Qed.
Hint Resolve binary_op_beq_refl : expr_beq.
Hint Rewrite binary_op_beq_refl : expr_beq.
Hint Resolve binary_op_beq_sound : expr_beq.
Fixpoint expr_beq a b :=
match a, b with
| Econst_int i1 ty1, Econst_int i2 ty2 => andb (beq_int i1 i2) (eqb_type ty1 ty2)
| Econst_float f1 ty1, Econst_float f2 ty2 => andb (beq_float_dec f1 f2) (eqb_type ty1 ty2)
| Econst_single f1 ty1, Econst_single f2 ty2 => andb (beq_float32_dec f1 f2) (eqb_type ty1 ty2)
| Econst_long l1 ty1, Econst_long l2 ty2 => andb (beq_long l1 l2) (eqb_type ty1 ty2)
| Evar id1 ty1, Evar id2 ty2
| Etempvar id1 ty1, Etempvar id2 ty2 => andb (BinPos.Pos.eqb id1 id2) (eqb_type ty1 ty2)
| Ederef e1 ty1, Ederef e2 ty2
| Eaddrof e1 ty1, Eaddrof e2 ty2
| Ecast e1 ty1, Ecast e2 ty2 => (andb (expr_beq e1 e2) (eqb_type ty1 ty2))
| Eunop op1 e1 ty1, Eunop op2 e2 ty2 => (andb (andb (unary_op_beq op1 op2) (expr_beq e1 e2)) (eqb_type ty1 ty2))
| Ebinop op1 e11 e21 ty1, Ebinop op2 e12 e22 ty2 => (andb (andb (andb (binary_op_beq op1 op2) (expr_beq e11 e12)) (eqb_type ty1 ty2)) (expr_beq e21 e22))
| Efield e1 id1 ty1, Efield e2 id2 ty2 => andb (andb (expr_beq e1 e2) (BinPos.Pos.eqb id1 id2)) (eqb_type ty1 ty2)
| _, _ => false
end.
Hint Rewrite Bool.andb_true_iff : expr_beq.
Hint Resolve eqb_type_true : expr_beq.
Hint Resolve BinPos.Peqb_true_eq : expr_beq.
Hint Rewrite BinPos.Pos.eqb_refl : expr_beq.
Hint Resolve BinPos.Pos.eqb_refl : expr_beq.
Hint Resolve eqb_type_refl : expr_beq.
Hint Rewrite eqb_type_refl : expr_beq.
Ltac solve_expr_beq_sound :=
try solve [simpl in *; try congruence]; try reflexivity;
simpl in *; autorewrite with expr_beq in *;
repeat match goal with
| [ H : _ /\ _  |- _] => destruct H
end;
try match goal with
| [ H : List.list_eqb ?r ?a ?b = _ |- _] => consider ( List.list_eqb r a b); intros
end;
try f_equal;
auto with expr_beq.

(* Specifications: *)
Lemma expr_beq_refl : forall a, expr_beq a a = true.
Hint Resolve expr_beq_refl : expr_beq.
Lemma expr_beq_spec : forall a b, expr_beq a b = true <-> a = b.
split; revert b. induction a; intros; match goal with [ |-  _ = ?b] => destruct b end; solve_expr_beq_sound.