src/HOL/Real_Vector_Spaces.thy
author haftmann
Tue Nov 19 10:05:53 2013 +0100 (2013-11-19)
changeset 54489 03ff4d1e6784
parent 54263 c4159fe6fa46
child 54703 499f92dc6e45
permissions -rw-r--r--
eliminiated neg_numeral in favour of - (numeral _)
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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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huffman@27553
   321
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
huffman@20504
   434
huffman@20504
   435
subsection {* Real normed vector spaces *}
huffman@20504
   436
hoelzl@51531
   437
class dist =
hoelzl@51531
   438
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
hoelzl@51531
   439
haftmann@29608
   440
class norm =
huffman@22636
   441
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   442
huffman@24520
   443
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   444
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   445
huffman@31289
   446
class dist_norm = dist + norm + minus +
huffman@31289
   447
  assumes dist_norm: "dist x y = norm (x - y)"
huffman@31289
   448
hoelzl@51531
   449
class open_dist = "open" + dist +
hoelzl@51531
   450
  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
   451
huffman@31492
   452
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
hoelzl@51002
   453
  assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   454
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@31586
   455
  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
hoelzl@51002
   456
begin
hoelzl@51002
   457
hoelzl@51002
   458
lemma norm_ge_zero [simp]: "0 \<le> norm x"
hoelzl@51002
   459
proof -
hoelzl@51002
   460
  have "0 = norm (x + -1 *\<^sub>R x)" 
hoelzl@51002
   461
    using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
hoelzl@51002
   462
  also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
hoelzl@51002
   463
  finally show ?thesis by simp
hoelzl@51002
   464
qed
hoelzl@51002
   465
hoelzl@51002
   466
end
huffman@20504
   467
haftmann@24588
   468
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   469
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   470
haftmann@24588
   471
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   472
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   473
haftmann@24588
   474
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   475
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   476
haftmann@24588
   477
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   478
huffman@22852
   479
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   480
proof
huffman@20554
   481
  fix x y :: 'a
huffman@20554
   482
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   483
    by (simp add: norm_mult)
huffman@22852
   484
next
huffman@22852
   485
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   486
    by (rule norm_mult)
huffman@22852
   487
  thus "norm (1::'a) = 1" by simp
huffman@20554
   488
qed
huffman@20554
   489
huffman@22852
   490
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   491
by simp
huffman@20504
   492
huffman@22852
   493
lemma zero_less_norm_iff [simp]:
huffman@22852
   494
  fixes x :: "'a::real_normed_vector"
huffman@22852
   495
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   496
by (simp add: order_less_le)
huffman@20504
   497
huffman@22852
   498
lemma norm_not_less_zero [simp]:
huffman@22852
   499
  fixes x :: "'a::real_normed_vector"
huffman@22852
   500
  shows "\<not> norm x < 0"
huffman@20828
   501
by (simp add: linorder_not_less)
huffman@20828
   502
huffman@22852
   503
lemma norm_le_zero_iff [simp]:
huffman@22852
   504
  fixes x :: "'a::real_normed_vector"
huffman@22852
   505
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   506
by (simp add: order_le_less)
huffman@20828
   507
huffman@20504
   508
lemma norm_minus_cancel [simp]:
huffman@20584
   509
  fixes x :: "'a::real_normed_vector"
huffman@20584
   510
  shows "norm (- x) = norm x"
huffman@20504
   511
proof -
huffman@21809
   512
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   513
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   514
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   515
    by (rule norm_scaleR)
huffman@20504
   516
  finally show ?thesis by simp
huffman@20504
   517
qed
huffman@20504
   518
huffman@20504
   519
lemma norm_minus_commute:
huffman@20584
   520
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   521
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   522
proof -
huffman@22898
   523
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   524
    by (rule norm_minus_cancel)
huffman@22898
   525
  thus ?thesis by simp
huffman@20504
   526
qed
huffman@20504
   527
huffman@20504
   528
lemma norm_triangle_ineq2:
huffman@20584
   529
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   530
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   531
proof -
huffman@20533
   532
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   533
    by (rule norm_triangle_ineq)
huffman@22898
   534
  thus ?thesis by simp
huffman@20504
   535
qed
huffman@20504
   536
huffman@20584
   537
lemma norm_triangle_ineq3:
huffman@20584
   538
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   539
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   540
apply (subst abs_le_iff)
huffman@20584
   541
apply auto
huffman@20584
   542
apply (rule norm_triangle_ineq2)
huffman@20584
   543
apply (subst norm_minus_commute)
huffman@20584
   544
apply (rule norm_triangle_ineq2)
huffman@20584
   545
done
huffman@20584
   546
huffman@20504
   547
lemma norm_triangle_ineq4:
huffman@20584
   548
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   549
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   550
proof -
huffman@22898
   551
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   552
    by (rule norm_triangle_ineq)
haftmann@54230
   553
  then show ?thesis by simp
huffman@22898
   554
qed
huffman@22898
   555
huffman@22898
   556
lemma norm_diff_ineq:
huffman@22898
   557
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   558
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   559
proof -
huffman@22898
   560
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   561
    by (rule norm_triangle_ineq2)
huffman@22898
   562
  thus ?thesis by simp
huffman@20504
   563
qed
huffman@20504
   564
huffman@20551
   565
lemma norm_diff_triangle_ineq:
huffman@20551
   566
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   567
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   568
proof -
huffman@20551
   569
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
haftmann@54230
   570
    by (simp add: algebra_simps)
huffman@20551
   571
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   572
    by (rule norm_triangle_ineq)
huffman@20551
   573
  finally show ?thesis .
