src/HOL/Real_Vector_Spaces.thy
author hoelzl
Tue Mar 26 12:20:57 2013 +0100 (2013-03-26)
changeset 51524 7cb5ac44ca9e
parent 51518 src/HOL/RealVector.thy@6a56b7088a6a
child 51531 f415febf4234
permissions -rw-r--r--
rename RealVector.thy to Real_Vector_Spaces.thy
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(*  Title:      HOL/Real_Vector_Spaces.thy
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    Author:     Brian Huffman
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*)
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header {* Vector Spaces and Algebras over the Reals *}
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theory Real_Vector_Spaces
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imports Metric_Spaces
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begin
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subsection {* Locale for additive functions *}
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locale additive =
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  fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
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  assumes add: "f (x + y) = f x + f y"
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begin
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lemma zero: "f 0 = 0"
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proof -
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  have "f 0 = f (0 + 0)" by simp
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  also have "\<dots> = f 0 + f 0" by (rule add)
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  finally show "f 0 = 0" by simp
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qed
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lemma minus: "f (- x) = - f x"
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proof -
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  have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
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  also have "\<dots> = - f x + f x" by (simp add: zero)
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  finally show "f (- x) = - f x" by (rule add_right_imp_eq)
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qed
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lemma diff: "f (x - y) = f x - f y"
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by (simp add: add minus diff_minus)
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lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
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apply (cases "finite A")
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apply (induct set: finite)
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apply (simp add: zero)
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apply (simp add: add)
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apply (simp add: zero)
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done
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end
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subsection {* Vector spaces *}
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locale vector_space =
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  fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
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  assumes scale_right_distrib [algebra_simps]:
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    "scale a (x + y) = scale a x + scale a y"
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  and scale_left_distrib [algebra_simps]:
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    "scale (a + b) x = scale a x + scale b x"
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  and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
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  and scale_one [simp]: "scale 1 x = x"
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begin
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lemma scale_left_commute:
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  "scale a (scale b x) = scale b (scale a x)"
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by (simp add: mult_commute)
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lemma scale_zero_left [simp]: "scale 0 x = 0"
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  and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
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  and scale_left_diff_distrib [algebra_simps]:
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        "scale (a - b) x = scale a x - scale b x"
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  and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
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proof -
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  interpret s: additive "\<lambda>a. scale a x"
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    proof qed (rule scale_left_distrib)
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  show "scale 0 x = 0" by (rule s.zero)
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  show "scale (- a) x = - (scale a x)" by (rule s.minus)
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  show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
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  show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum)
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qed
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lemma scale_zero_right [simp]: "scale a 0 = 0"
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  and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
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  and scale_right_diff_distrib [algebra_simps]:
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        "scale a (x - y) = scale a x - scale a y"
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  and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"
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proof -
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  interpret s: additive "\<lambda>x. scale a x"
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    proof qed (rule scale_right_distrib)
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  show "scale a 0 = 0" by (rule s.zero)
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  show "scale a (- x) = - (scale a x)" by (rule s.minus)
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  show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
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  show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)
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qed
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lemma scale_eq_0_iff [simp]:
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  "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
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proof cases
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  assume "a = 0" thus ?thesis by simp
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next
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  assume anz [simp]: "a \<noteq> 0"
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  { assume "scale a x = 0"
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    hence "scale (inverse a) (scale a x) = 0" by simp
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    hence "x = 0" by simp }
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  thus ?thesis by force
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qed
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lemma scale_left_imp_eq:
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  "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
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proof -
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  assume nonzero: "a \<noteq> 0"
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  assume "scale a x = scale a y"
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  hence "scale a (x - y) = 0"
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     by (simp add: scale_right_diff_distrib)
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  hence "x - y = 0" by (simp add: nonzero)
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  thus "x = y" by (simp only: right_minus_eq)
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qed
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lemma scale_right_imp_eq:
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  "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
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proof -
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  assume nonzero: "x \<noteq> 0"
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  assume "scale a x = scale b x"
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  hence "scale (a - b) x = 0"
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     by (simp add: scale_left_diff_distrib)
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  hence "a - b = 0" by (simp add: nonzero)
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  thus "a = b" by (simp only: right_minus_eq)
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qed
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lemma scale_cancel_left [simp]:
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  "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
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by (auto intro: scale_left_imp_eq)
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lemma scale_cancel_right [simp]:
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  "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
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by (auto intro: scale_right_imp_eq)
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end
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subsection {* Real vector spaces *}
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class scaleR =
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  fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
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begin
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abbreviation
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  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
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where
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  "x /\<^sub>R r == scaleR (inverse r) x"
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end
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class real_vector = scaleR + ab_group_add +
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  assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y"
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  and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x"
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  and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
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  and scaleR_one: "scaleR 1 x = x"
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interpretation real_vector:
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  vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
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apply unfold_locales
