Use fractional pointers with extensional variable

theories/logrel/F_mu_ref_conc/examples/queue/common.v

@@ -58,9 +58,43 @@ Section common.
     iIntros. iApply (own_update with "[$]"). by apply cmra_update_exclusive.
   Qed.
 
+  Lemma mapsto_full_to_frac l v : l ↦ᵢ v -∗ l ↦ᵢ{-} v.
+  Proof. iIntros. by iExists _. Qed.
+
+  Lemma mapsto_full_to_frac_2 l v : l ↦ᵢ v -∗ l ↦ᵢ{-} v ∗ l ↦ᵢ{-} v.
+  Proof.
+    iIntros. iDestruct (mapsto_full_to_frac with "[$]") as "H".
+    iDestruct (mapsto_frac_duplicable with "H") as "[H1 H2]".
+    iFrame.
+  Qed.
+
+  Lemma mapsto_full_to_frac_3 l v : l ↦ᵢ v -∗ (l ↦ᵢ{-} v ∗ l ↦ᵢ{-} v ∗ l ↦ᵢ{-} v).
+  Proof.
+    iIntros.
+    iDestruct (mapsto_full_to_frac_2 with "[$]") as "[H1 H2]".
+    iDestruct (mapsto_frac_duplicable with "H2") as "[H2 H3]".
+    iFrame.
+  Qed.
+
+  Lemma mapsto_agree_frac l q v1 v2 : l ↦ᵢ{q} v1 -∗ l ↦ᵢ{-} v2 -∗ ⌜v1 = v2⌝.
+  Proof.
+    iIntros "P1". iDestruct 1 as (q2) "P2". iApply (mapsto_agree with "P1 P2").
+  Qed.
+
+  Lemma mapsto_agree_frac_frac l v1 v2 : l ↦ᵢ{-} v1 -∗ l ↦ᵢ{-} v2 -∗ ⌜v1 = v2⌝.
+  Proof.
+    iDestruct 1 as (q1) "P1". iDestruct 1 as (q2) "P2".
+    iApply (mapsto_agree with "P1 P2").
+  Qed.
+
   (* Maybe commit this upstream. *)
   Lemma mapsto_exclusive l v1 v2 q : l ↦ᵢ v1 -∗ l ↦ᵢ{q} v2 -∗ False.
     iIntros "Hl1 Hl2". iDestruct (mapsto_valid_2 with "Hl1 Hl2") as %[]%Qp_not_plus_q_ge_1.
   Qed.
 
+  Lemma mapsto_exclusive_frac l v1 v2 : l ↦ᵢ v1 -∗ l ↦ᵢ{-} v2 -∗ False.
+    iIntros "Hl1 Hl2". iDestruct "Hl2" as (q) "Hl2".
+     iDestruct (mapsto_valid_2 with "Hl1 Hl2") as %[]%Qp_not_plus_q_ge_1.
+  Qed.
+
 End common.
\ No newline at end of file

theories/logrel/F_mu_ref_conc/rules.v

@@ -24,6 +24,7 @@ Global Opaque iris_invG.
 Notation "l ↦ᵢ{ q } v" := (mapsto (L:=loc) (V:=val) l q v)
   (at level 20, q at level 50, format "l  ↦ᵢ{ q }  v") : bi_scope.
 Notation "l ↦ᵢ v" := (mapsto (L:=loc) (V:=val) l 1 v) (at level 20) : bi_scope.
+Notation "l ↦ᵢ{-} v" := (∃ q, l ↦ᵢ{q} v)%I (at level 20, format "l  ↦ᵢ{-}  v") : bi_scope.
 
 Section lang_rules.
   Context `{heapIG Σ}.
@@ -74,6 +75,14 @@ Section lang_rules.
     iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
   Qed.
 
+  Lemma wp_load_frac E l v :
+  {{{ ▷ l ↦ᵢ{-} v }}} Load (Loc l) @ E {{{ RET v; l ↦ᵢ{-} v }}}.
+  Proof.
+    iIntros (Φ) ">Hl HΦ". iDestruct "Hl" as (q) "Hl".
+    iApply (wp_load with "Hl"). iNext. iIntros "Hl".
+    iApply "HΦ". by iExists q.
+  Qed.
+
   Lemma wp_store E l v' e v :
     IntoVal e v →
     {{{ ▷ l ↦ᵢ v' }}} Store (Loc l) e @ E
@@ -100,6 +109,16 @@ Section lang_rules.
     iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
   Qed.
 
+  Lemma wp_cas_fail_frac E l v' e1 v1 e2 v2 :
+    IntoVal e1 v1 → IntoVal e2 v2 → v' ≠ v1 →
+    {{{ ▷ l ↦ᵢ{-} v' }}} CAS (Loc l) e1 e2 @ E
+    {{{ RET (BoolV false); l ↦ᵢ{-} v' }}}.
+  Proof.
+    iIntros (<- <- ? Φ) ">Hl HΦ". iDestruct "Hl" as (q) "Hl".
+    iApply (wp_cas_fail with "Hl"); first done.
+    iNext. iIntros "Hl". iApply "HΦ". by iExists q.
+  Qed.
+
   Lemma wp_cas_suc E l e1 v1 e2 v2 :
     IntoVal e1 v1 → IntoVal e2 v2 →
     {{{ ▷ l ↦ᵢ v1 }}} CAS (Loc l) e1 e2 @ E
@@ -121,6 +140,12 @@ Section lang_rules.
     iNext; iIntros (v2 σ2 efs Hstep); inv_head_step. by iFrame.
   Qed.
 
+  Lemma mapsto_frac_duplicable l v :
+    l ↦ᵢ{-} v -∗ l ↦ᵢ{-} v ∗ l ↦ᵢ{-} v.
+  Proof.
+    iIntros "H"; iDestruct "H" as (?) "[Hl Hl']"; iSplitL "Hl"; eauto.
+  Qed.
+
   Local Ltac solve_exec_safe := intros; subst; do 3 eexists; econstructor; eauto.
   Local Ltac solve_exec_puredet := simpl; intros; by inv_head_step.
   Local Ltac solve_pure_exec :=