WIP list sort example

_CoqProject

 -Q theories osiris
 -arg -w -arg -notation-overridden,-redundant-canonical-projection,-several-object-files
+theories/utils/ordering.v
+theories/utils/permutation.v
 theories/typing/involutive.v
 theories/typing/side.v
 theories/typing/stype.v
 theories/encodings/encodable.v
+theories/encodings/list_enc.v
 theories/encodings/list.v
 theories/encodings/auth_excl.v
 theories/encodings/channel.v

theories/encodings/list_enc.v

+From iris.heap_lang Require Export proofmode notation.
+From iris.heap_lang Require Import assert.
+From osiris Require Export encodable.
+
+(** Immutable ML-style functional lists *)
+Fixpoint is_list_enc `{EncDec T} (hd : val) (xs : list T) : Prop :=
+  match xs with
+  | [] => hd = NONEV
+  | x :: xs => ∃ hd', hd = SOMEV (encode x, hd') ∧ is_list_enc hd' xs
+  end.
+
+Definition lnil : val := λ: <>, NONEV.
+Definition lcons : val := λ: "x" "l", SOME ("x", "l").
+
+Definition lhead : val := λ: "l",
+  match: "l" with
+    SOME "p" => Fst "p"
+  | NONE => assert: #false
+  end.
+Definition ltail : val := λ: "l",
+  match: "l" with
+    SOME "p" => Snd "p"
+  | NONE => assert: #false
+  end.
+
+Definition llookup : val :=
+  rec: "go" "n" "l" :=
+    if: "n" = #0 then lhead "l" else
+    let: "l" := ltail "l" in "go" ("n" - #1) "l".
+
+Definition linsert : val :=
+  rec: "go" "n" "x" "l" :=
+    if: "n" = #0 then let: "l" := ltail "l" in lcons "x" "l" else
+    let: "k" := ltail "l" in
+    let: "k" := "go" ("n" - #1) "x" "k" in
+    lcons (lhead "l") "k".
+
+Definition lreplicate : val :=
+  rec: "go" "n" "x" :=
+    if: "n" = #0 then lnil #() else
+    let: "l" := "go" ("n" - #1) "x" in lcons "x" "l".
+
+Definition llist_member : val :=
+  rec: "go" "x" "l" :=
+    match: "l" with
+      NONE => #false
+    | SOME "p" =>
+      if: Fst "p" = "x" then #true
+      else let: "l" := (Snd "p")  in "go" "x" "l"
+    end.
+
+Definition llength : val :=
+  rec: "go" "l" :=
+    match: "l" with
+      NONE => #0
+    | SOME "p" => #1 + "go" (Snd "p")
+    end.
+
+Definition lsnoc : val :=
+  rec: "go" "l" "x" :=
+    match: "l" with
+      SOME "p" => (lcons (Fst "p") ("go" (Snd "p") "x"))
+    | NONE => lcons "x" NONE
+    end.
+
+Section lists.
+Context `{heapG Σ}.
+Implicit Types i : nat.
+Context `{EncDec T}.
+Implicit Types v : val.
+
+Lemma lnil_spec :
+  {{{ True }}}
+    lnil #()
+  {{{ hd, RET hd; ⌜ is_list_enc hd [] ⌝ }}}.
+Proof. iIntros (Φ _) "HΦ". wp_lam. by iApply "HΦ". Qed.
+
+Lemma lcons_spec hd xs x :
+  {{{ ⌜ is_list_enc hd xs ⌝ }}}
+    lcons (encode x) hd
+  {{{ hd', RET hd'; ⌜ is_list_enc hd' (x :: xs) ⌝ }}}.
+Proof.
+  iIntros (Φ ?) "HΦ". wp_lam. wp_pures.
+  iApply "HΦ". simpl; eauto 10.
+Qed.
+
+Lemma lhead_spec hd x xs:
+  {{{ ⌜ is_list_enc hd (x :: xs) ⌝ }}}
+    lhead hd
+  {{{ RET encode x; True }}}.
+Proof. iIntros (Φ (hd'&->&?)) "HΦ". wp_lam. wp_pures. by iApply "HΦ". Qed.
+
+Lemma ltail_spec hd xs x :
+  {{{ ⌜ is_list_enc hd (x :: xs) ⌝ }}}
+    ltail hd
+  {{{ hd', RET hd'; ⌜ is_list_enc hd' xs ⌝ }}}.
+Proof. iIntros (Φ (hd'&->&?)) "HΦ". wp_lam. wp_pures. by iApply "HΦ". Qed.
