src/HOL/Real_Vector_Spaces.thy
author blanchet
Wed Feb 12 08:35:57 2014 +0100 (2014-02-12)
changeset 55415 05f5fdb8d093
parent 54890 cb892d835803
child 55719 cdddd073bff8
permissions -rw-r--r--
renamed 'nat_{case,rec}' to '{case,rec}_nat'
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(*  Title:      HOL/Real_Vector_Spaces.thy
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    Author:     Brian Huffman
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    Author:     Johannes Hölzl
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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 Real Topological_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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  using add [of x "- y"] by (simp add: 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 (- numeral w) = - numeral w"
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using of_real_of_int_eq [of "- 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
   322
huffman@20554
   323
huffman@20554
   324
subsection {* The Set of Real Numbers *}
huffman@20554
   325
haftmann@37767
   326
definition Reals :: "'a::real_algebra_1 set" where
haftmann@37767
   327
  "Reals = range of_real"
huffman@20554
   328
wenzelm@21210
   329
notation (xsymbols)
huffman@20554
   330
  Reals  ("\<real>")
huffman@20554
   331
huffman@21809
   332
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
huffman@20554
   333
by (simp add: Reals_def)
huffman@20554
   334
huffman@21809
   335
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
huffman@21809
   336
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
huffman@20718
   337
huffman@21809
   338
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
huffman@21809
   339
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
huffman@21809
   340
huffman@47108
   341
lemma Reals_numeral [simp]: "numeral w \<in> Reals"
huffman@47108
   342
by (subst of_real_numeral [symmetric], rule Reals_of_real)
huffman@47108
   343
huffman@20554
   344
lemma Reals_0 [simp]: "0 \<in> Reals"
huffman@20554
   345
apply (unfold Reals_def)
huffman@20554
   346
apply (rule range_eqI)
huffman@20554
   347
apply (rule of_real_0 [symmetric])
huffman@20554
   348
done
huffman@20554
   349
huffman@20554
   350
lemma Reals_1 [simp]: "1 \<in> Reals"
huffman@20554
   351
apply (unfold Reals_def)
huffman@20554
   352
apply (rule range_eqI)
huffman@20554
   353
apply (rule of_real_1 [symmetric])
huffman@20554
   354
done
huffman@20554
   355
huffman@20584
   356
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
huffman@20554
   357
apply (auto simp add: Reals_def)
huffman@20554
   358
apply (rule range_eqI)
huffman@20554
   359
apply (rule of_real_add [symmetric])
huffman@20554
   360
done
huffman@20554
   361
huffman@20584
   362
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
huffman@20584
   363
apply (auto simp add: Reals_def)
huffman@20584
   364
apply (rule range_eqI)
huffman@20584
   365
apply (rule of_real_minus [symmetric])
huffman@20584
   366
done
huffman@20584
   367
huffman@20584
   368
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
huffman@20584
   369
apply (auto simp add: Reals_def)
huffman@20584
   370
apply (rule range_eqI)
huffman@20584
   371
apply (rule of_real_diff [symmetric])
huffman@20584
   372
done
huffman@20584
   373
huffman@20584
   374
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
huffman@20554
   375
apply (auto simp add: Reals_def)
huffman@20554
   376
apply (rule range_eqI)
huffman@20554
   377
apply (rule of_real_mult [symmetric])
huffman@20554
   378
done
huffman@20554
   379
huffman@20584
   380
lemma nonzero_Reals_inverse:
huffman@20584
   381
  fixes a :: "'a::real_div_algebra"
huffman@20584
   382
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   383
apply (auto simp add: Reals_def)
huffman@20584
   384
apply (rule range_eqI)
huffman@20584
   385
apply (erule nonzero_of_real_inverse [symmetric])
huffman@20584
   386
done
huffman@20584
   387
huffman@20584
   388
lemma Reals_inverse [simp]:
haftmann@36409
   389
  fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
huffman@20584
   390
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   391
apply (auto simp add: Reals_def)
huffman@20584
   392
apply (rule range_eqI)
huffman@20584
   393
apply (rule of_real_inverse [symmetric])
huffman@20584
   394
done
huffman@20584
   395
huffman@20584
   396
lemma nonzero_Reals_divide:
huffman@20584
   397
  fixes a b :: "'a::real_field"
huffman@20584
   398
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   399
apply (auto simp add: Reals_def)
huffman@20584
   400
apply (rule range_eqI)
huffman@20584
   401
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   402
done
huffman@20584
   403
huffman@20584
   404
lemma Reals_divide [simp]:
haftmann@36409
   405
  fixes a b :: "'a::{real_field, field_inverse_zero}"
huffman@20584
   406
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   407
apply (auto simp add: Reals_def)
huffman@20584
   408
apply (rule range_eqI)
huffman@20584
   409
apply (rule of_real_divide [symmetric])
huffman@20584
   410
done
huffman@20584
   411
huffman@20722
   412
lemma Reals_power [simp]:
haftmann@31017
   413
  fixes a :: "'a::{real_algebra_1}"
huffman@20722
   414
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   415
apply (auto simp add: Reals_def)
huffman@20722
   416
apply (rule range_eqI)
huffman@20722
   417
apply (rule of_real_power [symmetric])
huffman@20722
   418
done
huffman@20722
   419
huffman@20554
   420
lemma Reals_cases [cases set: Reals]:
huffman@20554
   421
  assumes "q \<in> \<real>"
huffman@20554
   422
  obtains (of_real) r where "q = of_real r"
huffman@20554
   423
  unfolding Reals_def
huffman@20554
   424
proof -
huffman@20554
   425
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   426
  then obtain r where "q = of_real r" ..
huffman@20554
   427
  then show thesis ..
huffman@20554
   428
qed
huffman@20554
   429
huffman@20554
   430
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   431
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   432
  by (rule Reals_cases) auto
huffman@20554
   433
immler@54778
   434
subsection {* Ordered real vector spaces *}
immler@54778
   435
immler@54778
   436
class ordered_real_vector = real_vector + ordered_ab_group_add +
immler@54778
   437
  assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y"
immler@54778
   438
  assumes scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"
immler@54778
   439
begin
immler@54778
   440
immler@54778
   441
lemma scaleR_mono:
immler@54778
   442
  "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y"
immler@54778
   443
apply (erule scaleR_right_mono [THEN order_trans], assumption)
immler@54778
   444
apply (erule scaleR_left_mono, assumption)
immler@54778
   445
done
immler@54778
   446
immler@54778
   447
lemma scaleR_mono':
immler@54778
   448
  "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R d"
immler@54778
   449
  by (rule scaleR_mono) (auto intro: order.trans)
immler@54778
   450
immler@54785
   451
lemma pos_le_divideRI:
immler@54785
   452
  assumes "0 < c"
immler@54785
   453
  assumes "c *\<^sub>R a \<le> b"
immler@54785
   454
  shows "a \<le> b /\<^sub>R c"
immler@54785
   455
proof -
immler@54785
   456
  from scaleR_left_mono[OF assms(2)] assms(1)
immler@54785
   457
  have "c *\<^sub>R a /\<^sub>R c \<le> b /\<^sub>R c"
immler@54785
   458
    by simp
immler@54785
   459
  with assms show ?thesis
immler@54785
   460
    by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
immler@54785
   461
qed
immler@54785
   462
immler@54785
   463
lemma pos_le_divideR_eq:
immler@54785
   464
  assumes "0 < c"
immler@54785
   465
  shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b"
immler@54785
   466
proof rule
immler@54785
   467
  assume "a \<le> b /\<^sub>R c"
immler@54785
   468
  from scaleR_left_mono[OF this] assms
immler@54785
   469
  have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"
immler@54785
   470
    by simp
immler@54785
   471
  with assms show "c *\<^sub>R a \<le> b"
immler@54785
   472
    by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
immler@54785
   473
qed (rule pos_le_divideRI[OF assms])
immler@54785
   474
immler@54785
   475
lemma scaleR_image_atLeastAtMost:
immler@54785
   476
  "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"
immler@54785
   477
  apply (auto intro!: scaleR_left_mono)
immler@54785
   478
  apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI)
immler@54785
   479
  apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)
immler@54785
   480
  done
immler@54785
   481
immler@54778
   482
end
immler@54778
   483
immler@54778
   484
lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> 0 \<le> a *\<^sub>R x"
immler@54778
   485
  using scaleR_left_mono [of 0 x a]
immler@54778
   486
  by simp
immler@54778
   487
immler@54778
   488
lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> (x::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"
immler@54778
   489
  using scaleR_left_mono [of x 0 a] by simp
immler@54778
   490
immler@54778
   491
lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> a *\<^sub>R x \<le> 0"
immler@54778
   492
  using scaleR_right_mono [of a 0 x] by simp
immler@54778
   493
immler@54778
   494
lemma split_scaleR_neg_le: "(0 \<le> a & x \<le> 0) | (a \<le> 0 & 0 \<le> x) \<Longrightarrow>
immler@54778
   495
  a *\<^sub>R (x::'a::ordered_real_vector) \<le> 0"
immler@54778
   496
  by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)
immler@54778
   497
immler@54778
   498
lemma le_add_iff1:
immler@54778
   499
  fixes c d e::"'a::ordered_real_vector"
immler@54778
   500
  shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"
immler@54778
   501
  by (simp add: algebra_simps)
immler@54778
   502
immler@54778
   503
lemma le_add_iff2:
immler@54778
   504
  fixes c d e::"'a::ordered_real_vector"
immler@54778
   505
  shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"
immler@54778
   506
  by (simp add: algebra_simps)
immler@54778
   507
immler@54778
   508
lemma scaleR_left_mono_neg:
immler@54778
   509
  fixes a b::"'a::ordered_real_vector"
immler@54778
