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(* as published by the Free Software Foundation; either version 2.1   *)
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(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
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(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(** %\subsection*{ support :  seq\_equality\_facts.v }%*)
Set Implicit Arguments.
Unset Strict Implicit.
Require Export seq_equality.
Require Export concat_facts.
Require Export omit.
Require Export Classical_Prop.
(** - Some useful lemmas concerning seq_equal *)
intros.
red in H.
apply NNPP; intro.
assert (n < m \/ m < n).
apply nat_total_order; auto with algebra.
inversion_clear H1.
generalize (H n); clear H; intro.
inversion_clear H.
inversion_clear H1.
apply (lt_irrefl n); auto with algebra.
inversion_clear H1.
apply (lt_not_le n m); auto with algebra.
generalize (H m); clear H; intro.
inversion_clear H.
inversion_clear H1.
inversion_clear H.
apply (lt_irrefl m); auto with algebra.
inversion_clear H1.
apply (le_not_lt n m); auto with algebra.
Qed.
Hint Resolve seq_equal_length_equal: algebra.
intros.
red in H.
destruct i.
rename index into i.
destruct j.
rename index into j.
simpl in H0.
case (H i); clear H; intros.
inversion_clear H.
inversion_clear H1.
apply Trans with (v (Build_finiteT x)); auto with algebra.
apply Trans with (w (Build_finiteT x0)); auto with algebra.
inversion_clear H.
apply False_ind.
apply (le_not_lt n i); auto with algebra.
Qed.
Hint Resolve seq_equal_equal_elements: algebra.

(** - The interplay with other operations *)
Lemma seq_equal_head :
forall (A : Setoid) (n m : Nat) (v : seq (S n) A) (w : seq (S m) A),
seq_equal v w -> head v =' head w in _.