Hint Resolve f_pred_precise f_pred_positive.

Lemma key_hists_fail : forall h k v i h' j (Hfail : add_item_trace h k v i false h')
(Hk : k <> 0) (Hrep : repable_signed k),
exists rest, Znth j h' ([], []) = (fst (Znth j h ([], [])) ++ rest, snd (Znth j h ([], []))) /\
Forall (fun e => forall v, ~writes e v) (map snd rest).