Use proper linked lists instead of functional lists.

TODO: fix in-place merge function.

_CoqProject

 -Q theories actris
 -arg -w -arg -notation-overridden,-redundant-canonical-projection,-several-object-files
 theories/utils/auth_excl.v
-theories/utils/list.v
+theories/utils/flist.v
+theories/utils/llist.v
 theories/utils/compare.v
 theories/utils/contribution.v
 theories/channel/channel.v

theories/channel/channel.v

 From iris.heap_lang Require Import proofmode notation.
 From iris.heap_lang.lib Require Import spin_lock.
 From iris.algebra Require Import excl auth list.
-From actris.utils Require Import auth_excl list.
+From actris.utils Require Import auth_excl llist.
 Set Default Proof Using "Type".
 
 Inductive side := Left | Right.
@@ -11,8 +11,8 @@ Definition side_elim {A} (s : side) (l r : A) : A :=
 
 Definition new_chan : val :=
   λ: <>,
-     let: "l" := ref (lnil #()) in
-     let: "r" := ref (lnil #()) in
+     let: "l" := lnil #() in
+     let: "r" := lnil #() in
      let: "lk" := newlock #() in
      ((("l","r"),"lk"), (("r","l"),"lk")).
 
@@ -21,7 +21,7 @@ Definition send : val :=
     let: "lk" := Snd "c" in
     acquire "lk";;
     let: "l" := Fst (Fst "c") in
-    "l" <- lsnoc !"l" "v";;
+    lsnoc "l" "v";;
     release "lk".
 
 Definition try_recv : val :=
@@ -29,11 +29,7 @@ Definition try_recv : val :=
     let: "lk" := Snd "c" in
     acquire "lk";;
     let: "l" := Snd (Fst "c") in
-    let: "ret" :=
-      match: !"l" with
-        SOME "p" => "l" <- Snd "p";; SOME (Fst "p")
-      | NONE => NONE
-      end in
+    let: "ret" := if: lisnil "l" then NONE else SOME (lpop "l") in
     release "lk";; "ret".
 
 Definition recv : val :=
@@ -52,27 +48,32 @@ Definition chanΣ : gFunctors :=
 Instance subG_chanΣ {Σ} : subG chanΣ Σ → chanG Σ.
 Proof. solve_inG. Qed.
 
+(** MOVE TO IRIS *)
+Global Instance fst_atomic a v1 v2 : Atomic a (Fst (v1,v2)%V).
+Proof.
+  apply strongly_atomic_atomic, ectx_language_atomic;
+    [inversion 1; naive_solver
+    |apply ectxi_language_sub_redexes_are_values; intros [] **; naive_solver].
+Qed.
+
 Section channel.
   Context `{!heapG Σ, !chanG Σ} (N : namespace).
 
-  Definition is_list_ref (l : val) (xs : list val) : iProp Σ :=
-    (∃ l' : loc, ⌜l = #l'⌝ ∧ l' ↦ llist xs)%I.
-
   Record chan_name := Chan_name {
     chan_lock_name : gname;
     chan_l_name : gname;
     chan_r_name : gname
   }.
 
-  Definition chan_inv (γ : chan_name) (l r : val) : iProp Σ :=
+  Definition chan_inv (γ : chan_name) (l r : loc) : iProp Σ :=
     (∃ ls rs,
-      is_list_ref l ls ∗ own (chan_l_name γ) (● to_auth_excl ls) ∗
-      is_list_ref r rs ∗ own (chan_r_name γ) (● to_auth_excl rs))%I.
+      llist l ls ∗ own (chan_l_name γ) (● to_auth_excl ls) ∗
+      llist r rs ∗ own (chan_r_name γ) (● to_auth_excl rs))%I.
   Typeclasses Opaque chan_inv.
 
   Definition is_chan (γ : chan_name) (c1 c2 : val) : iProp Σ :=
-    (∃ l r lk : val,
-      ⌜ c1 = ((l, r), lk)%V ∧ c2 = ((r, l), lk)%V ⌝ ∗
+    (∃ (l r : loc) (lk : val),
+      ⌜ c1 = ((#l, #r), lk)%V ∧ c2 = ((#r, #l), lk)%V ⌝ ∗
       is_lock N (chan_lock_name γ) lk (chan_inv γ l r))%I.
 