huffman@20551
   574
qed
huffman@20551
   575
huffman@22857
   576
lemma abs_norm_cancel [simp]:
huffman@22857
   577
  fixes a :: "'a::real_normed_vector"
huffman@22857
   578
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   579
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   580
huffman@22880
   581
lemma norm_add_less:
huffman@22880
   582
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   583
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   584
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   585
huffman@22880
   586
lemma norm_mult_less:
huffman@22880
   587
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   588
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   589
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   590
apply (simp add: mult_strict_mono')
huffman@22880
   591
done
huffman@22880
   592
huffman@22857
   593
lemma norm_of_real [simp]:
huffman@22857
   594
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@31586
   595
unfolding of_real_def by simp
huffman@20560
   596
huffman@47108
   597
lemma norm_numeral [simp]:
huffman@47108
   598
  "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
huffman@47108
   599
by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
huffman@47108
   600
huffman@47108
   601
lemma norm_neg_numeral [simp]:
haftmann@54489
   602
  "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
huffman@47108
   603
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
huffman@22876
   604
huffman@22876
   605
lemma norm_of_int [simp]:
huffman@22876
   606
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   607
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   608
huffman@22876
   609
lemma norm_of_nat [simp]:
huffman@22876
   610
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   611
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   612
apply (subst norm_of_real, simp)
huffman@22876
   613
done
huffman@22876
   614
huffman@20504
   615
lemma nonzero_norm_inverse:
huffman@20504
   616
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   617
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   618
apply (rule inverse_unique [symmetric])
huffman@20504
   619
apply (simp add: norm_mult [symmetric])
huffman@20504
   620
done
huffman@20504
   621
huffman@20504
   622
lemma norm_inverse:
haftmann@36409
   623
  fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
huffman@20533
   624
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   625
apply (case_tac "a = 0", simp)
huffman@20504
   626
apply (erule nonzero_norm_inverse)
huffman@20504
   627
done
huffman@20504
   628
huffman@20584
   629
lemma nonzero_norm_divide:
huffman@20584
   630
  fixes a b :: "'a::real_normed_field"
huffman@20584
   631
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   632
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   633
huffman@20584
   634
lemma norm_divide:
haftmann@36409
   635
  fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
huffman@20584
   636
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   637
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   638
huffman@22852
   639
lemma norm_power_ineq:
haftmann@31017
   640
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@22852
   641
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   642
proof (induct n)
huffman@22852
   643
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   644
next
huffman@22852
   645
  case (Suc n)
huffman@22852
   646
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   647
    by (rule norm_mult_ineq)
huffman@22852
   648
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   649
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   650
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@30273
   651
    by simp
huffman@22852
   652
qed
huffman@22852
   653
huffman@20684
   654
lemma norm_power:
haftmann@31017
   655
  fixes x :: "'a::{real_normed_div_algebra}"
huffman@20684
   656
  shows "norm (x ^ n) = norm x ^ n"
huffman@30273
   657
by (induct n) (simp_all add: norm_mult)
huffman@20684
   658
hoelzl@51531
   659
hoelzl@51531
   660
subsection {* Metric spaces *}
hoelzl@51531
   661
hoelzl@51531
   662
class metric_space = open_dist +
hoelzl@51531
   663
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
hoelzl@51531
   664
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
hoelzl@51531
   665
begin
hoelzl@51531
   666
hoelzl@51531
   667
lemma dist_self [simp]: "dist x x = 0"
hoelzl@51531
   668
by simp
hoelzl@51531
   669
hoelzl@51531
   670
lemma zero_le_dist [simp]: "0 \<le> dist x y"
hoelzl@51531
   671
using dist_triangle2 [of x x y] by simp
hoelzl@51531
   672
hoelzl@51531
   673
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
hoelzl@51531
   674
by (simp add: less_le)
hoelzl@51531
   675
hoelzl@51531
   676
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
hoelzl@51531
   677
by (simp add: not_less)
hoelzl@51531
   678
hoelzl@51531
   679
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
hoelzl@51531
   680
by (simp add: le_less)
hoelzl@51531
   681
hoelzl@51531
   682
lemma dist_commute: "dist x y = dist y x"
hoelzl@51531
   683
proof (rule order_antisym)
hoelzl@51531
   684
  show "dist x y \<le> dist y x"
hoelzl@51531
   685
    using dist_triangle2 [of x y x] by simp
hoelzl@51531
   686
  show "dist y x \<le> dist x y"
hoelzl@51531
   687
    using dist_triangle2 [of y x y] by simp
hoelzl@51531
   688
qed
hoelzl@51531
   689
hoelzl@51531
   690
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
hoelzl@51531
   691
using dist_triangle2 [of x z y] by (simp add: dist_commute)
hoelzl@51531
   692
hoelzl@51531
   693
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
hoelzl@51531
   694
using dist_triangle2 [of x y a] by (simp add: dist_commute)
hoelzl@51531
   695
hoelzl@51531
   696
lemma dist_triangle_alt:
hoelzl@51531
   697
  shows "dist y z <= dist x y + dist x z"
hoelzl@51531
   698
by (rule dist_triangle3)
hoelzl@51531
   699
hoelzl@51531
   700
lemma dist_pos_lt:
hoelzl@51531
   701
  shows "x \<noteq> y ==> 0 < dist x y"
hoelzl@51531
   702
by (simp add: zero_less_dist_iff)
hoelzl@51531
   703
hoelzl@51531
   704
lemma dist_nz:
hoelzl@51531
   705
  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
hoelzl@51531
   706
by (simp add: zero_less_dist_iff)
hoelzl@51531
   707
hoelzl@51531
   708
lemma dist_triangle_le:
hoelzl@51531
   709
  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
hoelzl@51531
   710
by (rule order_trans [OF dist_triangle2])
hoelzl@51531
   711
hoelzl@51531
   712
lemma dist_triangle_lt:
hoelzl@51531
   713
  shows "dist x z + dist y z < e ==> dist x y < e"
hoelzl@51531
   714
by (rule le_less_trans [OF dist_triangle2])
hoelzl@51531
   715
hoelzl@51531
   716
lemma dist_triangle_half_l:
hoelzl@51531
   717
  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
hoelzl@51531
   718
by (rule dist_triangle_lt [where z=y], simp)
hoelzl@51531
   719
hoelzl@51531
   720
lemma dist_triangle_half_r:
hoelzl@51531
   721
  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
