--- a/src/HOL/Limits.thy Fri Jul 22 21:43:56 2016 +0200
+++ b/src/HOL/Limits.thy Fri Jul 22 23:55:47 2016 +0200
@@ -8,13 +8,13 @@
section \<open>Limits on Real Vector Spaces\<close>
theory Limits
-imports Real_Vector_Spaces
+ imports Real_Vector_Spaces
begin
subsection \<open>Filter going to infinity norm\<close>
-definition at_infinity :: "'a::real_normed_vector filter" where
- "at_infinity = (INF r. principal {x. r \<le> norm x})"
+definition at_infinity :: "'a::real_normed_vector filter"
+ where "at_infinity = (INF r. principal {x. r \<le> norm x})"
lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
unfolding at_infinity_def
@@ -22,21 +22,24 @@
(auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
corollary eventually_at_infinity_pos:
- "eventually p at_infinity \<longleftrightarrow> (\<exists>b. 0 < b \<and> (\<forall>x. norm x \<ge> b \<longrightarrow> p x))"
-apply (simp add: eventually_at_infinity, auto)
-apply (case_tac "b \<le> 0")
-using norm_ge_zero order_trans zero_less_one apply blast
-apply (force simp:)
-done
-
-lemma at_infinity_eq_at_top_bot:
- "(at_infinity :: real filter) = sup at_top at_bot"
+ "eventually p at_infinity \<longleftrightarrow> (\<exists>b. 0 < b \<and> (\<forall>x. norm x \<ge> b \<longrightarrow> p x))"
+ apply (simp add: eventually_at_infinity)
+ apply auto
+ apply (case_tac "b \<le> 0")
+ using norm_ge_zero order_trans zero_less_one apply blast
+ apply force
+ done
+
+lemma at_infinity_eq_at_top_bot: "(at_infinity :: real filter) = sup at_top at_bot"
apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
- eventually_at_top_linorder eventually_at_bot_linorder)
+ eventually_at_top_linorder eventually_at_bot_linorder)
apply safe
- apply (rule_tac x="b" in exI, simp)
- apply (rule_tac x="- b" in exI, simp)
- apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
+ apply (rule_tac x="b" in exI)
+ apply simp
+ apply (rule_tac x="- b" in exI)
+ apply simp
+ apply (rule_tac x="max (- Na) N" in exI)
+ apply (auto simp: abs_real_def)
done
lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
@@ -45,23 +48,21 @@
lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
unfolding at_infinity_eq_at_top_bot by simp
-lemma filterlim_at_top_imp_at_infinity:
- fixes f :: "_ \<Rightarrow> real"
- shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
+lemma filterlim_at_top_imp_at_infinity: "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
+ for f :: "_ \<Rightarrow> real"
by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
-lemma lim_infinity_imp_sequentially:
- "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
-by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
+lemma lim_infinity_imp_sequentially: "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
+ by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
subsubsection \<open>Boundedness\<close>
-definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
- Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
-
-abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
- "Bseq X \<equiv> Bfun X sequentially"
+definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool"
+ where Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
+
+abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool"
+ where "Bseq X \<equiv> Bfun X sequentially"
lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
@@ -71,11 +72,11 @@
lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
-lemma Bfun_def:
- "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
+lemma Bfun_def: "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
unfolding Bfun_metric_def norm_conv_dist
proof safe
- fix y K assume K: "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
+ fix y K
+ assume K: "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
by (intro always_eventually) (metis dist_commute dist_triangle)
with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
@@ -85,19 +86,19 @@
qed (force simp del: norm_conv_dist [symmetric])
lemma BfunI:
- assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
-unfolding Bfun_def
+ assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F"
+ shows "Bfun f F"
+ unfolding Bfun_def
proof (intro exI conjI allI)
show "0 < max K 1" by simp
-next
show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
- using K by (rule eventually_mono, simp)
+ using K by (rule eventually_mono) simp
qed
lemma BfunE:
assumes "Bfun f F"
obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
-using assms unfolding Bfun_def by blast
+ using assms unfolding Bfun_def by blast
lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
unfolding Cauchy_def Bfun_metric_def eventually_sequentially
@@ -124,57 +125,66 @@
lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
unfolding Bfun_def eventually_sequentially
proof safe
- fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
+ fix N K
+ assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
(auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
qed auto
-lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
-unfolding Bseq_def by auto
-
-lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
-by (simp add: Bseq_def)
-
-lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
-by (auto simp add: Bseq_def)
-
-lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
+lemma BseqE: "Bseq X \<Longrightarrow> (\<And>K. 0 < K \<Longrightarrow> \<forall>n. norm (X n) \<le> K \<Longrightarrow> Q) \<Longrightarrow> Q"
+ unfolding Bseq_def by auto
+
+lemma BseqD: "Bseq X \<Longrightarrow> \<exists>K. 0 < K \<and> (\<forall>n. norm (X n) \<le> K)"
+ by (simp add: Bseq_def)
+
+lemma BseqI: "0 < K \<Longrightarrow> \<forall>n. norm (X n) \<le> K \<Longrightarrow> Bseq X"
+ by (auto simp add: Bseq_def)
+
+lemma Bseq_bdd_above: "Bseq X \<Longrightarrow> bdd_above (range X)"
+ for X :: "nat \<Rightarrow> real"
proof (elim BseqE, intro bdd_aboveI2)
- fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
+ fix K n
+ assume "0 < K" "\<forall>n. norm (X n) \<le> K"
+ then show "X n \<le> K"
by (auto elim!: allE[of _ n])
qed
-lemma Bseq_bdd_above':
- "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
+lemma Bseq_bdd_above': "Bseq X \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
+ for X :: "nat \<Rightarrow> 'a :: real_normed_vector"
proof (elim BseqE, intro bdd_aboveI2)
- fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"
+ fix K n
+ assume "0 < K" "\<forall>n. norm (X n) \<le> K"
+ then show "norm (X n) \<le> K"
by (auto elim!: allE[of _ n])
qed
-lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
+lemma Bseq_bdd_below: "Bseq X \<Longrightarrow> bdd_below (range X)"
+ for X :: "nat \<Rightarrow> real"
proof (elim BseqE, intro bdd_belowI2)
- fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
+ fix K n
+ assume "0 < K" "\<forall>n. norm (X n) \<le> K"
+ then show "- K \<le> X n"
by (auto elim!: allE[of _ n])
qed
lemma Bseq_eventually_mono:
assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
- shows "Bseq f"
+ shows "Bseq f"
proof -
from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"
by (auto simp: eventually_at_top_linorder)
- moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)
+ moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K"
+ by (blast elim!: BseqE)
ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n
apply (cases "n < N")
- apply (rule max.coboundedI2, rule Max.coboundedI, auto) []
- apply (rule max.coboundedI1, force intro: order.trans[OF N K])
+ subgoal by (rule max.coboundedI2, rule Max.coboundedI) auto
+ subgoal by (rule max.coboundedI1) (force intro: order.trans[OF N K])
done
- thus ?thesis by (blast intro: BseqI')
+ then show ?thesis by (blast intro: BseqI')
qed
-lemma lemma_NBseq_def:
- "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
+lemma lemma_NBseq_def: "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
proof safe
fix K :: real
from reals_Archimedean2 obtain n :: nat where "K < real n" ..
@@ -188,47 +198,50 @@
using of_nat_0_less_iff by blast
qed
-text\<open>alternative definition for Bseq\<close>
-lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
-apply (simp add: Bseq_def)
-apply (simp (no_asm) add: lemma_NBseq_def)
-done
-
-lemma lemma_NBseq_def2:
- "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
-apply (subst lemma_NBseq_def, auto)
-apply (rule_tac x = "Suc N" in exI)
-apply (rule_tac [2] x = N in exI)
-apply (auto simp add: of_nat_Suc)
- prefer 2 apply (blast intro: order_less_imp_le)
-apply (drule_tac x = n in spec, simp)
-done
-
-(* yet another definition for Bseq *)
-lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
-by (simp add: Bseq_def lemma_NBseq_def2)
-
-subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
-
-text\<open>alternative formulation for boundedness\<close>
-lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
-apply (unfold Bseq_def, safe)
-apply (rule_tac [2] x = "k + norm x" in exI)
-apply (rule_tac x = K in exI, simp)
-apply (rule exI [where x = 0], auto)
-apply (erule order_less_le_trans, simp)
-apply (drule_tac x=n in spec)
-apply (drule order_trans [OF norm_triangle_ineq2])
-apply simp
-done
-
-text\<open>alternative formulation for boundedness\<close>
-lemma Bseq_iff3:
- "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
+text \<open>Alternative definition for \<open>Bseq\<close>.\<close>
+lemma Bseq_iff: "Bseq X \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
+ by (simp add: Bseq_def) (simp add: lemma_NBseq_def)
+
+lemma lemma_NBseq_def2: "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
+ apply (subst lemma_NBseq_def)
+ apply auto
+ apply (rule_tac x = "Suc N" in exI)
+ apply (rule_tac [2] x = N in exI)
+ apply auto
+ prefer 2 apply (blast intro: order_less_imp_le)
+ apply (drule_tac x = n in spec)
+ apply simp
+ done
+
+text \<open>Yet another definition for Bseq.\<close>
+lemma Bseq_iff1a: "Bseq X \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) < real (Suc N))"
+ by (simp add: Bseq_def lemma_NBseq_def2)
+
+subsubsection \<open>A Few More Equivalence Theorems for Boundedness\<close>
+
+text \<open>Alternative formulation for boundedness.\<close>
+lemma Bseq_iff2: "Bseq X \<longleftrightarrow> (\<exists>k > 0. \<exists>x. \<forall>n. norm (X n + - x) \<le> k)"
+ apply (unfold Bseq_def)
+ apply safe
+ apply (rule_tac [2] x = "k + norm x" in exI)
+ apply (rule_tac x = K in exI)
+ apply simp
+ apply (rule exI [where x = 0])
+ apply auto
+ apply (erule order_less_le_trans)
+ apply simp
+ apply (drule_tac x=n in spec)
+ apply (drule order_trans [OF norm_triangle_ineq2])
+ apply simp
+ done
+
+text \<open>Alternative formulation for boundedness.\<close>
+lemma Bseq_iff3: "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)"
+ (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P
- then obtain K
- where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
+ then obtain K where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K"
+ by (auto simp add: Bseq_def)
from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
by (auto intro: order_trans norm_triangle_ineq4)
@@ -236,129 +249,150 @@
by simp
with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
next
- assume ?Q then show ?P by (auto simp add: Bseq_iff2)
+ assume ?Q
+ then show ?P by (auto simp add: Bseq_iff2)
qed
-lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
-apply (simp add: Bseq_def)
-apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
-apply (drule_tac x = n in spec, arith)
-done
-
-
-subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
-
-lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
+lemma BseqI2: "\<forall>n. k \<le> f n \<and> f n \<le> K \<Longrightarrow> Bseq f"
+ for k K :: real
+ apply (simp add: Bseq_def)
+ apply (rule_tac x = "(\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI)
+ apply auto
+ apply (drule_tac x = n in spec)
+ apply arith
+ done
+
+
+subsubsection \<open>Upper Bounds and Lubs of Bounded Sequences\<close>
+
+lemma Bseq_minus_iff: "Bseq (\<lambda>n. - (X n) :: 'a::real_normed_vector) \<longleftrightarrow> Bseq X"
by (simp add: Bseq_def)
lemma Bseq_add:
- assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
- shows "Bseq (\<lambda>x. f x + c)"
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ assumes "Bseq f"
+ shows "Bseq (\<lambda>x. f x + c)"
proof -
- from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast
+ from assms obtain K where K: "\<And>x. norm (f x) \<le> K"
+ unfolding Bseq_def by blast
{
fix x :: nat
have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
also have "norm (f x) \<le> K" by (rule K)
finally have "norm (f x + c) \<le> K + norm c" by simp
}
- thus ?thesis by (rule BseqI')
+ then show ?thesis by (rule BseqI')
qed
-lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
+lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq f"
+ for f :: "nat \<Rightarrow> 'a::real_normed_vector"
using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
lemma Bseq_mult:
- assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"
- assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"
- shows "Bseq (\<lambda>x. f x * g x)"
+ fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
+ assumes "Bseq f" and "Bseq g"
+ shows "Bseq (\<lambda>x. f x * g x)"
proof -
- from assms obtain K1 K2 where K: "\<And>x. norm (f x) \<le> K1" "K1 > 0" "\<And>x. norm (g x) \<le> K2" "K2 > 0"
+ from assms obtain K1 K2 where K: "norm (f x) \<le> K1" "K1 > 0" "norm (g x) \<le> K2" "K2 > 0"
+ for x
unfolding Bseq_def by blast
- hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)
- thus ?thesis by (rule BseqI')
+ then have "norm (f x * g x) \<le> K1 * K2" for x
+ by (auto simp: norm_mult intro!: mult_mono)
+ then show ?thesis by (rule BseqI')
qed
lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
-lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
+lemma Bseq_cmult_iff:
+ fixes c :: "'a::real_normed_field"
+ assumes "c \<noteq> 0"
+ shows "Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
proof
- assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
- find_theorems "Bfun (\<lambda>_. ?c) _"
- from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
- with \<open>c \<noteq> 0\<close> show "Bseq f" by (simp add: divide_simps)
+ assume "Bseq (\<lambda>x. c * f x)"
+ with Bfun_const have "Bseq (\<lambda>x. inverse c * (c * f x))"
+ by (rule Bseq_mult)
+ with \<open>c \<noteq> 0\<close> show "Bseq f"
+ by (simp add: divide_simps)
qed (intro Bseq_mult Bfun_const)
-lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
+lemma Bseq_subseq: "Bseq f \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
+ for f :: "nat \<Rightarrow> 'a::real_normed_vector"
unfolding Bseq_def by auto
-lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
+lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq f"
+ for f :: "nat \<Rightarrow> 'a::real_normed_vector"
using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
lemma increasing_Bseq_subseq_iff:
- assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
- shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
+ assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a::real_normed_vector) \<le> norm (f y)" "subseq g"
+ shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
proof
assume "Bseq (\<lambda>x. f (g x))"
- then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
+ then obtain K where K: "\<And>x. norm (f (g x)) \<le> K"
+ unfolding Bseq_def by auto
{
fix x :: nat
from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
by (auto simp: filterlim_at_top eventually_at_top_linorder)
- hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
+ then have "norm (f x) \<le> norm (f (g y))"
+ using assms(1) by blast
also have "norm (f (g y)) \<le> K" by (rule K)
finally have "norm (f x) \<le> K" .
