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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.
intros.
apply
seq_equal_trans
with
(pred (S n))
(cast_seq (head v;; omit (Seqtl v) (Build_finiteT H)) Heq);
auto with algebra.
apply Map_eq_seq_equal; auto with algebra.
Qed.
Hint Resolve omit_tlelt'': algebra.
induction n.
intros.
destruct i as [| n].
2: inversion_clear Hi.
2: inversion_clear H.
apply Trans with (Seqtl (v ++ w)); auto with algebra.
apply Trans with (Seqtl v ++ w); auto with algebra.
intros.
destruct i as [| n0].
apply Trans with (Seqtl (v ++ w)); auto with algebra.
apply Trans with (Seqtl v ++ w); auto with algebra.
apply
Trans
with
(head (v ++ w);;
omit (Seqtl (v ++ w)) (Build_finiteT (lt_S_n _ (S (n + m)) Hi')));
auto with algebra.
apply
Trans with ((head v;; omit (Seqtl v) (Build_finiteT (lt_S_n _ _ Hi))) ++ w);
auto with algebra.
apply
Trans
with
(head v;;
omit (Seqtl (v ++ w)) (Build_finiteT (lt_S_n _ (S (n + m)) Hi')));
auto with algebra.
apply
Trans
with (head v;; omit (Seqtl v) (Build_finiteT (lt_S_n n0 (S n) Hi)) ++ w).
2: change
(head v;; omit (Seqtl v) (Build_finiteT (lt_S_n n0 (S n) Hi)) ++ w ='
(head v;; omit (Seqtl v) (Build_finiteT (lt_S_n n0 (S n) Hi))) ++ w
in seq (S (n + m)) A) in |- *.
2: apply cons_concat_special.
change
(head v;; omit (Seqtl (v ++ w)) (Build_finiteT (lt_S_n n0 (S (n + m)) Hi')) ='
head v;; omit (Seqtl v) (Build_finiteT (lt_S_n n0 (S n) Hi)) ++ w
in seq (S (n + m)) A) in |- *.
apply cons_comp; auto with algebra.
apply
Trans with (omit (Seqtl v ++ w) (Build_finiteT (lt_S_n n0 (S (n + m)) Hi'))).
change
(omit (Seqtl (v ++ w)) (Build_finiteT (lt_S_n n0 (S (n + m)) Hi')) ='
omit (Seqtl v ++ w) (Build_finiteT (lt_S_n n0 (S (n + m)) Hi'))
in seq (pred (S (n + m))) A) in |- *.
apply omit_comp; auto with algebra.
change
(omit (Seqtl v ++ w) (Build_finiteT (lt_S_n n0 (S (n + m)) Hi')) ='
omit (Seqtl v) (Build_finiteT (lt_S_n n0 (S n) Hi)) ++ w
in seq (pred (S n + m)) A) in |- *.
apply (IHn m A (Seqtl v) w).
Qed.
Hint Resolve omit_concat_first_part: algebra.
(** - The following also comes in two guises. It is again an equation between two sequences of equal but non-convertible length. The first formulation uses the external (not inherited from any setoid) equality seq_equal, the second does a cast_seq in the appropriate places *)
intros.
induction n.
intro j.
generalize (le_or_lt (pred (0 + m)) j).
intro p; inversion_clear p.
right; split; auto.
left.
exists H0.
exists (lt_comp H0 (sym_eq H)).
simpl in Hi', H, H0.
simpl in |- *.
apply Ap_comp; auto with algebra.
intro j.
generalize (le_or_lt (pred (S n + m)) j).
intro p; inversion_clear p.
right; split; auto.
rewrite H.
auto.
left.
exists H0.
exists (lt_comp H0 (sym_eq H)).
apply Trans with (omit (hdtl v ++ w) (Build_finiteT Hi') (Build_finiteT H0));
auto with algebra.
apply
Trans
with
((hdtl v ++ omit w (Build_finiteT Hi))
(Build_finiteT (lt_comp H0 (sym_eq H)))); auto with algebra.
unfold hdtl in |- *.
apply
Trans
with (omit (head v;; Seqtl v ++ w) (Build_finiteT Hi') (Build_finiteT H0));
auto with algebra.
apply
Trans
with
((head v;; Seqtl v ++ omit w (Build_finiteT Hi))
(Build_finiteT (lt_comp H0 (sym_eq H)))).
