diff --git a/theories/Core/Semantic/PERBasicLemmas.v b/theories/Core/Semantic/PERBasicLemmas.v
deleted file mode 100644
index 30399aca..00000000
--- a/theories/Core/Semantic/PERBasicLemmas.v
+++ /dev/null
@@ -1,354 +0,0 @@
-From Coq Require Import Lia PeanoNat Relation_Definitions RelationClasses.
-From Mcltt Require Import Axioms Base Evaluation LibTactics PERDefinitions Readback.
-Import Domain_Notations.
-
-(* Lemma rel_mod_eval_ex_pull : *)
-(*   forall (A : Type) (P : domain -> domain -> relation domain -> A -> Prop) {T p T' p'} R, *)
-(*     rel_mod_eval (fun a b R => exists x : A, P a b R x) T p T' p' R <-> *)
-(*       exists x : A, rel_mod_eval (fun a b R => P a b R x) T p T' p' R. *)
-(* Proof. *)
-(*   split; intros. *)
-(*   - destruct H. *)
-(*     destruct H1 as [? ?]. *)
-(*     eexists; econstructor; eauto. *)
-(*   - do 2 destruct H; econstructor; eauto. *)
-(* Qed. *)
-
-(* Lemma rel_mod_eval_simp_ex : *)
-(*   forall (A : Type) (P : domain -> domain -> relation domain -> Prop) (Q : domain -> domain -> relation domain -> A -> Prop) {T p T' p'} R, *)
-(*     rel_mod_eval (fun a b R => P a b R /\ exists x : A, Q a b R x) T p T' p' R <-> *)
-(*       exists x : A, rel_mod_eval (fun a b R => P a b R /\ Q a b R x) T p T' p' R. *)
-(* Proof. *)
-(*   split; intros. *)
-(*   - destruct H. *)
-(*     destruct H1 as [? [? ?]]. *)
-(*     eexists; econstructor; eauto. *)
-(*   - do 2 destruct H; econstructor; eauto. *)
-(*     firstorder. *)
-(* Qed. *)
-
-(* Lemma rel_mod_eval_simp_and : *)
-(*   forall (P : domain -> domain -> relation domain -> Prop) (Q : relation domain -> Prop) {T p T' p'} R, *)
-(*     rel_mod_eval (fun a b R => P a b R /\ Q R) T p T' p' R <-> *)
-(*       rel_mod_eval P T p T' p' R /\ Q R. *)
-(* Proof. *)
-(*   split; intros. *)
-(*   - destruct H. *)
-(*     destruct H1 as [? ?]. *)
-(*     split; try econstructor; eauto. *)
-(*   - do 2 destruct H; econstructor; eauto. *)
-(* Qed. *)
-
-Lemma per_bot_sym : forall m n,
-    {{ Dom m ≈ n ∈ per_bot }} ->
-    {{ Dom n ≈ m ∈ per_bot }}.
-Proof with solve [mauto].
-  intros * H s.
-  destruct (H s) as [? []]...
-Qed.
-
-#[export]
-Hint Resolve per_bot_sym : mcltt.
-
-Lemma per_bot_trans : forall m n l,
-    {{ Dom m ≈ n ∈ per_bot }} ->
-    {{ Dom n ≈ l ∈ per_bot }} ->
-    {{ Dom m ≈ l ∈ per_bot }}.
-Proof with solve [mauto].
-  intros * Hmn Hnl s.
-  destruct (Hmn s) as [? []].
-  destruct (Hnl s) as [? []].
-  functional_read_rewrite_clear...
-Qed.
-
-#[export]
-Hint Resolve per_bot_trans : mcltt.
-
-Lemma var_per_bot : forall {n},
-    {{ Dom !n ≈ !n ∈ per_bot }}.
-Proof.
-  intros n s. repeat econstructor.
-Qed.
-
-#[export]
-Hint Resolve var_per_bot : mcltt.
-
-Lemma per_top_sym : forall m n,
-    {{ Dom m ≈ n ∈ per_top }} ->
-    {{ Dom n ≈ m ∈ per_top }}.
-Proof with solve [mauto].
-  intros * H s.
-  destruct (H s) as [? []]...
-Qed.
-
-#[export]
-Hint Resolve per_top_sym : mcltt.