hoelzl@51531
   722
by (rule dist_triangle_half_l, simp_all add: dist_commute)
hoelzl@51531
   723
hoelzl@51531
   724
subclass topological_space
hoelzl@51531
   725
proof
hoelzl@51531
   726
  have "\<exists>e::real. 0 < e"
hoelzl@51531
   727
    by (fast intro: zero_less_one)
hoelzl@51531
   728
  then show "open UNIV"
hoelzl@51531
   729
    unfolding open_dist by simp
hoelzl@51531
   730
next
hoelzl@51531
   731
  fix S T assume "open S" "open T"
hoelzl@51531
   732
  then show "open (S \<inter> T)"
hoelzl@51531
   733
    unfolding open_dist
hoelzl@51531
   734
    apply clarify
hoelzl@51531
   735
    apply (drule (1) bspec)+
hoelzl@51531
   736
    apply (clarify, rename_tac r s)
hoelzl@51531
   737
    apply (rule_tac x="min r s" in exI, simp)
hoelzl@51531
   738
    done
hoelzl@51531
   739
next
hoelzl@51531
   740
  fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
hoelzl@51531
   741
    unfolding open_dist by fast
hoelzl@51531
   742
qed
hoelzl@51531
   743
hoelzl@51531
   744
lemma open_ball: "open {y. dist x y < d}"
hoelzl@51531
   745
proof (unfold open_dist, intro ballI)
hoelzl@51531
   746
  fix y assume *: "y \<in> {y. dist x y < d}"
hoelzl@51531
   747
  then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
hoelzl@51531
   748
    by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
hoelzl@51531
   749
qed
hoelzl@51531
   750
hoelzl@51531
   751
subclass first_countable_topology
hoelzl@51531
   752
proof
hoelzl@51531
   753
  fix x 
hoelzl@51531
   754
  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
   755
  proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
hoelzl@51531
   756
    fix S assume "open S" "x \<in> S"
wenzelm@53374
   757
    then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
hoelzl@51531
   758
      by (auto simp: open_dist subset_eq dist_commute)
hoelzl@51531
   759
    moreover
wenzelm@53374
   760
    from e obtain i where "inverse (Suc i) < e"
hoelzl@51531
   761
      by (auto dest!: reals_Archimedean)
hoelzl@51531
   762
    then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
hoelzl@51531
   763
      by auto
hoelzl@51531
   764
    ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
hoelzl@51531
   765
      by blast
hoelzl@51531
   766
  qed (auto intro: open_ball)
hoelzl@51531
   767
qed
hoelzl@51531
   768
hoelzl@51531
   769
end
hoelzl@51531
   770
hoelzl@51531
   771
instance metric_space \<subseteq> t2_space
hoelzl@51531
   772
proof
hoelzl@51531
   773
  fix x y :: "'a::metric_space"
hoelzl@51531
   774
  assume xy: "x \<noteq> y"
hoelzl@51531
   775
  let ?U = "{y'. dist x y' < dist x y / 2}"
hoelzl@51531
   776
  let ?V = "{x'. dist y x' < dist x y / 2}"
hoelzl@51531
   777
  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
   778
               \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
hoelzl@51531
   779
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
hoelzl@51531
   780
    using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
hoelzl@51531
   781
    using open_ball[of _ "dist x y / 2"] by auto
hoelzl@51531
   782
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
hoelzl@51531
   783
    by blast
hoelzl@51531
   784
qed
hoelzl@51531
   785
huffman@31289
   786
text {* Every normed vector space is a metric space. *}
huffman@31285
   787
huffman@31289
   788
instance real_normed_vector < metric_space
huffman@31289
   789
proof
huffman@31289
   790
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   791
    unfolding dist_norm by simp
huffman@31289
   792
next
huffman@31289
   793
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
huffman@31289
   794
    unfolding dist_norm
huffman@31289
   795
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
huffman@31289
   796
qed
huffman@31285
   797
huffman@31564
   798
subsection {* Class instances for real numbers *}
huffman@31564
   799
huffman@31564
   800
instantiation real :: real_normed_field
huffman@31564
   801
begin
huffman@31564
   802
hoelzl@51531
   803
definition dist_real_def:
hoelzl@51531
   804
  "dist x y = \<bar>x - y\<bar>"
hoelzl@51531
   805
haftmann@52381
   806
definition open_real_def [code del]:
hoelzl@51531
   807
  "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
hoelzl@51531
   808
huffman@31564
   809
definition real_norm_def [simp]:
huffman@31564
   810
  "norm r = \<bar>r\<bar>"
huffman@31564
   811
huffman@31564
   812
instance
huffman@31564
   813
apply (intro_classes, unfold real_norm_def real_scaleR_def)
huffman@31564
   814
apply (rule dist_real_def)
hoelzl@51531
   815
apply (rule open_real_def)
huffman@36795
   816
apply (simp add: sgn_real_def)
huffman@31564
   817
apply (rule abs_eq_0)
huffman@31564
   818
apply (rule abs_triangle_ineq)
huffman@31564
   819
apply (rule abs_mult)
huffman@31564
   820
apply (rule abs_mult)
huffman@31564
   821
done
huffman@31564
   822
huffman@31564
   823
end
huffman@31564
   824
haftmann@52381
   825
code_abort "open :: real set \<Rightarrow> bool"
haftmann@52381
   826
hoelzl@51531
   827
instance real :: linorder_topology
hoelzl@51531
   828
proof
hoelzl@51531
   829
  show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
hoelzl@51531
   830
  proof (rule ext, safe)
hoelzl@51531
   831
    fix S :: "real set" assume "open S"
wenzelm@53381
   832
    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
   833
      unfolding open_real_def bchoice_iff ..
hoelzl@51531
   834
    then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
hoelzl@51531
   835
      by (fastforce simp: dist_real_def)
hoelzl@51531
   836
    show "generate_topology (range lessThan \<union> range greaterThan) S"
hoelzl@51531
   837
      apply (subst *)
hoelzl@51531
   838
      apply (intro generate_topology_Union generate_topology.Int)
hoelzl@51531
   839
      apply (auto intro: generate_topology.Basis)
hoelzl@51531
   840
      done
hoelzl@51531
   841
  next
hoelzl@51531
   842
    fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
hoelzl@51531
   843
    moreover have "\<And>a::real. open {..<a}"
hoelzl@51531
   844
      unfolding open_real_def dist_real_def
hoelzl@51531
   845
    proof clarify
hoelzl@51531
   846
      fix x a :: real assume "x < a"
hoelzl@51531
   847
      hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
hoelzl@51531
   848
      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
hoelzl@51531
   849
    qed
hoelzl@51531
   850
    moreover have "\<And>a::real. open {a <..}"
hoelzl@51531
   851
      unfolding open_real_def dist_real_def
hoelzl@51531
   852
    proof clarify
hoelzl@51531
   853
      fix x a :: real assume "a < x"
hoelzl@51531
   854
      hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
hoelzl@51531
   855
      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
hoelzl@51531
   856
    qed
hoelzl@51531
   857
    ultimately show "open S"
hoelzl@51531
   858
      by induct auto
hoelzl@51531
   859
  qed
hoelzl@51531
   860
qed
hoelzl@51531
   861
hoelzl@51775
   862
instance real :: linear_continuum_topology ..