   510
  shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"
immler@54778
   511
  apply (drule scaleR_left_mono [of _ _ "- c"])
immler@54778
   512
  apply simp_all
immler@54778
   513
  done
immler@54778
   514
immler@54778
   515
lemma scaleR_right_mono_neg:
immler@54778
   516
  fixes c::"'a::ordered_real_vector"
immler@54778
   517
  shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"
immler@54778
   518
  apply (drule scaleR_right_mono [of _ _ "- c"])
immler@54778
   519
  apply simp_all
immler@54778
   520
  done
immler@54778
   521
immler@54778
   522
lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> (b::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"
immler@54778
   523
using scaleR_right_mono_neg [of a 0 b] by simp
immler@54778
   524
immler@54778
   525
lemma split_scaleR_pos_le:
immler@54778
   526
  fixes b::"'a::ordered_real_vector"
immler@54778
   527
  shows "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"
immler@54778
   528
  by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)
immler@54778
   529
immler@54778
   530
lemma zero_le_scaleR_iff:
immler@54778
   531
  fixes b::"'a::ordered_real_vector"
immler@54778
   532
  shows "0 \<le> a *\<^sub>R b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0" (is "?lhs = ?rhs")
immler@54778
   533
proof cases
immler@54778
   534
  assume "a \<noteq> 0"
immler@54778
   535
  show ?thesis
immler@54778
   536
  proof
immler@54778
   537
    assume lhs: ?lhs
immler@54778
   538
    {
immler@54778
   539
      assume "0 < a"
immler@54778
   540
      with lhs have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"
immler@54778
   541
        by (intro scaleR_mono) auto
immler@54778
   542
      hence ?rhs using `0 < a`
immler@54778
   543
        by simp
immler@54778
   544
    } moreover {
immler@54778
   545
      assume "0 > a"
immler@54778
   546
      with lhs have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"
immler@54778
   547
        by (intro scaleR_mono) auto
immler@54778
   548
      hence ?rhs using `0 > a`
immler@54778
   549
        by simp
immler@54778
   550
    } ultimately show ?rhs using `a \<noteq> 0` by arith
immler@54778
   551
  qed (auto simp: not_le `a \<noteq> 0` intro!: split_scaleR_pos_le)
immler@54778
   552
qed simp
immler@54778
   553
immler@54778
   554
lemma scaleR_le_0_iff:
immler@54778
   555
  fixes b::"'a::ordered_real_vector"
immler@54778
   556
  shows "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
immler@54778
   557
  by (insert zero_le_scaleR_iff [of "-a" b]) force
immler@54778
   558
immler@54778
   559
lemma scaleR_le_cancel_left:
immler@54778
   560
  fixes b::"'a::ordered_real_vector"
immler@54778
   561
  shows "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
immler@54778
   562
  by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg
immler@54778
   563
    dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])
immler@54778
   564
immler@54778
   565
lemma scaleR_le_cancel_left_pos:
immler@54778
   566
  fixes b::"'a::ordered_real_vector"
immler@54778
   567
  shows "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"
immler@54778
   568
  by (auto simp: scaleR_le_cancel_left)
immler@54778
   569
immler@54778
   570
lemma scaleR_le_cancel_left_neg:
immler@54778
   571
  fixes b::"'a::ordered_real_vector"
immler@54778
   572
  shows "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"
immler@54778
   573
  by (auto simp: scaleR_le_cancel_left)
immler@54778
   574
immler@54778
   575
lemma scaleR_left_le_one_le:
immler@54778
   576
  fixes x::"'a::ordered_real_vector" and a::real
immler@54778
   577
  shows "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"
immler@54778
   578
  using scaleR_right_mono[of a 1 x] by simp
immler@54778
   579
huffman@20504
   580
huffman@20504
   581
subsection {* Real normed vector spaces *}
huffman@20504
   582
hoelzl@51531
   583
class dist =
hoelzl@51531
   584
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@51531
   585
haftmann@29608
   586
class norm =
huffman@22636
   587
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   588
huffman@24520
   589
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   590
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   591
huffman@31289
   592
class dist_norm = dist + norm + minus +
huffman@31289
   593
  assumes dist_norm: "dist x y = norm (x - y)"
huffman@31289
   594
hoelzl@51531
   595
class open_dist = "open" + dist +
hoelzl@51531
   596
  assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
hoelzl@51531
   597
huffman@31492
   598
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
hoelzl@51002
   599
  assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   600
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@31586
   601
  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
hoelzl@51002
   602
begin
hoelzl@51002
   603
hoelzl@51002
   604
lemma norm_ge_zero [simp]: "0 \<le> norm x"
hoelzl@51002
   605
proof -
hoelzl@51002
   606
  have "0 = norm (x + -1 *\<^sub>R x)" 
hoelzl@51002
   607
    using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
hoelzl@51002
   608
  also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
hoelzl@51002
   609
  finally show ?thesis by simp
hoelzl@51002
   610
qed
hoelzl@51002
   611
hoelzl@51002
   612
end
huffman@20504
   613
haftmann@24588
   614
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   615
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   616
haftmann@24588
   617
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   618
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   619
haftmann@24588
   620
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   621
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   622
haftmann@24588
   623
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   624
huffman@22852
   625
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   626
proof
huffman@20554
   627
  fix x y :: 'a
huffman@20554
   628
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   629
    by (simp add: norm_mult)
huffman@22852
   630
next
huffman@22852
   631
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   632
    by (rule norm_mult)
huffman@22852
   633
  thus "norm (1::'a) = 1" by simp
huffman@20554
   634
qed
huffman@20554
   635
huffman@22852
   636
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   637
by simp
huffman@20504
   638
huffman@22852
   639
lemma zero_less_norm_iff [simp]:
huffman@22852
   640
  fixes x :: "'a::real_normed_vector"
huffman@22852
   641
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   642
by (simp add: order_less_le)
huffman@20504
   643
huffman@22852
   644
lemma norm_not_less_zero [simp]:
huffman@22852
   645
  fixes x :: "'a::real_normed_vector"
huffman@22852
   646
  shows "\<not> norm x < 0"
huffman@20828
   647
by (simp add: linorder_not_less)
huffman@20828
   648
huffman@22852
   649
lemma norm_le_zero_iff [simp]:
huffman@22852
   650
  fixes x :: "'a::real_normed_vector"
huffman@22852
   651
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   652
by (simp add: order_le_less)
huffman@20828
   653
huffman@20504
   654
lemma norm_minus_cancel [simp]:
huffman@20584
   655
  fixes x :: "'a::real_normed_vector"
huffman@20584
   656
  shows "norm (- x) = norm x"
huffman@20504
   657
proof -
huffman@21809
   658
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   659
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   660
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   661
    by (rule norm_scaleR)
huffman@20504
   662
  finally show ?thesis by simp
huffman@20504
   663
qed
huffman@20504
   664
huffman@20504
   665
lemma norm_minus_commute:
huffman@20584
   666
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   667
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   668
proof -
huffman@22898
   669
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   670
    by (rule norm_minus_cancel)
huffman@22898
   671
  thus ?thesis by simp
huffman@20504
   672
qed
huffman@20504
   673
huffman@20504
   674
lemma norm_triangle_ineq2:
huffman@20584
   675
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   676
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   677
proof -
huffman@20533
   678
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   679
    by (rule norm_triangle_ineq)
huffman@22898
   680
  thus ?thesis by simp
huffman@20504
   681
qed
huffman@20504
   682
huffman@20584
   683
lemma norm_triangle_ineq3:
huffman@20584
   684
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   685
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   686
apply (subst abs_le_iff)
huffman@20584
   687
apply auto
huffman@20584
   688
apply (rule norm_triangle_ineq2)
huffman@20584
   689
apply (subst norm_minus_commute)
huffman@20584
   690
apply (rule norm_triangle_ineq2)
huffman@20584
   691
done
huffman@20584
   692
huffman@20504
   693
lemma norm_triangle_ineq4:
huffman@20584
   694
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   695
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   696
proof -
huffman@22898
   697
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   698
    by (rule norm_triangle_ineq)
haftmann@54230
   699
  then show ?thesis by simp
huffman@22898
   700
qed
huffman@22898
   701
huffman@22898
   702
lemma norm_diff_ineq:
huffman@22898
   703
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   704
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   705
proof -
huffman@22898
   706
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   707
    by (rule norm_triangle_ineq2)
huffman@22898
   708
  thus ?thesis by simp
huffman@20504
   709
qed
huffman@20504
   710
huffman@20551
   711
lemma norm_diff_triangle_ineq:
huffman@20551
   712
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   713
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   714
proof -
huffman@20551
   715
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
haftmann@54230
   716
    by (simp add: algebra_simps)
huffman@20551
   717
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   718
    by (rule norm_triangle_ineq)
huffman@20551
   719
  finally show ?thesis .
