src/HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy

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explicit checks stable_finished_theory/stable_command allow parallel asynchronous command transactions;
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(* Title: HOL/Hahn_Banach/Hahn_Banach_Sup_Lemmas.thy Author: Gertrud Bauer, TU Munich *) header {* The supremum w.r.t.~the function order *} theory Hahn_Banach_Sup_Lemmas imports Function_Norm Zorn_Lemma begin text {* This section contains some lemmas that will be used in the proof of the Hahn-Banach Theorem. In this section the following context is presumed. Let @{text E} be a real vector space with a seminorm @{text p} on @{text E}. @{text F} is a subspace of @{text E} and @{text f} a linear form on @{text F}. We consider a chain @{text c} of norm-preserving extensions of @{text f}, such that @{text "\<Union>c = graph H h"}. We will show some properties about the limit function @{text h}, i.e.\ the supremum of the chain @{text c}. \medskip Let @{text c} be a chain of norm-preserving extensions of the function @{text f} and let @{text "graph H h"} be the supremum of @{text c}. Every element in @{text H} is member of one of the elements of the chain. *} lemmas [dest?] = chainD lemmas chainE2 [elim?] = chainD2 [elim_format] lemma some_H'h't: assumes M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and u: "graph H h = \<Union>c" and x: "x \<in> H" shows "\<exists>H' h'. graph H' h' \<in> c \<and> (x, h x) \<in> graph H' h' \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)" proof - from x have "(x, h x) \<in> graph H h" .. also from u have "\<dots> = \<Union>c" . finally obtain g where gc: "g \<in> c" and gh: "(x, h x) \<in> g" by blast from cM have "c \<subseteq> M" .. with gc have "g \<in> M" .. also from M have "\<dots> = norm_pres_extensions E p F f" . finally obtain H' and h' where g: "g = graph H' h'" and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'" "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x" .. from gc and g have "graph H' h' \<in> c" by (simp only:) moreover from gh and g have "(x, h x) \<in> graph H' h'" by (simp only:) ultimately show ?thesis using * by blast qed text {* \medskip Let @{text c} be a chain of norm-preserving extensions of the function @{text f} and let @{text "graph H h"} be the supremum of @{text c}. Every element in the domain @{text H} of the supremum function is member of the domain @{text H'} of some function @{text h'}, such that @{text h} extends @{text h'}. *} lemma some_H'h': assumes M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and u: "graph H h = \<Union>c" and x: "x \<in> H" shows "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)" proof - from M cM u x obtain H' h' where x_hx: "(x, h x) \<in> graph H' h'" and c: "graph H' h' \<in> c" and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'" "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x" by (rule some_H'h't [elim_format]) blast from x_hx have "x \<in> H'" .. moreover from cM u c have "graph H' h' \<subseteq> graph H h" by (simp only: chain_ball_Union_upper) ultimately show ?thesis using * by blast qed text {* \medskip Any two elements @{text x} and @{text y} in the domain @{text H} of the supremum function @{text h} are both in the domain @{text H'} of some function @{text h'}, such that @{text h} extends @{text h'}. *} lemma some_H'h'2: assumes M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and u: "graph H h = \<Union>c" and x: "x \<in> H" and y: "y \<in> H" shows "\<exists>H' h'. x \<in> H' \<and> y \<in> H' \<and> graph H' h' \<subseteq> graph H h \<and> linearform H' h' \<and> H' \<unlhd> E \<and> F \<unlhd> H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)" proof - txt {* @{text y} is in the domain @{text H''} of some function @{text h''}, such that @{text h} extends @{text h''}. *} from M cM u and y obtain H' h' where y_hy: "(y, h y) \<in> graph H' h'" and c': "graph H' h' \<in> c" and * : "linearform H' h'" "H' \<unlhd> E" "F \<unlhd> H'" "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x" by (rule some_H'h't [elim_format]) blast txt {* @{text x} is in the domain @{text H'} of some function @{text h'}, such that @{text h} extends @{text h'}. *} from M cM u and x obtain H'' h'' where x_hx: "(x, h x) \<in> graph H'' h''" and c'': "graph H'' h'' \<in> c" and ** : "linearform H'' h''" "H'' \<unlhd> E" "F \<unlhd> H''" "graph F f \<subseteq> graph H'' h''" "\<forall>x \<in> H''. h'' x \<le> p x" by (rule some_H'h't [elim_format]) blast txt {* Since both @{text h'} and @{text h''} are elements of the chain, @{text h''} is an extension of @{text h'} or vice versa. Thus both @{text x} and @{text y} are contained in the greater one. \label{cases1}*} from cM c'' c' have "graph H'' h'' \<subseteq> graph H' h' \<or> graph H' h' \<subseteq> graph H'' h''" (is "?case1 \<or> ?case2") .. then show ?thesis proof assume ?case1 have "(x, h x) \<in> graph H'' h''" by fact also have "\<dots> \<subseteq> graph H' h'" by fact finally have xh:"(x, h x) \<in> graph H' h'" . then have "x \<in> H'" .. moreover from y_hy have "y \<in> H'" .. moreover from cM u and c' have "graph H' h' \<subseteq> graph H h" by (simp only: chain_ball_Union_upper) ultimately show ?thesis using * by blast next assume ?case2 from x_hx have "x \<in> H''" .. moreover { have "(y, h y) \<in> graph H' h'" by (rule y_hy) also have "\<dots> \<subseteq> graph H'' h''" by fact finally have "(y, h y) \<in> graph H'' h''" . } then have "y \<in> H''" .. moreover from cM u and c'' have "graph H'' h'' \<subseteq> graph H h" by (simp only: chain_ball_Union_upper) ultimately show ?thesis using ** by blast qed qed text {* \medskip The relation induced by the graph of the supremum of a chain @{text c} is definite, i.~e.~t is the graph of a function. *} lemma sup_definite: assumes M_def: "M \<equiv> norm_pres_extensions E p F f" and cM: "c \<in> chain M" and xy: "(x, y) \<in> \<Union>c" and xz: "(x, z) \<in> \<Union>c" shows "z = y" proof - from cM have c: "c \<subseteq> M" .. from xy obtain G1 where xy': "(x, y) \<in> G1" and G1: "G1 \<in> c" .. from xz obtain G2 where xz': "(x, z) \<in> G2" and G2: "G2 \<in> c" .. from G1 c have "G1 \<in> M" .. then obtain H1 h1 where G1_rep: "G1 = graph H1 h1" unfolding M_def by blast from G2 c have "G2 \<in> M" .. then obtain H2 h2 where G2_rep: "G2 = graph H2 h2" unfolding M_def by blast txt {* @{text "G\<^sub>1"} is contained in @{text "G\<^sub>2"} or vice versa, since both @{text "G\<^sub>1"} and @{text "G\<^sub>2"} are members of @{text c}. \label{cases2}*} from cM G1 G2 have "G1 \<subseteq> G2 \<or> G2 \<subseteq> G1" (is "?case1 \<or> ?case2") .. then show ?thesis proof assume ?case1 with xy' G2_rep have "(x, y) \<in> graph H2 h2" by blast then have "y = h2 x" .. also from xz' G2_rep have "(x, z) \<in> graph H2 h2" by (simp only:) then have "z = h2 x" .. finally show ?thesis . next assume ?case2 with xz' G1_rep have "(x, z) \<in> graph H1 h1" by blast then have "z = h1 x" .. also from xy' G1_rep have "(x, y) \<in> graph H1 h1" by (simp only:) then have "y = h1 x" .. finally show ?thesis .. qed qed text {* \medskip The limit function @{text