ρ σ' ρ')
     with signature (@relation_equivalence env) ==> iff as rel_sub_morphism.
 Proof.
   intros R R' HRR'.
   split; intros []; econstructor; intuition.
 Qed.
 
-Lemma rel_exp_implies_rel_typ : forall {i A p A' p'},
-    rel_exp A p A' p' (per_univ i) ->
-    exists R, rel_typ i A p A' p' R.
+Lemma rel_exp_implies_rel_typ : forall {i A ρ A' ρ'},
+    rel_exp A ρ A' ρ' (per_univ i) ->
+    exists R, rel_typ i A ρ A' ρ' R.
 Proof.
   intros.
   destruct_by_head rel_exp.
@@ -30,9 +30,9 @@ Qed.
 #[export]
 Hint Resolve rel_exp_implies_rel_typ : mcltt.
 
-Lemma rel_typ_implies_rel_exp : forall {i A p A' p' R},
-    rel_typ i A p A' p' R ->
-    rel_exp A p A' p' (per_univ i).
+Lemma rel_typ_implies_rel_exp : forall {i A ρ A' ρ' R},
+    rel_typ i A ρ A' ρ' R ->
+    rel_exp A ρ A' ρ' (per_univ i).
 Proof.
   intros.
   destruct_by_head rel_typ.
diff --git a/theories/Core/Completeness/NatCases.v b/theories/Core/Completeness/NatCases.v
index ec429a5d..fc8ed9b6 100644
--- a/theories/Core/Completeness/NatCases.v
+++ b/theories/Core/Completeness/NatCases.v
@@ -8,8 +8,8 @@ Import Domain_Notations.
 Lemma rel_exp_of_nat_inversion : forall {Γ M M'},
     {{ Γ ⊨ M ≈ M' : ℕ }} ->
     exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}),
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-      rel_exp M p M' p' per_nat.
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+      rel_exp M ρ M' ρ' per_nat.
 Proof.
   intros * [env_relΓ].
   destruct_conjs.
@@ -24,8 +24,8 @@ Qed.
 
 Lemma rel_exp_of_nat : forall {Γ M M'},
     (exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}),
-      forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-        rel_exp M p M' p' per_nat) ->
+      forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+        rel_exp M ρ M' ρ' per_nat) ->
     {{ Γ ⊨ M ≈ M' : ℕ }}.
 Proof.
   intros * [env_relΓ].
@@ -145,8 +145,8 @@ Hint Resolve rel_exp_succ_cong : mcltt.
 Lemma rel_exp_of_sub_id_zero_inversion : forall {Γ M M' A},
     {{ Γ ⊨ M ≈ M' : A[Id,,zero] }} ->
     exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i,
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-    exists elem_rel, rel_typ i A d{{{ p ↦ zero }}} A d{{{ p' ↦ zero }}} elem_rel /\ rel_exp M p M' p' elem_rel.
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+    exists elem_rel, rel_typ i A d{{{ ρ ↦ zero }}} A d{{{ ρ' ↦ zero }}} elem_rel /\ rel_exp M ρ M' ρ' elem_rel.
 Proof.
   intros * [env_relΓ].
   destruct_conjs.
@@ -160,8 +160,8 @@ Qed.
 
 Lemma rel_exp_of_sub_id_zero : forall {Γ M M' A},
     (exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i,
-      forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-      exists elem_rel, rel_typ i A d{{{ p ↦ zero }}} A d{{{ p' ↦ zero }}} elem_rel /\ rel_exp M p M' p' elem_rel) ->
+      forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+      exists elem_rel, rel_typ i A d{{{ ρ ↦ zero }}} A d{{{ ρ' ↦ zero }}} elem_rel /\ rel_exp M ρ M' ρ' elem_rel) ->
     {{ Γ ⊨ M ≈ M' : A[Id,,zero] }}.
 Proof.
   intros * [env_relΓ].
@@ -182,8 +182,8 @@ Ltac eexists_rel_exp_of_sub_id_zero :=
 Lemma rel_exp_of_sub_wkwk_succ_var1_inversion : forall {Γ M M' A},
     {{ Γ, ℕ, A ⊨ M ≈ M' : A[Wk∘Wk,,succ(#1)] }} ->
     exists env_rel (_ : {{ EF Γ, ℕ, A ≈ Γ, ℕ, A ∈ per_ctx_env ↘ env_rel }}) i,
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-    exists elem_rel, rel_typ i A d{{{ p ↯ ↯ ↦ succ ~(p 1) }}} A d{{{ p' ↯ ↯ ↦ succ ~(p' 1) }}} elem_rel /\ rel_exp M p M' p' elem_rel.
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+    exists elem_rel, rel_typ i A d{{{ ρ ↯ ↯ ↦ succ ~(ρ 1) }}} A d{{{ ρ' ↯ ↯ ↦ succ ~(ρ' 1) }}} elem_rel /\ rel_exp M ρ M' ρ' elem_rel.
 Proof.
   intros * [env_relΓℕA].
   destruct_conjs.
@@ -198,11 +198,11 @@ Qed.
 Lemma rel_exp_of_sub_id_N : forall {Γ M M' N A},
     {{ Γ ⊨ N : ℕ }} ->
     (exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) i,
-      forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
+      forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
       exists n n',
-        {{ ⟦ N ⟧ p ↘ n }} /\
-          {{ ⟦ N ⟧ p' ↘ n' }} /\
-          exists elem_rel, rel_typ i A d{{{ p ↦ n }}} A d{{{ p' ↦ n' }}} elem_rel /\ rel_exp M p M' p' elem_rel) ->
+        {{ ⟦ N ⟧ ρ ↘ n }} /\
+          {{ ⟦ N ⟧ ρ' ↘ n' }} /\
+          exists elem_rel, rel_typ i A d{{{ ρ ↦ n }}} A d{{{ ρ' ↦ n' }}} elem_rel /\ rel_exp M ρ M' ρ' elem_rel) ->
     {{ Γ ⊨ M ≈ M' : A[Id,,N] }}.
 Proof.
   intros * []%rel_exp_of_nat_inversion [env_relΓ].
@@ -230,13 +230,13 @@ Lemma eval_natrec_sub_neut : forall {Γ env_relΓ σ Δ env_relΔ MZ MZ' MS MS'
     {{ Δ ⊨ MZ ≈ MZ' : A[Id,,zero] }} ->
     {{ Δ, ℕ, A ⊨ MS ≈ MS' : A[Wk∘Wk,,succ(#1)] }} ->
     {{ Dom m ≈ m' ∈ per_bot }} ->
-    (forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}) o o' mz mz',
-        {{ ⟦ σ ⟧s p ↘ o }} ->
-        {{ ⟦ σ ⟧s p' ↘ o' }} ->
-        {{ Dom o ≈ o' ∈ env_relΔ }} ->
-        {{ ⟦ MZ ⟧ o ↘ mz }} ->
-        {{ ⟦ MZ' ⟧ o' ↘ mz' }} ->
-        {{ Dom rec m under o return A | zero -> mz | succ -> MS end ≈ rec m' under p' return A'[q σ] | zero -> mz' | succ -> MS'[q (q σ)] end ∈ per_bot }}).
+    (forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}) ρσ ρ'σ' mz mz',
+        {{ ⟦ σ ⟧s ρ ↘ ρσ }} ->
+        {{ ⟦ σ ⟧s ρ' ↘ ρ'σ' }} ->
+        {{ Dom ρσ ≈ ρ'σ' ∈ env_relΔ }} ->
+        {{ ⟦ MZ ⟧ ρσ ↘ mz }} ->
+        {{ ⟦ MZ' ⟧ ρ'σ' ↘ mz' }} ->
+        {{ Dom rec m under ρσ return A | zero -> mz | succ -> MS end ≈ rec m' under ρ' return A'[q σ] | zero -> mz' | succ -> MS'[q (q σ)] end ∈ per_bot }}).
 Proof.
   intros * equiv_Γ_Γ equiv_Δ_Δ
              [env_relΔℕ]%rel_exp_of_typ_inversion
@@ -258,17 +258,23 @@ Proof.
   destruct_by_head rel_exp.
   destruct_conjs.
   functional_eval_rewrite_clear.
+  match goal with
+  | _: {{ ⟦ σ ⟧s ρ ↘ ~?ρ1 }},
+      _: {{ ⟦ σ ⟧s ρ' ↘ ~?ρ2 }} |- _ =>
+      rename ρ1 into ρσ;
+      rename ρ2 into ρ'σ
+  end.
   intro s.
   assert {{ Dom ⇑! ℕ s ≈ ⇑! ℕ s ∈ per_nat }} by mauto.
-  assert {{ Dom o ↦ ⇑! ℕ s ≈ o' ↦ ⇑! ℕ s ∈ env_relΔℕ }} as HinΔℕs by (apply_relation_equivalence; mauto).
+  assert {{ Dom ρσ ↦ ⇑! ℕ s ≈ ρ'σ ↦ ⇑! ℕ s ∈ env_relΔℕ }} as HinΔℕs by (apply_relation_equivalence; mauto).
   (on_all_hyp: fun H => destruct (H _ _ HinΔℕs)).
   assert {{ Dom succ (⇑! ℕ s) ≈ succ (⇑! ℕ s) ∈ per_nat }} by mauto.
-  assert {{ Dom o ↦ succ (⇑! ℕ s) ≈ o' ↦ succ (⇑! ℕ s) ∈ env_relΔℕ }} as HinΔℕsuccs by (apply_relation_equivalence; mauto).
+  assert {{ Dom ρσ ↦ succ (⇑! ℕ s) ≈ ρ'σ ↦ succ (⇑! ℕ s) ∈ env_relΔℕ }} as HinΔℕsuccs by (apply_relation_equivalence; mauto).
   (on_all_hyp: fun H => destruct (H _ _ HinΔℕsuccs)).
-  assert {{ Dom o ↦ ⇑! ℕ s ≈ o' ↦ ⇑! ℕ s ∈ env_relΔℕ }} as HinΔℕs' by (apply_relation_equivalence; mauto).
-  assert {{ Dom o ↦ succ (⇑! ℕ s) ≈ o' ↦ succ (⇑! ℕ s) ∈ env_relΔℕ }} as HinΔℕsuccs' by (apply_relation_equivalence; mauto).
+  assert {{ Dom ρσ ↦ ⇑! ℕ s ≈ ρ'σ ↦ ⇑! ℕ s ∈ env_relΔℕ }} as HinΔℕs' by (apply_relation_equivalence; mauto).
+  assert {{ Dom ρσ ↦ succ (⇑! ℕ s) ≈ ρ'σ ↦ succ (⇑! ℕ s) ∈ env_relΔℕ }} as HinΔℕsuccs' by (apply_relation_equivalence; mauto).
   assert {{ Dom zero ≈ zero ∈ per_nat }} by econstructor.
-  assert {{ Dom o ↦ zero ≈ o' ↦ zero ∈ env_relΔℕ }} as HinΔℕz by (apply_relation_equivalence; mauto).
+  assert {{ Dom ρσ ↦ zero ≈ ρ'σ ↦ zero ∈ env_relΔℕ }} as HinΔℕz by (apply_relation_equivalence; mauto).
   apply_relation_equivalence.
   (on_all_hyp: fun H => destruct (H _ _ HinΔℕs')).
   (on_all_hyp: fun H => destruct (H _ _ HinΔℕsuccs')).
@@ -280,12 +286,12 @@ Proof.
   rename m'0 into a'.
   rename a1 into asucc.
   rename m'1 into asucc'.
-  assert {{ Dom o ↦ ⇑! ℕ s ↦ ⇑! a (S s) ≈ o' ↦ ⇑! ℕ s ↦ ⇑! a' (S s) ∈ env_relΔℕA }} as HinΔℕA.
+  assert {{ Dom ρσ ↦ ⇑! ℕ s ↦ ⇑! a (S s) ≈ ρ'σ ↦ ⇑! ℕ s ↦ ⇑! a' (S s) ∈ env_relΔℕA }} as HinΔℕA.
   {
     apply_relation_equivalence; eexists; eauto.
     unfold drop_env.
-    repeat change (fun n => d{{{ ~?p ↦ ~?x ↦ ~?y }}} (S n)) with (fun n => d{{{ p ↦ x }}} n).
-    repeat change (d{{{ ~?p ↦ ~?x ↦ ~?y }}} 0) with y.
+    repeat change (fun n => d{{{ ~?ρσ ↦ ~?x ↦ ~?y }}} (S n)) with (fun n => d{{{ ρ ↦ x }}} n).
+    repeat change (d{{{ ~?ρσ ↦ ~?x ↦ ~?y }}} 0) with y.
     eapply per_bot_then_per_elem; mauto.
   }
   apply_relation_equivalence.
@@ -310,13 +316,13 @@ Corollary eval_natrec_neut : forall {Γ env_relΓ MZ MZ' MS MS' A A' i m m'},
     {{ Γ ⊨ MZ ≈ MZ' : A[Id,,zero] }} ->
     {{ Γ, ℕ, A ⊨ MS ≈ MS' : A[Wk∘Wk,,succ(#1)] }} ->
     {{ Dom m ≈ m' ∈ per_bot }} ->
-    (forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}) mz mz',
-        {{ ⟦ MZ ⟧ p ↘ mz }} ->
-        {{ ⟦ MZ' ⟧ p' ↘ mz' }} ->
-        {{ Dom rec m under p return A | zero -> mz | succ -> MS end ≈ rec m' under p' return A' | zero -> mz' | succ -> MS' end ∈ per_bot }}).
+    (forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}) mz mz',
+        {{ ⟦ MZ ⟧ ρ ↘ mz }} ->
+        {{ ⟦ MZ' ⟧ ρ' ↘ mz' }} ->
+        {{ Dom rec m under ρ return A | zero -> mz | succ -> MS end ≈ rec m' under ρ' return A' | zero -> mz' | succ -> MS' end ∈ per_bot }}).
 Proof.
   intros.
-  assert {{ Dom rec m under p return A | zero -> mz | succ -> MS end ≈ rec m' under p' return A'[q Id] | zero -> mz' | succ -> MS'[q (q Id)] end ∈ per_bot }} by (mauto using eval_natrec_sub_neut).
+  assert {{ Dom rec m under ρ return A | zero -> mz | succ -> MS end ≈ rec m' under ρ' return A'[q Id] | zero -> mz' | succ -> MS'[q (q Id)] end ∈ per_bot }} by (mauto using eval_natrec_sub_neut).
   etransitivity; [eassumption |].
   intros s.
   match_by_head per_bot ltac:(fun H => specialize (H s) as [? []]).
@@ -332,12 +338,12 @@ Lemma eval_natrec_rel : forall {Γ env_relΓ MZ MZ' MS MS' A A' i m m'},
     {{ Γ ⊨ MZ ≈ MZ' : A[Id,,zero] }} ->
     {{ Γ, ℕ, A ⊨ MS ≈ MS' : A[Wk∘Wk,,succ(#1)] }} ->
     {{ Dom m ≈ m' ∈ per_nat }} ->
-    (forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}),
+    (forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}),
       forall elem_rel,
-        rel_typ i A d{{{ p ↦ m }}} A d{{{ p' ↦ m' }}} elem_rel ->
+        rel_typ i A d{{{ ρ ↦ m }}} A d{{{ ρ' ↦ m' }}} elem_rel ->
         exists r r',
-          {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
-            {{ rec m' ⟦return A' | zero -> MZ' | succ -> MS' end⟧ p' ↘ r' }} /\
+          {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
+            {{ rec m' ⟦return A' | zero -> MZ' | succ -> MS' end⟧ ρ' ↘ r' }} /\
             {{ Dom r ≈ r' ∈ elem_rel }}).
 Proof.
   intros * equiv_Γ_Γ HA HMZ HMS equiv_m_m'.
@@ -367,17 +373,25 @@ Proof.
     (on_all_hyp_rev: destruct_rel_by_assumption env_relΓ).
     destruct_by_head rel_typ.
     invert_rel_typ_body.
-    assert {{ Dom p ↦ m ≈ p' ↦ m' ∈ env_relΓℕ }} by (apply_relation_equivalence; mauto).
+    match goal with
+    | _: env_relΓ ρ ?ρ0 |- _ =>
+        rename ρ0 into ρ'
+    end.
+    assert {{ Dom ρ ↦ m ≈ ρ' ↦ m' ∈ env_relΓℕ }} by (apply_relation_equivalence; mauto).
     (on_all_hyp: destruct_rel_by_assumption env_relΓℕ).
-    assert {{ Dom p ↦ m ≈ p' ↦ m' ∈ env_relΓℕ }} as HinΓℕ by (apply_relation_equivalence; mauto).
+    assert {{ Dom ρ ↦ m ≈ ρ' ↦ m' ∈ env_relΓℕ }} as HinΓℕ by (apply_relation_equivalence; mauto).
     apply_relation_equivalence.
     (on_all_hyp: fun H => directed destruct (H _ _ HinΓℕ)).
     destruct_by_head per_univ.
-    unshelve epose proof (IHequiv_m_m' _ _ equiv_p_p' _ _) as [? [? [? []]]]; shelve_unifiable; [solve [mauto] |].
+    unshelve epose proof (IHequiv_m_m' _ _ equiv_ρ_ρ' _ _) as [? [? [? []]]]; shelve_unifiable; [solve [mauto] |].
     handle_per_univ_elem_irrel.
-    rename x5 into r1.
-    rename x6 into r1'.
-    assert {{ Dom p ↦ m ↦ r1 ≈ p' ↦ m' ↦ r1' ∈ env_relΓℕA }} as HinΓℕA by (apply_relation_equivalence; mauto).
+    match goal with
+    | _: {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ ~?r0 }},
+        _: {{ rec m' ⟦return A' | zero -> MZ' | succ -> MS' end⟧ ρ' ↘ ~?r0' }} |- _ =>
+        rename r0 into rm;
+        rename r0' into rm'
+    end.
+    assert {{ Dom ρ ↦ m ↦ rm ≈ ρ' ↦ m' ↦ rm' ∈ env_relΓℕA }} as HinΓℕA by (apply_relation_equivalence; mauto).
     apply_relation_equivalence.
     (on_all_hyp: fun H => directed destruct (H _ _ HinΓℕA)).
     destruct_conjs.
@@ -385,15 +399,22 @@ Proof.
     destruct_by_head rel_exp.
     handle_per_univ_elem_irrel.
     do 2 eexists; mauto.
-  - rename n into m'.
+  - match goal with
+    | _: per_bot m ?n |- _ =>
+        rename n into m'
+    end.
     handle_per_ctx_env_irrel.
     match_by_head (per_ctx_env env_relΓℕ) invert_per_ctx_env.
     handle_per_ctx_env_irrel.
     (on_all_hyp: destruct_rel_by_assumption env_relΓ).
     (on_all_hyp_rev: destruct_rel_by_assumption env_relΓ).
     invert_rel_typ_body.
+    match goal with
+    | _: env_relΓ ρ ?ρ0 |- _ =>
+        rename ρ0 into ρ'
+    end.
     assert {{ Dom ⇑ ℕ m ≈ ⇑ ℕ m' ∈ per_nat }} by (econstructor; eassumption).
-    assert {{ Dom p ↦ ⇑ ℕ m ≈ p' ↦ ⇑ ℕ m' ∈ env_relΓℕ }} as HinΓℕ by (apply_relation_equivalence; mauto).
+    assert {{ Dom ρ ↦ ⇑ ℕ m ≈ ρ' ↦ ⇑ ℕ m' ∈ env_relΓℕ }} as HinΓℕ by (apply_relation_equivalence; mauto).
     apply_relation_equivalence.
     (on_all_hyp: fun H => directed destruct (H _ _ HinΓℕ)).
     destruct_by_head per_univ.
@@ -416,11 +437,11 @@ Lemma rel_exp_natrec_cong_rel_typ: forall {Γ A A' i M M' env_relΓ},
     {{ DF Γ ≈ Γ ∈ per_ctx_env ↘ env_relΓ }} ->
     {{ Γ, ℕ ⊨ A ≈ A' : Type@i }} ->
     {{ Γ ⊨ M ≈ M' : ℕ }} ->
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}) n n',
-      {{ ⟦ M ⟧ p ↘ n }} ->
-      {{ ⟦ M' ⟧ p' ↘ n' }} ->
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}) n n',
+      {{ ⟦ M ⟧ ρ ↘ n }} ->
+      {{ ⟦ M' ⟧ ρ' ↘ n' }} ->
       exists elem_rel,
-        rel_typ i A d{{{ p ↦ n }}} A d{{{ p' ↦ n' }}} elem_rel.
+        rel_typ i A d{{{ ρ ↦ n }}} A d{{{ ρ' ↦ n' }}} elem_rel.
 Proof.
   intros.
   assert {{ ⊨ Γ }} by (eexists; eauto).
@@ -457,9 +478,9 @@ Proof.
   intros.
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   functional_eval_rewrite_clear.
-  assert (exists elem_rel, rel_typ i A d{{{ p ↦ m }}} A d{{{ p' ↦ m' }}} elem_rel) as [elem_rel]
+  assert (exists elem_rel, rel_typ i A d{{{ ρ ↦ m }}} A d{{{ ρ' ↦ m' }}} elem_rel) as [elem_rel]
       by mauto using rel_exp_natrec_cong_rel_typ.
-  assert (exists elem_rel, rel_typ i A d{{{ p ↦ m }}} A d{{{ p' ↦ m'0 }}} elem_rel) as []
+  assert (exists elem_rel, rel_typ i A d{{{ ρ ↦ m }}} A d{{{ ρ' ↦ m'0 }}} elem_rel) as []
       by mauto using rel_exp_natrec_cong_rel_typ.
   do 2 eexists.
   repeat split; [eassumption | eassumption |].
@@ -468,8 +489,8 @@ Proof.
   destruct_by_head rel_typ.
   handle_per_univ_elem_irrel.
   assert (exists r r',
-             {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
-               {{ rec m'0 ⟦return A' | zero -> MZ' | succ -> MS' end⟧ p' ↘ r' }} /\
+             {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
+               {{ rec m'0 ⟦return A' | zero -> MZ' | succ -> MS' end⟧ ρ' ↘ r' }} /\
                {{ Dom r ≈ r' ∈ elem_rel }})
     by mauto using eval_natrec_rel.
   destruct_conjs.
@@ -486,16 +507,16 @@ Lemma eval_natrec_sub_rel : forall {Γ env_relΓ σ Δ env_relΔ MZ MZ' MS MS' A
     {{ Δ ⊨ MZ ≈ MZ' : A[Id,,zero] }} ->
     {{ Δ, ℕ, A ⊨ MS ≈ MS' : A[Wk∘Wk,,succ(#1)] }} ->
     {{ Dom m ≈ m' ∈ per_nat }} ->
-    (forall p p'
-        (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }})
+    (forall ρ ρ'
+        (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }})
         o o' elem_rel,
-        {{ ⟦ σ ⟧s p ↘ o }} ->
-        {{ ⟦ σ ⟧s p' ↘ o' }} ->
+        {{ ⟦ σ ⟧s ρ ↘ o }} ->
+        {{ ⟦ σ ⟧s ρ' ↘ o' }} ->
         {{ Dom o ≈ o' ∈ env_relΔ }} ->
         rel_typ i A d{{{ o ↦ m }}} A d{{{ o' ↦ m' }}} elem_rel ->
         exists r r',
           {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ o ↘ r }} /\
-            {{ rec m' ⟦return A'[q σ] | zero -> MZ'[σ] | succ -> MS'[q (q σ)] end⟧ p' ↘ r' }} /\
+            {{ rec m' ⟦return A'[q σ] | zero -> MZ'[σ] | succ -> MS'[q (q σ)] end⟧ ρ' ↘ r' }} /\
             {{ Dom r ≈ r' ∈ elem_rel }}).
 Proof.
   intros * equiv_Γ_Γ equiv_Δ_Δ HA HMZ HMS equiv_m_m'.
@@ -512,7 +533,10 @@ Proof.
     destruct_by_head rel_exp.
     handle_per_univ_elem_irrel.
     do 2 eexists; repeat split; only 1-2: econstructor; mauto.
-  - rename n into m'.
+  - match goal with
+    | _: per_nat m ?n |- _ =>
+        rename n into m'
+    end.
     assert {{ Δ, ℕ ⊨ A ≈ A : Type@i }} as []%rel_exp_of_typ_inversion by (etransitivity; mauto).
     apply rel_exp_of_sub_wkwk_succ_var1_inversion in HMS as [env_relΔℕA].
     destruct_conjs.
@@ -525,17 +549,27 @@ Proof.
     (on_all_hyp_rev: destruct_rel_by_assumption env_relΔ).
     destruct_by_head rel_typ.
     invert_rel_typ_body.
-    assert {{ Dom o ↦ m ≈ o' ↦ m' ∈ env_relΔℕ }} by (apply_relation_equivalence; mauto).
+    match goal with
+    | _: {{ ⟦ σ ⟧s ρ ↘ ~?ρ1 }},
+        _: {{ ⟦ σ ⟧s ρ' ↘ ~?ρ2 }} |- _ =>
+        rename ρ1 into ρσ;
+        rename ρ2 into ρ'σ
+    end.
+    assert {{ Dom ρσ ↦ m ≈ ρ'σ ↦ m' ∈ env_relΔℕ }} by (apply_relation_equivalence; mauto).
     (on_all_hyp: destruct_rel_by_assumption env_relΔℕ).
-    assert {{ Dom o ↦ m ≈ o' ↦ m' ∈ env_relΔℕ }} as HinΔℕ by (apply_relation_equivalence; mauto).
+    assert {{ Dom ρσ ↦ m ≈ ρ'σ ↦ m' ∈ env_relΔℕ }} as HinΔℕ by (apply_relation_equivalence; mauto).
     apply_relation_equivalence.
     (on_all_hyp: fun H => directed destruct (H _ _ HinΔℕ)).
     destruct_by_head per_univ.
-    unshelve epose proof (IHequiv_m_m' _ _ equiv_p_p' _ _) as [? [? [? []]]]; shelve_unifiable; only 4: solve [mauto]; eauto.
+    unshelve epose proof (IHequiv_m_m' _ _ equiv_ρ_ρ' _ _) as [? [? [? []]]]; shelve_unifiable; only 4: solve [mauto]; eauto.
     handle_per_univ_elem_irrel.
-    rename x5 into r1.
-    rename x6 into r1'.
-    assert {{ Dom o ↦ m ↦ r1 ≈ o' ↦ m' ↦ r1' ∈ env_relΔℕA }} as HinΔℕA by (apply_relation_equivalence; mauto).
+    match goal with
+    | _: {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ~_ ↘ ~?r0 }},
+        _: {{ rec m' ⟦return A'[q σ] | zero -> MZ'[σ] | succ -> MS'[q (q σ)] end⟧ ~_ ↘ ~?r0' }} |- _ =>
+        rename r0 into rm;
+        rename r0' into rm'
+    end.
+    assert {{ Dom ρσ ↦ m ↦ rm ≈ ρ'σ ↦ m' ↦ rm' ∈ env_relΔℕA }} as HinΔℕA by (apply_relation_equivalence; mauto).
     apply_relation_equivalence.
     (on_all_hyp: fun H => directed destruct (H _ _ HinΔℕA)).
     destruct_conjs.
@@ -543,15 +577,24 @@ Proof.
     destruct_by_head rel_exp.
     handle_per_univ_elem_irrel.
     do 2 eexists; repeat split; only 1-2: repeat econstructor; mauto.
-  - rename n into m'.
+  - match goal with
+    | _: per_bot m ?n |- _ =>
+        rename n into m'
+    end.
     handle_per_ctx_env_irrel.
     match_by_head (per_ctx_env env_relΔℕ) invert_per_ctx_env.
     handle_per_ctx_env_irrel.
     (on_all_hyp: destruct_rel_by_assumption env_relΔ).
     (on_all_hyp_rev: destruct_rel_by_assumption env_relΔ).
     invert_rel_typ_body.
+    match goal with
+    | _: {{ ⟦ σ ⟧s ρ ↘ ~?ρ1 }},
+        _: {{ ⟦ σ ⟧s ρ' ↘ ~?ρ2 }} |- _ =>
+        rename ρ1 into ρσ;
+        rename ρ2 into ρ'σ
+    end.
     assert {{ Dom ⇑ ℕ m ≈ ⇑ ℕ m' ∈ per_nat }} by (econstructor; eassumption).
-    assert {{ Dom o ↦ ⇑ ℕ m ≈ o' ↦ ⇑ ℕ m' ∈ env_relΔℕ }} as HinΔℕ by (apply_relation_equivalence; mauto).
+    assert {{ Dom ρσ ↦ ⇑ ℕ m ≈ ρ'σ ↦ ⇑ ℕ m' ∈ env_relΔℕ }} as HinΔℕ by (apply_relation_equivalence; mauto).
     apply_relation_equivalence.
     (on_all_hyp: fun H => directed destruct (H _ _ HinΔℕ)).
     destruct_by_head per_univ.
@@ -575,9 +618,9 @@ Lemma rel_exp_natrec_sub_rel_typ: forall {Γ σ Δ A i M env_relΓ},
     {{ Γ ⊨s σ : Δ }} ->
     {{ Δ, ℕ ⊨ A : Type@i }} ->
     {{ Δ ⊨ M : ℕ }} ->
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}),
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}),
     exists elem_rel,
-      rel_typ i {{{ A[σ,,M[σ]] }}} p {{{ A[σ,,M[σ]] }}} p' elem_rel.
+      rel_typ i {{{ A[σ,,M[σ]] }}} ρ {{{ A[σ,,M[σ]] }}} ρ' elem_rel.
 Proof.
   intros.
   assert {{ ⊨ Γ }} by (eexists; eauto).
@@ -605,7 +648,7 @@ Proof.
   handle_per_ctx_env_irrel.
   eexists_rel_exp.
   intros.
-  assert (exists elem_rel, rel_typ i {{{ A[σ,,M[σ]] }}} p {{{ A[σ,,M[σ]] }}} p' elem_rel) as [elem_rel]
+  assert (exists elem_rel, rel_typ i {{{ A[σ,,M[σ]] }}} ρ {{{ A[σ,,M[σ]] }}} ρ' elem_rel) as [elem_rel]
       by (eapply rel_exp_natrec_sub_rel_typ; only 5: eassumption; mauto; eexists; mauto).
   eexists.
   split; [eassumption |].
@@ -613,9 +656,20 @@ Proof.
   (on_all_hyp: destruct_rel_by_assumption env_relΔ).
   destruct_by_head rel_typ.
   invert_rel_typ_body.
+  match goal with
+  | _: {{ ⟦ σ ⟧s ~?ρ0 ↘ ~?ρσ0 }},
+      _: {{ ⟦ A ⟧ ρσ ↦ ~?m0 ↘ ~?a0 }},
+        _: {{ ⟦ A ⟧ ~?ρσ0 ↦ ~?m0' ↘ ~?a0' }} |- _ =>
+      rename ρ0 into ρ';
+      rename ρσ0 into ρ'σ;
+      rename a0 into a;
+      rename m0 into m;
+      rename a0' into a';
+      rename m0' into m'
+  end.
   enough (exists r r',
-             {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ o ↘ r }} /\
-               {{ rec m' ⟦return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end⟧ p' ↘ r' }} /\
+             {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρσ ↘ r }} /\
+               {{ rec m' ⟦return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end⟧ ρ' ↘ r' }} /\
                {{ Dom r ≈ r' ∈ elem_rel }})
     by (destruct_conjs; econstructor; mauto).
   mauto 4 using eval_natrec_sub_rel.
@@ -644,8 +698,12 @@ Proof.
   destruct_by_head rel_typ.
   invert_rel_typ_body.
   destruct_by_head rel_exp.
+  match goal with
+  | _: env_relΓ ρ ?ρ0 |- _ =>
+      rename ρ0 into ρ'
+  end.
   assert {{ Dom zero ≈ zero ∈ per_nat }} by econstructor.
-  assert {{ Dom p ↦ zero ≈ p' ↦ zero ∈ env_relΓℕ }} by (apply_relation_equivalence; eauto).
+  assert {{ Dom ρ ↦ zero ≈ ρ' ↦ zero ∈ env_relΓℕ }} by (apply_relation_equivalence; eauto).
   (on_all_hyp: destruct_rel_by_assumption env_relΓℕ).
   handle_per_univ_elem_irrel.
   eexists.
@@ -659,9 +717,9 @@ Lemma rel_exp_nat_beta_succ_rel_typ : forall {Γ env_relΓ A i M},
     {{ DF Γ ≈ Γ ∈ per_ctx_env ↘ env_relΓ }} ->
     {{ Γ, ℕ ⊨ A : Type@i }} ->
     {{ Γ ⊨ M : ℕ }} ->
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}),
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}),
     exists elem_rel,
-      rel_typ i {{{ A[Id,,succ M] }}} p {{{ A[Id,,succ M] }}} p' elem_rel.
+      rel_typ i {{{ A[Id,,succ M] }}} ρ {{{ A[Id,,succ M] }}} ρ' elem_rel.
 Proof.
   intros.
   assert {{ ⊨ Γ }} by (eexists; eauto).
@@ -690,16 +748,18 @@ Proof.
   intros.
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   assert (exists elem_rel,
-             rel_typ i {{{ A[Id,,succ M] }}} p {{{ A[Id,,succ M] }}} p' elem_rel) as [elem_rel]
+             rel_typ i {{{ A[Id,,succ M] }}} ρ {{{ A[Id,,succ M] }}} ρ' elem_rel) as [elem_rel]
       by (eapply rel_exp_nat_beta_succ_rel_typ; mauto).
   eexists; split; [eassumption |].
   destruct_by_head rel_typ.
   invert_rel_typ_body.
-  rename p'2 into p.
-  rename p'1 into p'.
+  match goal with
+  | _: env_relΓ ρ ?ρ0 |- _ =>
+      rename ρ0 into ρ'
+  end.
   assert (exists r r',
-             {{ rec succ m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
-               {{ rec succ m' ⟦return A | zero -> MZ | succ -> MS end⟧ p' ↘ r' }} /\
+             {{ rec succ m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
+               {{ rec succ m' ⟦return A | zero -> MZ | succ -> MS end⟧ ρ' ↘ r' }} /\
                {{ Dom r ≈ r' ∈ elem_rel }})
     by (eapply eval_natrec_rel; mauto).
   destruct_conjs.
diff --git a/theories/Core/Completeness/SubstitutionCases.v b/theories/Core/Completeness/SubstitutionCases.v
index 46d1ea2b..8dfcb085 100644
--- a/theories/Core/Completeness/SubstitutionCases.v
+++ b/theories/Core/Completeness/SubstitutionCases.v
@@ -205,7 +205,7 @@ Proof with mautosolve.
   destruct_conjs.
   eexists_rel_sub.
   intros.
-  assert (env_relΓ p' p) by (symmetry; eassumption).
+  assert (env_relΓ ρ' ρ) by (symmetry; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   econstructor; mauto; symmetry...
 Qed.
@@ -224,7 +224,7 @@ Proof with mautosolve.
   handle_per_ctx_env_irrel.
   eexists_rel_sub.
   intros.
-  assert (env_relΓ p' p') by (etransitivity; [symmetry |]; eassumption).
+  assert (env_relΓ ρ' ρ') by (etransitivity; [symmetry |]; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   functional_eval_rewrite_clear.
   econstructor; mauto; etransitivity...
diff --git a/theories/Core/Completeness/SubtypingCases.v b/theories/Core/Completeness/SubtypingCases.v
index 1ae31917..df63a3d4 100644
--- a/theories/Core/Completeness/SubtypingCases.v
+++ b/theories/Core/Completeness/SubtypingCases.v
@@ -89,38 +89,43 @@ Proof.
   destruct_by_head rel_exp.
   destruct_conjs.
   handle_per_univ_elem_irrel.
+  match goal with
+  | _: env_relΓ ρ ?ρ0 |- _ =>
+      assert_fails (unify ρ ρ0);
+      rename ρ0 into ρ'
+  end.
   rename x0 into head_rel.
 
