(* Author: Tobias Nipkow *)
section \<open>1-2 Brother Tree Implementation of Sets\<close>
theory Brother12_Set
imports
Cmp
Set_by_Ordered
begin
subsection \<open>Data Type and Operations\<close>
datatype 'a bro =
N0 |
N1 "'a bro" |
N2 "'a bro" 'a "'a bro" |
(* auxiliary constructors: *)
L2 'a |
N3 "'a bro" 'a "'a bro" 'a "'a bro"
fun inorder :: "'a bro \<Rightarrow> 'a list" where
"inorder N0 = []" |
"inorder (N1 t) = inorder t" |
"inorder (N2 l a r) = inorder l @ a # inorder r" |
"inorder (L2 a) = [a]" |
"inorder (N3 t1 a1 t2 a2 t3) = inorder t1 @ a1 # inorder t2 @ a2 # inorder t3"
fun isin :: "'a bro \<Rightarrow> 'a::cmp \<Rightarrow> bool" where
"isin N0 x = False" |
"isin (N1 t) x = isin t x" |
"isin (N2 l a r) x =
(case cmp x a of
LT \<Rightarrow> isin l x |
EQ \<Rightarrow> True |
GT \<Rightarrow> isin r x)"
fun n1 :: "'a bro \<Rightarrow> 'a bro" where
"n1 (L2 a) = N2 N0 a N0" |
"n1 (N3 t1 a1 t2 a2 t3) = N2 (N2 t1 a1 t2) a2 (N1 t3)" |
"n1 t = N1 t"
hide_const (open) insert
locale insert
begin
fun n2 :: "'a bro \<Rightarrow> 'a \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
"n2 (L2 a1) a2 t = N3 N0 a1 N0 a2 t" |
"n2 (N3 t1 a1 t2 a2 t3) a3 (N1 t4) = N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4)" |
"n2 (N3 t1 a1 t2 a2 t3) a3 t4 = N3 (N2 t1 a1 t2) a2 (N1 t3) a3 t4" |
"n2 t1 a1 (L2 a2) = N3 t1 a1 N0 a2 N0" |
"n2 (N1 t1) a1 (N3 t2 a2 t3 a3 t4) = N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4)" |
"n2 t1 a1 (N3 t2 a2 t3 a3 t4) = N3 t1 a1 (N1 t2) a2 (N2 t3 a3 t4)" |
"n2 t1 a t2 = N2 t1 a t2"
fun ins :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
"ins x N0 = L2 x" |
"ins x (N1 t) = n1 (ins x t)" |
"ins x (N2 l a r) =
(case cmp x a of
LT \<Rightarrow> n2 (ins x l) a r |
EQ \<Rightarrow> N2 l a r |
GT \<Rightarrow> n2 l a (ins x r))"
fun tree :: "'a bro \<Rightarrow> 'a bro" where
"tree (L2 a) = N2 N0 a N0" |
"tree (N3 t1 a1 t2 a2 t3) = N2 (N2 t1 a1 t2) a2 (N1 t3)" |
"tree t = t"
definition insert :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
"insert x t = tree(ins x t)"
end
locale delete
begin
fun n2 :: "'a bro \<Rightarrow> 'a \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
"n2 (N1 t1) a1 (N1 t2) = N1 (N2 t1 a1 t2)" |
"n2 (N1 (N1 t1)) a1 (N2 (N1 t2) a2 (N2 t3 a3 t4)) =
N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
"n2 (N1 (N1 t1)) a1 (N2 (N2 t2 a2 t3) a3 (N1 t4)) =
N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
"n2 (N1 (N1 t1)) a1 (N2 (N2 t2 a2 t3) a3 (N2 t4 a4 t5)) =
N2 (N2 (N1 t1) a1 (N2 t2 a2 t3)) a3 (N1 (N2 t4 a4 t5))" |
"n2 (N2 (N1 t1) a1 (N2 t2 a2 t3)) a3 (N1 (N1 t4)) =
N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
"n2 (N2 (N2 t1 a1 t2) a2 (N1 t3)) a3 (N1 (N1 t4)) =
N1 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4))" |