   Global Instance is_chan_persistent : Persistent (is_chan γ c1 c2).
@@ -91,20 +92,15 @@ Section channel.
   Proof.
     iIntros (Φ) "_ HΦ". rewrite /is_chan /chan_own.
     wp_lam.
-    wp_apply (lnil_spec with "[//]"); iIntros (ls). wp_alloc l as "Hl".
-    wp_apply (lnil_spec with "[//]"); iIntros (rs). wp_alloc r as "Hr".
+    wp_apply (lnil_spec with "[//]"); iIntros (l) "Hl".
+    wp_apply (lnil_spec with "[//]"); iIntros (r) "Hr".
     iMod (own_alloc (● (to_auth_excl []) ⋅ ◯ (to_auth_excl []))) as (lsγ) "[Hls Hls']".
     { by apply auth_both_valid. }
     iMod (own_alloc (● (to_auth_excl []) ⋅ ◯ (to_auth_excl []))) as (rsγ) "[Hrs Hrs']".
     { by apply auth_both_valid. }
-    iAssert (is_list_ref #l []) with "[Hl]" as "Hl".
-    { rewrite /is_list_ref; eauto. }
-    iAssert (is_list_ref #r []) with "[Hr]" as "Hr".
-    { rewrite /is_list_ref; eauto. }
     wp_apply (newlock_spec N (∃ ls rs,
-      is_list_ref #l ls ∗ own lsγ (● to_auth_excl ls) ∗
-      is_list_ref #r rs ∗ own rsγ (● to_auth_excl rs))%I
-                with "[Hl Hr Hls Hrs]").
+      llist l ls ∗ own lsγ (● to_auth_excl ls) ∗
+      llist r rs ∗ own rsγ (● to_auth_excl rs))%I with "[Hl Hr Hls Hrs]").
     { eauto 10 with iFrame. }
     iIntros (lk γlk) "#Hlk". wp_pures.
     iApply ("HΦ" $! _ _ (Chan_name γlk lsγ rsγ)); simpl.
@@ -113,9 +109,9 @@ Section channel.
 
   Lemma chan_inv_alt s γ l r :
     chan_inv γ l r ⊣⊢ ∃ ls rs,
-      is_list_ref (side_elim s l r) ls ∗
+      llist (side_elim s l r) ls ∗
       own (side_elim s chan_l_name chan_r_name γ) (● to_auth_excl ls) ∗
-      is_list_ref (side_elim s r l) rs ∗
+      llist (side_elim s r l) rs ∗
       own (side_elim s chan_r_name chan_l_name γ) (● to_auth_excl rs).
   Proof.
     destruct s; rewrite /chan_inv //=.
@@ -131,24 +127,20 @@ Section channel.
   Proof.
     iIntros "Hc HΦ". wp_lam; wp_pures.
     iDestruct "Hc" as (l r lk [-> ->]) "#Hlock"; wp_pures.
-    assert (side_elim s (l, r, lk)%V (r, l, lk)%V =
-      ((side_elim s l r, side_elim s r l), lk)%V) as -> by (by destruct s).
+    assert (side_elim s (#l, #r, lk)%V (#r, #l, lk)%V =
+      ((#(side_elim s l r), #(side_elim s r l)), lk)%V) as -> by (by destruct s).
     wp_apply (acquire_spec with "Hlock"); iIntros "[Hlocked Hinv]".
-    wp_pures.
     iDestruct (chan_inv_alt s with "Hinv") as
-      (vs ws) "(Href & Hvs & Href' & Hws)".
-    iDestruct "Href" as (ll Hslr) "Hll". rewrite Hslr.
-    wp_load.
-    wp_apply (lsnoc_spec with "[//]"); iIntros (_).
-    wp_bind (_ <- _)%E.
+      (vs ws) "(Hll & Hvs & Href' & Hws)".
+    wp_seq. wp_bind (Fst (_,_)%V)%E.
     iMod "HΦ" as (vs') "[Hchan HΦ]".
     iDestruct (excl_eq with "Hvs Hchan") as %<-%leibniz_equiv.
     iMod (excl_update _ _ _ (vs ++ [v]) with "Hvs Hchan") as "[Hvs Hchan]".
-    wp_store. iMod ("HΦ" with "Hchan") as "HΦ".
-    iModIntro.
+    wp_pures. iMod ("HΦ" with "Hchan") as "HΦ"; iModIntro.
+    wp_apply (lsnoc_spec with "Hll"); iIntros "Hll".
     wp_apply (release_spec with "[-HΦ $Hlock $Hlocked]"); last eauto.
     iApply (chan_inv_alt s).
-    rewrite /is_list_ref. eauto 20 with iFrame.
+    rewrite /llist. eauto 20 with iFrame.
   Qed.
 