huffman@20551
   720
qed
huffman@20551
   721
huffman@22857
   722
lemma abs_norm_cancel [simp]:
huffman@22857
   723
  fixes a :: "'a::real_normed_vector"
huffman@22857
   724
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   725
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   726
huffman@22880
   727
lemma norm_add_less:
huffman@22880
   728
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   729
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   730
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   731
huffman@22880
   732
lemma norm_mult_less:
huffman@22880
   733
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   734
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   735
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   736
apply (simp add: mult_strict_mono')
huffman@22880
   737
done
huffman@22880
   738
huffman@22857
   739
lemma norm_of_real [simp]:
huffman@22857
   740
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@31586
   741
unfolding of_real_def by simp
huffman@20560
   742
huffman@47108
   743
lemma norm_numeral [simp]:
huffman@47108
   744
  "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
huffman@47108
   745
by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
huffman@47108
   746
huffman@47108
   747
lemma norm_neg_numeral [simp]:
haftmann@54489
   748
  "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
huffman@47108
   749
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
huffman@22876
   750
huffman@22876
   751
lemma norm_of_int [simp]:
huffman@22876
   752
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   753
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   754
huffman@22876
   755
lemma norm_of_nat [simp]:
huffman@22876
   756
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   757
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   758
apply (subst norm_of_real, simp)
huffman@22876
   759
done
huffman@22876
   760
huffman@20504
   761
lemma nonzero_norm_inverse:
huffman@20504
   762
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   763
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   764
apply (rule inverse_unique [symmetric])
huffman@20504
   765
apply (simp add: norm_mult [symmetric])
huffman@20504
   766
done
huffman@20504
   767
huffman@20504
   768
lemma norm_inverse:
haftmann@36409
   769
  fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
huffman@20533
   770
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   771
apply (case_tac "a = 0", simp)
huffman@20504
   772
apply (erule nonzero_norm_inverse)
huffman@20504
   773
done
huffman@20504
   774
huffman@20584
   775
lemma nonzero_norm_divide:
huffman@20584
   776
  fixes a b :: "'a::real_normed_field"
huffman@20584
   777
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   778
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   779
huffman@20584
   780
lemma norm_divide:
haftmann@36409
   781
  fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
huffman@20584
   782
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   783
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   784
huffman@22852
   785
lemma norm_power_ineq:
haftmann@31017
   786
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@22852
   787
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   788
proof (induct n)
huffman@22852
   789
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   790
next
huffman@22852
   791
  case (Suc n)
huffman@22852
   792
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   793
    by (rule norm_mult_ineq)
huffman@22852
   794
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   795
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   796
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@30273
   797
    by simp
huffman@22852
   798
qed
huffman@22852
   799
huffman@20684
   800
lemma norm_power:
haftmann@31017
   801
  fixes x :: "'a::{real_normed_div_algebra}"
huffman@20684
   802
  shows "norm (x ^ n) = norm x ^ n"
huffman@30273
   803
by (induct n) (simp_all add: norm_mult)
huffman@20684
   804
hoelzl@51531
   805
hoelzl@51531
   806
subsection {* Metric spaces *}
hoelzl@51531
   807
hoelzl@51531
   808
class metric_space = open_dist +
hoelzl@51531
   809
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
hoelzl@51531
   810
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
hoelzl@51531
   811
begin
hoelzl@51531
   812
hoelzl@51531
   813
lemma dist_self [simp]: "dist x x = 0"
hoelzl@51531
   814
by simp
hoelzl@51531
   815
hoelzl@51531
   816
lemma zero_le_dist [simp]: "0 \<le> dist x y"
hoelzl@51531
   817
using dist_triangle2 [of x x y] by simp
hoelzl@51531
   818
hoelzl@51531
   819
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
hoelzl@51531
   820
by (simp add: less_le)
hoelzl@51531
   821
hoelzl@51531
   822
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
hoelzl@51531
   823
by (simp add: not_less)
hoelzl@51531
   824
hoelzl@51531
   825
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
hoelzl@51531
   826
by (simp add: le_less)
hoelzl@51531
   827
hoelzl@51531
   828
lemma dist_commute: "dist x y = dist y x"
hoelzl@51531
   829
proof (rule order_antisym)
hoelzl@51531
   830
  show "dist x y \<le> dist y x"
hoelzl@51531
   831
    using dist_triangle2 [of x y x] by simp
hoelzl@51531
   832
  show "dist y x \<le> dist x y"
hoelzl@51531
   833
    using dist_triangle2 [of y x y] by simp
hoelzl@51531
   834
qed
hoelzl@51531
   835
hoelzl@51531
   836
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
hoelzl@51531
   837
using dist_triangle2 [of x z y] by (simp add: dist_commute)
hoelzl@51531
   838
hoelzl@51531
   839
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
hoelzl@51531
   840
using dist_triangle2 [of x y a] by (simp add: dist_commute)
hoelzl@51531
   841
hoelzl@51531
   842
lemma dist_triangle_alt:
hoelzl@51531
   843
  shows "dist y z <= dist x y + dist x z"
hoelzl@51531
   844
by (rule dist_triangle3)
hoelzl@51531
   845
hoelzl@51531
   846
lemma dist_pos_lt:
hoelzl@51531
   847
  shows "x \<noteq> y ==> 0 < dist x y"
hoelzl@51531
   848
by (simp add: zero_less_dist_iff)
hoelzl@51531
   849
hoelzl@51531
   850
lemma dist_nz:
hoelzl@51531
   851
  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
hoelzl@51531
   852
by (simp add: zero_less_dist_iff)
hoelzl@51531
   853
hoelzl@51531
   854
lemma dist_triangle_le:
hoelzl@51531
   855
  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
hoelzl@51531
   856
by (rule order_trans [OF dist_triangle2])
hoelzl@51531
   857
hoelzl@51531
   858
lemma dist_triangle_lt:
hoelzl@51531
   859
  shows "dist x z + dist y z < e ==> dist x y < e"
hoelzl@51531
   860
by (rule le_less_trans [OF dist_triangle2])
hoelzl@51531
   861
hoelzl@51531
   862
lemma dist_triangle_half_l:
hoelzl@51531
   863
  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
hoelzl@51531
   864
by (rule dist_triangle_lt [where z=y], simp)
hoelzl@51531
   865
hoelzl@51531
   866
lemma dist_triangle_half_r:
hoelzl@51531
   867
  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