2: apply Ap_comp; auto with algebra.
2: apply toSeq; auto with algebra.
2: apply (cons_concat_special (head v) (Seqtl v) (omit w (Build_finiteT Hi))).
assert (n + m = S (pred (n + m))).
clear H0 j IHn w v A.
transitivity (S n + pred m); auto with arith.
transitivity (pred (S n + m)); auto with arith.
simpl in |- *.
apply S_pred with (n + i).
auto with arith.
apply
Trans
with
(cast_seq (head v;; omit (Seqtl v ++ w) (Build_finiteT (lt_S_n _ _ Hi')))
(sym_eq H1) (Build_finiteT H0)); fold (n + i) (n + m) in |- *.
apply Ap_comp; auto with algebra.
apply toSeq.
apply
Trans
with
(cast_seq
(head (head v;; Seqtl v ++ w);;
omit (Seqtl (head v;; Seqtl v ++ w))
(Build_finiteT (lt_S_n (n + i) (n + m) Hi')))
(sym_eq H1)).
apply (omit_tlelt' (head v;; Seqtl v ++ w) Hi' (lt_S_n _ _ Hi') (sym_eq H1));
auto with algebra.
apply cast_seq_comp; auto with algebra.
apply cons_comp; auto with algebra.
apply omit_comp; auto with algebra.
apply (Seqtl_cons_inv (head v) (Seqtl v ++ w)).
apply seq_equal_equal_elements; auto with algebra.
apply
seq_equal_trans
with
(w := head v;;
omit (Seqtl v ++ w) (Build_finiteT (lt_S_n (n + i) (n + m) Hi'))).
apply seq_equal_symm.
apply cast_seq_preserves_seq_equal.
apply seq_equal_cons; auto with algebra.
generalize (IHn (Seqtl v) (lt_S_n (n + i) (n + m) Hi')).
clear IHn; intro.
apply H2.
clear H2.
simpl in H.
transitivity (pred (S (n + pred m))); auto with arith.
Qed.
intros.
apply seq_equal_map_equal.
apply seq_equal_trans with (w := v ++ omit w (Build_finiteT Hi)).
apply omit_concat_second_part'; auto with algebra.
apply cast_seq_preserves_seq_equal.
Qed.
intros.
simpl in |- *.
elim (omit_removes v i j).
intros.
exists x; auto with algebra.
Qed.
destruct n.
intros.
apply False_ind; auto with algebra.
induction  n as [| n Hrecn].
simpl in |- *.
unfold head in |- *.
intros.
elim i.
intros.
apply SGROUP_comp; auto with algebra.
intros M.
generalize (Hrecn M); clear Hrecn; intros.
case i.
intro x; case x.
intro.
apply Trans with (head s +' sum (Seqtl s)); auto with algebra.
intros n0 l.
apply
Trans
with
(Seqtl s (Build_finiteT (lt_S_n _ _ l)) +' sum (omit s (Build_finiteT l))).
2: apply SGROUP_comp; auto with algebra.
apply
Trans
with
(Seqtl s (Build_finiteT (lt_S_n _ _ l)) +'
sum (head s;; omit (Seqtl s) (Build_finiteT (lt_S_n _ _ l))));
auto with algebra.
apply
Trans
with
(Seqtl s (Build_finiteT (lt_S_n _ _ l)) +'
(head s +' sum (omit (Seqtl s) (Build_finiteT (lt_S_n _ _ l)))));
auto with algebra.
apply
Trans
with
(head s +'
(Seqtl s (Build_finiteT (lt_S_n _ _ l)) +'
sum (omit (Seqtl s) (Build_finiteT (lt_S_n _ _ l)))));
auto with algebra.
apply Trans with (head s +' sum (Seqtl s)); auto with algebra.
Qed.

Lemma seqsum_min_elt_is_omitseq :
forall (n : Nat) (AG : abelian_group) (s : seq n AG) (i : fin n),
sum s +' (min s i) =' sum (omit s i) in _.