--- a/src/HOL/Real/HahnBanach/Aux.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/Aux.thy Thu Aug 22 20:49:43 2002 +0200
@@ -3,25 +3,15 @@
Author: Gertrud Bauer, TU Munich
*)
-header {* Auxiliary theorems *}
+header {* Auxiliary theorems *} (* FIXME clean *)
-theory Aux = Real + Zorn:
+theory Aux = Real + Bounds + Zorn:
text {* Some existing theorems are declared as extra introduction
or elimination rules, respectively. *}
-lemmas [intro?] = isLub_isUb
-lemmas [intro?] = chainD
-lemmas chainE2 = chainD2 [elim_format, standard]
-
-
-text {* \medskip Lemmas about sets. *}
-
-lemma Int_singletonD: "A \<inter> B = {v} \<Longrightarrow> x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> x = v"
- by (fast elim: equalityE)
-
-lemma set_less_imp_diff_not_empty: "H < E \<Longrightarrow> \<exists>x0 \<in> E. x0 \<notin> H"
- by (auto simp add: psubset_eq)
+lemmas [dest?] = chainD
+lemmas chainE2 [elim?] = chainD2 [elim_format, standard]
text{* \medskip Some lemmas about orders. *}
--- a/src/HOL/Real/HahnBanach/Bounds.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/Bounds.thy Thu Aug 22 20:49:43 2002 +0200
@@ -7,47 +7,71 @@
theory Bounds = Main + Real:
-text {*
- A supremum\footnote{The definition of the supremum is based on one
- in \url{http://isabelle.in.tum.de/library/HOL/HOL-Real/Lubs.html}}
- of an ordered set @{text B} w.~r.~t. @{text A} is defined as a least
- upper bound of @{text B}, which lies in @{text A}.
-*}
-
-text {*
- If a supremum exists, then @{text "Sup A B"} is equal to the
- supremum. *}
+locale lub =
+ fixes A and x
+ assumes least [intro?]: "(\<And>a. a \<in> A \<Longrightarrow> a \<le> b) \<Longrightarrow> x \<le> b"
+ and upper [intro?]: "a \<in> A \<Longrightarrow> a \<le> x"
+
+lemmas [elim?] = lub.least lub.upper
+
+syntax (xsymbols)
+ the_lub :: "'a::order set \<Rightarrow> 'a" ("\<Squnion>_" [90] 90)
constdefs
- is_Sup :: "('a::order) set \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool"
- "is_Sup A B x \<equiv> isLub A B x"
-
- Sup :: "('a::order) set \<Rightarrow> 'a set \<Rightarrow> 'a"
- "Sup A B \<equiv> Eps (is_Sup A B)"
+ the_lub :: "'a::order set \<Rightarrow> 'a"
+ "\<Squnion>A \<equiv> The (lub A)"
-text {*
- The supremum of @{text B} is less than any upper bound of
- @{text B}. *}
-
-lemma sup_le_ub: "isUb A B y \<Longrightarrow> is_Sup A B s \<Longrightarrow> s \<le> y"
- by (unfold is_Sup_def, rule isLub_le_isUb)
-
-text {* The supremum @{text B} is an upper bound for @{text B}. *}
+lemma the_lub_equality [elim?]:
+ includes lub
+ shows "\<Squnion>A = (x::'a::order)"
+proof (unfold the_lub_def)
+ from lub_axioms show "The (lub A) = x"
+ proof
+ fix x' assume lub': "lub A x'"
+ show "x' = x"
+ proof (rule order_antisym)
+ from lub' show "x' \<le> x"
+ proof
+ fix a assume "a \<in> A"
+ then show "a \<le> x" ..
+ qed
+ show "x \<le> x'"
+ proof
+ fix a assume "a \<in> A"
+ with lub' show "a \<le> x'" ..
+ qed
+ qed
+ qed
+qed
-lemma sup_ub: "y \<in> B \<Longrightarrow> is_Sup A B s \<Longrightarrow> y \<le> s"
- by (unfold is_Sup_def, rule isLubD2)
-
-text {*
- The supremum of a non-empty set @{text B} is greater than a lower
- bound of @{text B}. *}
+lemma the_lubI_ex:
+ assumes ex: "\<exists>x. lub A x"
+ shows "lub A (\<Squnion>A)"
+proof -
+ from ex obtain x where x: "lub A x" ..
+ also from x have [symmetric]: "\<Squnion>A = x" ..
+ finally show ?thesis .
+qed
-lemma sup_ub1:
- "\<forall>y \<in> B. a \<le> y \<Longrightarrow> is_Sup A B s \<Longrightarrow> x \<in> B \<Longrightarrow> a \<le> s"
-proof -
- assume "\<forall>y \<in> B. a \<le> y" "is_Sup A B s" "x \<in> B"
- have "a \<le> x" by (rule bspec)
- also have "x \<le> s" by (rule sup_ub)
- finally show "a \<le> s" .
+lemma lub_compat: "lub A x = isLub UNIV A x"
+proof -
+ have "isUb UNIV A = (\<lambda>x. A *<= x \<and> x \<in> UNIV)"
+ by (rule ext) (simp only: isUb_def)
+ then show ?thesis
+ by (simp only: lub_def isLub_def leastP_def setge_def setle_def) blast
qed
-
+
+lemma real_complete:
+ fixes A :: "real set"
+ assumes nonempty: "\<exists>a. a \<in> A"
+ and ex_upper: "\<exists>y. \<forall>a \<in> A. a \<le> y"
+ shows "\<exists>x. lub A x"
+proof -
+ from ex_upper have "\<exists>y. isUb UNIV A y"
+ by (unfold isUb_def setle_def) blast
+ with nonempty have "\<exists>x. isLub UNIV A x"
+ by (rule reals_complete)
+ then show ?thesis by (simp only: lub_compat)
+qed
+
end
--- a/src/HOL/Real/HahnBanach/FunctionNorm.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/FunctionNorm.thy Thu Aug 22 20:49:43 2002 +0200
@@ -11,7 +11,7 @@
text {*
A linear form @{text f} on a normed vector space @{text "(V, \<parallel>\<cdot>\<parallel>)"}
- is \emph{continuous}, iff it is bounded, i.~e.
+ is \emph{continuous}, iff it is bounded, i.e.
\begin{center}
@{text "\<exists>c \<in> R. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{center}
@@ -20,28 +20,21 @@
linear forms:
*}
-constdefs
- is_continuous ::
- "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
- "is_continuous V norm f \<equiv>
- is_linearform V f \<and> (\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x)"
+locale continuous = var V + norm_syntax + linearform +
+ assumes bounded: "\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
lemma continuousI [intro]:
- "is_linearform V f \<Longrightarrow> (\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * norm x)
- \<Longrightarrow> is_continuous V norm f"
- by (unfold is_continuous_def) blast
-
-lemma continuous_linearform [intro?]:
- "is_continuous V norm f \<Longrightarrow> is_linearform V f"
- by (unfold is_continuous_def) blast
-
-lemma continuous_bounded [intro?]:
- "is_continuous V norm f
- \<Longrightarrow> \<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
- by (unfold is_continuous_def) blast
+ includes norm_syntax + linearform
+ assumes r: "\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
+ shows "continuous V norm f"
+proof
+ show "linearform V f" .
+ from r have "\<exists>c. \<forall>x\<in>V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" by blast
+ then show "continuous_axioms V norm f" ..
+qed
-subsection{* The norm of a linear form *}
+subsection {* The norm of a linear form *}
text {*
The least real number @{text c} for which holds
@@ -62,174 +55,133 @@
element in this set. This element must be @{text "{} \<ge> 0"} so that
@{text function_norm} has the norm properties. Furthermore
it does not have to change the norm in all other cases, so it must
- be @{text 0}, as all other elements of are @{text "{} \<ge> 0"}.
+ be @{text 0}, as all other elements are @{text "{} \<ge> 0"}.
- Thus we define the set @{text B} the supremum is taken from as
+ Thus we define the set @{text B} where the supremum is taken from as
+ follows:
\begin{center}
@{text "{0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel>. x \<noteq> 0 \<and> x \<in> F}"}
\end{center}
-*}
-constdefs
- B :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a::{plus, minus, zero} \<Rightarrow> real) \<Rightarrow> real set"
- "B V norm f \<equiv>
- {0} \<union> {\<bar>f x\<bar> * inverse (norm x) | x. x \<noteq> 0 \<and> x \<in> V}"
-
-text {*
- @{text n} is the function norm of @{text f}, iff @{text n} is the
- supremum of @{text B}.
+ @{text function_norm} is equal to the supremum of @{text B}, if the
+ supremum exists (otherwise it is undefined).
*}
-constdefs
- is_function_norm ::
- "('a::{minus,plus,zero} \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> bool"
- "is_function_norm f V norm fn \<equiv> is_Sup UNIV (B V norm f) fn"
-
-text {*
- @{text function_norm} is equal to the supremum of @{text B}, if the
- supremum exists. Otherwise it is undefined.
-*}
+locale function_norm = norm_syntax +
+ fixes B
+ defines "B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> | x. x \<noteq> 0 \<and> x \<in> V}"
+ fixes function_norm ("\<parallel>_\<parallel>\<hyphen>_" [0, 1000] 999)
+ defines "\<parallel>f\<parallel>\<hyphen>V \<equiv> \<Squnion>(B V f)"
-constdefs
- function_norm :: "('a::{minus,plus,zero} \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real"
- "function_norm f V norm \<equiv> Sup UNIV (B V norm f)"
+lemma (in function_norm) B_not_empty [intro]: "0 \<in> B V f"
+ by (unfold B_def) simp
-syntax
- function_norm :: "('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" ("\<parallel>_\<parallel>_,_")
-
-lemma B_not_empty: "0 \<in> B V norm f"
- by (unfold B_def) blast
+locale (open) functional_vectorspace =
+ normed_vectorspace + function_norm +
+ fixes cont
+ defines "cont f \<equiv> continuous V norm f"
text {*
The following lemma states that every continuous linear form on a
normed space @{text "(V, \<parallel>\<cdot>\<parallel>)"} has a function norm.
*}
-lemma ex_fnorm [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> is_continuous V norm f
- \<Longrightarrow> is_function_norm f V norm \<parallel>f\<parallel>V,norm"
-proof (unfold function_norm_def is_function_norm_def
- is_continuous_def Sup_def, elim conjE, rule someI2_ex)
- assume "is_normed_vectorspace V norm"
- assume "is_linearform V f"
- and e: "\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
-
+lemma (in functional_vectorspace) function_norm_works:
+ includes continuous
+ shows "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)"
+proof -
txt {* The existence of the supremum is shown using the
- completeness of the reals. Completeness means, that
- every non-empty bounded set of reals has a
- supremum. *}
- show "\<exists>a. is_Sup UNIV (B V norm f) a"
- proof (unfold is_Sup_def, rule reals_complete)
-
+ completeness of the reals. Completeness means, that every
+ non-empty bounded set of reals has a supremum. *}
+ have "\<exists>a. lub (B V f) a"
+ proof (rule real_complete)
txt {* First we have to show that @{text B} is non-empty: *}
-
- show "\<exists>X. X \<in> B V norm f"
- proof
- show "0 \<in> (B V norm f)" by (unfold B_def) blast
- qed
+ have "0 \<in> B V f" ..
+ thus "\<exists>x. x \<in> B V f" ..
txt {* Then we have to show that @{text B} is bounded: *}
-
- from e show "\<exists>Y. isUb UNIV (B V norm f) Y"
- proof
-
+ show "\<exists>c. \<forall>y \<in> B V f. y \<le> c"
+ proof -
txt {* We know that @{text f} is bounded by some value @{text c}. *}
-
- fix c assume a: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
- def b \<equiv> "max c 0"
+ from bounded obtain c where c: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
- show "?thesis"
- proof (intro exI isUbI setleI ballI, unfold B_def,
- elim UnE CollectE exE conjE singletonE)
-
- txt {* To proof the thesis, we have to show that there is some
- constant @{text b}, such that @{text "y \<le> b"} for all
- @{text "y \<in> B"}. Due to the definition of @{text B} there are
- two cases for @{text "y \<in> B"}. If @{text "y = 0"} then
- @{text "y \<le> max c 0"}: *}
-
- fix y assume "y = (0::real)"
- show "y \<le> b" by (simp! add: le_maxI2)
-
- txt {* The second case is @{text "y = \<bar>f x\<bar> / \<parallel>x\<parallel>"} for some
- @{text "x \<in> V"} with @{text "x \<noteq> 0"}. *}
+ txt {* To prove the thesis, we have to show that there is some
+ @{text b}, such that @{text "y \<le> b"} for all @{text "y \<in>
+ B"}. Due to the definition of @{text B} there are two cases. *}
- next
- fix x y
- assume "x \<in> V" "x \<noteq> 0"
-
- txt {* The thesis follows by a short calculation using the
- fact that @{text f} is bounded. *}
+ def b \<equiv> "max c 0"
+ have "\<forall>y \<in> B V f. y \<le> b"
+ proof
+ fix y assume y: "y \<in> B V f"
+ show "y \<le> b"
+ proof cases
+ assume "y = 0"
+ thus ?thesis by (unfold b_def) arith
+ next
+ txt {* The second case is @{text "y = \<bar>f x\<bar> / \<parallel>x\<parallel>"} for some
+ @{text "x \<in> V"} with @{text "x \<noteq> 0"}. *}
+ assume "y \<noteq> 0"
+ with y obtain x where y_rep: "y = \<bar>f x\<bar> * inverse \<parallel>x\<parallel>"
+ and x: "x \<in> V" and neq: "x \<noteq> 0"
+ by (auto simp add: B_def real_divide_def)
+ from x neq have gt: "0 < \<parallel>x\<parallel>" ..
- assume "y = \<bar>f x\<bar> * inverse (norm x)"
- also have "... \<le> c * norm x * inverse (norm x)"
- proof (rule real_mult_le_le_mono2)
- show "0 \<le> inverse (norm x)"
- by (rule order_less_imp_le, rule real_inverse_gt_0,
- rule normed_vs_norm_gt_zero)
- from a show "\<bar>f x\<bar> \<le> c * norm x" ..
+ txt {* The thesis follows by a short calculation using the
+ fact that @{text f} is bounded. *}
+
+ note y_rep
+ also have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>"
+ proof (rule real_mult_le_le_mono2)
+ from c show "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
+ from gt have "0 < inverse \<parallel>x\<parallel>" by (rule real_inverse_gt_0)
+ thus "0 \<le> inverse \<parallel>x\<parallel>" by (rule order_less_imp_le)
+ qed
+ also have "\<dots> = c * (\<parallel>x\<parallel> * inverse \<parallel>x\<parallel>)"
+ by (rule real_mult_assoc)
+ also
+ from gt have "\<parallel>x\<parallel> \<noteq> 0" by simp
+ hence "\<parallel>x\<parallel> * inverse \<parallel>x\<parallel> = 1" by (simp add: real_mult_inv_right1)
+ also have "c * 1 \<le> b" by (simp add: b_def le_maxI1)
+ finally show "y \<le> b" .
qed
- also have "... = c * (norm x * inverse (norm x))"
- by (rule real_mult_assoc)
- also have "(norm x * inverse (norm x)) = (1::real)"
- proof (rule real_mult_inv_right1)
- show nz: "norm x \<noteq> 0"
- by (rule not_sym, rule lt_imp_not_eq,
- rule normed_vs_norm_gt_zero)
- qed
- also have "c * ... \<le> b " by (simp! add: le_maxI1)
- finally show "y \<le> b" .
- qed simp
+ qed
+ thus ?thesis ..
qed
qed
+ then show ?thesis
+ by (unfold function_norm_def) (rule the_lubI_ex)
+qed
+
+lemma (in functional_vectorspace) function_norm_ub [intro?]:
+ includes continuous
+ assumes b: "b \<in> B V f"
+ shows "b \<le> \<parallel>f\<parallel>\<hyphen>V"
+proof -
+ have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)" by (rule function_norm_works)
+ from this and b show ?thesis ..
+qed
+
+lemma (in functional_vectorspace) function_norm_least [intro?]:
+ includes continuous
+ assumes b: "\<And>b. b \<in> B V f \<Longrightarrow> b \<le> y"
+ shows "\<parallel>f\<parallel>\<hyphen>V \<le> y"
+proof -
+ have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)" by (rule function_norm_works)
+ from this and b show ?thesis ..
qed
text {* The norm of a continuous function is always @{text "\<ge> 0"}. *}
-lemma fnorm_ge_zero [intro?]:
- "is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm
- \<Longrightarrow> 0 \<le> \<parallel>f\<parallel>V,norm"
+lemma (in functional_vectorspace) function_norm_ge_zero [iff]:
+ includes continuous
+ shows "0 \<le> \<parallel>f\<parallel>\<hyphen>V"
proof -
- assume c: "is_continuous V norm f"
- and n: "is_normed_vectorspace V norm"
-
txt {* The function norm is defined as the supremum of @{text B}.
- So it is @{text "\<ge> 0"} if all elements in @{text B} are
- @{text "\<ge> 0"}, provided the supremum exists and @{text B} is not
- empty. *}
-
- show ?thesis
- proof (unfold function_norm_def, rule sup_ub1)
- show "\<forall>x \<in> (B V norm f). 0 \<le> x"
- proof (intro ballI, unfold B_def,
- elim UnE singletonE CollectE exE conjE)
- fix x r
- assume "x \<in> V" "x \<noteq> 0"
- and r: "r = \<bar>f x\<bar> * inverse (norm x)"
-
- have ge: "0 \<le> \<bar>f x\<bar>" by (simp! only: abs_ge_zero)
- have "0 \<le> inverse (norm x)"
- by (rule order_less_imp_le, rule real_inverse_gt_0, rule)(***
- proof (rule order_less_imp_le);
- show "0 < inverse (norm x)";
- proof (rule real_inverse_gt_zero);
- show "0 < norm x"; ..;
- qed;
- qed; ***)
- with ge show "0 \<le> r"
- by (simp only: r, rule real_le_mult_order1a)
- qed (simp!)
-
- txt {* Since @{text f} is continuous the function norm exists: *}
-
- have "is_function_norm f V norm \<parallel>f\<parallel>V,norm" ..
- thus "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"
- by (unfold is_function_norm_def function_norm_def)
-
- txt {* @{text B} is non-empty by construction: *}
-
- show "0 \<in> B V norm f" by (rule B_not_empty)
- qed
+ So it is @{text "\<ge> 0"} if all elements in @{text B} are @{text "\<ge>
+ 0"}, provided the supremum exists and @{text B} is not empty. *}
+ have "lub (B V f) (\<parallel>f\<parallel>\<hyphen>V)" by (rule function_norm_works)
+ moreover have "0 \<in> B V f" ..
+ ultimately show ?thesis ..
qed
text {*
@@ -239,141 +191,79 @@
\end{center}
*}
-lemma norm_fx_le_norm_f_norm_x:
- "is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm \<Longrightarrow> x \<in> V
- \<Longrightarrow> \<bar>f x\<bar> \<le> \<parallel>f\<parallel>V,norm * norm x"
-proof -
- assume "is_normed_vectorspace V norm" "x \<in> V"
- and c: "is_continuous V norm f"
- have v: "is_vectorspace V" ..
-
- txt{* The proof is by case analysis on @{text x}. *}
-
- show ?thesis
- proof cases
-
- txt {* For the case @{text "x = 0"} the thesis follows from the
- linearity of @{text f}: for every linear function holds
- @{text "f 0 = 0"}. *}
-
- assume "x = 0"
- have "\<bar>f x\<bar> = \<bar>f 0\<bar>" by (simp!)
- also from v continuous_linearform have "f 0 = 0" ..
- also note abs_zero
- also have "0 \<le> \<parallel>f\<parallel>V,norm * norm x"
- proof (rule real_le_mult_order1a)
- show "0 \<le> \<parallel>f\<parallel>V,norm" ..
- show "0 \<le> norm x" ..
+lemma (in functional_vectorspace) function_norm_le_cong:
+ includes continuous + vectorspace_linearform
+ assumes x: "x \<in> V"
+ shows "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
+proof cases
+ assume "x = 0"
+ then have "\<bar>f x\<bar> = \<bar>f 0\<bar>" by simp
+ also have "f 0 = 0" ..
+ also have "\<bar>\<dots>\<bar> = 0" by simp
+ also have "\<dots> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
+ proof (rule real_le_mult_order1a)
+ show "0 \<le> \<parallel>f\<parallel>\<hyphen>V" ..
+ show "0 \<le> norm x" ..
+ qed
+ finally show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>" .
+next
+ assume "x \<noteq> 0"
+ with x have neq: "\<parallel>x\<parallel> \<noteq> 0" by simp
+ then have "\<bar>f x\<bar> = (\<bar>f x\<bar> * inverse \<parallel>x\<parallel>) * \<parallel>x\<parallel>" by simp
+ also have "\<dots> \<le> \<parallel>f\<parallel>\<hyphen>V * \<parallel>x\<parallel>"
+ proof (rule real_mult_le_le_mono2)
+ from x show "0 \<le> \<parallel>x\<parallel>" ..
+ show "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> \<parallel>f\<parallel>\<hyphen>V"
+ proof
+ from x and neq show "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<in> B V f"
+ by (auto simp add: B_def real_divide_def)
qed
- finally
- show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>V,norm * norm x" .
-
- next
- assume "x \<noteq> 0"
- have n: "0 < norm x" ..
