(* Author: Tobias Nipkow *)
section \<open>Braun Trees\<close>
theory Braun_Tree
imports "HOL-Library.Tree_Real"
begin
text \<open>Braun Trees were studied by Braun and Rem~\<^cite>\<open>"BraunRem"\<close>
and later Hoogerwoord~\<^cite>\<open>"Hoogerwoord"\<close>.\<close>
fun braun :: "'a tree \<Rightarrow> bool" where
"braun Leaf = True" |
"braun (Node l x r) = ((size l = size r \<or> size l = size r + 1) \<and> braun l \<and> braun r)"
lemma braun_Node':
"braun (Node l x r) = (size r \<le> size l \<and> size l \<le> size r + 1 \<and> braun l \<and> braun r)"
by auto
text \<open>The shape of a Braun-tree is uniquely determined by its size:\<close>
lemma braun_unique: "\<lbrakk> braun (t1::unit tree); braun t2; size t1 = size t2 \<rbrakk> \<Longrightarrow> t1 = t2"
proof (induction t1 arbitrary: t2)
case Leaf thus ?case by simp
next
case (Node l1 _ r1)
from Node.prems(3) have "t2 \<noteq> Leaf" by auto
then obtain l2 x2 r2 where [simp]: "t2 = Node l2 x2 r2" by (meson neq_Leaf_iff)
with Node.prems have "size l1 = size l2 \<and> size r1 = size r2" by auto
thus ?case using Node.prems(1,2) Node.IH by auto
qed
text \<open>Braun trees are almost complete:\<close>
lemma acomplete_if_braun: "braun t \<Longrightarrow> acomplete t"
proof(induction t)
case Leaf show ?case by (simp add: acomplete_def)
next
case (Node l x r) thus ?case using acomplete_Node_if_wbal2 by force
qed
subsection \<open>Numbering Nodes\<close>
text \<open>We show that a tree is a Braun tree iff a parity-based
numbering (\<open>braun_indices\<close>) of nodes yields an interval of numbers.\<close>
fun braun_indices :: "'a tree \<Rightarrow> nat set" where
"braun_indices Leaf = {}" |
"braun_indices (Node l _ r) = {1} \<union> (*) 2 ` braun_indices l \<union> Suc ` (*) 2 ` braun_indices r"
lemma braun_indices1: "0 \<notin> braun_indices t"
by (induction t) auto
lemma finite_braun_indices: "finite(braun_indices t)"
by (induction t) auto
text "One direction:"
lemma braun_indices_if_braun: "braun t \<Longrightarrow> braun_indices t = {1..size t}"
proof(induction t)
case Leaf thus ?case by simp
next
have *: "(*) 2 ` {a..b} \<union> Suc ` (*) 2 ` {a..b} = {2*a..2*b+1}" (is "?l = ?r") for a b
proof
show "?l \<subseteq> ?r" by auto
next
have "\<exists>x2\<in>{a..b}. x \<in> {Suc (2*x2), 2*x2}" if *: "x \<in> {2*a .. 2*b+1}" for x
proof -
have "x div 2 \<in> {a..b}" using * by auto
moreover have "x \<in> {2 * (x div 2), Suc(2 * (x div 2))}" by auto
ultimately show ?thesis by blast
qed
thus "?r \<subseteq> ?l" by fastforce
qed
case (Node l x r)
hence "size l = size r \<or> size l = size r + 1" (is "?A \<or> ?B") by auto
thus ?case
proof
assume ?A
with Node show ?thesis by (auto simp: *)
next
assume ?B
with Node show ?thesis by (auto simp: * atLeastAtMostSuc_conv)
qed
qed
text "The other direction is more complicated. The following proof is due to Thomas Sewell."