   Lemma try_recv_spec Φ E γ c1 c2 s :
@@ -162,28 +154,27 @@ Section channel.
   Proof.
     iIntros "Hc HΦ". wp_lam; wp_pures.
     iDestruct "Hc" as (l r lk [-> ->]) "#Hlock"; wp_pures.
-    assert (side_elim s (r, l, lk)%V (l, r, lk)%V =
-      ((side_elim s r l, side_elim s l r), lk)%V) as -> by (by destruct s).
+    assert (side_elim s (#r, #l, lk)%V (#l, #r, lk)%V =
+      ((#(side_elim s r l), #(side_elim s l r)), lk)%V) as -> by (by destruct s).
     wp_apply (acquire_spec with "Hlock"); iIntros "[Hlocked Hinv]".
-    wp_pures.
     iDestruct (chan_inv_alt s with "Hinv")
-      as (vs ws) "(Href & Hvs & Href' & Hws)".
-    iDestruct "Href" as (ll Hslr) "Hll". rewrite Hslr.
-    wp_bind (! _)%E. destruct vs as [|v vs]; simpl.
-    - iDestruct "HΦ" as "[>HΦ _]". wp_load. iMod "HΦ"; iModIntro.
+      as (vs ws) "(Hll & Hvs & Href' & Hws)".
+    wp_seq. wp_bind (Fst (_,_)%V)%E. destruct vs as [|v vs]; simpl.
+    - iDestruct "HΦ" as "[>HΦ _]". wp_pures. iMod "HΦ"; iModIntro.
+      wp_apply (lisnil_spec with "Hll"); iIntros "Hll". wp_pures.
       wp_apply (release_spec with "[-HΦ $Hlocked $Hlock]").
       { iApply (chan_inv_alt s).
-        rewrite /is_list_ref. eauto 10 with iFrame. }
+        rewrite /llist. eauto 10 with iFrame. }
       iIntros (_). by wp_pures.
     - iDestruct "HΦ" as "[_ >HΦ]". iDestruct "HΦ" as (vs') "[Hvs' HΦ]".
       iDestruct (excl_eq with "Hvs Hvs'") as %<-%leibniz_equiv.
       iMod (excl_update _ _ _ vs with "Hvs Hvs'") as "[Hvs Hvs']".
-      wp_load.
-      iMod ("HΦ" with "[//] Hvs'") as "HΦ"; iModIntro.
-      wp_store; wp_pures.
+      wp_pures. iMod ("HΦ" with "[//] Hvs'") as "HΦ"; iModIntro.
+      wp_apply (lisnil_spec with "Hll"); iIntros "Hll".
+      wp_apply (lpop_spec with "Hll"); iIntros "Hll".
       wp_apply (release_spec with "[-HΦ $Hlocked $Hlock]").
       { iApply (chan_inv_alt s).
-        rewrite /is_list_ref. eauto 10 with iFrame. }
+        rewrite /llist. eauto 10 with iFrame. }
       iIntros (_). by wp_pures.
   Qed.
 

theories/examples/loop_sort.v

 From stdpp Require Import sorting.
 From actris.channel Require Import proto_channel.
 From iris.heap_lang Require Import proofmode notation.
-From actris.utils Require Import list.
 From actris.examples Require Import sort.
 
 Definition loop_sort_service : val :=

theories/examples/map.v

 From actris.channel Require Import proto_channel proofmode.
 From iris.heap_lang Require Import proofmode notation lib.spin_lock.
-From actris.utils Require Import list contribution.
+From actris.utils Require Import llist contribution.
 From iris.algebra Require Import gmultiset.
 
