Quellcode-Bibliothek

^© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Braun_Tree.thy Sprache: Isabelle

Original von: Isabelle^©

(* 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 \ bool" where
"braun Leaf = True" |
"braun (Node l x r) = ((size l = size r \ size l = size r + 1) \ braun l \ braun r)"

lemma braun_Node':
  "braun (Node l x r) = (size r \ size l \ size l \ size r + 1 \ braun l \ braun r)"
by auto

text \<open>The shape of a Braun-tree is uniquely determined by its size:\<close>

lemma braun_unique: "\ braun (t1::unit tree); braun t2; size t1 = size t2 \ \ t1 = t2"
proof (induction t1 arbitrary: t2)
  case Leaf thus ?case by simp
next
  case (Node l1 _ r1)
  from Node.prems(3) have "t2 \ 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 \ 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 \ 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 \ nat set" where
"braun_indices Leaf = {}" |
"braun_indices (Node l _ r) = {1} \ (*) 2 ` braun_indices l \ Suc ` (*) 2 ` braun_indices r"

lemma braun_indices1: "0 \ 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 \ braun_indices t = {1..size t}"
proof(induction t)
  case Leaf thus ?case by simp
next
  have *: "(*) 2 ` {a..b} \ Suc ` (*) 2 ` {a..b} = {2*a..2*b+1}" (is "?l = ?r") for a b
  proof
    show "?l \ ?r" by auto
  next
    have "\x2\{a..b}. x \ {Suc (2*x2), 2*x2}" if *: "x \ {2*a .. 2*b+1}" for x
    proof -
      have "x div 2 \ {a..b}" using * by auto
      moreover have "x \ {2 * (x div 2), Suc(2 * (x div 2))}" by auto
      ultimately show ?thesis by blast
    qed
    thus "?r \ ?l" by fastforce
  qed
  case (Node l x r)
  hence "size l = size r \ size l = size r + 1" (is "?A \ ?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 \ 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 \ 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} \ {i. even i}"
  shows "\m' n'. S = (\i. i * 2) ` {m'..n'}"
  apply (rule exI[where x="Suc m div 2"], rule exI[where x="n div 2"])
  apply (fastforce simp add: assms mult.commute)
  done

lemma odd_of_intvl_intvl:
  fixes S :: "nat set"
  assumes "S = {m..n} \ {i. odd i}"
  shows "\m' n'. S = Suc ` (\i. i * 2) ` {m'..n'}"
proof -
  have step1: "\m'. S = Suc ` ({m'..n - 1} \ {i. even i})"
    apply (rule_tac x="if n = 0 then 1 else m - 1" in exI)
    apply (auto simp: assms image_def elim!: oddE)
    done
  thus ?thesis
    by (metis even_of_intvl_intvl)
qed

lemma image_int_eq_image:
  "(\i \ S. f i \ T) \ (f ` S) \ T = f ` S"
  "(\i \ S. f i \ T) \ (f ` S) \ T = {}"
  by auto

lemma braun_indices1_le:
  "i \ braun_indices t \ Suc 0 \ i"
  using braun_indices1 not_less_eq_eq by blast

lemma braun_if_braun_indices: "braun_indices t = {1..size t} \ 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
  from Node.prems have eq: "{2 .. size t} = (\i. i * 2) ` braun_indices l \ Suc ` (\i. i * 2) ` braun_indices r"
    (is "?R = ?S \ ?T")
    apply clarsimp
    apply (drule_tac f="\S. S \ {2..}" in arg_cong)
    apply (simp add: t mult.commute Int_Un_distrib2 image_int_eq_image braun_indices1_le)
    done
  then have ST: "?S = ?R \ {i. even i}" "?T = ?R \ {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 sizes: "size l = size r \ size l = size r + 1"
    apply (clarsimp simp: t l r inj_image_mem_iff[OF inj_onI])
    apply (cases "even (size l)"; cases "even (size r)"; clarsimp elim!: oddE; fastforce)
    done
  from l r sizes show ?case
    by (clarsimp simp: Node.IH)
qed

lemma braun_iff_braun_indices: "braun t \ braun_indices t = {1..size t}"
using braun_if_braun_indices braun_indices_if_braun by blast

(* An older less appealing proof:
lemma Suc0_notin_double: "Suc 0 \<notin> ( * ) 2 ` A"
by(auto)

lemma zero_in_double_iff: "(0::nat) \<in> ( * ) 2 ` A \<longleftrightarrow> 0 \<in> A"
by(auto)

lemma Suc_in_Suc_image_iff: "Suc n \<in> Suc ` A \<longleftrightarrow> n \<in> A"
by(auto)

lemmas nat_in_image = Suc0_notin_double zero_in_double_iff Suc_in_Suc_image_iff

lemma disj_union_eq_iff:
  "\<lbrakk> L1 \<inter> R2 = {}; L2 \<inter> R1 = {} \<rbrakk> \<Longrightarrow> L1 \<union> R1 = L2 \<union> R2 \<longleftrightarrow> L1 = L2 \<and> R1 = R2"
by blast

lemma inj_braun_indices: "braun_indices t1 = braun_indices t2 \<Longrightarrow> t1 = (t2::unit tree)"
proof(induction t1 arbitrary: t2)
  case Leaf thus ?case using braun_indices.elims by blast
next
  case (Node l1 x1 r1)
  have "t2 \<noteq> Leaf"
  proof
    assume "t2 = Leaf"
    with Node.prems show False by simp
  qed
  thus ?case using Node
    by (auto simp: neq_Leaf_iff insert_ident nat_in_image braun_indices1
                  disj_union_eq_iff disj_evens_odds inj_image_eq_iff inj_def)
qed

text \<open>How many even/odd natural numbers are there between m and n?\<close>

lemma card_Icc_even_nat:
  "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_Icc_odd_nat: "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_Icc_even_nat[of "m+1" "n+1"] by simp
  finally show ?thesis .
qed

lemma compact_Icc_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_eq_if) presburger+
    thus ?thesis by blast
  qed
  thus "A \<subseteq> ?A" using assms
    by(auto simp: image_iff card_Icc_even_nat 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_eq_if)
  have 2: "even (2 * (j - 1) + m + m mod 2)" for j by presburger
  show "?A \<subseteq> A"
    apply(simp add: assms card_Icc_even_nat del: atLeastAtMost_iff One_nat_def)
    using 1 2 by blast
qed

lemma compact_Icc_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_Icc_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_Icc_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_Icc_odd)
  have "?a = ?b \<or> ?a = ?b+1 \<and> even m \<or> ?a+1 = ?b \<and> odd m"
    apply(simp add: Let_def mod2_eq_if
      card_Icc_even_nat[of m n, simplified A[symmetric]]
      card_Icc_odd_nat[of m n, simplified B[symmetric]] split!: if_splits)
    by linarith
  with ab A' B' show ?thesis by simp
qed

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: "( * ) 2 ` braun_indices t1 \<union> Suc ` ( * ) 2 ` braun_indices t2 =
     {2..size t1 + size t2 + 1}" using Node.prems
    by (simp add: insert_ident Icc_eq_insert_lb_nat nat_in_image braun_indices1)
  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 image_comp
           cong: image_cong_simp)
qed
*)

end

¤ Dauer der Verarbeitung: 0.4 Sekunden (vorverarbeitet) ¤

Download des Quellennavigators
Download des sprechenden Kalenders
in der Quellcodebibliothek suchen

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.