-
-Lemma per_top_trans : forall m n l,
-    {{ Dom m ≈ n ∈ per_top }} ->
-    {{ Dom n ≈ l ∈ per_top }} ->
-    {{ Dom m ≈ l ∈ per_top }}.
-Proof with solve [mauto].
-  intros * Hmn Hnl s.
-  destruct (Hmn s) as [? []].
-  destruct (Hnl s) as [? []].
-  functional_read_rewrite_clear...
-Qed.
-
-#[export]
-Hint Resolve per_top_trans : mcltt.
-
-Lemma per_bot_then_per_top : forall m m' a a' b b' c c',
-    {{ Dom m ≈ m' ∈ per_bot }} ->
-    {{ Dom ⇓ (⇑ a b) ⇑ c m ≈ ⇓ (⇑ a' b') ⇑ c' m' ∈ per_top }}.
-Proof.
-  intros * H s.
-  specialize (H s) as [? []].
-  eexists; split; constructor; eassumption.
-Qed.
-
-#[export]
-Hint Resolve per_bot_then_per_top : mcltt.
-
-Lemma per_top_typ_sym : forall m n,
-    {{ Dom m ≈ n ∈ per_top_typ }} ->
-    {{ Dom n ≈ m ∈ per_top_typ }}.
-Proof with solve [mauto].
-  intros * H s.
-  destruct (H s) as [? []]...
-Qed.
-
-#[export]
-Hint Resolve per_top_typ_sym : mcltt.
-
-Lemma per_top_typ_trans : forall m n l,
-    {{ Dom m ≈ n ∈ per_top_typ }} ->
-    {{ Dom n ≈ l ∈ per_top_typ }} ->
-    {{ Dom m ≈ l ∈ per_top_typ }}.
-Proof with solve [mauto].
-  intros * Hmn Hnl s.
-  destruct (Hmn s) as [? []].
-  destruct (Hnl s) as [? []].
-  functional_read_rewrite_clear...
-Qed.
-
-#[export]
-Hint Resolve per_top_typ_trans : mcltt.
-
-Lemma PER_per_top_typ : PER per_top_typ.
-Proof with solve [mauto].
-  econstructor...
-Qed.
-
-Lemma per_nat_sym : forall m n,
-    {{ Dom m ≈ n ∈ per_nat }} ->
-    {{ Dom n ≈ m ∈ per_nat }}.
-Proof with solve [mauto].
-  induction 1; econstructor...
-Qed.
-
-#[export]
-Hint Resolve per_nat_sym : mcltt.
-
-Lemma per_nat_trans : forall m n l,
-    {{ Dom m ≈ n ∈ per_nat }} ->
-    {{ Dom n ≈ l ∈ per_nat }} ->
-    {{ Dom m ≈ l ∈ per_nat }}.
-Proof with solve [mauto].
-  intros * H. gen l.
-  induction H; intros * H'; inversion_clear H'; econstructor...
-Qed.
-
-#[export]
-Hint Resolve per_nat_trans : mcltt.
-
-Lemma per_ne_sym : forall m n,
-    {{ Dom m ≈ n ∈ per_ne }} ->
-    {{ Dom n ≈ m ∈ per_ne }}.
-Proof with solve [mauto].
-  intros * [].
-  econstructor...
-Qed.
-
-#[export]
-Hint Resolve per_ne_sym : mcltt.
-
-Lemma per_ne_trans : forall m n l,
-    {{ Dom m ≈ n ∈ per_ne }} ->
-    {{ Dom n ≈ l ∈ per_ne }} ->
-    {{ Dom m ≈ l ∈ per_ne }}.
-Proof with solve [mauto].
-  intros * [] Hnl.
-  inversion_clear Hnl.
-  econstructor...
-Qed.
-
-#[export]
-Hint Resolve per_ne_trans : mcltt.
-
-Lemma per_univ_elem_right_irrel : forall i i' A B R B' R',
-    per_univ_elem i A B R ->
-    per_univ_elem i' A B' R' ->
-    R = R'.
-Proof.
-  intros * Horig.
-  remember A as A' in |- *.
-  gen A' B' R'.
-  induction Horig using per_univ_elem_ind; intros * Heq Hright;
-    try solve [induction Hright using per_univ_elem_ind; congruence].