 Definition map_worker : val :=
@@ -30,22 +30,25 @@ Definition start_map_service : val := λ: "n" "map",
 
 Definition par_map_server : val :=
   rec: "go" "n" "c" "xs" "ys" :=
-    if: "n" = #0 then "ys" else
+    if: "n" = #0 then #() else
     if: recv "c" then (* send item to map_worker *)
       if: lisnil "xs" then
         send "c" #false;;
         "go" ("n" - #1) "c" "xs" "ys"
       else
         send "c" #true;;
-        send "c" (lhead "xs");;
-        "go" "n" "c" (ltail "xs") "ys"
+        send "c" (lpop "xs");;
+        "go" "n" "c" "xs" "ys"
     else (* receive item from map_worker *)
       let: "zs" := recv "c" in
-      "go" "n" "c" "xs" (lapp "zs" "ys").
+      lprep "ys" "zs";;
+      "go" "n" "c" "xs" "ys".
 
 Definition par_map : val := λ: "n" "map" "xs",
   let: "c" := start_map_service "n" "map" in
-  par_map_server "n" "c" "xs" (lnil #()).
+  let: "ys" := lnil #() in
+  par_map_server "n" "c" "xs" "ys";;
+  "ys".
 
 Class mapG Σ A `{Countable A} := {
   map_contributionG :> contributionG Σ (gmultisetUR A);
@@ -62,7 +65,7 @@ Section map.
   Definition map_spec (vmap : val) : iProp Σ := (∀ x v,
     {{{ IA x v }}}
       vmap v
-    {{{ ws, RET (llist ws); [∗ list] y;w ∈ map x;ws, IB y w }}})%I.
+    {{{ l ws, RET #l; llist l ws ∗ [∗ list] y;w ∈ map x;ws, IB y w }}})%I.
 
   Definition map_worker_protocol_aux (rec : nat -d> gmultiset A -d> iProto Σ) :
       nat -d> gmultiset A -d> iProto Σ := λ i X,
@@ -72,7 +75,7 @@ Section map.
         <+>
       rec (pred i) X
      ) <{⌜ i ≠ 1 ∨ X = ∅ ⌝}&>
-         <?> x ws, MSG llist ws {{ ⌜ x ∈ X ⌝ ∧ [∗ list] y;w ∈ map x;ws, IB y w }};
+         <?> x (l : loc) ws, MSG #l {{ ⌜ x ∈ X ⌝ ∗ llist l ws ∗ [∗ list] y;w ∈ map x;ws, IB y w }};
          rec i (X ∖ {[ x ]}))%proto.
   Instance map_worker_protocol_aux_contractive : Contractive map_worker_protocol_aux.
   Proof. solve_proper_prepare. f_equiv. solve_proto_contractive. Qed.
@@ -109,7 +112,7 @@ Section map.
     rewrite left_id_L.
     wp_apply (release_spec with "[$Hlk $Hl Hc Hs]").
     { iExists (S i), _. iFrame. }
-    clear dependent i X. iIntros "Hu". wp_apply ("Hmap" with "HI"); iIntros (w) "HI".
+    clear dependent i X. iIntros "Hu". wp_apply ("Hmap" with "HI"); iIntros (l ws) "HI".
     wp_apply (acquire_spec with "[$Hlk $Hu]"); iIntros "[Hl H]".
     iDestruct "H" as (i X) "[Hs Hc]".
     iDestruct (@server_agree with "Hs Hγ")
@@ -156,38 +159,39 @@ Section map.
     wp_apply (start_map_workers_spec with "Hf [$Hlk $Hγs]"); auto.
   Qed.
 