}
- thus "Bseq f" by (rule BseqI')
-qed (insert Bseq_subseq[of f g], simp_all)
+ then show "Bseq f" by (rule BseqI')
+qed (use Bseq_subseq[of f g] in simp_all)
lemma nonneg_incseq_Bseq_subseq_iff:
- assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
- shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
+ fixes f :: "nat \<Rightarrow> real"
+ and g :: "nat \<Rightarrow> nat"
+ assumes "\<And>x. f x \<ge> 0" "incseq f" "subseq g"
+ shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
-lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
+lemma Bseq_eq_bounded: "range f \<subseteq> {a..b} \<Longrightarrow> Bseq f"
+ for a b :: real
apply (simp add: subset_eq)
apply (rule BseqI'[where K="max (norm a) (norm b)"])
apply (erule_tac x=n in allE)
apply auto
done
-lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
+lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> B \<Longrightarrow> Bseq X"
+ for B :: real
by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
-lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
+lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. B \<le> X i \<Longrightarrow> Bseq X"
+ for B :: real
by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
+
subsection \<open>Bounded Monotonic Sequences\<close>
-subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
+subsubsection \<open>A Bounded and Monotonic Sequence Converges\<close>
(* TODO: delete *)
(* FIXME: one use in NSA/HSEQ.thy *)
-lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
+lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n \<longrightarrow> X n = X m \<Longrightarrow> \<exists>L. X \<longlonglongrightarrow> L"
apply (rule_tac x="X m" in exI)
apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
unfolding eventually_sequentially
apply blast
done
+
subsection \<open>Convergence to Zero\<close>
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
-lemma ZfunI:
- "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
- unfolding Zfun_def by simp
-
-lemma ZfunD:
- "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
- unfolding Zfun_def by simp
-
-lemma Zfun_ssubst:
- "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
+lemma ZfunI: "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
+ by (simp add: Zfun_def)
+
+lemma ZfunD: "Zfun f F \<Longrightarrow> 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
+ by (simp add: Zfun_def)
+
+lemma Zfun_ssubst: "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
unfolding Zfun_def by (auto elim!: eventually_rev_mp)
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
@@ -369,28 +403,29 @@
lemma Zfun_imp_Zfun:
assumes f: "Zfun f F"
- assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
+ and g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
shows "Zfun (\<lambda>x. g x) F"
-proof (cases)
- assume K: "0 < K"
+proof (cases "0 < K")
+ case K: True
show ?thesis
proof (rule ZfunI)
- fix r::real assume "0 < r"
- hence "0 < r / K" using K by simp
+ fix r :: real
+ assume "0 < r"
+ then have "0 < r / K" using K by simp
then have "eventually (\<lambda>x. norm (f x) < r / K) F"
using ZfunD [OF f] by blast
with g show "eventually (\<lambda>x. norm (g x) < r) F"
proof eventually_elim
case (elim x)
- hence "norm (f x) * K < r"
+ then have "norm (f x) * K < r"
by (simp add: pos_less_divide_eq K)
- thus ?case
+ then show ?case
by (simp add: order_le_less_trans [OF elim(1)])
qed
qed
next
- assume "\<not> 0 < K"
- hence K: "K \<le> 0" by (simp only: not_less)
+ case False
+ then have K: "K \<le> 0" by (simp only: not_less)
show ?thesis
proof (rule ZfunI)
fix r :: real
@@ -406,15 +441,17 @@
qed
qed
-lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
- by (erule_tac K="1" in Zfun_imp_Zfun, simp)
+lemma Zfun_le: "Zfun g F \<Longrightarrow> \<forall>x. norm (f x) \<le> norm (g x) \<Longrightarrow> Zfun f F"
+ by (erule Zfun_imp_Zfun [where K = 1]) simp
lemma Zfun_add:
- assumes f: "Zfun f F" and g: "Zfun g F"
+ assumes f: "Zfun f F"
+ and g: "Zfun g F"
shows "Zfun (\<lambda>x. f x + g x) F"
proof (rule ZfunI)
- fix r::real assume "0 < r"
- hence r: "0 < r / 2" by simp
+ fix r :: real
+ assume "0 < r"
+ then have r: "0 < r / 2" by simp
have "eventually (\<lambda>x. norm (f x) < r/2) F"
using f r by (rule ZfunD)
moreover
@@ -436,14 +473,14 @@
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
unfolding Zfun_def by simp
-lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
+lemma Zfun_diff: "Zfun f F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
lemma (in bounded_linear) Zfun:
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f (g x)) F"
proof -
- obtain K where "\<And>x. norm (f x) \<le> norm x * K"
+ obtain K where "norm (f x) \<le> norm x * K" for x
using bounded by blast
then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
by simp
@@ -453,12 +490,13 @@
lemma (in bounded_bilinear) Zfun:
assumes f: "Zfun f F"
- assumes g: "Zfun g F"
+ and g: "Zfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
proof (rule ZfunI)
- fix r::real assume r: "0 < r"
+ fix r :: real
+ assume r: "0 < r"
obtain K where K: "0 < K"
- and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
+ and norm_le: "norm (x ** y) \<le> norm x * norm y * K" for x y
using pos_bounded by blast
from K have K': "0 < inverse K"
by (rule positive_imp_inverse_positive)
@@ -481,12 +519,10 @@
qed
qed
-lemma (in bounded_bilinear) Zfun_left:
- "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
+lemma (in bounded_bilinear) Zfun_left: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
by (rule bounded_linear_left [THEN bounded_linear.Zfun])
-lemma (in bounded_bilinear) Zfun_right:
- "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
+lemma (in bounded_bilinear) Zfun_right: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
by (rule bounded_linear_right [THEN bounded_linear.Zfun])
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
@@ -496,19 +532,22 @@
lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
by (simp only: tendsto_iff Zfun_def dist_norm)
-lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
- \<Longrightarrow> (g \<longlongrightarrow> 0) F"
+lemma tendsto_0_le:
+ "(f \<longlongrightarrow> 0) F \<Longrightarrow> eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F \<Longrightarrow> (g \<longlongrightarrow> 0) F"
by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
+
subsubsection \<open>Distance and norms\<close>
lemma tendsto_dist [tendsto_intros]:
- fixes l m :: "'a :: metric_space"
- assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
+ fixes l m :: "'a::metric_space"
+ assumes f: "(f \<longlongrightarrow> l) F"
+ and g: "(g \<longlongrightarrow> m) F"
shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
proof (rule tendstoI)
- fix e :: real assume "0 < e"
- hence e2: "0 < e/2" by simp
+ fix e :: real
+ assume "0 < e"
+ then have e2: "0 < e/2" by simp
from tendstoD [OF f e2] tendstoD [OF g e2]
show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
proof (eventually_elim)
@@ -516,9 +555,9 @@
then show "dist (dist (f x) (g x)) (dist l m) < e"
unfolding dist_real_def
using dist_triangle2 [of "f x" "g x" "l"]
- using dist_triangle2 [of "g x" "l" "m"]
- using dist_triangle3 [of "l" "m" "f x"]
- using dist_triangle [of "f x" "m" "g x"]
+ and dist_triangle2 [of "g x" "l" "m"]
+ and dist_triangle3 [of "l" "m" "f x"]
+ and dist_triangle [of "f x" "m" "g x"]
by arith
qed
qed
@@ -533,33 +572,28 @@
shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
unfolding continuous_on_def by (auto intro: tendsto_dist)
-lemma tendsto_norm [tendsto_intros]:
- "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
+lemma tendsto_norm [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
unfolding norm_conv_dist by (intro tendsto_intros)
-lemma continuous_norm [continuous_intros]:
- "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
+lemma continuous_norm [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
unfolding continuous_def by (rule tendsto_norm)
lemma continuous_on_norm [continuous_intros]:
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
unfolding continuous_on_def by (auto intro: tendsto_norm)
-lemma tendsto_norm_zero:
- "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
- by (drule tendsto_norm, simp)
-
-lemma tendsto_norm_zero_cancel:
- "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
+lemma tendsto_norm_zero: "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
+ by (drule tendsto_norm) simp
+
+lemma tendsto_norm_zero_cancel: "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
unfolding tendsto_iff dist_norm by simp
-lemma tendsto_norm_zero_iff:
- "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
+lemma tendsto_norm_zero_iff: "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
unfolding tendsto_iff dist_norm by simp
-lemma tendsto_rabs [tendsto_intros]:
- "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
- by (fold real_norm_def, rule tendsto_norm)
+lemma tendsto_rabs [tendsto_intros]: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
+ for l :: real
+ by (fold real_norm_def) (rule tendsto_norm)
lemma continuous_rabs [continuous_intros]:
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
@@ -569,17 +603,15 @@
"continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
-lemma tendsto_rabs_zero:
- "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
- by (fold real_norm_def, rule tendsto_norm_zero)
-
-lemma tendsto_rabs_zero_cancel:
- "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
- by (fold real_norm_def, rule tendsto_norm_zero_cancel)
-
-lemma tendsto_rabs_zero_iff:
- "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
- by (fold real_norm_def, rule tendsto_norm_zero_iff)
+lemma tendsto_rabs_zero: "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
+ by (fold real_norm_def) (rule tendsto_norm_zero)
+
+lemma tendsto_rabs_zero_cancel: "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
+ by (fold real_norm_def) (rule tendsto_norm_zero_cancel)
+
+lemma tendsto_rabs_zero_iff: "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
+ by (fold real_norm_def) (rule tendsto_norm_zero_iff)
+
subsection \<open>Topological Monoid\<close>
@@ -606,17 +638,22 @@
lemma tendsto_add_zero:
fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
- shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
- by (drule (1) tendsto_add, simp)
+ shows "(f \<longlongrightarrow> 0) F \<Longrightarrow> (g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
+ by (drule (1) tendsto_add) simp
lemma tendsto_setsum [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
shows "((\<lambda>x. \<Sum>i\<in>I. f i x) \<longlongrightarrow> (\<Sum>i\<in>I. a i)) F"
proof (cases "finite I")
- assume "finite I" thus ?thesis using assms
- by (induct, simp, simp add: tendsto_add)
-qed simp
+ case True
+ then show ?thesis
+ using assms by induct (simp_all add: tendsto_add)
+next
+ case False
+ then show ?thesis
+ by simp
+qed
lemma continuous_setsum [continuous_intros]:
fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
@@ -629,10 +666,13 @@
unfolding continuous_on_def by (auto intro: tendsto_setsum)
instance nat :: topological_comm_monoid_add
- proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
+ by standard
+ (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
instance int :: topological_comm_monoid_add
- proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
+ by standard
+ (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
+
subsubsection \<open>Topological group\<close>
@@ -640,7 +680,7 @@
assumes tendsto_uminus_nhds: "(uminus \<longlongrightarrow> - a) (nhds a)"
begin
-lemma tendsto_minus [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> -a) F"
+lemma tendsto_minus [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
by (rule filterlim_compose[OF tendsto_uminus_nhds])
end
@@ -649,29 +689,26 @@
instance topological_ab_group_add < topological_comm_monoid_add ..