hoelzl@51531
   868
by (rule dist_triangle_half_l, simp_all add: dist_commute)
hoelzl@51531
   869
hoelzl@51531
   870
subclass topological_space
hoelzl@51531
   871
proof
hoelzl@51531
   872
  have "\<exists>e::real. 0 < e"
hoelzl@51531
   873
    by (fast intro: zero_less_one)
hoelzl@51531
   874
  then show "open UNIV"
hoelzl@51531
   875
    unfolding open_dist by simp
hoelzl@51531
   876
next
hoelzl@51531
   877
  fix S T assume "open S" "open T"
hoelzl@51531
   878
  then show "open (S \<inter> T)"
hoelzl@51531
   879
    unfolding open_dist
hoelzl@51531
   880
    apply clarify
hoelzl@51531
   881
    apply (drule (1) bspec)+
hoelzl@51531
   882
    apply (clarify, rename_tac r s)
hoelzl@51531
   883
    apply (rule_tac x="min r s" in exI, simp)
hoelzl@51531
   884
    done
hoelzl@51531
   885
next
hoelzl@51531
   886
  fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
hoelzl@51531
   887
    unfolding open_dist by fast
hoelzl@51531
   888
qed
hoelzl@51531
   889
hoelzl@51531
   890
lemma open_ball: "open {y. dist x y < d}"
hoelzl@51531
   891
proof (unfold open_dist, intro ballI)
hoelzl@51531
   892
  fix y assume *: "y \<in> {y. dist x y < d}"
hoelzl@51531
   893
  then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
hoelzl@51531
   894
    by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
hoelzl@51531
   895
qed
hoelzl@51531
   896
hoelzl@51531
   897
subclass first_countable_topology
hoelzl@51531
   898
proof
hoelzl@51531
   899
  fix x 
hoelzl@51531
   900
  show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hoelzl@51531
   901
  proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
hoelzl@51531
   902
    fix S assume "open S" "x \<in> S"
wenzelm@53374
   903
    then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
hoelzl@51531
   904
      by (auto simp: open_dist subset_eq dist_commute)
hoelzl@51531
   905
    moreover
wenzelm@53374
   906
    from e obtain i where "inverse (Suc i) < e"
hoelzl@51531
   907
      by (auto dest!: reals_Archimedean)
hoelzl@51531
   908
    then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
hoelzl@51531
   909
      by auto
hoelzl@51531
   910
    ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
hoelzl@51531
   911
      by blast
hoelzl@51531
   912
  qed (auto intro: open_ball)
hoelzl@51531
   913
qed
hoelzl@51531
   914
hoelzl@51531
   915
end
hoelzl@51531
   916
hoelzl@51531
   917
instance metric_space \<subseteq> t2_space
hoelzl@51531
   918
proof
hoelzl@51531
   919
  fix x y :: "'a::metric_space"
hoelzl@51531
   920
  assume xy: "x \<noteq> y"
hoelzl@51531
   921
  let ?U = "{y'. dist x y' < dist x y / 2}"
hoelzl@51531
   922
  let ?V = "{x'. dist y x' < dist x y / 2}"
hoelzl@51531
   923
  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@51531
   924
               \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
hoelzl@51531
   925
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
hoelzl@51531
   926
    using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
hoelzl@51531
   927
    using open_ball[of _ "dist x y / 2"] by auto
hoelzl@51531
   928
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
hoelzl@51531
   929
    by blast
hoelzl@51531
   930
qed
hoelzl@51531
   931
huffman@31289
   932
text {* Every normed vector space is a metric space. *}
huffman@31285
   933
huffman@31289
   934
instance real_normed_vector < metric_space
huffman@31289
   935
proof
huffman@31289
   936
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   937
    unfolding dist_norm by simp
huffman@31289
   938
next
huffman@31289
   939
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
huffman@31289
   940
    unfolding dist_norm
huffman@31289
   941
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
huffman@31289
   942
qed
huffman@31285
   943
huffman@31564
   944
subsection {* Class instances for real numbers *}
huffman@31564
   945
huffman@31564
   946
instantiation real :: real_normed_field
huffman@31564
   947
begin
huffman@31564
   948
hoelzl@51531
   949
definition dist_real_def:
hoelzl@51531
   950
  "dist x y = \<bar>x - y\<bar>"
hoelzl@51531
   951
haftmann@52381
   952
definition open_real_def [code del]:
hoelzl@51531
   953
  "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
hoelzl@51531
   954
huffman@31564
   955
definition real_norm_def [simp]:
huffman@31564
   956
  "norm r = \<bar>r\<bar>"
huffman@31564
   957
huffman@31564
   958
instance
huffman@31564
   959
apply (intro_classes, unfold real_norm_def real_scaleR_def)
huffman@31564
   960
apply (rule dist_real_def)
hoelzl@51531
   961
apply (rule open_real_def)
huffman@36795
   962
apply (simp add: sgn_real_def)
huffman@31564
   963
apply (rule abs_eq_0)
huffman@31564
   964
apply (rule abs_triangle_ineq)
huffman@31564
   965
apply (rule abs_mult)
huffman@31564
   966
apply (rule abs_mult)
huffman@31564
   967
done
huffman@31564
   968
huffman@31564
   969
end
huffman@31564
   970
haftmann@54890
   971
declare [[code abort: "open :: real set \<Rightarrow> bool"]]
haftmann@52381
   972
hoelzl@51531
   973
instance real :: linorder_topology
hoelzl@51531
   974
proof
hoelzl@51531
   975
  show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
hoelzl@51531
   976
  proof (rule ext, safe)
hoelzl@51531
   977
    fix S :: "real set" assume "open S"
wenzelm@53381
   978
    then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"
wenzelm@53381
   979
      unfolding open_real_def bchoice_iff ..
hoelzl@51531
   980
    then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
hoelzl@51531
   981
      by (fastforce simp: dist_real_def)
hoelzl@51531
   982
    show "generate_topology (range lessThan \<union> range greaterThan) S"
hoelzl@51531
   983
      apply (subst *)
hoelzl@51531
   984
      apply (intro generate_topology_Union generate_topology.Int)
hoelzl@51531
   985
      apply (auto intro: generate_topology.Basis)
hoelzl@51531
   986
      done
hoelzl@51531
   987
  next
hoelzl@51531
   988
    fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
hoelzl@51531
   989
    moreover have "\<And>a::real. open {..<a}"
hoelzl@51531
   990
      unfolding open_real_def dist_real_def
hoelzl@51531
   991
    proof clarify
hoelzl@51531
   992
      fix x a :: real assume "x < a"
hoelzl@51531
   993
      hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
hoelzl@51531
   994
      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
hoelzl@51531
   995
    qed
hoelzl@51531
   996
    moreover have "\<And>a::real. open {a <..}"
hoelzl@51531
   997
      unfolding open_real_def dist_real_def
hoelzl@51531
   998
    proof clarify
hoelzl@51531
   999
      fix x a :: real assume "a < x"
hoelzl@51531
  1000
      hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
hoelzl@51531
  1001
      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
hoelzl@51531
  1002
    qed
hoelzl@51531
  1003
    ultimately show "open S"
hoelzl@51531
  1004
      by induct auto
hoelzl@51531
  1005
  qed
hoelzl@51531
  1006
qed
hoelzl@51531
  1007
hoelzl@51775
  1008
instance real :: linear_continuum_topology ..