-  assert (forall c c', head_rel p p' equiv_p_p' c c' -> env_relΓA' d{{{ p ↦ c }}} d{{{ p' ↦ c' }}}) as HΓA'
+  assert (forall c c', head_rel ρ ρ' equiv_ρ_ρ' c c' -> env_relΓA' d{{{ ρ ↦ c }}} d{{{ ρ' ↦ c' }}}) as HΓA'
       by (intros; apply_relation_equivalence; unshelve eexists; eassumption).
 
   (* The proofs for the next two assertions are basically the same *)
   exvar (relation domain)
-    ltac:(fun R => assert ({{ DF Π m0 p B ≈ Π m1 p' B ∈ per_univ_elem (Nat.max i k) ↘ R }})).
+    ltac:(fun R => assert ({{ DF Π m0 ρ B ≈ Π m1 ρ' B ∈ per_univ_elem (Nat.max i k) ↘ R }})).
   {
     intros.
     per_univ_elem_econstructor; [| | solve_refl].
     - etransitivity; [| symmetry]; mauto using per_univ_elem_cumu_max_left.
     - eapply rel_exp_pi_core; [| reflexivity].
       intros.
-      assert (env_relΓA' d{{{ p ↦ c }}} d{{{ p' ↦ c' }}}) as equiv_pc_p'c' by (apply HΓA'; intuition).
+      assert (env_relΓA' d{{{ ρ ↦ c }}} d{{{ ρ' ↦ c' }}}) as equiv_ρc_ρ'c' by (apply HΓA'; intuition).
       apply_relation_equivalence.
-      (on_all_hyp: fun H => destruct (H _ _ equiv_pc_p'c')).
+      (on_all_hyp: fun H => destruct (H _ _ equiv_ρc_ρ'c')).
       destruct_conjs.
       destruct_by_head rel_typ.
       econstructor; mauto using per_univ_elem_cumu_max_right.
   }
   exvar (relation domain)
-    ltac:(fun R => assert ({{ DF Π a0 p B' ≈ Π a p' B' ∈ per_univ_elem (Nat.max i k) ↘ R }})).
+    ltac:(fun R => assert ({{ DF Π a0 ρ B' ≈ Π a ρ' B' ∈ per_univ_elem (Nat.max i k) ↘ R }})).
   {
     intros.
     per_univ_elem_econstructor; [| | solve_refl].
     - etransitivity; [symmetry |]; mauto using per_univ_elem_cumu_max_left.
     - eapply rel_exp_pi_core; [| reflexivity].
       intros.
-      assert (env_relΓA' d{{{ p ↦ c }}} d{{{ p' ↦ c' }}}) as equiv_pc_p'c' by (apply HΓA'; intuition).
+      assert (env_relΓA' d{{{ ρ ↦ c }}} d{{{ ρ' ↦ c' }}}) as equiv_ρc_ρ'c' by (apply HΓA'; intuition).
       apply_relation_equivalence.
-      (on_all_hyp: fun H => destruct (H _ _ equiv_pc_p'c')).
+      (on_all_hyp: fun H => destruct (H _ _ equiv_ρc_ρ'c')).
       destruct_conjs.
       destruct_by_head rel_typ.
       econstructor; mauto using per_univ_elem_cumu_max_right.
@@ -131,9 +136,9 @@ Proof.
   econstructor; only 3-4: try (saturate_refl; mautosolve 2).
   - eauto using per_univ_elem_cumu_max_left.
   - intros.
-    assert (env_relΓA' d{{{ p ↦ c }}} d{{{ p' ↦ c' }}}) as equiv_pc_p'c' by (apply HΓA'; intuition).
+    assert (env_relΓA' d{{{ ρ ↦ c }}} d{{{ ρ' ↦ c' }}}) as equiv_ρc_ρ'c' by (apply HΓA'; intuition).
     apply_relation_equivalence.
-    (on_all_hyp: fun H => destruct (H _ _ equiv_pc_p'c')).
+    (on_all_hyp: fun H => destruct (H _ _ equiv_ρc_ρ'c')).
     destruct_conjs.
     destruct_by_head rel_exp.
     simplify_evals.
diff --git a/theories/Core/Completeness/TermStructureCases.v b/theories/Core/Completeness/TermStructureCases.v
index 3d6f5ba6..1b4ab845 100644
--- a/theories/Core/Completeness/TermStructureCases.v
+++ b/theories/Core/Completeness/TermStructureCases.v
@@ -15,11 +15,17 @@ Proof with mautosolve.
   handle_per_ctx_env_irrel.
   eexists_rel_exp.
   intros.
-  assert (env_relΓ p' p) by (symmetry; eassumption).
-  assert (env_relΓ p p) by (etransitivity; eassumption).
+  assert (env_relΓ ρ' ρ) by (symmetry; eassumption).
+  assert (env_relΓ ρ ρ) by (etransitivity; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   handle_per_univ_elem_irrel.
-  assert (env_relΔ o o0) by (etransitivity; [|symmetry; eassumption]; eassumption).
+  match goal with
+  | _: {{ ⟦ σ ⟧s ρ ↘ ~?ρ0 }},
+      _: {{ ⟦ σ ⟧s ρ' ↘ ~?ρ'0 }} |- _ =>
+      rename ρ0 into ρσ;
+      rename ρ'0 into ρ'σ
+  end.
+  assert (env_relΔ ρσ ρ'σ) by (etransitivity; [|symmetry; eassumption]; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΔ).
   destruct_by_head rel_typ.
   destruct_by_head rel_exp.
@@ -62,8 +68,8 @@ Proof with mautosolve.
   handle_per_ctx_env_irrel.
   eexists_rel_exp.
   intros.
-  assert (env_relΓ p' p) by (symmetry; eassumption).
-  assert (env_relΓ p p) by (etransitivity; eassumption).
+  assert (env_relΓ ρ' ρ) by (symmetry; eassumption).
+  assert (env_relΓ ρ ρ) by (etransitivity; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   handle_per_univ_elem_irrel.
   (on_all_hyp: destruct_rel_by_assumption env_relΓ').
@@ -90,7 +96,7 @@ Proof with mautosolve.
   handle_per_ctx_env_irrel.
   eexists_rel_exp.
   intros.
-  assert (env_relΓ p p) by (etransitivity; [| symmetry]; eassumption).
+  assert (env_relΓ ρ ρ) by (etransitivity; [| symmetry]; eassumption).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ).
   destruct_by_head per_univ.
   destruct_by_head rel_typ.
@@ -112,7 +118,7 @@ Proof with mautosolve.
   destruct_conjs.
   eexists_rel_exp.
   intros.
-  assert (env_relΓ p' p) by (symmetry; eauto).
+  assert (env_relΓ ρ' ρ) by (symmetry; eauto).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ); destruct_conjs.
   destruct_by_head rel_typ.
   destruct_by_head rel_exp.
@@ -136,8 +142,8 @@ Proof with mautosolve.
   handle_per_ctx_env_irrel.
   eexists_rel_exp.
   intros.
-  assert (env_relΓ p' p) by (symmetry; eauto).
-  assert (env_relΓ p' p') by (etransitivity; eauto).
+  assert (env_relΓ ρ' ρ) by (symmetry; eauto).
+  assert (env_relΓ ρ' ρ') by (etransitivity; eauto).
   (on_all_hyp: destruct_rel_by_assumption env_relΓ); destruct_conjs.
   destruct_by_head rel_typ.
   destruct_by_head rel_exp.
diff --git a/theories/Core/Completeness/UniverseCases.v b/theories/Core/Completeness/UniverseCases.v
index e0662c43..5835e22a 100644
--- a/theories/Core/Completeness/UniverseCases.v
+++ b/theories/Core/Completeness/UniverseCases.v
@@ -7,8 +7,8 @@ Import Domain_Notations.
 Lemma rel_exp_of_typ_inversion : forall {Γ A A' i},
     {{ Γ ⊨ A ≈ A' : Type@i }} ->
     exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}),
-    forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-      rel_exp A p A' p' (per_univ i).
+    forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+      rel_exp A ρ A' ρ' (per_univ i).
 Proof.
   intros * [env_relΓ].
   destruct_conjs.
@@ -23,8 +23,8 @@ Qed.
 
 Lemma rel_exp_of_typ : forall {Γ A A' i},
     (exists env_rel (_ : {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}),
-      forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_rel }}),
-        rel_exp A p A' p' (per_univ i)) ->
+      forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_rel }}),
+        rel_exp A ρ A' ρ' (per_univ i)) ->
     {{ Γ ⊨ A ≈ A' : Type@i }}.
 Proof.
   intros * [env_relΓ].
diff --git a/theories/Core/Semantic/Domain.v b/theories/Core/Semantic/Domain.v
index 4d2e19d7..5ee3ac7f 100644
--- a/theories/Core/Semantic/Domain.v
+++ b/theories/Core/Semantic/Domain.v
@@ -28,16 +28,16 @@ Derive NoConfusion for domain domain_ne domain_nf.
 Definition empty_env : env := fun x => d_zero.
 Arguments empty_env _ /.
 
-Definition extend_env (p : env) (d : domain) : env :=
+Definition extend_env (ρ : env) (d : domain) : env :=
   fun n =>
     match n with
     | 0 => d
-    | S n' => p n'
+    | S n' => ρ n'
     end.
 Arguments extend_env _ _ _ /.
 Transparent extend_env.
 
-Definition drop_env (p : env) : env := fun n => p (S n).
+Definition drop_env (ρ : env) : env := fun n => ρ (S n).
 Arguments drop_env _ _ /.
 Transparent drop_env.
 
@@ -51,27 +51,27 @@ Module Domain_Notations.
   Notation "( x )" := x (in custom domain at level 0, x custom domain at level 60) : mcltt_scope.
   Notation "~ x" := x (in custom domain at level 0, x constr at level 0) : mcltt_scope.
   Notation "x" := x (in custom domain at level 0, x global) : mcltt_scope.
-  Notation "'λ' p M" := (d_fn p M) (in custom domain at level 0, p custom domain at level 30, M custom exp at level 30) : mcltt_scope.
+  Notation "'λ' ρ M" := (d_fn ρ M) (in custom domain at level 0, ρ custom domain at level 30, M custom exp at level 30) : mcltt_scope.
   Notation "f x .. y" := (d_app .. (d_app f x) .. y) (in custom domain at level 40, f custom domain, x custom domain at next level, y custom domain at next level) : mcltt_scope.
   Notation "'ℕ'" := d_nat (in custom domain) : mcltt_scope.
   Notation "'𝕌' @ n" := (d_univ n) (in custom domain at level 0, n constr at level 0) : mcltt_scope.
-  Notation "'Π' a p B" := (d_pi a p B) (in custom domain at level 0, a custom domain at level 30, p custom domain at level 0, B custom exp at level 30) : mcltt_scope.
+  Notation "'Π' a ρ B" := (d_pi a ρ B) (in custom domain at level 0, a custom domain at level 30, ρ custom domain at level 0, B custom exp at level 30) : mcltt_scope.
   Notation "'zero'" := d_zero (in custom domain at level 0) : mcltt_scope.
   Notation "'succ' m" := (d_succ m) (in custom domain at level 30, m custom domain at level 30) : mcltt_scope.
-  Notation "'rec' m 'under' p 'return' P | 'zero' -> mz | 'succ' -> MS 'end'" := (d_natrec p P mz MS m) (in custom domain at level 0, P custom exp at level 60, mz custom domain at level 60, MS custom exp at level 60, p custom domain at level 60, m custom domain at level 60) : mcltt_scope.
+  Notation "'rec' m 'under' ρ 'return' P | 'zero' -> mz | 'succ' -> MS 'end'" := (d_natrec ρ P mz MS m) (in custom domain at level 0, P custom exp at level 60, mz custom domain at level 60, MS custom exp at level 60, ρ custom domain at level 60, m custom domain at level 60) : mcltt_scope.
   Notation "'!' n" := (d_var n) (in custom domain at level 0, n constr at level 0) : mcltt_scope.
   Notation "'⇑' a m" := (d_neut a m) (in custom domain at level 0, a custom domain at level 30, m custom domain at level 30) : mcltt_scope.
   Notation "'⇓' a m" := (d_dom a m) (in custom domain at level 0, a custom domain at level 30, m custom domain at level 30) : mcltt_scope.
   Notation "'⇑!' a n" := (d_neut a (d_var n)) (in custom domain at level 0, a custom domain at level 30, n constr at level 0) : mcltt_scope.
 
-  Notation "p ↦ m" := (extend_env p m) (in custom domain at level 20, left associativity) : mcltt_scope.
-  Notation "p '↯'" := (drop_env p) (in custom domain at level 10, p custom domain) : mcltt_scope.
+  Notation "ρ ↦ m" := (extend_env ρ m) (in custom domain at level 20, left associativity) : mcltt_scope.
+  Notation "ρ '↯'" := (drop_env ρ) (in custom domain at level 10, ρ custom domain) : mcltt_scope.
 End Domain_Notations.
 
 Import Domain_Notations.
 
-Proposition drop_env_extend_env_cancel : forall p a,
-    d{{{ (p ↦ a) ↯ }}} = p.
+Proposition drop_env_extend_env_cancel : forall ρ a,
+    d{{{ (ρ ↦ a) ↯ }}} = ρ.
 Proof.
   reflexivity.
 Qed.
diff --git a/theories/Core/Semantic/Evaluation/Definitions.v b/theories/Core/Semantic/Evaluation/Definitions.v
index 79d3cd22..87779978 100644
--- a/theories/Core/Semantic/Evaluation/Definitions.v
+++ b/theories/Core/Semantic/Evaluation/Definitions.v
@@ -2,85 +2,90 @@ From Mcltt Require Import Base.
 From Mcltt Require Export Domain.
 Import Domain_Notations.
 
-Reserved Notation "'⟦' M '⟧' p '↘' r" (in custom judg at level 80, M custom exp at level 99, p custom domain at level 99, r custom domain at level 99).
-Reserved Notation "'rec' m '⟦return' A | 'zero' -> MZ | 'succ' -> MS 'end⟧' p '↘' r" (in custom judg at level 80, m custom domain at level 99, A custom exp at level 99, MZ custom exp at level 99, MS custom exp at level 99, p custom domain at level 99, r custom domain at level 99).
+Reserved Notation "'⟦' M '⟧' ρ '↘' r" (in custom judg at level 80, M custom exp at level 99, ρ custom domain at level 99, r custom domain at level 99).
+Reserved Notation "'rec' m '⟦return' A | 'zero' -> MZ | 'succ' -> MS 'end⟧' ρ '↘' r" (in custom judg at level 80, m custom domain at level 99, A custom exp at level 99, MZ custom exp at level 99, MS custom exp at level 99, ρ custom domain at level 99, r custom domain at level 99).
 Reserved Notation "'$|' m '&' n '|↘' r" (in custom judg at level 80, m custom domain at level 99, n custom domain at level 99, r custom domain at level 99).
-Reserved Notation "'⟦' σ '⟧s' p '↘' p'" (in custom judg at level 80, σ custom exp at level 99, p custom domain at level 99, p' custom domain at level 99).
+Reserved Notation "'⟦' σ '⟧s' ρ '↘' ρσ" (in custom judg at level 80, σ custom exp at level 99, ρ custom domain at level 99, ρσ custom domain at level 99).
 
 Generalizable All Variables.
 
 Inductive eval_exp : exp -> env -> domain -> Prop :=
 | eval_exp_typ :
-  `( {{ ⟦ Type@i ⟧ p ↘ 𝕌@i }} )
+  `( {{ ⟦ Type@i ⟧ ρ ↘ 𝕌@i }} )
 | eval_exp_var :
-  `( {{ ⟦ # x ⟧ p ↘ ~(p x) }} )
+  `( {{ ⟦ # x ⟧ ρ ↘ ~(ρ x) }} )
 | eval_exp_nat :
-  `( {{ ⟦ ℕ ⟧ p ↘ ℕ }} )
+  `( {{ ⟦ ℕ ⟧ ρ ↘ ℕ }} )
 | eval_exp_zero :
-  `( {{ ⟦ zero ⟧ p ↘ zero }} )
+  `( {{ ⟦ zero ⟧ ρ ↘ zero }} )
 | eval_exp_succ :
-  `( {{ ⟦ M ⟧ p ↘ m }} ->
-     {{ ⟦ succ M ⟧ p ↘ succ m }} )
+  `( {{ ⟦ M ⟧ ρ ↘ m }} ->
+     {{ ⟦ succ M ⟧ ρ ↘ succ m }} )
 | eval_exp_natrec :
-  `( {{ ⟦ M ⟧ p ↘ m }} ->
-     {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} ->
-     {{ ⟦ rec M return A | zero -> MZ | succ -> MS end ⟧ p ↘ r }} )
+  `( {{ ⟦ M ⟧ ρ ↘ m }} ->
+     {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} ->
+     {{ ⟦ rec M return A | zero -> MZ | succ -> MS end ⟧ ρ ↘ r }} )
 | eval_exp_pi :
-  `( {{ ⟦ A ⟧ p ↘ a }} ->
-     {{ ⟦ Π A B ⟧ p ↘ Π a p B }} )
+  `( {{ ⟦ A ⟧ ρ ↘ a }} ->
+     {{ ⟦ Π A B ⟧ ρ ↘ Π a ρ B }} )
 | eval_exp_fn :
-  `( {{ ⟦ λ A M ⟧ p ↘ λ p M }} )
+  `( {{ ⟦ λ A M ⟧ ρ ↘ λ ρ M }} )
 | eval_exp_app :
-  `( {{ ⟦ M ⟧ p ↘ m }} ->
-     {{ ⟦ N ⟧ p ↘ n }} ->
+  `( {{ ⟦ M ⟧ ρ ↘ m }} ->
+     {{ ⟦ N ⟧ ρ ↘ n }} ->
      {{ $| m & n |↘ r }} ->
-     {{ ⟦ M N ⟧ p ↘ r }} )
+     {{ ⟦ M N ⟧ ρ ↘ r }} )
 | eval_exp_sub :
-  `( {{ ⟦ σ ⟧s p ↘ p' }} ->
-     {{ ⟦ M ⟧ p' ↘ m }} ->
-     {{ ⟦ M[σ] ⟧ p ↘ m }} )
-where "'⟦' e '⟧' p '↘' r" := (eval_exp e p r) (in custom judg)
+  `( {{ ⟦ σ ⟧s ρ ↘ ρ' }} ->
+     {{ ⟦ M ⟧ ρ' ↘ m }} ->
+     {{ ⟦ M[σ] ⟧ ρ ↘ m }} )
+where "'⟦' e '⟧' ρ '↘' r" := (eval_exp e ρ r) (in custom judg)
 with eval_natrec : exp -> exp -> exp -> domain -> env -> domain -> Prop :=
 | eval_natrec_zero :
-  `( {{ ⟦ MZ ⟧ p ↘ mz }} ->
-     {{ rec zero ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ mz }} )
+  `( {{ ⟦ MZ ⟧ ρ ↘ mz }} ->
+     {{ rec zero ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ mz }} )
 | eval_natrec_succ :
-  `( {{ rec b ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} ->
-     {{ ⟦ MS ⟧ p ↦ b ↦ r ↘ ms }} ->
-     {{ rec succ b ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ ms }} )
+  `( {{ rec b ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} ->
+     {{ ⟦ MS ⟧ ρ ↦ b ↦ r ↘ ms }} ->
+     {{ rec succ b ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ ms }} )
 | eval_natrec_neut :
-  `( {{ ⟦ MZ ⟧ p ↘ mz }} ->
-     {{ ⟦ A ⟧ p ↦ ⇑ ℕ m ↘ a }} ->
-     {{ rec ⇑ ℕ m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ ⇑ a (rec m under p return A | zero -> mz | succ -> MS end) }} )
-where "'rec' m '⟦return' A | 'zero' -> MZ | 'succ' -> MS 'end⟧' p '↘' r" := (eval_natrec A MZ MS m p r) (in custom judg)
+  `( {{ ⟦ MZ ⟧ ρ ↘ mz }} ->
+     {{ ⟦ A ⟧ ρ ↦ ⇑ ℕ m ↘ a }} ->
+     {{ rec ⇑ ℕ m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ ⇑ a (rec m under ρ return A | zero -> mz | succ -> MS end) }} )
+where "'rec' m '⟦return' A | 'zero' -> MZ | 'succ' -> MS 'end⟧' ρ '↘' r" := (eval_natrec A MZ MS m ρ r) (in custom judg)
 with eval_app : domain -> domain -> domain -> Prop :=
 | eval_app_fn :
-  `( {{ ⟦ M ⟧ p ↦ n ↘ m }} ->
-     {{ $| λ p M & n |↘ m }} )
+  `( {{ ⟦ M ⟧ ρ ↦ n ↘ m }} ->
+     {{ $| λ ρ M & n |↘ m }} )
 | eval_app_neut :
-  `( {{ ⟦ B ⟧ p ↦ n ↘ b }} ->
-     {{ $| ⇑ (Π a p B) m & n |↘ ⇑ b (m (⇓ a n)) }} )
+  `( {{ ⟦ B ⟧ ρ ↦ n ↘ b }} ->
+     {{ $| ⇑ (Π a ρ B) m & n |↘ ⇑ b (m (⇓ a n)) }} )
 where "'$|' m '&' n '|↘' r" := (eval_app m n r) (in custom judg)
 with eval_sub : sub -> env -> env -> Prop :=
 | eval_sub_id :
-  `( {{ ⟦ Id ⟧s p ↘ p }} )
+  `( {{ ⟦ Id ⟧s ρ ↘ ρ }} )
 | eval_sub_weaken :
-  `( {{ ⟦ Wk ⟧s p ↘ p↯ }} )
+  `( {{ ⟦ Wk ⟧s ρ ↘ ρ ↯ }} )
 | eval_sub_extend :
-  `( {{ ⟦ σ ⟧s p ↘ p' }} ->
-     {{ ⟦ M ⟧ p ↘ m }} ->
-     {{ ⟦ σ ,, M ⟧s p ↘ p' ↦ m }} )
+  `( {{ ⟦ σ ⟧s ρ ↘ ρσ }} ->
+     {{ ⟦ M ⟧ ρ ↘ m }} ->
+     {{ ⟦ σ ,, M ⟧s ρ ↘ ρσ ↦ m }} )
 | eval_sub_compose :
-  `( {{ ⟦ τ ⟧s p ↘ p' }} ->
-     {{ ⟦ σ ⟧s p' ↘ p'' }} ->
-     {{ ⟦ σ ∘ τ ⟧s p ↘ p'' }} )
-where "'⟦' σ '⟧s' p '↘' p'" := (eval_sub σ p p') (in custom judg)
+  `( {{ ⟦ τ ⟧s ρ ↘ ρτ }} ->
+     {{ ⟦ σ ⟧s ρτ ↘ ρτσ }} ->
+     {{ ⟦ σ ∘ τ ⟧s ρ ↘ ρτσ }} )
+where "'⟦' σ '⟧s' ρ '↘' ρσ" := (eval_sub σ ρ ρσ) (in custom judg)
 .
 
 Scheme eval_exp_mut_ind := Induction for eval_exp Sort Prop
 with eval_natrec_mut_ind := Induction for eval_natrec Sort Prop
 with eval_app_mut_ind := Induction for eval_app Sort Prop
 with eval_sub_mut_ind := Induction for eval_sub Sort Prop.
+Combined Scheme eval_mut_ind from
+  eval_exp_mut_ind,
+  eval_natrec_mut_ind,
+  eval_app_mut_ind,
+  eval_sub_mut_ind.
 
 #[export]
 Hint Constructors eval_exp eval_natrec eval_app eval_sub : mcltt.
diff --git a/theories/Core/Semantic/Evaluation/Lemmas.v b/theories/Core/Semantic/Evaluation/Lemmas.v
index a9586819..c7490d66 100644
--- a/theories/Core/Semantic/Evaluation/Lemmas.v
+++ b/theories/Core/Semantic/Evaluation/Lemmas.v
@@ -3,58 +3,62 @@ From Mcltt Require Import Base LibTactics Evaluation.Definitions.
 Import Domain_Notations.
 
 Section functional_eval.
-  Let functional_eval_exp_prop M p m1 (_ : {{ ⟦ M ⟧ p ↘ m1 }}) : Prop := forall m2 (Heval2: {{ ⟦ M ⟧ p ↘ m2 }}), m1 = m2.
-  Let functional_eval_natrec_prop A MZ MS m p r1 (_ : {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r1 }}) : Prop := forall r2 (Heval2 : {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r2 }}), r1 = r2.
-  Let functional_eval_app_prop m n r1 (_ : {{ $| m & n |↘ r1 }}) : Prop := forall r2 (Heval2 : {{ $| m & n |↘ r2 }}), r1 = r2.
-  Let functional_eval_sub_prop σ p p1 (_ : {{ ⟦ σ ⟧s p ↘ p1 }}) : Prop := forall p2 (Heval2 : {{ ⟦ σ ⟧s p ↘ p2 }}), p1 = p2.
-
-  #[local]
-  Ltac unfold_functional_eval :=
-    unfold functional_eval_exp_prop, functional_eval_natrec_prop, functional_eval_app_prop, functional_eval_sub_prop in *;
-    clear functional_eval_exp_prop functional_eval_natrec_prop functional_eval_app_prop functional_eval_sub_prop.
-
-  Lemma functional_eval_exp : forall M p m1 (Heval1 : {{ ⟦ M ⟧ p ↘ m1 }}), functional_eval_exp_prop M p m1 Heval1.
+  Lemma functional_eval :
+    (forall M ρ m1,
+        {{ ⟦ M ⟧ ρ ↘ m1 }} ->
+        forall m2,
+          {{ ⟦ M ⟧ ρ ↘ m2 }} ->
+          m1 = m2) /\
+      (forall A MZ MS m ρ r1,
+          {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r1 }} ->
+          forall r2,
+            {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r2 }} ->
+            r1 = r2) /\
+      (forall m n r1,
+          {{ $| m & n |↘ r1 }} ->
+          forall r2,
+            {{ $| m & n |↘ r2 }} ->
+            r1 = r2) /\
+      (forall σ ρ ρσ1,
+          {{ ⟦ σ ⟧s ρ ↘ ρσ1 }} ->
+          forall ρσ2,
+            {{ ⟦ σ ⟧s ρ ↘ ρσ2 }} ->
+            ρσ1 = ρσ2).
   Proof with ((on_all_hyp: fun H => erewrite H in *; eauto); solve [eauto]) using.
-    induction Heval1
-      using eval_exp_mut_ind
-      with (P0 := functional_eval_natrec_prop)
-           (P1 := functional_eval_app_prop)
-           (P2 := functional_eval_sub_prop);
-      unfold_functional_eval;
-      inversion_clear 1; do 2 f_equal...
+    apply eval_mut_ind; intros;
+      progressive_inversion; do 2 f_equal; try reflexivity...
   Qed.
 
-  Lemma functional_eval_natrec : forall A MZ MS m p r1 (Heval1 : {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r1 }}), functional_eval_natrec_prop A MZ MS m p r1 Heval1.
-  Proof with ((on_all_hyp: fun H => erewrite H in *; eauto); solve [eauto]) using.
-    induction Heval1
-      using eval_natrec_mut_ind
-      with (P := functional_eval_exp_prop)
-           (P1 := functional_eval_app_prop)
-           (P2 := functional_eval_sub_prop);
-      unfold_functional_eval;
-      inversion_clear 1; do 2 f_equal...
+  Corollary functional_eval_exp : forall M ρ m1 m2,
+      {{ ⟦ M ⟧ ρ ↘ m1 }} ->
+      {{ ⟦ M ⟧ ρ ↘ m2 }} ->
+      m1 = m2.
+  Proof.
+    pose proof functional_eval; firstorder.
   Qed.
 
-  Lemma functional_eval_app : forall m n r1 (Heval1 : {{ $| m & n |↘ r1 }}), functional_eval_app_prop m n r1 Heval1.
-  Proof with ((on_all_hyp: fun H => erewrite H in *; eauto); solve [eauto]) using.
-    induction Heval1
-      using eval_app_mut_ind
-      with (P := functional_eval_exp_prop)
-           (P0 := functional_eval_natrec_prop)
-           (P2 := functional_eval_sub_prop);
-      unfold_functional_eval;
-      inversion_clear 1; do 2 f_equal...
+  Corollary functional_eval_natrec : forall A MZ MS m ρ r1 r2,
+      {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r1 }} ->
+      {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r2 }} ->
+      r1 = r2.
+  Proof.
+    pose proof functional_eval; intuition.
   Qed.
 
-  Lemma functional_eval_sub : forall σ p p1 (Heval1 : {{ ⟦ σ ⟧s p ↘ p1 }}), functional_eval_sub_prop σ p p1 Heval1.
-  Proof with ((on_all_hyp: fun H => erewrite H in *; eauto); solve [eauto]) using.
-    induction Heval1
-      using eval_sub_mut_ind
-      with (P := functional_eval_exp_prop)
-           (P0 := functional_eval_natrec_prop)
-           (P1 := functional_eval_app_prop);
-      unfold_functional_eval;
-      inversion_clear 1; do 2 f_equal...
+  Corollary functional_eval_app : forall m n r1 r2,
+      {{ $| m & n |↘ r1 }} ->
+      {{ $| m & n |↘ r2 }} ->
+      r1 = r2.
+  Proof.
+    pose proof functional_eval; intuition.
+  Qed.
+
+  Corollary functional_eval_sub : forall σ ρ ρσ1 ρσ2,
+      {{ ⟦ σ ⟧s ρ ↘ ρσ1 }} ->
+      {{ ⟦ σ ⟧s ρ ↘ ρσ2 }} ->
+      ρσ1 = ρσ2.
+  Proof.
+    pose proof functional_eval; firstorder.
   Qed.
 End functional_eval.
 
