--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Sun Jun 04 00:09:04 2000 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy Sun Jun 04 19:39:29 2000 +0200
@@ -3,17 +3,17 @@
Author: Gertrud Bauer, TU Munich
*)
-header {* The Hahn-Banach Theorem *};
+header {* The Hahn-Banach Theorem *}
theory HahnBanach
- = HahnBanachSupLemmas + HahnBanachExtLemmas + ZornLemma:;
+ = HahnBanachSupLemmas + HahnBanachExtLemmas + ZornLemma:
text {*
We present the proof of two different versions of the Hahn-Banach
Theorem, closely following \cite[\S36]{Heuser:1986}.
-*};
+*}
-subsection {* The Hahn-Banach Theorem for vector spaces *};
+subsection {* The Hahn-Banach Theorem for vector spaces *}
text {* {\bf Theorem.} Let $f$ be a linear form defined on a subspace
$F$ of a real vector space $E$, such that $f$ is bounded by a seminorm
@@ -32,282 +32,282 @@
a norm-preserving way to a greater vector space $H_0$.
So $g$ cannot be maximal in $M$.
\bigskip
-*};
+*}
theorem HahnBanach: "[| is_vectorspace E; is_subspace F E;
is_seminorm E p; is_linearform F f; ALL x:F. f x <= p x |]
==> EX h. is_linearform E h & (ALL x:F. h x = f x)
- & (ALL x:E. h x <= p x)";
-proof -;
+ & (ALL x:E. h x <= p x)"
+proof -
-txt {* Let $E$ be a vector space, $F$ a subspace of $E$, $p$ a seminorm on $E$ and $f$ a linear form on $F$ such that $f$ is bounded by $p$. *};
+txt {* Let $E$ be a vector space, $F$ a subspace of $E$, $p$ a seminorm on $E$ and $f$ a linear form on $F$ such that $f$ is bounded by $p$. *}
assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
- "is_linearform F f" "ALL x:F. f x <= p x";
+ "is_linearform F f" "ALL x:F. f x <= p x"
-txt {* Define $M$ as the set of all norm-preserving extensions of $F$. *};
+txt {* Define $M$ as the set of all norm-preserving extensions of $F$. *}
- def M == "norm_pres_extensions E p F f";
- {;
- fix c; assume "c : chain M" "EX x. x:c";
+ def M == "norm_pres_extensions E p F f"
+ {
+ fix c assume "c : chain M" "EX x. x:c"
-txt {* Show that every non-empty chain $c$ in $M$ has an upper bound in $M$: $\Union c$ is greater that every element of the chain $c$, so $\Union c$ is an upper bound of $c$ that lies in $M$. *};
+txt {* Show that every non-empty chain $c$ in $M$ has an upper bound in $M$: $\Union c$ is greater that every element of the chain $c$, so $\Union c$ is an upper bound of $c$ that lies in $M$. *}
- have "Union c : M";
- proof (unfold M_def, rule norm_pres_extensionI);
+ have "Union c : M"
+ proof (unfold M_def, rule norm_pres_extensionI)
show "EX (H::'a set) h::'a => real. graph H h = Union c
& is_linearform H h
& is_subspace H E
& is_subspace F H
& graph F f <= graph H h
- & (ALL x::'a:H. h x <= p x)";
- proof (intro exI conjI);
- let ?H = "domain (Union c)";
- let ?h = "funct (Union c)";
+ & (ALL x::'a:H. h x <= p x)"
+ proof (intro exI conjI)
+ let ?H = "domain (Union c)"
+ let ?h = "funct (Union c)"
- show a: "graph ?H ?h = Union c";
- proof (rule graph_domain_funct);
- fix x y z; assume "(x, y) : Union c" "(x, z) : Union c";
- show "z = y"; by (rule sup_definite);
- qed;
- show "is_linearform ?H ?h";
- by (simp! add: sup_lf a);
- show "is_subspace ?H E";
- by (rule sup_subE [OF _ _ _ a]) (simp!)+;
- show "is_subspace F ?H";
- by (rule sup_supF [OF _ _ _ a]) (simp!)+;
- show "graph F f <= graph ?H ?h";
- by (rule sup_ext [OF _ _ _ a]) (simp!)+;
- show "ALL x::'a:?H. ?h x <= p x";
- by (rule sup_norm_pres [OF _ _ a]) (simp!)+;
- qed;
- qed;
- };
+ show a: "graph ?H ?h = Union c"
+ proof (rule graph_domain_funct)
+ fix x y z assume "(x, y) : Union c" "(x, z) : Union c"
+ show "z = y" by (rule sup_definite)
+ qed
+ show "is_linearform ?H ?h"
+ by (simp! add: sup_lf a)
+ show "is_subspace ?H E"
+ by (rule sup_subE [OF _ _ _ a]) (simp!)+
+ show "is_subspace F ?H"
+ by (rule sup_supF [OF _ _ _ a]) (simp!)+
+ show "graph F f <= graph ?H ?h"
+ by (rule sup_ext [OF _ _ _ a]) (simp!)+
+ show "ALL x::'a:?H. ?h x <= p x"
+ by (rule sup_norm_pres [OF _ _ a]) (simp!)+
+ qed
+ qed
+ }
-txt {* With Zorn's Lemma we can conclude that there is a maximal element $g$ in $M$. *};
+txt {* With Zorn's Lemma we can conclude that there is a maximal element $g$ in $M$. *}
- hence "EX g:M. ALL x:M. g <= x --> g = x";
- proof (rule Zorn's_Lemma);
- txt {* We show that $M$ is non-empty: *};
- have "graph F f : norm_pres_extensions E p F f";
- proof (rule norm_pres_extensionI2);
- have "is_vectorspace F"; ..;
- thus "is_subspace F F"; ..;
- qed (blast!)+;
- thus "graph F f : M"; by (simp!);
- qed;
- thus ?thesis;
- proof;
+ hence "EX g:M. ALL x:M. g <= x --> g = x"
+ proof (rule Zorn's_Lemma)
+ txt {* We show that $M$ is non-empty: *}
+ have "graph F f : norm_pres_extensions E p F f"
+ proof (rule norm_pres_extensionI2)
+ have "is_vectorspace F" ..
+ thus "is_subspace F F" ..
+ qed (blast!)+
+ thus "graph F f : M" by (simp!)