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apply (rule scaleR_add_right)
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apply (rule scaleR_add_left)
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apply (rule scaleR_scaleR)
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apply (rule scaleR_one)
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done
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text {* Recover original theorem names *}
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lemmas scaleR_left_commute = real_vector.scale_left_commute
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lemmas scaleR_zero_left = real_vector.scale_zero_left
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lemmas scaleR_minus_left = real_vector.scale_minus_left
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lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib
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lemmas scaleR_setsum_left = real_vector.scale_setsum_left
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lemmas scaleR_zero_right = real_vector.scale_zero_right
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lemmas scaleR_minus_right = real_vector.scale_minus_right
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lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib
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lemmas scaleR_setsum_right = real_vector.scale_setsum_right
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lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
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lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
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lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
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lemmas scaleR_cancel_left = real_vector.scale_cancel_left
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lemmas scaleR_cancel_right = real_vector.scale_cancel_right
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text {* Legacy names *}
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lemmas scaleR_left_distrib = scaleR_add_left
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lemmas scaleR_right_distrib = scaleR_add_right
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lemmas scaleR_left_diff_distrib = scaleR_diff_left
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lemmas scaleR_right_diff_distrib = scaleR_diff_right
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lemma scaleR_minus1_left [simp]:
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  fixes x :: "'a::real_vector"
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  shows "scaleR (-1) x = - x"
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  using scaleR_minus_left [of 1 x] by simp
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class real_algebra = real_vector + ring +
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  assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
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  and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
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class real_algebra_1 = real_algebra + ring_1
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class real_div_algebra = real_algebra_1 + division_ring
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class real_field = real_div_algebra + field
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instantiation real :: real_field
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begin
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definition
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  real_scaleR_def [simp]: "scaleR a x = a * x"
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instance proof
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qed (simp_all add: algebra_simps)
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end
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interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
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proof qed (rule scaleR_left_distrib)
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interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
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proof qed (rule scaleR_right_distrib)
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lemma nonzero_inverse_scaleR_distrib:
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  fixes x :: "'a::real_div_algebra" shows
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  "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
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by (rule inverse_unique, simp)
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lemma inverse_scaleR_distrib:
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  fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}"
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  shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
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apply (case_tac "a = 0", simp)
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apply (case_tac "x = 0", simp)
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apply (erule (1) nonzero_inverse_scaleR_distrib)
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done
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subsection {* Embedding of the Reals into any @{text real_algebra_1}:
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@{term of_real} *}
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definition
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  of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
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  "of_real r = scaleR r 1"
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lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
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by (simp add: of_real_def)
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lemma of_real_0 [simp]: "of_real 0 = 0"
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by (simp add: of_real_def)
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lemma of_real_1 [simp]: "of_real 1 = 1"
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by (simp add: of_real_def)
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lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
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by (simp add: of_real_def scaleR_left_distrib)
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lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
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by (simp add: of_real_def)
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lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
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by (simp add: of_real_def scaleR_left_diff_distrib)
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lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
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by (simp add: of_real_def mult_commute)
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lemma nonzero_of_real_inverse:
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  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
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   inverse (of_real x :: 'a::real_div_algebra)"
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by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
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lemma of_real_inverse [simp]:
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  "of_real (inverse x) =
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   inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
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by (simp add: of_real_def inverse_scaleR_distrib)
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lemma nonzero_of_real_divide:
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  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
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   (of_real x / of_real y :: 'a::real_field)"
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by (simp add: divide_inverse nonzero_of_real_inverse)
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lemma of_real_divide [simp]:
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  "of_real (x / y) =
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   (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
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by (simp add: divide_inverse)
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lemma of_real_power [simp]:
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  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
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by (induct n) simp_all
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lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
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by (simp add: of_real_def)
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lemma inj_of_real:
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  "inj of_real"
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  by (auto intro: injI)
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lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
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lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
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proof
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  fix r
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  show "of_real r = id r"
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    by (simp add: of_real_def)
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qed
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text{*Collapse nested embeddings*}
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lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
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by (induct n) auto
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lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
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by (cases z rule: int_diff_cases, simp)
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lemma of_real_numeral: "of_real (numeral w) = numeral w"
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using of_real_of_int_eq [of "numeral w"] by simp
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lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w"
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using of_real_of_int_eq [of "neg_numeral w"] by simp
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text{*Every real algebra has characteristic zero*}
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instance real_algebra_1 < ring_char_0
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proof
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  from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
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  then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
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qed
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instance real_field < field_char_0 ..