hoelzl@51518
   863
hoelzl@51531
   864
lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
hoelzl@51531
   865
lemmas open_real_lessThan = open_lessThan[where 'a=real]
hoelzl@51531
   866
lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
hoelzl@51531
   867
lemmas closed_real_atMost = closed_atMost[where 'a=real]
hoelzl@51531
   868
lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
hoelzl@51531
   869
lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
hoelzl@51531
   870
huffman@31446
   871
subsection {* Extra type constraints *}
huffman@31446
   872
huffman@31492
   873
text {* Only allow @{term "open"} in class @{text topological_space}. *}
huffman@31492
   874
huffman@31492
   875
setup {* Sign.add_const_constraint
huffman@31492
   876
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
huffman@31492
   877
huffman@31446
   878
text {* Only allow @{term dist} in class @{text metric_space}. *}
huffman@31446
   879
huffman@31446
   880
setup {* Sign.add_const_constraint
huffman@31446
   881
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
huffman@31446
   882
huffman@31446
   883
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
huffman@31446
   884
huffman@31446
   885
setup {* Sign.add_const_constraint
huffman@31446
   886
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
huffman@31446
   887
huffman@22972
   888
subsection {* Sign function *}
huffman@22972
   889
nipkow@24506
   890
lemma norm_sgn:
nipkow@24506
   891
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
huffman@31586
   892
by (simp add: sgn_div_norm)
huffman@22972
   893
nipkow@24506
   894
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   895
by (simp add: sgn_div_norm)
huffman@22972
   896
nipkow@24506
   897
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   898
by (simp add: sgn_div_norm)
huffman@22972
   899
nipkow@24506
   900
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   901
by (simp add: sgn_div_norm)
huffman@22972
   902
nipkow@24506
   903
lemma sgn_scaleR:
nipkow@24506
   904
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
huffman@31586
   905
by (simp add: sgn_div_norm mult_ac)
huffman@22973
   906
huffman@22972
   907
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   908
by (simp add: sgn_div_norm)
huffman@22972
   909
huffman@22972
   910
lemma sgn_of_real:
huffman@22972
   911
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   912
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   913
huffman@22973
   914
lemma sgn_mult:
huffman@22973
   915
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   916
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   917
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   918
huffman@22972
   919
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   920
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   921
huffman@22972
   922
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   923
unfolding real_sgn_eq by simp
huffman@22972
   924
huffman@22972
   925
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   926
unfolding real_sgn_eq by simp
huffman@22972
   927
hoelzl@51474
   928
lemma norm_conv_dist: "norm x = dist x 0"
hoelzl@51474
   929
  unfolding dist_norm by simp
huffman@22972
   930
huffman@22442
   931
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   932
huffman@53600
   933
locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
huffman@22442
   934
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@53600
   935
huffman@53600
   936
lemma linearI:
huffman@53600
   937
  assumes "\<And>x y. f (x + y) = f x + f y"
huffman@53600
   938
  assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
huffman@53600
   939
  shows "linear f"
huffman@53600
   940
  by default (rule assms)+
huffman@53600
   941
huffman@53600
   942
locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
huffman@22442
   943
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
   944
begin
huffman@22442
   945
huffman@27443
   946
lemma pos_bounded:
huffman@22442
   947
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   948
proof -
huffman@22442
   949
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   950
    using bounded by fast
huffman@22442
   951
  show ?thesis
huffman@22442
   952
  proof (intro exI impI conjI allI)
huffman@22442
   953
    show "0 < max 1 K"
huffman@22442
   954
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   955
  next
huffman@22442
   956
    fix x
huffman@22442
   957
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   958
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   959
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   960
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   961
  qed
huffman@22442
   962
qed
huffman@22442
   963
huffman@27443
   964
lemma nonneg_bounded:
huffman@22442
   965
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   966
proof -
huffman@22442
   967
  from pos_bounded
huffman@22442
   968
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   969
qed
huffman@22442
   970
huffman@27443
   971
end
huffman@27443
   972
huffman@44127
   973
lemma bounded_linear_intro:
huffman@44127
   974
  assumes "\<And>x y. f (x + y) = f x + f y"
huffman@44127
   975
  assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
huffman@44127
   976
  assumes "\<And>x. norm (f x) \<le> norm x * K"
huffman@44127
   977
  shows "bounded_linear f"
huffman@44127
   978
  by default (fast intro: assms)+
huffman@44127
   979
huffman@22442
   980
locale bounded_bilinear =
huffman@22442
   981
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   982
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   983
    (infixl "**" 70)
huffman@22442
   984
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   985
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   986
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   987
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   988
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
   989
begin
huffman@22442
   990
huffman@27443
   991
lemma pos_bounded:
huffman@22442
   992
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   993
apply (cut_tac bounded, erule exE)
huffman@22442
   994
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   995
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   996
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   997
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   998
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   999
done
huffman@22442
  1000
huffman@27443
  1001
lemma nonneg_bounded:
huffman@22442
  1002
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
  1003
proof -
huffman@22442
  1004
  from pos_bounded
huffman@22442
  1005
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
  1006
qed
huffman@22442
  1007
huffman@27443
  1008
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
  1009
by (rule additive.intro, rule add_right)
huffman@22442
  1010
huffman@27443
  1011
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
  1012
by (rule additive.intro, rule add_left)
huffman@22442
  1013
huffman@27443
  1014
lemma zero_left: "prod 0 b = 0"
huffman@22442
  1015
by (rule additive.zero [OF additive_left])
huffman@22442
  1016
huffman@27443
  1017
lemma zero_right: "prod a 0 = 0"
huffman@22442
  1018
by (rule additive.zero [OF additive_right])
huffman@22442
  1019
huffman@27443
  1020
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
  1021
by (rule additive.minus [OF additive_left])
huffman@22442
  1022
huffman@27443
  1023
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
  1024
by (rule additive.minus [OF additive_right])
huffman@22442
  1025
huffman@27443
  1026
lemma diff_left:
huffman@22442
  1027
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
