src/HOL/Data_Structures/Brother12_Set.thy
author nipkow
Sat Dec 05 16:13:28 2015 +0100 (2015-12-05)
changeset 61789 9ce1a397410a
parent 61784 21b34a2269e5
child 61792 8dd150a50acc
permissions -rw-r--r--
added Brother12_Map
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(* Author: Tobias Nipkow *)
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section \<open>A 1-2 Brother Tree Implementation of Sets\<close>
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theory Brother12_Set
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imports
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  Cmp
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  Set_by_Ordered
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begin
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subsection \<open>Data Type and Operations\<close>
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datatype 'a bro =
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  N0 |
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  N1 "'a bro" |
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  N2 "'a bro" 'a "'a bro" |
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  (* auxiliary constructors: *)
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  L2 'a |
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  N3 "'a bro" 'a "'a bro" 'a "'a bro"
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fun inorder :: "'a bro \<Rightarrow> 'a list" where
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"inorder N0 = []" |
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"inorder (N1 t) = inorder t" |
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"inorder (N2 l a r) = inorder l @ a # inorder r" |
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"inorder (L2 a) = [a]" |
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"inorder (N3 t1 a1 t2 a2 t3) = inorder t1 @ a1 # inorder t2 @ a2 # inorder t3"
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fun isin :: "'a bro \<Rightarrow> 'a::cmp \<Rightarrow> bool" where
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"isin N0 x = False" |
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"isin (N1 t) x = isin t x" |
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"isin (N2 l a r) x =
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  (case cmp x a of
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     LT \<Rightarrow> isin l x |
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     EQ \<Rightarrow> True |
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     GT \<Rightarrow> isin r x)"
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fun n1 :: "'a bro \<Rightarrow> 'a bro" where
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"n1 (L2 a) = N2 N0 a N0" |
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"n1 (N3 t1 a1 t2 a2 t3) = N2 (N2 t1 a1 t2) a2 (N1 t3)" |
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"n1 t = N1 t"
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hide_const (open) insert
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locale insert
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begin
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fun n2 :: "'a bro \<Rightarrow> 'a \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
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"n2 (L2 a1) a2 t = N3 N0 a1 N0 a2 t" |
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"n2 (N3 t1 a1 t2 a2 t3) a3 (N1 t4) = N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4)" |
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"n2 (N3 t1 a1 t2 a2 t3) a3 t4 = N3 (N2 t1 a1 t2) a2 (N1 t3) a3 t4" |
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"n2 t1 a1 (L2 a2) = N3 t1 a1 N0 a2 N0" |
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"n2 (N1 t1) a1 (N3 t2 a2 t3 a3 t4) = N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4)" |
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"n2 t1 a1 (N3 t2 a2 t3 a3 t4) = N3 t1 a1 (N1 t2) a2 (N2 t3 a3 t4)" |
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"n2 t1 a t2 = N2 t1 a t2"
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fun ins :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
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"ins x N0 = L2 x" |
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"ins x (N1 t) = n1 (ins x t)" |
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"ins x (N2 l a r) =
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  (case cmp x a of
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     LT \<Rightarrow> n2 (ins x l) a r |
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     EQ \<Rightarrow> N2 l a r |
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     GT \<Rightarrow> n2 l a (ins x r))"
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fun tree :: "'a bro \<Rightarrow> 'a bro" where
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"tree (L2 a) = N2 N0 a N0" |
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"tree (N3 t1 a1 t2 a2 t3) = N2 (N2 t1 a1 t2) a2 (N1 t3)" |
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"tree t = t"
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definition insert :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
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"insert x t = tree(ins x t)"
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end
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locale delete
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begin
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fun n2 :: "'a bro \<Rightarrow> 'a \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
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"n2 (N1 t1) a1 (N1 t2) = N1 (N2 t1 a1 t2)" |
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"n2 (N1 (N1 t1)) a1 (N2 (N1 t2) a2 (N2 t3 a3 t4)) =
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  N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
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"n2 (N1 (N1 t1)) a1 (N2 (N2 t2 a2 t3) a3 (N1 t4)) =
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  N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
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"n2 (N1 (N1 t1)) a1 (N2 (N2 t2 a2 t3) a3 (N2 t4 a4 t5)) =
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  N2 (N2 (N1 t1) a1 (N2 t2 a2 t3)) a3 (N1 (N2 t4 a4 t5))" |
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"n2 (N2 (N1 t1) a1 (N2 t2 a2 t3)) a3 (N1 (N1 t4)) =
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  N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
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"n2 (N2 (N2 t1 a1 t2) a2 (N1 t3)) a3 (N1 (N1 t4)) =