huffman@27553
   321
huffman@20554
   322
huffman@20554
   323
subsection {* The Set of Real Numbers *}
huffman@20554
   324
haftmann@37767
   325
definition Reals :: "'a::real_algebra_1 set" where
haftmann@37767
   326
  "Reals = range of_real"
huffman@20554
   327
wenzelm@21210
   328
notation (xsymbols)
huffman@20554
   329
  Reals  ("\<real>")
huffman@20554
   330
huffman@21809
   331
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
huffman@20554
   332
by (simp add: Reals_def)
huffman@20554
   333
huffman@21809
   334
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
huffman@21809
   335
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
huffman@20718
   336
huffman@21809
   337
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
huffman@21809
   338
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
huffman@21809
   339
huffman@47108
   340
lemma Reals_numeral [simp]: "numeral w \<in> Reals"
huffman@47108
   341
by (subst of_real_numeral [symmetric], rule Reals_of_real)
huffman@47108
   342
huffman@47108
   343
lemma Reals_neg_numeral [simp]: "neg_numeral w \<in> Reals"
huffman@47108
   344
by (subst of_real_neg_numeral [symmetric], rule Reals_of_real)
huffman@20718
   345
huffman@20554
   346
lemma Reals_0 [simp]: "0 \<in> Reals"
huffman@20554
   347
apply (unfold Reals_def)
huffman@20554
   348
apply (rule range_eqI)
huffman@20554
   349
apply (rule of_real_0 [symmetric])
huffman@20554
   350
done
huffman@20554
   351
huffman@20554
   352
lemma Reals_1 [simp]: "1 \<in> Reals"
huffman@20554
   353
apply (unfold Reals_def)
huffman@20554
   354
apply (rule range_eqI)
huffman@20554
   355
apply (rule of_real_1 [symmetric])
huffman@20554
   356
done
huffman@20554
   357
huffman@20584
   358
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
huffman@20554
   359
apply (auto simp add: Reals_def)
huffman@20554
   360
apply (rule range_eqI)
huffman@20554
   361
apply (rule of_real_add [symmetric])
huffman@20554
   362
done
huffman@20554
   363
huffman@20584
   364
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
huffman@20584
   365
apply (auto simp add: Reals_def)
huffman@20584
   366
apply (rule range_eqI)
huffman@20584
   367
apply (rule of_real_minus [symmetric])
huffman@20584
   368
done
huffman@20584
   369
huffman@20584
   370
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
huffman@20584
   371
apply (auto simp add: Reals_def)
huffman@20584
   372
apply (rule range_eqI)
huffman@20584
   373
apply (rule of_real_diff [symmetric])
huffman@20584
   374
done
huffman@20584
   375
huffman@20584
   376
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
huffman@20554
   377
apply (auto simp add: Reals_def)
huffman@20554
   378
apply (rule range_eqI)
huffman@20554
   379
apply (rule of_real_mult [symmetric])
huffman@20554
   380
done
huffman@20554
   381
huffman@20584
   382
lemma nonzero_Reals_inverse:
huffman@20584
   383
  fixes a :: "'a::real_div_algebra"
huffman@20584
   384
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   385
apply (auto simp add: Reals_def)
huffman@20584
   386
apply (rule range_eqI)
huffman@20584
   387
apply (erule nonzero_of_real_inverse [symmetric])
huffman@20584
   388
done
huffman@20584
   389
huffman@20584
   390
lemma Reals_inverse [simp]:
haftmann@36409
   391
  fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
huffman@20584
   392
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   393
apply (auto simp add: Reals_def)
huffman@20584
   394
apply (rule range_eqI)
huffman@20584
   395
apply (rule of_real_inverse [symmetric])
huffman@20584
   396
done
huffman@20584
   397
huffman@20584
   398
lemma nonzero_Reals_divide:
huffman@20584
   399
  fixes a b :: "'a::real_field"
huffman@20584
   400
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   401
apply (auto simp add: Reals_def)
huffman@20584
   402
apply (rule range_eqI)
huffman@20584
   403
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   404
done
huffman@20584
   405
huffman@20584
   406
lemma Reals_divide [simp]:
haftmann@36409
   407
  fixes a b :: "'a::{real_field, field_inverse_zero}"
huffman@20584
   408
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   409
apply (auto simp add: Reals_def)
huffman@20584
   410
apply (rule range_eqI)
huffman@20584
   411
