src/HOL/Data_Structures/AA_Set.thy
author nipkow
Thu Jul 07 18:08:02 2016 +0200 (2016-07-07)
changeset 63411 e051eea34990
parent 62526 347150095fd2
child 63636 6f38b7abb648
permissions -rw-r--r--
got rid of class cmp; added height-size proofs by Daniel Stuewe
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(*
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Author: Tobias Nipkow, Daniel Stüwe
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*)
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section \<open>AA Tree Implementation of Sets\<close>
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theory AA_Set
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imports
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  Isin2
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  Cmp
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begin
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type_synonym 'a aa_tree = "('a,nat) tree"
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fun lvl :: "'a aa_tree \<Rightarrow> nat" where
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"lvl Leaf = 0" |
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"lvl (Node lv _ _ _) = lv"
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fun invar :: "'a aa_tree \<Rightarrow> bool" where
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"invar Leaf = True" |
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"invar (Node h l a r) =
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 (invar l \<and> invar r \<and>
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  h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = Node h lr b rr \<and> h = lvl rr + 1)))"
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fun skew :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
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"skew (Node lva (Node lvb t1 b t2) a t3) =
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  (if lva = lvb then Node lva t1 b (Node lva t2 a t3) else Node lva (Node lvb t1 b t2) a t3)" |
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"skew t = t"
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fun split :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
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"split (Node lva t1 a (Node lvb t2 b (Node lvc t3 c t4))) =
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   (if lva = lvb \<and> lvb = lvc (* lva = lvc suffices *)
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    then Node (lva+1) (Node lva t1 a t2) b (Node lva t3 c t4)
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    else Node lva t1 a (Node lvb t2 b (Node lvc t3 c t4)))" |
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"split t = t"
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hide_const (open) insert
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fun insert :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
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"insert x Leaf = Node 1 Leaf x Leaf" |
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"insert x (Node lv t1 a t2) =
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  (case cmp x a of
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     LT \<Rightarrow> split (skew (Node lv (insert x t1) a t2)) |
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     GT \<Rightarrow> split (skew (Node lv t1 a (insert x t2))) |
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     EQ \<Rightarrow> Node lv t1 x t2)"
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fun sngl :: "'a aa_tree \<Rightarrow> bool" where
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"sngl Leaf = False" |
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"sngl (Node _ _ _ Leaf) = True" |
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"sngl (Node lva _ _ (Node lvb _ _ _)) = (lva > lvb)"
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definition adjust :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
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"adjust t =
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 (case t of
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  Node lv l x r \<Rightarrow>
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   (if lvl l >= lv-1 \<and> lvl r >= lv-1 then t else
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    if lvl r < lv-1 \<and> sngl l then skew (Node (lv-1) l x r) else
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    if lvl r < lv-1
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    then case l of
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           Node lva t1 a (Node lvb t2 b t3)
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             \<Rightarrow> Node (lvb+1) (Node lva t1 a t2) b (Node (lv-1) t3 x r) 
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    else
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    if lvl r < lv then split (Node (lv-1) l x r)
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    else
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      case r of
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        Node lvb t1 b t4 \<Rightarrow>
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          (case t1 of
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             Node lva t2 a t3
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               \<Rightarrow> Node (lva+1) (Node (lv-1) l x t2) a
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                    (split (Node (if sngl t1 then lva else lva+1) t3 b t4)))))"
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text{* In the paper, the last case of @{const adjust} is expressed with the help of an
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incorrect auxiliary function \texttt{nlvl}.
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Function @{text del_max} below is called \texttt{dellrg} in the paper.
