src/HOL/RealVector.thy
author wenzelm
Sun Mar 11 13:54:08 2012 +0100 (2012-03-11)
changeset 46868 6c250adbe101
parent 44937 22c0857b8aab
child 47108 2a1953f0d20d
permissions -rw-r--r--
eliminated old-fashioned 'constrains' element;
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(*  Title:      HOL/RealVector.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 RealVector
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imports RComplete
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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_number_of_eq:
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  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
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by (simp add: number_of_eq)
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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 ..
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subsection {* The Set of Real Numbers *}
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definition Reals :: "'a::real_algebra_1 set" where
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  "Reals = range of_real"
huffman@20554
   325
wenzelm@21210
   326
notation (xsymbols)
huffman@20554
   327
  Reals  ("\<real>")
huffman@20554
   328
huffman@21809
   329
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
huffman@20554
   330
by (simp add: Reals_def)
huffman@20554
   331
huffman@21809
   332
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
huffman@21809
   333
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
huffman@20718
   334
huffman@21809
   335
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
huffman@21809
   336
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
huffman@21809
   337
huffman@21809
   338
lemma Reals_number_of [simp]:
huffman@21809
   339
  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
huffman@21809
   340
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
huffman@20718
   341
huffman@20554
   342
lemma Reals_0 [simp]: "0 \<in> Reals"
huffman@20554
   343
apply (unfold Reals_def)
huffman@20554
   344
apply (rule range_eqI)
huffman@20554
   345
apply (rule of_real_0 [symmetric])
huffman@20554
   346
done
huffman@20554
   347
huffman@20554
   348
lemma Reals_1 [simp]: "1 \<in> Reals"
huffman@20554
   349
apply (unfold Reals_def)
huffman@20554
   350
apply (rule range_eqI)
huffman@20554
   351
apply (rule of_real_1 [symmetric])
huffman@20554
   352
done
huffman@20554
   353
huffman@20584
   354
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
huffman@20554
   355
apply (auto simp add: Reals_def)
huffman@20554
   356
apply (rule range_eqI)
huffman@20554
   357
apply (rule of_real_add [symmetric])
huffman@20554
   358
done
huffman@20554
   359
huffman@20584
   360
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
huffman@20584
   361
apply (auto simp add: Reals_def)
huffman@20584
   362
apply (rule range_eqI)
huffman@20584
   363
apply (rule of_real_minus [symmetric])
huffman@20584
   364
done
huffman@20584
   365
huffman@20584
   366
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
huffman@20584
   367
apply (auto simp add: Reals_def)
huffman@20584
   368
apply (rule range_eqI)
huffman@20584
   369
apply (rule of_real_diff [symmetric])
huffman@20584
   370
done
huffman@20584
   371
huffman@20584
   372
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
huffman@20554
   373
apply (auto simp add: Reals_def)
huffman@20554
   374
apply (rule range_eqI)
huffman@20554
   375
apply (rule of_real_mult [symmetric])
huffman@20554
   376
done
huffman@20554
   377
huffman@20584
   378
lemma nonzero_Reals_inverse:
huffman@20584
   379
  fixes a :: "'a::real_div_algebra"
huffman@20584
   380
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   381
apply (auto simp add: Reals_def)
huffman@20584
   382
apply (rule range_eqI)
huffman@20584
   383
apply (erule nonzero_of_real_inverse [symmetric])
huffman@20584
   384
done
huffman@20584
   385
huffman@20584
   386
lemma Reals_inverse [simp]:
haftmann@36409
   387
  fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
huffman@20584
   388
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   389
apply (auto simp add: Reals_def)
huffman@20584
   390
apply (rule range_eqI)
huffman@20584
   391
apply (rule of_real_inverse [symmetric])
huffman@20584
   392
done
huffman@20584
   393
huffman@20584
   394
lemma nonzero_Reals_divide:
huffman@20584
   395
  fixes a b :: "'a::real_field"
huffman@20584
   396
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   397
apply (auto simp add: Reals_def)
huffman@20584
   398
apply (rule range_eqI)
huffman@20584
   399
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   400
done
huffman@20584
   401
huffman@20584
   402
lemma Reals_divide [simp]:
haftmann@36409
   403
  fixes a b :: "'a::{real_field, field_inverse_zero}"
huffman@20584
   404
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   405
apply (auto simp add: Reals_def)
huffman@20584
   406
apply (rule range_eqI)
huffman@20584
   407
apply (rule of_real_divide [symmetric])
huffman@20584
   408
done
huffman@20584
   409
huffman@20722
   410
lemma Reals_power [simp]:
haftmann@31017
   411
  fixes a :: "'a::{real_algebra_1}"
huffman@20722
   412
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   413
apply (auto simp add: Reals_def)
huffman@20722
   414
apply (rule range_eqI)
huffman@20722
   415
apply (rule of_real_power [symmetric])
huffman@20722
   416
done
huffman@20722
   417
huffman@20554
   418
lemma Reals_cases [cases set: Reals]:
huffman@20554
   419
  assumes "q \<in> \<real>"
huffman@20554
   420
  obtains (of_real) r where "q = of_real r"
huffman@20554
   421
  unfolding Reals_def
huffman@20554
   422
proof -
huffman@20554
   423
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   424
  then obtain r where "q = of_real r" ..
