--- a/src/HOL/Library/Convex_Euclidean_Space.thy Fri Jun 12 00:45:28 2009 +0200
+++ b/src/HOL/Library/Convex_Euclidean_Space.thy Fri Jun 12 07:23:45 2009 +0200
@@ -1213,32 +1213,56 @@
qed
qed
+lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
+unfolding open_vector_def all_1
+by (auto simp add: dest_vec1_def)
+
+lemma tendsto_dest_vec1 [tendsto_intros]:
+ "(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
+ unfolding tendsto_def
+ apply clarify
+ apply (drule_tac x="dest_vec1 -` S" in spec)
+ apply (simp add: open_dest_vec1_vimage)
+ done
+
+lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
+ unfolding continuous_def by (rule tendsto_dest_vec1)
+
+(* TODO: move *)
+lemma compact_real_interval:
+ fixes a b :: real shows "compact {a..b}"
+proof -
+ have "continuous_on {vec1 a .. vec1 b} dest_vec1"
+ unfolding continuous_on
+ by (simp add: tendsto_dest_vec1 Lim_at_within Lim_ident_at)
+ moreover have "compact {vec1 a .. vec1 b}" by (rule compact_interval)
+ ultimately have "compact (dest_vec1 ` {vec1 a .. vec1 b})"
+ by (rule compact_continuous_image)
+ also have "dest_vec1 ` {vec1 a .. vec1 b} = {a..b}"
+ by (auto simp add: image_def Bex_def exists_vec1)
+ finally show ?thesis .
+qed
lemma compact_convex_combinations:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "(real ^ 'n::finite) set"
assumes "compact s" "compact t"
shows "compact { (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
proof-
- let ?X = "{ pastecart u w | u w. u \<in> {vec1 0 .. vec1 1} \<and> w \<in> { pastecart x y |x y. x \<in> s \<and> y \<in> t} }"
- let ?h = "(\<lambda>z. (1 - dest_vec1(fstcart z)) *s fstcart(sndcart z) + dest_vec1(fstcart z) *s sndcart(sndcart z))"
+ let ?X = "{0..1} \<times> s \<times> t"
+ let ?h = "(\<lambda>z. (1 - fst z) *s fst (snd z) + fst z *s snd (snd z))"
have *:"{ (1 - u) *s x + u *s y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
- apply(rule set_ext) unfolding image_iff mem_Collect_eq unfolding mem_interval_1 vec1_dest_vec1
- apply rule apply auto apply(rule_tac x="pastecart (vec1 u) (pastecart xa y)" in exI) apply simp
- apply(rule_tac x="vec1 u" in exI) apply(rule_tac x="pastecart xa y" in exI) by auto
- { fix u::"real^1" fix x y assume as:"0 \<le> dest_vec1 u" "dest_vec1 u \<le> 1" "x \<in> s" "y \<in> t"
- hence "continuous (at (pastecart u (pastecart x y)))
- (\<lambda>z. fstcart (sndcart z) - dest_vec1 (fstcart z) *s fstcart (sndcart z) +
- dest_vec1 (fstcart z) *s sndcart (sndcart z))"
- apply (auto intro!: continuous_add continuous_sub continuous_mul simp add: o_def vec1_dest_vec1)
- using linear_continuous_at linear_fstcart linear_sndcart linear_sndcart
- using linear_compose[unfolded o_def] by auto }
- hence "continuous_on {pastecart u w |u w. u \<in> {vec1 0..vec1 1} \<and> w \<in> {pastecart x y |x y. x \<in> s \<and> y \<in> t}}
- (\<lambda>z. (1 - dest_vec1 (fstcart z)) *s fstcart (sndcart z) + dest_vec1 (fstcart z) *s sndcart (sndcart z))"
- apply(rule_tac continuous_at_imp_continuous_on) unfolding mem_Collect_eq
- unfolding mem_interval_1 vec1_dest_vec1 by auto
- thus ?thesis unfolding * apply(rule compact_continuous_image)
- defer apply(rule compact_pastecart) defer apply(rule compact_pastecart)
- using compact_interval assms by auto
+ apply(rule set_ext) unfolding image_iff mem_Collect_eq
+ apply rule apply auto
+ apply (rule_tac x=u in rev_bexI, simp)
+ apply (erule rev_bexI, erule rev_bexI, simp)
+ by auto
+ have "continuous_on ({0..1} \<times> s \<times> t)
+ (\<lambda>z. (1 - fst z) *s fst (snd z) + fst z *s snd (snd z))"
+ unfolding continuous_on by (rule ballI) (intro tendsto_intros)
+ thus ?thesis unfolding *
+ apply (rule compact_continuous_image)
+ apply (intro compact_Times compact_real_interval assms)
+ done
qed
lemma compact_convex_hull: fixes s::"(real^'n::finite) set"
@@ -1888,7 +1912,9 @@
unfolding image_image[of "\<lambda>u. u *s x" "\<lambda>x. dest_vec1 x", THEN sym]
unfolding dest_vec1_inverval vec1_dest_vec1 by auto
have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
- apply(rule, rule continuous_vmul) unfolding o_def vec1_dest_vec1 apply(rule continuous_at_id) by(rule compact_interval)
+ apply(rule, rule continuous_vmul)
+ apply (rule continuous_dest_vec1)
+ apply(rule continuous_at_id) by(rule compact_interval)
moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *s x} \<inter> s \<noteq> {}" apply(rule not_disjointI[OF _ assms(2)])
unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *s x"
@@ -1925,12 +1951,13 @@
have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
- apply rule unfolding pi_def apply(rule continuous_mul) unfolding o_def
- apply(rule continuous_at_inv[unfolded o_def]) unfolding continuous_at_vec1_range[unfolded o_def]
- apply(rule,rule) apply(rule_tac x=e in exI) apply(rule,assumption,rule,rule)
- proof- fix e x y assume "0 < e" "norm (y - x::real^'n) < e"
- thus "\<bar>norm y - norm x\<bar> < e" using norm_triangle_ineq3[of y x] by auto
- qed(auto intro!:continuous_at_id)
+ apply rule unfolding pi_def
+ apply (rule continuous_mul)
+ apply (rule continuous_at_inv[unfolded o_def])
+ apply (rule continuous_at_norm)
+ apply simp
+ apply (rule continuous_at_id)
+ done
def sphere \<equiv> "{x::real^'n. norm x = 1}"
have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *s x) = pi x" unfolding pi_def sphere_def by auto
@@ -2015,7 +2042,7 @@
prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
fix x::"real^'n" assume as:"x \<in> cball 0 1"
thus "continuous (at x) (\<lambda>x. norm x *s surf (pi x))" proof(cases "x=0")
- case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_vec1_norm)
+ case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
next guess a using UNIV_witness[where 'a = 'n] ..
obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
@@ -2332,8 +2359,8 @@
lemma convex_on_bounded_continuous:
assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
- shows "continuous_on s (vec1 o f)"
- apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_vec1_range proof(rule,rule,rule)
+ shows "continuous_on s f"
+ apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
fix x e assume "x\<in>s" "(0::real) < e"
def B \<equiv> "abs b + 1"
have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
@@ -2398,7 +2425,7 @@
lemma convex_on_continuous:
assumes "open (s::(real^'n::finite) set)" "convex_on s f"
- shows "continuous_on s (vec1 \<circ> f)"
+ shows "continuous_on s f"
unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
note dimge1 = dimindex_ge_1[where 'a='n]
fix x assume "x\<in>s"
@@ -2428,9 +2455,9 @@
using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add: vector_component) }
thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm
by(auto simp add: vector_component_simps) qed
- hence "continuous_on (ball x d) (vec1 \<circ> f)" apply(rule_tac convex_on_bounded_continuous)
+ hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball) by auto
- thus "continuous (at x) (vec1 \<circ> f)" unfolding continuous_on_eq_continuous_at[OF open_ball] using `d>0` by auto qed
+ thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball] using `d>0` by auto qed
subsection {* Line segments, starlike sets etc. *)
(* Use the same overloading tricks as for intervals, so that *)
@@ -2975,7 +3002,8 @@
unfolding pathfinish_def linepath_def by auto
lemma continuous_linepath_at[intro]: "continuous (at x) (linepath a b)"
- unfolding linepath_def by(auto simp add: vec1_dest_vec1 o_def intro!: continuous_intros)
+ unfolding linepath_def
+ by (intro continuous_intros continuous_dest_vec1)
lemma continuous_on_linepath[intro]: "continuous_on s (linepath a b)"
using continuous_linepath_at by(auto intro!: continuous_at_imp_continuous_on)
@@ -3202,9 +3230,9 @@
have ***:"\<And>xa. (if xa = 0 then 0 else 1) \<noteq> 1 \<Longrightarrow> xa = 0" apply(rule ccontr) by auto
have **:"{x::real^'n. norm x = 1} = (\<lambda>x. (1/norm x) *s x) ` (UNIV - {0})" apply(rule set_ext,rule)
unfolding image_iff apply(rule_tac x=x in bexI) unfolding mem_Collect_eq norm_mul by(auto intro!: ***)
- have "continuous_on (UNIV - {0}) (vec1 \<circ> (\<lambda>x::real^'n. 1 / norm x))" unfolding o_def continuous_on_eq_continuous_within
+ have "continuous_on (UNIV - {0}) (\<lambda>x::real^'n. 1 / norm x)" unfolding o_def continuous_on_eq_continuous_within
apply(rule, rule continuous_at_within_inv[unfolded o_def inverse_eq_divide]) apply(rule continuous_at_within)
- apply(rule continuous_at_vec1_norm[unfolded o_def]) by auto
+ apply(rule continuous_at_norm[unfolded o_def]) by auto
thus ?thesis unfolding * ** using path_connected_punctured_universe[OF assms]
by(auto intro!: path_connected_continuous_image continuous_on_intros continuous_on_mul) qed
--- a/src/HOL/Library/Topology_Euclidean_Space.thy Fri Jun 12 00:45:28 2009 +0200
+++ b/src/HOL/Library/Topology_Euclidean_Space.thy Fri Jun 12 07:23:45 2009 +0200
@@ -6,7 +6,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Euclidean_Space
+imports SEQ Euclidean_Space Product_Vector
begin
declare fstcart_pastecart[simp] sndcart_pastecart[simp]
@@ -1245,6 +1245,9 @@
unfolding linear_conv_bounded_linear
by (rule bounded_linear.tendsto)
+lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
+ unfolding tendsto_def Limits.eventually_at_topological by fast
+
lemma Lim_const: "((\<lambda>x. a) ---> a) net"
by (rule tendsto_const)
@@ -1277,38 +1280,34 @@
lemma Lim_null_norm:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. vec1(norm(f x))) ---> 0) net"
- by (simp add: Lim dist_norm norm_vec1)
+ shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
+ by (simp add: Lim dist_norm)
lemma Lim_null_comparison:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "eventually (\<lambda>x. norm(f x) <= g x) net" "((\<lambda>x. vec1(g x)) ---> 0) net"
+ assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
shows "(f ---> 0) net"
proof(simp add: tendsto_iff, rule+)
fix e::real assume "0<e"
{ fix x
- assume "norm (f x) \<le> g x" "dist (vec1 (g x)) 0 < e"
- hence "dist (f x) 0 < e" unfolding vec_def using dist_vec1[of "g x" "0"]
- by (vector dist_norm norm_vec1 real_vector_norm_def dot_def vec1_def)
+ assume "norm (f x) \<le> g x" "dist (g x) 0 < e"
+ hence "dist (f x) 0 < e" by (simp add: dist_norm)
}
thus "eventually (\<lambda>x. dist (f x) 0 < e) net"
- using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (vec1 (g x)) 0 < e" net]
- using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (vec1 (g x)) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
+ using eventually_and[of "\<lambda>x. norm(f x) <= g x" "\<lambda>x. dist (g x) 0 < e" net]
+ using eventually_mono[of "(\<lambda>x. norm (f x) \<le> g x \<and> dist (g x) 0 < e)" "(\<lambda>x. dist (f x) 0 < e)" net]
using assms `e>0` unfolding tendsto_iff by auto
qed
-lemma Lim_component: "(f ---> l) net
- ==> ((\<lambda>a. vec1((f a :: real ^'n::finite)$i)) ---> vec1(l$i)) net"
+lemma Lim_component:
+ fixes f :: "'a \<Rightarrow> 'b::metric_space ^ 'n::finite"
+ shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $i) ---> l$i) net"
unfolding tendsto_iff
- apply (simp add: dist_norm vec1_sub[symmetric] norm_vec1 vector_minus_component[symmetric] del: vector_minus_component)
- apply (auto simp del: vector_minus_component)
- apply (erule_tac x=e in allE)
- apply clarify
- apply (erule eventually_rev_mono)
- apply (auto simp del: vector_minus_component)
- apply (rule order_le_less_trans)
- apply (rule component_le_norm)
- by auto
+ apply (clarify)
+ apply (drule spec, drule (1) mp)
+ apply (erule eventually_elim1)
+ apply (erule le_less_trans [OF dist_nth_le])
+ done
lemma Lim_transform_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
@@ -1504,12 +1503,6 @@
netlimit :: "'a::metric_space net \<Rightarrow> 'a" where
"netlimit net = (SOME a. \<forall>r>0. eventually (\<lambda>x. dist x a < r) net)"
-lemma dist_triangle3:
- fixes x y :: "'a::metric_space"
- shows "dist x y \<le> dist a x + dist a y"
-using dist_triangle2 [of x y a]
-by (simp add: dist_commute)
-
lemma netlimit_within:
assumes "\<not> trivial_limit (at a within S)"
shows "netlimit (at a within S) = a"
@@ -1694,14 +1687,14 @@
apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k \<and> (n - k) + k = n")
by metis arith
-lemma seq_harmonic: "((\<lambda>n. vec1(inverse (real n))) ---> 0) sequentially"
+lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
proof-
{ fix e::real assume "e>0"
hence "\<exists>N::nat. \<forall>n::nat\<ge>N. inverse (real n) < e"
using real_arch_inv[of e] apply auto apply(rule_tac x=n in exI)
- by (metis dlo_simps(4) le_imp_inverse_le linorder_not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
+ by (metis not_le le_imp_inverse_le not_less real_of_nat_gt_zero_cancel_iff real_of_nat_less_iff xt1(7))
}
- thus ?thesis unfolding Lim_sequentially dist_norm apply simp unfolding norm_vec1 by auto
+ thus ?thesis unfolding Lim_sequentially dist_norm by simp
qed
text{* More properties of closed balls. *}
@@ -1856,9 +1849,10 @@
apply (simp add: zero_less_dist_iff)
done
+(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball:
- fixes x :: "real ^ _" (* FIXME: generalize *)
- shows "interior(cball x e) = ball x e"
+ fixes x :: "'a::{real_normed_vector, perfect_space}"
+ shows "interior (cball x e) = ball x e"
proof(cases "e\<ge>0")
case False note cs = this
from cs have "ball x e = {}" using ball_empty[of e x] by auto moreover
@@ -1873,9 +1867,9 @@
{ fix S y assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
then obtain d where "d>0" and d:"\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S" unfolding open_dist by blast
- then obtain xa where xa:"dist y xa = d / 2" using vector_choose_dist[of "d/2" y] by auto
- hence xa_y:"xa \<noteq> y" using dist_nz[of y xa] using `d>0` by auto
- have "xa\<in>S" using d[THEN spec[where x=xa]] using xa apply(auto simp add: dist_commute) unfolding dist_nz[THEN sym] using xa_y by auto
+ then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
+ using perfect_choose_dist [of d] by auto
+ have "xa\<in>S" using d[THEN spec[where x=xa]] using xa by(auto simp add: dist_commute)
hence xa_cball:"xa \<in> cball x e" using as(1) by auto
hence "y \<in> ball x e" proof(cases "x = y")
@@ -1884,18 +1878,19 @@
thus "y \<in> ball x e" using `x = y ` by simp
next
case False
- have "dist (y + (d / 2 / dist y x) *s (y - x)) y < d" unfolding dist_norm
- using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by auto
- hence *:"y + (d / 2 / dist y x) *s (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
+ have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
+ using `d>0` norm_ge_zero[of "y - x"] `x \<noteq> y` by (auto simp add: norm_scaleR)
+ hence *:"y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e" using d as(1)[unfolded subset_eq] by blast
have "y - x \<noteq> 0" using `x \<noteq> y` by auto
hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
using `d>0` divide_pos_pos[of d "2*norm (y - x)"] by auto
- have "dist (y + (d / 2 / dist y x) *s (y - x)) x = norm (y + (d / (2 * norm (y - x))) *s y - (d / (2 * norm (y - x))) *s x - x)"
- by (auto simp add: dist_norm vector_ssub_ldistrib add_diff_eq)
- also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *s (y - x))"
- by (auto simp add: vector_sadd_rdistrib vector_smult_lid ring_simps vector_sadd_rdistrib vector_ssub_ldistrib)
- also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)" using ** by auto
+ have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x = norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
+ by (auto simp add: dist_norm algebra_simps)
+ also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
+ by (auto simp add: algebra_simps)
+ also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
+ using ** by (auto simp add: norm_scaleR)
also have "\<dots> = (dist y x) + d/2"using ** by (auto simp add: left_distrib dist_norm)
finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
thus "y \<in> ball x e" unfolding mem_ball using `d>0` by auto
@@ -1905,14 +1900,14 @@
qed
lemma frontier_ball:
- fixes a :: "real ^ _" (* FIXME: generalize *)
+ fixes a :: "'a::real_normed_vector"
shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
apply (simp add: frontier_def closure_ball interior_open open_ball order_less_imp_le)
apply (simp add: expand_set_eq)
by arith
lemma frontier_cball:
- fixes a :: "real ^ _" (* FIXME: generalize *)
+ fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "frontier(cball a e) = {x. dist a x = e}"