  1028
by (rule additive.diff [OF additive_left])
huffman@22442
  1029
huffman@27443
  1030
lemma diff_right:
huffman@22442
  1031
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
  1032
by (rule additive.diff [OF additive_right])
huffman@22442
  1033
huffman@27443
  1034
lemma bounded_linear_left:
huffman@22442
  1035
  "bounded_linear (\<lambda>a. a ** b)"
huffman@44127
  1036
apply (cut_tac bounded, safe)
huffman@44127
  1037
apply (rule_tac K="norm b * K" in bounded_linear_intro)
huffman@22442
  1038
apply (rule add_left)
huffman@22442
  1039
apply (rule scaleR_left)
huffman@22442
  1040
apply (simp add: mult_ac)
huffman@22442
  1041
done
huffman@22442
  1042
huffman@27443
  1043
lemma bounded_linear_right:
huffman@22442
  1044
  "bounded_linear (\<lambda>b. a ** b)"
huffman@44127
  1045
apply (cut_tac bounded, safe)
huffman@44127
  1046
apply (rule_tac K="norm a * K" in bounded_linear_intro)
huffman@22442
  1047
apply (rule add_right)
huffman@22442
  1048
apply (rule scaleR_right)
huffman@22442
  1049
apply (simp add: mult_ac)
huffman@22442
  1050
done
huffman@22442
  1051
huffman@27443
  1052
lemma prod_diff_prod:
huffman@22442
  1053
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
  1054
by (simp add: diff_left diff_right)
huffman@22442
  1055
huffman@27443
  1056
end
huffman@27443
  1057
hoelzl@51642
  1058
lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
hoelzl@51642
  1059
  by default (auto intro!: exI[of _ 1])
hoelzl@51642
  1060
hoelzl@51642
  1061
lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
hoelzl@51642
  1062
  by default (auto intro!: exI[of _ 1])
hoelzl@51642
  1063
hoelzl@51642
  1064
lemma bounded_linear_add:
hoelzl@51642
  1065
  assumes "bounded_linear f"
hoelzl@51642
  1066
  assumes "bounded_linear g"
hoelzl@51642
  1067
  shows "bounded_linear (\<lambda>x. f x + g x)"
hoelzl@51642
  1068
proof -
hoelzl@51642
  1069
  interpret f: bounded_linear f by fact
hoelzl@51642
  1070
  interpret g: bounded_linear g by fact
hoelzl@51642
  1071
  show ?thesis
hoelzl@51642
  1072
  proof
hoelzl@51642
  1073
    from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
hoelzl@51642
  1074
    from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
hoelzl@51642
  1075
    show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
hoelzl@51642
  1076
      using add_mono[OF Kf Kg]
hoelzl@51642
  1077
      by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
hoelzl@51642
  1078
  qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
hoelzl@51642
  1079
qed
hoelzl@51642
  1080
hoelzl@51642
  1081
lemma bounded_linear_minus:
hoelzl@51642
  1082
  assumes "bounded_linear f"
hoelzl@51642
  1083
  shows "bounded_linear (\<lambda>x. - f x)"
hoelzl@51642
  1084
proof -
hoelzl@51642
  1085
  interpret f: bounded_linear f by fact
hoelzl@51642
  1086
  show ?thesis apply (unfold_locales)
hoelzl@51642
  1087
    apply (simp add: f.add)
hoelzl@51642
  1088
    apply (simp add: f.scaleR)
hoelzl@51642
  1089
    apply (simp add: f.bounded)
hoelzl@51642
  1090
    done
hoelzl@51642
  1091
qed
hoelzl@51642
  1092
hoelzl@51642
  1093
lemma bounded_linear_compose:
hoelzl@51642
  1094
  assumes "bounded_linear f"
hoelzl@51642
  1095
  assumes "bounded_linear g"
hoelzl@51642
  1096
  shows "bounded_linear (\<lambda>x. f (g x))"
hoelzl@51642
  1097
proof -
hoelzl@51642
  1098
  interpret f: bounded_linear f by fact
hoelzl@51642
  1099
  interpret g: bounded_linear g by fact
hoelzl@51642
  1100
  show ?thesis proof (unfold_locales)
hoelzl@51642
  1101
    fix x y show "f (g (x + y)) = f (g x) + f (g y)"
hoelzl@51642
  1102
      by (simp only: f.add g.add)
hoelzl@51642
  1103
  next
hoelzl@51642
  1104
    fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
hoelzl@51642
  1105
      by (simp only: f.scaleR g.scaleR)
hoelzl@51642
  1106
  next
hoelzl@51642
  1107
    from f.pos_bounded
hoelzl@51642
  1108
    obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
hoelzl@51642
  1109
    from g.pos_bounded
hoelzl@51642
  1110
    obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
hoelzl@51642
  1111
    show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
hoelzl@51642
  1112
    proof (intro exI allI)
hoelzl@51642
  1113
      fix x
hoelzl@51642
  1114
      have "norm (f (g x)) \<le> norm (g x) * Kf"
hoelzl@51642
  1115
        using f .
hoelzl@51642
  1116
      also have "\<dots> \<le> (norm x * Kg) * Kf"
hoelzl@51642
  1117
        using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
hoelzl@51642
  1118
      also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
hoelzl@51642
  1119
        by (rule mult_assoc)
hoelzl@51642
  1120
      finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
hoelzl@51642
  1121
    qed
hoelzl@51642
  1122
  qed
hoelzl@51642
  1123
qed
hoelzl@51642
  1124
huffman@44282
  1125
lemma bounded_bilinear_mult:
huffman@44282
  1126
  "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
huffman@22442
  1127
apply (rule bounded_bilinear.intro)
webertj@49962
  1128
apply (rule distrib_right)
webertj@49962
  1129
apply (rule distrib_left)
huffman@22442
  1130
apply (rule mult_scaleR_left)
huffman@22442
  1131
apply (rule mult_scaleR_right)
huffman@22442
  1132
apply (rule_tac x="1" in exI)
huffman@22442
  1133
apply (simp add: norm_mult_ineq)
huffman@22442
  1134
done
huffman@22442
  1135
huffman@44282
  1136
lemma bounded_linear_mult_left:
huffman@44282
  1137
  "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
huffman@44282
  1138
  using bounded_bilinear_mult
huffman@44282
  1139
  by (rule bounded_bilinear.bounded_linear_left)
huffman@22442
  1140
huffman@44282
  1141
lemma bounded_linear_mult_right:
huffman@44282
  1142
  "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
huffman@44282
  1143
  using bounded_bilinear_mult
huffman@44282
  1144
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
  1145
hoelzl@51642
  1146
lemmas bounded_linear_mult_const =
hoelzl@51642
  1147
  bounded_linear_mult_left [THEN bounded_linear_compose]
hoelzl@51642
  1148
hoelzl@51642
  1149
lemmas bounded_linear_const_mult =
hoelzl@51642
  1150
  bounded_linear_mult_right [THEN bounded_linear_compose]
hoelzl@51642
  1151
huffman@44282
  1152
lemma bounded_linear_divide:
huffman@44282
  1153
  "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
huffman@44282
  1154
  unfolding divide_inverse by (rule bounded_linear_mult_left)
huffman@23120
  1155
huffman@44282
  1156
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
huffman@22442
  1157
apply (rule bounded_bilinear.intro)
huffman@22442
  1158
apply (rule scaleR_left_distrib)
huffman@22442
  1159
apply (rule scaleR_right_distrib)
huffman@22973
  1160
apply simp
huffman@22442
  1161
apply (rule scaleR_left_commute)
huffman@31586
  1162
apply (rule_tac x="1" in exI, simp)
huffman@22442
  1163
done
huffman@22442
  1164
huffman@44282
  1165
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
huffman@44282
  1166
  using bounded_bilinear_scaleR
huffman@44282
  1167
  by (rule bounded_bilinear.bounded_linear_left)
huffman@23127
  1168
huffman@44282
  1169
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
huffman@44282
  1170
  using bounded_bilinear_scaleR
huffman@44282
  1171
  by (rule bounded_bilinear.bounded_linear_right)
huffman@23127
  1172
huffman@44282
  1173
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
huffman@44282
  1174
  unfolding of_real_def by (rule bounded_linear_scaleR_left)
huffman@22625
  1175
hoelzl@51642
  1176
lemma real_bounded_linear:
hoelzl@51642
  1177
  fixes f :: "real \<Rightarrow> real"
hoelzl@51642
  1178
  shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
hoelzl@51642
  1179
proof -
hoelzl@51642
  1180
  { fix x assume "bounded_linear f"
hoelzl@51642
  1181
    then interpret bounded_linear f .
