Updated encoding specifications and separated examples and proofs

_CoqProject

@@ -5,4 +5,5 @@ theories/auth_excl.v
 theories/typing.v
 theories/channel.v
 theories/logrel.v
+theories/examples.v
 theories/encodable.v

theories/encodable.v

@@ -94,7 +94,7 @@ Section Encodings.
     (TSR' Receive
           (λ v', ⌜v' = 5⌝%I)
           (λ v', TEnd')).
-  
+
   Example ex_st2 : stype' (iProp Σ) :=
     TSR' Send
          (λ b, ⌜b = true⌝%I)
@@ -106,22 +106,23 @@ Section Encodings.
   Fixpoint stype'_to_stype (st : stype' (iProp Σ)) : stype val (iProp Σ) :=
     match st with
     | TEnd' => TEnd
-    | TSR' a P st => TSR a
-                         (λ v, match decode v with
-                               | Some v => P v
-                               | None => False
-                               end%I)
-                         (λ v, match decode v with
-                               | Some v => stype'_to_stype (st v)
-                               | None => TEnd
-                               end)
+    | TSR' a P st =>
+      TSR a
+          (λ v, match decode v with
+                | Some v => P v
+                | None => False
+                end%I)
+          (λ v, match decode v with
+                | Some v => stype'_to_stype (st v)
+                | None => TEnd
+                end)
     end.
   Global Instance: Params (@stype'_to_stype) 1.
   Global Arguments stype'_to_stype : simpl never.
 
   Lemma dual_stype'_comm st :
-    dual_stype (stype'_to_stype st) ≡ stype'_to_stype (dual_stype' st).
-  Proof. 
+    stype'_to_stype (dual_stype' st) ≡ dual_stype (stype'_to_stype st).
+  Proof.
     induction st.
     - by simpl.
     - unfold stype'_to_stype. simpl.
@@ -130,9 +131,9 @@ Section Encodings.
       + intros v. destruct (decode v); eauto.
   Qed.
 
-  Lemma dual_stype'_comm_eq st :
-    dual_stype (stype'_to_stype st) = stype'_to_stype (dual_stype' st).
-  Proof. Admitted.
+  Lemma stype_map_equiv {A B : ofeT} (f : A -n> B) (st st' : stype val A) :
+    st ≡ st' → stype_map f st ≡ stype_map f st'.
+  Proof. induction 1=>//. constructor=>//. by repeat f_equiv. Qed.
 
   Notation "⟦ c @ s : sτ ⟧{ γ }" := (interp_st N γ sτ c s)
     (at level 10, s at next level, sτ at next level, γ at next level,
@@ -149,47 +150,57 @@ Section Encodings.
     iNext.
     iIntros (c γ) "[Hl Hr]".
     iApply "HΦ".
-    iFrame. 
-    rewrite dual_stype'_comm_eq.    
     iFrame.
+    iDestruct "Hr" as "[Hown Hctx]".
+    iFrame.
+    unfold st_own. simpl.
+    iApply (own_mono with "Hown").
+    apply (auth_frag_mono).
+    apply Some_included.
+    left.
+    f_equiv.
+    f_equiv.
+    apply stype_map_equiv.
+    apply dual_stype'_comm.
   Qed.
 