-lemma continuous_minus [continuous_intros]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
- shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
+lemma continuous_minus [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
+ for f :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
unfolding continuous_def by (rule tendsto_minus)
-lemma continuous_on_minus [continuous_intros]:
- fixes f :: "_ \<Rightarrow> 'b::topological_group_add"
- shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
+lemma continuous_on_minus [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
+ for f :: "_ \<Rightarrow> 'b::topological_group_add"
unfolding continuous_on_def by (auto intro: tendsto_minus)
-lemma tendsto_minus_cancel:
- fixes a :: "'a::topological_group_add"
- shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
- by (drule tendsto_minus, simp)
+lemma tendsto_minus_cancel: "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
+ for a :: "'a::topological_group_add"
+ by (drule tendsto_minus) simp
lemma tendsto_minus_cancel_left:
- "(f \<longlongrightarrow> - (y::_::topological_group_add)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
+ "(f \<longlongrightarrow> - (y::_::topological_group_add)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
using tendsto_minus_cancel[of f "- y" F] tendsto_minus[of f "- y" F]
by auto
lemma tendsto_diff [tendsto_intros]:
fixes a b :: "'a::topological_group_add"
- shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
+ shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
lemma continuous_diff [continuous_intros]:
@@ -689,7 +726,8 @@
instance real_normed_vector < topological_ab_group_add
proof
- fix a b :: 'a show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
+ fix a b :: 'a
+ show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
unfolding tendsto_Zfun_iff add_diff_add
using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
by (intro Zfun_add)
@@ -702,32 +740,28 @@
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
+
subsubsection \<open>Linear operators and multiplication\<close>
-lemma linear_times:
- fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
+lemma linear_times: "linear (\<lambda>x. c * x)"
+ for c :: "'a::real_algebra"
by (auto simp: linearI distrib_left)
-lemma (in bounded_linear) tendsto:
- "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
+lemma (in bounded_linear) tendsto: "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
-lemma (in bounded_linear) continuous:
- "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
+lemma (in bounded_linear) continuous: "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
using tendsto[of g _ F] by (auto simp: continuous_def)
-lemma (in bounded_linear) continuous_on:
- "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
+lemma (in bounded_linear) continuous_on: "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
using tendsto[of g] by (auto simp: continuous_on_def)
-lemma (in bounded_linear) tendsto_zero:
- "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
- by (drule tendsto, simp only: zero)
+lemma (in bounded_linear) tendsto_zero: "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
+ by (drule tendsto) (simp only: zero)
lemma (in bounded_bilinear) tendsto:
- "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
- by (simp only: tendsto_Zfun_iff prod_diff_prod
- Zfun_add Zfun Zfun_left Zfun_right)
+ "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
+ by (simp only: tendsto_Zfun_iff prod_diff_prod Zfun_add Zfun Zfun_left Zfun_right)
lemma (in bounded_bilinear) continuous:
"continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
@@ -739,7 +773,7 @@
lemma (in bounded_bilinear) tendsto_zero:
assumes f: "(f \<longlongrightarrow> 0) F"
- assumes g: "(g \<longlongrightarrow> 0) F"
+ and g: "(g \<longlongrightarrow> 0) F"
shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
using tendsto [OF f g] by (simp add: zero_left)
@@ -760,15 +794,13 @@
lemmas tendsto_mult [tendsto_intros] =
bounded_bilinear.tendsto [OF bounded_bilinear_mult]
-lemma tendsto_mult_left:
- fixes c::"'a::real_normed_algebra"
- shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
-by (rule tendsto_mult [OF tendsto_const])
-
-lemma tendsto_mult_right:
- fixes c::"'a::real_normed_algebra"
- shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
-by (rule tendsto_mult [OF _ tendsto_const])
+lemma tendsto_mult_left: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
+ for c :: "'a::real_normed_algebra"
+ by (rule tendsto_mult [OF tendsto_const])
+
+lemma tendsto_mult_right: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
+ for c :: "'a::real_normed_algebra"
+ by (rule tendsto_mult [OF _ tendsto_const])
lemmas continuous_of_real [continuous_intros] =
bounded_linear.continuous [OF bounded_linear_of_real]
@@ -797,14 +829,12 @@
lemmas tendsto_mult_right_zero =
bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
-lemma tendsto_power [tendsto_intros]:
- fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
- shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
+lemma tendsto_power [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
+ for f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
by (induct n) (simp_all add: tendsto_mult)
-lemma continuous_power [continuous_intros]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
- shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
+lemma continuous_power [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
+ for f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
unfolding continuous_def by (rule tendsto_power)
lemma continuous_on_power [continuous_intros]:
@@ -817,9 +847,13 @@
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
proof (cases "finite S")
- assume "finite S" thus ?thesis using assms
- by (induct, simp, simp add: tendsto_mult)
-qed simp
+ case True
+ then show ?thesis using assms
+ by induct (simp_all add: tendsto_mult)
+next
+ case False
+ then show ?thesis by simp
+qed
lemma continuous_setprod [continuous_intros]:
fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
@@ -832,20 +866,20 @@
unfolding continuous_on_def by (auto intro: tendsto_setprod)
lemma tendsto_of_real_iff:
- "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
+ "((\<lambda>x. of_real (f x) :: 'a::real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
unfolding tendsto_iff by simp
lemma tendsto_add_const_iff:
- "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
+ "((\<lambda>x. c + f x :: 'a::real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
using tendsto_add[OF tendsto_const[of c], of f d]
- tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
+ and tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
subsubsection \<open>Inverse and division\<close>
lemma (in bounded_bilinear) Zfun_prod_Bfun:
assumes f: "Zfun f F"
- assumes g: "Bfun g F"
+ and g: "Bfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
proof -
obtain K where K: "0 \<le> K"
@@ -860,8 +894,7 @@
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
by (rule norm_le)
also have "\<dots> \<le> norm (f x) * B * K"
- by (intro mult_mono' order_refl norm_g norm_ge_zero
- mult_nonneg_nonneg K elim)
+ by (intro mult_mono' order_refl norm_g norm_ge_zero mult_nonneg_nonneg K elim)
also have "\<dots> = norm (f x) * (B * K)"
by (rule mult.assoc)
finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
@@ -872,14 +905,15 @@
lemma (in bounded_bilinear) Bfun_prod_Zfun:
assumes f: "Bfun f F"
- assumes g: "Zfun g F"
+ and g: "Zfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
lemma Bfun_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"
- apply (subst nonzero_norm_inverse, clarsimp)
+ shows "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
+ apply (subst nonzero_norm_inverse)
+ apply clarsimp
apply (erule (1) le_imp_inverse_le)
done
@@ -890,38 +924,40 @@
shows "Bfun (\<lambda>x. inverse (f x)) F"
proof -
from a have "0 < norm a" by simp
- hence "\<exists>r>0. r < norm a" by (rule dense)
- then obtain r where r1: "0 < r" and r2: "r < norm a" by blast
+ then have "\<exists>r>0. r < norm a" by (rule dense)
+ then obtain r where r1: "0 < r" and r2: "r < norm a"
+ by blast
have "eventually (\<lambda>x. dist (f x) a < r) F"
using tendstoD [OF f r1] by blast
- hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
+ then have "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
proof eventually_elim
case (elim x)
- hence 1: "norm (f x - a) < r"
+ then have 1: "norm (f x - a) < r"
by (simp add: dist_norm)
- hence 2: "f x \<noteq> 0" using r2 by auto
- hence "norm (inverse (f x)) = inverse (norm (f x))"
+ then have 2: "f x \<noteq> 0" using r2 by auto
+ then have "norm (inverse (f x)) = inverse (norm (f x))"
by (rule nonzero_norm_inverse)
also have "\<dots> \<le> inverse (norm a - r)"
proof (rule le_imp_inverse_le)
- show "0 < norm a - r" using r2 by simp
- next
+ show "0 < norm a - r"
+ using r2 by simp
have "norm a - norm (f x) \<le> norm (a - f x)"
by (rule norm_triangle_ineq2)
also have "\<dots> = norm (f x - a)"
by (rule norm_minus_commute)
also have "\<dots> < r" using 1 .
