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(* GNU General Public License for more details.                       *)
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(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  omit\_facts.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
From Algebra Require Field_facts.
Export Sets Categories Parts Sgroup_cat Monoid_cat Group_cat Abelian_group_cat Ring_cat.
Export Field_facts.
From Algebra Require Ring_facts.
Export Ring_facts.
From Algebra Require Export Field_cat.
Require Export seq_equality_facts.
Require Export sums.
Require Export mult_by_scalars.
destruct n.
simpl in |- *.
red in |- *; auto with algebra.
destruct n.
intros; (apply Trans with (empty_seq M); auto with algebra).
auto with algebra.
Qed.
Hint Resolve omit_head_is_seqtl: algebra.
simple induction n.
intros A v m.
case m.
intros.
apply Trans with (head v;; empty_seq A); auto with algebra.
intros.
inversion H.
inversion H1.
intros.
apply Trans with (head v;; omit (Seqtl v) (Build_finiteT (lt_S_n _ _ HS)));
auto with algebra.
change
(head v;; omit (Seqtl v) (Build_finiteT (lt_S_n m (S (S n0)) HS)) ='
head v;; omit (Seqtl v) (Build_finiteT H0) in seq (S (S n0)) A)
in |- *.
apply cons_comp; auto with algebra.
Qed.
Hint Resolve omit_tlelt: algebra.
(** - The above also holds for seqs of length (S n), but since the types become non-convertible then, we have to do a cast_seq or use seq_equal. *)
destruct n.
intros; inversion_clear Heq.
intros.
apply Trans with (head v;; omit (Seqtl v) (Build_finiteT H));
auto with algebra.
Qed.
Hint Resolve omit_tlelt': algebra.

Lemma omit_tlelt'' :
forall (n : Nat) (A : Setoid) (v : seq (S n) A) (m : Nat)
(HS : S m < S n) (H : m < n) (Heq : S (pred n) = n),
seq_equal (omit v (Build_finiteT HS))
(head v;; omit (Seqtl v) (Build_finiteT H)).