hoelzl@51518
  1009
hoelzl@51531
  1010
lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
hoelzl@51531
  1011
lemmas open_real_lessThan = open_lessThan[where 'a=real]
hoelzl@51531
  1012
lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
hoelzl@51531
  1013
lemmas closed_real_atMost = closed_atMost[where 'a=real]
hoelzl@51531
  1014
lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
hoelzl@51531
  1015
lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
hoelzl@51531
  1016
huffman@31446
  1017
subsection {* Extra type constraints *}
huffman@31446
  1018
huffman@31492
  1019
text {* Only allow @{term "open"} in class @{text topological_space}. *}
huffman@31492
  1020
huffman@31492
  1021
setup {* Sign.add_const_constraint
huffman@31492
  1022
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
huffman@31492
  1023
huffman@31446
  1024
text {* Only allow @{term dist} in class @{text metric_space}. *}
huffman@31446
  1025
huffman@31446
  1026
setup {* Sign.add_const_constraint
huffman@31446
  1027
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
huffman@31446
  1028
huffman@31446
  1029
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
huffman@31446
  1030
huffman@31446
  1031
setup {* Sign.add_const_constraint
huffman@31446
  1032
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
huffman@31446
  1033
huffman@22972
  1034
subsection {* Sign function *}
huffman@22972
  1035
nipkow@24506
  1036
lemma norm_sgn:
nipkow@24506
  1037
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
huffman@31586
  1038
by (simp add: sgn_div_norm)
huffman@22972
  1039
nipkow@24506
  1040
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
  1041
by (simp add: sgn_div_norm)
huffman@22972
  1042
nipkow@24506
  1043
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
  1044
by (simp add: sgn_div_norm)
huffman@22972
  1045
nipkow@24506
  1046
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
  1047
by (simp add: sgn_div_norm)
huffman@22972
  1048
nipkow@24506
  1049
lemma sgn_scaleR:
nipkow@24506
  1050
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
huffman@31586
  1051
by (simp add: sgn_div_norm mult_ac)
huffman@22973
  1052
huffman@22972
  1053
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
  1054
by (simp add: sgn_div_norm)
huffman@22972
  1055
huffman@22972
  1056
lemma sgn_of_real:
huffman@22972
  1057
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
  1058
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
  1059
huffman@22973
  1060
lemma sgn_mult:
huffman@22973
  1061
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
  1062
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
  1063
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
  1064
huffman@22972
  1065
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
  1066
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
  1067
huffman@22972
  1068
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
  1069
unfolding real_sgn_eq by simp
huffman@22972
  1070
huffman@22972
  1071
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
  1072
unfolding real_sgn_eq by simp
huffman@22972
  1073
hoelzl@51474
  1074
lemma norm_conv_dist: "norm x = dist x 0"
hoelzl@51474
  1075
  unfolding dist_norm by simp
huffman@22972
  1076
huffman@22442
  1077
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
  1078
huffman@53600
  1079
locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
huffman@22442
  1080
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@53600
  1081
huffman@53600
  1082
lemma linearI:
huffman@53600
  1083
  assumes "\<And>x y. f (x + y) = f x + f y"
huffman@53600
  1084
  assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
huffman@53600
  1085
  shows "linear f"
huffman@53600
  1086
  by default (rule assms)+
huffman@53600
  1087
huffman@53600
  1088
locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
huffman@22442
  1089
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
  1090
begin
huffman@22442
  1091
huffman@27443
  1092
lemma pos_bounded:
huffman@22442
  1093
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
  1094
proof -
huffman@22442
  1095
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
  1096
    using bounded by fast
huffman@22442
  1097
  show ?thesis
huffman@22442
  1098
  proof (intro exI impI conjI allI)
huffman@22442
  1099
    show "0 < max 1 K"
haftmann@54863
  1100
      by (rule order_less_le_trans [OF zero_less_one max.cobounded1])
huffman@22442
  1101
  next
huffman@22442
  1102
    fix x
huffman@22442
  1103
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
  1104
    also have "\<dots> \<le> norm x * max 1 K"
haftmann@54863
  1105
      by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero])
huffman@22442
  1106
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
  1107
  qed
huffman@22442
  1108
qed
huffman@22442
  1109
huffman@27443
  1110
lemma nonneg_bounded:
huffman@22442
  1111
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
  1112
proof -
huffman@22442
  1113
  from pos_bounded
huffman@22442
  1114
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
  1115
qed
huffman@22442
  1116
huffman@27443
  1117
end
huffman@27443
  1118
huffman@44127
  1119
lemma bounded_linear_intro:
huffman@44127
  1120
  assumes "\<And>x y. f (x + y) = f x + f y"
huffman@44127
  1121
  assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
huffman@44127
  1122
  assumes "\<And>x. norm (f x) \<le> norm x * K"
huffman@44127
  1123
  shows "bounded_linear f"
huffman@44127
  1124
  by default (fast intro: assms)+
huffman@44127
  1125
huffman@22442
  1126
locale bounded_bilinear =
huffman@22442
  1127
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
  1128
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
  1129
    (infixl "**" 70)
huffman@22442
  1130
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
  1131
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
  1132
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
  1133
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
  1134
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
  1135
begin
huffman@22442
  1136
huffman@27443
  1137
lemma pos_bounded:
huffman@22442
  1138
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
  1139
apply (cut_tac bounded, erule exE)
huffman@22442
  1140
apply (rule_tac x="max 1 K" in exI, safe)
haftmann@54863
  1141
apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])
huffman@22442
  1142
apply (drule spec, drule spec, erule order_trans)
haftmann@54863
  1143
apply (rule mult_left_mono [OF max.cobounded2])
huffman@22442
  1144
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
  1145
done
huffman@22442
  1146
huffman@27443
  1147
lemma nonneg_bounded:
huffman@22442
  1148
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
  1149
proof -
huffman@22442
  1150
  from pos_bounded
huffman@22442
  1151
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
  1152
qed
huffman@22442
  1153
huffman@27443
  1154
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
  1155
by (rule additive.intro, rule add_right)
huffman@22442
  1156
huffman@27443
  1157
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
  1158
by (rule additive.intro, rule add_left)
huffman@22442
  1159
huffman@27443
  1160
lemma zero_left: "prod 0 b = 0"
huffman@22442
  1161
by (rule additive.zero [OF additive_left])
huffman@22442
  1162
huffman@27443
  1163
lemma zero_right: "prod a 0 = 0"
huffman@22442
  1164
by (rule additive.zero [OF additive_right])
huffman@22442
  1165
huffman@27443
  1166
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
  1167
by (rule additive.minus [OF additive_left])
huffman@22442
  1168
huffman@27443
  1169
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
  1170
by (rule additive.minus [OF additive_right])
huffman@22442
  1171
huffman@27443
  1172
lemma diff_left:
huffman@22442
  1173
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
  1174
by (rule additive.diff [OF additive_left])
huffman@22442
  1175
huffman@27443
  1176
lemma diff_right:
huffman@22442
  1177
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
  1178
by (rule additive.diff [OF additive_right])
huffman@22442
  1179
huffman@27443
  1180
lemma bounded_linear_left:
huffman@22442
  1181
  "bounded_linear (\<lambda>a. a ** b)"
huffman@44127
  1182
apply (cut_tac bounded, safe)
huffman@44127
  1183
apply (rule_tac K="norm b * K" in bounded_linear_intro)
huffman@22442
  1184
apply (rule add_left)
huffman@22442
  1185
apply (rule scaleR_left)
huffman@22442
  1186
apply (simp add: mult_ac)
huffman@22442
  1187
done
huffman@22442
  1188
huffman@27443
  1189
lemma bounded_linear_right:
huffman@22442
  1190
  "bounded_linear (\<lambda>b. a ** b)"
huffman@44127
  1191
apply (cut_tac bounded, safe)
huffman@44127
  1192
apply (rule_tac K="norm a * K" in bounded_linear_intro)
huffman@22442
  1193
apply (rule add_right)
huffman@22442
  1194
apply (rule scaleR_right)
huffman@22442
  1195
apply (simp add: mult_ac)
huffman@22442
  1196
done
huffman@22442
  1197
huffman@27443
  1198
lemma prod_diff_prod:
huffman@22442
  1199
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
  1200
by (simp add: diff_left diff_right)
huffman@22442
  1201
huffman@27443
  1202
end
huffman@27443
  1203
hoelzl@51642
  1204
lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
hoelzl@51642
  1205
  by default (auto intro!: exI[of _ 1])
hoelzl@51642
  1206
hoelzl@51642
  1207
lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
hoelzl@51642
  1208
  by default (auto intro!: exI[of _ 1])
hoelzl@51642
  1209
hoelzl@51642
  1210
lemma bounded_linear_add:
hoelzl@51642
  1211
  assumes "bounded_linear f"
hoelzl@51642
  1212
  assumes "bounded_linear g"
hoelzl@51642
  1213
  shows "bounded_linear (\<lambda>x. f x + g x)"
hoelzl@51642
  1214
proof -
hoelzl@51642
  1215
  interpret f: bounded_linear f by fact
hoelzl@51642
  1216
  interpret g: bounded_linear g by fact
hoelzl@51642
  1217
  show ?thesis
hoelzl@51642
  1218
  proof
hoelzl@51642
  1219
    from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
hoelzl@51642
  1220
    from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
hoelzl@51642
  1221
    show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
hoelzl@51642
  1222
      using add_mono[OF Kf Kg]
hoelzl@51642
  1223
      by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
hoelzl@51642
  1224
  qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
hoelzl@51642
  1225
qed
hoelzl@51642
  1226
hoelzl@51642
  1227
lemma bounded_linear_minus:
hoelzl@51642
  1228
  assumes "bounded_linear f"
hoelzl@51642
  1229
  shows "bounded_linear (\<lambda>x. - f x)"
hoelzl@51642
  1230
proof -
hoelzl@51642
  1231
  interpret f: bounded_linear f by fact
hoelzl@51642
  1232
  show ?thesis apply (unfold_locales)
hoelzl@51642
  1233
    apply (simp add: f.add)
hoelzl@51642
  1234
    apply (simp add: f.scaleR)
hoelzl@51642
  1235
    apply (simp add: f.bounded)
hoelzl@51642
  1236
    done
hoelzl@51642
  1237
qed
hoelzl@51642
  1238
hoelzl@51642
  1239
lemma bounded_linear_compose:
hoelzl@51642
  1240
  assumes "bounded_linear f"
hoelzl@51642
  1241
  assumes "bounded_linear g"
hoelzl@51642
  1242
  shows "bounded_linear (\<lambda>x. f (g x))"
hoelzl@51642
  1243
proof -
hoelzl@51642
  1244
  interpret f: bounded_linear f by fact
hoelzl@51642
  1245
  interpret g: bounded_linear g by fact
hoelzl@51642
  1246
  show ?thesis proof (unfold_locales)
hoelzl@51642
  1247
    fix x y show "f (g (x + y)) = f (g x) + f (g y)"
hoelzl@51642
  1248
      by (simp only: f.add g.add)
hoelzl@51642
  1249
  next
hoelzl@51642
  1250
    fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
hoelzl@51642
  1251
      by (simp only: f.scaleR g.scaleR)
hoelzl@51642
  1252
  next
hoelzl@51642
  1253
    from f.pos_bounded
hoelzl@51642
  1254
    obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
hoelzl@51642
  1255
    from g.pos_bounded
hoelzl@51642
  1256
    obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
hoelzl@51642
  1257
    show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
hoelzl@51642
  1258
    proof (intro exI allI)
hoelzl@51642
  1259
      fix x
hoelzl@51642
  1260
      have "norm (f (g x)) \<le> norm (g x) * Kf"
hoelzl@51642
  1261
        using f .