- hence nz: "norm x \<noteq> 0"
- by (simp only: lt_imp_not_eq)
-
- txt {* For the case @{text "x \<noteq> 0"} we derive the following fact
- from the definition of the function norm:*}
-
- have l: "\<bar>f x\<bar> * inverse (norm x) \<le> \<parallel>f\<parallel>V,norm"
- proof (unfold function_norm_def, rule sup_ub)
- from ex_fnorm [OF _ c]
- show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"
- by (simp! add: is_function_norm_def function_norm_def)
- show "\<bar>f x\<bar> * inverse (norm x) \<in> B V norm f"
- by (unfold B_def, intro UnI2 CollectI exI [of _ x]
- conjI, simp)
- qed
-
- txt {* The thesis now follows by a short calculation: *}
-
- have "\<bar>f x\<bar> = \<bar>f x\<bar> * 1" by (simp!)
- also from nz have "1 = inverse (norm x) * norm x"
- by (simp add: real_mult_inv_left1)
- also have "\<bar>f x\<bar> * ... = \<bar>f x\<bar> * inverse (norm x) * norm x"
- by (simp! add: real_mult_assoc)
- also from n l have "... \<le> \<parallel>f\<parallel>V,norm * norm x"
- by (simp add: real_mult_le_le_mono2)
- finally show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>V,norm * norm x" .
qed
+ finally show ?thesis .
qed
text {*
\medskip The function norm is the least positive real number for
which the following inequation holds:
\begin{center}
- @{text "\<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
+ @{text "\<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
\end{center}
*}
-lemma fnorm_le_ub:
- "is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm \<Longrightarrow>
- \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x \<Longrightarrow> 0 \<le> c
- \<Longrightarrow> \<parallel>f\<parallel>V,norm \<le> c"
-proof (unfold function_norm_def)
- assume "is_normed_vectorspace V norm"
- assume c: "is_continuous V norm f"
- assume fb: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
- and "0 \<le> c"
-
- txt {* Suppose the inequation holds for some @{text "c \<ge> 0"}. If
- @{text c} is an upper bound of @{text B}, then @{text c} is greater
- than the function norm since the function norm is the least upper
- bound. *}
-
- show "Sup UNIV (B V norm f) \<le> c"
- proof (rule sup_le_ub)
- from ex_fnorm [OF _ c]
- show "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"
- by (simp! add: is_function_norm_def function_norm_def)
-
- txt {* @{text c} is an upper bound of @{text B}, i.e. every
- @{text "y \<in> B"} is less than @{text c}. *}
-
- show "isUb UNIV (B V norm f) c"
- proof (intro isUbI setleI ballI)
- fix y assume "y \<in> B V norm f"
- thus le: "y \<le> c"
- proof (unfold B_def, elim UnE CollectE exE conjE singletonE)
-
- txt {* The first case for @{text "y \<in> B"} is @{text "y = 0"}. *}
-
- assume "y = 0"
- show "y \<le> c" by (blast!)
-
- txt{* The second case is @{text "y = \<bar>f x\<bar> / \<parallel>x\<parallel>"} for some
- @{text "x \<in> V"} with @{text "x \<noteq> 0"}. *}
-
- next
- fix x
- assume "x \<in> V" "x \<noteq> 0"
-
- have lz: "0 < norm x"
- by (simp! add: normed_vs_norm_gt_zero)
-
- have nz: "norm x \<noteq> 0"
- proof (rule not_sym)
- from lz show "0 \<noteq> norm x"
- by (simp! add: order_less_imp_not_eq)
- qed
-
- from lz have "0 < inverse (norm x)"
- by (simp! add: real_inverse_gt_0)
- hence inverse_gez: "0 \<le> inverse (norm x)"
- by (rule order_less_imp_le)
-
- assume "y = \<bar>f x\<bar> * inverse (norm x)"
- also from inverse_gez have "... \<le> c * norm x * inverse (norm x)"
- proof (rule real_mult_le_le_mono2)
- show "\<bar>f x\<bar> \<le> c * norm x" by (rule bspec)
- qed
- also have "... \<le> c" by (simp add: nz real_mult_assoc)
- finally show ?thesis .
- qed
- qed blast
+lemma (in functional_vectorspace) function_norm_least [intro?]:
+ includes continuous
+ assumes ineq: "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" and ge: "0 \<le> c"
+ shows "\<parallel>f\<parallel>\<hyphen>V \<le> c"
+proof (rule function_norm_least)
+ fix b assume b: "b \<in> B V f"
+ show "b \<le> c"
+ proof cases
+ assume "b = 0"
+ with ge show ?thesis by simp
+ next
+ assume "b \<noteq> 0"
+ with b obtain x where b_rep: "b = \<bar>f x\<bar> * inverse \<parallel>x\<parallel>"
+ and x_neq: "x \<noteq> 0" and x: "x \<in> V"
+ by (auto simp add: B_def real_divide_def)
+ note b_rep
+ also have "\<bar>f x\<bar> * inverse \<parallel>x\<parallel> \<le> (c * \<parallel>x\<parallel>) * inverse \<parallel>x\<parallel>"
+ proof (rule real_mult_le_le_mono2)
+ have "0 < \<parallel>x\<parallel>" ..
+ then show "0 \<le> inverse \<parallel>x\<parallel>" by simp
+ from ineq and x show "\<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>" ..
+ qed
+ also have "\<dots> = c"
+ proof -
+ from x_neq and x have "\<parallel>x\<parallel> \<noteq> 0" by simp
+ then show ?thesis by simp
+ qed
+ finally show ?thesis .
qed
qed
+lemmas [iff?] =
+ functional_vectorspace.function_norm_ge_zero
+ functional_vectorspace.function_norm_le_cong
+ functional_vectorspace.function_norm_least
+
end
--- a/src/HOL/Real/HahnBanach/FunctionOrder.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/FunctionOrder.thy Thu Aug 22 20:49:43 2002 +0200
@@ -20,79 +20,78 @@
graph.
*}
-types 'a graph = "('a * real) set"
+types 'a graph = "('a \<times> real) set"
constdefs
- graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph "
+ graph :: "'a set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a graph"
"graph F f \<equiv> {(x, f x) | x. x \<in> F}"
-lemma graphI [intro?]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
+lemma graphI [intro]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
by (unfold graph_def) blast
-lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t\<in> (graph F f). t = (x, f x)"
+lemma graphI2 [intro?]: "x \<in> F \<Longrightarrow> \<exists>t \<in> graph F f. t = (x, f x)"
by (unfold graph_def) blast
-lemma graphD1 [intro?]: "(x, y) \<in> graph F f \<Longrightarrow> x \<in> F"
- by (unfold graph_def) blast
-
-lemma graphD2 [intro?]: "(x, y) \<in> graph H h \<Longrightarrow> y = h x"
+lemma graphE [elim?]:
+ "(x, y) \<in> graph F f \<Longrightarrow> (x \<in> F \<Longrightarrow> y = f x \<Longrightarrow> C) \<Longrightarrow> C"
by (unfold graph_def) blast
subsection {* Functions ordered by domain extension *}
-text {* A function @{text h'} is an extension of @{text h}, iff the
- graph of @{text h} is a subset of the graph of @{text h'}. *}
+text {*
+ A function @{text h'} is an extension of @{text h}, iff the graph of
+ @{text h} is a subset of the graph of @{text h'}.
+*}
lemma graph_extI:
"(\<And>x. x \<in> H \<Longrightarrow> h x = h' x) \<Longrightarrow> H \<subseteq> H'
- \<Longrightarrow> graph H h \<subseteq> graph H' h'"
+ \<Longrightarrow> graph H h \<subseteq> graph H' h'"
by (unfold graph_def) blast
-lemma graph_extD1 [intro?]:
+lemma graph_extD1 [dest?]:
"graph H h \<subseteq> graph H' h' \<Longrightarrow> x \<in> H \<Longrightarrow> h x = h' x"
by (unfold graph_def) blast
-lemma graph_extD2 [intro?]:
+lemma graph_extD2 [dest?]:
"graph H h \<subseteq> graph H' h' \<Longrightarrow> H \<subseteq> H'"
by (unfold graph_def) blast
+
subsection {* Domain and function of a graph *}
text {*
- The inverse functions to @{text graph} are @{text domain} and
- @{text funct}.
+ The inverse functions to @{text graph} are @{text domain} and @{text
+ funct}.
*}
constdefs
- domain :: "'a graph \<Rightarrow> 'a set"
+ "domain" :: "'a graph \<Rightarrow> 'a set"
"domain g \<equiv> {x. \<exists>y. (x, y) \<in> g}"
funct :: "'a graph \<Rightarrow> ('a \<Rightarrow> real)"
"funct g \<equiv> \<lambda>x. (SOME y. (x, y) \<in> g)"
-
text {*
The following lemma states that @{text g} is the graph of a function
if the relation induced by @{text g} is unique.
*}
lemma graph_domain_funct:
- "(\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y)
- \<Longrightarrow> graph (domain g) (funct g) = g"
-proof (unfold domain_def funct_def graph_def, auto)
+ assumes uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
+ shows "graph (domain g) (funct g) = g"
+proof (unfold domain_def funct_def graph_def, auto) (* FIXME !? *)
fix a b assume "(a, b) \<in> g"
show "(a, SOME y. (a, y) \<in> g) \<in> g" by (rule someI2)
show "\<exists>y. (a, y) \<in> g" ..
- assume uniq: "\<And>x y z. (x, y) \<in> g \<Longrightarrow> (x, z) \<in> g \<Longrightarrow> z = y"
show "b = (SOME y. (a, y) \<in> g)"
proof (rule some_equality [symmetric])
- fix y assume "(a, y) \<in> g" show "y = b" by (rule uniq)
+ fix y assume "(a, y) \<in> g"
+ show "y = b" by (rule uniq)
qed
qed
-
subsection {* Norm-preserving extensions of a function *}
text {*
@@ -105,39 +104,35 @@
constdefs
norm_pres_extensions ::
"'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
- \<Rightarrow> 'a graph set"
+ \<Rightarrow> 'a graph set"
"norm_pres_extensions E p F f
- \<equiv> {g. \<exists>H h. graph H h = g
- \<and> is_linearform H h
- \<and> is_subspace H E
- \<and> is_subspace F H
- \<and> graph F f \<subseteq> graph H h
- \<and> (\<forall>x \<in> H. h x \<le> p x)}"
+ \<equiv> {g. \<exists>H h. g = 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)}"
-lemma norm_pres_extension_D:
+lemma norm_pres_extensionE [elim]:
"g \<in> norm_pres_extensions E p F f
- \<Longrightarrow> \<exists>H h. graph H h = g
- \<and> is_linearform H h
- \<and> is_subspace H E
- \<and> is_subspace F H
- \<and> graph F f \<subseteq> graph H h
- \<and> (\<forall>x \<in> H. h x \<le> p x)"
+ \<Longrightarrow> (\<And>H h. g = graph H h \<Longrightarrow> linearform H h
+ \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H \<Longrightarrow> graph F f \<subseteq> graph H h
+ \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x \<Longrightarrow> C) \<Longrightarrow> C"
by (unfold norm_pres_extensions_def) blast
lemma norm_pres_extensionI2 [intro]:
- "is_linearform H h \<Longrightarrow> is_subspace H E \<Longrightarrow> is_subspace F H \<Longrightarrow>
- graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x
- \<Longrightarrow> (graph H h \<in> norm_pres_extensions E p F f)"
- by (unfold norm_pres_extensions_def) blast
+ "linearform H h \<Longrightarrow> H \<unlhd> E \<Longrightarrow> F \<unlhd> H
+ \<Longrightarrow> graph F f \<subseteq> graph H h \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x
+ \<Longrightarrow> graph H h \<in> norm_pres_extensions E p F f"
+ by (unfold norm_pres_extensions_def) blast
-lemma norm_pres_extensionI [intro]:
- "\<exists>H h. graph H h = g
- \<and> is_linearform H h
- \<and> is_subspace H E
- \<and> is_subspace F H
- \<and> graph F f \<subseteq> graph H h
- \<and> (\<forall>x \<in> H. h x \<le> p x)
- \<Longrightarrow> g \<in> norm_pres_extensions E p F f"
+lemma norm_pres_extensionI: (* FIXME ? *)
+ "\<exists>H h. g = 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) \<Longrightarrow> g \<in> norm_pres_extensions E p F f"
by (unfold norm_pres_extensions_def) blast
end
--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy Thu Aug 22 20:49:43 2002 +0200
@@ -49,251 +49,235 @@
\item Thus @{text g} can not be maximal. Contradiction!
\end{itemize}
-
\end{enumerate}
*}
theorem HahnBanach:
- "is_vectorspace E \<Longrightarrow> is_subspace F E \<Longrightarrow> is_seminorm E p
- \<Longrightarrow> is_linearform F f \<Longrightarrow> \<forall>x \<in> F. f x \<le> p x
- \<Longrightarrow> \<exists>h. is_linearform E h \<and> (\<forall>x \<in> F. h x = f x) \<and> (\<forall>x \<in> E. h x \<le> p x)"
+ includes vectorspace E + subvectorspace F E +
+ seminorm_vectorspace E p + linearform F f
+ assumes fp: "\<forall>x \<in> F. f x \<le> p x"
+ shows "\<exists>h. linearform E h \<and> (\<forall>x \<in> F. h x = f x) \<and> (\<forall>x \<in> E. h x \<le> p x)"
-- {* Let @{text E} be a vector space, @{text F} a subspace of @{text E}, @{text p} a seminorm on @{text E}, *}
-- {* and @{text f} a linear form on @{text F} such that @{text f} is bounded by @{text p}, *}
-- {* then @{text f} can be extended to a linear form @{text h} on @{text E} in a norm-preserving way. \skp *}
proof -
- assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
- and "is_linearform F f" "\<forall>x \<in> F. f x \<le> p x"
- -- {* Assume the context of the theorem. \skp *}
def M \<equiv> "norm_pres_extensions E p F f"
- -- {* Define @{text M} as the set of all norm-preserving extensions of @{text F}. \skp *}
+ hence M: "M = \<dots>" by (simp only:)
+ have F: "vectorspace F" ..
{
- fix c assume "c \<in> chain M" "\<exists>x. x \<in> c"
+ fix c assume cM: "c \<in> chain M" and ex: "\<exists>x. x \<in> c"
have "\<Union>c \<in> M"
- -- {* Show that every non-empty chain @{text c} of @{text M} has an upper bound in @{text M}: *}
- -- {* @{text "\<Union>c"} is greater than any element of the chain @{text c}, so it suffices to show @{text "\<Union>c \<in> M"}. *}
+ -- {* Show that every non-empty chain @{text c} of @{text M} has an upper bound in @{text M}: *}
+ -- {* @{text "\<Union>c"} is greater than any element of the chain @{text c}, so it suffices to show @{text "\<Union>c \<in> M"}. *}
proof (unfold M_def, rule norm_pres_extensionI)
- show "\<exists>H h. graph H h = \<Union>c
- \<and> is_linearform H h
- \<and> is_subspace H E
- \<and> is_subspace F H
- \<and> graph F f \<subseteq> graph H h
- \<and> (\<forall>x \<in> H. h x \<le> p x)"
- proof (intro exI conjI)
- let ?H = "domain (\<Union>c)"
- let ?h = "funct (\<Union>c)"
+ 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) \<in> \<Union>c" "(x, z) \<in> \<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 \<subseteq> graph ?H ?h"
- by (rule sup_ext, rule a) (simp!)+
- show "\<forall>x \<in> ?H. ?h x \<le> p x"
- by (rule sup_norm_pres, rule a) (simp!)+
+ have a: "graph ?H ?h = \<Union>c"
+ proof (rule graph_domain_funct)
+ fix x y z assume "(x, y) \<in> \<Union>c" and "(x, z) \<in> \<Union>c"
+ with M_def cM show "z = y" by (rule sup_definite)
qed
+ moreover from M cM a have "linearform ?H ?h"
+ by (rule sup_lf)
+ moreover from a M cM ex have "?H \<unlhd> E"
+ by (rule sup_subE)
+ moreover from a M cM ex have "F \<unlhd> ?H"
+ by (rule sup_supF)
+ moreover from a M cM ex have "graph F f \<subseteq> graph ?H ?h"
+ by (rule sup_ext)
+ moreover from a M cM have "\<forall>x \<in> ?H. ?h x \<le> p x"
+ by (rule sup_norm_pres)
+ ultimately show "\<exists>H h. \<Union>c = 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)" by blast
qed
-
}
hence "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
-- {* With Zorn's Lemma we can conclude that there is a maximal element in @{text M}. \skp *}
proof (rule Zorn's_Lemma)
- -- {* We show that @{text M} is non-empty: *}
- have "graph F f \<in> 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 \<in> M" by (simp!)
+ -- {* We show that @{text M} is non-empty: *}
+ show "graph F f \<in> M"
+ proof (unfold M_def, rule norm_pres_extensionI2)
+ show "linearform F f" .
+ show "F \<unlhd> E" .
+ from F show "F \<unlhd> F" by (rule vectorspace.subspace_refl)
+ show "graph F f \<subseteq> graph F f" ..
+ show "\<forall>x\<in>F. f x \<le> p x" .
+ qed
qed
- thus ?thesis
- proof
- fix g assume "g \<in> M" "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
- -- {* We consider such a maximal element @{text "g \<in> M"}. \skp *}
- obtain H h where "graph H h = g" "is_linearform H h"
- "is_subspace H E" "is_subspace F H" "graph F f \<subseteq> graph H h"
- "\<forall>x \<in> H. h x \<le> p x"
+ then obtain g where gM: "g \<in> M" and "\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x"
+ by blast
+ from gM [unfolded M_def] obtain H h where
+ g_rep: "g = graph H h"
+ and linearform: "linearform H h"
+ and HE: "H \<unlhd> E" and FH: "F \<unlhd> H"
+ and graphs: "graph F f \<subseteq> graph H h"
+ and hp: "\<forall>x \<in> H. h x \<le> p x" ..
-- {* @{text g} is a norm-preserving extension of @{text f}, in other words: *}
-- {* @{text g} is the graph of some linear form @{text h} defined on a subspace @{text H} of @{text E}, *}
-- {* and @{text h} is an extension of @{text f} that is again bounded by @{text p}. \skp *}
- proof -
- have "\<exists>H h. graph H h = g \<and> is_linearform H h
- \<and> is_subspace H E \<and> is_subspace F H
- \<and> graph F f \<subseteq> graph H h
- \<and> (\<forall>x \<in> H. h x \<le> p x)"
- by (simp! add: norm_pres_extension_D)
- with that show ?thesis by blast
- qed
- have h: "is_vectorspace H" ..
- have "H = E"
+ from HE have H: "vectorspace H"
+ by (rule subvectorspace.vectorspace)
+
+ have HE_eq: "H = E"
-- {* We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp *}
- proof (rule classical)
- assume "H \<noteq> E"
+ proof (rule classical)
+ assume neq: "H \<noteq> E"
-- {* Assume @{text h} is not defined on whole @{text E}. Then show that @{text h} can be extended *}
-- {* in a norm-preserving way to a function @{text h'} with the graph @{text g'}. \skp *}
- have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
- proof -
- obtain x' where "x' \<in> E" "x' \<notin> H"
- -- {* Pick @{text "x' \<in> E - H"}. \skp *}
- proof -
- have "\<exists>x' \<in> E. x' \<notin> H"
- proof (rule set_less_imp_diff_not_empty)
- have "H \<subseteq> E" ..
- thus "H \<subset> E" ..
- qed
- with that show ?thesis by blast
+ have "\<exists>g' \<in> M. g \<subseteq> g' \<and> g \<noteq> g'"
+ proof -
+ from HE have "H \<subseteq> E" ..
+ with neq obtain x' where x'E: "x' \<in> E" and "x' \<notin> H" by blast
+ obtain x': "x' \<noteq> 0"
+ proof
+ show "x' \<noteq> 0"
+ proof
+ assume "x' = 0"
+ with H have "x' \<in> H" by (simp only: vectorspace.zero)
+ then show False by contradiction
qed
- have x': "x' \<noteq> 0"
- proof (rule classical)
- presume "x' = 0"
- with h have "x' \<in> H" by simp
- thus ?thesis by contradiction
- qed blast
- def H' \<equiv> "H + lin x'"
+ qed
+
+ def H' \<equiv> "H + lin x'"
-- {* Define @{text H'} as the direct sum of @{text H} and the linear closure of @{text x'}. \skp *}
- obtain xi where "\<forall>y \<in> H. - p (y + x') - h y \<le> xi
- \<and> xi \<le> p (y + x') - h y"
+ have HH': "H \<unlhd> H'"
+ proof (unfold H'_def)
+ have "vectorspace (lin x')" ..
+ with H show "H \<unlhd> H + lin x'" ..
+ qed
+
+ obtain xi where
+ "\<forall>y \<in> H. - p (y + x') - h y \<le> xi
+ \<and> xi \<le> p (y + x') - h y"
-- {* Pick a real number @{text \<xi>} that fulfills certain inequations; this will *}
-- {* be used to establish that @{text h'} is a norm-preserving extension of @{text h}.
\label{ex-xi-use}\skp *}
+ proof -
+ from H have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
+ \<and> xi \<le> p (y + x') - h y"
+ proof (rule ex_xi)
+ fix u v assume u: "u \<in> H" and v: "v \<in> H"
+ with HE have uE: "u \<in> E" and vE: "v \<in> E" by auto
+ from H u v linearform have "h v - h u = h (v - u)"
+ by (simp add: vectorspace_linearform.diff)
+ also from hp and H u v have "\<dots> \<le> p (v - u)"
+ by (simp only: vectorspace.diff_closed)
+ also from x'E uE vE have "v - u = x' + - x' + v + - u"
+ by (simp add: diff_eq1)
+ also from x'E uE vE have "\<dots> = v + x' + - (u + x')"
+ by (simp add: add_ac)
+ also from x'E uE vE have "\<dots> = (v + x') - (u + x')"
+ by (simp add: diff_eq1)
+ also from x'E uE vE have "p \<dots> \<le> p (v + x') + p (u + x')"
+ by (simp add: diff_subadditive)
+ finally have "h v - h u \<le> p (v + x') + p (u + x')" .
