Commit 314e5f72 by Robbert Krebbers

### Finish `list_sort_elem` client proof.

parent 842726e2
 ... ... @@ -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
 ... ... @@ -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.
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