- finally show "norm a - r \<le> norm (f x)" by simp
+ finally show "norm a - r \<le> norm (f x)"
+ by simp
qed
finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
qed
- thus ?thesis by (rule BfunI)
+ then show ?thesis by (rule BfunI)
qed
lemma tendsto_inverse [tendsto_intros]:
fixes a :: "'a::real_normed_div_algebra"
assumes f: "(f \<longlongrightarrow> a) F"
- assumes a: "a \<noteq> 0"
+ and a: "a \<noteq> 0"
shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
proof -
from a have "0 < norm a" by simp
@@ -942,43 +978,49 @@
lemma continuous_inverse:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
- assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
+ assumes "continuous F f"
+ and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
shows "continuous F (\<lambda>x. inverse (f x))"
using assms unfolding continuous_def by (rule tendsto_inverse)
lemma continuous_at_within_inverse[continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
- assumes "continuous (at a within s) f" and "f a \<noteq> 0"
+ assumes "continuous (at a within s) f"
+ and "f a \<noteq> 0"
shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
using assms unfolding continuous_within by (rule tendsto_inverse)
lemma isCont_inverse[continuous_intros, simp]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
- assumes "isCont f a" and "f a \<noteq> 0"
+ assumes "isCont f a"
+ and "f a \<noteq> 0"
shows "isCont (\<lambda>x. inverse (f x)) a"
using assms unfolding continuous_at by (rule tendsto_inverse)
lemma continuous_on_inverse[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
- assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
+ assumes "continuous_on s f"
+ and "\<forall>x\<in>s. f x \<noteq> 0"
shows "continuous_on s (\<lambda>x. inverse (f x))"
using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
lemma tendsto_divide [tendsto_intros]:
fixes a b :: "'a::real_normed_field"
- shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
- \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
+ shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
by (simp add: tendsto_mult tendsto_inverse divide_inverse)
lemma continuous_divide:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
- assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
+ assumes "continuous F f"
+ and "continuous F g"
+ and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
shows "continuous F (\<lambda>x. (f x) / (g x))"
using assms unfolding continuous_def by (rule tendsto_divide)
lemma continuous_at_within_divide[continuous_intros]:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
- assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
+ assumes "continuous (at a within s) f" "continuous (at a within s) g"
+ and "g a \<noteq> 0"
shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
using assms unfolding continuous_within by (rule tendsto_divide)
@@ -990,36 +1032,40 @@
lemma continuous_on_divide[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
- assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
+ assumes "continuous_on s f" "continuous_on s g"
+ and "\<forall>x\<in>s. g x \<noteq> 0"
shows "continuous_on s (\<lambda>x. (f x) / (g x))"
using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
-lemma tendsto_sgn [tendsto_intros]:
- fixes l :: "'a::real_normed_vector"
- shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
+lemma tendsto_sgn [tendsto_intros]: "(f \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
+ for l :: "'a::real_normed_vector"
unfolding sgn_div_norm by (simp add: tendsto_intros)
lemma continuous_sgn:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
- assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
+ assumes "continuous F f"
+ and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
shows "continuous F (\<lambda>x. sgn (f x))"
using assms unfolding continuous_def by (rule tendsto_sgn)
lemma continuous_at_within_sgn[continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
- assumes "continuous (at a within s) f" and "f a \<noteq> 0"
+ assumes "continuous (at a within s) f"
+ and "f a \<noteq> 0"
shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
using assms unfolding continuous_within by (rule tendsto_sgn)
lemma isCont_sgn[continuous_intros]:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
- assumes "isCont f a" and "f a \<noteq> 0"
+ assumes "isCont f a"
+ and "f a \<noteq> 0"
shows "isCont (\<lambda>x. sgn (f x)) a"
using assms unfolding continuous_at by (rule tendsto_sgn)
lemma continuous_on_sgn[continuous_intros]:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
+ assumes "continuous_on s f"
+ and "\<forall>x\<in>s. f x \<noteq> 0"
shows "continuous_on s (\<lambda>x. sgn (f x))"
using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
@@ -1029,35 +1075,40 @@
shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
unfolding filterlim_iff eventually_at_infinity
proof safe
- fix P :: "'a \<Rightarrow> bool" and b
+ fix P :: "'a \<Rightarrow> bool"
+ fix b
assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
- and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
+ assume P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
have "max b (c + 1) > c" by auto
with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
by auto
then show "eventually (\<lambda>x. P (f x)) F"
proof eventually_elim
- fix x assume "max b (c + 1) \<le> norm (f x)"
+ case (elim x)
with P show "P (f x)" by auto
qed
qed force
lemma not_tendsto_and_filterlim_at_infinity:
+ fixes c :: "'a::real_normed_vector"
assumes "F \<noteq> bot"
- assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
- assumes "filterlim f at_infinity F"
- shows False
+ and "(f \<longlongrightarrow> c) F"
+ and "filterlim f at_infinity F"
+ shows False
proof -
from tendstoD[OF assms(2), of "1/2"]
- have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
- moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
- have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
+ have "eventually (\<lambda>x. dist (f x) c < 1/2) F"
+ by simp
+ moreover
+ from filterlim_at_infinity[of "norm c" f F] assms(3)
+ have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
ultimately have "eventually (\<lambda>x. False) F"
proof eventually_elim
- fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
- note B
+ fix x
+ assume A: "dist (f x) c < 1/2"
+ assume "norm (f x) \<ge> norm c + 1"
also have "norm (f x) = dist (f x) 0" by simp
- also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
+ also have "\<dots> \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
finally show False using A by simp
qed
with assms show False by simp
@@ -1065,83 +1116,97 @@
lemma filterlim_at_infinity_imp_not_convergent:
assumes "filterlim f at_infinity sequentially"
- shows "\<not>convergent f"
+ shows "\<not> convergent f"
by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
(simp_all add: convergent_LIMSEQ_iff)
lemma filterlim_at_infinity_imp_eventually_ne:
assumes "filterlim f at_infinity F"
- shows "eventually (\<lambda>z. f z \<noteq> c) F"
+ shows "eventually (\<lambda>z. f z \<noteq> c) F"
proof -
- have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
+ have "norm c + 1 > 0"
+ by (intro add_nonneg_pos) simp_all
with filterlim_at_infinity[OF order.refl, of f F] assms
- have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
- thus ?thesis by eventually_elim auto
+ have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F"
+ by blast
+ then show ?thesis
+ by eventually_elim auto
qed
lemma tendsto_of_nat [tendsto_intros]:
- "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
+ "filterlim (of_nat :: nat \<Rightarrow> 'a::real_normed_algebra_1) at_infinity sequentially"
proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
fix r :: real
assume r: "r > 0"
define n where "n = nat \<lceil>r\<rceil>"
- from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
+ from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r"
+ unfolding n_def by linarith
from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
- by eventually_elim (insert n, simp_all)
+ by eventually_elim (use n in simp_all)
qed
subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
text \<open>
-
-This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
-@{term "at_right x"} and also @{term "at_right 0"}.
-
+ This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
+ @{term "at_right x"} and also @{term "at_right 0"}.
\<close>
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
-lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
+lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d)"
+ for a d :: "'a::real_normed_vector"
by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
- (auto intro!: tendsto_eq_intros filterlim_ident)
-
-lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
+ (auto intro!: tendsto_eq_intros filterlim_ident)
+
+lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a)"
+ for a :: "'a::real_normed_vector"
by (rule filtermap_fun_inverse[where g=uminus])
- (auto intro!: tendsto_eq_intros filterlim_ident)
-
-lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
+ (auto intro!: tendsto_eq_intros filterlim_ident)
+
+lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d)"
+ for a d :: "'a::real_normed_vector"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
-lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
+lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d)"
+ for a d :: "real"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
-lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
+lemma at_right_to_0: "at_right a = filtermap (\<lambda>x. x + a) (at_right 0)"
+ for a :: real
using filtermap_at_right_shift[of "-a" 0] by simp
lemma filterlim_at_right_to_0:
- "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
+ "filterlim f F (at_right a) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
+ for a :: real
unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
lemma eventually_at_right_to_0:
- "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
+ "eventually P (at_right a) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
+ for a :: real
unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
-lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
+lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a)"
+ for a :: "'a::real_normed_vector"
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
-lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
+lemma at_left_minus: "at_left a = filtermap (\<lambda>x. - x) (at_right (- a))"
+ for a :: real
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
-lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
+lemma at_right_minus: "at_right a = filtermap (\<lambda>x. - x) (at_left (- a))"
+ for a :: real
by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
lemma filterlim_at_left_to_right:
- "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
+ "filterlim f F (at_left a) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
+ for a :: real
unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
lemma eventually_at_left_to_right:
- "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
+ "eventually P (at_left a) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
+ for a :: real
unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
@@ -1167,7 +1232,7 @@
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
- using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
+ and filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
by auto
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
@@ -1176,7 +1241,8 @@
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
proof safe
- fix Z :: real assume [arith]: "0 < Z"
+ fix Z :: real
+ assume [arith]: "0 < Z"
then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
@@ -1188,41 +1254,56 @@
shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
proof safe
- fix r :: real assume "0 < r"
+ fix r :: real
+ assume "0 < r"
show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
proof (intro exI[of _ "inverse (r / 2)"] allI impI)
fix x :: 'a
from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
also assume *: "inverse (r / 2) \<le> norm x"
finally show "norm (inverse x) < r"
- using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
+ using * \<open>0 < r\<close>
+ by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
qed
qed
lemma tendsto_add_filterlim_at_infinity:
- assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
- assumes "filterlim g at_infinity F"
- shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
+ fixes c :: "'b::real_normed_vector"
+ and F :: "'a filter"
+ assumes "(f \<longlongrightarrow> c) F"
+ and "filterlim g at_infinity F"
+ shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
proof (subst filterlim_at_infinity[OF order_refl], safe)
- fix r :: real assume r: "r > 0"
- from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
- hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
- moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
+ fix r :: real
+ assume r: "r > 0"
+ from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F"
+ by (rule tendsto_norm)
+ then have "eventually (\<lambda>x. norm (f x) < norm c + 1) F"
+ by (rule order_tendstoD) simp_all
+ moreover from r have "r + norm c + 1 > 0"
+ by (intro add_pos_nonneg) simp_all
with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
- unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
+ unfolding filterlim_at_infinity[OF order_refl]
+ by (elim allE[of _ "r + norm c + 1"]) simp_all
ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
proof eventually_elim
- fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
- from A B have "r \<le> norm (g x) - norm (f x)" by simp
- also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
- finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
+ fix x :: 'a
+ assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
+ from A B have "r \<le> norm (g x) - norm (f x)"
+ by simp
+ also have "norm (g x) - norm (f x) \<le> norm (g x + f x)"
+ by (rule norm_diff_ineq)
+ finally show "r \<le> norm (f x + g x)"
+ by (simp add: add_ac)
qed
qed
lemma tendsto_add_filterlim_at_infinity':
+ fixes c :: "'b::real_normed_vector"
+ and F :: "'a filter"
assumes "filterlim f at_infinity F"
- assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
- shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
+ and "(g \<longlongrightarrow> c) F"
+ shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
@@ -1272,7 +1353,8 @@
shows "filterlim inverse at_infinity (at (0::'a))"
unfolding filterlim_at_infinity[OF order_refl]
proof safe
- fix r :: real assume "0 < r"
+ fix r :: real
+ assume "0 < r"
then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
unfolding eventually_at norm_inverse
by (intro exI[of _ "inverse r"])
@@ -1290,7 +1372,7 @@
also have "\<dots> \<le> at 0"
using tendsto_inverse_0[where 'a='b]
by (auto intro!: exI[of _ 1]
- simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
+ simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
finally show "filtermap inverse (filtermap g F) \<le> at 0" .
next
assume "filtermap inverse (filtermap g F) \<le> at 0"
@@ -1301,36 +1383,40 @@
qed
lemma tendsto_mult_filterlim_at_infinity:
- assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
+ fixes c :: "'a::real_normed_field"
+ assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> 0"
assumes "filterlim g at_infinity F"
- shows "filterlim (\<lambda>x. f x * g x) at_infinity F"
+ shows "filterlim (\<lambda>x. f x * g x) at_infinity F"
proof -
have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
- hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
- unfolding filterlim_at using assms
+ then have "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
+ unfolding filterlim_at
+ using assms
by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
- thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
+ then show ?thesis
+ by (subst filterlim_inverse_at_iff[symmetric]) simp_all
qed
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
-lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
- by (rule filterlim_subseq) (auto simp: subseq_def)
-
-lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
+lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x) at_top sequentially"
+ for c :: nat
by (rule filterlim_subseq) (auto simp: subseq_def)
-lemma at_to_infinity:
- fixes x :: "'a :: {real_normed_field,field}"
- shows "(at (0::'a)) = filtermap inverse at_infinity"
+lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c) at_top sequentially"
+ for c :: nat
+ by (rule filterlim_subseq) (auto simp: subseq_def)
+
+lemma at_to_infinity: "(at (0::'a::{real_normed_field,field})) = filtermap inverse at_infinity"
proof (rule antisym)
have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
by (fact tendsto_inverse_0)
then show "filtermap inverse at_infinity \<le> at (0::'a)"
apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
- apply (rule_tac x="1" in exI, auto)
+ apply (rule_tac x="1" in exI)
+ apply auto
done
next
have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
@@ -1341,38 +1427,39 @@
qed
lemma lim_at_infinity_0:
- fixes l :: "'a :: {real_normed_field,field}"
- shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
-by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
+ fixes l :: "'a::{real_normed_field,field}"
+ shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f \<circ> inverse) \<longlongrightarrow> l) (at (0::'a))"
+ by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
lemma lim_zero_infinity:
- fixes l :: "'a :: {real_normed_field,field}"
+ fixes l :: "'a::{real_normed_field,field}"
shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
-by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
+ by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
text \<open>
-
-We only show rules for multiplication and addition when the functions are either against a real
-value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
-
+ We only show rules for multiplication and addition when the functions are either against a real
+ value or against infinity. Further rules are easy to derive by using @{thm
+ filterlim_uminus_at_top}.