hoelzl@51642
  1262
      also have "\<dots> \<le> (norm x * Kg) * Kf"
hoelzl@51642
  1263
        using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
hoelzl@51642
  1264
      also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
hoelzl@51642
  1265
        by (rule mult_assoc)
hoelzl@51642
  1266
      finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
hoelzl@51642
  1267
    qed
hoelzl@51642
  1268
  qed
hoelzl@51642
  1269
qed
hoelzl@51642
  1270
huffman@44282
  1271
lemma bounded_bilinear_mult:
huffman@44282
  1272
  "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
huffman@22442
  1273
apply (rule bounded_bilinear.intro)
webertj@49962
  1274
apply (rule distrib_right)
webertj@49962
  1275
apply (rule distrib_left)
huffman@22442
  1276
apply (rule mult_scaleR_left)
huffman@22442
  1277
apply (rule mult_scaleR_right)
huffman@22442
  1278
apply (rule_tac x="1" in exI)
huffman@22442
  1279
apply (simp add: norm_mult_ineq)
huffman@22442
  1280
done
huffman@22442
  1281
huffman@44282
  1282
lemma bounded_linear_mult_left:
huffman@44282
  1283
  "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
huffman@44282
  1284
  using bounded_bilinear_mult
huffman@44282
  1285
  by (rule bounded_bilinear.bounded_linear_left)
huffman@22442
  1286
huffman@44282
  1287
lemma bounded_linear_mult_right:
huffman@44282
  1288
  "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
huffman@44282
  1289
  using bounded_bilinear_mult
huffman@44282
  1290
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
  1291
hoelzl@51642
  1292
lemmas bounded_linear_mult_const =
hoelzl@51642
  1293
  bounded_linear_mult_left [THEN bounded_linear_compose]
hoelzl@51642
  1294
hoelzl@51642
  1295
lemmas bounded_linear_const_mult =
hoelzl@51642
  1296
  bounded_linear_mult_right [THEN bounded_linear_compose]
hoelzl@51642
  1297
huffman@44282
  1298
lemma bounded_linear_divide:
huffman@44282
  1299
  "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
huffman@44282
  1300
  unfolding divide_inverse by (rule bounded_linear_mult_left)
huffman@23120
  1301
huffman@44282
  1302
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
huffman@22442
  1303
apply (rule bounded_bilinear.intro)
huffman@22442
  1304
apply (rule scaleR_left_distrib)
huffman@22442
  1305
apply (rule scaleR_right_distrib)
huffman@22973
  1306
apply simp
huffman@22442
  1307
apply (rule scaleR_left_commute)
huffman@31586
  1308
apply (rule_tac x="1" in exI, simp)
huffman@22442
  1309
done
huffman@22442
  1310
huffman@44282
  1311
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
huffman@44282
  1312
  using bounded_bilinear_scaleR
huffman@44282
  1313
  by (rule bounded_bilinear.bounded_linear_left)
huffman@23127
  1314
huffman@44282
  1315
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
huffman@44282
  1316
  using bounded_bilinear_scaleR
huffman@44282
  1317
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
  1318
huffman@44282
  1319
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
huffman@44282
  1320
  unfolding of_real_def by (rule bounded_linear_scaleR_left)
huffman@22625
  1321
hoelzl@51642
  1322
lemma real_bounded_linear:
hoelzl@51642
  1323
  fixes f :: "real \<Rightarrow> real"
hoelzl@51642
  1324
  shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
hoelzl@51642
  1325
proof -
hoelzl@51642
  1326
  { fix x assume "bounded_linear f"
hoelzl@51642
  1327
    then interpret bounded_linear f .
hoelzl@51642
  1328
    from scaleR[of x 1] have "f x = x * f 1"
hoelzl@51642
  1329
      by simp }
hoelzl@51642
  1330
  then show ?thesis
hoelzl@51642
  1331
    by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
hoelzl@51642
  1332
qed
hoelzl@51642
  1333
huffman@44571
  1334
instance real_normed_algebra_1 \<subseteq> perfect_space
huffman@44571
  1335
proof
huffman@44571
  1336
  fix x::'a
huffman@44571
  1337
  show "\<not> open {x}"
huffman@44571
  1338
    unfolding open_dist dist_norm
huffman@44571
  1339
    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
huffman@44571
  1340
qed
huffman@44571
  1341
hoelzl@51531
  1342
subsection {* Filters and Limits on Metric Space *}
hoelzl@51531
  1343
hoelzl@51531
  1344
lemma eventually_nhds_metric:
hoelzl@51531
  1345
  fixes a :: "'a :: metric_space"
hoelzl@51531
  1346
  shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
hoelzl@51531
  1347
unfolding eventually_nhds open_dist
hoelzl@51531
  1348
apply safe
hoelzl@51531
  1349
apply fast
hoelzl@51531
  1350
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
hoelzl@51531
  1351
apply clarsimp
hoelzl@51531
  1352
apply (rule_tac x="d - dist x a" in exI, clarsimp)
hoelzl@51531
  1353
apply (simp only: less_diff_eq)
hoelzl@51531
  1354
apply (erule le_less_trans [OF dist_triangle])
hoelzl@51531
  1355
done
hoelzl@51531
  1356
hoelzl@51531
  1357
lemma eventually_at:
hoelzl@51641
  1358
  fixes a :: "'a :: metric_space"
hoelzl@51641
  1359
  shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
hoelzl@51641
  1360
  unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)
hoelzl@51531
  1361
hoelzl@51641
  1362
lemma eventually_at_le:
hoelzl@51641
  1363
  fixes a :: "'a::metric_space"
hoelzl@51641
  1364
  shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
hoelzl@51641
  1365
  unfolding eventually_at_filter eventually_nhds_metric
hoelzl@51641
  1366
  apply auto
hoelzl@51641
  1367
  apply (rule_tac x="d / 2" in exI)
hoelzl@51641
  1368
  apply auto
hoelzl@51641
  1369
  done
hoelzl@51531
  1370
hoelzl@51531
  1371
lemma tendstoI:
hoelzl@51531
  1372
  fixes l :: "'a :: metric_space"
hoelzl@51531
  1373
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
hoelzl@51531
  1374
  shows "(f ---> l) F"
hoelzl@51531
  1375
  apply (rule topological_tendstoI)
hoelzl@51531
  1376
  apply (simp add: open_dist)
hoelzl@51531
  1377
  apply (drule (1) bspec, clarify)
hoelzl@51531
  1378
  apply (drule assms)
hoelzl@51531
  1379
  apply (erule eventually_elim1, simp)
hoelzl@51531
  1380
  done
hoelzl@51531
  1381
hoelzl@51531
  1382
lemma tendstoD:
hoelzl@51531
  1383
  fixes l :: "'a :: metric_space"
hoelzl@51531
  1384
  shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
hoelzl@51531
  1385
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
hoelzl@51531
  1386
  apply (clarsimp simp add: open_dist)
hoelzl@51531
  1387
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
hoelzl@51531
  1388
  apply (simp only: less_diff_eq)
hoelzl@51531
  1389
  apply (erule le_less_trans [OF dist_triangle])
hoelzl@51531
  1390
  apply simp
hoelzl@51531
  1391
  apply simp
hoelzl@51531
  1392
  done
hoelzl@51531
  1393
hoelzl@51531
  1394
lemma tendsto_iff:
hoelzl@51531
  1395
  fixes l :: "'a :: metric_space"
hoelzl@51531
  1396
  shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
hoelzl@51531
  1397
  using tendstoI tendstoD by fast
hoelzl@51531
  1398
hoelzl@51531
  1399
lemma metric_tendsto_imp_tendsto:
hoelzl@51531
  1400
  fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
hoelzl@51531
  1401
  assumes f: "(f ---> a) F"
hoelzl@51531
  1402
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
hoelzl@51531
  1403
  shows "(g ---> b) F"
hoelzl@51531
  1404
proof (rule tendstoI)
hoelzl@51531
  1405
  fix e :: real assume "0 < e"
hoelzl@51531
  1406
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
hoelzl@51531
  1407
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
hoelzl@51531
  1408
    using le_less_trans by (rule eventually_elim2)
hoelzl@51531
  1409
qed
hoelzl@51531
  1410
hoelzl@51531
  1411
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
hoelzl@51531
  1412
  unfolding filterlim_at_top
hoelzl@51531
  1413
  apply (intro allI)
hoelzl@51531
  1414
  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
hoelzl@51531
  1415
  apply (auto simp: natceiling_le_eq)
hoelzl@51531
  1416
  done
hoelzl@51531
  1417
hoelzl@51531
  1418
subsubsection {* Limits of Sequences *}
hoelzl@51531
  1419
hoelzl@51531
  1420
lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
hoelzl@51531
  1421
  unfolding tendsto_iff eventually_sequentially ..
