diff --git a/theories/logrel_heaplang/lib/arrays.v b/theories/logrel_heaplang/lib/arrays.v
index a643581867a4a95c69894e279f809a326dcda493..ca03f7f1237dee4a7e364d2cae93852f4d1e0dc4 100644
--- a/theories/logrel_heaplang/lib/arrays.v
+++ b/theories/logrel_heaplang/lib/arrays.v
@@ -1,5 +1,4 @@
 From iris.algebra Require Import auth agree.
-From iris.base_logic.lib Require Import invariants.
 From iris.heap_lang Require Export lifting metatheory.
 From iris.heap_lang Require Import notation proofmode.
 
@@ -15,7 +14,7 @@ Ltac bool_decide H :=
     [rewrite  bool_decide_true; [|exact H]|rewrite  bool_decide_false; [|exact H]]
   end.
 
-Lemma Z_to_nat n: n >= 0 -> ∃ m: nat, n = m.
+Lemma Z_to_nat (n: Z): 0 ≤ n -> ∃ m: nat, n = m.
 Proof.
   intros; exists (Z.to_nat n); rewrite Z2Nat.id; lia.
 Qed.
@@ -46,7 +45,6 @@ Section Arrays.
     λ: "a" "i" "x", if: ((#-1) < "i") && ("i" < !"a") then (unsafe_set "a" "i" "x") else #().
 
   (* Verification *)
-
   Variable (N: namespace).
   Definition array (n: Z) (l: loc) : iProp Σ :=
     (∃ zs: list Z, ⌜n = length zs⌝ ∗ l ↦∗ (#n :: ((LitV ∘ LitInt) <$> zs)))%I.
@@ -143,7 +141,7 @@ Section Arrays.
   Qed.
 
   (** ** Specs *)
-  Lemma unsafe_arr_spec n: {{{ ⌜n >= 0⌝ }}} unsafe_arr #n {{{ (l: loc), RET #l; array n l }}}.
+  Lemma unsafe_arr_spec n: {{{ ⌜0 ≤ n⌝ }}} unsafe_arr #n {{{ (l: loc), RET #l; array n l }}}.
   Proof.
     iIntros (ϕ) "% Post". wp_lam. wp_pures. wp_bind (AllocN _ _).
     wp_apply (wp_allocN); eauto. lia.
@@ -176,13 +174,11 @@ Section Arrays.
     now iApply "Post".
   Qed.
 
-
-
   (** Semantic Types *)
   Definition Option (S: lty Σ) : lty Σ :=
-    Lty (fun v => (□ ⌜v = NONEV⌝ ∨ ∃ s, S s ∗ ⌜v = SOMEV s⌝)%I).
+    Lty (λ v, (□ ⌜v = NONEV⌝ ∨ ∃ s, S s ∗ ⌜v = SOMEV s⌝)%I).
   Definition Array : lty Σ :=
-    Lty (fun v => (□ ∃ (a: loc) (m: mnat), inv N (array m a) ∗ ⌜v = #a⌝)%I).
+    Lty (λ v, (□ ∃ (a: loc) (m: mnat), inv N (array m a) ∗ ⌜v = #a⌝)%I).
 
   Lemma typing_arr: ∅ ⊨ arr : lty_int → Option Array.
   Proof.
@@ -192,7 +188,7 @@ Section Arrays.
     bool_decide H; wp_pures.
     - iLeft. eauto.
     - wp_apply wp_fupd.
-      wp_apply (unsafe_arr_spec); eauto with lia.
+      wp_apply (unsafe_arr_spec with "[]"); eauto with lia.
       iIntros (l) "A".
       edestruct (Z_to_nat z) as [m ->]; try lia.
       wp_pures.
@@ -219,7 +215,6 @@ Section Arrays.
     - wp_pures. now iLeft.
   Qed.
 
-
   Lemma typing_set: ∅ ⊨ set : Array → lty_int → lty_int → lty_unit.
   Proof.
     iIntros (s) "!# _ /=". iApply wp_value.
@@ -238,8 +233,6 @@ Section Arrays.
     - now wp_pures.
   Qed.
 
-
-
   (** Copying Arrays *)
   Definition copy_rec : val :=
     rec: "f" "old" "new" "n" :=
@@ -249,9 +242,8 @@ Section Arrays.
   Definition copy : val :=
     λ: "a" "n", let: "b" := unsafe_arr "n" in copy_rec "a" "b" (!"a");; "b".
 