   Lemma send_st_enc_spec (A : Type) `{Encodable A} `{Decodable A} `{EncDec A}
-        st γ c s (P : A → iProp Σ) v w :
-    decode v = Some w →
+        st γ c s (P : A → iProp Σ) w :
     {{{ P w ∗ ⟦ c @ s : (stype'_to_stype (TSR' Send P st)) ⟧{γ} }}}
-      send c #s v
+      send c #s (encode w)
     {{{ RET #(); ⟦ c @ s : stype'_to_stype (st w) ⟧{γ} }}}.
   Proof.
-    intros Henc.
     iIntros (Φ) "[HP Hsend] HΦ".
     iApply (send_st_spec with "[HP Hsend]").
     simpl.
     iFrame.
-    by destruct (decode v); inversion Henc.
+    by rewrite encdec.
     iNext.
-    destruct (decode v); inversion Henc.
+    rewrite encdec.
     by iApply "HΦ".
   Qed.
-    
+
   Lemma recv_st_enc_spec (A : Type) `{EncDec A}
         st γ c s (P : A → iProp Σ) :
     {{{ ⟦ c @ s : (stype'_to_stype (TSR' Receive P st)) ⟧{γ} }}}
       recv c #s
-    {{{ v w, RET v; ⟦ c @ s : stype'_to_stype (st w) ⟧{γ} ∗ P w ∗
-        ⌜encode w = v⌝ }}}.
+    {{{ v, RET (encode v); ⟦ c @ s : stype'_to_stype (st v) ⟧{γ} ∗ P v }}}.
   Proof.
     iIntros (Φ) "Hrecv HΦ".
     iApply (recv_st_spec with "Hrecv").
-    iNext. iIntros (v). iSpecialize ("HΦ" $!v).
+    iNext. iIntros (v). (* iSpecialize ("HΦ" $!v). *)
     iIntros "[H HP]".
     iAssert ((∃ w, ⌜decode v = Some w⌝ ∗ P w)%I) with "[HP]" as (w Hw) "HP".
-    destruct (decode v). iExists a. by iFrame. iDestruct "HP" as %HP=>//.
-    assert (encode w = v). by apply decenc.
-    destruct (decode v); inversion Hw.
+    { destruct (decode v).
+      iExists a. by iFrame. iDestruct "HP" as %HP=>//. }
+    iSpecialize ("HΦ" $!w).
+    apply enc_dec in Hw. rewrite Hw.
     iApply "HΦ".
     iFrame.
-    iPureIntro. eauto.
+    apply enc_dec in Hw.
+    destruct (decode v).
+    - inversion Hw. subst. iApply "H".
+    - inversion Hw.
   Qed.
 
 End Encodings.
\ No newline at end of file

theories/examples.v

@@ -2,7 +2,6 @@ From iris.proofmode Require Import tactics.
 From iris.program_logic Require Export weakestpre.
 From iris.heap_lang Require Import proofmode notation.
 From osiris Require Import typing channel logrel.
-From iris.algebra Require Import list auth excl.
 From iris.base_logic Require Import invariants.
 
 Section Examples.
@@ -18,103 +17,21 @@ Section Examples.
      send "c" #Left #5;;
      recv "c" #Right)%E.
 
-  Lemma seq_proof :
-    {{{ True }}} seq_example {{{ v, RET v; ⌜v = #5⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /seq_example.
-    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (send_st_spec N with "[$Hstl //]").
-    iIntros "Hstl".
-    wp_apply (recv_st_spec _ with "Hstr").
-    iIntros (v) "[Hstr HP]".
-    by iApply "HΦ".
-  Qed.
-
   Definition par_example : expr :=
     (let: "c" := new_chan #() in
      Fork(send "c" #Left #5);;
      recv "c" #Right)%E.
 
-  Lemma par_proof :
-    {{{ True }}} par_example {{{ v, RET v; ⌜v = #5⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /par_example.
-    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (wp_fork with "[Hstl]").
-    - iNext. wp_apply (send_st_spec N with "[Hstl]"). by iFrame. eauto.
-    - wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
-      iIntros (v) "[Hstr HP]". by iApply "HΦ".
-  Qed.
-
   Definition par_2_example : expr :=
     (let: "c" := new_chan #() in
      Fork(let: "v" := recv "c" #Right in send "c" #Right ("v"+#2));;
      send "c" #Left #5;;recv "c" #Left)%E.
 
