--- a/src/HOL/Data_Structures/RBT.thy Fri Nov 27 18:59:47 2015 +0100
+++ b/src/HOL/Data_Structures/RBT.thy Fri Nov 27 19:00:27 2015 +0100
@@ -22,20 +22,20 @@
"bal t1 a1 (R (R t2 a2 t3) a3 t4) = R (B t1 a1 t2) a2 (B t3 a3 t4)" |
"bal t1 a t2 = B t1 a t2"
-fun red :: "'a rbt \<Rightarrow> 'a rbt" where
-"red Leaf = Leaf" |
-"red (Node _ l a r) = R l a r"
+fun paint :: "color \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
+"paint c Leaf = Leaf" |
+"paint c (Node _ l a r) = Node c l a r"
fun balL :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"balL (R t1 x t2) y t3 = R (B t1 x t2) y t3" |
"balL bl x (B t1 y t2) = bal bl x (R t1 y t2)" |
-"balL bl x (R (B t1 y t2) z t3) = R (B bl x t1) y (bal t2 z (red t3))" |
+"balL bl x (R (B t1 y t2) z t3) = R (B bl x t1) y (bal t2 z (paint Red t3))" |
"balL t1 x t2 = R t1 x t2"
fun balR :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"balR t1 x (R t2 y t3) = R t1 x (B t2 y t3)" |
"balR (B t1 x t2) y t3 = bal (R t1 x t2) y t3" |
-"balR (R t1 x (B t2 y t3)) z t4 = R (bal (red t1) x t2) y (B t3 z t4)" |
+"balR (R t1 x (B t2 y t3)) z t4 = R (bal (paint Red t1) x t2) y (B t3 z t4)" |
"balR t1 x t2 = R t1 x t2"
fun combine :: "'a rbt \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
--- a/src/HOL/Data_Structures/RBT_Map.thy Fri Nov 27 18:59:47 2015 +0100
+++ b/src/HOL/Data_Structures/RBT_Map.thy Fri Nov 27 19:00:27 2015 +0100
@@ -8,48 +8,61 @@
Lookup2
begin
-fun update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
-"update x y Leaf = R Leaf (x,y) Leaf" |
-"update x y (B l (a,b) r) = (case cmp x a of
- LT \<Rightarrow> bal (update x y l) (a,b) r |
- GT \<Rightarrow> bal l (a,b) (update x y r) |
+fun upd :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
+"upd x y Leaf = R Leaf (x,y) Leaf" |
+"upd x y (B l (a,b) r) = (case cmp x a of
+ LT \<Rightarrow> bal (upd x y l) (a,b) r |
+ GT \<Rightarrow> bal l (a,b) (upd x y r) |
EQ \<Rightarrow> B l (x,y) r)" |
-"update x y (R l (a,b) r) = (case cmp x a of
- LT \<Rightarrow> R (update x y l) (a,b) r |
- GT \<Rightarrow> R l (a,b) (update x y r) |
+"upd x y (R l (a,b) r) = (case cmp x a of
+ LT \<Rightarrow> R (upd x y l) (a,b) r |
+ GT \<Rightarrow> R l (a,b) (upd x y r) |
EQ \<Rightarrow> R l (x,y) r)"
-fun delete :: "'a::cmp \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
-and deleteL :: "'a::cmp \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
-and deleteR :: "'a::cmp \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
+definition update :: "'a::cmp \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
+"update x y t = paint Black (upd x y t)"
+
+fun del :: "'a::cmp \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
+and delL :: "'a::cmp \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
+and delR :: "'a::cmp \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
where
-"delete x Leaf = Leaf" |
-"delete x (Node c t1 (a,b) t2) = (case cmp x a of
- LT \<Rightarrow> deleteL x t1 (a,b) t2 |
- GT \<Rightarrow> deleteR x t1 (a,b) t2 |
+"del x Leaf = Leaf" |
+"del x (Node c t1 (a,b) t2) = (case cmp x a of
+ LT \<Rightarrow> delL x t1 (a,b) t2 |
+ GT \<Rightarrow> delR x t1 (a,b) t2 |
EQ \<Rightarrow> combine t1 t2)" |
-"deleteL x (B t1 a t2) b t3 = balL (delete x (B t1 a t2)) b t3" |
-"deleteL x t1 a t2 = R (delete x t1) a t2" |
-"deleteR x t1 a (B t2 b t3) = balR t1 a (delete x (B t2 b t3))" |