+ qed
+ thus ?thesis
+ proof
-txt {* We take this maximal element $g$. *};
+txt {* We take this maximal element $g$. *}
- fix g; assume "g:M" "ALL x:M. g <= x --> g = x";
- show ?thesis;
+ fix g assume "g:M" "ALL x:M. g <= x --> g = x"
+ show ?thesis
txt {* $g$ is a norm-preserving extension of $f$, that is: $g$
is the graph of a linear form $h$, defined on a subspace $H$ of
$E$, which is a superspace of $F$. $h$ is an extension of $f$
- and $h$ is again bounded by $p$. *};
+ and $h$ is again bounded by $p$. *}
obtain H h where "graph H h = g" "is_linearform H h"
"is_subspace H E" "is_subspace F H" "graph F f <= graph H h"
- "ALL x:H. h x <= p x";
- proof -;
+ "ALL x:H. h x <= p x"
+ proof -
have "EX H h. graph H h = g & is_linearform H h
& is_subspace H E & is_subspace F H
& graph F f <= graph H h
- & (ALL x:H. h x <= p x)"; by (simp! add: norm_pres_extension_D);
- thus ?thesis; by (elim exE conjE) rule;
- qed;
- have h: "is_vectorspace H"; ..;
+ & (ALL x:H. h x <= p x)" by (simp! add: norm_pres_extension_D)
+ thus ?thesis by (elim exE conjE) rule
+ qed
+ have h: "is_vectorspace H" ..
-txt {* We show that $h$ is defined on whole $E$ by classical contradiction. *};
+txt {* We show that $h$ is defined on whole $E$ by classical contradiction. *}
- have "H = E";
- proof (rule classical);
+ have "H = E"
+ proof (rule classical)
- txt {* Assume $h$ is not defined on whole $E$. *};
+ txt {* Assume $h$ is not defined on whole $E$. *}
- assume "H ~= E";
+ assume "H ~= E"
-txt {* Then show that $h$ can be extended in a norm-preserving way to a function $h_0$ with the graph $g_{h0}$. *};
+txt {* Then show that $h$ can be extended in a norm-preserving way to a function $h_0$ with the graph $g_{h0}$. *}
- have "EX g_h0 : M. g <= g_h0 & g ~= g_h0";
+ have "EX g_h0 : M. g <= g_h0 & g ~= g_h0"
- txt {* Consider $x_0 \in E \setminus H$. *};
+ txt {* Consider $x_0 \in E \setminus H$. *}
- obtain x0 where "x0:E" "x0~:H";
- proof -;
- have "EX x0:E. x0~:H";
- proof (rule set_less_imp_diff_not_empty);
- have "H <= E"; ..;
- thus "H < E"; ..;
- qed;
- thus ?thesis; by blast;
- qed;
- have x0: "x0 ~= 00";
- proof (rule classical);
- presume "x0 = 00";
- with h; have "x0:H"; by simp;
- thus ?thesis; by contradiction;
- qed blast;
+ obtain x0 where "x0:E" "x0~:H"
+ proof -
+ have "EX x0:E. x0~:H"
+ proof (rule set_less_imp_diff_not_empty)
+ have "H <= E" ..
+ thus "H < E" ..
+ qed
+ thus ?thesis by blast
+ qed
+ have x0: "x0 ~= 00"
+ proof (rule classical)
+ presume "x0 = 00"
+ with h have "x0:H" by simp
+ thus ?thesis by contradiction
+ qed blast
-txt {* Define $H_0$ as the direct sum of $H$ and the linear closure of $x_0$. *};
+txt {* Define $H_0$ as the direct sum of $H$ and the linear closure of $x_0$. *}
- def H0 == "H + lin x0";
- show ?thesis;
+ def H0 == "H + lin x0"
+ show ?thesis
txt {* Pick a real number $\xi$ that fulfills certain
inequations, which will be used to establish that $h_0$ is
- a norm-preserving extension of $h$. *};
+ a norm-preserving extension of $h$. *}
obtain xi where "ALL y:H. - p (y + x0) - h y <= xi
- & xi <= p (y + x0) - h y";
- proof -;
- from h; have "EX xi. ALL y:H. - p (y + x0) - h y <= xi
- & xi <= p (y + x0) - h y";
- proof (rule ex_xi);
- fix u v; assume "u:H" "v:H";
- from h; have "h v - h u = h (v - u)";
- by (simp! add: linearform_diff);
- also; have "... <= p (v - u)";
- by (simp!);
- also; have "v - u = x0 + - x0 + v + - u";
- by (simp! add: diff_eq1);
- also; have "... = v + x0 + - (u + x0)";
- by (simp!);
- also; have "... = (v + x0) - (u + x0)";
- by (simp! add: diff_eq1);
- also; have "p ... <= p (v + x0) + p (u + x0)";
- by (rule seminorm_diff_subadditive) (simp!)+;
- finally; have "h v - h u <= p (v + x0) + p (u + x0)"; .;
+ & xi <= p (y + x0) - h y"
+ proof -
+ from h have "EX xi. ALL y:H. - p (y + x0) - h y <= xi
+ & xi <= p (y + x0) - h y"
+ proof (rule ex_xi)
+ fix u v assume "u:H" "v:H"
+ from h have "h v - h u = h (v - u)"
+ by (simp! add: linearform_diff)
+ also have "... <= p (v - u)"
+ by (simp!)
+ also have "v - u = x0 + - x0 + v + - u"
+ by (simp! add: diff_eq1)
+ also have "... = v + x0 + - (u + x0)"
+ by (simp!)
+ also have "... = (v + x0) - (u + x0)"
+ by (simp! add: diff_eq1)
+ also have "p ... <= p (v + x0) + p (u + x0)"
+ by (rule seminorm_diff_subadditive) (simp!)+
+ finally have "h v - h u <= p (v + x0) + p (u + x0)" .
