--- a/src/HOL/Analysis/Brouwer_Fixpoint.thy Thu Jul 12 17:23:38 2018 +0100
+++ b/src/HOL/Analysis/Brouwer_Fixpoint.thy Fri Jul 13 12:40:35 2018 +0200
@@ -18,40 +18,6 @@
imports Path_Connected Homeomorphism
begin
-subsection \<open>Unit cubes\<close>
-
-(* FIXME mv euclidean topological space *)
-definition unit_cube :: "'a::euclidean_space set"
- where "unit_cube = {x. \<forall>i\<in>Basis. 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1}"
-
-lemma mem_unit_cube: "x \<in> unit_cube \<longleftrightarrow> (\<forall>i\<in>Basis. 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
- unfolding unit_cube_def by simp
-
-lemma bounded_unit_cube: "bounded unit_cube"
- unfolding bounded_def
-proof (intro exI ballI)
- fix y :: 'a assume y: "y \<in> unit_cube"
- have "dist 0 y = norm y" by (rule dist_0_norm)
- also have "\<dots> = norm (\<Sum>i\<in>Basis. (y \<bullet> i) *\<^sub>R i)" unfolding euclidean_representation ..
- also have "\<dots> \<le> (\<Sum>i\<in>Basis. norm ((y \<bullet> i) *\<^sub>R i))" by (rule norm_sum)
- also have "\<dots> \<le> (\<Sum>i::'a\<in>Basis. 1)"
- by (rule sum_mono, simp add: y [unfolded mem_unit_cube])
- finally show "dist 0 y \<le> (\<Sum>i::'a\<in>Basis. 1)" .
-qed
-
-lemma closed_unit_cube: "closed unit_cube"
- unfolding unit_cube_def Collect_ball_eq Collect_conj_eq
- by (rule closed_INT, auto intro!: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma compact_unit_cube: "compact unit_cube" (is "compact ?C")
- unfolding compact_eq_seq_compact_metric
- using bounded_unit_cube closed_unit_cube
- by (rule bounded_closed_imp_seq_compact)
-
-lemma convex_unit_cube: "convex unit_cube"
- by (rule is_interval_convex) (fastforce simp add: is_interval_def mem_unit_cube)
-
-
(* FIXME mv topology euclidean space *)
subsection \<open>Retractions\<close>
@@ -3126,44 +3092,44 @@
subsection \<open>Brouwer's fixed point theorem\<close>
-text \<open>We start proving Brouwer's fixed point theorem for unit cubes.\<close>
+text \<open>We start proving Brouwer's fixed point theorem for the unit cube = \<open>cbox 0 One\<close>.\<close>
lemma brouwer_cube:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
- assumes "continuous_on unit_cube f"
- and "f ` unit_cube \<subseteq> unit_cube"
- shows "\<exists>x\<in>unit_cube. f x = x"
+ assumes "continuous_on (cbox 0 One) f"
+ and "f ` cbox 0 One \<subseteq> cbox 0 One"
+ shows "\<exists>x\<in>cbox 0 One. f x = x"
proof (rule ccontr)
define n where "n = DIM('a)"
have n: "1 \<le> n" "0 < n" "n \<noteq> 0"
unfolding n_def by (auto simp: Suc_le_eq DIM_positive)
assume "\<not> ?thesis"
- then have *: "\<not> (\<exists>x\<in>unit_cube. f x - x = 0)"
+ then have *: "\<not> (\<exists>x\<in>cbox 0 One. f x - x = 0)"
by auto
obtain d where
- d: "d > 0" "\<And>x. x \<in> unit_cube \<Longrightarrow> d \<le> norm (f x - x)"
- apply (rule brouwer_compactness_lemma[OF compact_unit_cube _ *])
+ d: "d > 0" "\<And>x. x \<in> cbox 0 One \<Longrightarrow> d \<le> norm (f x - x)"
+ apply (rule brouwer_compactness_lemma[OF compact_cbox _ *])
apply (rule continuous_intros assms)+
apply blast
done
- have *: "\<forall>x. x \<in> unit_cube \<longrightarrow> f x \<in> unit_cube"
- "\<forall>x. x \<in> (unit_cube::'a set) \<longrightarrow> (\<forall>i\<in>Basis. True \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
+ have *: "\<forall>x. x \<in> cbox 0 One \<longrightarrow> f x \<in> cbox 0 One"
+ "\<forall>x. x \<in> (cbox 0 One::'a set) \<longrightarrow> (\<forall>i\<in>Basis. True \<longrightarrow> 0 \<le> x \<bullet> i \<and> x \<bullet> i \<le> 1)"
using assms(2)[unfolded image_subset_iff Ball_def]
- unfolding mem_unit_cube
+ unfolding cbox_def
by auto
