src/HOL/Real/HahnBanach/Linearform.thy
 changeset 10687 c186279eecea parent 9408 d3d56e1d2ec1 child 11701 3d51fbf81c17
     1.1 --- a/src/HOL/Real/HahnBanach/Linearform.thy	Sat Dec 16 21:41:14 2000 +0100
1.2 +++ b/src/HOL/Real/HahnBanach/Linearform.thy	Sat Dec 16 21:41:51 2000 +0100
1.3 @@ -7,63 +7,65 @@
1.4
1.5  theory Linearform = VectorSpace:
1.6
1.7 -text{* A \emph{linear form} is a function on a vector
1.8 -space into the reals that is additive and multiplicative. *}
1.9 +text {*
1.10 +  A \emph{linear form} is a function on a vector space into the reals
1.11 +  that is additive and multiplicative.
1.12 +*}
1.13
1.14  constdefs
1.15 -  is_linearform :: "['a::{plus, minus, zero} set, 'a => real] => bool"
1.16 -  "is_linearform V f ==
1.17 +  is_linearform :: "'a::{plus, minus, zero} set \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> bool"
1.18 +  "is_linearform V f \<equiv>
1.19        (\<forall>x \<in> V. \<forall>y \<in> V. f (x + y) = f x + f y) \<and>
1.20 -      (\<forall>x \<in> V. \<forall>a. f (a \<cdot> x) = a * (f x))"
1.21 +      (\<forall>x \<in> V. \<forall>a. f (a \<cdot> x) = a * (f x))"
1.22
1.23 -lemma is_linearformI [intro]:
1.24 -  "[| !! x y. [| x \<in> V; y \<in> V |] ==> f (x + y) = f x + f y;
1.25 -    !! x c. x \<in> V ==> f (c \<cdot> x) = c * f x |]
1.26 - ==> is_linearform V f"
1.27 - by (unfold is_linearform_def) force
1.28 +lemma is_linearformI [intro]:
1.29 +  "(\<And>x y. x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y) \<Longrightarrow>
1.30 +    (\<And>x c. x \<in> V \<Longrightarrow> f (c \<cdot> x) = c * f x)
1.31 + \<Longrightarrow> is_linearform V f"
1.32 + by (unfold is_linearform_def) blast
1.33
1.35 -  "[| is_linearform V f; x \<in> V; y \<in> V |] ==> f (x + y) = f x + f y"
1.36 -  by (unfold is_linearform_def) fast
1.38 +  "is_linearform V f \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x + y) = f x + f y"
1.39 +  by (unfold is_linearform_def) blast
1.40
1.41 -lemma linearform_mult [intro?]:
1.42 -  "[| is_linearform V f; x \<in> V |] ==>  f (a \<cdot> x) = a * (f x)"
1.43 -  by (unfold is_linearform_def) fast
1.44 +lemma linearform_mult [intro?]:
1.45 +  "is_linearform V f \<Longrightarrow> x \<in> V \<Longrightarrow>  f (a \<cdot> x) = a * (f x)"
1.46 +  by (unfold is_linearform_def) blast
1.47
1.48  lemma linearform_neg [intro?]:
1.49 -  "[|  is_vectorspace V; is_linearform V f; x \<in> V|]
1.50 -  ==> f (- x) = - f x"
1.51 -proof -
1.52 -  assume "is_linearform V f" "is_vectorspace V" "x \<in> V"
1.53 +  "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> x \<in> V
1.54 +  \<Longrightarrow> f (- x) = - f x"
1.55 +proof -
1.56 +  assume "is_linearform V f"  "is_vectorspace V"  "x \<in> V"
1.57    have "f (- x) = f ((- #1) \<cdot> x)" by (simp! add: negate_eq1)
1.58    also have "... = (- #1) * (f x)" by (rule linearform_mult)
1.59    also have "... = - (f x)" by (simp!)
1.60    finally show ?thesis .
1.61  qed
1.62
1.63 -lemma linearform_diff [intro?]:
1.64 -  "[| is_vectorspace V; is_linearform V f; x \<in> V; y \<in> V |]
1.65 -  ==> f (x - y) = f x - f y"
1.66 +lemma linearform_diff [intro?]:
1.67 +  "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> x \<in> V \<Longrightarrow> y \<in> V
1.68 +  \<Longrightarrow> f (x - y) = f x - f y"
1.69  proof -
1.70 -  assume "is_vectorspace V" "is_linearform V f" "x \<in> V" "y \<in> V"
1.71 +  assume "is_vectorspace V"  "is_linearform V f"  "x \<in> V"  "y \<in> V"
1.72    have "f (x - y) = f (x + - y)" by (simp! only: diff_eq1)
1.73 -  also have "... = f x + f (- y)"
1.74 +  also have "... = f x + f (- y)"
1.76    also have "f (- y) = - f y" by (rule linearform_neg)
1.77    finally show "f (x - y) = f x - f y" by (simp!)
1.78  qed
1.79
1.80 -text{* Every linear form yields $0$ for the $\zero$ vector.*}
1.81 +text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
1.82
1.83 -lemma linearform_zero [intro?, simp]:
1.84 -  "[| is_vectorspace V; is_linearform V f |] ==> f 0 = #0"
1.85 -proof -
1.86 -  assume "is_vectorspace V" "is_linearform V f"
1.87 +lemma linearform_zero [intro?, simp]:
1.88 +  "is_vectorspace V \<Longrightarrow> is_linearform V f \<Longrightarrow> f 0 = #0"
1.89 +proof -
1.90 +  assume "is_vectorspace V"  "is_linearform V f"
1.91    have "f 0 = f (0 - 0)" by (simp!)
1.92 -  also have "... = f 0 - f 0"
1.93 +  also have "... = f 0 - f 0"
1.94      by (rule linearform_diff) (simp!)+
1.95    also have "... = #0" by simp
1.96    finally show "f 0 = #0" .
1.97 -qed
1.98 +qed
1.99
1.100 -end
1.101 \ No newline at end of file
1.102 +end