+
+Lemma llookup_spec i hd xs x :
+  xs !! i = Some x →
+  {{{ ⌜ is_list_enc hd xs ⌝ }}}
+    llookup #i hd
+  {{{ RET encode x; True }}}.
+Proof.
+  iIntros (Hi Φ Hl) "HΦ".
+  iInduction xs as [|x' xs] "IH" forall (hd i Hi Hl);
+    destruct i as [|i]=> //; simplify_eq/=.
+  { wp_lam. wp_pures. by iApply (lhead_spec with "[//]"). }
+  wp_lam. wp_pures.
+  wp_apply (ltail_spec with "[//]"); iIntros (hd' ?). wp_let.
+  wp_op. rewrite Nat2Z.inj_succ -Z.add_1_l Z.add_simpl_l. by iApply "IH".
+Qed.
+
+Lemma linsert_spec i hd xs x :
+  is_Some (xs !! i) →
+  {{{ ⌜ is_list_enc hd xs ⌝ }}}
+    linsert #i (encode x) hd
+  {{{ hd', RET hd'; ⌜ is_list_enc hd' (<[i:=x]> xs) ⌝ }}}.
+Proof.
+  iIntros ([w Hi] Φ Hl) "HΦ".
+  iInduction xs as [|x' xs] "IH" forall (hd i Hi Hl Φ);
+    destruct i as [|i]=> //; simplify_eq/=.
+  { wp_lam. wp_pures. wp_apply (ltail_spec with "[//]"). iIntros (hd' ?).
+    wp_let. by iApply (lcons_spec with "[//]"). }
+  wp_lam; wp_pures.
+  wp_apply (ltail_spec with "[//]"); iIntros (hd' ?). wp_let.
+  wp_op. rewrite Nat2Z.inj_succ -Z.add_1_l Z.add_simpl_l.
+  wp_apply ("IH" with "[//] [//]"). iIntros (hd'' ?). wp_let.
+  wp_apply (lhead_spec with "[//]"); iIntros "_".
+  by wp_apply (lcons_spec with "[//]").
+Qed.
+
+Lemma llist_member_spec hd xs x :
+  {{{ ⌜ is_list_enc hd xs ⌝ }}}
+    llist_member (encode x) hd
+  {{{ RET #(bool_decide (x ∈ xs)); True }}}.
+Proof.
+  iIntros (Φ Hl) "HΦ".
+  iInduction xs as [|x' xs] "IH" forall (hd Hl); simplify_eq/=.
+  { wp_lam; wp_pures. by iApply "HΦ". }
+  destruct Hl as (hd'&->&?). wp_lam; wp_pures.
+  destruct (bool_decide_reflect (encode x' = encode x)) as [Heq|?]; wp_if.
+  { rewrite -Heq. inversion Heq. rewrite (bool_decide_true (_ ∈ _ :: _)); last by left. by iApply "HΦ". }
+  wp_proj. wp_let. iApply ("IH" with "[//]").
+  destruct (bool_decide_reflect (x ∈ xs)).
+  - by rewrite bool_decide_true; last by right.
+  - by rewrite bool_decide_false ?elem_of_cons; last naive_solver.
+Qed.
+
+Lemma lreplicate_spec i x :
+  {{{ True }}}
+    lreplicate #i (encode x)
+  {{{ hd, RET hd; ⌜ is_list_enc hd (replicate i x) ⌝ }}}.
+Proof.
+  iIntros (Φ _) "HΦ". iInduction i as [|i] "IH" forall (Φ); simpl.
+  { wp_lam; wp_pures.
+    iApply (lnil_spec with "[//]"). iApply "HΦ". }
+  wp_lam; wp_pures.
+  rewrite Nat2Z.inj_succ -Z.add_1_l Z.add_simpl_l.
+  wp_apply "IH"; iIntros (hd' Hl). wp_let. by iApply (lcons_spec with "[//]").
+Qed.
+
+Lemma llength_spec hd xs :
+  {{{ ⌜ is_list_enc hd xs ⌝ }}} llength hd {{{ RET #(length xs); True }}}.
+Proof.
+  iIntros (Φ Hl) "HΦ".
+  iInduction xs as [|x' xs] "IH" forall (hd Hl Φ); simplify_eq/=.
+  { wp_lam. wp_match. by iApply "HΦ". }
+  destruct Hl as (hd'&->&?). wp_lam. wp_match. wp_proj.