obtain label :: "'a \<Rightarrow> 'a \<Rightarrow> nat" where label [rule_format]:
"\<forall>x. \<forall>i\<in>Basis. label x i \<le> 1"
- "\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> x \<bullet> i = 0 \<longrightarrow> label x i = 0"
- "\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> x \<bullet> i = 1 \<longrightarrow> label x i = 1"
- "\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> label x i = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i"
- "\<forall>x. \<forall>i\<in>Basis. x \<in> unit_cube \<and> label x i = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i"
+ "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> x \<bullet> i = 0 \<longrightarrow> label x i = 0"
+ "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> x \<bullet> i = 1 \<longrightarrow> label x i = 1"
+ "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> label x i = 0 \<longrightarrow> x \<bullet> i \<le> f x \<bullet> i"
+ "\<forall>x. \<forall>i\<in>Basis. x \<in> cbox 0 One \<and> label x i = 1 \<longrightarrow> f x \<bullet> i \<le> x \<bullet> i"
using kuhn_labelling_lemma[OF *] by auto
note label = this [rule_format]
- have lem1: "\<forall>x\<in>unit_cube. \<forall>y\<in>unit_cube. \<forall>i\<in>Basis. label x i \<noteq> label y i \<longrightarrow>
+ have lem1: "\<forall>x\<in>cbox 0 One. \<forall>y\<in>cbox 0 One. \<forall>i\<in>Basis. label x i \<noteq> label y i \<longrightarrow>
\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)"
proof safe
fix x y :: 'a
- assume x: "x \<in> unit_cube" and y: "y \<in> unit_cube"
+ assume x: "x \<in> cbox 0 One" and y: "y \<in> cbox 0 One"
fix i
assume i: "label x i \<noteq> label y i" "i \<in> Basis"
have *: "\<And>x y fx fy :: real. x \<le> fx \<and> fy \<le> y \<or> fx \<le> x \<and> y \<le> fy \<Longrightarrow>
@@ -3188,18 +3154,18 @@
finally show "\<bar>f x \<bullet> i - x \<bullet> i\<bar> \<le> norm (f y - f x) + norm (y - x)"
unfolding inner_simps .
qed
- have "\<exists>e>0. \<forall>x\<in>unit_cube. \<forall>y\<in>unit_cube. \<forall>z\<in>unit_cube. \<forall>i\<in>Basis.
+ have "\<exists>e>0. \<forall>x\<in>cbox 0 One. \<forall>y\<in>cbox 0 One. \<forall>z\<in>cbox 0 One. \<forall>i\<in>Basis.
norm (x - z) < e \<longrightarrow> norm (y - z) < e \<longrightarrow> label x i \<noteq> label y i \<longrightarrow>
\<bar>(f(z) - z)\<bullet>i\<bar> < d / (real n)"
proof -
have d': "d / real n / 8 > 0"
using d(1) by (simp add: n_def DIM_positive)
- have *: "uniformly_continuous_on unit_cube f"
- by (rule compact_uniformly_continuous[OF assms(1) compact_unit_cube])
+ have *: "uniformly_continuous_on (cbox 0 One) f"
+ by (rule compact_uniformly_continuous[OF assms(1) compact_cbox])
obtain e where e:
"e > 0"
- "\<And>x x'. x \<in> unit_cube \<Longrightarrow>
- x' \<in> unit_cube \<Longrightarrow>
+ "\<And>x x'. x \<in> cbox 0 One \<Longrightarrow>
+ x' \<in> cbox 0 One \<Longrightarrow>
norm (x' - x) < e \<Longrightarrow>
norm (f x' - f x) < d / real n / 8"
using *[unfolded uniformly_continuous_on_def,rule_format,OF d']
@@ -3211,7 +3177,7 @@
using d' e by auto
fix x y z i
assume as:
- "x \<in> unit_cube" "y \<in> unit_cube" "z \<in> unit_cube"
+ "x \<in> cbox 0 One" "y \<in> cbox 0 One" "z \<in> cbox 0 One"
"norm (x - z) < min (e / 2) (d / real n / 8)"
"norm (y - z) < min (e / 2) (d / real n / 8)"
"label x i \<noteq> label y i"
@@ -3249,9 +3215,9 @@
then
obtain e where e:
"e > 0"
- "\<And>x y z i. x \<in> unit_cube \<Longrightarrow>
- y \<in> unit_cube \<Longrightarrow>
- z \<in> unit_cube \<Longrightarrow>
+ "\<And>x y z i. x \<in> cbox 0 One \<Longrightarrow>
+ y \<in> cbox 0 One \<Longrightarrow>
+ z \<in> cbox 0 One \<Longrightarrow>
