--- a/src/HOL/Data_Structures/Balance.thy Tue Sep 24 16:17:37 2019 +0200
+++ b/src/HOL/Data_Structures/Balance.thy Tue Sep 24 17:36:14 2019 +0200
@@ -40,7 +40,7 @@
lemma bal_inorder:
"\<lbrakk> n \<le> length xs; bal n xs = (t,zs) \<rbrakk>
- \<Longrightarrow> inorder t = take n xs \<and> zs = drop n xs"
+ \<Longrightarrow> xs = inorder t @ zs \<and> size t = n"
proof(induction n xs arbitrary: t zs rule: bal.induct)
case (1 n xs) show ?case
proof cases
@@ -53,24 +53,18 @@
b2: "bal ?m' (tl ys) = (r,zs)" and
t: "t = \<langle>l, hd ys, r\<rangle>"
by(auto simp: Let_def bal_simps split: prod.splits)
- have IH1: "inorder l = take ?m xs \<and> ys = drop ?m xs"
+ have IH1: "xs = inorder l @ ys \<and> size l = ?m"
using b1 "1.prems"(1) by(intro "1.IH"(1)) auto
- have IH2: "inorder r = take ?m' (tl ys) \<and> zs = drop ?m' (tl ys)"
+ have IH2: "tl ys = inorder r @ zs \<and> size r = ?m'"
using b1 b2 IH1 "1.prems"(1) by(intro "1.IH"(2)) auto
- have "drop (n div 2) xs \<noteq> []" using "1.prems"(1) by simp
- hence "hd (drop ?m xs) # take ?m' (tl (drop ?m xs)) = take (?m' + 1) (drop ?m xs)"
- by (metis Suc_eq_plus1 take_Suc)
- hence *: "inorder t = take n xs" using t IH1 IH2
- using take_add[of ?m "?m'+1" xs] by(simp)
- have "n - n div 2 + n div 2 = n" by simp
- hence "zs = drop n xs" using IH1 IH2 by (simp add: drop_Suc[symmetric])
- thus ?thesis using * by blast
+ show ?thesis using t IH1 IH2 "1.prems"(1) hd_Cons_tl[of ys] by fastforce
qed
qed
corollary inorder_bal_list[simp]:
"n \<le> length xs \<Longrightarrow> inorder(bal_list n xs) = take n xs"
-unfolding bal_list_def by (metis bal_inorder eq_fst_iff)
+unfolding bal_list_def
+by (metis (mono_tags) prod.collapse[of "bal n xs"] append_eq_conv_conj bal_inorder length_inorder)
corollary inorder_balance_list[simp]: "inorder(balance_list xs) = xs"
by(simp add: balance_list_def)
@@ -83,18 +77,18 @@
by(simp add: balance_tree_def inorder_bal_tree)
-text\<open>The size lemmas below do not require the precondition @{prop"n \<le> length xs"}
+text\<open>The length/size lemmas below do not require the precondition @{prop"n \<le> length xs"}
(or @{prop"n \<le> size t"}) that they come with. They could take advantage of the fact
that @{term "bal xs n"} yields a result even if @{prop "n > length xs"}.
In that case the result will contain one or more occurrences of @{term "hd []"}.
However, this is counter-intuitive and does not reflect the execution
in an eager functional language.\<close>
-lemma size_bal: "\<lbrakk> n \<le> length xs; bal n xs = (t,zs) \<rbrakk> \<Longrightarrow> size t = n \<and> length zs = length xs - n"
-by (metis bal_inorder length_drop length_inorder length_take min.absorb2)
+lemma bal_length: "\<lbrakk> n \<le> length xs; bal n xs = (t,zs) \<rbrakk> \<Longrightarrow> length zs = length xs - n"
+using bal_inorder by fastforce
corollary size_bal_list[simp]: "n \<le> length xs \<Longrightarrow> size(bal_list n xs) = n"
-unfolding bal_list_def by (metis prod.collapse size_bal)
+unfolding bal_list_def using bal_inorder prod.exhaust_sel by blast
corollary size_balance_list[simp]: "size(balance_list xs) = length xs"
by (simp add: balance_list_def)
@@ -105,10 +99,6 @@
corollary size_balance_tree[simp]: "size(balance_tree t) = size t"
by(simp add: balance_tree_def)
-lemma pre_rec2: "\<lbrakk> n \<le> length xs; bal (n div 2) xs = (l, ys) \<rbrakk>
- \<Longrightarrow> (n - 1 - n div 2) \<le> length(tl ys)"
-using size_bal[of "n div 2" xs l ys] by simp
-
lemma min_height_bal:
"\<lbrakk> n \<le> length xs; bal n xs = (t,zs) \<rbrakk> \<Longrightarrow> min_height t = nat(\<lfloor>log 2 (n + 1)\<rfloor>)"
proof(induction n xs arbitrary: t zs rule: bal.induct)
@@ -118,23 +108,24 @@
assume "n = 0" thus ?thesis using "1.prems"(2) by (simp add: bal_simps)
next
assume [arith]: "n \<noteq> 0"
+ let ?m = "n div 2" let ?m' = "n - 1 - ?m"
from "1.prems" obtain l r ys where
- b1: "bal (n div 2) xs = (l,ys)" and
- b2: "bal (n - 1 - n div 2) (tl ys) = (r,zs)" and
+ b1: "bal ?m xs = (l,ys)" and
+ b2: "bal ?m' (tl ys) = (r,zs)" and
