(* Author: Tobias Nipkow *)
section "Splay Tree Implementation of Maps"
theory Splay_Map
imports
Splay_Set
Map_by_Ordered
begin
function splay :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
"splay x Leaf = Leaf" |
"x = fst a \<Longrightarrow> splay x (Node t1 a t2) = Node t1 a t2" |
"x = fst a \<Longrightarrow> x < fst b \<Longrightarrow> splay x (Node (Node t1 a t2) b t3) = Node t1 a (Node t2 b t3)" |
"x < fst a \<Longrightarrow> splay x (Node Leaf a t) = Node Leaf a t" |
"x < fst a \<Longrightarrow> x < fst b \<Longrightarrow> splay x (Node (Node Leaf a t1) b t2) = Node Leaf a (Node t1 b t2)" |
"x < fst a \<Longrightarrow> x < fst b \<Longrightarrow> t1 \<noteq> Leaf \<Longrightarrow>
splay x (Node (Node t1 a t2) b t3) =
(case splay x t1 of Node t11 y t12 \<Rightarrow> Node t11 y (Node t12 a (Node t2 b t3)))" |
"fst a < x \<Longrightarrow> x < fst b \<Longrightarrow> splay x (Node (Node t1 a Leaf) b t2) = Node t1 a (Node Leaf b t2)" |
"fst a < x \<Longrightarrow> x < fst b \<Longrightarrow> t2 \<noteq> Leaf \<Longrightarrow>
splay x (Node (Node t1 a t2) b t3) =
(case splay x t2 of Node t21 y t22 \<Rightarrow> Node (Node t1 a t21) y (Node t22 b t3))" |
"fst a < x \<Longrightarrow> x = fst b \<Longrightarrow> splay x (Node t1 a (Node t2 b t3)) = Node (Node t1 a t2) b t3" |
"fst a < x \<Longrightarrow> splay x (Node t a Leaf) = Node t a Leaf" |
"fst a < x \<Longrightarrow> x < fst b \<Longrightarrow> t2 \<noteq> Leaf \<Longrightarrow>
splay x (Node t1 a (Node t2 b t3)) =
(case splay x t2 of Node t21 y t22 \<Rightarrow> Node (Node t1 a t21) y (Node t22 b t3))" |
"fst a < x \<Longrightarrow> x < fst b \<Longrightarrow> splay x (Node t1 a (Node Leaf b t2)) = Node (Node t1 a Leaf) b t2" |
"fst a < x \<Longrightarrow> fst b < x \<Longrightarrow> splay x (Node t1 a (Node t2 b Leaf)) = Node (Node t1 a t2) b Leaf" |
"fst a < x \<Longrightarrow> fst b < x \<Longrightarrow> t3 \<noteq> Leaf \<Longrightarrow>
splay x (Node t1 a (Node t2 b t3)) =
(case splay x t3 of Node t31 y t32 \<Rightarrow> Node (Node (Node t1 a t2) b t31) y t32)"
apply(atomize_elim)
apply(auto)
(* 1 subgoal *)
apply (subst (asm) neq_Leaf_iff)
apply(auto)
apply (metis tree.exhaust surj_pair less_linear)+
done
termination splay
by lexicographic_order
lemma splay_code: "splay (x::_::cmp) t = (case t of Leaf \<Rightarrow> Leaf |
Node al a ar \<Rightarrow> (case cmp x (fst a) of
EQ \<Rightarrow> t |
LT \<Rightarrow> (case al of
Leaf \<Rightarrow> t |
Node bl b br \<Rightarrow> (case cmp x (fst b) of
EQ \<Rightarrow> Node bl b (Node br a ar) |
LT \<Rightarrow> if bl = Leaf then Node bl b (Node br a ar)
else case splay x bl of
Node bll y blr \<Rightarrow> Node bll y (Node blr b (Node br a ar)) |
GT \<Rightarrow> if br = Leaf then Node bl b (Node br a ar)
else case splay x br of
Node brl y brr \<Rightarrow> Node (Node bl b brl) y (Node brr a ar))) |
GT \<Rightarrow> (case ar of
Leaf \<Rightarrow> t |
Node bl b br \<Rightarrow> (case cmp x (fst b) of
EQ \<Rightarrow> Node (Node al a bl) b br |
LT \<Rightarrow> if bl = Leaf then Node (Node al a bl) b br
else case splay x bl of
Node bll y blr \<Rightarrow> Node (Node al a bll) y (Node blr b br) |
GT \<Rightarrow> if br=Leaf then Node (Node al a bl) b br
else case splay x br of
Node bll y blr \<Rightarrow> Node (Node (Node al a bl) b bll) y blr))))"
by(auto cong: case_tree_cong split: tree.split)
definition lookup :: "('a*'b)tree \<Rightarrow> 'a::linorder \<Rightarrow> 'b option" where "lookup t x =
(case splay x t of Leaf \<Rightarrow> None | Node _ (a,b) _ \<Rightarrow> if x=a then Some b else None)"
hide_const (open) insert