"n2 (N2 (N2 t1 a1 t2) a2 (N2 t3 a3 t4)) a5 (N1 (N1 t5)) =
N2 (N1 (N2 t1 a1 t2)) a2 (N2 (N2 t3 a3 t4) a5 (N1 t5))" |
"n2 t1 a1 t2 = N2 t1 a1 t2"
fun del_min :: "'a bro \<Rightarrow> ('a \<times> 'a bro) option" where
"del_min N0 = None" |
"del_min (N1 t) =
(case del_min t of
None \<Rightarrow> None |
Some (a, t') \<Rightarrow> Some (a, N1 t'))" |
"del_min (N2 t1 a t2) =
(case del_min t1 of
None \<Rightarrow> Some (a, N1 t2) |
Some (b, t1') \<Rightarrow> Some (b, n2 t1' a t2))"
fun del :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
"del _ N0 = N0" |
"del x (N1 t) = N1 (del x t)" |
"del x (N2 l a r) =
(case cmp x a of
LT \<Rightarrow> n2 (del x l) a r |
GT \<Rightarrow> n2 l a (del x r) |
EQ \<Rightarrow> (case del_min r of
None \<Rightarrow> N1 l |
Some (b, r') \<Rightarrow> n2 l b r'))"
fun tree :: "'a bro \<Rightarrow> 'a bro" where
"tree (N1 t) = t" |
"tree t = t"
definition delete :: "'a::cmp \<Rightarrow> 'a bro \<Rightarrow> 'a bro" where
"delete a t = tree (del a t)"
end
subsection \<open>Invariants\<close>
fun B :: "nat \<Rightarrow> 'a bro set"
and U :: "nat \<Rightarrow> 'a bro set" where
"B 0 = {N0}" |
"B (Suc h) = { N2 t1 a t2 | t1 a t2.
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}" |
"U 0 = {}" |
"U (Suc h) = N1 ` B h"
abbreviation "T h \<equiv> B h \<union> U h"
fun Bp :: "nat \<Rightarrow> 'a bro set" where
"Bp 0 = B 0 \<union> L2 ` UNIV" |
"Bp (Suc 0) = B (Suc 0) \<union> {N3 N0 a N0 b N0|a b. True}" |
"Bp (Suc(Suc h)) = B (Suc(Suc h)) \<union>
{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)}"
fun Um :: "nat \<Rightarrow> 'a bro set" where
"Um 0 = {}" |
"Um (Suc h) = N1 ` T h"
subsection "Functional Correctness Proofs"
subsubsection "Proofs for isin"
lemma
"t \<in> T h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
by(induction h arbitrary: t) (fastforce simp: elems_simps1 split: if_splits)+
lemma isin_set: "t \<in> T h \<Longrightarrow>
sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
by(induction h arbitrary: t) (auto simp: elems_simps2 split: if_splits)
subsubsection "Proofs for insertion"
lemma inorder_n1: "inorder(n1 t) = inorder t"
by(induction t rule: n1.induct) (auto simp: sorted_lems)
context insert
begin
lemma inorder_n2: "inorder(n2 l a r) = inorder l @ a # inorder r"
by(cases "(l,a,r)" rule: n2.cases) (auto simp: sorted_lems)
lemma inorder_tree: "inorder(tree t) = inorder t"
by(cases t) auto
lemma inorder_ins: "t \<in> T h \<Longrightarrow>
sorted(inorder t) \<Longrightarrow> inorder(ins a t) = ins_list a (inorder t)"
by(induction h arbitrary: t) (auto simp: ins_list_simps inorder_n1 inorder_n2)
lemma inorder_insert: "t \<in> T h \<Longrightarrow>
sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