apply (rule of_real_divide [symmetric])
huffman@20584
   412
done
huffman@20584
   413
huffman@20722
   414
lemma Reals_power [simp]:
haftmann@31017
   415
  fixes a :: "'a::{real_algebra_1}"
huffman@20722
   416
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   417
apply (auto simp add: Reals_def)
huffman@20722
   418
apply (rule range_eqI)
huffman@20722
   419
apply (rule of_real_power [symmetric])
huffman@20722
   420
done
huffman@20722
   421
huffman@20554
   422
lemma Reals_cases [cases set: Reals]:
huffman@20554
   423
  assumes "q \<in> \<real>"
huffman@20554
   424
  obtains (of_real) r where "q = of_real r"
huffman@20554
   425
  unfolding Reals_def
huffman@20554
   426
proof -
huffman@20554
   427
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   428
  then obtain r where "q = of_real r" ..
huffman@20554
   429
  then show thesis ..
huffman@20554
   430
qed
huffman@20554
   431
huffman@20554
   432
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   433
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   434
  by (rule Reals_cases) auto
huffman@20554
   435
huffman@20504
   436
huffman@20504
   437
subsection {* Real normed vector spaces *}
huffman@20504
   438
haftmann@29608
   439
class norm =
huffman@22636
   440
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   441
huffman@24520
   442
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   443
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   444
huffman@31289
   445
class dist_norm = dist + norm + minus +
huffman@31289
   446
  assumes dist_norm: "dist x y = norm (x - y)"
huffman@31289
   447
huffman@31492
   448
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
hoelzl@51002
   449
  assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   450
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@31586
   451
  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
hoelzl@51002
   452
begin
hoelzl@51002
   453
hoelzl@51002
   454
lemma norm_ge_zero [simp]: "0 \<le> norm x"
hoelzl@51002
   455
proof -
hoelzl@51002
   456
  have "0 = norm (x + -1 *\<^sub>R x)" 
hoelzl@51002
   457
    using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
hoelzl@51002
   458
  also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
hoelzl@51002
   459
  finally show ?thesis by simp
hoelzl@51002
   460
qed
hoelzl@51002
   461
hoelzl@51002
   462
end
huffman@20504
   463
haftmann@24588
   464
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   465
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   466
haftmann@24588
   467
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   468
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   469
haftmann@24588
   470
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   471
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   472
haftmann@24588
   473
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   474
huffman@22852
   475
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   476
proof
huffman@20554
   477
  fix x y :: 'a
huffman@20554
   478
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   479
    by (simp add: norm_mult)
huffman@22852
   480
next
huffman@22852
   481
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   482
    by (rule norm_mult)
huffman@22852
   483
  thus "norm (1::'a) = 1" by simp
huffman@20554
   484
qed
huffman@20554
   485
huffman@22852
   486
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   487
by simp
huffman@20504
   488
huffman@22852
   489
lemma zero_less_norm_iff [simp]:
huffman@22852
   490
  fixes x :: "'a::real_normed_vector"
huffman@22852
   491
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   492
by (simp add: order_less_le)
huffman@20504
   493
huffman@22852
   494
lemma norm_not_less_zero [simp]:
huffman@22852
   495
  fixes x :: "'a::real_normed_vector"
huffman@22852
   496
  shows "\<not> norm x < 0"
huffman@20828
   497
by (simp add: linorder_not_less)
huffman@20828
   498
huffman@22852
   499
lemma norm_le_zero_iff [simp]:
huffman@22852
   500
  fixes x :: "'a::real_normed_vector"
huffman@22852
   501
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   502
by (simp add: order_le_less)
huffman@20828
   503
huffman@20504
   504
lemma norm_minus_cancel [simp]:
huffman@20584
   505
