Minor cleanup of Treiber stack graph proof

theories/examples/stack/proof_treiber_graph.v

@@ -270,7 +270,7 @@ Definition in_sloc E T n (on' : option loc) : vProp :=
 Definition in_slocs E T S : vProp :=
   [∗ list] n ; on' ∈ S ; (next_nodes S), in_sloc E T n on'.
 
-Definition all_slocs T (LVs : list (option loc * view)) : vProp :=
+Definition all_slocs (LVs : list (option loc * view)) : vProp :=
   (* TODO: use read-only atomic points-to [ROPtsTo] here instead of fractions. *)
   [∗ list] onV ∈ LVs,
     if onV is (Some n, V)
@@ -279,9 +279,9 @@ Definition all_slocs T (LVs : list (option loc * view)) : vProp :=
     else emp%I
     .
 
-#[global] Instance all_slocs_objective T LVs : Objective (all_slocs T LVs).
+#[global] Instance all_slocs_objective LVs : Objective (all_slocs LVs).
 Proof. apply big_sepL_objective => _ [[n|] V]; apply _. Qed.
-#[global] Instance all_slocs_timeless T LVs : Timeless (all_slocs T LVs).
+#[global] Instance all_slocs_timeless LVs : Timeless (all_slocs LVs).
 Proof. apply big_sepL_timeless => _ [[n|] V]; apply _. Qed.
 
 (* Locs of the Stack: Head, popped nodes, and nodes *)
@@ -296,7 +296,7 @@ Definition StackLocs_no_exist s (E : event_list) γh T S t0 LVs Vh : vProp :=
   ∗ ⌜ HeadValues T LVs' ∧ HeadUnpopped S ζh
     ∧ PushHistory E LVs' ∧ StackTopView LVs' Vh ⌝
   (* anything that is ever in the stack nodes *)
-  ∗ all_slocs T LVs'
+  ∗ all_slocs LVs'
   (* the actual stack: all resources are at the top's write view *)
   ∗ @{Vh} in_slocs E T S
   .
@@ -619,11 +619,9 @@ Proof.
   iIntros "(H & #Hs & As & Is)". eauto with iFrame.
 Qed.
 
-Lemma all_slocs_node_next_access T LVs n Vn :
+Lemma all_slocs_node_next_access LVs n Vn :
   (Some n, Vn) ∈ LVs →
-  all_slocs T LVs -∗
-    all_slocs T LVs
-    ∗ ∃ q on, @{Vn} (n >> 0) ↦{q} #(oloc_to_lit on).
+  all_slocs LVs -∗ all_slocs LVs ∗ ∃ q on, @{Vn} (n >> 0) ↦{q} #(oloc_to_lit on).
 Proof.
   iIntros ([i Inn]%elem_of_list_lookup) "As".
   rewrite {1}/all_slocs (big_sepL_lookup_acc _ _ _ _ Inn).
@@ -633,17 +631,10 @@ Proof.
   - eauto with iFrame.
 Qed.
 
-Lemma all_slocs_app T LVs LVs' :
-  all_slocs T (LVs ++ LVs') ⊣⊢ all_slocs T LVs ∗ all_slocs T LVs'.
+Lemma all_slocs_app LVs LVs' :
+  all_slocs (LVs ++ LVs') ⊣⊢ all_slocs LVs ∗ all_slocs LVs'.
 Proof. by rewrite /all_slocs -big_sepL_app. Qed.
 
-Lemma all_slocs_mono T T' LVs :
-  T ⊆ T' → all_slocs T LVs ⊢ all_slocs T' LVs.
-Proof.
-  intros SUB. apply big_sepL_mono => i [[n|] V] Eqon; [|done].
-  iDestruct 1 as (q on) "Hn". iExists q, on. iFrame.
-Qed.
-
 Lemma in_slocs_node_access E T S i n :
   S !! i = Some n →
   in_slocs E T S -∗
@@ -1022,7 +1013,7 @@ Proof.
   iDestruct (view_at_elim with "sV1 sH1") as "sH1'".
   iDestruct (AtomicPtsTo_AtomicSeen_latest_1 with "H sH1'") as %Subζ0.
 
-  set Pr : vProp := (@{Vb} all_slocs T2 (LVs2 ++ [(hd_error S2, Vh2)]))%I.
+  set Pr : vProp := (@{Vb} all_slocs (LVs2 ++ [(hd_error S2, Vh2)]))%I.
 