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  N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
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"n2 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4)) a5 (N1 (N1 t5)) =
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  N2 (N1 (N2 t1 a1 t2)) a2 (N2 (N2 t3 a3 t4) a5 (N1 t5))" |
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"n2 t1 a1 t2 = N2 t1 a1 t2"
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fun del_min :: "'a bro \<Rightarrow> ('a \<times> 'a bro) option" where
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"del_min N0 = None" |
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"del_min (N1 t) =
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  (case del_min t of
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     None \<Rightarrow> None |
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     Some (a, t') \<Rightarrow> Some (a, N1 t'))" |
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"del_min (N2 t1 a t2) =
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  (case del_min t1 of
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     None \<Rightarrow> Some (a, N1 t2) |
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     Some (b, t1') \<Rightarrow> Some (b, n2 t1' a t2))"
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fun del :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
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"del _ N0         = N0" |
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"del x (N1 t)     = N1 (del x t)" |
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"del x (N2 l a r) =
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  (case cmp x a of
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     LT \<Rightarrow> n2 (del x l) a r |
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     GT \<Rightarrow> n2 l a (del x r) |
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     EQ \<Rightarrow> (case del_min r of
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              None \<Rightarrow> N1 l |
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              Some (b, r') \<Rightarrow> n2 l b r'))"
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fun tree :: "'a bro \<Rightarrow> 'a bro" where
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"tree (N1 t) = t" |
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"tree t = t"
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definition delete :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
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"delete a t = tree (del a t)"
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end
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subsection \<open>Invariants\<close>
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fun B :: "nat \<Rightarrow> 'a bro set"
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and U :: "nat \<Rightarrow> 'a bro set" where
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"B 0 = {N0}" |
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"B (Suc h) = { N2 t1 a t2 | t1 a t2. 
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  t1 \<in> B h \<union> U h \<and> t2 \<in> B h \<or> t1 \<in> B h \<and> t2 \<in> B h \<union> U h}" |
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"U 0 = {}" |
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"U (Suc h) = N1 ` B h"
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abbreviation "T h \<equiv> B h \<union> U h"
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fun Bp :: "nat \<Rightarrow> 'a bro set" where
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"Bp 0 = B 0 \<union> L2 ` UNIV" |
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"Bp (Suc 0) = B (Suc 0) \<union> {N3 N0 a N0 b N0|a b. True}" |
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"Bp (Suc(Suc h)) = B (Suc(Suc h)) \<union>
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  {N3 t1 a t2 b t3 | t1 a t2 b t3. t1 \<in> B (Suc h) \<and> t2 \<in> U (Suc h) \<and> t3 \<in> B (Suc h)}"
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fun Um :: "nat \<Rightarrow> 'a bro set" where
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"Um 0 = {}" |
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"Um (Suc h) = N1 ` T h"
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subsection "Functional Correctness Proofs"
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subsubsection "Proofs for isin"
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lemma
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  "t \<in> T h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
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by(induction h arbitrary: t) (fastforce simp: elems_simps1 split: if_splits)+
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lemma isin_set: "t \<in> T h \<Longrightarrow>
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  sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
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by(induction h arbitrary: t) (auto simp: elems_simps2 split: if_splits)
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subsubsection "Proofs for insertion"
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lemma inorder_n1: "inorder(n1 t) = inorder t"
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by(induction t rule: n1.induct) (auto simp: sorted_lems)
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context insert
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begin
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lemma inorder_n2: "inorder(n2 l a r) = inorder l @ a # inorder r"
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by(cases "(l,a,r)" rule: n2.cases) (auto simp: sorted_lems)
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lemma inorder_tree: "inorder(tree t) = inorder t"
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by(cases t) auto
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lemma inorder_ins: "t \<in> T h \<Longrightarrow>
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  sorted(inorder t) \<Longrightarrow> inorder(ins a t) = ins_list a (inorder t)"
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by(induction h arbitrary: t) (auto simp: ins_list_simps inorder_n1 inorder_n2)
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lemma inorder_insert: "t \<in> T h \<Longrightarrow>
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  sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
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by(simp add: insert_def inorder_ins inorder_tree)
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end