+ then show "- p (u + x') - h u \<le> p (v + x') - h v"
+ by simp
+ qed
+ then show ?thesis ..
+ qed
+
+ def h' \<equiv> "\<lambda>x. let (y, a) =
+ SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H in h y + a * xi"
+ -- {* Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp *}
+
+ have "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
+ -- {* @{text h'} is an extension of @{text h} \dots \skp *}
+ proof
+ show "g \<subseteq> graph H' h'"
proof -
- from h have "\<exists>xi. \<forall>y \<in> H. - p (y + x') - h y \<le> xi
- \<and> xi \<le> p (y + x') - h y"
- proof (rule ex_xi)
- fix u v assume "u \<in> H" "v \<in> H"
- from h have "h v - h u = h (v - u)"
- by (simp! add: linearform_diff)
- also have "... \<le> p (v - u)"
- by (simp!)
- also have "v - u = x' + - x' + v + - u"
- by (simp! add: diff_eq1)
- also have "... = v + x' + - (u + x')"
- by (simp!)
- also have "... = (v + x') - (u + x')"
- by (simp! add: diff_eq1)
- also have "p ... \<le> p (v + x') + p (u + x')"
- by (rule seminorm_diff_subadditive) (simp_all!)
- finally have "h v - h u \<le> p (v + x') + p (u + x')" .
-
- thus "- p (u + x') - h u \<le> p (v + x') - h v"
- by (rule real_diff_ineq_swap)
+ have "graph H h \<subseteq> graph H' h'"
+ proof (rule graph_extI)
+ fix t assume t: "t \<in> H"
+ have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
+ by (rule decomp_H'_H)
+ with h'_def show "h t = h' t" by (simp add: Let_def)
+ next
+ from HH' show "H \<subseteq> H'" ..
qed
- thus ?thesis ..
+ with g_rep show ?thesis by (simp only:)
qed
- def h' \<equiv> "\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H
- in h y + a * xi"
- -- {* Define the extension @{text h'} of @{text h} to @{text H'} using @{text \<xi>}. \skp *}
- show ?thesis
- proof
- show "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
- -- {* Show that @{text h'} is an extension of @{text h} \dots \skp *}
+ show "g \<noteq> graph H' h'"
+ proof -
+ have "graph H h \<noteq> graph H' h'"
proof
- show "g \<subseteq> graph H' h'"
- proof -
- have "graph H h \<subseteq> graph H' h'"
- proof (rule graph_extI)
- fix t assume "t \<in> H"
- have "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H)
- = (t, 0)"
- by (rule decomp_H'_H) (assumption+, rule x')
- thus "h t = h' t" by (simp! add: Let_def)
- next
- show "H \<subseteq> H'"
- proof (rule subspace_subset)
- show "is_subspace H H'"
- proof (unfold H'_def, rule subspace_vs_sum1)
- show "is_vectorspace H" ..
- show "is_vectorspace (lin x')" ..
- qed
- qed
- qed
- thus ?thesis by (simp!)
+ assume eq: "graph H h = graph H' h'"
+ have "x' \<in> H'"
+ proof (unfold H'_def, rule)
+ from H show "0 \<in> H" by (rule vectorspace.zero)
+ from x'E show "x' \<in> lin x'" by (rule x_lin_x)
+ from x'E show "x' = 0 + x'" by simp
qed
- show "g \<noteq> graph H' h'"
- proof -
- have "graph H h \<noteq> graph H' h'"
- proof
- assume e: "graph H h = graph H' h'"
- have "x' \<in> H'"
- proof (unfold H'_def, rule vs_sumI)
- show "x' = 0 + x'" by (simp!)
- from h show "0 \<in> H" ..
- show "x' \<in> lin x'" by (rule x_lin_x)
- qed
- hence "(x', h' x') \<in> graph H' h'" ..
- with e have "(x', h' x') \<in> graph H h" by simp
- hence "x' \<in> H" ..
- thus False by contradiction
- qed
- thus ?thesis by (simp!)
- qed
+ hence "(x', h' x') \<in> graph H' h'" ..
+ with eq have "(x', h' x') \<in> graph H h" by (simp only:)
+ hence "x' \<in> H" ..
+ thus False by contradiction
qed
- show "graph H' h' \<in> M"
- -- {* and @{text h'} is norm-preserving. \skp *}
- proof -
- have "graph H' h' \<in> norm_pres_extensions E p F f"
- proof (rule norm_pres_extensionI2)
- show "is_linearform H' h'"
- by (rule h'_lf) (simp! add: x')+
- show "is_subspace H' E"
- by (unfold H'_def)
- (rule vs_sum_subspace [OF _ lin_subspace])
- have "is_subspace F H" .
- also from h lin_vs
- have [folded H'_def]: "is_subspace H (H + lin x')" ..
- finally (subspace_trans [OF _ h])
- show f_h': "is_subspace F H'" .
-
- show "graph F f \<subseteq> graph H' h'"
- proof (rule graph_extI)
- fix x assume "x \<in> 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 \<cdot> x' \<and> y \<in> H)"
- by (rule decomp_H'_H [symmetric]) (simp! add: x')+
- also have
- "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)
- in h y + a * xi) = h' x" by (simp!)
- finally show "f x = h' x" .
- next
- from f_h' show "F \<subseteq> H'" ..
- qed
-
- show "\<forall>x \<in> H'. h' x \<le> p x"
- by (rule h'_norm_pres) (assumption+, rule x')
- qed
- thus "graph H' h' \<in> M" by (simp!)
- qed
+ with g_rep show ?thesis by simp
qed
qed
- hence "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
- -- {* So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \skp *}
- thus "H = E" by contradiction
+ moreover have "graph H' h' \<in> M"
+ -- {* and @{text h'} is norm-preserving. \skp *}
+ proof (unfold M_def)
+ show "graph H' h' \<in> norm_pres_extensions E p F f"
+ proof (rule norm_pres_extensionI2)
+ show "linearform H' h'" by (rule h'_lf)
+ show "H' \<unlhd> E"
+ proof (unfold H'_def, rule)
+ show "H \<unlhd> E" .
+ show "vectorspace E" .
+ from x'E show "lin x' \<unlhd> E" ..
+ qed
+ have "F \<unlhd> H" .
+ from H this HH' show FH': "F \<unlhd> H'"
+ by (rule vectorspace.subspace_trans)
+ show "graph F f \<subseteq> graph H' h'"
+ proof (rule graph_extI)
+ fix x assume x: "x \<in> F"
+ with graphs have "f x = h x" ..
+ also have "\<dots> = h x + 0 * xi" by simp
+ also have "\<dots> = (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 \<cdot> x' \<and> y \<in> H)"
+ proof (rule decomp_H'_H [symmetric])
+ from FH x show "x \<in> H" ..
+ from x' show "x' \<noteq> 0" .
+ qed
+ also have
+ "(let (y, a) = (SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H)
+ in h y + a * xi) = h' x" by (simp only: h'_def)
+ finally show "f x = h' x" .
+ next
+ from FH' show "F \<subseteq> H'" ..
+ qed
+ show "\<forall>x \<in> H'. h' x \<le> p x" by (rule h'_norm_pres)
+ qed
+ qed
+ ultimately show ?thesis ..
qed
- thus "\<exists>h. is_linearform E h \<and> (\<forall>x \<in> F. h x = f x)
- \<and> (\<forall>x \<in> E. h x \<le> p x)"
- proof (intro exI conjI)
- assume eq: "H = E"
- from eq show "is_linearform E h" by (simp!)
- show "\<forall>x \<in> F. h x = f x"
- proof
- fix x assume "x \<in> F" have "f x = h x " ..
- thus "h x = f x" ..
- qed
- from eq show "\<forall>x \<in> E. h x \<le> p x" by (blast!)
- qed
+ hence "\<not> (\<forall>x \<in> M. g \<subseteq> x \<longrightarrow> g = x)" by simp
+ -- {* So the graph @{text g} of @{text h} cannot be maximal. Contradiction! \skp *}
+ then show "H = E" by contradiction
qed
+
+ from HE_eq and linearform have "linearform E h"
+ by (simp only:)
+ moreover have "\<forall>x \<in> F. h x = f x"
+ proof
+ fix x assume "x \<in> F"
+ with graphs have "f x = h x" ..
+ then show "h x = f x" ..
+ qed
+ moreover from HE_eq and hp have "\<forall>x \<in> E. h x \<le> p x"
+ by (simp only:)
+ ultimately show ?thesis by blast
qed
@@ -314,26 +298,28 @@
*}
theorem abs_HahnBanach:
- "is_vectorspace E \<Longrightarrow> is_subspace F E \<Longrightarrow> is_linearform F f
- \<Longrightarrow> is_seminorm E p \<Longrightarrow> \<forall>x \<in> F. \<bar>f x\<bar> \<le> p x
- \<Longrightarrow> \<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
+ includes vectorspace E + subvectorspace F E +
+ linearform F f + seminorm_vectorspace E p
+ assumes fp: "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
+ shows "\<exists>g. linearform E g
+ \<and> (\<forall>x \<in> F. g x = f x)
\<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"
proof -
-assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p"
-"is_linearform F f" "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
-have "\<forall>x \<in> F. f x \<le> p x" by (rule abs_ineq_iff [THEN iffD1])
-hence "\<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
- \<and> (\<forall>x \<in> E. g x \<le> p x)"
-by (simp! only: HahnBanach)
-thus ?thesis
-proof (elim exE conjE)
-fix g assume "is_linearform E g" "\<forall>x \<in> F. g x = f x"
- "\<forall>x \<in> E. g x \<le> p x"
-hence "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
- by (simp! add: abs_ineq_iff [OF subspace_refl])
-thus ?thesis by (intro exI conjI)
+ have "\<exists>g. linearform E g \<and> (\<forall>x \<in> F. g x = f x)
+ \<and> (\<forall>x \<in> E. g x \<le> p x)"
+ proof (rule HahnBanach)
+ show "\<forall>x \<in> F. f x \<le> p x"
+ by (rule abs_ineq_iff [THEN iffD1])
+ qed
+ then obtain g where * : "linearform E g" "\<forall>x \<in> F. g x = f x"
+ and "\<forall>x \<in> E. g x \<le> p x" by blast
+ have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
+ proof (rule abs_ineq_iff [THEN iffD2])
+ show "E \<unlhd> E" ..
+ qed
+ with * show ?thesis by blast
qed
-qed
+
subsection {* The Hahn-Banach Theorem for normed spaces *}
@@ -344,126 +330,108 @@
*}
theorem norm_HahnBanach:
- "is_normed_vectorspace E norm \<Longrightarrow> is_subspace F E
- \<Longrightarrow> is_linearform F f \<Longrightarrow> is_continuous F norm f
- \<Longrightarrow> \<exists>g. is_linearform E g
- \<and> is_continuous E norm g
+ includes functional_vectorspace E + subvectorspace F E +
+ linearform F f + continuous F norm f
+ shows "\<exists>g. linearform E g
+ \<and> continuous E norm g
\<and> (\<forall>x \<in> F. g x = f x)
- \<and> \<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
-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" ..
-hence f_norm: "is_normed_vectorspace F norm" ..
-
-txt{*
- We define a function @{text p} on @{text E} as follows:
- @{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
-*}
-
-def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>F,norm * norm x"
-
-txt {* @{text p} is a seminorm on @{text E}: *}
-
-have q: "is_seminorm E p"
-proof
-fix x y a assume "x \<in> E" "y \<in> E"
-
-txt {* @{text p} is positive definite: *}
-
-show "0 \<le> p x"
-proof (unfold p_def, rule real_le_mult_order1a)
- from f_cont f_norm show "0 \<le> \<parallel>f\<parallel>F,norm" ..
- show "0 \<le> norm x" ..
-qed
-
-txt {* @{text p} is absolutely homogenous: *}
-
-show "p (a \<cdot> x) = \<bar>a\<bar> * p x"
-proof -
- have "p (a \<cdot> x) = \<parallel>f\<parallel>F,norm * norm (a \<cdot> x)"
- by (simp!)
- also have "norm (a \<cdot> x) = \<bar>a\<bar> * norm x"
- by (rule normed_vs_norm_abs_homogenous)
- also have "\<parallel>f\<parallel>F,norm * (\<bar>a\<bar> * norm x )
- = \<bar>a\<bar> * (\<parallel>f\<parallel>F,norm * norm x)"
- by (simp! only: real_mult_left_commute)
- also have "... = \<bar>a\<bar> * p x" by (simp!)
- finally show ?thesis .
-qed
-
-txt {* Furthermore, @{text p} is subadditive: *}
-
-show "p (x + y) \<le> p x + p y"
+ \<and> \<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
proof -
- have "p (x + y) = \<parallel>f\<parallel>F,norm * norm (x + y)"
- by (simp!)
- also
- have "... \<le> \<parallel>f\<parallel>F,norm * (norm x + norm y)"
- proof (rule real_mult_le_le_mono1a)
- from f_cont f_norm show "0 \<le> \<parallel>f\<parallel>F,norm" ..
- show "norm (x + y) \<le> norm x + norm y" ..
- qed
- also have "... = \<parallel>f\<parallel>F,norm * norm x
- + \<parallel>f\<parallel>F,norm * norm y"
- by (simp! only: real_add_mult_distrib2)
- finally show ?thesis by (simp!)
-qed
-qed
+ have E: "vectorspace E" .
+ have E_norm: "normed_vectorspace E norm" ..
+ have FE: "F \<unlhd> E" .
+ have F: "vectorspace F" ..
+ have linearform: "linearform F f" .
+ have F_norm: "normed_vectorspace F norm" ..
+
+ txt {* We define a function @{text p} on @{text E} as follows:
+ @{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"} *}
+ def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
+
+ txt {* @{text p} is a seminorm on @{text E}: *}
+ have q: "seminorm E p"
+ proof
+ fix x y a assume x: "x \<in> E" and y: "y \<in> E"
-txt {* @{text f} is bounded by @{text p}. *}
+ txt {* @{text p} is positive definite: *}
+ show "0 \<le> p x"
+ proof (unfold p_def, rule real_le_mult_order1a)
+ show "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
+ apply (unfold function_norm_def B_def)
+ using normed_vectorspace.axioms [OF F_norm] ..
+ from x show "0 \<le> \<parallel>x\<parallel>" ..
+ qed
+
+ txt {* @{text p} is absolutely homogenous: *}
-have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
-proof
-fix x assume "x \<in> F"
- from f_norm show "\<bar>f x\<bar> \<le> p x"
- by (simp! add: norm_fx_le_norm_f_norm_x)
-qed
+ show "p (a \<cdot> x) = \<bar>a\<bar> * p x"
+ proof -
+ have "p (a \<cdot> x) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>a \<cdot> x\<parallel>"
+ by (simp only: p_def)
+ also from x have "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"
+ by (rule abs_homogenous)
+ also have "\<parallel>f\<parallel>\<hyphen>F * (\<bar>a\<bar> * \<parallel>x\<parallel>) = \<bar>a\<bar> * (\<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>)"
+ by simp
+ also have "\<dots> = \<bar>a\<bar> * p x"
+ by (simp only: p_def)
+ finally show ?thesis .
+ qed
+
+ txt {* Furthermore, @{text p} is subadditive: *}
-txt {*
- Using the fact that @{text p} is a seminorm and @{text f} is bounded
- by @{text p} we can apply the Hahn-Banach Theorem for real vector
- spaces. So @{text f} can be extended in a norm-preserving way to
- some function @{text g} on the whole vector space @{text E}.
-*}
+ show "p (x + y) \<le> p x + p y"
+ proof -
+ have "p (x + y) = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x + y\<parallel>"
+ by (simp only: p_def)
+ also have "\<dots> \<le> \<parallel>f\<parallel>\<hyphen>F * (\<parallel>x\<parallel> + \<parallel>y\<parallel>)"
+ proof (rule real_mult_le_le_mono1a)
+ show "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
+ apply (unfold function_norm_def B_def)
+ using normed_vectorspace.axioms [OF F_norm] .. (* FIXME *)
+ from x y show "\<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>" ..
+ qed
+ also have "\<dots> = \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel> + \<parallel>f\<parallel>\<hyphen>F * \<parallel>y\<parallel>"
+ by (simp only: real_add_mult_distrib2)
+ also have "\<dots> = p x + p y"
+ by (simp only: p_def)
+ finally show ?thesis .
+ qed
+ qed
-with e f q
-have "\<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
- \<and> (\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x)"
-by (simp! add: abs_HahnBanach)
+ txt {* @{text f} is bounded by @{text p}. *}
-thus ?thesis
-proof (elim exE conjE)
-fix g
-assume "is_linearform E g" and a: "\<forall>x \<in> F. g x = f x"
- and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
+ have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
+ proof
+ fix x assume "x \<in> F"
+ show "\<bar>f x\<bar> \<le> p x"
+ apply (unfold p_def function_norm_def B_def)
+ using normed_vectorspace.axioms [OF F_norm] .. (* FIXME *)
+ qed
-show "\<exists>g. is_linearform E g
- \<and> is_continuous E norm g
- \<and> (\<forall>x \<in> F. g x = f x)
- \<and> \<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
-proof (intro exI conjI)
+ txt {* Using the fact that @{text p} is a seminorm and @{text f} is bounded
+ by @{text p} we can apply the Hahn-Banach Theorem for real vector
+ spaces. So @{text f} can be extended in a norm-preserving way to
+ some function @{text g} on the whole vector space @{text E}. *}
-txt {*
- We furthermore have to show that @{text g} is also continuous:
-*}
+ with E FE linearform q obtain g where
+ linearformE: "linearform E g"
+ and a: "\<forall>x \<in> F. g x = f x"
+ and b: "\<forall>x \<in> E. \<bar>g x\<bar> \<le> p x"
+ by (rule abs_HahnBanach [elim_format]) rules
- show g_cont: "is_continuous E norm g"
+ txt {* We furthermore have to show that @{text g} is also continuous: *}
+
+ have g_cont: "continuous E norm g" using linearformE
proof
fix x assume "x \<in> E"
- with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
- by (simp add: p_def)
+ with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
+ by (simp only: p_def)
qed
- txt {*
- To complete the proof, we show that
- @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. \label{order_antisym} *}
+ txt {* To complete the proof, we show that @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. *}
- show "\<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
- (is "?L = ?R")
+ have "\<parallel>g\<parallel>\<hyphen>E = \<parallel>f\<parallel>\<hyphen>F"
proof (rule order_antisym)
-
txt {*
First we show @{text "\<parallel>g\<parallel> \<le> \<parallel>f\<parallel>"}. The function norm @{text
"\<parallel>g\<parallel>"} is defined as the smallest @{text "c \<in> \<real>"} such that
@@ -480,40 +448,51 @@
\end{center}
*}
- have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
+ have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
proof
fix x assume "x \<in> E"
- show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
- by (simp!)
- qed
-
- with g_cont e_norm show "?L \<le> ?R"
- proof (rule fnorm_le_ub)
- from f_cont f_norm show "0 \<le> \<parallel>f\<parallel>F,norm" ..
+ with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>\<hyphen>F * \<parallel>x\<parallel>"
+ by (simp only: p_def)
qed
-
- txt{* The other direction is achieved by a similar
- argument. *}
+ show "\<parallel>g\<parallel>\<hyphen>E \<le> \<parallel>f\<parallel>\<hyphen>F"
+ apply (unfold function_norm_def B_def)
+ apply rule
+ apply (rule normed_vectorspace.axioms [OF E_norm])+
+ apply (rule continuous.axioms [OF g_cont])+
+ apply (rule b [unfolded p_def function_norm_def B_def])
+ using normed_vectorspace.axioms [OF F_norm] .. (* FIXME *)
- have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>E,norm * norm x"
+ txt {* The other direction is achieved by a similar argument. *}
+
+ have ** : "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
proof
- fix x assume "x \<in> F"
+ fix x assume x: "x \<in> F"
from a have "g x = f x" ..
- hence "\<bar>f x\<bar> = \<bar>g x\<bar>" by simp
- also from g_cont
- have "... \<le> \<parallel>g\<parallel>E,norm * norm x"
- proof (rule norm_fx_le_norm_f_norm_x)
- show "x \<in> E" ..
+ hence "\<bar>f x\<bar> = \<bar>g x\<bar>" by (simp only:)
+ also have "\<dots> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>"
+ apply (unfold function_norm_def B_def)
+ apply rule
+ apply (rule normed_vectorspace.axioms [OF E_norm])+
+ apply (rule continuous.axioms [OF g_cont])+
+ proof -
+ from FE x show "x \<in> E" ..
qed
- finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>E,norm * norm x" .
+ finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>\<hyphen>E * \<parallel>x\<parallel>" .
qed
- thus "?R \<le> ?L"
- proof (rule fnorm_le_ub [OF f_cont f_norm])
- from g_cont show "0 \<le> \<parallel>g\<parallel>E,norm" ..
- qed
+ show "\<parallel>f\<parallel>\<hyphen>F \<le> \<parallel>g\<parallel>\<hyphen>E"
+ apply (unfold function_norm_def B_def)
+ apply rule
+ apply (rule normed_vectorspace.axioms [OF F_norm])+
+ apply assumption+
+ apply (rule ** [unfolded function_norm_def B_def])
+ apply rule
+ apply assumption+
+ apply (rule continuous.axioms [OF g_cont])+
+ done (* FIXME *)
qed
-qed
-qed
+
+ with linearformE a g_cont show ?thesis
+ by blast
qed
end
--- a/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Thu Aug 22 20:49:43 2002 +0200
@@ -17,17 +17,15 @@
an element in @{text "E - H"}. @{text H} is extended to the direct
sum @{text "H' = H + lin x\<^sub>0"}, so for any @{text "x \<in> H'"}
the decomposition of @{text "x = y + a \<cdot> x"} with @{text "y \<in> H"} is
- unique. @{text h'} is defined on @{text H'} by
- @{text "h' x = h y + a \<cdot> \<xi>"} for a certain @{text \<xi>}.
