Admitted.

Lemma update_partially_expanded_key: forall k j j' v key_chars,
j = j'+k-8 ->
upd_Znth (j' + k) (map Vint (pow_fun GrowKeyByOne (Z.to_nat j) (key_bytes_to_key_words key_chars))
++ repeat_op_table (60 - j) Vundef id) v
= (map Vint (pow_fun GrowKeyByOne (Z.to_nat (j + 1)) (key_bytes_to_key_words key_chars))
++ repeat_op_table (60 - (j + 1)) Vundef id).