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subsubsection \<open>Proofs for deletion\<close>
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context delete
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begin
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lemma inorder_tree: "inorder(tree t) = inorder t"
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by(cases t) auto
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lemma inorder_n2: "inorder(n2 l a r) = inorder l @ a # inorder r"
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by(induction l a r rule: n2.induct) (auto)
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lemma inorder_del_min:
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shows "t \<in> B h \<Longrightarrow> (del_min t = None \<longleftrightarrow> inorder t = []) \<and>
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  (del_min t = Some(a,t') \<longrightarrow> inorder t = a # inorder t')"
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and "t \<in> U h \<Longrightarrow> (del_min t = None \<longleftrightarrow> inorder t = []) \<and>
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  (del_min t = Some(a,t') \<longrightarrow> inorder t = a # inorder t')"
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by(induction h arbitrary: t a t') (auto simp: inorder_n2 split: option.splits)
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lemma inorder_del:
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  "t \<in> B h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(del a t) = del_list a (inorder t)"
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  "t \<in> U h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(del a t) = del_list a (inorder t)"
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by(induction h arbitrary: t)
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  (auto simp: del_list_simps inorder_n2 inorder_del_min split: option.splits)
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end
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subsection \<open>Invariant Proofs\<close>
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subsubsection \<open>Proofs for insertion\<close>
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lemma n1_type: "t \<in> Bp h \<Longrightarrow> n1 t \<in> T (Suc h)"
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by(cases h rule: Bp.cases) auto
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context insert
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begin
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lemma tree_type1: "t \<in> Bp h \<Longrightarrow> tree t \<in> B h \<union> B (Suc h)"
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by(cases h rule: Bp.cases) auto
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lemma tree_type2: "t \<in> T h \<Longrightarrow> tree t \<in> T h"
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by(cases h) auto
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lemma n2_type:
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  "(t1 \<in> Bp h \<and> t2 \<in> T h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h)) \<and>
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   (t1 \<in> T h \<and> t2 \<in> Bp h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h))"
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apply(cases h rule: Bp.cases)
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apply (auto)[2]
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apply(rule conjI impI | erule conjE exE imageE | simp | erule disjE)+
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done
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lemma Bp_if_B: "t \<in> B h \<Longrightarrow> t \<in> Bp h"
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by (cases h rule: Bp.cases) simp_all
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text{* An automatic proof: *}
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lemma
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  "(t \<in> B h \<longrightarrow> ins x t \<in> Bp h) \<and> (t \<in> U h \<longrightarrow> ins x t \<in> T h)"
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apply(induction h arbitrary: t)
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 apply (simp)
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apply (fastforce simp: Bp_if_B n2_type dest: n1_type)
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done
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text{* A detailed proof: *}
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lemma ins_type:
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shows "t \<in> B h \<Longrightarrow> ins x t \<in> Bp h" and "t \<in> U h \<Longrightarrow> ins x t \<in> T h"
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proof(induction h arbitrary: t)
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  case 0
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  { case 1 thus ?case by simp
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  next
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    case 2 thus ?case by simp }
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next
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  case (Suc h)
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  { case 1
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    then obtain t1 a t2 where [simp]: "t = N2 t1 a t2" and
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      t1: "t1 \<in> T h" and t2: "t2 \<in> T h" and t12: "t1 \<in> B h \<or> t2 \<in> B h"
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      by auto
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    { assume "x < a"
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      hence "?case \<longleftrightarrow> n2 (ins x t1) a t2 \<in> Bp (Suc h)" by simp
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      also have "\<dots>"
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      proof cases
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        assume "t1 \<in> B h"
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        with t2 show ?thesis by (simp add: Suc.IH(1) n2_type)
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      next
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        assume "t1 \<notin> B h"
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        hence 1: "t1 \<in> U h" and 2: "t2 \<in> B h" using t1 t12 by auto
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        show ?thesis by (metis Suc.IH(2)[OF 1] Bp_if_B[OF 2] n2_type)
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      qed
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      finally have ?case .
