--- a/src/HOL/Library/RBT.thy Tue Mar 02 15:39:15 2010 +0100
+++ b/src/HOL/Library/RBT.thy Wed Mar 03 08:28:33 2010 +0100
@@ -11,135 +11,151 @@
begin
datatype color = R | B
-datatype ('a,'b)"rbt" = Empty | Tr color "('a,'b)rbt" 'a 'b "('a,'b)rbt"
+datatype ('a, 'b) rbt = Empty | Branch color "('a, 'b) rbt" 'a 'b "('a, 'b) rbt"
+
+lemma rbt_cases:
+ obtains (Empty) "t = Empty"
+ | (Red) l k v r where "t = Branch R l k v r"
+ | (Black) l k v r where "t = Branch B l k v r"
+proof (cases t)
+ case Empty with that show thesis by blast
+next
+ case (Branch c) with that show thesis by (cases c) blast+
+qed
+
+text {* Content of a tree *}
+
+primrec entries
+where
+ "entries Empty = []"
+| "entries (Branch _ l k v r) = entries l @ (k,v) # entries r"
text {* Search tree properties *}
-primrec
- pin_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> bool"
+primrec entry_in_tree :: "'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
where
- "pin_tree k v Empty = False"
-| "pin_tree k v (Tr c l x y r) = (k = x \<and> v = y \<or> pin_tree k v l \<or> pin_tree k v r)"
+ "entry_in_tree k v Empty = False"
+| "entry_in_tree k v (Branch c l x y r) \<longleftrightarrow> k = x \<and> v = y \<or> entry_in_tree k v l \<or> entry_in_tree k v r"
-primrec
- keys :: "('k,'v) rbt \<Rightarrow> 'k set"
+primrec keys :: "('k, 'v) rbt \<Rightarrow> 'k set"
where
"keys Empty = {}"
-| "keys (Tr _ l k _ r) = { k } \<union> keys l \<union> keys r"
+| "keys (Branch _ l k _ r) = { k } \<union> keys l \<union> keys r"
-lemma pint_keys: "pin_tree k v t \<Longrightarrow> k \<in> keys t" by (induct t) auto
+lemma entry_in_tree_keys:
+ "entry_in_tree k v t \<Longrightarrow> k \<in> keys t"
+ by (induct t) auto
-primrec tlt :: "'a\<Colon>order \<Rightarrow> ('a,'b) rbt \<Rightarrow> bool"
+definition tree_less :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"
where
- "tlt k Empty = True"
-| "tlt k (Tr c lt kt v rt) = (kt < k \<and> tlt k lt \<and> tlt k rt)"
+ tree_less_prop: "tree_less k t \<longleftrightarrow> (\<forall>x\<in>keys t. x < k)"
+
+abbreviation tree_less_symbol (infix "|\<guillemotleft>" 50)
+where "t |\<guillemotleft> x \<equiv> tree_less x t"
-abbreviation tllt (infix "|\<guillemotleft>" 50)
-where "t |\<guillemotleft> x == tlt x t"
+definition tree_greater :: "'a\<Colon>order \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50)
+where
+ tree_greater_prop: "tree_greater k t = (\<forall>x\<in>keys t. k < x)"
-primrec tgt :: "'a\<Colon>order \<Rightarrow> ('a,'b) rbt \<Rightarrow> bool" (infix "\<guillemotleft>|" 50)
-where
- "tgt k Empty = True"
-| "tgt k (Tr c lt kt v rt) = (k < kt \<and> tgt k lt \<and> tgt k rt)"
+lemma tree_less_simps [simp]:
+ "tree_less k Empty = True"
+ "tree_less k (Branch c lt kt v rt) \<longleftrightarrow> kt < k \<and> tree_less k lt \<and> tree_less k rt"
+ by (auto simp add: tree_less_prop)
-lemma tlt_prop: "(t |\<guillemotleft> k) = (\<forall>x\<in>keys t. x < k)" by (induct t) auto
-lemma tgt_prop: "(k \<guillemotleft>| t) = (\<forall>x\<in>keys t. k < x)" by (induct t) auto
-lemmas tlgt_props = tlt_prop tgt_prop
+lemma tree_greater_simps [simp]:
+ "tree_greater k Empty = True"
+ "tree_greater k (Branch c lt kt v rt) \<longleftrightarrow> k < kt \<and> tree_greater k lt \<and> tree_greater k rt"
+ by (auto simp add: tree_greater_prop)
-lemmas tgt_nit = tgt_prop pint_keys
-lemmas tlt_nit = tlt_prop pint_keys
+lemmas tree_ord_props = tree_less_prop tree_greater_prop
-lemma tlt_trans: "\<lbrakk> t |\<guillemotleft> x; x < y \<rbrakk> \<Longrightarrow> t |\<guillemotleft> y"
- and tgt_trans: "\<lbrakk> x < y; y \<guillemotleft>| t\<rbrakk> \<Longrightarrow> x \<guillemotleft>| t"
-by (auto simp: tlgt_props)
-
+lemmas tree_greater_nit = tree_greater_prop entry_in_tree_keys
+lemmas tree_less_nit = tree_less_prop entry_in_tree_keys
-primrec st :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
-where
- "st Empty = True"
-| "st (Tr c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> st l \<and> st r)"
+lemma tree_less_trans: "t |\<guillemotleft> x \<Longrightarrow> x < y \<Longrightarrow> t |\<guillemotleft> y"
+ and tree_greater_trans: "x < y \<Longrightarrow> y \<guillemotleft>| t \<Longrightarrow> x \<guillemotleft>| t"
+by (auto simp: tree_ord_props)
-primrec map_of :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
+primrec sorted :: "('a::linorder, 'b) rbt \<Rightarrow> bool"
where
- "map_of Empty k = None"
-| "map_of (Tr _ l x y r) k = (if k < x then map_of l k else if x < k then map_of r k else Some y)"
+ "sorted Empty = True"
+| "sorted (Branch c l k v r) = (l |\<guillemotleft> k \<and> k \<guillemotleft>| r \<and> sorted l \<and> sorted r)"
-lemma map_of_tlt[simp]: "t |\<guillemotleft> k \<Longrightarrow> map_of t k = None"
+primrec lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b"
+where
+ "lookup Empty k = None"
+| "lookup (Branch _ l x y r) k = (if k < x then lookup l k else if x < k then lookup r k else Some y)"
+
+lemma lookup_tree_less[simp]: "t |\<guillemotleft> k \<Longrightarrow> lookup t k = None"
by (induct t) auto
-lemma map_of_tgt[simp]: "k \<guillemotleft>| t \<Longrightarrow> map_of t k = None"
+lemma lookup_tree_greater[simp]: "k \<guillemotleft>| t \<Longrightarrow> lookup t k = None"
by (induct t) auto
-lemma mapof_keys: "st t \<Longrightarrow> dom (map_of t) = keys t"
-by (induct t) (auto simp: dom_def tgt_prop tlt_prop)
+lemma lookup_keys: "sorted t \<Longrightarrow> dom (lookup t) = keys t"
+by (induct t) (auto simp: dom_def tree_greater_prop tree_less_prop)
-lemma mapof_pit: "st t \<Longrightarrow> (map_of t k = Some v) = pin_tree k v t"
-by (induct t) (auto simp: tlt_prop tgt_prop pint_keys)
+lemma lookup_pit: "sorted t \<Longrightarrow> (lookup t k = Some v) = entry_in_tree k v t"
+by (induct t) (auto simp: tree_less_prop tree_greater_prop entry_in_tree_keys)
-lemma map_of_Empty: "map_of Empty = empty"
+lemma lookup_Empty: "lookup Empty = empty"
by (rule ext) simp
(* a kind of extensionality *)
-lemma mapof_from_pit:
- assumes st: "st t1" "st t2"
- and eq: "\<And>v. pin_tree (k\<Colon>'a\<Colon>linorder) v t1 = pin_tree k v t2"
- shows "map_of t1 k = map_of t2 k"
-proof (cases "map_of t1 k")
+lemma lookup_from_pit:
+ assumes sorted: "sorted t1" "sorted t2"
+ and eq: "\<And>v. entry_in_tree (k\<Colon>'a\<Colon>linorder) v t1 = entry_in_tree k v t2"
+ shows "lookup t1 k = lookup t2 k"
+proof (cases "lookup t1 k")
case None
- then have "\<And>v. \<not> pin_tree k v t1"
- by (simp add: mapof_pit[symmetric] st)
+ then have "\<And>v. \<not> entry_in_tree k v t1"
+ by (simp add: lookup_pit[symmetric] sorted)
with None show ?thesis
- by (cases "map_of t2 k") (auto simp: mapof_pit st eq)
+ by (cases "lookup t2 k") (auto simp: lookup_pit sorted eq)
next
case (Some a)
then show ?thesis
- apply (cases "map_of t2 k")
- apply (auto simp: mapof_pit st eq)
- by (auto simp add: mapof_pit[symmetric] st Some)
+ apply (cases "lookup t2 k")
+ apply (auto simp: lookup_pit sorted eq)
+ by (auto simp add: lookup_pit[symmetric] sorted Some)
qed
subsection {* Red-black properties *}
-primrec treec :: "('a,'b) rbt \<Rightarrow> color"
+primrec color_of :: "('a, 'b) rbt \<Rightarrow> color"
where
- "treec Empty = B"
-| "treec (Tr c _ _ _ _) = c"
+ "color_of Empty = B"
+| "color_of (Branch c _ _ _ _) = c"
-primrec inv1 :: "('a,'b) rbt \<Rightarrow> bool"
+primrec bheight :: "('a,'b) rbt \<Rightarrow> nat"
+where
+ "bheight Empty = 0"
+| "bheight (Branch c lt k v rt) = (if c = B then Suc (bheight lt) else bheight lt)"
+
+primrec inv1 :: "('a, 'b) rbt \<Rightarrow> bool"
where
"inv1 Empty = True"
-| "inv1 (Tr c lt k v rt) = (inv1 lt \<and> inv1 rt \<and> (c = B \<or> treec lt = B \<and> treec rt = B))"
+| "inv1 (Branch c lt k v rt) \<longleftrightarrow> inv1 lt \<and> inv1 rt \<and> (c = B \<or> color_of lt = B \<and> color_of rt = B)"
-(* Weaker version *)
-primrec inv1l :: "('a,'b) rbt \<Rightarrow> bool"
+primrec inv1l :: "('a, 'b) rbt \<Rightarrow> bool" -- {* Weaker version *}
where
"inv1l Empty = True"
-| "inv1l (Tr c l k v r) = (inv1 l \<and> inv1 r)"
+| "inv1l (Branch c l k v r) = (inv1 l \<and> inv1 r)"
lemma [simp]: "inv1 t \<Longrightarrow> inv1l t" by (cases t) simp+
-primrec bh :: "('a,'b) rbt \<Rightarrow> nat"
-where
- "bh Empty = 0"
-| "bh (Tr c lt k v rt) = (if c = B then Suc (bh lt) else bh lt)"
-
-primrec inv2 :: "('a,'b) rbt \<Rightarrow> bool"
+primrec inv2 :: "('a, 'b) rbt \<Rightarrow> bool"
where
"inv2 Empty = True"
-| "inv2 (Tr c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bh lt = bh rt)"
+| "inv2 (Branch c lt k v rt) = (inv2 lt \<and> inv2 rt \<and> bheight lt = bheight rt)"
-definition
- "isrbt t = (inv1 t \<and> inv2 t \<and> treec t = B \<and> st t)"
-
-lemma isrbt_st[simp]: "isrbt t \<Longrightarrow> st t" by (simp add: isrbt_def)
+definition is_rbt :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where
+ "is_rbt t \<longleftrightarrow> inv1 t \<and> inv2 t \<and> color_of t = B \<and> sorted t"
