New lemmas connected with the reals and infinite series

--- a/src/HOL/Real.thy	Mon Oct 05 16:41:06 2009 +0100
+++ b/src/HOL/Real.thy	Mon Oct 05 17:27:46 2009 +0100
@@ -2,4 +2,28 @@
 imports RComplete RealVector
 begin
 
+lemma field_le_epsilon:
+  fixes x y :: "'a:: {number_ring,division_by_zero,ordered_field}"
+  assumes e: "(!!e. 0 < e ==> x \<le> y + e)"
+  shows "x \<le> y"
+proof (rule ccontr)
+  assume xy: "\<not> x \<le> y"
+  hence "(x-y)/2 > 0"
+    by (metis half_gt_zero le_iff_diff_le_0 linorder_not_le local.xy)
+  hence "x \<le> y + (x - y) / 2"
+    by (rule e [of "(x-y)/2"])
+  also have "... = (x - y + 2*y)/2"
+    by auto
+       (metis add_less_cancel_left add_numeral_0_right class_semiring.add_c xy e
+           diff_add_cancel gt_half_sum less_half_sum linorder_not_le number_of_Pls)
+  also have "... = (x + y) / 2" 
+    by auto
+  also have "... < x" using xy 
+    by auto
+  finally have "x<x" .
+  thus False
+    by auto 
+qed
+
+
 end

--- a/src/HOL/SEQ.thy	Mon Oct 05 16:41:06 2009 +0100
+++ b/src/HOL/SEQ.thy	Mon Oct 05 17:27:46 2009 +0100
@@ -193,6 +193,9 @@
 
 subsection {* Limits of Sequences *}
 
+lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
+  by simp
+
 lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> (X ---> L) sequentially"
 unfolding LIMSEQ_def tendsto_iff eventually_sequentially ..
 
@@ -315,6 +318,39 @@
   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
 by (rule mult.LIMSEQ)
 
+lemma increasing_LIMSEQ:
+  fixes f :: "nat \<Rightarrow> real"
+  assumes inc: "!!n. f n \<le> f (Suc n)"
+      and bdd: "!!n. f n \<le> l"
+      and en: "!!e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
+  shows "f ----> l"
+proof (auto simp add: LIMSEQ_def)
+  fix e :: real
+  assume e: "0 < e"
+  then obtain N where "l \<le> f N + e/2"
+    by (metis half_gt_zero e en that)
+  hence N: "l < f N + e" using e
+    by simp
+  { fix k
+    have [simp]: "!!n. \<bar>f n - l\<bar> = l - f n"
+      by (simp add: bdd) 
+    have "\<bar>f (N+k) - l\<bar> < e"
+    proof (induct k)
+      case 0 show ?case using N
+	by simp   
+    next
+      case (Suc k) thus ?case using N inc [of "N+k"]
+	by simp
+    qed 
+  } note 1 = this
+  { fix n
+    have "N \<le> n \<Longrightarrow> \<bar>f n - l\<bar> < e" using 1 [of "n-N"]
+      by simp 
+  } note [intro] = this
+  show " \<exists>no. \<forall>n\<ge>no. dist (f n) l < e"
+    by (auto simp add: dist_real_def) 
+  qed
+
 lemma Bseq_inverse_lemma:
   fixes x :: "'a::real_normed_div_algebra"
   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"

--- a/src/HOL/Series.thy	Mon Oct 05 16:41:06 2009 +0100
+++ b/src/HOL/Series.thy	Mon Oct 05 17:27:46 2009 +0100
@@ -32,6 +32,9 @@
   "\<Sum>i. b" == "CONST suminf (%i. b)"
 
 
+lemma [trans]: "f=g ==> g sums z ==> f sums z"
+  by simp
+
 lemma sumr_diff_mult_const:
  "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
 by (simp add: diff_minus setsum_addf real_of_nat_def)