Commit 9295617e authored by Felipe Cerqueira's avatar Felipe Cerqueira

Add lemma about concatenation

parent e7d2a65b
......@@ -52,6 +52,29 @@ Section BigCatLemmas.
[by apply/andP; split | by rewrite eq_fun_ord_to_nat].
Qed.
Lemma bigcat_nat_uniq :
forall (T: eqType) n1 n2 (F: nat -> list T),
(forall i, uniq (F i)) ->
(forall x i1 i2,
x \in (F i1) -> x \in (F i2) -> i1 = i2) ->
uniq (\cat_(n1 <= i < n2) (F i)).
Proof.
intros T n1 n2 f SINGLE UNIQ.
case (leqP n1 n2) => [LE | GT]; last by rewrite big_geq // ltnW.
rewrite -[n2](addKn n1).
rewrite -addnBA //; set delta := n2 - n1.
induction delta; first by rewrite addn0 big_geq.
rewrite addnS big_nat_recr /=; last by apply leq_addr.
rewrite cat_uniq; apply/andP; split; first by apply IHdelta.
apply /andP; split; last by apply SINGLE.
rewrite -all_predC; apply/allP; intros x INx.
simpl; apply/negP; unfold not; intro BUG.
apply mem_bigcat_nat_exists in BUG.
move: BUG => [i [IN /andP [_ LTi]]].
apply UNIQ with (i1 := i) in INx; last by done.
by rewrite INx ltnn in LTi.
Qed.
Lemma mem_bigcat_ord_exists :
forall (T: eqType) x n (f: 'I_n -> list T),
x \in \cat_(i < n) (f i) ->
......
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