huffman@20554
   425
  then show thesis ..
huffman@20554
   426
qed
huffman@20554
   427
huffman@20554
   428
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   429
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   430
  by (rule Reals_cases) auto
huffman@20554
   431
huffman@20504
   432
huffman@31413
   433
subsection {* Topological spaces *}
huffman@31413
   434
huffman@31492
   435
class "open" =
huffman@31494
   436
  fixes "open" :: "'a set \<Rightarrow> bool"
huffman@31490
   437
huffman@31492
   438
class topological_space = "open" +
huffman@31492
   439
  assumes open_UNIV [simp, intro]: "open UNIV"
huffman@31492
   440
  assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
huffman@31492
   441
  assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
huffman@31490
   442
begin
huffman@31490
   443
huffman@31490
   444
definition
huffman@31490
   445
  closed :: "'a set \<Rightarrow> bool" where
huffman@31490
   446
  "closed S \<longleftrightarrow> open (- S)"
huffman@31490
   447
huffman@31490
   448
lemma open_empty [intro, simp]: "open {}"
huffman@31490
   449
  using open_Union [of "{}"] by simp
huffman@31490
   450
huffman@31490
   451
lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
huffman@31490
   452
  using open_Union [of "{S, T}"] by simp
huffman@31490
   453
huffman@31490
   454
lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
hoelzl@44937
   455
  unfolding SUP_def by (rule open_Union) auto
hoelzl@44937
   456
hoelzl@44937
   457
lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
hoelzl@44937
   458
  by (induct set: finite) auto
huffman@31490
   459
huffman@31490
   460
lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
hoelzl@44937
   461
  unfolding INF_def by (rule open_Inter) auto
huffman@31490
   462
huffman@31490
   463
lemma closed_empty [intro, simp]:  "closed {}"
huffman@31490
   464
  unfolding closed_def by simp
huffman@31490
   465
huffman@31490
   466
lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
huffman@31490
   467
  unfolding closed_def by auto
huffman@31490
   468
huffman@31490
   469
lemma closed_UNIV [intro, simp]: "closed UNIV"
huffman@31490
   470
  unfolding closed_def by simp
huffman@31490
   471
huffman@31490
   472
lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
huffman@31490
   473
  unfolding closed_def by auto
huffman@31490
   474
huffman@31490
   475
lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
huffman@31490
   476
  unfolding closed_def by auto
huffman@31490
   477
hoelzl@44937
   478
lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
hoelzl@44937
   479
  unfolding closed_def uminus_Inf by auto
hoelzl@44937
   480
hoelzl@44937
   481
lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
huffman@31490
   482
  by (induct set: finite) auto
huffman@31490
   483
hoelzl@44937
   484
lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
hoelzl@44937
   485
  unfolding SUP_def by (rule closed_Union) auto
huffman@31490
   486
huffman@31490
   487
lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
huffman@31490
   488
  unfolding closed_def by simp
huffman@31490
   489
huffman@31490
   490
lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
huffman@31490
   491
  unfolding closed_def by simp
huffman@31490
   492
huffman@31490
   493
lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
huffman@31490
   494
  unfolding closed_open Diff_eq by (rule open_Int)
huffman@31490
   495
huffman@31490
   496
lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
huffman@31490
   497
  unfolding open_closed Diff_eq by (rule closed_Int)
huffman@31490
   498
huffman@31490
   499
lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
huffman@31490
   500
  unfolding closed_open .
huffman@31490
   501
huffman@31490
   502
lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
huffman@31490
   503
  unfolding open_closed .