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The latter is incorrect for two reasons: \texttt{dellrg} is meant to delete the largest
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element but recurses on the left instead of the right subtree; the invariant
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is not restored.*}
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fun del_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
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"del_max (Node lv l a Leaf) = (l,a)" |
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"del_max (Node lv l a r) = (let (r',b) = del_max r in (adjust(Node lv l a r'), b))"
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fun delete :: "'a::linorder \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
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"delete _ Leaf = Leaf" |
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"delete x (Node lv l a r) =
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  (case cmp x a of
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     LT \<Rightarrow> adjust (Node lv (delete x l) a r) |
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     GT \<Rightarrow> adjust (Node lv l a (delete x r)) |
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     EQ \<Rightarrow> (if l = Leaf then r
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            else let (l',b) = del_max l in adjust (Node lv l' b r)))"
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fun pre_adjust where
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"pre_adjust (Node lv l a r) = (invar l \<and> invar r \<and>
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    ((lv = lvl l + 1 \<and> (lv = lvl r + 1 \<or> lv = lvl r + 2 \<or> lv = lvl r \<and> sngl r)) \<or>
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     (lv = lvl l + 2 \<and> (lv = lvl r + 1 \<or> lv = lvl r \<and> sngl r))))"
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declare pre_adjust.simps [simp del]
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subsection "Auxiliary Proofs"
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lemma split_case: "split t = (case t of
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  Node lvx a x (Node lvy b y (Node lvz c z d)) \<Rightarrow>
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   (if lvx = lvy \<and> lvy = lvz
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    then Node (lvx+1) (Node lvx a x b) y (Node lvx c z d)
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    else t)
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  | t \<Rightarrow> t)"
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by(auto split: tree.split)
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lemma skew_case: "skew t = (case t of
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  Node lvx (Node lvy a y b) x c \<Rightarrow>
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  (if lvx = lvy then Node lvx a y (Node lvx b x c) else t)
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 | t \<Rightarrow> t)"
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by(auto split: tree.split)
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lemma lvl_0_iff: "invar t \<Longrightarrow> lvl t = 0 \<longleftrightarrow> t = Leaf"
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by(cases t) auto
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lemma lvl_Suc_iff: "lvl t = Suc n \<longleftrightarrow> (\<exists> l a r. t = Node (Suc n) l a r)"
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by(cases t) auto
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lemma lvl_skew: "lvl (skew t) = lvl t"
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by(cases t rule: skew.cases) auto
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lemma lvl_split: "lvl (split t) = lvl t \<or> lvl (split t) = lvl t + 1 \<and> sngl (split t)"
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by(cases t rule: split.cases) auto
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lemma invar_2Nodes:"invar (Node lv l x (Node rlv rl rx rr)) =
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     (invar l \<and> invar \<langle>rlv, rl, rx, rr\<rangle> \<and> lv = Suc (lvl l) \<and>
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     (lv = Suc rlv \<or> rlv = lv \<and> lv = Suc (lvl rr)))"
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by simp
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lemma invar_NodeLeaf[simp]:
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  "invar (Node lv l x Leaf) = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
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by simp
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lemma sngl_if_invar: "invar (Node n l a r) \<Longrightarrow> n = lvl r \<Longrightarrow> sngl r"
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by(cases r rule: sngl.cases) clarsimp+
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subsection "Invariance"
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subsubsection "Proofs for insert"
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lemma lvl_insert_aux:
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  "lvl (insert x t) = lvl t \<or> lvl (insert x t) = lvl t + 1 \<and> sngl (insert x t)"
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apply(induction t)
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apply (auto simp: lvl_skew)
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apply (metis Suc_eq_plus1 lvl.simps(2) lvl_split lvl_skew)+
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done
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lemma lvl_insert: obtains
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  (Same) "lvl (insert x t) = lvl t" |
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  (Incr) "lvl (insert x t) = lvl t + 1" "sngl (insert x t)"
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using lvl_insert_aux by blast
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lemma lvl_insert_sngl: "invar t \<Longrightarrow> sngl t \<Longrightarrow> lvl(insert x t) = lvl t"
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proof (induction t rule: insert.induct)
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  case (2 x lv t1 a t2)
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  consider (LT) "x < a" | (GT) "x > a" | (EQ) "x = a" 
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    using less_linear by blast 