-"deleteR x t1 a t2 = R t1 a (delete x t2)"
+"delL x (B t1 a t2) b t3 = balL (del x (B t1 a t2)) b t3" |
+"delL x t1 a t2 = R (del x t1) a t2" |
+"delR x t1 a (B t2 b t3) = balR t1 a (del x (B t2 b t3))" |
+"delR x t1 a t2 = R t1 a (del x t2)"
+
+definition delete :: "'a::cmp \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
+"delete x t = paint Black (del x t)"
subsection "Functional Correctness Proofs"
+lemma inorder_upd:
+ "sorted1(inorder t) \<Longrightarrow> inorder(upd x y t) = upd_list x y (inorder t)"
+by(induction x y t rule: upd.induct)
+ (auto simp: upd_list_simps inorder_bal)
+
lemma inorder_update:
"sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
-by(induction x y t rule: update.induct)
- (auto simp: upd_list_simps inorder_bal)
+by(simp add: update_def inorder_upd inorder_paint)
+lemma inorder_del:
+ "sorted1(inorder t1) \<Longrightarrow> inorder(del x t1) = del_list x (inorder t1)" and
+ "sorted1(inorder t1) \<Longrightarrow> inorder(delL x t1 a t2) =
+ del_list x (inorder t1) @ a # inorder t2" and
+ "sorted1(inorder t2) \<Longrightarrow> inorder(delR x t1 a t2) =
+ inorder t1 @ a # del_list x (inorder t2)"
+by(induction x t1 and x t1 a t2 and x t1 a t2 rule: del_delL_delR.induct)
+ (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
lemma inorder_delete:
- "sorted1(inorder t1) \<Longrightarrow> inorder(delete x t1) = del_list x (inorder t1)" and
- "sorted1(inorder t1) \<Longrightarrow> inorder(deleteL x t1 a t2) =
- del_list x (inorder t1) @ a # inorder t2" and
- "sorted1(inorder t2) \<Longrightarrow> inorder(deleteR x t1 a t2) =
- inorder t1 @ a # del_list x (inorder t2)"
-by(induction x t1 and x t1 a t2 and x t1 a t2 rule: delete_deleteL_deleteR.induct)
- (auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
+ "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
+by(simp add: delete_def inorder_del inorder_paint)
interpretation Map_by_Ordered
where empty = Leaf and lookup = lookup and update = update and delete = delete
--- a/src/HOL/Data_Structures/RBT_Set.thy Fri Nov 27 18:59:47 2015 +0100
+++ b/src/HOL/Data_Structures/RBT_Set.thy Fri Nov 27 19:00:27 2015 +0100
@@ -9,70 +9,84 @@
Isin2
begin
-fun insert :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
-"insert x Leaf = R Leaf x Leaf" |
-"insert x (B l a r) =
+fun ins :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
+"ins x Leaf = R Leaf x Leaf" |
+"ins x (B l a r) =
(case cmp x a of
- LT \<Rightarrow> bal (insert x l) a r |
- GT \<Rightarrow> bal l a (insert x r) |
+ LT \<Rightarrow> bal (ins x l) a r |
+ GT \<Rightarrow> bal l a (ins x r) |
EQ \<Rightarrow> B l a r)" |
-"insert x (R l a r) =
+"ins x (R l a r) =
(case cmp x a of
- LT \<Rightarrow> R (insert x l) a r |
- GT \<Rightarrow> R l a (insert x r) |
+ LT \<Rightarrow> R (ins x l) a r |
+ GT \<Rightarrow> R l a (ins x r) |
EQ \<Rightarrow> R l a r)"
-fun delete :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
-and deleteL :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
-and deleteR :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
+definition insert :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
+"insert x t = paint Black (ins x t)"
+
+fun del :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
+and delL :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
+and delR :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt"
where
-"delete x Leaf = Leaf" |
