src/HOL/Library/Tree_Real.thy
 author haftmann Wed Jul 18 20:51:21 2018 +0200 (11 months ago) changeset 68658 16cc1161ad7f parent 68484 59793df7f853 child 68998 818898556504 permissions -rw-r--r--
tuned equation
1 (* Author: Tobias Nipkow *)
3 theory Tree_Real
4 imports
5   Complex_Main
6   Tree
7 begin
9 text \<open>
10   This theory is separate from @{theory "HOL-Library.Tree"} because the former is discrete and
11   builds on @{theory Main} whereas this theory builds on @{theory Complex_Main}.
12 \<close>
15 lemma size1_height_log: "log 2 (size1 t) \<le> height t"
16 by (simp add: log2_of_power_le size1_height)
18 lemma min_height_size1_log: "min_height t \<le> log 2 (size1 t)"
19 by (simp add: le_log2_of_power min_height_size1)
21 lemma size1_log_if_complete: "complete t \<Longrightarrow> height t = log 2 (size1 t)"
22 by (simp add: log2_of_power_eq size1_if_complete)
24 lemma min_height_size1_log_if_incomplete:
25   "\<not> complete t \<Longrightarrow> min_height t < log 2 (size1 t)"
26 by (simp add: less_log2_of_power min_height_size1_if_incomplete)
29 lemma min_height_balanced: assumes "balanced t"
30 shows "min_height t = nat(floor(log 2 (size1 t)))"
31 proof cases
32   assume *: "complete t"
33   hence "size1 t = 2 ^ min_height t"
34     by (simp add: complete_iff_height size1_if_complete)
35   from log2_of_power_eq[OF this] show ?thesis by linarith
36 next
37   assume *: "\<not> complete t"
38   hence "height t = min_height t + 1"
39     using assms min_height_le_height[of t]
40     by(auto simp add: balanced_def complete_iff_height)
41   hence "size1 t < 2 ^ (min_height t + 1)"
42     by (metis * size1_height_if_incomplete)
43   hence "log 2 (size1 t) < min_height t + 1"
44     using log2_of_power_less size1_ge0 by blast
45   thus ?thesis using min_height_size1_log[of t] by linarith
46 qed
48 lemma height_balanced: assumes "balanced t"
49 shows "height t = nat(ceiling(log 2 (size1 t)))"
50 proof cases
51   assume *: "complete t"
52   hence "size1 t = 2 ^ height t"
54   from log2_of_power_eq[OF this] show ?thesis
55     by linarith
56 next
57   assume *: "\<not> complete t"
58   hence **: "height t = min_height t + 1"
59     using assms min_height_le_height[of t]
60     by(auto simp add: balanced_def complete_iff_height)
61   hence "size1 t \<le> 2 ^ (min_height t + 1)" by (metis size1_height)
62   from  log2_of_power_le[OF this size1_ge0] min_height_size1_log_if_incomplete[OF *] **
63   show ?thesis by linarith
64 qed
66 lemma balanced_Node_if_wbal1:
67 assumes "balanced l" "balanced r" "size l = size r + 1"
68 shows "balanced \<langle>l, x, r\<rangle>"
69 proof -
70   from assms(3) have [simp]: "size1 l = size1 r + 1" by(simp add: size1_def)
71   have "nat \<lceil>log 2 (1 + size1 r)\<rceil> \<ge> nat \<lceil>log 2 (size1 r)\<rceil>"
72     by(rule nat_mono[OF ceiling_mono]) simp
73   hence 1: "height(Node l x r) = nat \<lceil>log 2 (1 + size1 r)\<rceil> + 1"
74     using height_balanced[OF assms(1)] height_balanced[OF assms(2)]
75     by (simp del: nat_ceiling_le_eq add: max_def)
76   have "nat \<lfloor>log 2 (1 + size1 r)\<rfloor> \<ge> nat \<lfloor>log 2 (size1 r)\<rfloor>"
77     by(rule nat_mono[OF floor_mono]) simp
78   hence 2: "min_height(Node l x r) = nat \<lfloor>log 2 (size1 r)\<rfloor> + 1"
79     using min_height_balanced[OF assms(1)] min_height_balanced[OF assms(2)]
80     by (simp)
81   have "size1 r \<ge> 1" by(simp add: size1_def)
82   then obtain i where i: "2 ^ i \<le> size1 r" "size1 r < 2 ^ (i + 1)"
83     using ex_power_ivl1[of 2 "size1 r"] by auto
84   hence i1: "2 ^ i < size1 r + 1" "size1 r + 1 \<le> 2 ^ (i + 1)" by auto
85   from 1 2 floor_log_nat_eq_if[OF i] ceiling_log_nat_eq_if[OF i1]
87 qed
89 lemma balanced_sym: "balanced \<langle>l, x, r\<rangle> \<Longrightarrow> balanced \<langle>r, y, l\<rangle>"
90 by(auto simp: balanced_def)
92 lemma balanced_Node_if_wbal2:
93 assumes "balanced l" "balanced r" "abs(int(size l) - int(size r)) \<le> 1"
94 shows "balanced \<langle>l, x, r\<rangle>"
95 proof -
96   have "size l = size r \<or> (size l = size r + 1 \<or> size r = size l + 1)" (is "?A \<or> ?B")
97     using assms(3) by linarith
98   thus ?thesis
99   proof
100     assume "?A"
101     thus ?thesis using assms(1,2)
102       apply(simp add: balanced_def min_def max_def)
103       by (metis assms(1,2) balanced_optimal le_antisym le_less)
104   next
105     assume "?B"
106     thus ?thesis
107       by (meson assms(1,2) balanced_sym balanced_Node_if_wbal1)
108   qed
109 qed
111 lemma balanced_if_wbalanced: "wbalanced t \<Longrightarrow> balanced t"
112 proof(induction t)
113   case Leaf show ?case by (simp add: balanced_def)
114 next
115   case (Node l x r)
116   thus ?case by(simp add: balanced_Node_if_wbal2)
117 qed
119 end