huffman@31490
   504
huffman@31490
   505
end
huffman@31413
   506
huffman@31413
   507
huffman@31289
   508
subsection {* Metric spaces *}
huffman@31289
   509
huffman@31289
   510
class dist =
huffman@31289
   511
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
huffman@31289
   512
huffman@31492
   513
class open_dist = "open" + dist +
huffman@31492
   514
  assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31413
   515
huffman@31492
   516
class metric_space = open_dist +
huffman@31289
   517
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   518
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
huffman@31289
   519
begin
huffman@31289
   520
huffman@31289
   521
lemma dist_self [simp]: "dist x x = 0"
huffman@31289
   522
by simp
huffman@31289
   523
huffman@31289
   524
lemma zero_le_dist [simp]: "0 \<le> dist x y"
huffman@31289
   525
using dist_triangle2 [of x x y] by simp
huffman@31289
   526
huffman@31289
   527
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
huffman@31289
   528
by (simp add: less_le)
huffman@31289
   529
huffman@31289
   530
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
huffman@31289
   531
by (simp add: not_less)
huffman@31289
   532
huffman@31289
   533
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
huffman@31289
   534
by (simp add: le_less)
huffman@31289
   535
huffman@31289
   536
lemma dist_commute: "dist x y = dist y x"
huffman@31289
   537
proof (rule order_antisym)
huffman@31289
   538
  show "dist x y \<le> dist y x"
huffman@31289
   539
    using dist_triangle2 [of x y x] by simp
huffman@31289
   540
  show "dist y x \<le> dist x y"
huffman@31289
   541
    using dist_triangle2 [of y x y] by simp
huffman@31289
   542
qed
huffman@31289
   543
huffman@31289
   544
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
huffman@31289
   545
using dist_triangle2 [of x z y] by (simp add: dist_commute)
huffman@31289
   546
huffman@31565
   547
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
huffman@31565
   548
using dist_triangle2 [of x y a] by (simp add: dist_commute)
huffman@31565
   549
hoelzl@41969
   550
lemma dist_triangle_alt:
hoelzl@41969
   551
  shows "dist y z <= dist x y + dist x z"
hoelzl@41969
   552
by (rule dist_triangle3)
hoelzl@41969
   553
hoelzl@41969
   554
lemma dist_pos_lt:
hoelzl@41969
   555
  shows "x \<noteq> y ==> 0 < dist x y"
hoelzl@41969
   556
by (simp add: zero_less_dist_iff)
hoelzl@41969
   557
hoelzl@41969
   558
lemma dist_nz:
hoelzl@41969
   559
  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
hoelzl@41969
   560
by (simp add: zero_less_dist_iff)
hoelzl@41969
   561
hoelzl@41969
   562
lemma dist_triangle_le:
hoelzl@41969
   563
  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
hoelzl@41969
   564
by (rule order_trans [OF dist_triangle2])
hoelzl@41969
   565
hoelzl@41969
   566
lemma dist_triangle_lt:
hoelzl@41969
   567
  shows "dist x z + dist y z < e ==> dist x y < e"
hoelzl@41969
   568
by (rule le_less_trans [OF dist_triangle2])
hoelzl@41969
   569
hoelzl@41969
   570
lemma dist_triangle_half_l:
hoelzl@41969
   571
  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
hoelzl@41969
   572
by (rule dist_triangle_lt [where z=y], simp)
hoelzl@41969
   573
hoelzl@41969
   574
lemma dist_triangle_half_r:
hoelzl@41969
   575
  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
hoelzl@41969
   576
by (rule dist_triangle_half_l, simp_all add: dist_commute)
hoelzl@41969
   577
huffman@31413
   578
subclass topological_space
huffman@31413
   579
proof
huffman@31413
   580
  have "\<exists>e::real. 0 < e"
huffman@31413
   581
    by (fast intro: zero_less_one)
huffman@31492
   582
  then show "open UNIV"
huffman@31492
   583
    unfolding open_dist by simp
huffman@31413
   584
next
huffman@31492
   585
  fix S T assume "open S" "open T"
huffman@31492
   586
  then show "open (S \<inter> T)"
huffman@31492
   587
    unfolding open_dist
huffman@31413
   588
    apply clarify
huffman@31413
   589
    apply (drule (1) bspec)+
huffman@31413
   590
    apply (clarify, rename_tac r s)
huffman@31413
   591
    apply (rule_tac x="min r s" in exI, simp)
huffman@31413
   592
    done
huffman@31413
   593
next
huffman@31492
   594
  fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
huffman@31492
   595
    unfolding open_dist by fast
huffman@31413
   596
qed
huffman@31413
   597
hoelzl@41969
   598
lemma (in metric_space) open_ball: "open {y. dist x y < d}"
hoelzl@41969
   599
proof (unfold open_dist, intro ballI)
hoelzl@41969
   600
  fix y assume *: "y \<in> {y. dist x y < d}"
hoelzl@41969
   601
  then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
hoelzl@41969
   602
    by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
hoelzl@41969
   603
qed
hoelzl@41969
   604
huffman@31289
   605
end
huffman@31289
   606
huffman@31289
   607
huffman@20504
   608
subsection {* Real normed vector spaces *}
huffman@20504
   609
haftmann@29608
   610
class norm =
huffman@22636
   611
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   612
huffman@24520
   613
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   614
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   615
huffman@31289
   616
class dist_norm = dist + norm + minus +
huffman@31289
   617
  assumes dist_norm: "dist x y = norm (x - y)"