-"delete x (Node _ l a r) =
+"del x Leaf = Leaf" |
+"del x (Node _ l a r) =
(case cmp x a of
- LT \<Rightarrow> deleteL x l a r |
- GT \<Rightarrow> deleteR x l a r |
+ LT \<Rightarrow> delL x l a r |
+ GT \<Rightarrow> delR x l a r |
EQ \<Rightarrow> combine l r)" |
-"deleteL x (B t1 a t2) b t3 = balL (delete x (B t1 a t2)) b t3" |
-"deleteL x l a r = R (delete x l) a r" |
-"deleteR x t1 a (B t2 b t3) = balR t1 a (delete x (B t2 b t3))" |
-"deleteR x l a r = R l a (delete x r)"
+"delL x (B t1 a t2) b t3 = balL (del x (B t1 a t2)) b t3" |
+"delL x l a r = R (del x l) a r" |
+"delR x t1 a (B t2 b t3) = balR t1 a (del x (B t2 b t3))" |
+"delR x l a r = R l a (del x r)"
+
+definition delete :: "'a::cmp \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
+"delete x t = paint Black (del x t)"
subsection "Functional Correctness Proofs"
+lemma inorder_paint: "inorder(paint c t) = inorder t"
+by(induction t) (auto)
+
lemma inorder_bal:
"inorder(bal l a r) = inorder l @ a # inorder r"
by(induction l a r rule: bal.induct) (auto)
+lemma inorder_ins:
+ "sorted(inorder t) \<Longrightarrow> inorder(ins x t) = ins_list x (inorder t)"
+by(induction x t rule: ins.induct) (auto simp: ins_list_simps inorder_bal)
+
lemma inorder_insert:
- "sorted(inorder t) \<Longrightarrow> inorder(insert a t) = ins_list a (inorder t)"
-by(induction a t rule: insert.induct) (auto simp: ins_list_simps inorder_bal)
-
-lemma inorder_red: "inorder(red t) = inorder t"
-by(induction t) (auto)
+ "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
+by (simp add: insert_def inorder_ins inorder_paint)
lemma inorder_balL:
"inorder(balL l a r) = inorder l @ a # inorder r"
-by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_red)
+by(induction l a r rule: balL.induct)(auto simp: inorder_bal inorder_paint)
lemma inorder_balR:
"inorder(balR l a r) = inorder l @ a # inorder r"
-by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_red)
+by(induction l a r rule: balR.induct) (auto simp: inorder_bal inorder_paint)
lemma inorder_combine:
"inorder(combine l r) = inorder l @ inorder r"
by(induction l r rule: combine.induct)
(auto simp: inorder_balL inorder_balR split: tree.split color.split)
-lemma inorder_delete:
- "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
- "sorted(inorder l) \<Longrightarrow> inorder(deleteL x l a r) =
+lemma inorder_del:
+ "sorted(inorder t) \<Longrightarrow> inorder(del x t) = del_list x (inorder t)"
+ "sorted(inorder l) \<Longrightarrow> inorder(delL x l a r) =
del_list x (inorder l) @ a # inorder r"
- "sorted(inorder r) \<Longrightarrow> inorder(deleteR x l a r) =
+ "sorted(inorder r) \<Longrightarrow> inorder(delR x l a r) =
inorder l @ a # del_list x (inorder r)"
-by(induction x t and x l a r and x l a r rule: delete_deleteL_deleteR.induct)
+by(induction x t and x l a r and x l a r rule: del_delL_delR.induct)
(auto simp: del_list_simps inorder_combine inorder_balL inorder_balR)
+lemma inorder_delete:
+ "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
+by (auto simp: delete_def inorder_del inorder_paint)
+
interpretation Set_by_Ordered
where empty = Leaf and isin = isin and insert = insert and delete = delete
@@ -84,7 +98,7 @@
next
case 3 thus ?case by(simp add: inorder_insert)
next
- case 4 thus ?case by(simp add: inorder_delete(1))
-qed (rule TrueI)+
+ case 4 thus ?case by(simp add: inorder_delete)
+qed auto
end