apply (simp add: frontier_def interior_cball closed_cball closure_closed order_less_imp_le)
apply (simp add: expand_set_eq)
@@ -1924,20 +1919,20 @@
lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)
lemma cball_eq_sing:
- fixes x :: "real ^ _" (* FIXME: generalize *)
+ fixes x :: "'a::perfect_space"
shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
-proof-
- { assume as:"\<forall>xa. (dist x xa \<le> e) = (xa = x)"
- hence "e \<ge> 0" apply (erule_tac x=x in allE) by auto
- then obtain y where y:"dist x y = e" using vector_choose_dist[of e] by auto
- hence "e = 0" using as apply(erule_tac x=y in allE) by auto
- }
- thus ?thesis unfolding expand_set_eq mem_cball by (auto simp add: dist_nz)
-qed
+proof (rule linorder_cases)
+ assume e: "0 < e"
+ obtain a where "a \<noteq> x" "dist a x < e"
+ using perfect_choose_dist [OF e] by auto
+ hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
+ with e show ?thesis by (auto simp add: expand_set_eq)
+qed auto
lemma cball_sing:
- fixes x :: "real ^ _" (* FIXME: generalize *)
- shows "e = 0 ==> cball x e = {x}" by (simp add: cball_eq_sing)
+ fixes x :: "'a::metric_space"
+ shows "e = 0 ==> cball x e = {x}"
+ by (auto simp add: expand_set_eq)
text{* For points in the interior, localization of limits makes no difference. *}
@@ -2123,26 +2118,26 @@
text{* Some theorems on sups and infs using the notion "bounded". *}
-lemma bounded_vec1:
+lemma bounded_real:
fixes S :: "real set"
- shows "bounded(vec1 ` S) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
- by (simp add: bounded_iff forall_vec1 norm_vec1 vec1_in_image_vec1)
-
-lemma bounded_has_rsup: assumes "bounded(vec1 ` S)" "S \<noteq> {}"
+ shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x <= a)"
+ by (simp add: bounded_iff)
+
+lemma bounded_has_rsup: assumes "bounded S" "S \<noteq> {}"
shows "\<forall>x\<in>S. x <= rsup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> rsup S <= b"
proof
fix x assume "x\<in>S"
- from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_vec1 by auto
+ from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
hence *:"S *<= a" using setleI[of S a] by (metis abs_le_interval_iff mem_def)
- thus "x \<le> rsup S" using rsup[OF `S\<noteq>{}`] using assms(1)[unfolded bounded_vec1] using isLubD2[of UNIV S "rsup S" x] using `x\<in>S` by auto
+ thus "x \<le> rsup S" using rsup[OF `S\<noteq>{}`] using assms(1)[unfolded bounded_real] using isLubD2[of UNIV S "rsup S" x] using `x\<in>S` by auto
next
show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> rsup S \<le> b" using assms
using rsup[of S, unfolded isLub_def isUb_def leastP_def setle_def setge_def]
- apply (auto simp add: bounded_vec1)
+ apply (auto simp add: bounded_real)
by (auto simp add: isLub_def isUb_def leastP_def setle_def setge_def)
qed
-lemma rsup_insert: assumes "bounded (vec1 ` S)"
+lemma rsup_insert: assumes "bounded S"
shows "rsup(insert x S) = (if S = {} then x else max x (rsup S))"
proof(cases "S={}")
case True thus ?thesis using rsup_finite_in[of "{x}"] by auto
@@ -2168,17 +2163,17 @@
by simp
lemma bounded_has_rinf:
- assumes "bounded(vec1 ` S)" "S \<noteq> {}"
+ assumes "bounded S" "S \<noteq> {}"
shows "\<forall>x\<in>S. x >= rinf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S >= b"
proof
fix x assume "x\<in>S"
- from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_vec1 by auto
+ from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
hence *:"- a <=* S" using setgeI[of S "-a"] unfolding abs_le_interval_iff by auto
thus "x \<ge> rinf S" using rinf[OF `S\<noteq>{}`] using isGlbD2[of UNIV S "rinf S" x] using `x\<in>S` by auto
next
show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> rinf S \<ge> b" using assms
using rinf[of S, unfolded isGlb_def isLb_def greatestP_def setle_def setge_def]
- apply (auto simp add: bounded_vec1)
+ apply (auto simp add: bounded_real)
by (auto simp add: isGlb_def isLb_def greatestP_def setle_def setge_def)
qed
@@ -2189,7 +2184,7 @@
apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
done
-lemma rinf_insert: assumes "bounded (vec1 ` S)"
+lemma rinf_insert: assumes "bounded S"
shows "rinf(insert x S) = (if S = {} then x else min x (rinf S))" (is "?lhs = ?rhs")
proof(cases "S={}")
case True thus ?thesis using rinf_finite_in[of "{x}"] by auto
@@ -2217,8 +2212,8 @@
definition
compact :: "'a::metric_space set \<Rightarrow> bool" where (* TODO: generalize *)
"compact S \<longleftrightarrow>
- (\<forall>f. (\<forall>n::nat. f n \<in> S) \<longrightarrow>
- (\<exists>l\<in>S. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f o r) ---> l) sequentially))"
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
+ (\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"
text {*
A metric space (or topological vector space) is said to have the
@@ -2227,44 +2222,43 @@
class heine_borel =
assumes bounded_imp_convergent_subsequence:
- "bounded s \<Longrightarrow> \<forall>n::nat. f n \<in> s
- \<Longrightarrow> \<exists>l r. (\<forall>m n. m < n --> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially"
+ "bounded s \<Longrightarrow> \<forall>n. f n \<in> s
+ \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
lemma bounded_closed_imp_compact:
fixes s::"'a::heine_borel set"
assumes "bounded s" and "closed s" shows "compact s"
proof (unfold compact_def, clarify)
fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
- obtain l r where r: "\<forall>m n. m < n \<longrightarrow> r m < r n" and l: "((f \<circ> r) ---> l) sequentially"
+ obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
using bounded_imp_convergent_subsequence [OF `bounded s` `\<forall>n. f n \<in> s`] by auto
from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
have "l \<in> s" using `closed s` fr l
unfolding closed_sequential_limits by blast
- show "\<exists>l\<in>s. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially"
+ show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
using `l \<in> s` r l by blast
qed
-lemma monotone_bigger: fixes r::"nat\<Rightarrow>nat"
- assumes "\<forall>m n::nat. m < n --> r m < r n"
- shows "n \<le> r n"
+lemma subseq_bigger: assumes "subseq r" shows "n \<le> r n"
proof(induct n)
show "0 \<le> r 0" by auto
next
fix n assume "n \<le> r n"
- moreover have "r n < r (Suc n)" using assms by auto
+ moreover have "r n < r (Suc n)"
+ using assms [unfolded subseq_def] by auto
ultimately show "Suc n \<le> r (Suc n)" by auto
qed
-lemma eventually_subsequence:
- assumes r: "\<forall>m n. m < n \<longrightarrow> r m < r n"
+lemma eventually_subseq:
+ assumes r: "subseq r"
shows "eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
unfolding eventually_sequentially
-by (metis monotone_bigger [OF r] le_trans)
-
-lemma lim_subsequence:
- fixes l :: "'a::metric_space" (* TODO: generalize *)
- shows "\<forall>m n. m < n \<longrightarrow> r m < r n \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
-unfolding Lim_sequentially by (simp, metis monotone_bigger le_trans)
+by (metis subseq_bigger [OF r] le_trans)
+
+lemma lim_subseq:
+ "subseq r \<Longrightarrow> (s ---> l) sequentially \<Longrightarrow> ((s o r) ---> l) sequentially"
+unfolding tendsto_def eventually_sequentially o_def
+by (metis subseq_bigger le_trans)
lemma num_Axiom: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
unfolding Ex1_def
@@ -2280,9 +2274,8 @@
apply (simp)
done
-
lemma convergent_bounded_increasing: fixes s ::"nat\<Rightarrow>real"
- assumes "\<forall>m n. m \<le> n --> s m \<le> s n" and "\<forall>n. abs(s n) \<le> b"
+ assumes "incseq s" and "\<forall>n. abs(s n) \<le> b"
shows "\<exists> l. \<forall>e::real>0. \<exists> N. \<forall>n \<ge> N. abs(s n - l) < e"
proof-
have "isUb UNIV (range s) b" using assms(2) and abs_le_D1 unfolding isUb_def and setle_def by auto
@@ -2292,27 +2285,27 @@
obtain N where "N\<ge>n" and n:"\<bar>s N - t\<bar> \<ge> e" using as[THEN spec[where x=n]] by auto
have "t \<ge> s N" using isLub_isUb[OF t, unfolded isUb_def setle_def] by auto
with n have "s N \<le> t - e" using `e>0` by auto
- hence "s n \<le> t - e" using assms(1)[THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto }
+ hence "s n \<le> t - e" using assms(1)[unfolded incseq_def, THEN spec[where x=n], THEN spec[where x=N]] using `n\<le>N` by auto }
hence "isUb UNIV (range s) (t - e)" unfolding isUb_def and setle_def by auto
hence False using isLub_le_isUb[OF t, of "t - e"] and `e>0` by auto }
thus ?thesis by blast
qed
lemma convergent_bounded_monotone: fixes s::"nat \<Rightarrow> real"
- assumes "\<forall>n. abs(s n) \<le> b" and "(\<forall>m n. m \<le> n --> s m \<le> s n) \<or> (\<forall>m n. m \<le> n --> s n \<le> s m)"
+ assumes "\<forall>n. abs(s n) \<le> b" and "monoseq s"
shows "\<exists>l. \<forall>e::real>0. \<exists>N. \<forall>n\<ge>N. abs(s n - l) < e"
using convergent_bounded_increasing[of s b] assms using convergent_bounded_increasing[of "\<lambda>n. - s n" b]
+ unfolding monoseq_def incseq_def
apply auto unfolding minus_add_distrib[THEN sym, unfolded diff_minus[THEN sym]]
unfolding abs_minus_cancel by(rule_tac x="-l" in exI)auto
lemma compact_real_lemma:
assumes "\<forall>n::nat. abs(s n) \<le> b"
- shows "\<exists>(l::real) r. (\<forall>m n::nat. m < n --> r m < r n) \<and> ((s \<circ> r) ---> l) sequentially"
+ shows "\<exists>(l::real) r. subseq r \<and> ((s \<circ> r) ---> l) sequentially"
proof-
- obtain r where r:"\<forall>m n::nat. m < n \<longrightarrow> r m < r n"
- "(\<forall>m n. m \<le> n \<longrightarrow> s (r m) \<le> s (r n)) \<or> (\<forall>m n. m \<le> n \<longrightarrow> s (r n) \<le> s (r m))"
- using seq_monosub[of s] by (auto simp add: subseq_def monoseq_def)
- thus ?thesis using convergent_bounded_monotone[of "s o r" b] and assms
+ obtain r where r:"subseq r" "monoseq (\<lambda>n. s (r n))"
+ using seq_monosub[of s] by auto
+ thus ?thesis using convergent_bounded_monotone[of "\<lambda>n. s (r n)" b] and assms
unfolding tendsto_iff dist_norm eventually_sequentially by auto
qed
@@ -2323,9 +2316,9 @@
then obtain b where b: "\<forall>n. abs (f n) \<le> b"
unfolding bounded_iff by auto
obtain l :: real and r :: "nat \<Rightarrow> nat" where
- r: "\<forall>m n. m < n \<longrightarrow> r m < r n" and l: "((f \<circ> r) ---> l) sequentially"
+ r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
using compact_real_lemma [OF b] by auto
- thus "\<exists>l r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially"
+ thus "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
by auto
qed
@@ -2342,28 +2335,29 @@
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n::finite"
assumes "bounded s" and "\<forall>n. f n \<in> s"
shows "\<forall>d.