-  subst.
-  invert_per_univ_elem Hright; try congruence.
-  specialize (IHHorig _ _ _ eq_refl equiv_a_a').
-  subst.
-  extensionality f.
-  extensionality f'.
-  rewrite H2, H10.
-  extensionality c.
-  extensionality c'.
-  extensionality equiv_c_c'.
-  destruct_rel_mod_eval.
-  functional_eval_rewrite_clear.
-  specialize (H12 _ _ _ eq_refl H5).
-  congruence.
-Qed.
-
-Ltac per_univ_elem_right_irrel_rewrite1 :=
-  match goal with
-  | H1 : {{ DF ~?A ≈ ~?B ∈ per_univ_elem ?i ↘ ?R1 }}, H2 : {{ DF ~?A ≈ ~?B' ∈ per_univ_elem ?i ↘ ?R2 }} |- _ =>
-      clean replace R2 with R1 by (eauto using per_univ_elem_right_irrel)
-  end.
-Ltac per_univ_elem_right_irrel_rewrite := repeat per_univ_elem_right_irrel_rewrite1.  
-
-Lemma per_univ_elem_sym : forall i A B R,
-    per_univ_elem i A B R ->
-    per_univ_elem i B A R /\ (forall a b, {{ Dom a ≈ b ∈ R }} -> {{ Dom b ≈ a ∈ R }}).
-Proof.
-  induction 1 using per_univ_elem_ind; subst.
-  - split.
-    + apply per_univ_elem_core_univ'; trivial.
-    + intros * [].
-      specialize (H1 _ _ _ H0).
-      firstorder.
-  - split.
-    + econstructor.
-    + intros; mauto.
-  - destruct IHper_univ_elem as [? ?].
-    setoid_rewrite H2.
-    split.
-    + per_univ_elem_econstructor; eauto.
-      intros.
-      assert (equiv_c'_c : in_rel c' c) by firstorder.
-      assert (equiv_c_c : in_rel c c) by (etransitivity; eassumption).
-      destruct_rel_mod_eval.
-      econstructor; eauto.
-      functional_eval_rewrite_clear.
-      per_univ_elem_right_irrel_rewrite.
-      assumption.
-    + assert (forall a b : domain,
-                 (forall (c c' : domain) (equiv_c_c' : in_rel c c'), rel_mod_app (out_rel c c' equiv_c_c') a c b c') ->
-                 (forall (c c' : domain) (equiv_c_c' : in_rel c c'), rel_mod_app (out_rel c c' equiv_c_c') b c a c')).
-      {
-        intros.
-        assert (equiv_c'_c : in_rel c' c) by firstorder.
-        assert (equiv_c_c : in_rel c c) by (etransitivity; eassumption).
-        destruct_rel_mod_eval.
-        destruct_rel_mod_app.
-        functional_eval_rewrite_clear.
-        per_univ_elem_right_irrel_rewrite.
-        econstructor; eauto.
-      }
-      firstorder.
-  - split.
-    + econstructor; mauto.
-    + intros; mauto.
-Qed.
-
-Lemma per_univ_elem_left_irrel : forall i i' A B R A' R',
-    per_univ_elem i A B R ->
-    per_univ_elem i' A' B R' ->
-    R = R'.
-Proof.
-  intros * []%per_univ_elem_sym []%per_univ_elem_sym.
-  eauto using per_univ_elem_right_irrel.
-Qed.
-
-Lemma per_univ_elem_cross_irrel : forall i i' A B R B' R',
-    per_univ_elem i A B R ->
-    per_univ_elem i' B' A R' ->
-    R = R'.
-Proof.
-  intros * ? []%per_univ_elem_sym.
-  eauto using per_univ_elem_right_irrel.
-Qed.
-
-Ltac do_per_univ_elem_irrel_rewrite1 :=
-  match goal with
-    | H1 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i ↘ ?R1 }},
-        H2 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
-        clean replace R2 with R1 by (eauto using per_univ_elem_right_irrel)
-    | H1 : {{ DF ~_ ≈ ~?B ∈ per_univ_elem ?i ↘ ?R1 }},
-        H2 : {{ DF ~_ ≈ ~?B ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
-        clean replace R2 with R1 by (eauto using per_univ_elem_left_irrel)
-    | 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 *)
-        clean replace R2 with R1 by (symmetry; eauto using per_univ_elem_cross_irrel)
-    end.
