--- a/src/HOL/IsaMakefile Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/IsaMakefile Sat Dec 16 21:41:51 2000 +0100
@@ -152,10 +152,8 @@
Real/HahnBanach/Linearform.thy Real/HahnBanach/NormedSpace.thy \
Real/HahnBanach/README.html Real/HahnBanach/ROOT.ML \
Real/HahnBanach/Subspace.thy Real/HahnBanach/VectorSpace.thy \
- Real/HahnBanach/ZornLemma.thy Real/HahnBanach/document/notation.tex \
- Real/HahnBanach/document/bbb.sty Real/HahnBanach/document/root.bib \
- Real/HahnBanach/document/root.tex \
- Real/HahnBanach/document/notation.tex
+ Real/HahnBanach/ZornLemma.thy Real/HahnBanach/document/root.bib \
+ Real/HahnBanach/document/root.tex
@cd Real; $(ISATOOL) usedir $(OUT)/HOL-Real HahnBanach
--- a/src/HOL/Real/HahnBanach/Aux.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/Aux.thy Sat Dec 16 21:41:51 2000 +0100
@@ -11,139 +11,141 @@
or elimination rules, respectively. *}
lemmas [intro?] = isLub_isUb
-lemmas [intro?] = chainD
+lemmas [intro?] = chainD
lemmas chainE2 = chainD2 [elim_format, standard]
-text_raw {* \medskip *}
-text{* Lemmas about sets. *}
-lemma Int_singletonD: "[| A \<inter> B = {v}; x \<in> A; x \<in> B |] ==> x = v"
+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 ==> \<exists>x0 \<in> E. x0 \<notin> H"
- by (force simp add: psubset_eq)
+lemma set_less_imp_diff_not_empty: "H < E \<Longrightarrow> \<exists>x0 \<in> E. x0 \<notin> H"
+ by (auto simp add: psubset_eq)
+
-text_raw {* \medskip *}
-text{* Some lemmas about orders. *}
+text{* \medskip Some lemmas about orders. *}
-lemma lt_imp_not_eq: "x < (y::'a::order) ==> x \<noteq> y"
+lemma lt_imp_not_eq: "x < (y::'a::order) \<Longrightarrow> x \<noteq> y"
by (simp add: order_less_le)
-lemma le_noteq_imp_less:
- "[| x <= (r::'a::order); x \<noteq> r |] ==> x < r"
+lemma le_noteq_imp_less:
+ "x \<le> (r::'a::order) \<Longrightarrow> x \<noteq> r \<Longrightarrow> x < r"
proof -
- assume "x <= r" and ne:"x \<noteq> r"
- hence "x < r | x = r" by (simp add: order_le_less)
+ assume "x \<le> r" and ne:"x \<noteq> r"
+ hence "x < r \<or> x = r" by (simp add: order_le_less)
with ne show ?thesis by simp
qed
-text_raw {* \medskip *}
-text{* Some lemmas for the reals. *}
-lemma real_add_minus_eq: "x - y = (#0::real) ==> x = y"
+text {* \medskip Some lemmas for the reals. *}
+
+lemma real_add_minus_eq: "x - y = (#0::real) \<Longrightarrow> x = y"
by simp
-lemma abs_minus_one: "abs (- (#1::real)) = #1"
+lemma abs_minus_one: "abs (- (#1::real)) = #1"
by simp
-lemma real_mult_le_le_mono1a:
- "[| (#0::real) <= z; x <= y |] ==> z * x <= z * y"
+lemma real_mult_le_le_mono1a:
+ "(#0::real) \<le> z \<Longrightarrow> x \<le> y \<Longrightarrow> z * x \<le> z * y"
proof -
- assume z: "(#0::real) <= z" and "x <= y"
- hence "x < y | x = y" by (force simp add: order_le_less)
+ assume z: "(#0::real) \<le> z" and "x \<le> y"
+ hence "x < y \<or> x = y" by (auto simp add: order_le_less)
thus ?thesis
- proof (elim disjE)
+ proof
assume "x < y" show ?thesis by (rule real_mult_le_less_mono2) simp
- next
+ next
assume "x = y" thus ?thesis by simp
qed
qed
-lemma real_mult_le_le_mono2:
- "[| (#0::real) <= z; x <= y |] ==> x * z <= y * z"
+lemma real_mult_le_le_mono2:
+ "(#0::real) \<le> z \<Longrightarrow> x \<le> y \<Longrightarrow> x * z \<le> y * z"
proof -
- assume "(#0::real) <= z" "x <= y"
- hence "x < y | x = y" by (force simp add: order_le_less)
+ assume "(#0::real) \<le> z" "x \<le> y"
+ hence "x < y \<or> x = y" by (auto simp add: order_le_less)
thus ?thesis
- proof (elim disjE)
- assume "x < y" show ?thesis by (rule real_mult_le_less_mono1) simp
- next
- assume "x = y" thus ?thesis by simp
+ proof
+ assume "x < y"
+ show ?thesis by (rule real_mult_le_less_mono1) (simp!)
+ next
+ assume "x = y"
+ thus ?thesis by simp
qed
qed
-lemma real_mult_less_le_anti:
- "[| z < (#0::real); x <= y |] ==> z * y <= z * x"
+lemma real_mult_less_le_anti:
+ "z < (#0::real) \<Longrightarrow> x \<le> y \<Longrightarrow> z * y \<le> z * x"
proof -
- assume "z < #0" "x <= y"
+ assume "z < #0" "x \<le> y"
hence "#0 < - z" by simp
- hence "#0 <= - z" by (rule real_less_imp_le)
- hence "x * (- z) <= y * (- z)"
+ hence "#0 \<le> - z" by (rule real_less_imp_le)
+ hence "x * (- z) \<le> y * (- z)"
by (rule real_mult_le_le_mono2)
- hence "- (x * z) <= - (y * z)"
+ hence "- (x * z) \<le> - (y * z)"
by (simp only: real_minus_mult_eq2)
thus ?thesis by (simp only: real_mult_commute)
qed
-lemma real_mult_less_le_mono:
- "[| (#0::real) < z; x <= y |] ==> z * x <= z * y"
-proof -
- assume "#0 < z" "x <= y"
- have "#0 <= z" by (rule real_less_imp_le)
- hence "x * z <= y * z"
+lemma real_mult_less_le_mono:
+ "(#0::real) < z \<Longrightarrow> x \<le> y \<Longrightarrow> z * x \<le> z * y"
+proof -
+ assume "#0 < z" "x \<le> y"
+ have "#0 \<le> z" by (rule real_less_imp_le)
+ hence "x * z \<le> y * z"
by (rule real_mult_le_le_mono2)
thus ?thesis by (simp only: real_mult_commute)
qed
-lemma real_inverse_gt_zero1: "#0 < (x::real) ==> #0 < inverse x"
-proof -
+lemma real_inverse_gt_zero1: "#0 < (x::real) \<Longrightarrow> #0 < inverse x"
+proof -
assume "#0 < x"
have "0 < x" by simp
hence "0 < inverse x" by (rule real_inverse_gt_zero)
thus ?thesis by simp
qed
-lemma real_mult_inv_right1: "(x::real) \<noteq> #0 ==> x * inverse x = #1"
+lemma real_mult_inv_right1: "(x::real) \<noteq> #0 \<Longrightarrow> x * inverse x = #1"
by simp
-lemma real_mult_inv_left1: "(x::real) \<noteq> #0 ==> inverse x * x = #1"
+lemma real_mult_inv_left1: "(x::real) \<noteq> #0 \<Longrightarrow> inverse x * x = #1"
by simp
-lemma real_le_mult_order1a:
- "[| (#0::real) <= x; #0 <= y |] ==> #0 <= x * y"
+lemma real_le_mult_order1a:
+ "(#0::real) \<le> x \<Longrightarrow> #0 \<le> y \<Longrightarrow> #0 \<le> x * y"
proof -
- assume "#0 <= x" "#0 <= y"
- have "[|0 <= x; 0 <= y|] ==> 0 <= x * y"
+ assume "#0 \<le> x" "#0 \<le> y"
+ have "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x * y"
by (rule real_le_mult_order)
thus ?thesis by (simp!)
qed
-lemma real_mult_diff_distrib:
+lemma real_mult_diff_distrib:
"a * (- x - (y::real)) = - a * x - a * y"
proof -
have "- x - y = - x + - y" by simp
- also have "a * ... = a * - x + a * - y"
+ also have "a * ... = a * - x + a * - y"
by (simp only: real_add_mult_distrib2)
- also have "... = - a * x - a * y"
+ also have "... = - a * x - a * y"
by simp
finally show ?thesis .
qed
lemma real_mult_diff_distrib2: "a * (x - (y::real)) = a * x - a * y"
-proof -
+proof -
have "x - y = x + - y" by simp
- also have "a * ... = a * x + a * - y"
+ also have "a * ... = a * x + a * - y"
by (simp only: real_add_mult_distrib2)
- also have "... = a * x - a * y"
+ also have "... = a * x - a * y"
by simp
finally show ?thesis .
qed
-lemma real_minus_le: "- (x::real) <= y ==> - y <= x"
+lemma real_minus_le: "- (x::real) \<le> y \<Longrightarrow> - y \<le> x"
by simp
-lemma real_diff_ineq_swap:
- "(d::real) - b <= c + a ==> - a - b <= c - d"
+lemma real_diff_ineq_swap:
+ "(d::real) - b \<le> c + a \<Longrightarrow> - a - b \<le> c - d"
by simp
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/Bounds.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/Bounds.thy Sat Dec 16 21:41:51 2000 +0100
@@ -6,128 +6,48 @@
header {* Bounds *}
theory Bounds = Main + Real:
-(*<*)
-subsection {* The sets of lower and upper bounds *}
-constdefs
- is_LowerBound :: "('a::order) set => 'a set => 'a => bool"
- "is_LowerBound A B == \<lambda>x. x \<in> A \<and> (\<forall>y \<in> B. x <= y)"
+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}.
+*}
- LowerBounds :: "('a::order) set => 'a set => 'a set"
- "LowerBounds A B == Collect (is_LowerBound A B)"
-
- is_UpperBound :: "('a::order) set => 'a set => 'a => bool"
- "is_UpperBound A B == \<lambda>x. x \<in> A \<and> (\<forall>y \<in> B. y <= x)"
-
- UpperBounds :: "('a::order) set => 'a set => 'a set"
- "UpperBounds A B == Collect (is_UpperBound A B)"
-
-syntax
- "_UPPERS" :: "[pttrn, 'a set, 'a => bool] => 'a set"
- ("(3UPPER'_BOUNDS _:_./ _)" 10)
- "_UPPERS_U" :: "[pttrn, 'a => bool] => 'a set"
- ("(3UPPER'_BOUNDS _./ _)" 10)
- "_LOWERS" :: "[pttrn, 'a set, 'a => bool] => 'a set"
- ("(3LOWER'_BOUNDS _:_./ _)" 10)
- "_LOWERS_U" :: "[pttrn, 'a => bool] => 'a set"
- ("(3LOWER'_BOUNDS _./ _)" 10)
-
-translations
- "UPPER_BOUNDS x:A. P" == "UpperBounds A (Collect (\<lambda>x. P))"
- "UPPER_BOUNDS x. P" == "UPPER_BOUNDS x:UNIV. P"
- "LOWER_BOUNDS x:A. P" == "LowerBounds A (Collect (\<lambda>x. P))"
- "LOWER_BOUNDS x. P" == "LOWER_BOUNDS x:UNIV. P"
-
-
-subsection {* Least and greatest elements *}
+text {*
+ If a supremum exists, then @{text "Sup A B"} is equal to the
+ supremum. *}
constdefs
- is_Least :: "('a::order) set => 'a => bool"
- "is_Least B == is_LowerBound B B"
-
- Least :: "('a::order) set => 'a"
- "Least B == Eps (is_Least B)"
-
- is_Greatest :: "('a::order) set => 'a => bool"
- "is_Greatest B == is_UpperBound B B"
-
- Greatest :: "('a::order) set => 'a"
- "Greatest B == Eps (is_Greatest B)"
+ is_Sup :: "('a::order) set \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool"
+ "is_Sup A B x \<equiv> isLub A B x"
-syntax
- "_LEAST" :: "[pttrn, 'a => bool] => 'a"
- ("(3LLEAST _./ _)" 10)
- "_GREATEST" :: "[pttrn, 'a => bool] => 'a"
- ("(3GREATEST _./ _)" 10)
-
-translations
- "LLEAST x. P" == "Least {x. P}"
- "GREATEST x. P" == "Greatest {x. P}"
-
-
-subsection {* Infimum and Supremum *}
-(*>*)
-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 $B$ w.~r.~t. $A$ is defined as a least upper bound of
- $B$, which lies in $A$.
-*}
-
-text{* If a supremum exists, then $\idt{Sup}\ap A\ap B$
-is equal to the supremum. *}
+ Sup :: "('a::order) set \<Rightarrow> 'a set \<Rightarrow> 'a"
+ "Sup A B \<equiv> Eps (is_Sup A B)"
-constdefs
- is_Sup :: "('a::order) set => 'a set => 'a => bool"
- "is_Sup A B x == isLub A B x"
-
- Sup :: "('a::order) set => 'a set => 'a"
- "Sup A B == Eps (is_Sup A B)"
-(*<*)
-constdefs
- is_Inf :: "('a::order) set => 'a set => 'a => bool"
- "is_Inf A B x == x \<in> A \<and> is_Greatest (LowerBounds A B) x"
-
- Inf :: "('a::order) set => 'a set => 'a"
- "Inf A B == Eps (is_Inf A B)"
+text {*
+ The supremum of @{text B} is less than any upper bound of
+ @{text B}. *}
-syntax
- "_SUP" :: "[pttrn, 'a set, 'a => bool] => 'a set"
- ("(3SUP _:_./ _)" 10)
- "_SUP_U" :: "[pttrn, 'a => bool] => 'a set"
- ("(3SUP _./ _)" 10)
- "_INF" :: "[pttrn, 'a set, 'a => bool] => 'a set"
- ("(3INF _:_./ _)" 10)
- "_INF_U" :: "[pttrn, 'a => bool] => 'a set"
- ("(3INF _./ _)" 10)
-
-translations
- "SUP x:A. P" == "Sup A (Collect (\<lambda>x. P))"
- "SUP x. P" == "SUP x:UNIV. P"
- "INF x:A. P" == "Inf A (Collect (\<lambda>x. P))"
- "INF x. P" == "INF x:UNIV. P"
-(*>*)
-text{* The supremum of $B$ is less than any upper bound
-of $B$.*}
-
-lemma sup_le_ub: "isUb A B y ==> is_Sup A B s ==> s <= y"
+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 $B$ is an upper bound for $B$. *}
+text {* The supremum @{text B} is an upper bound for @{text B}. *}
-lemma sup_ub: "y \<in> B ==> is_Sup A B s ==> y <= s"
+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 $B$ is greater
-than a lower bound of $B$. *}
+text {*
+ The supremum of a non-empty set @{text B} is greater than a lower
+ bound of @{text B}. *}
lemma sup_ub1:
- "[| \<forall>y \<in> B. a <= y; is_Sup A B s; x \<in> B |] ==> a <= s"
+ "\<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 <= y" "is_Sup A B s" "x \<in> B"
- have "a <= x" by (rule bspec)
- also have "x <= s" by (rule sup_ub)
- finally show "a <= s" .
+ 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" .
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/FunctionNorm.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/FunctionNorm.thy Sat Dec 16 21:41:51 2000 +0100
@@ -9,377 +9,374 @@
subsection {* Continuous linear forms*}
-text{* A linear form $f$ on a normed vector space $(V, \norm{\cdot})$
-is \emph{continuous}, iff it is bounded, i.~e.
-\[\Ex {c\in R}{\All {x\in V} {|f\ap x| \leq c \cdot \norm x}}\]
-In our application no other functions than linear forms are considered,
-so we can define continuous linear forms as bounded linear forms:
+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.
+ \begin{center}
+ @{text "\<exists>c \<in> R. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
+ \end{center}
+ In our application no other functions than linear forms are
+ considered, so we can define continuous linear forms as bounded
+ linear forms:
*}
constdefs
is_continuous ::
- "['a::{plus, minus, zero} set, 'a => real, 'a => real] => bool"
- "is_continuous V norm f ==
- is_linearform V f \<and> (\<exists>c. \<forall>x \<in> V. |f x| <= c * norm x)"
+ "'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)"
-lemma continuousI [intro]:
- "[| is_linearform V f; !! x. x \<in> V ==> |f x| <= c * norm x |]
- ==> is_continuous V norm f"
+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"
proof (unfold is_continuous_def, intro exI conjI ballI)
- assume r: "!! x. x \<in> V ==> |f x| <= c * norm x"
- fix x assume "x \<in> V" show "|f x| <= c * norm x" by (rule r)
+ assume r: "\<And>x. x \<in> V \<Longrightarrow> \<bar>f x\<bar> \<le> c * norm x"
+ fix x assume "x \<in> V" show "\<bar>f x\<bar> \<le> c * norm x" by (rule r)
qed
-
-lemma continuous_linearform [intro?]:
- "is_continuous V norm f ==> is_linearform V f"
- by (unfold is_continuous_def) force
+
+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
- ==> \<exists>c. \<forall>x \<in> V. |f x| <= c * norm x"
- by (unfold is_continuous_def) force
+ "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
subsection{* The norm of a linear form *}
-text{* The least real number $c$ for which holds
-\[\All {x\in V}{|f\ap x| \leq c \cdot \norm x}\]
-is called the \emph{norm} of $f$.