+ unique. @{text h'} is defined on @{text H'} by @{text "h' x = h y +
+ a \<cdot> \<xi>"} for a certain @{text \<xi>}.
Subsequently we show some properties of this extension @{text h'} of
@{text h}.
-*}
-text {*
- This lemma will be used to show the existence of a linear extension
- of @{text f} (see page \pageref{ex-xi-use}). It is a consequence of
- the completeness of @{text \<real>}. To show
+ \medskip This lemma will be used to show the existence of a linear
+ extension of @{text f} (see page \pageref{ex-xi-use}). It is a
+ consequence of the completeness of @{text \<real>}. To show
\begin{center}
\begin{tabular}{l}
@{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"}
@@ -42,307 +40,227 @@
*}
lemma ex_xi:
- "is_vectorspace F \<Longrightarrow> (\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v)
- \<Longrightarrow> \<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
+ includes vectorspace F
+ assumes r: "\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> b v"
+ shows "\<exists>xi::real. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
proof -
- assume vs: "is_vectorspace F"
- assume r: "(\<And>u v. u \<in> F \<Longrightarrow> v \<in> F \<Longrightarrow> a u \<le> (b v::real))"
-
txt {* From the completeness of the reals follows:
- The set @{text "S = {a u. u \<in> F}"} has a supremum, if
- it is non-empty and has an upper bound. *}
-
- let ?S = "{a u :: real | u. u \<in> F}"
-
- have "\<exists>xi. isLub UNIV ?S xi"
- proof (rule reals_complete)
-
- txt {* The set @{text S} is non-empty, since @{text "a 0 \<in> S"}: *}
-
- from vs have "a 0 \<in> ?S" by blast
- thus "\<exists>X. X \<in> ?S" ..
-
- txt {* @{text "b 0"} is an upper bound of @{text S}: *}
-
- show "\<exists>Y. isUb UNIV ?S Y"
- proof
- show "isUb UNIV ?S (b 0)"
- proof (intro isUbI setleI ballI)
- show "b 0 \<in> UNIV" ..
- next
-
- txt {* Every element @{text "y \<in> S"} is less than @{text "b 0"}: *}
+ The set @{text "S = {a u. u \<in> F}"} has a supremum, if it is
+ non-empty and has an upper bound. *}
- fix y assume y: "y \<in> ?S"
- from y have "\<exists>u \<in> F. y = a u" by fast
- thus "y \<le> b 0"
- proof
- fix u assume "u \<in> F"
- assume "y = a u"
- also have "a u \<le> b 0" by (rule r) (simp!)+
- finally show ?thesis .
- qed
- qed
+ let ?S = "{a u | u. u \<in> F}"
+ have "\<exists>xi. lub ?S xi"
+ proof (rule real_complete)
+ have "a 0 \<in> ?S" by blast
+ then show "\<exists>X. X \<in> ?S" ..
+ have "\<forall>y \<in> ?S. y \<le> b 0"
+ proof
+ fix y assume y: "y \<in> ?S"
+ then obtain u where u: "u \<in> F" and y: "y = a u" by blast
+ from u and zero have "a u \<le> b 0" by (rule r)
+ with y show "y \<le> b 0" by (simp only:)
qed
+ then show "\<exists>u. \<forall>y \<in> ?S. y \<le> u" ..
qed
-
- thus "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y"
- proof (elim exE)
- fix xi assume "isLub UNIV ?S xi"
- show ?thesis
- proof (intro exI conjI ballI)
-
- txt {* For all @{text "y \<in> F"} holds @{text "a y \<le> \<xi>"}: *}
-
- fix y assume y: "y \<in> F"
- show "a y \<le> xi"
- proof (rule isUbD)
- show "isUb UNIV ?S xi" ..
- qed (blast!)
- next
-
- txt {* For all @{text "y \<in> F"} holds @{text "\<xi> \<le> b y"}: *}
-
- fix y assume "y \<in> F"
- show "xi \<le> b y"
- proof (intro isLub_le_isUb isUbI setleI)
- show "b y \<in> UNIV" ..
- show "\<forall>ya \<in> ?S. ya \<le> b y"
- proof
- fix au assume au: "au \<in> ?S "
- hence "\<exists>u \<in> F. au = a u" by fast
- thus "au \<le> b y"
- proof
- fix u assume "u \<in> F" assume "au = a u"
- also have "... \<le> b y" by (rule r)
- finally show ?thesis .
- qed
- qed
- qed
+ then obtain xi where xi: "lub ?S xi" ..
+ {
+ fix y assume "y \<in> F"
+ then have "a y \<in> ?S" by blast
+ with xi have "a y \<le> xi" by (rule lub.upper)
+ } moreover {
+ fix y assume y: "y \<in> F"
+ from xi have "xi \<le> b y"
+ proof (rule lub.least)
+ fix au assume "au \<in> ?S"
+ then obtain u where u: "u \<in> F" and au: "au = a u" by blast
+ from u y have "a u \<le> b y" by (rule r)
+ with au show "au \<le> b y" by (simp only:)
qed
- qed
+ } ultimately show "\<exists>xi. \<forall>y \<in> F. a y \<le> xi \<and> xi \<le> b y" by blast
qed
text {*
- \medskip The function @{text h'} is defined as a
- @{text "h' x = h y + a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a
- linear extension of @{text h} to @{text H'}. *}
+ \medskip The function @{text h'} is defined as a @{text "h' x = h y
+ + a \<cdot> \<xi>"} where @{text "x = y + a \<cdot> \<xi>"} is a linear extension of
+ @{text h} to @{text H'}.
+*}
lemma h'_lf:
- "h' \<equiv> \<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi
- \<Longrightarrow> H' \<equiv> H + lin x0 \<Longrightarrow> is_subspace H E \<Longrightarrow> is_linearform H h \<Longrightarrow> x0 \<notin> H
- \<Longrightarrow> x0 \<in> E \<Longrightarrow> x0 \<noteq> 0 \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_linearform H' h'"
-proof -
- assume h'_def:
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
- in h y + a * xi)"
+ includes var H + var h + var E
+ assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
+ SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
and H'_def: "H' \<equiv> H + lin x0"
- and vs: "is_subspace H E" "is_linearform H h" "x0 \<notin> H"
- "x0 \<noteq> 0" "x0 \<in> E" "is_vectorspace E"
-
- have h': "is_vectorspace H'"
- proof (unfold H'_def, rule vs_sum_vs)
- show "is_subspace (lin x0) E" ..
+ and HE: "H \<unlhd> E"
+ includes linearform H h
+ assumes x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
+ includes vectorspace E
+ shows "linearform H' h'"
+proof
+ have H': "vectorspace H'"
+ proof (unfold H'_def)
+ have "x0 \<in> E" .
+ then have "lin x0 \<unlhd> E" ..
+ with HE show "vectorspace (H + lin x0)" ..
qed
-
- show ?thesis
- proof
+ {
fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
-
- txt {* We now have to show that @{text h'} is additive, i.~e.\
- @{text "h' (x\<^sub>1 + x\<^sub>2) = h' x\<^sub>1 + h' x\<^sub>2"} for
- @{text "x\<^sub>1, x\<^sub>2 \<in> H"}. *}
+ show "h' (x1 + x2) = h' x1 + h' x2"
+ proof -
+ from H' x1 x2 have "x1 + x2 \<in> H'"
+ by (rule vectorspace.add_closed)
+ with x1 x2 obtain y y1 y2 a a1 a2 where
+ x1x2: "x1 + x2 = y + a \<cdot> x0" and y: "y \<in> H"
+ and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
+ and x2_rep: "x2 = y2 + a2 \<cdot> x0" and y2: "y2 \<in> H"
+ by (unfold H'_def sum_def lin_def) blast
- have x1x2: "x1 + x2 \<in> H'"
- by (rule vs_add_closed, rule h')
- from x1
- have ex_x1: "\<exists>y1 a1. x1 = y1 + a1 \<cdot> x0 \<and> y1 \<in> H"
- by (unfold H'_def vs_sum_def lin_def) fast
- from x2
- have ex_x2: "\<exists>y2 a2. x2 = y2 + a2 \<cdot> x0 \<and> y2 \<in> H"
- by (unfold H'_def vs_sum_def lin_def) fast
- from x1x2
- have ex_x1x2: "\<exists>y a. x1 + x2 = y + a \<cdot> x0 \<and> y \<in> H"
- by (unfold H'_def vs_sum_def lin_def) fast
-
- from ex_x1 ex_x2 ex_x1x2
- show "h' (x1 + x2) = h' x1 + h' x2"
- proof (elim exE conjE)
- fix y1 y2 y a1 a2 a
- assume y1: "x1 = y1 + a1 \<cdot> x0" and y1': "y1 \<in> H"
- and y2: "x2 = y2 + a2 \<cdot> x0" and y2': "y2 \<in> H"
- and y: "x1 + x2 = y + a \<cdot> x0" and y': "y \<in> H"
- txt {* \label{decomp-H-use}*}
- have ya: "y1 + y2 = y \<and> a1 + a2 = a"
- proof (rule decomp_H')
- show "y1 + y2 + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0"
- by (simp! add: vs_add_mult_distrib2 [of E])
- show "y1 + y2 \<in> H" ..
+ have ya: "y1 + y2 = y \<and> a1 + a2 = a" using _ HE _ y x0
+ proof (rule decomp_H') txt_raw {* \label{decomp-H-use} *}
+ from HE y1 y2 show "y1 + y2 \<in> H"
+ by (rule subspace.add_closed)
+ from x0 and HE y y1 y2
+ have "x0 \<in> E" "y \<in> E" "y1 \<in> E" "y2 \<in> E" by auto
+ with x1_rep x2_rep have "(y1 + y2) + (a1 + a2) \<cdot> x0 = x1 + x2"
+ by (simp add: add_ac add_mult_distrib2)
+ also note x1x2
+ finally show "(y1 + y2) + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0" .
qed
+ from h'_def x1x2 _ HE y x0
have "h' (x1 + x2) = h y + a * xi"
by (rule h'_definite)
- also have "... = h (y1 + y2) + (a1 + a2) * xi"
- by (simp add: ya)
- also from vs y1' y2'
- have "... = h y1 + h y2 + a1 * xi + a2 * xi"
- by (simp add: linearform_add [of H]
- real_add_mult_distrib)
- also have "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
+ also have "\<dots> = h (y1 + y2) + (a1 + a2) * xi"
+ by (simp only: ya)
+ also from y1 y2 have "h (y1 + y2) = h y1 + h y2"
by simp
- also have "h y1 + a1 * xi = h' x1"
+ also have "\<dots> + (a1 + a2) * xi = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
+ by (simp add: real_add_mult_distrib)
+ also from h'_def x1_rep _ HE y1 x0
+ have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
- also have "h y2 + a2 * xi = h' x2"
+ also from h'_def x2_rep _ HE y2 x0
+ have "h y2 + a2 * xi = h' x2"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
-
- txt {* We further have to show that @{text h'} is multiplicative,
- i.~e.\ @{text "h' (c \<cdot> x\<^sub>1) = c \<cdot> h' x\<^sub>1"} for @{text "x \<in> H"}
- and @{text "c \<in> \<real>"}. *}
-
next
- fix c x1 assume x1: "x1 \<in> H'"
- have ax1: "c \<cdot> x1 \<in> H'"
- by (rule vs_mult_closed, rule h')
- from x1
- have ex_x: "\<And>x. x\<in> H' \<Longrightarrow> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
- by (unfold H'_def vs_sum_def lin_def) fast
+ fix x1 c assume x1: "x1 \<in> H'"
+ show "h' (c \<cdot> x1) = c * (h' x1)"
+ proof -
+ from H' x1 have ax1: "c \<cdot> x1 \<in> H'"
+ by (rule vectorspace.mult_closed)
+ with x1 obtain y a y1 a1 where
+ cx1_rep: "c \<cdot> x1 = y + a \<cdot> x0" and y: "y \<in> H"
+ and x1_rep: "x1 = y1 + a1 \<cdot> x0" and y1: "y1 \<in> H"
+ by (unfold H'_def sum_def lin_def) blast
- from x1 have ex_x1: "\<exists>y1 a1. x1 = y1 + a1 \<cdot> x0 \<and> y1 \<in> H"
- by (unfold H'_def vs_sum_def lin_def) fast
- with ex_x [of "c \<cdot> x1", OF ax1]
- show "h' (c \<cdot> x1) = c * (h' x1)"
- proof (elim exE conjE)
- fix y1 y a1 a
- assume y1: "x1 = y1 + a1 \<cdot> x0" and y1': "y1 \<in> H"
- and y: "c \<cdot> x1 = y + a \<cdot> x0" and y': "y \<in> H"
-
- have ya: "c \<cdot> y1 = y \<and> c * a1 = a"
+ have ya: "c \<cdot> y1 = y \<and> c * a1 = a" using _ HE _ y x0
proof (rule decomp_H')
- show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0"
- by (simp! add: vs_add_mult_distrib1)
- show "c \<cdot> y1 \<in> H" ..
+ from HE y1 show "c \<cdot> y1 \<in> H"
+ by (rule subspace.mult_closed)
+ from x0 and HE y y1
+ have "x0 \<in> E" "y \<in> E" "y1 \<in> E" by auto
+ with x1_rep have "c \<cdot> y1 + (c * a1) \<cdot> x0 = c \<cdot> x1"
+ by (simp add: mult_assoc add_mult_distrib1)
+ also note cx1_rep
+ finally show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0" .
qed
- have "h' (c \<cdot> x1) = h y + a * xi"
+ from h'_def cx1_rep _ HE y x0 have "h' (c \<cdot> x1) = h y + a * xi"
by (rule h'_definite)
- also have "... = h (c \<cdot> y1) + (c * a1) * xi"
- by (simp add: ya)
- also from vs y1' have "... = c * h y1 + c * a1 * xi"
- by (simp add: linearform_mult [of H])
- also from vs y1' have "... = c * (h y1 + a1 * xi)"
- by (simp add: real_add_mult_distrib2 real_mult_assoc)
- also have "h y1 + a1 * xi = h' x1"
+ also have "\<dots> = h (c \<cdot> y1) + (c * a1) * xi"
+ by (simp only: ya)
+ also from y1 have "h (c \<cdot> y1) = c * h y1"
+ by simp
+ also have "\<dots> + (c * a1) * xi = c * (h y1 + a1 * xi)"
+ by (simp only: real_add_mult_distrib2)
+ also from h'_def x1_rep _ HE y1 x0 have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
- qed
+ }
qed
text {* \medskip The linear extension @{text h'} of @{text h}
-is bounded by the seminorm @{text p}. *}
+ is bounded by the seminorm @{text p}. *}
lemma h'_norm_pres:
- "h' \<equiv> \<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi
- \<Longrightarrow> H' \<equiv> H + lin x0 \<Longrightarrow> x0 \<notin> H \<Longrightarrow> x0 \<in> E \<Longrightarrow> x0 \<noteq> 0 \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_subspace H E \<Longrightarrow> is_seminorm E p \<Longrightarrow> is_linearform H h
- \<Longrightarrow> \<forall>y \<in> H. h y \<le> p y
- \<Longrightarrow> \<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y
- \<Longrightarrow> \<forall>x \<in> H'. h' x \<le> p x"
-proof
- assume h'_def:
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
- in (h y) + a * xi)"
+ includes var H + var h + var E
+ assumes h'_def: "h' \<equiv> \<lambda>x. let (y, a) =
+ SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H in h y + a * xi"
and H'_def: "H' \<equiv> H + lin x0"
- and vs: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" "is_vectorspace E"
- "is_subspace H E" "is_seminorm E p" "is_linearform H h"
- and a: "\<forall>y \<in> H. h y \<le> p y"
- presume a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"
- presume a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya"
- fix x assume "x \<in> H'"
- have ex_x:
- "\<And>x. x \<in> H' \<Longrightarrow> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
- by (unfold H'_def vs_sum_def lin_def) fast
- have "\<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
- by (rule ex_x)
- thus "h' x \<le> p x"
- proof (elim exE conjE)
- fix y a assume x: "x = y + a \<cdot> x0" and y: "y \<in> H"
- have "h' x = h y + a * xi"
+ and x0: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0"
+ includes vectorspace E + subvectorspace H E +
+ seminorm E p + linearform H h
+ assumes a: "\<forall>y \<in> H. h y \<le> p y"
+ and a': "\<forall>y \<in> H. - p (y + x0) - h y \<le> xi \<and> xi \<le> p (y + x0) - h y"
+ shows "\<forall>x \<in> H'. h' x \<le> p x"
+proof
+ fix x assume x': "x \<in> H'"
+ show "h' x \<le> p x"
+ proof -
+ from a' have a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya \<le> xi"
+ and a2: "\<forall>ya \<in> H. xi \<le> p (ya + x0) - h ya" by auto
+ from x' obtain y a where
+ x_rep: "x = y + a \<cdot> x0" and y: "y \<in> H"
+ by (unfold H'_def sum_def lin_def) blast
+ from y have y': "y \<in> E" ..
+ from y have ay: "inverse a \<cdot> y \<in> H" by simp
+
+ from h'_def x_rep _ _ y x0 have "h' x = h y + a * xi"
by (rule h'_definite)
-
- txt {* Now we show @{text "h y + a \<cdot> \<xi> \<le> p (y + a \<cdot> x\<^sub>0)"}
- by case analysis on @{text a}. *}
-
- also have "... \<le> p (y + a \<cdot> x0)"
+ also have "\<dots> \<le> p (y + a \<cdot> x0)"
proof (rule linorder_cases)
-
assume z: "a = 0"
- with vs y a show ?thesis by simp
-
- txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}
- with @{text ya} taken as @{text "y / a"}: *}
-
+ then have "h y + a * xi = h y" by simp
+ also from a y have "\<dots> \<le> p y" ..
+ also from x0 y' z have "p y = p (y + a \<cdot> x0)" by simp
+ finally show ?thesis .
next
+ txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}
+ with @{text ya} taken as @{text "y / a"}: *}
assume lz: "a < 0" hence nz: "a \<noteq> 0" by simp
- from a1
- have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi"
- by (rule bspec) (simp!)
-
- txt {* The thesis for this case now follows by a short
- calculation. *}
- hence "a * xi \<le> a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
- by (rule real_mult_less_le_anti [OF lz])
- also
- have "... = - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
+ from a1 ay
+ have "- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y) \<le> xi" ..
+ with lz have "a * xi \<le>
+ a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
+ by (rule real_mult_less_le_anti)
+ also have "\<dots> =
+ - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
by (rule real_mult_diff_distrib)
- also from lz vs y
- have "- a * (p (inverse a \<cdot> y + x0)) = p (a \<cdot> (inverse a \<cdot> y + x0))"
- by (simp add: seminorm_abs_homogenous abs_minus_eqI2)
- also from nz vs y have "... = p (y + a \<cdot> x0)"
- by (simp add: vs_add_mult_distrib1)
- also from nz vs y have "a * (h (inverse a \<cdot> y)) = h y"
- by (simp add: linearform_mult [symmetric])
+ also from lz x0 y' have "- a * (p (inverse a \<cdot> y + x0)) =
+ p (a \<cdot> (inverse a \<cdot> y + x0))"
+ by (simp add: abs_homogenous abs_minus_eqI2)
+ also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
+ by (simp add: add_mult_distrib1 mult_assoc [symmetric])
+ also from nz y have "a * (h (inverse a \<cdot> y)) = h y"
+ by simp
finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
-
- hence "h y + a * xi \<le> h y + p (y + a \<cdot> x0) - h y"
- by (simp add: real_add_left_cancel_le)
- thus ?thesis by simp
-
+ then show ?thesis by simp
+ next
txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"}
with @{text ya} taken as @{text "y / a"}: *}
-
- next
assume gz: "0 < a" hence nz: "a \<noteq> 0" by simp
- from a2 have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)"
- by (rule bspec) (simp!)
-
- txt {* The thesis for this case follows by a short
- calculation: *}
-
- with gz
- have "a * xi \<le> a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
+ from a2 ay
+ have "xi \<le> p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)" ..
+ with gz have "a * xi \<le>
+ a * (p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
by (rule real_mult_less_le_mono)
also have "... = a * p (inverse a \<cdot> y + x0) - a * h (inverse a \<cdot> y)"
by (rule real_mult_diff_distrib2)
- also from gz vs y
+ also from gz x0 y'
have "a * p (inverse a \<cdot> y + x0) = p (a \<cdot> (inverse a \<cdot> y + x0))"
- by (simp add: seminorm_abs_homogenous abs_eqI2)
- also from nz vs y have "... = p (y + a \<cdot> x0)"
- by (simp add: vs_add_mult_distrib1)
- also from nz vs y have "a * h (inverse a \<cdot> y) = h y"
- by (simp add: linearform_mult [symmetric])
+ by (simp add: abs_homogenous abs_eqI2)
+ also from nz x0 y' have "\<dots> = p (y + a \<cdot> x0)"
+ by (simp add: add_mult_distrib1 mult_assoc [symmetric])
+ also from nz y have "a * h (inverse a \<cdot> y) = h y"
+ by simp
finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
-
- hence "h y + a * xi \<le> h y + (p (y + a \<cdot> x0) - h y)"
- by (simp add: real_add_left_cancel_le)
- thus ?thesis by simp
+ then show ?thesis by simp
qed
- also from x have "... = p x" by simp
+ also from x_rep have "\<dots> = p x" by (simp only:)
finally show ?thesis .
qed
-qed blast+
+qed
end
--- a/src/HOL/Real/HahnBanach/HahnBanachSupLemmas.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanachSupLemmas.thy Thu Aug 22 20:49:43 2002 +0200
@@ -13,62 +13,43 @@
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}.