+  wp_apply ("IH" $! hd' with "[//]"); iIntros "_ /=". wp_op.
+  rewrite (Nat2Z.inj_add 1). by iApply "HΦ".
+Qed.
+
+Lemma lsnoc_spec hd xs x :
+  {{{ ⌜ is_list_enc hd xs ⌝ }}}
+    lsnoc hd (encode x)
+  {{{ hd', RET hd'; ⌜ is_list_enc hd' (xs++[x]) ⌝ }}}.
+Proof.
+  iIntros (Φ Hvs) "HΦ".
+  iInduction xs as [|x' xs] "IH" forall (hd Hvs Φ).
+  - simplify_eq/=.
+    wp_rec.
+    wp_pures.
+    wp_lam.
+    wp_pures.
+    iApply "HΦ". simpl. eauto 10.
+  - destruct Hvs as [vs' [-> Hvs']].
+    wp_rec.
+    wp_pures.
+    wp_bind (lsnoc vs' (encode x)).
+    iApply "IH".
+    eauto.
+    iNext.
+    iIntros (hd' Hhd').
+    wp_pures.
+    iApply lcons_spec.
+    eauto.
+    iApply "HΦ".
+Qed.
+
+End lists.
\ No newline at end of file

theories/examples/list_sort_example.v

+From iris.proofmode Require Import tactics.
+From iris.program_logic Require Export weakestpre.
+From iris.heap_lang Require Import proofmode notation.
+From osiris.typing Require Import side stype.
+From osiris.encodings Require Import list_enc channel stype_enc.
+From osiris.utils Require Import ordering permutation. 
+From iris.base_logic Require Import invariants.
+Require Import Coq.Lists.List.
+Require Import Omega.
+
+Section ListSortExample.
+  Context `{!heapG Σ} `{logrelG val Σ} (N : namespace).  
+  
+  Definition lsort : val :=
+    λ: "hd", "hd".
+
+  Definition sorted (xs : list Z) : Prop :=
+    ordered (≤) xs.
+
+  Definition sorted_hd (hd : val) (xs : list Z) : Prop :=
+    is_list_enc hd xs ∧ sorted xs.
+
+  Definition sorted_of (xs ys : list Z) : Prop :=
+    sorted xs ∧ permutation xs ys.
+
+  Definition sorted_of_hd (hd : val) (xs ys : list Z) : Prop :=
+    is_list_enc hd xs ∧ sorted_of xs ys.
+
+  Definition is_list_enc_ref (l : val) (xs : list Z) : iProp Σ :=
+    (∃ l':loc, ∃ hd : val, ⌜l = #l'⌝ ∧ l' ↦ hd ∗ ⌜is_list_enc hd xs⌝)%I.
+
+  Definition sorted_ref (hd : val) (xs : list Z) : iProp Σ :=
+    (is_list_enc_ref hd xs ∗ ⌜sorted xs⌝)%I.
+
+  Definition sorted_of_ref (hd : val) (xs ys : list Z) : iProp Σ :=
+    (is_list_enc_ref hd xs ∗ ⌜sorted_of xs ys⌝)%I.
+
+  Definition lsplit : val :=
+    λ: "xs",
+      let: "hd" := lhead !"xs" in
+      let: "ys" := lcons "hd" NONEV in
+      let: "zs" := ltail !"xs" in 
+    (ref "ys",ref "zs").
+
+   Lemma lsplit_spec (hdx : val) (xs : list Z) :
+    {{{ is_list_enc_ref hdx xs ∗ (∃ x xs', ⌜xs = x::xs'⌝)}}}
+      lsplit hdx
+    {{{ hdy hdz ys zs, RET (hdy,hdz);
+        is_list_enc_ref hdx xs ∗ is_list_enc_ref hdy ys ∗ is_list_enc_ref hdz zs ∗ ⌜xs = ys++zs⌝ }}}.
+   Proof.
+     iIntros (Φ) "[Hhd Heq] HΦ".
+     iDestruct "Heq" as %(x & xs' & ->).
+     iDestruct "Hhd" as (l hd ->) "[Hlxs Hhd]".
+     iDestruct "Hhd" as %Hhd.
+     wp_lam.
+     wp_load.
+     wp_apply (lhead_spec (T := Z) hd)=> //. 
+     iIntros "_".
+     wp_pures.
+     wp_apply (lcons_spec (T := Z) _ [])=> //. 
+     iIntros (ys Hys).
+     wp_load.
+     wp_apply (ltail_spec (T := Z))=> //.