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    }
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    moreover
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    { assume "a < x"
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      hence "?case \<longleftrightarrow> n2 t1 a (ins x t2) \<in> Bp (Suc h)" by simp
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      also have "\<dots>"
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      proof cases
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        assume "t2 \<in> B h"
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        with t1 show ?thesis by (simp add: Suc.IH(1) n2_type)
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      next
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        assume "t2 \<notin> B h"
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        hence 1: "t1 \<in> B h" and 2: "t2 \<in> U h" using t2 t12 by auto
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        show ?thesis by (metis Bp_if_B[OF 1] Suc.IH(2)[OF 2] n2_type)
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      qed
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    }
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    moreover 
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    { assume "x = a"
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      from 1 have "t \<in> Bp (Suc h)" by(rule Bp_if_B)
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      hence "?case" using `x = a` by simp
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    }
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    ultimately show ?case by auto
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  next
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    case 2 thus ?case using Suc(1) n1_type by fastforce }
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qed
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lemma insert_type:
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  "t \<in> T h \<Longrightarrow> insert x t \<in> T h \<union> T (Suc h)"
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unfolding insert_def by (metis Un_iff ins_type tree_type1 tree_type2)
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end
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subsubsection "Proofs for deletion"
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lemma B_simps[simp]: 
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  "N1 t \<in> B h = False"
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  "L2 y \<in> B h = False"
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  "(N3 t1 a1 t2 a2 t3) \<in> B h = False"
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  "N0 \<in> B h \<longleftrightarrow> h = 0"
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by (cases h, auto)+
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context delete
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begin
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lemma n2_type1:
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  "\<lbrakk>t1 \<in> Um h; t2 \<in> B h\<rbrakk> \<Longrightarrow> n2 t1 a t2 \<in> T (Suc h)"
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apply(cases h rule: Bp.cases)
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apply auto[2]
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apply(erule exE bexE conjE imageE | simp | erule disjE)+
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done
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lemma n2_type2:
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  "\<lbrakk>t1 \<in> B h ; t2 \<in> Um h \<rbrakk> \<Longrightarrow> n2 t1 a t2 \<in> T (Suc h)"
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apply(cases h rule: Bp.cases)
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apply auto[2]
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apply(erule exE bexE conjE imageE | simp | erule disjE)+
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done
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lemma n2_type3:
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  "\<lbrakk>t1 \<in> T h ; t2 \<in> T h \<rbrakk> \<Longrightarrow> n2 t1 a t2 \<in> T (Suc h)"
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apply(cases h rule: Bp.cases)
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apply auto[2]
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apply(erule exE bexE conjE imageE | simp | erule disjE)+
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done
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lemma del_minNoneN0: "\<lbrakk>t \<in> B h; del_min t = None\<rbrakk> \<Longrightarrow>  t = N0"
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by (cases t) (auto split: option.splits)
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lemma del_minNoneN1 : "\<lbrakk>t \<in> U h; del_min t = None\<rbrakk> \<Longrightarrow> t = N1 N0"
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by (cases h) (auto simp: del_minNoneN0  split: option.splits)
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lemma del_min_type:
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  "t \<in> B h \<Longrightarrow> del_min t = Some (a, t') \<Longrightarrow> t' \<in> T h"
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  "t \<in> U h \<Longrightarrow> del_min t = Some (a, t') \<Longrightarrow> t' \<in> Um h"
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proof (induction h arbitrary: t a t')
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  case (Suc h)
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  { case 1
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    then obtain t1 a t2 where [simp]: "t = N2 t1 a t2" and
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      t12: "t1 \<in> T h" "t2 \<in> T h" "t1 \<in> B h \<or> t2 \<in> B h"
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      by auto
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    show ?case
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    proof (cases "del_min t1")
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      case None
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      show ?thesis
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      proof cases
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        assume "t1 \<in> B h"
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        with del_minNoneN0[OF this None] 1 show ?thesis by(auto)
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      next
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        assume "t1 \<notin> B h"
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        thus ?thesis using 1 None by (auto)
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      qed
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    next
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      case [simp]: (Some bt')
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      obtain b t1' where [simp]: "bt' = (b,t1')" by fastforce
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      show ?thesis
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      proof cases
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        assume "t1 \<in> B h"
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        from Suc.IH(1)[OF this] 1 have "t1' \<in> T h" by simp
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        from n2_type3[OF this t12(2)] 1 show ?thesis by auto
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      next
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        assume "t1 \<notin> B h"
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        hence t1: "t1 \<in> U h" and t2: "t2 \<in> B h" using t12 by auto
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        from Suc.IH(2)[OF t1] have "t1' \<in> Um h" by simp
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        from n2_type1[OF this t2] 1 show ?thesis by auto
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      qed
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    qed
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  }
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  { case 2
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    then obtain t1 where [simp]: "t = N1 t1" and t1: "t1 \<in> B h" by auto
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    show ?case
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    proof (cases "del_min t1")
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      case None
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      with del_minNoneN0[OF t1 None] 2 show ?thesis by(auto)
