(* 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{BraunRem}
and later Hoogerwoord~\cite{Hoogerwoord}.\<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 balanced:\<close>
lemma balanced_if_braun: "braun t \<Longrightarrow> balanced t"
proof(induction t)
case Leaf show ?case by (simp add: balanced_def)
next
case (Node l x r)
have "size l = size r \<or> size l = size r + 1" (is "?A \<or> ?B")
using Node.prems by simp
thus ?case
proof
assume "?A"
thus ?thesis using Node
apply(simp add: balanced_def min_def max_def)
by (metis Node.IH balanced_optimal le_antisym le_refl)
next
assume "?B"
thus ?thesis using Node by(intro balanced_Node_if_wbal1) auto
qed
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>
abbreviation double :: "nat \<Rightarrow> nat" where
"double \<equiv> (*) 2"
abbreviation double1 :: "nat \<Rightarrow> nat" where
"double1 \<equiv> \<lambda>n. Suc(2*n)"
fun braun_indices :: "'a tree \<Rightarrow> nat set" where
"braun_indices Leaf = {}" |
"braun_indices (Node l _ r) = {1} \<union> double ` braun_indices l \<union> double1 ` braun_indices r"
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 *: "double ` {a..b} \<union> double1 ` {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 \<open>How many even/odd natural numbers are there between m and n?\<close>
lemma card_atLeastAtMost_even:
"card {i \<in> {m..n::nat}. even i} = (n+1-m + (m+1) mod 2) div 2" (is "?l m n = ?r m n")
proof(induction "n+1 - m" arbitrary: n m)
case 0 thus ?case by simp
next
case Suc
have "m \<le> n" using Suc(2) by arith
hence "{m..n} = insert m {m+1..n}" by auto
hence "?l m n = card {i \<in> insert m {m+1..n}. even i}" by simp
also have "\<dots> = ?r m n" (is "?l = ?r")
proof (cases)
assume "even m"
hence "{i \<in> insert m {m+1..n}. even i} = insert m {i \<in> {m+1..n}. even i}" by auto
hence "?l = card {i \<in> {m+1..n}. even i} + 1" by simp
also have "\<dots> = (n-m + (m+2) mod 2) div 2 + 1" using Suc(1)[of n "m+1"] Suc(2) by simp
also have "\<dots> = ?r" using \<open>even m\<close> \<open>m \<le> n\<close> by auto
finally show ?thesis .
next
assume "odd m"
hence "{i \<in> insert m {m+1..n}. even i} = {i \<in> {m+1..n}. even i}" by auto
hence "?l = card ..." by simp
also have "\<dots> = (n-m + (m+2) mod 2) div 2" using Suc(1)[of n "m+1"] Suc(2) by simp
also have "\<dots> = ?r" using \<open>odd m\<close> \<open>m \<le> n\<close> even_iff_mod_2_eq_zero[of m] by simp
finally show ?thesis .
qed
finally show ?case .
qed
lemma card_atLeastAtMost_odd: "card {i \<in> {m..n::nat}. odd i} = (n+1-m + m mod 2) div 2"
proof -
let ?A = "{i \<in> {m..n}. odd i}"
let ?B = "{i \<in> {m+1..n+1}. even i}"
have "card ?A = card (Suc ` ?A)" by (simp add: card_image)
also have "Suc ` ?A = ?B" using Suc_le_D by(force simp: image_iff)
also have "card ?B = (n+1-m + (m) mod 2) div 2"
using card_atLeastAtMost_even[of "m+1" "n+1"] by simp
finally show ?thesis .