- thus "- p (u + x0) - h u <= p (v + x0) - h v";
- by (rule real_diff_ineq_swap);
- qed;
- thus ?thesis; by rule rule;
- qed;
+ thus "- p (u + x0) - h u <= p (v + x0) - h v"
+ by (rule real_diff_ineq_swap)
+ qed
+ thus ?thesis by rule rule
+ qed
-txt {* Define the extension $h_0$ of $h$ to $H_0$ using $\xi$. *};
+txt {* Define the extension $h_0$ of $h$ to $H_0$ using $\xi$. *}
def h0 == "\\<lambda>x. let (y,a) = SOME (y, a). x = y + a (*) x0
& y:H
- in (h y) + a * xi";
- show ?thesis;
- proof;
+ in (h y) + a * xi"
+ show ?thesis
+ proof
-txt {* Show that $h_0$ is an extension of $h$ *};
+txt {* Show that $h_0$ is an extension of $h$ *}
- show "g <= graph H0 h0 & g ~= graph H0 h0";
- proof;
- show "g <= graph H0 h0";
- proof -;
- have "graph H h <= graph H0 h0";
- proof (rule graph_extI);
- fix t; assume "t:H";
+ show "g <= graph H0 h0 & g ~= graph H0 h0"
+ proof
+ show "g <= graph H0 h0"
+ proof -
+ have "graph H h <= graph H0 h0"
+ proof (rule graph_extI)
+ fix t assume "t:H"
have "(SOME (y, a). t = y + a (*) x0 & y : H)
- = (t,#0)";
- by (rule decomp_H0_H [OF _ _ _ _ _ x0]);
- thus "h t = h0 t"; by (simp! add: Let_def);
- next;
- show "H <= H0";
- proof (rule subspace_subset);
- show "is_subspace H H0";
- proof (unfold H0_def, rule subspace_vs_sum1);
- show "is_vectorspace H"; ..;
- show "is_vectorspace (lin x0)"; ..;
- qed;
- qed;
- qed;
- thus ?thesis; by (simp!);
- qed;
- show "g ~= graph H0 h0";
- proof -;
- have "graph H h ~= graph H0 h0";
- proof;
- assume e: "graph H h = graph H0 h0";
- have "x0 : H0";
- proof (unfold H0_def, rule vs_sumI);
- show "x0 = 00 + x0"; by (simp!);
- from h; show "00 : H"; ..;
- show "x0 : lin x0"; by (rule x_lin_x);
- qed;
- hence "(x0, h0 x0) : graph H0 h0"; ..;
- with e; have "(x0, h0 x0) : graph H h"; by simp;
- hence "x0 : H"; ..;
- thus False; by contradiction;
- qed;
- thus ?thesis; by (simp!);
- qed;
- qed;
+ = (t,#0)"
+ by (rule decomp_H0_H [OF _ _ _ _ _ x0])
+ thus "h t = h0 t" by (simp! add: Let_def)
+ next
+ show "H <= H0"
+ proof (rule subspace_subset)
+ show "is_subspace H H0"
+ proof (unfold H0_def, rule subspace_vs_sum1)
+ show "is_vectorspace H" ..
+ show "is_vectorspace (lin x0)" ..
+ qed
+ qed
+ qed
+ thus ?thesis by (simp!)
+ qed
+ show "g ~= graph H0 h0"
+ proof -
+ have "graph H h ~= graph H0 h0"
+ proof
+ assume e: "graph H h = graph H0 h0"
+ have "x0 : H0"
+ proof (unfold H0_def, rule vs_sumI)
+ show "x0 = 00 + x0" by (simp!)
+ from h show "00 : H" ..
+ show "x0 : lin x0" by (rule x_lin_x)
+ qed
+ hence "(x0, h0 x0) : graph H0 h0" ..
+ with e have "(x0, h0 x0) : graph H h" by simp
+ hence "x0 : H" ..
+ thus False by contradiction
+ qed
+ thus ?thesis by (simp!)
+ qed
+ qed
-txt {* and $h_0$ is norm-preserving. *};
+txt {* and $h_0$ is norm-preserving. *}
- show "graph H0 h0 : M";
- proof -;
- have "graph H0 h0 : norm_pres_extensions E p F f";
- proof (rule norm_pres_extensionI2);
- show "is_linearform H0 h0";
- by (rule h0_lf [OF _ _ _ _ _ _ x0]) (simp!)+;
- show "is_subspace H0 E";
- by (unfold H0_def) (rule vs_sum_subspace [OF _ lin_subspace]);
- have "is_subspace F H"; .;
- also; from h lin_vs;
- have [fold H0_def]: "is_subspace H (H + lin x0)"; ..;
- finally (subspace_trans [OF _ h]);
- show f_h0: "is_subspace F H0"; .;
+ show "graph H0 h0 : M"
+ proof -
+ have "graph H0 h0 : norm_pres_extensions E p F f"
+ proof (rule norm_pres_extensionI2)
+ show "is_linearform H0 h0"
+ by (rule h0_lf [OF _ _ _ _ _ _ x0]) (simp!)+
+ show "is_subspace H0 E"
+ by (unfold H0_def) (rule vs_sum_subspace [OF _ lin_subspace])
+ have "is_subspace F H" .
+ also from h lin_vs
+ have [fold H0_def]: "is_subspace H (H + lin x0)" ..
+ finally (subspace_trans [OF _ h])
+ show f_h0: "is_subspace F H0" .
- show "graph F f <= graph H0 h0";
- proof (rule graph_extI);
- fix x; assume "x:F";
- have "f x = h x"; ..;
- also; have " ... = h x + #0 * xi"; by simp;
- also; have "... = (let (y,a) = (x, #0) in h y + a * xi)";
- by (simp add: Let_def);
- also; have
- "(x, #0) = (SOME (y, a). x = y + a (*) x0 & y : H)";
- by (rule decomp_H0_H [RS sym, OF _ _ _ _ _ x0]) (simp!)+;
- also; have
+ show "graph F f <= graph H0 h0"
+ proof (rule graph_extI)
+ fix x assume "x:F"
+ have "f x = h x" ..
+ also have " ... = h x + #0 * xi" by simp
+ also have "... = (let (y,a) = (x, #0) in h y + a * xi)"
+ by (simp add: Let_def)
+ also have
+ "(x, #0) = (SOME (y, a). x = y + a (*) x0 & y : H)"
+ by (rule decomp_H0_H [RS sym, OF _ _ _ _ _ x0]) (simp!)+
+ also have
"(let (y,a) = (SOME (y,a). x = y + a (*) x0 & y : H)
in h y + a * xi)
- = h0 x"; by (simp!);
- finally; show "f x = h0 x"; .;
- next;
- from f_h0; show "F <= H0"; ..;
- qed;
+ = h0 x" by (simp!)
+ finally show "f x = h0 x" .
+ next
+ from f_h0 show "F <= H0" ..
+ qed
- show "ALL x:H0. h0 x <= p x";
- by (rule h0_norm_pres [OF _ _ _ _ x0]);
- qed;
- thus "graph H0 h0 : M"; by (simp!);
- qed;
- qed;
- qed;
- qed;
+ show "ALL x:H0. h0 x <= p x"
+ by (rule h0_norm_pres [OF _ _ _ _ x0])
+ qed
+ thus "graph H0 h0 : M" by (simp!)
