rattac.
inversion H; clear H; subst.
Local Open Scope nat_scope.
Ltac arithNormalize_step :=
repeat rewrite mult_succ_r in *;
repeat rewrite mult_plus_distr_r in *;
repeat rewrite mult_plus_distr_l in *;
repeat rewrite mult_minus_distr_r in *;
repeat rewrite mult_minus_distr_l in *;
repeat rewrite plus_assoc in *;
repeat rewrite mult_assoc in *.
Ltac arithNormalize := repeat arithNormalize_step.
Ltac arithSimplify :=
match goal with
| [|- _ + ?x = _ + ?x] => f_equal
| [|- _ + (?x1 * ?x2) = _ + (?x2 * ?x1) ] => f_equal
| [|- _ + (?x1 * ?x2 * ?x3) = _ + (?x1 * ?x3 * ?x2) ] => f_equal
| [|- _ + (?x1 * ?x2 * ?x3 * ?x4 * ?x5 * ?x6) = _ + (?x1 * ?x3 * ?x2 * ?x4 * ?x5 * ?x6) ] => f_equal
| [|- _ * ?x1 = _ * ?x1] => f_equal
| [|- _ * ?x = _ ] => rewrite mult_comm; repeat rewrite mult_assoc; arithSimplify
| [|- _ + ?x = _ ] => rewrite plus_comm; repeat rewrite plus_assoc; arithSimplify
| [|- _ * ?x1 <= _ * ?x1] => apply mult_le_compat; auto
| [|- _ * ?x1 <= _ * ?x1] => apply plus_le_compat; auto
| [|- _ * ?x <= _ ] => rewrite mult_comm; repeat rewrite mult_assoc; arithSimplify
| [|- _ + ?x <= _ ] => rewrite plus_comm; repeat rewrite plus_assoc; arithSimplify
end.
arithNormalize.
arithSimplify.
arithSimplify.
do 5 arithSimplify.
do 5 arithSimplify.
Qed.
Local Open Scope rat_scope.

(* Specifications: *)
Theorem ratMult_assoc : forall r1 r2 r3,
r1 * r2 * r3 == r1 * (r2 * r3).