+text {*
+ The least real number @{text c} for which holds
+ \begin{center}
+ @{text "\<forall>x \<in> V. \<bar>f x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
+ \end{center}
+ is called the \emph{norm} of @{text f}.
-For non-trivial vector spaces $V \neq \{\zero\}$ the norm can be defined as
-\[\fnorm {f} =\sup_{x\neq\zero}\frac{|f\ap x|}{\norm x} \]
+ For non-trivial vector spaces @{text "V \<noteq> {0}"} the norm can be
+ defined as
+ \begin{center}
+ @{text "\<parallel>f\<parallel> = \<sup>x \<noteq> 0. \<bar>f x\<bar> / \<parallel>x\<parallel>"}
+ \end{center}
-For the case $V = \{\zero\}$ the supremum would be taken from an
-empty set. Since $\bbbR$ is unbounded, there would be no supremum. To
-avoid this situation it must be guaranteed that there is an element in
-this set. This element must be ${} \ge 0$ so that
-$\idt{function{\dsh}norm}$ has the norm properties. Furthermore it
-does not have to change the norm in all other cases, so it must be
-$0$, as all other elements of are ${} \ge 0$.
+ For the case @{text "V = {0}"} the supremum would be taken from an
+ empty set. Since @{text \<real>} is unbounded, there would be no supremum.
+ To avoid this situation it must be guaranteed that there is an
+ 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"}.
-Thus we define the set $B$ the supremum is taken from as
-\[
-\{ 0 \} \Union \left\{ \frac{|f\ap x|}{\norm x} \dt x\neq \zero \And x\in F\right\}
- \]
+ Thus we define the set @{text B} the supremum is taken from as
+ \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}.
*}
constdefs
- B :: "[ 'a set, 'a => real, 'a::{plus, minus, zero} => real ] => real set"
- "B V norm f ==
- {#0} \<union> {|f x| * inverse (norm x) | x. x \<noteq> 0 \<and> x \<in> V}"
+ 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{* $n$ is the function norm of $f$, iff
-$n$ is the supremum of $B$.
+text {*
+ @{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} => real, 'a set, 'a => real] => real => bool"
- "is_function_norm f V norm fn == is_Sup UNIV (B V norm f) fn"
+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)"
-text{* $\idt{function{\dsh}norm}$ is equal to the supremum of $B$,
-if the supremum exists. Otherwise it is undefined. *}
-
-constdefs
- function_norm :: " ['a::{minus,plus,zero} => real, 'a set, 'a => real] => real"
- "function_norm f V norm == Sup UNIV (B V norm f)"
-
-syntax
- function_norm :: " ['a => real, 'a set, 'a => real] => real" ("\<parallel>_\<parallel>_,_")
+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, force)
-
-text {* The following lemma states that every continuous linear form
-on a normed space $(V, \norm{\cdot})$ has a function norm. *}
+ by (unfold B_def) blast
-lemma ex_fnorm [intro?]:
- "[| is_normed_vectorspace V norm; is_continuous V norm f|]
- ==> is_function_norm f V norm \<parallel>f\<parallel>V,norm"
-proof (unfold function_norm_def is_function_norm_def
+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. |f x| <= c * norm x"
+ assume "is_linearform V f"
+ and e: "\<exists>c. \<forall>x \<in> V. \<bar>f x\<bar> \<le> c * norm x"
- txt {* The existence of the supremum is shown using the
+ 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
+ every non-empty bounded set of reals has a
supremum. *}
- show "\<exists>a. is_Sup UNIV (B V norm f) a"
+ show "\<exists>a. is_Sup UNIV (B V norm f) a"
proof (unfold is_Sup_def, rule reals_complete)
- txt {* First we have to show that $B$ is non-empty: *}
+ txt {* First we have to show that @{text B} is non-empty: *}
- show "\<exists>X. X \<in> B V norm f"
- proof (intro exI)
- show "#0 \<in> (B V norm f)" by (unfold B_def, force)
+ show "\<exists>X. X \<in> B V norm f"
+ proof
+ show "#0 \<in> (B V norm f)" by (unfold B_def) blast
qed
- txt {* Then we have to show that $B$ is bounded: *}
+ txt {* Then we have to show that @{text B} is bounded: *}
from e show "\<exists>Y. isUb UNIV (B V norm f) Y"
proof
- txt {* We know that $f$ is bounded by some value $c$. *}
-
- fix c assume a: "\<forall>x \<in> V. |f x| <= c * norm x"
- def b == "max c #0"
+ 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"
show "?thesis"
- proof (intro exI isUbI setleI ballI, unfold B_def,
- elim UnE CollectE exE conjE singletonE)
+ 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 $b$, such that $y \leq b$ for all $y\in B$.
- Due to the definition of $B$ there are two cases for
- $y\in B$. If $y = 0$ then $y \leq \idt{max}\ap c\ap 0$: *}
+ 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 <= b" by (simp! add: le_maxI2)
+ fix y assume "y = (#0::real)"
+ show "y \<le> b" by (simp! add: le_maxI2)
- txt{* The second case is
- $y = {|f\ap x|}/{\norm x}$ for some
- $x\in V$ with $x \neq \zero$. *}
+ 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 y
- assume "x \<in> V" "x \<noteq> 0" (***
+ 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. *}
- have ge: "#0 <= inverse (norm x)";
- by (rule real_less_imp_le, rule real_inverse_gt_zero,
- rule normed_vs_norm_gt_zero); ( ***
- proof (rule real_less_imp_le);
- show "#0 < inverse (norm x)";
- proof (rule real_inverse_gt_zero);
- show "#0 < norm x"; ..;
- qed;
- qed; *** )
- have nz: "norm x \<noteq> #0"
- by (rule not_sym, rule lt_imp_not_eq,
- rule normed_vs_norm_gt_zero) (***
- proof (rule not_sym);
- show "#0 \<noteq> norm x";
- proof (rule lt_imp_not_eq);
- show "#0 < norm x"; ..;
- qed;
- qed; ***)***)
-
- txt {* The thesis follows by a short calculation using the
- fact that $f$ is bounded. *}
-
- assume "y = |f x| * inverse (norm x)"
- also have "... <= c * norm x * inverse (norm x)"
+ 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 <= inverse (norm x)"
- by (rule real_less_imp_le, rule real_inverse_gt_zero1,
+ show "#0 \<le> inverse (norm x)"
+ by (rule real_less_imp_le, rule real_inverse_gt_zero1,
rule normed_vs_norm_gt_zero)
- from a show "|f x| <= c * norm x" ..
+ from a show "\<bar>f x\<bar> \<le> c * norm x" ..
qed
- also have "... = c * (norm x * inverse (norm x))"
+ also have "... = c * (norm x * inverse (norm x))"
by (rule real_mult_assoc)
- also have "(norm x * inverse (norm x)) = (#1::real)"
+ 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,
+ 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 * ... <= b " by (simp! add: le_maxI1)
- finally show "y <= b" .
+ also have "c * ... \<le> b " by (simp! add: le_maxI1)
+ finally show "y \<le> b" .
qed simp
qed
qed
qed
-text {* The norm of a continuous function is always $\geq 0$. *}
+text {* The norm of a continuous function is always @{text "\<ge> 0"}. *}
-lemma fnorm_ge_zero [intro?]:
- "[| is_continuous V norm f; is_normed_vectorspace V norm |]
- ==> #0 <= \<parallel>f\<parallel>V,norm"
+lemma fnorm_ge_zero [intro?]:
+ "is_continuous V norm f \<Longrightarrow> is_normed_vectorspace V norm
+ \<Longrightarrow> #0 \<le> \<parallel>f\<parallel>V,norm"
proof -
- assume c: "is_continuous V norm f"
+ assume c: "is_continuous V norm f"
and n: "is_normed_vectorspace V norm"
- txt {* The function norm is defined as the supremum of $B$.
- So it is $\geq 0$ if all elements in $B$ are $\geq 0$, provided
- the supremum exists and $B$ is not empty. *}
+ 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
+ show ?thesis
proof (unfold function_norm_def, rule sup_ub1)
- show "\<forall>x \<in> (B V norm f). #0 <= x"
+ show "\<forall>x \<in> (B V norm f). #0 \<le> x"
proof (intro ballI, unfold B_def,
- elim UnE singletonE CollectE exE conjE)
+ elim UnE singletonE CollectE exE conjE)
fix x r
- assume "x \<in> V" "x \<noteq> 0"
- and r: "r = |f x| * inverse (norm x)"
+ assume "x \<in> V" "x \<noteq> 0"
+ and r: "r = \<bar>f x\<bar> * inverse (norm x)"
- have ge: "#0 <= |f x|" by (simp! only: abs_ge_zero)
- have "#0 <= 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 real_less_imp_le, rule real_inverse_gt_zero1, rule)(***
proof (rule real_less_imp_le);
- show "#0 < inverse (norm x)";
+ show "#0 < inverse (norm x)";
proof (rule real_inverse_gt_zero);
show "#0 < norm x"; ..;
qed;
qed; ***)
- with ge show "#0 <= r"
+ with ge show "#0 \<le> r"
by (simp only: r, rule real_le_mult_order1a)
qed (simp!)
- txt {* Since $f$ is continuous the function norm exists: *}
+ 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))"
+ thus "is_Sup UNIV (B V norm f) (Sup UNIV (B V norm f))"
by (unfold is_function_norm_def function_norm_def)
- txt {* $B$ is non-empty by construction: *}
+ txt {* @{text B} is non-empty by construction: *}
show "#0 \<in> B V norm f" by (rule B_not_empty)
qed
qed
-
-text{* \medskip The fundamental property of function norms is:
-\begin{matharray}{l}
-| f\ap x | \leq {\fnorm {f}} \cdot {\norm x}
-\end{matharray}
+
+text {*
+ \medskip The fundamental property of function norms is:
+ \begin{center}
+ @{text "\<bar>f x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
+ \end{center}
*}
-lemma norm_fx_le_norm_f_norm_x:
- "[| is_continuous V norm f; is_normed_vectorspace V norm; x \<in> V |]
- ==> |f x| <= \<parallel>f\<parallel>V,norm * norm x"
-proof -
- assume "is_normed_vectorspace V norm" "x \<in> V"
+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 $x$. *}
-
+ txt{* The proof is by case analysis on @{text x}. *}
+
show ?thesis
proof cases
- txt {* For the case $x = \zero$ the thesis follows
- from the linearity of $f$: for every linear function
- holds $f\ap \zero = 0$. *}
+ 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 "|f x| = |f 0|" by (simp!)
+ 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 <= \<parallel>f\<parallel>V,norm * norm x"
+ also have "#0 \<le> \<parallel>f\<parallel>V,norm * norm x"
proof (rule real_le_mult_order1a)
- show "#0 <= \<parallel>f\<parallel>V,norm" ..
- show "#0 <= norm x" ..
+ show "#0 \<le> \<parallel>f\<parallel>V,norm" ..
+ show "#0 \<le> norm x" ..
qed
- finally
- show "|f x| <= \<parallel>f\<parallel>V,norm * norm x" .
+ 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"
+ hence nz: "norm x \<noteq> #0"
by (simp only: lt_imp_not_eq)
- txt {* For the case $x\neq \zero$ we derive the following
- fact from the definition of the function norm:*}
+ txt {* For the case @{text "x \<noteq> 0"} we derive the following fact
+ from the definition of the function norm:*}
- have l: "|f x| * inverse (norm x) <= \<parallel>f\<parallel>V,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 "|f x| * inverse (norm x) \<in> B V norm f"
+ 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 "|f x| = |f x| * #1" by (simp!)
- also from nz have "#1 = inverse (norm x) * norm x"
+ 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 "|f x| * ... = |f x| * inverse (norm x) * norm x"
+ 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 "... <= \<parallel>f\<parallel>V,norm * norm x"
+ also from n l have "... \<le> \<parallel>f\<parallel>V,norm * norm x"
by (simp add: real_mult_le_le_mono2)
- finally show "|f x| <= \<parallel>f\<parallel>V,norm * norm x" .
+ finally show "\<bar>f x\<bar> \<le> \<parallel>f\<parallel>V,norm * norm x" .
qed
qed
-text{* \medskip The function norm is the least positive real number for
-which the following inequation holds:
-\begin{matharray}{l}
-| f\ap x | \leq c \cdot {\norm x}
-\end{matharray}
+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>"}
+ \end{center}
*}
-lemma fnorm_le_ub:
- "[| is_continuous V norm f; is_normed_vectorspace V norm;
- \<forall>x \<in> V. |f x| <= c * norm x; #0 <= c |]
- ==> \<parallel>f\<parallel>V,norm <= c"
+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 "is_normed_vectorspace V norm"
assume c: "is_continuous V norm f"
- assume fb: "\<forall>x \<in> V. |f x| <= c * norm x"
- and "#0 <= c"
+ 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 $c\geq 0$.
- If $c$ is an upper bound of $B$, then $c$ is greater
- than the function norm since the function norm is the
- least upper bound.
- *}
+ 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) <= c"
+ 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 {* $c$ is an upper bound of $B$, i.e. every
- $y\in B$ is less than $c$. *}
+ 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 "isUb UNIV (B V norm f) c"
+ 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 <= c"
+ thus le: "y \<le> c"
proof (unfold B_def, elim UnE CollectE exE conjE singletonE)
- txt {* The first case for $y\in B$ is $y=0$. *}
+ txt {* The first case for @{text "y \<in> B"} is @{text "y = 0"}. *}
assume "y = #0"
- show "y <= c" by (force!)
+ show "y \<le> c" by (blast!)
- txt{* The second case is
- $y = {|f\ap x|}/{\norm x}$ for some
- $x\in V$ with $x \neq \zero$. *}
+ 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"
+ fix x
+ assume "x \<in> V" "x \<noteq> 0"
- have lz: "#0 < norm x"
+ have lz: "#0 < norm x"
by (simp! add: normed_vs_norm_gt_zero)
-
- have nz: "norm x \<noteq> #0"
+
+ 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_zero1)
- hence inverse_gez: "#0 <= inverse (norm x)"
+
+ from lz have "#0 < inverse (norm x)"
+ by (simp! add: real_inverse_gt_zero1)
+ hence inverse_gez: "#0 \<le> inverse (norm x)"
by (rule real_less_imp_le)
- assume "y = |f x| * inverse (norm x)"
- also from inverse_gez have "... <= c * norm x * inverse (norm x)"
- proof (rule real_mult_le_le_mono2)
- show "|f x| <= c * norm x" by (rule bspec)
- qed
- also have "... <= c" by (simp add: nz real_mult_assoc)
- finally show ?thesis .
+ 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 force
+ qed blast
qed
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/FunctionOrder.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/FunctionOrder.thy Sat Dec 16 21:41:51 2000 +0100
@@ -9,75 +9,82 @@
subsection {* The graph of a function *}
-text{* We define the \emph{graph} of a (real) function $f$ with
-domain $F$ as the set
-\[
-\{(x, f\ap x). \ap x \in F\}
-\]
-So we are modeling partial functions by specifying the domain and
-the mapping function. We use the term ``function'' also for its graph.
+text {*
+ We define the \emph{graph} of a (real) function @{text f} with
+ domain @{text F} as the set
+ \begin{center}
+ @{text "{(x, f x). x \<in> F}"}
+ \end{center}
+ So we are modeling partial functions by specifying the domain and
+ the mapping function. We use the term ``function'' also for its
+ graph.
*}
types 'a graph = "('a * real) set"
constdefs
- graph :: "['a set, 'a => real] => 'a graph "
- "graph F f == {(x, f x) | x. x \<in> F}"
+ 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 ==> (x, f x) \<in> graph F f"
- by (unfold graph_def, intro CollectI exI) force
+lemma graphI [intro?]: "x \<in> F \<Longrightarrow> (x, f x) \<in> graph F f"
+ by (unfold graph_def, intro CollectI exI) blast
-lemma graphI2 [intro?]: "x \<in> F ==> \<exists>t\<in> (graph F f). t = (x, f x)"
- by (unfold graph_def, force)
+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 ==> x \<in> F"
- by (unfold graph_def, elim CollectE exE) force
+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 ==> y = h x"
- by (unfold graph_def, elim CollectE exE) force
+lemma graphD2 [intro?]: "(x, y) \<in> graph H h \<Longrightarrow> y = h x"
+ by (unfold graph_def) blast
+
subsection {* Functions ordered by domain extension *}
-text{* A function $h'$ is an extension of $h$, iff the graph of
-$h$ is a subset of the graph of $h'$.*}
+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:
- "[| !! x. x \<in> H ==> h x = h' x; H <= H'|]
- ==> graph H h <= graph H' h'"
- by (unfold graph_def, force)
+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'"
+ by (unfold graph_def) blast
-lemma graph_extD1 [intro?]:
- "[| graph H h <= graph H' h'; x \<in> H |] ==> h x = h' x"
- by (unfold graph_def, force)
+lemma graph_extD1 [intro?]:
+ "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?]:
- "[| graph H h <= graph H' h' |] ==> H <= H'"
- by (unfold graph_def, force)
+lemma graph_extD2 [intro?]:
+ "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 $\idt{graph}$ are $\idt{domain}$ and
-$\idt{funct}$.*}
+text {*
+ The inverse functions to @{text graph} are @{text domain} and
+ @{text funct}.