@@ -64,13 +68,13 @@ Hint Resolve functional_eval_exp functional_eval_natrec functional_eval_app func
 Ltac functional_eval_rewrite_clear1 :=
   let tactic_error o1 o2 := fail 3 "functional_eval equality between" o1 "and" o2 "cannot be solved by mauto" in
   match goal with
-  | H1 : {{ ⟦ ~?M ⟧ ~?p ↘ ~?m1 }}, H2 : {{ ⟦ ~?M ⟧ ~?p ↘ ~?m2 }} |- _ =>
+  | H1 : {{ ⟦ ~?M ⟧ ~?ρ ↘ ~?m1 }}, H2 : {{ ⟦ ~?M ⟧ ~?ρ ↘ ~?m2 }} |- _ =>
       clean replace m2 with m1 by first [solve [mauto 2] | tactic_error m2 m1]; clear H2
   | H1 : {{ $| ~?m & ~?n |↘ ~?r1 }}, H2 : {{ $| ~?m & ~?n |↘ ~?r2 }} |- _ =>
       clean replace r2 with r1 by first [solve [mauto 2] | tactic_error r2 r1]; clear H2
-  | H1 : {{ rec ~?m ⟦return ~?A | zero -> ~?MZ | succ -> ~?MS end⟧ ~?p ↘ ~?r1 }}, H2 : {{ rec ~?m ⟦return ~?A | zero -> ~?MZ | succ -> ~?MS end⟧ ~?p ↘ ~?r2 }} |- _ =>
+  | H1 : {{ rec ~?m ⟦return ~?A | zero -> ~?MZ | succ -> ~?MS end⟧ ~?ρ ↘ ~?r1 }}, H2 : {{ rec ~?m ⟦return ~?A | zero -> ~?MZ | succ -> ~?MS end⟧ ~?ρ ↘ ~?r2 }} |- _ =>
       clean replace r2 with r1 by first [solve [mauto 2] | tactic_error r2 r1]; clear H2
-  | H1 : {{ ⟦ ~?σ ⟧s ~?p ↘ ~?p1 }}, H2 : {{ ⟦ ~?σ ⟧s ~?p ↘ ~?p2 }} |- _ =>
-      clean replace p2 with p1 by first [solve [mauto 2] | tactic_error p2 p1]; clear H2
+  | H1 : {{ ⟦ ~?σ ⟧s ~?ρ ↘ ~?ρσ1 }}, H2 : {{ ⟦ ~?σ ⟧s ~?ρ ↘ ~?ρσ2 }} |- _ =>
+      clean replace ρσ2 with ρσ1 by first [solve [mauto 2] | tactic_error ρσ2 ρσ1]; clear H2
   end.
 Ltac functional_eval_rewrite_clear := repeat functional_eval_rewrite_clear1.
diff --git a/theories/Core/Semantic/Evaluation/Tactics.v b/theories/Core/Semantic/Evaluation/Tactics.v
index 1a1694f5..5669f3b2 100644
--- a/theories/Core/Semantic/Evaluation/Tactics.v
+++ b/theories/Core/Semantic/Evaluation/Tactics.v
@@ -4,7 +4,9 @@ Import Domain_Notations.
 Ltac simplify_evals :=
   functional_eval_rewrite_clear;
   clear_dups;
-  repeat (match_by_head eval_exp ltac:(fun H => directed inversion H; subst; clear H)
-          || match_by_head eval_sub ltac:(fun H => directed inversion H; subst; clear H));
+  repeat (match_by_head eval_exp ltac:(fun H => directed dependent destruction H)
+          || match_by_head eval_app ltac:(fun H => directed dependent destruction H)
+          || match_by_head eval_natrec ltac:(fun H => directed dependent destruction H)
+          || match_by_head eval_sub ltac:(fun H => directed dependent destruction H));
   functional_eval_rewrite_clear;
   clear_dups.
diff --git a/theories/Core/Semantic/NbE.v b/theories/Core/Semantic/NbE.v
index 87c3cc14..16b5ced3 100644
--- a/theories/Core/Semantic/NbE.v
+++ b/theories/Core/Semantic/NbE.v
@@ -7,18 +7,18 @@ Generalizable All Variables.
 Inductive initial_env : ctx -> env -> Prop :=
 | initial_env_nil : initial_env nil empty_env
 | initial_env_cons :
-  `( initial_env Γ p ->
-     {{ ⟦ A ⟧ p ↘ a }} ->
-     initial_env (A :: Γ) d{{{ p ↦ ⇑! a (length Γ) }}}).
+  `( initial_env Γ ρ ->
+     {{ ⟦ A ⟧ ρ ↘ a }} ->
+     initial_env (A :: Γ) d{{{ ρ ↦ ⇑! a (length Γ) }}}).
 
 #[export]
 Hint Constructors initial_env : mcltt.
 
-Lemma functional_initial_env : forall Γ p,
-    initial_env Γ p ->
-    forall p',
-      initial_env Γ p' ->
-      p = p'.
+Lemma functional_initial_env : forall Γ ρ,
+    initial_env Γ ρ ->
+    forall ρ',
+      initial_env Γ ρ' ->
+      ρ = ρ'.
 Proof.
   induction 1; intros ? Hother; inversion_clear Hother; eauto.
   erewrite IHinitial_env in *; try eassumption;
@@ -32,16 +32,16 @@ Hint Resolve functional_initial_env : mcltt.
 Ltac functional_initial_env_rewrite_clear1 :=
   let tactic_error o1 o2 := fail 3 "functional_initial_env equality between" o1 "and" o2 "cannot be solved by mauto" in
   match goal with
-  | H1 : initial_env ?G ?p, H2 : initial_env ?G ?p' |- _ =>
-      clean replace p' with p by first [solve [mauto 2] | tactic_error p' p]; clear H2
+  | H1 : initial_env ?G ?ρ, H2 : initial_env ?G ?ρ' |- _ =>
+      clean replace ρ' with ρ by first [solve [mauto 2] | tactic_error ρ' ρ]; clear H2
   end.
 Ltac functional_initial_env_rewrite_clear := repeat functional_initial_env_rewrite_clear1.
 
 Inductive nbe : ctx -> exp -> typ -> nf -> Prop :=
 | nbe_run :
-  `( initial_env Γ p ->
-     {{ ⟦ A ⟧ p ↘ a }} ->
-     {{ ⟦ M ⟧ p ↘ m }} ->
+  `( initial_env Γ ρ ->
+     {{ ⟦ A ⟧ ρ ↘ a }} ->
+     {{ ⟦ M ⟧ ρ ↘ m }} ->
      {{ Rnf ⇓ a m in (length Γ) ↘ w }} ->
      nbe Γ M A w ).
 
@@ -117,8 +117,8 @@ Hint Resolve functional_nbe_of_typ : mcltt.
 
 Inductive nbe_ty : ctx -> typ -> nf -> Prop :=
 | nbe_ty_run :
-  `( initial_env Γ p ->
-     {{ ⟦ M ⟧ p ↘ m }} ->
+  `( initial_env Γ ρ ->
+     {{ ⟦ M ⟧ ρ ↘ m }} ->
      {{ Rtyp m in (length Γ) ↘ W }} ->
      nbe_ty Γ M W ).
 
diff --git a/theories/Core/Semantic/PER/Definitions.v b/theories/Core/Semantic/PER/Definitions.v
index 8b34c85b..3e38aafa 100644
--- a/theories/Core/Semantic/PER/Definitions.v
+++ b/theories/Core/Semantic/PER/Definitions.v
@@ -18,7 +18,7 @@ Generalizable All Variables.
 
 (** Helper Bundles *)
 (* Related modulo evaluation *)
-Inductive rel_mod_eval (R : relation domain -> domain -> domain -> Prop) A p A' p' R' : Prop := mk_rel_mod_eval : forall a a', {{ ⟦ A ⟧ p ↘ a }} -> {{ ⟦ A' ⟧ p' ↘ a' }} -> {{ DF a ≈ a' ∈ R ↘ R' }} -> rel_mod_eval R A p A' p' R'.
+Inductive rel_mod_eval (R : relation domain -> domain -> domain -> Prop) A ρ A' ρ' R' : Prop := mk_rel_mod_eval : forall a a', {{ ⟦ A ⟧ ρ ↘ a }} -> {{ ⟦ A' ⟧ ρ' ↘ a' }} -> {{ DF a ≈ a' ∈ R ↘ R' }} -> rel_mod_eval R A ρ A' ρ' R'.
 #[global]
 Arguments mk_rel_mod_eval {_ _ _ _ _ _}.
 #[export]
@@ -102,9 +102,9 @@ Section Per_univ_elem_core_def.
          (equiv_a_a' : {{ DF a ≈ a' ∈ per_univ_elem_core ↘ in_rel}}),
           PER in_rel ->
           (forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}),
-              rel_mod_eval per_univ_elem_core B d{{{ p ↦ c }}} B' d{{{ p' ↦ c' }}} (out_rel equiv_c_c')) ->
+              rel_mod_eval per_univ_elem_core B d{{{ ρ ↦ c }}} B' d{{{ ρ' ↦ c' }}} (out_rel equiv_c_c')) ->
           (elem_rel <~> fun f f' => forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_mod_app f c f' c' (out_rel equiv_c_c')) ->
-          {{ DF Π a p B ≈ Π a' p' B' ∈ per_univ_elem_core ↘ elem_rel }} }
+          {{ DF Π a ρ B ≈ Π a' ρ' B' ∈ per_univ_elem_core ↘ elem_rel }} }
   | per_univ_elem_core_neut :
     `{ forall (elem_rel : relation domain),
           {{ Dom b ≈ b' ∈ per_bot }} ->
@@ -122,16 +122,16 @@ Section Per_univ_elem_core_def.
           (elem_rel <~> per_nat) ->
           motive elem_rel d{{{ ℕ }}} d{{{ ℕ }}})
       (case_Pi :
-        forall {A p B A' p' B' in_rel}
+        forall {a ρ B a' ρ' B' in_rel}
            (out_rel : forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), relation domain)
            {elem_rel},
-          {{ DF A ≈ A' ∈ per_univ_elem_core ↘ in_rel }} ->
-          motive in_rel A A' ->
+          {{ DF a ≈ a' ∈ per_univ_elem_core ↘ in_rel }} ->
+          motive in_rel a a' ->
           PER in_rel ->
           (forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}),
-              rel_mod_eval (fun R x y => {{ DF x ≈ y ∈ per_univ_elem_core ↘ R }} /\ motive R x y) B d{{{ p ↦ c }}} B' d{{{ p' ↦ c' }}} (out_rel equiv_c_c')) ->
+              rel_mod_eval (fun R x y => {{ DF x ≈ y ∈ per_univ_elem_core ↘ R }} /\ motive R x y) B d{{{ ρ ↦ c }}} B' d{{{ ρ' ↦ c' }}} (out_rel equiv_c_c')) ->
           (elem_rel <~> fun f f' => forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_mod_app f c f' c' (out_rel equiv_c_c')) ->
-          motive elem_rel d{{{ Π A p B }}} d{{{ Π A' p' B' }}})
+          motive elem_rel d{{{ Π a ρ B }}} d{{{ Π a' ρ' B' }}})
       (case_ne : forall {a b a' b' elem_rel},
           {{ Dom b ≈ b' ∈ per_bot }} ->
           (elem_rel <~> per_ne) ->
@@ -193,16 +193,16 @@ Section Per_univ_elem_ind_def.
           (elem_rel <~> per_nat) ->
           motive i elem_rel d{{{ ℕ }}} d{{{ ℕ }}})
       (case_Pi :
-        forall i {A p B A' p' B' in_rel}
+        forall i {a ρ B a' ρ' B' in_rel}
            (out_rel : forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), relation domain)
            {elem_rel},
-          {{ DF A ≈ A' ∈ per_univ_elem i ↘ in_rel }} ->
-          motive i in_rel A A' ->
+          {{ DF a ≈ a' ∈ per_univ_elem i ↘ in_rel }} ->
+          motive i in_rel a a' ->
           PER in_rel ->
           (forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}),
-              rel_mod_eval (fun R x y => {{ DF x ≈ y ∈ per_univ_elem i ↘ R }} /\ motive i R x y) B d{{{ p ↦ c }}} B' d{{{ p' ↦ c' }}} (out_rel equiv_c_c')) ->
+              rel_mod_eval (fun R x y => {{ DF x ≈ y ∈ per_univ_elem i ↘ R }} /\ motive i R x y) B d{{{ ρ ↦ c }}} B' d{{{ ρ' ↦ c' }}} (out_rel equiv_c_c')) ->
           (elem_rel <~> fun f f' => forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_mod_app f c f' c' (out_rel equiv_c_c')) ->
-          motive i elem_rel d{{{ Π A p B }}} d{{{ Π A' p' B' }}})
+          motive i elem_rel d{{{ Π a ρ B }}} d{{{ Π a' ρ' B' }}})
       (case_ne : forall i {a b a' b' elem_rel},
           {{ Dom b ≈ b' ∈ per_bot }} ->
           (elem_rel <~> per_ne) ->
@@ -244,12 +244,12 @@ Inductive per_subtyp : nat -> domain -> domain -> Prop :=
         {{ DF a ≈ a' ∈ per_univ_elem i ↘ in_rel }} ->
         (forall c c' b b',
             {{ Dom c ≈ c' ∈ in_rel }} ->
-            {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
-            {{ ⟦ B' ⟧ p' ↦ c' ↘ b' }} ->
+            {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
+            {{ ⟦ B' ⟧ ρ' ↦ c' ↘ b' }} ->
             {{ Sub b <: b' at i }}) ->
-        {{ DF Π a p B ≈ Π a p B ∈ per_univ_elem i ↘ elem_rel }} ->
-        {{ DF Π a' p' B' ≈ Π a' p' B' ∈ per_univ_elem i ↘ elem_rel' }} ->
-        {{ Sub Π a p B <: Π a' p' B' at i }})
+        {{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
+        {{ DF Π a' ρ' B' ≈ Π a' ρ' B' ∈ per_univ_elem i ↘ elem_rel' }} ->
+        {{ Sub Π a ρ B <: Π a' ρ' B' at i }})
 where "'Sub' a <: b 'at' i" := (per_subtyp i a b) (in custom judg) : type_scope.
 
 #[export]
@@ -257,7 +257,7 @@ where "'Sub' a <: b 'at' i" := (per_subtyp i a b) (in custom judg) : type_scope.
 
 (** Context/Environment PER *)
 
-Definition rel_typ (i : nat) (A : typ) (p : env) (A' : typ) (p' : env) R' := rel_mod_eval (per_univ_elem i) A p A' p' R'.
+Definition rel_typ i A ρ A' ρ' R' := rel_mod_eval (per_univ_elem i) A ρ A' ρ' R'.
 Arguments rel_typ _ _ _ _ _ _ /.
 #[export]
 Hint Transparent rel_typ : mcltt.
@@ -267,19 +267,19 @@ Hint Unfold rel_typ : mcltt.
 Inductive per_ctx_env : relation env -> ctx -> ctx -> Prop :=
 | per_ctx_env_nil :
   `{ forall env_rel,
-        (env_rel <~> fun p p' => True) ->
+        (env_rel <~> fun ρ ρ' => True) ->
         {{ EF ⋅ ≈ ⋅ ∈ per_ctx_env ↘ env_rel }} }
 | per_ctx_env_cons :
   `{ forall tail_rel
-        (head_rel : forall {p p'} (equiv_p_p' : {{ Dom p ≈ p' ∈ tail_rel }}), relation domain)
+        (head_rel : forall {ρ ρ'} (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ tail_rel }}), relation domain)
         env_rel
         (equiv_Γ_Γ' : {{ EF Γ ≈ Γ' ∈ per_ctx_env ↘ tail_rel }}),
         PER tail_rel ->
-        (forall {p p'} (equiv_p_p' : {{ Dom p ≈ p' ∈ tail_rel }}),
-            rel_typ i A p A' p' (head_rel equiv_p_p')) ->
-        (env_rel <~> fun p p' =>
-             exists (equiv_p_drop_p'_drop : {{ Dom p ↯ ≈ p' ↯ ∈ tail_rel }}),
-               {{ Dom ~(p 0) ≈ ~(p' 0) ∈ head_rel equiv_p_drop_p'_drop }}) ->
+        (forall {ρ ρ'} (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ tail_rel }}),
+            rel_typ i A ρ A' ρ' (head_rel equiv_ρ_ρ')) ->
+        (env_rel <~> fun ρ ρ' =>
+             exists (equiv_ρ_drop_ρ'_drop : {{ Dom ρ ↯ ≈ ρ' ↯ ∈ tail_rel }}),
+               {{ Dom ~(ρ 0) ≈ ~(ρ' 0) ∈ head_rel equiv_ρ_drop_ρ'_drop }}) ->
         {{ EF Γ, A ≈ Γ', A' ∈ per_ctx_env ↘ env_rel }} }
 .
 #[export]
@@ -294,7 +294,6 @@ Hint Unfold valid_ctx : mcltt.
 
 Reserved Notation "'SubE' Γ <: Δ" (in custom judg at level 90, Γ custom exp, Δ custom exp).
 
-
 Inductive per_ctx_subtyp : ctx -> ctx -> Prop :=
 | per_ctx_subtyp_nil :
   {{ SubE ⋅ <: ⋅ }}
@@ -302,10 +301,10 @@ Inductive per_ctx_subtyp : ctx -> ctx -> Prop :=
   `{ forall tail_rel env_rel env_rel',
         {{ SubE Γ <: Γ' }} ->
         {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ tail_rel }} ->
-        (forall p p' a a'
-           (equiv_p_p' : {{ Dom p ≈ p' ∈ tail_rel }}),
-            {{ ⟦ A ⟧ p ↘ a }} ->
-            {{ ⟦ A' ⟧ p' ↘ a' }} ->
+        (forall ρ ρ' a a'
+           (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ tail_rel }}),
+            {{ ⟦ A ⟧ ρ ↘ a }} ->
+            {{ ⟦ A' ⟧ ρ' ↘ a' }} ->
             {{ Sub a <: a' at i }}) ->
         {{ EF Γ , A ≈ Γ , A ∈ per_ctx_env ↘ env_rel }} ->
         {{ EF Γ' , A' ≈ Γ' , A' ∈ per_ctx_env ↘ env_rel' }} ->
diff --git a/theories/Core/Semantic/PER/Lemmas.v b/theories/Core/Semantic/PER/Lemmas.v
index 5b8583d9..01ec4fed 100644
--- a/theories/Core/Semantic/PER/Lemmas.v
+++ b/theories/Core/Semantic/PER/Lemmas.v
@@ -4,7 +4,7 @@ From Mcltt Require Import Base LibTactics.
 From Mcltt.Core Require Import PER.Definitions PER.CoreTactics.
 Import Domain_Notations.
 
-Add Parametric Morphism R0 `(R0_morphism : Proper _ ((@relation_equivalence domain) ==> (@relation_equivalence domain)) R0) A p A' p' : (rel_mod_eval R0 A p A' p')
+Add Parametric Morphism R0 `(R0_morphism : Proper _ ((@relation_equivalence domain) ==> (@relation_equivalence domain)) R0) A ρ A' ρ' : (rel_mod_eval R0 A ρ A' ρ')
     with signature (@relation_equivalence domain) ==> iff as rel_mod_eval_morphism.
 Proof.
   split; intros []; econstructor; try eassumption;
@@ -227,7 +227,7 @@ Proof with mautosolve.
   reflexivity.
 Qed.
 
-Add Parametric Morphism i A p A' p' : (rel_typ i A p A' p')
+Add Parametric Morphism i A ρ A' ρ' : (rel_typ i A ρ A' ρ')
     with signature (@relation_equivalence domain) ==> iff as rel_typ_morphism.
 Proof.
   intros * HRR'.
@@ -401,24 +401,24 @@ Qed.
 Ltac do_per_univ_elem_irrel_assert1 :=
   let tactic_error o1 o2 := fail 2 "per_univ_elem_irrel biconditional between" o1 "and" o2 "cannot be solved" in
   match goal with
-  | H1 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i ↘ ?R1 }},
-      H2 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
+  | H1 : {{ DF ~?a ≈ ~_ ∈ per_univ_elem ?i ↘ ?R1 }},
+      H2 : {{ DF ~?a ≈ ~_ ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
       assert_fails (unify R1 R2);
       match goal with
       | H : R1 <~> R2 |- _ => fail 1
       | H : R2 <~> R1 |- _ => fail 1
       | _ => assert (R1 <~> R2) by (eapply per_univ_elem_right_irrel; [apply H1 | apply H2]) || tactic_error R1 R2
       end
-  | H1 : {{ DF ~_ ≈ ~?B ∈ per_univ_elem ?i ↘ ?R1 }},
-      H2 : {{ DF ~_ ≈ ~?B ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
+  | H1 : {{ DF ~_ ≈ ~?b ∈ per_univ_elem ?i ↘ ?R1 }},
+      H2 : {{ DF ~_ ≈ ~?b ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
       assert_fails (unify R1 R2);
       match goal with
       | H : R1 <~> R2 |- _ => fail 1
       | H : R2 <~> R1 |- _ => fail 1
       | _ => assert (R1 <~> R2) by (eapply per_univ_elem_left_irrel; [apply H1 | apply H2]) || tactic_error R1 R2
       end
-  | H1 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i ↘ ?R1 }},
-      H2 : {{ DF ~_ ≈ ~?A ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
+  | H1 : {{ DF ~?a ≈ ~_ ∈ per_univ_elem ?i ↘ ?R1 }},
+      H2 : {{ DF ~_ ≈ ~?a ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
       (* Order matters less here as H1 and H2 cannot be exchanged *)
       assert_fails (unify R1 R2);
       match goal with
@@ -437,15 +437,15 @@ Ltac handle_per_univ_elem_irrel :=
   apply_relation_equivalence;
   clear_dups.
 
-Lemma per_univ_elem_trans : forall i R A1 A2,
-    per_univ_elem i R A1 A2 ->
-    (forall j A3,
-        per_univ_elem j R A2 A3 ->
-        per_univ_elem i R A1 A3) /\
-      (forall a1 a2 a3,
-          R a1 a2 ->
-          R a2 a3 ->
-          R a1 a3).
+Lemma per_univ_elem_trans : forall i R a1 a2,
+    per_univ_elem i R a1 a2 ->
+    (forall j a3,
+        per_univ_elem j R a2 a3 ->
+        per_univ_elem i R a1 a3) /\
+      (forall m1 m2 m3,
+          R m1 m2 ->
+          R m2 m3 ->
+          R m1 m3).
 Proof with (basic_per_univ_elem_econstructor; mautosolve).
   induction 1 using per_univ_elem_ind;
     [> split;
@@ -486,30 +486,30 @@ Proof with (basic_per_univ_elem_econstructor; mautosolve).
     idtac...
 Qed.
 
-Corollary per_univ_trans : forall i j R A1 A2 A3,
-    per_univ_elem i R A1 A2 ->
-    per_univ_elem j R A2 A3 ->
-    per_univ_elem i R A1 A3.
+Corollary per_univ_trans : forall i j R a1 a2 a3,
+    per_univ_elem i R a1 a2 ->
+    per_univ_elem j R a2 a3 ->
+    per_univ_elem i R a1 a3.
 Proof.
   intros * ?%per_univ_elem_trans.
   firstorder.
 Qed.
 
-Corollary per_univ_trans' : forall i j A1 A2 A3,
-    {{ Dom A1 ≈ A2 ∈ per_univ i }} ->
-    {{ Dom A2 ≈ A3 ∈ per_univ j }} ->
-    {{ Dom A1 ≈ A3 ∈ per_univ i }}.
+Corollary per_univ_trans' : forall i j a1 a2 a3,
+    {{ Dom a1 ≈ a2 ∈ per_univ i }} ->
+    {{ Dom a2 ≈ a3 ∈ per_univ j }} ->
+    {{ Dom a1 ≈ a3 ∈ per_univ i }}.
 Proof.
   intros * [? ?] [? ?].
   handle_per_univ_elem_irrel.
   firstorder mauto using per_univ_trans.
 Qed.
 
-Corollary per_elem_trans : forall i R A1 A2 a1 a2 a3,
-    per_univ_elem i R A1 A2 ->
-    R a1 a2 ->
-    R a2 a3 ->
-    R a1 a3.
+Corollary per_elem_trans : forall i R a1 a2 m1 m2 m3,
+    per_univ_elem i R a1 a2 ->
+    R m1 m2 ->
+    R m2 m3 ->
+    R m1 m3.
 Proof.
   intros * ?% per_univ_elem_trans.
   firstorder.
@@ -532,7 +532,7 @@ Proof.
 Qed.
 
 #[export]
-Instance per_elem_PER {i R A B} `(H : per_univ_elem i R A B) : PER R.
+Instance per_elem_PER {i R a b} `(H : per_univ_elem i R a b) : PER R.
 Proof.
   split.
   - eauto using (per_elem_sym _ _ _ _ _ _ H).
@@ -541,15 +541,15 @@ Qed.
 
 (* This lemma gets rid of the unnecessary PER premise. *)
 Lemma per_univ_elem_pi' :
-  forall i a a' p B p' B'
+  forall i a a' ρ B ρ' B'
     (in_rel : relation domain)
     (out_rel : forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), relation domain)
     elem_rel,
     {{ DF a ≈ a' ∈ per_univ_elem i ↘ in_rel}} ->
     (forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}),
-        rel_mod_eval (per_univ_elem i) B d{{{ p ↦ c }}} B' d{{{ p' ↦ c' }}} (out_rel equiv_c_c')) ->
+        rel_mod_eval (per_univ_elem i) B d{{{ ρ ↦ c }}} B' d{{{ ρ' ↦ c' }}} (out_rel equiv_c_c')) ->
     (elem_rel <~> fun f f' => forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_mod_app f c f' c' (out_rel equiv_c_c')) ->
-    {{ DF Π a p B ≈ Π a' p' B' ∈ per_univ_elem i ↘ elem_rel }}.
+    {{ DF Π a ρ B ≈ Π a' ρ' B' ∈ per_univ_elem i ↘ elem_rel }}.
 Proof.
   intros.
   basic_per_univ_elem_econstructor; eauto.
@@ -562,12 +562,12 @@ Ltac per_univ_elem_econstructor :=
 #[export]
 Hint Resolve per_univ_elem_pi' : mcltt.
 
-Lemma per_univ_elem_pi_clean_inversion : forall {i j a a' in_rel p p' B B' elem_rel},
+Lemma per_univ_elem_pi_clean_inversion : forall {i j a a' in_rel ρ ρ' B B' elem_rel},
     {{ DF a ≈ a' ∈ per_univ_elem i ↘ in_rel }} ->
-    {{ DF Π a p B ≈ Π a' p' B' ∈ per_univ_elem j ↘ elem_rel }} ->
+    {{ DF Π a ρ B ≈ Π a' ρ' B' ∈ per_univ_elem j ↘ elem_rel }} ->
     exists (out_rel : forall {c c'} (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), relation domain),
       (forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}),
-          rel_mod_eval (per_univ_elem j) B d{{{ p ↦ c }}} B' d{{{ p' ↦ c' }}} (out_rel equiv_c_c')) /\
+          rel_mod_eval (per_univ_elem j) B d{{{ ρ ↦ c }}} B' d{{{ ρ' ↦ c' }}} (out_rel equiv_c_c')) /\
         (elem_rel <~> fun f f' => forall c c' (equiv_c_c' : {{ Dom c ≈ c' ∈ in_rel }}), rel_mod_app f c f' c' (out_rel equiv_c_c')).
 Proof.
   intros * Ha HΠ.
@@ -577,7 +577,7 @@ Proof.
   split.
   - instantiate (1 := fun c c' (equiv_c_c' : in_rel c c') m m' =>
                         forall R,
-                          rel_typ j B d{{{ p ↦ c }}} B' d{{{ p' ↦ c' }}} R ->
+                          rel_typ j B d{{{ ρ ↦ c }}} B' d{{{ ρ' ↦ c' }}} R ->
                           R m m').
     intros.
     assert (in_rel0 c c') by intuition.
@@ -682,12 +682,12 @@ Proof.
     intuition.
 Qed.
 
-Lemma per_elem_subtyping_gen : forall A B i A' B' R R' a b,
-    {{ Sub A <: B at i }} ->
-    {{ DF A ≈ A' ∈ per_univ_elem i ↘ R }} ->
-    {{ DF B ≈ B' ∈ per_univ_elem i ↘ R' }} ->
-    R a b ->
-    R' a b.
+Lemma per_elem_subtyping_gen : forall a b i a' b' R R' m n,
+    {{ Sub a <: b at i }} ->
+    {{ DF a ≈ a' ∈ per_univ_elem i ↘ R }} ->
+    {{ DF b ≈ b' ∈ per_univ_elem i ↘ R' }} ->
+    R m n ->
+    R' m n.
 Proof.
   intros.
   eapply per_elem_subtyping; saturate_refl; try eassumption.
@@ -701,7 +701,7 @@ Proof.
     subst;
     mauto;
     destruct_all.
-  assert ({{ DF Π A p B ≈ Π A' p' B' ∈ per_univ_elem i ↘ elem_rel }})
+  assert ({{ DF Π a ρ B ≈ Π a' ρ' B' ∈ per_univ_elem i ↘ elem_rel }})
     by (eapply per_univ_elem_pi'; eauto; intros; destruct_rel_mod_eval; mauto).
   saturate_refl.
   econstructor; eauto.
@@ -726,11 +726,11 @@ Qed.
 #[export]
 Hint Resolve per_subtyp_refl2 : mcltt.
 
-Lemma per_subtyp_trans : forall A1 A2 i,
-    {{ Sub A1 <: A2 at i }} ->
-    forall A3,
-      {{ Sub A2 <: A3 at i }} ->
-      {{ Sub A1 <: A3 at i }}.
+Lemma per_subtyp_trans : forall a1 a2 i,
+    {{ Sub a1 <: a2 at i }} ->
+    forall a3,
+      {{ Sub a2 <: a3 at i }} ->
+      {{ Sub a1 <: a3 at i }}.
 Proof.
   induction 1; intros ? Hsub; simpl in *.
   1-3: progressive_inversion; mauto.
@@ -756,39 +756,39 @@ Proof.
   eauto using per_subtyp_trans.
 Qed.
 
-Lemma per_subtyp_transp : forall A B i A' B' R R',
-    {{ Sub A <: B at i }} ->
-    {{ DF A ≈ A' ∈ per_univ_elem i ↘ R }} ->
-    {{ DF B ≈ B' ∈ per_univ_elem i ↘ R' }} ->
-    {{ Sub A' <: B' at i }}.
+Lemma per_subtyp_transp : forall a b i a' b' R R',
+    {{ Sub a <: b at i }} ->
+    {{ DF a ≈ a' ∈ per_univ_elem i ↘ R }} ->
+    {{ DF b ≈ b' ∈ per_univ_elem i ↘ R' }} ->
+    {{ Sub a' <: b' at i }}.
 Proof.
   mauto using per_subtyp_refl1, per_subtyp_refl2.
 Qed.
 
-Lemma per_subtyp_cumu : forall A1 A2 i,
-    {{ Sub A1 <: A2 at i }} ->
+Lemma per_subtyp_cumu : forall a1 a2 i,
+    {{ Sub a1 <: a2 at i }} ->
     forall j,
       i <= j ->
-      {{ Sub A1 <: A2 at j }}.
+      {{ Sub a1 <: a2 at j }}.
 Proof.
   induction 1; intros; econstructor; mauto.
   lia.
 Qed.
 
 #[export]
- Hint Resolve per_subtyp_cumu : mcltt.
+Hint Resolve per_subtyp_cumu : mcltt.
 
-Lemma per_subtyp_cumu_left : forall A1 A2 i j,
-    {{ Sub A1 <: A2 at i }} ->
-    {{ Sub A1 <: A2 at max i j }}.
+Lemma per_subtyp_cumu_left : forall a1 a2 i j,
+    {{ Sub a1 <: a2 at i }} ->
+    {{ Sub a1 <: a2 at max i j }}.
 Proof.
   intros. eapply per_subtyp_cumu; try eassumption.
   lia.
 Qed.
 
-Lemma per_subtyp_cumu_right : forall A1 A2 i j,
-    {{ Sub A1 <: A2 at i }} ->
-    {{ Sub A1 <: A2 at max j i }}.
+Lemma per_subtyp_cumu_right : forall a1 a2 i j,
+    {{ Sub a1 <: a2 at i }} ->
+    {{ Sub a1 <: a2 at max j i }}.
 Proof.
   intros. eapply per_subtyp_cumu; try eassumption.
   lia.
@@ -823,35 +823,34 @@ Proof with (destruct_rel_typ; handle_per_univ_elem_irrel; eexists; intuition).
     apply_relation_equivalence;
     try reflexivity.
   specialize (IHHorig _ _ equiv_Γ_Γ'0).
-  intros p p'.
+  intros ρ ρ'.
   split; intros [].
-  - assert (equiv0_p_drop_p'_drop : {{ Dom p ↯ ≈ p' ↯ ∈ tail_rel0 }}) by intuition...
-  - assert (equiv_p_drop_p'_drop : {{ Dom p ↯ ≈ p' ↯ ∈ tail_rel }}) by intuition...
+  - assert {{ Dom ρ ↯ ≈ ρ' ↯ ∈ tail_rel0 }} by intuition...
+  - assert {{ Dom ρ ↯ ≈ ρ' ↯ ∈ tail_rel }} by intuition...
 Qed.
 
 Lemma per_ctx_env_sym : forall Γ Δ R,
     {{ DF Γ ≈ Δ ∈ per_ctx_env ↘ R }} ->
     {{ DF Δ ≈ Γ ∈ per_ctx_env ↘ R }} /\
-      (forall o p,
-          {{ Dom o ≈ p ∈ R }} ->
-          {{ Dom p ≈ o ∈ R }}).
+      (forall ρ ρ',
+          {{ Dom ρ ≈ ρ' ∈ R }} ->
+          {{ Dom ρ' ≈ ρ ∈ R }}).
 Proof with solve [intuition].
   simpl.
   induction 1; split; simpl in *; destruct_conjs; try econstructor; intuition;
     pose proof (@relation_equivalence_pointwise env).
-  - assert (tail_rel p' p) by eauto.
-    assert (tail_rel p p) by (etransitivity; eassumption).
+  - assert (tail_rel ρ' ρ) by eauto.
+    assert (tail_rel ρ ρ) by (etransitivity; eassumption).
     destruct_rel_mod_eval.
     handle_per_univ_elem_irrel.
     econstructor; eauto.
     symmetry...
   - apply_relation_equivalence.
     destruct_conjs.
-    assert (tail_rel d{{{ p ↯ }}} d{{{ o ↯ }}}) by eauto.
-    assert (tail_rel d{{{ p ↯ }}} d{{{ p ↯ }}}) by (etransitivity; eassumption).
+    assert (tail_rel d{{{ ρ' ↯ }}} d{{{ ρ ↯ }}}) by eauto.
+    assert (tail_rel d{{{ ρ ↯ }}} d{{{ ρ ↯ }}}) by (etransitivity; eassumption).
     destruct_rel_mod_eval.
-    eexists.
-    handle_per_univ_elem_irrel; symmetry; intuition.
+    eexists; symmetry; handle_per_univ_elem_irrel; intuition.
 Qed.
 
 Corollary per_ctx_sym : forall Γ Δ R,
@@ -862,10 +861,10 @@ Proof.
   firstorder.
 Qed.
 