-lemma rbt_cases:
- obtains (Empty) "t = Empty"
- | (Red) l k v r where "t = Tr R l k v r"
- | (Black) l k v r where "t = Tr B l k v r"
-by (cases t, simp) (case_tac "color", auto)
+lemma is_rbt_sorted [simp]:
+ "is_rbt t \<Longrightarrow> sorted t" by (simp add: is_rbt_def)
-theorem Empty_isrbt[simp]: "isrbt Empty"
-unfolding isrbt_def by simp
+theorem Empty_is_rbt [simp]:
+ "is_rbt Empty" by (simp add: is_rbt_def)
subsection {* Insertion *}
@@ -147,80 +163,80 @@
fun (* slow, due to massive case splitting *)
balance :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "balance (Tr R a w x b) s t (Tr R c y z d) = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
- "balance (Tr R (Tr R a w x b) s t c) y z d = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
- "balance (Tr R a w x (Tr R b s t c)) y z d = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
- "balance a w x (Tr R b s t (Tr R c y z d)) = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
- "balance a w x (Tr R (Tr R b s t c) y z d) = Tr R (Tr B a w x b) s t (Tr B c y z d)" |
- "balance a s t b = Tr B a s t b"
+ "balance (Branch R a w x b) s t (Branch R c y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+ "balance (Branch R (Branch R a w x b) s t c) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+ "balance (Branch R a w x (Branch R b s t c)) y z d = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+ "balance a w x (Branch R b s t (Branch R c y z d)) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+ "balance a w x (Branch R (Branch R b s t c) y z d) = Branch R (Branch B a w x b) s t (Branch B c y z d)" |
+ "balance a s t b = Branch B a s t b"
lemma balance_inv1: "\<lbrakk>inv1l l; inv1l r\<rbrakk> \<Longrightarrow> inv1 (balance l k v r)"
by (induct l k v r rule: balance.induct) auto
-lemma balance_bh: "bh l = bh r \<Longrightarrow> bh (balance l k v r) = Suc (bh l)"
+lemma balance_bheight: "bheight l = bheight r \<Longrightarrow> bheight (balance l k v r) = Suc (bheight l)"
by (induct l k v r rule: balance.induct) auto
lemma balance_inv2:
- assumes "inv2 l" "inv2 r" "bh l = bh r"
+ assumes "inv2 l" "inv2 r" "bheight l = bheight r"
shows "inv2 (balance l k v r)"
using assms
by (induct l k v r rule: balance.induct) auto
-lemma balance_tgt[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)"
+lemma balance_tree_greater[simp]: "(v \<guillemotleft>| balance a k x b) = (v \<guillemotleft>| a \<and> v \<guillemotleft>| b \<and> v < k)"
by (induct a k x b rule: balance.induct) auto
-lemma balance_tlt[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
+lemma balance_tree_less[simp]: "(balance a k x b |\<guillemotleft> v) = (a |\<guillemotleft> v \<and> b |\<guillemotleft> v \<and> k < v)"
by (induct a k x b rule: balance.induct) auto
-lemma balance_st:
+lemma balance_sorted:
fixes k :: "'a::linorder"
- assumes "st l" "st r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
- shows "st (balance l k v r)"
+ assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+ shows "sorted (balance l k v r)"
using assms proof (induct l k v r rule: balance.induct)
case ("2_2" a x w b y t c z s va vb vd vc)
- hence "y < z \<and> z \<guillemotleft>| Tr B va vb vd vc"
- by (auto simp add: tlgt_props)
- hence "tgt y (Tr B va vb vd vc)" by (blast dest: tgt_trans)
+ hence "y < z \<and> z \<guillemotleft>| Branch B va vb vd vc"
+ by (auto simp add: tree_ord_props)
+ hence "tree_greater y (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
with "2_2" show ?case by simp
next
case ("3_2" va vb vd vc x w b y s c z)
- from "3_2" have "x < y \<and> tlt x (Tr B va vb vd vc)"
- by (simp add: tlt.simps tgt.simps)
- hence "tlt y (Tr B va vb vd vc)" by (blast dest: tlt_trans)
+ from "3_2" have "x < y \<and> tree_less x (Branch B va vb vd vc)"
+ by simp
+ hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
with "3_2" show ?case by simp
next
case ("3_3" x w b y s c z t va vb vd vc)
- from "3_3" have "y < z \<and> tgt z (Tr B va vb vd vc)" by simp
- hence "tgt y (Tr B va vb vd vc)" by (blast dest: tgt_trans)
+ from "3_3" have "y < z \<and> tree_greater z (Branch B va vb vd vc)" by simp
+ hence "tree_greater y (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
with "3_3" show ?case by simp
next
case ("3_4" vd ve vg vf x w b y s c z t va vb vii vc)
- hence "x < y \<and> tlt x (Tr B vd ve vg vf)" by simp
- hence 1: "tlt y (Tr B vd ve vg vf)" by (blast dest: tlt_trans)
- from "3_4" have "y < z \<and> tgt z (Tr B va vb vii vc)" by simp
- hence "tgt y (Tr B va vb vii vc)" by (blast dest: tgt_trans)
+ hence "x < y \<and> tree_less x (Branch B vd ve vg vf)" by simp
+ hence 1: "tree_less y (Branch B vd ve vg vf)" by (blast dest: tree_less_trans)
+ from "3_4" have "y < z \<and> tree_greater z (Branch B va vb vii vc)" by simp
+ hence "tree_greater y (Branch B va vb vii vc)" by (blast dest: tree_greater_trans)
with 1 "3_4" show ?case by simp
next
case ("4_2" va vb vd vc x w b y s c z t dd)
- hence "x < y \<and> tlt x (Tr B va vb vd vc)" by simp
- hence "tlt y (Tr B va vb vd vc)" by (blast dest: tlt_trans)
+ hence "x < y \<and> tree_less x (Branch B va vb vd vc)" by simp
+ hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
with "4_2" show ?case by simp
next
case ("5_2" x w b y s c z t va vb vd vc)
- hence "y < z \<and> tgt z (Tr B va vb vd vc)" by simp
- hence "tgt y (Tr B va vb vd vc)" by (blast dest: tgt_trans)
+ hence "y < z \<and> tree_greater z (Branch B va vb vd vc)" by simp
+ hence "tree_greater y (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
with "5_2" show ?case by simp
next
case ("5_3" va vb vd vc x w b y s c z t)
- hence "x < y \<and> tlt x (Tr B va vb vd vc)" by simp
- hence "tlt y (Tr B va vb vd vc)" by (blast dest: tlt_trans)
+ hence "x < y \<and> tree_less x (Branch B va vb vd vc)" by simp
+ hence "tree_less y (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
with "5_3" show ?case by simp
next
case ("5_4" va vb vg vc x w b y s c z t vd ve vii vf)
- hence "x < y \<and> tlt x (Tr B va vb vg vc)" by simp
- hence 1: "tlt y (Tr B va vb vg vc)" by (blast dest: tlt_trans)
- from "5_4" have "y < z \<and> tgt z (Tr B vd ve vii vf)" by simp
- hence "tgt y (Tr B vd ve vii vf)" by (blast dest: tgt_trans)
+ hence "x < y \<and> tree_less x (Branch B va vb vg vc)" by simp
+ hence 1: "tree_less y (Branch B va vb vg vc)" by (blast dest: tree_less_trans)
+ from "5_4" have "y < z \<and> tree_greater z (Branch B vd ve vii vf)" by simp
+ hence "tree_greater y (Branch B vd ve vii vf)" by (blast dest: tree_greater_trans)
with 1 "5_4" show ?case by simp
qed simp+
@@ -229,62 +245,62 @@
by (induct l k v r rule: balance.induct) auto
lemma balance_pit:
- "pin_tree k x (balance l v y r) = (pin_tree k x l \<or> k = v \<and> x = y \<or> pin_tree k x r)"
+ "entry_in_tree k x (balance l v y r) = (entry_in_tree k x l \<or> k = v \<and> x = y \<or> entry_in_tree k x r)"
by (induct l v y r rule: balance.induct) auto
-lemma map_of_balance[simp]:
+lemma lookup_balance[simp]:
fixes k :: "'a::linorder"
-assumes "st l" "st r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
-shows "map_of (balance l k v r) x = map_of (Tr B l k v r) x"
-by (rule mapof_from_pit) (auto simp:assms balance_pit balance_st)
+assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+shows "lookup (balance l k v r) x = lookup (Branch B l k v r) x"
+by (rule lookup_from_pit) (auto simp:assms balance_pit balance_sorted)
primrec paint :: "color \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
"paint c Empty = Empty"
-| "paint c (Tr _ l k v r) = Tr c l k v r"
+| "paint c (Branch _ l k v r) = Branch c l k v r"
lemma paint_inv1l[simp]: "inv1l t \<Longrightarrow> inv1l (paint c t)" by (cases t) auto
lemma paint_inv1[simp]: "inv1l t \<Longrightarrow> inv1 (paint B t)" by (cases t) auto
lemma paint_inv2[simp]: "inv2 t \<Longrightarrow> inv2 (paint c t)" by (cases t) auto
-lemma paint_treec[simp]: "treec (paint B t) = B" by (cases t) auto
-lemma paint_st[simp]: "st t \<Longrightarrow> st (paint c t)" by (cases t) auto
-lemma paint_pit[simp]: "pin_tree k x (paint c t) = pin_tree k x t" by (cases t) auto
-lemma paint_mapof[simp]: "map_of (paint c t) = map_of t" by (rule ext) (cases t, auto)
-lemma paint_tgt[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
-lemma paint_tlt[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
+lemma paint_color_of[simp]: "color_of (paint B t) = B" by (cases t) auto
+lemma paint_sorted[simp]: "sorted t \<Longrightarrow> sorted (paint c t)" by (cases t) auto
+lemma paint_pit[simp]: "entry_in_tree k x (paint c t) = entry_in_tree k x t" by (cases t) auto
+lemma paint_lookup[simp]: "lookup (paint c t) = lookup t" by (rule ext) (cases t, auto)
+lemma paint_tree_greater[simp]: "(v \<guillemotleft>| paint c t) = (v \<guillemotleft>| t)" by (cases t) auto
+lemma paint_tree_less[simp]: "(paint c t |\<guillemotleft> v) = (t |\<guillemotleft> v)" by (cases t) auto
fun