+ qed
+ qed
+ qed
+ qed
-txt {* So the graph $g$ of $h$ cannot be maximal. Contradiction. *};
+txt {* So the graph $g$ of $h$ cannot be maximal. Contradiction. *}
- hence "~ (ALL x:M. g <= x --> g = x)"; by simp;
- thus ?thesis; by contradiction;
- qed;
+ hence "~ (ALL x:M. g <= x --> g = x)" by simp
+ thus ?thesis by contradiction
+ qed
txt {* Now we have a linear extension $h$ of $f$ to $E$ that is
-bounded by $p$. *};
+bounded by $p$. *}
thus "EX h. is_linearform E h & (ALL x:F. h x = f x)
- & (ALL x:E. h x <= p x)";
- proof (intro exI conjI);
- assume eq: "H = E";
- from eq; show "is_linearform E h"; by (simp!);
- show "ALL x:F. h x = f x";
- proof (intro ballI, rule sym);
- fix x; assume "x:F"; show "f x = h x "; ..;
- qed;
- from eq; show "ALL x:E. h x <= p x"; by (force!);
- qed;
- qed;
- qed;
-qed;
+ & (ALL x:E. h x <= p x)"
+ proof (intro exI conjI)
+ assume eq: "H = E"
+ from eq show "is_linearform E h" by (simp!)
+ show "ALL x:F. h x = f x"
+ proof (intro ballI, rule sym)
+ fix x assume "x:F" show "f x = h x " ..
+ qed
+ from eq show "ALL x:E. h x <= p x" by (force!)
+ qed
+ qed
+ qed
+qed
(*
theorem HahnBanach:
"[| is_vectorspace E; is_subspace F E; is_seminorm E p;
@@ -320,69 +320,69 @@
"is_linearform F f" "ALL x:F. f x <= p x";
txt{* We define $M$ to be the set of all linear extensions
- of $f$ to superspaces of $F$, which are bounded by $p$. *};
+ of $f$ to superspaces of $F$, which are bounded by $p$. *}
- def M == "norm_pres_extensions E p F f";
+ def M == "norm_pres_extensions E p F f"
- txt{* We show that $M$ is non-empty: *};
+ txt{* We show that $M$ is non-empty: *}
- have aM: "graph F f : norm_pres_extensions E p F f";
- proof (rule norm_pres_extensionI2);
- have "is_vectorspace F"; ..;
- thus "is_subspace F F"; ..;
- qed (blast!)+;
+ have aM: "graph F f : norm_pres_extensions E p F f"
+ proof (rule norm_pres_extensionI2)
+ have "is_vectorspace F" ..
+ thus "is_subspace F F" ..
+ qed (blast!)+
- subsubsect {* Existence of a limit function *};
+ subsubsect {* Existence of a limit function *}
txt {* For every non-empty chain of norm-preserving extensions
the union of all functions in the chain is again a norm-preserving
extension. (The union is an upper bound for all elements.
It is even the least upper bound, because every upper bound of $M$
- is also an upper bound for $\Union c$, as $\Union c\in M$) *};
+ is also an upper bound for $\Union c$, as $\Union c\in M$) *}
- {;
- fix c; assume "c:chain M" "EX x. x:c";
- have "Union c : M";
+ {
+ fix c assume "c:chain M" "EX x. x:c"
+ have "Union c : M"
- proof (unfold M_def, rule norm_pres_extensionI);
+ proof (unfold M_def, rule norm_pres_extensionI)
show "EX (H::'a set) h::'a => real. graph H h = Union c
& is_linearform H h
& is_subspace H E
& is_subspace F H
& graph F f <= graph H h
- & (ALL x::'a:H. h x <= p x)";
- proof (intro exI conjI);
- let ?H = "domain (Union c)";
- let ?h = "funct (Union c)";
+ & (ALL x::'a:H. h x <= p x)"
+ proof (intro exI conjI)
+ let ?H = "domain (Union c)"
+ let ?h = "funct (Union c)"
- show a: "graph ?H ?h = Union c";
- proof (rule graph_domain_funct);
- fix x y z; assume "(x, y) : Union c" "(x, z) : Union c";
- show "z = y"; by (rule sup_definite);
- qed;
- show "is_linearform ?H ?h";
- by (simp! add: sup_lf a);
- show "is_subspace ?H E";
- by (rule sup_subE, rule a) (simp!)+;
- show "is_subspace F ?H";
- by (rule sup_supF, rule a) (simp!)+;
- show "graph F f <= graph ?H ?h";
- by (rule sup_ext, rule a) (simp!)+;
- show "ALL x::'a:?H. ?h x <= p x";
- by (rule sup_norm_pres, rule a) (simp!)+;
- qed;
- qed;
- };
+ show a: "graph ?H ?h = Union c"
+ proof (rule graph_domain_funct)
+ fix x y z assume "(x, y) : Union c" "(x, z) : Union c"
+ show "z = y" by (rule sup_definite)
+ qed
+ show "is_linearform ?H ?h"
+ by (simp! add: sup_lf a)
+ show "is_subspace ?H E"
+ by (rule sup_subE, rule a) (simp!)+
+ show "is_subspace F ?H"
+ by (rule sup_supF, rule a) (simp!)+
+ show "graph F f <= graph ?H ?h"
+ by (rule sup_ext, rule a) (simp!)+
+ show "ALL x::'a:?H. ?h x <= p x"
+ by (rule sup_norm_pres, rule a) (simp!)+
+ qed
+ qed
+ }
txt {* According to Zorn's Lemma there is
- a maximal norm-preserving extension $g\in M$. *};
+ a maximal norm-preserving extension $g\in M$. *}
- with aM; have bex_g: "EX g:M. ALL x:M. g <= x --> g = x";
- by (simp! add: Zorn's_Lemma);
+ with aM have bex_g: "EX g:M. ALL x:M. g <= x --> g = x"
+ by (simp! add: Zorn's_Lemma)
- thus ?thesis;
- proof;
- fix g; assume g: "g:M" "ALL x:M. g <= x --> g = x";
+ thus ?thesis
+ proof
+ fix g assume g: "g:M" "ALL x:M. g <= x --> g = x"
have ex_Hh:
"EX H h. graph H h = g
@@ -390,145 +390,145 @@
& is_subspace H E
& is_subspace F H
& graph F f <= graph H h
- & (ALL x:H. h x <= p x) ";
- by (simp! add: norm_pres_extension_D);
- thus ?thesis;
- proof (elim exE conjE, intro exI);
- fix H h;
+ & (ALL x:H. h x <= p x) "
+ by (simp! add: norm_pres_extension_D)
+ thus ?thesis
+ proof (elim exE conjE, intro exI)
+ fix H h
assume "graph H h = g" "is_linearform (H::'a set) h"
"is_subspace H E" "is_subspace F H"
and h_ext: "graph F f <= graph H h"
- and h_bound: "ALL x:H. h x <= p x";
+ and h_bound: "ALL x:H. h x <= p x"
- have h: "is_vectorspace H"; ..;
- have f: "is_vectorspace F"; ..;
+ have h: "is_vectorspace H" ..