- \<exists>l r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
+ \<exists>l r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
proof
fix d::"'n set" have "finite d" by simp
- thus "\<exists>l::'a ^ 'n. \<exists>r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
+ thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
- proof(induct d) case empty thus ?case by auto
+ proof(induct d) case empty thus ?case unfolding subseq_def by auto
next case (insert k d)
have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component)
- obtain l1::"'a^'n" and r1 where r1:"\<forall>n m::nat. m < n \<longrightarrow> r1 m < r1 n" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
+ obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
using insert(3) by auto
have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
- obtain l2 r2 where r2:"\<forall>m n::nat. m < n \<longrightarrow> r2 m < r2 n" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
+ obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
- def r \<equiv> "r1 \<circ> r2" have r:"\<forall>m n. m < n \<longrightarrow> r m < r n" unfolding r_def o_def using r1 and r2 by auto
+ def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
+ using r1 and r2 unfolding r_def o_def subseq_def by auto
moreover
def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
{ fix e::real assume "e>0"
from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
- by (rule eventually_subsequence)
+ by (rule eventually_subseq)
have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
using N1' N2 by (rule eventually_elim2, simp add: l_def r_def)
}
@@ -2375,7 +2369,7 @@
proof
fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
- then obtain l r where r: "\<forall>n m::nat. m < n --> r m < r n"
+ then obtain l r where r: "subseq r"
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
using compact_lemma [OF s f] by blast
let ?d = "UNIV::'b set"
@@ -2396,7 +2390,55 @@
by (rule eventually_elim1)
}
hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
- with r show "\<exists>l r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f \<circ> r) ---> l) sequentially" by auto
+ with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
+qed
+
+lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
+unfolding bounded_def
+apply clarify
+apply (rule_tac x="a" in exI)
+apply (rule_tac x="e" in exI)
+apply clarsimp
+apply (drule (1) bspec)
+apply (simp add: dist_Pair_Pair)
+apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
+done
+
+lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
+unfolding bounded_def
+apply clarify
+apply (rule_tac x="b" in exI)
+apply (rule_tac x="e" in exI)
+apply clarsimp
+apply (drule (1) bspec)
+apply (simp add: dist_Pair_Pair)
+apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
+done
+
+instance "*" :: (heine_borel, heine_borel) heine_borel
+proof
+ fix s :: "('a * 'b) set" and f :: "nat \<Rightarrow> 'a * 'b"
+ assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
+ from s have s1: "bounded (fst ` s)" by (rule bounded_fst)
+ from f have f1: "\<forall>n. fst (f n) \<in> fst ` s" by simp
+ obtain l1 r1 where r1: "subseq r1"
+ and l1: "((\<lambda>n. fst (f (r1 n))) ---> l1) sequentially"
+ using bounded_imp_convergent_subsequence [OF s1 f1]
+ unfolding o_def by fast
+ from s have s2: "bounded (snd ` s)" by (rule bounded_snd)
+ from f have f2: "\<forall>n. snd (f (r1 n)) \<in> snd ` s" by simp
+ obtain l2 r2 where r2: "subseq r2"
+ and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
+ using bounded_imp_convergent_subsequence [OF s2 f2]
+ unfolding o_def by fast
+ have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
+ using lim_subseq [OF r2 l1] unfolding o_def .
+ have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
+ using tendsto_Pair [OF l1' l2] unfolding o_def by simp
+ have r: "subseq (r1 \<circ> r2)"
+ using r1 r2 unfolding subseq_def by simp
+ show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
+ using l r by fast
qed
subsection{* Completeness. *}
@@ -2461,14 +2503,9 @@
lemma compact_imp_complete: assumes "compact s" shows "complete s"
proof-
{ fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
- from as(1) obtain l r where lr: "l\<in>s" "(\<forall>m n. m < n \<longrightarrow> r m < r n)" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast
-
- { fix n :: nat have lr':"n \<le> r n"
- proof (induct n)
- show "0 \<le> r 0" using lr(2) by blast
- next fix na assume "na \<le> r na" moreover have "na < Suc na \<longrightarrow> r na < r (Suc na)" using lr(2) by blast
- ultimately show "Suc na \<le> r (Suc na)" by auto
- qed } note lr' = this
+ from as(1) obtain l r where lr: "l\<in>s" "subseq r" "((f \<circ> r) ---> l) sequentially" using assms unfolding compact_def by blast
+
+ note lr' = subseq_bigger [OF lr(2)]
{ fix e::real assume "e>0"
from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using `e>0` apply (erule_tac x="e/2" in allE) by auto
@@ -2575,12 +2612,12 @@
thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
qed }
hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
- then obtain l r where "l\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto
+ then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded compact_def, THEN spec[where x=x]] by auto
from this(3) have "Cauchy (x \<circ> r)" using convergent_imp_cauchy by auto
then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using `e>0` by auto
show False
using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
- using r[THEN spec[where x=N], THEN spec[where x="N+1"]]
+ using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
qed
@@ -2599,7 +2636,7 @@
then obtain f where f:"\<forall>n::nat. f n \<in> s \<and> (\<forall>xa\<in>t. \<not> ball (f n) (inverse (real (n + 1))) \<subseteq> xa)"
using choice[of "\<lambda>n::nat. \<lambda>x. x\<in>s \<and> (\<forall>xa\<in>t. \<not> ball x (inverse (real (n + 1))) \<subseteq> xa)"] by auto
- then obtain l r where l:"l\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"((f \<circ> r) ---> l) sequentially"
+ then obtain l r where l:"l\<in>s" and r:"subseq r" and lr:"((f \<circ> r) ---> l) sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x=f]] by auto
obtain b where "l\<in>b" "b\<in>t" using assms(2) and l by auto
@@ -2612,7 +2649,7 @@
obtain N2::nat where N2:"N2>0" "inverse (real N2) < e /2" using real_arch_inv[of "e/2"] and `e>0` by auto
have N2':"inverse (real (r (N1 + N2) +1 )) < e/2"
apply(rule order_less_trans) apply(rule less_imp_inverse_less) using N2
- using monotone_bigger[OF r, of "N1 + N2"] by auto
+ using subseq_bigger[OF r, of "N1 + N2"] by auto
def x \<equiv> "(f (r (N1 + N2)))"
have x:"\<not> ball x (inverse (real (r (N1 + N2) + 1))) \<subseteq> b" unfolding x_def
@@ -2926,7 +2963,7 @@
by blast
lemma compact_sing [simp]: "compact {a}"
- unfolding compact_def o_def
+ unfolding compact_def o_def subseq_def
by (auto simp add: tendsto_const)
lemma compact_cball[simp]:
@@ -2987,7 +3024,7 @@
from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
- then obtain l r where lr:"l\<in>s 0" "\<forall>m n. m < n \<longrightarrow> r m < r n" "((x \<circ> r) ---> l) sequentially"
+ then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
unfolding compact_def apply(erule_tac x=x in allE) using x using assms(3) by blast
{ fix n::nat
@@ -2995,7 +3032,7 @@
with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding Lim_sequentially by auto
hence "dist ((x \<circ> r) (max N n)) l < e" by auto
moreover
- have "r (max N n) \<ge> n" using lr(2) using monotone_bigger[of r "max N n"] by auto
+ have "r (max N n) \<ge> n" using lr(2) using subseq_bigger[of r "max N n"] by auto
hence "(x \<circ> r) (max N n) \<in> s n"
using x apply(erule_tac x=n in allE)
using x apply(erule_tac x="r (max N n)" in allE)
@@ -3701,13 +3738,11 @@
qed