by(simp add: insert_def inorder_ins inorder_tree)
end
subsubsection \<open>Proofs for deletion\<close>
context delete
begin
lemma inorder_tree: "inorder(tree t) = inorder t"
by(cases t) auto
lemma inorder_n2: "inorder(n2 l a r) = inorder l @ a # inorder r"
by(induction l a r rule: n2.induct) (auto)
lemma inorder_del_min:
"t \<in> T h \<Longrightarrow> (del_min t = None \<longleftrightarrow> inorder t = []) \<and>
(del_min t = Some(a,t') \<longrightarrow> inorder t = a # inorder t')"
by(induction h arbitrary: t a t') (auto simp: inorder_n2 split: option.splits)
lemma inorder_del:
"t \<in> T h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)"
by(induction h arbitrary: t) (auto simp: del_list_simps inorder_n2
inorder_del_min[OF UnI1] inorder_del_min[OF UnI2] split: option.splits)
lemma inorder_delete:
"t \<in> T h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
by(simp add: delete_def inorder_del inorder_tree)
end
subsection \<open>Invariant Proofs\<close>
subsubsection \<open>Proofs for insertion\<close>
lemma n1_type: "t \<in> Bp h \<Longrightarrow> n1 t \<in> T (Suc h)"
by(cases h rule: Bp.cases) auto
context insert
begin
lemma tree_type: "t \<in> Bp h \<Longrightarrow> tree t \<in> B h \<union> B (Suc h)"
by(cases h rule: Bp.cases) auto
lemma n2_type:
"(t1 \<in> Bp h \<and> t2 \<in> T h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h)) \<and>
(t1 \<in> T h \<and> t2 \<in> Bp h \<longrightarrow> n2 t1 a t2 \<in> Bp (Suc h))"
apply(cases h rule: Bp.cases)
apply (auto)[2]
apply(rule conjI impI | erule conjE exE imageE | simp | erule disjE)+
done
lemma Bp_if_B: "t \<in> B h \<Longrightarrow> t \<in> Bp h"
by (cases h rule: Bp.cases) simp_all
text{* An automatic proof: *}
lemma
"(t \<in> B h \<longrightarrow> ins x t \<in> Bp h) \<and> (t \<in> U h \<longrightarrow> ins x t \<in> T h)"
apply(induction h arbitrary: t)
apply (simp)
apply (fastforce simp: Bp_if_B n2_type dest: n1_type)
done
text{* A detailed proof: *}
lemma ins_type:
shows "t \<in> B h \<Longrightarrow> ins x t \<in> Bp h" and "t \<in> U h \<Longrightarrow> ins x t \<in> T h"
proof(induction h arbitrary: t)
case 0
{ case 1 thus ?case by simp
next
case 2 thus ?case by simp }
next
case (Suc h)
{ case 1
then obtain t1 a t2 where [simp]: "t = N2 t1 a t2" and
t1: "t1 \<in> T h" and t2: "t2 \<in> T h" and t12: "t1 \<in> B h \<or> t2 \<in> B h"
by auto
{ assume "x < a"
hence "?case \<longleftrightarrow> n2 (ins x t1) a t2 \<in> Bp (Suc h)" by simp
also have "\<dots>"
proof cases
assume "t1 \<in> B h"
with t2 show ?thesis by (simp add: Suc.IH(1) n2_type)
next
assume "t1 \<notin> B h"
hence 1: "t1 \<in> U h" and 2: "t2 \<in> B h" using t1 t12 by auto
show ?thesis by (metis Suc.IH(2)[OF 1] Bp_if_B[OF 2] n2_type)
qed
finally have ?case .