fun update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
"update x y t = (if t = Leaf then Node Leaf (x,y) Leaf
else case splay x t of
Node l a r \<Rightarrow> if x = fst a then Node l (x,y) r
else if x < fst a then Node l (x,y) (Node Leaf a r) else Node (Node l a Leaf) (x,y) r)"
definition delete :: "'a::linorder \<Rightarrow> ('a*'b) tree \<Rightarrow> ('a*'b) tree" where
"delete x t = (if t = Leaf then Leaf
else case splay x t of Node l a r \<Rightarrow>
if x = fst a
then if l = Leaf then r else case splay_max l of Node l' m r' \<Rightarrow> Node l' m r
else Node l a r)"
subsection "Functional Correctness Proofs"
lemma splay_Leaf_iff: "(splay x t = Leaf) = (t = Leaf)"
by(induction x t rule: splay.induct) (auto split: tree.splits)
subsubsection "Proofs for lookup"
lemma splay_map_of_inorder:
"splay x t = Node l a r \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
map_of (inorder t) x = (if x = fst a then Some(snd a) else None)"
by(induction x t arbitrary: l a r rule: splay.induct)
(auto simp: map_of_simps splay_Leaf_iff split: tree.splits)
lemma lookup_eq:
"sorted1(inorder t) \<Longrightarrow> lookup t x = map_of (inorder t) x"
by(auto simp: lookup_def splay_Leaf_iff splay_map_of_inorder split: tree.split)
subsubsection "Proofs for update"
lemma inorder_splay: "inorder(splay x t) = inorder t"
by(induction x t rule: splay.induct)
(auto simp: neq_Leaf_iff split: tree.split)
lemma sorted_splay:
"sorted1(inorder t) \<Longrightarrow> splay x t = Node l a r \<Longrightarrow>
sorted(map fst (inorder l) @ x # map fst (inorder r))"
unfolding inorder_splay[of x t, symmetric]
by(induction x t arbitrary: l a r rule: splay.induct)
(auto simp: sorted_lems sorted_Cons_le sorted_snoc_le splay_Leaf_iff split: tree.splits)
(* more upd_list lemmas; unify with basic set? *)
lemma upd_list_Cons:
"sorted1 ((x,y) # xs) \<Longrightarrow> upd_list x y xs = (x,y) # xs"
by (induction xs) auto
lemma upd_list_snoc:
"sorted1 (xs @ [(x,y)]) \<Longrightarrow> upd_list x y xs = xs @ [(x,y)]"
by(induction xs) (auto simp add: sorted_mid_iff2)
lemma inorder_update:
"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
using inorder_splay[of x t, symmetric] sorted_splay[of t x]
by(auto simp: upd_list_simps upd_list_Cons upd_list_snoc neq_Leaf_iff split: tree.split)
subsubsection "Proofs for delete"
(* more del_simp lemmas; unify with basic set? *)
lemma del_list_notin_Cons: "sorted (x # map fst xs) \<Longrightarrow> del_list x xs = xs"
by(induction xs)(auto simp: sorted_Cons_iff)
lemma del_list_sorted_app:
"sorted(map fst xs @ [x]) \<Longrightarrow> del_list x (xs @ ys) = xs @ del_list x ys"
by (induction xs) (auto simp: sorted_mid_iff2)
lemma inorder_splay_maxD:
"splay_max t = Node l a r \<Longrightarrow> sorted1(inorder t) \<Longrightarrow>
inorder l @ [a] = inorder t \<and> r = Leaf"
by(induction t arbitrary: l a r rule: splay_max.induct)
(auto simp: sorted_lems splay_max_Leaf_iff split: tree.splits if_splits)
lemma inorder_delete:
"sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
using inorder_splay[of x t, symmetric] sorted_splay[of t x]
by (auto simp: del_list_simps del_list_sorted_app delete_def
del_list_notin_Cons inorder_splay_maxD splay_Leaf_iff splay_max_Leaf_iff
split: tree.splits)
subsubsection "Overall Correctness"
interpretation Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update
and delete = delete and inorder = inorder and inv = "\<lambda>_. True"
proof (standard, goal_cases)
case 2 thus ?case by(simp add: lookup_eq)
next
case 3 thus ?case by(simp add: inorder_update del: update.simps)
next
case 4 thus ?case by(simp add: inorder_delete)
qed auto
end