-  Lemma par_2_proof :
-    {{{ True }}}
-      par_2_example
-    {{{ (v:Z), RET #v; ⌜7 ≤ v⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /par_2_example.
-    wp_apply (new_chan_st_spec N
-      (TSR Send
-           (λ v, ⌜v = #5⌝%I)
-           (λ v, TSR Receive
-                     (λ v',
-                      (∃ w, ⌜v = LitV $ LitInt $ w⌝ ∧
-                       ∃ w', ⌜v' = LitV $ LitInt $ w' ∧ w' >= w+2⌝)%I)
-                     (λ v', TEnd))))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (wp_fork with "[Hstr]").
-    - iNext.
-      wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
-      iIntros (v) "[Hstr HP]".
-      iDestruct "HP" as %->.
-      wp_apply (send_st_spec N with "[Hstr]"). iFrame; eauto 10 with iFrame.
-      eauto.
-    - wp_apply (send_st_spec _ with "[Hstl]"). by iFrame.
-      iIntros "Hstl".
-      wp_apply (recv_st_spec _ with "[Hstl]"). by iFrame.
-      iIntros (v) "[Hstl HP]".
-      iDestruct "HP" as %[w [HP [w' [-> HQ']]]].
-      iApply "HΦ".
-      iPureIntro. simplify_eq. lia.
-  Qed.
-
   Definition heaplet_example : expr :=
     (let: "c" := new_chan #() in
      Fork(send "c" #Left (ref #5));;
      !(recv "c" #Right))%E.
 
-  Lemma heaplet_proof :
-    {{{ True }}} heaplet_example {{{ v l, RET v; ⌜v = #5⌝ ∗ l ↦ v }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /heaplet_example.
-    wp_apply (new_chan_st_spec N
-      (TSR Send (λ v, (∃ l, ⌜v = LitV $ LitLoc $ l⌝ ∧ (l ↦ #5))%I)
-                (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (wp_fork with "[Hstl]").
-    - iNext.
-      wp_apply (wp_alloc)=> //.
-      iIntros (l) "HP".
-      wp_apply (send_st_spec N with "[Hstl HP]"). eauto 10 with iFrame.
-      eauto.
-    - wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
-      iIntros (v) "[Hstr HP]".
-      iDestruct "HP" as (l ->) "HP".
-      wp_load.
-      iApply "HΦ".
-      iFrame. eauto.
-  Qed.
-
   Definition channel_example : expr :=
     (let: "c" := new_chan #() in
      Fork(
@@ -124,101 +41,20 @@ Section Examples.
      recv "c'" #Right
     )%E.
 
-  Lemma channel_proof :
-    {{{ True }}} channel_example {{{ v, RET v; ⌜v = #5⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /channel_example.
-    wp_apply (new_chan_st_spec N
-      (TSR Send (λ v, ∃ γ, ⟦ v @ Right :
-        (TSR Receive
-             (λ v : val, ⌜v = #5⌝)
-             (λ _ : val, TEnd)) ⟧{γ})%I
-                (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_pures.
-    wp_apply (wp_fork with "[Hstl]").
-    - iNext.
-      wp_apply (new_chan_st_spec N
-        (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-      iIntros (c' γ') "[Hstl' Hstr']"=> /=.
-      wp_apply (send_st_spec N with "[Hstl Hstr']").
-      iFrame. iExists γ'. iFrame.
-      iIntros "Hstl".
-      wp_apply (send_st_spec N with "[Hstl']").
-      iFrame. eauto. eauto.
-    - wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
-      iIntros (v) "[Hstr Hstr']".
-      iDestruct "Hstr'" as (γ') "Hstr'".
-      wp_apply (recv_st_spec _ with "[Hstr']").
-      iApply "Hstr'".
-      iIntros (v') "[Hstr' HP]".
-      by iApply "HΦ".
-  Qed.
-
   Definition bad_seq_example : expr :=
     (let: "c" := new_chan #() in
      let: "v" := recv "c" #Right in
      send "c" #Left #5;; "v")%E.
 
-  Lemma bad_seq_proof :
-    {{{ True }}} bad_seq_example {{{ v, RET v; ⌜v = #5⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /bad_seq_example.
-    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (recv_st_spec _ with "Hstr").
-    iIntros (v) "[Hstr HP]".
-    wp_apply (send_st_spec N with "[Hstl]"). by iFrame.
-    iIntros "Hstl".
-    wp_pures.
-    by iApply "HΦ".
-  Qed.
-
   Definition bad_par_example : expr :=
     (let: "c" := new_chan #() in
      Fork(#());;
      recv "c" #Right)%E.
 