}
moreover
{ assume "a < x"
hence "?case \<longleftrightarrow> n2 t1 a (ins x t2) \<in> Bp (Suc h)" by simp
also have "\<dots>"
proof cases
assume "t2 \<in> B h"
with t1 show ?thesis by (simp add: Suc.IH(1) n2_type)
next
assume "t2 \<notin> B h"
hence 1: "t1 \<in> B h" and 2: "t2 \<in> U h" using t2 t12 by auto
show ?thesis by (metis Bp_if_B[OF 1] Suc.IH(2)[OF 2] n2_type)
qed
}
moreover
{ assume "x = a"
from 1 have "t \<in> Bp (Suc h)" by(rule Bp_if_B)
hence "?case" using `x = a` by simp
}
ultimately show ?case by auto
next
case 2 thus ?case using Suc(1) n1_type by fastforce }
qed
lemma insert_type:
"t \<in> B h \<Longrightarrow> insert x t \<in> B h \<union> B (Suc h)"
unfolding insert_def by (metis ins_type(1) tree_type)
end
subsubsection "Proofs for deletion"
lemma B_simps[simp]:
"N1 t \<in> B h = False"
"L2 y \<in> B h = False"
"(N3 t1 a1 t2 a2 t3) \<in> B h = False"
"N0 \<in> B h \<longleftrightarrow> h = 0"
by (cases h, auto)+
context delete
begin
lemma n2_type1:
"\<lbrakk>t1 \<in> Um h; t2 \<in> B h\<rbrakk> \<Longrightarrow> n2 t1 a t2 \<in> T (Suc h)"
apply(cases h rule: Bp.cases)
apply auto[2]
apply(erule exE bexE conjE imageE | simp | erule disjE)+
done
lemma n2_type2:
"\<lbrakk>t1 \<in> B h ; t2 \<in> Um h \<rbrakk> \<Longrightarrow> n2 t1 a t2 \<in> T (Suc h)"
apply(cases h rule: Bp.cases)
apply auto[2]
apply(erule exE bexE conjE imageE | simp | erule disjE)+
done
lemma n2_type3:
"\<lbrakk>t1 \<in> T h ; t2 \<in> T h \<rbrakk> \<Longrightarrow> n2 t1 a t2 \<in> T (Suc h)"
apply(cases h rule: Bp.cases)
apply auto[2]
apply(erule exE bexE conjE imageE | simp | erule disjE)+
done
lemma del_minNoneN0: "\<lbrakk>t \<in> B h; del_min t = None\<rbrakk> \<Longrightarrow> t = N0"
by (cases t) (auto split: option.splits)
lemma del_minNoneN1 : "\<lbrakk>t \<in> U h; del_min t = None\<rbrakk> \<Longrightarrow> t = N1 N0"
by (cases h) (auto simp: del_minNoneN0 split: option.splits)
lemma del_min_type:
"t \<in> B h \<Longrightarrow> del_min t = Some (a, t') \<Longrightarrow> t' \<in> T h"
"t \<in> U h \<Longrightarrow> del_min t = Some (a, t') \<Longrightarrow> t' \<in> Um h"
proof (induction h arbitrary: t a t')
case (Suc h)
{ case 1
then obtain t1 a t2 where [simp]: "t = N2 t1 a t2" and
t12: "t1 \<in> T h" "t2 \<in> T h" "t1 \<in> B h \<or> t2 \<in> B h"
by auto
show ?case
proof (cases "del_min t1")
case None
show ?thesis
proof cases
assume "t1 \<in> B h"
with del_minNoneN0[OF this None] 1 show ?thesis by(auto)
next
assume "t1 \<notin> B h"
thus ?thesis using 1 None by (auto)
qed
next
case [simp]: (Some bt')
obtain b t1' where [simp]: "bt' = (b,t1')" by fastforce
show ?thesis
proof cases
assume "t1 \<in> B h"
from Suc.IH(1)[OF this] 1 have "t1' \<in> T h" by simp
from n2_type3[OF this t12(2)] 1 show ?thesis by auto
next
assume "t1 \<notin> B h"
hence t1: "t1 \<in> U h" and t2: "t2 \<in> B h" using t12 by auto
from Suc.IH(2)[OF t1] have "t1' \<in> Um h" by simp