hoelzl@51642
  1182
    from scaleR[of x 1] have "f x = x * f 1"
hoelzl@51642
  1183
      by simp }
hoelzl@51642
  1184
  then show ?thesis
hoelzl@51642
  1185
    by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
hoelzl@51642
  1186
qed
hoelzl@51642
  1187
huffman@44571
  1188
instance real_normed_algebra_1 \<subseteq> perfect_space
huffman@44571
  1189
proof
huffman@44571
  1190
  fix x::'a
huffman@44571
  1191
  show "\<not> open {x}"
huffman@44571
  1192
    unfolding open_dist dist_norm
huffman@44571
  1193
    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
huffman@44571
  1194
qed
huffman@44571
  1195
hoelzl@51531
  1196
subsection {* Filters and Limits on Metric Space *}
hoelzl@51531
  1197
hoelzl@51531
  1198
lemma eventually_nhds_metric:
hoelzl@51531
  1199
  fixes a :: "'a :: metric_space"
hoelzl@51531
  1200
  shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
hoelzl@51531
  1201
unfolding eventually_nhds open_dist
hoelzl@51531
  1202
apply safe
hoelzl@51531
  1203
apply fast
hoelzl@51531
  1204
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
hoelzl@51531
  1205
apply clarsimp
hoelzl@51531
  1206
apply (rule_tac x="d - dist x a" in exI, clarsimp)
hoelzl@51531
  1207
apply (simp only: less_diff_eq)
hoelzl@51531
  1208
apply (erule le_less_trans [OF dist_triangle])
hoelzl@51531
  1209
done
hoelzl@51531
  1210
hoelzl@51531
  1211
lemma eventually_at:
hoelzl@51641
  1212
  fixes a :: "'a :: metric_space"
hoelzl@51641
  1213
  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
  1214
  unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)
hoelzl@51531
  1215
hoelzl@51641
  1216
lemma eventually_at_le:
hoelzl@51641
  1217
  fixes a :: "'a::metric_space"
hoelzl@51641
  1218
  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
  1219
  unfolding eventually_at_filter eventually_nhds_metric
hoelzl@51641
  1220
  apply auto
hoelzl@51641
  1221
  apply (rule_tac x="d / 2" in exI)
hoelzl@51641
  1222
  apply auto
hoelzl@51641
  1223
  done
hoelzl@51531
  1224
hoelzl@51531
  1225
lemma tendstoI:
hoelzl@51531
  1226
  fixes l :: "'a :: metric_space"
hoelzl@51531
  1227
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
hoelzl@51531
  1228
  shows "(f ---> l) F"
hoelzl@51531
  1229
  apply (rule topological_tendstoI)
hoelzl@51531
  1230
  apply (simp add: open_dist)
hoelzl@51531
  1231
  apply (drule (1) bspec, clarify)
hoelzl@51531
  1232
  apply (drule assms)
hoelzl@51531
  1233
  apply (erule eventually_elim1, simp)
hoelzl@51531
  1234
  done
hoelzl@51531
  1235
hoelzl@51531
  1236
lemma tendstoD:
hoelzl@51531
  1237
  fixes l :: "'a :: metric_space"
hoelzl@51531
  1238
  shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
hoelzl@51531
  1239
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
hoelzl@51531
  1240
  apply (clarsimp simp add: open_dist)
hoelzl@51531
  1241
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
hoelzl@51531
  1242
  apply (simp only: less_diff_eq)
hoelzl@51531
  1243
  apply (erule le_less_trans [OF dist_triangle])
hoelzl@51531
  1244
  apply simp
hoelzl@51531
  1245
  apply simp
hoelzl@51531
  1246
  done
hoelzl@51531
  1247
hoelzl@51531
  1248
lemma tendsto_iff:
hoelzl@51531
  1249
  fixes l :: "'a :: metric_space"
hoelzl@51531
  1250
  shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
hoelzl@51531
  1251
  using tendstoI tendstoD by fast
hoelzl@51531
  1252
hoelzl@51531
  1253
lemma metric_tendsto_imp_tendsto:
hoelzl@51531
  1254
  fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
hoelzl@51531
  1255
  assumes f: "(f ---> a) F"
hoelzl@51531
  1256
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
hoelzl@51531
  1257
  shows "(g ---> b) F"
hoelzl@51531
  1258
proof (rule tendstoI)
hoelzl@51531
  1259
  fix e :: real assume "0 < e"
hoelzl@51531
  1260
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
hoelzl@51531
  1261
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
hoelzl@51531
  1262
    using le_less_trans by (rule eventually_elim2)
hoelzl@51531
  1263
qed
hoelzl@51531
  1264
hoelzl@51531
  1265
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
hoelzl@51531
  1266
  unfolding filterlim_at_top
hoelzl@51531
  1267
  apply (intro allI)
hoelzl@51531
  1268
  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
hoelzl@51531
  1269
  apply (auto simp: natceiling_le_eq)
hoelzl@51531
  1270
  done
hoelzl@51531
  1271
hoelzl@51531
  1272
subsubsection {* Limits of Sequences *}
hoelzl@51531
  1273
hoelzl@51531
  1274
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
  1275
  unfolding tendsto_iff eventually_sequentially ..