hoelzl@51531
  1422
hoelzl@51531
  1423
lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
hoelzl@51531
  1424
  unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
hoelzl@51531
  1425
hoelzl@51531
  1426
lemma metric_LIMSEQ_I:
hoelzl@51531
  1427
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
hoelzl@51531
  1428
by (simp add: LIMSEQ_def)
hoelzl@51531
  1429
hoelzl@51531
  1430
lemma metric_LIMSEQ_D:
hoelzl@51531
  1431
  "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
hoelzl@51531
  1432
by (simp add: LIMSEQ_def)
hoelzl@51531
  1433
hoelzl@51531
  1434
hoelzl@51531
  1435
subsubsection {* Limits of Functions *}
hoelzl@51531
  1436
hoelzl@51531
  1437
lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
hoelzl@51531
  1438
     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
hoelzl@51531
  1439
        --> dist (f x) L < r)"
hoelzl@51641
  1440
  unfolding tendsto_iff eventually_at by simp
hoelzl@51531
  1441
hoelzl@51531
  1442
lemma metric_LIM_I:
hoelzl@51531
  1443
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
hoelzl@51531
  1444
    \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
hoelzl@51531
  1445
by (simp add: LIM_def)
hoelzl@51531
  1446
hoelzl@51531
  1447
lemma metric_LIM_D:
hoelzl@51531
  1448
  "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
hoelzl@51531
  1449
    \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
hoelzl@51531
  1450
by (simp add: LIM_def)
hoelzl@51531
  1451
hoelzl@51531
  1452
lemma metric_LIM_imp_LIM:
hoelzl@51531
  1453
  assumes f: "f -- a --> (l::'a::metric_space)"
hoelzl@51531
  1454
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
hoelzl@51531
  1455
  shows "g -- a --> (m::'b::metric_space)"
hoelzl@51531
  1456
  by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
hoelzl@51531
  1457
hoelzl@51531
  1458
lemma metric_LIM_equal2:
hoelzl@51531
  1459
  assumes 1: "0 < R"
hoelzl@51531
  1460
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
hoelzl@51531
  1461
  shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
hoelzl@51531
  1462
apply (rule topological_tendstoI)
hoelzl@51531
  1463
apply (drule (2) topological_tendstoD)
hoelzl@51531
  1464
apply (simp add: eventually_at, safe)
hoelzl@51531
  1465
apply (rule_tac x="min d R" in exI, safe)
hoelzl@51531
  1466
apply (simp add: 1)
hoelzl@51531
  1467
apply (simp add: 2)
hoelzl@51531
  1468
done
hoelzl@51531
  1469
hoelzl@51531
  1470
lemma metric_LIM_compose2:
hoelzl@51531
  1471
  assumes f: "f -- (a::'a::metric_space) --> b"
hoelzl@51531
  1472
  assumes g: "g -- b --> c"
hoelzl@51531
  1473
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
hoelzl@51531
  1474
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51641
  1475
  using inj
hoelzl@51641
  1476
  by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
hoelzl@51531
  1477
hoelzl@51531
  1478
lemma metric_isCont_LIM_compose2:
hoelzl@51531
  1479
  fixes f :: "'a :: metric_space \<Rightarrow> _"
hoelzl@51531
  1480
  assumes f [unfolded isCont_def]: "isCont f a"
hoelzl@51531
  1481
  assumes g: "g -- f a --> l"
hoelzl@51531
  1482
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
hoelzl@51531
  1483
  shows "(\<lambda>x. g (f x)) -- a --> l"
hoelzl@51531
  1484
by (rule metric_LIM_compose2 [OF f g inj])
hoelzl@51531
  1485
hoelzl@51531
  1486
subsection {* Complete metric spaces *}
hoelzl@51531
  1487
hoelzl@51531
  1488
subsection {* Cauchy sequences *}
hoelzl@51531
  1489
hoelzl@51531
  1490
definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
hoelzl@51531
  1491
  "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
hoelzl@51531
  1492
hoelzl@51531
  1493
subsection {* Cauchy Sequences *}
hoelzl@51531
  1494
hoelzl@51531
  1495
lemma metric_CauchyI:
hoelzl@51531
  1496
  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
hoelzl@51531
  1497
  by (simp add: Cauchy_def)
hoelzl@51531
  1498
hoelzl@51531
  1499
lemma metric_CauchyD:
hoelzl@51531
  1500
  "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
hoelzl@51531
  1501
  by (simp add: Cauchy_def)
hoelzl@51531
  1502
hoelzl@51531
  1503
lemma metric_Cauchy_iff2:
hoelzl@51531
  1504
  "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
hoelzl@51531
  1505
apply (simp add: Cauchy_def, auto)
hoelzl@51531
  1506
apply (drule reals_Archimedean, safe)
hoelzl@51531
  1507
apply (drule_tac x = n in spec, auto)
hoelzl@51531
  1508
apply (rule_tac x = M in exI, auto)
hoelzl@51531
  1509
apply (drule_tac x = m in spec, simp)
hoelzl@51531
  1510
apply (drule_tac x = na in spec, auto)
hoelzl@51531
  1511
done
hoelzl@51531
  1512
hoelzl@51531
  1513
lemma Cauchy_iff2:
hoelzl@51531
  1514
  "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
hoelzl@51531
  1515
  unfolding metric_Cauchy_iff2 dist_real_def ..
hoelzl@51531
  1516
hoelzl@51531
  1517
lemma Cauchy_subseq_Cauchy:
hoelzl@51531
  1518
  "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
hoelzl@51531
  1519
apply (auto simp add: Cauchy_def)
hoelzl@51531
  1520
apply (drule_tac x=e in spec, clarify)
hoelzl@51531
  1521
apply (rule_tac x=M in exI, clarify)
hoelzl@51531
  1522
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
hoelzl@51531
  1523
done
hoelzl@51531
  1524
hoelzl@51531
  1525
theorem LIMSEQ_imp_Cauchy:
hoelzl@51531
  1526
  assumes X: "X ----> a" shows "Cauchy X"
hoelzl@51531
  1527
proof (rule metric_CauchyI)
hoelzl@51531
  1528
  fix e::real assume "0 < e"
hoelzl@51531
  1529
  hence "0 < e/2" by simp
hoelzl@51531
  1530
  with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
hoelzl@51531
  1531
  then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
hoelzl@51531
  1532
  show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
hoelzl@51531
  1533
  proof (intro exI allI impI)
hoelzl@51531
  1534
    fix m assume "N \<le> m"
hoelzl@51531
  1535
    hence m: "dist (X m) a < e/2" using N by fast
hoelzl@51531
  1536
    fix n assume "N \<le> n"
hoelzl@51531
  1537
    hence n: "dist (X n) a < e/2" using N by fast
hoelzl@51531
  1538
    have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
hoelzl@51531
  1539
      by (rule dist_triangle2)
hoelzl@51531
  1540
    also from m n have "\<dots> < e" by simp
hoelzl@51531
  1541
    finally show "dist (X m) (X n) < e" .
