Weaken the associativity axiom of the Dra class.

Now, associativity needs only to be established in case the elements are
valid and their compositions are defined. This is very much like the notion
of separation algebras I had in my PhD thesis (Def 4.2.1). The Dra to Ra
construction still easily works out.

theories/algebra/dra.v

@@ -12,7 +12,8 @@ Record DraMixin A `{Equiv A, Core A, Disjoint A, Op A, Valid A} := {
   mixin_dra_op_valid x y : ✓ x → ✓ y → x ⊥ y → ✓ (x ⋅ y);
   mixin_dra_core_valid x : ✓ x → ✓ core x;
   (* monoid *)
-  mixin_dra_assoc : Assoc (≡) (⋅);
+  mixin_dra_assoc x y z :
+    ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⋅ (y ⋅ z) ≡ (x ⋅ y) ⋅ z;
   mixin_dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z;
   mixin_dra_disjoint_move_l x y z :
     ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z;
@@ -62,7 +63,8 @@ Section dra_mixin.
   Proof. apply (mixin_dra_op_valid _ (dra_mixin A)). Qed.
   Lemma dra_core_valid x : ✓ x → ✓ core x.
   Proof. apply (mixin_dra_core_valid _ (dra_mixin A)). Qed.
-  Global Instance dra_assoc : Assoc (≡) (@op A _).
+  Lemma dra_assoc x y z :
+    ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⋅ (y ⋅ z) ≡ (x ⋅ y) ⋅ z.
   Proof. apply (mixin_dra_assoc _ (dra_mixin A)). Qed.
   Lemma dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z.
   Proof. apply (mixin_dra_disjoint_ll _ (dra_mixin A)). Qed.
@@ -161,7 +163,7 @@ Proof.
   - intros ?? [??]; naive_solver.
   - intros [x px ?] [y py ?] [z pz ?]; split; simpl;
       [intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl
-      |intros; by rewrite assoc].
+      |intuition eauto using dra_assoc, dra_disjoint_rl].
   - intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
   - intros [x px ?]; split;
       naive_solver eauto using dra_core_l, dra_core_disjoint_l.

theories/algebra/sts.v

@@ -244,7 +244,7 @@ Proof.
       match goal with H : closed _ _ |- _ => destruct H end; set_solver.
   - intros []; naive_solver eauto using closed_up, closed_up_set,
       elem_of_up, elem_of_up_set with sts.
-  - intros [] [] []; constructor; rewrite ?assoc; auto with sts.
+  - intros [] [] [] _ _ _ _ _; constructor; rewrite ?assoc; auto with sts.
   - destruct 4; inversion_clear 1; constructor; auto with sts.
   - destruct 4; inversion_clear 1; constructor; auto with sts.
   - destruct 1; constructor; auto with sts.