  fixes x :: "'a::real_normed_vector"
huffman@20584
   506
  shows "norm (- x) = norm x"
huffman@20504
   507
proof -
huffman@21809
   508
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   509
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   510
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   511
    by (rule norm_scaleR)
huffman@20504
   512
  finally show ?thesis by simp
huffman@20504
   513
qed
huffman@20504
   514
huffman@20504
   515
lemma norm_minus_commute:
huffman@20584
   516
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   517
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   518
proof -
huffman@22898
   519
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   520
    by (rule norm_minus_cancel)
huffman@22898
   521
  thus ?thesis by simp
huffman@20504
   522
qed
huffman@20504
   523
huffman@20504
   524
lemma norm_triangle_ineq2:
huffman@20584
   525
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   526
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   527
proof -
huffman@20533
   528
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   529
    by (rule norm_triangle_ineq)
huffman@22898
   530
  thus ?thesis by simp
huffman@20504
   531
qed
huffman@20504
   532
huffman@20584
   533
lemma norm_triangle_ineq3:
huffman@20584
   534
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   535
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   536
apply (subst abs_le_iff)
huffman@20584
   537
apply auto
huffman@20584
   538
apply (rule norm_triangle_ineq2)
huffman@20584
   539
apply (subst norm_minus_commute)
huffman@20584
   540
apply (rule norm_triangle_ineq2)
huffman@20584
   541
done
huffman@20584
   542
huffman@20504
   543
lemma norm_triangle_ineq4:
huffman@20584
   544
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   545
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   546
proof -
huffman@22898
   547
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   548
    by (rule norm_triangle_ineq)
huffman@22898
   549
  thus ?thesis
huffman@22898
   550
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   551
qed
huffman@22898
   552
huffman@22898
   553
lemma norm_diff_ineq:
huffman@22898
   554
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   555
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   556
proof -
huffman@22898
   557
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   558
    by (rule norm_triangle_ineq2)
huffman@22898
   559
  thus ?thesis by simp
huffman@20504
   560
qed
huffman@20504
   561
huffman@20551
   562
lemma norm_diff_triangle_ineq:
huffman@20551
   563
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   564
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   565
proof -
huffman@20551
   566
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   567
    by (simp add: diff_minus add_ac)
huffman@20551
   568
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   569
    by (rule norm_triangle_ineq)
huffman@20551
   570
  finally show ?thesis .
huffman@20551
   571
qed
huffman@20551
   572
huffman@22857
   573
lemma abs_norm_cancel [simp]:
huffman@22857
   574
  fixes a :: "'a::real_normed_vector"
huffman@22857
   575
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   576
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   577
huffman@22880
   578
lemma norm_add_less:
huffman@22880
   579
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   580
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   581
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   582
huffman@22880
   583
lemma norm_mult_less:
huffman@22880
   584
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   585
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   586
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   587
apply (simp add: mult_strict_mono')
huffman@22880
   588
done
huffman@22880
   589
huffman@22857
   590
lemma norm_of_real [simp]:
huffman@22857
   591
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@31586
   592
unfolding of_real_def by simp
huffman@20560
   593
huffman@47108
   594
lemma norm_numeral [simp]:
huffman@47108
   595
  "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
huffman@47108
   596
by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