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  thus ?case proof cases
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    case LT
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    thus ?thesis using 2 by (auto simp add: skew_case split_case split: tree.splits)
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  next
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    case GT
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    thus ?thesis using 2 proof (cases t1)
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      case Node
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      thus ?thesis using 2 GT  
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        apply (auto simp add: skew_case split_case split: tree.splits)
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        by (metis less_not_refl2 lvl.simps(2) lvl_insert_aux n_not_Suc_n sngl.simps(3))+ 
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    qed (auto simp add: lvl_0_iff)
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  qed simp
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qed simp
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lemma skew_invar: "invar t \<Longrightarrow> skew t = t"
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by(cases t rule: skew.cases) auto
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lemma split_invar: "invar t \<Longrightarrow> split t = t"
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by(cases t rule: split.cases) clarsimp+
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lemma invar_NodeL:
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  "\<lbrakk> invar(Node n l x r); invar l'; lvl l' = lvl l \<rbrakk> \<Longrightarrow> invar(Node n l' x r)"
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by(auto)
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lemma invar_NodeR:
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  "\<lbrakk> invar(Node n l x r); n = lvl r + 1; invar r'; lvl r' = lvl r \<rbrakk> \<Longrightarrow> invar(Node n l x r')"
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by(auto)
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lemma invar_NodeR2:
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  "\<lbrakk> invar(Node n l x r); sngl r'; n = lvl r + 1; invar r'; lvl r' = n \<rbrakk> \<Longrightarrow> invar(Node n l x r')"
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by(cases r' rule: sngl.cases) clarsimp+
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lemma lvl_insert_incr_iff: "(lvl(insert a t) = lvl t + 1) \<longleftrightarrow>
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  (EX l x r. insert a t = Node (lvl t + 1) l x r \<and> lvl l = lvl r)"
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apply(cases t)
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apply(auto simp add: skew_case split_case split: if_splits)
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apply(auto split: tree.splits if_splits)
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done
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lemma invar_insert: "invar t \<Longrightarrow> invar(insert a t)"
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proof(induction t)
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  case (Node n l x r)
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  hence il: "invar l" and ir: "invar r" by auto
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  note N = Node
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  let ?t = "Node n l x r"
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  have "a < x \<or> a = x \<or> x < a" by auto
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  moreover
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  { assume "a < x"
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    note iil = Node.IH(1)[OF il]
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    have ?case
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    proof (cases rule: lvl_insert[of a l])
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      case (Same) thus ?thesis
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        using \<open>a<x\<close> invar_NodeL[OF Node.prems iil Same]
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        by (simp add: skew_invar split_invar del: invar.simps)
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    next
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      case (Incr)
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      then obtain t1 w t2 where ial[simp]: "insert a l = Node n t1 w t2"
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        using Node.prems by (auto simp: lvl_Suc_iff)
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      have l12: "lvl t1 = lvl t2"
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        by (metis Incr(1) ial lvl_insert_incr_iff tree.inject)
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      have "insert a ?t = split(skew(Node n (insert a l) x r))"
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        by(simp add: \<open>a<x\<close>)
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      also have "skew(Node n (insert a l) x r) = Node n t1 w (Node n t2 x r)"
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        by(simp)
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      also have "invar(split \<dots>)"
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      proof (cases r)
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        case Leaf
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        hence "l = Leaf" using Node.prems by(auto simp: lvl_0_iff)
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        thus ?thesis using Leaf ial by simp
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      next
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        case [simp]: (Node m t3 y t4)
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        show ?thesis (*using N(3) iil l12 by(auto)*)
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        proof cases
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          assume "m = n" thus ?thesis using N(3) iil by(auto)
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        next
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          assume "m \<noteq> n" thus ?thesis using N(3) iil l12 by(auto)
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        qed
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      qed
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      finally show ?thesis .