lemma continuous_open_preimage_univ:
- fixes f :: "real ^ _ \<Rightarrow> real ^ _" (* FIXME: generalize *)
- shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
+ "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto
lemma continuous_closed_preimage_univ:
- fixes f :: "real ^ _ \<Rightarrow> real ^ _" (* FIXME: generalize *)
- shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
+ "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto
text{* Equality of continuous functions on closure and related results. *}
@@ -3830,7 +3865,7 @@
unfolding vector_sneg_minus1 by auto
lemma open_translation:
- fixes s :: "(real ^ _) set" (* FIXME: generalize *)
+ fixes s :: "'a::real_normed_vector set"
assumes "open s" shows "open((\<lambda>x. a + x) ` s)"
proof-
{ fix x have "continuous (at x) (\<lambda>x. x - a)" using continuous_sub[of "at x" "\<lambda>x. x" "\<lambda>x. a"] continuous_at_id[of x] continuous_const[of "at x" a] by auto }
@@ -3875,13 +3910,13 @@
proof-
{ fix x assume x:"\<forall>n::nat. x n \<in> f ` s"
then obtain y where y:"\<forall>n. y n \<in> s \<and> x n = f (y n)" unfolding image_iff Bex_def using choice[of "\<lambda>n xa. xa \<in> s \<and> x n = f xa"] by auto
- then obtain l r where "l\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
+ then obtain l r where "l\<in>s" and r:"subseq r" and lr:"((y \<circ> r) ---> l) sequentially" using assms(2)[unfolded compact_def, THEN spec[where x=y]] by auto
{ fix e::real assume "e>0"
then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' l < d \<longrightarrow> dist (f x') (f l) < e" using assms(1)[unfolded continuous_on_def, THEN bspec[where x=l], OF `l\<in>s`] by auto
then obtain N::nat where N:"\<forall>n\<ge>N. dist ((y \<circ> r) n) l < d" using lr[unfolded Lim_sequentially, THEN spec[where x=d]] by auto
{ fix n::nat assume "n\<ge>N" hence "dist ((x \<circ> r) n) (f l) < e" using N[THEN spec[where x=n]] d[THEN bspec[where x="y (r n)"]] y[THEN spec[where x="r n"]] by auto }
hence "\<exists>N. \<forall>n\<ge>N. dist ((x \<circ> r) n) (f l) < e" by auto }
- hence "\<exists>l\<in>f ` s. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((x \<circ> r) ---> l) sequentially" unfolding Lim_sequentially using r lr `l\<in>s` by auto }
+ hence "\<exists>l\<in>f ` s. \<exists>r. subseq r \<and> ((x \<circ> r) ---> l) sequentially" unfolding Lim_sequentially using r lr `l\<in>s` by auto }
thus ?thesis unfolding compact_def by auto
qed
@@ -3938,7 +3973,8 @@
text{* Continuity of inverse function on compact domain. *}
lemma continuous_on_inverse:
- fixes f :: "real ^ _ \<Rightarrow> real ^ _"
+ fixes f :: "'a::heine_borel \<Rightarrow> 'b::heine_borel"
+ (* TODO: can this be generalized more? *)
assumes "continuous_on s f" "compact s" "\<forall>x \<in> s. g (f x) = x"
shows "continuous_on (f ` s) g"
proof-
@@ -4050,68 +4086,52 @@
subsection{* Topological stuff lifted from and dropped to R *}
-lemma open_vec1:
- fixes s :: "real set" shows
- "open(vec1 ` s) \<longleftrightarrow>
- (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
- unfolding open_dist apply simp unfolding forall_vec1 dist_vec1 vec1_in_image_vec1 by simp
-
-lemma islimpt_approachable_vec1:
+lemma open_real:
fixes s :: "real set" shows
- "(vec1 x) islimpt (vec1 ` s) \<longleftrightarrow>
- (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
- by (auto simp add: islimpt_approachable dist_vec1 vec1_eq)
-
-lemma closed_vec1:
- fixes s :: "real set" shows
- "closed (vec1 ` s) \<longleftrightarrow>
+ "open s \<longleftrightarrow>
+ (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
+ unfolding open_dist dist_norm by simp
+
+lemma islimpt_approachable_real:
+ fixes s :: "real set"
+ shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> abs(x' - x) < e)"
+ unfolding islimpt_approachable dist_norm by simp
+
+lemma closed_real:
+ fixes s :: "real set"
+ shows "closed s \<longleftrightarrow>
(\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> abs(x' - x) < e)
--> x \<in> s)"
- unfolding closed_limpt islimpt_approachable forall_vec1 apply simp
- unfolding dist_vec1 vec1_in_image_vec1 abs_minus_commute by auto
-
-lemma continuous_at_vec1_range:
- fixes f :: "real ^ _ \<Rightarrow> real"
- shows "continuous (at x) (vec1 o f) \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
+ unfolding closed_limpt islimpt_approachable dist_norm by simp
+
+lemma continuous_at_real_range:
+ fixes f :: "'a::real_normed_vector \<Rightarrow> real"
+ shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
\<forall>x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
- unfolding continuous_at unfolding Lim_at apply simp unfolding dist_vec1 unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
+ unfolding continuous_at unfolding Lim_at
+ unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
apply(erule_tac x=e in allE) by auto
-lemma continuous_on_vec1_range:
+lemma continuous_on_real_range:
fixes f :: "'a::real_normed_vector \<Rightarrow> real"
- shows "continuous_on s (vec1 o f) \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
- unfolding continuous_on_def apply (simp del: dist_commute) unfolding dist_vec1 unfolding dist_norm ..
-
-lemma continuous_at_vec1_norm:
- fixes x :: "real ^ _"
- shows "continuous (at x) (vec1 o norm)"
- unfolding continuous_at_vec1_range using real_abs_sub_norm order_le_less_trans by blast
-
-lemma continuous_on_vec1_norm:
- fixes s :: "(real ^ _) set"
- shows "continuous_on s (vec1 o norm)"
-unfolding continuous_on_vec1_range norm_vec1[THEN sym] by (metis norm_vec1 order_le_less_trans real_abs_sub_norm)
-
-lemma continuous_at_vec1_component:
- shows "continuous (at (a::real^'a::finite)) (\<lambda> x. vec1(x$i))"
-proof-
- { fix e::real and x assume "0 < dist x a" "dist x a < e" "e>0"
- hence "\<bar>x $ i - a $ i\<bar> < e" using component_le_norm[of "x - a" i] unfolding dist_norm by auto }
- thus ?thesis unfolding continuous_at tendsto_iff eventually_at dist_vec1 by auto
-qed
-
-lemma continuous_on_vec1_component:
- shows "continuous_on s (\<lambda> x::real^'a::finite. vec1(x$i))"
-proof-
- { fix e::real and x xa assume "x\<in>s" "e>0" "xa\<in>s" "0 < norm (xa - x) \<and> norm (xa - x) < e"
- hence "\<bar>xa $ i - x $ i\<bar> < e" using component_le_norm[of "xa - x" i] by auto }
- thus ?thesis unfolding continuous_on Lim_within dist_vec1 unfolding dist_norm by auto
-qed
-
-lemma continuous_at_vec1_infnorm:
- "continuous (at x) (vec1 o infnorm)"
- unfolding continuous_at Lim_at o_def unfolding dist_vec1 unfolding dist_norm
+ shows "continuous_on s f \<longleftrightarrow> (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d --> abs(f x' - f x) < e))"
+ unfolding continuous_on_def dist_norm by simp
+
+lemma continuous_at_norm: "continuous (at x) norm"
+ unfolding continuous_at by (intro tendsto_intros)
+
+lemma continuous_on_norm: "continuous_on s norm"
+unfolding continuous_on by (intro ballI tendsto_intros)
+
+lemma continuous_at_component: "continuous (at a) (\<lambda>x. x $ i)"
+unfolding continuous_at by (intro tendsto_intros)
+
+lemma continuous_on_component: "continuous_on s (\<lambda>x. x $ i)"
+unfolding continuous_on by (intro ballI tendsto_intros)
+
+lemma continuous_at_infnorm: "continuous (at x) infnorm"
+ unfolding continuous_at Lim_at o_def unfolding dist_norm
apply auto apply (rule_tac x=e in exI) apply auto
using order_trans[OF real_abs_sub_infnorm infnorm_le_norm, of _ x] by (metis xt1(7))
@@ -4119,23 +4139,23 @@
lemma compact_attains_sup:
fixes s :: "real set"
- assumes "compact (vec1 ` s)" "s \<noteq> {}"
+ assumes "compact s" "s \<noteq> {}"
shows "\<exists>x \<in> s. \<forall>y \<in> s. y \<le> x"
proof-
- from assms(1) have a:"bounded (vec1 ` s)" "closed (vec1 ` s)" unfolding compact_eq_bounded_closed by auto
+ from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
