--- a/src/HOL/Data_Structures/Balance.thy Sat Sep 10 14:11:04 2016 +0200
+++ b/src/HOL/Data_Structures/Balance.thy Sun Sep 11 18:12:05 2016 +0200
@@ -23,6 +23,16 @@
definition balance_tree :: "'a tree \<Rightarrow> 'a tree" where
"balance_tree = balance_list o inorder"
+
+lemma bal_simps:
+ "bal xs 0 = (Leaf, xs)"
+ "n > 0 \<Longrightarrow>
+ bal xs n =
+ (let m = n div 2;
+ (l, ys) = Balance.bal xs m;
+ (r, zs) = Balance.bal (tl ys) (n-1-m)
+ in (Node l (hd ys) r, zs))"
+by(simp_all add: bal.simps)
lemma bal_inorder:
"\<lbrakk> bal xs n = (t,ys); n \<le> length xs \<rbrakk>
@@ -30,7 +40,7 @@
proof(induction xs n arbitrary: t ys rule: bal.induct)
case (1 xs n) show ?case
proof cases
- assume "n = 0" thus ?thesis using 1 by (simp add: bal.simps)
+ assume "n = 0" thus ?thesis using 1 by (simp add: bal_simps)
next
assume [arith]: "n \<noteq> 0"
let ?n1 = "n div 2" let ?n2 = "n - 1 - ?n1"
@@ -38,7 +48,7 @@
b1: "bal xs ?n1 = (l,xs')" and
b2: "bal (tl xs') ?n2 = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
- using bal.simps[of xs n] by(auto simp: Let_def split: prod.splits)
+ by(auto simp: Let_def bal_simps split: prod.splits)
have IH1: "inorder l = take ?n1 xs \<and> xs' = drop ?n1 xs"
using b1 "1.prems" by(intro "1.IH"(1)) auto
have IH2: "inorder r = take ?n2 (tl xs') \<and> ys = drop ?n2 (tl xs')"
@@ -63,14 +73,14 @@
case (1 xs n) show ?case
proof cases
assume "n = 0" thus ?thesis
- using "1.prems" by (simp add: floorlog_def bal.simps)
+ using "1.prems" by (simp add: floorlog_def bal_simps)
next
assume [arith]: "n \<noteq> 0"
from "1.prems" obtain l r xs' where
b1: "bal xs (n div 2) = (l,xs')" and
b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
- using bal.simps[of xs n] by(auto simp: Let_def split: prod.splits)
+ by(auto simp: bal_simps Let_def split: prod.splits)
let ?log1 = "floorlog 2 (n div 2)"
let ?log2 = "floorlog 2 (n - 1 - n div 2)"
have IH1: "height l = ?log1" using "1.IH"(1) b1 by simp
@@ -90,14 +100,14 @@
case (1 xs n) show ?case
proof cases
assume "n = 0" thus ?thesis
- using "1.prems" by (simp add: floorlog_def bal.simps)
+ using "1.prems" by (simp add: floorlog_def bal_simps)
next
assume [arith]: "n \<noteq> 0"
from "1.prems" obtain l r xs' where
b1: "bal xs (n div 2) = (l,xs')" and
b2: "bal (tl xs') (n - 1 - n div 2) = (r,ys)" and
t: "t = \<langle>l, hd xs', r\<rangle>"
- using bal.simps[of xs n] by(auto simp: Let_def split: prod.splits)
+ by(auto simp: bal_simps Let_def split: prod.splits)
let ?log1 = "floorlog 2 (n div 2 + 1) - 1"
let ?log2 = "floorlog 2 (n - 1 - n div 2 + 1) - 1"
let ?log2' = "floorlog 2 (n - n div 2) - 1"