\<close>
lemma filterlim_tendsto_pos_mult_at_top:
- assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
- assumes g: "LIM x F. g x :> at_top"
+ assumes f: "(f \<longlongrightarrow> c) F"
+ and c: "0 < c"
+ and g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x * g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
- fix Z :: real assume "0 < Z"
+ fix Z :: real
+ assume "0 < Z"
from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
- simp: dist_real_def abs_real_def split: if_split_asm)
+ simp: dist_real_def abs_real_def split: if_split_asm)
moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
unfolding filterlim_at_top by auto
ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
proof eventually_elim
- fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
+ case (elim x)
with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
by (intro mult_mono) (auto simp: zero_le_divide_iff)
with \<open>0 < c\<close> show "Z \<le> f x * g x"
@@ -1382,18 +1469,19 @@
lemma filterlim_at_top_mult_at_top:
assumes f: "LIM x F. f x :> at_top"
- assumes g: "LIM x F. g x :> at_top"
+ and g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x * g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
- fix Z :: real assume "0 < Z"
+ fix Z :: real
+ assume "0 < Z"
from f have "eventually (\<lambda>x. 1 \<le> f x) F"
unfolding filterlim_at_top by auto
moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
unfolding filterlim_at_top by auto
ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
proof eventually_elim
- fix x assume "1 \<le> f x" "Z \<le> g x"
+ case (elim x)
with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
by (intro mult_mono) (auto simp: zero_le_divide_iff)
then show "Z \<le> f x * g x"
@@ -1402,25 +1490,32 @@
qed
lemma filterlim_tendsto_pos_mult_at_bot:
- assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
+ fixes c :: real
+ assumes "(f \<longlongrightarrow> c) F" "0 < c" "filterlim g at_bot F"
shows "LIM x F. f x * g x :> at_bot"
using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
unfolding filterlim_uminus_at_bot by simp
lemma filterlim_tendsto_neg_mult_at_bot:
- assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
+ fixes c :: real
+ assumes c: "(f \<longlongrightarrow> c) F" "c < 0" and g: "filterlim g at_top F"
shows "LIM x F. f x * g x :> at_bot"
using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
lemma filterlim_pow_at_top:
fixes f :: "real \<Rightarrow> real"
- assumes "0 < n" and f: "LIM x F. f x :> at_top"
+ assumes "0 < n"
+ and f: "LIM x F. f x :> at_top"
shows "LIM x F. (f x)^n :: real :> at_top"
-using \<open>0 < n\<close> proof (induct n)
+ using \<open>0 < n\<close>
+proof (induct n)
+ case 0
+ then show ?case by simp
+next
case (Suc n) with f show ?case
by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
-qed simp
+qed
lemma filterlim_pow_at_bot_even:
fixes f :: "real \<Rightarrow> real"
@@ -1434,11 +1529,12 @@
lemma filterlim_tendsto_add_at_top:
assumes f: "(f \<longlongrightarrow> c) F"
- assumes g: "LIM x F. g x :> at_top"
+ and g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x + g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
- fix Z :: real assume "0 < Z"
+ fix Z :: real
+ assume "0 < Z"
from f have "eventually (\<lambda>x. c - 1 < f x) F"
by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
@@ -1450,18 +1546,19 @@
lemma LIM_at_top_divide:
fixes f g :: "'a \<Rightarrow> real"
assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
- assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
+ and g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
shows "LIM x F. f x / g x :> at_top"
unfolding divide_inverse
by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
lemma filterlim_at_top_add_at_top:
assumes f: "LIM x F. f x :> at_top"
- assumes g: "LIM x F. g x :> at_top"
+ and g: "LIM x F. g x :> at_top"
shows "LIM x F. (f x + g x :: real) :> at_top"
unfolding filterlim_at_top_gt[where c=0]
proof safe
- fix Z :: real assume "0 < Z"
+ fix Z :: real
+ assume "0 < Z"
from f have "eventually (\<lambda>x. 0 \<le> f x) F"
unfolding filterlim_at_top by auto
moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
@@ -1473,34 +1570,43 @@
lemma tendsto_divide_0:
fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
assumes f: "(f \<longlongrightarrow> c) F"
- assumes g: "LIM x F. g x :> at_infinity"
+ and g: "LIM x F. g x :> at_infinity"
shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
- using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
+ using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]]
+ by (simp add: divide_inverse)
lemma linear_plus_1_le_power:
fixes x :: real
assumes x: "0 \<le> x"
shows "real n * x + 1 \<le> (x + 1) ^ n"
proof (induct n)
+ case 0
+ then show ?case by simp
+next
case (Suc n)
- have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
- by (simp add: field_simps of_nat_Suc x)
+ from x have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
+ by (simp add: field_simps)
also have "\<dots> \<le> (x + 1)^Suc n"
using Suc x by (simp add: mult_left_mono)
finally show ?case .
-qed simp
+qed
lemma filterlim_realpow_sequentially_gt1:
fixes x :: "'a :: real_normed_div_algebra"
assumes x[arith]: "1 < norm x"
shows "LIM n sequentially. x ^ n :> at_infinity"
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
- fix y :: real assume "0 < y"
+ fix y :: real
+ assume "0 < y"
have "0 < norm x - 1" by simp
- then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
- also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
- also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
- also have "\<dots> = norm x ^ N" by simp
+ then obtain N :: nat where "y < real N * (norm x - 1)"
+ by (blast dest: reals_Archimedean3)
+ also have "\<dots> \<le> real N * (norm x - 1) + 1"
+ by simp
+ also have "\<dots> \<le> (norm x - 1 + 1) ^ N"
+ by (rule linear_plus_1_le_power) simp
+ also have "\<dots> = norm x ^ N"
+ by simp
finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
by (metis order_less_le_trans power_increasing order_less_imp_le x)
then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
@@ -1512,48 +1618,48 @@
subsection \<open>Floor and Ceiling\<close>
lemma eventually_floor_less:
- fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
+ fixes f :: "'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
assumes f: "(f \<longlongrightarrow> l) F"
- assumes l: "l \<notin> \<int>"
+ and l: "l \<notin> \<int>"
shows "\<forall>\<^sub>F x in F. of_int (floor l) < f x"
by (intro order_tendstoD[OF f]) (metis Ints_of_int antisym_conv2 floor_correct l)
lemma eventually_less_ceiling:
- fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
+ fixes f :: "'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
assumes f: "(f \<longlongrightarrow> l) F"
- assumes l: "l \<notin> \<int>"
+ and l: "l \<notin> \<int>"
shows "\<forall>\<^sub>F x in F. f x < of_int (ceiling l)"
by (intro order_tendstoD[OF f]) (metis Ints_of_int l le_of_int_ceiling less_le)
lemma eventually_floor_eq:
- fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
+ fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
assumes f: "(f \<longlongrightarrow> l) F"
- assumes l: "l \<notin> \<int>"
+ and l: "l \<notin> \<int>"
shows "\<forall>\<^sub>F x in F. floor (f x) = floor l"
using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)
lemma eventually_ceiling_eq:
- fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
+ fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
assumes f: "(f \<longlongrightarrow> l) F"
- assumes l: "l \<notin> \<int>"
+ and l: "l \<notin> \<int>"
shows "\<forall>\<^sub>F x in F. ceiling (f x) = ceiling l"
using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)
lemma tendsto_of_int_floor:
- fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
+ fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
assumes "(f \<longlongrightarrow> l) F"
- assumes "l \<notin> \<int>"
- shows "((\<lambda>x. of_int (floor (f x))::'c::{ring_1, topological_space}) \<longlongrightarrow> of_int (floor l)) F"
+ and "l \<notin> \<int>"
+ shows "((\<lambda>x. of_int (floor (f x)) :: 'c::{ring_1,topological_space}) \<longlongrightarrow> of_int (floor l)) F"
using eventually_floor_eq[OF assms]
by (simp add: eventually_mono topological_tendstoI)
lemma tendsto_of_int_ceiling:
- fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
+ fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
assumes "(f \<longlongrightarrow> l) F"
- assumes "l \<notin> \<int>"
- shows "((\<lambda>x. of_int (ceiling (f x))::'c::{ring_1, topological_space}) \<longlongrightarrow> of_int (ceiling l)) F"
+ and "l \<notin> \<int>"
+ shows "((\<lambda>x. of_int (ceiling (f x)):: 'c::{ring_1,topological_space}) \<longlongrightarrow> of_int (ceiling l)) F"
using eventually_ceiling_eq[OF assms]
by (simp add: eventually_mono topological_tendstoI)
@@ -1580,69 +1686,64 @@
shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
unfolding lim_sequentially dist_norm ..
