Finish `list_sort_elem` client proof.

_CoqProject

@@ -12,3 +12,4 @@ theories/examples/list_sort.v
 theories/examples/list_sort_instances.v
 theories/examples/list_sort_elem.v
 theories/examples/loop_sort.v
+theories/examples/list_sort_elem_client.v

theories/examples/list_sort_elem_client.v

@@ -15,67 +15,70 @@ Definition send_all : val :=
 Definition recv_all : val :=
   rec: "go" "c" :=
     if: recv "c"
-    then lcons (recv "c") ("go" "c")
+    then let: "x" := recv "c" in lcons "x" ("go" "c")
     else lnil #().
 
 Definition list_sort_elem_client : val :=
   λ: "cmp" "xs",
     let: "c" := start_chan list_sort_elem_service_top in
     send "c" "cmp";;
-    send_all "c" ("xs");;
+    send_all "c" "xs";;
     recv_all "c".
 
 Section list_sort_elem_client.
   Context `{!heapG Σ, !proto_chanG Σ} (N : namespace).
+  Context {A} (I : A → val → iProp Σ) (R : relation A) `{!RelDecision R, !Total R}.
 
-  Lemma send_all_spec 
-    {A : Type} (I : A → val → iProp Σ) R `{!RelDecision R, !Total R} 
-    (xs : list A) (vs : list val) c :
-    {{{ ([∗ list] x;v ∈ xs ; vs, I x v) ∗
-        c ↣ head_protocol I R [] @ N }}}
+  Lemma send_all_spec c p xs' xs vs :
+    {{{ c ↣ head_protocol I R xs' <++> p @ N ∗ [∗ list] x;v ∈ xs;vs, I x v }}}
       send_all c (val_encode vs)
-    {{{ RET #(); 
-        ([∗ list] x;v ∈ xs ; vs, I x v) ∗
-        c ↣ tail_protocol I R (zip xs vs) [] @ N }}}.
-  Proof. Admitted.
+    {{{ RET #(); c ↣ tail_protocol I R (xs' ++ xs) [] <++> p @ N }}}.
+  Proof.
+    iIntros (Φ) "[Hc HI] HΦ".
+    iInduction xs as [|x xs] "IH" forall (xs' vs); destruct vs as [|v vs]=>//.
+    { wp_lam; wp_pures. wp_select. iApply "HΦ". rewrite right_id_L. iFrame. }
+    iDestruct "HI" as "[HIxv HI]". wp_lam; wp_pures.
+    wp_select. wp_send with "[$HIxv]". wp_apply ("IH" with "Hc HI").
+    by rewrite -assoc_L.
+  Qed.
 
-  Lemma recv_all_spec 
-    {A : Type} (I : A → val → iProp Σ) R `{!RelDecision R, !Total R} 
-    (xs : list A) (vs : list val) c :
-    {{{ ([∗ list] x;v ∈ xs ; vs, I x v) ∗
-        c ↣ tail_protocol I R (zip xs vs) [] @ N }}}
+  Lemma recv_all_spec c p xs ys :
+    Sorted R ys →
+    {{{ c ↣ tail_protocol I R xs ys <++> p @ N }}}
       recv_all c
-    {{{ ys ws, RET (val_encode ws);
-        c ↣ END @ N ∗
-        ⌜Sorted R ys⌝ ∗ ⌜ys ≡ₚ xs⌝ ∗
-        [∗ list] y;w ∈ ys;ws, I y w
+    {{{ ys' ws, RET (val_encode ws);
+        ⌜Sorted R (ys ++ ys')⌝ ∗ ⌜ys ++ ys' ≡ₚ xs⌝ ∗
+        c ↣ p @ N ∗ [∗ list] y;w ∈ ys';ws, I y w
     }}}.
-  Proof. Admitted.
+  Proof.
+    iIntros (Hsort Φ) "Hc HΦ".
+    iLöb as "IH" forall (xs ys Φ Hsort).
+    wp_lam. wp_branch as "_" "Hperm"; wp_pures.
+    - wp_recv (y v) as (Htl) "HIxv".
+      wp_apply ("IH" with "[] Hc"); first by auto using Sorted_snoc.
+      iIntros (ys' ws). rewrite -!assoc_L. iDestruct 1 as (??) "[Hc HI]".
+      wp_apply (lcons_spec (A:=val) with "[//]"); iIntros (_).
+      iApply ("HΦ" $! (y :: ys')). simpl; iFrame; auto.
+    - wp_apply (lnil_spec with "[//]"); iIntros (_).
+      iApply ("HΦ" $! [] []); rewrite /= right_id_L; by iFrame.
+  Qed.
 
-  Lemma list_sort_client_spec {A} (I : A → val → iProp Σ) R
-       `{!RelDecision R, !Total R} cmp (vs : list val) (xs : list A) :
+  Lemma list_sort_client_spec cmp vs xs :
     cmp_spec I R cmp -∗
     {{{ [∗ list] x;v ∈ xs;vs, I x v }}}
       list_sort_elem_client cmp (val_encode vs)
     {{{ ys ws, RET (val_encode ws); ⌜Sorted R ys⌝ ∗ ⌜ys ≡ₚ xs⌝ ∗
                [∗ list] y;w ∈ ys;ws, I y w }}}.
   Proof.
-    iIntros "#Hcmp !>" (Φ) "Hinterp HΦ". wp_lam.
-    wp_apply (start_chan_proto_spec N list_sort_elem_top_protocol); iIntros (c) "Hc".
+    iIntros "#Hcmp !>" (Φ) "HI HΦ". wp_lam.
+    wp_apply (start_chan_proto_spec N (list_sort_elem_top_protocol <++> END)%proto);
+      iIntros (c) "Hc".
     { wp_apply (list_sort_elem_service_top_spec N with "Hc"); auto. }
-    wp_pures.
     rewrite /list_sort_elem_top_protocol.
-    simpl.
-    wp_send with "[$Hcmp]".
-    wp_pures.
-    wp_apply (send_all_spec with "[Hinterp Hc]")=> //.
-    iFrame.
-    iIntros "Hc".
-    wp_pures.
-    wp_apply (recv_all_spec with "[Hc]")=>//.
-    iIntros (ys ws) "(Hc & Hsorted & Hperm & Hinterp)".
-    iApply "HΦ".
-    iFrame.
+    wp_send (A I R) with "[$Hcmp]".
+    wp_apply (send_all_spec with "[$HI $Hc]"); iIntros "Hc".
+    wp_apply (recv_all_spec _ _ _ [] with "[$Hc]"); auto.
+    iIntros (ys ws) "/=". iDestruct 1 as (??) "[_ HI]".
+    iApply "HΦ"; auto.
   Qed.
-
-End list_sort_elem_client.
\ No newline at end of file
+End list_sort_elem_client.