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hivert merged 1 commit into from
Jul 5, 2026
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Add ordered trees #21

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36 changes: 36 additions & 0 deletions theories/Combi/ordtree.v
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Original file line number Diff line number Diff line change
Expand Up @@ -273,3 +273,39 @@ exists (size f).-1.
have {eqmax} -> : (size f).-1.+1 = (size f) by case: f eqmax {alld1}.
by apply/all_pred1P/allP => /= t {}/alld1; rewrite depth_tree_eq1.
Qed.


Fixpoint line_ordtree n :=
if n is n0.+1 then OrdNode [:: line_ordtree n0] else OrdNode [::].

Lemma size_line_ordtree n : size_ordtree (line_ordtree n) = n.+1.
Proof. by elim: n => // n /= ->; rewrite addn0. Qed.
Lemma depth_line_ordtree n : depth_ordtree (line_ordtree n) = n.+1.
Proof. by elim: n => // n /= ->; rewrite maxn0. Qed.
Lemma depth_le_size t :
depth_ordtree t <= size_ordtree t
?= iff (t == line_ordtree (size_ordtree t).-1).
Proof.
have dlesz t1 : depth_ordtree t1 <= size_ordtree t1.
elim/indtree: t1 {t} => /=; elim=> [// |t f IHf Ht] /=.
have /Ht ledt : t \in t :: f by rewrite inE eqxx.
have {Ht} /IHf : forall t, t \in f -> depth_ordtree t <= size_ordtree t.
by move=> t0 t0in; apply: Ht; rewrite inE t0in orbT.
rewrite !ltnS -(leq_add2l (size_ordtree t)) => /(leq_trans _); apply.
by rewrite geq_max leq_addl andbT (leq_trans ledt) // leq_addr.
split; first exact: dlesz.
apply/eqP/eqP => [|->]; last by rewrite size_line_ordtree depth_line_ordtree.
elim/indtree: t => /= [[// | t f]] IHf [] Heq.
have /= := dlesz (OrdNode f); rewrite ltnS.
rewrite -(leq_add2l (size_ordtree t)) -Heq.
rewrite leq_max => /orP[]; first last.
by rewrite -{2}(add0n (foldr _ _ _)) leq_add2r leqNgt size_ordtree_pos.
move/leq_trans => /(_ _ (dlesz t)).
rewrite -{2}(addn0 (size_ordtree t)) leq_add2l leqn0 => H0.
move: Heq; have {H0} -> : f = [::].
case: f H0 {IHf} => //= t1 f.
by rewrite -leqn0 geq_max leqNgt depth_ordtree_pos.
rewrite /= maxn0 addn0 => /[dup] + ->.
have {}/IHf/[apply] : t \in t :: f by rewrite inE eqxx.
by case: size_ordtree (size_ordtree_pos t) => //= n _ ->.
Qed.
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