+     iIntros (zs Hzs).
+     wp_alloc lzs as "Hlzs".
+     wp_alloc lys as "Hlys".
+     wp_pures.
+     iApply "HΦ".
+     iFrame.
+     rewrite /is_list_enc_ref.
+     iSplitL "Hlxs". eauto 10 with iFrame.
+     iSplitL "Hlys". eauto 10 with iFrame.
+     iSplitL "Hlzs". eauto 10 with iFrame.
+     eauto.
+   Qed.
+
+  Definition compare_vals : val :=
+    λ: "x" "y", "x" ≤ "y".
+
+  Lemma compare_vals_spec (x y : Z) :
+    {{{ True%I }}}
+      compare_vals (encode x) (encode y)
+    {{{ RET (encode (bool_decide (x ≤ y))); True%I }}}.
+  Proof.
+    iIntros (Φ) "_ HΦ".
+    wp_lam.
+    wp_pures.
+    by iApply "HΦ".
+  Qed.
+
+  Definition lmerge : val :=
+    rec: "go" "hys" "hzs" :=
+      if: llength "hys" = #0
+      then "hzs"
+      else if: llength "hzs" = #0
+           then "hys"
+           else let: "y" := lhead "hys" in
+                let: "z" := lhead "hzs" in
+                if: (compare_vals "y" "z")
+                then lcons "y" ("go" (ltail "hys") "hzs")
+                else lcons "z" ("go" "hys" (ltail "hzs")).
+
+  Lemma lmerge_spec hdy hdz (ys zs : list Z) :
+    {{{ ⌜sorted_hd hdy ys⌝ ∗ ⌜sorted_hd hdz zs⌝ }}}
+      lmerge hdy hdz
+    {{{ hxs xs, RET hxs; ⌜sorted_of_hd hxs xs (ys++zs)⌝ }}}.
+  Proof.
+    revert hdy hdz ys zs.
+    iLöb as "IH".
+    iIntros (hdy hdz ys zs).
+    iIntros (Φ [[Hhdy Hhdy_sorted] [Hhdz Hhdz_sorted]]) "HΦ".
+    wp_lam. 
+    wp_pures.
+    wp_apply (llength_spec (T:=Z))=> //.
+    iIntros "_".
+    destruct ys as [|y ys].
+    {
+      simpl. wp_pures.
+      iApply ("HΦ" $!hdz zs) => //. simpl.
+      iSplit=> //.
+      iPureIntro. rewrite /sorted_of. split=> //. apply permutation_refl.
+    }
+    wp_pures.
+    wp_bind (llength _).
+    wp_apply (llength_spec (T:=Z))=> //.
+    iIntros "_".
+    destruct zs as [|z zs].
+    {
+      simpl. wp_pures.
+      iApply ("HΦ" $!hdy (y::ys)) => //. repeat rewrite app_nil_r. simpl.
+      iSplit=> //.
+      iPureIntro. rewrite /sorted_of. split=> //. apply permutation_refl.
+    }
+    wp_pures.
+    wp_apply (lhead_spec (T:=Z))=> //.
+    iIntros "_".
+    wp_apply (lhead_spec (T:=Z))=> //.
+    iIntros "_".
+    wp_pures.
+    wp_apply (compare_vals_spec)=> //.
+    iIntros "_".
+    wp_pures.
+    destruct (bool_decide (y ≤ z)).
+    - wp_pures.
+      wp_apply (ltail_spec (T:=Z))=> //.
+      iIntros (hdy' Hhdy').
+      wp_apply ("IH")=> //. rewrite /sorted_hd. iSplit; iSplit=>//. iPureIntro. eapply ordered_cons=> //.
+      iIntros (hxs xs [Hhxs [Hhxs_sorted Hhxs_of]]).
+      wp_apply (lcons_spec (T:=Z))=> //.
+      iIntros (hd Hhd).
+      iApply ("HΦ").
+      iPureIntro.
+      admit.
+    - wp_pures.
+      wp_apply (ltail_spec (T:=Z))=> //.
+      iIntros (hdz' Hhdz').
+      wp_apply ("IH")=> //. rewrite /sorted_hd. iSplit; iSplit=>//. iPureIntro. eapply ordered_cons=> //.
+      iIntros (hxs xs [Hhxs [Hhxs_sorted Hhxs_of]]).
+      wp_apply (lcons_spec (T:=Z))=> //.
+      iIntros (hd Hhd).
+      iApply ("HΦ").
+      iPureIntro.