-Corollary per_env_sym : forall Γ Δ R o p,
+Corollary per_env_sym : forall Γ Δ R ρ ρ',
     {{ DF Γ ≈ Δ ∈ per_ctx_env ↘ R }} ->
-    {{ Dom o ≈ p ∈ R }} ->
-    {{ Dom p ≈ o ∈ R }}.
+    {{ Dom ρ ≈ ρ' ∈ R }} ->
+    {{ Dom ρ' ≈ ρ ∈ R }}.
 Proof.
   intros * ?%per_ctx_env_sym.
   firstorder.
@@ -933,10 +932,10 @@ Lemma per_ctx_env_trans : forall Γ1 Γ2 R,
     (forall Γ3,
         {{ DF Γ2 ≈ Γ3 ∈ per_ctx_env ↘ R }} ->
         {{ DF Γ1 ≈ Γ3 ∈ per_ctx_env ↘ R }}) /\
-      (forall p1 p2 p3,
-          {{ Dom p1 ≈ p2 ∈ R }} ->
-          {{ Dom p2 ≈ p3 ∈ R }} ->
-          {{ Dom p1 ≈ p3 ∈ R }}).
+      (forall ρ1 ρ2 ρ3,
+          {{ Dom ρ1 ≈ ρ2 ∈ R }} ->
+          {{ Dom ρ2 ≈ ρ3 ∈ R }} ->
+          {{ Dom ρ1 ≈ ρ3 ∈ R }}).
 Proof with solve [eauto using per_univ_trans].
   simpl.
   induction 1; subst;
@@ -949,15 +948,15 @@ Proof with solve [eauto using per_univ_trans].
   - econstructor; only 4: reflexivity; eauto.
     + apply_relation_equivalence. intuition.
     + intros.
-      assert (tail_rel p p) by intuition.
-      assert (tail_rel0 p p') by intuition.
+      assert (tail_rel ρ ρ) by intuition.
+      assert (tail_rel0 ρ ρ') by intuition.
       destruct_rel_typ.
       handle_per_univ_elem_irrel.
       econstructor; intuition.
       (* This one cannot be replaced with `etransitivity` as we need different `i`s. *)
       eapply per_univ_trans; [| eassumption]; eassumption.
   - destruct_conjs.
-    assert (tail_rel d{{{ p1 ↯ }}} d{{{ p3 ↯ }}}) by eauto.
+    assert (tail_rel d{{{ ρ1 ↯ }}} d{{{ ρ3 ↯ }}}) by eauto.
     destruct_rel_typ.
     handle_per_univ_elem_irrel.
     eexists.
@@ -974,11 +973,11 @@ Proof.
   firstorder.
 Qed.
 
-Corollary per_env_trans : forall Γ1 Γ2 R p1 p2 p3,
+Corollary per_env_trans : forall Γ1 Γ2 R ρ1 ρ2 ρ3,
     {{ DF Γ1 ≈ Γ2 ∈ per_ctx_env ↘ R }} ->
-    {{ Dom p1 ≈ p2 ∈ R }} ->
-    {{ Dom p2 ≈ p3 ∈ R }} ->
-    {{ Dom p1 ≈ p3 ∈ R }}.
+    {{ Dom ρ1 ≈ ρ2 ∈ R }} ->
+    {{ Dom ρ2 ≈ ρ3 ∈ R }} ->
+    {{ Dom ρ1 ≈ ρ3 ∈ R }}.
 Proof.
   intros * ?% per_ctx_env_trans.
   firstorder.
@@ -1002,14 +1001,14 @@ Qed.
 
 (* This lemma removes the PER argument *)
 Lemma per_ctx_env_cons' : forall {Γ Γ' i A A' tail_rel}
-                             (head_rel : forall {p p'} (equiv_p_p' : {{ Dom p ≈ p' ∈ tail_rel }}), relation domain)
+                             (head_rel : forall {ρ ρ'} (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ tail_rel }}), relation domain)
                              env_rel,
     {{ EF Γ ≈ Γ' ∈ per_ctx_env ↘ tail_rel }} ->
-    (forall {p p'} (equiv_p_p' : {{ Dom p ≈ p' ∈ tail_rel }}),
-        rel_typ i A p A' p' (head_rel equiv_p_p')) ->
-    (env_rel <~> fun p p' =>
-         exists (equiv_p_drop_p'_drop : {{ Dom p ↯ ≈ p' ↯ ∈ tail_rel }}),
-           {{ Dom ~(p 0) ≈ ~(p' 0) ∈ head_rel equiv_p_drop_p'_drop }}) ->
+    (forall {ρ ρ'} (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ tail_rel }}),
+        rel_typ i A ρ A' ρ' (head_rel equiv_ρ_ρ')) ->
+    (env_rel <~> fun ρ ρ' =>
+         exists (equiv_ρ_drop_ρ'_drop : {{ Dom ρ ↯ ≈ ρ' ↯ ∈ tail_rel }}),
+           {{ Dom ~(ρ 0) ≈ ~(ρ' 0) ∈ head_rel equiv_ρ_drop_ρ'_drop }}) ->
     {{ EF Γ, A ≈ Γ', A' ∈ per_ctx_env ↘ env_rel }}.
 Proof.
   intros.
@@ -1026,12 +1025,12 @@ Ltac per_ctx_env_econstructor :=
 Lemma per_ctx_env_cons_clean_inversion : forall {Γ Γ' env_relΓ A A' env_relΓA},
     {{ EF Γ ≈ Γ' ∈ per_ctx_env ↘ env_relΓ }} ->
     {{ EF Γ, A ≈ Γ', A' ∈ per_ctx_env ↘ env_relΓA }} -> 
-    exists i (head_rel : forall {p p'} (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}), relation domain),
-      (forall p p' (equiv_p_p' : {{ Dom p ≈ p' ∈ env_relΓ }}),
-          rel_typ i A p A' p' (head_rel equiv_p_p')) /\
-        (env_relΓA <~> fun p p' =>
-             exists (equiv_p_drop_p'_drop : {{ Dom p ↯ ≈ p' ↯ ∈ env_relΓ }}),
-               {{ Dom ~(p 0) ≈ ~(p' 0) ∈ head_rel equiv_p_drop_p'_drop }}).
+    exists i (head_rel : forall {ρ ρ'} (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}), relation domain),
+      (forall ρ ρ' (equiv_ρ_ρ' : {{ Dom ρ ≈ ρ' ∈ env_relΓ }}),
+          rel_typ i A ρ A' ρ' (head_rel equiv_ρ_ρ')) /\
+        (env_relΓA <~> fun ρ ρ' =>
+             exists (equiv_ρ_drop_ρ'_drop : {{ Dom ρ ↯ ≈ ρ' ↯ ∈ env_relΓ }}),
+               {{ Dom ~(ρ 0) ≈ ~(ρ' 0) ∈ head_rel equiv_ρ_drop_ρ'_drop }}).
 Proof with intuition.
   intros * HΓ HΓA.
   inversion HΓA; subst.
@@ -1039,20 +1038,20 @@ Proof with intuition.
   eexists.
   eexists.
   split; intros.
-  - instantiate (1 := fun p p' (equiv_p_p' : env_relΓ p p') m m' =>
+  - instantiate (1 := fun ρ ρ' (equiv_ρ_ρ' : env_relΓ ρ ρ') m m' =>
                         forall R,
-                          rel_typ i A p A' p' R ->
+                          rel_typ i A ρ A' ρ' R ->
                           {{ Dom m ≈ m' ∈ R }}).
-    assert (tail_rel p p') by intuition.
+    assert (tail_rel ρ ρ') by intuition.
     (on_all_hyp: destruct_rel_by_assumption tail_rel).
     econstructor; eauto.
     apply -> per_univ_elem_morphism_iff; eauto.
     split; intros...
     destruct_by_head rel_typ.
     handle_per_univ_elem_irrel...
-  - intros o o'.
+  - intros ρ ρ'.
     split; intros; destruct_conjs;
-      assert {{ Dom o ↯ ≈ o' ↯ ∈ tail_rel }} by intuition;
+      assert {{ Dom ρ ↯ ≈ ρ' ↯ ∈ tail_rel }} by intuition;
       (on_all_hyp: destruct_rel_by_assumption tail_rel);
       unshelve eexists; intros...
     destruct_by_head rel_typ.
@@ -1084,11 +1083,11 @@ Qed.
 
 Lemma per_ctx_env_subtyping : forall Γ Δ,
     {{ SubE Γ <: Δ }} ->
-    forall R R' p p',
+    forall R R' ρ ρ',
       {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ R }} ->
       {{ EF Δ ≈ Δ ∈ per_ctx_env ↘ R' }} ->
-      R p p' ->
-      R' p p'.
+      R ρ ρ' ->
+      R' ρ ρ'.
 Proof.
   induction 1; intros;
     handle_per_ctx_env_irrel;
@@ -1097,7 +1096,7 @@ Proof.
     trivial.
 
   destruct_all.
-  assert {{ Dom p ↯ ≈ p' ↯ ∈ tail_rel0 }} by intuition.
+  assert {{ Dom ρ ↯ ≈ ρ' ↯ ∈ tail_rel0 }} by intuition.
   unshelve eexists; [eassumption |].
   destruct_rel_typ.
   eapply per_elem_subtyping with (i := max x (max i0 i)); try eassumption.
@@ -1147,7 +1146,7 @@ Proof.
   - firstorder.
   - instantiate (1 := max i i0).
     intros.
-    assert {{ Dom p ≈ p' ∈ tail_rel0 }} by (eapply per_ctx_env_subtyping; revgoals; eassumption).
+    assert {{ Dom ρ ≈ ρ' ∈ tail_rel0 }} by (eapply per_ctx_env_subtyping; revgoals; eassumption).
     saturate_refl_for tail_rel.
     destruct_rel_typ.
     handle_per_univ_elem_irrel.
diff --git a/theories/Core/Semantic/Readback/Definitions.v b/theories/Core/Semantic/Readback/Definitions.v
index 35f51a7a..77ff2073 100644
--- a/theories/Core/Semantic/Readback/Definitions.v
+++ b/theories/Core/Semantic/Readback/Definitions.v
@@ -26,10 +26,10 @@ Inductive read_nf : nat -> domain_nf -> nf -> Prop :=
 
      (* Nf of eta-expanded body *)
      {{ $| m & ⇑! a s |↘ m' }} ->
-     {{ ⟦ B ⟧ p ↦ ⇑! a s ↘ b }} ->
+     {{ ⟦ B ⟧ ρ ↦ ⇑! a s ↘ b }} ->
      {{ Rnf ⇓ b m' in S s ↘ M }} ->
 
-     {{ Rnf ⇓ (Π a p B) m in s ↘ λ A M }} )
+     {{ Rnf ⇓ (Π a ρ B) m in s ↘ λ A M }} )
 | read_nf_neut :
   `( {{ Rne m in s ↘ M }} ->
      {{ Rnf ⇓ (⇑ a b) (⇑ c m) in s ↘ ⇑ M }} )
@@ -43,22 +43,22 @@ with read_ne : nat -> domain_ne -> ne -> Prop :=
      {{ Rne m n in s ↘ M N }} )
 | read_ne_natrec :
   (* Nf of motive *)
-  `( {{ ⟦ B ⟧ p ↦ ⇑! ℕ s ↘ b }} ->
+  `( {{ ⟦ B ⟧ ρ ↦ ⇑! ℕ s ↘ b }} ->
      {{ Rtyp b in S s ↘ B' }} ->
 
      (* Nf of mz *)
-     {{ ⟦ B ⟧ p ↦ zero ↘ bz }} ->
+     {{ ⟦ B ⟧ ρ ↦ zero ↘ bz }} ->
      {{ Rnf ⇓ bz mz in s ↘ MZ }} ->
 
      (* Nf of MS *)
-     {{ ⟦ B ⟧ p ↦ succ (⇑! ℕ s) ↘ bs }} ->
-     {{ ⟦ MS ⟧ p ↦ ⇑! ℕ s ↦ ⇑! b (S s) ↘ ms }} ->
+     {{ ⟦ B ⟧ ρ ↦ succ (⇑! ℕ s) ↘ bs }} ->
+     {{ ⟦ MS ⟧ ρ ↦ ⇑! ℕ s ↦ ⇑! b (S s) ↘ ms }} ->
      {{ Rnf ⇓ bs ms in S (S s) ↘ MS' }} ->
 
      (* Ne of m *)
      {{ Rne m in s ↘ M }} ->
 
-     {{ Rne rec m under p return B | zero -> mz | succ -> MS end in s ↘ rec M return B' | zero -> MZ | succ -> MS' end }} )
+     {{ Rne rec m under ρ return B | zero -> mz | succ -> MS end in s ↘ rec M return B' | zero -> MZ | succ -> MS' end }} )
 where "'Rne' m 'in' s ↘ M" := (read_ne s m M) (in custom judg) : type_scope
 with read_typ : nat -> domain -> nf -> Prop :=
 | read_typ_univ :
@@ -70,10 +70,10 @@ with read_typ : nat -> domain -> nf -> Prop :=
   `( {{ Rtyp a in s ↘ A }} ->
 
      (* Nf of ret type *)
-     {{ ⟦ B ⟧ p ↦ ⇑! a s ↘ b }} ->
+     {{ ⟦ B ⟧ ρ ↦ ⇑! a s ↘ b }} ->
      {{ Rtyp b in S s ↘ B' }} ->
 
-     {{ Rtyp Π a p B in s ↘ Π A B' }})
+     {{ Rtyp Π a ρ B in s ↘ Π A B' }})
 | read_typ_neut :
   `( {{ Rne b in s ↘ B }} ->
      {{ Rtyp ⇑ a b in s ↘ ⇑ B }})
@@ -83,6 +83,10 @@ where "'Rtyp' m 'in' s ↘ M" := (read_typ s m M) (in custom judg) : type_scope
 Scheme read_nf_mut_ind := Induction for read_nf Sort Prop
 with read_ne_mut_ind := Induction for read_ne Sort Prop
 with read_typ_mut_ind := Induction for read_typ Sort Prop.
+Combined Scheme read_mut_ind from
+  read_nf_mut_ind,
+  read_ne_mut_ind,
+  read_typ_mut_ind.
 
 #[export]
 Hint Constructors read_nf read_ne read_typ : mcltt.
diff --git a/theories/Core/Semantic/Readback/Lemmas.v b/theories/Core/Semantic/Readback/Lemmas.v
index 4b13188a..58d146d4 100644
--- a/theories/Core/Semantic/Readback/Lemmas.v
+++ b/theories/Core/Semantic/Readback/Lemmas.v
@@ -4,44 +4,48 @@ From Mcltt.Core Require Import Evaluation Readback.Definitions.
 Import Domain_Notations.
 
 Section functional_read.
-  Let functional_read_nf_prop s m M1 (_ : {{ Rnf m in s ↘ M1 }}) : Prop := forall M2 (Hread2: {{ Rnf m in s ↘ M2 }}), M1 = M2.
-  Let functional_read_ne_prop s m M1 (_ : {{ Rne m in s ↘ M1 }}) : Prop := forall M2 (Hread2 : {{ Rne m in s ↘ M2 }}), M1 = M2.
-  Let functional_read_typ_prop s m M1 (_ : {{ Rtyp m in s ↘ M1 }}) : Prop := forall M2 (Hread2 : {{ Rtyp m in s ↘ M2 }}), M1 = M2.
-
-  #[local]
-  Ltac unfold_functional_read := unfold functional_read_nf_prop, functional_read_ne_prop, functional_read_typ_prop in *.
-
-  Lemma functional_read_nf : forall s m M1 (Hread1: {{ Rnf m in s ↘ M1 }}), functional_read_nf_prop s m M1 Hread1.
+  Lemma functional_read :
+    (forall s m M1,
+        {{ Rnf m in s ↘ M1 }} ->
+        forall M2,
+          {{ Rnf m in s ↘ M2 }} ->
+          M1 = M2) /\
+      (forall s m M1,
+          {{ Rne m in s ↘ M1 }} ->
+          forall M2,
+            {{ Rne m in s ↘ M2 }} ->
+            M1 = M2) /\
+      (forall s m M1,
+          {{ Rtyp m in s ↘ M1 }} ->
+          forall M2,
+            {{ Rtyp m in s ↘ M2 }} ->
+            M1 = M2).
   Proof with (functional_eval_rewrite_clear; f_equal; solve [eauto]) using.
-    intros *.
-    induction Hread1
-      using read_nf_mut_ind
-      with (P0 := functional_read_ne_prop)
-           (P1 := functional_read_typ_prop);
-      unfold_functional_read;
-      inversion_clear 1...
+    apply read_mut_ind; intros; progressive_inversion...
   Qed.
 
-  Lemma functional_read_ne : forall s m M1 (Hread1 : {{ Rne m in s ↘ M1 }}), functional_read_ne_prop s m M1 Hread1.
-  Proof with (functional_eval_rewrite_clear; f_equal; solve [eauto]) using.
-    intros *.
-    induction Hread1
-      using read_ne_mut_ind
-      with (P := functional_read_nf_prop)
-           (P1 := functional_read_typ_prop);
-      unfold_functional_read;
-      inversion_clear 1...
+  Corollary functional_read_nf : forall s m M1 M2,
+      {{ Rnf m in s ↘ M1 }} ->
+      {{ Rnf m in s ↘ M2 }} ->
+      M1 = M2.
+  Proof.
+    pose proof functional_read; firstorder.
   Qed.
 
-  Lemma functional_read_typ : forall s m M1 (Hread1 : {{ Rtyp m in s ↘ M1 }}), functional_read_typ_prop s m M1 Hread1.
-  Proof with (functional_eval_rewrite_clear; f_equal; solve [eauto]) using.
-    intros *.
-    induction Hread1
-      using read_typ_mut_ind
-      with (P := functional_read_nf_prop)
-           (P0 := functional_read_ne_prop);
-      unfold_functional_read;
-      inversion_clear 1...
+  Lemma functional_read_ne : forall s m M1 M2,
+      {{ Rne m in s ↘ M1 }} ->
+      {{ Rne m in s ↘ M2 }} ->
+      M1 = M2.
+  Proof.
+    pose proof functional_read; firstorder.
+  Qed.
+
+  Lemma functional_read_typ : forall s m M1 M2,
+      {{ Rtyp m in s ↘ M1 }} ->
+      {{ Rtyp m in s ↘ M2 }} ->
+      M1 = M2.
+  Proof.
+    pose proof functional_read; firstorder.
   Qed.
 End functional_read.
 
diff --git a/theories/Core/Semantic/Realizability.v b/theories/Core/Semantic/Realizability.v
index 40d8c753..f1b53320 100644
--- a/theories/Core/Semantic/Realizability.v
+++ b/theories/Core/Semantic/Realizability.v
@@ -37,7 +37,7 @@ Proof with (solve [try (try (eexists; split); econstructor); mauto]).
     specialize (H1 s) as [? []]...
   - destruct IHHunivelem as [? []].
     intro s.
-    assert {{ Dom ⇑! A s ≈ ⇑! A' s ∈ in_rel }} by eauto using var_per_bot.
+    assert {{ Dom ⇑! a s ≈ ⇑! a' s ∈ in_rel }} by eauto using var_per_bot.
     destruct_rel_mod_eval.
     specialize (H9 (S s)) as [? []].
     specialize (H2 s) as [? []]...
@@ -45,16 +45,26 @@ Proof with (solve [try (try (eexists; split); econstructor); mauto]).
     destruct_conjs.
     destruct_rel_mod_eval.
     econstructor; try solve [econstructor; eauto].
-    enough ({{ Dom c ⇓ A c0 ≈ c' ⇓ A' c0' ∈ per_bot }}) by eauto.
+    enough ({{ Dom c ⇓ a c0 ≈ c' ⇓ a' c0' ∈ per_bot }}) by eauto.
     intro s.
     specialize (H3 s) as [? []].
     specialize (H5 _ _ equiv_c0_c0' s) as [? []]...
   - destruct_conjs.
     intro s.
-    assert {{ Dom ⇑! A s ≈ ⇑! A' s ∈ in_rel }} by eauto using var_per_bot.
+    assert {{ Dom ⇑! a s ≈ ⇑! a' s ∈ in_rel }} by eauto using var_per_bot.
     destruct_rel_mod_eval.
     destruct_rel_mod_app.
-    assert {{ Dom ⇓ a fa ≈ ⇓ a' f'a' ∈ per_top }} by eauto.
+    match goal with
+    | _: {{ $| ~?f0 & ⇑! a s |↘ ~_ }},
+        _: {{ $| ~?f0' & ⇑! a' s |↘ ~_ }},
+          _: {{ ⟦ B ⟧ ρ ↦ ⇑! a s ↘ ~?b0 }},
+            _: {{ ⟦ B' ⟧ ρ' ↦ ⇑! a' s ↘ ~?b0' }} |- _ =>
+        rename f0 into f;
+        rename f0' into f';
+        rename b0 into b;
+        rename b0' into b'
+    end.
+    assert {{ Dom ⇓ b fa ≈ ⇓ b' f'a' ∈ per_top }} by eauto.
     specialize (H2 s) as [? []].
     specialize (H16 (S s)) as [? []]...
   - intro s.
@@ -97,7 +107,7 @@ Hint Resolve per_elem_then_per_top : mcltt.
 
 Lemma per_ctx_then_per_env_initial_env : forall {Γ Γ' env_rel},
     {{ EF Γ ≈ Γ' ∈ per_ctx_env ↘ env_rel }} ->
-    exists p p', initial_env Γ p /\ initial_env Γ' p' /\ {{ Dom p ≈ p' ∈ env_rel }}.
+    exists ρ ρ', initial_env Γ ρ /\ initial_env Γ' ρ' /\ {{ Dom ρ ≈ ρ' ∈ env_rel }}.
 Proof.
   induction 1.
   - do 2 eexists; intuition.
@@ -111,9 +121,9 @@ Proof.
     eexists; eauto.
 Qed.
 
-Lemma var_per_elem : forall {A B i R} n,
-    {{ DF A ≈ B ∈ per_univ_elem i ↘ R }} ->
-    {{ Dom ⇑! A n ≈ ⇑! B n ∈ R }}.
+Lemma var_per_elem : forall {a b i R} n,
+    {{ DF a ≈ b ∈ per_univ_elem i ↘ R }} ->
+    {{ Dom ⇑! a n ≈ ⇑! b n ∈ R }}.
 Proof.
   intros.
   eapply per_bot_then_per_elem; mauto.
diff --git a/theories/Core/Soundness/FunctionCases.v b/theories/Core/Soundness/FunctionCases.v
index c88d2884..6fe838af 100644
--- a/theories/Core/Soundness/FunctionCases.v
+++ b/theories/Core/Soundness/FunctionCases.v
@@ -77,18 +77,19 @@ Proof.
     handle_functional_glu_univ_elem.
     autorewrite with mcltt.
     eassumption.
-  - rename a0 into c.
-    rename equiv_a into equiv_c.
-    assert {{ Dom ρ ↦ c ≈ ρ ↦ c ∈ env_relΓA }} as HrelΓA by (apply_relation_equivalence; mautosolve 2).
+  - assert {{ Dom ρ ↦ m ≈ ρ ↦ m ∈ env_relΓA }} as HrelΓA by (apply_relation_equivalence; mautosolve 2).
     apply_relation_equivalence.
     (on_all_hyp: fun H => destruct (H _ _ HrelΓA)).
     destruct_by_head per_univ.
     functional_eval_rewrite_clear.
-    rename m0 into b.
-    assert {{ Δ' ⊢ m : A[σ][τ] }} by mauto 3 using glu_univ_elem_trm_escape.
-    assert {{ DG b ∈ glu_univ_elem i ↘ OP c equiv_c ↘ OEl c equiv_c }} by mauto 2.
+    match goal with
+    | _: {{ ⟦ B ⟧ ρ ↦ m ↘ ~?a }} |- _ =>
+        rename a into b
+    end.
+    assert {{ Δ' ⊢ M : A[σ][τ] }} by mauto 3 using glu_univ_elem_trm_escape.
+    assert {{ DG b ∈ glu_univ_elem i ↘ OP m equiv_m ↘ OEl m equiv_m }} by mauto 2.
     erewrite <- @sub_decompose_q_typ; mauto 3.
-    assert {{ Δ' ⊢s (σ∘τ),,m ® ρ ↦ c ∈ cons_glu_sub_pred i Γ A SbΓ }} as Hconspred by mauto 2.
+    assert {{ Δ' ⊢s (σ∘τ),,M ® ρ ↦ m ∈ cons_glu_sub_pred i Γ A SbΓ }} as Hconspred by mauto 2.
     (on_all_hyp: fun H => destruct (H _ _ _ Hconspred)).
     simplify_evals.
     match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H).
@@ -184,23 +185,17 @@ Proof.
     handle_per_univ_elem_irrel.
     econstructor; mauto 3.
   - eapply glu_univ_elem_typ_monotone; mauto 3.
-  - match goal with
-    | H: {{ Dom ~?a ≈ ~?a ∈ in_rel }}, _: {{ ~_ ⊢ ~?M : ~_ ® ~?a ∈ Ela }} |- _ =>
-        rename a into c;
-        rename H into equiv_c;
-        rename M into N
-    end.
-    assert {{ Dom ρ ↦ c ≈ ρ ↦ c ∈ env_relΓA }} as HrelΓA by (apply_relation_equivalence; mautosolve 2).
+  - assert {{ Dom ρ ↦ n ≈ ρ ↦ n ∈ env_relΓA }} as HrelΓA by (apply_relation_equivalence; mautosolve 2).
     destruct_rel_mod_eval.
     apply_relation_equivalence.
     (on_all_hyp: fun H => destruct (H _ _ HrelΓA) as [? [[] []]]).
     handle_per_univ_elem_irrel.
     eexists; split; mauto 3.
     match goal with
-    | _: {{ ⟦ B ⟧ ρ ↦ c ↘ ~?a }} |- _ =>
+    | _: {{ ⟦ B ⟧ ρ ↦ n ↘ ~?a }} |- _ =>
         rename a into b
     end.
-    assert {{ DG b ∈ glu_univ_elem i ↘ OP c equiv_c ↘ OEl c equiv_c }} by mauto 3.
+    assert {{ DG b ∈ glu_univ_elem i ↘ OP n equiv_n ↘ OEl n equiv_n }} by mauto 3.
     assert {{ Δ0 ⊢s σ0 : Δ }} by mauto 3.
     assert {{ Δ0 ⊢ N : A[σ][σ0] }} by mauto 2 using glu_univ_elem_trm_escape.
     erewrite <- @sub_decompose_q_typ; mauto 2.
@@ -221,8 +216,8 @@ Proof.
       assert {{ Δ0 ⊢ B[q (σ∘σ0)∘(Id,,N)] ≈ B[σ∘σ0,,N] : Type@i }} as <- by mauto 2.
       transitivity {{{ M[q (σ∘σ0)∘(Id,,N)] }}}; mauto 3.
     }
-    assert {{ Δ0 ⊢ N : A[σ][σ0] ® c ∈ Ela }} by mauto 3.
-    assert {{ Δ0 ⊢s (σ∘σ0),,N ® ρ ↦ c ∈ SbΓA }} as HSbΓA by (unfold SbΓA; mauto 2).
+    assert {{ Δ0 ⊢ N : A[σ][σ0] ® n ∈ Ela }} by mauto 3.
+    assert {{ Δ0 ⊢s (σ∘σ0),,N ® ρ ↦ n ∈ SbΓA }} as HSbΓA by (unfold SbΓA; mauto 2).
     (on_all_hyp: fun H => destruct (H _ _ _ HSbΓA)).
     simplify_evals.
     match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H).
@@ -284,15 +279,23 @@ Proof.
   handle_functional_glu_univ_elem.
   match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
   inversion_clear_by_head pi_glu_exp_pred.
-  rename m0 into a.
-  rename m1 into n.
+  match goal with
+  | _: {{ ⟦ A ⟧ ρ ↘ ~?a' }},
+      _: {{ ⟦ N ⟧ ρ ↘ ~?n' }} |- _ =>
+      rename a' into a;
+      rename n' into n
+  end.
   assert {{ Dom a ≈ a ∈ per_univ i }} as [] by mauto 3.
   handle_per_univ_elem_irrel.
   assert {{ Dom n ≈ n ∈ in_rel }} by (eapply glu_univ_elem_per_elem; revgoals; eassumption).
   (on_all_hyp: destruct_rel_by_assumption in_rel).
   simplify_evals.
-  rename a0 into b.
-  rename fa into mn.
+  match goal with
+  | _: {{ ⟦ B ⟧ ρ ↦ n ↘ ~?b' }},
+      _: {{ $| m & n |↘ ~?mn' }} |- _ =>
+      rename b' into b;
+      rename mn' into mn
+  end.
   eapply mk_glu_rel_exp_with_sub''; mauto 3.
   intros.
   match_by_head1 (in_rel n n) ltac:(fun H => rename H into equiv_n).
diff --git a/theories/Core/Soundness/LogicalRelation.v b/theories/Core/Soundness/LogicalRelation.v
index ffffd117..5665881b 100644
--- a/theories/Core/Soundness/LogicalRelation.v
+++ b/theories/Core/Soundness/LogicalRelation.v
@@ -1 +1 @@
-From Mcltt.Core.Soundness Require Export LogicalRelation.Core LogicalRelation.Lemmas.
+From Mcltt.Core.Soundness.LogicalRelation Require Export Core Lemmas.
diff --git a/theories/Core/Soundness/LogicalRelation/Core.v b/theories/Core/Soundness/LogicalRelation/Core.v
index 96b8b8b5..6756faba 100644
--- a/theories/Core/Soundness/LogicalRelation/Core.v
+++ b/theories/Core/Soundness/LogicalRelation/Core.v
@@ -1 +1 @@
-From Mcltt.Core.Soundness Require Export LogicalRelation.Definitions LogicalRelation.CoreLemmas.
+From Mcltt.Core.Soundness.LogicalRelation Require Export CoreLemmas CoreTactics Definitions.
diff --git a/theories/Core/Soundness/LogicalRelation/CoreLemmas.v b/theories/Core/Soundness/LogicalRelation/CoreLemmas.v
index 30f23e87..976d2247 100644
--- a/theories/Core/Soundness/LogicalRelation/CoreLemmas.v
+++ b/theories/Core/Soundness/LogicalRelation/CoreLemmas.v
@@ -6,8 +6,8 @@ From Mcltt.Core.Soundness Require Import LogicalRelation.Definitions LogicalRela
 From Mcltt.Core.Soundness Require Export Weakening.Lemmas.
 Import Domain_Notations.
 
-Lemma glu_nat_per_nat : forall Γ m a,
-    glu_nat Γ m a ->
+Lemma glu_nat_per_nat : forall Γ M a,
+    glu_nat Γ M a ->
     {{ Dom a ≈ a ∈ per_nat }}.
 Proof.
   induction 1; mauto.
@@ -16,19 +16,19 @@ Qed.
 #[local]
 Hint Resolve glu_nat_per_nat : mcltt.
 
-Lemma glu_nat_escape : forall Γ m a,
-    glu_nat Γ m a ->
+Lemma glu_nat_escape : forall Γ M a,
+    glu_nat Γ M a ->
     {{ ⊢ Γ }} ->
-    {{ Γ ⊢ m : ℕ }}.
+    {{ Γ ⊢ M : ℕ }}.
 Proof.
   induction 1; intros;
     try match goal with
     | H : _ |- _ => solve [gen_presup H; mauto]
     end.
   assert {{ Γ ⊢w Id : Γ }} by mauto.
-  match_by_head (per_bot c c) ltac:(fun H => specialize (H (length Γ)) as [L []]).
+  match_by_head (per_bot m m) ltac:(fun H => specialize (H (length Γ)) as [M' []]).
   clear_dups.
-  assert {{ Γ ⊢ m[Id] ≈ L : ℕ }} by mauto.
+  assert {{ Γ ⊢ M[Id] ≈ M' : ℕ }} by mauto.
   gen_presups.
   mauto.
 Qed.
@@ -36,10 +36,10 @@ Qed.
 #[export]
 Hint Resolve glu_nat_escape : mcltt.
 
-Lemma glu_nat_resp_ctx_eq : forall Γ m a Δ,
-    glu_nat Γ m a ->
+Lemma glu_nat_resp_ctx_eq : forall Γ M a Δ,
+    glu_nat Γ M a ->
     {{ ⊢ Γ ≈ Δ }} ->
-    glu_nat Δ m a.
+    glu_nat Δ M a.
 Proof.
   induction 1; intros; mauto.
 Qed.
@@ -53,16 +53,16 @@ Proof.
   split; mauto using glu_nat_resp_ctx_eq.
 Qed.
 
-Lemma glu_nat_resp_exp_eq : forall Γ m a,
-    glu_nat Γ m a ->
-    forall m',
-    {{ Γ ⊢ m ≈ m' : ℕ }} ->
-    glu_nat Γ m' a.
+Lemma glu_nat_resp_exp_eq : forall Γ M a,
+    glu_nat Γ M a ->
+    forall M',
+    {{ Γ ⊢ M ≈ M' : ℕ }} ->
+    glu_nat Γ M' a.
 Proof.
   induction 1; intros; mauto.
   econstructor; trivial.
   intros.
-  transitivity {{{ m[σ] }}}; mauto.
+  transitivity {{{ M[σ] }}}; mauto.
 Qed.
 
 #[local]
@@ -74,18 +74,18 @@ Proof.
   split; mauto using glu_nat_resp_exp_eq.
 Qed.
 
-Lemma glu_nat_readback : forall Γ m a,
-    glu_nat Γ m a ->
-    forall Δ σ b,
+Lemma glu_nat_readback : forall Γ M a,
+    glu_nat Γ M a ->
+    forall Δ σ M',
       {{ Δ ⊢w σ : Γ }} ->
-      {{ Rnf ⇓ ℕ a in length Δ ↘ b }} ->
-      {{ Δ ⊢ m [ σ ] ≈ b : ℕ }}.
+      {{ Rnf ⇓ ℕ a in length Δ ↘ M' }} ->
+      {{ Δ ⊢ M[σ] ≈ M' : ℕ }}.
 Proof.
   induction 1; intros; progressive_inversion; gen_presups.
   - transitivity {{{ zero[σ] }}}; mauto.
-  - assert {{ Δ ⊢ m'[σ] ≈ M : ℕ }} by mauto.
-    transitivity {{{ (succ m')[σ] }}}; mauto 3.
-    transitivity {{{ succ m'[σ] }}}; mauto 4.
+  - assert {{ Δ ⊢ M'[σ] ≈ M0 : ℕ }} by mauto.
+    transitivity {{{ (succ M')[σ] }}}; mauto 3.
+    transitivity {{{ succ M'[σ] }}}; mauto 4.
   - mauto.
 Qed.
 
