simple induction n.
intros.
red in |- *.
inversion i0.
inversion in_range_prf.
intros.
case i.
intro x; case x.
intro l.
apply distinct_comp with (Seqtl v); auto with algebra.
intros n1 l.
apply
distinct_comp with (head v;; omit (Seqtl v) (Build_finiteT (lt_S_n _ _ l)));
auto with algebra.
unfold distinct in H, H0.
unfold distinct in |- *.
intros i0 j.
case i0.
case j.
clear x.
intro x; case x.
intros p x'; case x'.
intros p' H'.
simpl in H'.
absurd (0 = 0); auto.
intros n2 p' q; clear q.
red in |- *; red in H0; intro.
generalize
(omit_removes (Seqtl v) (Build_finiteT (lt_S_n _ _ l))
(Build_finiteT (lt_S_n _ _ p'))).
intro k; inversion_clear k.
destruct x0.
apply
H0
with
(Build_finiteT (lt_O_Sn (S n0)))
(Build_finiteT (lt_n_S _ _ in_range_prf)); auto with algebra.
simpl in |- *.
auto.
apply Trans with (head v); auto with algebra.
apply Trans with (Seqtl v (Build_finiteT in_range_prf)); auto with algebra.
apply
Trans
with
((head v;; omit (Seqtl v) (Build_finiteT (lt_S_n n1 (S n0) l)))
(Build_finiteT p')); auto with algebra.
intros n2 l1 x1.
case x1.
intros l2 H2.
clear H2.
cut
(~
omit (Seqtl v) (Build_finiteT (lt_S_n n1 (S n0) l))
(Build_finiteT (lt_S_n _ _ l1)) =' head v in _).
unfold head in |- *.
auto with algebra.
red in |- *.
intro.
generalize
(omit_removes (Seqtl v) (Build_finiteT (lt_S_n n1 (S n0) l))
(Build_finiteT (lt_S_n n2 n0 l1))).
intro.
inversion_clear X.
destruct x0.
cut (Seqtl v (Build_finiteT in_range_prf) =' head v in _).
intros.
cut (v (Build_finiteT (lt_n_S _ _ in_range_prf)) =' head v in _).
unfold head in |- *.
intros.
cut
(~
Build_finiteT (lt_n_S index (S n0) in_range_prf) ='
Build_finiteT (lt_O_Sn (S n0)) in fin _).
intro.
red in H0.
apply
H0
with
(Build_finiteT (lt_n_S index (S n0) in_range_prf))
(Build_finiteT (lt_O_Sn (S n0))); auto.
simpl in |- *.
auto.
auto with algebra.
apply
Trans
with
(omit (Seqtl v) (Build_finiteT (lt_S_n n1 (S n0) l))
(Build_finiteT (lt_S_n n2 n0 l1))); auto with algebra.
intros n3 l0 H2.
simpl in H2.
cut
(~
omit (Seqtl v) (Build_finiteT (lt_S_n n1 (S n0) l))
(Build_finiteT (lt_S_n _ _ l0)) ='
omit (Seqtl v) (Build_finiteT (lt_S_n n1 (S n0) l))
(Build_finiteT (lt_S_n _ _ l1)) in _).
unfold not in |- *; auto with algebra.
apply H; auto with algebra.
intros.
change (~ Seqtl v i1 =' Seqtl v j0 in _) in |- *.
apply Seqtl_preserves_distinct; auto with algebra.
Qed.
Hint Resolve omit_preserves_distinct: algebra.

Lemma distinct_omit_removes_all :
forall (A : Setoid) (n : Nat) (v : seq (S n) A),
distinct v -> forall (i : fin (S n)) (j : fin n), ~ omit v i j =' v i in _.