ins :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "ins f k v Empty = Tr R Empty k v Empty" |
- "ins f k v (Tr B l x y r) = (if k < x then balance (ins f k v l) x y r
+ "ins f k v Empty = Branch R Empty k v Empty" |
+ "ins f k v (Branch B l x y r) = (if k < x then balance (ins f k v l) x y r
else if k > x then balance l x y (ins f k v r)
- else Tr B l x (f k y v) r)" |
- "ins f k v (Tr R l x y r) = (if k < x then Tr R (ins f k v l) x y r
- else if k > x then Tr R l x y (ins f k v r)
- else Tr R l x (f k y v) r)"
+ else Branch B l x (f k y v) r)" |
+ "ins f k v (Branch R l x y r) = (if k < x then Branch R (ins f k v l) x y r
+ else if k > x then Branch R l x y (ins f k v r)
+ else Branch R l x (f k y v) r)"
lemma ins_inv1_inv2:
assumes "inv1 t" "inv2 t"
- shows "inv2 (ins f k x t)" "bh (ins f k x t) = bh t"
- "treec t = B \<Longrightarrow> inv1 (ins f k x t)" "inv1l (ins f k x t)"
+ shows "inv2 (ins f k x t)" "bheight (ins f k x t) = bheight t"
+ "color_of t = B \<Longrightarrow> inv1 (ins f k x t)" "inv1l (ins f k x t)"
using assms
- by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bh)
+ by (induct f k x t rule: ins.induct) (auto simp: balance_inv1 balance_inv2 balance_bheight)
-lemma ins_tgt[simp]: "(v \<guillemotleft>| ins f k x t) = (v \<guillemotleft>| t \<and> k > v)"
+lemma ins_tree_greater[simp]: "(v \<guillemotleft>| ins f k x t) = (v \<guillemotleft>| t \<and> k > v)"
by (induct f k x t rule: ins.induct) auto
-lemma ins_tlt[simp]: "(ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
+lemma ins_tree_less[simp]: "(ins f k x t |\<guillemotleft> v) = (t |\<guillemotleft> v \<and> k < v)"
by (induct f k x t rule: ins.induct) auto
-lemma ins_st[simp]: "st t \<Longrightarrow> st (ins f k x t)"
- by (induct f k x t rule: ins.induct) (auto simp: balance_st)
+lemma ins_sorted[simp]: "sorted t \<Longrightarrow> sorted (ins f k x t)"
+ by (induct f k x t rule: ins.induct) (auto simp: balance_sorted)
lemma keys_ins: "keys (ins f k v t) = { k } \<union> keys t"
by (induct f k v t rule: ins.induct) auto
-lemma map_of_ins:
+lemma lookup_ins:
fixes k :: "'a::linorder"
- assumes "st t"
- shows "map_of (ins f k v t) x = ((map_of t)(k |-> case map_of t k of None \<Rightarrow> v
+ assumes "sorted t"
+ shows "lookup (ins f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v
| Some w \<Rightarrow> f k w v)) x"
using assms by (induct f k v t rule: ins.induct) auto
@@ -293,98 +309,97 @@
where
"insertwithkey f k v t = paint B (ins f k v t)"
-lemma insertwk_st: "st t \<Longrightarrow> st (insertwithkey f k x t)"
+lemma insertwk_sorted: "sorted t \<Longrightarrow> sorted (insertwithkey f k x t)"
by (auto simp: insertwithkey_def)
-theorem insertwk_isrbt:
- assumes inv: "isrbt t"
- shows "isrbt (insertwithkey f k x t)"
+theorem insertwk_is_rbt:
+ assumes inv: "is_rbt t"
+ shows "is_rbt (insertwithkey f k x t)"
using assms
-unfolding insertwithkey_def isrbt_def
+unfolding insertwithkey_def is_rbt_def
by (auto simp: ins_inv1_inv2)
-lemma map_of_insertwk:
- assumes "st t"
- shows "map_of (insertwithkey f k v t) x = ((map_of t)(k |-> case map_of t k of None \<Rightarrow> v
+lemma lookup_insertwk:
+ assumes "sorted t"
+ shows "lookup (insertwithkey f k v t) x = ((lookup t)(k |-> case lookup t k of None \<Rightarrow> v
| Some w \<Rightarrow> f k w v)) x"
unfolding insertwithkey_def using assms
-by (simp add:map_of_ins)
+by (simp add:lookup_ins)
definition
insertw_def: "insertwith f = insertwithkey (\<lambda>_. f)"
-lemma insertw_st: "st t \<Longrightarrow> st (insertwith f k v t)" by (simp add: insertwk_st insertw_def)
-theorem insertw_isrbt: "isrbt t \<Longrightarrow> isrbt (insertwith f k v t)" by (simp add: insertwk_isrbt insertw_def)
+lemma insertw_sorted: "sorted t \<Longrightarrow> sorted (insertwith f k v t)" by (simp add: insertwk_sorted insertw_def)
+theorem insertw_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insertwith f k v t)" by (simp add: insertwk_is_rbt insertw_def)
-lemma map_of_insertw:
- assumes "isrbt t"
- shows "map_of (insertwith f k v t) = (map_of t)(k \<mapsto> (if k:dom (map_of t) then f (the (map_of t k)) v else v))"
+lemma lookup_insertw:
+ assumes "is_rbt t"
+ shows "lookup (insertwith f k v t) = (lookup t)(k \<mapsto> (if k:dom (lookup t) then f (the (lookup t k)) v else v))"
using assms
unfolding insertw_def
-by (rule_tac ext) (cases "map_of t k", auto simp:map_of_insertwk dom_def)
-
+by (rule_tac ext) (cases "lookup t k", auto simp:lookup_insertwk dom_def)
-definition
- "insrt k v t = insertwithkey (\<lambda>_ _ nv. nv) k v t"
+definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" where
+ "insert k v t = insertwithkey (\<lambda>_ _ nv. nv) k v t"
-lemma insrt_st: "st t \<Longrightarrow> st (insrt k v t)" by (simp add: insertwk_st insrt_def)
-theorem insrt_isrbt: "isrbt t \<Longrightarrow> isrbt (insrt k v t)" by (simp add: insertwk_isrbt insrt_def)
+lemma insert_sorted: "sorted t \<Longrightarrow> sorted (insert k v t)" by (simp add: insertwk_sorted insert_def)
+theorem insert_is_rbt: "is_rbt t \<Longrightarrow> is_rbt (insert k v t)" by (simp add: insertwk_is_rbt insert_def)
-lemma map_of_insert:
- assumes "isrbt t"
- shows "map_of (insrt k v t) = (map_of t)(k\<mapsto>v)"
-unfolding insrt_def
+lemma lookup_insert:
+ assumes "is_rbt t"
+ shows "lookup (insert k v t) = (lookup t)(k\<mapsto>v)"
+unfolding insert_def
using assms
-by (rule_tac ext) (simp add: map_of_insertwk split:option.split)
+by (rule_tac ext) (simp add: lookup_insertwk split:option.split)
subsection {* Deletion *}
-lemma bh_paintR'[simp]: "treec t = B \<Longrightarrow> bh (paint R t) = bh t - 1"
+lemma bheight_paintR'[simp]: "color_of t = B \<Longrightarrow> bheight (paint R t) = bheight t - 1"
by (cases t rule: rbt_cases) auto
fun
balleft :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "balleft (Tr R a k x b) s y c = Tr R (Tr B a k x b) s y c" |
- "balleft bl k x (Tr B a s y b) = balance bl k x (Tr R a s y b)" |
- "balleft bl k x (Tr R (Tr B a s y b) t z c) = Tr R (Tr B bl k x a) s y (balance b t z (paint R c))" |
+ "balleft (Branch R a k x b) s y c = Branch R (Branch B a k x b) s y c" |
+ "balleft bl k x (Branch B a s y b) = balance bl k x (Branch R a s y b)" |
+ "balleft bl k x (Branch R (Branch B a s y b) t z c) = Branch R (Branch B bl k x a) s y (balance b t z (paint R c))" |
"balleft t k x s = Empty"
lemma balleft_inv2_with_inv1:
- assumes "inv2 lt" "inv2 rt" "bh lt + 1 = bh rt" "inv1 rt"
- shows "bh (balleft lt k v rt) = bh lt + 1"
+ assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "inv1 rt"
+ shows "bheight (balleft lt k v rt) = bheight lt + 1"
and "inv2 (balleft lt k v rt)"
using assms
-by (induct lt k v rt rule: balleft.induct) (auto simp: balance_inv2 balance_bh)
+by (induct lt k v rt rule: balleft.induct) (auto simp: balance_inv2 balance_bheight)
lemma balleft_inv2_app:
- assumes "inv2 lt" "inv2 rt" "bh lt + 1 = bh rt" "treec rt = B"
+ assumes "inv2 lt" "inv2 rt" "bheight lt + 1 = bheight rt" "color_of rt = B"
shows "inv2 (balleft lt k v rt)"
- "bh (balleft lt k v rt) = bh rt"
+ "bheight (balleft lt k v rt) = bheight rt"
using assms
-by (induct lt k v rt rule: balleft.induct) (auto simp add: balance_inv2 balance_bh)+
+by (induct lt k v rt rule: balleft.induct) (auto simp add: balance_inv2 balance_bheight)+
-lemma balleft_inv1: "\<lbrakk>inv1l a; inv1 b; treec b = B\<rbrakk> \<Longrightarrow> inv1 (balleft a k x b)"
+lemma balleft_inv1: "\<lbrakk>inv1l a; inv1 b; color_of b = B\<rbrakk> \<Longrightarrow> inv1 (balleft a k x b)"
by (induct a k x b rule: balleft.induct) (simp add: balance_inv1)+
lemma balleft_inv1l: "\<lbrakk> inv1l lt; inv1 rt \<rbrakk> \<Longrightarrow> inv1l (balleft lt k x rt)"
by (induct lt k x rt rule: balleft.induct) (auto simp: balance_inv1)
-lemma balleft_st: "\<lbrakk> st l; st r; tlt k l; tgt k r \<rbrakk> \<Longrightarrow> st (balleft l k v r)"
+lemma balleft_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balleft l k v r)"
apply (induct l k v r rule: balleft.induct)
-apply (auto simp: balance_st)
-apply (unfold tgt_prop tlt_prop)
+apply (auto simp: balance_sorted)
+apply (unfold tree_greater_prop tree_less_prop)
by force+
-lemma balleft_tgt:
+lemma balleft_tree_greater:
fixes k :: "'a::order"
assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x"
shows "k \<guillemotleft>| balleft a x t b"
using assms
by (induct a x t b rule: balleft.induct) auto
-lemma balleft_tlt:
+lemma balleft_tree_less:
fixes k :: "'a::order"
assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k"
shows "balleft a x t b |\<guillemotleft> k"
@@ -392,52 +407,52 @@
by (induct a x t b rule: balleft.induct) auto
lemma balleft_pit:
- assumes "inv1l l" "inv1 r" "bh l + 1 = bh r"
- shows "pin_tree k v (balleft l a b r) = (pin_tree k v l \<or> k = a \<and> v = b \<or> pin_tree k v r)"
+ assumes "inv1l l" "inv1 r" "bheight l + 1 = bheight r"
+ shows "entry_in_tree k v (balleft l a b r) = (entry_in_tree k v l \<or> k = a \<and> v = b \<or> entry_in_tree k v r)"
using assms
