repeat rewrite sub_square_identity; auto with arith.
repeat rewrite square_recompose; rewrite mult_minus_distr_l.
apply minus_eq_decompose; try rewrite hyp_sqrt; ring.
Qed.
End sqrt2_decrease.
Hint Resolve lt_le_weak comparison2: sqrt.

Theorem sqrt2_not_rational :
forall p q : nat, q <> 0 -> p * p = 2 * (q * q) -> False.