huffman@31289
   618
huffman@31492
   619
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
haftmann@24588
   620
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
haftmann@25062
   621
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   622
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@31586
   623
  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@20504
   624
haftmann@24588
   625
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   626
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   627
haftmann@24588
   628
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   629
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   630
haftmann@24588
   631
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   632
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   633
haftmann@24588
   634
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   635
huffman@22852
   636
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   637
proof
huffman@20554
   638
  fix x y :: 'a
huffman@20554
   639
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   640
    by (simp add: norm_mult)
huffman@22852
   641
next
huffman@22852
   642
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   643
    by (rule norm_mult)
huffman@22852
   644
  thus "norm (1::'a) = 1" by simp
huffman@20554
   645
qed
huffman@20554
   646
huffman@22852
   647
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   648
by simp
huffman@20504
   649
huffman@22852
   650
lemma zero_less_norm_iff [simp]:
huffman@22852
   651
  fixes x :: "'a::real_normed_vector"
huffman@22852
   652
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   653
by (simp add: order_less_le)
huffman@20504
   654
huffman@22852
   655
lemma norm_not_less_zero [simp]:
huffman@22852
   656
  fixes x :: "'a::real_normed_vector"
huffman@22852
   657
  shows "\<not> norm x < 0"
huffman@20828
   658
by (simp add: linorder_not_less)
huffman@20828
   659
huffman@22852
   660
lemma norm_le_zero_iff [simp]:
huffman@22852
   661
  fixes x :: "'a::real_normed_vector"
huffman@22852
   662
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   663
by (simp add: order_le_less)
huffman@20828
   664
huffman@20504
   665
lemma norm_minus_cancel [simp]:
huffman@20584
   666
  fixes x :: "'a::real_normed_vector"
huffman@20584
   667
  shows "norm (- x) = norm x"
huffman@20504
   668
proof -
huffman@21809
   669
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   670
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   671
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   672
    by (rule norm_scaleR)
huffman@20504
   673
  finally show ?thesis by simp
huffman@20504
   674
qed
huffman@20504
   675
huffman@20504
   676
lemma norm_minus_commute:
huffman@20584
   677
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   678
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   679
proof -
huffman@22898
   680
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   681
    by (rule norm_minus_cancel)
huffman@22898
   682
  thus ?thesis by simp
huffman@20504
   683
qed
huffman@20504
   684
huffman@20504
   685
lemma norm_triangle_ineq2:
huffman@20584
   686
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   687
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   688
proof -
huffman@20533
   689
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   690
    by (rule norm_triangle_ineq)
huffman@22898
   691
  thus ?thesis by simp
huffman@20504
   692
qed
huffman@20504
   693
huffman@20584
   694
lemma norm_triangle_ineq3:
huffman@20584
   695
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   696
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   697
apply (subst abs_le_iff)
huffman@20584
   698
apply auto
huffman@20584
   699
apply (rule norm_triangle_ineq2)
huffman@20584
   700
apply (subst norm_minus_commute)
huffman@20584
   701
apply (rule norm_triangle_ineq2)
huffman@20584
   702
done
huffman@20584
   703
huffman@20504
   704
lemma norm_triangle_ineq4:
huffman@20584
   705
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   706
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   707
proof -
huffman@22898
   708
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   709
    by (rule norm_triangle_ineq)
huffman@22898
   710
  thus ?thesis
huffman@22898
   711
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   712
qed
huffman@22898
   713
huffman@22898
   714
lemma norm_diff_ineq:
huffman@22898
   715
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   716
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   717
proof -
huffman@22898
   718
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   719
    by (rule norm_triangle_ineq2)
huffman@22898
   720
  thus ?thesis by simp
huffman@20504
   721
qed
huffman@20504
   722
huffman@20551
   723
lemma norm_diff_triangle_ineq:
huffman@20551
   724
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   725
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   726
proof -
huffman@20551
   727
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   728
    by (simp add: diff_minus add_ac)
huffman@20551
   729
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   730
    by (rule norm_triangle_ineq)
huffman@20551
   731
  finally show ?thesis .