-  Lemma bad_par_proof :
-    {{{ True }}} bad_par_example {{{ v, RET v; ⌜v = #5⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /bad_par_example.
-    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (wp_fork with "[Hstl]").
-    - iNext. wp_finish. eauto.
-    - wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
-      iIntros (v) "[Hstr HP]". by iApply "HΦ".
-  Qed.
-
   Definition bad_interleave_example : expr :=
     (let: "c" := new_chan #() in
      let: "c'" := new_chan #() in
      Fork(recv "c" #Right;; send "c'" #Right #5);;
      recv "c'" #Left;; send "c" #Left #5)%E.
 
-  Lemma bad_interleave_proof :
-    {{{ True }}} bad_interleave_example {{{ v, RET v; ⌜v = #()⌝ }}}.
-  Proof.
-    iIntros (Φ H) "HΦ".
-    rewrite /bad_interleave_example.
-    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-    iIntros (c γ) "[Hstl Hstr]"=> /=.
-    wp_apply (new_chan_st_spec N (TSR Receive (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
-    iIntros (c' γ') "[Hstl' Hstr']"=> /=.
-    wp_apply (wp_fork with "[Hstr Hstr']").
-    - iNext. wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
-      iIntros (v) "[Hstr HP]".
-      wp_apply (send_st_spec _ with "[Hstr']"). iFrame. eauto.
-      eauto.
-    - wp_apply (recv_st_spec _ with "[Hstl']"). by iFrame.
-      iIntros (v) "[Hstl' HP]".
-      wp_apply (send_st_spec _ with "[Hstl]"). iFrame. eauto.
-      iIntros "Hstl".
-      by iApply "HΦ".
-  Qed.
-  
 End Examples.
\ No newline at end of file

theories/encodings_examples.v → theories/examples_encoding_proofs.v

@@ -5,15 +5,15 @@ From osiris Require Import typing channel logrel.
 From iris.algebra Require Import list auth excl.
 From iris.base_logic Require Import invariants.
 From osiris Require Import encodable.
+From osiris Require Import examples.
 
 Section Encodings_Examples.
   Context `{!heapG Σ} {N : namespace}.
   Context `{!logrelG val Σ}.
 
-  Definition seq_example : expr :=
-    (let: "c" := new_chan #() in
-     send "c" #Left #5;;
-     recv "c" #Right)%E.
+  Notation "⟦ c @ s : sτ ⟧{ γ }" := (interp_st N γ (stype'_to_stype sτ) c s)
+    (at level 10, s at next level, sτ at next level, γ at next level,
+     format "⟦  c  @  s  :  sτ  ⟧{ γ }").
 
   Lemma seq_proof :
     {{{ True }}} seq_example {{{ v, RET v; ⌜v = #5⌝ }}}.
@@ -26,18 +26,12 @@ Section Encodings_Examples.
     wp_apply (send_st_enc_spec N Z with "[Hstl]")=> //. by iFrame.
     iIntros "Hstl".
     wp_apply (recv_st_enc_spec N Z with "[Hstr]"). iFrame.
-    iIntros (v w) "[Hstr [HP Heq]]".
+    iIntros (v) "[Hstr HP]".
     iApply "HΦ".
-    iDestruct "Heq" as %<-.
     iDestruct "HP" as %->.    
     eauto.
   Qed.
 