huffman@47108
   597
huffman@47108
   598
lemma norm_neg_numeral [simp]:
huffman@47108
   599
  "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w"
huffman@47108
   600
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
huffman@22876
   601
huffman@22876
   602
lemma norm_of_int [simp]:
huffman@22876
   603
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   604
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   605
huffman@22876
   606
lemma norm_of_nat [simp]:
huffman@22876
   607
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   608
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   609
apply (subst norm_of_real, simp)
huffman@22876
   610
done
huffman@22876
   611
huffman@20504
   612
lemma nonzero_norm_inverse:
huffman@20504
   613
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   614
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   615
apply (rule inverse_unique [symmetric])
huffman@20504
   616
apply (simp add: norm_mult [symmetric])
huffman@20504
   617
done
huffman@20504
   618
huffman@20504
   619
lemma norm_inverse:
haftmann@36409
   620
  fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
huffman@20533
   621
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   622
apply (case_tac "a = 0", simp)
huffman@20504
   623
apply (erule nonzero_norm_inverse)
huffman@20504
   624
done
huffman@20504
   625
huffman@20584
   626
lemma nonzero_norm_divide:
huffman@20584
   627
  fixes a b :: "'a::real_normed_field"
huffman@20584
   628
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   629
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   630
huffman@20584
   631
lemma norm_divide:
haftmann@36409
   632
  fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
huffman@20584
   633
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   634
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   635
huffman@22852
   636
lemma norm_power_ineq:
haftmann@31017
   637
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@22852
   638
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   639
proof (induct n)
huffman@22852
   640
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   641
next
huffman@22852
   642
  case (Suc n)
huffman@22852
   643
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   644
    by (rule norm_mult_ineq)
huffman@22852
   645
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   646
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   647
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@30273
   648
    by simp
huffman@22852
   649
qed
huffman@22852
   650
huffman@20684
   651
lemma norm_power:
haftmann@31017
   652
  fixes x :: "'a::{real_normed_div_algebra}"
huffman@20684
   653
  shows "norm (x ^ n) = norm x ^ n"
huffman@30273
   654
by (induct n) (simp_all add: norm_mult)
huffman@20684
   655
huffman@31289
   656
text {* Every normed vector space is a metric space. *}
huffman@31285
   657
huffman@31289
   658
instance real_normed_vector < metric_space
huffman@31289
   659
proof
huffman@31289
   660
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   661
    unfolding dist_norm by simp
huffman@31289
   662
next
huffman@31289
   663
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
huffman@31289
   664
    unfolding dist_norm
huffman@31289
   665
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
huffman@31289
   666
qed
huffman@31285
   667
huffman@31564
   668
subsection {* Class instances for real numbers *}
huffman@31564
   669
huffman@31564
   670
instantiation real :: real_normed_field
huffman@31564
   671
begin
huffman@31564
   672
huffman@31564
   673
definition real_norm_def [simp]:
huffman@31564
   674
  "norm r = \<bar>r\<bar>"
huffman@31564
   675
huffman@31564
   676
instance
huffman@31564
   677
apply (intro_classes, unfold real_norm_def real_scaleR_def)
huffman@31564
   678
apply (rule dist_real_def)
huffman@36795
   679
apply (simp add: sgn_real_def)
huffman@31564
   680
apply (rule abs_eq_0)
huffman@31564
   681
apply (rule abs_triangle_ineq)
huffman@31564
   682
apply (rule abs_mult)
huffman@31564
   683
apply (rule abs_mult)
huffman@31564
   684
done
huffman@31564
   685
huffman@31564
   686
end
huffman@31564
   687
hoelzl@51518
   688
instance real :: linear_continuum_topology ..
