intuition.
assert (Nat.log2 x < S (Nat.log2 x)).
omega.
eapply lt_le_trans.
eapply mult_lt_compat.
eapply H0.
eapply H0.
eapply S_log_square_lt.
trivial.
Qed.

Theorem poly_lt_exp_ge_6 :
forall c x,
x >= (Nat.pow 2 c) ->
x >= (Nat.pow 2 6) ->
Nat.pow x c < Nat.pow 2 x.