-*}
+ 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}.
-text {*
- 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
+ \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.
*}
lemma some_H'h't:
- "M = norm_pres_extensions E p F f \<Longrightarrow> c \<in> chain M \<Longrightarrow>
- graph H h = \<Union>c \<Longrightarrow> x \<in> H
- \<Longrightarrow> \<exists>H' h'. graph H' h' \<in> c \<and> (x, h x) \<in> graph H' h'
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
+ 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 -
- assume m: "M = norm_pres_extensions E p F f" and "c \<in> chain M"
- and u: "graph H h = \<Union>c" "x \<in> H"
+ 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
- have h: "(x, h x) \<in> graph H h" ..
- with u have "(x, h x) \<in> \<Union>c" by simp
- hence ex1: "\<exists>g \<in> c. (x, h x) \<in> g"
- by (simp only: Union_iff)
- thus ?thesis
- proof (elim bexE)
- fix g assume g: "g \<in> c" "(x, h x) \<in> g"
- have "c \<subseteq> M" by (rule chainD2)
- hence "g \<in> M" ..
- hence "g \<in> norm_pres_extensions E p F f" by (simp only: m)
- hence "\<exists>H' h'. graph H' h' = g
- \<and> is_linearform H' h'
- \<and> is_subspace H' E
- \<and> is_subspace F H'
- \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule norm_pres_extension_D)
- thus ?thesis
- proof (elim exE conjE)
- fix H' h'
- assume "graph H' h' = g" "is_linearform H' h'"
- "is_subspace H' E" "is_subspace F H'"
- "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x"
- show ?thesis
- proof (intro exI conjI)
- show "graph H' h' \<in> c" by (simp!)
- show "(x, h x) \<in> graph H' h'" by (simp!)
- qed
- qed
- qed
+ 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
@@ -78,35 +59,26 @@
*}
lemma some_H'h':
- "M = norm_pres_extensions E p F f \<Longrightarrow> c \<in> chain M \<Longrightarrow>
- graph H h = \<Union>c \<Longrightarrow> x \<in> H
- \<Longrightarrow> \<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E \<and> is_subspace F H'
- \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
+ 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 -
- assume "M = norm_pres_extensions E p F f" and cM: "c \<in> chain M"
- and u: "graph H h = \<Union>c" "x \<in> H"
-
- have "\<exists>H' h'. graph H' h' \<in> c \<and> (x, h x) \<in> graph H' h'
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h't)
- thus ?thesis
- proof (elim exE conjE)
- fix H' h' assume "(x, h x) \<in> graph H' h'" "graph H' h' \<in> c"
- "is_linearform H' h'" "is_subspace H' E" "is_subspace F H'"
+ 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"
- show ?thesis
- proof (intro exI conjI)
- show "x \<in> H'" by (rule graphD1)
- from cM u show "graph H' h' \<subseteq> graph H h"
- by (simp! only: chain_ball_Union_upper)
- qed
- qed
+ 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
@@ -115,136 +87,116 @@
*}
lemma some_H'h'2:
- "M = norm_pres_extensions E p F f \<Longrightarrow> c \<in> chain M \<Longrightarrow>
- graph H h = \<Union>c \<Longrightarrow> x \<in> H \<Longrightarrow> y \<in> H
- \<Longrightarrow> \<exists>H' h'. x \<in> H' \<and> y \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E \<and> is_subspace F H'
- \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
+ 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 -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
- "graph H h = \<Union>c" "x \<in> H" "y \<in> H"
-
- txt {*
- @{text x} is in the domain @{text H'} of some function @{text h'},
- such that @{text h} extends @{text h'}. *}
-
- have e1: "\<exists>H' h'. graph H' h' \<in> c \<and> (x, h x) \<in> graph H' h'
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h't)
-
txt {* @{text y} is in the domain @{text H''} of some function @{text h''},
such that @{text h} extends @{text h''}. *}
- have e2: "\<exists>H'' h''. graph H'' h'' \<in> c \<and> (y, h y) \<in> graph H'' h''
- \<and> is_linearform H'' h'' \<and> is_subspace H'' E
- \<and> is_subspace F H'' \<and> graph F f \<subseteq> graph H'' h''
- \<and> (\<forall>x \<in> H''. h'' x \<le> p x)"
- by (rule some_H'h't)
+ 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 e1 e2 show ?thesis
- proof (elim exE conjE)
- fix H' h' assume "(y, h y) \<in> graph H' h'" "graph H' h' \<in> c"
- "is_linearform H' h'" "is_subspace H' E" "is_subspace F H'"
- "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x \<le> p x"
+ 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
- fix H'' h'' assume "(x, h x) \<in> graph H'' h''" "graph H'' h'' \<in> c"
- "is_linearform H'' h''" "is_subspace H'' E" "is_subspace F H''"
- "graph F f \<subseteq> graph H'' h''" "\<forall>x \<in> H''. h'' x \<le> p x"
-
- 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}*}
+ 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}*}
- have "graph H'' h'' \<subseteq> graph H' h' \<or> graph H' h' \<subseteq> graph H'' h''"
- (is "?case1 \<or> ?case2")
- by (rule chainD)
- thus ?thesis
- proof
- assume ?case1
- show ?thesis
- proof (intro exI conjI)
- have "(x, h x) \<in> graph H'' h''" .
- also have "... \<subseteq> graph H' h'" .
- finally have xh:"(x, h x) \<in> graph H' h'" .
- thus x: "x \<in> H'" ..
- show y: "y \<in> H'" ..
- show "graph H' h' \<subseteq> graph H h"
- by (simp! only: chain_ball_Union_upper)
- qed
- next
- assume ?case2
- show ?thesis
- proof (intro exI conjI)
- show x: "x \<in> H''" ..
- have "(y, h y) \<in> graph H' h'" by (simp!)
- also have "... \<subseteq> graph H'' h''" .
- finally have yh: "(y, h y) \<in> graph H'' h''" .
- thus y: "y \<in> H''" ..
- show "graph H'' h'' \<subseteq> graph H h"
- by (simp! only: chain_ball_Union_upper)
- qed
- qed
+ from cM 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''" .
+ also have "... \<subseteq> graph H' h'" .
+ 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 {
+ from y_hy have "(y, h y) \<in> graph H' h'" .
+ also have "\<dots> \<subseteq> graph H'' h''" .
+ 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. *}
+ chain @{text c} is definite, i.~e.~t is the graph of a function.
+*}
lemma sup_definite:
- "M \<equiv> norm_pres_extensions E p F f \<Longrightarrow> c \<in> chain M \<Longrightarrow>
- (x, y) \<in> \<Union>c \<Longrightarrow> (x, z) \<in> \<Union>c \<Longrightarrow> z = y"
+ 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 -
- assume "c \<in> chain M" "M \<equiv> norm_pres_extensions E p F f"
- "(x, y) \<in> \<Union>c" "(x, z) \<in> \<Union>c"
- thus ?thesis
- proof (elim UnionE chainE2)
+ 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" ..
- txt {* Since both @{text "(x, y) \<in> \<Union>c"} and @{text "(x, z) \<in> \<Union>c"}
- they are members of some graphs @{text "G\<^sub>1"} and @{text
- "G\<^sub>2"}, resp., such that both @{text "G\<^sub>1"} and @{text
- "G\<^sub>2"} are members of @{text c}.*}
+ from G1 c have "G1 \<in> M" ..
+ then obtain H1 h1 where G1_rep: "G1 = graph H1 h1"
+ by (unfold M_def) blast
- fix G1 G2 assume
- "(x, y) \<in> G1" "G1 \<in> c" "(x, z) \<in> G2" "G2 \<in> c" "c \<subseteq> M"
+ from G2 c have "G2 \<in> M" ..
+ then obtain H2 h2 where G2_rep: "G2 = graph H2 h2"
+ by (unfold M_def) blast
- have "G1 \<in> M" ..
- hence e1: "\<exists>H1 h1. graph H1 h1 = G1"
- by (blast! dest: norm_pres_extension_D)
- have "G2 \<in> M" ..
- hence e2: "\<exists>H2 h2. graph H2 h2 = G2"
- by (blast! dest: norm_pres_extension_D)
- from e1 e2 show ?thesis
- proof (elim exE)
- fix H1 h1 H2 h2
- assume "graph H1 h1 = G1" "graph H2 h2 = G2"
-
- 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}*}
+ 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}*}
- have "G1 \<subseteq> G2 \<or> G2 \<subseteq> G1" (is "?case1 \<or> ?case2") ..
- thus ?thesis
- proof
- assume ?case1
- have "(x, y) \<in> graph H2 h2" by (blast!)
- hence "y = h2 x" ..
- also have "(x, z) \<in> graph H2 h2" by (simp!)
- hence "z = h2 x" ..
- finally show ?thesis .
- next
- assume ?case2
- have "(x, y) \<in> graph H1 h1" by (simp!)
- hence "y = h1 x" ..
- also have "(x, z) \<in> graph H1 h1" by (blast!)
- hence "z = h1 x" ..
- finally show ?thesis .
- qed
- qed
+ 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
+ hence "y = h2 x" ..
+ also
+ from xz' G2_rep have "(x, z) \<in> graph H2 h2" by (simp only:)
+ hence "z = h2 x" ..
+ finally show ?thesis .
+ next
+ assume ?case2
+ with xz' G1_rep have "(x, z) \<in> graph H1 h1" by blast
+ hence "z = h1 x" ..
+ also
+ from xy' G1_rep have "(x, y) \<in> graph H1 h1" by (simp only:)
+ hence "y = h1 x" ..
+ finally show ?thesis ..
qed
qed
@@ -258,58 +210,48 @@
*}
lemma sup_lf:
- "M = norm_pres_extensions E p F f \<Longrightarrow> c \<in> chain M \<Longrightarrow>
- graph H h = \<Union>c \<Longrightarrow> is_linearform H h"
-proof -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
- "graph H h = \<Union>c"
-
- show "is_linearform H h"
- proof
- fix x y assume "x \<in> H" "y \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> y \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h'2)
-
- txt {* We have to show that @{text h} is additive. *}
+ 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
- thus "h (x + y) = h x + h y"
- proof (elim exE conjE)
- fix H' h' assume "x \<in> H'" "y \<in> H'"
- and b: "graph H' h' \<subseteq> graph H h"
- and "is_linearform H' h'" "is_subspace H' E"
- have "h' (x + y) = h' x + h' y"
- by (rule linearform_add)
- also have "h' x = h x" ..
- also have "h' y = h y" ..
- also have "x + y \<in> H'" ..
- with b have "h' (x + y) = h (x + y)" ..
- finally show ?thesis .
- qed
- next
- fix a x assume "x \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h')
+ 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
- txt{* We have to show that @{text h} is multiplicative. *}
-
- thus "h (a \<cdot> x) = a * h x"
- proof (elim exE conjE)
- fix H' h' assume "x \<in> H'"
- and b: "graph H' h' \<subseteq> graph H h"
- and "is_linearform H' h'" "is_subspace H' E"
- have "h' (a \<cdot> x) = a * h' x"
- by (rule linearform_mult)
- also have "h' x = h x" ..
- also have "a \<cdot> x \<in> H'" ..
- with b have "h' (a \<cdot> x) = h (a \<cdot> x)" ..
- finally show ?thesis .
- qed
+ 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
@@ -321,37 +263,22 @@
*}
lemma sup_ext:
- "graph H h = \<Union>c \<Longrightarrow> M = norm_pres_extensions E p F f \<Longrightarrow>
- c \<in> chain M \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> graph F f \<subseteq> graph H h"
+ 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 -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
- "graph H h = \<Union>c"
- assume "\<exists>x. x \<in> c"
- thus ?thesis
- proof
- fix x assume "x \<in> c"
- have "c \<subseteq> M" by (rule chainD2)
- hence "x \<in> M" ..
- hence "x \<in> norm_pres_extensions E p F f" by (simp!)
-
- hence "\<exists>G g. graph G g = x
- \<and> is_linearform G g
- \<and> is_subspace G E
- \<and> is_subspace F G
- \<and> graph F f \<subseteq> graph G g
- \<and> (\<forall>x \<in> G. g x \<le> p x)"
- by (simp! add: norm_pres_extension_D)
-
- thus ?thesis
- proof (elim exE conjE)
- fix G g assume "graph F f \<subseteq> graph G g"
- also assume "graph G g = x"
- also have "... \<in> c" .
- hence "x \<subseteq> \<Union>c" by fast
- also have [symmetric]: "graph H h = \<Union>c" .
- finally show ?thesis .
- qed
- qed
+ 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 {*
@@ -362,32 +289,21 @@
*}
lemma sup_supF:
- "graph H h = \<Union>c \<Longrightarrow> M = norm_pres_extensions E p F f \<Longrightarrow>
- c \<in> chain M \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> is_subspace F E \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_subspace F H"
-proof -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M" "\<exists>x. x \<in> c"
- "graph H h = \<Union>c" "is_subspace F E" "is_vectorspace E"
-
- show ?thesis
- proof
- show "0 \<in> F" ..
- show "F \<subseteq> H"
- proof (rule graph_extD2)
- show "graph F f \<subseteq> graph H h"
- by (rule sup_ext)
- qed
- show "\<forall>x \<in> F. \<forall>y \<in> F. x + y \<in> F"
- proof (intro ballI)
- fix x y assume "x \<in> F" "y \<in> F"
- show "x + y \<in> F" by (simp!)
- qed
- show "\<forall>x \<in> F. \<forall>a. a \<cdot> x \<in> F"
- proof (intro ballI allI)
- fix x a assume "x\<in>F"
- show "a \<cdot> x \<in> F" by (simp!)
- qed
- qed
+ 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 {*
@@ -396,81 +312,53 @@
*}
lemma sup_subE:
- "graph H h = \<Union>c \<Longrightarrow> M = norm_pres_extensions E p F f \<Longrightarrow>
- c \<in> chain M \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> is_subspace F E \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_subspace H E"
-proof -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M" "\<exists>x. x \<in> c"
- "graph H h = \<Union>c" "is_subspace F E" "is_vectorspace E"
- show ?thesis
+ 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 subvectorspace.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
-
- txt {* The @{text 0} element is in @{text H}, as @{text F} is a
- subset of @{text H}: *}
-
- have "0 \<in> F" ..
- also have "is_subspace F H" by (rule sup_supF)
- hence "F \<subseteq> H" ..
- finally show "0 \<in> H" .
-
- txt {* @{text H} is a subset of @{text E}: *}
-
- show "H \<subseteq> E"
- proof
- fix x assume "x \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h')
- thus "x \<in> E"
- proof (elim exE conjE)
- fix H' h' assume "x \<in> H'" "is_subspace H' E"
- have "H' \<subseteq> E" ..
- thus "x \<in> E" ..
- qed
- qed
-
- txt {* @{text H} is closed under addition: *}
-
- show "\<forall>x \<in> H. \<forall>y \<in> H. x + y \<in> H"
- proof (intro ballI)
- fix x y assume "x \<in> H" "y \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> y \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h'2)
- thus "x + y \<in> H"
- proof (elim exE conjE)
- fix H' h'
- assume "x \<in> H'" "y \<in> H'" "is_subspace H' E"
- "graph H' h' \<subseteq> graph H h"
- have "x + y \<in> H'" ..
- also have "H' \<subseteq> H" ..
- finally show ?thesis .
- qed
- qed
-
- txt {* @{text H} is closed under scalar multiplication: *}
-
- show "\<forall>x \<in> H. \<forall>a. a \<cdot> x \<in> H"
- proof (intro ballI allI)
- fix x a assume "x \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E
- \<and> is_subspace F H' \<and> graph F f \<subseteq> graph H' h'
- \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h')
- thus "a \<cdot> x \<in> H"
- proof (elim exE conjE)
- fix H' h'
- assume "x \<in> H'" "is_subspace H' E" "graph H' h' \<subseteq> graph H h"
- have "a \<cdot> x \<in> H'" ..
- also have "H' \<subseteq> H" ..
- finally show ?thesis .
- qed
- qed
+ fix x assume "x \<in> H"
+ with M cM graph
+ obtain H' 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
@@ -480,28 +368,21 @@
*}
lemma sup_norm_pres:
- "graph H h = \<Union>c \<Longrightarrow> M = norm_pres_extensions E p F f \<Longrightarrow>
- c \<in> chain M \<Longrightarrow> \<forall>x \<in> H. h x \<le> p x"
+ 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
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
- "graph H h = \<Union>c"
fix x assume "x \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
- \<and> is_linearform H' h' \<and> is_subspace H' E \<and> is_subspace F H'
- \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
- by (rule some_H'h')
- thus "h x \<le> p x"
- proof (elim exE conjE)
- fix H' h'
- assume "x \<in> H'" "graph H' h' \<subseteq> graph H 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"
- have [symmetric]: "h' x = h x" ..
- also from a have "h' x \<le> p x " ..
- finally show ?thesis .
- qed
+ 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_HahnBanach}
@@ -516,45 +397,41 @@
*}
lemma abs_ineq_iff:
- "is_subspace H E \<Longrightarrow> is_vectorspace E \<Longrightarrow> is_seminorm E p \<Longrightarrow>
- is_linearform H h
- \<Longrightarrow> (\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x) = (\<forall>x \<in> H. h x \<le> p x)"
- (concl is "?L = ?R")
-proof -
- assume "is_subspace H E" "is_vectorspace E" "is_seminorm E p"
- "is_linearform H h"
- have h: "is_vectorspace H" ..
- show ?thesis
- proof
+ includes subvectorspace H E + seminorm E p + 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
+ have h: "vectorspace H" by (rule vectorspace)
+ {
assume l: ?L
show ?R
proof
fix x assume x: "x \<in> H"
- have "h x \<le> \<bar>h x\<bar>" by (rule abs_ge_self)
- also from l have "... \<le> p x" ..
+ 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 \<in> H"
+ 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
show "- p x \<le> h x"
proof (rule real_minus_le)
- from h have "- h x = h (- x)"
- by (rule linearform_neg [symmetric])
- also from r have "... \<le> p (- x)" by (simp!)
- also have "... = p x"
- proof (rule seminorm_minus)
+ have "linearform H h" .
+ from h this x have "- h x = h (- x)"
+ by (rule vectorspace_linearform.neg [symmetric])
+ also from r x have "\<dots> \<le> p (- x)" by simp
+ also have "\<dots> = p x"
+ proof (rule seminorm_vectorspace.minus)
show "x \<in> E" ..
qed
finally show "- h x \<le> p x" .
qed
- from r show "h x \<le> p x" ..
+ from r x show "h x \<le> p x" ..
qed
- qed
+ }
qed
end
--- a/src/HOL/Real/HahnBanach/Linearform.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/Linearform.thy Thu Aug 22 20:49:43 2002 +0200
@@ -12,60 +12,43 @@
that is additive and multiplicative.
*}
-constdefs
- is_linearform :: "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
- "is_linearform V f \<equiv>
- (\<forall>x \<in> V. \<forall>y \<in> V. f (x + y) = f x + f y) \<and>
- (\<forall>x \<in> V. \<forall>a. f (a \<cdot> x) = a * (f x))"
+locale linearform = var V + var f +
+ assumes add [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"
+ and mult [iff]: "x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * f x"
-lemma is_linearformI [intro]:
- "(\<And>x y. x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y) \<Longrightarrow>
- (\<And>x c. x \<in> V \<Longrightarrow> f (c \<cdot> x) = c * f x)
- \<Longrightarrow> is_linearform V f"
- by (unfold is_linearform_def) blast
+locale (open) vectorspace_linearform =
+ vectorspace + linearform
-lemma linearform_add [intro?]:
- "is_linearform V f \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"
- by (unfold is_linearform_def) blast
-
-lemma linearform_mult [intro?]:
- "is_linearform V f \<Longrightarrow> x \<in> V \<Longrightarrow> f (a \<cdot> x) = a * (f x)"
- by (unfold is_linearform_def) blast
-
-lemma linearform_neg [intro?]:
- "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> x \<in> V
- \<Longrightarrow> f (- x) = - f x"
+lemma (in vectorspace_linearform) neg [iff]:
+ "x \<in> V \<Longrightarrow> f (- x) = - f x"
proof -
- assume "is_linearform V f" "is_vectorspace V" "x \<in> V"
- have "f (- x) = f ((- 1) \<cdot> x)" by (simp! add: negate_eq1)
- also have "... = (- 1) * (f x)" by (rule linearform_mult)
- also have "... = - (f x)" by (simp!)
+ assume x: "x \<in> V"
+ hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
+ also from x have "... = (- 1) * (f x)" by (rule mult)
+ also from x have "... = - (f x)" by simp
finally show ?thesis .
qed
-lemma linearform_diff [intro?]:
- "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
- \<Longrightarrow> f (x - y) = f x - f y"
+lemma (in vectorspace_linearform) diff [iff]:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
proof -
- assume "is_vectorspace V" "is_linearform V f" "x \<in> V" "y \<in> V"
- have "f (x - y) = f (x + - y)" by (simp! only: diff_eq1)
- also have "... = f x + f (- y)"
- by (rule linearform_add) (simp!)+
- also have "f (- y) = - f y" by (rule linearform_neg)
- finally show "f (x - y) = f x - f y" by (simp!)
+ assume x: "x \<in> V" and y: "y \<in> V"
+ hence "x - y = x + - y" by (rule diff_eq1)
+ also have "f ... = f x + f (- y)"
+ by (rule add) (simp_all add: x y)
+ also from y have "f (- y) = - f y" by (rule neg)
+ finally show ?thesis by simp
qed
text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
-lemma linearform_zero [intro?, simp]:
- "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> f 0 = 0"
+lemma (in vectorspace_linearform) linearform_zero [iff]:
+ "f 0 = 0"
proof -
- assume "is_vectorspace V" "is_linearform V f"
- have "f 0 = f (0 - 0)" by (simp!)