by (induct l k v r rule: balleft.induct) (auto simp: balance_pit)
fun
balright :: "('a,'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
- "balright a k x (Tr R b s y c) = Tr R a k x (Tr B b s y c)" |
- "balright (Tr B a k x b) s y bl = balance (Tr R a k x b) s y bl" |
- "balright (Tr R a k x (Tr B b s y c)) t z bl = Tr R (balance (paint R a) k x b) s y (Tr B c t z bl)" |
+ "balright a k x (Branch R b s y c) = Branch R a k x (Branch B b s y c)" |
+ "balright (Branch B a k x b) s y bl = balance (Branch R a k x b) s y bl" |
+ "balright (Branch R a k x (Branch B b s y c)) t z bl = Branch R (balance (paint R a) k x b) s y (Branch B c t z bl)" |
"balright t k x s = Empty"
lemma balright_inv2_with_inv1:
- assumes "inv2 lt" "inv2 rt" "bh lt = bh rt + 1" "inv1 lt"
- shows "inv2 (balright lt k v rt) \<and> bh (balright lt k v rt) = bh lt"
+ assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt + 1" "inv1 lt"
+ shows "inv2 (balright lt k v rt) \<and> bheight (balright lt k v rt) = bheight lt"
using assms
-by (induct lt k v rt rule: balright.induct) (auto simp: balance_inv2 balance_bh)
+by (induct lt k v rt rule: balright.induct) (auto simp: balance_inv2 balance_bheight)
-lemma balright_inv1: "\<lbrakk>inv1 a; inv1l b; treec a = B\<rbrakk> \<Longrightarrow> inv1 (balright a k x b)"
+lemma balright_inv1: "\<lbrakk>inv1 a; inv1l b; color_of a = B\<rbrakk> \<Longrightarrow> inv1 (balright a k x b)"
by (induct a k x b rule: balright.induct) (simp add: balance_inv1)+
lemma balright_inv1l: "\<lbrakk> inv1 lt; inv1l rt \<rbrakk> \<Longrightarrow>inv1l (balright lt k x rt)"
by (induct lt k x rt rule: balright.induct) (auto simp: balance_inv1)
-lemma balright_st: "\<lbrakk> st l; st r; tlt k l; tgt k r \<rbrakk> \<Longrightarrow> st (balright l k v r)"
+lemma balright_sorted: "\<lbrakk> sorted l; sorted r; tree_less k l; tree_greater k r \<rbrakk> \<Longrightarrow> sorted (balright l k v r)"
apply (induct l k v r rule: balright.induct)
-apply (auto simp:balance_st)
-apply (unfold tlt_prop tgt_prop)
+apply (auto simp:balance_sorted)
+apply (unfold tree_less_prop tree_greater_prop)
by force+
-lemma balright_tgt:
+lemma balright_tree_greater:
fixes k :: "'a::order"
assumes "k \<guillemotleft>| a" "k \<guillemotleft>| b" "k < x"
shows "k \<guillemotleft>| balright a x t b"
using assms by (induct a x t b rule: balright.induct) auto
-lemma balright_tlt:
+lemma balright_tree_less:
fixes k :: "'a::order"
assumes "a |\<guillemotleft> k" "b |\<guillemotleft> k" "x < k"
shows "balright a x t b |\<guillemotleft> k"
using assms by (induct a x t b rule: balright.induct) auto
lemma balright_pit:
- assumes "inv1 l" "inv1l r" "bh l = bh r + 1" "inv2 l" "inv2 r"
- shows "pin_tree x y (balright l k v r) = (pin_tree x y l \<or> x = k \<and> y = v \<or> pin_tree x y r)"
+ assumes "inv1 l" "inv1l r" "bheight l = bheight r + 1" "inv2 l" "inv2 r"
+ shows "entry_in_tree x y (balright l k v r) = (entry_in_tree x y l \<or> x = k \<and> y = v \<or> entry_in_tree x y r)"
using assms by (induct l k v r rule: balright.induct) (auto simp: balance_pit)
@@ -448,50 +463,50 @@
where
"app Empty x = x"
| "app x Empty = x"
-| "app (Tr R a k x b) (Tr R c s y d) = (case (app b c) of
- Tr R b2 t z c2 \<Rightarrow> (Tr R (Tr R a k x b2) t z (Tr R c2 s y d)) |
- bc \<Rightarrow> Tr R a k x (Tr R bc s y d))"
-| "app (Tr B a k x b) (Tr B c s y d) = (case (app b c) of
- Tr R b2 t z c2 \<Rightarrow> Tr R (Tr B a k x b2) t z (Tr B c2 s y d) |
- bc \<Rightarrow> balleft a k x (Tr B bc s y d))"
-| "app a (Tr R b k x c) = Tr R (app a b) k x c"
-| "app (Tr R a k x b) c = Tr R a k x (app b c)"
+| "app (Branch R a k x b) (Branch R c s y d) = (case (app b c) of
+ Branch R b2 t z c2 \<Rightarrow> (Branch R (Branch R a k x b2) t z (Branch R c2 s y d)) |
+ bc \<Rightarrow> Branch R a k x (Branch R bc s y d))"
+| "app (Branch B a k x b) (Branch B c s y d) = (case (app b c) of
+ Branch R b2 t z c2 \<Rightarrow> Branch R (Branch B a k x b2) t z (Branch B c2 s y d) |
+ bc \<Rightarrow> balleft a k x (Branch B bc s y d))"
+| "app a (Branch R b k x c) = Branch R (app a b) k x c"
+| "app (Branch R a k x b) c = Branch R a k x (app b c)"
lemma app_inv2:
- assumes "inv2 lt" "inv2 rt" "bh lt = bh rt"
- shows "bh (app lt rt) = bh lt" "inv2 (app lt rt)"
+ assumes "inv2 lt" "inv2 rt" "bheight lt = bheight rt"
+ shows "bheight (app lt rt) = bheight lt" "inv2 (app lt rt)"
using assms
by (induct lt rt rule: app.induct)
(auto simp: balleft_inv2_app split: rbt.splits color.splits)
lemma app_inv1:
assumes "inv1 lt" "inv1 rt"
- shows "treec lt = B \<Longrightarrow> treec rt = B \<Longrightarrow> inv1 (app lt rt)"
+ shows "color_of lt = B \<Longrightarrow> color_of rt = B \<Longrightarrow> inv1 (app lt rt)"
"inv1l (app lt rt)"
using assms
by (induct lt rt rule: app.induct)
(auto simp: balleft_inv1 split: rbt.splits color.splits)
-lemma app_tgt[simp]:
+lemma app_tree_greater[simp]:
fixes k :: "'a::linorder"
assumes "k \<guillemotleft>| l" "k \<guillemotleft>| r"
shows "k \<guillemotleft>| app l r"
using assms
by (induct l r rule: app.induct)
- (auto simp: balleft_tgt split:rbt.splits color.splits)
+ (auto simp: balleft_tree_greater split:rbt.splits color.splits)
-lemma app_tlt[simp]:
+lemma app_tree_less[simp]:
fixes k :: "'a::linorder"
assumes "l |\<guillemotleft> k" "r |\<guillemotleft> k"
shows "app l r |\<guillemotleft> k"
using assms
by (induct l r rule: app.induct)
- (auto simp: balleft_tlt split:rbt.splits color.splits)
+ (auto simp: balleft_tree_less split:rbt.splits color.splits)
-lemma app_st:
+lemma app_sorted:
fixes k :: "'a::linorder"
- assumes "st l" "st r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
- shows "st (app l r)"
+ assumes "sorted l" "sorted r" "l |\<guillemotleft> k" "k \<guillemotleft>| r"
+ shows "sorted (app l r)"
using assms proof (induct l r rule: app.induct)
case (3 a x v b c y w d)
hence ineqs: "a |\<guillemotleft> x" "x \<guillemotleft>| b" "b |\<guillemotleft> k" "k \<guillemotleft>| c" "c |\<guillemotleft> y" "y \<guillemotleft>| d"
@@ -500,55 +515,55 @@
show ?case
apply (cases "app b c" rule: rbt_cases)
apply auto
- by (metis app_tgt app_tlt ineqs ineqs tlt.simps(2) tgt.simps(2) tgt_trans tlt_trans)+
+ by (metis app_tree_greater app_tree_less ineqs ineqs tree_less_simps(2) tree_greater_simps(2) tree_greater_trans tree_less_trans)+
next
case (4 a x v b c y w d)
- hence "x < k \<and> tgt k c" by simp
- hence "tgt x c" by (blast dest: tgt_trans)
- with 4 have 2: "tgt x (app b c)" by (simp add: app_tgt)
- from 4 have "k < y \<and> tlt k b" by simp
- hence "tlt y b" by (blast dest: tlt_trans)
- with 4 have 3: "tlt y (app b c)" by (simp add: app_tlt)
+ hence "x < k \<and> tree_greater k c" by simp
+ hence "tree_greater x c" by (blast dest: tree_greater_trans)
+ with 4 have 2: "tree_greater x (app b c)" by (simp add: app_tree_greater)
+ from 4 have "k < y \<and> tree_less k b" by simp
+ hence "tree_less y b" by (blast dest: tree_less_trans)
+ with 4 have 3: "tree_less y (app b c)" by (simp add: app_tree_less)
show ?case
proof (cases "app b c" rule: rbt_cases)
case Empty
- from 4 have "x < y \<and> tgt y d" by auto
- hence "tgt x d" by (blast dest: tgt_trans)
- with 4 Empty have "st a" and "st (Tr B Empty y w d)" and "tlt x a" and "tgt x (Tr B Empty y w d)" by auto
- with Empty show ?thesis by (simp add: balleft_st)
+ from 4 have "x < y \<and> tree_greater y d" by auto
+ hence "tree_greater x d" by (blast dest: tree_greater_trans)
+ with 4 Empty have "sorted a" and "sorted (Branch B Empty y w d)" and "tree_less x a" and "tree_greater x (Branch B Empty y w d)" by auto
+ with Empty show ?thesis by (simp add: balleft_sorted)
next
case (Red lta va ka rta)
- with 2 4 have "x < va \<and> tlt x a" by simp
- hence 5: "tlt va a" by (blast dest: tlt_trans)
- from Red 3 4 have "va < y \<and> tgt y d" by simp
- hence "tgt va d" by (blast dest: tgt_trans)
+ with 2 4 have "x < va \<and> tree_less x a" by simp
+ hence 5: "tree_less va a" by (blast dest: tree_less_trans)
+ from Red 3 4 have "va < y \<and> tree_greater y d" by simp
+ hence "tree_greater va d" by (blast dest: tree_greater_trans)
with Red 2 3 4 5 show ?thesis by simp
next
case (Black lta va ka rta)
- from 4 have "x < y \<and> tgt y d" by auto
- hence "tgt x d" by (blast dest: tgt_trans)
- with Black 2 3 4 have "st a" and "st (Tr B (app b c) y w d)" and "tlt x a" and "tgt x (Tr B (app b c) y w d)" by auto
- with Black show ?thesis by (simp add: balleft_st)
+ from 4 have "x < y \<and> tree_greater y d" by auto
+ hence "tree_greater x d" by (blast dest: tree_greater_trans)
+ with Black 2 3 4 have "sorted a" and "sorted (Branch B (app b c) y w d)" and "tree_less x a" and "tree_greater x (Branch B (app b c) y w d)" by auto
+ with Black show ?thesis by (simp add: balleft_sorted)
qed
next
case (5 va vb vd vc b x w c)
- hence "k < x \<and> tlt k (Tr B va vb vd vc)" by simp
- hence "tlt x (Tr B va vb vd vc)" by (blast dest: tlt_trans)
- with 5 show ?case by (simp add: app_tlt)