-
-Ltac do_per_univ_elem_irrel_rewrite :=
-  repeat do_per_univ_elem_irrel_rewrite1.
-
-Ltac per_univ_elem_irrel_rewrite :=
-  functional_eval_rewrite_clear;
-  do_per_univ_elem_irrel_rewrite;
-  clear_dups.
-
-Lemma per_univ_elem_trans : forall i A1 A2 R,
-    per_univ_elem i A1 A2 R ->
-    forall j A3,
-    per_univ_elem j A2 A3 R ->
-    per_univ_elem i A1 A3 R /\ (forall a1 a2 a3, R a1 a2 -> R a2 a3 -> R a1 a3).
-Proof with solve [mauto].
-  induction 1 using per_univ_elem_ind; intros * HT2;
-    invert_per_univ_elem HT2; clear HT2.
-  - split; mauto.
-    intros * [] [].
-    per_univ_elem_irrel_rewrite.
-    destruct (H1 _ _ _ H0 _ _ H2) as [? ?].
-    econstructor...
-  - split; try econstructor...
-  - per_univ_elem_irrel_rewrite.
-    rename in_rel0 into in_rel.
-    specialize (IHper_univ_elem _ _ equiv_a_a') as [? _].
-    split.
-    + per_univ_elem_econstructor; mauto.
-      intros.
-      assert (equiv_c_c : in_rel c c) by (etransitivity; [ | symmetry]; eassumption).
-      destruct_rel_mod_eval.
-      per_univ_elem_irrel_rewrite.
-      firstorder (econstructor; eauto).
-    + setoid_rewrite H2.
-      intros.
-      assert (equiv_c'_c' : in_rel c' c') by (etransitivity; [symmetry | ]; eassumption).
-      destruct_rel_mod_eval.
-      destruct_rel_mod_app.
-      per_univ_elem_irrel_rewrite.
-      firstorder (econstructor; eauto).
-  - split; try per_univ_elem_econstructor...
-Qed.
-
-Lemma per_univ_elem_cumu : forall {i a0 a1 R},
-    {{ DF a0 ≈ a1 ∈ per_univ_elem i ↘ R }} ->
-    {{ DF a0 ≈ a1 ∈ per_univ_elem (S i) ↘ R }}.
-Proof.
-  simpl.
-  induction 1 using per_univ_elem_ind; subst.
-  - eapply per_univ_elem_core_univ'.
-    lia.
-  - per_univ_elem_econstructor.
-  - per_univ_elem_econstructor; mauto.
-    intros.
-    destruct_rel_mod_eval.
-    econstructor; mauto.
-  - per_univ_elem_econstructor; mauto.
-Qed.
diff --git a/theories/Core/Semantic/PERLemmas.v b/theories/Core/Semantic/PERLemmas.v
index 552bbad1..858203b4 100644
--- a/theories/Core/Semantic/PERLemmas.v
+++ b/theories/Core/Semantic/PERLemmas.v
@@ -1,8 +1,343 @@
 From Coq Require Import Lia PeanoNat Relation_Definitions RelationClasses.
-From Mcltt Require Import Axioms Base Evaluation LibTactics PERDefinitions.
-From Mcltt Require Export PERBasicLemmas.
+From Mcltt Require Import Axioms Base Evaluation LibTactics PER.Definitions Readback.
 Import Domain_Notations.
 
+Lemma per_bot_sym : forall m n,
+    {{ Dom m ≈ n ∈ per_bot }} ->
+    {{ Dom n ≈ m ∈ per_bot }}.
+Proof with solve [eauto].
+  intros * H s.
+  destruct (H s) as [? []]...
+Qed.
+
+#[export]
+Hint Resolve per_bot_sym : mcltt.
+
+Lemma per_bot_trans : forall m n l,
+    {{ Dom m ≈ n ∈ per_bot }} ->
+    {{ Dom n ≈ l ∈ per_bot }} ->
+    {{ Dom m ≈ l ∈ per_bot }}.
+Proof with solve [eauto].
+  intros * Hmn Hnl s.
+  destruct (Hmn s) as [? []].