{ fix e::real assume as: "\<forall>x\<in>s. x \<le> rsup s" "rsup s \<notin> s" "0 < e" "\<forall>x'\<in>s. x' = rsup s \<or> \<not> rsup s - x' < e"
have "isLub UNIV s (rsup s)" using rsup[OF assms(2)] unfolding setle_def using as(1) by auto
moreover have "isUb UNIV s (rsup s - e)" unfolding isUb_def unfolding setle_def using as(4,2) by auto
ultimately have False using isLub_le_isUb[of UNIV s "rsup s" "rsup s - e"] using `e>0` by auto }
- thus ?thesis using bounded_has_rsup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_vec1, THEN spec[where x="rsup s"]]
+ thus ?thesis using bounded_has_rsup(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rsup s"]]
apply(rule_tac x="rsup s" in bexI) by auto
qed
lemma compact_attains_inf:
fixes s :: "real set"
- assumes "compact (vec1 ` s)" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
+ assumes "compact s" "s \<noteq> {}" shows "\<exists>x \<in> s. \<forall>y \<in> s. x \<le> y"
proof-
- from assms(1) have a:"bounded (vec1 ` s)" "closed (vec1 ` s)" unfolding compact_eq_bounded_closed by auto
+ from assms(1) have a:"bounded s" "closed s" unfolding compact_eq_bounded_closed by auto
{ fix e::real assume as: "\<forall>x\<in>s. x \<ge> rinf s" "rinf s \<notin> s" "0 < e"
"\<forall>x'\<in>s. x' = rinf s \<or> \<not> abs (x' - rinf s) < e"
have "isGlb UNIV s (rinf s)" using rinf[OF assms(2)] unfolding setge_def using as(1) by auto
@@ -4145,43 +4165,40 @@
have "rinf s + e \<le> x" using as(4)[THEN bspec[where x=x]] using as(2) `x\<in>s` unfolding * by auto }
hence "isLb UNIV s (rinf s + e)" unfolding isLb_def and setge_def by auto
ultimately have False using isGlb_le_isLb[of UNIV s "rinf s" "rinf s + e"] using `e>0` by auto }
- thus ?thesis using bounded_has_rinf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_vec1, THEN spec[where x="rinf s"]]
+ thus ?thesis using bounded_has_rinf(1)[OF a(1) assms(2)] using a(2)[unfolded closed_real, THEN spec[where x="rinf s"]]
apply(rule_tac x="rinf s" in bexI) by auto
qed
lemma continuous_attains_sup:
fixes f :: "'a::metric_space \<Rightarrow> real"
- shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s (vec1 o f)
+ shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
==> (\<exists>x \<in> s. \<forall>y \<in> s. f y \<le> f x)"
using compact_attains_sup[of "f ` s"]
- using compact_continuous_image[of s "vec1 \<circ> f"] unfolding image_compose by auto
+ using compact_continuous_image[of s f] by auto
lemma continuous_attains_inf:
fixes f :: "'a::metric_space \<Rightarrow> real"
- shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s (vec1 o f)
+ shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f
\<Longrightarrow> (\<exists>x \<in> s. \<forall>y \<in> s. f x \<le> f y)"
using compact_attains_inf[of "f ` s"]
- using compact_continuous_image[of s "vec1 \<circ> f"] unfolding image_compose by auto
+ using compact_continuous_image[of s f] by auto
lemma distance_attains_sup:
- fixes s :: "(real ^ _) set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a y \<le> dist a x"
-proof-
- { fix x assume "x\<in>s" fix e::real assume "e>0"
- { fix x' assume "x'\<in>s" and as:"norm (x' - x) < e"
- hence "\<bar>norm (x' - a) - norm (x - a)\<bar> < e"
- using real_abs_sub_norm[of "x' - a" "x - a"] by auto }
- hence "\<exists>d>0. \<forall>x'\<in>s. norm (x' - x) < d \<longrightarrow> \<bar>dist x' a - dist x a\<bar> < e" using `e>0` unfolding dist_norm by auto }
- thus ?thesis using assms
- using continuous_attains_sup[of s "\<lambda>x. dist a x"]
- unfolding continuous_on_vec1_range by (auto simp add: dist_commute)
+proof (rule continuous_attains_sup [OF assms])
+ { fix x assume "x\<in>s"
+ have "(dist a ---> dist a x) (at x within s)"
+ by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
+ }
+ thus "continuous_on s (dist a)"
+ unfolding continuous_on ..
qed
text{* For *minimal* distance, we only need closure, not compactness. *}
lemma distance_attains_inf:
- fixes a :: "real ^ _" (* FIXME: generalize *)
+ fixes a :: "'a::heine_borel"
assumes "closed s" "s \<noteq> {}"
shows "\<exists>x \<in> s. \<forall>y \<in> s. dist a x \<le> dist a y"
proof-
@@ -4192,54 +4209,60 @@
moreover
{ fix x assume "x\<in>?B"
fix e::real assume "e>0"
- { fix x' assume "x'\<in>?B" and as:"norm (x' - x) < e"
- hence "\<bar>norm (x' - a) - norm (x - a)\<bar> < e"
- using real_abs_sub_norm[of "x' - a" "x - a"] by auto }
- hence "\<exists>d>0. \<forall>x'\<in>?B. norm (x' - x) < d \<longrightarrow> \<bar>dist x' a - dist x a\<bar> < e" using `e>0` unfolding dist_norm by auto }
- hence "continuous_on (cball a (dist b a) \<inter> s) (vec1 \<circ> dist a)" unfolding continuous_on_vec1_range
- by (auto simp add: dist_commute)
- moreover have "compact ?B" using compact_cball[of a "dist b a"] unfolding compact_eq_bounded_closed using bounded_Int and closed_Int and assms(1) by auto
- ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y" using continuous_attains_inf[of ?B "dist a"] by fastsimp
+ { fix x' assume "x'\<in>?B" and as:"dist x' x < e"
+ from as have "\<bar>dist a x' - dist a x\<bar> < e"
+ unfolding abs_less_iff minus_diff_eq
+ using dist_triangle2 [of a x' x]
+ using dist_triangle [of a x x']
+ by arith
+ }
+ hence "\<exists>d>0. \<forall>x'\<in>?B. dist x' x < d \<longrightarrow> \<bar>dist a x' - dist a x\<bar> < e"
+ using `e>0` by auto
+ }
+ hence "continuous_on (cball a (dist b a) \<inter> s) (dist a)"
+ unfolding continuous_on Lim_within dist_norm real_norm_def
+ by fast
+ moreover have "compact ?B"
+ using compact_cball[of a "dist b a"]
+ unfolding compact_eq_bounded_closed
+ using bounded_Int and closed_Int and assms(1) by auto
+ ultimately obtain x where "x\<in>cball a (dist b a) \<inter> s" "\<forall>y\<in>cball a (dist b a) \<inter> s. dist a x \<le> dist a y"
+ using continuous_attains_inf[of ?B "dist a"] by fastsimp
thus ?thesis by fastsimp
qed
subsection{* We can now extend limit compositions to consider the scalar multiplier. *}
-lemma Lim_mul:
+lemma Lim_mul [tendsto_intros]:
fixes f :: "'a \<Rightarrow> real ^ _"
- assumes "((vec1 o c) ---> vec1 d) net" "(f ---> l) net"
+ assumes "(c ---> d) net" "(f ---> l) net"
shows "((\<lambda>x. c(x) *s f x) ---> (d *s l)) net"
-proof-
- have "bilinear (\<lambda>x. op *s (dest_vec1 (x::real^1)))" unfolding bilinear_def linear_def
- unfolding dest_vec1_add dest_vec1_cmul
- apply vector apply auto unfolding semiring_class.right_distrib semiring_class.left_distrib by auto
- thus ?thesis using Lim_bilinear[OF assms, of "\<lambda>x y. (dest_vec1 x) *s y"] by auto
-qed
+ unfolding smult_conv_scaleR using assms by (rule scaleR.tendsto)
lemma Lim_vmul:
fixes c :: "'a \<Rightarrow> real"
- shows "((vec1 o c) ---> vec1 d) net ==> ((\<lambda>x. c(x) *s v) ---> d *s v) net"
+ shows "(c ---> d) net ==> ((\<lambda>x. c(x) *s v) ---> d *s v) net"
using Lim_mul[of c d net "\<lambda>x. v" v] using Lim_const[of v] by auto
lemma continuous_vmul:
fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous net (vec1 o c) ==> continuous net (\<lambda>x. c(x) *s v)"
+ shows "continuous net c ==> continuous net (\<lambda>x. c(x) *s v)"
unfolding continuous_def using Lim_vmul[of c] by auto
lemma continuous_mul:
fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous net (vec1 o c) \<Longrightarrow> continuous net f
+ shows "continuous net c \<Longrightarrow> continuous net f
==> continuous net (\<lambda>x. c(x) *s f x) "
unfolding continuous_def using Lim_mul[of c] by auto
lemma continuous_on_vmul:
fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous_on s (vec1 o c) ==> continuous_on s (\<lambda>x. c(x) *s v)"
+ shows "continuous_on s c ==> continuous_on s (\<lambda>x. c(x) *s v)"
unfolding continuous_on_eq_continuous_within using continuous_vmul[of _ c] by auto
lemma continuous_on_mul:
fixes c :: "'a::metric_space \<Rightarrow> real"
- shows "continuous_on s (vec1 o c) \<Longrightarrow> continuous_on s f