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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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    note iir = Node.IH(2)[OF ir]
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    from \<open>invar ?t\<close> have "n = lvl r \<or> n = lvl r + 1" by auto
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    hence ?case
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    proof
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      assume 0: "n = lvl r"
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      have "insert a ?t = split(skew(Node n l x (insert a r)))"
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        using \<open>a>x\<close> by(auto)
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      also have "skew(Node n l x (insert a r)) = Node n l x (insert a r)"
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        using Node.prems by(simp add: skew_case split: tree.split)
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      also have "invar(split \<dots>)"
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      proof -
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        from lvl_insert_sngl[OF ir sngl_if_invar[OF \<open>invar ?t\<close> 0], of a]
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        obtain t1 y t2 where iar: "insert a r = Node n t1 y t2"
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          using Node.prems 0 by (auto simp: lvl_Suc_iff)
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        from Node.prems iar 0 iir
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        show ?thesis by (auto simp: split_case split: tree.splits)
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      qed
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      finally show ?thesis .
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    next
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      assume 1: "n = lvl r + 1"
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      hence "sngl ?t" by(cases r) auto
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      show ?thesis
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      proof (cases rule: lvl_insert[of a r])
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        case (Same)
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        show ?thesis using \<open>x<a\<close> il ir invar_NodeR[OF Node.prems 1 iir Same]
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          by (auto simp add: skew_invar split_invar)
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      next
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        case (Incr)
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        thus ?thesis using invar_NodeR2[OF `invar ?t` Incr(2) 1 iir] 1 \<open>x < a\<close>
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          by (auto simp add: skew_invar split_invar split: if_splits)
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      qed
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    qed
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  }
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  moreover { assume "a = x" hence ?case using Node.prems by auto }
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  ultimately show ?case by blast
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qed simp
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subsubsection "Proofs for delete"
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lemma invarL: "ASSUMPTION(invar \<langle>lv, l, a, r\<rangle>) \<Longrightarrow> invar l"
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by(simp add: ASSUMPTION_def)
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lemma invarR: "ASSUMPTION(invar \<langle>lv, l, a, r\<rangle>) \<Longrightarrow> invar r"
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by(simp add: ASSUMPTION_def)
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lemma sngl_NodeI:
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  "sngl (Node lv l a r) \<Longrightarrow> sngl (Node lv l' a' r)"
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by(cases r) (simp_all)
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declare invarL[simp] invarR[simp]
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lemma pre_cases:
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assumes "pre_adjust (Node lv l x r)"
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obtains
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 (tSngl) "invar l \<and> invar r \<and>
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    lv = Suc (lvl r) \<and> lvl l = lvl r" |
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 (tDouble) "invar l \<and> invar r \<and>
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    lv = lvl r \<and> Suc (lvl l) = lvl r \<and> sngl r " |