-  Definition par_2_example : expr :=
-    (let: "c" := new_chan #() in
-     Fork(let: "v" := recv "c" #Right in send "c" #Right ("v"+#4));;
-     send "c" #Left #5;;recv "c" #Left)%E.
-
   Lemma par_2_proof :
     {{{ True }}}
       par_2_example
@@ -55,26 +49,19 @@ Section Encodings_Examples.
     wp_apply (wp_fork with "[Hstr]").
     - iNext.
       wp_apply (recv_st_enc_spec N Z with "[Hstr]"). by iFrame.
-      iIntros (v w) "[Hstr HP]".
-      iDestruct "HP" as %[HP <-].
+      iIntros (v) "[Hstr HP]".
       wp_apply (send_st_enc_spec N Z with "[Hstr]")=> //.
       iFrame; eauto 10 with iFrame.
-      iPureIntro. lia.
       eauto.
     - wp_apply (send_st_enc_spec N Z with "[Hstl]")=> //. by iFrame.
       iIntros "Hstl".
       wp_apply (recv_st_enc_spec N Z with "[Hstl]"). by iFrame.
-      iIntros (v w) "[Hstl HP]".
-      iDestruct "HP" as %[HP <-].
+      iIntros (v) "[Hstl HP]".
       iApply "HΦ".
+      iDestruct "HP" as %HP.
       iPureIntro. lia.
   Qed.
 
-  Definition heaplet_example : expr :=
-    (let: "c" := new_chan #() in
-     Fork(send "c" #Left (ref #5));;
-     !(recv "c" #Right))%E.
-
   Lemma heaplet_proof :
     {{{ True }}} heaplet_example {{{ v l, RET v; ⌜v = #5⌝ ∗ l ↦ v }}}.
   Proof.
@@ -91,27 +78,12 @@ Section Encodings_Examples.
       wp_apply (send_st_enc_spec N loc with "[Hstl HP]")=> //. by iFrame.
       eauto.
     - wp_apply (recv_st_enc_spec N loc with "[Hstr]"). iFrame.
-      iIntros (v w) "[Hstr HP]".
-      iDestruct "HP" as "[HP Henc]".
-      iDestruct "Henc" as %<-.
+      iIntros (v) "[Hstr HP]".
       wp_load.
       iApply "HΦ".
       iFrame. eauto.
   Qed.
 
-  Definition channel_example : expr :=
-    (let: "c" := new_chan #() in
-     Fork(
-         let: "c'" := new_chan #() in
-         send "c" #Left ("c'");; send "c'" #Left #5);;
-         let: "c'" := recv "c" #Right in
-         recv "c'" #Right
-    )%E.
-
-  Notation "⟦ c @ s : sτ ⟧{ γ }" := (interp_st N γ (stype'_to_stype sτ) c s)
-    (at level 10, s at next level, sτ at next level, γ at next level,
-     format "⟦  c  @  s  :  sτ  ⟧{ γ }").
-
   Lemma channel_proof :
     {{{ True }}} channel_example {{{ v, RET v; ⌜v = #5⌝ }}}.
   Proof.
@@ -136,12 +108,10 @@ Section Encodings_Examples.
       wp_apply (send_st_enc_spec N Z with "[Hstl']")=> //.
       iFrame. eauto. eauto.
     - wp_apply (recv_st_enc_spec N val with "[Hstr]")=> //.
-      iIntros (v w) "[Hstr [Hstr' Henc]]".
-      iDestruct "Henc" as %<-.
+      iIntros (v) "[Hstr Hstr']".
       iDestruct "Hstr'" as (γ') "Hstr'".
       wp_apply (recv_st_enc_spec N Z with "[Hstr']")=> //.
-      iIntros (v' w') "[Hstr' [HP Henc]]".
-      iDestruct "Henc" as %<-.
+      iIntros (v') "[Hstr' HP]".
       iDestruct "HP" as %<-.
       by iApply "HΦ".
   Qed.

