introduce notation for value comparison

theories/heap_lang/lang.v

@@ -184,6 +184,10 @@ Fixpoint val_for_compare (v : val) : val :=
   | RecV f x e => RecV BAnon BAnon (Val $ LitV $ LitUnit)
   end.
 
+(** Now we can define how to compare values. *)
+Notation "v1 =ᵥ v2" := (val_for_compare v1 = val_for_compare v2) (at level 70) : stdpp_scope.
+Notation "v1 ≠ᵥ v2" := (val_for_compare v1 ≠ val_for_compare v2) (at level 70) : stdpp_scope.
+
 (** The state: heaps of vals. *)
 Record state : Type := {
   heap: gmap loc val;
@@ -518,7 +522,7 @@ Definition bin_op_eval_bool (op : bin_op) (b1 b2 : bool) : option base_lit :=
 
 Definition bin_op_eval (op : bin_op) (v1 v2 : val) : option val :=
   if decide (op = EqOp) then
-    Some $ LitV $ LitBool $ bool_decide (val_for_compare v1 = val_for_compare v2)
+    Some $ LitV $ LitBool $ bool_decide (v1 =ᵥ v2)
   else
     match v1, v2 with
     | LitV (LitInt n1), LitV (LitInt n2) => LitV <$> bin_op_eval_int op n1 n2
@@ -636,13 +640,13 @@ Inductive head_step : expr → state → list observation → expr → state →
   | CasFailS l v1 v2 vl σ :
      vals_cas_compare_safe vl v1 →
      σ.(heap) !! l = Some vl →
-     val_for_compare vl ≠ val_for_compare v1 →
+     vl ≠ᵥ v1 →
      head_step (CAS (Val $ LitV $ LitLoc l) (Val v1) (Val v2)) σ []
                (Val $ LitV $ LitBool false) σ []
   | CasSucS l v1 v2 vl σ :
      vals_cas_compare_safe vl v1 →
      σ.(heap) !! l = Some vl →
-     val_for_compare vl = val_for_compare v1 →
+     vl =ᵥ v1 →
      head_step (CAS (Val $ LitV $ LitLoc l) (Val v1) (Val v2)) σ
                []
                (Val $ LitV $ LitBool true) (state_upd_heap <[l:=v2]> σ)