- also have "... = f 0 - f 0"
- by (rule linearform_diff) (simp!)+
- also have "... = 0" by simp
- finally show "f 0 = 0" .
+ have "f 0 = f (0 - 0)" by simp
+ also have "\<dots> = f 0 - f 0" by (rule diff) simp_all
+ also have "\<dots> = 0" by simp
+ finally show ?thesis .
qed
end
--- a/src/HOL/Real/HahnBanach/NormedSpace.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/NormedSpace.thy Thu Aug 22 20:49:43 2002 +0200
@@ -15,59 +15,40 @@
definite, absolute homogenous and subadditive.
*}
-constdefs
- is_seminorm :: "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
- "is_seminorm V norm \<equiv> \<forall>x \<in> V. \<forall>y \<in> V. \<forall>a.
- 0 \<le> norm x
- \<and> norm (a \<cdot> x) = \<bar>a\<bar> * norm x
- \<and> norm (x + y) \<le> norm x + norm y"
+locale norm_syntax =
+ fixes norm :: "'a \<Rightarrow> real" ("\<parallel>_\<parallel>")
-lemma is_seminormI [intro]:
- "(\<And>x y a. x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> 0 \<le> norm x) \<Longrightarrow>
- (\<And>x a. x \<in> V \<Longrightarrow> norm (a \<cdot> x) = \<bar>a\<bar> * norm x) \<Longrightarrow>
- (\<And>x y. x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> norm (x + y) \<le> norm x + norm y)
- \<Longrightarrow> is_seminorm V norm"
- by (unfold is_seminorm_def) auto
+locale seminorm = var V + norm_syntax +
+ assumes ge_zero [iff?]: "x \<in> V \<Longrightarrow> 0 \<le> \<parallel>x\<parallel>"
+ and abs_homogenous [iff?]: "x \<in> V \<Longrightarrow> \<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"
+ and subadditive [iff?]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> \<parallel>x + y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>"
-lemma seminorm_ge_zero [intro?]:
- "is_seminorm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> 0 \<le> norm x"
- by (unfold is_seminorm_def) blast
+locale (open) seminorm_vectorspace =
+ seminorm + vectorspace
-lemma seminorm_abs_homogenous:
- "is_seminorm V norm \<Longrightarrow> x \<in> V
- \<Longrightarrow> norm (a \<cdot> x) = \<bar>a\<bar> * norm x"
- by (unfold is_seminorm_def) blast
-
-lemma seminorm_subadditive:
- "is_seminorm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
- \<Longrightarrow> norm (x + y) \<le> norm x + norm y"
- by (unfold is_seminorm_def) blast
-
-lemma seminorm_diff_subadditive:
- "is_seminorm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> is_vectorspace V
- \<Longrightarrow> norm (x - y) \<le> norm x + norm y"
+lemma (in seminorm_vectorspace) diff_subadditive:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> \<parallel>x - y\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>y\<parallel>"
proof -
- assume "is_seminorm V norm" "x \<in> V" "y \<in> V" "is_vectorspace V"
- have "norm (x - y) = norm (x + - 1 \<cdot> y)"
- by (simp! add: diff_eq2 negate_eq2a)
- also have "... \<le> norm x + norm (- 1 \<cdot> y)"
- by (simp! add: seminorm_subadditive)
- also have "norm (- 1 \<cdot> y) = \<bar>- 1\<bar> * norm y"
- by (rule seminorm_abs_homogenous)
- also have "\<bar>- 1\<bar> = (1::real)" by (rule abs_minus_one)
- finally show "norm (x - y) \<le> norm x + norm y" by simp
+ assume x: "x \<in> V" and y: "y \<in> V"
+ hence "x - y = x + - 1 \<cdot> y"
+ by (simp add: diff_eq2 negate_eq2a)
+ also from x y have "\<parallel>\<dots>\<parallel> \<le> \<parallel>x\<parallel> + \<parallel>- 1 \<cdot> y\<parallel>"
+ by (simp add: subadditive)
+ also from y have "\<parallel>- 1 \<cdot> y\<parallel> = \<bar>- 1\<bar> * \<parallel>y\<parallel>"
+ by (rule abs_homogenous)
+ also have "\<dots> = \<parallel>y\<parallel>" by simp
+ finally show ?thesis .
qed
-lemma seminorm_minus:
- "is_seminorm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> is_vectorspace V
- \<Longrightarrow> norm (- x) = norm x"
+lemma (in seminorm_vectorspace) minus:
+ "x \<in> V \<Longrightarrow> \<parallel>- x\<parallel> = \<parallel>x\<parallel>"
proof -
- assume "is_seminorm V norm" "x \<in> V" "is_vectorspace V"
- have "norm (- x) = norm (- 1 \<cdot> x)" by (simp! only: negate_eq1)
- also have "... = \<bar>- 1\<bar> * norm x"
- by (rule seminorm_abs_homogenous)
- also have "\<bar>- 1\<bar> = (1::real)" by (rule abs_minus_one)
- finally show "norm (- x) = norm x" by simp
+ assume x: "x \<in> V"
+ hence "- x = - 1 \<cdot> x" by (simp only: negate_eq1)
+ also from x have "\<parallel>\<dots>\<parallel> = \<bar>- 1\<bar> * \<parallel>x\<parallel>"
+ by (rule abs_homogenous)
+ also have "\<dots> = \<parallel>x\<parallel>" by simp
+ finally show ?thesis .
qed
@@ -78,110 +59,46 @@
@{text 0} vector to @{text 0}.
*}
-constdefs
- is_norm :: "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
- "is_norm V norm \<equiv> \<forall>x \<in> V. is_seminorm V norm
- \<and> (norm x = 0) = (x = 0)"
-
-lemma is_normI [intro]:
- "\<forall>x \<in> V. is_seminorm V norm \<and> (norm x = 0) = (x = 0)
- \<Longrightarrow> is_norm V norm" by (simp only: is_norm_def)
-
-lemma norm_is_seminorm [intro?]:
- "is_norm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> is_seminorm V norm"
- by (unfold is_norm_def) blast
-
-lemma norm_zero_iff:
- "is_norm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> (norm x = 0) = (x = 0)"
- by (unfold is_norm_def) blast
-
-lemma norm_ge_zero [intro?]:
- "is_norm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> 0 \<le> norm x"
- by (unfold is_norm_def is_seminorm_def) blast
+locale norm = seminorm +
+ assumes zero_iff [iff]: "x \<in> V \<Longrightarrow> (\<parallel>x\<parallel> = 0) = (x = 0)"
subsection {* Normed vector spaces *}
-text{* A vector space together with a norm is called
-a \emph{normed space}. *}
-
-constdefs
- is_normed_vectorspace ::
- "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
- "is_normed_vectorspace V norm \<equiv>
- is_vectorspace V \<and> is_norm V norm"
+text {*
+ A vector space together with a norm is called a \emph{normed
+ space}.
+*}
-lemma normed_vsI [intro]:
- "is_vectorspace V \<Longrightarrow> is_norm V norm
- \<Longrightarrow> is_normed_vectorspace V norm"
- by (unfold is_normed_vectorspace_def) blast
-
-lemma normed_vs_vs [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> is_vectorspace V"
- by (unfold is_normed_vectorspace_def) blast
+locale normed_vectorspace = vectorspace + seminorm_vectorspace + norm
-lemma normed_vs_norm [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> is_norm V norm"
- by (unfold is_normed_vectorspace_def) blast
-
-lemma normed_vs_norm_ge_zero [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> x \<in> V \<Longrightarrow> 0 \<le> norm x"
- by (unfold is_normed_vectorspace_def) (fast elim: norm_ge_zero)
-
-lemma normed_vs_norm_gt_zero [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> 0 < norm x"
-proof (unfold is_normed_vectorspace_def, elim conjE)
- assume "x \<in> V" "x \<noteq> 0" "is_vectorspace V" "is_norm V norm"
- have "0 \<le> norm x" ..
- also have "0 \<noteq> norm x"
+lemma (in normed_vectorspace) gt_zero [intro?]:
+ "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> 0 < \<parallel>x\<parallel>"
+proof -
+ assume x: "x \<in> V" and neq: "x \<noteq> 0"
+ from x have "0 \<le> \<parallel>x\<parallel>" ..
+ also have [symmetric]: "\<dots> \<noteq> 0"
proof
- presume "norm x = 0"
- also have "?this = (x = 0)" by (rule norm_zero_iff)
- finally have "x = 0" .
- thus "False" by contradiction
- qed (rule sym)
- finally show "0 < norm x" .
+ assume "\<parallel>x\<parallel> = 0"
+ with x have "x = 0" by simp
+ with neq show False by contradiction
+ qed
+ finally show ?thesis .
qed
-lemma normed_vs_norm_abs_homogenous [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> x \<in> V
- \<Longrightarrow> norm (a \<cdot> x) = \<bar>a\<bar> * norm x"
- by (rule seminorm_abs_homogenous, rule norm_is_seminorm,
- rule normed_vs_norm)
-
-lemma normed_vs_norm_subadditive [intro?]:
- "is_normed_vectorspace V norm \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
- \<Longrightarrow> norm (x + y) \<le> norm x + norm y"
- by (rule seminorm_subadditive, rule norm_is_seminorm,
- rule normed_vs_norm)
-
-text{* Any subspace of a normed vector space is again a
-normed vectorspace.*}
+text {*
+ Any subspace of a normed vector space is again a normed vectorspace.
+*}
lemma subspace_normed_vs [intro?]:
- "is_vectorspace E \<Longrightarrow> is_subspace F E \<Longrightarrow>
- is_normed_vectorspace E norm \<Longrightarrow> is_normed_vectorspace F norm"
-proof (rule normed_vsI)
- assume "is_subspace F E" "is_vectorspace E"
- "is_normed_vectorspace E norm"
- show "is_vectorspace F" ..
- show "is_norm F norm"
- proof (intro is_normI ballI conjI)
- show "is_seminorm F norm"
- proof
- fix x y a presume "x \<in> E"
- show "0 \<le> norm x" ..
- show "norm (a \<cdot> x) = \<bar>a\<bar> * norm x" ..
- presume "y \<in> E"
- show "norm (x + y) \<le> norm x + norm y" ..
- qed (simp!)+
-
- fix x assume "x \<in> F"
- show "(norm x = 0) = (x = 0)"
- proof (rule norm_zero_iff)
- show "is_norm E norm" ..
- qed (simp!)
- qed
+ includes subvectorspace F E + normed_vectorspace E
+ shows "normed_vectorspace F norm"
+proof
+ show "vectorspace F" by (rule vectorspace)
+ have "seminorm E norm" . with subset show "seminorm F norm"
+ by (simp add: seminorm_def)
+ have "norm_axioms E norm" . with subset show "norm_axioms F norm"
+ by (simp add: norm_axioms_def)
qed
end
--- a/src/HOL/Real/HahnBanach/Subspace.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Thu Aug 22 20:49:43 2002 +0200
@@ -16,122 +16,109 @@
and scalar multiplication.
*}
-constdefs
- is_subspace :: "'a::{plus, minus, zero} set \<Rightarrow> 'a set \<Rightarrow> bool"
- "is_subspace U V \<equiv> U \<noteq> {} \<and> U \<subseteq> V
- \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x \<in> U)"
+locale subspace = var U + var V +
+ assumes non_empty [iff, intro]: "U \<noteq> {}"
+ and subset [iff]: "U \<subseteq> V"
+ and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
+ and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
-lemma subspaceI [intro]:
- "0 \<in> U \<Longrightarrow> U \<subseteq> V \<Longrightarrow> \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U) \<Longrightarrow>
- \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U
- \<Longrightarrow> is_subspace U V"
-proof (unfold is_subspace_def, intro conjI)
- assume "0 \<in> U" thus "U \<noteq> {}" by fast
-qed (simp+)
+syntax (symbols)
+ subspace :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "\<unlhd>" 50)
-lemma subspace_not_empty [intro?]: "is_subspace U V \<Longrightarrow> U \<noteq> {}"
- by (unfold is_subspace_def) blast
+lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"
+ by (rule subspace.subset)
-lemma subspace_subset [intro?]: "is_subspace U V \<Longrightarrow> U \<subseteq> V"
- by (unfold is_subspace_def) blast
+lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"
+ using subset by blast
-lemma subspace_subsetD [simp, intro?]:
- "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
- by (unfold is_subspace_def) blast
+lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
+ by (rule subspace.subsetD)
-lemma subspace_add_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
- by (unfold is_subspace_def) blast
+lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"
+ by (rule subspace.subsetD)
+
-lemma subspace_mult_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
- by (unfold is_subspace_def) blast
+locale (open) subvectorspace =
+ subspace + vectorspace
-lemma subspace_diff_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> y \<in> U
- \<Longrightarrow> x - y \<in> U"
+lemma (in subvectorspace) diff_closed [iff]:
+ "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x - y \<in> U"
by (simp add: diff_eq1 negate_eq1)
-text {* Similar as for linear spaces, the existence of the
-zero element in every subspace follows from the non-emptiness
-of the carrier set and by vector space laws.*}
-lemma zero_in_subspace [intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> 0 \<in> U"
+text {*
+ \medskip Similar as for linear spaces, the existence of the zero
+ element in every subspace follows from the non-emptiness of the
+ carrier set and by vector space laws.
+*}
+
+lemma (in subvectorspace) zero [intro]: "0 \<in> U"
proof -
- assume "is_subspace U V" and v: "is_vectorspace V"
- have "U \<noteq> {}" ..
- hence "\<exists>x. x \<in> U" by blast
- thus ?thesis
- proof
- fix x assume u: "x \<in> U"
- hence "x \<in> V" by (simp!)
- with v have "0 = x - x" by (simp!)
- also have "... \<in> U" by (rule subspace_diff_closed)
- finally show ?thesis .
- qed
+ have "U \<noteq> {}" by (rule U_V.non_empty)
+ then obtain x where x: "x \<in> U" by blast
+ hence "x \<in> V" .. hence "0 = x - x" by simp
+ also have "... \<in> U" by (rule U_V.diff_closed)
+ finally show ?thesis .
qed
-lemma subspace_neg_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> - x \<in> U"
+lemma (in subvectorspace) neg_closed [iff]: "x \<in> U \<Longrightarrow> - x \<in> U"
by (simp add: negate_eq1)
+
text {* \medskip Further derived laws: every subspace is a vector space. *}
-lemma subspace_vs [intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_vectorspace U"
-proof -
- assume "is_subspace U V" "is_vectorspace V"
- show ?thesis
- proof
- show "0 \<in> U" ..
- show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" by (simp!)
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" by (simp!)
- show "\<forall>x \<in> U. - x = - 1 \<cdot> x" by (simp! add: negate_eq1)
- show "\<forall>x \<in> U. \<forall>y \<in> U. x - y = x + - y"
- by (simp! add: diff_eq1)
- qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+
+lemma (in subvectorspace) vectorspace [iff]:
+ "vectorspace U"
+proof
+ show "U \<noteq> {}" ..
+ fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"
+ fix a b :: real
+ from x y show "x + y \<in> U" by simp
+ from x show "a \<cdot> x \<in> U" by simp
+ from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)
+ from x y show "x + y = y + x" by (simp add: add_ac)
+ from x show "x - x = 0" by simp
+ from x show "0 + x = x" by simp
+ from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)
+ from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)
+ from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)
+ from x show "1 \<cdot> x = x" by simp
+ from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)
+ from x y show "x - y = x + - y" by (simp add: diff_eq1)
qed
+
text {* The subspace relation is reflexive. *}
-lemma subspace_refl [intro]: "is_vectorspace V \<Longrightarrow> is_subspace V V"
+lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"
proof
- assume "is_vectorspace V"
- show "0 \<in> V" ..
+ show "V \<noteq> {}" ..
show "V \<subseteq> V" ..
- show "\<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V" by (simp!)
- show "\<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V" by (simp!)
+ fix x y assume x: "x \<in> V" and y: "y \<in> V"
+ fix a :: real
+ from x y show "x + y \<in> V" by simp
+ from x show "a \<cdot> x \<in> V" by simp
qed
text {* The subspace relation is transitive. *}
-lemma subspace_trans:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_subspace V W
- \<Longrightarrow> is_subspace U W"
+lemma (in vectorspace) subspace_trans [trans]:
+ "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"
proof
- assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
- show "0 \<in> U" ..
-
- have "U \<subseteq> V" ..
- also have "V \<subseteq> W" ..
- finally show "U \<subseteq> W" .
-
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
- proof (intro ballI)
- fix x y assume "x \<in> U" "y \<in> U"
- show "x + y \<in> U" by (simp!)
+ assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"
+ from uv show "U \<noteq> {}" by (rule subspace.non_empty)
+ show "U \<subseteq> W"
+ proof -
+ from uv have "U \<subseteq> V" by (rule subspace.subset)
+ also from vw have "V \<subseteq> W" by (rule subspace.subset)
+ finally show ?thesis .
qed
-
- show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
- proof (intro ballI allI)
- fix x a assume "x \<in> U"
- show "a \<cdot> x \<in> U" by (simp!)
- qed
+ fix x y assume x: "x \<in> U" and y: "y \<in> U"
+ from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)
+ from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)
qed
-
subsection {* Linear closure *}
text {*
@@ -140,73 +127,75 @@
*}
constdefs
- lin :: "('a::{minus,plus,zero}) \<Rightarrow> 'a set"
+ lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set"
"lin x \<equiv> {a \<cdot> x | a. True}"
-lemma linD: "(x \<in> lin v) = (\<exists>a::real. x = a \<cdot> v)"
- by (unfold lin_def) fast
+lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"
+ by (unfold lin_def) blast
-lemma linI [intro?]: "a \<cdot> x0 \<in> lin x0"
- by (unfold lin_def) fast
+lemma linI' [iff]: "a \<cdot> x \<in> lin x"
+ by (unfold lin_def) blast
+
+lemma linE [elim]:
+ "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"
+ by (unfold lin_def) blast
+
text {* Every vector is contained in its linear closure. *}
-lemma x_lin_x: "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x \<in> lin x"
-proof (unfold lin_def, intro CollectI exI conjI)
- assume "is_vectorspace V" "x \<in> V"
- show "x = 1 \<cdot> x" by (simp!)
-qed simp
+lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"
+proof -
+ assume "x \<in> V"
+ hence "x = 1 \<cdot> x" by simp
+ also have "\<dots> \<in> lin x" ..
+ finally show ?thesis .
+qed
+
+lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"
+proof
+ assume "x \<in> V"
+ thus "0 = 0 \<cdot> x" by simp
+qed
text {* Any linear closure is a subspace. *}
-lemma lin_subspace [intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_subspace (lin x) V"
+lemma (in vectorspace) lin_subspace [intro]:
+ "x \<in> V \<Longrightarrow> lin x \<unlhd> V"
proof
- assume "is_vectorspace V" "x \<in> V"
- show "0 \<in> lin x"
- proof (unfold lin_def, intro CollectI exI conjI)
- show "0 = (0::real) \<cdot> x" by (simp!)
- qed simp
-
+ assume x: "x \<in> V"
+ thus "lin x \<noteq> {}" by (auto simp add: x_lin_x)
show "lin x \<subseteq> V"
- proof (unfold lin_def, intro subsetI, elim CollectE exE conjE)
- fix xa a assume "xa = a \<cdot> x"
- show "xa \<in> V" by (simp!)
+ proof
+ fix x' assume "x' \<in> lin x"
+ then obtain a where "x' = a \<cdot> x" ..
+ with x show "x' \<in> V" by simp
qed
-
- show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x"
- proof (intro ballI)
- fix x1 x2 assume "x1 \<in> lin x" "x2 \<in> lin x"
- thus "x1 + x2 \<in> lin x"
- proof (unfold lin_def, elim CollectE exE conjE,
- intro CollectI exI conjI)
- fix a1 a2 assume "x1 = a1 \<cdot> x" "x2 = a2 \<cdot> x"
- show "x1 + x2 = (a1 + a2) \<cdot> x"
- by (simp! add: vs_add_mult_distrib2)
- qed simp
+ fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"
+ show "x' + x'' \<in> lin x"
+ proof -
+ from x' obtain a' where "x' = a' \<cdot> x" ..
+ moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..
+ ultimately have "x' + x'' = (a' + a'') \<cdot> x"
+ using x by (simp add: distrib)
+ also have "\<dots> \<in> lin x" ..
+ finally show ?thesis .
qed
-
- show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x"
- proof (intro ballI allI)
- fix x1 a assume "x1 \<in> lin x"
- thus "a \<cdot> x1 \<in> lin x"
- proof (unfold lin_def, elim CollectE exE conjE,
- intro CollectI exI conjI)
- fix a1 assume "x1 = a1 \<cdot> x"
- show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
- qed simp
+ fix a :: real
+ show "a \<cdot> x' \<in> lin x"
+ proof -
+ from x' obtain a' where "x' = a' \<cdot> x" ..
+ with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)
+ also have "\<dots> \<in> lin x" ..
+ finally show ?thesis .
qed
qed
+
text {* Any linear closure is a vector space. *}
-lemma lin_vs [intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_vectorspace (lin x)"
-proof (rule subspace_vs)
- assume "is_vectorspace V" "x \<in> V"
- show "is_subspace (lin x) V" ..