+*}
constdefs
- domain :: "'a graph => 'a set"
- "domain g == {x. \<exists>y. (x, y) \<in> g}"
+ domain :: "'a graph \<Rightarrow> 'a set"
+ "domain g \<equiv> {x. \<exists>y. (x, y) \<in> g}"
- funct :: "'a graph => ('a => real)"
- "funct g == \<lambda>x. (SOME 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 $g$ is the graph of a function
-if the relation induced by $g$ is unique. *}
+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:
- "(!!x y z. (x, y) \<in> g ==> (x, z) \<in> g ==> z = y)
- ==> graph (domain g) (funct g) = g"
+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)
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: "!!x y z. (x, y) \<in> g ==> (x, z) \<in> g ==> z = y"
+ 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)
@@ -88,47 +95,49 @@
subsection {* Norm-preserving extensions of a function *}
-text {* Given a linear form $f$ on the space $F$ and a seminorm $p$ on
-$E$. The set of all linear extensions of $f$, to superspaces $H$ of
-$F$, which are bounded by $p$, is defined as follows. *}
-
+text {*
+ Given a linear form @{text f} on the space @{text F} and a seminorm
+ @{text p} on @{text E}. The set of all linear extensions of @{text
+ f}, to superspaces @{text H} of @{text F}, which are bounded by
+ @{text p}, is defined as follows.
+*}
constdefs
- norm_pres_extensions ::
- "['a::{plus, minus, zero} set, 'a => real, 'a set, 'a => real]
- => 'a graph set"
- "norm_pres_extensions E p F f
- == {g. \<exists>H h. graph H h = g
- \<and> is_linearform H h
+ norm_pres_extensions ::
+ "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a set \<Rightarrow> ('a \<Rightarrow> real)
+ \<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 <= graph H h
- \<and> (\<forall>x \<in> H. h x <= p x)}"
-
-lemma norm_pres_extension_D:
+ \<and> graph F f \<subseteq> graph H h
+ \<and> (\<forall>x \<in> H. h x \<le> p x)}"
+
+lemma norm_pres_extension_D:
"g \<in> norm_pres_extensions E p F f
- ==> \<exists>H h. graph H h = g
- \<and> is_linearform H h
+ \<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 <= graph H h
- \<and> (\<forall>x \<in> H. h x <= p x)"
- by (unfold norm_pres_extensions_def) force
+ \<and> graph F f \<subseteq> graph H h
+ \<and> (\<forall>x \<in> H. h x \<le> p x)"
+ by (unfold norm_pres_extensions_def) blast
-lemma norm_pres_extensionI2 [intro]:
- "[| is_linearform H h; is_subspace H E; is_subspace F H;
- graph F f <= graph H h; \<forall>x \<in> H. h x <= p x |]
- ==> (graph H h \<in> norm_pres_extensions E p F f)"
- by (unfold norm_pres_extensions_def) force
+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
-lemma norm_pres_extensionI [intro]:
- "\<exists>H h. graph H h = g
- \<and> is_linearform H h
+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 <= graph H h
- \<and> (\<forall>x \<in> H. h x <= p x)
- ==> g \<in> norm_pres_extensions E p F f"
- by (unfold norm_pres_extensions_def) force
+ \<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
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/HahnBanach.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy Sat Dec 16 21:41:51 2000 +0100
@@ -5,152 +5,142 @@
header {* The Hahn-Banach Theorem *}
-theory HahnBanach = HahnBanachLemmas:
+theory HahnBanach = HahnBanachLemmas:
text {*
- We present the proof of two different versions of the Hahn-Banach
+ We present the proof of two different versions of the Hahn-Banach
Theorem, closely following \cite[\S36]{Heuser:1986}.
*}
subsection {* The Hahn-Banach Theorem for vector spaces *}
text {*
-{\bf Hahn-Banach Theorem.}\quad
- Let $F$ be a subspace of a real vector space $E$, let $p$ be a semi-norm on
- $E$, and $f$ be a linear form defined on $F$ such that $f$ is bounded by
- $p$, i.e. $\All {x\in F} f\ap x \leq p\ap x$. Then $f$ can be extended to
- a linear form $h$ on $E$ such that $h$ is norm-preserving, i.e. $h$ is also
- bounded by $p$.
+ \textbf{Hahn-Banach Theorem.} Let @{text F} be a subspace of a real
+ vector space @{text E}, let @{text p} be a semi-norm on @{text E},
+ and @{text f} be a linear form defined on @{text F} such that @{text
+ f} is bounded by @{text p}, i.e. @{text "\<forall>x \<in> F. f x \<le> p x"}. Then
+ @{text f} can be extended to a linear form @{text h} on @{text E}
+ such that @{text h} is norm-preserving, i.e. @{text h} is also
+ bounded by @{text p}.
+
+ \bigskip
+ \textbf{Proof Sketch.}
+ \begin{enumerate}
+
+ \item Define @{text M} as the set of norm-preserving extensions of
+ @{text f} to subspaces of @{text E}. The linear forms in @{text M}
+ are ordered by domain extension.
-\bigskip
-{\bf Proof Sketch.}
-\begin{enumerate}
-\item Define $M$ as the set of norm-preserving extensions of $f$ to subspaces
- of $E$. The linear forms in $M$ are ordered by domain extension.
-\item We show that every non-empty chain in $M$ has an upper bound in $M$.
-\item With Zorn's Lemma we conclude that there is a maximal function $g$ in
- $M$.
-\item The domain $H$ of $g$ is the whole space $E$, as shown by classical
- contradiction:
-\begin{itemize}
-\item Assuming $g$ is not defined on whole $E$, it can still be extended in a
- norm-preserving way to a super-space $H'$ of $H$.
-\item Thus $g$ can not be maximal. Contradiction!
-\end{itemize}
-\end{enumerate}
-\bigskip
+ \item We show that every non-empty chain in @{text M} has an upper
+ bound in @{text M}.
+
+ \item With Zorn's Lemma we conclude that there is a maximal function
+ @{text g} in @{text M}.
+
+ \item The domain @{text H} of @{text g} is the whole space @{text
+ E}, as shown by classical contradiction:
+
+ \begin{itemize}
+
+ \item Assuming @{text g} is not defined on whole @{text E}, it can
+ still be extended in a norm-preserving way to a super-space @{text
+ H'} of @{text H}.
+
+ \item Thus @{text g} can not be maximal. Contradiction!
+
+ \end{itemize}
+
+ \end{enumerate}
*}
-(*
-text {* {\bf Theorem.} Let $f$ be a linear form defined on a subspace
- $F$ of a real vector space $E$, such that $f$ is bounded by a seminorm
- $p$.
-
- Then $f$ can be extended to a linear form $h$ on $E$ that is again
- bounded by $p$.
-
- \bigskip{\bf Proof Outline.}
- First we define the set $M$ of all norm-preserving extensions of $f$.
- We show that every chain in $M$ has an upper bound in $M$.
- With Zorn's lemma we can conclude that $M$ has a maximal element $g$.
- We further show by contradiction that the domain $H$ of $g$ is the whole
- vector space $E$.
- If $H \neq E$, then $g$ can be extended in
- a norm-preserving way to a greater vector space $H_0$.
- So $g$ cannot be maximal in $M$.
- \bigskip
-*}
-*)
-
theorem HahnBanach:
- "[| is_vectorspace E; is_subspace F E; is_seminorm E p;
- is_linearform F f; \<forall>x \<in> F. f x <= p x |]
- ==> \<exists>h. is_linearform E h \<and> (\<forall>x \<in> F. h x = f x)
- \<and> (\<forall>x \<in> E. h x <= p x)"
- -- {* Let $E$ be a vector space, $F$ a subspace of $E$, $p$ a seminorm on $E$, *}
- -- {* and $f$ a linear form on $F$ such that $f$ is bounded by $p$, *}
- -- {* then $f$ can be extended to a linear form $h$ on $E$ in a norm-preserving way. \skp *}
+ "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)"
+ -- {* 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 <= p x"
+ 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 == "norm_pres_extensions E p F f"
- -- {* Define $M$ as the set of all norm-preserving extensions of $F$. \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 *}
{
- fix c assume "c \<in> chain M" "\<exists>x. x \<in> c"
+ fix c assume "c \<in> chain M" "\<exists>x. x \<in> c"
have "\<Union>c \<in> M"
- -- {* Show that every non-empty chain $c$ of $M$ has an upper bound in $M$: *}
- -- {* $\Union c$ is greater than any element of the chain $c$, so it suffices to show $\Union c \in M$. *}
+ -- {* 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> 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 <= p x)"
+ \<and> (\<forall>x \<in> H. h x \<le> p x)"
proof (intro exI conjI)
let ?H = "domain (\<Union>c)"
let ?h = "funct (\<Union>c)"
- show a: "graph ?H ?h = \<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"
+ 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"
+ show "is_linearform ?H ?h"
by (simp! add: sup_lf a)
- show "is_subspace ?H E"
+ show "is_subspace ?H E"
by (rule sup_subE, rule a) (simp!)+
- show "is_subspace F ?H"
+ show "is_subspace F ?H"
by (rule sup_supF, rule a) (simp!)+
- show "graph F f \<subseteq> graph ?H ?h"
+ show "graph F f \<subseteq> graph ?H ?h"
by (rule sup_ext, rule a) (simp!)+
- show "\<forall>x \<in> ?H. ?h x <= p x"
+ show "\<forall>x \<in> ?H. ?h x \<le> p x"
by (rule sup_norm_pres, rule a) (simp!)+
qed
qed
}
- hence "\<exists>g \<in> M. \<forall>x \<in> M. g \<subseteq> x --> g = x"
- -- {* With Zorn's Lemma we can conclude that there is a maximal element in $M$.\skp *}
+ 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 $M$ is non-empty: *}
+ -- {* 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!)+
+ qed (blast!)+
thus "graph F f \<in> M" by (simp!)
qed
thus ?thesis
proof
- fix g assume "g \<in> M" "\<forall>x \<in> M. g \<subseteq> x --> g = x"
- -- {* We consider such a maximal element $g \in M$. \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 <= p x"
- -- {* $g$ is a norm-preserving extension of $f$, in other words: *}
- -- {* $g$ is the graph of some linear form $h$ defined on a subspace $H$ of $E$, *}
- -- {* and $h$ is an extension of $f$ that is again bounded by $p$. \skp *}
+ 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"
+ -- {* @{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
+ 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 <= p x)"
+ \<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"
- -- {* We show that $h$ is defined on whole $E$ by classical contradiction. \skp *}
+ -- {* We show that @{text h} is defined on whole @{text E} by classical contradiction. \skp *}
proof (rule classical)
assume "H \<noteq> E"
- -- {* Assume $h$ is not defined on whole $E$. Then show that $h$ can be extended *}
- -- {* in a norm-preserving way to a function $h'$ with the graph $g'$. \skp *}
+ -- {* 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 $x' \in E \setminus H$. \skp *}
+ 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)
@@ -165,21 +155,21 @@
with h have "x' \<in> H" by simp
thus ?thesis by contradiction
qed blast
- def H' == "H + lin x'"
- -- {* Define $H'$ as the direct sum of $H$ and the linear closure of $x'$. \skp *}
- obtain xi where "\<forall>y \<in> H. - p (y + x') - h y <= xi
- \<and> xi <= p (y + x') - h y"
- -- {* Pick a real number $\xi$ that fulfills certain inequations; this will *}
- -- {* be used to establish that $h'$ is a norm-preserving extension of $h$.
+ 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"
+ -- {* 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 <= xi
- \<and> xi <= p (y + x') - h y"
+ 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"
+ 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 "... <= p (v - u)"
+ also have "... \<le> p (v - u)"
by (simp!)
also have "v - u = x' + - x' + v + - u"
by (simp! add: diff_eq1)
@@ -187,29 +177,29 @@
by (simp!)
also have "... = (v + x') - (u + x')"
by (simp! add: diff_eq1)
- also have "p ... <= p (v + x') + p (u + x')"
+ also have "p ... \<le> p (v + x') + p (u + x')"
by (rule seminorm_diff_subadditive) (simp_all!)
- finally have "h v - h u <= p (v + x') + p (u + x')" .
+ finally have "h v - h u \<le> p (v + x') + p (u + x')" .
- thus "- p (u + x') - h u <= p (v + x') - h v"
+ thus "- p (u + x') - h u \<le> p (v + x') - h v"
by (rule real_diff_ineq_swap)
qed
thus ?thesis ..
qed
- def h' == "\<lambda>x. let (y,a) = SOME (y,a). x = y + a \<cdot> x' \<and> y \<in> 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"
- -- {* Define the extension $h'$ of $h$ to $H'$ using $\xi$. \skp *}
+ -- {* 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 $h'$ is an extension of $h$ \dots \skp *}
+ show "g \<subseteq> graph H' h' \<and> g \<noteq> graph H' h'"
+ -- {* Show that @{text h'} is an extension of @{text h} \dots \skp *}
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"
+ 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')
@@ -223,7 +213,7 @@
show "is_vectorspace (lin x')" ..
qed
qed
- qed
+ qed
thus ?thesis by (simp!)
qed
show "g \<noteq> graph H' h'"
@@ -231,7 +221,7 @@
have "graph H h \<noteq> graph H' h'"
proof
assume e: "graph H h = graph H' h'"
- have "x' \<in> H'"
+ have "x' \<in> H'"
proof (unfold H'_def, rule vs_sumI)
show "x' = 0 + x'" by (simp!)
from h show "0 \<in> H" ..
@@ -245,63 +235,63 @@
thus ?thesis by (simp!)
qed
qed
- show "graph H' h' \<in> M"
- -- {* and $h'$ is norm-preserving. \skp *}
+ 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)
+ 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
+ also from h lin_vs
have [folded H'_def]: "is_subspace H (H + lin x')" ..
- finally (subspace_trans [OF _ h])
+ 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)"
+ also
+ have "... = (let (y, a) = (x, #0) in h y + a * xi)"
by (simp add: Let_def)
- also have
+ 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)
+ 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 <= p x"
+
+ 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
qed
qed
- hence "\<not> (\<forall>x \<in> M. g \<subseteq> x --> g = x)" by simp
- -- {* So the graph $g$ of $h$ cannot be maximal. Contradiction! \skp *}
+ 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
qed
- thus "\<exists>h. is_linearform E h \<and> (\<forall>x \<in> F. h x = f x)
- \<and> (\<forall>x \<in> E. h x <= p x)"
+ 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" ..
+ 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 <= p x" by (force!)
+ from eq show "\<forall>x \<in> E. h x \<le> p x" by (blast!)
qed
qed
qed
@@ -309,34 +299,37 @@
subsection {* Alternative formulation *}
-text {* The following alternative formulation of the Hahn-Banach
-Theorem\label{abs-HahnBanach} uses the fact that for a real linear form
-$f$ and a seminorm $p$ the
-following inequations are equivalent:\footnote{This was shown in lemma
-$\idt{abs{\dsh}ineq{\dsh}iff}$ (see page \pageref{abs-ineq-iff}).}
-\begin{matharray}{ll}
-\forall x\in H.\ap |h\ap x|\leq p\ap x& {\rm and}\\
-\forall x\in H.\ap h\ap x\leq p\ap x\\
-\end{matharray}
+text {*
+ The following alternative formulation of the Hahn-Banach
+ Theorem\label{abs-HahnBanach} uses the fact that for a real linear
+ form @{text f} and a seminorm @{text p} the following inequations
+ are equivalent:\footnote{This was shown in lemma @{thm [source]
+ abs_ineq_iff} (see page \pageref{abs-ineq-iff}).}
+ \begin{center}
+ \begin{tabular}{lll}
+ @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
+ @{text "\<forall>x \<in> H. h x \<le> p x"} \\
+ \end{tabular}
+ \end{center}
*}
theorem abs_HahnBanach:
-"[| is_vectorspace E; is_subspace F E; is_linearform F f;
-is_seminorm E p; \<forall>x \<in> F. |f x| <= p x |]
-==> \<exists>g. is_linearform E g \<and> (\<forall>x \<in> F. g x = f x)
- \<and> (\<forall>x \<in> E. |g x| <= p x)"
+ "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)
+ \<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. |f x| <= p x"
-have "\<forall>x \<in> F. f x <= 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 <= p x)"
+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
+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 <= p x"
-hence "\<forall>x \<in> E. |g x| <= p x"
+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)
qed
@@ -344,170 +337,179 @@
subsection {* The Hahn-Banach Theorem for normed spaces *}
-text {* Every continuous linear form $f$ on a subspace $F$ of a
-norm space $E$, can be extended to a continuous linear form $g$ on
-$E$ such that $\fnorm{f} = \fnorm {g}$. *}
+text {*
+ Every continuous linear form @{text f} on a subspace @{text F} of a
+ norm space @{text E}, can be extended to a continuous linear form
+ @{text g} on @{text E} such that @{text "\<parallel>f\<parallel> = \<parallel>g\<parallel>"}.