from n2_type1[OF this t2] 1 show ?thesis by auto
qed
qed
}
{ case 2
then obtain t1 where [simp]: "t = N1 t1" and t1: "t1 \<in> B h" by auto
show ?case
proof (cases "del_min t1")
case None
with del_minNoneN0[OF t1 None] 2 show ?thesis by(auto)
next
case [simp]: (Some bt')
obtain b t1' where [simp]: "bt' = (b,t1')" by fastforce
from Suc.IH(1)[OF t1] have "t1' \<in> T h" by simp
thus ?thesis using 2 by auto
qed
}
qed auto
lemma del_type:
"t \<in> B h \<Longrightarrow> del x t \<in> T h"
"t \<in> U h \<Longrightarrow> del x t \<in> Um h"
proof (induction h arbitrary: x t)
case (Suc h)
{ case 1
then obtain l a r where [simp]: "t = N2 l a r" and
lr: "l \<in> T h" "r \<in> T h" "l \<in> B h \<or> r \<in> B h" by auto
{ assume "x < a"
have ?case
proof cases
assume "l \<in> B h"
from n2_type3[OF Suc.IH(1)[OF this] lr(2)]
show ?thesis using `x<a` by(simp)
next
assume "l \<notin> B h"
hence "l \<in> U h" "r \<in> B h" using lr by auto
from n2_type1[OF Suc.IH(2)[OF this(1)] this(2)]
show ?thesis using `x<a` by(simp)
qed
} moreover
{ assume "x > a"
have ?case
proof cases
assume "r \<in> B h"
from n2_type3[OF lr(1) Suc.IH(1)[OF this]]
show ?thesis using `x>a` by(simp)
next
assume "r \<notin> B h"
hence "l \<in> B h" "r \<in> U h" using lr by auto
from n2_type2[OF this(1) Suc.IH(2)[OF this(2)]]
show ?thesis using `x>a` by(simp)
qed
} moreover
{ assume [simp]: "x=a"
have ?case
proof (cases "del_min r")
case None
show ?thesis
proof cases
assume "r \<in> B h"
with del_minNoneN0[OF this None] lr show ?thesis by(simp)
next
assume "r \<notin> B h"
hence "r \<in> U h" using lr by auto
with del_minNoneN1[OF this None] lr(3) show ?thesis by (simp)
qed
next
case [simp]: (Some br')
obtain b r' where [simp]: "br' = (b,r')" by fastforce
show ?thesis
proof cases
assume "r \<in> B h"
from del_min_type(1)[OF this] n2_type3[OF lr(1)]
show ?thesis by simp
next
assume "r \<notin> B h"
hence "l \<in> B h" and "r \<in> U h" using lr by auto
from del_min_type(2)[OF this(2)] n2_type2[OF this(1)]
show ?thesis by simp
qed
qed
} ultimately show ?case by auto
}
{ case 2 with Suc.IH(1) show ?case by auto }
qed auto
lemma tree_type: "t \<in> T (h+1) \<Longrightarrow> tree t : B (h+1) \<union> B h"
by(auto)
lemma delete_type: "t \<in> B h \<Longrightarrow> delete x t \<in> B h \<union> B(h-1)"
unfolding delete_def
by (cases h) (simp, metis del_type(1) tree_type Suc_eq_plus1 diff_Suc_1)
end
subsection "Overall correctness"
interpretation Set_by_Ordered
where empty = N0 and isin = isin and insert = insert.insert
and delete = delete.delete and inorder = inorder and inv = "\<lambda>t. \<exists>h. t \<in> B h"
proof (standard, goal_cases)
case 2 thus ?case by(auto intro!: isin_set)
next
case 3 thus ?case by(auto intro!: insert.inorder_insert)
next
case 4 thus ?case by(auto intro!: delete.inorder_delete)
next
case 6 thus ?case using insert.insert_type by blast
next
case 7 thus ?case using delete.delete_type by blast
qed auto
end