+ hence "k < x \<and> tree_less k (Branch B va vb vd vc)" by simp
+ hence "tree_less x (Branch B va vb vd vc)" by (blast dest: tree_less_trans)
+ with 5 show ?case by (simp add: app_tree_less)
next
case (6 a x v b va vb vd vc)
- hence "x < k \<and> tgt k (Tr B va vb vd vc)" by simp
- hence "tgt x (Tr B va vb vd vc)" by (blast dest: tgt_trans)
- with 6 show ?case by (simp add: app_tgt)
+ hence "x < k \<and> tree_greater k (Branch B va vb vd vc)" by simp
+ hence "tree_greater x (Branch B va vb vd vc)" by (blast dest: tree_greater_trans)
+ with 6 show ?case by (simp add: app_tree_greater)
qed simp+
lemma app_pit:
- assumes "inv2 l" "inv2 r" "bh l = bh r" "inv1 l" "inv1 r"
- shows "pin_tree k v (app l r) = (pin_tree k v l \<or> pin_tree k v r)"
+ assumes "inv2 l" "inv2 r" "bheight l = bheight r" "inv1 l" "inv1 r"
+ shows "entry_in_tree k v (app l r) = (entry_in_tree k v l \<or> entry_in_tree k v r)"
using assms
proof (induct l r rule: app.induct)
case (4 _ _ _ b c)
- hence a: "bh (app b c) = bh b" by (simp add: app_inv2)
+ hence a: "bheight (app b c) = bheight b" by (simp add: app_inv2)
from 4 have b: "inv1l (app b c)" by (simp add: app_inv1)
show ?case
@@ -570,21 +585,21 @@
del :: "('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
"del x Empty = Empty" |
- "del x (Tr c a y s b) = (if x < y then delformLeft x a y s b else (if x > y then delformRight x a y s b else app a b))" |
- "delformLeft x (Tr B lt z v rt) y s b = balleft (del x (Tr B lt z v rt)) y s b" |
- "delformLeft x a y s b = Tr R (del x a) y s b" |
- "delformRight x a y s (Tr B lt z v rt) = balright a y s (del x (Tr B lt z v rt))" |
- "delformRight x a y s b = Tr R a y s (del x b)"
+ "del x (Branch c a y s b) = (if x < y then delformLeft x a y s b else (if x > y then delformRight x a y s b else app a b))" |
+ "delformLeft x (Branch B lt z v rt) y s b = balleft (del x (Branch B lt z v rt)) y s b" |
+ "delformLeft x a y s b = Branch R (del x a) y s b" |
+ "delformRight x a y s (Branch B lt z v rt) = balright a y s (del x (Branch B lt z v rt))" |
+ "delformRight x a y s b = Branch R a y s (del x b)"
lemma
assumes "inv2 lt" "inv1 lt"
shows
- "\<lbrakk>inv2 rt; bh lt = bh rt; inv1 rt\<rbrakk> \<Longrightarrow>
- inv2 (delformLeft x lt k v rt) \<and> bh (delformLeft x lt k v rt) = bh lt \<and> (treec lt = B \<and> treec rt = B \<and> inv1 (delformLeft x lt k v rt) \<or> (treec lt \<noteq> B \<or> treec rt \<noteq> B) \<and> inv1l (delformLeft x lt k v rt))"
- and "\<lbrakk>inv2 rt; bh lt = bh rt; inv1 rt\<rbrakk> \<Longrightarrow>
- inv2 (delformRight x lt k v rt) \<and> bh (delformRight x lt k v rt) = bh lt \<and> (treec lt = B \<and> treec rt = B \<and> inv1 (delformRight x lt k v rt) \<or> (treec lt \<noteq> B \<or> treec rt \<noteq> B) \<and> inv1l (delformRight x lt k v rt))"
- and del_inv1_inv2: "inv2 (del x lt) \<and> (treec lt = R \<and> bh (del x lt) = bh lt \<and> inv1 (del x lt)
- \<or> treec lt = B \<and> bh (del x lt) = bh lt - 1 \<and> inv1l (del x lt))"
+ "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
+ inv2 (delformLeft x lt k v rt) \<and> bheight (delformLeft x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (delformLeft x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (delformLeft x lt k v rt))"
+ and "\<lbrakk>inv2 rt; bheight lt = bheight rt; inv1 rt\<rbrakk> \<Longrightarrow>
+ inv2 (delformRight x lt k v rt) \<and> bheight (delformRight x lt k v rt) = bheight lt \<and> (color_of lt = B \<and> color_of rt = B \<and> inv1 (delformRight x lt k v rt) \<or> (color_of lt \<noteq> B \<or> color_of rt \<noteq> B) \<and> inv1l (delformRight x lt k v rt))"
+ and del_inv1_inv2: "inv2 (del x lt) \<and> (color_of lt = R \<and> bheight (del x lt) = bheight lt \<and> inv1 (del x lt)
+ \<or> color_of lt = B \<and> bheight (del x lt) = bheight lt - 1 \<and> inv1l (del x lt))"
using assms
proof (induct x lt k v rt and x lt k v rt and x lt rule: delformLeft_delformRight_del.induct)
case (2 y c _ y')
@@ -601,55 +616,55 @@
qed
next
case (3 y lt z v rta y' ss bb)
- thus ?case by (cases "treec (Tr B lt z v rta) = B \<and> treec bb = B") (simp add: balleft_inv2_with_inv1 balleft_inv1 balleft_inv1l)+
+ thus ?case by (cases "color_of (Branch B lt z v rta) = B \<and> color_of bb = B") (simp add: balleft_inv2_with_inv1 balleft_inv1 balleft_inv1l)+
next
case (5 y a y' ss lt z v rta)
- thus ?case by (cases "treec a = B \<and> treec (Tr B lt z v rta) = B") (simp add: balright_inv2_with_inv1 balright_inv1 balright_inv1l)+
+ thus ?case by (cases "color_of a = B \<and> color_of (Branch B lt z v rta) = B") (simp add: balright_inv2_with_inv1 balright_inv1 balright_inv1l)+
next
- case ("6_1" y a y' ss) thus ?case by (cases "treec a = B \<and> treec Empty = B") simp+
+ case ("6_1" y a y' ss) thus ?case by (cases "color_of a = B \<and> color_of Empty = B") simp+
qed auto
lemma
- delformLeft_tlt: "\<lbrakk>tlt v lt; tlt v rt; k < v\<rbrakk> \<Longrightarrow> tlt v (delformLeft x lt k y rt)"
- and delformRight_tlt: "\<lbrakk>tlt v lt; tlt v rt; k < v\<rbrakk> \<Longrightarrow> tlt v (delformRight x lt k y rt)"
- and del_tlt: "tlt v lt \<Longrightarrow> tlt v (del x lt)"
+ delformLeft_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (delformLeft x lt k y rt)"
+ and delformRight_tree_less: "\<lbrakk>tree_less v lt; tree_less v rt; k < v\<rbrakk> \<Longrightarrow> tree_less v (delformRight x lt k y rt)"
+ and del_tree_less: "tree_less v lt \<Longrightarrow> tree_less v (del x lt)"
by (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
- (auto simp: balleft_tlt balright_tlt)
+ (auto simp: balleft_tree_less balright_tree_less)
-lemma delformLeft_tgt: "\<lbrakk>tgt v lt; tgt v rt; k > v\<rbrakk> \<Longrightarrow> tgt v (delformLeft x lt k y rt)"
- and delformRight_tgt: "\<lbrakk>tgt v lt; tgt v rt; k > v\<rbrakk> \<Longrightarrow> tgt v (delformRight x lt k y rt)"
- and del_tgt: "tgt v lt \<Longrightarrow> tgt v (del x lt)"
+lemma delformLeft_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (delformLeft x lt k y rt)"
+ and delformRight_tree_greater: "\<lbrakk>tree_greater v lt; tree_greater v rt; k > v\<rbrakk> \<Longrightarrow> tree_greater v (delformRight x lt k y rt)"
+ and del_tree_greater: "tree_greater v lt \<Longrightarrow> tree_greater v (del x lt)"
by (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
- (auto simp: balleft_tgt balright_tgt)
+ (auto simp: balleft_tree_greater balright_tree_greater)
-lemma "\<lbrakk>st lt; st rt; tlt k lt; tgt k rt\<rbrakk> \<Longrightarrow> st (delformLeft x lt k y rt)"
- and "\<lbrakk>st lt; st rt; tlt k lt; tgt k rt\<rbrakk> \<Longrightarrow> st (delformRight x lt k y rt)"
- and del_st: "st lt \<Longrightarrow> st (del x lt)"
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (delformLeft x lt k y rt)"
+ and "\<lbrakk>sorted lt; sorted rt; tree_less k lt; tree_greater k rt\<rbrakk> \<Longrightarrow> sorted (delformRight x lt k y rt)"
+ and del_sorted: "sorted lt \<Longrightarrow> sorted (del x lt)"
proof (induct x lt k y rt and x lt k y rt and x lt rule: delformLeft_delformRight_del.induct)
case (3 x lta zz v rta yy ss bb)
- from 3 have "tlt yy (Tr B lta zz v rta)" by simp
- hence "tlt yy (del x (Tr B lta zz v rta))" by (rule del_tlt)
- with 3 show ?case by (simp add: balleft_st)
+ from 3 have "tree_less yy (Branch B lta zz v rta)" by simp
+ hence "tree_less yy (del x (Branch B lta zz v rta))" by (rule del_tree_less)
+ with 3 show ?case by (simp add: balleft_sorted)
next
case ("4_2" x vaa vbb vdd vc yy ss bb)
- hence "tlt yy (Tr R vaa vbb vdd vc)" by simp
- hence "tlt yy (del x (Tr R vaa vbb vdd vc))" by (rule del_tlt)
+ hence "tree_less yy (Branch R vaa vbb vdd vc)" by simp
+ hence "tree_less yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_less)
with "4_2" show ?case by simp
next
case (5 x aa yy ss lta zz v rta)
- hence "tgt yy (Tr B lta zz v rta)" by simp
- hence "tgt yy (del x (Tr B lta zz v rta))" by (rule del_tgt)
- with 5 show ?case by (simp add: balright_st)
+ hence "tree_greater yy (Branch B lta zz v rta)" by simp
+ hence "tree_greater yy (del x (Branch B lta zz v rta))" by (rule del_tree_greater)
+ with 5 show ?case by (simp add: balright_sorted)
next
case ("6_2" x aa yy ss vaa vbb vdd vc)
- hence "tgt yy (Tr R vaa vbb vdd vc)" by simp
- hence "tgt yy (del x (Tr R vaa vbb vdd vc))" by (rule del_tgt)
+ hence "tree_greater yy (Branch R vaa vbb vdd vc)" by simp
+ hence "tree_greater yy (del x (Branch R vaa vbb vdd vc))" by (rule del_tree_greater)
with "6_2" show ?case by simp
-qed (auto simp: app_st)
+qed (auto simp: app_sorted)
-lemma "\<lbrakk>st lt; st rt; tlt kt lt; tgt kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bh lt = bh rt; x < kt\<rbrakk> \<Longrightarrow> pin_tree k v (delformLeft x lt kt y rt) = (False \<or> (x \<noteq> k \<and> pin_tree k v (Tr c lt kt y rt)))"
- and "\<lbrakk>st lt; st rt; tlt kt lt; tgt kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bh lt = bh rt; x > kt\<rbrakk> \<Longrightarrow> pin_tree k v (delformRight x lt kt y rt) = (False \<or> (x \<noteq> k \<and> pin_tree k v (Tr c lt kt y rt)))"