+  destruct (Hnl s) as [? []].
+  functional_read_rewrite_clear...
+Qed.
+
+#[export]
+Hint Resolve per_bot_trans : mcltt.
+
+Lemma var_per_bot : forall {n},
+    {{ Dom !n ≈ !n ∈ per_bot }}.
+Proof.
+  intros n s. repeat econstructor.
+Qed.
+
+#[export]
+Hint Resolve var_per_bot : mcltt.
+
+Lemma per_top_sym : forall m n,
+    {{ Dom m ≈ n ∈ per_top }} ->
+    {{ Dom n ≈ m ∈ per_top }}.
+Proof with solve [eauto].
+  intros * H s.
+  destruct (H s) as [? []]...
+Qed.
+
+#[export]
+Hint Resolve per_top_sym : mcltt.
+
+Lemma per_top_trans : forall m n l,
+    {{ Dom m ≈ n ∈ per_top }} ->
+    {{ Dom n ≈ l ∈ per_top }} ->
+    {{ Dom m ≈ l ∈ per_top }}.
+Proof with solve [eauto].
+  intros * Hmn Hnl s.
+  destruct (Hmn s) as [? []].
+  destruct (Hnl s) as [? []].
+  functional_read_rewrite_clear...
+Qed.
+
+#[export]
+Hint Resolve per_top_trans : mcltt.
+
+Lemma per_bot_then_per_top : forall m m' a a' b b' c c',
+    {{ Dom m ≈ m' ∈ per_bot }} ->
+    {{ Dom ⇓ (⇑ a b) ⇑ c m ≈ ⇓ (⇑ a' b') ⇑ c' m' ∈ per_top }}.
+Proof.
+  intros * H s.
+  specialize (H s) as [? []].
+  eexists; split; constructor; eassumption.
+Qed.
+
+#[export]
+Hint Resolve per_bot_then_per_top : mcltt.
+
+Lemma per_top_typ_sym : forall m n,
+    {{ Dom m ≈ n ∈ per_top_typ }} ->
+    {{ Dom n ≈ m ∈ per_top_typ }}.
+Proof with solve [eauto].
+  intros * H s.
+  destruct (H s) as [? []]...
+Qed.
+
+#[export]
+Hint Resolve per_top_typ_sym : mcltt.
+
+Lemma per_top_typ_trans : forall m n l,
+    {{ Dom m ≈ n ∈ per_top_typ }} ->
+    {{ Dom n ≈ l ∈ per_top_typ }} ->
+    {{ Dom m ≈ l ∈ per_top_typ }}.
+Proof with solve [eauto].
+  intros * Hmn Hnl s.
+  destruct (Hmn s) as [? []].
+  destruct (Hnl s) as [? []].
+  functional_read_rewrite_clear...
+Qed.
+
+#[export]
+Hint Resolve per_top_typ_trans : mcltt.
+
+Lemma PER_per_top_typ : PER per_top_typ.
+Proof with solve [mauto].
+  econstructor...
+Qed.
+
+Lemma per_nat_sym : forall m n,
+    {{ Dom m ≈ n ∈ per_nat }} ->
+    {{ Dom n ≈ m ∈ per_nat }}.
+Proof with solve [mauto].
+  induction 1; econstructor...
+Qed.
+
+#[export]
+Hint Resolve per_nat_sym : mcltt.
+
+Lemma per_nat_trans : forall m n l,
+    {{ Dom m ≈ n ∈ per_nat }} ->
+    {{ Dom n ≈ l ∈ per_nat }} ->
+    {{ Dom m ≈ l ∈ per_nat }}.
+Proof with solve [mauto].
+  intros * H. gen l.
+  induction H; intros * H'; inversion_clear H'; econstructor...
+Qed.
+
+#[export]
+Hint Resolve per_nat_trans : mcltt.
+
+Lemma per_ne_sym : forall m n,
+    {{ Dom m ≈ n ∈ per_ne }} ->
+    {{ Dom n ≈ m ∈ per_ne }}.
+Proof with solve [mauto].
+  intros * [].
+  econstructor...
+Qed.
+
+#[export]
+Hint Resolve per_ne_sym : mcltt.