-lemma LIMSEQ_I:
- fixes L :: "'a::real_normed_vector"
- shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
-by (simp add: LIMSEQ_iff)
-
-lemma LIMSEQ_D:
- fixes L :: "'a::real_normed_vector"
- shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
-by (simp add: LIMSEQ_iff)
-
-lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
+lemma LIMSEQ_I: "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
+ for L :: "'a::real_normed_vector"
+ by (simp add: LIMSEQ_iff)
+
+lemma LIMSEQ_D: "X \<longlonglongrightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
+ for L :: "'a::real_normed_vector"
+ by (simp add: LIMSEQ_iff)
+
+lemma LIMSEQ_linear: "X \<longlonglongrightarrow> x \<Longrightarrow> l > 0 \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
unfolding tendsto_def eventually_sequentially
by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
-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"
-apply (subst nonzero_norm_inverse, clarsimp)
-apply (erule (1) le_imp_inverse_le)
-done
-
-lemma Bseq_inverse:
- fixes a :: "'a::real_normed_div_algebra"
- shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
+lemma Bseq_inverse_lemma: "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
+ for x :: "'a::real_normed_div_algebra"
+ apply (subst nonzero_norm_inverse, clarsimp)
+ apply (erule (1) le_imp_inverse_le)
+ done
+
+lemma Bseq_inverse: "X \<longlonglongrightarrow> a \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
+ for a :: "'a::real_normed_div_algebra"
by (rule Bfun_inverse)
-text\<open>Transformation of limit.\<close>
-
-lemma Lim_transform:
- fixes a b :: "'a::real_normed_vector"
- shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
+
+text \<open>Transformation of limit.\<close>
+
+lemma Lim_transform: "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
+ for a b :: "'a::real_normed_vector"
using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
-lemma Lim_transform2:
- fixes a b :: "'a::real_normed_vector"
- shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
+lemma Lim_transform2: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (g \<longlongrightarrow> a) F"
+ for a b :: "'a::real_normed_vector"
by (erule Lim_transform) (simp add: tendsto_minus_cancel)
-proposition Lim_transform_eq:
- fixes a :: "'a::real_normed_vector"
- shows "((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F \<longleftrightarrow> (g \<longlongrightarrow> a) F"
-using Lim_transform Lim_transform2 by blast
+proposition Lim_transform_eq: "((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F \<longleftrightarrow> (g \<longlongrightarrow> a) F"
+ for a :: "'a::real_normed_vector"
+ using Lim_transform Lim_transform2 by blast
lemma Lim_transform_eventually:
"eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
- apply (erule (1) eventually_elim2, simp)
+ apply (erule (1) eventually_elim2)
+ apply simp
done
lemma Lim_transform_within:
assumes "(f \<longlongrightarrow> l) (at x within S)"
and "0 < d"
- and "\<And>x'. \<lbrakk>x'\<in>S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
+ and "\<And>x'. x'\<in>S \<Longrightarrow> 0 < dist x' x \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x'"
shows "(g \<longlongrightarrow> l) (at x within S)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at x within S)"
using assms by (auto simp: eventually_at)
- show "(f \<longlongrightarrow> l) (at x within S)" by fact
+ show "(f \<longlongrightarrow> l) (at x within S)"
+ by fact
qed
-text\<open>Common case assuming being away from some crucial point like 0.\<close>
-
+text \<open>Common case assuming being away from some crucial point like 0.\<close>
lemma Lim_transform_away_within:
fixes a b :: "'a::t1_space"
assumes "a \<noteq> b"
@@ -1650,26 +1751,26 @@
and "(f \<longlongrightarrow> l) (at a within S)"
shows "(g \<longlongrightarrow> l) (at a within S)"
proof (rule Lim_transform_eventually)
- show "(f \<longlongrightarrow> l) (at a within S)" by fact
+ show "(f \<longlongrightarrow> l) (at a within S)"
+ by fact
show "eventually (\<lambda>x. f x = g x) (at a within S)"
unfolding eventually_at_topological
- by (rule exI [where x="- {b}"], simp add: open_Compl assms)
+ by (rule exI [where x="- {b}"]) (simp add: open_Compl assms)
qed
lemma Lim_transform_away_at:
fixes a b :: "'a::t1_space"
- assumes ab: "a\<noteq>b"
+ assumes ab: "a \<noteq> b"
and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
and fl: "(f \<longlongrightarrow> l) (at a)"
shows "(g \<longlongrightarrow> l) (at a)"
using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
-text\<open>Alternatively, within an open set.\<close>
-
+text \<open>Alternatively, within an open set.\<close>
lemma Lim_transform_within_open:
assumes "(f \<longlongrightarrow> l) (at a within T)"
and "open s" and "a \<in> s"
- and "\<And>x. \<lbrakk>x\<in>s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
+ and "\<And>x. x\<in>s \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x = g x"
shows "(g \<longlongrightarrow> l) (at a within T)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at a within T)"
@@ -1678,7 +1779,8 @@
show "(f \<longlongrightarrow> l) (at a within T)" by fact
qed
-text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
+
+text \<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
@@ -1697,35 +1799,32 @@
shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
unfolding tendsto_def eventually_at_topological
using assms by simp
-text\<open>An unbounded sequence's inverse tends to 0\<close>
-
-lemma LIMSEQ_inverse_zero:
- "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
+
+text \<open>An unbounded sequence's inverse tends to 0.\<close>
+lemma LIMSEQ_inverse_zero: "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
apply (rule filterlim_compose[OF tendsto_inverse_0])
apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
done
-text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
-
-lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
+text \<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity.\<close>
+lemma LIMSEQ_inverse_real_of_nat: "(\<lambda>n. inverse (real (Suc n))) \<longlonglongrightarrow> 0"
by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
- filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
-
-text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
-infinity is now easily proved\<close>
-
-lemma LIMSEQ_inverse_real_of_nat_add:
- "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
+ filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
+
+text \<open>
+ The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
+ infinity is now easily proved.
+\<close>
+
+lemma LIMSEQ_inverse_real_of_nat_add: "(\<lambda>n. r + inverse (real (Suc n))) \<longlonglongrightarrow> r"
using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
-lemma LIMSEQ_inverse_real_of_nat_add_minus:
- "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
+lemma LIMSEQ_inverse_real_of_nat_add_minus: "(\<lambda>n. r + -inverse (real (Suc n))) \<longlonglongrightarrow> r"
using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
by auto
-lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
- "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
+lemma LIMSEQ_inverse_real_of_nat_add_minus_mult: "(\<lambda>n. r * (1 + - inverse (real (Suc n)))) \<longlonglongrightarrow> r"
using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
by auto
@@ -1735,46 +1834,57 @@
lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
- using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
+ using eventually_gt_at_top[of "0::nat"]
+ by eventually_elim (simp add: field_simps)
have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
by (intro tendsto_add tendsto_const lim_inverse_n)
- thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
+ then show "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1"
+ by simp
qed
lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
- of_nat n / of_nat (Suc n)) sequentially"
+ of_nat n / of_nat (Suc n)) sequentially"
using eventually_gt_at_top[of "0::nat"]
by eventually_elim (simp add: field_simps del: of_nat_Suc)
have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
- thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
+ then show "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1"
+ by simp
qed
+
subsection \<open>Convergence on sequences\<close>
lemma convergent_cong:
assumes "eventually (\<lambda>x. f x = g x) sequentially"
- shows "convergent f \<longleftrightarrow> convergent g"
- unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
+ shows "convergent f \<longleftrightarrow> convergent g"
+ unfolding convergent_def
+ by (subst filterlim_cong[OF refl refl assms]) (rule refl)
lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
by (auto simp: convergent_def LIMSEQ_Suc_iff)
lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
-proof (induction m arbitrary: f)
+proof (induct m arbitrary: f)
+ case 0
+ then show ?case by simp
+next
case (Suc m)
- have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
- also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
- also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
+ have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))"
+ by simp
+ also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))"
+ by (rule convergent_Suc_iff)
+ also have "\<dots> \<longleftrightarrow> convergent f"
+ by (rule Suc)
finally show ?case .
-qed simp_all
+qed
lemma convergent_add:
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
assumes "convergent (\<lambda>n. X n)"
- assumes "convergent (\<lambda>n. Y n)"
+ and "convergent (\<lambda>n. Y n)"
shows "convergent (\<lambda>n. X n + Y n)"
using assms unfolding convergent_def by (blast intro: tendsto_add)
@@ -1783,9 +1893,14 @@
assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
proof (cases "finite A")
- case True from this and assms show ?thesis
- by (induct A set: finite) (simp_all add: convergent_const convergent_add)
-qed (simp add: convergent_const)
+ case True
+ then show ?thesis
+ using assms by (induct A set: finite) (simp_all add: convergent_const convergent_add)
+next
+ case False
+ then show ?thesis
+ by (simp add: convergent_const)
+qed
lemma (in bounded_linear) convergent:
assumes "convergent (\<lambda>n. X n)"
@@ -1793,17 +1908,18 @@
using assms unfolding convergent_def by (blast intro: tendsto)
lemma (in bounded_bilinear) convergent:
- assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
+ assumes "convergent (\<lambda>n. X n)"
+ and "convergent (\<lambda>n. Y n)"
shows "convergent (\<lambda>n. X n ** Y n)"
using assms unfolding convergent_def by (blast intro: tendsto)
-lemma convergent_minus_iff:
- fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
-apply (simp add: convergent_def)
-apply (auto dest: tendsto_minus)
-apply (drule tendsto_minus, auto)
-done
+lemma convergent_minus_iff: "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
+ for X :: "nat \<Rightarrow> 'a::real_normed_vector"
+ apply (simp add: convergent_def)
+ apply (auto dest: tendsto_minus)
+ apply (drule tendsto_minus)
+ apply auto
+ done
lemma convergent_diff:
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
@@ -1814,57 +1930,64 @@
lemma convergent_norm:
assumes "convergent f"
- shows "convergent (\<lambda>n. norm (f n))"
+ shows "convergent (\<lambda>n. norm (f n))"
proof -
- from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
- hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
- thus ?thesis by (auto simp: convergent_def)
+ from assms have "f \<longlonglongrightarrow> lim f"
+ by (simp add: convergent_LIMSEQ_iff)
+ then have "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)"
+ by (rule tendsto_norm)
+ then show ?thesis
+ by (auto simp: convergent_def)
qed
lemma convergent_of_real:
- "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
+ "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a::real_normed_algebra_1)"
unfolding convergent_def by (blast intro!: tendsto_of_real)
lemma convergent_add_const_iff:
- "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
+ "convergent (\<lambda>n. c + f n :: 'a::real_normed_vector) \<longleftrightarrow> convergent f"
proof
assume "convergent (\<lambda>n. c + f n)"
- from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
+ from convergent_diff[OF this convergent_const[of c]] show "convergent f"
+ by simp
next
assume "convergent f"
- from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
+ from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)"
+ by simp
qed
lemma convergent_add_const_right_iff:
- "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
+ "convergent (\<lambda>n. f n + c :: 'a::real_normed_vector) \<longleftrightarrow> convergent f"
using convergent_add_const_iff[of c f] by (simp add: add_ac)
lemma convergent_diff_const_right_iff:
- "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
+ "convergent (\<lambda>n. f n - c :: 'a::real_normed_vector) \<longleftrightarrow> convergent f"
using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
lemma convergent_mult:
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
assumes "convergent (\<lambda>n. X n)"
- assumes "convergent (\<lambda>n. Y n)"
+ and "convergent (\<lambda>n. Y n)"
shows "convergent (\<lambda>n. X n * Y n)"
using assms unfolding convergent_def by (blast intro: tendsto_mult)
lemma convergent_mult_const_iff:
assumes "c \<noteq> 0"
- shows "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
+ shows "convergent (\<lambda>n. c * f n :: 'a::real_normed_field) \<longleftrightarrow> convergent f"
proof
assume "convergent (\<lambda>n. c * f n)"
from assms convergent_mult[OF this convergent_const[of "inverse c"]]
show "convergent f" by (simp add: field_simps)
next
assume "convergent f"
- from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
+ from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)"
+ by simp
qed
lemma convergent_mult_const_right_iff:
+ fixes c :: "'a::real_normed_field"
assumes "c \<noteq> 0"
- shows "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
+ shows "convergent (\<lambda>n. f n * c) \<longleftrightarrow> convergent f"
using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
@@ -1874,60 +1997,66 @@
text \<open>A monotone sequence converges to its least upper bound.\<close>
lemma LIMSEQ_incseq_SUP:
- fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
+ fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder,linorder_topology}"
assumes u: "bdd_above (range X)"
- assumes X: "incseq X"
+ and X: "incseq X"
shows "X \<longlonglongrightarrow> (SUP i. X i)"
by (rule order_tendstoI)
- (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
+ (auto simp: eventually_sequentially u less_cSUP_iff
+ intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
lemma LIMSEQ_decseq_INF:
fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
assumes u: "bdd_below (range X)"
- assumes X: "decseq X"
+ and X: "decseq X"
shows "X \<longlonglongrightarrow> (INF i. X i)"
by (rule order_tendstoI)
- (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
-
-text\<open>Main monotonicity theorem\<close>
-
-lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
- by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
-
-lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
+ (auto simp: eventually_sequentially u cINF_less_iff
+ intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
+
+text \<open>Main monotonicity theorem.\<close>
+
+lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent X"
+ for X :: "nat \<Rightarrow> real"
+ by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP
+ dest: Bseq_bdd_above Bseq_bdd_below)
+
+lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent X"
+ for X :: "nat \<Rightarrow> real"
by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
-lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
+lemma monoseq_imp_convergent_iff_Bseq: "monoseq f \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
+ for f :: "nat \<Rightarrow> real"
using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
lemma Bseq_monoseq_convergent'_inc:
- "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
+ fixes f :: "nat \<Rightarrow> real"
+ shows "Bseq (\<lambda>n. f (n + M)) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
by (subst convergent_ignore_initial_segment [symmetric, of _ M])
(auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
lemma Bseq_monoseq_convergent'_dec:
- "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
+ fixes f :: "nat \<Rightarrow> real"
+ shows "Bseq (\<lambda>n. f (n + M)) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
by (subst convergent_ignore_initial_segment [symmetric, of _ M])
- (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
-
-lemma Cauchy_iff:
- fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
+ (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
+
+lemma Cauchy_iff: "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
+ for X :: "nat \<Rightarrow> 'a::real_normed_vector"
unfolding Cauchy_def dist_norm ..