-qed
-
+lemma (in vectorspace) lin_vectorspace [intro]:
+ "x \<in> V \<Longrightarrow> vectorspace (lin x)"
+ by (rule subvectorspace.vectorspace) (rule lin_subspace)
subsection {* Sum of two vectorspaces *}
@@ -219,101 +208,92 @@
instance set :: (plus) plus ..
defs (overloaded)
- vs_sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}"
+ sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}"
-lemma vs_sumD:
- "(x \<in> U + V) = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
- by (unfold vs_sum_def) fast
+lemma sumE [elim]:
+ "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
+ by (unfold sum_def) blast
-lemmas vs_sumE = vs_sumD [THEN iffD1, elim_format, standard]
+lemma sumI [intro]:
+ "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
+ by (unfold sum_def) blast
-lemma vs_sumI [intro?]:
- "x \<in> U \<Longrightarrow> y \<in> V \<Longrightarrow> t = x + y \<Longrightarrow> t \<in> U + V"
- by (unfold vs_sum_def) fast
+lemma sumI' [intro]:
+ "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
+ by (unfold sum_def) blast
text {* @{text U} is a subspace of @{text "U + V"}. *}
-lemma subspace_vs_sum1 [intro?]:
- "is_vectorspace U \<Longrightarrow> is_vectorspace V
- \<Longrightarrow> is_subspace U (U + V)"
+lemma subspace_sum1 [iff]:
+ includes vectorspace U + vectorspace V
+ shows "U \<unlhd> U + V"
proof
- assume "is_vectorspace U" "is_vectorspace V"
- show "0 \<in> U" ..
+ show "U \<noteq> {}" ..
show "U \<subseteq> U + V"
- proof (intro subsetI vs_sumI)
- fix x assume "x \<in> U"
- show "x = x + 0" by (simp!)
- show "0 \<in> V" by (simp!)
+ proof
+ fix x assume x: "x \<in> U"
+ moreover have "0 \<in> V" ..
+ ultimately have "x + 0 \<in> U + V" ..
+ with x show "x \<in> U + V" by simp
qed
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
- proof (intro ballI)
- fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
- qed
- show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
- proof (intro ballI allI)
- fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
- qed
+ fix x y assume x: "x \<in> U" and "y \<in> U"
+ thus "x + y \<in> U" by simp
+ from x show "\<And>a. a \<cdot> x \<in> U" by simp
qed
-text{* The sum of two subspaces is again a subspace.*}
+text {* The sum of two subspaces is again a subspace. *}
-lemma vs_sum_subspace [intro?]:
- "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_subspace (U + V) E"
+lemma sum_subspace [intro?]:
+ includes subvectorspace U E + subvectorspace V E
+ shows "U + V \<unlhd> E"
proof
- assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
- show "0 \<in> U + V"
- proof (intro vs_sumI)
+ have "0 \<in> U + V"
+ proof
show "0 \<in> U" ..
show "0 \<in> V" ..
- show "(0::'a) = 0 + 0" by (simp!)
+ show "(0::'a) = 0 + 0" by simp
qed
-
+ thus "U + V \<noteq> {}" by blast
show "U + V \<subseteq> E"
- proof (intro subsetI, elim vs_sumE bexE)
- fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
- show "x \<in> E" by (simp!)
+ proof
+ fix x assume "x \<in> U + V"
+ then obtain u v where x: "x = u + v" and
+ u: "u \<in> U" and v: "v \<in> V" ..
+ have "U \<unlhd> E" . with u have "u \<in> E" ..
+ moreover have "V \<unlhd> E" . with v have "v \<in> E" ..
+ ultimately show "x \<in> E" using x by simp
qed
-
- show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
- proof (intro ballI)
- fix x y assume "x \<in> U + V" "y \<in> U + V"
- thus "x + y \<in> U + V"
- proof (elim vs_sumE bexE, intro vs_sumI)
- fix ux vx uy vy
- assume "ux \<in> U" "vx \<in> V" "x = ux + vx"
- and "uy \<in> U" "vy \<in> V" "y = uy + vy"
- show "x + y = (ux + uy) + (vx + vy)" by (simp!)
- qed (simp_all!)
+ fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
+ show "x + y \<in> U + V"
+ proof -
+ from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
+ moreover
+ from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..
+ ultimately
+ have "ux + uy \<in> U"
+ and "vx + vy \<in> V"
+ and "x + y = (ux + uy) + (vx + vy)"
+ using x y by (simp_all add: add_ac)
+ thus ?thesis ..
qed
-
- show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
- proof (intro ballI allI)
- fix x a assume "x \<in> U + V"
- thus "a \<cdot> x \<in> U + V"
- proof (elim vs_sumE bexE, intro vs_sumI)
- fix a x u v assume "u \<in> U" "v \<in> V" "x = u + v"
- show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)"
- by (simp! add: vs_add_mult_distrib1)
- qed (simp_all!)
+ fix a show "a \<cdot> x \<in> U + V"
+ proof -
+ from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
+ hence "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
+ and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)
+ thus ?thesis ..
qed
qed
text{* The sum of two subspaces is a vectorspace. *}
-lemma vs_sum_vs [intro?]:
- "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_vectorspace (U + V)"
-proof (rule subspace_vs)
- assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
- show "is_subspace (U + V) E" ..
-qed
-
+lemma sum_vs [intro?]:
+ "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
+ by (rule subvectorspace.vectorspace) (rule sum_subspace)
subsection {* Direct sums *}
-
text {*
The sum of @{text U} and @{text V} is called \emph{direct}, iff the
zero element is the only common element of @{text U} and @{text
@@ -323,31 +303,35 @@
*}
lemma decomp:
- "is_vectorspace E \<Longrightarrow> is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow>
- U \<inter> V = {0} \<Longrightarrow> u1 \<in> U \<Longrightarrow> u2 \<in> U \<Longrightarrow> v1 \<in> V \<Longrightarrow> v2 \<in> V \<Longrightarrow>
- u1 + v1 = u2 + v2 \<Longrightarrow> u1 = u2 \<and> v1 = v2"
+ includes vectorspace E + subspace U E + subspace V E
+ assumes direct: "U \<inter> V = {0}"
+ and u1: "u1 \<in> U" and u2: "u2 \<in> U"
+ and v1: "v1 \<in> V" and v2: "v2 \<in> V"
+ and sum: "u1 + v1 = u2 + v2"
+ shows "u1 = u2 \<and> v1 = v2"
proof
- assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"
- "U \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> V"
- "u1 + v1 = u2 + v2"
- have eq: "u1 - u2 = v2 - v1" by (simp! add: vs_add_diff_swap)
- have u: "u1 - u2 \<in> U" by (simp!)
- with eq have v': "v2 - v1 \<in> U" by simp
- have v: "v2 - v1 \<in> V" by (simp!)
- with eq have u': "u1 - u2 \<in> V" by simp
+ have U: "vectorspace U" by (rule subvectorspace.vectorspace)
+ have V: "vectorspace V" by (rule subvectorspace.vectorspace)
+ from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
+ by (simp add: add_diff_swap)
+ from u1 u2 have u: "u1 - u2 \<in> U"
+ by (rule vectorspace.diff_closed [OF U])
+ with eq have v': "v2 - v1 \<in> U" by (simp only:)
+ from v2 v1 have v: "v2 - v1 \<in> V"
+ by (rule vectorspace.diff_closed [OF V])
+ with eq have u': " u1 - u2 \<in> V" by (simp only:)
show "u1 = u2"
- proof (rule vs_add_minus_eq)
- show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
+ proof (rule add_minus_eq)
show "u1 \<in> E" ..
show "u2 \<in> E" ..
+ from u u' and direct show "u1 - u2 = 0" by blast
qed
-
show "v1 = v2"
- proof (rule vs_add_minus_eq [symmetric])
- show "v2 - v1 = 0" by (rule Int_singletonD [OF _ v' v])
+ proof (rule add_minus_eq [symmetric])
show "v1 \<in> E" ..
show "v2 \<in> E" ..
+ from v v' and direct show "v2 - v1 = 0" by blast
qed
qed
@@ -361,58 +345,48 @@
*}
lemma decomp_H':
- "is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow> y1 \<in> H \<Longrightarrow> y2 \<in> H \<Longrightarrow>
- x' \<notin> H \<Longrightarrow> x' \<in> E \<Longrightarrow> x' \<noteq> 0 \<Longrightarrow> y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'
- \<Longrightarrow> y1 = y2 \<and> a1 = a2"
+ includes vectorspace E + subvectorspace H E
+ assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"
+ and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
+ shows "y1 = y2 \<and> a1 = a2"
proof
- assume "is_vectorspace E" and h: "is_subspace H E"
- and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
-
have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
proof (rule decomp)
show "a1 \<cdot> x' \<in> lin x'" ..
show "a2 \<cdot> x' \<in> lin x'" ..
- show "H \<inter> (lin x') = {0}"
+ show "H \<inter> lin x' = {0}"
proof
show "H \<inter> lin x' \<subseteq> {0}"
- proof (intro subsetI, elim IntE, rule singleton_iff [THEN iffD2])
- fix x assume "x \<in> H" "x \<in> lin x'"
- thus "x = 0"
- proof (unfold lin_def, elim CollectE exE conjE)
- fix a assume "x = a \<cdot> x'"
- show ?thesis
- proof cases
- assume "a = (0::real)" show ?thesis by (simp!)
- next
- assume "a \<noteq> (0::real)"
- from h have "inverse a \<cdot> a \<cdot> x' \<in> H"
- by (rule subspace_mult_closed) (simp!)
- also have "inverse a \<cdot> a \<cdot> x' = x'" by (simp!)
- finally have "x' \<in> H" .
- thus ?thesis by contradiction
- qed
- qed
+ proof
+ fix x assume x: "x \<in> H \<inter> lin x'"
+ then obtain a where xx': "x = a \<cdot> x'"
+ by blast
+ have "x = 0"
+ proof cases
+ assume "a = 0"
+ with xx' and x' show ?thesis by simp
+ next
+ assume a: "a \<noteq> 0"
+ from x have "x \<in> H" ..
+ with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp
+ with a and x' have "x' \<in> H" by (simp add: mult_assoc2)
+ thus ?thesis by contradiction
+ qed
+ thus "x \<in> {0}" ..
qed
show "{0} \<subseteq> H \<inter> lin x'"
proof -
- have "0 \<in> H \<inter> lin x'"
- proof (rule IntI)
- show "0 \<in> H" ..
- from lin_vs show "0 \<in> lin x'" ..
- qed
- thus ?thesis by simp
+ have "0 \<in> H" ..
+ moreover have "0 \<in> lin x'" ..
+ ultimately show ?thesis by blast
qed
qed
- show "is_subspace (lin x') E" ..
+ show "lin x' \<unlhd> E" ..
qed
-
- from c show "y1 = y2" by simp
-
- show "a1 = a2"
- proof (rule vs_mult_right_cancel [THEN iffD1])
- from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
- qed
+ thus "y1 = y2" ..
+ from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..
+ with x' show "a1 = a2" by (simp add: mult_right_cancel)
qed
text {*
@@ -423,18 +397,20 @@
*}
lemma decomp_H'_H:
- "is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow> t \<in> H \<Longrightarrow> x' \<notin> H \<Longrightarrow> x' \<in> E
- \<Longrightarrow> x' \<noteq> 0
- \<Longrightarrow> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (0::real))"
-proof (rule, unfold split_tupled_all)
- assume "is_vectorspace E" "is_subspace H E" "t \<in> H" "x' \<notin> H" "x' \<in> E"
- "x' \<noteq> 0"
- have h: "is_vectorspace H" ..
- fix y a presume t1: "t = y + a \<cdot> x'" and "y \<in> H"
- have "y = t \<and> a = (0::real)"
- by (rule decomp_H') (auto!)
- thus "(y, a) = (t, (0::real))" by (simp!)
-qed (simp_all!)
+ includes vectorspace E + subvectorspace H E
+ assumes t: "t \<in> H"
+ and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
+proof (rule, simp_all only: split_paired_all split_conv)
+ from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp
+ fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"
+ have "y = t \<and> a = 0"
+ proof (rule decomp_H')
+ from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp
+ from ya show "y \<in> H" ..
+ qed
+ with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp
+qed
text {*
The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}
@@ -443,42 +419,41 @@
*}
lemma h'_definite:
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
- in (h y) + a * xi) \<Longrightarrow>
- x = y + a \<cdot> x' \<Longrightarrow> is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow>
- y \<in> H \<Longrightarrow> x' \<notin> H \<Longrightarrow> x' \<in> E \<Longrightarrow> x' \<noteq> 0
- \<Longrightarrow> h' x = h y + a * xi"
+ includes var H
+ assumes h'_def:
+ "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
+ in (h y) + a * xi)"
+ and x: "x = y + a \<cdot> x'"
+ includes vectorspace E + subvectorspace H E
+ assumes y: "y \<in> H"
+ and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ shows "h' x = h y + a * xi"
proof -
- assume
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
- in (h y) + a * xi)"
- "x = y + a \<cdot> x'" "is_vectorspace E" "is_subspace H E"
- "y \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- hence "x \<in> H + (lin x')"
- by (auto simp add: vs_sum_def lin_def)
- have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
+ from x y x' have "x \<in> H + lin x'" by auto
+ have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
proof
- show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
- by (blast!)
- next
- fix xa ya
- assume "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) xa"
- "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) ya"
- show "xa = ya"
+ from x y show "\<exists>p. ?P p" by blast
+ fix p q assume p: "?P p" and q: "?P q"
+ show "p = q"
proof -
- show "fst xa = fst ya \<and> snd xa = snd ya \<Longrightarrow> xa = ya"
- by (simp add: Pair_fst_snd_eq)
- have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H"
- by (auto!)
- have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H"
- by (auto!)
- from x y show "fst xa = fst ya \<and> snd xa = snd ya"
- by (elim conjE) (rule decomp_H', (simp!)+)
+ from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"
+ by (cases p) simp
+ from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"
+ by (cases q) simp
+ have "fst p = fst q \<and> snd p = snd q"
+ proof (rule decomp_H')
+ from xp show "fst p \<in> H" ..
+ from xq show "fst q \<in> H" ..
+ from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"
+ by simp
+ apply_end assumption+
+ qed
+ thus ?thesis by (cases p, cases q) simp
qed
qed
hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
- by (rule some1_equality) (blast!)
- thus "h' x = h y + a * xi" by (simp! add: Let_def)
+ by (rule some1_equality) (simp add: x y)
+ with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)
qed
end
--- a/src/HOL/Real/HahnBanach/VectorSpace.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/VectorSpace.thy Thu Aug 22 20:49:43 2002 +0200
@@ -5,7 +5,7 @@
header {* Vector spaces *}
-theory VectorSpace = Bounds + Aux:
+theory VectorSpace = Aux:
subsection {* Signature *}
@@ -35,459 +35,379 @@
the neutral element of scalar multiplication.
*}
-constdefs
- is_vectorspace :: "('a::{plus, minus, zero}) set \<Rightarrow> bool"
- "is_vectorspace V \<equiv> V \<noteq> {}
- \<and> (\<forall>x \<in> V. \<forall>y \<in> V. \<forall>z \<in> V. \<forall>a b.
- x + y \<in> V
- \<and> a \<cdot> x \<in> V
- \<and> (x + y) + z = x + (y + z)
- \<and> x + y = y + x
- \<and> x - x = 0
- \<and> 0 + x = x
- \<and> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y
- \<and> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x
- \<and> (a * b) \<cdot> x = a \<cdot> b \<cdot> x
- \<and> 1 \<cdot> x = x
- \<and> - x = (- 1) \<cdot> x
- \<and> x - y = x + - y)"
-
-
-text {* \medskip The corresponding introduction rule is:*}
-
-lemma vsI [intro]:
- "0 \<in> V \<Longrightarrow>
- \<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V \<Longrightarrow>
- \<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V \<Longrightarrow>
- \<forall>x \<in> V. \<forall>y \<in> V. \<forall>z \<in> V. (x + y) + z = x + (y + z) \<Longrightarrow>
- \<forall>x \<in> V. \<forall>y \<in> V. x + y = y + x \<Longrightarrow>
- \<forall>x \<in> V. x - x = 0 \<Longrightarrow>
- \<forall>x \<in> V. 0 + x = x \<Longrightarrow>
- \<forall>x \<in> V. \<forall>y \<in> V. \<forall>a. a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y \<Longrightarrow>
- \<forall>x \<in> V. \<forall>a b. (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x \<Longrightarrow>
- \<forall>x \<in> V. \<forall>a b. (a * b) \<cdot> x = a \<cdot> b \<cdot> x \<Longrightarrow>
- \<forall>x \<in> V. 1 \<cdot> x = x \<Longrightarrow>
- \<forall>x \<in> V. - x = (- 1) \<cdot> x \<Longrightarrow>
- \<forall>x \<in> V. \<forall>y \<in> V. x - y = x + - y \<Longrightarrow> is_vectorspace V"
- by (unfold is_vectorspace_def) auto
+locale vectorspace = var V +
+ assumes non_empty [iff, intro?]: "V \<noteq> {}"
+ and add_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y \<in> V"
+ and mult_closed [iff]: "x \<in> V \<Longrightarrow> a \<cdot> x \<in> V"
+ and add_assoc: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y) + z = x + (y + z)"
+ and add_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = y + x"
+ and diff_self [simp]: "x \<in> V \<Longrightarrow> x - x = 0"
+ and add_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 + x = x"
+ and add_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
+ and add_mult_distrib2: "x \<in> V \<Longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
+ and mult_assoc: "x \<in> V \<Longrightarrow> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"
+ and mult_1 [simp]: "x \<in> V \<Longrightarrow> 1 \<cdot> x = x"
+ and negate_eq1: "x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"
+ and diff_eq1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"
-text {* \medskip The corresponding destruction rules are: *}
-
-lemma negate_eq1:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"
- by (unfold is_vectorspace_def) simp
-
-lemma diff_eq1:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"
- by (unfold is_vectorspace_def) simp
-
-lemma negate_eq2:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
- by (unfold is_vectorspace_def) simp
-
-lemma negate_eq2a:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
- by (unfold is_vectorspace_def) simp
+lemma (in vectorspace) negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
+ by (rule negate_eq1 [symmetric])
-lemma diff_eq2:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_not_empty [intro?]: "is_vectorspace V \<Longrightarrow> (V \<noteq> {})"
- by (unfold is_vectorspace_def) simp
+lemma (in vectorspace) negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
+ by (simp add: negate_eq1)
-lemma vs_add_closed [simp, intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y \<in> V"
- by (unfold is_vectorspace_def) simp
+lemma (in vectorspace) diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
+ by (rule diff_eq1 [symmetric])
-lemma vs_mult_closed [simp, intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> a \<cdot> x \<in> V"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_diff_closed [simp, intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
+lemma (in vectorspace) diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
by (simp add: diff_eq1 negate_eq1)
-lemma vs_neg_closed [simp, intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - x \<in> V"
+lemma (in vectorspace) neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
by (simp add: negate_eq1)
-lemma vs_add_assoc [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> (x + y) + z = x + (y + z)"
- by (unfold is_vectorspace_def) blast
-
-lemma vs_add_commute [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> y + x = x + y"
- by (unfold is_vectorspace_def) blast
-
-lemma vs_add_left_commute [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> x + (y + z) = y + (x + z)"
+lemma (in vectorspace) add_left_commute:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"
proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
+ assume xyz: "x \<in> V" "y \<in> V" "z \<in> V"
hence "x + (y + z) = (x + y) + z"
- by (simp only: vs_add_assoc)
- also have "... = (y + x) + z" by (simp! only: vs_add_commute)
- also have "... = y + (x + z)" by (simp! only: vs_add_assoc)
+ by (simp only: add_assoc)
+ also from xyz have "... = (y + x) + z" by (simp only: add_commute)
+ also from xyz have "... = y + (x + z)" by (simp only: add_assoc)
finally show ?thesis .
qed
-theorems vs_add_ac = vs_add_assoc vs_add_commute vs_add_left_commute
+theorems (in vectorspace) add_ac =
+ add_assoc add_commute add_left_commute
-lemma vs_diff_self [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x - x = 0"
- by (unfold is_vectorspace_def) simp
text {* The existence of the zero element of a vector space
-follows from the non-emptiness of carrier set. *}
+ follows from the non-emptiness of carrier set. *}
-lemma zero_in_vs [simp, intro]: "is_vectorspace V \<Longrightarrow> 0 \<in> V"
+lemma (in vectorspace) zero [iff]: "0 \<in> V"
proof -
- assume "is_vectorspace V"
- have "V \<noteq> {}" ..
- then obtain x where "x \<in> V" by blast
- have "0 = x - x" by (simp!)
- also have "... \<in> V" by (simp! only: vs_diff_closed)
+ from non_empty obtain x where x: "x \<in> V" by blast
+ then have "0 = x - x" by (rule diff_self [symmetric])
+ also from x have "... \<in> V" by (rule diff_closed)
finally show ?thesis .
qed
-lemma vs_add_zero_left [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> 0 + x = x"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_add_zero_right [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x + 0 = x"
+lemma (in vectorspace) add_zero_right [simp]:
+ "x \<in> V \<Longrightarrow> x + 0 = x"
proof -
- assume "is_vectorspace V" "x \<in> V"
- hence "x + 0 = 0 + x" by simp
- also have "... = x" by (simp!)