+ shows "continuous_on s c \<Longrightarrow> continuous_on s f
==> continuous_on s (\<lambda>x. c(x) *s f x)"
unfolding continuous_on_eq_continuous_within using continuous_mul[of _ c] by auto
@@ -4247,59 +4270,27 @@
lemma Lim_inv:
fixes f :: "'a \<Rightarrow> real"
- assumes "((vec1 o f) ---> vec1 l) (net::'a net)" "l \<noteq> 0"
- shows "((vec1 o inverse o f) ---> vec1(inverse l)) net"
-proof -
- { fix e::real assume "e>0"
- let ?d = "min (\<bar>l\<bar> / 2) (l\<twosuperior> * e / 2)"
- have "0 < ?d" using `l\<noteq>0` `e>0` mult_pos_pos[of "l^2" "e/2"] by auto
- with assms(1) have "eventually (\<lambda>x. dist ((vec1 o f) x) (vec1 l) < ?d) net"
- by (rule tendstoD)
- moreover
- { fix x assume "dist ((vec1 o f) x) (vec1 l) < ?d"
- hence *:"\<bar>f x - l\<bar> < min (\<bar>l\<bar> / 2) (l\<twosuperior> * e / 2)" unfolding o_def dist_vec1 by auto
- hence fx0:"f x \<noteq> 0" using `l \<noteq> 0` by auto
- hence fxl0: "(f x) * l \<noteq> 0" using `l \<noteq> 0` by auto
- from * have **:"\<bar>f x - l\<bar> < l\<twosuperior> * e / 2" by auto
- have "\<bar>f x\<bar> * 2 \<ge> \<bar>l\<bar>" using * by (auto simp del: less_divide_eq_number_of1)
- hence "\<bar>f x\<bar> * 2 * \<bar>l\<bar> \<ge> \<bar>l\<bar> * \<bar>l\<bar>" unfolding mult_le_cancel_right by auto
- hence "\<bar>f x * l\<bar> * 2 \<ge> \<bar>l\<bar>^2" unfolding real_mult_commute and power2_eq_square by auto
- hence ***:"inverse \<bar>f x * l\<bar> \<le> inverse (l\<twosuperior> / 2)" using fxl0
- using le_imp_inverse_le[of "l^2 / 2" "\<bar>f x * l\<bar>"] by auto
-
- have "dist ((vec1 \<circ> inverse \<circ> f) x) (vec1 (inverse l)) < e" unfolding o_def unfolding dist_vec1
- unfolding inverse_diff_inverse[OF fx0 `l\<noteq>0`] apply simp
- unfolding mult_commute[of "inverse (f x)"]
- unfolding real_divide_def[THEN sym]
- unfolding divide_divide_eq_left
- unfolding nonzero_abs_divide[OF fxl0]
- using mult_less_le_imp_less[OF **, of "inverse \<bar>f x * l\<bar>", of "inverse (l^2 / 2)"] using *** using fx0 `l\<noteq>0`
- unfolding inverse_eq_divide using `e>0` by auto
- }
- ultimately
- have "eventually (\<lambda>x. dist ((vec1 o inverse o f) x) (vec1(inverse l)) < e) net"
- by (auto elim: eventually_rev_mono)
- }
- thus ?thesis unfolding tendsto_iff by auto
-qed
+ assumes "(f ---> l) (net::'a net)" "l \<noteq> 0"
+ shows "((inverse o f) ---> inverse l) net"
+ unfolding o_def using assms by (rule tendsto_inverse)
lemma continuous_inv:
fixes f :: "'a::metric_space \<Rightarrow> real"
- shows "continuous net (vec1 o f) \<Longrightarrow> f(netlimit net) \<noteq> 0
- ==> continuous net (vec1 o inverse o f)"
+ shows "continuous net f \<Longrightarrow> f(netlimit net) \<noteq> 0
+ ==> continuous net (inverse o f)"
unfolding continuous_def using Lim_inv by auto
lemma continuous_at_within_inv:
- fixes f :: "real ^ _ \<Rightarrow> real"
- assumes "continuous (at a within s) (vec1 o f)" "f a \<noteq> 0"
- shows "continuous (at a within s) (vec1 o inverse o f)"
- using assms unfolding continuous_within o_apply
- by (rule Lim_inv)
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
+ assumes "continuous (at a within s) f" "f a \<noteq> 0"
+ shows "continuous (at a within s) (inverse o f)"
+ using assms unfolding continuous_within o_def
+ by (intro tendsto_intros)
lemma continuous_at_inv:
- fixes f :: "real ^ _ \<Rightarrow> real"
- shows "continuous (at a) (vec1 o f) \<Longrightarrow> f a \<noteq> 0
- ==> continuous (at a) (vec1 o inverse o f) "
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_field"
+ shows "continuous (at a) f \<Longrightarrow> f a \<noteq> 0
+ ==> continuous (at a) (inverse o f) "
using within_UNIV[THEN sym, of "at a"] using continuous_at_within_inv[of a UNIV] by auto
subsection{* Preservation properties for pasted sets. *}
@@ -4316,6 +4307,16 @@
thus ?thesis unfolding bounded_iff by auto
qed
+lemma bounded_Times:
+ assumes "bounded s" "bounded t" shows "bounded (s \<times> t)"
+proof-
+ obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
+ using assms [unfolded bounded_def] by auto
+ then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<twosuperior> + b\<twosuperior>)"
+ by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
+ thus ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
+qed
+
lemma closed_pastecart:
fixes s :: "(real ^ 'a::finite) set" (* FIXME: generalize *)
assumes "closed s" "closed t"
@@ -4343,6 +4344,26 @@
shows "compact s \<Longrightarrow> compact t ==> compact {pastecart x y | x y . x \<in> s \<and> y \<in> t}"
unfolding compact_eq_bounded_closed using bounded_pastecart[of s t] closed_pastecart[of s t] by auto
+lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
+by (induct x) simp
+
+lemma compact_Times: "compact s \<Longrightarrow> compact t \<Longrightarrow> compact (s \<times> t)"
+unfolding compact_def
+apply clarify
+apply (drule_tac x="fst \<circ> f" in spec)
+apply (drule mp, simp add: mem_Times_iff)
+apply (clarify, rename_tac l1 r1)
+apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
+apply (drule mp, simp add: mem_Times_iff)
+apply (clarify, rename_tac l2 r2)
+apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
+apply (rule_tac x="r1 \<circ> r2" in exI)
+apply (rule conjI, simp add: subseq_def)
+apply (drule_tac r=r2 in lim_subseq [COMP swap_prems_rl], assumption)
+apply (drule (1) tendsto_Pair) back
+apply (simp add: o_def)
+done
+
text{* Hence some useful properties follow quite easily. *}
lemma compact_scaleR_image:
@@ -4366,30 +4387,27 @@
using compact_scaleR_image [OF assms, of "- 1"] by auto
lemma compact_sums:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "'a::real_normed_vector set"
assumes "compact s" "compact t" shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
proof-
- have *:"{x + y | x y. x \<in> s \<and> y \<in> t} =(\<lambda>z. fstcart z + sndcart z) ` {pastecart x y | x y. x \<in> s \<and> y \<in> t}"
- apply auto unfolding image_iff apply(rule_tac x="pastecart xa y" in bexI) unfolding fstcart_pastecart sndcart_pastecart by auto
- have "linear (\<lambda>z::real^('a + 'a). fstcart z + sndcart z)" unfolding linear_def
- unfolding fstcart_add sndcart_add apply auto
- unfolding vector_add_ldistrib fstcart_cmul[THEN sym] sndcart_cmul[THEN sym] by auto
- hence "continuous_on {pastecart x y |x y. x \<in> s \<and> y \<in> t} (\<lambda>z. fstcart z + sndcart z)"
- using continuous_at_imp_continuous_on linear_continuous_at by auto
- thus ?thesis unfolding * using compact_continuous_image compact_pastecart[OF assms] by auto
+ have *:"{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
+ apply auto unfolding image_iff apply(rule_tac x="(xa, y)" in bexI) by auto
+ have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
+ unfolding continuous_on by (rule ballI) (intro tendsto_intros)
+ thus ?thesis unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed
lemma compact_differences:
- fixes s t :: "(real ^ 'a::finite) set"
+ fixes s t :: "'a::real_normed_vector set"
assumes "compact s" "compact t" shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
proof-
- have "{x - y | x y::real^'a. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
+ have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
apply auto apply(rule_tac x= xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
thus ?thesis using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed
lemma compact_translation:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "compact s" shows "compact ((\<lambda>x. a + x) ` s)"
proof-
have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s" by auto
@@ -4407,14 +4425,14 @@
text{* Hence we get the following. *}
lemma compact_sup_maxdistance:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. norm(u - v) \<le> norm(x - y)"
proof-
have "{x - y | x y . x\<in>s \<and> y\<in>s} \<noteq> {}" using `s \<noteq> {}` by auto