-  Lemma par_map_server_spec n c vs xs ws X ys :
+  Lemma par_map_server_spec n c l k vs xs ws X ys :
     (n = 0 → X = ∅ ∧ xs = []) →
     {{{
+      llist l vs ∗ llist k ws ∗
       c ↣ map_worker_protocol n X @ N ∗
       ([∗ list] x;v ∈ xs;vs, IA x v) ∗ ([∗ list] y;w ∈ ys;ws, IB y w)
     }}}
-      par_map_server #n c (llist vs) (llist ws)
-    {{{ ys' ws', RET (llist ws');
-       ⌜ys' ≡ₚ (xs ++ elements X) ≫= map⌝ ∗ [∗ list] y;w ∈ ys' ++ ys;ws', IB y w
+      par_map_server #n c #l #k
+    {{{ ys' ws', RET #();
+       ⌜ys' ≡ₚ (xs ++ elements X) ≫= map⌝ ∗
+       llist k ws' ∗ [∗ list] y;w ∈ ys' ++ ys;ws', IB y w
     }}}.
   Proof.
-    iIntros (Hn Φ) "(Hc & HIA & HIB) HΦ".
-    iLöb as "IH" forall (n vs xs ws X ys Hn Φ); wp_rec; wp_pures; simpl.
+    iIntros (Hn Φ) "(Hl & Hk & Hc & HIA & HIB) HΦ".
+    iLöb as "IH" forall (n l vs xs ws X ys Hn Φ); wp_rec; wp_pures; simpl.
     case_bool_decide; wp_pures; simplify_eq/=.
     { destruct Hn as [-> ->]; first lia.
-      iApply ("HΦ" $! []); simpl; auto. }
+      iApply ("HΦ" $! []); simpl; auto with iFrame. }
     destruct n as [|n]=> //=. wp_branch as %?|%_; wp_pures.
-    - wp_apply (lisnil_spec with "[//]"); iIntros (_).
+    - wp_apply (lisnil_spec with "Hl"); iIntros "Hl".
       destruct vs as [|v vs], xs as [|x xs]; csimpl; try done; wp_pures.
       + wp_select. wp_pures. rewrite Nat2Z.inj_succ Z.sub_1_r Z.pred_succ.
-        iApply ("IH" with "[%] Hc [//] [$] HΦ"); naive_solver.
+        iApply ("IH" with "[%] Hl Hk Hc [//] [$] HΦ"); naive_solver.
       + iDestruct "HIA" as "[HIAx HIA]". wp_select.
-        wp_apply (lhead_spec with "[//]"); iIntros (_).
+        wp_apply (lpop_spec with "Hl"); iIntros "Hl".
         wp_send with "[$HIAx]".
-        wp_apply (ltail_spec with "[//]"); iIntros (_).
-        wp_apply ("IH" with "[] Hc HIA HIB"); first done.
+        wp_apply ("IH" with "[] Hl Hk Hc HIA HIB"); first done.
         iIntros (ys' ws').
         rewrite gmultiset_elements_disj_union gmultiset_elements_singleton.
         rewrite assoc_L -(comm _ [x]). iApply "HΦ".
-    - wp_recv (x w) as (Hx) "HIBx".
-      wp_apply (lapp_spec with "[//]"); iIntros (_).
-      wp_apply ("IH" $! _ _ _ _ _ (_ ++ _) with "[] Hc HIA [HIBx HIB]"); first done.
+    - wp_recv (x l' w) as (Hx) "[Hl' HIBx]".
+      wp_apply (lprep_spec with "[$Hk $Hl']"); iIntros "[Hk _]".
+      wp_apply ("IH" $! _ _ _ _ _ _ (_ ++ _) with "[] Hl Hk Hc HIA [HIBx HIB]"); first done.
       { simpl; iFrame. }
       iIntros (ys' ws'); iDestruct 1 as (Hys) "HIB"; simplify_eq/=.
       iApply ("HΦ" $! (ys' ++ map x)). iSplit.
@@ -198,17 +202,18 @@ Section map.
       + by rewrite -assoc_L.
   Qed.
 