-lemma CauchyI:
- fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
-by (simp add: Cauchy_iff)
-
-lemma CauchyD:
- fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
-by (simp add: Cauchy_iff)
+lemma CauchyI: "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
+ for X :: "nat \<Rightarrow> 'a::real_normed_vector"
+ by (simp add: Cauchy_iff)
+
+lemma CauchyD: "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
+ for X :: "nat \<Rightarrow> 'a::real_normed_vector"
+ by (simp add: Cauchy_iff)
lemma incseq_convergent:
fixes X :: "nat \<Rightarrow> real"
- assumes "incseq X" and "\<forall>i. X i \<le> B"
+ assumes "incseq X"
+ and "\<forall>i. X i \<le> B"
obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
proof atomize_elim
from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
@@ -1939,7 +2068,8 @@
lemma decseq_convergent:
fixes X :: "nat \<Rightarrow> real"
- assumes "decseq X" and "\<forall>i. B \<le> X i"
+ assumes "decseq X"
+ and "\<forall>i. B \<le> X i"
obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
proof atomize_elim
from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
@@ -1949,69 +2079,85 @@
by (auto intro!: exI[of _ L] decseq_le)
qed
+
subsubsection \<open>Cauchy Sequences are Bounded\<close>
-text\<open>A Cauchy sequence is bounded -- this is the standard
- proof mechanization rather than the nonstandard proof\<close>
-
-lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
- ==> \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
-apply (clarify, drule spec, drule (1) mp)
-apply (simp only: norm_minus_commute)
-apply (drule order_le_less_trans [OF norm_triangle_ineq2])
-apply simp
-done
+text \<open>
+ A Cauchy sequence is bounded -- this is the standard
+ proof mechanization rather than the nonstandard proof.
+\<close>
+
+lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real) \<Longrightarrow>
+ \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
+ apply clarify
+ apply (drule spec)
+ apply (drule (1) mp)
+ apply (simp only: norm_minus_commute)
+ apply (drule order_le_less_trans [OF norm_triangle_ineq2])
+ apply simp
+ done
+
subsection \<open>Power Sequences\<close>
-text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
-"x<1"}. Proof will use (NS) Cauchy equivalence for convergence and
- also fact that bounded and monotonic sequence converges.\<close>
-
-lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
-apply (simp add: Bseq_def)
-apply (rule_tac x = 1 in exI)
-apply (simp add: power_abs)
-apply (auto dest: power_mono)
-done
-
-lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
-apply (clarify intro!: mono_SucI2)
-apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
-done
-
-lemma convergent_realpow:
- "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
-by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
-
-lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
+text \<open>
+ The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
+ "x<1"}. Proof will use (NS) Cauchy equivalence for convergence and
+ also fact that bounded and monotonic sequence converges.
+\<close>
+
+lemma Bseq_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> Bseq (\<lambda>n. x ^ n)"
+ for x :: real
+ apply (simp add: Bseq_def)
+ apply (rule_tac x = 1 in exI)
+ apply (simp add: power_abs)
+ apply (auto dest: power_mono)
+ done
+
+lemma monoseq_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> monoseq (\<lambda>n. x ^ n)"
+ for x :: real
+ apply (clarify intro!: mono_SucI2)
+ apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing)
+ apply auto
+ done
+
+lemma convergent_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> convergent (\<lambda>n. x ^ n)"
+ for x :: real
+ by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
+
+lemma LIMSEQ_inverse_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
+ for x :: real
by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
lemma LIMSEQ_realpow_zero:
- "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
-proof cases
- assume "0 \<le> x" and "x \<noteq> 0"
- hence x0: "0 < x" by simp
- assume x1: "x < 1"
- from x0 x1 have "1 < inverse x"
- by (rule one_less_inverse)
- hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
+ fixes x :: real
+ assumes "0 \<le> x" "x < 1"
+ shows "(\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
+proof (cases "x = 0")
+ case False
+ with \<open>0 \<le> x\<close> have x0: "0 < x" by simp
+ then have "1 < inverse x"
+ using \<open>x < 1\<close> by (rule one_less_inverse)
+ then have "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
by (rule LIMSEQ_inverse_realpow_zero)
- thus ?thesis by (simp add: power_inverse)
-qed (rule LIMSEQ_imp_Suc, simp)
-
-lemma LIMSEQ_power_zero:
- fixes x :: "'a::{real_normed_algebra_1}"
- shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
-apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
-apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
-apply (simp add: power_abs norm_power_ineq)
-done
+ then show ?thesis by (simp add: power_inverse)
+next
+ case True
+ show ?thesis
+ by (rule LIMSEQ_imp_Suc) (simp add: True)
+qed
+
+lemma LIMSEQ_power_zero: "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
+ for x :: "'a::real_normed_algebra_1"
+ apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
+ apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
+ apply (simp add: power_abs norm_power_ineq)
+ done
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
-text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
+text \<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}.\<close>
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
@@ -2022,92 +2168,81 @@
subsection \<open>Limits of Functions\<close>
-lemma LIM_eq:
- fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
- shows "f \<midarrow>a\<rightarrow> L =
- (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
-by (simp add: LIM_def dist_norm)
+lemma LIM_eq: "f \<midarrow>a\<rightarrow> L = (\<forall>r>0. \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r)"
+ for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
+ by (simp add: LIM_def dist_norm)
lemma LIM_I:
- fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
- shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
- ==> f \<midarrow>a\<rightarrow> L"
-by (simp add: LIM_eq)
-
-lemma LIM_D:
- fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
- shows "[| f \<midarrow>a\<rightarrow> L; 0<r |]
- ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
-by (simp add: LIM_eq)
-
-lemma LIM_offset:
- fixes a :: "'a::real_normed_vector"
- shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
- unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
-
-lemma LIM_offset_zero:
- fixes a :: "'a::real_normed_vector"
- shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
-by (drule_tac k="a" in LIM_offset, simp add: add.commute)
-
-lemma LIM_offset_zero_cancel:
- fixes a :: "'a::real_normed_vector"
- shows "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
-by (drule_tac k="- a" in LIM_offset, simp)
-
-lemma LIM_offset_zero_iff:
- fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
- shows "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
+ "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r) \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
+ for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
+ by (simp add: LIM_eq)
+
+lemma LIM_D: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
+ for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
+ by (simp add: LIM_eq)
+
+lemma LIM_offset: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
+ for a :: "'a::real_normed_vector"
+ by (simp add: filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap)
+
+lemma LIM_offset_zero: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
+ for a :: "'a::real_normed_vector"
+ by (drule LIM_offset [where k = a]) (simp add: add.commute)
+
+lemma LIM_offset_zero_cancel: "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
+ for a :: "'a::real_normed_vector"
+ by (drule LIM_offset [where k = "- a"]) simp
+
+lemma LIM_offset_zero_iff: "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
+ for f :: "'a :: real_normed_vector \<Rightarrow> _"
using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
-lemma LIM_zero:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
- shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
-unfolding tendsto_iff dist_norm by simp
+lemma LIM_zero: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
+ for f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
+ unfolding tendsto_iff dist_norm by simp
lemma LIM_zero_cancel:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
unfolding tendsto_iff dist_norm by simp
-lemma LIM_zero_iff:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
-unfolding tendsto_iff dist_norm by simp
+lemma LIM_zero_iff: "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
+ for f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ unfolding tendsto_iff dist_norm by simp
lemma LIM_imp_LIM:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
assumes f: "f \<midarrow>a\<rightarrow> l"
- assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
+ and le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
shows "g \<midarrow>a\<rightarrow> m"
- by (rule metric_LIM_imp_LIM [OF f],
- simp add: dist_norm le)
+ by (rule metric_LIM_imp_LIM [OF f]) (simp add: dist_norm le)
lemma LIM_equal2:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
- assumes 1: "0 < R"
- assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
+ assumes "0 < R"
+ and "\<And>x. x \<noteq> a \<Longrightarrow> norm (x - a) < R \<Longrightarrow> f x = g x"
shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
-by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
+ by (rule metric_LIM_equal2 [OF assms]) (simp_all add: dist_norm)
lemma LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f: "f \<midarrow>a\<rightarrow> b"
- assumes g: "g \<midarrow>b\<rightarrow> c"
- assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
+ and g: "g \<midarrow>b\<rightarrow> c"
+ and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
-by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
+ by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
lemma real_LIM_sandwich_zero:
fixes f g :: "'a::topological_space \<Rightarrow> real"
assumes f: "f \<midarrow>a\<rightarrow> 0"
- assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
- assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
+ and 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
+ and 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
shows "g \<midarrow>a\<rightarrow> 0"
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
- fix x assume x: "x \<noteq> a"
- have "norm (g x - 0) = g x" by (simp add: 1 x)
+ fix x
+ assume x: "x \<noteq> a"
+ with 1 have "norm (g x - 0) = g x" by simp
also have "g x \<le> f x" by (rule 2 [OF x])
also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
@@ -2117,61 +2252,50 @@
subsection \<open>Continuity\<close>
-lemma LIM_isCont_iff:
- fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
- shows "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
-by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
-
-lemma isCont_iff:
- fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
- shows "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
-by (simp add: isCont_def LIM_isCont_iff)
+lemma LIM_isCont_iff: "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
+ for f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
+ by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
+
+lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
+ for f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
+ by (simp add: isCont_def LIM_isCont_iff)
lemma isCont_LIM_compose2:
fixes a :: "'a::real_normed_vector"
assumes f [unfolded isCont_def]: "isCont f a"
- assumes g: "g \<midarrow>f a\<rightarrow> l"
- assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
+ and g: "g \<midarrow>f a\<rightarrow> l"
+ and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
-by (rule LIM_compose2 [OF f g inj])
-
-
-lemma isCont_norm [simp]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
- shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
+ by (rule LIM_compose2 [OF f g inj])
+
+lemma isCont_norm [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
+ for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
by (fact continuous_norm)
-lemma isCont_rabs [simp]:
- fixes f :: "'a::t2_space \<Rightarrow> real"
- shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
+lemma isCont_rabs [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
+ for f :: "'a::t2_space \<Rightarrow> real"
by (fact continuous_rabs)
-lemma isCont_add [simp]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
- shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
+lemma isCont_add [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
+ for f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
by (fact continuous_add)
-lemma isCont_minus [simp]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
- shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
+lemma isCont_minus [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
+ for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
by (fact continuous_minus)
-lemma isCont_diff [simp]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
- shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
+lemma isCont_diff [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
+ for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
by (fact continuous_diff)
-lemma isCont_mult [simp]:
- fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
- shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
+lemma isCont_mult [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
+ for f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
by (fact continuous_mult)
-lemma (in bounded_linear) isCont:
- "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
+lemma (in bounded_linear) isCont: "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
by (fact continuous)
-lemma (in bounded_bilinear) isCont:
- "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
+lemma (in bounded_bilinear) isCont: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
by (fact continuous)
lemmas isCont_scaleR [simp] =
@@ -2180,16 +2304,15 @@
lemmas isCont_of_real [simp] =