hoelzl@51531
  1276
hoelzl@51531
  1277
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
  1278
  unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
hoelzl@51531
  1279
hoelzl@51531
  1280
lemma metric_LIMSEQ_I:
hoelzl@51531
  1281
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
hoelzl@51531
  1282
by (simp add: LIMSEQ_def)
hoelzl@51531
  1283
hoelzl@51531
  1284
lemma metric_LIMSEQ_D:
hoelzl@51531
  1285
  "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
hoelzl@51531
  1286
by (simp add: LIMSEQ_def)
hoelzl@51531
  1287
hoelzl@51531
  1288
hoelzl@51531
  1289
subsubsection {* Limits of Functions *}
hoelzl@51531
  1290
hoelzl@51531
  1291
lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
hoelzl@51531
  1292
     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
hoelzl@51531
  1293
        --> dist (f x) L < r)"
hoelzl@51641
  1294
  unfolding tendsto_iff eventually_at by simp
hoelzl@51531
  1295
hoelzl@51531
  1296
lemma metric_LIM_I:
hoelzl@51531
  1297
  "(\<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
  1298
    \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
hoelzl@51531
  1299
by (simp add: LIM_def)
hoelzl@51531
  1300
hoelzl@51531
  1301
lemma metric_LIM_D:
hoelzl@51531
  1302
  "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
hoelzl@51531
  1303
    \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
hoelzl@51531
  1304
by (simp add: LIM_def)
hoelzl@51531
  1305
hoelzl@51531
  1306
lemma metric_LIM_imp_LIM:
hoelzl@51531
  1307
  assumes f: "f -- a --> (l::'a::metric_space)"
hoelzl@51531
  1308
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
hoelzl@51531
  1309
  shows "g -- a --> (m::'b::metric_space)"
hoelzl@51531
  1310
  by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
hoelzl@51531
  1311
hoelzl@51531
  1312
lemma metric_LIM_equal2:
hoelzl@51531
  1313
  assumes 1: "0 < R"
hoelzl@51531
  1314
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
hoelzl@51531
  1315
  shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
hoelzl@51531
  1316
apply (rule topological_tendstoI)
hoelzl@51531
  1317
apply (drule (2) topological_tendstoD)
hoelzl@51531
  1318
apply (simp add: eventually_at, safe)
hoelzl@51531
  1319
apply (rule_tac x="min d R" in exI, safe)
hoelzl@51531
  1320
apply (simp add: 1)
hoelzl@51531
  1321
apply (simp add: 2)
hoelzl@51531
  1322
done
hoelzl@51531
  1323
hoelzl@51531
  1324
lemma metric_LIM_compose2:
hoelzl@51531
  1325
  assumes f: "f -- (a::'a::metric_space) --> b"
hoelzl@51531
  1326
  assumes g: "g -- b --> c"
hoelzl@51531
  1327
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
hoelzl@51531
  1328
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51641
  1329
  using inj
hoelzl@51641
  1330
  by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
hoelzl@51531
  1331
hoelzl@51531
  1332
lemma metric_isCont_LIM_compose2:
hoelzl@51531
  1333
  fixes f :: "'a :: metric_space \<Rightarrow> _"
hoelzl@51531
  1334
  assumes f [unfolded isCont_def]: "isCont f a"
hoelzl@51531
  1335
  assumes g: "g -- f a --> l"
hoelzl@51531
  1336
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
hoelzl@51531
  1337
  shows "(\<lambda>x. g (f x)) -- a --> l"
hoelzl@51531
  1338
by (rule metric_LIM_compose2 [OF f g inj])
hoelzl@51531
  1339
hoelzl@51531
  1340
subsection {* Complete metric spaces *}
hoelzl@51531
  1341
hoelzl@51531
  1342
subsection {* Cauchy sequences *}
hoelzl@51531
  1343
hoelzl@51531
  1344
definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
hoelzl@51531
  1345
  "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
hoelzl@51531
  1346
hoelzl@51531
  1347
subsection {* Cauchy Sequences *}
hoelzl@51531
  1348
hoelzl@51531
  1349
lemma metric_CauchyI:
hoelzl@51531
  1350
  "(\<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
  1351
  by (simp add: Cauchy_def)
hoelzl@51531
  1352
hoelzl@51531
  1353
lemma metric_CauchyD:
hoelzl@51531
  1354
  "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
hoelzl@51531
  1355
  by (simp add: Cauchy_def)
hoelzl@51531
  1356
hoelzl@51531
  1357
lemma metric_Cauchy_iff2:
hoelzl@51531
  1358
  "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
  1359
apply (simp add: Cauchy_def, auto)
hoelzl@51531
  1360
apply (drule reals_Archimedean, safe)
hoelzl@51531
  1361
apply (drule_tac x = n in spec, auto)
hoelzl@51531
  1362
apply (rule_tac x = M in exI, auto)
hoelzl@51531
  1363
apply (drule_tac x = m in spec, simp)
hoelzl@51531
  1364
apply (drule_tac x = na in spec, auto)
hoelzl@51531
  1365
done
hoelzl@51531
  1366
hoelzl@51531
  1367
lemma Cauchy_iff2:
hoelzl@51531
  1368
  "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
  1369
  unfolding metric_Cauchy_iff2 dist_real_def ..
hoelzl@51531
  1370
hoelzl@51531
  1371
lemma Cauchy_subseq_Cauchy:
hoelzl@51531
  1372
  "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
hoelzl@51531
  1373
apply (auto simp add: Cauchy_def)
hoelzl@51531
  1374
apply (drule_tac x=e in spec, clarify)
hoelzl@51531
  1375
apply (rule_tac x=M in exI, clarify)
hoelzl@51531
  1376
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
hoelzl@51531
  1377
done
hoelzl@51531
  1378
hoelzl@51531
  1379
theorem LIMSEQ_imp_Cauchy:
hoelzl@51531
  1380
  assumes X: "X ----> a" shows "Cauchy X"
hoelzl@51531
  1381
proof (rule metric_CauchyI)
hoelzl@51531
  1382
  fix e::real assume "0 < e"
hoelzl@51531
  1383
  hence "0 < e/2" by simp
hoelzl@51531
  1384
  with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
hoelzl@51531
  1385
  then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
hoelzl@51531
  1386
  show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
hoelzl@51531
  1387
  proof (intro exI allI impI)
hoelzl@51531
  1388
    fix m assume "N \<le> m"
hoelzl@51531
  1389
    hence m: "dist (X m) a < e/2" using N by fast
hoelzl@51531
  1390
    fix n assume "N \<le> n"
hoelzl@51531
  1391
    hence n: "dist (X n) a < e/2" using N by fast
hoelzl@51531
  1392
    have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
hoelzl@51531
  1393
      by (rule dist_triangle2)
hoelzl@51531
  1394
    also from m n have "\<dots> < e" by simp
hoelzl@51531
  1395
    finally show "dist (X m) (X n) < e" .