-  Lemma par_map_spec n vmap vs xs :
+  Lemma par_map_spec n vmap l vs xs :
     0 < n →
     map_spec vmap -∗
-    {{{ [∗ list] x;v ∈ xs;vs, IA x v }}}
-      par_map #n vmap (llist vs)
-    {{{ ys ws, RET (llist ws); ⌜ys ≡ₚ xs ≫= map⌝ ∗ [∗ list] y;w ∈ ys;ws, IB y w }}}.
+    {{{ llist l vs ∗ [∗ list] x;v ∈ xs;vs, IA x v }}}
+      par_map #n vmap #l
+    {{{ k ys ws, RET #k; ⌜ys ≡ₚ xs ≫= map⌝ ∗ llist k ws ∗ [∗ list] y;w ∈ ys;ws, IB y w }}}.
   Proof.
-    iIntros (?) "#Hmap !>"; iIntros (Φ) "HI HΦ". wp_lam; wp_pures.
+    iIntros (?) "#Hmap !>"; iIntros (Φ) "[Hl HI] HΦ". wp_lam; wp_pures.
     wp_apply (start_map_service_spec with "Hmap [//]"); iIntros (c) "Hc".
-    wp_pures. wp_apply (lnil_spec with "[//]"); iIntros (_).
-    wp_apply (par_map_server_spec _ _ _ _ [] ∅ [] with "[$Hc $HI //]"); first lia.
-    iIntros (ys ws). rewrite /= gmultiset_elements_empty !right_id_L . iApply "HΦ".
+    wp_pures. wp_apply (lnil_spec with "[//]"); iIntros (k) "Hk".
+    wp_apply (par_map_server_spec _ _ _ _ _ _ [] ∅ [] with "[$Hl $Hk $Hc $HI //]"); first lia.
+    iIntros (ys ws) "H". rewrite /= gmultiset_elements_empty !right_id_L .
+    wp_pures. by iApply "HΦ".
   Qed.
 End map.

theories/examples/map_reduce.v

 From stdpp Require Import sorting.
 From actris.channel Require Import proto_channel proofmode.
 From iris.heap_lang Require Import proofmode notation.
-From actris.utils Require Import list compare contribution.
+From actris.utils Require Import llist compare contribution.
 From actris.examples Require Import map sort_elem sort_elem_client.
 From iris.algebra Require Import gmultiset.
 From Coq Require Import SetoidPermutation.
@@ -40,8 +40,8 @@ Definition par_map_reduce_map_server : val :=
         "go" ("n" - #1) "cmap" "csort" "xs"
       else
         send "cmap" #true;;
-        send "cmap" (lhead "xs");;
-        "go" "n" "cmap" "csort" (ltail "xs")
+        send "cmap" (lpop "xs");;
+        "go" "n" "cmap" "csort" "xs"
     else (* receive item from mapper *)
       let: "zs" := recv "cmap" in
       send_all "csort" "zs";;
@@ -49,15 +49,15 @@ Definition par_map_reduce_map_server : val :=
 
 Definition par_map_reduce_collect : val :=
   rec: "go" "csort" "i" "ys" :=
-    if: ~recv "csort" then (("i", "ys"), NONE) else
+    if: ~recv "csort" then NONE else
     let: "jy" := recv "csort" in
     let: "j" := Fst "jy" in let: "y" := Snd "jy" in
-    if: "i" = "j" then "go" "csort" "j" (lcons "y" "ys") else
-    (("i", "ys"), SOME ("j", "y")).
+    if: "i" = "j" then lcons "y" "ys";; "go" "csort" "j" "ys" else
+    SOME ("j", "y").
 
 Definition par_map_reduce_reduce_server : val :=
   rec: "go" "n" "csort" "cred" "acc" "zs" :=
-    if: "n" = #0 then "zs" else
+    if: "n" = #0 then #() else
     if: recv "cred" then (* Send item to mapper *)
       match: "acc" with
         NONE =>
@@ -65,15 +65,16 @@ Definition par_map_reduce_reduce_server : val :=
          send "cred" #false;; "go" ("n" - #1) "csort" "cred" NONE "zs"
       | SOME "acc" =>
          (* Read subsequent items with the same key *)
-         let: "grp" := par_map_reduce_collect "csort"
-           (Fst "acc") (lcons (Snd "acc") (lnil #())) in
+         let: "ys" := lcons (Snd "acc") (lnil #()) in
+         let: "new_acc" := par_map_reduce_collect "csort" (Fst "acc") "ys" in
          send "cred" #true;;
-         send "cred" (Fst "grp");;
-         "go" "n" "csort" "cred" (Snd "grp") "zs"
+         send "cred" (Fst "acc", "ys");;
+         "go" "n" "csort" "cred" "new_acc" "zs"
       end
     else (* receive item from mapper *)
       let: "zs'" := recv "cred" in
-      "go" "n" "csort" "cred" "acc" (lapp "zs'" "zs").
+      lprep "zs" "zs'";;
+      "go" "n" "csort" "cred" "acc" "zs".
 