+    admit.
+  Admitted.
+
+  Definition lmerge_ref : val :=
+    λ: "lxs" "hys" "hzs",
+      "lxs" <- lmerge "hys" "hzs".
+      
+  Lemma lmerge_ref_spec (ldx hdy hdz : val) xs ys zs ys' zs' :
+    {{{ ⌜sorted_of_hd hdy ys' ys⌝ ∗
+        ⌜sorted_of_hd hdz zs' zs⌝ ∗
+        ⌜xs = ys++zs⌝ ∗
+        is_list_enc_ref ldx xs }}}
+      lmerge_ref ldx hdy hdz
+    {{{ xs', RET #(); sorted_of_ref ldx xs' xs }}}.
+  Proof. 
+    iIntros (Φ) "HPre HΦ".
+    iDestruct "HPre" as ([Hhdy [Hys_sorted Hys_perm]]
+                         [Hhdz [Hzs_sorted Hzs_perm]] Heq)
+                         "Hldx".
+    iDestruct "Hldx" as (l hd ->) "[Hl Hldx]".
+    wp_lam.
+    wp_apply (lmerge_spec hdy hdz ys' zs')=> //.
+    iIntros (hxs xs') "Hhxs".
+    iDestruct "Hhxs" as %(Hhxs & Hhys & HHzs).
+    wp_store.
+    iApply ("HΦ" $!xs'). 
+    rewrite /sorted_of_ref /sorted_of /is_list_enc_ref.
+    iSplit.
+    - eauto 10 with iFrame.
+    - rewrite Heq. iPureIntro.
+      split=> //.
+      eapply permutation_trans=> //.
+      apply permutation_app3=> //.
+  Qed.
+
+  Definition list_sort_service : val :=
+    rec: "go" "c" :=
+      let: "xs" := recv "c" #Right in
+      if: llength (!"xs") ≤ #1
+      then send "c" #Right "xs"
+      else let: "ys_zs" := lsplit "xs" in
+           let: "ys" := Fst "ys_zs" in
+           let: "zs" := Snd "ys_zs" in
+           let: "cy" := new_chan #() in Fork("go" "cy");;
+           let: "cz" := new_chan #() in Fork("go" "cz");;
+           send "cy" #Left "ys";;
+           send "cz" #Left "zs";;
+           let: "ys" := recv "cy" #Left in
+           let: "zs" := recv "cz" #Left in
+           lmerge_ref "xs" !"ys" !"zs";;
+           send "c" #Right "xs".
+
+  Lemma list_sort_service_spec γ c xs :
+    {{{ ⟦ c @ Right : (<?> hd @ is_list_enc_ref hd xs,
+                          <!> hd' @
+                          ⌜hd = hd'⌝ ∗ (∃ ys : list Z, sorted_of_ref hd' ys xs),
+                          TEnd) ⟧{N,γ} }}}
+      list_sort_service c
+    {{{ RET #(); ⟦ c @ Right : TEnd ⟧{N,γ} }}}.
+  Proof.
+    revert γ c xs.
+    iLöb as "IH".
+    iIntros (γ c xs Φ) "Hstr HΦ".
+    wp_lam.
+    wp_apply (recv_st_spec N val with "Hstr").
+    iIntros (v) "Hstr".
+    iDestruct "Hstr" as "[Hstr HP]".
+    wp_pures.
+    iDestruct "HP" as (l hd' ->) "[Hl Hhd']".
+    wp_load.
+    iDestruct ("Hhd'") as %Hhd'.
+    destruct xs.
+    + rewrite /is_list_ref.
+      wp_apply (llength_spec (T:=Z))=> //. 
+      iIntros "_".
+      wp_pures.
+      wp_apply (send_st_spec N val with "[Hl Hstr]")=> //. iFrame.
+      iSplit=> //.
+      rewrite /(sorted_of_ref #l). rewrite /(is_list_enc_ref #l).
+      iExists _.
+      iSplit; first by eauto 10 with iFrame.
+      iPureIntro.
+      split; constructor.
+      iIntros "Hstr".
+      by iApply "HΦ".
+    + destruct xs.
+      * rewrite /is_list_ref.
+        wp_apply (llength_spec (T:=Z))=> //. 
+        iIntros "_".
+        wp_pures.
+        wp_apply (send_st_spec N val with "[Hl Hstr]")=> //. iFrame.
+        iSplit=> //.
+        rewrite /(sorted_of_ref #l). rewrite /(is_list_enc_ref #l).
+        iExists _.