-
   Lemma copy_rec_spec (a1 a2: loc) (n1 n2 n: Z):
-    {{{ inv N (body n1 a1) ∗ body n2 a2 ∗ ⌜n <= n1⌝ ∗ ⌜n <= n2⌝ }}}
+    {{{ inv N (body n1 a1) ∗ body n2 a2 ∗ ⌜n ≤ n1⌝ ∗ ⌜n ≤ n2⌝ }}}
       copy_rec #a1 #a2 #n
     {{{ RET #(); body n2 a2 }}}.
   Proof.
@@ -275,9 +267,8 @@ Section Arrays.
     - wp_pures; iApply "Post"; eauto.
   Qed.
 
-
   Lemma copy_spec (a: loc) (m n: Z):
-    {{{ inv N (array m a) ∗ ⌜0 <= m <= n⌝}}}
+    {{{ inv N (array m a) ∗ ⌜0 ≤ m ≤ n⌝}}}
       copy #a #n
     {{{ (b: loc), RET #b; array n b }}}.
   Proof.
@@ -291,6 +282,4 @@ Section Arrays.
     { repeat iSplit; eauto with lia. iApply array_body_inv; eauto. }
     iIntros "?". wp_pures. iApply "Post". by iApply "Bclose".
   Qed.
-
-
 End Arrays.
diff --git a/theories/logrel_heaplang/lib/vectors.v b/theories/logrel_heaplang/lib/vectors.v
index 22e8f88bf446cb02e8b0bbb1c1a9ab9e7f631570..c253e2bc2ff5c1df473dd35814a74fc4672add49 100644
--- a/theories/logrel_heaplang/lib/vectors.v
+++ b/theories/logrel_heaplang/lib/vectors.v
@@ -6,9 +6,8 @@ From iris_examples.logrel_heaplang.lib Require Import invariants arrays.
 
 Set Default Proof Using "Type".
 
- Section Vectors.
-
-  (* vec = ref array *)
+(* vec = ref array *)
+Section Vectors.
 
   (* API *)
   Definition use : val :=
@@ -32,8 +31,7 @@ Set Default Proof Using "Type".
       "b" + !"a" + #1 <- "x";;
       if: CAS "v" "a" "b" then #() else "f" "v" "x".
 
-
-    (* History *)
+  (* History *)
   Definition Hist := gmap loc nat.
   Definition HistUR  := gmapUR loc (agreeR (leibnizO nat)).
   Definition hist : Hist → HistUR := fmap (λ v: nat, to_agree (v : leibnizO nat)).
@@ -43,6 +41,7 @@ Set Default Proof Using "Type".
             !inG Σ (authR mnatUR), (* maximum length *)
             !inG Σ (authR HistUR) (* history *)
            }.
+  Variable (N: namespace).
 
   (* Maximum Length *)
   Global Instance auth_mnat_persistent (n: mnat) γ: Persistent (own γ (◯ n)).
@@ -58,7 +57,7 @@ Set Default Proof Using "Type".
   Qed.
 
   Lemma auth_mnat_agree γ (m n : mnat):
-    own γ (● m) -∗ own γ (◯ n) -∗ ⌜n <= m⌝.
+    own γ (● m) -∗ own γ (◯ n) -∗ ⌜n ≤ m⌝.
   Proof.
     iIntros "Hγ● Hγ◯".
     iDestruct (own_valid_2 with "Hγ● Hγ◯") as "%".
@@ -75,13 +74,13 @@ Set Default Proof Using "Type".
   Qed.
 
   Lemma auth_mnat_previous γ (n m: mnat):
-    n <= m -> own γ (◯ m) -∗ own γ (◯ n).
+    n ≤ m → own γ (◯ m) -∗ own γ (◯ n).
   Proof.
     intros H. eapply own_mono. eapply (auth_frag_mono n m), mnat_included. lia.
   Qed.
 
   Lemma auth_mnat_update γ (n p: mnat):
-    n <= p -> own γ (● n) ==∗ own γ (● p).
+    n ≤ p → own γ (● n) ==∗ own γ (● p).
   Proof.
     iIntros (H) "Hγ●".
     iMod (auth_mnat_fork with "Hγ●") as "[Hγ● Hγ◯]"; eauto.
@@ -89,9 +88,6 @@ Set Default Proof Using "Type".
     eapply auth_update, mnat_local_update; lia.
   Qed.
 
-
-
-
   Section hist.
     (** Conversion to hists and back *)
     Lemma hist_valid h : ✓ hist h.
@@ -131,9 +127,8 @@ Set Default Proof Using "Type".
     eauto with iFrame.
   Qed.
 
