
diff -r f6360c0c531b -r 3529946fca19 src/HOL/Data_Structures/Braun_Tree.thy
--- a/src/HOL/Data_Structures/Braun_Tree.thy	Tue Mar 18 21:39:42 2025 +0000
+++ b/src/HOL/Data_Structures/Braun_Tree.thy	Wed Mar 19 22:18:52 2025 +0000
@@ -3,19 +3,19 @@
 section \<open>Braun Trees\<close>
 
 theory Braun_Tree
-imports "HOL-Library.Tree_Real"
+  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)"
+  "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
+  by auto
 
 text \<open>The shape of a Braun-tree is uniquely determined by its size:\<close>
 
@@ -45,14 +45,14 @@
 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"
+  "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
+  by (induction t) auto
 
 lemma finite_braun_indices: "finite(braun_indices t)"
-by (induction t) auto
+  by (induction t) auto
 
 text "One direction:"
 
@@ -87,7 +87,7 @@
 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
+  using double_not_eq_Suc_double by auto
 
 lemma card_braun_indices: "card (braun_indices t) = size t"
 proof (induction t)
@@ -96,7 +96,7 @@
   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)
+        card_insert_if disj_evens_odds card_image inj_on_def braun_indices1)
 qed
 
 lemma braun_indices_intvl_base_1:
@@ -120,19 +120,20 @@
   fixes S :: "nat set"
   assumes "S = {m..n} \<inter> {i. even i}"
   shows "\<exists>m' n'. S = (\<lambda>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
+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 step1: "\<exists>m'. S = Suc ` ({m'..n - 1} \<inter> {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
+  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
@@ -148,186 +149,35 @@
 
 lemma braun_if_braun_indices: "braun_indices t = {1..size t} \<Longrightarrow> braun t"
 proof(induction t)
-case Leaf
+  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} = (\<lambda>i. i * 2) ` braun_indices l \<union> Suc ` (\<lambda>i. i * 2) ` braun_indices r"
+  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")
-    apply clarsimp
-    apply (drule_tac f="\<lambda>S. S \<inter> {2..}" in arg_cong)
-    apply (simp add: t mult.commute Int_Un_distrib2 image_int_eq_image braun_indices1_le)
-    done
+    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])
+        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])
+        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 \<or> 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
+  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
-
-(* 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
-*)
+  using braun_if_braun_indices braun_indices_if_braun by blast
 
 end
\ No newline at end of file