+        iSplit; first by eauto 10 with iFrame.
+        iPureIntro.
+        split; [ constructor | apply permutation_cons; constructor ].
+        iIntros "Hstr".
+        by iApply "HΦ".
+      * wp_apply (llength_spec (T:=Z))=> //. 
+        iIntros "_".
+        wp_pures.
+        assert (bool_decide (S (S (length xs)) ≤ 1) = false).
+        { apply bool_decide_false. lia. }
+        rewrite H0.
+        wp_pures.
+        wp_apply (lsplit_spec with "[Hl]").
+        rewrite /(is_list_enc_ref (encode #l)). eauto 10 with iFrame.
+        iIntros (hdy hdz ys zs) "(Hhdx & Hhdy & Hhdz & Heq)".
+        wp_pures.
+        iDestruct "Hhdy" as (ly hdy' ->) "[Hhdy Hhdys]".
+        iDestruct "Hhdys" as %Hhdys.
+        iDestruct "Hhdz" as (lz hdz' ->) "[Hhdz Hhdzs]".
+        iDestruct "Hhdzs" as %Hhdzs.
+        iDestruct "Heq" as %->.
+        wp_apply (new_chan_st_spec N
+                 (<!> hd @ is_list_enc_ref hd ys,
+                  <?> hd' @ (⌜hd = hd'⌝ ∗ ∃ ys',
+                                 sorted_of_ref hd' ys' ys)%I, TEnd))=>//;
+        iIntros (cy γy) "[Hstly Hstry]".
+        wp_apply (wp_fork with "[Hstry]").
+        { iApply ("IH" with "Hstry"). iNext. iNext. iIntros "Hstry". done. }
+        wp_apply (new_chan_st_spec N
+                 (<!> hd @ is_list_enc_ref hd zs,
+                  <?> hd' @ (⌜hd = hd'⌝ ∗ ∃ zs',
+                                 sorted_of_ref hd' zs' zs)%I, TEnd))=>//;
+        iIntros (cz γz) "[Hstlz Hstrz]".
+        wp_apply (wp_fork with "[Hstrz]").
+        { iApply ("IH" with "Hstrz"). iNext. iNext. iIntros "Hstrz". done. }
+        wp_apply (send_st_spec N val with "[Hhdy Hstly]"). iFrame.
+        { rewrite /is_list_enc_ref.
+          iExists _, _. iSplit=> //. iFrame.
+          eauto 10 with iFrame. } 
+        iIntros "Hstly".
+        wp_apply (send_st_spec N val with "[Hhdz Hstlz]"). iFrame.
+        { rewrite /is_list_enc_ref.
+          iExists _, _. iSplit=> //. iFrame.
+          eauto 10 with iFrame. } 
+        iIntros "Hstlz".
+        wp_apply (recv_st_spec N val with "Hstly").
+        iIntros (hdy) "[Hstly Hys']".
+        iDestruct "Hys'" as (<-) "Hys'_sorted".
+        iDestruct "Hys'_sorted" as (ys') "[Hys' Hys'_sorted]".
+        iDestruct "Hys'_sorted" as %Hys'_sorted.
+        iDestruct "Hys'" as (ly' Hy' ->) "[Hlys' Hys']".
+        iDestruct "Hys'" as %Hys'.
+        wp_apply (recv_st_spec N val with "Hstlz").
+        iIntros (hdz) "[Hstlz Hzs']".
+        iDestruct "Hzs'" as (<-) "Hzs'_sorted".
+        iDestruct "Hzs'_sorted" as (zs') "[Hzs' Hzs'_sorted]".
+        iDestruct "Hzs'_sorted" as %Hzs'_sorted.
+        iDestruct "Hzs'" as (lz' Hz' ->) "[Hlzs' Hzs']".
+        iDestruct "Hzs'" as %Hzs'.
+        wp_load.
+        wp_load.
+        wp_apply (lmerge_ref_spec with "[Hhdx]")=> //.        
+        iFrame. rewrite /sorted_hd. eauto 10 with iFrame.
+        iIntros (xs') "Hxs'_sorted". 
+        wp_apply (send_st_spec N val with "[Hstr Hxs'_sorted]"). iFrame.
+        eauto 10 with iFrame.
+        iIntros "Hstr".
+        by iApply "HΦ".
+  Qed.
+
+  Definition list_sort : val :=
+    λ: "xs",
+    let: "c" := new_chan #() in
+    Fork(list_sort_service "c");;
+    send "c" #Left "xs";;