+
+Lemma per_ne_trans : forall m n l,
+    {{ Dom m ≈ n ∈ per_ne }} ->
+    {{ Dom n ≈ l ∈ per_ne }} ->
+    {{ Dom m ≈ l ∈ per_ne }}.
+Proof with solve [mauto].
+  intros * [] Hnl.
+  inversion_clear Hnl.
+  econstructor...
+Qed.
+
+#[export]
+Hint Resolve per_ne_trans : mcltt.
+
+Lemma per_univ_elem_right_irrel : forall i i' A B R B' R',
+    per_univ_elem i A B R ->
+    per_univ_elem i' A B' R' ->
+    R = R'.
+Proof.
+  intros * Horig.
+  remember A as A' in |- *.
+  gen A' B' R'.
+  induction Horig using per_univ_elem_ind; intros * Heq Hright;
+    try solve [induction Hright using per_univ_elem_ind; congruence].
+  subst.
+  invert_per_univ_elem Hright; try congruence.
+  specialize (IHHorig _ _ _ eq_refl equiv_a_a').
+  subst.
+  extensionality f.
+  extensionality f'.
+  rewrite H2, H10.
+  extensionality c.
+  extensionality c'.
+  extensionality equiv_c_c'.
+  destruct_rel_mod_eval.
+  functional_eval_rewrite_clear.
+  specialize (H12 _ _ _ eq_refl H5).
+  congruence.
+Qed.
+
+Ltac per_univ_elem_right_irrel_rewrite1 :=
+  match goal with
+  | H1 : {{ DF ~?A ≈ ~?B ∈ per_univ_elem ?i ↘ ?R1 }}, H2 : {{ DF ~?A ≈ ~?B' ∈ per_univ_elem ?i ↘ ?R2 }} |- _ =>
+      clean replace R2 with R1 by (eauto using per_univ_elem_right_irrel)
+  end.
+Ltac per_univ_elem_right_irrel_rewrite := repeat per_univ_elem_right_irrel_rewrite1.  
+
+Lemma per_univ_elem_sym : forall i A B R,
+    per_univ_elem i A B R ->
+    per_univ_elem i B A R /\ (forall a b, {{ Dom a ≈ b ∈ R }} -> {{ Dom b ≈ a ∈ R }}).
+Proof with solve [try econstructor; mauto].
+  induction 1 using per_univ_elem_ind; subst.
+  - split.
+    + apply per_univ_elem_core_univ'...
+    + intros * [].
+      specialize (H1 _ _ _ H0) as []...
+  - split...
+  - destruct IHper_univ_elem as [? ?].
+    setoid_rewrite H2.
+    split.
+    + per_univ_elem_econstructor; eauto.
+      intros.
+      assert (in_rel c' c) by firstorder.
+      assert (in_rel c c) by (etransitivity; eassumption).
+      destruct_rel_mod_eval.
+      functional_eval_rewrite_clear.
+      per_univ_elem_right_irrel_rewrite...
+    + enough (forall a b : domain,
+                 (forall (c c' : domain) (equiv_c_c' : in_rel c c'), rel_mod_app (out_rel c c' equiv_c_c') a c b c') ->
+                 (forall (c c' : domain) (equiv_c_c' : in_rel c c'), rel_mod_app (out_rel c c' equiv_c_c') b c a c')) by firstorder.
+      intros.
+      assert (in_rel c' c) by firstorder.
+      assert (in_rel c c) by (etransitivity; eassumption).
+      destruct_rel_mod_eval.
+      destruct_rel_mod_app.
+      functional_eval_rewrite_clear.
+      per_univ_elem_right_irrel_rewrite...
+  - split...
+Qed.
+
+Corollary per_univ_sym : forall i A B R,
+    per_univ_elem i A B R ->
+    per_univ_elem i B A R.
+Proof.
+  intros * HS.
+  apply per_univ_elem_sym; assumption.
+Qed.
+
+Corollary per_elem_sym : forall i A B a b R,
+    per_univ_elem i A B R ->
+    R a b -> R b a.
+Proof.
+  intros * HS.
+  eapply per_univ_elem_sym; eassumption.
+Qed.
+
+Lemma per_univ_elem_left_irrel : forall i i' A B R A' R',
+    per_univ_elem i A B R ->
+    per_univ_elem i' A' B R' ->
+    R = R'.