 Definition cmpZfst : val := λ: "x" "y", Fst "x" ≤ Fst "y".
 
@@ -86,7 +87,8 @@ Definition par_map_reduce : val := λ: "n" "map" "red" "xs",
   (* We need the first sorted element in the loop to compare subsequent elements *)
   if: ~recv "csort" then lnil #() else (* Handle the empty case *)
   let: "jy" := recv "csort" in
-  par_map_reduce_reduce_server "n" "csort" "cred" (SOME "jy") (lnil #()).
+  let: "zs" := lnil #() in
+  par_map_reduce_reduce_server "n" "csort" "cred" (SOME "jy") "zs";; "zs".
 
 
 (** Properties about the functional version *)
@@ -220,7 +222,8 @@ Section mapper.
   Definition IZB (iy : Z * B) (w : val) : iProp Σ :=
     (∃ w', ⌜ w = (#iy.1, w')%V ⌝ ∧ IB iy.1 iy.2 w')%I.
   Definition IZBs (iys : Z * list B) (w : val) : iProp Σ :=
-    (∃ ws, ⌜ w = (#iys.1, llist ws)%V ⌝ ∧ [∗ list] y;w'∈iys.2;ws, IB iys.1 y w')%I.
+    (∃ (l : loc) ws,
+      ⌜ w = (#iys.1, #l)%V ⌝ ∗ llist l ws ∗ [∗ list] y;w'∈iys.2;ws, IB iys.1 y w')%I.
   Definition RZB : relation (Z * B) := prod_relation (≤)%Z (λ _ _ : B, True).
 
   Instance RZB_dec : RelDecision RZB.
@@ -238,42 +241,42 @@ Section mapper.
     repeat case_bool_decide=> //; unfold RZB, prod_relation in *; naive_solver.
   Qed.
 
-  Lemma par_map_reduce_map_server_spec n cmap csort vs xs X ys :
+  Lemma par_map_reduce_map_server_spec n cmap csort l vs xs X ys :
     (n = 0 → X = ∅ ∧ xs = []) →
     {{{
+      llist l vs ∗
       cmap ↣ map_worker_protocol IA IZB map n (X : gmultiset A) @ N ∗
       csort ↣ sort_elem_head_protocol IZB RZB ys @ N ∗
       ([∗ list] x;v ∈ xs;vs, IA x v)
     }}}
-      par_map_reduce_map_server #n cmap csort (llist vs)
+      par_map_reduce_map_server #n cmap csort #l
     {{{ ys', RET #();
       ⌜ys' ≡ₚ (xs ++ elements X) ≫= map⌝ ∗
       csort ↣ sort_elem_head_protocol IZB RZB (ys ++ ys') @ N
     }}}.
   Proof.
-    iIntros (Hn Φ) "(Hcmap & Hcsort & HIA) HΦ".
+    iIntros (Hn Φ) "(Hl & Hcmap & Hcsort & HIA) HΦ".
     iLöb as "IH" forall (n vs xs X ys Hn Φ); wp_rec; wp_pures; simpl.
     case_bool_decide; wp_pures; simplify_eq/=.
     { destruct Hn as [-> ->]; first lia.
       iApply ("HΦ" $! []). rewrite right_id_L. auto. }
     destruct n as [|n]=> //=. wp_branch as %?|%_; wp_pures.
-    - wp_apply (lisnil_spec with "[//]"); iIntros (_).
+    - wp_apply (lisnil_spec with "Hl"); iIntros "Hl".
       destruct vs as [|v vs], xs as [|x xs]; csimpl; try done; wp_pures.
       + wp_select. wp_pures. rewrite Nat2Z.inj_succ Z.sub_1_r Z.pred_succ.
-        iApply ("IH" $! _ _ [] with "[%] Hcmap Hcsort [//] HΦ"); naive_solver.
+        iApply ("IH" $! _ _ [] with "[%] Hl Hcmap Hcsort [//] HΦ"); naive_solver.
       + iDestruct "HIA" as "[HIAx HIA]". wp_select.
-        wp_apply (lhead_spec with "[//]"); iIntros (_).
+        wp_apply (lpop_spec with "Hl"); iIntros "Hl".
         wp_send with "[$HIAx]".
-        wp_apply (ltail_spec with "[//]"); iIntros (_).
-        wp_apply ("IH" with "[%] Hcmap Hcsort HIA"); first done.
+        wp_apply ("IH" with "[%] Hl Hcmap Hcsort HIA"); first done.
         iIntros (ys').
         rewrite gmultiset_elements_disj_union gmultiset_elements_singleton.
         rewrite assoc_L -(comm _ [x]). iApply "HΦ".
-    - wp_recv (x w) as (Hx) "HIBfx".
+    - wp_recv (x k ws) as (Hx) "[Hk HIBfx]".
       rewrite -(right_id END%proto _ (sort_elem_head_protocol _ _ _)).
-      wp_apply (send_all_spec with "[$Hcsort $HIBfx]"); iIntros "Hcsort".
+      wp_apply (send_all_spec with "[$Hk $Hcsort $HIBfx]"); iIntros "Hcsort".
       rewrite right_id.
-      wp_apply ("IH" with "[] Hcmap Hcsort HIA"); first done.
+      wp_apply ("IH" with "[] Hl Hcmap Hcsort HIA"); first done.
       iIntros (ys'). iDestruct 1 as (Hys) "Hcsort"; simplify_eq/=.
       rewrite -assoc_L. iApply ("HΦ" $! (map x ++ ys') with "[$Hcsort]").
       iPureIntro. rewrite (gmultiset_disj_union_difference {[ x ]} X)
@@ -282,37 +285,39 @@ Section mapper.
       by rewrite gmultiset_elements_singleton assoc_L bind_app -Hys /= right_id_L comm.
   Qed.
 
