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(* 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.
intros.
red in H.
case (H 0); clear H; intros.
inversion_clear H; inversion_clear H0.
apply Trans with (v (Build_finiteT x)); auto with algebra.
apply Trans with (w (Build_finiteT x0)); auto with algebra.
inversion_clear H; inversion_clear H0.
Qed.
Hint Resolve seq_equal_head: algebra.
intros.
destruct n.
red in |- *.
intro.
right.
split; simpl in |- *; auto with arith.
assert (0 = m).
apply seq_equal_length_equal with A v w; auto with algebra.
rewrite <- H0.
simpl in |- *; auto with arith.
destruct m as [| n0].
assert (S n = 0).
apply seq_equal_length_equal with A v w; auto with algebra.
inversion_clear H0.
red in H.
red in |- *.
simpl in |- *.
intro; case (H (S i)); clear H; intros.
inversion_clear H; inversion_clear H0.
left.
exists (lt_S_n _ _ x).
exists (lt_S_n _ _ x0).
apply Trans with (v (Build_finiteT x)); auto with algebra.
apply Trans with (w (Build_finiteT x0)); auto with algebra.
right.
inversion_clear H.
split; auto with arith.
Qed.
Hint Resolve seq_equal_seqtl: algebra.
intros.
assert (n =' m in _).
apply (seq_equal_length_equal (v:=v) (w:=w)); auto with algebra.
red in |- *.
intro.
destruct i as [| n0].
left.
exists (lt_O_Sn n).
exists (lt_O_Sn m).
simpl in |- *.
auto.
case (le_or_lt (S n) (S n0)).
intro.
right.
split; auto.
rewrite <- H1.
auto.
left.
exists H2.
assert (S n0 < S m).
rewrite <- H1.
auto.
exists H3.
simpl in |- *.
apply seq_equal_equal_elements; auto with algebra.
Qed.
Hint Resolve seq_equal_cons: algebra.
intros.
generalize (seq_equal_length_equal H).
generalize (seq_equal_length_equal H0).
intros.
red in |- *.
intro.
generalize (le_or_lt (n + p) i).
intro.
case H3.
right.
split; try tauto.
rewrite <- H1.
rewrite <- H2.
tauto.
intro.
left.
exists H4.
assert (i < m + q).
rewrite <- H1.
rewrite <- H2.
tauto.
exists H5.
generalize (le_or_lt n i).
intro.
case H6; clear H6; intro.
2: assert (i < m).
2: rewrite <- H2.
2: auto.
2: apply Trans with (v (Build_finiteT H6)); auto with algebra.
2: apply Trans with (w (Build_finiteT H7)); auto with algebra.
assert (exists k : Nat, i =' n + k in _).
exists (i - n).
simpl in |- *; symmetry  in |- *.
auto with arith.
inversion_clear H7.
assert (n + x < n + p).
rewrite <- H8.
auto.
assert (m + x < m + q).
rewrite <- H2.
rewrite <- H1.
auto.
apply Trans with ((v ++ a) (Build_finiteT H7)); auto with algebra.
apply Trans with ((w ++ b) (Build_finiteT H9)).
2: apply Ap_comp; auto with algebra.
2: simpl in |- *.
2: symmetry  in |- *.
2: rewrite <- H2.
2: auto.
assert (x < p).
apply plus_lt_reg_l with n; auto with algebra.
apply Trans with (a (Build_finiteT H10)); auto with algebra.
assert (x < q).
rewrite <- H1.
auto.
apply Trans with (b (Build_finiteT H11)); auto with algebra.
Qed.
Hint Resolve seq_equal_concat: algebra.
intros.
assert (n =' m in _).
apply seq_equal_length_equal with A v w; auto with algebra.
simpl in |- *.
red in |- *.
split; intro.
simpl in |- *; simpl in H1.
inversion_clear H1.
destruct x0.
rename index into i.
case (H i).
intro.
inversion_clear H1.
inversion_clear H3.
exists (Build_finiteT x1).
apply Trans with (v (Build_finiteT in_range_prf)); auto with algebra.
intro.
inversion_clear H1.
apply False_ind; auto with algebra.
apply (le_not_lt n i); auto with algebra.
simpl in |- *; simpl in H1.
inversion_clear H1.
destruct x0.
rename index into i.
case (H i).
intro.
inversion_clear H1.
inversion_clear H3.
exists (Build_finiteT x0).
apply Trans with (w (Build_finiteT in_range_prf)); auto with algebra.
intro.
inversion_clear H1.
apply False_ind; auto with algebra.
apply (le_not_lt m i); auto with algebra.
Qed.

Lemma seq_equal_omit :
forall (A : Setoid) (n m : Nat) (v : seq n A) (w : seq m A)
(i : fin n) (j : fin m),
index i = index j -> seq_equal v w -> seq_equal (omit v i) (omit w j).