+*}
theorem norm_HahnBanach:
-"[| is_normed_vectorspace E norm; is_subspace F E;
-is_linearform F f; is_continuous F norm f |]
-==> \<exists>g. is_linearform E g
- \<and> is_continuous E norm g
- \<and> (\<forall>x \<in> F. g x = f x)
+ "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
+ \<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: "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 $p$ on $E$ as follows:
-\begin{matharray}{l}
-p \: x = \fnorm f \cdot \norm x\\
-\end{matharray}
+txt{*
+ We define a function @{text p} on @{text E} as follows:
+ @{text "p x = \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
*}
-def p == "\<lambda>x. \<parallel>f\<parallel>F,norm * norm x"
+def p \<equiv> "\<lambda>x. \<parallel>f\<parallel>F,norm * norm x"
-txt{* $p$ is a seminorm on $E$: *}
+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"
+fix x y a assume "x \<in> E" "y \<in> E"
-txt{* $p$ is positive definite: *}
+txt {* @{text p} is positive definite: *}
-show "#0 <= p x"
+show "#0 \<le> p x"
proof (unfold p_def, rule real_le_mult_order1a)
- from f_cont f_norm show "#0 <= \<parallel>f\<parallel>F,norm" ..
- show "#0 <= norm x" ..
+ from f_cont f_norm show "#0 \<le> \<parallel>f\<parallel>F,norm" ..
+ show "#0 \<le> norm x" ..
qed
-txt{* $p$ is absolutely homogenous: *}
+txt {* @{text p} is absolutely homogenous: *}
-show "p (a \<cdot> x) = |a| * p x"
-proof -
+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) = |a| * norm x"
+ 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 * ( |a| * norm x )
- = |a| * (\<parallel>f\<parallel>F,norm * norm x)"
+ 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 "... = |a| * p x" by (simp!)
+ also have "... = \<bar>a\<bar> * p x" by (simp!)
finally show ?thesis .
qed
-txt{* Furthermore, $p$ is subadditive: *}
+txt {* Furthermore, @{text p} is subadditive: *}
-show "p (x + y) <= p x + p y"
+show "p (x + y) \<le> p x + p y"
proof -
have "p (x + y) = \<parallel>f\<parallel>F,norm * norm (x + y)"
by (simp!)
- also
- have "... <= \<parallel>f\<parallel>F,norm * (norm x + norm y)"
+ 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 <= \<parallel>f\<parallel>F,norm" ..
- show "norm (x + y) <= norm x + norm y" ..
+ 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
+ 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
-txt{* $f$ is bounded by $p$. *}
+txt {* @{text f} is bounded by @{text p}. *}
-have "\<forall>x \<in> F. |f x| <= p x"
+have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> p x"
proof
fix x assume "x \<in> F"
- from f_norm show "|f x| <= p x"
+ from f_norm show "\<bar>f x\<bar> \<le> p x"
by (simp! add: norm_fx_le_norm_f_norm_x)
qed
-txt{* Using the fact that $p$ is a seminorm and
-$f$ is bounded by $p$ we can apply the Hahn-Banach Theorem
-for real vector spaces.
-So $f$ can be extended in a norm-preserving way to some function
-$g$ on the whole vector space $E$. *}
+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}.
+*}
-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. |g x| <= p x)"
+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)
thus ?thesis
-proof (elim exE conjE)
+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. |g x| <= p x"
+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"
-show "\<exists>g. is_linearform E g
- \<and> is_continuous E norm g
- \<and> (\<forall>x \<in> F. g x = f x)
+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{* We furthermore have to show that
-$g$ is also continuous: *}
+txt {*
+ We furthermore have to show that @{text g} is also continuous:
+*}
show g_cont: "is_continuous E norm g"
proof
fix x assume "x \<in> E"
- with b show "|g x| <= \<parallel>f\<parallel>F,norm * norm x"
- by (simp add: p_def)
- qed
+ with b show "\<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
+ by (simp add: p_def)
+ qed
- txt {* To complete the proof, we show that
- $\fnorm g = \fnorm f$. \label{order_antisym} *}
+ txt {*
+ To complete the proof, we show that
+ @{text "\<parallel>g\<parallel> = \<parallel>f\<parallel>"}. \label{order_antisym} *}
show "\<parallel>g\<parallel>E,norm = \<parallel>f\<parallel>F,norm"
(is "?L = ?R")
proof (rule order_antisym)
- txt{* First we show $\fnorm g \leq \fnorm f$. The function norm
- $\fnorm g$ is defined as the smallest $c\in\bbbR$ such that
- \begin{matharray}{l}
- \All {x\in E} {|g\ap x| \leq c \cdot \norm x}
- \end{matharray}
- Furthermore holds
- \begin{matharray}{l}
- \All {x\in E} {|g\ap x| \leq \fnorm f \cdot \norm x}
- \end{matharray}
+ 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
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> c \<cdot> \<parallel>x\<parallel>"}
+ \end{tabular}
+ \end{center}
+ \noindent Furthermore holds
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel> \<cdot> \<parallel>x\<parallel>"}
+ \end{tabular}
+ \end{center}
*}
-
- have "\<forall>x \<in> E. |g x| <= \<parallel>f\<parallel>F,norm * norm x"
+
+ have "\<forall>x \<in> E. \<bar>g x\<bar> \<le> \<parallel>f\<parallel>F,norm * norm x"
proof
- fix x assume "x \<in> E"
- show "|g x| <= \<parallel>f\<parallel>F,norm * norm x"
+ 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 <= ?R"
+ with g_cont e_norm show "?L \<le> ?R"
proof (rule fnorm_le_ub)
- from f_cont f_norm show "#0 <= \<parallel>f\<parallel>F,norm" ..
+ from f_cont f_norm show "#0 \<le> \<parallel>f\<parallel>F,norm" ..
qed
- txt{* The other direction is achieved by a similar
+ txt{* The other direction is achieved by a similar
argument. *}
- have "\<forall>x \<in> F. |f x| <= \<parallel>g\<parallel>E,norm * norm x"
+ have "\<forall>x \<in> F. \<bar>f x\<bar> \<le> \<parallel>g\<parallel>E,norm * norm x"
proof
- fix x assume "x \<in> F"
+ fix x assume "x \<in> F"
from a have "g x = f x" ..
- hence "|f x| = |g x|" by simp
+ hence "\<bar>f x\<bar> = \<bar>g x\<bar>" by simp
also from g_cont
- have "... <= \<parallel>g\<parallel>E,norm * norm x"
+ have "... \<le> \<parallel>g\<parallel>E,norm * norm x"
proof (rule norm_fx_le_norm_f_norm_x)
show "x \<in> E" ..
qed
- finally show "|f x| <= \<parallel>g\<parallel>E,norm * norm x" .
- qed
- thus "?R <= ?L"
+ finally show "\<bar>f x\<bar> \<le> \<parallel>g\<parallel>E,norm * norm x" .
+ qed
+ thus "?R \<le> ?L"
proof (rule fnorm_le_ub [OF f_cont f_norm])
- from g_cont show "#0 <= \<parallel>g\<parallel>E,norm" ..
+ from g_cont show "#0 \<le> \<parallel>g\<parallel>E,norm" ..
qed
qed
qed
--- a/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy Sat Dec 16 21:41:51 2000 +0100
@@ -7,155 +7,162 @@
theory HahnBanachExtLemmas = FunctionNorm:
-text{* In this section the following context is presumed.
-Let $E$ be a real vector space with a
-seminorm $q$ on $E$. $F$ is a subspace of $E$ and $f$ a linear
-function on $F$. We consider a subspace $H$ of $E$ that is a
-superspace of $F$ and a linear form $h$ on $H$. $H$ is a not equal
-to $E$ and $x_0$ is an element in $E \backslash H$.
-$H$ is extended to the direct sum $H' = H + \idt{lin}\ap x_0$, so for
-any $x\in H'$ the decomposition of $x = y + a \mult x$
-with $y\in H$ is unique. $h'$ is defined on $H'$ by
-$h'\ap x = h\ap y + a \cdot \xi$ for a certain $\xi$.
+text {*
+ In this section the following context is presumed. Let @{text E} be
+ a real vector space with a seminorm @{text q} on @{text E}. @{text
+ F} is a subspace of @{text E} and @{text f} a linear function on
+ @{text F}. We consider a subspace @{text H} of @{text E} that is a
+ superspace of @{text F} and a linear form @{text h} on @{text
+ H}. @{text H} is a not equal to @{text E} and @{text "x\<^sub>0"} is
+ 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>}.
-Subsequently we show some properties of this extension $h'$ of $h$.
-*}
-
+ 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 $f$ (see page \pageref{ex-xi-use}).
-It is a consequence
-of the completeness of $\bbbR$. To show
-\begin{matharray}{l}
-\Ex{\xi}{\All {y\in F}{a\ap y \leq \xi \land \xi \leq b\ap y}}
-\end{matharray}
-it suffices to show that
-\begin{matharray}{l} \All
-{u\in F}{\All {v\in F}{a\ap u \leq b \ap v}}
-\end{matharray} *}
+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
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<exists>\<xi>. \<forall>y \<in> F. a y \<le> \<xi> \<and> \<xi> \<le> b y"}
+ \end{tabular}
+ \end{center}
+ \noindent it suffices to show that
+ \begin{center}
+ \begin{tabular}{l}
+ @{text "\<forall>u \<in> F. \<forall>v \<in> F. a u \<le> b v"}
+ \end{tabular}
+ \end{center}
+*}
-lemma ex_xi:
- "[| is_vectorspace F; !! u v. [| u \<in> F; v \<in> F |] ==> a u <= b v |]
- ==> \<exists>xi::real. \<forall>y \<in> F. a y <= xi \<and> xi <= b y"
+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"
proof -
assume vs: "is_vectorspace F"
- assume r: "(!! u v. [| u \<in> F; v \<in> F |] ==> a u <= (b v::real))"
+ 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 $S = \{a\: u\dt\: u\in F\}$ has a supremum, if
+ 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"
+ have "\<exists>xi. isLub UNIV ?S xi"
proof (rule reals_complete)
-
- txt {* The set $S$ is non-empty, since $a\ap\zero \in S$: *}
- from vs have "a 0 \<in> ?S" by force
+ 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 {* $b\ap \zero$ is an upper bound of $S$: *}
+ txt {* @{text "b 0"} is an upper bound of @{text S}: *}
- show "\<exists>Y. isUb UNIV ?S Y"
- proof
+ 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 $y\in S$ is less than $b\ap \zero$: *}
+ txt {* Every element @{text "y \<in> S"} is less than @{text "b 0"}: *}
- fix y assume y: "y \<in> ?S"
+ fix y assume y: "y \<in> ?S"
from y have "\<exists>u \<in> F. y = a u" by fast
- thus "y <= b 0"
+ thus "y \<le> b 0"
proof
- fix u assume "u \<in> F"
+ fix u assume "u \<in> F"
assume "y = a u"
- also have "a u <= b 0" by (rule r) (simp!)+
+ also have "a u \<le> b 0" by (rule r) (simp!)+
finally show ?thesis .
qed
qed
qed
qed
- thus "\<exists>xi. \<forall>y \<in> F. a y <= xi \<and> xi <= b y"
+ 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"
+ fix xi assume "isLub UNIV ?S xi"
show ?thesis
- proof (intro exI conjI ballI)
-
- txt {* For all $y\in F$ holds $a\ap y \leq \xi$: *}
-
+ 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 <= xi"
- proof (rule isUbD)
+ show "a y \<le> xi"
+ proof (rule isUbD)
show "isUb UNIV ?S xi" ..
- qed (force!)
+ qed (blast!)
next
- txt {* For all $y\in F$ holds $\xi\leq b\ap y$: *}
+ txt {* For all @{text "y \<in> F"} holds @{text "\<xi> \<le> b y"}: *}
fix y assume "y \<in> F"
- show "xi <= b y"
+ show "xi \<le> b y"
proof (intro isLub_le_isUb isUbI setleI)
show "b y \<in> UNIV" ..
- show "\<forall>ya \<in> ?S. ya <= b y"
+ 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 <= b y"
+ thus "au \<le> b y"
proof
- fix u assume "u \<in> F" assume "au = a u"
- also have "... <= b y" by (rule r)
+ fix u assume "u \<in> F" assume "au = a u"
+ also have "... \<le> b y" by (rule r)
finally show ?thesis .
qed
qed
- qed
+ qed
qed
qed
qed
-text{* \medskip The function $h'$ is defined as a
-$h'\ap x = h\ap y + a\cdot \xi$ where $x = y + a\mult \xi$
-is a linear extension of $h$ to $H'$. *}
+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'}. *}
-lemma h'_lf:
- "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
- in h y + a * xi);
- H' == H + lin x0; is_subspace H E; is_linearform H h; x0 \<notin> H;
- x0 \<in> E; x0 \<noteq> 0; is_vectorspace E |]
- ==> is_linearform H' 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' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
+ 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)"
- and H'_def: "H' == 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"
+ 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'"
+ have h': "is_vectorspace H'"
proof (unfold H'_def, rule vs_sum_vs)
show "is_subspace (lin x0) E" ..
- qed
+ qed
show ?thesis
proof
- fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
+ fix x1 x2 assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"
- txt{* We now have to show that $h'$ is additive, i.~e.\
- $h' \ap (x_1\plus x_2) = h'\ap x_1 + h'\ap x_2$
- for $x_1, x_2\in H$. *}
+ 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"}. *}
- 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"
+ 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"
+ 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
+ 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
@@ -164,181 +171,178 @@
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"
+ 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"
+ have ya: "y1 + y2 = y \<and> a1 + a2 = a"
proof (rule decomp_H')
- show "y1 + y2 + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0"
+ 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" ..
qed
have "h' (x1 + x2) = h y + a * xi"
- by (rule h'_definite)
- also have "... = h (y1 + y2) + (a1 + a2) * 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]
+ 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 "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)"
by simp
also have "h y1 + a1 * xi = h' x1"
by (rule h'_definite [symmetric])
- also have "h y2 + a2 * xi = h' x2"
+ also have "h y2 + a2 * xi = h' x2"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
-
- txt{* We further have to show that $h'$ is multiplicative,
- i.~e.\ $h'\ap (c \mult x_1) = c \cdot h'\ap x_1$
- for $x\in H$ and $c\in \bbbR$.
- *}
- next
- fix c x1 assume x1: "x1 \<in> H'"
+ 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: "!! x. x\<in> H' ==> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> 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
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)"
+ show "h' (c \<cdot> x1) = c * (h' x1)"
proof (elim exE conjE)
- fix y1 y a1 a
+ 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"
+ and y: "c \<cdot> x1 = y + a \<cdot> x0" and y': "y \<in> H"
- have ya: "c \<cdot> y1 = y \<and> c * a1 = a"
- proof (rule decomp_H')
- show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0"
+ have ya: "c \<cdot> y1 = y \<and> c * a1 = a"
+ 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" ..
qed
- have "h' (c \<cdot> x1) = h y + a * xi"
- by (rule h'_definite)
+ 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 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"
by (rule h'_definite [symmetric])
finally show ?thesis .
qed
qed
qed
-text{* \medskip The linear extension $h'$ of $h$
-is bounded by the seminorm $p$. *}
+text {* \medskip The linear extension @{text h'} of @{text h}
+is bounded by the seminorm @{text p}. *}
lemma h'_norm_pres:
- "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
- in h y + a * xi);
- H' == H + lin x0; x0 \<notin> H; x0 \<in> E; x0 \<noteq> 0; is_vectorspace E;
- is_subspace H E; is_seminorm E p; is_linearform H h;
- \<forall>y \<in> H. h y <= p y;
- \<forall>y \<in> H. - p (y + x0) - h y <= xi \<and> xi <= p (y + x0) - h y |]
- ==> \<forall>x \<in> H'. h' x <= p x"
-proof
- assume h'_def:
- "h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H
+ "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)"
- and H'_def: "H' == 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 <= p y"
- presume a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya <= xi"
- presume a2: "\<forall>ya \<in> H. xi <= p (ya + x0) - h ya"
- fix x assume "x \<in> H'"
- have ex_x:
- "!! x. x \<in> H' ==> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H"
+ 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 <= p 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"
by (rule h'_definite)
- txt{* Now we show
- $h\ap y + a \cdot \xi\leq p\ap (y\plus a \mult x_0)$
- by case analysis on $a$. *}
+ 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 "... <= p (y + a \<cdot> x0)"
+ also have "... \<le> p (y + a \<cdot> x0)"
proof (rule linorder_cases)
- assume z: "a = #0"
+ assume z: "a = #0"
with vs y a show ?thesis by simp
- txt {* In the case $a < 0$, we use $a_1$ with $\idt{ya}$
- taken as $y/a$: *}
+ txt {* In the case @{text "a < 0"}, we use @{text "a\<^sub>1"}
+ with @{text ya} taken as @{text "y / a"}: *}
next
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) <= xi"
+ 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 <= a * (- p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y))"
+ 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
+ also
have "... = - a * (p (inverse a \<cdot> y + x0)) - a * (h (inverse a \<cdot> y))"
by (rule real_mult_diff_distrib)
- also from lz vs y
+ 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])
- finally have "a * xi <= p (y + a \<cdot> x0) - h y" .
+ finally have "a * xi \<le> p (y + a \<cdot> x0) - h y" .