hoelzl@51531
  1542
  qed
hoelzl@51531
  1543
qed
hoelzl@51531
  1544
hoelzl@51531
  1545
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
hoelzl@51531
  1546
unfolding convergent_def
hoelzl@51531
  1547
by (erule exE, erule LIMSEQ_imp_Cauchy)
hoelzl@51531
  1548
hoelzl@51531
  1549
subsubsection {* Cauchy Sequences are Convergent *}
hoelzl@51531
  1550
hoelzl@51531
  1551
class complete_space = metric_space +
hoelzl@51531
  1552
  assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
hoelzl@51531
  1553
hoelzl@51531
  1554
lemma Cauchy_convergent_iff:
hoelzl@51531
  1555
  fixes X :: "nat \<Rightarrow> 'a::complete_space"
hoelzl@51531
  1556
  shows "Cauchy X = convergent X"
hoelzl@51531
  1557
by (fast intro: Cauchy_convergent convergent_Cauchy)
hoelzl@51531
  1558
hoelzl@51531
  1559
subsection {* The set of real numbers is a complete metric space *}
hoelzl@51531
  1560
hoelzl@51531
  1561
text {*
hoelzl@51531
  1562
Proof that Cauchy sequences converge based on the one from
wenzelm@54703
  1563
@{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
hoelzl@51531
  1564
*}
hoelzl@51531
  1565
hoelzl@51531
  1566
text {*
hoelzl@51531
  1567
  If sequence @{term "X"} is Cauchy, then its limit is the lub of
hoelzl@51531
  1568
  @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
hoelzl@51531
  1569
*}
hoelzl@51531
  1570
hoelzl@51531
  1571
lemma increasing_LIMSEQ:
hoelzl@51531
  1572
  fixes f :: "nat \<Rightarrow> real"
hoelzl@51531
  1573
  assumes inc: "\<And>n. f n \<le> f (Suc n)"
hoelzl@51531
  1574
      and bdd: "\<And>n. f n \<le> l"
hoelzl@51531
  1575
      and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
hoelzl@51531
  1576
  shows "f ----> l"
hoelzl@51531
  1577
proof (rule increasing_tendsto)
hoelzl@51531
  1578
  fix x assume "x < l"
hoelzl@51531
  1579
  with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
hoelzl@51531
  1580
    by auto
hoelzl@51531
  1581
  from en[OF `0 < e`] obtain n where "l - e \<le> f n"
hoelzl@51531
  1582
    by (auto simp: field_simps)
hoelzl@51531
  1583
  with `e < l - x` `0 < e` have "x < f n" by simp
hoelzl@51531
  1584
  with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
hoelzl@51531
  1585
    by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
hoelzl@51531
  1586
qed (insert bdd, auto)
hoelzl@51531
  1587
hoelzl@51531
  1588
lemma real_Cauchy_convergent:
hoelzl@51531
  1589
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51531
  1590
  assumes X: "Cauchy X"
hoelzl@51531
  1591
  shows "convergent X"
hoelzl@51531
  1592
proof -
hoelzl@51531
  1593
  def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
hoelzl@51531
  1594
  then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
hoelzl@51531
  1595
hoelzl@51531
  1596
  { fix N x assume N: "\<forall>n\<ge>N. X n < x"
hoelzl@51531
  1597
  fix y::real assume "y \<in> S"
hoelzl@51531
  1598
  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
hoelzl@51531
  1599
    by (simp add: S_def)
hoelzl@51531
  1600
  then obtain M where "\<forall>n\<ge>M. y < X n" ..
hoelzl@51531
  1601
  hence "y < X (max M N)" by simp
hoelzl@51531
  1602
  also have "\<dots> < x" using N by simp
hoelzl@54263
  1603
  finally have "y \<le> x"
hoelzl@54263
  1604
    by (rule order_less_imp_le) }
hoelzl@51531
  1605
  note bound_isUb = this 
hoelzl@51531
  1606
hoelzl@51531
  1607
  obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
hoelzl@51531
  1608
    using X[THEN metric_CauchyD, OF zero_less_one] by auto
hoelzl@51531
  1609
  hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
hoelzl@54263
  1610
  have [simp]: "S \<noteq> {}"
hoelzl@54263
  1611
  proof (intro exI ex_in_conv[THEN iffD1])
hoelzl@51531
  1612
    from N have "\<forall>n\<ge>N. X N - 1 < X n"
hoelzl@51531
  1613
      by (simp add: abs_diff_less_iff dist_real_def)
hoelzl@51531
  1614
    thus "X N - 1 \<in> S" by (rule mem_S)
hoelzl@51531
  1615
  qed
hoelzl@54263
  1616
  have [simp]: "bdd_above S"
hoelzl@51531
  1617
  proof
hoelzl@51531
  1618
    from N have "\<forall>n\<ge>N. X n < X N + 1"
hoelzl@51531
  1619
      by (simp add: abs_diff_less_iff dist_real_def)
hoelzl@54263
  1620
    thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
hoelzl@51531
  1621
      by (rule bound_isUb)
hoelzl@51531
  1622
  qed
hoelzl@54263
  1623
  have "X ----> Sup S"
hoelzl@51531
  1624
  proof (rule metric_LIMSEQ_I)
hoelzl@51531
  1625
  fix r::real assume "0 < r"
hoelzl@51531
  1626
  hence r: "0 < r/2" by simp
hoelzl@51531
  1627
  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
hoelzl@51531
  1628
    using metric_CauchyD [OF X r] by auto
hoelzl@51531
  1629
  hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
hoelzl@51531
  1630
  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
hoelzl@51531
  1631
    by (simp only: dist_real_def abs_diff_less_iff)
hoelzl@51531
  1632
hoelzl@51531
  1633
  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
hoelzl@51531
  1634
  hence "X N - r/2 \<in> S" by (rule mem_S)
hoelzl@54263
  1635
  hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
hoelzl@51531
  1636
hoelzl@51531
  1637
  from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
hoelzl@54263
  1638
  from bound_isUb[OF this]
hoelzl@54263
  1639
  have 2: "Sup S \<le> X N + r/2"
hoelzl@54263
  1640
    by (intro cSup_least) simp_all
hoelzl@51531
  1641
hoelzl@54263
  1642
  show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
hoelzl@51531
  1643
  proof (intro exI allI impI)
hoelzl@51531
  1644
    fix n assume n: "N \<le> n"
hoelzl@51531
  1645
    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
hoelzl@54263
  1646
    thus "dist (X n) (Sup S) < r" using 1 2
hoelzl@51531
  1647
      by (simp add: abs_diff_less_iff dist_real_def)
hoelzl@51531
  1648
  qed
hoelzl@51531
  1649
  qed
hoelzl@51531
  1650
  then show ?thesis unfolding convergent_def by auto
hoelzl@51531
  1651
qed
hoelzl@51531
  1652
hoelzl@51531
  1653
instance real :: complete_space
hoelzl@51531
  1654
  by intro_classes (rule real_Cauchy_convergent)
hoelzl@51531
  1655
hoelzl@51531
  1656
class banach = real_normed_vector + complete_space
hoelzl@51531
  1657
hoelzl@51531
  1658
instance real :: banach by default
hoelzl@51531
  1659
hoelzl@51531
  1660
lemma tendsto_at_topI_sequentially:
hoelzl@51531
  1661
  fixes f :: "real \<Rightarrow> real"
hoelzl@51531
  1662
  assumes mono: "mono f"
hoelzl@51531
  1663
  assumes limseq: "(\<lambda>n. f (real n)) ----> y"
hoelzl@51531
  1664
  shows "(f ---> y) at_top"
hoelzl@51531
  1665
proof (rule tendstoI)
hoelzl@51531
  1666
  fix e :: real assume "0 < e"
hoelzl@51531
  1667
  with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
hoelzl@51531
  1668
    by (auto simp: LIMSEQ_def dist_real_def)
hoelzl@51531
  1669
  { fix x :: real
wenzelm@53381
  1670
    obtain n where "x \<le> real_of_nat n"
wenzelm@53381
  1671
      using ex_le_of_nat[of x] ..
hoelzl@51531
  1672
    note monoD[OF mono this]
hoelzl@51531
  1673
    also have "f (real_of_nat n) \<le> y"
hoelzl@51531
  1674
      by (rule LIMSEQ_le_const[OF limseq])
hoelzl@51531
  1675
         (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
hoelzl@51531
  1676
    finally have "f x \<le> y" . }
hoelzl@51531
  1677
  note le = this
hoelzl@51531
  1678
  have "eventually (\<lambda>x. real N \<le> x) at_top"
hoelzl@51531
  1679
    by (rule eventually_ge_at_top)
hoelzl@51531
  1680
  then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
hoelzl@51531
  1681
  proof eventually_elim
hoelzl@51531
  1682
    fix x assume N': "real N \<le> x"
hoelzl@51531
  1683
    with N[of N] le have "y - f (real N) < e" by auto
hoelzl@51531
  1684
    moreover note monoD[OF mono N']
hoelzl@51531
  1685
    ultimately show "dist (f x) y < e"
hoelzl@51531
  1686
      using le[of x] by (auto simp: dist_real_def field_simps)
hoelzl@51531
  1687
  qed
hoelzl@51531
  1688
qed
hoelzl@51531
  1689
huffman@20504
  1690
end