bounded_linear.isCont [OF bounded_linear_of_real]
-lemma isCont_power [simp]:
- fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
- shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
+lemma isCont_power [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
+ for f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
by (fact continuous_power)
-lemma isCont_setsum [simp]:
- fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
- shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
+lemma isCont_setsum [simp]: "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
+ for f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
by (auto intro: continuous_setsum)
+
subsection \<open>Uniform Continuity\<close>
lemma uniformly_continuous_on_def:
@@ -2200,37 +2323,39 @@
uniformity_dist filterlim_INF filterlim_principal eventually_inf_principal
by (force simp: Ball_def uniformity_dist[symmetric] eventually_uniformity_metric)
-abbreviation isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
- "isUCont f \<equiv> uniformly_continuous_on UNIV f"
-
-lemma isUCont_def: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
+abbreviation isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool"
+ where "isUCont f \<equiv> uniformly_continuous_on UNIV f"
+
+lemma isUCont_def: "isUCont f \<longleftrightarrow> (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
by (auto simp: uniformly_continuous_on_def dist_commute)
-lemma isUCont_isCont: "isUCont f ==> isCont f x"
+lemma isUCont_isCont: "isUCont f \<Longrightarrow> isCont f x"
by (drule uniformly_continuous_imp_continuous) (simp add: continuous_on_eq_continuous_at)
lemma uniformly_continuous_on_Cauchy:
- fixes f::"'a::metric_space \<Rightarrow> 'b::metric_space"
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
assumes "uniformly_continuous_on S f" "Cauchy X" "\<And>n. X n \<in> S"
shows "Cauchy (\<lambda>n. f (X n))"
using assms
- unfolding uniformly_continuous_on_def
- apply -
+ apply (simp only: uniformly_continuous_on_def)
apply (rule metric_CauchyI)
- apply (drule_tac x=e in spec, safe)
- apply (drule_tac e=d in metric_CauchyD, safe)
- apply (rule_tac x=M in exI, simp)
+ apply (drule_tac x=e in spec)
+ apply safe
+ apply (drule_tac e=d in metric_CauchyD)
+ apply safe
+ apply (rule_tac x=M in exI)
+ apply simp
done
-lemma isUCont_Cauchy:
- "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
+lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all
lemma (in bounded_linear) isUCont: "isUCont f"
-unfolding isUCont_def dist_norm
+ unfolding isUCont_def dist_norm
proof (intro allI impI)
- fix r::real assume r: "0 < r"
- obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
+ fix r :: real
+ assume r: "0 < r"
+ obtain K where K: "0 < K" and norm_le: "norm (f x) \<le> norm x * K" for x
using pos_bounded by blast
show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
proof (rule exI, safe)
@@ -2246,7 +2371,7 @@
qed
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
-by (rule isUCont [THEN isUCont_Cauchy])
+ by (rule isUCont [THEN isUCont_Cauchy])
lemma LIM_less_bound:
fixes f :: "real \<Rightarrow> real"
@@ -2268,16 +2393,21 @@
proof -
have "incseq f" unfolding incseq_Suc_iff by fact
have "decseq g" unfolding decseq_Suc_iff by fact
-
- { fix n
- from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
- with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
+ have "f n \<le> g 0" for n
+ proof -
+ from \<open>decseq g\<close> have "g n \<le> g 0"
+ by (rule decseqD) simp
+ with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] show ?thesis
+ by auto
+ qed
then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
using incseq_convergent[OF \<open>incseq f\<close>] by auto
- moreover
- { fix n
+ moreover have "f 0 \<le> g n" for n
+ proof -
from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
- with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
+ with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] show ?thesis
+ by simp
+ qed
then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
using decseq_convergent[OF \<open>decseq g\<close>] by auto
moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
@@ -2287,8 +2417,8 @@
lemma Bolzano[consumes 1, case_names trans local]:
fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
assumes [arith]: "a \<le> b"
- assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
- assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
+ and trans: "\<And>a b c. P a b \<Longrightarrow> P b c \<Longrightarrow> a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> P a c"
+ and local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
shows "P a b"
proof -
define bisect where "bisect =
@@ -2298,57 +2428,73 @@
and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
by (simp_all add: l_def u_def bisect_def split: prod.split)
- { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
+ have [simp]: "l n \<le> u n" for n by (induct n) auto
have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
proof (safe intro!: nested_sequence_unique)
- fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
+ show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" for n
+ by (induct n) auto
next
- { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
- then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
+ have "l n - u n = (a - b) / 2^n" for n
+ by (induct n) (auto simp: field_simps)
+ then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0"
+ by (simp add: LIMSEQ_divide_realpow_zero)
qed fact
- then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x" by auto
- obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
+ then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x"
+ by auto
+ obtain d where "0 < d" and d: "a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b" for a b
using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
show "P a b"
proof (rule ccontr)
assume "\<not> P a b"
- { fix n have "\<not> P (l n) (u n)"
- proof (induct n)
- case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
- qed (simp add: \<open>\<not> P a b\<close>) }
+ have "\<not> P (l n) (u n)" for n
+ proof (induct n)
+ case 0
+ then show ?case
+ by (simp add: \<open>\<not> P a b\<close>)
+ next
+ case (Suc n)
+ with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case
+ by auto
+ qed
moreover
- { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
+ {
+ have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
proof eventually_elim
- fix n assume "x - d / 2 < l n" "u n < x + d / 2"
+ case (elim n)
from add_strict_mono[OF this] have "u n - l n < d" by simp
with x show "P (l n) (u n)" by (rule d)
- qed }
+ qed
+ }
ultimately show False by simp
qed
qed
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
proof (cases "a \<le> b", rule compactI)
- fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
+ fix C
+ assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
define T where "T = {a .. b}"
from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
proof (induct rule: Bolzano)
case (trans a b c)
- then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
- from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
- by (auto simp: *)
+ then have *: "{a..c} = {a..b} \<union> {b..c}"
+ by auto
+ with trans obtain C1 C2
+ where "C1\<subseteq>C" "finite C1" "{a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C" "finite C2" "{b..c} \<subseteq> \<Union>C2"
+ by auto
with trans show ?case
unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
next
case (local x)
- then have "x \<in> \<Union>C" using C by auto
- with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
+ with C have "x \<in> \<Union>C" by auto
+ with C(2) obtain c where "x \<in> c" "open c" "c \<in> C"
+ by auto
then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
with \<open>c \<in> C\<close> show ?case
@@ -2378,17 +2524,18 @@
qed
lemma open_Collect_positive:
- fixes f :: "'a::t2_space \<Rightarrow> real"
- assumes f: "continuous_on s f"
- shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
- using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
- by (auto simp: Int_def field_simps)
+ fixes f :: "'a::t2_space \<Rightarrow> real"
+ assumes f: "continuous_on s f"
+ shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
+ using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
+ by (auto simp: Int_def field_simps)
lemma open_Collect_less_Int:
- fixes f g :: "'a::t2_space \<Rightarrow> real"
- assumes f: "continuous_on s f" and g: "continuous_on s g"
- shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
- using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
+ fixes f g :: "'a::t2_space \<Rightarrow> real"
+ assumes f: "continuous_on s f"
+ and g: "continuous_on s g"
+ shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
+ using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
subsection \<open>Boundedness of continuous functions\<close>
@@ -2399,14 +2546,14 @@
fixes f :: "real \<Rightarrow> 'a::linorder_topology"
shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
- using continuous_attains_sup[of "{a .. b}" f]
+ using continuous_attains_sup[of "{a..b}" f]
by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
lemma isCont_eq_Lb:
fixes f :: "real \<Rightarrow> 'a::linorder_topology"
shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
\<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
- using continuous_attains_inf[of "{a .. b}" f]
+ using continuous_attains_inf[of "{a..b}" f]
by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
lemma isCont_bounded:
@@ -2421,21 +2568,23 @@
using isCont_eq_Ub[of a b f] by auto
(*HOL style here: object-level formulations*)
-lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
- (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
- --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
+lemma IVT_objl:
+ "(f a \<le> y \<and> y \<le> f b \<and> a \<le> b \<and> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x)) \<longrightarrow>
+ (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y)"
+ for a y :: real
by (blast intro: IVT)
-lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
- (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
- --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
+lemma IVT2_objl:
+ "(f b \<le> y \<and> y \<le> f a \<and> a \<le> b \<and> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x)) \<longrightarrow>
+ (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y)"
+ for b y :: real
by (blast intro: IVT2)
lemma isCont_Lb_Ub:
fixes f :: "real \<Rightarrow> real"
assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
- (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
+ (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
proof -
obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
using isCont_eq_Ub[OF assms] by auto
@@ -2446,22 +2595,26 @@
apply (rule_tac x="f L" in exI)
apply (rule_tac x="f M" in exI)
apply (cases "L \<le> M")
- apply (simp, metis order_trans)
- apply (simp, metis order_trans)
+ apply simp
+ apply (metis order_trans)
+ apply simp
+ apply (metis order_trans)
done
qed
-text\<open>Continuity of inverse function\<close>
+text \<open>Continuity of inverse function.\<close>
lemma isCont_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes d: "0 < d"
- and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
- and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
+ and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
+ and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
shows "isCont g (f x)"
proof -
- let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
+ let ?A = "f (x - d)"
+ let ?B = "f (x + d)"
+ let ?D = "{x - d..x + d}"
have f: "continuous_on ?D f"
using cont by (intro continuous_at_imp_continuous_on ballI) auto
@@ -2483,45 +2636,42 @@
qed
lemma isCont_inverse_function2:
- fixes f g :: "real \<Rightarrow> real" shows
- "\<lbrakk>a < x; x < b;
- \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
- \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
- \<Longrightarrow> isCont g (f x)"
-apply (rule isCont_inverse_function
- [where f=f and d="min (x - a) (b - x)"])
-apply (simp_all add: abs_le_iff)
-done
+ fixes f g :: "real \<Rightarrow> real"
+ shows
+ "a < x \<Longrightarrow> x < b \<Longrightarrow>
+ \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z \<Longrightarrow>
+ \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z \<Longrightarrow> isCont g (f x)"
+ apply (rule isCont_inverse_function [where f=f and d="min (x - a) (b - x)"])
+ apply (simp_all add: abs_le_iff)
+ done
(* need to rename second isCont_inverse *)
-
lemma isCont_inv_fun:
fixes f g :: "real \<Rightarrow> real"
- shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
- \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
- ==> isCont g (f x)"
-by (rule isCont_inverse_function)
-
-text\<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110\<close>
-lemma LIM_fun_gt_zero:
- fixes f :: "real \<Rightarrow> real"
- shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
-apply (drule (1) LIM_D, clarify)
-apply (rule_tac x = s in exI)
-apply (simp add: abs_less_iff)
-done
-
-lemma LIM_fun_less_zero:
- fixes f :: "real \<Rightarrow> real"
- shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
-apply (drule LIM_D [where r="-l"], simp, clarify)
-apply (rule_tac x = s in exI)
-apply (simp add: abs_less_iff)
-done
-
-lemma LIM_fun_not_zero:
- fixes f :: "real \<Rightarrow> real"
- shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
+ shows "0 < d \<Longrightarrow> (\<forall>z. \<bar>z - x\<bar> \<le> d \<longrightarrow> g (f z) = z) \<Longrightarrow>
+ \<forall>z. \<bar>z - x\<bar> \<le> d \<longrightarrow> isCont f z \<Longrightarrow> isCont g (f x)"
+ by (rule isCont_inverse_function)
+
+text \<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110.\<close>
+lemma LIM_fun_gt_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
+ for f :: "real \<Rightarrow> real"
+ apply (drule (1) LIM_D)
+ apply clarify
+ apply (rule_tac x = s in exI)
+ apply (simp add: abs_less_iff)
+ done
+
+lemma LIM_fun_less_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
+ for f :: "real \<Rightarrow> real"
+ apply (drule LIM_D [where r="-l"])
+ apply simp
+ apply clarify
+ apply (rule_tac x = s in exI)
+ apply (simp add: abs_less_iff)
+ done
+
+lemma LIM_fun_not_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
+ for f :: "real \<Rightarrow> real"
using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
end