then obtain x where x:"x\<in>{x - y |x y. x \<in> s \<and> y \<in> s}" "\<forall>y\<in>{x - y |x y. x \<in> s \<and> y \<in> s}. norm y \<le> norm x"
using compact_differences[OF assms(1) assms(1)]
- using distance_attains_sup[unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
+ using distance_attains_sup[where 'a="'a", unfolded dist_norm, of "{x - y | x y . x\<in>s \<and> y\<in>s}" 0] by(auto simp add: norm_minus_cancel)
from x(1) obtain a b where "a\<in>s" "b\<in>s" "x = a - b" by auto
thus ?thesis using x(2)[unfolded `x = a - b`] by blast
qed
@@ -4458,11 +4476,11 @@
using diameter_bounded by blast
lemma diameter_compact_attained:
- fixes s :: "(real ^ _) set"
+ fixes s :: "'a::real_normed_vector set"
assumes "compact s" "s \<noteq> {}"
shows "\<exists>x\<in>s. \<exists>y\<in>s. (norm(x - y) = diameter s)"
proof-
- have b:"bounded s" using assms(1) compact_eq_bounded_closed by auto
+ have b:"bounded s" using assms(1) by (rule compact_imp_bounded)
then obtain x y where xys:"x\<in>s" "y\<in>s" and xy:"\<forall>u\<in>s. \<forall>v\<in>s. norm (u - v) \<le> norm (x - y)" using compact_sup_maxdistance[OF assms] by auto
hence "diameter s \<le> norm (x - y)" using rsup_le[of "{norm (x - y) |x y. x \<in> s \<and> y \<in> s}" "norm (x - y)"]
unfolding setle_def and diameter_def by auto
@@ -4523,10 +4541,10 @@
{ fix x l assume as:"\<forall>n. x n \<in> ?S" "(x ---> l) sequentially"
from as(1) obtain f where f:"\<forall>n. x n = fst (f n) + snd (f n)" "\<forall>n. fst (f n) \<in> s" "\<forall>n. snd (f n) \<in> t"
using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> s \<and> snd y \<in> t"] by auto
- obtain l' r where "l'\<in>s" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
+ obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
- using Lim_sub[OF lim_subsequence[OF r as(2)] lr] and f(1) unfolding o_def by auto
+ using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
hence "l - l' \<in> t"
using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
using f(3) by auto
@@ -4600,7 +4618,7 @@
subsection{* Separation between points and sets. *}
lemma separate_point_closed:
- fixes s :: "(real ^ _) set" (* FIXME: generalize *)
+ fixes s :: "'a::heine_borel set"
shows "closed s \<Longrightarrow> a \<notin> s ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
proof(cases "s = {}")
case True
@@ -4613,7 +4631,8 @@
qed
lemma separate_compact_closed:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "'a::{heine_borel, real_normed_vector} set"
+ (* TODO: does this generalize to heine_borel? *)
assumes "compact s" and "closed t" and "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
proof-
@@ -4629,7 +4648,7 @@
qed
lemma separate_closed_compact:
- fixes s t :: "(real ^ _) set"
+ fixes s t :: "'a::{heine_borel, real_normed_vector} set"
assumes "closed s" and "compact t" and "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
proof-
@@ -4965,8 +4984,8 @@
then obtain N::nat where "inverse (real (N + 1)) < e" by auto
hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto }
- hence "((vec1 \<circ> (\<lambda>n. inverse (real n + 1))) ---> vec1 0) sequentially"
- unfolding Lim_sequentially by(auto simp add: dist_vec1)
+ hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
+ unfolding Lim_sequentially by(auto simp add: dist_norm)
hence "(f ---> x) sequentially" unfolding f_def
using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *s ((1 / 2) *s (a + b) - x)" 0 sequentially x]
using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *s (a + b) - x)"] by auto }
@@ -5320,7 +5339,7 @@
have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq) }
hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
- then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] by auto
+ then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
thus ?thesis unfolding Lim_sequentially apply(rule_tac x="vec1 l" in exI)
unfolding dist_norm unfolding abs_dest_vec1 and dest_vec1_sub by auto
qed
@@ -5940,9 +5959,10 @@
subsection{* Edelstein fixed point theorem. *}
lemma edelstein_fix:
+ fixes s :: "'a::real_normed_vector set"
assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
- shows "\<exists>! x::real^'a::finite\<in>s. g x = x"
+ shows "\<exists>! x\<in>s. g x = x"
proof(cases "\<exists>x\<in>s. g x \<noteq> x")
obtain x where "x\<in>s" using s(2) by auto
case False hence g:"\<forall>x\<in>s. g x = x" by auto
@@ -5985,26 +6005,23 @@
qed
qed } note distf = this
- def h \<equiv> "\<lambda>n. pastecart (f n x) (f n y)"
- let ?s2 = "{pastecart x y |x y. x \<in> s \<and> y \<in> s}"
- obtain l r where "l\<in>?s2" and r:"\<forall>m n. m < n \<longrightarrow> r m < r n" and lr:"((h \<circ> r) ---> l) sequentially"
- using compact_pastecart[OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
+ def h \<equiv> "\<lambda>n. (f n x, f n y)"
+ let ?s2 = "s \<times> s"
+ obtain l r where "l\<in>?s2" and r:"subseq r" and lr:"((h \<circ> r) ---> l) sequentially"
+ using compact_Times [OF s(1) s(1), unfolded compact_def, THEN spec[where x=h]] unfolding h_def
using fs[OF `x\<in>s`] and fs[OF `y\<in>s`] by blast
- def a \<equiv> "fstcart l" def b \<equiv> "sndcart l"
- have lab:"l = pastecart a b" unfolding a_def b_def and pastecart_fst_snd by simp
+ def a \<equiv> "fst l" def b \<equiv> "snd l"
+ have lab:"l = (a, b)" unfolding a_def b_def by simp
have [simp]:"a\<in>s" "b\<in>s" unfolding a_def b_def using `l\<in>?s2` by auto
- have "continuous_on (UNIV :: (real ^ _) set) fstcart"
- and "continuous_on (UNIV :: (real ^ _) set) sndcart"
- using linear_continuous_on using linear_fstcart and linear_sndcart by auto
- hence lima:"((fstcart \<circ> (h \<circ> r)) ---> a) sequentially" and limb:"((sndcart \<circ> (h \<circ> r)) ---> b) sequentially"
- unfolding atomize_conj unfolding continuous_on_sequentially
- apply(erule_tac x="h \<circ> r" in allE) apply(erule_tac x="h \<circ> r" in allE) using lr
- unfolding o_def and h_def a_def b_def by auto
+ have lima:"((fst \<circ> (h \<circ> r)) ---> a) sequentially"
+ and limb:"((snd \<circ> (h \<circ> r)) ---> b) sequentially"
+ using lr
+ unfolding o_def a_def b_def by (simp_all add: tendsto_intros)
{ fix n::nat
- have *:"\<And>fx fy (x::real^_) y. dist fx fy \<le> dist x y \<Longrightarrow> \<not> (dist (fx - fy) (a - b) < dist a b - dist x y)" unfolding dist_norm by norm
- { fix x y ::"real^'a"
+ have *:"\<And>fx fy (x::'a) y. dist fx fy \<le> dist x y \<Longrightarrow> \<not> (dist (fx - fy) (a - b) < dist a b - dist x y)" unfolding dist_norm by norm
+ { fix x y :: 'a
have "dist (-x) (-y) = dist x y" unfolding dist_norm
using norm_minus_cancel[of "x - y"] by (auto simp add: uminus_add_conv_diff) } note ** = this
@@ -6021,7 +6038,7 @@
moreover
have "dist (f (r (Na + Nb + n)) x - f (r (Na + Nb + n)) y) (a - b) \<ge> dist a b - dist (f n x) (f n y)"
using distf[of n "r (Na+Nb+n)", OF _ `x\<in>s` `y\<in>s`]
- using monotone_bigger[OF r, of "Na+Nb+n"]
+ using subseq_bigger[OF r, of "Na+Nb+n"]
using *[of "f (r (Na + Nb + n)) x" "f (r (Na + Nb + n)) y" "f n x" "f n y"] by auto
ultimately have False by simp
}
@@ -6049,8 +6066,8 @@
have "dist y x < e \<longrightarrow> dist (g y) (g x) < e" using dist by fastsimp }
hence "continuous_on s g" unfolding continuous_on_def by auto
- hence "((sndcart \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
- apply (rule allE[where x="\<lambda>n. (fstcart \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
+ hence "((snd \<circ> h \<circ> r) ---> g a) sequentially" unfolding continuous_on_sequentially
+ apply (rule allE[where x="\<lambda>n. (fst \<circ> h \<circ> r) n"]) apply (erule ballE[where x=a])
using lima unfolding h_def o_def using fs[OF `x\<in>s`] by (auto simp add: y_def)
hence "g a = a" using Lim_unique[OF trivial_limit_sequentially limb, of "g a"]
unfolding `a=b` and o_assoc by auto