@@ -97,23 +97,23 @@ Ltac simpl_glu_rel :=
   destruct_all;
   gen_presups.
 
-Lemma glu_univ_elem_univ_lvl : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ T,
-      {{ Γ ⊢ T ® P }} ->
-      {{ Γ ⊢ T : Type@i }}.
+Lemma glu_univ_elem_univ_lvl : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A,
+      {{ Γ ⊢ A ® P }} ->
+      {{ Γ ⊢ A : Type@i }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
     simpl_glu_rel; trivial.
 Qed.
 
-Lemma glu_univ_elem_typ_resp_exp_eq : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ T T',
-      {{ Γ ⊢ T ® P }} ->
-      {{ Γ ⊢ T ≈ T' : Type@i }} ->
-      {{ Γ ⊢ T' ® P }}.
+Lemma glu_univ_elem_typ_resp_exp_eq : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A A',
+      {{ Γ ⊢ A ® P }} ->
+      {{ Γ ⊢ A ≈ A' : Type@i }} ->
+      {{ Γ ⊢ A' ® P }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
@@ -121,29 +121,31 @@ Proof.
 
   split; [trivial |].
   intros.
-  assert {{ Δ ⊢ T[σ] ≈ V : Type@i }}; mauto.
+  transitivity {{{ A[σ] }}}; mauto 4.
 Qed.
 
-Add Parametric Morphism i P El A (H : glu_univ_elem i P El A) Γ : (P Γ)
-    with signature wf_exp_eq Γ {{{Type@i}}} ==> iff as glu_univ_elem_typ_morphism_iff1.
+Add Parametric Morphism i P El a (H : glu_univ_elem i P El a) Γ : (P Γ)
+    with signature wf_exp_eq Γ {{{ Type@i }}} ==> iff as glu_univ_elem_typ_morphism_iff1.
 Proof.
   split; intros; eapply glu_univ_elem_typ_resp_exp_eq; mauto 2.
 Qed.
 
-Lemma glu_univ_elem_trm_resp_typ_exp_eq : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a T',
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ Γ ⊢ T ≈ T' : Type@i }} ->
-      {{ Γ ⊢ t : T' ® a ∈ El }}.
+Lemma glu_univ_elem_trm_resp_typ_exp_eq : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ M A m A',
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ Γ ⊢ A ≈ A' : Type@i }} ->
+      {{ Γ ⊢ M : A' ® m ∈ El }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
     simpl_glu_rel; repeat split; intros; mauto 3;
-    [firstorder | | assert {{ Δ ⊢ M[σ] ≈ V : Type@i }} by mauto 3 |]; mauto.
+    [firstorder | | transitivity {{{ A[σ] }}}; mauto 4 | assert {{ Δ ⊢ A[σ] ≈ A'[σ] : Type@i }}; mauto 3].
+
+  econstructor; mauto 3.
 Qed.
 
-Add Parametric Morphism i P El A (H : glu_univ_elem i P El A) Γ : (El Γ)
+Add Parametric Morphism i P El a (H : glu_univ_elem i P El a) Γ : (El Γ)
     with signature wf_exp_eq Γ {{{Type@i}}} ==> eq ==> eq ==> iff as glu_univ_elem_trm_morphism_iff1.
 Proof.
   split; intros;
@@ -151,12 +153,12 @@ Proof.
     mauto 2.
 Qed.
 
-Lemma glu_univ_elem_typ_resp_ctx_eq : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ T Δ,
-      {{ Γ ⊢ T ® P }} ->
+Lemma glu_univ_elem_typ_resp_ctx_eq : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A Δ,
+      {{ Γ ⊢ A ® P }} ->
       {{ ⊢ Γ ≈ Δ }} ->
-      {{ Δ ⊢ T ® P }}.
+      {{ Δ ⊢ A ® P }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
@@ -164,7 +166,7 @@ Proof.
     econstructor; mauto 4.
 Qed.
 
-Add Parametric Morphism i P El A (H : glu_univ_elem i P El A) : P
+Add Parametric Morphism i P El a (H : glu_univ_elem i P El a) : P
     with signature wf_ctx_eq ==> eq ==> iff as glu_univ_elem_typ_morphism_iff2.
 Proof.
   intros. split; intros;
@@ -172,12 +174,12 @@ Proof.
     mauto 2.
 Qed.
 
-Lemma glu_univ_elem_trm_resp_ctx_eq : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ T M m Δ,
-      {{ Γ ⊢ M : T ® m ∈ El }} ->
+Lemma glu_univ_elem_trm_resp_ctx_eq : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A M m Δ,
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
       {{ ⊢ Γ ≈ Δ }} ->
-      {{ Δ ⊢ M : T ® m ∈ El }}.
+      {{ Δ ⊢ M : A ® m ∈ El }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
@@ -191,7 +193,7 @@ Proof.
   - split; mauto 4.
 Qed.
 
-Add Parametric Morphism i P El A (H : glu_univ_elem i P El A) : El
+Add Parametric Morphism i P El a (H : glu_univ_elem i P El a) : El
     with signature wf_ctx_eq ==> eq ==> eq ==> eq ==> iff as glu_univ_elem_trm_morphism_iff2.
 Proof.
   intros. split; intros;
@@ -199,30 +201,31 @@ Proof.
     mauto 2.
 Qed.
 
-Lemma glu_nat_resp_wk' : forall Γ m a,
-    glu_nat Γ m a ->
+Lemma glu_nat_resp_wk' : forall Γ M a,
+    glu_nat Γ M a ->
     forall Δ σ,
-      {{ Γ ⊢ m : ℕ }} ->
+      {{ Γ ⊢ M : ℕ }} ->
       {{ Δ ⊢w σ : Γ }} ->
-      glu_nat Δ {{{ m[σ] }}} a.
+      glu_nat Δ {{{ M[σ] }}} a.
 Proof.
   induction 1; intros; gen_presups.
   - econstructor.
     transitivity {{{ zero[σ] }}}; mauto.
   - econstructor; [ |mauto].
-    transitivity {{{ (succ m')[σ] }}}; mauto 4.
+    transitivity {{{ (succ M')[σ] }}}; mauto 4.
   - econstructor; trivial.
-    intros. gen_presups.
-    assert {{ Δ0 ⊢w σ ∘ σ0 : Γ }} by mauto.
-    assert {{ Δ0 ⊢ m[σ ∘ σ0] ≈ v : ℕ }} by mauto 4.
-    transitivity {{{ m[σ ∘ σ0] }}}; mauto 4.
+    intros Δ' τ M' **.
+    gen_presups.
+    assert {{ Δ' ⊢w σ∘τ : Γ }} by mauto.
+    assert {{ Δ' ⊢ M[σ∘τ] ≈ M' : ℕ }} by mauto 4.
+    transitivity {{{ M[σ∘τ] }}}; mauto 4.
 Qed.
 
-Lemma glu_nat_resp_wk : forall Γ m a,
-    glu_nat Γ m a ->
+Lemma glu_nat_resp_wk : forall Γ M a,
+    glu_nat Γ M a ->
     forall Δ σ,
       {{ Δ ⊢w σ : Γ }} ->
-      glu_nat Δ {{{ m[σ] }}} a.
+      glu_nat Δ {{{ M[σ] }}} a.
 Proof.
   mauto using glu_nat_resp_wk'.
 Qed.
@@ -230,20 +233,20 @@ Qed.
 #[export]
 Hint Resolve glu_nat_resp_wk : mcltt.
 
-Lemma glu_univ_elem_trm_escape : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a,
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ Γ ⊢ t : T }}.
+Lemma glu_univ_elem_trm_escape : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ M A m,
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ Γ ⊢ M : A }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
     simpl_glu_rel; mauto 4.
 Qed.
 
-Lemma glu_univ_elem_per_univ : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    {{ Dom A ≈ A ∈ per_univ i }}.
+Lemma glu_univ_elem_per_univ : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    {{ Dom a ≈ a ∈ per_univ i }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros; eexists;
@@ -258,18 +261,18 @@ Qed.
 #[export]
 Hint Resolve glu_univ_elem_per_univ : mcltt.
 
-Lemma glu_univ_elem_per_elem : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a R,
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ DF A ≈ A ∈ per_univ_elem i ↘ R }} ->
-      {{ Dom a ≈ a ∈ R }}.
+Lemma glu_univ_elem_per_elem : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ M A m R,
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ DF a ≈ a ∈ per_univ_elem i ↘ R }} ->
+      {{ Dom m ≈ m ∈ R }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
     try do 2 match_by_head1 per_univ_elem invert_per_univ_elem;
     simpl_glu_rel;
-    try fold (per_univ j a a);
+    try fold (per_univ j m m);
     mauto 4.
 
   intros.
@@ -281,11 +284,11 @@ Proof.
   econstructor; firstorder eauto.
 Qed.
 
-Lemma glu_univ_elem_trm_typ : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a,
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ Γ ⊢ T ® P }}.
+Lemma glu_univ_elem_trm_typ : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ M A m,
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ Γ ⊢ A ® P }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
@@ -294,26 +297,26 @@ Proof.
 
   econstructor; eauto.
   match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
-  intros.
+  intros ? ? N n ? ? equiv_n.
   destruct_rel_mod_eval.
-  edestruct H12 as [? []]; eauto.
+  enough (exists mn : domain, {{ $| m & n |↘ mn }} /\  {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl n equiv_n }}) as [? []]; eauto 3.
 Qed.
 
-Lemma glu_univ_elem_trm_univ_lvl : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a,
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ Γ ⊢ T : Type@i }}.
+Lemma glu_univ_elem_trm_univ_lvl : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ M A m,
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ Γ ⊢ A : Type@i }}.
 Proof.
   intros. eapply glu_univ_elem_univ_lvl; [| eapply glu_univ_elem_trm_typ]; eassumption.
 Qed.
 
-Lemma glu_univ_elem_trm_resp_exp_eq : forall i P El A,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a t',
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ Γ ⊢ t ≈ t' : T }} ->
-      {{ Γ ⊢ t' : T ® a ∈ El }}.
+Lemma glu_univ_elem_trm_resp_exp_eq : forall i P El a,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A M m M',
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ Γ ⊢ M ≈ M' : A }} ->
+      {{ Γ ⊢ M' : A ® m ∈ El }}.
 Proof.
   simpl.
   induction 1 using glu_univ_elem_ind; intros;
@@ -324,26 +327,25 @@ Proof.
     eapply glu_univ_elem_typ_resp_exp_eq; mauto.
 
   - econstructor; eauto.
-    invert_per_univ_elem H3.
+    match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
     intros.
     destruct_rel_mod_eval.
-    assert {{ Δ ⊢ m' : IT [σ]}} by eauto using glu_univ_elem_trm_escape.
-    edestruct H13 as [? []]; eauto.
+    assert {{ Δ ⊢ N : IT[σ]}} by eauto using glu_univ_elem_trm_escape.
+    assert (exists mn : domain, {{ $| m & n |↘ mn }} /\  {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl n equiv_n }}) as [? []] by intuition.
     eexists; split; eauto.
-    enough {{ Δ ⊢ m[σ] m' ≈ t'[σ] m' : OT[σ,,m'] }} by eauto.
-    assert {{ Γ ⊢ m ≈ t' : Π IT OT }} as Hty by mauto.
+    enough {{ Δ ⊢ M[σ] N ≈ M'[σ] N : OT[σ,,N] }} by eauto.
+    assert {{ Γ ⊢ M ≈ M' : Π IT OT }} as Hty by mauto.
     eassert {{ Δ ⊢ IT[σ] : Type@_ }} by mauto 3.
     eapply wf_exp_eq_sub_cong with (Γ := Δ) in Hty; [| eapply sub_eq_refl; mauto 3].
     autorewrite with mcltt in Hty.
-    eapply wf_exp_eq_app_cong with (N := m') (N' := m') in Hty; try pi_univ_level_tac; [|mauto 2].
+    eapply wf_exp_eq_app_cong with (N := N) (N' := N) in Hty; try pi_univ_level_tac; [|mauto 2].
     autorewrite with mcltt in Hty.
     eassumption.
   - intros.
-    assert {{ Δ ⊢ m[σ] ≈ t'[σ] : M[σ] }} by mauto 4.
-    mauto 4.
+    enough {{ Δ ⊢ M[σ] ≈ M'[σ] : A[σ] }}; mauto 4.
 Qed.
 
-Add Parametric Morphism i P El A (H : glu_univ_elem i P El A) Γ T : (El Γ T)
+Add Parametric Morphism i P El a (H : glu_univ_elem i P El a) Γ T : (El Γ T)
     with signature wf_exp_eq Γ T ==> eq ==> iff as glu_univ_elem_trm_morphism_iff3.
 Proof.
   split; intros;
@@ -412,7 +414,7 @@ Ltac apply_predicate_equivalence :=
 Add Parametric Morphism i : (glu_univ_elem i)
     with signature glu_typ_pred_equivalence ==> glu_exp_pred_equivalence ==> eq ==> iff as simple_glu_univ_elem_morphism_iff.
 Proof with mautosolve.
-  intros P P' HPP' El El' HElEl' a.
+  intros P P' ? El El'.
   split; intro Horig; [gen El' P' | gen El P];
     induction Horig using glu_univ_elem_ind; unshelve glu_univ_elem_econstructor;
     try (etransitivity; [symmetry + idtac|]; eassumption); eauto.
@@ -489,31 +491,28 @@ Proof.
   handle_per_univ_elem_irrel.
   split; [intros Γ C | intros Γ M C m].
   - split; intros []; econstructor; intuition;
-      rename a0 into c;
-      [rename equiv_a into equiv_c; assert (equiv_c' : in_rel c c) by intuition
-      | rename equiv_a into equiv_c'; assert (equiv_c : in_rel0 c c) by intuition];
+      [rename equiv_m into equiv0_m; assert (equiv_m : in_rel m m) by intuition
+      | assert (equiv0_m : in_rel0 m m) by intuition ];
       destruct_rel_mod_eval;
-      rename a0 into b;
-      assert ((OP c equiv_c' <∙> OP0 c equiv_c) /\ (OEl c equiv_c' <∙> OEl0 c equiv_c)) as [] by mauto 3;
+      functional_eval_rewrite_clear;
+      assert ((OP m equiv_m <∙> OP0 m equiv0_m) /\ (OEl m equiv_m <∙> OEl0 m equiv0_m)) as [] by mauto 3;
       intuition.
   - split; intros []; econstructor; intuition;
-      clear m;
-      rename b into m;
-      [rename equiv_b into equiv_m; assert (equiv_m' : in_rel m m) by intuition
-      | rename equiv_b into equiv_m'; assert (equiv_m : in_rel0 m m) by intuition];
+      [rename equiv_n into equiv0_n; assert (equiv_n : in_rel n n) by intuition
+      | assert (equiv0_n : in_rel0 n n) by intuition];
       destruct_rel_mod_eval;
-      [assert (exists am : domain, {{ $| a0 & m |↘ am }} /\ {{ Δ ⊢ m0[σ] m' : OT[σ,,m'] ® am ∈ OEl m equiv_m' }}) by intuition
-      | assert (exists am : domain, {{ $| a0 & m |↘ am }} /\ {{ Δ ⊢ m0[σ] m' : OT[σ,,m'] ® am ∈ OEl0 m equiv_m }}) by intuition];
+      [assert (exists m0n, {{ $| m0 & n |↘ m0n }} /\ {{ Δ ⊢ M0[σ] N : OT[σ,,N] ® m0n ∈ OEl n equiv_n }}) by intuition
+      | assert (exists m0n, {{ $| m0 & n |↘ m0n }} /\ {{ Δ ⊢ M0[σ] N : OT[σ,,N] ® m0n ∈ OEl0 n equiv0_n }}) by intuition];
       destruct_conjs;
-      assert ((OP m equiv_m' <∙> OP0 m equiv_m) /\ (OEl m equiv_m' <∙> OEl0 m equiv_m)) as [] by mauto 3;
+      assert ((OP n equiv_n <∙> OP0 n equiv0_n) /\ (OEl n equiv_n <∙> OEl0 n equiv0_n)) as [] by mauto 3;
       eexists; split; intuition.
 Qed.
 
 Ltac apply_functional_glu_univ_elem1 :=
   let tactic_error o1 o2 := fail 2 "functional_glu_univ_elem biconditional between" o1 "and" o2 "cannot be solved" in
   match goal with
-  | H1 : {{ DG ~?A ∈ glu_univ_elem ?i ↘ ?P1 ↘ ?El1 }},
-      H2 : {{ DG ~?A ∈ glu_univ_elem ?i' ↘ ?P2 ↘ ?El2 }} |- _ =>
+  | H1 : {{ DG ~?a ∈ glu_univ_elem ?i ↘ ?P1 ↘ ?El1 }},
+      H2 : {{ DG ~?a ∈ glu_univ_elem ?i' ↘ ?P2 ↘ ?El2 }} |- _ =>
       unify i i';
       assert_fails (unify P1 P2; unify El1 El2);
       match goal with
@@ -536,18 +535,18 @@ Ltac handle_functional_glu_univ_elem :=
   apply_predicate_equivalence;
   clear_dups.
 
-Lemma glu_univ_elem_pi_clean_inversion1 : forall {i a p B in_rel typ_rel el_rel},
+Lemma glu_univ_elem_pi_clean_inversion1 : forall {i a ρ B in_rel P El},
   {{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
-  {{ DG Π a p B ∈ glu_univ_elem i ↘ typ_rel ↘ el_rel }} ->
+  {{ DG Π a ρ B ∈ glu_univ_elem i ↘ P ↘ El }} ->
   exists IP IEl (OP : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_typ_pred)
      (OEl : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_exp_pred) elem_rel,
       {{ DG a ∈ glu_univ_elem i ↘ IP ↘ IEl }} /\
         (forall c (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) b,
-            {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+            {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
             {{ DG b ∈ glu_univ_elem i ↘ OP _ equiv_c ↘ OEl _ equiv_c }}) /\
-        {{ DF Π a p B ≈ Π a p B ∈ per_univ_elem i ↘ elem_rel }} /\
-        (typ_rel <∙> pi_glu_typ_pred i in_rel IP IEl OP) /\
-        (el_rel <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl).
+        {{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} /\
+        (P <∙> pi_glu_typ_pred i in_rel IP IEl OP) /\
+        (El <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl).
 Proof.
   intros *.
   simpl.
@@ -559,11 +558,11 @@ Proof.
   repeat split.
   1,3: eassumption.
   1: instantiate (1 := fun c equiv_c Γ A M m => forall (b : domain) Pb Elb,
-                          {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+                          {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
                           {{ DG b ∈ glu_univ_elem i ↘ Pb ↘ Elb }} ->
                           {{ Γ ⊢ M : A ® m ∈ Elb }}).
   1: instantiate (1 := fun c equiv_c Γ A => forall (b : domain) Pb Elb,
-                          {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+                          {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
                           {{ DG b ∈ glu_univ_elem i ↘ Pb ↘ Elb }} ->
                           {{ Γ ⊢ A ® Pb }}).
   2-5: intros []; econstructor; mauto.
@@ -575,31 +574,27 @@ Proof.
       split; intros; handle_functional_glu_univ_elem; intuition.
     + intros ? ?.
       split; [intros; handle_functional_glu_univ_elem |]; intuition.
-  - rename a0 into c.
-    rename equiv_a into equiv_c.
-    assert {{ Dom c ≈ c ∈ in_rel0 }} as equiv0_c by intuition.
-    assert {{ DG b ∈ glu_univ_elem i ↘ OP c equiv0_c ↘ OEl c equiv0_c }} by mauto 3.
+  - assert {{ Dom m ≈ m ∈ in_rel0 }} as equiv0_m by intuition.
+    assert {{ DG b ∈ glu_univ_elem i ↘ OP m equiv0_m ↘ OEl m equiv0_m }} by mauto 3.
     handle_functional_glu_univ_elem.
     intuition.
   - match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
     destruct_rel_mod_eval.
     intuition.
-  - rename b into c.
-    rename equiv_b into equiv_c.
-    assert {{ Dom c ≈ c ∈ in_rel0 }} as equiv0_c by intuition.
-    assert (exists ac : domain, {{ $| a0 & c |↘ ac }} /\ {{ Δ ⊢ m[σ] m' : OT[σ,,m'] ® ac ∈ OEl c equiv0_c }}) by mauto 3.
+  - assert {{ Dom n ≈ n ∈ in_rel0 }} as equiv0_n by intuition.
+    assert (exists mn : domain, {{ $| m & n |↘ mn }} /\ {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl n equiv0_n }}) by mauto 3.
     destruct_conjs.
     eexists.
     intuition.
-    assert {{ DG b ∈ glu_univ_elem i ↘ OP c equiv0_c ↘ OEl c equiv0_c }} by mauto 3.
+    assert {{ DG b ∈ glu_univ_elem i ↘ OP n equiv0_n ↘ OEl n equiv0_n }} by mauto 3.
     handle_functional_glu_univ_elem.
     intuition.
-  - assert (exists ab : domain,
-               {{ $| a0 & b |↘ ab }} /\
-                 (forall (b0 : domain) (Pb : glu_typ_pred) (Elb : glu_exp_pred),
-                     {{ ⟦ B ⟧ p ↦ b ↘ b0 }} ->
-                     {{ DG b0 ∈ glu_univ_elem i ↘ Pb ↘ Elb }} ->
-                     {{ Δ ⊢ m[σ] m' : OT[σ,,m'] ® ab ∈ Elb }})) by intuition.
+  - assert (exists mn : domain,
+               {{ $| m & n |↘ mn }} /\
+                 (forall (b : domain) (Pb : glu_typ_pred) (Elb : glu_exp_pred),
+                     {{ ⟦ B ⟧ ρ ↦ n ↘ b }} ->
+                     {{ DG b ∈ glu_univ_elem i ↘ Pb ↘ Elb }} ->
+                     {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ Elb }})) by intuition.
     destruct_conjs.
     match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
     handle_per_univ_elem_irrel.
@@ -610,18 +605,18 @@ Proof.
     intuition.
 Qed.
 
-Lemma glu_univ_elem_pi_clean_inversion2 : forall {i a p B in_rel IP IEl typ_rel el_rel},
+Lemma glu_univ_elem_pi_clean_inversion2 : forall {i a ρ B in_rel IP IEl P El},
   {{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
   {{ DG a ∈ glu_univ_elem i ↘ IP ↘ IEl }} ->
-  {{ DG Π a p B ∈ glu_univ_elem i ↘ typ_rel ↘ el_rel }} ->
+  {{ DG Π a ρ B ∈ glu_univ_elem i ↘ P ↘ El }} ->
   exists (OP : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_typ_pred)
      (OEl : forall c (equiv_c_c : {{ Dom c ≈ c ∈ in_rel }}), glu_exp_pred) elem_rel,
     (forall c (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) b,
-        {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+        {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
         {{ DG b ∈ glu_univ_elem i ↘ OP _ equiv_c ↘ OEl _ equiv_c }}) /\
-      {{ DF Π a p B ≈ Π a p B ∈ per_univ_elem i ↘ elem_rel }} /\
-      (typ_rel <∙> pi_glu_typ_pred i in_rel IP IEl OP) /\
-      (el_rel <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl).
+      {{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} /\
+      (P <∙> pi_glu_typ_pred i in_rel IP IEl OP) /\
+      (El <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl).
 Proof.
   intros *.
   simpl.
@@ -680,7 +675,7 @@ Qed.
 Add Parametric Morphism i R : (glu_univ_elem i)
     with signature glu_typ_pred_equivalence ==> glu_exp_pred_equivalence ==> per_univ_elem i R ==> iff as glu_univ_elem_morphism_iff'.
 Proof with mautosolve.
-  intros P P' HPP' El El' HElEl' a a' Haa'.
+  intros P P' HPP' El El' HElEl' **.
   rewrite HPP', HElEl'.
   split; intros; eapply glu_univ_elem_resp_per_univ; mauto.
   symmetry; mauto.
@@ -717,12 +712,12 @@ Proof.
     + saturate_refl; eassumption.
     + instantiate (1 := fun (c : domain) (equiv_c : in_rel c c) Γ A M m =>
                           forall b P El,
-                            {{ ⟦ B' ⟧ p' ↦ c ↘ b }} ->
+                            {{ ⟦ B' ⟧ ρ' ↦ c ↘ b }} ->
                             glu_univ_elem i P El b ->
                             {{ Γ ⊢ M : A ® m ∈ El }}).
       instantiate (1 := fun (c : domain) (equiv_c : in_rel c c) Γ A =>
                           forall b P El,
-                            {{ ⟦ B' ⟧ p' ↦ c ↘ b }} ->
+                            {{ ⟦ B' ⟧ ρ' ↦ c ↘ b }} ->
                             glu_univ_elem i P El b ->
                             {{ Γ ⊢ A ® P }}).
       intros.
@@ -730,7 +725,7 @@ Proof.
       handle_per_univ_elem_irrel.
       rewrite simple_glu_univ_elem_morphism_iff; try (eassumption + reflexivity);
         split; intros; handle_functional_glu_univ_elem; intuition.
-    + enough {{ DF Π A p B ≈ Π A' p' B' ∈ per_univ_elem i ↘ elem_rel }} by (etransitivity; [symmetry |]; eassumption).
+    + enough {{ DF Π a ρ B ≈ Π a' ρ' B' ∈ per_univ_elem i ↘ elem_rel }} by (etransitivity; [symmetry |]; eassumption).
       per_univ_elem_econstructor; mauto.
       intros.
       (on_all_hyp: destruct_rel_by_assumption in_rel).
@@ -833,23 +828,25 @@ Proof.
       deepexec H1 ltac:(fun H => pose proof H).
       autorewrite with mcltt in *.
       repeat eexists; eauto.
-      assert {{ Δ0 ⊢s σ0,, m' : Δ, ~ (a_sub IT σ) }}. {
+      assert {{ Δ0 ⊢s σ0,,N : Δ, ~ (a_sub IT σ) }}.
+      {
         econstructor; mauto 2.
         bulky_rewrite.
       }
-      assert {{Δ0 ⊢ (m [σ][σ0]) m' ≈ (m [σ ∘ σ0]) m' : OT [σ ∘ σ0,, m']}}. {
+      assert {{Δ0 ⊢ M[σ][σ0] N ≈ M[σ∘σ0] N : OT [σ∘σ0,,N]}}.
+      {
         rewrite <- @sub_eq_q_sigma_id_extend; mauto 4.
         rewrite <- @exp_eq_sub_compose_typ; mauto 3;
           [eapply wf_exp_eq_app_cong' |];
           mauto 4.
         symmetry.
         bulky_rewrite_in H4.
-        assert {{ Δ0 ⊢ Π IT[σ ∘ σ0] (OT[q (σ ∘ σ0)]) ≈ (Π IT OT)[σ ∘ σ0] : Type@(S (max i4 i)) }} by mauto.
+        assert {{ Δ0 ⊢ Π IT[σ∘σ0] (OT[q (σ∘σ0)]) ≈ (Π IT OT)[σ∘σ0] : Type@(S (max i4 i)) }} by mauto.
         eapply wf_exp_eq_conv'; mauto 4.
       }
 
       bulky_rewrite.
-      edestruct H10 with (b := b) as [? []];
+      edestruct H10 with (n := n) as [? []];
         simplify_evals; [| | eassumption];
         mauto.
 
diff --git a/theories/Core/Soundness/LogicalRelation/CoreTactics.v b/theories/Core/Soundness/LogicalRelation/CoreTactics.v
index 87da9f2c..2b85a547 100644
--- a/theories/Core/Soundness/LogicalRelation/CoreTactics.v
+++ b/theories/Core/Soundness/LogicalRelation/CoreTactics.v
@@ -1,5 +1,6 @@
 From Equations Require Import Equations.
 
+From Mcltt Require Import Base LibTactics.
 From Mcltt.Core.Soundness Require Import LogicalRelation.Definitions.
 
 Ltac basic_invert_glu_univ_elem H :=
@@ -12,3 +13,15 @@ Ltac basic_glu_univ_elem_econstructor :=
   progress simp glu_univ_elem;
   econstructor;
   try rewrite <- glu_univ_elem_equation_1 in *.
+
+Ltac invert_glu_rel1 :=
+  match goal with
+  | H : pi_glu_typ_pred _ _ _ _ _ _ _ |- _ =>
+      progressive_invert H
+  | H : pi_glu_exp_pred _ _ _ _ _ _ _ _ _ _ |- _ =>
+      progressive_invert H
+  | H : neut_glu_typ_pred _ _ _ _ |- _ =>
+      progressive_invert H
+  | H : neut_glu_exp_pred _ _ _ _ _ _ |- _ =>
+      progressive_invert H
+  end.
diff --git a/theories/Core/Soundness/LogicalRelation/Definitions.v b/theories/Core/Soundness/LogicalRelation/Definitions.v
index b5c44a16..9802d177 100644
--- a/theories/Core/Soundness/LogicalRelation/Definitions.v
+++ b/theories/Core/Soundness/LogicalRelation/Definitions.v
@@ -31,84 +31,84 @@ Notation "'EG' A ∈ R ↘ Sb " := (R Sb A : ((Prop : (Type : Type)) : (Type : T
 
 Inductive glu_nat : ctx -> exp -> domain -> Prop :=
 | glu_nat_zero :
-  `{ {{ Γ ⊢ m ≈ zero : ℕ }} ->
-     glu_nat Γ m d{{{ zero }}} }
+  `{ {{ Γ ⊢ M ≈ zero : ℕ }} ->
+     glu_nat Γ M d{{{ zero }}} }
 | glu_nat_succ :
-  `{ {{ Γ ⊢ m ≈ succ m' : ℕ }} ->
-     glu_nat Γ m' a ->
-     glu_nat Γ m d{{{ succ a }}} }
+  `{ {{ Γ ⊢ M ≈ succ M' : ℕ }} ->
+     glu_nat Γ M' m' ->
+     glu_nat Γ M d{{{ succ m' }}} }
 | glu_nat_neut :
-  `{ per_bot c c ->
-     (forall {Δ σ v}, {{ Δ ⊢w σ : Γ }} -> {{ Rne c in length Δ ↘ v }} -> {{ Δ ⊢ m[σ] ≈ v : ℕ }}) ->
-     glu_nat Γ m d{{{ ⇑ ℕ c }}} }.
+  `{ per_bot m m ->
+     (forall {Δ σ M'}, {{ Δ ⊢w σ : Γ }} -> {{ Rne m in length Δ ↘ M' }} -> {{ Δ ⊢ M[σ] ≈ M' : ℕ }}) ->
+     glu_nat Γ M d{{{ ⇑ ℕ m }}} }.
 
 #[export]
 Hint Constructors glu_nat : mcltt.
 
-Definition nat_glu_typ_pred i : glu_typ_pred := fun Γ M => {{ Γ ⊢ M ≈ ℕ : Type@i }}.
-Arguments nat_glu_typ_pred i Γ M/.
+Definition nat_glu_typ_pred i : glu_typ_pred := fun Γ A => {{ Γ ⊢ A ≈ ℕ : Type@i }}.
+Arguments nat_glu_typ_pred i Γ A/.
 
-Definition nat_glu_exp_pred i : glu_exp_pred := fun Γ M m a => {{ Γ ⊢ M ® nat_glu_typ_pred i }} /\ glu_nat Γ m a.
-Arguments nat_glu_exp_pred i Γ m M a/.
+Definition nat_glu_exp_pred i : glu_exp_pred := fun Γ A M m => {{ Γ ⊢ A ® nat_glu_typ_pred i }} /\ glu_nat Γ M m.
+Arguments nat_glu_exp_pred i Γ A M m/.
 
-Definition neut_glu_typ_pred i C : glu_typ_pred :=
-  fun Γ M => {{ Γ ⊢ M : Type@i }} /\
-            (forall Δ σ V, {{ Δ ⊢w σ : Γ }} -> {{ Rne C in length Δ ↘ V }} -> {{ Δ ⊢ M[σ] ≈ V : Type@i }}).
-Arguments neut_glu_typ_pred i C Γ M/.
+Definition neut_glu_typ_pred i a : glu_typ_pred :=
+  fun Γ A => {{ Γ ⊢ A : Type@i }} /\
+            (forall Δ σ A', {{ Δ ⊢w σ : Γ }} -> {{ Rne a in length Δ ↘ A' }} -> {{ Δ ⊢ A[σ] ≈ A' : Type@i }}).
+Arguments neut_glu_typ_pred i a Γ A/.
 