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 (rDown) "invar l \<and> invar r \<and>
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    lv = Suc (Suc (lvl r)) \<and>  lv = Suc (lvl l)" |
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 (lDown_tSngl) "invar l \<and> invar r \<and>
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    lv = Suc (lvl r) \<and> lv = Suc (Suc (lvl l))" |
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 (lDown_tDouble) "invar l \<and> invar r \<and>
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    lv = lvl r \<and> lv = Suc (Suc (lvl l)) \<and> sngl r"
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using assms unfolding pre_adjust.simps
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by auto
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declare invar.simps(2)[simp del] invar_2Nodes[simp add]
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lemma invar_adjust:
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  assumes pre: "pre_adjust (Node lv l a r)"
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  shows  "invar(adjust (Node lv l a r))"
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using pre proof (cases rule: pre_cases)
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  case (tDouble) thus ?thesis unfolding adjust_def by (cases r) (auto simp: invar.simps(2)) 
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next 
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   323
  case (rDown)
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  from rDown obtain llv ll la lr where l: "l = Node llv ll la lr" by (cases l) auto
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  from rDown show ?thesis unfolding adjust_def by (auto simp: l invar.simps(2) split: tree.splits)
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next
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  case (lDown_tDouble)
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  from lDown_tDouble obtain rlv rr ra rl where r: "r = Node rlv rl ra rr" by (cases r) auto
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  from lDown_tDouble and r obtain rrlv rrr rra rrl where
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    rr :"rr = Node rrlv rrr rra rrl" by (cases rr) auto
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  from  lDown_tDouble show ?thesis unfolding adjust_def r rr
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    apply (cases rl) apply (auto simp add: invar.simps(2))
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    using lDown_tDouble by (auto simp: split_case lvl_0_iff  elim:lvl.elims split: tree.split)
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qed (auto simp: split_case invar.simps(2) adjust_def split: tree.splits)
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lemma lvl_adjust:
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  assumes "pre_adjust (Node lv l a r)"
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  shows "lv = lvl (adjust(Node lv l a r)) \<or> lv = lvl (adjust(Node lv l a r)) + 1"
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using assms(1) proof(cases rule: pre_cases)
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  case lDown_tSngl thus ?thesis
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    using lvl_split[of "\<langle>lvl r, l, a, r\<rangle>"] by (auto simp: adjust_def)
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next
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  case lDown_tDouble thus ?thesis
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    by (auto simp: adjust_def invar.simps(2) split: tree.split)
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qed (auto simp: adjust_def split: tree.splits)
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lemma sngl_adjust: assumes "pre_adjust (Node lv l a r)"
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  "sngl \<langle>lv, l, a, r\<rangle>" "lv = lvl (adjust \<langle>lv, l, a, r\<rangle>)"
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  shows "sngl (adjust \<langle>lv, l, a, r\<rangle>)" 
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using assms proof (cases rule: pre_cases)
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  case rDown
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  thus ?thesis using assms(2,3) unfolding adjust_def
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    by (auto simp add: skew_case) (auto split: tree.split)
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qed (auto simp: adjust_def skew_case split_case split: tree.split)
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   355
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   356
definition "post_del t t' ==
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  invar t' \<and>
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   358
  (lvl t' = lvl t \<or> lvl t' + 1 = lvl t) \<and>
