--- a/src/HOL/Data_Structures/Balance.thy Thu Oct 06 11:27:28 2016 +0200
+++ b/src/HOL/Data_Structures/Balance.thy Thu Oct 06 14:13:46 2016 +0200
@@ -11,9 +11,10 @@
(* mv *)
text \<open>The lemmas about \<open>floor\<close> and \<open>ceiling\<close> of \<open>log 2\<close> should be generalized
-from 2 to \<open>n\<close> and should be made executable. \<close>
+from 2 to \<open>n\<close> and should be made executable. In the end they should be moved
+to theory \<open>Log_Nat\<close> and \<open>floorlog\<close> should be replaced.\<close>
-lemma floor_log_nat: fixes b n k :: nat
+lemma floor_log_nat_ivl: fixes b n k :: nat
assumes "b \<ge> 2" "b^n \<le> k" "k < b^(n+1)"
shows "floor (log b (real k)) = int(n)"
proof -
@@ -32,7 +33,7 @@
qed
qed
-lemma ceil_log_nat: fixes b n k :: nat
+lemma ceil_log_nat_ivl: fixes b n k :: nat
assumes "b \<ge> 2" "b^n < k" "k \<le> b^(n+1)"
shows "ceiling (log b (real k)) = int(n)+1"
proof(rule ceiling_eq)
@@ -47,47 +48,6 @@
using assms(1,2) by(simp add: log_le_iff add_ac)
qed
-lemma ex_power_ivl1: fixes b k :: nat assumes "b \<ge> 2"
-shows "k \<ge> 1 \<Longrightarrow> \<exists>n. b^n \<le> k \<and> k < b^(n+1)" (is "_ \<Longrightarrow> \<exists>n. ?P k n")
-proof(induction k)
- case 0 thus ?case by simp
-next
- case (Suc k)
- show ?case
- proof cases
- assume "k=0"
- hence "?P (Suc k) 0"
- using assms by simp
- thus ?case ..
- next
- assume "k\<noteq>0"
- with Suc obtain n where IH: "?P k n" by auto
- show ?case
- proof (cases "k = b^(n+1) - 1")
- case True
- hence "?P (Suc k) (n+1)" using assms
- by (simp add: not_less_eq_eq[symmetric])
- thus ?thesis ..
- next
- case False
- hence "?P (Suc k) n" using IH by auto
- thus ?thesis ..
- qed
- qed
-qed
-
-lemma ex_power_ivl2: fixes b k :: nat assumes "b \<ge> 2" "(k::nat) \<ge> 2"
-shows "\<exists>n. b^n < k \<and> k \<le> b^(n+1)"
-proof -
- have "1 \<le> k - 1"
- using assms(2) by arith
- from ex_power_ivl1[OF assms(1) this]
- obtain n where "b ^ n \<le> k - 1 \<and> k - 1 < b ^ (n + 1)" ..
- hence "b^n < k \<and> k \<le> b^(n+1)"
- using assms by auto
- thus ?thesis ..
-qed
-
lemma ceil_log2_div2: assumes "n \<ge> 2"
shows "ceiling(log 2 (real n)) = ceiling(log 2 ((n-1) div 2 + 1)) + 1"
proof cases
@@ -107,7 +67,7 @@
finally have *: "n \<le> \<dots>" .
have "2^(i+1) < n"
using i(1) by (auto simp add: less_Suc_eq_0_disj)
- from ceil_log_nat[OF _ this *] ceil_log_nat[OF _ i]
+ from ceil_log_nat_ivl[OF _ this *] ceil_log_nat_ivl[OF _ i]
show ?thesis by simp
qed
@@ -130,7 +90,7 @@
finally have *: "2^(i+1) \<le> \<dots>" .
have "n < 2^(i+1+1)"
using i(2) by simp
- from floor_log_nat[OF _ * this] floor_log_nat[OF _ i]
+ from floor_log_nat_ivl[OF _ * this] floor_log_nat_ivl[OF _ i]
show ?thesis by simp
qed
--- a/src/HOL/Power.thy Thu Oct 06 11:27:28 2016 +0200
+++ b/src/HOL/Power.thy Thu Oct 06 14:13:46 2016 +0200
@@ -792,6 +792,44 @@
qed
qed
+lemma ex_power_ivl1: fixes b k :: nat assumes "b \<ge> 2"
+shows "k \<ge> 1 \<Longrightarrow> \<exists>n. b^n \<le> k \<and> k < b^(n+1)" (is "_ \<Longrightarrow> \<exists>n. ?P k n")
+proof(induction k)
+ case 0 thus ?case by simp
+next
+ case (Suc k)
+ show ?case
+ proof cases
+ assume "k=0"
+ hence "?P (Suc k) 0" using assms by simp
+ thus ?case ..
+ next
+ assume "k\<noteq>0"
+ with Suc obtain n where IH: "?P k n" by auto
+ show ?case
+ proof (cases "k = b^(n+1) - 1")
+ case True
+ hence "?P (Suc k) (n+1)" using assms
+ by (simp add: power_less_power_Suc)
+ thus ?thesis ..
+ next
+ case False
+ hence "?P (Suc k) n" using IH by auto
+ thus ?thesis ..
+ qed
+ qed
+qed
+
+lemma ex_power_ivl2: fixes b k :: nat assumes "b \<ge> 2" "k \<ge> 2"
+shows "\<exists>n. b^n < k \<and> k \<le> b^(n+1)"
+proof -
+ have "1 \<le> k - 1" using assms(2) by arith
+ from ex_power_ivl1[OF assms(1) this]
+ obtain n where "b ^ n \<le> k - 1 \<and> k - 1 < b ^ (n + 1)" ..
+ hence "b^n < k \<and> k \<le> b^(n+1)" using assms by auto
+ thus ?thesis ..
+qed
+
subsubsection \<open>Cardinality of the Powerset\<close>