lemma disj_evens_odds: "(*) 2 ` A \<inter> Suc ` (*) 2 ` B = {}"
using double_not_eq_Suc_double by auto
lemma card_braun_indices: "card (braun_indices t) = size t"
proof (induction t)
case Leaf thus ?case by simp
next
case Node
thus ?case
by(auto simp: UNION_singleton_eq_range finite_braun_indices card_Un_disjoint
card_insert_if disj_evens_odds card_image inj_on_def braun_indices1)
qed
lemma braun_indices_intvl_base_1:
assumes bi: "braun_indices t = {m..n}"
shows "{m..n} = {1..size t}"
proof (cases "t = Leaf")
case True then show ?thesis using bi by simp
next
case False
note eqs = eqset_imp_iff[OF bi]
from eqs[of 0] have 0: "0 < m"
by (simp add: braun_indices1)
from eqs[of 1] have 1: "m \<le> 1"
by (cases t; simp add: False)
from 0 1 have eq1: "m = 1" by simp
from card_braun_indices[of t] show ?thesis
by (simp add: bi eq1)
qed
lemma even_of_intvl_intvl:
fixes S :: "nat set"
assumes "S = {m..n} \<inter> {i. even i}"
shows "\<exists>m' n'. S = (\<lambda>i. i * 2) ` {m'..n'}"
proof -
have "S = (\<lambda>i. i * 2) ` {Suc m div 2..n div 2}"
by (fastforce simp add: assms mult.commute)
then show ?thesis
by blast
qed
lemma odd_of_intvl_intvl:
fixes S :: "nat set"
assumes "S = {m..n} \<inter> {i. odd i}"
shows "\<exists>m' n'. S = Suc ` (\<lambda>i. i * 2) ` {m'..n'}"
proof -
have "S = Suc ` ({if n = 0 then 1 else m - 1..n - 1} \<inter> Collect even)"
by (auto simp: assms image_def elim!: oddE)
thus ?thesis
by (metis even_of_intvl_intvl)
qed
lemma image_int_eq_image:
"(\<forall>i \<in> S. f i \<in> T) \<Longrightarrow> (f ` S) \<inter> T = f ` S"
"(\<forall>i \<in> S. f i \<notin> T) \<Longrightarrow> (f ` S) \<inter> T = {}"
by auto
lemma braun_indices1_le:
"i \<in> braun_indices t \<Longrightarrow> Suc 0 \<le> i"
using braun_indices1 not_less_eq_eq by blast
lemma braun_if_braun_indices: "braun_indices t = {1..size t} \<Longrightarrow> braun t"
proof(induction t)
case Leaf
then show ?case by simp
next
case (Node l x r)
obtain t where t: "t = Node l x r" by simp
then have "insert (Suc 0) ((*) 2 ` braun_indices l \<union> Suc ` (*) 2 ` braun_indices r) \<inter> {2..}
= {Suc 0..Suc (size l + size r)} \<inter> {2..}"
by (metis Node.prems One_nat_def Suc_eq_plus1 Un_insert_left braun_indices.simps(2)
sup_bot_left tree.size(4))
then have eq: "{2 .. size t} = (\<lambda>i. i * 2) ` braun_indices l \<union> Suc ` (\<lambda>i. i * 2) ` braun_indices r"
(is "?R = ?S \<union> ?T")
by (simp add: t mult.commute Int_Un_distrib2 image_int_eq_image braun_indices1_le)
then have ST: "?S = ?R \<inter> {i. even i}" "?T = ?R \<inter> {i. odd i}"
by (simp_all add: Int_Un_distrib2 image_int_eq_image)
from ST have l: "braun_indices l = {1 .. size l}"
by (fastforce dest: braun_indices_intvl_base_1 dest!: even_of_intvl_intvl
simp: mult.commute inj_image_eq_iff[OF inj_onI])
from ST have r: "braun_indices r = {1 .. size r}"
by (fastforce dest: braun_indices_intvl_base_1 dest!: odd_of_intvl_intvl
simp: mult.commute inj_image_eq_iff[OF inj_onI])
note STa = ST[THEN eqset_imp_iff, THEN iffD2]
note STb = STa[of "size t"] STa[of "size t - 1"]
then have "size l = size r \<or> size l = size r + 1"
using t l r by atomize auto
with l r show ?case
by (clarsimp simp: Node.IH)
qed
lemma braun_iff_braun_indices: "braun t \<longleftrightarrow> braun_indices t = {1..size t}"
using braun_if_braun_indices braun_indices_if_braun by blast
end