(* Author: Tobias Nipkow *)
section \<open>Red-Black Trees\<close>
theory RBT
imports Tree2
begin
datatype color = Red | Black
type_synonym 'a rbt = "('a*color)tree"
abbreviation R where "R l a r \<equiv> Node l (a, Red) r"
abbreviation B where "B l a r \<equiv> Node l (a, Black) r"
fun baliL :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"baliL (R (R t1 a t2) b t3) c t4 = R (B t1 a t2) b (B t3 c t4)" |
"baliL (R t1 a (R t2 b t3)) c t4 = R (B t1 a t2) b (B t3 c t4)" |
"baliL t1 a t2 = B t1 a t2"
fun baliR :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"baliR t1 a (R t2 b (R t3 c t4)) = R (B t1 a t2) b (B t3 c t4)" |
"baliR t1 a (R (R t2 b t3) c t4) = R (B t1 a t2) b (B t3 c t4)" |
"baliR t1 a t2 = B t1 a t2"
fun paint :: "color \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"paint c Leaf = Leaf" |
"paint c (Node l (a,_) r) = Node l (a,c) r"
fun baldL :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"baldL (R t1 a t2) b t3 = R (B t1 a t2) b t3" |
"baldL t1 a (B t2 b t3) = baliR t1 a (R t2 b t3)" |
"baldL t1 a (R (B t2 b t3) c t4) = R (B t1 a t2) b (baliR t3 c (paint Red t4))" |
"baldL t1 a t2 = R t1 a t2"
fun baldR :: "'a rbt \<Rightarrow> 'a \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"baldR t1 a (R t2 b t3) = R t1 a (B t2 b t3)" |
"baldR (B t1 a t2) b t3 = baliL (R t1 a t2) b t3" |
"baldR (R t1 a (B t2 b t3)) c t4 = R (baliL (paint Red t1) a t2) b (B t3 c t4)" |
"baldR t1 a t2 = R t1 a t2"
fun app :: "'a rbt \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
"app Leaf t = t" |
"app t Leaf = t" |
"app (R t1 a t2) (R t3 c t4) =
(case app t2 t3 of
R u2 b u3 \<Rightarrow> (R (R t1 a u2) b (R u3 c t4)) |
t23 \<Rightarrow> R t1 a (R t23 c t4))" |
"app (B t1 a t2) (B t3 c t4) =
(case app t2 t3 of
R u2 b u3 \<Rightarrow> R (B t1 a u2) b (B u3 c t4) |
t23 \<Rightarrow> baldL t1 a (B t23 c t4))" |
"app t1 (R t2 a t3) = R (app t1 t2) a t3" |
"app (R t1 a t2) t3 = R t1 a (app t2 t3)"
end