+ assume x: "x \<in> V"
+ from this and zero have "x + 0 = 0 + x" by (rule add_commute)
+ also from x have "... = x" by (rule add_zero_left)
finally show ?thesis .
qed
-lemma vs_add_mult_distrib1:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
- \<Longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
- by (unfold is_vectorspace_def) simp
+lemma (in vectorspace) mult_assoc2:
+ "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
+ by (simp only: mult_assoc)
-lemma vs_add_mult_distrib2:
- "is_vectorspace V \<Longrightarrow> x \<in> V
- \<Longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_mult_assoc:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_mult_assoc2 [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
- by (simp only: vs_mult_assoc)
+lemma (in vectorspace) diff_mult_distrib1:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
+ by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2)
-lemma vs_mult_1 [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> 1 \<cdot> x = x"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_diff_mult_distrib1:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
- \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
- by (simp add: diff_eq1 negate_eq1 vs_add_mult_distrib1)
-
-lemma vs_diff_mult_distrib2:
- "is_vectorspace V \<Longrightarrow> x \<in> V
- \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
+lemma (in vectorspace) diff_mult_distrib2:
+ "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
proof -
- assume "is_vectorspace V" "x \<in> V"
+ assume x: "x \<in> V"
have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
- by (unfold real_diff_def, simp)
+ by (simp add: real_diff_def)
also have "... = a \<cdot> x + (- b) \<cdot> x"
- by (rule vs_add_mult_distrib2)
- also have "... = a \<cdot> x + - (b \<cdot> x)"
- by (simp! add: negate_eq1)
- also have "... = a \<cdot> x - (b \<cdot> x)"
- by (simp! add: diff_eq1)
+ by (rule add_mult_distrib2)
+ also from x have "... = a \<cdot> x + - (b \<cdot> x)"
+ by (simp add: negate_eq1 mult_assoc2)
+ also from x have "... = a \<cdot> x - (b \<cdot> x)"
+ by (simp add: diff_eq1)
finally show ?thesis .
qed
+lemmas (in vectorspace) distrib =
+ add_mult_distrib1 add_mult_distrib2
+ diff_mult_distrib1 diff_mult_distrib2
+
text {* \medskip Further derived laws: *}
-lemma vs_mult_zero_left [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
+lemma (in vectorspace) mult_zero_left [simp]:
+ "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
proof -
- assume "is_vectorspace V" "x \<in> V"
- have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp
+ assume x: "x \<in> V"
+ have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp
also have "... = (1 + - 1) \<cdot> x" by simp
also have "... = 1 \<cdot> x + (- 1) \<cdot> x"
- by (rule vs_add_mult_distrib2)
- also have "... = x + (- 1) \<cdot> x" by (simp!)
- also have "... = x + - x" by (simp! add: negate_eq2a)
- also have "... = x - x" by (simp! add: diff_eq2)
- also have "... = 0" by (simp!)
+ by (rule add_mult_distrib2)
+ also from x have "... = x + (- 1) \<cdot> x" by simp
+ also from x have "... = x + - x" by (simp add: negate_eq2a)
+ also from x have "... = x - x" by (simp add: diff_eq2)
+ also from x have "... = 0" by simp
finally show ?thesis .
qed
-lemma vs_mult_zero_right [simp]:
- "is_vectorspace (V:: 'a::{plus, minus, zero} set)
- \<Longrightarrow> a \<cdot> 0 = (0::'a)"
+lemma (in vectorspace) mult_zero_right [simp]:
+ "a \<cdot> 0 = (0::'a)"
proof -
- assume "is_vectorspace V"
- have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by (simp!)
+ have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by simp
also have "... = a \<cdot> 0 - a \<cdot> 0"
- by (rule vs_diff_mult_distrib1) (simp!)+
- also have "... = 0" by (simp!)
+ by (rule diff_mult_distrib1) simp_all
+ also have "... = 0" by simp
finally show ?thesis .
qed
-lemma vs_minus_mult_cancel [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
- by (simp add: negate_eq1)
+lemma (in vectorspace) minus_mult_cancel [simp]:
+ "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
+ by (simp add: negate_eq1 mult_assoc2)
-lemma vs_add_minus_left_eq_diff:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
+lemma (in vectorspace) add_minus_left_eq_diff:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V"
- hence "- x + y = y + - x"
- by (simp add: vs_add_commute)
- also have "... = y - x" by (simp! add: diff_eq1)
+ assume xy: "x \<in> V" "y \<in> V"
+ hence "- x + y = y + - x" by (simp add: add_commute)
+ also from xy have "... = y - x" by (simp add: diff_eq1)
finally show ?thesis .
qed
-lemma vs_add_minus [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x + - x = 0"
- by (simp! add: diff_eq2)
+lemma (in vectorspace) add_minus [simp]:
+ "x \<in> V \<Longrightarrow> x + - x = 0"
+ by (simp add: diff_eq2)
-lemma vs_add_minus_left [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - x + x = 0"
- by (simp! add: diff_eq2)
+lemma (in vectorspace) add_minus_left [simp]:
+ "x \<in> V \<Longrightarrow> - x + x = 0"
+ by (simp add: diff_eq2 add_commute)
-lemma vs_minus_minus [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - (- x) = x"
+lemma (in vectorspace) minus_minus [simp]:
+ "x \<in> V \<Longrightarrow> - (- x) = x"
+ by (simp add: negate_eq1 mult_assoc2)
+
+lemma (in vectorspace) minus_zero [simp]:
+ "- (0::'a) = 0"
by (simp add: negate_eq1)
-lemma vs_minus_zero [simp]:
- "is_vectorspace (V::'a::{plus, minus, zero} set) \<Longrightarrow> - (0::'a) = 0"
- by (simp add: negate_eq1)
-
-lemma vs_minus_zero_iff [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> (- x = 0) = (x = 0)"
- (concl is "?L = ?R")
-proof -
- assume "is_vectorspace V" "x \<in> V"
- show "?L = ?R"
- proof
- have "x = - (- x)" by (simp! add: vs_minus_minus)
- also assume ?L
- also have "- ... = 0" by (rule vs_minus_zero)
- finally show ?R .
- qed (simp!)
+lemma (in vectorspace) minus_zero_iff [simp]:
+ "x \<in> V \<Longrightarrow> (- x = 0) = (x = 0)"
+proof
+ assume x: "x \<in> V"
+ {
+ from x have "x = - (- x)" by (simp add: minus_minus)
+ also assume "- x = 0"
+ also have "- ... = 0" by (rule minus_zero)
+ finally show "x = 0" .
+ next
+ assume "x = 0"
+ then show "- x = 0" by simp
+ }
qed
-lemma vs_add_minus_cancel [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
- by (simp add: vs_add_assoc [symmetric] del: vs_add_commute)
+lemma (in vectorspace) add_minus_cancel [simp]:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
+ by (simp add: add_assoc [symmetric] del: add_commute)
-lemma vs_minus_add_cancel [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
- by (simp add: vs_add_assoc [symmetric] del: vs_add_commute)
+lemma (in vectorspace) minus_add_cancel [simp]:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
+ by (simp add: add_assoc [symmetric] del: add_commute)
-lemma vs_minus_add_distrib [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
- \<Longrightarrow> - (x + y) = - x + - y"
- by (simp add: negate_eq1 vs_add_mult_distrib1)
+lemma (in vectorspace) minus_add_distrib [simp]:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"
+ by (simp add: negate_eq1 add_mult_distrib1)
-lemma vs_diff_zero [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x - 0 = x"
+lemma (in vectorspace) diff_zero [simp]:
+ "x \<in> V \<Longrightarrow> x - 0 = x"
+ by (simp add: diff_eq1)
+
+lemma (in vectorspace) diff_zero_right [simp]:
+ "x \<in> V \<Longrightarrow> 0 - x = - x"
by (simp add: diff_eq1)
-lemma vs_diff_zero_right [simp]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> 0 - x = - x"
- by (simp add:diff_eq1)
-
-lemma vs_add_left_cancel:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> (x + y = x + z) = (y = z)"
- (concl is "?L = ?R")
+lemma (in vectorspace) add_left_cancel:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y = x + z) = (y = z)"
proof
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
- have "y = 0 + y" by (simp!)
- also have "... = - x + x + y" by (simp!)
- also have "... = - x + (x + y)"
- by (simp! only: vs_add_assoc vs_neg_closed)
- also assume "x + y = x + z"
- also have "- x + (x + z) = - x + x + z"
- by (simp! only: vs_add_assoc [symmetric] vs_neg_closed)
- also have "... = z" by (simp!)
- finally show ?R .
-qed blast
+ assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
+ {
+ from y have "y = 0 + y" by simp
+ also from x y have "... = (- x + x) + y" by simp
+ also from x y have "... = - x + (x + y)"
+ by (simp add: add_assoc neg_closed)
+ also assume "x + y = x + z"
+ also from x z have "- x + (x + z) = - x + x + z"
+ by (simp add: add_assoc [symmetric] neg_closed)
+ also from x z have "... = z" by simp
+ finally show "y = z" .
+ next
+ assume "y = z"
+ then show "x + y = x + z" by (simp only:)
+ }
+qed
-lemma vs_add_right_cancel:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> (y + x = z + x) = (y = z)"
- by (simp only: vs_add_commute vs_add_left_cancel)
+lemma (in vectorspace) add_right_cancel:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"
+ by (simp only: add_commute add_left_cancel)
-lemma vs_add_assoc_cong:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"
- by (simp only: vs_add_assoc [symmetric])
+lemma (in vectorspace) add_assoc_cong:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V
+ \<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"
+ by (simp only: add_assoc [symmetric])
-lemma vs_mult_left_commute:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> x \<cdot> y \<cdot> z = y \<cdot> x \<cdot> z"
- by (simp add: real_mult_commute)
+lemma (in vectorspace) mult_left_commute:
+ "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
+ by (simp add: real_mult_commute mult_assoc2)
-lemma vs_mult_zero_uniq:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> a \<cdot> x = 0 \<Longrightarrow> a = 0"
+lemma (in vectorspace) mult_zero_uniq:
+ "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> a \<cdot> x = 0 \<Longrightarrow> a = 0"
proof (rule classical)
- assume "is_vectorspace V" "x \<in> V" "a \<cdot> x = 0" "x \<noteq> 0"
- assume "a \<noteq> 0"
- have "x = (inverse a * a) \<cdot> x" by (simp!)
- also have "... = inverse a \<cdot> (a \<cdot> x)" by (rule vs_mult_assoc)
- also have "... = inverse a \<cdot> 0" by (simp!)
- also have "... = 0" by (simp!)
+ assume a: "a \<noteq> 0"
+ assume x: "x \<in> V" "x \<noteq> 0" and ax: "a \<cdot> x = 0"
+ from x a have "x = (inverse a * a) \<cdot> x" by simp
+ also have "... = inverse a \<cdot> (a \<cdot> x)" by (rule mult_assoc)
+ also from ax have "... = inverse a \<cdot> 0" by simp
+ also have "... = 0" by simp
finally have "x = 0" .
thus "a = 0" by contradiction
qed
-lemma vs_mult_left_cancel:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<noteq> 0 \<Longrightarrow>
- (a \<cdot> x = a \<cdot> y) = (x = y)"
- (concl is "?L = ?R")
+lemma (in vectorspace) mult_left_cancel:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (a \<cdot> x = a \<cdot> y) = (x = y)"
proof
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "a \<noteq> 0"
- have "x = 1 \<cdot> x" by (simp!)
- also have "... = (inverse a * a) \<cdot> x" by (simp!)
- also have "... = inverse a \<cdot> (a \<cdot> x)"
- by (simp! only: vs_mult_assoc)
- also assume ?L
- also have "inverse a \<cdot> ... = y" by (simp!)
- finally show ?R .
-qed simp
+ assume x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
+ from x have "x = 1 \<cdot> x" by simp
+ also from a have "... = (inverse a * a) \<cdot> x" by simp
+ also from x have "... = inverse a \<cdot> (a \<cdot> x)"
+ by (simp only: mult_assoc)
+ also assume "a \<cdot> x = a \<cdot> y"
+ also from a y have "inverse a \<cdot> ... = y"
+ by (simp add: mult_assoc2)
+ finally show "x = y" .
+next
+ assume "x = y"
+ then show "a \<cdot> x = a \<cdot> y" by (simp only:)
+qed
-lemma vs_mult_right_cancel: (*** forward ***)
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x \<noteq> 0
- \<Longrightarrow> (a \<cdot> x = b \<cdot> x) = (a = b)" (concl is "?L = ?R")
+lemma (in vectorspace) mult_right_cancel:
+ "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> (a \<cdot> x = b \<cdot> x) = (a = b)"
proof
- assume v: "is_vectorspace V" "x \<in> V" "x \<noteq> 0"
- have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
- by (simp! add: vs_diff_mult_distrib2)
- also assume ?L hence "a \<cdot> x - b \<cdot> x = 0" by (simp!)
- finally have "(a - b) \<cdot> x = 0" .
- from v this have "a - b = 0" by (rule vs_mult_zero_uniq)
- thus "a = b" by simp
-qed simp
+ assume x: "x \<in> V" and neq: "x \<noteq> 0"
+ {
+ from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
+ by (simp add: diff_mult_distrib2)
+ also assume "a \<cdot> x = b \<cdot> x"
+ with x have "a \<cdot> x - b \<cdot> x = 0" by simp
+ finally have "(a - b) \<cdot> x = 0" .
+ with x neq have "a - b = 0" by (rule mult_zero_uniq)
+ thus "a = b" by simp
+ next
+ assume "a = b"
+ then show "a \<cdot> x = b \<cdot> x" by (simp only:)
+ }
+qed
-lemma vs_eq_diff_eq:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow>
- (x = z - y) = (x + y = z)"
- (concl is "?L = ?R" )
-proof -
- assume vs: "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
- show "?L = ?R"
- proof
- assume ?L
+lemma (in vectorspace) eq_diff_eq:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x = z - y) = (x + y = z)"
+proof
+ assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
+ {
+ assume "x = z - y"
hence "x + y = z - y + y" by simp
- also have "... = z + - y + y" by (simp! add: diff_eq1)
+ also from y z have "... = z + - y + y"
+ by (simp add: diff_eq1)
also have "... = z + (- y + y)"
- by (rule vs_add_assoc) (simp!)+
- also from vs have "... = z + 0"
- by (simp only: vs_add_minus_left)
- also from vs have "... = z" by (simp only: vs_add_zero_right)
- finally show ?R .
+ by (rule add_assoc) (simp_all add: y z)
+ also from y z have "... = z + 0"
+ by (simp only: add_minus_left)
+ also from z have "... = z"
+ by (simp only: add_zero_right)
+ finally show "x + y = z" .
next
- assume ?R
+ assume "x + y = z"
hence "z - y = (x + y) - y" by simp
- also from vs have "... = x + y + - y"
+ also from x y have "... = x + y + - y"
by (simp add: diff_eq1)
also have "... = x + (y + - y)"
- by (rule vs_add_assoc) (simp!)+
- also have "... = x" by (simp!)
- finally show ?L by (rule sym)
- qed
+ by (rule add_assoc) (simp_all add: x y)
+ also from x y have "... = x" by simp
+ finally show "x = z - y" ..
+ }
qed
-lemma vs_add_minus_eq_minus:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = 0 \<Longrightarrow> x = - y"
+lemma (in vectorspace) add_minus_eq_minus:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = 0 \<Longrightarrow> x = - y"
proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V"
- have "x = (- y + y) + x" by (simp!)
- also have "... = - y + (x + y)" by (simp!)
+ assume x: "x \<in> V" and y: "y \<in> V"
+ from x y have "x = (- y + y) + x" by simp
+ also from x y have "... = - y + (x + y)" by (simp add: add_ac)
also assume "x + y = 0"
- also have "- y + 0 = - y" by (simp!)
+ also from y have "- y + 0 = - y" by simp
finally show "x = - y" .
qed
-lemma vs_add_minus_eq:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = 0 \<Longrightarrow> x = y"
+lemma (in vectorspace) add_minus_eq:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = 0 \<Longrightarrow> x = y"
proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "x - y = 0"
+ assume x: "x \<in> V" and y: "y \<in> V"
assume "x - y = 0"
- hence e: "x + - y = 0" by (simp! add: diff_eq1)
- with _ _ _ have "x = - (- y)"
- by (rule vs_add_minus_eq_minus) (simp!)+
- thus "x = y" by (simp!)
+ with x y have eq: "x + - y = 0" by (simp add: diff_eq1)
+ with _ _ have "x = - (- y)"
+ by (rule add_minus_eq_minus) (simp_all add: x y)
+ with x y show "x = y" by simp
qed
-lemma vs_add_diff_swap:
- "is_vectorspace V \<Longrightarrow> a \<in> V \<Longrightarrow> b \<in> V \<Longrightarrow> c \<in> V \<Longrightarrow> d \<in> V \<Longrightarrow> a + b = c + d
- \<Longrightarrow> a - c = d - b"
+lemma (in vectorspace) add_diff_swap:
+ "a \<in> V \<Longrightarrow> b \<in> V \<Longrightarrow> c \<in> V \<Longrightarrow> d \<in> V \<Longrightarrow> a + b = c + d
+ \<Longrightarrow> a - c = d - b"
proof -
- assume vs: "is_vectorspace V" "a \<in> V" "b \<in> V" "c \<in> V" "d \<in> V"
+ assume vs: "a \<in> V" "b \<in> V" "c \<in> V" "d \<in> V"
and eq: "a + b = c + d"
- have "- c + (a + b) = - c + (c + d)"
- by (simp! add: vs_add_left_cancel)
- also have "... = d" by (rule vs_minus_add_cancel)
+ then have "- c + (a + b) = - c + (c + d)"
+ by (simp add: add_left_cancel)
+ also have "... = d" by (rule minus_add_cancel)
finally have eq: "- c + (a + b) = d" .
from vs have "a - c = (- c + (a + b)) + - b"
- by (simp add: vs_add_ac diff_eq1)
- also from eq have "... = d + - b"
- by (simp! add: vs_add_right_cancel)
- also have "... = d - b" by (simp! add: diff_eq2)
+ by (simp add: add_ac diff_eq1)
+ also from vs eq have "... = d + - b"
+ by (simp add: add_right_cancel)
+ also from vs have "... = d - b" by (simp add: diff_eq2)
finally show "a - c = d - b" .
qed
-lemma vs_add_cancel_21:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> u \<in> V
- \<Longrightarrow> (x + (y + z) = y + u) = ((x + z) = u)"
- (concl is "?L = ?R")
-proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V" "u \<in> V"
- show "?L = ?R"
- proof
- have "x + z = - y + y + (x + z)" by (simp!)
+lemma (in vectorspace) vs_add_cancel_21:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> u \<in> V
+ \<Longrightarrow> (x + (y + z) = y + u) = (x + z = u)"
+proof
+ assume vs: "x \<in> V" "y \<in> V" "z \<in> V" "u \<in> V"
+ {
+ from vs have "x + z = - y + y + (x + z)" by simp
also have "... = - y + (y + (x + z))"
- by (rule vs_add_assoc) (simp!)+
- also have "y + (x + z) = x + (y + z)" by (simp!)
- also assume ?L
- also have "- y + (y + u) = u" by (simp!)
- finally show ?R .
- qed (simp! only: vs_add_left_commute [of V x])
+ by (rule add_assoc) (simp_all add: vs)
+ also from vs have "y + (x + z) = x + (y + z)"
+ by (simp add: add_ac)
+ also assume "x + (y + z) = y + u"
+ also from vs have "- y + (y + u) = u" by simp
+ finally show "x + z = u" .
+ next
+ assume "x + z = u"
+ with vs show "x + (y + z) = y + u"
+ by (simp only: add_left_commute [of x])
+ }
qed
-lemma vs_add_cancel_end:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V
- \<Longrightarrow> (x + (y + z) = y) = (x = - z)"
- (concl is "?L = ?R" )
-proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
- show "?L = ?R"
- proof
- assume l: ?L
- have "x + z = 0"
- proof (rule vs_add_left_cancel [THEN iffD1])
- have "y + (x + z) = x + (y + z)" by (simp!)
- also note l
- also have "y = y + 0" by (simp!)
- finally show "y + (x + z) = y + 0" .
- qed (simp!)+
- thus "x = - z" by (simp! add: vs_add_minus_eq_minus)
+lemma (in vectorspace) add_cancel_end:
+ "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + (y + z) = y) = (x = - z)"
+proof
+ assume vs: "x \<in> V" "y \<in> V" "z \<in> V"
+ {
+ assume "x + (y + z) = y"
+ with vs have "(x + z) + y = 0 + y"
+ by (simp add: add_ac)
+ with vs have "x + z = 0"
+ by (simp only: add_right_cancel add_closed zero)
+ with vs show "x = - z" by (simp add: add_minus_eq_minus)
next
- assume r: ?R
+ assume eq: "x = - z"
hence "x + (y + z) = - z + (y + z)" by simp
also have "... = y + (- z + z)"
- by (simp! only: vs_add_left_commute)
- also have "... = y" by (simp!)
- finally show ?L .
- qed
+ by (rule add_left_commute) (simp_all add: vs)
+ also from vs have "... = y" by simp
+ finally show "x + (y + z) = y" .
+ }
qed
end
--- a/src/HOL/Real/HahnBanach/ZornLemma.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/ZornLemma.thy Thu Aug 22 20:49:43 2002 +0200
@@ -19,23 +19,22 @@
*}
theorem Zorn's_Lemma:
- "(\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S) \<Longrightarrow> a \<in> S
- \<Longrightarrow> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z"
+ assumes r: "\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S"
+ and aS: "a \<in> S"
+ shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z"
proof (rule Zorn_Lemma2)
txt_raw {* \footnote{See
\url{http://isabelle.in.tum.de/library/HOL/HOL-Real/Zorn.html}} \isanewline *}
- assume r: "\<And>c. c \<in> chain S \<Longrightarrow> \<exists>x. x \<in> c \<Longrightarrow> \<Union>c \<in> S"
- assume aS: "a \<in> S"
show "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
proof
fix c assume "c \<in> chain S"
show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
proof cases
-
+
txt {* If @{text c} is an empty chain, then every element in
@{text S} is an upper bound of @{text c}. *}
- assume "c = {}"
+ assume "c = {}"
with aS show ?thesis by fast
txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper
@@ -43,12 +42,12 @@
next
assume c: "c \<noteq> {}"
- show ?thesis
- proof
+ show ?thesis
+ proof
show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast
- show "\<Union>c \<in> S"
+ show "\<Union>c \<in> S"
proof (rule r)
- from c show "\<exists>x. x \<in> c" by fast
+ from c show "\<exists>x. x \<in> c" by fast
qed
qed
qed