huffman@20551
   732
qed
huffman@20551
   733
huffman@22857
   734
lemma abs_norm_cancel [simp]:
huffman@22857
   735
  fixes a :: "'a::real_normed_vector"
huffman@22857
   736
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   737
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   738
huffman@22880
   739
lemma norm_add_less:
huffman@22880
   740
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   741
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   742
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   743
huffman@22880
   744
lemma norm_mult_less:
huffman@22880
   745
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   746
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   747
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   748
apply (simp add: mult_strict_mono')
huffman@22880
   749
done
huffman@22880
   750
huffman@22857
   751
lemma norm_of_real [simp]:
huffman@22857
   752
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@31586
   753
unfolding of_real_def by simp
huffman@20560
   754
huffman@22876
   755
lemma norm_number_of [simp]:
huffman@22876
   756
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
huffman@22876
   757
    = \<bar>number_of w\<bar>"
huffman@22876
   758
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
huffman@22876
   759
huffman@22876
   760
lemma norm_of_int [simp]:
huffman@22876
   761
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   762
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   763
huffman@22876
   764
lemma norm_of_nat [simp]:
huffman@22876
   765
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   766
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   767
apply (subst norm_of_real, simp)
huffman@22876
   768
done
huffman@22876
   769
huffman@20504
   770
lemma nonzero_norm_inverse:
huffman@20504
   771
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   772
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   773
apply (rule inverse_unique [symmetric])
huffman@20504
   774
apply (simp add: norm_mult [symmetric])
huffman@20504
   775
done
huffman@20504
   776
huffman@20504
   777
lemma norm_inverse:
haftmann@36409
   778
  fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
huffman@20533
   779
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   780
apply (case_tac "a = 0", simp)
huffman@20504
   781
apply (erule nonzero_norm_inverse)
huffman@20504
   782
done
huffman@20504
   783
huffman@20584
   784
lemma nonzero_norm_divide:
huffman@20584
   785
  fixes a b :: "'a::real_normed_field"
huffman@20584
   786
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   787
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   788
huffman@20584
   789
lemma norm_divide:
haftmann@36409
   790
  fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
huffman@20584
   791
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   792
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   793
huffman@22852
   794
lemma norm_power_ineq:
haftmann@31017
   795
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@22852
   796
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   797
proof (induct n)
huffman@22852
   798
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   799
next
huffman@22852
   800
  case (Suc n)
huffman@22852
   801
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   802
    by (rule norm_mult_ineq)
huffman@22852
   803
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   804
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   805
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@30273
   806
    by simp
huffman@22852
   807
qed
huffman@22852
   808
huffman@20684
   809
lemma norm_power:
haftmann@31017
   810
  fixes x :: "'a::{real_normed_div_algebra}"
huffman@20684
   811
  shows "norm (x ^ n) = norm x ^ n"
huffman@30273
   812
by (induct n) (simp_all add: norm_mult)
huffman@20684
   813
huffman@31289
   814
text {* Every normed vector space is a metric space. *}
huffman@31285
   815
huffman@31289
   816
instance real_normed_vector < metric_space
huffman@31289
   817
proof
huffman@31289
   818
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   819
    unfolding dist_norm by simp
huffman@31289
   820
next
huffman@31289
   821
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
huffman@31289
   822
    unfolding dist_norm
huffman@31289
   823
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
huffman@31289
   824
qed
huffman@31285
   825
huffman@31564
   826
huffman@31564
   827
subsection {* Class instances for real numbers *}
huffman@31564
   828
huffman@31564
   829
instantiation real :: real_normed_field
huffman@31564
   830
begin
huffman@31564
   831
huffman@31564
   832
definition real_norm_def [simp]:
huffman@31564
   833
  "norm r = \<bar>r\<bar>"
huffman@31564
   834
huffman@31564
   835
definition dist_real_def:
huffman@31564
   836
  "dist x y = \<bar>x - y\<bar>"
huffman@31564
   837
haftmann@37767
   838
definition open_real_def:
huffman@31564
   839
  "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31564
   840
huffman@31564
   841
instance
huffman@31564
   842
apply (intro_classes, unfold real_norm_def real_scaleR_def)
huffman@31564
   843
apply (rule dist_real_def)
huffman@31564
   844
apply (rule open_real_def)
huffman@36795
   845
apply (simp add: sgn_real_def)
huffman@31564
   846
apply (rule abs_ge_zero)
huffman@31564
   847
apply (rule abs_eq_0)
huffman@31564
   848
apply (rule abs_triangle_ineq)
huffman@31564
   849
apply (rule abs_mult)
huffman@31564
   850
apply (rule abs_mult)
huffman@31564
   851
done
huffman@31564
   852
huffman@31564
   853
end
huffman@31564
   854
huffman@31564
   855
lemma open_real_lessThan [simp]:
huffman@31564
   856
  fixes a :: real shows "open {..<a}"
huffman@31564
   857
unfolding open_real_def dist_real_def
huffman@31564
   858
proof (clarify)
huffman@31564
   859
  fix x assume "x < a"
huffman@31564
   860
  hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
huffman@31564
   861
  thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
huffman@31564
   862
qed
huffman@31564
   863
huffman@31564
   864
lemma open_real_greaterThan [simp]:
huffman@31564
   865
  fixes a :: real shows "open {a<..}"
huffman@31564
   866
unfolding open_real_def dist_real_def
huffman@31564
   867
proof (clarify)
huffman@31564
   868
  fix x assume "a < x"
huffman@31564
   869
  hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
huffman@31564
   870
  thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
huffman@31564
   871
qed
huffman@31564
   872
huffman@31564
   873
lemma open_real_greaterThanLessThan [simp]:
huffman@31564
   874
  fixes a b :: real shows "open {a<..<b}"
huffman@31564
   875
proof -
huffman@31564
   876
  have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
huffman@31564
   877
  thus "open {a<..<b}" by (simp add: open_Int)
huffman@31564
   878
qed
huffman@31564
   879
huffman@31567
   880
lemma closed_real_atMost [simp]: 
huffman@31567
   881
  fixes a :: real shows "closed {..a}"
huffman@31567
   882
unfolding closed_open by simp
huffman@31567
   883
huffman@31567
   884
lemma closed_real_atLeast [simp]:
huffman@31567
   885
  fixes a :: real shows "closed {a..}"
huffman@31567
   886
unfolding closed_open by simp
huffman@31567
   887
huffman@31567
   888
lemma closed_real_atLeastAtMost [simp]:
huffman@31567
   889
  fixes a b :: real shows "closed {a..b}"
huffman@31567
   890
proof -
huffman@31567
   891
  have "{a..b} = {a..} \<inter> {..b}" by auto
huffman@31567
   892
  thus "closed {a..b}" by (simp add: closed_Int)
huffman@31567
   893
qed
huffman@31567
   894
huffman@31564
   895
huffman@31446
   896
subsection {* Extra type constraints *}
huffman@31446
   897
huffman@31492
   898
text {* Only allow @{term "open"} in class @{text topological_space}. *}
huffman@31492
   899
huffman@31492
   900
setup {* Sign.add_const_constraint
huffman@31492
   901
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
huffman@31492
   902
huffman@31446
   903
text {* Only allow @{term dist} in class @{text metric_space}. *}
huffman@31446
   904
huffman@31446
   905
setup {* Sign.add_const_constraint
huffman@31446
   906
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
huffman@31446
   907
huffman@31446
   908
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
huffman@31446
   909
huffman@31446
   910
setup {* Sign.add_const_constraint
huffman@31446
   911
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
huffman@31446
   912
huffman@31285
   913
huffman@22972
   914
subsection {* Sign function *}
huffman@22972
   915
nipkow@24506
   916
lemma norm_sgn:
nipkow@24506
   917
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
huffman@31586
   918
by (simp add: sgn_div_norm)
huffman@22972
   919
nipkow@24506
   920
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   921
by (simp add: sgn_div_norm)
huffman@22972
   922
nipkow@24506
   923
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   924
by (simp add: sgn_div_norm)
huffman@22972
   925
nipkow@24506
   926
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   927
by (simp add: sgn_div_norm)
huffman@22972
   928
nipkow@24506
   929
lemma sgn_scaleR:
nipkow@24506
   930
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
huffman@31586
   931
by (simp add: sgn_div_norm mult_ac)
huffman@22973
   932
huffman@22972
   933
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   934
by (simp add: sgn_div_norm)
huffman@22972
   935
huffman@22972
   936
lemma sgn_of_real:
huffman@22972
   937
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   938
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   939
huffman@22973
   940
lemma sgn_mult:
huffman@22973
   941
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   942
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   943
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   944
huffman@22972
   945
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   946
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   947
huffman@22972
   948
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   949
unfolding real_sgn_eq by simp
huffman@22972
   950
huffman@22972
   951
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   952
unfolding real_sgn_eq by simp
huffman@22972
   953
huffman@22972
   954
huffman@22442
   955
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   956
wenzelm@46868
   957
locale bounded_linear = additive f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
huffman@22442
   958
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   959
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
   960
begin
huffman@22442
   961
huffman@27443
   962
lemma pos_bounded:
huffman@22442
   963
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   964
proof -
huffman@22442
   965
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   966
    using bounded by fast
huffman@22442
   967
  show ?thesis
huffman@22442
   968
  proof (intro exI impI conjI allI)
huffman@22442
   969
    show "0 < max 1 K"
huffman@22442
   970
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   971
  next
huffman@22442
   972
    fix x
huffman@22442
   973
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   974
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   975
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   976
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   977
  qed
huffman@22442
   978
qed
huffman@22442
   979
huffman@27443
   980
lemma nonneg_bounded:
huffman@22442
   981
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   982
proof -
huffman@22442
   983
  from pos_bounded
huffman@22442
   984
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   985
qed
huffman@22442
   986
huffman@27443
   987
end
huffman@27443
   988
huffman@44127
   989
lemma bounded_linear_intro:
huffman@44127
   990
  assumes "\<And>x y. f (x + y) = f x + f y"
huffman@44127
   991
  assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
huffman@44127
   992
  assumes "\<And>x. norm (f x) \<le> norm x * K"
huffman@44127
   993
  shows "bounded_linear f"
huffman@44127
   994
  by default (fast intro: assms)+
huffman@44127
   995
huffman@22442
   996
locale bounded_bilinear =
huffman@22442
   997
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   998
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   999
    (infixl "**" 70)
huffman@22442
  1000
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
  1001
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
  1002
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
  1003
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
  1004
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
  1005
begin
huffman@22442
  1006
huffman@27443
  1007
lemma pos_bounded:
huffman@22442
  1008
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
  1009
apply (cut_tac bounded, erule exE)
huffman@22442
  1010
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