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   359
  (lvl t' = lvl t \<and> sngl t \<longrightarrow> sngl t')"
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   360
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   361
lemma pre_adj_if_postR:
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   362
  "invar\<langle>lv, l, a, r\<rangle> \<Longrightarrow> post_del r r' \<Longrightarrow> pre_adjust \<langle>lv, l, a, r'\<rangle>"
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   363
by(cases "sngl r")
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   364
  (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
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   365
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   366
lemma pre_adj_if_postL:
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  "invar\<langle>lv, l, a, r\<rangle> \<Longrightarrow> post_del l l' \<Longrightarrow> pre_adjust \<langle>lv, l', b, r\<rangle>"
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by(cases "sngl r")
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  (auto simp: pre_adjust.simps post_del_def invar.simps(2) elim: sngl.elims)
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   370
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   371
lemma post_del_adjL:
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  "\<lbrakk> invar\<langle>lv, l, a, r\<rangle>; pre_adjust \<langle>lv, l', b, r\<rangle> \<rbrakk>
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  \<Longrightarrow> post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l', b, r\<rangle>)"
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unfolding post_del_def
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by (metis invar_adjust lvl_adjust sngl_NodeI sngl_adjust lvl.simps(2))
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   376
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   377
lemma post_del_adjR:
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assumes "invar\<langle>lv, l, a, r\<rangle>" "pre_adjust \<langle>lv, l, a, r'\<rangle>" "post_del r r'"
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shows "post_del \<langle>lv, l, a, r\<rangle> (adjust \<langle>lv, l, a, r'\<rangle>)"
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   380
proof(unfold post_del_def, safe del: disjCI)
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   381
  let ?t = "\<langle>lv, l, a, r\<rangle>"
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   382
  let ?t' = "adjust \<langle>lv, l, a, r'\<rangle>"
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   383
  show "invar ?t'" by(rule invar_adjust[OF assms(2)])
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   384
  show "lvl ?t' = lvl ?t \<or> lvl ?t' + 1 = lvl ?t"
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   385
    using lvl_adjust[OF assms(2)] by auto
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   386
  show "sngl ?t'" if as: "lvl ?t' = lvl ?t" "sngl ?t"
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   387
  proof -
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   388
    have s: "sngl \<langle>lv, l, a, r'\<rangle>"
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   389
    proof(cases r')
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   390
      case Leaf thus ?thesis by simp
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   391
    next
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   392
      case Node thus ?thesis using as(2) assms(1,3)
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   393
      by (cases r) (auto simp: post_del_def)
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   394
    qed
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   395
    show ?thesis using as(1) sngl_adjust[OF assms(2) s] by simp
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   396
  qed
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   397
qed
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   398
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   399
declare prod.splits[split]
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   400
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   401
theorem post_del_max:
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   402
 "\<lbrakk> invar t; (t', x) = del_max t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> post_del t t'"
nipkow@62496
   403
proof (induction t arbitrary: t' rule: del_max.induct)
nipkow@62496
   404
  case (2 lv l a lvr rl ra rr)
nipkow@62496
   405
  let ?r =  "\<langle>lvr, rl, ra, rr\<rangle>"
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   406
  let ?t = "\<langle>lv, l, a, ?r\<rangle>"
nipkow@62496
   407
  from "2.prems"(2) obtain r' where r': "(r', x) = del_max ?r"
nipkow@62496
   408
    and [simp]: "t' = adjust \<langle>lv, l, a, r'\<rangle>" by auto
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   409