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    next
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      case [simp]: (Some bt')
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      obtain b t1' where [simp]: "bt' = (b,t1')" by fastforce
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      from Suc.IH(1)[OF t1] have "t1' \<in> T h" by simp
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      thus ?thesis using 2 by auto
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    qed
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  }
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qed auto
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lemma del_type:
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  "t \<in> B h \<Longrightarrow> del x t \<in> T h"
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  "t \<in> U h \<Longrightarrow> del x t \<in> Um h"
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proof (induction h arbitrary: x t)
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  case (Suc h)
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  { case 1
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   400
    then obtain l a r where [simp]: "t = N2 l a r" and
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      lr: "l \<in> T h" "r \<in> T h" "l \<in> B h \<or> r \<in> B h" by auto
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   402
    { assume "x < a"
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   403
      have ?case
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   404
      proof cases
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   405
        assume "l \<in> B h"
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   406
        from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
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   407
        show ?thesis using `x<a` by(simp)
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   408
      next
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   409
        assume "l \<notin> B h"
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   410
        hence "l \<in> U h" "r \<in> B h" using lr by auto
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   411
        from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
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        show ?thesis using `x<a` by(simp)
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   413
      qed
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   414
    } moreover
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   415
    { assume "x > a"
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   416
      have ?case
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   417
      proof cases
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   418
        assume "r \<in> B h"
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   419
        from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
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   420
        show ?thesis using `x>a` by(simp)
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   421
      next
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   422
        assume "r \<notin> B h"
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   423
        hence "l \<in> B h" "r \<in> U h" using lr by auto
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   424
        from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
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        show ?thesis using `x>a` by(simp)
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   426
      qed
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   427
    } moreover
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   428
    { assume [simp]: "x=a"
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   429
      have ?case
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   430
      proof (cases "del_min r")
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   431
        case None
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   432
        show ?thesis
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   433
        proof cases
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   434
          assume "r \<in> B h"
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   435
          with del_minNoneN0[OF this None] lr show ?thesis by(simp)
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   436
        next
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   437
          assume "r \<notin> B h"
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   438
          hence "r \<in> U h" using lr by auto
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   439
          with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
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   440
        qed
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   441
      next
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   442
        case [simp]: (Some br')
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   443
        obtain b r' where [simp]: "br' = (b,r')" by fastforce
nipkow@61784
   444
        show ?thesis
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   445
        proof cases
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   446
          assume "r \<in> B h"
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   447
          from del_min_type(1)[OF this] n2_type3[OF lr(1)]
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   448
          show ?thesis by simp
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   449
        next
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   450
          assume "r \<notin> B h"
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   451
          hence "l \<in> B h" and "r \<in> U h" using lr by auto
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   452
          from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
nipkow@61784
   453
          show ?thesis by simp
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   454
        qed
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   455
      qed
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   456
    } ultimately show ?case by auto
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   457
  }
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   458
  { case 2 with Suc.IH(1) show ?case by auto }
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   459
qed auto
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   460
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   461
lemma tree_type:
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   462
  "t \<in> Um (Suc h) \<Longrightarrow> tree t : T h"
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   463
  "t \<in> T (Suc h) \<Longrightarrow> tree t : T h \<union> T(h+1)"
nipkow@61784
   464
by(auto)
nipkow@61784
   465
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   466
lemma delete_type:
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   467
  "t \<in> T h \<Longrightarrow> delete x t \<in> T h \<union> T(h-1)"
nipkow@61784
   468
unfolding delete_def
nipkow@61784
   469
by (cases h) (simp, metis del_type tree_type Un_iff Suc_eq_plus1 diff_Suc_1)
nipkow@61784
   470
nipkow@61784
   471
end
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   472
nipkow@61789
   473
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   474
subsection "Overall correctness"
nipkow@61784
   475
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   476
interpretation Set_by_Ordered
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   477
where empty = N0 and isin = isin and insert = insert.insert
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   478
and delete = delete.delete and inorder = inorder and inv = "\<lambda>t. \<exists>h. t \<in> T h"
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   479
proof (standard, goal_cases)
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   480
  case 2 thus ?case by(auto intro!: isin_set)
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   481
next
nipkow@61784
   482
  case 3 thus ?case by(auto intro!: insert.inorder_insert)
nipkow@61784
   483
next
nipkow@61784
   484
  case 4 thus ?case
nipkow@61784
   485
    by(auto simp: delete.delete_def delete.inorder_tree delete.inorder_del)
nipkow@61784
   486
next
nipkow@61784
   487
  case 6 thus ?case using insert.insert_type by blast
nipkow@61784
   488
next
nipkow@61784
   489
  case 7 thus ?case using delete.delete_type by blast
nipkow@61784
   490
qed auto
nipkow@61784
   491
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   492
end