src/HOL/Library/Tree_Real.thy
 author Manuel Eberl Mon Mar 26 16:14:16 2018 +0200 (18 months ago) changeset 67951 655aa11359dc parent 67399 eab6ce8368fa child 68484 59793df7f853 permissions -rw-r--r--
Removed some uses of deprecated _tac methods. (Patch from Viorel Preoteasa)
1 (* Author: Tobias Nipkow *)
3 theory Tree_Real
4 imports
5   Complex_Main
6   Tree
7 begin
9 text \<open>This theory is separate from @{theory Tree} because the former is discrete and builds on
10 @{theory Main} whereas this theory builds on @{theory Complex_Main}.\<close>
13 lemma size1_height_log: "log 2 (size1 t) \<le> height t"
14 by (simp add: log2_of_power_le size1_height)
16 lemma min_height_size1_log: "min_height t \<le> log 2 (size1 t)"
17 by (simp add: le_log2_of_power min_height_size1)
19 lemma size1_log_if_complete: "complete t \<Longrightarrow> height t = log 2 (size1 t)"
20 by (simp add: log2_of_power_eq size1_if_complete)
22 lemma min_height_size1_log_if_incomplete:
23   "\<not> complete t \<Longrightarrow> min_height t < log 2 (size1 t)"
24 by (simp add: less_log2_of_power min_height_size1_if_incomplete)
27 lemma min_height_balanced: assumes "balanced t"
28 shows "min_height t = nat(floor(log 2 (size1 t)))"
29 proof cases
30   assume *: "complete t"
31   hence "size1 t = 2 ^ min_height t"
32     by (simp add: complete_iff_height size1_if_complete)
33   from log2_of_power_eq[OF this] show ?thesis by linarith
34 next
35   assume *: "\<not> complete t"
36   hence "height t = min_height t + 1"
37     using assms min_height_le_height[of t]
38     by(auto simp add: balanced_def complete_iff_height)
39   hence "size1 t < 2 ^ (min_height t + 1)"
40     by (metis * size1_height_if_incomplete)
41   hence "log 2 (size1 t) < min_height t + 1"
42     using log2_of_power_less size1_ge0 by blast
43   thus ?thesis using min_height_size1_log[of t] by linarith
44 qed
46 lemma height_balanced: assumes "balanced t"
47 shows "height t = nat(ceiling(log 2 (size1 t)))"
48 proof cases
49   assume *: "complete t"
50   hence "size1 t = 2 ^ height t"
51     by (simp add: size1_if_complete)
52   from log2_of_power_eq[OF this] show ?thesis
53     by linarith
54 next
55   assume *: "\<not> complete t"
56   hence **: "height t = min_height t + 1"
57     using assms min_height_le_height[of t]
58     by(auto simp add: balanced_def complete_iff_height)
59   hence "size1 t \<le> 2 ^ (min_height t + 1)" by (metis size1_height)
60   from  log2_of_power_le[OF this size1_ge0] min_height_size1_log_if_incomplete[OF *] **
61   show ?thesis by linarith
62 qed
64 lemma balanced_Node_if_wbal1:
65 assumes "balanced l" "balanced r" "size l = size r + 1"
66 shows "balanced \<langle>l, x, r\<rangle>"
67 proof -
68   from assms(3) have [simp]: "size1 l = size1 r + 1" by(simp add: size1_def)
69   have "nat \<lceil>log 2 (1 + size1 r)\<rceil> \<ge> nat \<lceil>log 2 (size1 r)\<rceil>"
70     by(rule nat_mono[OF ceiling_mono]) simp
71   hence 1: "height(Node l x r) = nat \<lceil>log 2 (1 + size1 r)\<rceil> + 1"
72     using height_balanced[OF assms(1)] height_balanced[OF assms(2)]
73     by (simp del: nat_ceiling_le_eq add: max_def)
74   have "nat \<lfloor>log 2 (1 + size1 r)\<rfloor> \<ge> nat \<lfloor>log 2 (size1 r)\<rfloor>"
75     by(rule nat_mono[OF floor_mono]) simp
76   hence 2: "min_height(Node l x r) = nat \<lfloor>log 2 (size1 r)\<rfloor> + 1"
77     using min_height_balanced[OF assms(1)] min_height_balanced[OF assms(2)]
78     by (simp)
79   have "size1 r \<ge> 1" by(simp add: size1_def)
80   then obtain i where i: "2 ^ i \<le> size1 r" "size1 r < 2 ^ (i + 1)"
81     using ex_power_ivl1[of 2 "size1 r"] by auto
82   hence i1: "2 ^ i < size1 r + 1" "size1 r + 1 \<le> 2 ^ (i + 1)" by auto
83   from 1 2 floor_log_nat_eq_if[OF i] ceiling_log_nat_eq_if[OF i1]
84   show ?thesis by(simp add:balanced_def)
85 qed
87 lemma balanced_sym: "balanced \<langle>l, x, r\<rangle> \<Longrightarrow> balanced \<langle>r, y, l\<rangle>"
88 by(auto simp: balanced_def)
90 lemma balanced_Node_if_wbal2:
91 assumes "balanced l" "balanced r" "abs(int(size l) - int(size r)) \<le> 1"
92 shows "balanced \<langle>l, x, r\<rangle>"
93 proof -
94   have "size l = size r \<or> (size l = size r + 1 \<or> size r = size l + 1)" (is "?A \<or> ?B")
95     using assms(3) by linarith
96   thus ?thesis
97   proof
98     assume "?A"
99     thus ?thesis using assms(1,2)
100       apply(simp add: balanced_def min_def max_def)
101       by (metis assms(1,2) balanced_optimal le_antisym le_less)
102   next
103     assume "?B"
104     thus ?thesis
105       by (meson assms(1,2) balanced_sym balanced_Node_if_wbal1)
106   qed
107 qed
109 lemma balanced_if_wbalanced: "wbalanced t \<Longrightarrow> balanced t"
110 proof(induction t)
111   case Leaf show ?case by (simp add: balanced_def)
112 next
113   case (Node l x r)
114   thus ?case by(simp add: balanced_Node_if_wbal2)
115 qed
117 end