+ have f: "is_vectorspace F" ..
-subsubsect {* The domain of the limit function *};
+subsubsect {* The domain of the limit function *}
-have eq: "H = E";
-proof (rule classical);
+have eq: "H = E"
+proof (rule classical)
-txt {* Assume that the domain of the supremum is not $E$, *};
+txt {* Assume that the domain of the supremum is not $E$, *}
- assume "H ~= E";
- have "H <= E"; ..;
- hence "H < E"; ..;
+ assume "H ~= E"
+ have "H <= E" ..
+ hence "H < E" ..
- txt{* then there exists an element $x_0$ in $E \setminus H$: *};
+ txt{* then there exists an element $x_0$ in $E \setminus H$: *}
- hence "EX x0:E. x0~:H";
- by (rule set_less_imp_diff_not_empty);
+ hence "EX x0:E. x0~:H"
+ by (rule set_less_imp_diff_not_empty)
txt {* We get that $h$ can be extended in a
- norm-preserving way to some $H_0$. *};
+ norm-preserving way to some $H_0$. *}
hence "EX H0 h0. g <= graph H0 h0 & g ~= graph H0 h0
- & graph H0 h0 : M";
- proof;
- fix x0; assume "x0:E" "x0~:H";
+ & graph H0 h0 : M"
+ proof
+ fix x0 assume "x0:E" "x0~:H"
- have x0: "x0 ~= 00";
- proof (rule classical);
- presume "x0 = 00";
- with h; have "x0:H"; by simp;
- thus ?thesis; by contradiction;
- qed blast;
+ have x0: "x0 ~= 00"
+ proof (rule classical)
+ presume "x0 = 00"
+ with h have "x0:H" by simp
+ thus ?thesis by contradiction
+ qed blast
txt {* Define $H_0$ as the (direct) sum of H and the
- linear closure of $x_0$. \label{ex-xi-use}*};
+ linear closure of $x_0$. \label{ex-xi-use}*}
- def H0 == "H + lin x0";
+ def H0 == "H + lin x0"
- from h; have xi: "EX xi. ALL y:H. - p (y + x0) - h y <= xi
- & xi <= p (y + x0) - h y";
- proof (rule ex_xi);
- fix u v; assume "u:H" "v:H";
- from h; have "h v - h u = h (v - u)";
- by (simp! add: linearform_diff);
- also; from h_bound; have "... <= p (v - u)";
- by (simp!);
- also; have "v - u = x0 + - x0 + v + - u";
- by (simp! add: diff_eq1);
- also; have "... = v + x0 + - (u + x0)";
- by (simp!);
- also; have "... = (v + x0) - (u + x0)";
- by (simp! add: diff_eq1);
- also; have "p ... <= p (v + x0) + p (u + x0)";
- by (rule seminorm_diff_subadditive) (simp!)+;
- finally; have "h v - h u <= p (v + x0) + p (u + x0)"; .;
+ from h have xi: "EX xi. ALL y:H. - p (y + x0) - h y <= xi
+ & xi <= p (y + x0) - h y"
+ proof (rule ex_xi)
+ fix u v assume "u:H" "v:H"
+ from h have "h v - h u = h (v - u)"
+ by (simp! add: linearform_diff)
+ also from h_bound have "... <= p (v - u)"
+ by (simp!)
+ also have "v - u = x0 + - x0 + v + - u"
+ by (simp! add: diff_eq1)
+ also have "... = v + x0 + - (u + x0)"
+ by (simp!)
+ also have "... = (v + x0) - (u + x0)"
+ by (simp! add: diff_eq1)
+ also have "p ... <= p (v + x0) + p (u + x0)"
+ by (rule seminorm_diff_subadditive) (simp!)+
+ finally have "h v - h u <= p (v + x0) + p (u + x0)" .
- thus "- p (u + x0) - h u <= p (v + x0) - h v";
- by (rule real_diff_ineq_swap);
- qed;
+ thus "- p (u + x0) - h u <= p (v + x0) - h v"
+ by (rule real_diff_ineq_swap)
+ qed
hence "EX h0. g <= graph H0 h0 & g ~= graph H0 h0
- & graph H0 h0 : M";
- proof (elim exE, intro exI conjI);
- fix xi;
+ & graph H0 h0 : M"
+ proof (elim exE, intro exI conjI)
+ fix xi
assume a: "ALL y:H. - p (y + x0) - h y <= xi
- & xi <= p (y + x0) - h y";
+ & xi <= p (y + x0) - h y"
txt {* Define $h_0$ as the canonical linear extension
- of $h$ on $H_0$:*};
+ of $h$ on $H_0$:*}
def h0 ==
"\\<lambda>x. let (y,a) = SOME (y, a). x = y + a ( * ) x0 & y:H
- in (h y) + a * xi";
+ in (h y) + a * xi"
txt {* We get that the graph of $h_0$ extends that of
- $h$. *};
+ $h$. *}
- have "graph H h <= graph H0 h0";
- proof (rule graph_extI);
- fix t; assume "t:H";
- have "(SOME (y, a). t = y + a ( * ) x0 & y : H) = (t,#0)";
- by (rule decomp_H0_H, rule x0);
- thus "h t = h0 t"; by (simp! add: Let_def);
- next;
- show "H <= H0";
- proof (rule subspace_subset);
- show "is_subspace H H0";
- proof (unfold H0_def, rule subspace_vs_sum1);
- show "is_vectorspace H"; ..;
- show "is_vectorspace (lin x0)"; ..;
- qed;
- qed;
- qed;
- thus "g <= graph H0 h0"; by (simp!);
+ have "graph H h <= graph H0 h0"
+ proof (rule graph_extI)
+ fix t assume "t:H"
+ have "(SOME (y, a). t = y + a ( * ) x0 & y : H) = (t,#0)"
+ by (rule decomp_H0_H, rule x0)
+ thus "h t = h0 t" by (simp! add: Let_def)
+ next
+ show "H <= H0"
+ proof (rule subspace_subset)
+ show "is_subspace H H0"
+ proof (unfold H0_def, rule subspace_vs_sum1)
+ show "is_vectorspace H" ..