  1011
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
  1012
apply (drule spec, drule spec, erule order_trans)
huffman@22442
  1013
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
  1014
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
  1015
done
huffman@22442
  1016
huffman@27443
  1017
lemma nonneg_bounded:
huffman@22442
  1018
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
  1019
proof -
huffman@22442
  1020
  from pos_bounded
huffman@22442
  1021
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
  1022
qed
huffman@22442
  1023
huffman@27443
  1024
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
  1025
by (rule additive.intro, rule add_right)
huffman@22442
  1026
huffman@27443
  1027
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
  1028
by (rule additive.intro, rule add_left)
huffman@22442
  1029
huffman@27443
  1030
lemma zero_left: "prod 0 b = 0"
huffman@22442
  1031
by (rule additive.zero [OF additive_left])
huffman@22442
  1032
huffman@27443
  1033
lemma zero_right: "prod a 0 = 0"
huffman@22442
  1034
by (rule additive.zero [OF additive_right])
huffman@22442
  1035
huffman@27443
  1036
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
  1037
by (rule additive.minus [OF additive_left])
huffman@22442
  1038
huffman@27443
  1039
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
  1040
by (rule additive.minus [OF additive_right])
huffman@22442
  1041
huffman@27443
  1042
lemma diff_left:
huffman@22442
  1043
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
  1044
by (rule additive.diff [OF additive_left])
huffman@22442
  1045
huffman@27443
  1046
lemma diff_right:
huffman@22442
  1047
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
  1048
by (rule additive.diff [OF additive_right])
huffman@22442
  1049
huffman@27443
  1050
lemma bounded_linear_left:
huffman@22442
  1051
  "bounded_linear (\<lambda>a. a ** b)"
huffman@44127
  1052
apply (cut_tac bounded, safe)
huffman@44127
  1053
apply (rule_tac K="norm b * K" in bounded_linear_intro)
huffman@22442
  1054
apply (rule add_left)
huffman@22442
  1055
apply (rule scaleR_left)
huffman@22442
  1056
apply (simp add: mult_ac)
huffman@22442
  1057
done
huffman@22442
  1058
huffman@27443
  1059
lemma bounded_linear_right:
huffman@22442
  1060
  "bounded_linear (\<lambda>b. a ** b)"
huffman@44127
  1061
apply (cut_tac bounded, safe)
huffman@44127
  1062
apply (rule_tac K="norm a * K" in bounded_linear_intro)
huffman@22442
  1063
apply (rule add_right)
huffman@22442
  1064
apply (rule scaleR_right)
huffman@22442
  1065
apply (simp add: mult_ac)
huffman@22442
  1066
done
huffman@22442
  1067
huffman@27443
  1068
lemma prod_diff_prod:
huffman@22442
  1069
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
  1070
by (simp add: diff_left diff_right)
huffman@22442
  1071
huffman@27443
  1072
end
huffman@27443
  1073
huffman@44282
  1074
lemma bounded_bilinear_mult:
huffman@44282
  1075
  "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
huffman@22442
  1076
apply (rule bounded_bilinear.intro)
huffman@22442
  1077
apply (rule left_distrib)
huffman@22442
  1078
apply (rule right_distrib)
huffman@22442
  1079
apply (rule mult_scaleR_left)
huffman@22442
  1080
apply (rule mult_scaleR_right)
huffman@22442
  1081
apply (rule_tac x="1" in exI)
huffman@22442
  1082
apply (simp add: norm_mult_ineq)
huffman@22442
  1083
done
huffman@22442
  1084
huffman@44282
  1085
lemma bounded_linear_mult_left:
huffman@44282
  1086
  "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
huffman@44282
  1087
  using bounded_bilinear_mult
huffman@44282
  1088
  by (rule bounded_bilinear.bounded_linear_left)
huffman@22442
  1089
huffman@44282
  1090
lemma bounded_linear_mult_right:
huffman@44282
  1091
  "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
huffman@44282
  1092
  using bounded_bilinear_mult
huffman@44282
  1093
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
  1094
huffman@44282
  1095
lemma bounded_linear_divide:
huffman@44282
  1096
  "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
huffman@44282
  1097
  unfolding divide_inverse by (rule bounded_linear_mult_left)
huffman@23120
  1098
huffman@44282
  1099
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
huffman@22442
  1100
apply (rule bounded_bilinear.intro)
huffman@22442
  1101
apply (rule scaleR_left_distrib)
huffman@22442
  1102
apply (rule scaleR_right_distrib)
huffman@22973
  1103
apply simp
huffman@22442
  1104
apply (rule scaleR_left_commute)
huffman@31586
  1105
apply (rule_tac x="1" in exI, simp)
huffman@22442
  1106
done
huffman@22442
  1107
huffman@44282
  1108
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
huffman@44282
  1109
  using bounded_bilinear_scaleR
huffman@44282
  1110
  by (rule bounded_bilinear.bounded_linear_left)
huffman@23127
  1111
huffman@44282
  1112
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
huffman@44282
  1113
  using bounded_bilinear_scaleR
huffman@44282
  1114
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
  1115
huffman@44282
  1116
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
huffman@44282
  1117
  unfolding of_real_def by (rule bounded_linear_scaleR_left)
huffman@22625
  1118
hoelzl@41969
  1119
subsection{* Hausdorff and other separation properties *}
hoelzl@41969
  1120
hoelzl@41969
  1121
class t0_space = topological_space +
hoelzl@41969
  1122
  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
hoelzl@41969
  1123
hoelzl@41969
  1124
class t1_space = topological_space +
hoelzl@41969
  1125
  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
hoelzl@41969
  1126
hoelzl@41969
  1127
instance t1_space \<subseteq> t0_space
hoelzl@41969
  1128
proof qed (fast dest: t1_space)
hoelzl@41969
  1129
hoelzl@41969
  1130
lemma separation_t1:
hoelzl@41969
  1131
  fixes x y :: "'a::t1_space"
hoelzl@41969
  1132
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
hoelzl@41969
  1133
  using t1_space[of x y] by blast
hoelzl@41969
  1134
hoelzl@41969
  1135
lemma closed_singleton:
hoelzl@41969
  1136
  fixes a :: "'a::t1_space"
hoelzl@41969
  1137
  shows "closed {a}"
hoelzl@41969
  1138
proof -
hoelzl@41969
  1139
  let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
hoelzl@41969
  1140
  have "open ?T" by (simp add: open_Union)
hoelzl@41969
  1141
  also have "?T = - {a}"
hoelzl@41969
  1142
    by (simp add: set_eq_iff separation_t1, auto)
hoelzl@41969
  1143
  finally show "closed {a}" unfolding closed_def .