+Proof.
+  intros * []%per_univ_elem_sym []%per_univ_elem_sym.
+  eauto using per_univ_elem_right_irrel.
+Qed.
+
+Lemma per_univ_elem_cross_irrel : forall i i' A B R B' R',
+    per_univ_elem i A B R ->
+    per_univ_elem i' B' A R' ->
+    R = R'.
+Proof.
+  intros * ? []%per_univ_elem_sym.
+  eauto using per_univ_elem_right_irrel.
+Qed.
+
+Ltac do_per_univ_elem_irrel_rewrite1 :=
+  match goal with
+    | H1 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i ↘ ?R1 }},
+        H2 : {{ DF ~?A ≈ ~_ ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
+        clean replace R2 with R1 by (eauto using per_univ_elem_right_irrel)
+    | H1 : {{ DF ~_ ≈ ~?B ∈ per_univ_elem ?i ↘ ?R1 }},
+        H2 : {{ DF ~_ ≈ ~?B ∈ per_univ_elem ?i' ↘ ?R2 }} |- _ =>
+        clean replace R2 with R1 by (eauto using per_univ_elem_left_irrel)
+    | 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 *)
+        clean replace R2 with R1 by (symmetry; eauto using per_univ_elem_cross_irrel)
+    end.
+
+Ltac do_per_univ_elem_irrel_rewrite :=
+  repeat do_per_univ_elem_irrel_rewrite1.
+
+Ltac per_univ_elem_irrel_rewrite :=
+  functional_eval_rewrite_clear;
+  do_per_univ_elem_irrel_rewrite;
+  clear_dups.
+
+Lemma per_univ_elem_trans : forall i A1 A2 R,
+    per_univ_elem i A1 A2 R ->
+    (forall j A3,
+        per_univ_elem j A2 A3 R ->
+        per_univ_elem i A1 A3 R) /\
+      (forall a1 a2 a3, R a1 a2 -> R a2 a3 -> R a1 a3).
+Proof with solve [mauto| firstorder (econstructor; eauto)].
+  induction 1 using per_univ_elem_ind;
+    [> split;
+       [ intros * HT2; invert_per_univ_elem HT2; clear HT2
+       | intros * HTR1 HTR2 ] ..]; mauto.
+  - destruct HTR1, HTR2.
+    per_univ_elem_irrel_rewrite.
+    specialize (H1 _ _ _ H2) as [].
+    specialize (H0 _ _ H3)...
+  - idtac...
+  - per_univ_elem_irrel_rewrite.
+    rename in_rel0 into in_rel.
+    destruct IHper_univ_elem as [? _].
+    per_univ_elem_econstructor; mauto.
+    intros.
+    assert (in_rel c c) by (etransitivity; [ | symmetry]; eassumption).
+    destruct_rel_mod_eval.
+    per_univ_elem_irrel_rewrite...
+  - destruct IHper_univ_elem as [? ?].
+    rewrite H2 in *.
+    intros.
+    assert (in_rel c' c') by (etransitivity; [symmetry | ]; eassumption).
+    destruct_rel_mod_eval.
+    destruct_rel_mod_app.
+    per_univ_elem_irrel_rewrite...
+  - try per_univ_elem_econstructor...
+Qed.
+
+Corollary per_univ_trans : forall i j A1 A2 A3 R,
+    per_univ_elem i A1 A2 R ->
+    per_univ_elem j A2 A3 R ->
+    per_univ_elem i A1 A3 R.
+Proof.
+  intros * HT1.
+  apply per_univ_elem_trans; assumption.
+Qed.
+
+Corollary per_elem_trans : forall i A1 A2 a1 a2 a3 R,
+    per_univ_elem i A1 A2 R ->
+    R a1 a2 -> R a2 a3 -> R a1 a3.
+Proof.
+  intros * HT1.
+  eapply per_univ_elem_trans; eassumption.
+Qed.
+
+Lemma per_univ_elem_cumu : forall {i a0 a1 R},
+    {{ DF a0 ≈ a1 ∈ per_univ_elem i ↘ R }} ->
+    {{ DF a0 ≈ a1 ∈ per_univ_elem (S i) ↘ R }}.
+Proof with solve [mauto].
+  simpl.