+ show "is_vectorspace (lin x0)" ..
+ qed
+ qed
+ qed
+ thus "g <= graph H0 h0" by (simp!)
- txt {* Apparently $h_0$ is not equal to $h$. *};
+ txt {* Apparently $h_0$ is not equal to $h$. *}
- have "graph H h ~= graph H0 h0";
- proof;
- assume e: "graph H h = graph H0 h0";
- have "x0 : H0";
- proof (unfold H0_def, rule vs_sumI);
- show "x0 = 00 + x0"; by (simp!);
- from h; show "00 : H"; ..;
- show "x0 : lin x0"; by (rule x_lin_x);
- qed;
- hence "(x0, h0 x0) : graph H0 h0"; ..;
- with e; have "(x0, h0 x0) : graph H h"; by simp;
- hence "x0 : H"; ..;
- thus False; by contradiction;
- qed;
- thus "g ~= graph H0 h0"; by (simp!);
+ have "graph H h ~= graph H0 h0"
+ proof
+ assume e: "graph H h = graph H0 h0"
+ have "x0 : H0"
+ proof (unfold H0_def, rule vs_sumI)
+ show "x0 = 00 + x0" by (simp!)
+ from h show "00 : H" ..
+ show "x0 : lin x0" by (rule x_lin_x)
+ qed
+ hence "(x0, h0 x0) : graph H0 h0" ..
+ with e have "(x0, h0 x0) : graph H h" by simp
+ hence "x0 : H" ..
+ thus False by contradiction
+ qed
+ thus "g ~= graph H0 h0" by (simp!)
txt {* Furthermore $h_0$ is a norm-preserving extension
- of $f$. *};
+ of $f$. *}
- have "graph H0 h0 : norm_pres_extensions E p F f";
- proof (rule norm_pres_extensionI2);
- show "is_linearform H0 h0";
- by (rule h0_lf, rule x0) (simp!)+;
- show "is_subspace H0 E";
+ have "graph H0 h0 : norm_pres_extensions E p F f"
+ proof (rule norm_pres_extensionI2)
+ show "is_linearform H0 h0"
+ by (rule h0_lf, rule x0) (simp!)+
+ show "is_subspace H0 E"
by (unfold H0_def, rule vs_sum_subspace,
- rule lin_subspace);
+ rule lin_subspace)
- have "is_subspace F H"; .;
- also; from h lin_vs;
- have [fold H0_def]: "is_subspace H (H + lin x0)"; ..;
- finally (subspace_trans [OF _ h]);
- show f_h0: "is_subspace F H0"; .; (***
+ have "is_subspace F H" .
+ also from h lin_vs
+ have [fold H0_def]: "is_subspace H (H + lin x0)" ..
+ finally (subspace_trans [OF _ h])
+ show f_h0: "is_subspace F H0" . (***
backwards:
show f_h0: "is_subspace F H0"; .;
proof (rule subspace_trans [of F H H0]);
@@ -537,63 +537,63 @@
thus "is_subspace H H0"; by (unfold H0_def);
qed; ***)
- show "graph F f <= graph H0 h0";
- proof (rule graph_extI);
- fix x; assume "x:F";
- have "f x = h x"; ..;
- also; have " ... = h x + #0 * xi"; by simp;
- also; have "... = (let (y,a) = (x, #0) in h y + a * xi)";
- by (simp add: Let_def);
- also; have
- "(x, #0) = (SOME (y, a). x = y + a ( * ) x0 & y : H)";
- by (rule decomp_H0_H [RS sym], rule x0) (simp!)+;
- also; have
+ show "graph F f <= graph H0 h0"
+ proof (rule graph_extI)
+ fix x assume "x:F"
+ have "f x = h x" ..
+ also have " ... = h x + #0 * xi" by simp
+ also have "... = (let (y,a) = (x, #0) in h y + a * xi)"
+ by (simp add: Let_def)
+ also have
+ "(x, #0) = (SOME (y, a). x = y + a ( * ) x0 & y : H)"
+ by (rule decomp_H0_H [RS sym], rule x0) (simp!)+
+ also have
"(let (y,a) = (SOME (y,a). x = y + a ( * ) x0 & y : H)
in h y + a * xi)
- = h0 x"; by (simp!);
- finally; show "f x = h0 x"; .;
- next;
- from f_h0; show "F <= H0"; ..;
- qed;
+ = h0 x" by (simp!)
+ finally show "f x = h0 x" .
+ next
+ from f_h0 show "F <= H0" ..
+ qed
- show "ALL x:H0. h0 x <= p x";
- by (rule h0_norm_pres, rule x0) (assumption | simp!)+;
- qed;
- thus "graph H0 h0 : M"; by (simp!);
- qed;
- thus ?thesis; ..;
- qed;
+ show "ALL x:H0. h0 x <= p x"
+ by (rule h0_norm_pres, rule x0) (assumption | simp!)+
+ qed
+ thus "graph H0 h0 : M" by (simp!)
+ qed
+ thus ?thesis ..
+ qed
txt {* We have shown that $h$ can still be extended to
some $h_0$, in contradiction to the assumption that
- $h$ is a maximal element. *};
+ $h$ is a maximal element. *}
- hence "EX x:M. g <= x & g ~= x";
- by (elim exE conjE, intro bexI conjI);
- hence "~ (ALL x:M. g <= x --> g = x)"; by simp;
- thus ?thesis; by contradiction;
-qed;
+ hence "EX x:M. g <= x & g ~= x"
+ by (elim exE conjE, intro bexI conjI)
+ hence "~ (ALL x:M. g <= x --> g = x)" by simp
+ thus ?thesis by contradiction
+qed
-txt{* It follows $H = E$, and the thesis can be shown. *};
+txt{* It follows $H = E$, and the thesis can be shown. *}
show "is_linearform E h & (ALL x:F. h x = f x)
- & (ALL x:E. h x <= p x)";
-proof (intro conjI);
- from eq; show "is_linearform E h"; by (simp!);
- show "ALL x:F. h x = f x";
- proof (intro ballI, rule sym);
- fix x; assume "x:F"; show "f x = h x "; ..;
- qed;
- from eq; show "ALL x:E. h x <= p x"; by (force!);
-qed;
+ & (ALL x:E. h x <= p x)"
+proof (intro conjI)
+ from eq show "is_linearform E h" by (simp!)
+ show "ALL x:F. h x = f x"
+ proof (intro ballI, rule sym)
+ fix x assume "x:F" show "f x = h x " ..