- and del_pit: "\<lbrakk>st t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> pin_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> pin_tree k v t))"
+lemma "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x < kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (delformLeft x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+ and "\<lbrakk>sorted lt; sorted rt; tree_less kt lt; tree_greater kt rt; inv1 lt; inv1 rt; inv2 lt; inv2 rt; bheight lt = bheight rt; x > kt\<rbrakk> \<Longrightarrow> entry_in_tree k v (delformRight x lt kt y rt) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v (Branch c lt kt y rt)))"
+ and del_pit: "\<lbrakk>sorted t; inv1 t; inv2 t\<rbrakk> \<Longrightarrow> entry_in_tree k v (del x t) = (False \<or> (x \<noteq> k \<and> entry_in_tree k v t))"
proof (induct x lt kt y rt and x lt kt y rt and x t rule: delformLeft_delformRight_del.induct)
case (2 xx c aa yy ss bb)
have "xx = yy \<or> xx < yy \<or> xx > yy" by auto
@@ -657,68 +672,68 @@
assume "xx = yy"
with 2 show ?thesis proof (cases "xx = k")
case True
- from 2 `xx = yy` `xx = k` have "st (Tr c aa yy ss bb) \<and> k = yy" by simp
- hence "\<not> pin_tree k v aa" "\<not> pin_tree k v bb" by (auto simp: tlt_nit tgt_prop)
+ from 2 `xx = yy` `xx = k` have "sorted (Branch c aa yy ss bb) \<and> k = yy" by simp
+ hence "\<not> entry_in_tree k v aa" "\<not> entry_in_tree k v bb" by (auto simp: tree_less_nit tree_greater_prop)
with `xx = yy` 2 `xx = k` show ?thesis by (simp add: app_pit)
qed (simp add: app_pit)
qed simp+
next
case (3 xx lta zz vv rta yy ss bb)
- def mt[simp]: mt == "Tr B lta zz vv rta"
+ def mt[simp]: mt == "Branch B lta zz vv rta"
from 3 have "inv2 mt \<and> inv1 mt" by simp
- hence "inv2 (del xx mt) \<and> (treec mt = R \<and> bh (del xx mt) = bh mt \<and> inv1 (del xx mt) \<or> treec mt = B \<and> bh (del xx mt) = bh mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
- with 3 have 4: "pin_tree k v (delformLeft xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> pin_tree k v mt \<or> (k = yy \<and> v = ss) \<or> pin_tree k v bb)" by (simp add: balleft_pit)
+ hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
+ with 3 have 4: "entry_in_tree k v (delformLeft xx mt yy ss bb) = (False \<or> xx \<noteq> k \<and> entry_in_tree k v mt \<or> (k = yy \<and> v = ss) \<or> entry_in_tree k v bb)" by (simp add: balleft_pit)
thus ?case proof (cases "xx = k")
case True
- from 3 True have "tgt yy bb \<and> yy > k" by simp
- hence "tgt k bb" by (blast dest: tgt_trans)
- with 3 4 True show ?thesis by (auto simp: tgt_nit)
+ from 3 True have "tree_greater yy bb \<and> yy > k" by simp
+ hence "tree_greater k bb" by (blast dest: tree_greater_trans)
+ with 3 4 True show ?thesis by (auto simp: tree_greater_nit)
qed auto
next
case ("4_1" xx yy ss bb)
show ?case proof (cases "xx = k")
case True
- with "4_1" have "tgt yy bb \<and> k < yy" by simp
- hence "tgt k bb" by (blast dest: tgt_trans)
+ with "4_1" have "tree_greater yy bb \<and> k < yy" by simp
+ hence "tree_greater k bb" by (blast dest: tree_greater_trans)
with "4_1" `xx = k`
- have "pin_tree k v (Tr R Empty yy ss bb) = pin_tree k v Empty" by (auto simp: tgt_nit)
+ have "entry_in_tree k v (Branch R Empty yy ss bb) = entry_in_tree k v Empty" by (auto simp: tree_greater_nit)
thus ?thesis by auto
qed simp+
next
case ("4_2" xx vaa vbb vdd vc yy ss bb)
thus ?case proof (cases "xx = k")
case True
- with "4_2" have "k < yy \<and> tgt yy bb" by simp
- hence "tgt k bb" by (blast dest: tgt_trans)
- with True "4_2" show ?thesis by (auto simp: tgt_nit)
+ with "4_2" have "k < yy \<and> tree_greater yy bb" by simp
+ hence "tree_greater k bb" by (blast dest: tree_greater_trans)
+ with True "4_2" show ?thesis by (auto simp: tree_greater_nit)
qed simp
next
case (5 xx aa yy ss lta zz vv rta)
- def mt[simp]: mt == "Tr B lta zz vv rta"
+ def mt[simp]: mt == "Branch B lta zz vv rta"
from 5 have "inv2 mt \<and> inv1 mt" by simp
- hence "inv2 (del xx mt) \<and> (treec mt = R \<and> bh (del xx mt) = bh mt \<and> inv1 (del xx mt) \<or> treec mt = B \<and> bh (del xx mt) = bh mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
- with 5 have 3: "pin_tree k v (delformRight xx aa yy ss mt) = (pin_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> pin_tree k v mt)" by (simp add: balright_pit)
+ hence "inv2 (del xx mt) \<and> (color_of mt = R \<and> bheight (del xx mt) = bheight mt \<and> inv1 (del xx mt) \<or> color_of mt = B \<and> bheight (del xx mt) = bheight mt - 1 \<and> inv1l (del xx mt))" by (blast dest: del_inv1_inv2)
+ with 5 have 3: "entry_in_tree k v (delformRight xx aa yy ss mt) = (entry_in_tree k v aa \<or> (k = yy \<and> v = ss) \<or> False \<or> xx \<noteq> k \<and> entry_in_tree k v mt)" by (simp add: balright_pit)
thus ?case proof (cases "xx = k")
case True
- from 5 True have "tlt yy aa \<and> yy < k" by simp
- hence "tlt k aa" by (blast dest: tlt_trans)
- with 3 5 True show ?thesis by (auto simp: tlt_nit)
+ from 5 True have "tree_less yy aa \<and> yy < k" by simp
+ hence "tree_less k aa" by (blast dest: tree_less_trans)
+ with 3 5 True show ?thesis by (auto simp: tree_less_nit)
qed auto
next
case ("6_1" xx aa yy ss)
show ?case proof (cases "xx = k")
case True
- with "6_1" have "tlt yy aa \<and> k > yy" by simp
- hence "tlt k aa" by (blast dest: tlt_trans)
- with "6_1" `xx = k` show ?thesis by (auto simp: tlt_nit)
+ with "6_1" have "tree_less yy aa \<and> k > yy" by simp
+ hence "tree_less k aa" by (blast dest: tree_less_trans)
+ with "6_1" `xx = k` show ?thesis by (auto simp: tree_less_nit)
qed simp
next
case ("6_2" xx aa yy ss vaa vbb vdd vc)
thus ?case proof (cases "xx = k")
case True
- with "6_2" have "k > yy \<and> tlt yy aa" by simp
- hence "tlt k aa" by (blast dest: tlt_trans)
- with True "6_2" show ?thesis by (auto simp: tlt_nit)
+ with "6_2" have "k > yy \<and> tree_less yy aa" by simp
+ hence "tree_less k aa" by (blast dest: tree_less_trans)
+ with True "6_2" show ?thesis by (auto simp: tree_less_nit)
qed simp
qed simp
@@ -726,36 +741,36 @@
definition delete where
delete_def: "delete k t = paint B (del k t)"
-theorem delete_isrbt[simp]: assumes "isrbt t" shows "isrbt (delete k t)"
+theorem delete_is_rbt[simp]: assumes "is_rbt t" shows "is_rbt (delete k t)"
proof -
- from assms have "inv2 t" and "inv1 t" unfolding isrbt_def by auto
- hence "inv2 (del k t) \<and> (treec t = R \<and> bh (del k t) = bh t \<and> inv1 (del k t) \<or> treec t = B \<and> bh (del k t) = bh t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
- hence "inv2 (del k t) \<and> inv1l (del k t)" by (cases "treec t") auto
+ from assms have "inv2 t" and "inv1 t" unfolding is_rbt_def by auto
+ hence "inv2 (del k t) \<and> (color_of t = R \<and> bheight (del k t) = bheight t \<and> inv1 (del k t) \<or> color_of t = B \<and> bheight (del k t) = bheight t - 1 \<and> inv1l (del k t))" by (rule del_inv1_inv2)
+ hence "inv2 (del k t) \<and> inv1l (del k t)" by (cases "color_of t") auto
with assms show ?thesis
- unfolding isrbt_def delete_def
- by (auto intro: paint_st del_st)
+ unfolding is_rbt_def delete_def
+ by (auto intro: paint_sorted del_sorted)
qed
lemma delete_pit:
- assumes "isrbt t"
- shows "pin_tree k v (delete x t) = (x \<noteq> k \<and> pin_tree k v t)"
- using assms unfolding isrbt_def delete_def
+ assumes "is_rbt t"
+ shows "entry_in_tree k v (delete x t) = (x \<noteq> k \<and> entry_in_tree k v t)"
+ using assms unfolding is_rbt_def delete_def
by (auto simp: del_pit)
-lemma map_of_delete:
- assumes isrbt: "isrbt t"
- shows "map_of (delete k t) = (map_of t)|`(-{k})"
+lemma lookup_delete:
+ assumes is_rbt: "is_rbt t"
+ shows "lookup (delete k t) = (lookup t)|`(-{k})"
proof
fix x
- show "map_of (delete k t) x = (map_of t |` (-{k})) x"
+ show "lookup (delete k t) x = (lookup t |` (-{k})) x"
proof (cases "x = k")
assume "x = k"
- with isrbt show ?thesis
- by (cases "map_of (delete k t) k") (auto simp: mapof_pit delete_pit)
+ with is_rbt show ?thesis
+ by (cases "lookup (delete k t) k") (auto simp: lookup_pit delete_pit)
next
assume "x \<noteq> k"
thus ?thesis
- by auto (metis isrbt delete_isrbt delete_pit isrbt_st mapof_from_pit)
+ by auto (metis is_rbt delete_is_rbt delete_pit is_rbt_sorted lookup_from_pit)
qed
qed
@@ -765,43 +780,43 @@
unionwithkey :: "('a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
"unionwithkey f t Empty = t"
-| "unionwithkey f t (Tr c lt k v rt) = unionwithkey f (unionwithkey f (insertwithkey f k v t) lt) rt"
+| "unionwithkey f t (Branch c lt k v rt) = unionwithkey f (unionwithkey f (insertwithkey f k v t) lt) rt"
-lemma unionwk_st: "st lt \<Longrightarrow> st (unionwithkey f lt rt)"
- by (induct rt arbitrary: lt) (auto simp: insertwk_st)
-theorem unionwk_isrbt[simp]: "isrbt lt \<Longrightarrow> isrbt (unionwithkey f lt rt)"
- by (induct rt arbitrary: lt) (simp add: insertwk_isrbt)+