hoelzl@51518
   689
huffman@31446
   690
subsection {* Extra type constraints *}
huffman@31446
   691
huffman@31492
   692
text {* Only allow @{term "open"} in class @{text topological_space}. *}
huffman@31492
   693
huffman@31492
   694
setup {* Sign.add_const_constraint
huffman@31492
   695
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
huffman@31492
   696
huffman@31446
   697
text {* Only allow @{term dist} in class @{text metric_space}. *}
huffman@31446
   698
huffman@31446
   699
setup {* Sign.add_const_constraint
huffman@31446
   700
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
huffman@31446
   701
huffman@31446
   702
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
huffman@31446
   703
huffman@31446
   704
setup {* Sign.add_const_constraint
huffman@31446
   705
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
huffman@31446
   706
huffman@22972
   707
subsection {* Sign function *}
huffman@22972
   708
nipkow@24506
   709
lemma norm_sgn:
nipkow@24506
   710
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
huffman@31586
   711
by (simp add: sgn_div_norm)
huffman@22972
   712
nipkow@24506
   713
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   714
by (simp add: sgn_div_norm)
huffman@22972
   715
nipkow@24506
   716
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   717
by (simp add: sgn_div_norm)
huffman@22972
   718
nipkow@24506
   719
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   720
by (simp add: sgn_div_norm)
huffman@22972
   721
nipkow@24506
   722
lemma sgn_scaleR:
nipkow@24506
   723
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
huffman@31586
   724
by (simp add: sgn_div_norm mult_ac)
huffman@22973
   725
huffman@22972
   726
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   727
by (simp add: sgn_div_norm)
huffman@22972
   728
huffman@22972
   729
lemma sgn_of_real:
huffman@22972
   730
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   731
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   732
huffman@22973
   733
lemma sgn_mult:
huffman@22973
   734
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   735
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   736
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   737
huffman@22972
   738
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   739
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   740
huffman@22972
   741
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   742
unfolding real_sgn_eq by simp
huffman@22972
   743
huffman@22972
   744
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   745
unfolding real_sgn_eq by simp
huffman@22972
   746
hoelzl@51474
   747
lemma norm_conv_dist: "norm x = dist x 0"
hoelzl@51474
   748
  unfolding dist_norm by simp
huffman@22972
   749
huffman@22442
   750
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   751
wenzelm@46868
   752
locale bounded_linear = additive f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
huffman@22442
   753
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   754
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
   755
begin
huffman@22442
   756
huffman@27443
   757
lemma pos_bounded:
huffman@22442
   758
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   759
proof -
huffman@22442
   760
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   761
    using bounded by fast
huffman@22442
   762
  show ?thesis
huffman@22442
   763
  proof (intro exI impI conjI allI)
huffman@22442
   764
    show "0 < max 1 K"
huffman@22442
   765
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   766
  next
huffman@22442
   767
    fix x
huffman@22442
   768
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   769
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   770
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   771
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   772
  qed
huffman@22442
   773
qed
huffman@22442
   774
huffman@27443
   775
lemma nonneg_bounded:
huffman@22442
   776
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   777
proof -
huffman@22442
   778
  from pos_bounded
huffman@22442
   779
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   780
qed
huffman@22442
   781
huffman@27443
   782
end
huffman@27443
   783
huffman@44127
   784
lemma bounded_linear_intro:
huffman@44127
   785
  assumes "\<And>x y. f (x + y) = f x + f y"
huffman@44127
   786
  assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
huffman@44127
   787
  assumes "\<And>x. norm (f x) \<le> norm x * K"
huffman@44127
   788
  shows "bounded_linear f"
huffman@44127
   789
  by default (fast intro: assms)+
huffman@44127
   790
huffman@22442
   791
locale bounded_bilinear =
huffman@22442
   792
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   793
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   794
    (infixl "**" 70)
huffman@22442
   795
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   796
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   797
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   798
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   799
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
   800
begin
huffman@22442
   801
huffman@27443
   802
lemma pos_bounded:
huffman@22442
   803
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   804
apply (cut_tac bounded, erule exE)
huffman@22442
   805
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   806
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   807