-Inductive neut_glu_exp_pred i C : glu_exp_pred :=
+Variant neut_glu_exp_pred i a : glu_exp_pred :=
 | mk_neut_glu_exp_pred :
-  `{ {{ Γ ⊢ M ® neut_glu_typ_pred i C }} ->
-     {{ Γ ⊢ m : M }} ->
-       {{ Dom c ≈ c ∈ per_bot }} ->
-       (forall Δ σ V v, {{ Δ ⊢w σ : Γ }} ->
-                      {{ Rne C in length Δ ↘ V }} ->
-                      {{ Rne c in length Δ ↘ v }} ->
-                      {{ Δ ⊢ m[σ] ≈ v : M[σ] }}) ->
-       {{ Γ ⊢ m : M ® ⇑ A c ∈ neut_glu_exp_pred i C }} }.
-
-Inductive pi_glu_typ_pred i
+  `{ {{ Γ ⊢ A ® neut_glu_typ_pred i a }} ->
+     {{ Γ ⊢ M : A }} ->
+     {{ Dom m ≈ m ∈ per_bot }} ->
+     (forall Δ σ (* A' *) M', {{ Δ ⊢w σ : Γ }} ->
+                    (* {{ Rne a in length Δ ↘ A' }} -> *)
+                    {{ Rne m in length Δ ↘ M' }} ->
+                    {{ Δ ⊢ M[σ] ≈ M' : A[σ] }}) ->
+     {{ Γ ⊢ M : A ® ⇑ b m ∈ neut_glu_exp_pred i a }} }.
+
+Variant pi_glu_typ_pred i
   (IR : relation domain)
   (IP : glu_typ_pred)
   (IEl : glu_exp_pred)
   (OP : forall c (equiv_c : {{ Dom c ≈ c ∈ IR }}), glu_typ_pred) : glu_typ_pred :=
 | mk_pi_glu_typ_pred :
-    `{ {{ Γ ⊢ M ≈ Π IT OT : Type@i }} ->
-       {{ Γ ⊢ IT : Type@i }} ->
-       {{ Γ , IT ⊢ OT : Type@i }} ->
-       (forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
-       (forall Δ σ m a,
-           {{ Δ ⊢w σ : Γ }} ->
-           {{ Δ ⊢ m : IT[σ] ® a ∈ IEl }} ->
-           forall (equiv_a : {{ Dom a ≈ a ∈ IR }}),
-             {{ Δ ⊢ OT[σ,,m] ® OP _ equiv_a }}) ->
-       {{ Γ ⊢ M ® pi_glu_typ_pred i IR IP IEl OP }} }.
-
-Inductive pi_glu_exp_pred i
+  `{ {{ Γ ⊢ A ≈ Π IT OT : Type@i }} ->
+     {{ Γ ⊢ IT : Type@i }} ->
+     {{ Γ , IT ⊢ OT : Type@i }} ->
+     (forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
+     (forall Δ σ M m,
+         {{ Δ ⊢w σ : Γ }} ->
+         {{ Δ ⊢ M : IT[σ] ® m ∈ IEl }} ->
+         forall (equiv_m : {{ Dom m ≈ m ∈ IR }}),
+           {{ Δ ⊢ OT[σ,,M] ® OP _ equiv_m }}) ->
+     {{ Γ ⊢ A ® pi_glu_typ_pred i IR IP IEl OP }} }.
+
+Variant pi_glu_exp_pred i
   (IR : relation domain)
   (IP : glu_typ_pred)
   (IEl : glu_exp_pred)
   (elem_rel : relation domain)
   (OEl : forall c (equiv_c : {{ Dom c ≈ c ∈ IR }}), glu_exp_pred): glu_exp_pred :=
 | mk_pi_glu_exp_pred :
-  `{ {{ Γ ⊢ m : M }} ->
-     {{ Dom a ≈ a ∈ elem_rel }} ->
-     {{ Γ ⊢ M ≈ Π IT OT : Type@i }} ->
-       {{ Γ ⊢ IT : Type@i }} ->
+  `{ {{ Γ ⊢ M : A }} ->
+     {{ Dom m ≈ m ∈ elem_rel }} ->
+     {{ Γ ⊢ A ≈ Π IT OT : Type@i }} ->
+     {{ Γ ⊢ IT : Type@i }} ->
      {{ Γ , IT ⊢ OT : Type@i }} ->
      (forall Δ σ, {{ Δ ⊢w σ : Γ }} -> {{ Δ ⊢ IT[σ] ® IP }}) ->
-     (forall Δ σ m' b,
+     (forall Δ σ N n,
          {{ Δ ⊢w σ : Γ }} ->
-         {{ Δ ⊢ m' : IT[ σ ] ® b ∈ IEl }} ->
-         forall (equiv_b : {{ Dom b ≈ b ∈ IR }}),
-         exists ab, {{ $| a & b |↘ ab }} /\ {{ Δ ⊢ m[σ] m' : OT[σ,,m'] ® ab ∈ OEl _ equiv_b }}) ->
-     {{ Γ ⊢ m : M ® a ∈ pi_glu_exp_pred i IR IP IEl elem_rel OEl }} }.
+         {{ Δ ⊢ N : IT[σ] ® n ∈ IEl }} ->
+         forall (equiv_n : {{ Dom n ≈ n ∈ IR }}),
+         exists mn, {{ $| m & n |↘ mn }} /\ {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl _ equiv_n }}) ->
+     {{ Γ ⊢ M : A ® m ∈ pi_glu_exp_pred i IR IP IEl elem_rel OEl }} }.
 
 #[export]
 Hint Constructors neut_glu_exp_pred pi_glu_typ_pred pi_glu_exp_pred : mcltt.
 
-Definition univ_glu_typ_pred j i : glu_typ_pred := fun Γ T => {{ Γ ⊢ T ≈ Type@j :  Type@i }}.
-Arguments univ_glu_typ_pred j i Γ T/.
+Definition univ_glu_typ_pred j i : glu_typ_pred := fun Γ A => {{ Γ ⊢ A ≈ Type@j :  Type@i }}.
+Arguments univ_glu_typ_pred j i Γ A/.
 Transparent univ_glu_typ_pred.
 
 Section Gluing.
@@ -117,13 +117,13 @@ Section Gluing.
       (glu_univ_typ_rec : forall {j}, j < i -> domain -> glu_typ_pred).
 
   Definition univ_glu_exp_pred' {j} (lt_j_i : j < i) : glu_exp_pred :=
-    fun Γ M m A =>
-      {{ Γ ⊢ m : M }} /\
-        {{ Γ ⊢ M ≈ Type@j : Type@i }} /\
-        {{ Γ ⊢ m ® glu_univ_typ_rec lt_j_i A }}.
+    fun Γ A M m =>
+      {{ Γ ⊢ M : A }} /\
+        {{ Γ ⊢ A ≈ Type@j : Type@i }} /\
+        {{ Γ ⊢ M ® glu_univ_typ_rec lt_j_i m }}.
 
   #[global]
-  Arguments univ_glu_exp_pred' {j} lt_j_i Γ m M A/.
+  Arguments univ_glu_exp_pred' {j} lt_j_i Γ A M m/.
 
   Inductive glu_univ_elem_core : glu_typ_pred -> glu_exp_pred -> domain -> Prop :=
   | glu_univ_elem_core_univ :
@@ -150,12 +150,12 @@ Section Gluing.
           {{ DG a ∈ glu_univ_elem_core ↘ IP ↘ IEl }} ->
           {{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
           (forall {c} (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) b,
-              {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+              {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
               {{ DG b ∈ glu_univ_elem_core ↘ OP _ equiv_c ↘ OEl _ equiv_c }}) ->
-          {{ DF Π a p B ≈ Π a p B ∈ per_univ_elem i ↘ elem_rel }} ->
+          {{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
           typ_rel <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
           el_rel <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
-          {{ DG Π a p B ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
+          {{ DG Π a ρ B ∈ glu_univ_elem_core ↘ typ_rel ↘ el_rel }} }
 
   | glu_univ_elem_core_neut :
     `{ forall typ_rel el_rel,
@@ -169,63 +169,63 @@ End Gluing.
 Hint Constructors glu_univ_elem_core : mcltt.
 
 Equations glu_univ_elem (i : nat) : glu_typ_pred -> glu_exp_pred -> domain -> Prop by wf i :=
-| i => glu_univ_elem_core i (fun j lt_j_i A Γ M => exists P El, {{ DG A ∈ glu_univ_elem j ↘ P ↘ El }} /\ {{ Γ ⊢ M ® P }}).
+| i => glu_univ_elem_core i (fun j lt_j_i a Γ A => exists P El, {{ DG a ∈ glu_univ_elem j ↘ P ↘ El }} /\ {{ Γ ⊢ A ® P }}).
 
-Definition glu_univ_typ (i : nat) (A : domain) : glu_typ_pred :=
-  fun Γ M => exists P El, {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} /\ {{ Γ ⊢ M ® P }}.
-Arguments glu_univ_typ i A Γ M/.
+Definition glu_univ_typ (i : nat) (a : domain) : glu_typ_pred :=
+  fun Γ A => exists P El, {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} /\ {{ Γ ⊢ A ® P }}.
+Arguments glu_univ_typ i a Γ A/.
 
 Definition univ_glu_exp_pred j i : glu_exp_pred :=
-    fun Γ M m A =>
-      {{ Γ ⊢ m : M }} /\ {{ Γ ⊢ M ≈ Type@j : Type@i }} /\
-        {{ Γ ⊢ m ® glu_univ_typ j A }}.
-Arguments univ_glu_exp_pred j i Γ t T a/.
+    fun Γ A M m =>
+      {{ Γ ⊢ M : A }} /\ {{ Γ ⊢ A ≈ Type@j : Type@i }} /\
+        {{ Γ ⊢ M ® glu_univ_typ j m }}.
+Arguments univ_glu_exp_pred j i Γ A M m/.
 
 Section GluingInduction.
   Hypothesis
     (motive : nat -> glu_typ_pred -> glu_exp_pred -> domain -> Prop)
 
       (case_univ :
-        forall i (j : nat)
-          (typ_rel : glu_typ_pred) (el_rel : glu_exp_pred) (lt_j_i : j < i),
-          (forall P El A, {{ DG A ∈ glu_univ_elem j ↘ P ↘ El }} -> motive j P El A) ->
-          typ_rel <∙> univ_glu_typ_pred j i ->
-          el_rel <∙> univ_glu_exp_pred j i ->
-          motive i typ_rel el_rel d{{{ 𝕌 @ j }}})
+        forall i j
+          (P : glu_typ_pred) (El : glu_exp_pred) (lt_j_i : j < i),
+          (forall P' El' a, {{ DG a ∈ glu_univ_elem j ↘ P' ↘ El' }} -> motive j P' El' a) ->
+          P <∙> univ_glu_typ_pred j i ->
+          El <∙> univ_glu_exp_pred j i ->
+          motive i P El d{{{ 𝕌 @ j }}})
 
       (case_nat :
-        forall i (typ_rel : glu_typ_pred) (el_rel : glu_exp_pred),
-          typ_rel <∙> nat_glu_typ_pred i ->
-          el_rel <∙> nat_glu_exp_pred i ->
-          motive i typ_rel el_rel d{{{ ℕ }}})
+        forall i (P : glu_typ_pred) (El : glu_exp_pred),
+          P <∙> nat_glu_typ_pred i ->
+          El <∙> nat_glu_exp_pred i ->
+          motive i P El d{{{ ℕ }}})
 
       (case_pi :
-        forall i (a : domain) (B : typ) (p : env) (in_rel : relation domain) (IP : glu_typ_pred)
+        forall i a B (ρ : env) (in_rel : relation domain) (IP : glu_typ_pred)
           (IEl : glu_exp_pred) (OP : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_typ_pred)
-          (OEl : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_exp_pred) (typ_rel : glu_typ_pred) (el_rel : glu_exp_pred)
+          (OEl : forall c : domain, {{ Dom c ≈ c ∈ in_rel }} -> glu_exp_pred) (P : glu_typ_pred) (El : glu_exp_pred)
           (elem_rel : relation domain),
           {{ DG a ∈ glu_univ_elem i ↘ IP ↘ IEl }} ->
           motive i IP IEl a ->
           {{ DF a ≈ a ∈ per_univ_elem i ↘ in_rel }} ->
           (forall (c : domain) (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) (b : domain),
-              {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+              {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
               {{ DG b ∈ glu_univ_elem i ↘ OP c equiv_c ↘ OEl c equiv_c }}) ->
           (forall (c : domain) (equiv_c : {{ Dom c ≈ c ∈ in_rel }}) (b : domain),
-              {{ ⟦ B ⟧ p ↦ c ↘ b }} ->
+              {{ ⟦ B ⟧ ρ ↦ c ↘ b }} ->
               motive i (OP c equiv_c) (OEl c equiv_c) b) ->
-          {{ DF Π a p B ≈ Π a p B ∈ per_univ_elem i ↘ elem_rel }} ->
-          typ_rel <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
-          el_rel <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
-          motive i typ_rel el_rel d{{{ Π a p B }}})
+          {{ DF Π a ρ B ≈ Π a ρ B ∈ per_univ_elem i ↘ elem_rel }} ->
+          P <∙> pi_glu_typ_pred i in_rel IP IEl OP ->
+          El <∙> pi_glu_exp_pred i in_rel IP IEl elem_rel OEl ->
+          motive i P El d{{{ Π a ρ B }}})
 
       (case_neut :
-        forall i (b : domain_ne) (a : domain)
-          (typ_rel : glu_typ_pred)
-          (el_rel : glu_exp_pred),
+        forall i b a
+          (P : glu_typ_pred)
+          (El : glu_exp_pred),
           {{ Dom b ≈ b ∈ per_bot }} ->
-          typ_rel <∙> neut_glu_typ_pred i b ->
-          el_rel <∙> neut_glu_exp_pred i b ->
-          motive i typ_rel el_rel d{{{ ⇑ a b }}})
+          P <∙> neut_glu_typ_pred i b ->
+          El <∙> neut_glu_exp_pred i b ->
+          motive i P El d{{{ ⇑ a b }}})
   .
 
   #[local]
@@ -237,13 +237,13 @@ Section GluingInduction.
   | i, P, El, a, H =>
       glu_univ_elem_core_ind
         i
-        (fun j lt_j_i A Γ M => glu_univ_typ j A Γ M)
+        (fun j lt_j_i a Γ A => glu_univ_typ j a Γ A)
         (motive i)
-        (fun j typ_rel el_rel lt_j_i Hty Hel =>
-           case_univ i j typ_rel el_rel lt_j_i
-             (fun P El A H => glu_univ_elem_ind j P El A H)
-             Hty
-             Hel)
+        (fun j P' El' lt_j_i HP' HEl' =>
+           case_univ i j P' El' lt_j_i
+             (fun P'' El'' A H => glu_univ_elem_ind j P'' El'' A H)
+             HP'
+             HEl')
         (case_nat i)
         _ (* (case_pi i) *)
         (case_neut i)
@@ -254,50 +254,36 @@ Section GluingInduction.
   Qed.
 End GluingInduction.
 
-Inductive glu_elem_bot i A Γ T t c : Prop :=
+Variant glu_elem_bot i a Γ A M m : Prop :=
 | glu_elem_bot_make : forall P El,
-    {{ Γ ⊢ t : T }} ->
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    {{ Γ ⊢ T ® P }} ->
-    {{ Dom c ≈ c ∈ per_bot }} ->
-    (forall Δ σ w, {{ Δ ⊢w σ : Γ }} -> {{ Rne c in length Δ ↘ w }} -> {{ Δ ⊢ t [ σ ] ≈ w : T [ σ ] }}) ->
-    {{ Γ ⊢ t : T ® c ∈ glu_elem_bot i A }}.
+    {{ Γ ⊢ M : A }} ->
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    {{ Γ ⊢ A ® P }} ->
+    {{ Dom m ≈ m ∈ per_bot }} ->
+    (forall Δ σ M', {{ Δ ⊢w σ : Γ }} -> {{ Rne m in length Δ ↘ M' }} -> {{ Δ ⊢ M[σ] ≈ M' : A[σ] }}) ->
+    {{ Γ ⊢ M : A ® m ∈ glu_elem_bot i a }}.
 #[export]
-  Hint Constructors glu_elem_bot : mcltt.
-
+Hint Constructors glu_elem_bot : mcltt.
 
-Inductive glu_elem_top i A Γ T t a : Prop :=
+Variant glu_elem_top i a Γ A M m : Prop :=
 | glu_elem_top_make : forall P El,
-    {{ Γ ⊢ t : T }} ->
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    {{ Γ ⊢ T ® P }} ->
-    {{ Dom ⇓ A a ≈ ⇓ A a ∈ per_top }} ->
-    (forall Δ σ w, {{ Δ ⊢w σ : Γ }} -> {{ Rnf ⇓ A a in length Δ ↘ w }} -> {{ Δ ⊢ t [ σ ] ≈ w : T [ σ ] }}) ->
-    {{ Γ ⊢ t : T ® a ∈ glu_elem_top i A }}.
+    {{ Γ ⊢ M : A }} ->
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    {{ Γ ⊢ A ® P }} ->
+    {{ Dom ⇓ a m ≈ ⇓ a m ∈ per_top }} ->
+    (forall Δ σ w, {{ Δ ⊢w σ : Γ }} -> {{ Rnf ⇓ a m in length Δ ↘ w }} -> {{ Δ ⊢ M[σ] ≈ w : A[σ] }}) ->
+    {{ Γ ⊢ M : A ® m ∈ glu_elem_top i a }}.
 #[export]
-  Hint Constructors glu_elem_top : mcltt.
-
+Hint Constructors glu_elem_top : mcltt.
 
-Inductive glu_typ_top i A Γ T : Prop :=
+Variant glu_typ_top i a Γ A : Prop :=
 | glu_typ_top_make :
-    {{ Γ ⊢ T : Type@i }} ->
-    {{ Dom A ≈ A ∈ per_top_typ }} ->
-    (forall Δ σ W, {{ Δ ⊢w σ : Γ }} -> {{ Rtyp A in length Δ ↘ W }} -> {{ Δ ⊢ T [ σ ] ≈ W : Type@i }}) ->
-    {{ Γ ⊢ T ® glu_typ_top i A }}.
+    {{ Γ ⊢ A : Type@i }} ->
+    {{ Dom a ≈ a ∈ per_top_typ }} ->
+    (forall Δ σ A', {{ Δ ⊢w σ : Γ }} -> {{ Rtyp a in length Δ ↘ A' }} -> {{ Δ ⊢ A[σ] ≈ A' : Type@i }}) ->
+    {{ Γ ⊢ A ® glu_typ_top i a }}.
 #[export]
-  Hint Constructors glu_typ_top : mcltt.
-
-Ltac invert_glu_rel1 :=
-  match goal with
-  | H : pi_glu_typ_pred _ _ _ _ _ _ _ |- _ =>
-      progressive_invert H
-  | H : pi_glu_exp_pred _ _ _ _ _ _ _ _ _ _ |- _ =>
-      progressive_invert H
-  | H : neut_glu_typ_pred _ _ _ _ |- _ =>
-      progressive_invert H
-  | H : neut_glu_exp_pred _ _ _ _ _ _ |- _ =>
-      progressive_invert H
-  end.
+Hint Constructors glu_typ_top : mcltt.
 
 Variant glu_rel_typ_with_sub i Δ A σ ρ : Prop :=
 | mk_glu_rel_typ_with_sub :
@@ -319,7 +305,10 @@ Variant cons_glu_sub_pred i Γ A (TSb : glu_sub_pred) : glu_sub_pred :=
         {{ Δ ⊢s σ : Γ, A }} ->
         {{ ⟦ A ⟧ ρ ↯ ↘ a }} ->
         {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
-        {{ Δ ⊢ #0[σ] : A[Wk∘σ] ® ~(ρ 0) ∈ El }} ->
+        (* Here we use [A[Wk][σ]] instead of [A[Wk∘σ]]
+           as syntactic judgement derived from that is
+           a more direct consequence of [{{ Γ, A ⊢ #0 : A[Wk] }}] *)
+        {{ Δ ⊢ #0[σ] : A[Wk][σ] ® ~(ρ 0) ∈ El }} ->
         {{ Δ ⊢s Wk ∘ σ ® ρ ↯ ∈ TSb }} ->
         {{ Δ ⊢s σ ® ρ ∈ cons_glu_sub_pred i Γ A TSb }} }.
 
diff --git a/theories/Core/Soundness/LogicalRelation/Lemmas.v b/theories/Core/Soundness/LogicalRelation/Lemmas.v
index 576339b7..b23b245d 100644
--- a/theories/Core/Soundness/LogicalRelation/Lemmas.v
+++ b/theories/Core/Soundness/LogicalRelation/Lemmas.v
@@ -187,15 +187,13 @@ Section glu_univ_elem_cumulativity.
       + assert {{ Δ ⊢ IT[σ] ® IP }} by mauto.
         enough (forall Γ A, {{ Γ ⊢ A ® IP }} -> {{ Γ ⊢ A ® IP' }}) by mauto 4.
         eapply proj1...
-      + rename a0 into c.
-        rename equiv_a into equiv_c.
-        match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
+      + match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
         apply_relation_equivalence.
         destruct_rel_mod_eval.
         handle_per_univ_elem_irrel.
-        assert (forall Γ A, {{ Γ ⊢ A ® OP c equiv_c }} -> {{ Γ ⊢ A ® OP' c equiv_c }}) by (eapply proj1; mauto).
-        enough {{ Δ ⊢ OT[σ,,m] ® OP c equiv_c }} by mauto.
-        enough {{ Δ ⊢ m : IT[σ] ® c ∈ IEl }} by mauto.
+        assert (forall Γ A, {{ Γ ⊢ A ® OP m equiv_m }} -> {{ Γ ⊢ A ® OP' m equiv_m }}) by (eapply proj1; mauto).
+        enough {{ Δ ⊢ OT[σ,,M] ® OP m equiv_m }} by mauto.
+        enough {{ Δ ⊢ M : IT[σ] ® m ∈ IEl }} by mauto.
         eapply IHHglu...
     - rename x into IP'.
       rename x0 into IEl'.
@@ -206,18 +204,16 @@ Section glu_univ_elem_cumulativity.
       econstructor; intros; mauto 4.
       + enough (forall Γ A, {{ Γ ⊢ A ® IP }} -> {{ Γ ⊢ A ® IP' }}) by mauto 4.
         eapply proj1...
-      + rename b into c.
-        rename equiv_b into equiv_c.
-        match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
+      + match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
         handle_per_univ_elem_irrel.
         destruct_rel_mod_eval.
         destruct_rel_mod_app.
         handle_per_univ_elem_irrel.
         eexists; split; mauto 4.
-        assert (forall Γ A M m, {{ Γ ⊢ M : A ® m ∈ OEl c equiv_c }} -> {{ Γ ⊢ M : A ® m ∈ OEl' c equiv_c }}) by (eapply proj1, proj2; mauto 4).
-        enough {{ Δ ⊢ m[σ] m' : OT[σ,,m'] ® fa ∈ OEl c equiv_c }} by mauto 4.
-        assert {{ Δ ⊢ m' : IT[σ] ® c ∈ IEl }} by (eapply IHHglu; mauto 3).
-        assert (exists ac, {{ $| a0 & c |↘ ac }} /\ {{ Δ ⊢ m[σ] m' : OT[σ,,m'] ® ac ∈ OEl c equiv_c }}) by mauto 4.
+        assert (forall Γ A M m, {{ Γ ⊢ M : A ® m ∈ OEl n equiv_n }} -> {{ Γ ⊢ M : A ® m ∈ OEl' n equiv_n }}) by (eapply proj1, proj2; mauto 4).
+        enough {{ Δ ⊢ M[σ] N : OT[σ,,N] ® fa ∈ OEl n equiv_n }} by mauto 4.
+        assert {{ Δ ⊢ N : IT[σ] ® n ∈ IEl }} by (eapply IHHglu; mauto 3).
+        assert (exists mn, {{ $| m & n |↘ mn }} /\ {{ Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl n equiv_n }}) by mauto 4.
         destruct_conjs.
         functional_eval_rewrite_clear...
     - rename x into IP'.
@@ -228,29 +224,30 @@ Section glu_univ_elem_cumulativity.
       destruct_by_head pi_glu_exp_pred.
       handle_per_univ_elem_irrel.
       econstructor; intros; mauto.
-      rename b into c.
-      rename equiv_b into equiv_c.
       match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
       handle_per_univ_elem_irrel.
       destruct_rel_mod_eval.
       destruct_rel_mod_app.
       handle_per_univ_elem_irrel.
-      rename a1 into b.
+      match goal with
+      | _: {{ ⟦ B ⟧ ρ ↦ n ↘ ~?a }} |- _ =>
+          rename a into b
+      end.
       eexists; split; mauto 4.
-      assert (forall Γ A M m, {{ Γ ⊢ A ® OP c equiv_c }} -> {{ Γ ⊢ M : A ® m ∈ OEl' c equiv_c }} -> {{ Γ ⊢ M : A ® m ∈ OEl c equiv_c }}) by (eapply proj2, proj2; eauto 3).
-      assert {{ Δ ⊢ OT[σ,,m'] ® OP c equiv_c }} by mauto 4.
-      enough {{ Δ ⊢ m[σ] m' : OT[σ,,m'] ® fa ∈ OEl' c equiv_c }} by mauto 4.
-      assert {{ Δ ⊢ m' : IT[σ] ® c ∈ IEl' }} by (eapply IHHglu; mauto 3).
+      assert (forall Γ A M m, {{ Γ ⊢ A ® OP n equiv_n }} -> {{ Γ ⊢ M : A ® m ∈ OEl' n equiv_n }} -> {{ Γ ⊢ M : A ® m ∈ OEl n equiv_n }}) by (eapply proj2, proj2; eauto 3).
+      assert {{ Δ ⊢ OT[σ,,N] ® OP n equiv_n }} by mauto 4.
+      enough {{ Δ ⊢ M[σ] N : OT[σ,,N] ® fa ∈ OEl' n equiv_n }} by mauto 4.
+      assert {{ Δ ⊢ N : IT[σ] ® n ∈ IEl' }} by (eapply IHHglu; mauto 3).
       assert {{ Δ ⊢ IT[σ] ® IP' }} by (eapply glu_univ_elem_trm_typ; mauto 2).
       assert {{ Δ ⊢ IT0[σ] ® IP' }} by mauto 2.
       assert {{ Δ ⊢ IT[σ] ≈ IT0[σ] : Type@j }} as HITeq by mauto 2.
-      assert {{ Δ ⊢ m' : IT0[σ] ® c ∈ IEl' }} by (rewrite <- HITeq; mauto 3).
-      assert (exists ac, {{ $| a0 & c |↘ ac }} /\ {{ Δ ⊢ m[σ] m' : OT0[σ,,m'] ® ac ∈ OEl' c equiv_c }}) by mauto 3.
+      assert {{ Δ ⊢ N : IT0[σ] ® n ∈ IEl' }} by (rewrite <- HITeq; mauto 3).
+      assert (exists mn, {{ $| m & n |↘ mn }} /\ {{ Δ ⊢ M[σ] N : OT0[σ,,N] ® mn ∈ OEl' n equiv_n }}) by mauto 3.
       destruct_conjs.
       functional_eval_rewrite_clear.
-      assert {{ DG b ∈ glu_univ_elem j ↘ OP' c equiv_c ↘ OEl' c equiv_c }} by mauto 3.
-      assert {{ Δ ⊢ OT0[σ,,m'] ® OP' c equiv_c }} by (eapply glu_univ_elem_trm_typ; mauto 3).
-      enough {{ Δ ⊢ OT[σ,,m'] ≈ OT0[σ,,m'] : Type@j }} as ->...
+      assert {{ DG b ∈ glu_univ_elem j ↘ OP' n equiv_n ↘ OEl' n equiv_n }} by mauto 3.
+      assert {{ Δ ⊢ OT0[σ,,N] ® OP' n equiv_n }} by (eapply glu_univ_elem_trm_typ; mauto 3).
+      enough {{ Δ ⊢ OT[σ,,N] ≈ OT0[σ,,N] : Type@j }} as ->...
     - destruct_by_head neut_glu_exp_pred.
       econstructor; mauto.
       destruct_by_head neut_glu_typ_pred.
@@ -412,7 +409,7 @@ Proof.
     mauto 3.
   - destruct_by_head pi_glu_typ_pred.
     rename x into IP. rename x0 into IEl. rename x1 into OP. rename x2 into OEl.
-    rename M0 into M'. rename IT0 into IT'. rename OT0 into OT'.
+    rename A0 into A'. rename IT0 into IT'. rename OT0 into OT'.
     rename x3 into OP'. rename x4 into OEl'.
     assert {{ Γ ⊢ IT ® IP }}.
     {
@@ -437,8 +434,12 @@ Proof.
     apply_relation_equivalence.
     destruct_rel_mod_eval.
     handle_per_univ_elem_irrel.
-    rename a1 into b.
-    rename a3 into b'.
+    match goal with
+    | _: {{ ⟦ B ⟧ ρ ↦ ⇑! a (length Γ) ↘ ~?a0 }},
+        _: {{ ⟦ B' ⟧ ρ' ↦ ⇑! a' (length Γ) ↘ ~?a0' }} |- _ =>
+        rename a0 into b;
+        rename a0' into b'
+    end.
     assert {{ DG b ∈ glu_univ_elem i ↘ OP d{{{ ⇑! a (length Γ) }}} equiv_len_len ↘ OEl d{{{ ⇑! a (length Γ) }}} equiv_len_len }} by mauto 4.
     assert {{ DG b' ∈ glu_univ_elem i ↘ OP' d{{{ ⇑! a' (length Γ) }}} equiv_len'_len' ↘ OEl' d{{{ ⇑! a' (length Γ) }}} equiv_len'_len' }} by mauto 4.
     assert {{ Γ, IT' ⊢ OT ® OP d{{{ ⇑! a (length Γ) }}} equiv_len_len }}.
@@ -483,18 +484,14 @@ Proof.
     handle_functional_glu_univ_elem;
     repeat invert_glu_rel1;
     try solve [simpl in *; intuition mauto 3].
-  - rename A into a''.
-    rename M into A.
-    rename m into M.
-    clear_dups.
-    match_by_head1 (per_bot b b') ltac:(fun H => destruct (H (length Γ)) as [V []]).
+  - match_by_head1 (per_bot b b') ltac:(fun H => destruct (H (length Γ)) as [V []]).
     econstructor; mauto 3.
     + econstructor; mauto 3.
     + intros.
       match_by_head1 (per_bot b b') ltac:(fun H => destruct (H (length Δ)) as [? []]).
       functional_read_rewrite_clear.
       assert {{ Δ ⊢ A[σ] ⊆ A'[σ] }} by mauto 3.
-      assert {{ Δ ⊢ M[σ] ≈ v : A[σ] }} by mauto 3.
+      assert {{ Δ ⊢ M[σ] ≈ M' : A[σ] }} by mauto 3.
       mauto 3.
   - simpl in *.
     destruct_conjs.
@@ -502,20 +499,16 @@ Proof.
     assert (exists P El, {{ DG m ∈ glu_univ_elem j ↘ P ↘ El }}) as [? [?]] by mauto 3.
     do 2 eexists; split; mauto 3.
     eapply glu_univ_elem_typ_cumu_ge; revgoals; mautosolve 3.
-  - rename M into A.
-    rename M0 into A'.
-    rename m into M.
+  - rename A0 into A'.
     rename IT0 into IT'. rename OT0 into OT'.
     rename x into IP. rename x0 into IEl.
     rename x1 into OP. rename x2 into OEl.
     rename x3 into OP'. rename x4 into OEl'.
     handle_per_univ_elem_irrel.
     econstructor; mauto 3.
-    + enough {{ Sub Π a p B <: Π a' p' B' at i }} by (eapply per_elem_subtyping; try eassumption).
+    + enough {{ Sub Π a ρ B <: Π a' ρ' B' at i }} by (eapply per_elem_subtyping; try eassumption).
       econstructor; mauto 3.
     + intros.
-      rename b into c.
-      rename equiv_b into equiv_c.
       assert {{ Γ ⊢w Id : Γ }} by mauto 3.
       assert {{ Γ ⊢ IT ® IP }}.
       {
@@ -530,20 +523,24 @@ Proof.
         autorewrite with mcltt in *; mauto 3.
       }
       assert {{ Δ ⊢ IT[σ] ≈ IT'[σ] : Type@i }} by mauto 4 using glu_univ_elem_per_univ_typ_escape.
-      assert {{ Δ ⊢ m' : IT[σ] ® c ∈ IEl }} by (simpl; bulky_rewrite1; eassumption).
-      assert (exists ac : domain, {{ $| a0 & c |↘ ac }} /\ OEl c equiv_c Δ (a_sub OT {{{ σ,, m' }}}) {{{ ~ (a_sub M σ) m' }}} ac) by mauto 3.
+      assert {{ Δ ⊢ N : IT[σ] ® n ∈ IEl }} by (simpl; bulky_rewrite1; eassumption).
+      assert (exists mn : domain, {{ $| m & n |↘ mn }} /\ {{Δ ⊢ M[σ] N : OT[σ,,N] ® mn ∈ OEl n equiv_n }}) by mauto 3.
       destruct_conjs.
       eexists; split; mauto 3.
       match_by_head per_univ_elem ltac:(fun H => directed invert_per_univ_elem H).
       destruct_rel_mod_eval.
       handle_per_univ_elem_irrel.
-      rename a1 into b.
-      rename a2 into b'.
-      assert {{ DG b ∈ glu_univ_elem i ↘ OP c equiv_c ↘ OEl c equiv_c }} by mauto 3.
-      assert {{ DG b' ∈ glu_univ_elem i ↘ OP' c equiv_c ↘ OEl' c equiv_c }} by mauto 3.
-      assert {{ Δ ⊢ OT[σ,,m'] ® OP c equiv_c }} by (eapply glu_univ_elem_trm_typ; mauto 3).
+      match goal with
+      | _: {{ ⟦ B ⟧ ρ ↦ n ↘ ~?a }},
+          _: {{ ⟦ B' ⟧ ρ' ↦ n ↘ ~?a' }} |- _ =>
+          rename a into b;
+          rename a' into b'
+      end.
+      assert {{ DG b ∈ glu_univ_elem i ↘ OP n equiv_n ↘ OEl n equiv_n }} by mauto 3.
+      assert {{ DG b' ∈ glu_univ_elem i ↘ OP' n equiv_n ↘ OEl' n equiv_n }} by mauto 3.
+      assert {{ Δ ⊢ OT[σ,,N] ® OP n equiv_n }} by (eapply glu_univ_elem_trm_typ; mauto 3).
       assert {{ Sub b <: b' at i }} by mauto 3.
-      assert {{ Δ ⊢ OT[σ,,m'] ⊆ OT'[σ,,m'] }} by mauto 3.
+      assert {{ Δ ⊢ OT[σ,,N] ⊆ OT'[σ,,N] }} by mauto 3.
       intuition.
 Qed.
 