+lemma unionwk_sorted: "sorted lt \<Longrightarrow> sorted (unionwithkey f lt rt)"
+ by (induct rt arbitrary: lt) (auto simp: insertwk_sorted)
+theorem unionwk_is_rbt[simp]: "is_rbt lt \<Longrightarrow> is_rbt (unionwithkey f lt rt)"
+ by (induct rt arbitrary: lt) (simp add: insertwk_is_rbt)+
definition
unionwith where
"unionwith f = unionwithkey (\<lambda>_. f)"
-theorem unionw_isrbt: "isrbt lt \<Longrightarrow> isrbt (unionwith f lt rt)" unfolding unionwith_def by simp
+theorem unionw_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (unionwith f lt rt)" unfolding unionwith_def by simp
definition union where
"union = unionwithkey (%_ _ rv. rv)"
-theorem union_isrbt: "isrbt lt \<Longrightarrow> isrbt (union lt rt)" unfolding union_def by simp
+theorem union_is_rbt: "is_rbt lt \<Longrightarrow> is_rbt (union lt rt)" unfolding union_def by simp
-lemma union_Tr[simp]:
- "union t (Tr c lt k v rt) = union (union (insrt k v t) lt) rt"
- unfolding union_def insrt_def
+lemma union_Branch[simp]:
+ "union t (Branch c lt k v rt) = union (union (insert k v t) lt) rt"
+ unfolding union_def insert_def
by simp
-lemma map_of_union:
- assumes "isrbt s" "st t"
- shows "map_of (union s t) = map_of s ++ map_of t"
+lemma lookup_union:
+ assumes "is_rbt s" "sorted t"
+ shows "lookup (union s t) = lookup s ++ lookup t"
using assms
proof (induct t arbitrary: s)
case Empty thus ?case by (auto simp: union_def)
next
- case (Tr c l k v r s)
- hence strl: "st r" "st l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
+ case (Branch c l k v r s)
+ hence sortedrl: "sorted r" "sorted l" "l |\<guillemotleft> k" "k \<guillemotleft>| r" by auto
- have meq: "map_of s(k \<mapsto> v) ++ map_of l ++ map_of r =
- map_of s ++
- (\<lambda>a. if a < k then map_of l a
- else if k < a then map_of r a else Some v)" (is "?m1 = ?m2")
+ have meq: "lookup s(k \<mapsto> v) ++ lookup l ++ lookup r =
+ lookup s ++
+ (\<lambda>a. if a < k then lookup l a
+ else if k < a then lookup r a else Some v)" (is "?m1 = ?m2")
proof (rule ext)
fix a
@@ -809,7 +824,7 @@
thus "?m1 a = ?m2 a"
proof (elim disjE)
assume "k < a"
- with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule tlt_trans)
+ with `l |\<guillemotleft> k` have "l |\<guillemotleft> a" by (rule tree_less_trans)
with `k < a` show ?thesis
by (auto simp: map_add_def split: option.splits)
next
@@ -818,20 +833,20 @@
show ?thesis by (auto simp: map_add_def)
next
assume "a < k"
- from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule tgt_trans)
+ from this `k \<guillemotleft>| r` have "a \<guillemotleft>| r" by (rule tree_greater_trans)
with `a < k` show ?thesis
by (auto simp: map_add_def split: option.splits)
qed
qed
- from Tr
+ from Branch
have IHs:
- "map_of (union (union (insrt k v s) l) r) = map_of (union (insrt k v s) l) ++ map_of r"
- "map_of (union (insrt k v s) l) = map_of (insrt k v s) ++ map_of l"
- by (auto intro: union_isrbt insrt_isrbt)
+ "lookup (union (union (insert k v s) l) r) = lookup (union (insert k v s) l) ++ lookup r"
+ "lookup (union (insert k v s) l) = lookup (insert k v s) ++ lookup l"
+ by (auto intro: union_is_rbt insert_is_rbt)
with meq show ?case
- by (auto simp: map_of_insert[OF Tr(3)])
+ by (auto simp: lookup_insert[OF Branch(3)])
qed
subsection {* Adjust *}
@@ -840,33 +855,33 @@
adjustwithkey :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'b) rbt"
where
"adjustwithkey f k Empty = Empty"
-| "adjustwithkey f k (Tr c lt x v rt) = (if k < x then (Tr c (adjustwithkey f k lt) x v rt) else if k > x then (Tr c lt x v (adjustwithkey f k rt)) else (Tr c lt x (f x v) rt))"
+| "adjustwithkey f k (Branch c lt x v rt) = (if k < x then (Branch c (adjustwithkey f k lt) x v rt) else if k > x then (Branch c lt x v (adjustwithkey f k rt)) else (Branch c lt x (f x v) rt))"
-lemma adjustwk_treec: "treec (adjustwithkey f k t) = treec t" by (induct t) simp+
-lemma adjustwk_inv1: "inv1 (adjustwithkey f k t) = inv1 t" by (induct t) (simp add: adjustwk_treec)+
-lemma adjustwk_inv2: "inv2 (adjustwithkey f k t) = inv2 t" "bh (adjustwithkey f k t) = bh t" by (induct t) simp+
-lemma adjustwk_tgt: "tgt k (adjustwithkey f kk t) = tgt k t" by (induct t) simp+
-lemma adjustwk_tlt: "tlt k (adjustwithkey f kk t) = tlt k t" by (induct t) simp+
-lemma adjustwk_st: "st (adjustwithkey f k t) = st t" by (induct t) (simp add: adjustwk_tlt adjustwk_tgt)+
+lemma adjustwk_color_of: "color_of (adjustwithkey f k t) = color_of t" by (induct t) simp+
+lemma adjustwk_inv1: "inv1 (adjustwithkey f k t) = inv1 t" by (induct t) (simp add: adjustwk_color_of)+
+lemma adjustwk_inv2: "inv2 (adjustwithkey f k t) = inv2 t" "bheight (adjustwithkey f k t) = bheight t" by (induct t) simp+
+lemma adjustwk_tree_greater: "tree_greater k (adjustwithkey f kk t) = tree_greater k t" by (induct t) simp+
+lemma adjustwk_tree_less: "tree_less k (adjustwithkey f kk t) = tree_less k t" by (induct t) simp+
+lemma adjustwk_sorted: "sorted (adjustwithkey f k t) = sorted t" by (induct t) (simp add: adjustwk_tree_less adjustwk_tree_greater)+
-theorem adjustwk_isrbt[simp]: "isrbt (adjustwithkey f k t) = isrbt t"
-unfolding isrbt_def by (simp add: adjustwk_inv2 adjustwk_treec adjustwk_st adjustwk_inv1 )
+theorem adjustwk_is_rbt[simp]: "is_rbt (adjustwithkey f k t) = is_rbt t"
+unfolding is_rbt_def by (simp add: adjustwk_inv2 adjustwk_color_of adjustwk_sorted adjustwk_inv1 )
theorem adjustwithkey_map[simp]:
- "map_of (adjustwithkey f k t) x =
- (if x = k then case map_of t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f k y)
- else map_of t x)"
+ "lookup (adjustwithkey f k t) x =
+ (if x = k then case lookup t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f k y)
+ else lookup t x)"
by (induct t arbitrary: x) (auto split:option.splits)
definition adjust where
"adjust f = adjustwithkey (\<lambda>_. f)"
-theorem adjust_isrbt[simp]: "isrbt (adjust f k t) = isrbt t" unfolding adjust_def by simp
+theorem adjust_is_rbt[simp]: "is_rbt (adjust f k t) = is_rbt t" unfolding adjust_def by simp
theorem adjust_map[simp]:
- "map_of (adjust f k t) x =
- (if x = k then case map_of t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f y)
- else map_of t x)"
+ "lookup (adjust f k t) x =
+ (if x = k then case lookup t x of None \<Rightarrow> None | Some y \<Rightarrow> Some (f y)
+ else lookup t x)"
unfolding adjust_def by simp
subsection {* Map *}
@@ -875,27 +890,27 @@
mapwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> ('a,'c) rbt"
where
"mapwithkey f Empty = Empty"
-| "mapwithkey f (Tr c lt k v rt) = Tr c (mapwithkey f lt) k (f k v) (mapwithkey f rt)"
+| "mapwithkey f (Branch c lt k v rt) = Branch c (mapwithkey f lt) k (f k v) (mapwithkey f rt)"
theorem mapwk_keys[simp]: "keys (mapwithkey f t) = keys t" by (induct t) auto
-lemma mapwk_tgt: "tgt k (mapwithkey f t) = tgt k t" by (induct t) simp+
-lemma mapwk_tlt: "tlt k (mapwithkey f t) = tlt k t" by (induct t) simp+
-lemma mapwk_st: "st (mapwithkey f t) = st t" by (induct t) (simp add: mapwk_tlt mapwk_tgt)+
-lemma mapwk_treec: "treec (mapwithkey f t) = treec t" by (induct t) simp+
-lemma mapwk_inv1: "inv1 (mapwithkey f t) = inv1 t" by (induct t) (simp add: mapwk_treec)+
-lemma mapwk_inv2: "inv2 (mapwithkey f t) = inv2 t" "bh (mapwithkey f t) = bh t" by (induct t) simp+
-theorem mapwk_isrbt[simp]: "isrbt (mapwithkey f t) = isrbt t"
-unfolding isrbt_def by (simp add: mapwk_inv1 mapwk_inv2 mapwk_st mapwk_treec)
+lemma mapwk_tree_greater: "tree_greater k (mapwithkey f t) = tree_greater k t" by (induct t) simp+
+lemma mapwk_tree_less: "tree_less k (mapwithkey f t) = tree_less k t" by (induct t) simp+
+lemma mapwk_sorted: "sorted (mapwithkey f t) = sorted t" by (induct t) (simp add: mapwk_tree_less mapwk_tree_greater)+
+lemma mapwk_color_of: "color_of (mapwithkey f t) = color_of t" by (induct t) simp+
+lemma mapwk_inv1: "inv1 (mapwithkey f t) = inv1 t" by (induct t) (simp add: mapwk_color_of)+
+lemma mapwk_inv2: "inv2 (mapwithkey f t) = inv2 t" "bheight (mapwithkey f t) = bheight t" by (induct t) simp+
+theorem mapwk_is_rbt[simp]: "is_rbt (mapwithkey f t) = is_rbt t"
+unfolding is_rbt_def by (simp add: mapwk_inv1 mapwk_inv2 mapwk_sorted mapwk_color_of)
-theorem map_of_mapwk[simp]: "map_of (mapwithkey f t) x = Option.map (f x) (map_of t x)"
+theorem lookup_mapwk[simp]: "lookup (mapwithkey f t) x = Option.map (f x) (lookup t x)"
by (induct t) auto
definition map
where map_def: "map f == mapwithkey (\<lambda>_. f)"
theorem map_keys[simp]: "keys (map f t) = keys t" unfolding map_def by simp
-theorem map_isrbt[simp]: "isrbt (map f t) = isrbt t" unfolding map_def by simp
-theorem map_of_map[simp]: "map_of (map f t) = Option.map f o map_of t"
+theorem map_is_rbt[simp]: "is_rbt (map f t) = is_rbt t" unfolding map_def by simp
+theorem lookup_map[simp]: "lookup (map f t) = Option.map f o lookup t"
by (rule ext) (simp add:map_def)
subsection {* Fold *}
@@ -906,62 +921,57 @@
foldwithkey :: "('a::linorder \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a,'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c"