qed
lemma mod2_iff: "x mod 2 = (if even x then 0 else 1)"
by (simp add: odd_iff_mod_2_eq_one)
lemma compact_ivl_even: assumes "A = {i \<in> {m..n}. even i}"
shows "A = (\<lambda>j. 2*(j-1) + m + m mod 2) ` {1..card A}" (is "_ = ?A")
proof
let ?a = "(n+1-m + (m+1) mod 2) div 2"
have "\<exists>j \<in> {1..?a}. i = 2*(j-1) + m + m mod 2" if *: "i \<in> {m..n}" "even i" for i
proof -
let ?j = "(i - (m + m mod 2)) div 2 + 1"
have "?j \<in> {1..?a} \<and> i = 2*(?j-1) + m + m mod 2" using * by(auto simp: mod2_iff) presburger+
thus ?thesis by blast
qed
thus "A \<subseteq> ?A" using assms
by(auto simp: image_iff card_atLeastAtMost_even simp del: atLeastAtMost_iff)
next
let ?a = "(n+1-m + (m+1) mod 2) div 2"
have 1: "2 * (j - 1) + m + m mod 2 \<in> {m..n}" if *: "j \<in> {1..?a}" for j
using * by(auto simp: mod2_iff)
have 2: "even (2 * (j - 1) + m + m mod 2)" for j by presburger
show "?A \<subseteq> A"
apply(simp add: assms card_atLeastAtMost_even del: atLeastAtMost_iff One_nat_def)
using 1 2 by blast
qed
lemma compact_ivl_odd:
assumes "B = {i \<in> {m..n}. odd i}" shows "B = (\<lambda>i. 2*(i-1) + m + (m+1) mod 2) ` {1..card B}"
proof -
define A :: " nat set" where "A = Suc ` B"
have "A = {i \<in> {m+1..n+1}. even i}"
using Suc_le_D by(force simp add: A_def assms image_iff)
from compact_ivl_even[OF this]
have "A = Suc ` (\<lambda>i. 2 * (i - 1) + m + (m + 1) mod 2) ` {1..card A}"
by (simp add: image_comp o_def)
hence B: "B = (\<lambda>i. 2 * (i - 1) + m + (m + 1) mod 2) ` {1..card A}"
using A_def by (simp add: inj_image_eq_iff)
have "card A = card B" by (metis A_def bij_betw_Suc bij_betw_same_card)
with B show ?thesis by simp
qed
lemma even_odd_decomp: assumes "\<forall>x \<in> A. even x" "\<forall>x \<in> B. odd x" "A \<union> B = {m..n}"
shows "(let a = card A; b = card B in
a + b = n+1-m \<and>
A = (\<lambda>i. 2*(i-1) + m + m mod 2) ` {1..a} \<and>
B = (\<lambda>i. 2*(i-1) + m + (m+1) mod 2) ` {1..b} \<and>
(a = b \<or> a = b+1 \<and> even m \<or> a+1 = b \<and> odd m))"
proof -
let ?a = "card A" let ?b = "card B"
have "finite A \<and> finite B"
by (metis \<open>A \<union> B = {m..n}\<close> finite_Un finite_atLeastAtMost)
hence ab: "?a + ?b = Suc n - m"
by (metis Int_emptyI assms card_Un_disjoint card_atLeastAtMost)
have A: "A = {i \<in> {m..n}. even i}" using assms by auto
hence A': "A = (\<lambda>i. 2*(i-1) + m + m mod 2) ` {1..?a}" by(rule compact_ivl_even)
have B: "B = {i \<in> {m..n}. odd i}" using assms by auto
hence B': "B = (\<lambda>i. 2*(i-1) + m + (m+1) mod 2) ` {1..?b}" by(rule compact_ivl_odd)
have "?a = ?b \<or> ?a = ?b+1 \<and> even m \<or> ?a+1 = ?b \<and> odd m"
apply(simp add: Let_def mod2_iff
card_atLeastAtMost_even[of m n, simplified A[symmetric]]
card_atLeastAtMost_odd[of m n, simplified B[symmetric]] split!: if_splits)
by linarith
with ab A' B' show ?thesis by simp
qed
lemma braun_indices1: "i \<in> braun_indices t \<Longrightarrow> i \<ge> 1"
by (induction t arbitrary: i) auto
lemma finite_braun_indices: "finite(braun_indices t)"
by (induction t) auto
lemma evens_odds_disj: "double ` braun_indices A \<inter> double1 ` 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 evens_odds_disj card_image inj_on_def dest: braun_indices1)
qed
lemma eq: "insert (Suc 0) M = {Suc 0..n} \<Longrightarrow> Suc 0 \<notin> M \<Longrightarrow> M = {2..n}"
by (metis Suc_n_not_le_n atLeastAtMost_iff atLeastAtMost_insertL insertI1 insert_ident numeral_2_eq_2)
lemma inj_on_Suc: "inj_on f N \<Longrightarrow> inj_on (\<lambda>n. Suc(f n)) N"
by (simp add: inj_on_def)
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 t1 x2 t2)
have 1: "i > 0 \<Longrightarrow> Suc(Suc(2 * (i - Suc 0))) = 2*i" for i::nat by(simp add: algebra_simps)
have 2: "i > 0 \<Longrightarrow> 2 * (i - Suc 0) + 3 = 2*i + 1" for i::nat by(simp add: algebra_simps)
have 3: "double ` braun_indices t1 \<union> double1 ` braun_indices t2 =
{2..size t1 + size t2 + 1}" using Node.prems braun_indices1[of 0 t2]
apply simp
apply(drule eq)
apply auto
done
thus ?case using Node.IH even_odd_decomp[OF _ _ 3]
by(simp add: card_image inj_on_def card_braun_indices Let_def 1 2 inj_image_eq_iff
cong: image_cong_strong)
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