@@ -678,10 +675,10 @@ Hint Resolve glu_rel_typ_with_sub_implies_glu_rel_exp_with_sub : mcltt.
 
 Lemma glu_ctx_env_sub_resp_ctx_eq : forall {Γ Sb},
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
-    forall {Δ Δ' σ p},
-      {{ Δ ⊢s σ ® p ∈ Sb }} ->
+    forall {Δ Δ' σ ρ},
+      {{ Δ ⊢s σ ® ρ ∈ Sb }} ->
       {{ ⊢ Δ ≈ Δ' }} ->
-      {{ Δ' ⊢s σ ® p ∈ Sb }}.
+      {{ Δ' ⊢s σ ® ρ ∈ Sb }}.
 Proof.
   induction 1; intros * HSb Hctxeq;
     apply_predicate_equivalence;
@@ -702,10 +699,10 @@ Qed.
 
 Lemma glu_ctx_env_sub_resp_sub_eq : forall {Γ Sb},
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
-    forall {Δ σ σ' p},
-      {{ Δ ⊢s σ ® p ∈ Sb }} ->
+    forall {Δ σ σ' ρ},
+      {{ Δ ⊢s σ ® ρ ∈ Sb }} ->
       {{ Δ ⊢s σ ≈ σ' : Γ }} ->
-      {{ Δ ⊢s σ' ® p ∈ Sb }}.
+      {{ Δ ⊢s σ' ® ρ ∈ Sb }}.
 Proof.
   induction 1; intros * HSb Hsubeq;
     apply_predicate_equivalence;
@@ -716,10 +713,8 @@ Proof.
   destruct_by_head cons_glu_sub_pred.
   econstructor; mauto 4.
   assert {{ Γ, A ⊢s Wk : Γ }} by mauto 3.
-  assert {{ Δ ⊢s Wk∘σ ≈ Wk∘σ' : Γ }} by mauto 3.
-  assert {{ Δ ⊢ A[Wk∘σ] ≈ A[Wk∘σ'] : Type@i }} as <- by mauto 3.
-  assert {{ Δ ⊢ #0[σ] ≈ #0[σ'] : A[Wk][σ] }} by mauto 3.
-  assert {{ Δ ⊢ #0[σ] ≈ #0[σ'] : A[Wk∘σ] }} as <- by mauto 3.
+  assert {{ Δ ⊢ A[Wk][σ] ≈ A[Wk][σ'] : Type@i }} as <- by mauto 3.
+  assert {{ Δ ⊢ #0[σ] ≈ #0[σ'] : A[Wk][σ] }} as <- by mauto 3.
   eassumption.
 Qed.
 
@@ -743,11 +738,8 @@ Proof.
   assert {{ Δ ⊢s Wk∘σ' : Γ }} by mauto 3.
   assert {{ Δ ⊢s Wk∘σ ≈ Wk∘σ' : Γ }} by mauto 3.
   econstructor; mauto 3.
-  - assert {{ Δ ⊢ #0[σ] ≈ #0[σ'] : A[Wk][σ] }} by mauto 3.
-    assert {{ Δ ⊢ A[Wk][σ] ≈ A[Wk∘σ] : Type@i }} by mauto 3.
-    assert {{ Δ ⊢ A[Wk][σ] ≈ A[Wk∘σ'] : Type@i }} by mauto 3.
-    assert {{ Δ ⊢ #0[σ] ≈ #0[σ'] : A[Wk∘σ'] }} as <- by mauto 3.
-    assert {{ Δ ⊢ A[Wk∘σ] ≈ A[Wk∘σ'] : Type@i }} as <-; mauto 2.
+  - assert {{ Δ ⊢ A[Wk][σ] ≈ A[Wk][σ'] : Type@i }} as <- by mauto 4.
+    assert {{ Δ ⊢ #0[σ] ≈ #0[σ'] : A[Wk][σ] }} as <-; mauto 3.
   - assert {{ Δ ⊢s Wk∘σ ≈ Wk∘σ' : Γ }} as <-; eassumption.
 Qed.
 
@@ -757,14 +749,14 @@ Proof.
   split; mauto using cons_glu_sub_pred_resp_wf_sub_eq.
 Qed.
 
-Lemma glu_ctx_env_per_env : forall {Γ Sb env_rel Δ σ p},
+Lemma glu_ctx_env_per_env : forall {Γ Sb env_rel Δ σ ρ},
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
     {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }} ->
-    {{ Δ ⊢s σ ® p ∈ Sb }} ->
-    {{ Dom p ≈ p ∈ env_rel }}.
+    {{ Δ ⊢s σ ® ρ ∈ Sb }} ->
+    {{ Dom ρ ≈ ρ ∈ env_rel }}.
 Proof.
   intros * Hglu Hper.
-  gen p σ Δ env_rel.
+  gen ρ σ Δ env_rel.
   induction Hglu; intros;
     invert_per_ctx_env Hper;
     apply_predicate_equivalence;
@@ -773,7 +765,7 @@ Proof.
 
   rename i0 into j.
   inversion_clear_by_head cons_glu_sub_pred.
-  assert {{ Dom p ↯ ≈ p ↯ ∈ tail_rel }} by intuition.
+  assert {{ Dom ρ ↯ ≈ ρ ↯ ∈ tail_rel }} by intuition.
   destruct_rel_typ.
   handle_per_univ_elem_irrel.
   assert (exists P' El', {{ DG a ∈ glu_univ_elem (max i j) ↘ P' ↘ El' }}) as [? []] by mauto 3 using glu_univ_elem_cumu_max_left.
@@ -790,8 +782,8 @@ Qed.
 
 Lemma glu_ctx_env_sub_escape : forall {Γ Sb},
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
-    forall Δ σ p,
-      {{ Δ ⊢s σ ® p ∈ Sb }} ->
+    forall Δ σ ρ,
+      {{ Δ ⊢s σ ® ρ ∈ Sb }} ->
       {{ Δ ⊢s σ : Γ }}.
 Proof.
   induction 1; intros;
@@ -840,12 +832,12 @@ Proof.
   rename i1 into k.
   inversion HΓΓ' as [|? ? l]; subst.
   assert (TSb -∙> TSb') by intuition.
-  intros Δ σ p [].
+  intros Δ σ ρ [].
   saturate_refl_for per_ctx_env.
-  assert {{ Dom ρ ↯ ≈ ρ ↯ ∈ tail_rel }}
+  assert {{ Dom ρ0 ↯ ≈ ρ0 ↯ ∈ tail_rel }}
     by (eapply glu_ctx_env_per_env; [| | eassumption]; eassumption).
-  assert {{ Δ0 ⊢s Wk∘σ0 ® ρ ↯ ∈ TSb' }} by intuition.
-  assert (glu_rel_typ_with_sub j Δ0 A' {{{ Wk∘σ0 }}} d{{{ ρ ↯ }}}) as [] by mauto 3.
+  assert {{ Δ0 ⊢s Wk∘σ0 ® ρ0 ↯ ∈ TSb' }} by intuition.
+  assert (glu_rel_typ_with_sub j Δ0 A' {{{ Wk∘σ0 }}} d{{{ ρ0 ↯ }}}) as [] by mauto 3.
   destruct_rel_typ.
   handle_functional_glu_univ_elem.
   rename a0 into a'.
@@ -855,13 +847,16 @@ Proof.
   assert (exists Pmax Elmax, {{ DG a ∈ glu_univ_elem (max i l) ↘ Pmax ↘ Elmax }}) as [Pmax [Elmax]]
       by mauto using glu_univ_elem_cumu_max_left.
   assert (i <= max i l) by lia.
-  assert {{ Δ0 ⊢ #0[σ0] : A'[Wk∘σ0] ® ~(ρ 0) ∈ Elmax }}.
+  assert {{ Δ0 ⊢ #0[σ0] : A'[Wk][σ0] ® ~(ρ0 0) ∈ Elmax }}.
   {
-    assert {{ Δ0 ⊢s Wk∘σ0 : Γ }} by mauto 4.
-    assert {{ Δ0 ⊢ A'[Wk∘σ0] ≈ A[Wk∘σ0] : Type@(max i l) }} as -> by mauto 4 using lift_exp_eq_max_right.
+    assert {{ Γ, A ⊢s Wk : Γ }} by mauto 3.
+    assert {{ Δ0 ⊢ A[Wk][σ0] ≈ A'[Wk][σ0] : Type@l }} by mauto 3.
+    assert {{ Δ0 ⊢ A[Wk][σ0] ≈ A'[Wk][σ0] : Type@(max i l) }} as <- by mauto 2 using lift_exp_eq_max_right.
     eapply glu_univ_elem_exp_cumu_ge; mauto 3.
   }
   eapply glu_univ_elem_exp_conv; [eexists | | | |]; intuition.
+  autorewrite with mcltt.
+  eassumption.
 Qed.
 
 Corollary functional_glu_ctx_env : forall {Γ Sb Sb'},
@@ -923,15 +918,15 @@ Ltac invert_glu_ctx_env H :=
    destruct H as [? [? []]])
   + (inversion H; subst).
 
-Lemma glu_ctx_env_subtyp_sub_if : forall Γ Γ' Sb Sb' Δ σ p,
+Lemma glu_ctx_env_subtyp_sub_if : forall Γ Γ' Sb Sb' Δ σ ρ,
     {{ ⊢ Γ ⊆ Γ' }} ->
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
     {{ EG Γ' ∈ glu_ctx_env ↘ Sb' }} ->
-    {{ Δ ⊢s σ ® p ∈ Sb }} ->
-    {{ Δ ⊢s σ ® p ∈ Sb' }}.
+    {{ Δ ⊢s σ ® ρ ∈ Sb }} ->
+    {{ Δ ⊢s σ ® ρ ∈ Sb' }}.
 Proof.
   intros * Hsubtyp Hglu Hglu'.
-  gen p σ Δ. gen Sb' Sb.
+  gen ρ σ Δ. gen Sb' Sb.
   induction Hsubtyp; intros;
     invert_glu_ctx_env Hglu;
     invert_glu_ctx_env Hglu';
@@ -958,14 +953,16 @@ Proof.
   destruct_by_head rel_exp.
   handle_functional_glu_ctx_env.
   eapply glu_univ_elem_per_subtyp_trm_conv; mauto 3.
+  autorewrite with mcltt.
+  eassumption.
 Qed.
 
 Lemma glu_ctx_env_sub_monotone : forall Γ Sb,
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
-    forall Δ' σ Δ τ p,
-      {{ Δ ⊢s τ ® p ∈ Sb }} ->
+    forall Δ' σ Δ τ ρ,
+      {{ Δ ⊢s τ ® ρ ∈ Sb }} ->
       {{ Δ' ⊢w σ : Δ }} ->
-      {{ Δ' ⊢s τ ∘ σ ® p ∈ Sb }}.
+      {{ Δ' ⊢s τ ∘ σ ® ρ ∈ Sb }}.
 Proof.
   induction 1; intros * HSb Hσ;
     apply_predicate_equivalence;
@@ -974,17 +971,12 @@ Proof.
 
   destruct_by_head cons_glu_sub_pred.
   econstructor; mauto 3.
-  - assert {{ Δ' ⊢ #0[σ0][σ] : A[Wk∘σ0][σ] ® ~(ρ 0) ∈ El }} by (eapply glu_univ_elem_exp_monotone; mauto 3).
+  - assert {{ Δ' ⊢ #0[σ0][σ] : A[Wk][σ0][σ] ® ~(ρ 0) ∈ El }} by (eapply glu_univ_elem_exp_monotone; mauto 3).
     assert {{ Γ, A ⊢ #0 : A[Wk] }} by mauto 3.
-    assert {{ Δ ⊢ A[Wk∘σ0] : Type@i }} by (eapply glu_univ_elem_trm_univ_lvl; mauto 3).
-    assert {{ ⊢ Γ, A }} by mauto 3.
-    assert {{ Γ,A ⊢s Wk : Γ }} by mauto 3.
+    assert {{ Γ, A ⊢s Wk : Γ }} by mauto 3.
+    assert {{ Δ' ⊢ #0[σ0∘σ] ≈ #0[σ0][σ] : A[Wk][σ0∘σ] }} as -> by mauto 3.
     assert {{ Δ' ⊢s σ : Δ }} by mauto 3.
-    assert {{ Δ' ⊢ A[Wk][σ0∘σ] ≈ A[Wk∘(σ0∘σ)] : Type@i }} as <- by mauto 3.
-    assert {{ Δ' ⊢ #0[σ0][σ] ≈ #0[σ0∘σ] : A[Wk][σ0∘σ] }} as <- by mauto 3.
     assert {{ Δ' ⊢ A[Wk][σ0][σ] ≈ A[Wk][σ0∘σ] : Type@i }} as <- by mauto 3.
-    assert {{ Δ ⊢ A[Wk∘σ0] ≈ A[Wk][σ0] : Type@i }} by mauto 3.
-    assert {{ Δ' ⊢ A[Wk∘σ0][σ] ≈ A[Wk][σ0][σ] : Type@i }} as <- by mauto 3.
     eassumption.
   - assert {{ Δ' ⊢s σ : Δ }} by mauto 3.
     assert {{ Γ, A ⊢s Wk : Γ }} by mauto 3.
@@ -1004,8 +996,9 @@ Proof.
   intros.
   assert {{ Δ ⊢s σ : Γ }} by mauto 2.
   assert {{ Δ ⊢ M : A[σ] }} by mauto 2 using glu_univ_elem_trm_escape.
+  assert {{ Δ ⊢s σ,,M : Γ, A }} by mauto 2.
   econstructor; mauto 3;
-    assert {{ Δ ⊢s Wk∘(σ,,M) ≈ σ : Γ }} as ->; mauto 3.
+    autorewrite with mcltt; mauto 3.
 
   assert {{ Δ ⊢ #0[σ,,M] ≈ M : A[σ] }} as ->; mauto 2.
 Qed.
@@ -1013,29 +1006,31 @@ Qed.
 #[export]
 Hint Resolve cons_glu_sub_pred_helper : mcltt.
 
-Lemma initial_env_glu_rel_exp : forall {Γ p Sb},
-    initial_env Γ p ->
+Lemma initial_env_glu_rel_exp : forall {Γ ρ Sb},
+    initial_env Γ ρ ->
     {{ EG Γ ∈ glu_ctx_env ↘ Sb }} ->
-    {{ Γ ⊢s Id ® p ∈ Sb }}.
+    {{ Γ ⊢s Id ® ρ ∈ Sb }}.
 Proof.
   intros * Hinit HΓ.
-  gen p.
+  gen ρ.
   induction HΓ; intros * Hinit;
     dependent destruction Hinit;
     apply_predicate_equivalence;
     try solve [econstructor; mauto].
 
-  assert (glu_rel_typ_with_sub i Γ A {{{ Id }}} p) as [] by mauto.
+  assert (glu_rel_typ_with_sub i Γ A {{{ Id }}} ρ) as [] by mauto.
   functional_eval_rewrite_clear.
   econstructor; mauto.
-  - eapply realize_glu_elem_bot; mauto.
+  - match goal with
+    | H: P Γ {{{ A[Id] }}} |- _ =>
+        bulky_rewrite_in H
+    end.
+    eapply realize_glu_elem_bot; mauto.
     assert {{ ⊢ Γ }} by mauto 3.
-    assert {{ Γ ⊢ A[Id] : Type@i }} by mauto 3.
-    assert {{ ⊢ Γ, A[Id] ≈ Γ, A }} as <- by mauto 4.
-    assert {{ Γ, A[Id] ⊢ A[Wk] : Type@i }} by mauto 4.
-    assert {{ Γ, A[Id] ⊢s Wk : Γ }} by mauto 4.
-    assert {{ Γ, A[Id] ⊢ A[Id][Wk] ≈ A[Wk∘Id] : Type@i }} as <- by (transitivity {{{ A[Wk] }}}; mauto 4).
-    assert {{ Γ, A[Id] ⊢ #0 ≈ #0[Id] : A[Id][Wk] }} as <- by mauto.
+    assert {{ Γ, A ⊢s Wk : Γ }} by mauto 4.
+    assert {{ Γ, A ⊢ A[Wk] : Type@i }} by mauto 4.
+    assert {{ Γ, A ⊢ A[Wk] ≈ A[Wk][Id] : Type@i }} as <- by mauto 3.
+    assert {{ Γ, A ⊢ #0 ≈ #0[Id] : A[Wk] }} as <- by mauto.
     eapply var_glu_elem_bot; mauto.
   - assert {{ Γ, A ⊢s Wk : Γ }} by mauto 4.
     assert {{ Γ, A ⊢s Wk∘Id : Γ }} by mauto 4.
@@ -1141,10 +1136,10 @@ Proof.
   intros * [Sb].
   destruct_conjs.
   assert (exists env_rel, {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_rel }}) as [env_rel] by mauto 3.
-  assert (exists p p', initial_env Γ p /\ initial_env Γ p' /\ {{ Dom p ≈ p' ∈ env_rel }}) as [p] by mauto using per_ctx_then_per_env_initial_env.
+  assert (exists ρ ρ', initial_env Γ ρ /\ initial_env Γ ρ' /\ {{ Dom ρ ≈ ρ' ∈ env_rel }}) as [ρ] by mauto using per_ctx_then_per_env_initial_env.
   destruct_conjs.
   functional_initial_env_rewrite_clear.
-  assert {{ Γ ⊢s Id ® p ∈ Sb }} by (eapply initial_env_glu_rel_exp; mauto 3).
+  assert {{ Γ ⊢s Id ® ρ ∈ Sb }} by (eapply initial_env_glu_rel_exp; mauto 3).
   destruct_glu_rel_exp_with_sub.
   enough {{ Γ ⊢ M[Id] : A[Id] }} as HId; mauto 3 using glu_univ_elem_trm_escape.
 Qed.
@@ -1216,10 +1211,10 @@ Proof.
   intros * [SbΓ [SbΔ]].
   destruct_conjs.
   assert (exists env_relΓ, {{ EF Γ ≈ Γ ∈ per_ctx_env ↘ env_relΓ }}) as [env_relΓ] by mauto 3.
-  assert (exists p p', initial_env Γ p /\ initial_env Γ p' /\ {{ Dom p ≈ p' ∈ env_relΓ }}) as [p] by mauto 3 using per_ctx_then_per_env_initial_env.
+  assert (exists ρ ρ', initial_env Γ ρ /\ initial_env Γ ρ' /\ {{ Dom ρ ≈ ρ' ∈ env_relΓ }}) as [ρ] by mauto 3 using per_ctx_then_per_env_initial_env.
   destruct_conjs.
   functional_initial_env_rewrite_clear.
-  assert {{ Γ ⊢s Id ® p ∈ SbΓ }} by (eapply initial_env_glu_rel_exp; mauto 3).
+  assert {{ Γ ⊢s Id ® ρ ∈ SbΓ }} by (eapply initial_env_glu_rel_exp; mauto 3).
   destruct_glu_rel_sub_with_sub.
   mauto 3.
 Qed.
diff --git a/theories/Core/Soundness/NatCases.v b/theories/Core/Soundness/NatCases.v
index c8e55c7a..49919751 100644
--- a/theories/Core/Soundness/NatCases.v
+++ b/theories/Core/Soundness/NatCases.v
@@ -131,18 +131,18 @@ Qed.
 #[export]
 Hint Resolve glu_rel_sub_extend_nat : mcltt.
 
-Lemma glu_rel_exp_natrec_zero_helper : forall {i Γ SbΓ A MZ MS Δ M σ p am P El},
+Lemma glu_rel_exp_natrec_zero_helper : forall {i Γ SbΓ A MZ MS Δ M σ ρ am P El},
     {{ EG Γ ∈ glu_ctx_env ↘ SbΓ }} ->
     {{ Γ, ℕ ⊢ A : Type@i }} ->
     {{ Γ ⊩ A[Id,,zero] : Type@i }} ->
     {{ Γ ⊩ MZ : A[Id,,zero] }} ->
     {{ Γ, ℕ, A ⊢ MS : A[Wk∘Wk,,succ #1] }} ->
     {{ Δ ⊢ M ≈ zero : ℕ }} ->
-    {{ Δ ⊢s σ ® p ∈ SbΓ }} ->
-    {{ ⟦ A ⟧ p ↦ zero ↘ am }} ->
+    {{ Δ ⊢s σ ® ρ ∈ SbΓ }} ->
+    {{ ⟦ A ⟧ ρ ↦ zero ↘ am }} ->
     {{ DG am ∈ glu_univ_elem i ↘ P ↘ El }} ->
     exists r,
-      {{ rec zero ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
+      {{ rec zero ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
         {{ Δ ⊢ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,,M] ® r ∈ El }}.
 Proof.
   intros * ? ? ? HMZ **.
@@ -153,7 +153,6 @@ Proof.
   assert {{ EG Γ, ℕ ∈ glu_ctx_env ↘ SbΓℕ }} by (invert_glu_rel_exp Hℕ; econstructor; mauto 3; reflexivity).
   destruct_glu_rel_exp_with_sub.
   simplify_evals.
-  rename p'0 into p.
   rename m into mz.
   eexists mz; split; mauto 3.
   handle_functional_glu_univ_elem.
@@ -201,7 +200,7 @@ Qed.
 #[local]
 Hint Resolve cons_glu_sub_pred_nat_helper : mcltt.
 
-Lemma glu_rel_exp_natrec_succ_helper : forall {i Γ SbΓ A MZ MS Δ M M' m' σ p am P El},
+Lemma glu_rel_exp_natrec_succ_helper : forall {i Γ SbΓ A MZ MS Δ M M' m' σ ρ am P El},
     {{ EG Γ ∈ glu_ctx_env ↘ SbΓ }} ->
     {{ Γ, ℕ ⊩ A : Type@i }} ->
     {{ Γ ⊢ MZ : A[Id,,zero] }} ->
@@ -209,18 +208,18 @@ Lemma glu_rel_exp_natrec_succ_helper : forall {i Γ SbΓ A MZ MS Δ M M' m' σ p
     {{ Γ, ℕ, A ⊩ MS : A[Wk∘Wk,,succ #1] }} ->
     {{ Δ ⊢ M ≈ succ M' : ℕ }} ->
     glu_nat Δ M' m' ->
-    (forall σ p am P El,
-        {{ Δ ⊢s σ ® p ∈ SbΓ }} ->
-        {{ ⟦ A ⟧ p ↦ m' ↘ am }} ->
+    (forall σ ρ am P El,
+        {{ Δ ⊢s σ ® ρ ∈ SbΓ }} ->
+        {{ ⟦ A ⟧ ρ ↦ m' ↘ am }} ->
         {{ DG am ∈ glu_univ_elem i ↘ P ↘ El }} ->
         exists r,
-          {{ rec m' ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
+          {{ rec m' ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
             {{ Δ ⊢ rec M' return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,,M'] ® r ∈ El }}) ->
-    {{ Δ ⊢s σ ® p ∈ SbΓ }} ->
-    {{ ⟦ A ⟧ p ↦ succ m' ↘ am }} ->
+    {{ Δ ⊢s σ ® ρ ∈ SbΓ }} ->
+    {{ ⟦ A ⟧ ρ ↦ succ m' ↘ am }} ->
     {{ DG am ∈ glu_univ_elem i ↘ P ↘ El }} ->
     exists r,
-      {{ rec succ m' ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
+      {{ rec succ m' ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
         {{ Δ ⊢ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,,M] ® r ∈ El }}.
 Proof.
   intros * ? HA ? ? HMS **.
@@ -234,7 +233,7 @@ Proof.
   assert {{ EG Γ, ℕ, A ∈ glu_ctx_env ↘ SbΓℕA }} by (econstructor; mauto 3; reflexivity).
   assert {{ Γ, ℕ, A ⊢ MS : A[Wk∘Wk,,succ #1] }} by mauto 2.
   invert_glu_rel_exp HMS.
-  assert {{ Δ ⊢s σ,,M' ® p ↦ m' ∈ SbΓℕ }} by (unfold SbΓℕ; mauto 3).
+  assert {{ Δ ⊢s σ,,M' ® ρ ↦ m' ∈ SbΓℕ }} by (unfold SbΓℕ; mauto 3).
   destruct_glu_rel_exp_with_sub.
   simplify_evals.
   match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H).
@@ -242,7 +241,7 @@ Proof.
   unfold univ_glu_exp_pred' in *.
   destruct_conjs.
   match goal with
-  | _: {{ ⟦ A ⟧ p ↦ m' ↘ ~?m }}, _: {{ DG ~?m ∈ glu_univ_elem i ↘ ?P ↘ ?El }} |- _ =>
+  | _: {{ ⟦ A ⟧ ρ ↦ m' ↘ ~?m }}, _: {{ DG ~?m ∈ glu_univ_elem i ↘ ?P ↘ ?El }} |- _ =>
       rename m into am';
       rename P into P';
       rename El into El'
@@ -268,9 +267,9 @@ Proof.
   assert {{ Δ, ℕ, A[q σ] ⊢s Wk∘Wk,,succ #1 : Δ, ℕ }} by mauto 3.
   assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }} by (rewrite @exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1; mauto 3).
   pose (R := {{{ rec M' return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end }}}).
-  assert (exists r, {{ rec m' ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\ {{ Δ ⊢ R : A[σ,,M'] ® r ∈ El' }}) as [r' []] by mauto 3.
+  assert (exists r, {{ rec m' ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\ {{ Δ ⊢ R : A[σ,,M'] ® r ∈ El' }}) as [r' []] by mauto 3.
   assert {{ Δ ⊢ R : A[σ,,M'] }} by (erewrite <- @exp_eq_elim_sub_rhs_typ; mauto 3).
-  assert {{ Δ ⊢s σ,,M',,R ® p ↦ m' ↦ r' ∈ SbΓℕA }} by (unfold SbΓℕA; mauto 3).
+  assert {{ Δ ⊢s σ,,M',,R ® ρ ↦ m' ↦ r' ∈ SbΓℕA }} by (unfold SbΓℕA; mauto 3).
   destruct_glu_rel_exp_with_sub.
   simplify_evals.
   match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H).
@@ -280,7 +279,7 @@ Proof.
   destruct_conjs.
   handle_functional_glu_univ_elem.
   match goal with
-  | _: {{ ⟦ MS ⟧ p ↦ m' ↦ r' ↘ ~?m }} |- _ =>
+  | _: {{ ⟦ MS ⟧ ρ ↦ m' ↦ r' ↘ ~?m }} |- _ =>
       rename m into ms
   end.
   exists ms; split; mauto 3.
@@ -349,7 +348,7 @@ Qed.
 #[local]
 Hint Resolve cons_glu_sub_pred_q_nat_helper : mcltt.
 
-Lemma glu_rel_exp_natrec_neut_helper : forall {i Γ SbΓ A MZ MS Δ M m σ p am P El},
+Lemma glu_rel_exp_natrec_neut_helper : forall {i Γ SbΓ A MZ MS Δ M m σ ρ am P El},
     {{ EG Γ ∈ glu_ctx_env ↘ SbΓ }} ->
     {{ Γ, ℕ ⊩ A : Type@i }} ->
     {{ Γ ⊩ A[Id,,zero] : Type@i }} ->
@@ -358,11 +357,11 @@ Lemma glu_rel_exp_natrec_neut_helper : forall {i Γ SbΓ A MZ MS Δ M m σ p am
     {{ Γ, ℕ, A ⊩ MS : A[Wk∘Wk,,succ #1] }} ->
     {{ Dom m ≈ m ∈ per_bot }} ->
     (forall Δ' τ V, {{ Δ' ⊢w τ : Δ }} -> {{ Rne m in length Δ' ↘ V }} -> {{ Δ' ⊢ M[τ] ≈ V : ℕ }}) ->
-    {{ Δ ⊢s σ ® p ∈ SbΓ }} ->
-    {{ ⟦ A ⟧ p ↦ ⇑ ℕ m ↘ am }} ->
+    {{ Δ ⊢s σ ® ρ ∈ SbΓ }} ->
+    {{ ⟦ A ⟧ ρ ↦ ⇑ ℕ m ↘ am }} ->
     {{ DG am ∈ glu_univ_elem i ↘ P ↘ El }} ->
     exists r,
-      {{ rec ⇑ ℕ m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
+      {{ rec ⇑ ℕ m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
         {{ Δ ⊢ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,,M] ® r ∈ El }}.
 Proof.
   intros * ? HA ? HMZ ? HMS **.
@@ -377,7 +376,7 @@ Proof.
   invert_glu_rel_exp HA.
   pose (SbΓℕA := cons_glu_sub_pred i {{{ Γ, ℕ }}} A SbΓℕ).
   assert {{ EG Γ, ℕ, A ∈ glu_ctx_env ↘ SbΓℕA }} by (econstructor; mauto 3; reflexivity).
-  assert {{ Δ ⊢s σ,,M ® p ↦ ⇑ ℕ m ∈ SbΓℕ }} by (unfold SbΓℕ; mauto 3).
+  assert {{ Δ ⊢s σ,,M ® ρ ↦ ⇑ ℕ m ∈ SbΓℕ }} by (unfold SbΓℕ; mauto 3).
   assert {{ Γ, ℕ, A ⊢ MS : A[Wk∘Wk,,succ #1] }} by mauto 2.
   invert_glu_rel_exp HMS.
   destruct_glu_rel_exp_with_sub.
@@ -388,8 +387,8 @@ Proof.
   destruct_conjs.
   handle_functional_glu_univ_elem.
   match goal with
-  | _: {{ ⟦ MZ ⟧ ~?p0 ↘ ~?m }}, _: {{ ⟦ A ⟧ ~?p0 ↦ zero ↘ ~?a }} |- _ =>
-      rename p0 into p;
+  | _: {{ ⟦ MZ ⟧ ~?ρ0 ↘ ~?m }}, _: {{ ⟦ A ⟧ ~?ρ0 ↦ zero ↘ ~?a }} |- _ =>
+      rename ρ0 into ρ;
       rename m into mz;
       rename a into az
   end.
@@ -414,7 +413,7 @@ Proof.
   assert {{ Δ, ℕ, A[q σ] ⊢s Wk∘Wk,,succ #1 : Δ, ℕ }} by mauto 3.
   assert {{ Δ, ℕ, A[q σ] ⊢ MS[q (q σ)] : A[q σ][Wk∘Wk,,succ #1] }} by (rewrite @exp_eq_typ_q_sigma_then_weak_weak_extend_succ_var_1; mauto 3).
   pose (R := {{{ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end }}}).
-  enough {{ Δ ⊢ R : A[σ,,M] ® rec m under p return A | zero -> mz | succ -> MS end ∈ glu_elem_bot i am }} by (eapply realize_glu_elem_bot; mauto 3).
+  enough {{ Δ ⊢ R : A[σ,,M] ® rec m under ρ return A | zero -> mz | succ -> MS end ∈ glu_elem_bot i am }} by (eapply realize_glu_elem_bot; mauto 3).
   econstructor; mauto 3.
   - erewrite <- @exp_eq_elim_sub_rhs_typ; mauto 3.
   - assert {{ Δ ⊢ MZ[σ] : A[Id,,zero][σ] ® mz ∈ glu_elem_top i az }} as [] by (eapply realize_glu_elem_top; eassumption).
@@ -427,21 +426,21 @@ Proof.
     match_by_head (per_ctx_env env_relΓℕA) ltac:(fun H => invert_per_ctx_env H).
     match_by_head (per_ctx_env env_relΓℕ) ltac:(fun H => invert_per_ctx_env H).
     intros s.
-    enough (exists r, {{ Rne rec m under p return A | zero -> mz | succ -> MS end in s ↘ r }}) as [] by (eexists; split; eassumption).
-    assert {{ Dom p ≈ p ∈ env_relΓ }} by (eapply glu_ctx_env_per_env; revgoals; eassumption).
+    enough (exists r, {{ Rne rec m under ρ return A | zero -> mz | succ -> MS end in s ↘ r }}) as [] by (eexists; split; eassumption).
+    assert {{ Dom ρ ≈ ρ ∈ env_relΓ }} by (eapply glu_ctx_env_per_env; revgoals; eassumption).
     destruct_rel_typ.
     invert_rel_typ_body.
     assert {{ Dom ! s ≈ ! s ∈ per_bot }} by mauto 3.
-    assert {{ Dom p ↦ ⇑! ℕ s ≈ p ↦ ⇑! ℕ s ∈ env_relΓℕ }} by (apply_relation_equivalence; unshelve eexists; simpl; intuition).
-    assert {{ Dom p ↦ succ ⇑! ℕ s ≈ p ↦ succ ⇑! ℕ s ∈ env_relΓℕ }} by (apply_relation_equivalence; unshelve eexists; simpl; intuition).
+    assert {{ Dom ρ ↦ ⇑! ℕ s ≈ ρ ↦ ⇑! ℕ s ∈ env_relΓℕ }} by (apply_relation_equivalence; unshelve eexists; simpl; intuition).
+    assert {{ Dom ρ ↦ succ ⇑! ℕ s ≈ ρ ↦ succ ⇑! ℕ s ∈ env_relΓℕ }} by (apply_relation_equivalence; unshelve eexists; simpl; intuition).
     destruct_rel_typ.
     invert_rel_typ_body.
     match goal with
-    | _: {{ ⟦ A ⟧ p ↦ ⇑! ℕ s ↘ ~?a }}, _: {{ ⟦ A ⟧ p ↦ (succ ⇑! ℕ s) ↘ ~?a' }} |- _ =>
+    | _: {{ ⟦ A ⟧ ρ ↦ ⇑! ℕ s ↘ ~?a }}, _: {{ ⟦ A ⟧ ρ ↦ (succ ⇑! ℕ s) ↘ ~?a' }} |- _ =>
         rename a into as'; (* We cannot use [as] as a name *)
         rename a' into asucc
     end.
-    assert {{ Dom p ↦ ⇑! ℕ s ↦ ⇑! as' (S s) ≈ p ↦ ⇑! ℕ s ↦ ⇑! as' (S s) ∈ env_relΓℕA }} as HΓℕA
+    assert {{ Dom ρ ↦ ⇑! ℕ s ↦ ⇑! as' (S s) ≈ ρ ↦ ⇑! ℕ s ↦ ⇑! as' (S s) ∈ env_relΓℕA }} as HΓℕA
         by (apply_relation_equivalence; unshelve eexists; simpl; intuition; eapply per_bot_then_per_elem; mauto 3).
     apply_relation_equivalence.
     (on_all_hyp_rev: fun H => destruct (H _ _ HΓℕA)).
@@ -451,7 +450,7 @@ Proof.
     destruct_by_head rel_exp.
     functional_eval_rewrite_clear.
     match goal with
-    | _: {{ ⟦ MS ⟧ p ↦ ⇑! ℕ s ↦ ⇑! as' (S s) ↘ ~?m }} |- _ =>
+    | _: {{ ⟦ MS ⟧ ρ ↦ ⇑! ℕ s ↦ ⇑! as' (S s) ↘ ~?m }} |- _ =>
         rename m into ms
     end.
     assert {{ Dom as' ≈ as' ∈ per_top_typ }} as [? []]%(fun {a} (f : per_top_typ a a) => f (S s)) by mauto 3.
@@ -465,8 +464,8 @@ Proof.
     assert {{ ⊢ Δ', ℕ }} by mauto 3.
     assert {{ Δ' ⊢s τ : Δ }} by mauto 3.
     assert {{ Δ' ⊢s σ∘τ : Γ }} by mauto 3.
-    assert {{ Δ' ⊢s σ∘τ ® p ∈ SbΓ }} by (eapply glu_ctx_env_sub_monotone; eassumption).
-    assert {{ Δ', ℕ ⊢s q (σ∘τ) ® p ↦ ⇑! ℕ (length Δ') ∈ SbΓℕ }} by (unfold SbΓℕ; mauto 3).
+    assert {{ Δ' ⊢s σ∘τ ® ρ ∈ SbΓ }} by (eapply glu_ctx_env_sub_monotone; eassumption).
+    assert {{ Δ', ℕ ⊢s q (σ∘τ) ® ρ ↦ ⇑! ℕ (length Δ') ∈ SbΓℕ }} by (unfold SbΓℕ; mauto 3).
     destruct_glu_rel_exp_with_sub.
     simplify_evals.
     match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H).
@@ -477,11 +476,11 @@ Proof.
     match_by_head read_ne ltac:(fun H => directed inversion_clear H).
     handle_functional_glu_univ_elem.
     match goal with
-    | _: {{ ⟦ A ⟧ ~?p' ↦ ⇑! ℕ (length Δ') ↘ ~?a }} |- _ =>
-        rename p' into p;
+    | _: {{ ⟦ A ⟧ ~?ρ' ↦ ⇑! ℕ (length Δ') ↘ ~?a }} |- _ =>
+        rename ρ' into ρ;
         rename a into aΔ'
     end.
-    assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢s q (q (σ∘τ)) ® p ↦ ⇑! ℕ (length Δ') ↦ ⇑! aΔ' (length {{{ Δ', ℕ }}}) ∈ SbΓℕA }}
+    assert {{ Δ', ℕ, A[q (σ∘τ)] ⊢s q (q (σ∘τ)) ® ρ ↦ ⇑! ℕ (length Δ') ↦ ⇑! aΔ' (length {{{ Δ', ℕ }}}) ∈ SbΓℕA }}
       by (unfold SbΓℕA; mauto 3).
     destruct_glu_rel_exp_with_sub.
     simplify_evals.
@@ -492,12 +491,12 @@ Proof.
     clear_dups.
     handle_functional_glu_univ_elem.
     match goal with
-    | _: {{ ⟦ A ⟧ ~?p' ↦ succ (⇑! ℕ (length Δ')) ↘ ~?a }},
+    | _: {{ ⟦ A ⟧ ~?ρ' ↦ succ (⇑! ℕ (length Δ')) ↘ ~?a }},
         _: {{ Rtyp aΔ' in S (length Δ') ↘ ~?A }},
         _: {{ Rnf ⇓ az mz in length Δ' ↘ ~?MZ }},
             _: {{ Rne m in length Δ' ↘ ~?M }} |- _ =>
         rename A into A';
-        rename p' into p;
+        rename ρ' into ρ;
         rename a into asucc;
         rename MZ into MZ';
         rename M into M'
@@ -574,12 +573,12 @@ Lemma glu_rel_exp_natrec_helper : forall {i Γ SbΓ A MZ MS},
     {{ Γ, ℕ, A ⊩ MS : A[Wk∘Wk,,succ #1] }} ->
     forall {Δ M m},
       glu_nat Δ M m ->
-      forall {σ p am P El},
-        {{ Δ ⊢s σ ® p ∈ SbΓ }} ->
-        {{ ⟦ A ⟧ p ↦ m ↘ am }} ->
+      forall {σ ρ am P El},
+        {{ Δ ⊢s σ ® ρ ∈ SbΓ }} ->
+        {{ ⟦ A ⟧ ρ ↦ m ↘ am }} ->
         {{ DG am ∈ glu_univ_elem i ↘ P ↘ El }} ->
         exists r,
-          {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} /\
+          {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ ρ ↘ r }} /\
             {{ Δ ⊢ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,,M] ® r ∈ El }}.
 Proof.
   intros * ? HA ? ?.
@@ -607,7 +606,7 @@ Proof.
     assert {{ Γ, ℕ, A ⊩ #1 : ℕ }} by mauto 3.
     mauto.
   }
-  induction 1; intros; rename m into M; rename Γ0 into Δ.
+  induction 1; intros; rename Γ0 into Δ.
   - (* glu_nat_zero *)
     mauto 4 using glu_rel_exp_natrec_zero_helper.
   - (* glu_nat_succ *)
diff --git a/theories/Core/Soundness/Realizability.v b/theories/Core/Soundness/Realizability.v
index a8564a35..ae0be94c 100644
--- a/theories/Core/Soundness/Realizability.v
+++ b/theories/Core/Soundness/Realizability.v
@@ -8,16 +8,16 @@ Import Domain_Notations.
 