where
"foldwithkey f Empty v = v"
-| "foldwithkey f (Tr c lt k x rt) v = foldwithkey f rt (f k x (foldwithkey f lt v))"
+| "foldwithkey f (Branch c lt k x rt) v = foldwithkey f rt (f k x (foldwithkey f lt v))"
-primrec alist_of
-where
- "alist_of Empty = []"
-| "alist_of (Tr _ l k v r) = alist_of l @ (k,v) # alist_of r"
-
-lemma map_of_alist_of_aux: "st (Tr c t1 k v t2) \<Longrightarrow> RBT.map_of (Tr c t1 k v t2) = RBT.map_of t2 ++ [k\<mapsto>v] ++ RBT.map_of t1"
+lemma lookup_entries_aux: "sorted (Branch c t1 k v t2) \<Longrightarrow> RBT.lookup (Branch c t1 k v t2) = RBT.lookup t2 ++ [k\<mapsto>v] ++ RBT.lookup t1"
proof (rule ext)
fix x
- assume ST: "st (Tr c t1 k v t2)"
- let ?thesis = "RBT.map_of (Tr c t1 k v t2) x = (RBT.map_of t2 ++ [k \<mapsto> v] ++ RBT.map_of t1) x"
+ assume SORTED: "sorted (Branch c t1 k v t2)"
+ let ?thesis = "RBT.lookup (Branch c t1 k v t2) x = (RBT.lookup t2 ++ [k \<mapsto> v] ++ RBT.lookup t1) x"
- have DOM_T1: "!!k'. k'\<in>dom (RBT.map_of t1) \<Longrightarrow> k>k'"
+ have DOM_T1: "!!k'. k'\<in>dom (RBT.lookup t1) \<Longrightarrow> k>k'"
proof -
fix k'
- from ST have "t1 |\<guillemotleft> k" by simp
- with tlt_prop have "\<forall>k'\<in>keys t1. k>k'" by auto
- moreover assume "k'\<in>dom (RBT.map_of t1)"
- ultimately show "k>k'" using RBT.mapof_keys ST by auto
+ from SORTED have "t1 |\<guillemotleft> k" by simp
+ with tree_less_prop have "\<forall>k'\<in>keys t1. k>k'" by auto
+ moreover assume "k'\<in>dom (RBT.lookup t1)"
+ ultimately show "k>k'" using RBT.lookup_keys SORTED by auto
qed
- have DOM_T2: "!!k'. k'\<in>dom (RBT.map_of t2) \<Longrightarrow> k<k'"
+ have DOM_T2: "!!k'. k'\<in>dom (RBT.lookup t2) \<Longrightarrow> k<k'"
proof -
fix k'
- from ST have "k \<guillemotleft>| t2" by simp
- with tgt_prop have "\<forall>k'\<in>keys t2. k<k'" by auto
- moreover assume "k'\<in>dom (RBT.map_of t2)"
- ultimately show "k<k'" using RBT.mapof_keys ST by auto
+ from SORTED have "k \<guillemotleft>| t2" by simp
+ with tree_greater_prop have "\<forall>k'\<in>keys t2. k<k'" by auto
+ moreover assume "k'\<in>dom (RBT.lookup t2)"
+ ultimately show "k<k'" using RBT.lookup_keys SORTED by auto
qed
{
assume C: "x<k"
- hence "RBT.map_of (Tr c t1 k v t2) x = RBT.map_of t1 x" by simp
+ hence "RBT.lookup (Branch c t1 k v t2) x = RBT.lookup t1 x" by simp
moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
- moreover have "x\<notin>dom (RBT.map_of t2)" proof
- assume "x\<in>dom (RBT.map_of t2)"
+ moreover have "x\<notin>dom (RBT.lookup t2)" proof
+ assume "x\<in>dom (RBT.lookup t2)"
with DOM_T2 have "k<x" by blast
with C show False by simp
qed
ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
} moreover {
assume [simp]: "x=k"
- hence "RBT.map_of (Tr c t1 k v t2) x = [k \<mapsto> v] x" by simp
- moreover have "x\<notin>dom (RBT.map_of t1)" proof
- assume "x\<in>dom (RBT.map_of t1)"
+ hence "RBT.lookup (Branch c t1 k v t2) x = [k \<mapsto> v] x" by simp
+ moreover have "x\<notin>dom (RBT.lookup t1)" proof
+ assume "x\<in>dom (RBT.lookup t1)"
with DOM_T1 have "k>x" by blast
thus False by simp
qed
ultimately have ?thesis by (simp add: map_add_upd_left map_add_dom_app_simps)
} moreover {
assume C: "x>k"
- hence "RBT.map_of (Tr c t1 k v t2) x = RBT.map_of t2 x" by (simp add: less_not_sym[of k x])
+ hence "RBT.lookup (Branch c t1 k v t2) x = RBT.lookup t2 x" by (simp add: less_not_sym[of k x])
moreover from C have "x\<notin>dom [k\<mapsto>v]" by simp
- moreover have "x\<notin>dom (RBT.map_of t1)" proof
- assume "x\<in>dom (RBT.map_of t1)"
+ moreover have "x\<notin>dom (RBT.lookup t1)" proof
+ assume "x\<in>dom (RBT.lookup t1)"
with DOM_T1 have "k>x" by simp
with C show False by simp
qed
@@ -969,35 +979,38 @@
} ultimately show ?thesis using less_linear by blast
qed
-lemma map_of_alist_of:
- shows "st t \<Longrightarrow> Map.map_of (alist_of t) = map_of t"
+lemma map_of_entries:
+ shows "sorted t \<Longrightarrow> map_of (entries t) = lookup t"
proof (induct t)
- case Empty thus ?case by (simp add: RBT.map_of_Empty)
+ case Empty thus ?case by (simp add: RBT.lookup_Empty)
next
- case (Tr c t1 k v t2)
- hence "Map.map_of (alist_of (Tr c t1 k v t2)) = RBT.map_of t2 ++ [k \<mapsto> v] ++ RBT.map_of t1" by simp
- also note map_of_alist_of_aux[OF Tr.prems,symmetric]
+ case (Branch c t1 k v t2)
+ hence "map_of (entries (Branch c t1 k v t2)) = RBT.lookup t2 ++ [k \<mapsto> v] ++ RBT.lookup t1" by simp
+ also note lookup_entries_aux [OF Branch.prems,symmetric]
finally show ?case .
qed
-lemma fold_alist_fold:
- "foldwithkey f t x = foldl (\<lambda>x (k,v). f k v x) x (alist_of t)"
+lemma fold_entries_fold:
+ "foldwithkey f t x = foldl (\<lambda>x (k,v). f k v x) x (entries t)"
by (induct t arbitrary: x) auto
-lemma alist_pit[simp]: "(k, v) \<in> set (alist_of t) = pin_tree k v t"
+lemma entries_pit[simp]: "(k, v) \<in> set (entries t) = entry_in_tree k v t"
by (induct t) auto
-lemma sorted_alist:
- "st t \<Longrightarrow> sorted (List.map fst (alist_of t))"
+lemma sorted_entries:
+ "sorted t \<Longrightarrow> List.sorted (List.map fst (entries t))"
by (induct t)
- (force simp: sorted_append sorted_Cons tlgt_props
- dest!:pint_keys)+
+ (force simp: sorted_append sorted_Cons tree_ord_props
+ dest!: entry_in_tree_keys)+
-lemma distinct_alist:
- "st t \<Longrightarrow> distinct (List.map fst (alist_of t))"
+lemma distinct_entries:
+ "sorted t \<Longrightarrow> distinct (List.map fst (entries t))"
by (induct t)
- (force simp: sorted_append sorted_Cons tlgt_props
- dest!:pint_keys)+
+ (force simp: sorted_append sorted_Cons tree_ord_props
+ dest!: entry_in_tree_keys)+
+
+hide (open) const Empty insert delete entries lookup map fold union adjust sorted
+
(*>*)
text {*
@@ -1010,20 +1023,20 @@
text {*
The type @{typ "('k, 'v) rbt"} denotes red-black trees with keys of
type @{typ "'k"} and values of type @{typ "'v"}. To function
- properly, the key type must belong to the @{text "linorder"} class.
+ properly, the key type musorted belong to the @{text "linorder"} class.
A value @{term t} of this type is a valid red-black tree if it
- satisfies the invariant @{text "isrbt t"}.
+ satisfies the invariant @{text "is_rbt t"}.
This theory provides lemmas to prove that the invariant is
satisfied throughout the computation.
- The interpretation function @{const "map_of"} returns the partial
+ The interpretation function @{const "RBT.lookup"} returns the partial
map represented by a red-black tree:
- @{term_type[display] "map_of"}
+ @{term_type[display] "RBT.lookup"}
This function should be used for reasoning about the semantics of the RBT
operations. Furthermore, it implements the lookup functionality for
- the data structure: It is executable and the lookup is performed in
+ the data sortedructure: It is executable and the lookup is performed in
$O(\log n)$.
*}
@@ -1032,19 +1045,19 @@
text {*
Currently, the following operations are supported:
- @{term_type[display] "Empty"}
+ @{term_type[display] "RBT.Empty"}
Returns the empty tree. $O(1)$
- @{term_type[display] "insrt"}
+ @{term_type[display] "RBT.insert"}
Updates the map at a given position. $O(\log n)$
- @{term_type[display] "delete"}
+ @{term_type[display] "RBT.delete"}
Deletes a map entry at a given position. $O(\log n)$
- @{term_type[display] "union"}
+ @{term_type[display] "RBT.union"}
Forms the union of two trees, preferring entries from the first one.
- @{term_type[display] "map"}
+ @{term_type[display] "RBT.map"}
Maps a function over the values of a map. $O(n)$
*}
@@ -1053,47 +1066,47 @@
text {*
\noindent
- @{thm Empty_isrbt}\hfill(@{text "Empty_isrbt"})
+ @{thm Empty_is_rbt}\hfill(@{text "Empty_is_rbt"})
\noindent
- @{thm insrt_isrbt}\hfill(@{text "insrt_isrbt"})
+ @{thm insert_is_rbt}\hfill(@{text "insert_is_rbt"})
\noindent
- @{thm delete_isrbt}\hfill(@{text "delete_isrbt"})
+ @{thm delete_is_rbt}\hfill(@{text "delete_is_rbt"})
\noindent
- @{thm union_isrbt}\hfill(@{text "union_isrbt"})
+ @{thm union_is_rbt}\hfill(@{text "union_is_rbt"})
\noindent
- @{thm map_isrbt}\hfill(@{text "map_isrbt"})
+ @{thm map_is_rbt}\hfill(@{text "map_is_rbt"})
*}
subsection {* Map Semantics *}
text {*
\noindent
- \underline{@{text "map_of_Empty"}}
- @{thm[display] map_of_Empty}
+ \underline{@{text "lookup_Empty"}}
+ @{thm[display] lookup_Empty}
\vspace{1ex}
\noindent
- \underline{@{text "map_of_insert"}}
- @{thm[display] map_of_insert}
+ \underline{@{text "lookup_insert"}}
+ @{thm[display] lookup_insert}
\vspace{1ex}
\noindent
- \underline{@{text "map_of_delete"}}
- @{thm[display] map_of_delete}
+ \underline{@{text "lookup_delete"}}
+ @{thm[display] lookup_delete}
\vspace{1ex}
\noindent
- \underline{@{text "map_of_union"}}
- @{thm[display] map_of_union}
+ \underline{@{text "lookup_union"}}
+ @{thm[display] lookup_union}
\vspace{1ex}
\noindent
- \underline{@{text "map_of_map"}}
- @{thm[display] map_of_map}
+ \underline{@{text "lookup_map"}}
+ @{thm[display] lookup_map}
\vspace{1ex}
*}