- hence "h y + a * xi <= h y + 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
- txt {* In the case $a > 0$, we use $a_2$ with $\idt{ya}$
- taken as $y/a$: *}
+ txt {* In the case @{text "a > 0"}, we use @{text "a\<^sub>2"}
+ with @{text ya} taken as @{text "y / a"}: *}
- next
+ next
assume gz: "#0 < a" hence nz: "a \<noteq> #0" by simp
- from a2 have "xi <= p (inverse a \<cdot> y + x0) - h (inverse a \<cdot> y)"
+ 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 <= a * (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
+ by (rule real_mult_diff_distrib2)
+ also from gz 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_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])
- finally have "a * xi <= p (y + a \<cdot> x0) - h y" .
-
- hence "h y + a * xi <= h y + (p (y + a \<cdot> x0) - h y)"
+ by (simp add: linearform_mult [symmetric])
+ 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
qed
also from x have "... = p x" by simp
finally show ?thesis .
qed
-qed blast+
+qed blast+
end
--- a/src/HOL/Real/HahnBanach/HahnBanachLemmas.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/HahnBanachLemmas.thy Sat Dec 16 21:41:51 2000 +0100
@@ -1,3 +1,4 @@
-
+(*<*)
theory HahnBanachLemmas = HahnBanachSupLemmas + HahnBanachExtLemmas:
end
+(*>*)
\ No newline at end of file
--- a/src/HOL/Real/HahnBanach/HahnBanachSupLemmas.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/HahnBanachSupLemmas.thy Sat Dec 16 21:41:51 2000 +0100
@@ -7,57 +7,59 @@
theory HahnBanachSupLemmas = FunctionNorm + ZornLemma:
-text{* This section contains some lemmas that will be used in the
-proof of the Hahn-Banach Theorem.
-In this section the following context is presumed.
-Let $E$ be a real vector space with a seminorm $p$ on $E$.
-$F$ is a subspace of $E$ and $f$ a linear form on $F$. We
-consider a chain $c$ of norm-preserving extensions of $f$, such that
-$\Union c = \idt{graph}\ap H\ap h$.
-We will show some properties about the limit function $h$,
-i.e.\ the supremum of the chain $c$.
-*}
+text {*
+ This section contains some lemmas that will be used in the proof of
+ the Hahn-Banach Theorem. In this section the following context is
+ presumed. Let @{text E} be a real vector space with a seminorm
+ @{text p} on @{text E}. @{text F} is a subspace of @{text E} and
+ @{text f} a linear form on @{text F}. We consider a chain @{text c}
+ of norm-preserving extensions of @{text f}, such that
+ @{text "\<Union>c = graph H h"}. We will show some properties about the
+ limit function @{text h}, i.e.\ the supremum of the chain @{text c}.
+*}
-text{* Let $c$ be a chain of norm-preserving extensions of the
-function $f$ and let $\idt{graph}\ap H\ap h$ be the supremum of $c$.
-Every element in $H$ is member of
-one of the elements of the chain. *}
+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
+ elements of the chain.
+*}
lemma some_H'h't:
- "[| M = norm_pres_extensions E p F f; c \<in> chain M;
- graph H h = \<Union>c; x \<in> H |]
- ==> \<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 <= p x)"
+ "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)"
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"
+ and u: "graph H h = \<Union>c" "x \<in> H"
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"
+ 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"
+ 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
+ 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 <= p x)"
+ \<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 <= p x"
- show ?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!)
@@ -67,93 +69,99 @@
qed
-text{* \medskip Let $c$ be a chain of norm-preserving extensions of the
-function $f$ and let $\idt{graph}\ap H\ap h$ be the supremum of $c$.
-Every element in the domain $H$ of the supremum function is member of
-the domain $H'$ of some function $h'$, such that $h$ extends $h'$.
+text {*
+ \medskip Let @{text c} be a chain of norm-preserving extensions of
+ the function @{text f} and let @{text "graph H h"} be the supremum
+ of @{text c}. Every element in the domain @{text H} of the supremum
+ function is member of the domain @{text H'} of some function @{text
+ h'}, such that @{text h} extends @{text h'}.
*}
-lemma some_H'h':
- "[| M = norm_pres_extensions E p F f; c \<in> chain M;
- graph H h = \<Union>c; x \<in> H |]
- ==> \<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H h
+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 <= p x)"
+ \<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"
+ 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 <= p x)"
+ 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
+ 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'"
- "graph F f \<subseteq> graph H' h'" "\<forall>x \<in> H'. h' x <= p x"
+ 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"
show ?thesis
proof (intro exI conjI)
show "x \<in> H'" by (rule graphD1)
- from cM u show "graph H' h' \<subseteq> graph H h"
+ from cM u show "graph H' h' \<subseteq> graph H h"
by (simp! only: chain_ball_Union_upper)
qed
qed
qed
-text{* \medskip Any two elements $x$ and $y$ in the domain $H$ of the
-supremum function $h$ are both in the domain $H'$ of some function
-$h'$, such that $h$ extends $h'$. *}
+text {*
+ \medskip Any two elements @{text x} and @{text y} in the domain
+ @{text H} of the supremum function @{text h} are both in the domain
+ @{text H'} of some function @{text h'}, such that @{text h} extends
+ @{text h'}.
+*}
-lemma some_H'h'2:
- "[| M = norm_pres_extensions E p F f; c \<in> chain M;
- graph H h = \<Union>c; x \<in> H; y \<in> H |]
- ==> \<exists>H' h'. x \<in> H' \<and> y \<in> H' \<and> graph H' h' \<subseteq> graph H h
+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 <= p x)"
+ \<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"
+ 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 {* $x$ is in the domain $H'$ of some function $h'$,
- such that $h$ extends $h'$. *}
+ 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 <= p x)"
+ \<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 {* $y$ is in the domain $H''$ of some function $h''$,
- such that $h$ extends $h''$. *}
+ 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 <= p x)"
+ \<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 e1 e2 show ?thesis
+ 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 <= p x"
+ 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"
- 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 <= p x"
+ 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 $h'$ and $h''$ are elements of the chain,
- $h''$ is an extension of $h'$ or vice versa. Thus both
- $x$ and $y$ 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' | graph H' h' \<subseteq> graph H'' h''"
- (is "?case1 | ?case2")
+ 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
+ proof
assume ?case1
show ?thesis
proof (intro exI conjI)
@@ -183,44 +191,47 @@
-text{* \medskip The relation induced by the graph of the supremum
-of a chain $c$ is definite, i.~e.~t is the graph of a function. *}
+text {*
+ \medskip The relation induced by the graph of the supremum of a
+ chain @{text c} is definite, i.~e.~t is the graph of a function. *}
-lemma sup_definite:
- "[| M == norm_pres_extensions E p F f; c \<in> chain M;
- (x, y) \<in> \<Union>c; (x, z) \<in> \<Union>c |] ==> z = y"
-proof -
- assume "c \<in> chain M" "M == norm_pres_extensions E p F f"
- "(x, y) \<in> \<Union>c" "(x, z) \<in> \<Union>c"
+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"
+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)
- txt{* Since both $(x, y) \in \Union c$ and $(x, z) \in \Union c$
- they are members of some graphs $G_1$ and $G_2$, resp., such that
- both $G_1$ and $G_2$ are members of $c$.*}
+ 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}.*}
fix G1 G2 assume
- "(x, y) \<in> G1" "G1 \<in> c" "(x, z) \<in> G2" "G2 \<in> c" "c \<subseteq> M"
+ "(x, y) \<in> G1" "G1 \<in> c" "(x, z) \<in> G2" "G2 \<in> c" "c \<subseteq> M"
have "G1 \<in> M" ..
- hence e1: "\<exists>H1 h1. graph H1 h1 = G1"
- by (force! dest: norm_pres_extension_D)
+ 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 (force! dest: norm_pres_extension_D)
- from e1 e2 show ?thesis
+ 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"
+ fix H1 h1 H2 h2
+ assume "graph H1 h1 = G1" "graph H2 h2 = G2"
- txt{* $G_1$ is contained in $G_2$ or vice versa,
- since both $G_1$ and $G_2$ are members of $c$. \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 | G2 \<subseteq> G1" (is "?case1 | ?case2") ..
+ 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 (force!)
+ 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" ..
@@ -229,7 +240,7 @@
assume ?case2
have "(x, y) \<in> graph H1 h1" by (simp!)
hence "y = h1 x" ..
- also have "(x, z) \<in> graph H1 h1" by (force!)
+ also have "(x, z) \<in> graph H1 h1" by (blast!)
hence "z = h1 x" ..
finally show ?thesis .
qed
@@ -237,36 +248,39 @@
qed
qed
-text{* \medskip The limit function $h$ is linear. Every element $x$ in the
-domain of $h$ is in the domain of a function $h'$ in the chain of norm
-preserving extensions. Furthermore, $h$ is an extension of $h'$ so
-the function values of $x$ are identical for $h'$ and $h$. Finally, the
-function $h'$ is linear by construction of $M$. *}
+text {*
+ \medskip The limit function @{text h} is linear. Every element
+ @{text x} in the domain of @{text h} is in the domain of a function
+ @{text h'} in the chain of norm preserving extensions. Furthermore,
+ @{text h} is an extension of @{text h'} so the function values of
+ @{text x} are identical for @{text h'} and @{text h}. Finally, the
+ function @{text h'} is linear by construction of @{text M}.
+*}
-lemma sup_lf:
- "[| M = norm_pres_extensions E p F f; c \<in> chain M;
- graph H h = \<Union>c |] ==> is_linearform H h"
-proof -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
+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
+ 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 <= p x)"
+ \<and> (\<forall>x \<in> H'. h' x \<le> p x)"
by (rule some_H'h'2)
- txt {* We have to show that $h$ is additive. *}
+ txt {* We have to show that @{text h} is additive. *}
- thus "h (x + y) = h x + h y"
+ 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"
+ 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" ..
@@ -274,22 +288,22 @@
with b have "h' (x + y) = h (x + y)" ..
finally show ?thesis .
qed
- next
+ next
fix a x assume "x \<in> H"
- have "\<exists>H' h'. x \<in> H' \<and> graph H' h' \<subseteq> graph H 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 <= p x)"
+ \<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')
- txt{* We have to show that $h$ is multiplicative. *}
+ 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"
+ 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'" ..
@@ -299,36 +313,37 @@
qed
qed
-text{* \medskip The limit of a non-empty chain of norm
-preserving extensions of $f$ is an extension of $f$,
-since every element of the chain is an extension
-of $f$ and the supremum is an extension
-for every element of the chain.*}
+text {*
+ \medskip The limit of a non-empty chain of norm preserving
+ extensions of @{text f} is an extension of @{text f}, since every
+ element of the chain is an extension of @{text f} and the supremum
+ is an extension for every element of the chain.
+*}
lemma sup_ext:
- "[| graph H h = \<Union>c; M = norm_pres_extensions E p F f;
- c \<in> chain M; \<exists>x. x \<in> c |] ==> graph F f \<subseteq> graph H h"
+ "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"
proof -
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
+ 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
+ thus ?thesis
proof
- fix x assume "x \<in> c"
+ 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_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 <= p x)"
+ \<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)
+ 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" .
@@ -339,30 +354,32 @@
qed
qed
-text{* \medskip The domain $H$ of the limit function is a superspace of $F$,
-since $F$ is a subset of $H$. The existence of the $\zero$ element in
-$F$ and the closure properties follow from the fact that $F$ is a
-vector space. *}
+text {*
+ \medskip The domain @{text H} of the limit function is a superspace
+ of @{text F}, since @{text F} is a subset of @{text H}. The
+ existence of the @{text 0} element in @{text F} and the closure
+ properties follow from the fact that @{text F} is a vector space.
+*}
-lemma sup_supF:
- "[| graph H h = \<Union>c; M = norm_pres_extensions E p F f;
- c \<in> chain M; \<exists>x. x \<in> c; is_subspace F E; is_vectorspace E |]
- ==> 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"
+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
+ show ?thesis
proof
show "0 \<in> F" ..
- show "F \<subseteq> H"
+ 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 "\<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"
@@ -373,166 +390,171 @@
qed
qed
-text{* \medskip The domain $H$ of the limit function is a subspace
-of $E$. *}
+text {*
+ \medskip The domain @{text H} of the limit function is a subspace of
+ @{text E}.
+*}
-lemma sup_subE:
- "[| graph H h = \<Union>c; M = norm_pres_extensions E p F f;
- c \<in> chain M; \<exists>x. x \<in> c; is_subspace F E; is_vectorspace E |]
- ==> 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
+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
proof
-
- txt {* The $\zero$ element is in $H$, as $F$ is a subset
- of $H$: *}
+
+ 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)
+ also have "is_subspace F H" by (rule sup_supF)
hence "F \<subseteq> H" ..
finally show "0 \<in> H" .
- txt{* $H$ is a subset of $E$: *}
+ txt {* @{text H} is a subset of @{text E}: *}
- show "H \<subseteq> 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 <= p x)"
- by (rule some_H'h')
- thus "x \<in> E"
+ \<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"
+ fix H' h' assume "x \<in> H'" "is_subspace H' E"
have "H' \<subseteq> E" ..
- thus "x \<in> E" ..
+ thus "x \<in> E" ..
qed
qed
- txt{* $H$ is closed under addition: *}
+ 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 <= 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"
+ 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 .
+ also have "H' \<subseteq> H" ..
+ finally show ?thesis .
qed
- qed
+ qed
- txt{* $H$ is closed under scalar multiplication: *}
+ 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"
+ 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 <= p x)"
- by (rule some_H'h')
- thus "a \<cdot> x \<in> 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"
+ 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 .
+ finally show ?thesis .
qed
qed
qed
qed
-text {* \medskip The limit function is bounded by
-the norm $p$ as well, since all elements in the chain are
-bounded by $p$.
+text {*
+ \medskip The limit function is bounded by the norm @{text p} as
+ well, since all elements in the chain are bounded by @{text p}.
*}
lemma sup_norm_pres:
- "[| graph H h = \<Union>c; M = norm_pres_extensions E p F f;
- c \<in> chain M |] ==> \<forall>x \<in> H. h x <= p x"
+ "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"
proof
- assume "M = norm_pres_extensions E p F f" "c \<in> chain M"
+ 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
+ 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 <= p x)"
+ \<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 <= p x"
+ thus "h x \<le> p x"
proof (elim exE conjE)
- fix H' h'
- assume "x \<in> H'" "graph H' h' \<subseteq> graph H h"
- and a: "\<forall>x \<in> H'. h' x <= p x"
+ fix H' h'
+ assume "x \<in> H'" "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 <= p x " ..
- finally show ?thesis .
+ also from a have "h' x \<le> p x " ..
+ finally show ?thesis .
qed
qed
-text{* \medskip The following lemma is a property of linear forms on
-real vector spaces. It will be used for the lemma
-$\idt{abs{\dsh}HahnBanach}$ (see page \pageref{abs-HahnBanach}). \label{abs-ineq-iff}
-For real vector spaces the following inequations are equivalent:
-\begin{matharray}{ll}
-\forall x\in H.\ap |h\ap x|\leq p\ap x& {\rm and}\\
-\forall x\in H.\ap h\ap x\leq p\ap x\\
-\end{matharray}
+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}
+ (see page \pageref{abs-HahnBanach}). \label{abs-ineq-iff} For real
+ vector spaces the following inequations are equivalent:
+ \begin{center}
+ \begin{tabular}{lll}
+ @{text "\<forall>x \<in> H. \<bar>h x\<bar> \<le> p x"} & and &
+ @{text "\<forall>x \<in> H. h x \<le> p x"} \\
+ \end{tabular}
+ \end{center}
*}
-lemma abs_ineq_iff:
- "[| is_subspace H E; is_vectorspace E; is_seminorm E p;
- is_linearform H h |]
- ==> (\<forall>x \<in> H. |h x| <= p x) = (\<forall>x \<in> H. h x <= p x)"
+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"
+ assume "is_subspace H E" "is_vectorspace E" "is_seminorm E p"
"is_linearform H h"
have h: "is_vectorspace H" ..
show ?thesis
- proof
+ proof
assume l: ?L
show ?R
proof
fix x assume x: "x \<in> H"
- have "h x <= |h x|" by (rule abs_ge_self)
- also from l have "... <= p x" ..
- finally show "h x <= p x" .
+ have "h x \<le> \<bar>h x\<bar>" by (rule abs_ge_self)
+ also from l have "... \<le> p x" ..
+ finally show "h x \<le> p x" .
qed
next
assume r: ?R
show ?L
- proof
+ proof
fix x assume "x \<in> H"
- show "!! a b :: real. [| - a <= b; b <= a |] ==> |b| <= a"
+ show "\<And>a b :: real. - a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> \<bar>b\<bar> \<le> a"
by arith
- show "- p x <= h x"
+ show "- p x \<le> h x"
proof (rule real_minus_le)
- from h have "- h x = h (- x)"
+ from h have "- h x = h (- x)"
by (rule linearform_neg [symmetric])
- also from r have "... <= p (- x)" by (simp!)
- also have "... = p x"
+ also from r have "... \<le> p (- x)" by (simp!)
+ also have "... = p x"
proof (rule seminorm_minus)
show "x \<in> E" ..
qed
- finally show "- h x <= p x" .
+ finally show "- h x \<le> p x" .
qed
- from r show "h x <= p x" ..