theories/examples_proofs.v

+From iris.proofmode Require Import tactics.
+From iris.program_logic Require Export weakestpre.
+From iris.heap_lang Require Import proofmode notation.
+From osiris Require Import typing channel logrel.
+From iris.base_logic Require Import invariants.
+From osiris Require Import examples.
+
+Section ExampleProofs.
+  Context `{!heapG Σ} (N : namespace).
+  Context `{!logrelG val Σ}.
+
+  Notation "⟦ c @ s : sτ ⟧{ γ }" := (interp_st N γ sτ c s)
+    (at level 10, s at next level, sτ at next level, γ at next level,
+     format "⟦  c  @  s  :  sτ  ⟧{ γ }").
+
+  Lemma seq_proof :
+    {{{ True }}} seq_example {{{ v, RET v; ⌜v = #5⌝ }}}.
+  Proof.
+    iIntros (Φ H) "HΦ".
+    rewrite /seq_example.
+    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
+    iIntros (c γ) "[Hstl Hstr]"=> /=.
+    wp_apply (send_st_spec N with "[$Hstl //]").
+    iIntros "Hstl".
+    wp_apply (recv_st_spec _ with "Hstr").
+    iIntros (v) "[Hstr HP]".
+    by iApply "HΦ".
+  Qed.
+
+  Lemma par_proof :
+    {{{ True }}} par_example {{{ v, RET v; ⌜v = #5⌝ }}}.
+  Proof.
+    iIntros (Φ H) "HΦ".
+    rewrite /par_example.
+    wp_apply (new_chan_st_spec N (TSR Send (λ v, ⌜v = #5⌝%I) (λ v, TEnd)))=> //.
+    iIntros (c γ) "[Hstl Hstr]"=> /=.
+    wp_apply (wp_fork with "[Hstl]").
+    - iNext. wp_apply (send_st_spec N with "[Hstl]"). by iFrame. eauto.
+    - wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
+      iIntros (v) "[Hstr HP]". by iApply "HΦ".
+  Qed.
+
+  Lemma par_2_proof :
+    {{{ True }}}
+      par_2_example
+    {{{ (v:Z), RET #v; ⌜7 ≤ v⌝ }}}.
+  Proof.
+    iIntros (Φ H) "HΦ".
+    rewrite /par_2_example.
+    wp_apply (new_chan_st_spec N
+      (TSR Send
+           (λ v, ⌜v = #5⌝%I)
+           (λ v, TSR Receive
+                     (λ v',
+                      (∃ w, ⌜v = LitV $ LitInt $ w⌝ ∧
+                       ∃ w', ⌜v' = LitV $ LitInt $ w' ∧ w' >= w+2⌝)%I)
+                     (λ v', TEnd))))=> //.
+    iIntros (c γ) "[Hstl Hstr]"=> /=.
+    wp_apply (wp_fork with "[Hstr]").
+    - iNext.
+      wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
+      iIntros (v) "[Hstr HP]".
+      iDestruct "HP" as %->.
+      wp_apply (send_st_spec N with "[Hstr]"). iFrame; eauto 10 with iFrame.
+      eauto.
+    - wp_apply (send_st_spec _ with "[Hstl]"). by iFrame.
+      iIntros "Hstl".
+      wp_apply (recv_st_spec _ with "[Hstl]"). by iFrame.
+      iIntros (v) "[Hstl HP]".
+      iDestruct "HP" as %[w [HP [w' [-> HQ']]]].
+      iApply "HΦ".
+      iPureIntro. simplify_eq. lia.
+  Qed.
+
+  Lemma heaplet_proof :
+    {{{ True }}} heaplet_example {{{ v l, RET v; ⌜v = #5⌝ ∗ l ↦ v }}}.
+  Proof.
+    iIntros (Φ H) "HΦ".
+    rewrite /heaplet_example.
+    wp_apply (new_chan_st_spec N
+      (TSR Send (λ v, (∃ l, ⌜v = LitV $ LitLoc $ l⌝ ∧ (l ↦ #5))%I)
+                (λ v, TEnd)))=> //.
+    iIntros (c γ) "[Hstl Hstr]"=> /=.
+    wp_apply (wp_fork with "[Hstl]").
+    - iNext.
+      wp_apply (wp_alloc)=> //.
+      iIntros (l) "HP".
+      wp_apply (send_st_spec N with "[Hstl HP]"). eauto 10 with iFrame.
+      eauto.
+    - wp_apply (recv_st_spec _ with "[Hstr]"). by iFrame.
+      iIntros (v) "[Hstr HP]".
+      iDestruct "HP" as (l ->) "HP".
+      wp_load.
+      iApply "HΦ".
+      iFrame. eauto.
+  Qed.
+
+  Lemma channel_proof :
+    {{{ True }}} channel_example {{{ v, RET v; ⌜v = #5⌝ }}}.