author nipkow Wed, 26 Feb 2020 23:49:16 +0100 changeset 71486 0e1b9b308d8f parent 71485 29e297fd5473 child 71487 059c55b61734
simplified proofs
```--- a/src/HOL/Data_Structures/AVL_Set.thy	Wed Feb 26 19:50:04 2020 +0100
+++ b/src/HOL/Data_Structures/AVL_Set.thy	Wed Feb 26 23:49:16 2020 +0100
@@ -483,38 +483,28 @@
} ultimately show ?case using C by blast
qed

-lemma fib_alt_induct [consumes 1, case_names 1 2 rec]:
-  assumes "n > 0" "P 1" "P 2" "\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc n) \<Longrightarrow> P (Suc (Suc n))"
-  shows   "P n"
-  using assms(1)
-proof (induction n rule: fib.induct)
-  case (3 n)
-  thus ?case using assms by (cases n) (auto simp: eval_nat_numeral)
-qed (insert assms, auto)
-
text \<open>An exponential lower bound for \<^const>\<open>fib\<close>:\<close>

lemma fib_lowerbound:
defines "\<phi> \<equiv> (1 + sqrt 5) / 2"
-  defines "c \<equiv> 1 / \<phi> ^ 2"
-  assumes "n > 0"
-  shows   "real (fib n) \<ge> c * \<phi> ^ n"
-proof -
-  have "\<phi> > 1" by (simp add: \<phi>_def)
-  hence "c > 0" by (simp add: c_def)
-  from \<open>n > 0\<close> show ?thesis
-  proof (induction n rule: fib_alt_induct)
-    case (rec n)
-    have "c * \<phi> ^ Suc (Suc n) = \<phi> ^ 2 * (c * \<phi> ^ n)"
-      by (simp add: field_simps power2_eq_square)
-    also have "\<dots> \<le> (\<phi> + 1) * (c * \<phi> ^ n)"
-      by (rule mult_right_mono) (insert \<open>c > 0\<close>, simp_all add: \<phi>_def power2_eq_square field_simps)
-    also have "\<dots> = c * \<phi> ^ Suc n + c * \<phi> ^ n"
+  shows "real (fib(n+2)) \<ge> \<phi> ^ n"
+proof (induction n rule: fib.induct)
+  case 1
+  then show ?case by simp
+next
+  case 2
+  then show ?case by (simp add: \<phi>_def real_le_lsqrt)
+next
+  case (3 n) term ?case
+  have "\<phi> ^ Suc (Suc n) = \<phi> ^ 2 * \<phi> ^ n"
+    by (simp add: field_simps power2_eq_square)
+  also have "\<dots> = (\<phi> + 1) * \<phi> ^ n"
+    by (simp_all add: \<phi>_def power2_eq_square field_simps)
+  also have "\<dots> = \<phi> ^ Suc n + \<phi> ^ n"
-    also have "\<dots> \<le> real (fib (Suc n)) + real (fib n)"
-    finally show ?case by simp
-  qed (insert \<open>\<phi> > 1\<close>, simp_all add: c_def power2_eq_square eval_nat_numeral)
+  also have "\<dots> \<le> real (fib (Suc n + 2)) + real (fib (n + 2))"
+  finally show ?case by simp
qed

text \<open>The size of an AVL tree is (at least) exponential in its height:\<close>
@@ -524,11 +514,8 @@
assumes "avl t"
shows   "\<phi> ^ (height t) \<le> size1 t"
proof -