i \<in> Basis \<Longrightarrow>
norm (x - z) < e \<and> norm (y - z) < e \<and> label x i \<noteq> label y i \<Longrightarrow>
\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
@@ -3290,9 +3256,9 @@
using label(1)[OF b'']
by auto
{ fix x :: "nat \<Rightarrow> nat" and i assume "\<forall>i<n. x i \<le> p" "i < n" "x i = p \<or> x i = 0"
- then have "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> (unit_cube::'a set)"
+ then have "(\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<in> (cbox 0 One::'a set)"
using b'_Basis
- by (auto simp: mem_unit_cube inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) }
+ by (auto simp: cbox_def inner_simps bij_betw_def zero_le_divide_iff divide_le_eq_1) }
note cube = this
have q2: "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow>
(label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)"
@@ -3314,8 +3280,8 @@
have "\<forall>i\<in>Basis. q (b' i) \<in> {0..p}"
using q(1) b'
by (auto intro: less_imp_le simp: bij_betw_def)
- then have "z \<in> unit_cube"
- unfolding z_def mem_unit_cube
+ then have "z \<in> cbox 0 One"
+ unfolding z_def cbox_def
using b'_Basis
by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1)
then have d_fz_z: "d \<le> norm (f z - z)"
@@ -3353,18 +3319,18 @@
using q(1)[rule_format,OF b'_im]
apply (auto simp: Suc_le_eq)
done
- then have "r' \<in> unit_cube"
- unfolding r'_def mem_unit_cube
+ then have "r' \<in> cbox 0 One"
+ unfolding r'_def cbox_def
using b'_Basis
by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1)
define s' :: 'a where "s' = (\<Sum>i\<in>Basis. (real (s (b' i)) / real p) *\<^sub>R i)"
have "\<And>i. i \<in> Basis \<Longrightarrow> s (b' i) \<le> p"
using b'_im q(1) rs(2) by fastforce
- then have "s' \<in> unit_cube"
- unfolding s'_def mem_unit_cube
+ then have "s' \<in> cbox 0 One"
+ unfolding s'_def cbox_def
using b'_Basis by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1)
- have "z \<in> unit_cube"
- unfolding z_def mem_unit_cube
+ have "z \<in> cbox 0 One"
+ unfolding z_def cbox_def
using b'_Basis q(1)[rule_format,OF b'_im] \<open>p > 0\<close>
by (auto simp: bij_betw_def zero_le_divide_iff divide_le_eq_1 less_imp_le)
{
@@ -3394,7 +3360,7 @@
then have "\<bar>(f z - z) \<bullet> i\<bar> < d / real n"
using rs(3) i
unfolding r'_def[symmetric] s'_def[symmetric] o_def bb'
- by (intro e(2)[OF \<open>r'\<in>unit_cube\<close> \<open>s'\<in>unit_cube\<close> \<open>z\<in>unit_cube\<close>]) auto
+ by (intro e(2)[OF \<open>r'\<in>cbox 0 One\<close> \<open>s'\<in>cbox 0 One\<close> \<open>z\<in>cbox 0 One\<close>]) auto
then show False
using i by auto
qed
@@ -3410,7 +3376,7 @@
and "f ` S \<subseteq> S"
obtains x where "x \<in> S" and "f x = x"
proof -
- let ?U = "unit_cube :: 'a set"
+ let ?U = "cbox 0 One :: 'a set"
have "\<Sum>Basis /\<^sub>R 2 \<in> interior ?U"
proof (rule interiorI)
let ?I = "(\<Inter>i\<in>Basis. {x::'a. 0 < x \<bullet> i} \<inter> {x. x \<bullet> i < 1})"
@@ -3418,12 +3384,12 @@
by (intro open_INT finite_Basis ballI open_Int, auto intro: open_Collect_less simp: continuous_on_inner continuous_on_const continuous_on_id)
show "\<Sum>Basis /\<^sub>R 2 \<in> ?I"
by simp
- show "?I \<subseteq> unit_cube"
- unfolding unit_cube_def by force
+ show "?I \<subseteq> cbox 0 One"
+ unfolding cbox_def by force
qed
then have *: "interior ?U \<noteq> {}" by fast
have *: "?U homeomorphic S"
- using homeomorphic_convex_compact[OF convex_unit_cube compact_unit_cube * assms(2,1,3)] .
+ using homeomorphic_convex_compact[OF convex_box(1) compact_cbox * assms(2,1,3)] .
have "\<forall>f. continuous_on ?U f \<and> f ` ?U \<subseteq> ?U \<longrightarrow>
(\<exists>x\<in>?U. f x = x)"
using brouwer_cube by auto