  from  "2.IH"[OF _ r'] \<open>invar ?t\<close> have post: "post_del ?r r'" by simp
nipkow@62496
   410
  note preR = pre_adj_if_postR[OF \<open>invar ?t\<close> post]
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   411
  show ?case by (simp add: post_del_adjR[OF "2.prems"(1) preR post])
nipkow@62496
   412
qed (auto simp: post_del_def)
nipkow@62496
   413
nipkow@62496
   414
theorem post_delete: "invar t \<Longrightarrow> post_del t (delete x t)"
nipkow@62496
   415
proof (induction t)
nipkow@62496
   416
  case (Node lv l a r)
nipkow@62496
   417
nipkow@62496
   418
  let ?l' = "delete x l" and ?r' = "delete x r"
nipkow@62496
   419
  let ?t = "Node lv l a r" let ?t' = "delete x ?t"
nipkow@62496
   420
nipkow@62496
   421
  from Node.prems have inv_l: "invar l" and inv_r: "invar r" by (auto)
nipkow@62496
   422
nipkow@62496
   423
  note post_l' = Node.IH(1)[OF inv_l]
nipkow@62496
   424
  note preL = pre_adj_if_postL[OF Node.prems post_l']
nipkow@62496
   425
nipkow@62496
   426
  note post_r' = Node.IH(2)[OF inv_r]
nipkow@62496
   427
  note preR = pre_adj_if_postR[OF Node.prems post_r']
nipkow@62496
   428
nipkow@62496
   429
  show ?case
nipkow@62496
   430
  proof (cases rule: linorder_cases[of x a])
nipkow@62496
   431
    case less
nipkow@62496
   432
    thus ?thesis using Node.prems by (simp add: post_del_adjL preL)
nipkow@62496
   433
  next
nipkow@62496
   434
    case greater
nipkow@62496
   435
    thus ?thesis using Node.prems by (simp add: post_del_adjR preR post_r')
nipkow@62496
   436
  next
nipkow@62496
   437
    case equal
nipkow@62496
   438
    show ?thesis
nipkow@62496
   439
    proof cases
nipkow@62496
   440
      assume "l = Leaf" thus ?thesis using equal Node.prems
nipkow@62496
   441
        by(auto simp: post_del_def invar.simps(2))
nipkow@62496
   442
    next
nipkow@62496
   443
      assume "l \<noteq> Leaf" thus ?thesis using equal
nipkow@62496
   444
        by simp (metis Node.prems inv_l post_del_adjL post_del_max pre_adj_if_postL)
nipkow@62496
   445
    qed
nipkow@62496
   446
  qed
nipkow@62496
   447
qed (simp add: post_del_def)
nipkow@62496
   448
nipkow@62496
   449
declare invar_2Nodes[simp del]
nipkow@62496
   450
nipkow@61793
   451
nipkow@61793
   452
subsection "Functional Correctness"
nipkow@61793
   453
nipkow@62496
   454
nipkow@61793
   455
subsubsection "Proofs for insert"
nipkow@61793
   456
nipkow@61793
   457
lemma inorder_split: "inorder(split t) = inorder t"
nipkow@61793
   458
by(cases t rule: split.cases) (auto)
nipkow@61793
   459
nipkow@61793
   460
lemma inorder_skew: "inorder(skew t) = inorder t"
nipkow@61793
   461
by(cases t rule: skew.cases) (auto)
nipkow@61793
   462
nipkow@61793
   463
lemma inorder_insert:
nipkow@61793
   464
  "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
nipkow@61793
   465
by(induction t) (auto simp: ins_list_simps inorder_split inorder_skew)
nipkow@61793
   466
nipkow@62496
   467
nipkow@61793
   468
subsubsection "Proofs for delete"
nipkow@61793
   469
nipkow@62496
   470
lemma inorder_adjust: "t \<noteq> Leaf \<Longrightarrow> pre_adjust t \<Longrightarrow> inorder(adjust t) = inorder t"
nipkow@62526
   471
by(cases t)
nipkow@62496
   472
  (auto simp: adjust_def inorder_skew inorder_split invar.simps(2) pre_adjust.simps
nipkow@62496
   473
     split: tree.splits)
nipkow@62496
   474
nipkow@61793
   475
lemma del_maxD:
nipkow@62496
   476
  "\<lbrakk> del_max t = (t',x); t \<noteq> Leaf; invar t \<rbrakk> \<Longrightarrow> inorder t' @ [x] = inorder t"
nipkow@61793
   477
by(induction t arbitrary: t' rule: del_max.induct)
nipkow@62496
   478
  (auto simp: sorted_lems inorder_adjust pre_adj_if_postR post_del_max split: prod.splits)
nipkow@61793
   479
nipkow@61793
   480
lemma inorder_delete:
nipkow@62496
   481
  "invar t \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
nipkow@61793
   482
by(induction t)
nipkow@62496
   483
  (auto simp: del_list_simps inorder_adjust pre_adj_if_postL pre_adj_if_postR 
nipkow@62496
   484
              post_del_max post_delete del_maxD split: prod.splits)
nipkow@61793
   485
nipkow@62496
   486
interpretation I: Set_by_Ordered
nipkow@61793
   487
where empty = Leaf and isin = isin and insert = insert and delete = delete
nipkow@62496
   488
and inorder = inorder and inv = invar
nipkow@61793
   489
proof (standard, goal_cases)
nipkow@61793
   490
  case 1 show ?case by simp
nipkow@61793
   491
next
nipkow@61793
   492
  case 2 thus ?case by(simp add: isin_set)
nipkow@61793
   493
next
nipkow@61793
   494
  case 3 thus ?case by(simp add: inorder_insert)
nipkow@61793
   495
next
nipkow@61793
   496
  case 4 thus ?case by(simp add: inorder_delete)
nipkow@62496
   497
next
nipkow@62496
   498
  case 5 thus ?case by(simp)
nipkow@62496
   499
next
nipkow@62496
   500
  case 6 thus ?case by(simp add: invar_insert)
nipkow@62496
   501
next
nipkow@62496
   502
  case 7 thus ?case using post_delete by(auto simp: post_del_def)
nipkow@62496
   503
qed
nipkow@61793
   504
nipkow@62390
   505
end