+ from r show "h x \<le> p x" ..
qed
qed
-qed
+qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/Linearform.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/Linearform.thy Sat Dec 16 21:41:51 2000 +0100
@@ -7,63 +7,65 @@
theory Linearform = VectorSpace:
-text{* A \emph{linear form} is a function on a vector
-space into the reals that is additive and multiplicative. *}
+text {*
+ A \emph{linear form} is a function on a vector space into the reals
+ that is additive and multiplicative.
+*}
constdefs
- is_linearform :: "['a::{plus, minus, zero} set, 'a => real] => bool"
- "is_linearform V f ==
+ 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))"
+ (\<forall>x \<in> V. \<forall>a. f (a \<cdot> x) = a * (f x))"
-lemma is_linearformI [intro]:
- "[| !! x y. [| x \<in> V; y \<in> V |] ==> f (x + y) = f x + f y;
- !! x c. x \<in> V ==> f (c \<cdot> x) = c * f x |]
- ==> is_linearform V f"
- by (unfold is_linearform_def) force
+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
-lemma linearform_add [intro?]:
- "[| is_linearform V f; x \<in> V; y \<in> V |] ==> f (x + y) = f x + f y"
- by (unfold is_linearform_def) fast
+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; x \<in> V |] ==> f (a \<cdot> x) = a * (f x)"
- by (unfold is_linearform_def) fast
+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; is_linearform V f; x \<in> V|]
- ==> f (- x) = - f x"
-proof -
- assume "is_linearform V f" "is_vectorspace V" "x \<in> V"
+ "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> 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!)
finally show ?thesis .
qed
-lemma linearform_diff [intro?]:
- "[| is_vectorspace V; is_linearform V f; x \<in> V; y \<in> V |]
- ==> f (x - y) = f x - f y"
+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"
proof -
- assume "is_vectorspace V" "is_linearform V f" "x \<in> V" "y \<in> V"
+ 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)"
+ 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!)
qed
-text{* Every linear form yields $0$ for the $\zero$ vector.*}
+text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
-lemma linearform_zero [intro?, simp]:
- "[| is_vectorspace V; is_linearform V f |] ==> f 0 = #0"
-proof -
- assume "is_vectorspace V" "is_linearform V f"
+lemma linearform_zero [intro?, simp]:
+ "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> 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"
+ also have "... = f 0 - f 0"
by (rule linearform_diff) (simp!)+
also have "... = #0" by simp
finally show "f 0 = #0" .
-qed
+qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/NormedSpace.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/NormedSpace.thy Sat Dec 16 21:41:51 2000 +0100
@@ -7,96 +7,97 @@
theory NormedSpace = Subspace:
-syntax
- abs :: "real \<Rightarrow> real" ("|_|")
-
subsection {* Quasinorms *}
-text{* A \emph{seminorm} $\norm{\cdot}$ is a function on a real vector
-space into the reals that has the following properties: It is positive
-definite, absolute homogenous and subadditive. *}
+text {*
+ A \emph{seminorm} @{text "\<parallel>\<cdot>\<parallel>"} is a function on a real vector space
+ into the reals that has the following properties: it is positive
+ definite, absolute homogenous and subadditive.
+*}
constdefs
- is_seminorm :: "['a::{plus, minus, zero} set, 'a => real] => bool"
- "is_seminorm V norm == \<forall>x \<in> V. \<forall>y \<in> V. \<forall>a.
- #0 <= norm x
- \<and> norm (a \<cdot> x) = |a| * norm x
- \<and> norm (x + y) <= norm x + norm y"
+ 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"
-lemma is_seminormI [intro]:
- "[| !! x y a. [| x \<in> V; y \<in> V|] ==> #0 <= norm x;
- !! x a. x \<in> V ==> norm (a \<cdot> x) = |a| * norm x;
- !! x y. [|x \<in> V; y \<in> V |] ==> norm (x + y) <= norm x + norm y |]
- ==> is_seminorm V norm"
- by (unfold is_seminorm_def, force)
+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
lemma seminorm_ge_zero [intro?]:
- "[| is_seminorm V norm; x \<in> V |] ==> #0 <= norm x"
- by (unfold is_seminorm_def, force)
+ "is_seminorm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> #0 \<le> norm x"
+ by (unfold is_seminorm_def) blast
-lemma seminorm_abs_homogenous:
- "[| is_seminorm V norm; x \<in> V |]
- ==> norm (a \<cdot> x) = |a| * norm x"
- by (unfold is_seminorm_def, force)
+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; x \<in> V; y \<in> V |]
- ==> norm (x + y) <= norm x + norm y"
- by (unfold is_seminorm_def, force)
+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; x \<in> V; y \<in> V; is_vectorspace V |]
- ==> norm (x - y) <= norm x + norm y"
+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"
proof -
- assume "is_seminorm V norm" "x \<in> V" "y \<in> V" "is_vectorspace V"
- have "norm (x - y) = norm (x + - #1 \<cdot> y)"
+ 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 "... <= norm x + norm (- #1 \<cdot> y)"
+ also have "... \<le> norm x + norm (- #1 \<cdot> y)"
by (simp! add: seminorm_subadditive)
- also have "norm (- #1 \<cdot> y) = |- #1| * norm y"
+ also have "norm (- #1 \<cdot> y) = \<bar>- #1\<bar> * norm y"
by (rule seminorm_abs_homogenous)
- also have "|- #1| = (#1::real)" by (rule abs_minus_one)
- finally show "norm (x - y) <= norm x + norm y" by simp
+ also have "\<bar>- #1\<bar> = (#1::real)" by (rule abs_minus_one)
+ finally show "norm (x - y) \<le> norm x + norm y" by simp
qed
-lemma seminorm_minus:
- "[| is_seminorm V norm; x \<in> V; is_vectorspace V |]
- ==> norm (- x) = norm x"
+lemma seminorm_minus:
+ "is_seminorm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> is_vectorspace V
+ \<Longrightarrow> norm (- x) = norm x"
proof -
- assume "is_seminorm V norm" "x \<in> V" "is_vectorspace V"
+ 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 "... = |- #1| * norm x"
+ also have "... = \<bar>- #1\<bar> * norm x"
by (rule seminorm_abs_homogenous)
- also have "|- #1| = (#1::real)" by (rule abs_minus_one)
+ also have "\<bar>- #1\<bar> = (#1::real)" by (rule abs_minus_one)
finally show "norm (- x) = norm x" by simp
qed
subsection {* Norms *}
-text{* A \emph{norm} $\norm{\cdot}$ is a seminorm that maps only the
-$\zero$ vector to $0$. *}
+text {*
+ A \emph{norm} @{text "\<parallel>\<cdot>\<parallel>"} is a seminorm that maps only the
+ @{text 0} vector to @{text 0}.
+*}
constdefs
- is_norm :: "['a::{plus, minus, zero} set, 'a => real] => bool"
- "is_norm V norm == \<forall>x \<in> V. is_seminorm V norm
+ 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)
- ==> is_norm V norm" by (simp only: is_norm_def)
+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; x \<in> V |] ==> is_seminorm V norm"
- by (unfold is_norm_def, force)
+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; x \<in> V |] ==> (norm x = #0) = (x = 0)"
- by (unfold is_norm_def, force)
+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; x \<in> V |] ==> #0 <= norm x"
- by (unfold is_norm_def is_seminorm_def, force)
+ "is_norm V norm \<Longrightarrow> x \<in> V \<Longrightarrow> #0 \<le> norm x"
+ by (unfold is_norm_def is_seminorm_def) blast
subsection {* Normed vector spaces *}
@@ -105,33 +106,33 @@
a \emph{normed space}. *}
constdefs
- is_normed_vectorspace ::
- "['a::{plus, minus, zero} set, 'a => real] => bool"
- "is_normed_vectorspace V norm ==
+ 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"
-lemma normed_vsI [intro]:
- "[| is_vectorspace V; is_norm V norm |]
- ==> is_normed_vectorspace V norm"
- by (unfold is_normed_vectorspace_def) blast
+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 ==> is_vectorspace V"
- by (unfold is_normed_vectorspace_def) force
+lemma normed_vs_vs [intro?]:
+ "is_normed_vectorspace V norm \<Longrightarrow> is_vectorspace V"
+ by (unfold is_normed_vectorspace_def) blast
-lemma normed_vs_norm [intro?]:
- "is_normed_vectorspace V norm ==> is_norm V norm"
- by (unfold is_normed_vectorspace_def, elim conjE)
+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; x \<in> V |] ==> #0 <= norm x"
- by (unfold is_normed_vectorspace_def, rule, elim conjE)
+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; x \<in> V; x \<noteq> 0 |] ==> #0 < norm x"
+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 <= norm x" ..
+ 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"
proof
presume "norm x = #0"
@@ -142,45 +143,45 @@
finally show "#0 < norm x" .
qed
-lemma normed_vs_norm_abs_homogenous [intro?]:
- "[| is_normed_vectorspace V norm; x \<in> V |]
- ==> norm (a \<cdot> x) = |a| * norm x"
- by (rule seminorm_abs_homogenous, rule norm_is_seminorm,
+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; x \<in> V; y \<in> V |]
- ==> norm (x + y) <= norm x + norm y"
- by (rule seminorm_subadditive, rule norm_is_seminorm,
+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
+text{* Any subspace of a normed vector space is again a
normed vectorspace.*}
-lemma subspace_normed_vs [intro?]:
- "[| is_vectorspace E; is_subspace F E;
- is_normed_vectorspace E norm |] ==> is_normed_vectorspace F norm"
+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"
+ assume "is_subspace F E" "is_vectorspace E"
"is_normed_vectorspace E norm"
show "is_vectorspace F" ..
- show "is_norm F norm"
+ show "is_norm F norm"
proof (intro is_normI ballI conjI)
- show "is_seminorm F norm"
+ show "is_seminorm F norm"
proof
fix x y a presume "x \<in> E"
- show "#0 <= norm x" ..
- show "norm (a \<cdot> x) = |a| * norm x" ..
+ show "#0 \<le> norm x" ..
+ show "norm (a \<cdot> x) = \<bar>a\<bar> * norm x" ..
presume "y \<in> E"
- show "norm (x + y) <= norm x + norm y" ..
+ show "norm (x + y) \<le> norm x + norm y" ..
qed (simp!)+
fix x assume "x \<in> F"
- show "(norm x = #0) = (x = 0)"
+ show "(norm x = #0) = (x = 0)"
proof (rule norm_zero_iff)
show "is_norm E norm" ..
qed (simp!)
qed
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/Subspace.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Sat Dec 16 21:41:51 2000 +0100
@@ -3,7 +3,6 @@
Author: Gertrud Bauer, TU Munich
*)
-
header {* Subspaces *}
theory Subspace = VectorSpace:
@@ -11,59 +10,61 @@
subsection {* Definition *}
-text {* A non-empty subset $U$ of a vector space $V$ is a
-\emph{subspace} of $V$, iff $U$ is closed under addition and
-scalar multiplication. *}
+text {*
+ A non-empty subset @{text U} of a vector space @{text V} is a
+ \emph{subspace} of @{text V}, iff @{text U} is closed under addition
+ and scalar multiplication.
+*}
-constdefs
- is_subspace :: "['a::{plus, minus, zero} set, 'a set] => bool"
- "is_subspace U V == U \<noteq> {} \<and> U <= V
- \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x\<in> U)"
+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)"
-lemma subspaceI [intro]:
- "[| 0 \<in> U; U <= V; \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U);
- \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U |]
- ==> is_subspace U V"
-proof (unfold is_subspace_def, intro conjI)
+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+)
-lemma subspace_not_empty [intro?]: "is_subspace U V ==> U \<noteq> {}"
- by (unfold is_subspace_def) simp
+lemma subspace_not_empty [intro?]: "is_subspace U V \<Longrightarrow> U \<noteq> {}"
+ by (unfold is_subspace_def) blast
-lemma subspace_subset [intro?]: "is_subspace U V ==> U <= V"
- by (unfold is_subspace_def) simp
+lemma subspace_subset [intro?]: "is_subspace U V \<Longrightarrow> U \<subseteq> V"
+ by (unfold is_subspace_def) blast
-lemma subspace_subsetD [simp, intro?]:
- "[| is_subspace U V; x \<in> U |] ==> x \<in> V"
- by (unfold is_subspace_def) force
+lemma subspace_subsetD [simp, intro?]:
+ "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
+ by (unfold is_subspace_def) blast
-lemma subspace_add_closed [simp, intro?]:
- "[| is_subspace U V; x \<in> U; y \<in> U |] ==> x + y \<in> U"
- by (unfold is_subspace_def) simp
+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 subspace_mult_closed [simp, intro?]:
- "[| is_subspace U V; x \<in> U |] ==> a \<cdot> x \<in> U"
- by (unfold is_subspace_def) simp
+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
-lemma subspace_diff_closed [simp, intro?]:
- "[| is_subspace U V; is_vectorspace V; x \<in> U; y \<in> U |]
- ==> x - y \<in> U"
- by (simp! add: diff_eq1 negate_eq1)
+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"
+ 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
+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; is_vectorspace V |] ==> 0 \<in> U"
-proof -
+ "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> 0 \<in> U"
+proof -
assume "is_subspace U V" and v: "is_vectorspace V"
have "U \<noteq> {}" ..
- hence "\<exists>x. x \<in> U" by force
- thus ?thesis
- proof
- fix x assume u: "x \<in> U"
+ 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)
@@ -71,55 +72,54 @@
qed
qed
-lemma subspace_neg_closed [simp, intro?]:
- "[| is_subspace U V; is_vectorspace V; x \<in> U |] ==> - x \<in> U"
+lemma subspace_neg_closed [simp, intro?]:
+ "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> - x \<in> U"
by (simp add: negate_eq1)
-text_raw {* \medskip *}
-text {* Further derived laws: every subspace is a vector space. *}
+text {* \medskip Further derived laws: every subspace is a vector space. *}
lemma subspace_vs [intro?]:
- "[| is_subspace U V; is_vectorspace V |] ==> is_vectorspace U"
+ "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_vectorspace U"
proof -
- assume "is_subspace U V" "is_vectorspace V"
+ assume "is_subspace U V" "is_vectorspace V"
show ?thesis
- proof
+ 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"
+ 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)+
qed
text {* The subspace relation is reflexive. *}
-lemma subspace_refl [intro]: "is_vectorspace V ==> is_subspace V V"
-proof
+lemma subspace_refl [intro]: "is_vectorspace V \<Longrightarrow> is_subspace V V"
+proof
assume "is_vectorspace V"
show "0 \<in> V" ..
- show "V <= V" ..
+ 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!)
qed
text {* The subspace relation is transitive. *}
-lemma subspace_trans:
- "[| is_subspace U V; is_vectorspace V; is_subspace V W |]
- ==> is_subspace U W"
-proof
- assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
+lemma subspace_trans:
+ "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_subspace V W
+ \<Longrightarrow> is_subspace U W"
+proof
+ assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
show "0 \<in> U" ..
- have "U <= V" ..
- also have "V <= W" ..
- finally show "U <= W" .
+ 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"
+ 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"
+ fix x y assume "x \<in> U" "y \<in> U"
show "x + y \<in> U" by (simp!)
qed
@@ -134,12 +134,14 @@
subsection {* Linear closure *}
-text {* The \emph{linear closure} of a vector $x$ is the set of all
-scalar multiples of $x$. *}
+text {*
+ The \emph{linear closure} of a vector @{text x} is the set of all
+ scalar multiples of @{text x}.
+*}
constdefs
- lin :: "('a::{minus,plus,zero}) => 'a set"
- "lin x == {a \<cdot> x | a. True}"
+ 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
@@ -149,59 +151,59 @@
text {* Every vector is contained in its linear closure. *}
-lemma x_lin_x: "[| is_vectorspace V; x \<in> V |] ==> x \<in> lin x"
+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"
+ assume "is_vectorspace V" "x \<in> V"
show "x = #1 \<cdot> x" by (simp!)
qed simp
text {* Any linear closure is a subspace. *}
-lemma lin_subspace [intro?]:
- "[| is_vectorspace V; x \<in> V |] ==> is_subspace (lin x) V"
+lemma lin_subspace [intro?]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_subspace (lin x) V"
proof
- assume "is_vectorspace V" "x \<in> V"
- show "0 \<in> lin x"
+ 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
- show "lin x <= V"
- proof (unfold lin_def, intro subsetI, elim CollectE exE conjE)
- fix xa a assume "xa = a \<cdot> 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!)
qed
- show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x"
+ 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"
+ 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,
+ 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"
+ 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
qed
- show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x"
+ 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"
+ 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
- qed
+ qed
qed
text {* Any linear closure is a vector space. *}
-lemma lin_vs [intro?]:
- "[| is_vectorspace V; x \<in> V |] ==> is_vectorspace (lin x)"
+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"
+ assume "is_vectorspace V" "x \<in> V"
show "is_subspace (lin x) V" ..
qed
@@ -209,49 +211,45 @@
subsection {* Sum of two vectorspaces *}
-text {* The \emph{sum} of two vectorspaces $U$ and $V$ is the set of
-all sums of elements from $U$ and $V$. *}
+text {*
+ The \emph{sum} of two vectorspaces @{text U} and @{text V} is the
+ set of all sums of elements from @{text U} and @{text V}.
+*}
instance set :: (plus) plus ..