-
   Lemma hist_update γ h a m:
-    (h !! a = None) ->
+    (h !! a = None) →
     hist_all γ h ==∗ hist_all γ (<[a := m]> h) ∗ hist_entry γ a m.
   Proof.
     intros H. iIntros "Hγ●".
@@ -161,12 +156,6 @@ Set Default Proof Using "Type".
     now eapply hist_singleton_included in H.
   Qed.
 
-
-
-
-  Variable (N: namespace).
-
-
   Definition Vec_inv γ1 γ2 (l: loc) : iProp Σ :=
     (∃  (h: Hist) (a: loc) (m: mnat),
         l ↦ #a ∗
@@ -178,7 +167,6 @@ Set Default Proof Using "Type".
   Definition Vec γ1 γ2 l := inv N (Vec_inv γ1 γ2 l).
   Definition Idx (γ1: gname) (i: mnat) := own γ1 (◯ (S i: mnat)).
 
-
   Lemma Vec_project_array γ1 γ2 a m l:
     hist_entry γ2 a m -∗ Vec γ1 γ2 l -∗ inv N (array m a).
   Proof.
@@ -194,18 +182,13 @@ Set Default Proof Using "Type".
     iExists h, b, n. iFrame. now iApply "Hst".
   Qed.
 
-
   (* Function Verification *)
-
   Definition ι γ1 γ2 : val := PairV (#(Z.pos γ1)) (#(Z.pos γ2)).
-
-  Definition β γ1 γ2 : lty Σ := Lty (fun v => ⌜v = ι γ1 γ2⌝)%I.
-
+  Definition β γ1 γ2 : lty Σ := Lty (λ v, ⌜v = ι γ1 γ2⌝)%I.
   Definition Index (S: lty Σ) : lty Σ :=
-    Lty (fun v => (□ ∃ γ1 γ2 (i: mnat), (∀ v, S v ∗-∗ β γ1 γ2 v) ∗ ⌜v = #i⌝ ∗ Idx γ1 i)%I).
+    Lty (λ v, (□ ∃ γ1 γ2 (i: mnat), (∀ v, S v ∗-∗ β γ1 γ2 v) ∗ ⌜v = #i⌝ ∗ Idx γ1 i)%I).
   Definition Vector (S: lty Σ) : lty Σ :=
-    Lty (fun v => (□ ∃ γ1 γ2 (l: loc), (∀ v, S v ∗-∗ β γ1 γ2 v) ∗  ⌜v = #l⌝ ∗ Vec γ1 γ2 l)%I).
-
+    Lty (λ v, (□ ∃ γ1 γ2 (l: loc), (∀ v, S v ∗-∗ β γ1 γ2 v) ∗  ⌜v = #l⌝ ∗ Vec γ1 γ2 l)%I).
 
   Lemma typing_use: ∅ ⊨ use : ∀ T,  Array N → (∀ S, Vector S → T) → T.
   Proof. unfold use.
@@ -232,8 +215,6 @@ Set Default Proof Using "Type".
       iExists γ1, γ2, l. repeat iSplit; eauto.
   Qed.
 
-
-
   Lemma typing_idx: ∅ ⊨ idx : (∀ S, Vector S → lty_int → Option (Index S)).
   Proof. unfold idx.
     iIntros (s) "!# T /=". iApply wp_value.
@@ -259,7 +240,6 @@ Set Default Proof Using "Type".
     iApply auth_mnat_previous; eauto. lia.
   Qed.
 
-
   Lemma typing_get: ∅ ⊨ get : (∀ S, Vector S → Index S → lty_int).
   Proof. unfold get.
     iIntros (s) "!# T /=". iApply wp_value.
@@ -279,8 +259,6 @@ Set Default Proof Using "Type".
     iApply Vec_project_array; eauto. iPureIntro; lia.
   Qed.
 
-
-
   Lemma typing_set: ∅ ⊨ set : (∀ S, Vector S → Index S → lty_int → lty_unit).
   Proof. unfold set.
     iIntros (s) "!# T /=". iApply wp_value.
@@ -302,7 +280,6 @@ Set Default Proof Using "Type".
     iApply Vec_project_array; eauto. iPureIntro; lia.
   Qed.
 
-
   Lemma typing_app: ∅ ⊨ app : (∀ S, Vector S → lty_int → lty_unit).
   Proof.
     iIntros (s) "!# #T /=".
@@ -353,5 +330,4 @@ Set Default Proof Using "Type".
       iApply "IH". iModIntro. iExists γ1, γ2, l. repeat iSplit; eauto.
       iModIntro. iExists n. done.
   Qed.
-
 End Vectors.