+ qed
+ from eq show "ALL x:E. h x <= p x" by (force!)
+qed
-qed;
-qed;
-qed;
+qed
+qed
+qed
*)
-subsection {* Alternative formulation *};
+subsection {* Alternative formulation *}
text {* The following alternative formulation of the Hahn-Banach
Theorem\label{abs-HahnBanach} uses the fact that for a real linear form
@@ -604,35 +604,35 @@
\forall x\in H.\ap |h\ap x|\leq p\ap x& {\rm and}\\
\forall x\in H.\ap h\ap x\leq p\ap x\\
\end{matharray}
-*};
+*}
theorem abs_HahnBanach:
"[| is_vectorspace E; is_subspace F E; is_linearform F f;
is_seminorm E p; ALL x:F. abs (f x) <= p x |]
==> EX g. is_linearform E g & (ALL x:F. g x = f x)
- & (ALL x:E. abs (g x) <= p x)";
-proof -;
+ & (ALL x:E. abs (g x) <= p x)"
+proof -
assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
- "is_linearform F f" "ALL x:F. abs (f x) <= p x";
- have "ALL x:F. f x <= p x"; by (rule abs_ineq_iff [RS iffD1]);
+ "is_linearform F f" "ALL x:F. abs (f x) <= p x"
+ have "ALL x:F. f x <= p x" by (rule abs_ineq_iff [RS iffD1])
hence "EX g. is_linearform E g & (ALL x:F. g x = f x)
- & (ALL x:E. g x <= p x)";
- by (simp! only: HahnBanach);
- thus ?thesis;
- proof (elim exE conjE);
- fix g; assume "is_linearform E g" "ALL x:F. g x = f x"
- "ALL x:E. g x <= p x";
- hence "ALL x:E. abs (g x) <= p x";
- by (simp! add: abs_ineq_iff [OF subspace_refl]);
- thus ?thesis; by (intro exI conjI);
- qed;
-qed;
+ & (ALL x:E. g x <= p x)"
+ by (simp! only: HahnBanach)
+ thus ?thesis
+ proof (elim exE conjE)
+ fix g assume "is_linearform E g" "ALL x:F. g x = f x"
+ "ALL x:E. g x <= p x"
+ hence "ALL x:E. abs (g x) <= p x"
+ by (simp! add: abs_ineq_iff [OF subspace_refl])
+ thus ?thesis by (intro exI conjI)
+ qed
+qed
-subsection {* The Hahn-Banach Theorem for normed spaces *};
+subsection {* The Hahn-Banach Theorem for normed spaces *}
text {* Every continuous linear form $f$ on a subspace $F$ of a
norm space $E$, can be extended to a continuous linear form $g$ on
-$E$ such that $\fnorm{f} = \fnorm {g}$. *};
+$E$ such that $\fnorm{f} = \fnorm {g}$. *}
theorem norm_HahnBanach:
"[| is_normed_vectorspace E norm; is_subspace F E;
@@ -640,119 +640,119 @@
==> EX g. is_linearform E g
& is_continuous E norm g
& (ALL x:F. g x = f x)
- & function_norm E norm g = function_norm F norm f";
-proof -;
- assume e_norm: "is_normed_vectorspace E norm";
- assume f: "is_subspace F E" "is_linearform F f";
- assume f_cont: "is_continuous F norm f";
- have e: "is_vectorspace E"; ..;
- with _; have f_norm: "is_normed_vectorspace F norm"; ..;
+ & function_norm E norm g = function_norm F norm f"
+proof -
+ assume e_norm: "is_normed_vectorspace E norm"
+ assume f: "is_subspace F E" "is_linearform F f"
+ assume f_cont: "is_continuous F norm f"
+ have e: "is_vectorspace E" ..
+ with _ have f_norm: "is_normed_vectorspace F norm" ..
txt{* We define a function $p$ on $E$ as follows:
\begin{matharray}{l}
p \: x = \fnorm f \cdot \norm x\\
\end{matharray}
- *};
+ *}
- def p == "\\<lambda>x. function_norm F norm f * norm x";
+ def p == "\\<lambda>x. function_norm F norm f * norm x"
- txt{* $p$ is a seminorm on $E$: *};
+ txt{* $p$ is a seminorm on $E$: *}
- have q: "is_seminorm E p";
- proof;
- fix x y a; assume "x:E" "y:E";
+ have q: "is_seminorm E p"
+ proof
+ fix x y a assume "x:E" "y:E"
- txt{* $p$ is positive definite: *};
+ txt{* $p$ is positive definite: *}
- show "#0 <= p x";
- proof (unfold p_def, rule real_le_mult_order1a);
- from _ f_norm; show "#0 <= function_norm F norm f"; ..;
- show "#0 <= norm x"; ..;
- qed;
+ show "#0 <= p x"
+ proof (unfold p_def, rule real_le_mult_order1a)
+ from _ f_norm show "#0 <= function_norm F norm f" ..
+ show "#0 <= norm x" ..
+ qed
- txt{* $p$ is absolutely homogenous: *};
+ txt{* $p$ is absolutely homogenous: *}
- show "p (a (*) x) = abs a * p x";
- proof -;
- have "p (a (*) x) = function_norm F norm f * norm (a (*) x)";
- by (simp!);
- also; have "norm (a (*) x) = abs a * norm x";
- by (rule normed_vs_norm_abs_homogenous);
- also; have "function_norm F norm f * (abs a * norm x)
- = abs a * (function_norm F norm f * norm x)";
- by (simp! only: real_mult_left_commute);
- also; have "... = abs a * p x"; by (simp!);
- finally; show ?thesis; .;
- qed;
+ show "p (a (*) x) = abs a * p x"
+ proof -
+ have "p (a (*) x) = function_norm F norm f * norm (a (*) x)"
+ by (simp!)
+ also have "norm (a (*) x) = abs a * norm x"
+ by (rule normed_vs_norm_abs_homogenous)
+ also have "function_norm F norm f * (abs a * norm x)
+ = abs a * (function_norm F norm f * norm x)"
+ by (simp! only: real_mult_left_commute)
+ also have "... = abs a * p x" by (simp!)
+ finally show ?thesis .
