Definition shorter_ss_l (ss1 ss2 : sseq) (o i: nat) : Prop :=
forall (j : nat), j <= i-o -> ss1.[i-j] = ss2.[i-j].
Definition shorter_ss_u (ss1 ss2 : sseq) (i d: nat) : Prop :=
forall (j : nat), ss1.[i+j+d] = ss2.[i+j].
Definition shorter_ss (ss1 ss2 : sseq) (o i d: nat) : Prop :=
shorter_ss_l ss1 ss2 o i /\ shorter_ss_u ss1 ss2 i d.

Lemma shorter_ss_prop1 :
forall (T : trans) (len i d : nat) (ss1 ss2 : sseq),
path T ss1 0 len -> ss1.[i] = ss1.[i+d] ->
shorter_ss ss1 ss2 0 i d ->
path T ss2 0 (len-d).