 Open Scope list_scope.
 
-Lemma wf_ctx_sub_ctx_lookup : forall n T Γ,
-    {{ #n : T ∈ Γ }} ->
+Lemma wf_ctx_sub_ctx_lookup : forall n A Γ,
+    {{ #n : A ∈ Γ }} ->
     forall Δ,
       {{ ⊢ Δ ⊆ Γ}} ->
-      exists Δ1 T0 Δ2 T',
-        Δ = Δ1 ++ T0 :: Δ2 /\
+      exists Δ1 A0 Δ2 A',
+        Δ = Δ1 ++ A0 :: Δ2 /\
           n = length Δ1 /\
-          T' = iter (S n) (fun T => {{{T [ Wk ]}}}) T0 /\
-          {{ #n : T' ∈ Δ }} /\
-          {{ Δ ⊢ T' ⊆ T }}.
+          A' = iter (S n) (fun A => {{{ A[Wk] }}}) A0 /\
+          {{ #n : A' ∈ Δ }} /\
+          {{ Δ ⊢ A' ⊆ A }}.
 Proof.
   induction 1; intros; progressive_inversion.
   - exists nil.
@@ -27,9 +27,8 @@ Proof.
     repeat eexists; mauto 4.
 Qed.
 
-
-Lemma var_arith : forall Γ1 Γ2 (T : typ),
-    length (Γ1 ++ T :: Γ2) - length Γ2 - 1 = length Γ1.
+Lemma var_arith : forall Γ1 Γ2 (A : typ),
+    length (Γ1 ++ A :: Γ2) - length Γ2 - 1 = length Γ1.
 Proof.
   intros.
   rewrite List.app_length. simpl.
@@ -38,9 +37,9 @@ Qed.
 
 Lemma var_weaken_gen : forall Δ σ Γ,
     {{ Δ ⊢w σ : Γ }} ->
-    forall Γ1 Γ2 T,
-      Γ = Γ1 ++ T :: Γ2 ->
-      {{Δ ⊢ (# (length Γ1)) [σ] ≈ # (length Δ - length Γ2 - 1) : ~(iter (S (length Γ1)) (fun T => {{{T [ Wk ]}}}) T) [ σ ] }}.
+    forall Γ1 Γ2 A0,
+      Γ = Γ1 ++ A0 :: Γ2 ->
+      {{ Δ ⊢ #(length Γ1)[σ] ≈ #(length Δ - length Γ2 - 1) : ~(iter (S (length Γ1)) (fun A => {{{ A[Wk] }}}) A0)[σ] }}.
 Proof.
   induction 1; intros; subst; gen_presups.
   - pose proof (app_ctx_vlookup _ _ _ _ HΔ eq_refl) as Hvar.
@@ -48,16 +47,16 @@ Proof.
     clear_dups.
     apply wf_sub_id_inversion in Hτ.
     pose proof (wf_ctx_sub_length _ _ Hτ).
-    transitivity {{{#(length Γ1) [Id]}}}; [mauto 3 |].
-    replace (length Γ) with (length (Γ1 ++ {{{ Γ2, T }}})) by eassumption.
+    transitivity {{{ #(length Γ1)[Id] }}}; [mauto 3 |].
+    replace (length Γ) with (length (Γ1 ++ {{{ Γ2, A0 }}})) by eassumption.
     rewrite var_arith, H.
     bulky_rewrite.
   - pose proof (app_ctx_vlookup _ _ _ _ HΔ0 eq_refl) as Hvar.
-    pose proof (app_ctx_lookup Γ1 T0 Γ2 _ eq_refl).
+    pose proof (app_ctx_lookup Γ1 A0 Γ2 _ eq_refl).
     gen_presup Hvar.
     clear_dups.
-    assert {{ ⊢ Δ', T }} by mauto 3.
-    assert {{ Δ', T ⊢s Wk : ~ (Γ1 ++ {{{ Γ2, T0 }}}) }} by mauto 3.
+    assert {{ ⊢ Δ', A }} by mauto 3.
+    assert {{ Δ', A ⊢s Wk : ~ (Γ1 ++ {{{ Γ2, A0 }}}) }} by mauto 3.
     transitivity {{{ #(length Γ1)[Wk∘τ] }}}; [mauto 3 |].
     rewrite H1.
     etransitivity; [eapply wf_exp_eq_sub_compose; mauto 3 |].
@@ -76,16 +75,16 @@ Proof.
       * eapply wf_exp_eq_var_weaken; [mauto 3|]; eauto.
       * mauto 4.
     + eapply wf_exp_eq_subtyp'.
-      * eapply IHweakening with (Γ1 := T :: _).
+      * eapply IHweakening with (Γ1 := A :: _).
         reflexivity.
       * eapply wf_subtyp_subst; [ |mauto 3].
         simpl. eapply wf_subtyp_subst; mauto 3.
 Qed.
 
-Lemma var_glu_elem_bot : forall A i P El Γ T,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    {{ Γ ⊢ T ® P }} ->
-    {{ Γ, T ⊢ #0 : T[Wk] ® ! (length Γ) ∈ glu_elem_bot i A }}.
+Lemma var_glu_elem_bot : forall a i P El Γ A,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    {{ Γ ⊢ A ® P }} ->
+    {{ Γ, A ⊢ #0 : A[Wk] ® !(length Γ) ∈ glu_elem_bot i a }}.
 Proof.
   intros. saturate_glu_info.
   econstructor; mauto 4.
@@ -98,19 +97,26 @@ Qed.
 #[local]
 Hint Rewrite -> wf_sub_eq_extend_compose using mauto 4 : mcltt.
 
-Theorem realize_glu_univ_elem_gen : forall A i P El,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    (forall Γ T R, {{ DF A ≈ A ∈ per_univ_elem i ↘ R }} ->
-              {{ Γ ⊢ T ® P }} ->
-              {{ Γ ⊢ T ® glu_typ_top i A }}) /\
-      (forall Γ t T c, {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-                  {{ Γ ⊢ t : T ® c ∈ glu_elem_bot i A }} ->
-                  {{ Γ ⊢ t : T ® ⇑ A c ∈ El }}) /\
-      (forall Γ t T a R, {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-                    {{ Γ ⊢ t : T ® a ∈ El }} ->
-                    {{ DF A ≈ A ∈ per_univ_elem i ↘ R }} ->
-                    {{ Dom a ≈ a ∈ R }} ->
-                    {{ Γ ⊢ t : T ® a ∈ glu_elem_top i A }}).
+Theorem realize_glu_univ_elem_gen : forall a i P El,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    (forall Γ A R,
+        {{ DF a ≈ a ∈ per_univ_elem i ↘ R }} ->
+        {{ Γ ⊢ A ® P }} ->
+        {{ Γ ⊢ A ® glu_typ_top i a }}) /\
+      (forall Γ M A m,
+          (* We repeat this to get the relation between [a] and [P]
+             more easily after applying [induction 1.] *)
+          {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+          {{ Γ ⊢ M : A ® m ∈ glu_elem_bot i a }} ->
+          {{ Γ ⊢ M : A ® ⇑ a m ∈ El }}) /\
+      (forall Γ M A m R,
+          (* We repeat this to get the relation between [a] and [P]
+             more easily after applying [induction 1.] *)
+          {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+          {{ Γ ⊢ M : A ® m ∈ El }} ->
+          {{ DF a ≈ a ∈ per_univ_elem i ↘ R }} ->
+          {{ Dom m ≈ m ∈ R }} ->
+          {{ Γ ⊢ M : A ® m ∈ glu_elem_top i a }}).
 Proof.
   simpl. induction 1 using glu_univ_elem_ind.
   all:split; [| split]; intros;
@@ -121,7 +127,7 @@ Proof.
     destruct_all.
   - econstructor; eauto; intros.
     progressive_inversion.
-    transitivity {{{Type@j [ σ ]}}}; mauto 4.
+    transitivity {{{ Type@j[σ] }}}; mauto 4.
   - handle_functional_glu_univ_elem.
     match_by_head glu_univ_elem invert_glu_univ_elem.
     clear_dups.
@@ -139,9 +145,8 @@ Proof.
   - deepexec glu_univ_elem_per_univ ltac:(fun H => pose proof H).
     firstorder.
     specialize (H _ _ _ H10) as [? []].
-    econstructor; eauto.
+    econstructor; mauto 3.
     + apply_equiv_left. trivial.
-    + mauto 2.
     + intros.
       saturate_weakening_escape.
       deepexec H ltac:(fun H => destruct H).
@@ -152,7 +157,7 @@ Proof.
 
   - econstructor; eauto; intros.
     progressive_inversion.
-    transitivity {{{ℕ [ σ ]}}}; mauto 3.
+    transitivity {{{ ℕ[σ] }}}; mauto 3.
   - handle_functional_glu_univ_elem.
     match_by_head glu_univ_elem invert_glu_univ_elem.
     apply_equiv_left.
@@ -161,13 +166,12 @@ Proof.
 
     intros.
     saturate_weakening_escape.
-    assert {{ Δ ⊢ T [σ] ≈ ℕ[σ] : Type @ i }} by mauto 3.
+    assert {{ Δ ⊢ A[σ] ≈ ℕ[σ] : Type @ i }} by mauto 3.
     rewrite <- wf_exp_eq_nat_sub; try eassumption.
     mauto 3.
-  - econstructor; eauto.
-    + rewrite H2. mauto 3.
+  - econstructor; mauto 3.
+    + bulky_rewrite. mauto 3.
     + apply_equiv_left. trivial.
-    + mauto 2.
     + intros.
       saturate_weakening_escape.
       bulky_rewrite.
@@ -180,23 +184,23 @@ Proof.
     + gen_presups. trivial.
     + saturate_weakening_escape.
       assert {{ Γ ⊢w Id : Γ }} by mauto 4.
-      pose proof (H13 _ _ H16).
-      specialize (H13 _ _ H19).
-      assert {{ Γ ⊢ IT[Id] ≈ IT : Type@i}} by mauto 3.
-      bulky_rewrite_in H13.
-      progressive_invert H17.
-      destruct (H9 _ _ _ H0 H13) as [].
+      assert {{ Δ ⊢ IT[σ] ® IP }} by mauto 3.
+      assert (IP Γ {{{ IT[Id] }}}) as HITId by mauto 3.
+      bulky_rewrite_in HITId.
+      assert {{ Γ ⊢ IT[Id] ≈ IT : Type@i }} by mauto 3.
+      dir_inversion_clear_by_head read_typ.
+      assert {{ Γ ⊢ IT ® glu_typ_top i a }} as [] by mauto 3.
       bulky_rewrite.
       simpl. apply wf_exp_eq_pi_cong'; [firstorder |].
       pose proof (var_per_elem (length Δ) H0).
       destruct_rel_mod_eval.
       simplify_evals.
       destruct (H2 _ ltac:(eassumption) _ ltac:(eassumption)) as [? []].
-      specialize (H10 _ _ _ _ ltac:(trivial) (var_glu_elem_bot _ _ _ _ _ _ ltac:(eassumption) ltac:(eassumption))).
-      autorewrite with mcltt in H10.
-      specialize (H14 {{{Δ, IT[σ]}}} {{{σ ∘ Wk}}} _ _ ltac:(mauto) ltac:(eassumption) ltac:(eassumption)).
+      assert (IEl {{{ Δ, IT[σ] }}} {{{ IT[σ][Wk] }}} {{{ #0 }}} d{{{ ⇑! a (length Δ) }}}) by mauto 3 using var_glu_elem_bot.
+      autorewrite with mcltt in H31.
+      specialize (H14 {{{ Δ, IT[σ] }}} {{{ σ∘Wk }}} _ _ ltac:(mauto) ltac:(eassumption) ltac:(eassumption)).
       specialize (H8 _ _ _ ltac:(eassumption) ltac:(eassumption)) as [].
-      etransitivity; [| eapply H30]; mauto 3.
+      etransitivity; [| eapply H33]; mauto 3.
   - handle_functional_glu_univ_elem.
     apply_equiv_left.
     invert_glu_rel1.
@@ -210,18 +214,18 @@ Proof.
     simplify_evals.
     eexists; repeat split; mauto 3.
     eapply H2; eauto.
-    assert {{ Δ ⊢ t [σ] : M[σ] }} by mauto 3.
+    assert {{ Δ ⊢ M[σ] : A[σ] }} by mauto 3.
     bulky_rewrite_in H23.
     unshelve (econstructor; eauto).
     + trivial.
-    + eassert {{ Δ ⊢ ~ (a_sub t σ) m' : ~_ }} by (eapply wf_app'; eassumption).
+    + eassert {{ Δ ⊢ M[σ] N : ~_ }} by (eapply wf_app'; eassumption).
       autorewrite with mcltt in H25.
       trivial.
     + mauto using domain_app_per.
     + intros.
       saturate_weakening_escape.
       progressive_invert H26.
-      destruct (H15 _ _ _ _ _ ltac:(eassumption) ltac:(eassumption) ltac:(eassumption) equiv_b).
+      destruct (H15 _ _ _ _ _ ltac:(eassumption) ltac:(eassumption) ltac:(eassumption) equiv_n).
       handle_functional_glu_univ_elem.
       autorewrite with mcltt.
 
@@ -232,7 +236,7 @@ Proof.
         rewrite <- @exp_eq_sub_compose_typ; mauto 3;
           [eapply wf_exp_eq_app_cong' |].
         -- specialize (H12 _ {{{σ ∘ σ0}}} _ ltac:(mauto 3) ltac:(eassumption)).
-           rewrite wf_exp_eq_sub_compose with (M := t) in H12; mauto 3.
+           rewrite wf_exp_eq_sub_compose with (M := M) in H12; mauto 3.
            bulky_rewrite_in H12.
         -- rewrite <- @exp_eq_sub_compose_typ; mauto 3.
         -- econstructor; mauto 3.
@@ -254,15 +258,15 @@ Proof.
       assert {{ Γ ⊢w Id : Γ }} by mauto 4.
       pose proof (H10 _ _ H24).
       specialize (H10 _ _ H19).
-      assert {{ Γ ⊢ IT[Id] ≈ IT : Type@i}} by mauto 3.
+      assert {{ Γ ⊢ IT[Id] ≈ IT : Type@i }} by mauto 3.
       bulky_rewrite_in H25.
       destruct (H11 _ _ _ ltac:(eassumption) ltac:(eassumption)) as [].
       specialize (H29 _ _ _ H19 H9).
       rewrite H5 in *.
       autorewrite with mcltt.
-      eassert {{ Δ ⊢ m[σ] : ~_ }} by (mauto 2).
+      eassert {{ Δ ⊢ M[σ] : ~_ }} by (mauto 2).
       autorewrite with mcltt in H30.
-      rewrite @wf_exp_eq_pi_eta' with (M := {{{m[σ]}}}); [| trivial].
+      rewrite @wf_exp_eq_pi_eta' with (M := {{{ M[σ] }}}); [| trivial].
       cbn [nf_to_exp].
       eapply wf_exp_eq_fn_cong'; eauto.
 
@@ -282,7 +286,7 @@ Proof.
       do 2 (rewrite wf_exp_eq_sub_id in H40; mauto 3).
       etransitivity; [|eassumption].
       simpl.
-      assert {{ Δ, IT[σ] ⊢ # 0 : IT[σ ∘ Wk] }} by (rewrite <- @exp_eq_sub_compose_typ; mauto 3).
+      assert {{ Δ, IT[σ] ⊢ #0 : IT[σ∘Wk] }} by (rewrite <- @exp_eq_sub_compose_typ; mauto 3).
       rewrite <- @sub_eq_q_sigma_id_extend; mauto 4.
       rewrite <- @exp_eq_sub_compose_typ; mauto 2.
       2:eapply sub_q; mauto 4.
@@ -309,12 +313,11 @@ Proof.
       firstorder.
 Qed.
 
-
-Corollary realize_glu_typ_top : forall A i P El,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ T,
-      {{ Γ ⊢ T ® P }} ->
-      {{ Γ ⊢ T ® glu_typ_top i A }}.
+Corollary realize_glu_typ_top : forall a i P El,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A,
+      {{ Γ ⊢ A ® P }} ->
+      {{ Γ ⊢ A ® glu_typ_top i a }}.
 Proof.
   intros.
   pose proof H.
@@ -323,21 +326,21 @@ Proof.
   eapply realize_glu_univ_elem_gen; eauto.
 Qed.
 
-Theorem realize_glu_elem_bot : forall A i P El,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T c,
-      {{ Γ ⊢ t : T ® c ∈ glu_elem_bot i A }} ->
-      {{ Γ ⊢ t : T ® ⇑ A c ∈ El }}.
+Theorem realize_glu_elem_bot : forall a i P El,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A M m,
+      {{ Γ ⊢ M : A ® m ∈ glu_elem_bot i a }} ->
+      {{ Γ ⊢ M : A ® ⇑ a m ∈ El }}.
 Proof.
   intros.
   eapply realize_glu_univ_elem_gen; eauto.
 Qed.
 
-Theorem realize_glu_elem_top : forall A i P El,
-    {{ DG A ∈ glu_univ_elem i ↘ P ↘ El }} ->
-    forall Γ t T a,
-      {{ Γ ⊢ t : T ® a ∈ El }} ->
-      {{ Γ ⊢ t : T ® a ∈ glu_elem_top i A }}.
+Theorem realize_glu_elem_top : forall a i P El,
+    {{ DG a ∈ glu_univ_elem i ↘ P ↘ El }} ->
+    forall Γ A M m,
+      {{ Γ ⊢ M : A ® m ∈ El }} ->
+      {{ Γ ⊢ M : A ® m ∈ glu_elem_top i a }}.
 Proof.
   intros.
   pose proof H.
diff --git a/theories/Core/Soundness/SubstitutionCases.v b/theories/Core/Soundness/SubstitutionCases.v
index d52b7788..5fd963f7 100644
--- a/theories/Core/Soundness/SubstitutionCases.v
+++ b/theories/Core/Soundness/SubstitutionCases.v
@@ -134,10 +134,9 @@ Proof.
     assert {{ Δ0 ⊢s Wk∘((σ,,M)∘σ0) ≈ σ∘σ0 : Δ }} by mauto 3.
     econstructor; mauto 4.
     + assert {{ Δ, A ⊢s Wk : Δ }} by mauto 4.
-      assert {{ Δ0 ⊢ A[Wk][(σ,,M)∘σ0] ≈ A[Wk∘((σ,,M)∘σ0)] : Type@i }} by mauto 4.
-      assert {{ Δ0 ⊢ A[Wk∘((σ,,M)∘σ0)] ≈ A[σ∘σ0] : Type@i }} by mauto 3.
-      assert {{ Δ0 ⊢ A[σ∘σ0] ≈ A[σ][σ0] : Type@i }} by mauto 4.
-      assert {{ Δ0 ⊢ A[Wk∘((σ,,M)∘σ0)] ≈ A[σ][σ0] : Type@i }} as -> by mauto 3.
+      assert {{ Δ0 ⊢ A[Wk][(σ,,M)∘σ0] ≈ A[Wk][(σ∘σ0),,M[σ0]] : Type@i }} as -> by mauto 3.
+      assert {{ Δ0 ⊢ A[Wk][(σ∘σ0),,M[σ0]] ≈ A[σ∘σ0] : Type@i }} as -> by mauto 3.
+      assert {{ Δ0 ⊢ A[σ∘σ0] ≈ A[σ][σ0] : Type@i }} as -> by mauto 3.
       assert {{ Δ, A ⊢ #0 : A[Wk] }} by mauto 3.
       assert {{ Δ0 ⊢ #0[(σ,,M)∘σ0] ≈ #0[(σ∘σ0),,M[σ0]] : A[Wk][(σ,,M)∘σ0] }} by mauto 3.
       assert {{ Δ0 ⊢ #0[(σ,,M)∘σ0] ≈ #0[(σ∘σ0),,M[σ0]] : A[σ][σ0] }} as -> by mauto 3.
diff --git a/theories/Core/Soundness/TermStructureCases.v b/theories/Core/Soundness/TermStructureCases.v
index deb89c73..b0467bb1 100644
--- a/theories/Core/Soundness/TermStructureCases.v
+++ b/theories/Core/Soundness/TermStructureCases.v
@@ -53,9 +53,6 @@ Proof.
     intros.
     destruct_by_head cons_glu_sub_pred.
     econstructor; mauto.
-    enough {{ Δ ⊢ A[Wk][σ] ≈ A[Wk∘σ] : Type@i }} as -> by eassumption.
-    assert {{ Γ, A ⊢s Wk : Γ }} by mauto 3.
-    mauto 3.
   - assert {{ Γ ⊩ #n : A }} as Hn by mauto.
     assert (exists i, {{ Γ ⊢ A : Type@i }}) as [j] by (gen_presups; mauto 3).
     invert_glu_rel_exp Hn.
diff --git a/theories/Core/Soundness/Weakening/Definition.v b/theories/Core/Soundness/Weakening/Definition.v
index 2acabf81..46a678ea 100644
--- a/theories/Core/Soundness/Weakening/Definition.v
+++ b/theories/Core/Soundness/Weakening/Definition.v
@@ -10,7 +10,7 @@ Inductive weakening : ctx -> sub -> ctx -> Prop :=
   `( {{ Γ ⊢s σ ≈ Id : Δ }} ->
      {{ Γ ⊢w σ : Δ }} )
 | wk_p :
-  `( {{ Γ ⊢w τ : Δ', T }} ->
+  `( {{ Γ ⊢w τ : Δ', A }} ->
      {{ ⊢ Δ' ⊆ Δ }} ->
      {{ Γ ⊢s σ ≈ Wk ∘ τ : Δ }} ->
      {{ Γ ⊢w σ : Δ }} )
diff --git a/theories/Core/Soundness/Weakening/Lemmas.v b/theories/Core/Soundness/Weakening/Lemmas.v
index bcdaa1e1..88cce9de 100644
--- a/theories/Core/Soundness/Weakening/Lemmas.v
+++ b/theories/Core/Soundness/Weakening/Lemmas.v
@@ -93,9 +93,9 @@ Proof.
   mauto.
 Qed.
 
-Lemma weakening_wk : forall Γ T,
-    {{ ⊢ Γ , T }} ->
-    {{ Γ , T ⊢w Wk : Γ }}.
+Lemma weakening_wk : forall Γ A,
+    {{ ⊢ Γ, A }} ->
+    {{ Γ, A ⊢w Wk : Γ }}.
 Proof.
   intros.
   econstructor; mautosolve.
diff --git a/theories/Extraction/TypeCheck.v b/theories/Extraction/TypeCheck.v
index 317e840c..316e978e 100644
--- a/theories/Extraction/TypeCheck.v
+++ b/theories/Extraction/TypeCheck.v
@@ -50,7 +50,13 @@ Section type_check.
   | _               => inright _
   .
 
-  Extraction Inline get_level_of_type_nf.
+  (* Don't forget to use 9th bit of [Extraction Flag] (for example, [Set Extraction Flag 1007.]) *)
+  Equations get_subterms_of_type_pi (A : nf) : { B & { C | A = n{{{ Π B C }}} } } + { forall B C, A <> n{{{ Π B C }}} } :=
+  | n{{{ Π B C }}} => pureo (existT _ B (exist _ C _))
+  | _              => inright _
+  .
+
+  Extraction Inline get_level_of_type_nf get_subterms_of_type_pi.
 
   Inductive type_check_order : exp -> Prop :=
   | tc_ti : forall {A}, type_infer_order A -> type_check_order A
@@ -153,8 +159,6 @@ Section type_check.
     | _ => idtac
     end.
 
-  Definition inspect {A : Set} (a : A) : { b : A | a = b } := (exist _ a eq_refl).
-
   #[tactic="impl_obl_tac",derive(equations=no,eliminator=no)]
   Equations type_check G A (HA : (exists i, {{ G ⊢ A : Type@i }})) M (H : type_check_order M) : { {{ G ⊢a M ⟸ A }} } + { ~ {{ G ⊢a M ⟸ A }} } by struct H :=
   | G, A, HA, M, H =>
@@ -192,13 +196,12 @@ Section type_check.
         let*o (exist _ B' _) := type_infer {{{ G, A' }}} _ M' _ while _ in
         let (A'', _) := nbe_ty_impl G A' _ in
         pureo (exist _ n{{{ Π A'' B' }}} _)
-    | {{{ M' N' }}} with type_infer G _ M' _ => {
-      | inleft (exist _ (nf_pi A B) _) =>
-          let*b->o _ := type_check G A _ N' _ while _ in
-          let (B', _) := nbe_ty_impl G {{{ B[Id,,N'] }}} _ in
-          pureo (exist _ B' _)
-      | _ => inright _
-      }
+    | {{{ M' N' }}} =>
+        let*o (exist _ C _) := type_infer G _ M' _ while _ in
+        let*o (existT _ A (exist _ B _)) := get_subterms_of_type_pi C while _ in
+        let*b->o _ := type_check G A _ N' _ while _ in
+        let (B', _) := nbe_ty_impl G {{{ B[Id,,N'] }}} _ in
+        pureo (exist _ B' _)
     | {{{ #x }}} =>
         let*o (exist _ A _) := lookup G _ x while _ in
         let (A', _) := nbe_ty_impl G A _ in
@@ -302,9 +305,9 @@ Section type_check.
     functional_alg_type_infer_rewrite_clear.
     progressive_inversion.
     assert {{ G ⊢ A : ~n{{{ Type@i }}} }} by mauto 4 using alg_type_infer_sound.
-    assert {{ G, ~(A : exp) ⊢ B : ~n{{{ Type@j }}} }} by mauto 4 using alg_type_infer_sound.
+    assert {{ G, ~(A : exp) ⊢ s : ~n{{{ Type@j }}} }} by mauto 4 using alg_type_infer_sound.
     assert {{ G ⊢ N' : A }} by mauto 3 using alg_type_check_sound.
-    assert {{ G ⊢ B[Id,,N'] : ~n{{{ Type@j }}} }} as [? []]%soundness_ty by mauto 3.
+    assert {{ G ⊢ s[Id,,N'] : ~n{{{ Type@j }}} }} as [? []]%soundness_ty by mauto 3.
     mauto 3 using nbe_ty_order_sound.
   Qed.
 
@@ -314,20 +317,13 @@ Section type_check.
     progressive_inversion.
     split; [mauto 3 |].
     assert {{ G ⊢ A : ~n{{{ Type@i }}} }} by mauto 4 using alg_type_infer_sound.
-    assert {{ G, ~(A : exp) ⊢ B : ~n{{{ Type@j }}} }} by mauto 4 using alg_type_infer_sound.
-    assert {{ G ⊢ B[Id,,N'] ≈ B' : Type@j }} by (eapply soundness_ty'; mauto 4 using alg_type_check_sound).
+    assert {{ G, ~(A : exp) ⊢ s : ~n{{{ Type@j }}} }} by mauto 4 using alg_type_infer_sound.
+    assert {{ G ⊢ s[Id,,N'] ≈ B' : Type@j }} by (eapply soundness_ty'; mauto 4 using alg_type_check_sound).
     assert (user_exp B') by trivial using user_exp_nf.
     assert (exists k, {{ G ⊢a B' ⟹ Type@k }} /\ k <= j) as [? []] by (gen_presups; mauto 3).
     firstorder.
   Qed.
 
-  Next Obligation. (* False *)
-    clear_defs.
-    functional_alg_type_infer_rewrite_clear.
-    progressive_inversion.
-    intuition.
-  Qed.
-
   Next Obligation. (* {{ ⊢ G, B }} *)
     clear_defs.
     assert {{ G ⊢ B : Type@i }} by mauto 4 using alg_type_infer_sound.