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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  distinct\_facts.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export omit_facts.
Require Export seq_set_seq.
Require Export distinct.
intros.
red in |- *.
intro; case i; clear i; intro i; case i; clear i.
intros hi j; case j; clear j; intro j; case j; clear j.
intros.
simpl in H1.
absurd (0 = 0); auto.
intros.
simpl in |- *.
apply H0; auto with algebra.
intros i hi j; case j; clear j; intro j; case j; clear j.
intros.
simpl in |- *.
red in |- *; intro; red in H0.
apply (H0 (Build_finiteT (lt_S_n _ _ hi))); auto with algebra.
intros j hj hij.
red in H.
simpl in |- *.
apply H; auto with algebra.
Qed.
Hint Resolve distinct_cons: algebra.
unfold distinct in |- *.
simple induction n.
simpl in |- *.
auto with algebra.
intros.
generalize H1; clear H1.
case i; case j.
intro x; case x.
intros l x0; case x0.
intros l0 H1.
simpl in H1.
absurd (0 = 0); auto.
intros n1 l0.
intro q; clear q.
red in |- *; red in H0.
intro.
cut
(v (Build_finiteT (lt_n_S _ _ l0)) =' v (Build_finiteT (lt_n_S _ _ l)) in _).
intro.
apply
H0
with
(Build_finiteT (lt_n_S (S n1) (pred (S n0)) l0))
(Build_finiteT (lt_n_S 0 (pred (S n0)) l)); auto with algebra.
simpl in |- *.
auto.
apply Trans with (Seqtl v (Build_finiteT l0)); auto with algebra.
intros n1 l x0; case x0.
intros l0 H1.
clear H1.
red in |- *; intro.
cut
(v (Build_finiteT (lt_n_S _ _ l0)) =' v (Build_finiteT (lt_n_S _ _ l)) in _).
intro.
red in H0.
apply H0 with (Build_finiteT (lt_n_S _ _ l0)) (Build_finiteT (lt_n_S _ _ l));
auto with algebra.
simpl in |- *.
auto.
apply Trans with (Seqtl v (Build_finiteT l0)); auto with algebra.
intros n2 l0.
simpl in |- *.
intros.
apply H0; auto with algebra.
Qed.
Hint Resolve Seqtl_preserves_distinct: algebra.

Lemma omit_preserves_distinct :
forall (A : Setoid) (n : Nat) (v : seq (S n) A),
distinct v -> forall i : fin (S n), distinct (omit v i).