+  induction 1 using per_univ_elem_ind; subst.
+  - eapply per_univ_elem_core_univ'.
+    lia.
+  - per_univ_elem_econstructor.
+  - per_univ_elem_econstructor; mauto.
+    intros.
+    destruct_rel_mod_eval.
+    econstructor...
+  - per_univ_elem_econstructor...
+Qed.
+
 Lemma per_ctx_env_right_irrel : forall Γ Δ Δ' R R',
     {{ DF Γ ≈ Δ ∈ per_ctx_env ↘ R }} ->
     {{ DF Γ ≈ Δ' ∈ per_ctx_env ↘ R' }} ->
@@ -37,7 +372,7 @@ Proof with solve [eauto].
     destruct_rel_mod_eval.
     per_univ_elem_irrel_rewrite.
     econstructor; eauto.
-    apply per_univ_elem_sym...
+    apply per_univ_sym...
   - subst.
     firstorder.
     assert (tail_rel d{{{ p ↯ }}} d{{{ o ↯ }}}) by (unfold Symmetric in *; eauto).
@@ -45,7 +380,7 @@ Proof with solve [eauto].
     destruct_rel_mod_eval.
     eexists.
     per_univ_elem_irrel_rewrite.
-    eapply per_univ_elem_sym...
+    eapply per_elem_sym...
 Qed.
 
 Lemma per_ctx_env_left_irrel : forall Γ Γ' Δ R R',
@@ -90,30 +425,30 @@ Ltac per_ctx_env_irrel_rewrite :=
 
 Lemma per_ctx_env_trans : forall Γ1 Γ2 R,
     {{ DF Γ1 ≈ Γ2 ∈ per_ctx_env ↘ 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 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 }}).
 Proof with solve [eauto].
   simpl.
-  induction 1; intros * HTrans23; inversion HTrans23; subst; eauto.
-  per_ctx_env_irrel_rewrite.
-  rename tail_rel0 into tail_rel.
-  split.
-  - econstructor; eauto.
+  induction 1; subst;
+    [> split; [intros * HT2; inversion HT2; subst; eauto; clear HT2 | intros * HRT1 HRT2; eauto] ..];
+    per_ctx_env_irrel_rewrite.
+  - rename tail_rel0 into tail_rel.
+    econstructor; eauto.
     + eapply IHper_ctx_env...
     + unfold rel_typ in *.
       intros.
-      assert (tail_rel p p) by (etransitivity; [ | symmetry]; eassumption).
+      assert (tail_rel p p) by (etransitivity; [| symmetry]; eassumption).
       destruct_rel_mod_eval.
-      econstructor; eauto.
       per_univ_elem_irrel_rewrite.
-      eapply proj1, per_univ_elem_trans; [|eassumption]...
-  - intros * [] [].
-    specialize (IHper_ctx_env _ equiv_Γ_Γ'0) as [].
-    assert (tail_rel d{{{ p1 ↯ }}} d{{{ p3 ↯ }}}) by eauto.
+      econstructor; eauto.
+      eapply per_univ_trans; [| eassumption]...
+  - destruct HRT1, HRT2, IHper_ctx_env.
+    assert (tail_rel d{{{ p1 ↯ }}} d{{{ p3 ↯ }}}) by mauto.
     eexists.
     unfold rel_typ in *.
     destruct_rel_mod_eval.
     per_univ_elem_irrel_rewrite.
-    eapply per_univ_elem_trans...
+    eapply per_elem_trans...
 Qed.
diff --git a/theories/Core/Semantic/Realizability.v b/theories/Core/Semantic/Realizability.v
index f0671a15..528907f1 100644
--- a/theories/Core/Semantic/Realizability.v
+++ b/theories/Core/Semantic/Realizability.v
@@ -1,7 +1,7 @@
 From Coq Require Import Lia PeanoNat Relation_Definitions.
 From Equations Require Import Equations.
-From Mcltt Require Import Base Evaluation LibTactics PERBasicLemmas.
-From Mcltt Require Export PERDefinitions PERBasicLemmas.
+From Mcltt Require Import Base Evaluation LibTactics Readback.
+From Mcltt Require Export PER.
 Import Domain_Notations.
 
 Lemma per_nat_then_per_top : forall {n m},