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   808
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   809
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   810
done
huffman@22442
   811
huffman@27443
   812
lemma nonneg_bounded:
huffman@22442
   813
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   814
proof -
huffman@22442
   815
  from pos_bounded
huffman@22442
   816
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   817
qed
huffman@22442
   818
huffman@27443
   819
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
   820
by (rule additive.intro, rule add_right)
huffman@22442
   821
huffman@27443
   822
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
   823
by (rule additive.intro, rule add_left)
huffman@22442
   824
huffman@27443
   825
lemma zero_left: "prod 0 b = 0"
huffman@22442
   826
by (rule additive.zero [OF additive_left])
huffman@22442
   827
huffman@27443
   828
lemma zero_right: "prod a 0 = 0"
huffman@22442
   829
by (rule additive.zero [OF additive_right])
huffman@22442
   830
huffman@27443
   831
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
   832
by (rule additive.minus [OF additive_left])
huffman@22442
   833
huffman@27443
   834
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
   835
by (rule additive.minus [OF additive_right])
huffman@22442
   836
huffman@27443
   837
lemma diff_left:
huffman@22442
   838
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
   839
by (rule additive.diff [OF additive_left])
huffman@22442
   840
huffman@27443
   841
lemma diff_right:
huffman@22442
   842
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
   843
by (rule additive.diff [OF additive_right])
huffman@22442
   844
huffman@27443
   845
lemma bounded_linear_left:
huffman@22442
   846
  "bounded_linear (\<lambda>a. a ** b)"
huffman@44127
   847
apply (cut_tac bounded, safe)
huffman@44127
   848
apply (rule_tac K="norm b * K" in bounded_linear_intro)
huffman@22442
   849
apply (rule add_left)
huffman@22442
   850
apply (rule scaleR_left)
huffman@22442
   851
apply (simp add: mult_ac)
huffman@22442
   852
done
huffman@22442
   853
huffman@27443
   854
lemma bounded_linear_right:
huffman@22442
   855
  "bounded_linear (\<lambda>b. a ** b)"
huffman@44127
   856
apply (cut_tac bounded, safe)
huffman@44127
   857
apply (rule_tac K="norm a * K" in bounded_linear_intro)
huffman@22442
   858
apply (rule add_right)
huffman@22442
   859
apply (rule scaleR_right)
huffman@22442
   860
apply (simp add: mult_ac)
huffman@22442
   861
done
huffman@22442
   862
huffman@27443
   863
lemma prod_diff_prod:
huffman@22442
   864
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
   865
by (simp add: diff_left diff_right)
huffman@22442
   866
huffman@27443
   867
end
huffman@27443
   868
huffman@44282
   869
lemma bounded_bilinear_mult:
huffman@44282
   870
  "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
huffman@22442
   871
apply (rule bounded_bilinear.intro)
webertj@49962
   872
apply (rule distrib_right)
webertj@49962
   873
apply (rule distrib_left)
huffman@22442
   874
apply (rule mult_scaleR_left)
huffman@22442
   875
apply (rule mult_scaleR_right)
huffman@22442
   876
apply (rule_tac x="1" in exI)
huffman@22442
   877
apply (simp add: norm_mult_ineq)
huffman@22442
   878
done
huffman@22442
   879
huffman@44282
   880
lemma bounded_linear_mult_left:
huffman@44282
   881
  "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
huffman@44282
   882
  using bounded_bilinear_mult
huffman@44282
   883
  by (rule bounded_bilinear.bounded_linear_left)
huffman@22442
   884
huffman@44282
   885
lemma bounded_linear_mult_right:
huffman@44282
   886
  "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
huffman@44282
   887
  using bounded_bilinear_mult
huffman@44282
   888
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
   889
huffman@44282
   890
lemma bounded_linear_divide:
huffman@44282
   891
  "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
huffman@44282
   892
  unfolding divide_inverse by (rule bounded_linear_mult_left)
huffman@23120
   893
huffman@44282
   894
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
huffman@22442
   895
apply (rule bounded_bilinear.intro)
huffman@22442
   896
apply (rule scaleR_left_distrib)
huffman@22442
   897
apply (rule scaleR_right_distrib)
huffman@22973
   898
apply simp
huffman@22442
   899
apply (rule scaleR_left_commute)
huffman@31586
   900
apply (rule_tac x="1" in exI, simp)
huffman@22442
   901
done
huffman@22442
   902
huffman@44282
   903
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
huffman@44282
   904
  using bounded_bilinear_scaleR
huffman@44282
   905
  by (rule bounded_bilinear.bounded_linear_left)
huffman@23127
   906
huffman@44282
   907
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
huffman@44282
   908
  using bounded_bilinear_scaleR
huffman@44282
   909
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
   910
huffman@44282
   911
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
huffman@44282
   912
  unfolding of_real_def by (rule bounded_linear_scaleR_left)
huffman@22625
   913
huffman@44571
   914
instance real_normed_algebra_1 \<subseteq> perfect_space
huffman@44571
   915
proof
huffman@44571
   916
  fix x::'a
huffman@44571
   917
  show "\<not> open {x}"
huffman@44571
   918
    unfolding open_dist dist_norm
huffman@44571
   919
    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
huffman@44571
   920
qed
huffman@44571
   921
huffman@20504
   922
end