Some tweaks to `mset`.

- Make use of the fact that https://gitlab.mpi-sws.org/FP/iris-coq/issues/184
  has been fixed.
- Fix indentation.
- Always name hypotheses.
- Make use of `set_solver` to discharge set related stuff.
- Don't import stuff we don't need.

theories/lib/mset.v

 From iris.heap_lang Require Export proofmode notation.
-From iris.heap_lang Require Import spin_lock assert par.
-From iris.algebra Require Import frac.
-From iris.base_logic.lib Require Import fractional.
-
+From iris.heap_lang Require Import assert.
 
 Section Sets.
-
   Fixpoint is_list (hd : val) (xs : list val) : Prop :=
     match xs with
     | [] => hd = NONEV
     | x :: xs => ∃ hd', hd = SOMEV (x,hd') ∧ is_list hd' xs
-  end.
+    end.
 
   Definition is_set (hd : val) (X : gset val) : Prop :=
-    ∃ l : list val, is_list hd l ∧ NoDup l ∧ of_list l ≡ X.
+    ∃ l : list val, is_list hd l ∧ NoDup l ∧ of_list l = X.
 
   Definition newset : val := λ: <>, NONEV.
 
@@ -27,35 +23,28 @@ Section Sets.
           let: "y" := (Snd "p")  in "member" "x" "y"
       end.
 
-  Definition set_add : val :=
-    λ: "x" "xs",
+  Definition set_add : val := λ: "x" "xs",
     if: set_member "x" "xs"
     then assert #false
     else SOME ("x", "xs").
-
 End Sets.
 
 Section MutableSet.
-
   Definition is_set_mut `{heapG Σ} (v : val) (X : gset val) : iProp Σ
     := (∃ (l : loc) hd, ⌜ v = #l ⌝ ∗ l ↦ hd ∗ ⌜ is_set hd X ⌝)%I.
 
   Definition mset_create : val := λ: <>, ref (newset #()).
 
-  Definition mset_member : val :=
-    λ: "x" "xs",
+  Definition mset_member : val := λ: "x" "xs",
     let: "v" := ! "xs" in
     set_member "x" "v".
 
-  Definition mset_add : val :=
-    λ: "x" "xs",
+  Definition mset_add : val := λ: "x" "xs",
     let: "v" := ! "xs" in
     "xs" <- set_add "x" "v".
 
-  Definition mset_clear : val :=
-    λ: "xs",
+  Definition mset_clear : val := λ: "xs",
     "xs" <-  newset #().
-
 End MutableSet.
 
 Section Sets_wp.
@@ -64,9 +53,8 @@ Section Sets_wp.
   Lemma set_newset_spec:
     {{{ True }}} newset #() {{{ v, RET v; ⌜is_set v ∅⌝ }}}.
   Proof.
-    iIntros (Φ) "_ H".  wp_lam.
-    iApply "H".  iExists []. iSimpl. iSplit. done.
-    iSplit. iPureIntro. apply NoDup_nil_2. done.
+    iIntros (Φ) "_ HΦ". wp_lam.
+    iApply "HΦ". iExists []. simpl; eauto using NoDup_nil_2.
   Qed.
 
   Lemma set_member_spec (x: val) (xs: val) (X:  gset val):
@@ -74,52 +62,35 @@ Section Sets_wp.
       set_member x xs
     {{{ RET #(bool_decide (x ∈ X)); True }}}.
   Proof.
-    iIntros (Φ) "% HPost".
-    unfold is_set in a.
-    destruct a as [l [HL [HDup HX]]].
-    iRevert (HL HX). iInduction l as [|hd tl] "IH" forall (xs X);
-    iIntros "% %"; simpl in a; subst; do 2 wp_lam.
-    - wp_case. wp_seq.
-      rewrite of_list_nil  in a0. apply leibniz_equiv in a0; subst.
-      case_bool_decide; first inversion H0. by iApply "HPost".
-    - destruct a as [hd' [HXs HL]].
-      rewrite of_list_cons in a0; simplify_eq /=.
-      wp_case. wp_let. wp_proj.
-      destruct (decide (hd = x)); simplify_eq /=.
-      + assert (BinOp EqOp x x = #true) as ->.  admit.
-        wp_if. assert (x ∈ ({[x]} ∪ (of_list tl : gset val))) by set_solver.
-        case_bool_decide; last by exfalso. by iApply "HPost".
-      + assert (BinOp EqOp hd x = #false) as ->. admit.
-        wp_if. wp_proj. wp_let. apply NoDup_cons_12 in HDup.
-        iApply ("IH" $! _ hd' (of_list tl) with "[HPost]");
-          [| done| done].
+    iIntros (Φ (l & Hl & _ & HX)) "HΦ".
+    iRevert (Hl HX). iInduction l as [|hd tl] "IH" forall (xs X); simpl.
+    - iIntros (-> <-) "/=". do 2 wp_lam. wp_case. wp_seq.
+      rewrite bool_decide_false; last by set_solver. by iApply "HΦ".
+    - iIntros ([hd' [-> Hl]] <-) "/=". do 2 wp_lam. wp_case. wp_let. wp_proj.
+      wp_op. destruct (bool_decide_reflect (hd = x)) as [->|?]; wp_if.
+      + rewrite bool_decide_true; last set_solver. by iApply "HΦ".
+      + wp_proj. wp_let.
+        iApply ("IH" $! hd' (of_list tl) with "[HΦ] [//] [//]").
         iNext; iIntros. case_bool_decide.
-        * case_bool_decide; last by set_solver. by iApply "HPost".
-        * case_bool_decide; first by set_solver. by iApply "HPost".
-  Admitted.
+        * rewrite bool_decide_true; last set_solver. by iApply "HΦ".
+        * rewrite bool_decide_false; last set_solver. by iApply "HΦ".
+  Qed.
 
   Lemma set_add_spec (x: val) (xs: val) (X:  gset val):
     {{{ ⌜is_set xs X⌝ ∧ ⌜x ∉ X⌝ }}}
       set_add x xs
-    {{{ v, RET v; (⌜is_set v ({[x]} ∪ X) ⌝)}}}.
+    {{{ v, RET v; ⌜is_set v ({[x]} ∪ X) ⌝ }}}.
   Proof.
-    iIntros (Φ) "[% %] HPost".
-    do 2 wp_lam.
-    wp_bind ((set_member x) xs).
-    iApply (set_member_spec _ _ X); first done.
-    iNext. iIntros (v). case_bool_decide; first by destruct H1.
-    wp_if. iApply "HPost".
-    destruct H0 as [l [HL [HD HX]]].
-    unfold is_set; simplify_eq.
-    iPureIntro. exists (x :: l).
-    split. simpl. exists xs. done.
-    split. apply NoDup_cons_2. by apply not_elem_of_of_list in H1.
-    done. done.
+    iIntros (Φ [Hset ?]) "HΦ". do 2 wp_lam.
+    wp_bind (set_member x xs).
+    iApply (set_member_spec _ _ X with "[//]").
+    iNext. iIntros "_". case_bool_decide; first by set_solver.
+    wp_if. iApply "HΦ". iPureIntro.
+    destruct Hset as (l & Hl & ? & <-). exists (x :: l).
+    split_and!; [|constructor|]; set_solver.
   Qed.
-
 End Sets_wp.
 
-
 Section MutableSet_wp.
   Context `{heapG Σ}.