+ qed
- txt{* Furthermore, $p$ is subadditive: *};
+ txt{* Furthermore, $p$ is subadditive: *}
- show "p (x + y) <= p x + p y";
- proof -;
- have "p (x + y) = function_norm F norm f * norm (x + y)";
- by (simp!);
- also;
- have "... <= function_norm F norm f * (norm x + norm y)";
- proof (rule real_mult_le_le_mono1a);
- from _ f_norm; show "#0 <= function_norm F norm f"; ..;
- show "norm (x + y) <= norm x + norm y"; ..;
- qed;
- also; have "... = function_norm F norm f * norm x
- + function_norm F norm f * norm y";
- by (simp! only: real_add_mult_distrib2);
- finally; show ?thesis; by (simp!);
- qed;
- qed;
+ show "p (x + y) <= p x + p y"
+ proof -
+ have "p (x + y) = function_norm F norm f * norm (x + y)"
+ by (simp!)
+ also
+ have "... <= function_norm F norm f * (norm x + norm y)"
+ proof (rule real_mult_le_le_mono1a)
+ from _ f_norm show "#0 <= function_norm F norm f" ..
+ show "norm (x + y) <= norm x + norm y" ..
+ qed
+ also have "... = function_norm F norm f * norm x
+ + function_norm F norm f * norm y"
+ by (simp! only: real_add_mult_distrib2)
+ finally show ?thesis by (simp!)
+ qed
+ qed
- txt{* $f$ is bounded by $p$. *};
+ txt{* $f$ is bounded by $p$. *}
- have "ALL x:F. abs (f x) <= p x";
- proof;
- fix x; assume "x:F";
- from f_norm; show "abs (f x) <= p x";
- by (simp! add: norm_fx_le_norm_f_norm_x);
- qed;
+ have "ALL x:F. abs (f x) <= p x"
+ proof
+ fix x assume "x:F"
+ from f_norm show "abs (f x) <= p x"
+ by (simp! add: norm_fx_le_norm_f_norm_x)
+ qed
txt{* Using the fact that $p$ is a seminorm and
$f$ is bounded by $p$ we can apply the Hahn-Banach Theorem
for real vector spaces.
So $f$ can be extended in a norm-preserving way to some function
- $g$ on the whole vector space $E$. *};
+ $g$ on the whole vector space $E$. *}
- with e f q;
+ with e f q
have "EX g. is_linearform E g & (ALL x:F. g x = f x)
- & (ALL x:E. abs (g x) <= p x)";
- by (simp! add: abs_HahnBanach);
+ & (ALL x:E. abs (g x) <= p x)"
+ by (simp! add: abs_HahnBanach)
- thus ?thesis;
- proof (elim exE conjE);
- fix g;
+ thus ?thesis
+ proof (elim exE conjE)
+ fix g
assume "is_linearform E g" and a: "ALL x:F. g x = f x"
- and b: "ALL x:E. abs (g x) <= p x";
+ and b: "ALL x:E. abs (g x) <= p x"
show "EX g. is_linearform E g
& is_continuous E norm g
& (ALL x:F. g x = f x)
- & function_norm E norm g = function_norm F norm f";
- proof (intro exI conjI);
+ & function_norm E norm g = function_norm F norm f"
+ proof (intro exI conjI)
txt{* We furthermore have to show that
- $g$ is also continuous: *};
+ $g$ is also continuous: *}
- show g_cont: "is_continuous E norm g";
- proof;
- fix x; assume "x:E";
- from b [RS bspec, OF this];
- show "abs (g x) <= function_norm F norm f * norm x";
- by (unfold p_def);
- qed;
+ show g_cont: "is_continuous E norm g"
+ proof
+ fix x assume "x:E"
+ from b [RS bspec, OF this]
+ show "abs (g x) <= function_norm F norm f * norm x"
+ by (unfold p_def)
+ qed
txt {* To complete the proof, we show that
- $\fnorm g = \fnorm f$. \label{order_antisym} *};
+ $\fnorm g = \fnorm f$. \label{order_antisym} *}
show "function_norm E norm g = function_norm F norm f"
- (is "?L = ?R");
- proof (rule order_antisym);
+ (is "?L = ?R")
+ proof (rule order_antisym)
txt{* First we show $\fnorm g \leq \fnorm f$. The function norm
$\fnorm g$ is defined as the smallest $c\in\bbbR$ such that
@@ -763,42 +763,42 @@
\begin{matharray}{l}
\All {x\in E} {|g\ap x| \leq \fnorm f \cdot \norm x}
\end{matharray}
- *};
+ *}
- have "ALL x:E. abs (g x) <= function_norm F norm f * norm x";
- proof;
- fix x; assume "x:E";
- show "abs (g x) <= function_norm F norm f * norm x";
- by (simp!);
- qed;
+ have "ALL x:E. abs (g x) <= function_norm F norm f * norm x"
+ proof
+ fix x assume "x:E"
+ show "abs (g x) <= function_norm F norm f * norm x"
+ by (simp!)
+ qed
- with _ g_cont; show "?L <= ?R";
- proof (rule fnorm_le_ub);
- from f_cont f_norm; show "#0 <= function_norm F norm f"; ..;
- qed;
+ with _ g_cont show "?L <= ?R"
+ proof (rule fnorm_le_ub)
+ from f_cont f_norm show "#0 <= function_norm F norm f" ..
+ qed
txt{* The other direction is achieved by a similar
- argument. *};
+ argument. *}
- have "ALL x:F. abs (f x) <= function_norm E norm g * norm x";
- proof;
- fix x; assume "x : F";
- from a; have "g x = f x"; ..;
- hence "abs (f x) = abs (g x)"; by simp;
- also; from _ _ g_cont;
- have "... <= function_norm E norm g * norm x";
- proof (rule norm_fx_le_norm_f_norm_x);
- show "x:E"; ..;
- qed;
- finally; show "abs (f x) <= function_norm E norm g * norm x"; .;
- qed;
- thus "?R <= ?L";
- proof (rule fnorm_le_ub [OF f_norm f_cont]);
- from g_cont; show "#0 <= function_norm E norm g"; ..;
- qed;
- qed;
- qed;
- qed;
-qed;
+ have "ALL x:F. abs (f x) <= function_norm E norm g * norm x"
+ proof
+ fix x assume "x : F"
+ from a have "g x = f x" ..
+ hence "abs (f x) = abs (g x)" by simp
+ also from _ _ g_cont
+ have "... <= function_norm E norm g * norm x"
+ proof (rule norm_fx_le_norm_f_norm_x)
+ show "x:E" ..
+ qed
+ finally show "abs (f x) <= function_norm E norm g * norm x" .
+ qed
+ thus "?R <= ?L"
+ proof (rule fnorm_le_ub [OF f_norm f_cont])
+ from g_cont show "#0 <= function_norm E norm g" ..
+ qed
+ qed
+ qed
+ qed
+qed
-end;
+end
\ No newline at end of file