hoelzl@41969
  1144
qed
hoelzl@41969
  1145
hoelzl@41969
  1146
lemma closed_insert [simp]:
hoelzl@41969
  1147
  fixes a :: "'a::t1_space"
hoelzl@41969
  1148
  assumes "closed S" shows "closed (insert a S)"
hoelzl@41969
  1149
proof -
hoelzl@41969
  1150
  from closed_singleton assms
hoelzl@41969
  1151
  have "closed ({a} \<union> S)" by (rule closed_Un)
hoelzl@41969
  1152
  thus "closed (insert a S)" by simp
hoelzl@41969
  1153
qed
hoelzl@41969
  1154
hoelzl@41969
  1155
lemma finite_imp_closed:
hoelzl@41969
  1156
  fixes S :: "'a::t1_space set"
hoelzl@41969
  1157
  shows "finite S \<Longrightarrow> closed S"
hoelzl@41969
  1158
by (induct set: finite, simp_all)
hoelzl@41969
  1159
hoelzl@41969
  1160
text {* T2 spaces are also known as Hausdorff spaces. *}
hoelzl@41969
  1161
hoelzl@41969
  1162
class t2_space = topological_space +
hoelzl@41969
  1163
  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
hoelzl@41969
  1164
hoelzl@41969
  1165
instance t2_space \<subseteq> t1_space
hoelzl@41969
  1166
proof qed (fast dest: hausdorff)
hoelzl@41969
  1167
hoelzl@41969
  1168
instance metric_space \<subseteq> t2_space
hoelzl@41969
  1169
proof
hoelzl@41969
  1170
  fix x y :: "'a::metric_space"
hoelzl@41969
  1171
  assume xy: "x \<noteq> y"
hoelzl@41969
  1172
  let ?U = "{y'. dist x y' < dist x y / 2}"
hoelzl@41969
  1173
  let ?V = "{x'. dist y x' < dist x y / 2}"
hoelzl@41969
  1174
  have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
hoelzl@41969
  1175
               \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
hoelzl@41969
  1176
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
hoelzl@41969
  1177
    using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
hoelzl@41969
  1178
    using open_ball[of _ "dist x y / 2"] by auto
hoelzl@41969
  1179
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
hoelzl@41969
  1180
    by blast
hoelzl@41969
  1181
qed
hoelzl@41969
  1182
hoelzl@41969
  1183
lemma separation_t2:
hoelzl@41969
  1184
  fixes x y :: "'a::t2_space"
hoelzl@41969
  1185
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
hoelzl@41969
  1186
  using hausdorff[of x y] by blast
hoelzl@41969
  1187
hoelzl@41969
  1188
lemma separation_t0:
hoelzl@41969
  1189
  fixes x y :: "'a::t0_space"
hoelzl@41969
  1190
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
hoelzl@41969
  1191
  using t0_space[of x y] by blast
hoelzl@41969
  1192
huffman@44571
  1193
text {* A perfect space is a topological space with no isolated points. *}
huffman@44571
  1194
huffman@44571
  1195
class perfect_space = topological_space +
huffman@44571
  1196
  assumes not_open_singleton: "\<not> open {x}"
huffman@44571
  1197
huffman@44571
  1198
instance real_normed_algebra_1 \<subseteq> perfect_space
huffman@44571
  1199
proof
huffman@44571
  1200
  fix x::'a
huffman@44571
  1201
  show "\<not> open {x}"
huffman@44571
  1202
    unfolding open_dist dist_norm
huffman@44571
  1203
    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
huffman@44571
  1204
qed
huffman@44571
  1205
huffman@20504
  1206
end