   (* CAS with possible pointer comparison *)
   wp_apply (AtomicSeen_CON_CAS _ _ _ _ _ _ _ (oloc_to_lit on') #n _ ∅ Pr
@@ -1032,7 +1023,7 @@ Proof.
   { iSplit; [done|]. iIntros "!>". destruct on' as [n'|]; [|done].
     simpl. iIntros "As !>".
     (* acquire a fraction of n' *)
-    rewrite /Pr (all_slocs_node_next_access _ _ n' V0); last first.
+    rewrite /Pr (all_slocs_node_next_access _ n' V0); last first.
     { eapply lookup_weaken in Eqth1; [|apply Subζ0]. simpl in Eqth1.
       clear -Eqth1. apply toHeadHist_lookup_Some in Eqth1 as (_ & on' & Eq & Eq1).
       destruct on' as [n''|]; [|done]. inversion Eq. subst n''.
@@ -1049,7 +1040,7 @@ Proof.
       assert (on = Some l') as ->.
       { clear -Eq. destruct on; by inversion Eq. } clear Eq.
       apply elem_of_list_lookup_2 in Eq2.
-      rewrite (all_slocs_node_next_access _ _ _ _ Eq2).
+      rewrite (all_slocs_node_next_access _ _ _ Eq2).
       iDestruct "As" as "[As En]". iDestruct "En" as (q' on) "En".
       rewrite view_at_view_at (shift_0 l') (own_loc_na_own_loc_prim l').
       iDestruct (view_at_subjectively with "En") as (C) "En".
@@ -1134,7 +1125,7 @@ Proof.
   { iIntros "{#}" (InT2). destruct (HV2 n) as [_ In2].
     specialize (In2 InT2). apply elem_of_list_fmap in In2 as ([n' Vn] & Eq & In2).
     simpl in Eq. subst n'.
-    rewrite (all_slocs_node_next_access _ _ _ _ In2). clear.
+    rewrite (all_slocs_node_next_access _ _ _ In2). clear.
     iDestruct "As" as "[_ Hn']". iDestruct "Hn'" as (q' on) "Hn'".
     iDestruct (view_at_intro with "Hn") as (V') "[_ Hn]".
     rewrite !view_at_view_at.
@@ -1265,16 +1256,14 @@ Proof.
 
   iCombine "Hn Hd" as "Hnd".
   set LVs' := (LVs2 ++ [(hd_error S2,Vr)]) ++ [(Some n, Vw)].
-  iAssert (@{Vb ⊔ V2} (all_slocs T' LVs' ∗ @{Vw} in_slocs E' T' S'))%I
+  iAssert (@{Vb ⊔ V2} (all_slocs LVs' ∗ @{Vw} in_slocs E' T' S'))%I
     with "[As Is Hnd SYE]" as "[As Is]".
   { iDestruct (view_at_mono _ Vw with "Hnd") as "Hnd"; [solve_lat|eauto|].
-    iDestruct "Hnd" as "[[Hn1 Hn2] Hd]". iSplitL "As Hn1".
-    - rewrite (all_slocs_app T'). iSplitL "As".
-      { iClear "#". rewrite (all_slocs_mono _ _ _ LeT2). by iFrame. }
+    iDestruct "Hnd" as "[[Hn1 Hn2] Hd]". iClear "#". iSplitL "As Hn1".
+    - rewrite(all_slocs_app (_ ++ _)). iSplitL "As"; [by iFrame "As"|].
       rewrite /all_slocs /=. iSplitL; [|done]. iExists _, (hd_error S2).
       rewrite view_at_view_at /=. iFrame "Hn1".
-    - iClear "#".
-      rewrite 2!view_at_view_at (in_sloc_cons E'). iSplitR "Is".
+    - rewrite 2!view_at_view_at (in_sloc_cons E'). iSplitR "Is".
       + iExists v, psId. rewrite /= /in_sloc.
         iSplit. { iPureIntro. by rewrite lookup_insert. }
         iSplitL "Hn2". { iExists _. by iFrame "Hn2". }
@@ -1720,7 +1709,7 @@ Proof.
   iDestruct (view_at_elim with "sV1 sH1") as "sH1'".
   set Pr : vProp :=
     ((n >> next) ↦{q} #(oloc_to_lit on')
-    ∗ @{Vb} all_slocs T2 (LVs2 ++ [(hd_error S2, Vh2)]))%I.
+    ∗ @{Vb} all_slocs (LVs2 ++ [(hd_error S2, Vh2)]))%I.
   wp_apply (AtomicSeen_CON_CAS_live_loc _ _ _ _ _ _ _ n #(oloc_to_lit on') Pr _ ∅
               with "[$sH1' $H $sV1 HRE $oN $As]"); [done|done|done|solve_ndisj|..].
   { intros t v NE. rewrite lookup_fmap_Some'.
@@ -1737,7 +1726,7 @@ Proof.
       assert (on = Some l') as ->.
       { clear -Eq. destruct on; by inversion Eq. } clear Eq.
       apply elem_of_list_lookup_2 in Eq2.
-      rewrite (all_slocs_node_next_access _ _ _ _ Eq2).
+      rewrite (all_slocs_node_next_access _ _ _ Eq2).
       iDestruct "As" as "[As En]". iDestruct "En" as (q' on) "En".
       rewrite view_at_view_at shift_0 own_loc_na_own_loc_prim.
       iDestruct (view_at_subjectively with "En") as (C) "En".
@@ -1820,7 +1809,7 @@ Proof.
     iDestruct (own_loc_na_agree with "oN Hnn") as %Eq. iPureIntro.
     inversion Eq as [Eq']. by apply oloc_to_lit_inj in Eq'. }
 
-  iAssert (@{Vb} all_slocs T2 ((LVs2 ++ [(Some n, V2)]) ++ [(hd_error S2', Vw)]))%I
+  iAssert (@{Vb} all_slocs ((LVs2 ++ [(Some n, V2)]) ++ [(hd_error S2', Vw)]))%I
     with "[Hnn HSh As]" as "As".
   { rewrite {2}/all_slocs big_sepL_app big_sepL_singleton.
     iSplitL "As"; [by iFrame|].