-defs vs_sum_def:
- "U + V == {u + v | u v. u \<in> U \<and> v \<in> V}" (***
+defs (overloaded)
+ vs_sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}"
-constdefs
- vs_sum ::
- "['a::{plus, minus, zero} set, 'a set] => 'a set" (infixl "+" 65)
- "vs_sum U V == {x. \<exists>u \<in> U. \<exists>v \<in> V. x = u + v}";
-***)
-
-lemma vs_sumD:
+lemma vs_sumD:
"x \<in> U + V = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
by (unfold vs_sum_def) fast
lemmas vs_sumE = vs_sumD [THEN iffD1, elim_format, standard]
-lemma vs_sumI [intro?]:
- "[| x \<in> U; y \<in> V; t= x + y |] ==> t \<in> U + V"
+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
-text{* $U$ is a subspace of $U + V$. *}
+text {* @{text U} is a subspace of @{text "U + V"}. *}
-lemma subspace_vs_sum1 [intro?]:
- "[| is_vectorspace U; is_vectorspace V |]
- ==> is_subspace U (U + V)"
-proof
- assume "is_vectorspace U" "is_vectorspace V"
+lemma subspace_vs_sum1 [intro?]:
+ "is_vectorspace U \<Longrightarrow> is_vectorspace V
+ \<Longrightarrow> is_subspace U (U + V)"
+proof
+ assume "is_vectorspace U" "is_vectorspace V"
show "0 \<in> U" ..
- show "U <= U + V"
+ 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!)
qed
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
+ 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!)
+ 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"
+ 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
@@ -259,34 +257,34 @@
text{* The sum of two subspaces is again a subspace.*}
-lemma vs_sum_subspace [intro?]:
- "[| is_subspace U E; is_subspace V E; is_vectorspace E |]
- ==> is_subspace (U + V) E"
-proof
- assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
+lemma vs_sum_subspace [intro?]:
+ "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
+ \<Longrightarrow> is_subspace (U + V) E"
+proof
+ assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
show "0 \<in> U + V"
proof (intro vs_sumI)
show "0 \<in> U" ..
show "0 \<in> V" ..
show "(0::'a) = 0 + 0" by (simp!)
qed
-
- show "U + V <= E"
+
+ 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"
+ fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
show "x \<in> E" 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"
+ 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"
+ 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!)+
+ qed (simp_all!)
qed
show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
@@ -294,20 +292,20 @@
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)"
+ 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!)+
+ qed (simp_all!)
qed
qed
text{* The sum of two subspaces is a vectorspace. *}
-lemma vs_sum_vs [intro?]:
- "[| is_subspace U E; is_subspace V E; is_vectorspace E |]
- ==> is_vectorspace (U + V)"
+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"
+ assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
show "is_subspace (U + V) E" ..
qed
@@ -316,28 +314,31 @@
subsection {* Direct sums *}
-text {* The sum of $U$ and $V$ is called \emph{direct}, iff the zero
-element is the only common element of $U$ and $V$. For every element
-$x$ of the direct sum of $U$ and $V$ the decomposition in
-$x = u + v$ with $u \in U$ and $v \in V$ is unique.*}
+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
+ V}. For every element @{text x} of the direct sum of @{text U} and
+ @{text V} the decomposition in @{text "x = u + v"} with
+ @{text "u \<in> U"} and @{text "v \<in> V"} is unique.
+*}
-lemma decomp:
- "[| 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 |]
- ==> 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"
+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"
+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!)
+ 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
-
+
show "u1 = u2"
proof (rule vs_add_minus_eq)
- show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
+ show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
show "u1 \<in> E" ..
show "u2 \<in> E" ..
qed
@@ -350,30 +351,33 @@
qed
qed
-text {* An application of the previous lemma will be used in the proof
-of the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
-element $y + a \mult x_0$ of the direct sum of a vectorspace $H$ and
-the linear closure of $x_0$ the components $y \in H$ and $a$ are
-uniquely determined. *}
+text {*
+ An application of the previous lemma will be used in the proof of
+ the Hahn-Banach Theorem (see page \pageref{decomp-H-use}): for any
+ element @{text "y + a \<cdot> x\<^sub>0"} of the direct sum of a
+ vectorspace @{text H} and the linear closure of @{text "x\<^sub>0"}
+ the components @{text "y \<in> H"} and @{text a} are uniquely
+ determined.
+*}
-lemma decomp_H':
- "[| is_vectorspace E; is_subspace H E; y1 \<in> H; y2 \<in> H;
- x' \<notin> H; x' \<in> E; x' \<noteq> 0; y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x' |]
- ==> y1 = y2 \<and> a1 = a2"
+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"
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"
+ 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'" ..
+ 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' <= {0}"
+ 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'"
+ 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'"
@@ -381,8 +385,8 @@
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"
+ 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" .
@@ -390,87 +394,91 @@
qed
qed
qed
- show "{0} <= H \<inter> lin x'"
+ 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 \<inter> lin x'"
+ proof (rule IntI)
+ show "0 \<in> H" ..
+ from lin_vs show "0 \<in> lin x'" ..
+ qed
+ thus ?thesis by simp
qed
qed
show "is_subspace (lin x') E" ..
qed
-
+
from c show "y1 = y2" by simp
-
- show "a1 = a2"
+
+ show "a1 = a2"
proof (rule vs_mult_right_cancel [THEN iffD1])
from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
qed
qed
-text {* Since for any element $y + a \mult x'$ of the direct sum
-of a vectorspace $H$ and the linear closure of $x'$ the components
-$y\in H$ and $a$ are unique, it follows from $y\in H$ that
-$a = 0$.*}
+text {*
+ Since for any element @{text "y + a \<cdot> x'"} of the direct sum of a
+ vectorspace @{text H} and the linear closure of @{text x'} the
+ components @{text "y \<in> H"} and @{text a} are unique, it follows from
+ @{text "y \<in> H"} that @{text "a = 0"}.
+*}
-lemma decomp_H'_H:
- "[| is_vectorspace E; is_subspace H E; t \<in> H; x' \<notin> H; x' \<in> E;
- x' \<noteq> 0 |]
- ==> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (#0::real))"
+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"
+ 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') (assumption | (simp!))+
+ have "y = t \<and> a = (#0::real)"
+ by (rule decomp_H') (auto!)
thus "(y, a) = (t, (#0::real))" by (simp!)
-qed (simp!)+
+qed (simp_all!)
-text {* The components $y\in H$ and $a$ in $y \plus a \mult x'$
-are unique, so the function $h'$ defined by
-$h' (y \plus a \mult x') = h y + a \cdot \xi$ is definite. *}
+text {*
+ The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}
+ are unique, so the function @{text h'} defined by
+ @{text "h' (y + a \<cdot> x') = h y + a \<cdot> \<xi>"} is definite.
+*}
lemma h'_definite:
- "[| h' == (\<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 |]
- ==> h' x = h y + a * xi"
-proof -
- assume
- "h' == (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
+ "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"
+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"
- have "x \<in> H + (lin x')"
- by (simp! add: vs_sum_def lin_def) force+
- have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
+ "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)"
proof
show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
- by (force!)
+ 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"
+ show "xa = ya"
proof -
- show "fst xa = fst ya \<and> snd xa = snd ya ==> xa = ya"
+ 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 (force!)
- have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H"
- by (force!)
- from x y show "fst xa = fst ya \<and> snd xa = snd ya"
+ 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!)+)
qed
qed
- hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
- by (rule some1_equality) (force!)
+ 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)
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/VectorSpace.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/VectorSpace.thy Sat Dec 16 21:41:51 2000 +0100
@@ -9,127 +9,124 @@
subsection {* Signature *}
-text {* For the definition of real vector spaces a type $\alpha$
-of the sort $\{ \idt{plus}, \idt{minus}, \idt{zero}\}$ is considered, on which a
-real scalar multiplication $\mult$ is defined. *}
+text {*
+ For the definition of real vector spaces a type @{typ 'a} of the
+ sort @{text "{plus, minus, zero}"} is considered, on which a real
+ scalar multiplication @{text \<cdot>} is declared.
+*}
consts
- prod :: "[real, 'a::{plus, minus, zero}] => 'a" (infixr "'(*')" 70)
+ prod :: "real \<Rightarrow> 'a::{plus, minus, zero} \<Rightarrow> 'a" (infixr "'(*')" 70)
syntax (symbols)
- prod :: "[real, 'a] => 'a" (infixr "\<cdot>" 70)
+ prod :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "\<cdot>" 70)
subsection {* Vector space laws *}
-text {* A \emph{vector space} is a non-empty set $V$ of elements from
- $\alpha$ with the following vector space laws: The set $V$ is closed
- under addition and scalar multiplication, addition is associative
- and commutative; $\minus x$ is the inverse of $x$ w.~r.~t.~addition
- and $0$ is the neutral element of addition. Addition and
- multiplication are distributive; scalar multiplication is
- associative and the real number $1$ is the neutral element of scalar
- multiplication.
+text {*
+ A \emph{vector space} is a non-empty set @{text V} of elements from
+ @{typ 'a} with the following vector space laws: The set @{text V} is
+ closed under addition and scalar multiplication, addition is
+ associative and commutative; @{text "- x"} is the inverse of @{text
+ x} w.~r.~t.~addition and @{text 0} is the neutral element of
+ addition. Addition and multiplication are distributive; scalar
+ multiplication is associative and the real number @{text "#1"} is
+ the neutral element of scalar multiplication.
*}
constdefs
- is_vectorspace :: "('a::{plus, minus, zero}) set => bool"
- "is_vectorspace V == V \<noteq> {}
+ 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
+ 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)"
+ \<and> x - y = x + - y)"
-text_raw {* \medskip *}
-text {* The corresponding introduction rule is:*}
+
+text {* \medskip The corresponding introduction rule is:*}
lemma vsI [intro]:
- "[| 0 \<in> V;
- \<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V;
- \<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V;
- \<forall>x \<in> V. \<forall>y \<in> V. \<forall>z \<in> V. (x + y) + z = x + (y + z);
- \<forall>x \<in> V. \<forall>y \<in> V. x + y = y + x;
- \<forall>x \<in> V. x - x = 0;
- \<forall>x \<in> V. 0 + x = x;
- \<forall>x \<in> V. \<forall>y \<in> V. \<forall>a. a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y;
- \<forall>x \<in> V. \<forall>a b. (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x;
- \<forall>x \<in> V. \<forall>a b. (a * b) \<cdot> x = a \<cdot> b \<cdot> x;
- \<forall>x \<in> V. #1 \<cdot> x = x;
- \<forall>x \<in> V. - x = (- #1) \<cdot> x;
- \<forall>x \<in> V. \<forall>y \<in> V. x - y = x + - y |] ==> is_vectorspace V"
-proof (unfold is_vectorspace_def, intro conjI ballI allI)
- fix x y z
- assume "x \<in> V" "y \<in> V" "z \<in> V"
- "\<forall>x \<in> V. \<forall>y \<in> V. \<forall>z \<in> V. x + y + z = x + (y + z)"
- thus "x + y + z = x + (y + z)" by blast
-qed force+
+ "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
-text_raw {* \medskip *}
-text {* The corresponding destruction rules are: *}
+text {* \medskip The corresponding destruction rules are: *}
-lemma negate_eq1:
- "[| is_vectorspace V; x \<in> V |] ==> - x = (- #1) \<cdot> x"
+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 diff_eq1:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> 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_eq2:
- "[| is_vectorspace V; x \<in> V |] ==> (- #1) \<cdot> x = - x"
+lemma negate_eq2a:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> #-1 \<cdot> x = - x"
by (unfold is_vectorspace_def) simp
-lemma negate_eq2a:
- "[| is_vectorspace V; x \<in> V |] ==> #-1 \<cdot> x = - x"
+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 diff_eq2:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> x + - y = x - y"
- by (unfold is_vectorspace_def) simp
+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 vs_not_empty [intro?]: "is_vectorspace V ==> (V \<noteq> {})"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_add_closed [simp, intro?]:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> x + y \<in> V"
+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_mult_closed [simp, intro?]:
- "[| is_vectorspace V; x \<in> V |] ==> a \<cdot> x \<in> V"
- by (unfold is_vectorspace_def) simp
-
-lemma vs_diff_closed [simp, intro?]:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> x - y \<in> V"
+lemma vs_diff_closed [simp, intro?]:
+ "is_vectorspace V \<Longrightarrow> 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; x \<in> V |] ==> - x \<in> V"
+lemma vs_neg_closed [simp, intro?]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - x \<in> V"
by (simp add: negate_eq1)
-lemma vs_add_assoc [simp]:
- "[| is_vectorspace V; x \<in> V; y \<in> V; z \<in> V |]
- ==> (x + y) + z = x + (y + z)"
- by (unfold is_vectorspace_def) fast
+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; x \<in> V; y \<in> V |] ==> y + x = x + y"
- by (unfold is_vectorspace_def) simp
+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; x \<in> V; y \<in> V; z \<in> V |]
- ==> x + (y + z) = y + (x + z)"
+ "is_vectorspace V \<Longrightarrow> 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"
- hence "x + (y + z) = (x + y) + z"
+ assume "is_vectorspace V" "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)
@@ -138,94 +135,93 @@
theorems vs_add_ac = vs_add_assoc vs_add_commute vs_add_left_commute
-lemma vs_diff_self [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> x - x = 0"
+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. *}
-lemma zero_in_vs [simp, intro]: "is_vectorspace V ==> 0 \<in> V"
-proof -
+lemma zero_in_vs [simp, intro]: "is_vectorspace V \<Longrightarrow> 0 \<in> V"
+proof -
assume "is_vectorspace V"
have "V \<noteq> {}" ..
- hence "\<exists>x. x \<in> V" by force
- thus ?thesis
- proof
- fix x assume "x \<in> V"
+ hence "\<exists>x. x \<in> V" by blast
+ thus ?thesis
+ proof
+ fix x assume "x \<in> V"
have "0 = x - x" by (simp!)
also have "... \<in> V" by (simp! only: vs_diff_closed)
finally show ?thesis .
qed
qed
-lemma vs_add_zero_left [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> 0 + x = x"
+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; x \<in> V |] ==> x + 0 = x"
+lemma vs_add_zero_right [simp]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x + 0 = x"
proof -
- assume "is_vectorspace V" "x \<in> V"
+ assume "is_vectorspace V" "x \<in> V"
hence "x + 0 = 0 + x" by simp
also have "... = x" by (simp!)
finally show ?thesis .
qed
-lemma vs_add_mult_distrib1:
- "[| is_vectorspace V; x \<in> V; y \<in> V |]
- ==> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
+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 vs_add_mult_distrib2:
- "[| is_vectorspace V; x \<in> V |]
- ==> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
+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; x \<in> V |] ==> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"
+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; x \<in> V |] ==> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
+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 vs_mult_1 [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> #1 \<cdot> x = x"
+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; x \<in> V; y \<in> V |]
- ==> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
+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; x \<in> V |]
- ==> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
+lemma vs_diff_mult_distrib2:
+ "is_vectorspace V \<Longrightarrow> x \<in> V
+ \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
proof -
- assume "is_vectorspace V" "x \<in> V"
- have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
+ assume "is_vectorspace V" "x \<in> V"
+ have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
by (unfold real_diff_def, simp)
- also have "... = a \<cdot> x + (- b) \<cdot> x"
+ also have "... = a \<cdot> x + (- b) \<cdot> x"
by (rule vs_add_mult_distrib2)
- also have "... = a \<cdot> x + - (b \<cdot> x)"
+ also have "... = a \<cdot> x + - (b \<cdot> x)"
by (simp! add: negate_eq1)
- also have "... = a \<cdot> x - (b \<cdot> x)"
+ also have "... = a \<cdot> x - (b \<cdot> x)"
by (simp! add: diff_eq1)
finally show ?thesis .
qed
-(*text_raw {* \paragraph {Further derived laws.} *}*)
-text_raw {* \medskip *}
-text{* Further derived laws: *}
+
+text {* \medskip Further derived laws: *}
-lemma vs_mult_zero_left [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> #0 \<cdot> x = 0"
+lemma vs_mult_zero_left [simp]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> #0 \<cdot> x = 0"
proof -
- assume "is_vectorspace V" "x \<in> V"
+ assume "is_vectorspace V" "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"
+ 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)
@@ -234,9 +230,9 @@
finally show ?thesis .
qed
-lemma vs_mult_zero_right [simp]:
- "[| is_vectorspace (V:: 'a::{plus, minus, zero} set) |]
- ==> a \<cdot> 0 = (0::'a)"
+lemma vs_mult_zero_right [simp]:
+ "is_vectorspace (V:: 'a::{plus, minus, zero} set)
+ \<Longrightarrow> a \<cdot> 0 = (0::'a)"
proof -
assume "is_vectorspace V"
have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by (simp!)