h} is linear. Every element @{text x} in the domain of @{text h} is in the domain of a function @{text h'} in the chain of norm preserving extensions. Furthermore, @{text h} is an extension of @{text h'} so the function values of @{text x} are identical for @{text h'} and @{text h}. Finally, the function @{text h'} is linear by construction of @{text M}. *} lemma sup_lf: assumes M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and u: "graph H h = \<Union>c" shows "linearform H h" proof fix x y assume x: "x \<in> H" and y: "y \<in> H" with M cM u obtain H' h' where x': "x \<in> H'" and y': "y \<in> H'" and b: "graph H' h' \<subseteq> graph H h" and linearform: "linearform H' h'" and subspace: "H' \<unlhd> E" by (rule some_H'h'2 [elim_format]) blast show "h (x + y) = h x + h y" proof - from linearform x' y' have "h' (x + y) = h' x + h' y" by (rule linearform.add) also from b x' have "h' x = h x" .. also from b y' have "h' y = h y" .. also from subspace x' y' have "x + y \<in> H'" by (rule subspace.add_closed) with b have "h' (x + y) = h (x + y)" .. finally show ?thesis . qed next fix x a assume x: "x \<in> H" with M cM u obtain H' h' where x': "x \<in> H'" and b: "graph H' h' \<subseteq> graph H h" and linearform: "linearform H' h'" and subspace: "H' \<unlhd> E" by (rule some_H'h' [elim_format]) blast show "h (a \<cdot> x) = a * h x" proof - from linearform x' have "h' (a \<cdot> x) = a * h' x" by (rule linearform.mult) also from b x' have "h' x = h x" .. also from subspace x' have "a \<cdot> x \<in> H'" by (rule subspace.mult_closed) with b have "h' (a \<cdot> x) = h (a \<cdot> x)" .. finally show ?thesis . qed qed text {* \medskip The limit of a non-empty chain of norm preserving extensions of @{text f} is an extension of @{text f}, since every element of the chain is an extension of @{text f} and the supremum is an extension for every element of the chain. *} lemma sup_ext: assumes graph: "graph H h = \<Union>c" and M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and ex: "\<exists>x. x \<in> c" shows "graph F f \<subseteq> graph H h" proof - from ex obtain x where xc: "x \<in> c" .. from cM have "c \<subseteq> M" .. with xc have "x \<in> M" .. with M have "x \<in> norm_pres_extensions E p F f" by (simp only:) then obtain G g where "x = graph G g" and "graph F f \<subseteq> graph G g" .. then have "graph F f \<subseteq> x" by (simp only:) also from xc have "\<dots> \<subseteq> \<Union>c" by blast also from graph have "\<dots> = graph H h" .. finally show ?thesis . qed text {* \medskip The domain @{text H} of the limit function is a superspace of @{text F}, since @{text F} is a subset of @{text H}. The existence of the @{text 0} element in @{text F} and the closure properties follow from the fact that @{text F} is a vector space. *} lemma sup_supF: assumes graph: "graph H h = \<Union>c" and M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and ex: "\<exists>x. x \<in> c" and FE: "F \<unlhd> E" shows "F \<unlhd> H" proof from FE show "F \<noteq> {}" by (rule subspace.non_empty) from graph M cM ex have "graph F f \<subseteq> graph H h" by (rule sup_ext) then show "F \<subseteq> H" .. fix x y assume "x \<in> F" and "y \<in> F" with FE show "x + y \<in> F" by (rule subspace.add_closed) next fix x a assume "x \<in> F" with FE show "a \<cdot> x \<in> F" by (rule subspace.mult_closed) qed text {* \medskip The domain @{text H} of the limit function is a subspace of @{text E}. *} lemma sup_subE: assumes graph: "graph H h = \<Union>c" and M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" and ex: "\<exists>x. x \<in> c" and FE: "F \<unlhd> E" and E: "vectorspace E" shows "H \<unlhd> E" proof show "H \<noteq> {}" proof - from FE E have "0 \<in> F" by (rule subspace.zero) also from graph M cM ex FE have "F \<unlhd> H" by (rule sup_supF) then have "F \<subseteq> H" .. finally show ?thesis by blast qed show "H \<subseteq> E" proof fix x assume "x \<in> H" with M cM graph obtain H' where x: "x \<in> H'" and H'E: "H' \<unlhd> E" by (rule some_H'h' [elim_format]) blast from H'E have "H' \<subseteq> E" .. with x show "x \<in> E" .. qed fix x y assume x: "x \<in> H" and y: "y \<in> H" show "x + y \<in> H" proof - from M cM graph x y obtain H' h' where x': "x \<in> H'" and y': "y \<in> H'" and H'E: "H' \<unlhd> E" and graphs: "graph H' h' \<subseteq> graph H h" by (rule some_H'h'2 [elim_format]) blast from H'E x' y' have "x + y \<in> H'" by (rule subspace.add_closed) also from graphs have "H' \<subseteq> H" .. finally show ?thesis . qed next fix x a assume x: "x \<in> H" show "a \<cdot> x \<in> H" proof - from M cM graph x obtain H' h' where x': "x \<in> H'" and H'E: "H' \<unlhd> E" and graphs: "graph H' h' \<subseteq> graph H h" by (rule some_H'h' [elim_format]) blast from H'E x' have "a \<cdot> x \<in> H'" by (rule subspace.mult_closed) also from graphs have "H' \<subseteq> H" .. finally show ?thesis . qed qed text {* \medskip The limit function is bounded by the norm @{text p} as well, since all elements in the chain are bounded by @{text p}. *} lemma sup_norm_pres: assumes graph: "graph H h = \<Union>c" and M: "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M" shows "\<forall>x \<in> H. h x \<le> p x" proof fix x assume "x \<in> H" with M cM graph obtain H' h' where x': "x \<in> H'" and graphs: "graph H' h' \<subseteq> graph H h" and a: "\<forall>x \<in> H'. h' x \<le> p x" by (rule some_H'h' [elim_format]) blast from graphs x' have [symmetric]: "h' x = h x" .. also from a x' have "h' x \<le> p x " .. finally show "h x \<le> p x" . qed text {* \medskip The following lemma is a property of linear forms on real vector spaces. It will be used for the lemma @{text abs_Hahn_Banach} (see page \pageref{abs-Hahn_Banach}). \label{abs-ineq-iff} For real vector spaces the following inequations are equivalent: \begin{center} \begin{tabular}{lll} @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and & @{text "\<forall>x \<in> H. h x \<le> p x"} \\ \end{tabular} \end{center} *} lemma abs_ineq_iff: assumes "subspace H E" and "vectorspace E" and "seminorm E p" and "linearform H h" shows "(\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x) = (\<forall>x \<in> H. h x \<le> p x)" (is "?L = ?R") proof interpret subspace H E by fact interpret vectorspace E by fact interpret seminorm E p by fact interpret linearform H h by fact have H: "vectorspace H" using `vectorspace E` .. { assume l: ?L show ?R proof fix x assume x: "x \<in> H" have "h x \<le> \<bar>h x\<bar>" by arith also from l x have "\<dots> \<le> p x" .. finally show "h x \<le> p x" . qed next assume r: ?R show ?L proof fix x assume x: "x \<in> H" show "\<And>a b :: real. - a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> \<bar>b\<bar> \<le> a" by arith from `linearform H h` and H x have "- h x = h (- x)" by (rule linearform.neg [symmetric]) also from H x have "- x \<in> H" by (rule vectorspace.neg_closed) with r have "h (- x) \<le> p (- x)" .. also have "\<dots> = p x" using `seminorm E p` `vectorspace E` proof (rule seminorm.minus) from x show "x \<in> E" .. qed finally have "- h x \<le> p x" . then show "- p x \<le> h x" by simp from r x show "h x \<le> p x" .. qed } qed end