-  Lemma par_map_reduce_collect_spec csort iys iys_sorted i ys ws :
+  Lemma par_map_reduce_collect_spec csort iys iys_sorted i l ys ws :
     let acc := from_option (λ '(i,y,w), [(i,y)]) [] in
     let accv := from_option (λ '(i,y,w), SOMEV (#(i:Z),w)) NONEV in
     ys ≠ [] →
     Sorted RZB (iys_sorted ++ ((i,) <$> ys)) →
     i ∉ iys_sorted.*1 →
     {{{
+      llist l (reverse ws) ∗
       csort ↣ sort_elem_tail_protocol IZB RZB iys (iys_sorted ++ ((i,) <$> ys)) @ N ∗
       [∗ list] y;w ∈ ys;ws, IB i y w
     }}}
-      par_map_reduce_collect csort #i (llist (reverse ws))
-    {{{ ys' ws' miy, RET ((#i,llist (reverse ws')),accv miy);
+      par_map_reduce_collect csort #i #l
+    {{{ ys' ws' miy, RET accv miy;
       ⌜ Sorted RZB ((iys_sorted ++ ((i,) <$> ys ++ ys')) ++ acc miy) ⌝ ∗
       ⌜ from_option (λ '(j,_,_), i ≠ j ∧ j ∉ iys_sorted.*1)
                     (iys_sorted ++ ((i,) <$> ys ++ ys') ≡ₚ iys) miy ⌝ ∗
+      llist l (reverse ws') ∗
       csort ↣ from_option (λ _, sort_elem_tail_protocol IZB RZB iys
         ((iys_sorted ++ ((i,) <$> ys ++ ys')) ++ acc miy)) END%proto miy @ N ∗
       ([∗ list] y;w ∈ ys ++ ys';ws', IB i y w) ∗
       from_option (λ '(i,y,w), IB i y w) True miy
     }}}.
   Proof.
-    iIntros (acc accv Hys Hsort Hi Φ) "[Hcsort HIB] HΦ".
+    iIntros (acc accv Hys Hsort Hi Φ) "(Hl & Hcsort & HIB) HΦ".
     iLöb as "IH" forall (ys ws Hys Hsort Hi Φ); wp_rec; wp_pures; simpl.
     wp_branch as %_|%Hperm; last first; wp_pures.
-    { iApply ("HΦ" $! [] _ None with "[$Hcsort HIB]"); simpl.
-     iEval (rewrite !right_id_L); auto. }