@@ -246,41 +242,41 @@
finally show ?thesis .
qed
-lemma vs_minus_mult_cancel [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> (- a) \<cdot> - x = a \<cdot> x"
+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 vs_add_minus_left_eq_diff:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> - x + y = y - x"
-proof -
- assume "is_vectorspace V" "x \<in> V" "y \<in> V"
- hence "- x + y = y + - x"
+lemma vs_add_minus_left_eq_diff:
+ "is_vectorspace V \<Longrightarrow> 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)
finally show ?thesis .
qed
-lemma vs_add_minus [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> x + - x = 0"
+lemma vs_add_minus [simp]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x + - x = 0"
by (simp! add: diff_eq2)
-lemma vs_add_minus_left [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> - x + x = 0"
+lemma vs_add_minus_left [simp]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - x + x = 0"
by (simp! add: diff_eq2)
-lemma vs_minus_minus [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> - (- x) = x"
+lemma vs_minus_minus [simp]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> - (- x) = x"
by (simp add: negate_eq1)
-lemma vs_minus_zero [simp]:
- "is_vectorspace (V::'a::{plus, minus, zero} set) ==> - (0::'a) = 0"
+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; x \<in> V |] ==> (- x = 0) = (x = 0)"
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> (- x = 0) = (x = 0)"
(concl is "?L = ?R")
proof -
- assume "is_vectorspace V" "x \<in> V"
+ assume "is_vectorspace V" "x \<in> V"
show "?L = ?R"
proof
have "x = - (- x)" by (simp! add: vs_minus_minus)
@@ -290,81 +286,81 @@
qed (simp!)
qed
-lemma vs_add_minus_cancel [simp]:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> x + (- x + y) = y"
- by (simp add: vs_add_assoc [symmetric] del: vs_add_commute)
+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 vs_minus_add_cancel [simp]:
- "[| is_vectorspace V; x \<in> V; y \<in> V |] ==> - x + (x + y) = y"
- by (simp add: vs_add_assoc [symmetric] del: vs_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 vs_minus_add_distrib [simp]:
- "[| is_vectorspace V; x \<in> V; y \<in> V |]
- ==> - (x + y) = - x + - y"
+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 vs_diff_zero [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> x - 0 = x"
- by (simp add: diff_eq1)
+lemma vs_diff_zero [simp]:
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x - 0 = x"
+ by (simp add: diff_eq1)
-lemma vs_diff_zero_right [simp]:
- "[| is_vectorspace V; x \<in> V |] ==> 0 - x = - x"
+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; x \<in> V; y \<in> V; z \<in> V |]
- ==> (x + y = x + z) = (y = z)"
+ "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")
proof
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
+ 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)"
+ 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"
+ 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 force
+qed blast
-lemma vs_add_right_cancel:
- "[| is_vectorspace V; x \<in> V; y \<in> V; z \<in> V |]
- ==> (y + x = z + x) = (y = z)"
+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 vs_add_assoc_cong:
- "[| is_vectorspace V; x \<in> V; y \<in> V; x' \<in> V; y' \<in> V; z \<in> V |]
- ==> x + y = x' + y' ==> x + (y + z) = x' + (y' + z)"
- by (simp only: vs_add_assoc [symmetric])
+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 vs_mult_left_commute:
- "[| is_vectorspace V; x \<in> V; y \<in> V; z \<in> V |]
- ==> x \<cdot> y \<cdot> z = y \<cdot> x \<cdot> z"
+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 vs_mult_zero_uniq:
- "[| is_vectorspace V; x \<in> V; x \<noteq> 0; a \<cdot> x = 0 |] ==> a = 0"
+ "is_vectorspace V \<Longrightarrow> 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 "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!)
finally have "x = 0" .
- thus "a = 0" by contradiction
+ thus "a = 0" by contradiction
qed
-lemma vs_mult_left_cancel:
- "[| is_vectorspace V; x \<in> V; y \<in> V; a \<noteq> #0 |] ==>
+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")
proof
- assume "is_vectorspace V" "x \<in> V" "y \<in> V" "a \<noteq> #0"
+ 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)"
+ 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!)
@@ -372,51 +368,51 @@
qed simp
lemma vs_mult_right_cancel: (*** forward ***)
- "[| is_vectorspace V; x \<in> V; x \<noteq> 0 |]
- ==> (a \<cdot> x = b \<cdot> x) = (a = b)" (concl is "?L = ?R")
+ "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x \<noteq> 0
+ \<Longrightarrow> (a \<cdot> x = b \<cdot> x) = (a = b)" (concl is "?L = ?R")
proof
- assume v: "is_vectorspace V" "x \<in> V" "x \<noteq> 0"
- have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
+ 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)
+ from v this have "a - b = 0" by (rule vs_mult_zero_uniq)
thus "a = b" by simp
-qed simp
+qed simp
-lemma vs_eq_diff_eq:
- "[| is_vectorspace V; x \<in> V; y \<in> V; z \<in> V |] ==>
+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" )
+ (concl is "?L = ?R" )
proof -
- assume vs: "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
- show "?L = ?R"
+ assume vs: "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
+ show "?L = ?R"
proof
assume ?L
hence "x + y = z - y + y" by simp
also have "... = z + - y + y" by (simp! add: diff_eq1)
- also have "... = z + (- y + y)"
+ also have "... = z + (- y + y)"
by (rule vs_add_assoc) (simp!)+
- also from vs have "... = z + 0"
+ 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 .
next
assume ?R
hence "z - y = (x + y) - y" by simp
- also from vs have "... = x + y + - y"
+ also from vs have "... = x + y + - y"
by (simp add: diff_eq1)
- also have "... = x + (y + - y)"
+ also have "... = x + (y + - y)"
by (rule vs_add_assoc) (simp!)+
also have "... = x" by (simp!)
finally show ?L by (rule sym)
qed
qed
-lemma vs_add_minus_eq_minus:
- "[| is_vectorspace V; x \<in> V; y \<in> V; x + y = 0 |] ==> x = - y"
+lemma vs_add_minus_eq_minus:
+ "is_vectorspace V \<Longrightarrow> 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"
+ assume "is_vectorspace V" "x \<in> V" "y \<in> V"
have "x = (- y + y) + x" by (simp!)
also have "... = - y + (x + y)" by (simp!)
also assume "x + y = 0"
@@ -424,41 +420,41 @@
finally show "x = - y" .
qed
-lemma vs_add_minus_eq:
- "[| is_vectorspace V; x \<in> V; y \<in> V; x - y = 0 |] ==> x = y"
+lemma vs_add_minus_eq:
+ "is_vectorspace V \<Longrightarrow> 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 "is_vectorspace V" "x \<in> V" "y \<in> V" "x - y = 0"
assume "x - y = 0"
hence e: "x + - y = 0" by (simp! add: diff_eq1)
- with _ _ _ have "x = - (- y)"
+ with _ _ _ have "x = - (- y)"
by (rule vs_add_minus_eq_minus) (simp!)+
thus "x = y" by (simp!)
qed
lemma vs_add_diff_swap:
- "[| is_vectorspace V; a \<in> V; b \<in> V; c \<in> V; d \<in> V; a + b = c + d |]
- ==> a - c = d - b"
-proof -
- assume vs: "is_vectorspace V" "a \<in> V" "b \<in> V" "c \<in> V" "d \<in> V"
+ "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"
+proof -
+ assume vs: "is_vectorspace V" "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)"
+ have "- c + (a + b) = - c + (c + d)"
by (simp! add: vs_add_left_cancel)
also have "... = d" by (rule vs_minus_add_cancel)
finally have eq: "- c + (a + b) = d" .
- from vs have "a - c = (- c + (a + b)) + - b"
+ from vs have "a - c = (- c + (a + b)) + - b"
by (simp add: vs_add_ac diff_eq1)
- also from eq have "... = d + - b"
+ also from eq have "... = d + - b"
by (simp! add: vs_add_right_cancel)
also have "... = d - b" by (simp! add: diff_eq2)
finally show "a - c = d - b" .
qed
-lemma vs_add_cancel_21:
- "[| is_vectorspace V; x \<in> V; y \<in> V; z \<in> V; u \<in> V |]
- ==> (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"
+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!)
@@ -471,16 +467,16 @@
qed (simp! only: vs_add_left_commute [of V x])
qed
-lemma vs_add_cancel_end:
- "[| is_vectorspace V; x \<in> V; y \<in> V; z \<in> V |]
- ==> (x + (y + z) = y) = (x = - z)"
+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"
+ assume "is_vectorspace V" "x \<in> V" "y \<in> V" "z \<in> V"
show "?L = ?R"
proof
assume l: ?L
- have "x + z = 0"
+ have "x + z = 0"
proof (rule vs_add_left_cancel [THEN iffD1])
have "y + (x + z) = x + (y + z)" by (simp!)
also note l
@@ -490,12 +486,12 @@
thus "x = - z" by (simp! add: vs_add_minus_eq_minus)
next
assume r: ?R
- hence "x + (y + z) = - z + (y + z)" by simp
- also have "... = y + (- z + 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
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/ZornLemma.thy Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/ZornLemma.thy Sat Dec 16 21:41:51 2000 +0100
@@ -7,49 +7,52 @@
theory ZornLemma = Aux + Zorn:
-text {* Zorn's Lemmas states: if every linear ordered subset of an
-ordered set $S$ has an upper bound in $S$, then there exists a maximal
-element in $S$. In our application, $S$ is a set of sets ordered by
-set inclusion. Since the union of a chain of sets is an upper bound
-for all elements of the chain, the conditions of Zorn's lemma can be
-modified: if $S$ is non-empty, it suffices to show that for every
-non-empty chain $c$ in $S$ the union of $c$ also lies in $S$. *}
+text {*
+ Zorn's Lemmas states: if every linear ordered subset of an ordered
+ set @{text S} has an upper bound in @{text S}, then there exists a
+ maximal element in @{text S}. In our application, @{text S} is a
+ set of sets ordered by set inclusion. Since the union of a chain of
+ sets is an upper bound for all elements of the chain, the conditions
+ of Zorn's lemma can be modified: if @{text S} is non-empty, it
+ suffices to show that for every non-empty chain @{text c} in @{text
+ S} the union of @{text c} also lies in @{text S}.
+*}
-theorem Zorn's_Lemma:
- "(!!c. c: chain S ==> EX x. x:c ==> Union c : S) ==> a:S
- ==> EX y: S. ALL z: S. y <= z --> y = z"
+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"
proof (rule Zorn_Lemma2)
txt_raw {* \footnote{See
\url{http://isabelle.in.tum.de/library/HOL/HOL-Real/Zorn.html}} \isanewline *}
- assume r: "!!c. c: chain S ==> EX x. x:c ==> Union c : S"
- assume aS: "a:S"
- show "ALL c:chain S. EX y:S. ALL z:c. z <= y"
+ 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:chain S"
- show "EX y:S. ALL z:c. z <= y"
+ fix c assume "c \<in> chain S"
+ show "\<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
proof cases
- txt{* If $c$ is an empty chain, then every element
- in $S$ is an upper bound of $c$. *}
+ 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 $c$ is non-empty, then $\Union c$
- is an upper bound of $c$, lying in $S$. *}
+ txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper
+ bound of @{text c}, lying in @{text S}. *}
next
- assume c: "c~={}"
+ assume c: "c \<noteq> {}"
show ?thesis
proof
- show "ALL z:c. z <= Union c" by fast
- show "Union c : S"
+ show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast
+ show "\<Union>c \<in> S"
proof (rule r)
- from c show "EX x. x:c" by fast
+ from c show "\<exists>x. x \<in> c" by fast
qed
qed
qed
qed
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Real/HahnBanach/document/bbb.sty Sat Dec 16 21:41:14 2000 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,24 +0,0 @@
-%
-% home made blackboard-bold symbols: B C D E F G H I J K L M N O P Q R T U Z
-%
-
-\def\bbbB{{{\rm I}\mkern-3.8mu{\rm B}}}
-\def\bbbC{{{\rm C}\mkern-15mu{\phantom{\rm t}\vrule}\mkern9mu}}
-\def\bbbD{{{\rm I}\mkern-3.8mu{\rm D}}}
-\def\bbbE{{{\rm I}\mkern-3.8mu{\rm E}}}
-\def\bbbF{{{\rm I}\mkern-3.8mu{\rm F}}}
-\def\bbbG{{{\rm G}\mkern-16mu{\phantom{\rm t}\vrule}\mkern10mu}}
-\def\bbbH{{{\rm I}\mkern-3.8mu{\rm H}}}
-\def\bbbI{{{\rm I}\mkern-12mu{\phantom{\rm t}\vrule}\mkern6mu}}
-\def\bbbJ{{{\rm J}\mkern-12mu{\phantom{\rm t}\vrule}\mkern6mu}}
-\def\bbbK{{{\rm I}\mkern-3.8mu{\rm K}}}
-\def\bbbL{{{\rm I}\mkern-3.8mu{\rm L}}}
-\def\bbbM{{{\rm I}\mkern-3.8mu{\rm M}}}
-\def\bbbN{{{\rm I}\mkern-3.8mu{\rm N}}}
-\def\bbbO{{{\rm O}\mkern-16mu{\phantom{\rm t}\vrule}\mkern10mu}}
-\def\bbbP{{{\rm I}\mkern-3.8mu{\rm P}}}
-\def\bbbQ{{{\rm Q}\mkern-16mu{\phantom{\rm t}\vrule}\mkern10mu}}
-\def\bbbR{{{\rm I}\mkern-3.8mu{\rm R}}}
-\def\bbbT{{{\rm T}\mkern-16mu{\phantom{\rm t}\vrule}\mkern10mu}}
-\def\bbbU{{{\rm U}\mkern-15mu{\phantom{\rm t}\vrule}\mkern9mu}}
-\def\bbbZ{{{\sf Z}\mkern-7.5mu{\sf Z}}}
--- a/src/HOL/Real/HahnBanach/document/notation.tex Sat Dec 16 21:41:14 2000 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,43 +0,0 @@
-
-\renewcommand{\isamarkupheader}[1]{\section{#1}}
-\newcommand{\isasymprod}{\isamath{\mult}}
-\newcommand{\isasymzero}{\isamath{\zero}}
-
-\newcommand{\idt}[1]{{\mathord{\mathit{#1}}}}
-\newcommand{\var}[1]{{?\!#1}}
-\DeclareMathSymbol{\dshsym}{\mathalpha}{letters}{"2D}
-\newcommand{\dsh}{\dshsym}
-
-\newenvironment{matharray}[1]{\[\begin{array}{#1}}{\end{array}\]}
-
-\newcommand{\skp}{\smallskip}
-
-\newcommand{\To}{\to}
-\newcommand{\dt}{{\mathpunct.}}
-\newcommand{\Ex}[1]{\exists #1\dt\;}
-\newcommand{\Forall}{\forall}
-\newcommand{\All}[1]{\Forall #1\dt\;}
-\newcommand{\Eq}{\mathbin{\,\equiv\,}}
-\newcommand{\Impl}{\Rightarrow}
-\newcommand{\And}{\;\land\;}
-\newcommand{\Or}{\;\lor\;}
-\newcommand{\Le}{\leq}
-\newcommand{\Lt}{\lt}
-\newcommand{\lam}[1]{\mathop{\lambda} #1\dt\;}
-%\newcommand{\ap}{\mathpalette{\mathbin{\!}}{\mathbin{\!}}{\mathbin{}}{\mathbin{}}}
-\newcommand{\ap}{\mathpalette{\mathbin{}}{\mathbin{}}{\mathbin{}}{\mathbin{}}}
-\newcommand{\Union}{\bigcup}
-\newcommand{\Un}{\cup}
-\newcommand{\Int}{\cap}
-
-\newcommand{\norm}[1]{\left\|#1\right\|}
-\newcommand{\fnorm}[1]{\left\|#1\right\|}
-\newcommand{\zero}{0}
-\newcommand{\plus}{\mathbin{\mathbf +}}
-\newcommand{\minus}{\mathbin{\mathbf -}}
-\newcommand{\mult}{\cdot}
-
-%%% Local Variables:
-%%% mode: latex
-%%% TeX-master: "root"
-%%% End:
--- a/src/HOL/Real/HahnBanach/document/root.tex Sat Dec 16 21:41:14 2000 +0100
+++ b/src/HOL/Real/HahnBanach/document/root.tex Sat Dec 16 21:41:51 2000 +0100
@@ -1,12 +1,16 @@
+
\documentclass[10pt,a4paper,twoside]{article}
-%\documentclass[11pt,a4paper,twoside]{article}
\usepackage{latexsym,theorem}
-\usepackage{isabelle,isabellesym,bbb}
+\usepackage{isabelle,isabellesym}
\usepackage{pdfsetup} %last one!
+
+\isabellestyle{it}
\urlstyle{rm}
-\input{notation}
+\newcommand{\isasymsup}{\isamath{\sup\,}}
+\newcommand{\skp}{\smallskip}
+
\begin{document}
@@ -50,27 +54,27 @@
\clearpage
\part {Basic Notions}
-\input{Bounds.tex}
-\input{Aux.tex}
-\input{VectorSpace.tex}
-\input{Subspace.tex}
-\input{NormedSpace.tex}
-\input{Linearform.tex}
-\input{FunctionOrder.tex}
-\input{FunctionNorm.tex}
-\input{ZornLemma.tex}
+\input{Bounds}
+\input{Aux}
+\input{VectorSpace}
+\input{Subspace}
+\input{NormedSpace}
+\input{Linearform}
+\input{FunctionOrder}
+\input{FunctionNorm}
+\input{ZornLemma}
\clearpage
\part {Lemmas for the Proof}
-\input{HahnBanachSupLemmas.tex}
-\input{HahnBanachExtLemmas.tex}
-\input{HahnBanachLemmas.tex}
+\input{HahnBanachSupLemmas}
+\input{HahnBanachExtLemmas}
+\input{HahnBanachLemmas}
\clearpage
\part {The Main Proof}
-\input{HahnBanach.tex}
+\input{HahnBanach}
\bibliographystyle{abbrv}
\bibliography{root}