t: "t = \<langle>l, hd ys, r\<rangle>"
by(auto simp: bal_simps Let_def split: prod.splits)
- let ?log1 = "nat (floor(log 2 (n div 2 + 1)))"
- let ?log2 = "nat (floor(log 2 (n - 1 - n div 2 + 1)))"
- have IH1: "min_height l = ?log1" using "1.IH"(1) b1 "1.prems"(1) by simp
- have IH2: "min_height r = ?log2"
- using "1.prems"(1) size_bal[OF _ b1] size_bal[OF _ b2] b1 b2 by(intro "1.IH"(2)) auto
+ let ?hl = "nat (floor(log 2 (?m + 1)))"
+ let ?hr = "nat (floor(log 2 (?m' + 1)))"
+ have IH1: "min_height l = ?hl" using "1.IH"(1) b1 "1.prems"(1) by simp
+ have IH2: "min_height r = ?hr"
+ using "1.prems"(1) bal_length[OF _ b1] b1 b2 by(intro "1.IH"(2)) auto
have "(n+1) div 2 \<ge> 1" by arith
hence 0: "log 2 ((n+1) div 2) \<ge> 0" by simp
- have "n - 1 - n div 2 + 1 \<le> n div 2 + 1" by arith
- hence le: "?log2 \<le> ?log1" by(simp add: nat_mono floor_mono)
- have "min_height t = min ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
- also have "\<dots> = ?log2 + 1" using le by (simp add: min_absorb2)
- also have "n - 1 - n div 2 + 1 = (n+1) div 2" by linarith
+ have "?m' \<le> ?m" by arith
+ hence le: "?hr \<le> ?hl" by(simp add: nat_mono floor_mono)
+ have "min_height t = min ?hl ?hr + 1" by (simp add: t IH1 IH2)
+ also have "\<dots> = ?hr + 1" using le by (simp add: min_absorb2)
+ also have "?m' + 1 = (n+1) div 2" by linarith
also have "nat (floor(log 2 ((n+1) div 2))) + 1
= nat (floor(log 2 ((n+1) div 2) + 1))"
using 0 by linarith
@@ -153,25 +144,24 @@
using "1.prems" by (simp add: bal_simps)
next
assume [arith]: "n \<noteq> 0"
+ let ?m = "n div 2" let ?m' = "n - 1 - ?m"
from "1.prems" obtain l r ys where
- b1: "bal (n div 2) xs = (l,ys)" and
- b2: "bal (n - 1 - n div 2) (tl ys) = (r,zs)" and
+ b1: "bal ?m xs = (l,ys)" and
+ b2: "bal ?m' (tl ys) = (r,zs)" and
t: "t = \<langle>l, hd ys, r\<rangle>"
by(auto simp: bal_simps Let_def split: prod.splits)
- let ?log1 = "nat \<lceil>log 2 (n div 2 + 1)\<rceil>"
- let ?log2 = "nat \<lceil>log 2 (n - 1 - n div 2 + 1)\<rceil>"
- have 1: "n div 2 \<le> length xs" using "1.prems"(1) by linarith
- have 2: "n - 1 - n div 2 \<le> length (tl ys)" using "1.prems"(1) size_bal[OF 1 b1] by simp
- have IH1: "height l = ?log1" using "1.IH"(1) b1 "1.prems"(1) by simp
- have IH2: "height r = ?log2"
- using b1 b2 size_bal[OF _ b1] size_bal[OF _ b2] "1.prems"(1) by(intro "1.IH"(2)) auto
- have 0: "log 2 (n div 2 + 1) \<ge> 0" by simp
- have "n - 1 - n div 2 + 1 \<le> n div 2 + 1" by arith
- hence le: "?log2 \<le> ?log1"
+ let ?hl = "nat \<lceil>log 2 (?m + 1)\<rceil>"
+ let ?hr = "nat \<lceil>log 2 (?m' + 1)\<rceil>"
+ have IH1: "height l = ?hl" using "1.IH"(1) b1 "1.prems"(1) by simp
+ have IH2: "height r = ?hr"
+ using b1 b2 bal_length[OF _ b1] "1.prems"(1) by(intro "1.IH"(2)) auto
+ have 0: "log 2 (?m + 1) \<ge> 0" by simp
+ have "?m' \<le> ?m" by arith
+ hence le: "?hr \<le> ?hl"
by(simp add: nat_mono ceiling_mono del: nat_ceiling_le_eq)
- have "height t = max ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
- also have "\<dots> = ?log1 + 1" using le by (simp add: max_absorb1)
- also have "\<dots> = nat \<lceil>log 2 (n div 2 + 1) + 1\<rceil>" using 0 by linarith
+ have "height t = max ?hl ?hr + 1" by (simp add: t IH1 IH2)
+ also have "\<dots> = ?hl + 1" using le by (simp add: max_absorb1)
+ also have "\<dots> = nat \<lceil>log 2 (?m + 1) + 1\<rceil>" using 0 by linarith
also have "\<dots> = nat \<lceil>log 2 (n + 1)\<rceil>"
using ceiling_log2_div2[of "n+1"] by (simp)
finally show ?thesis .
@@ -229,8 +219,8 @@
by(auto simp add: bal_simps Let_def split: prod.splits)
have l: "wbalanced l" using "1.IH"(1)[OF \<open>n\<noteq>0\<close> refl _ rec1] "1.prems"(1) by linarith
have "wbalanced r"
- using rec1 rec2 pre_rec2[OF "1.prems"(1) rec1] by(intro "1.IH"(2)) auto
- with l t size_bal[OF _ rec1] size_bal[OF _ rec2] "1.prems"(1)
+ using rec1 rec2 bal_length[OF _ rec1] "1.prems"(1) by(intro "1.IH"(2)) auto
+ with l t bal_length[OF _ rec1] "1.prems"(1) bal_inorder[OF _ rec1] bal_inorder[OF _ rec2]
show ?thesis by auto
qed
qed