hoelzl@51531
  1396
  qed
hoelzl@51531
  1397
qed
hoelzl@51531
  1398
hoelzl@51531
  1399
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
hoelzl@51531
  1400
unfolding convergent_def
hoelzl@51531
  1401
by (erule exE, erule LIMSEQ_imp_Cauchy)
hoelzl@51531
  1402
hoelzl@51531
  1403
subsubsection {* Cauchy Sequences are Convergent *}
hoelzl@51531
  1404
hoelzl@51531
  1405
class complete_space = metric_space +
hoelzl@51531
  1406
  assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
hoelzl@51531
  1407
hoelzl@51531
  1408
lemma Cauchy_convergent_iff:
hoelzl@51531
  1409
  fixes X :: "nat \<Rightarrow> 'a::complete_space"
hoelzl@51531
  1410
  shows "Cauchy X = convergent X"
hoelzl@51531
  1411
by (fast intro: Cauchy_convergent convergent_Cauchy)
hoelzl@51531
  1412
hoelzl@51531
  1413
subsection {* The set of real numbers is a complete metric space *}
hoelzl@51531
  1414
hoelzl@51531
  1415
text {*
hoelzl@51531
  1416
Proof that Cauchy sequences converge based on the one from
hoelzl@51531
  1417
http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
hoelzl@51531
  1418
*}
hoelzl@51531
  1419
hoelzl@51531
  1420
text {*
hoelzl@51531
  1421
  If sequence @{term "X"} is Cauchy, then its limit is the lub of
hoelzl@51531
  1422
  @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
hoelzl@51531
  1423
*}
hoelzl@51531
  1424
hoelzl@51531
  1425
lemma increasing_LIMSEQ:
hoelzl@51531
  1426
  fixes f :: "nat \<Rightarrow> real"
hoelzl@51531
  1427
  assumes inc: "\<And>n. f n \<le> f (Suc n)"
hoelzl@51531
  1428
      and bdd: "\<And>n. f n \<le> l"
hoelzl@51531
  1429
      and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
hoelzl@51531
  1430
  shows "f ----> l"
hoelzl@51531
  1431
proof (rule increasing_tendsto)
hoelzl@51531
  1432
  fix x assume "x < l"
hoelzl@51531
  1433
  with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
hoelzl@51531
  1434
    by auto
hoelzl@51531
  1435
  from en[OF `0 < e`] obtain n where "l - e \<le> f n"
hoelzl@51531
  1436
    by (auto simp: field_simps)
hoelzl@51531
  1437
  with `e < l - x` `0 < e` have "x < f n" by simp
hoelzl@51531
  1438
  with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
hoelzl@51531
  1439
    by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
hoelzl@51531
  1440
qed (insert bdd, auto)
hoelzl@51531
  1441
hoelzl@51531
  1442
lemma real_Cauchy_convergent:
hoelzl@51531
  1443
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51531
  1444
  assumes X: "Cauchy X"
hoelzl@51531
  1445
  shows "convergent X"
hoelzl@51531
  1446
proof -
hoelzl@51531
  1447
  def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
hoelzl@51531
  1448
  then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
hoelzl@51531
  1449
hoelzl@51531
  1450
  { fix N x assume N: "\<forall>n\<ge>N. X n < x"
hoelzl@51531
  1451
  fix y::real assume "y \<in> S"
hoelzl@51531
  1452
  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
hoelzl@51531
  1453
    by (simp add: S_def)
hoelzl@51531
  1454
  then obtain M where "\<forall>n\<ge>M. y < X n" ..
hoelzl@51531
  1455
  hence "y < X (max M N)" by simp
hoelzl@51531
  1456
  also have "\<dots> < x" using N by simp
hoelzl@54263
  1457
  finally have "y \<le> x"
hoelzl@54263
  1458
    by (rule order_less_imp_le) }
hoelzl@51531
  1459
  note bound_isUb = this 
hoelzl@51531
  1460
hoelzl@51531
  1461
  obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
hoelzl@51531
  1462
    using X[THEN metric_CauchyD, OF zero_less_one] by auto
hoelzl@51531
  1463
  hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
hoelzl@54263
  1464
  have [simp]: "S \<noteq> {}"
hoelzl@54263
  1465
  proof (intro exI ex_in_conv[THEN iffD1])
hoelzl@51531
  1466
    from N have "\<forall>n\<ge>N. X N - 1 < X n"
hoelzl@51531
  1467
      by (simp add: abs_diff_less_iff dist_real_def)
hoelzl@51531
  1468
    thus "X N - 1 \<in> S" by (rule mem_S)
hoelzl@51531
  1469
  qed
hoelzl@54263
  1470
  have [simp]: "bdd_above S"
hoelzl@51531
  1471
  proof
hoelzl@51531
  1472
    from N have "\<forall>n\<ge>N. X n < X N + 1"
hoelzl@51531
  1473
      by (simp add: abs_diff_less_iff dist_real_def)
hoelzl@54263
  1474
    thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
hoelzl@51531
  1475
      by (rule bound_isUb)
hoelzl@51531
  1476
  qed
hoelzl@54263
  1477
  have "X ----> Sup S"
hoelzl@51531
  1478
  proof (rule metric_LIMSEQ_I)
hoelzl@51531
  1479
  fix r::real assume "0 < r"
hoelzl@51531
  1480
  hence r: "0 < r/2" by simp
hoelzl@51531
  1481
  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
hoelzl@51531
  1482
    using metric_CauchyD [OF X r] by auto
hoelzl@51531
  1483
  hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
hoelzl@51531
  1484
  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
hoelzl@51531
  1485
    by (simp only: dist_real_def abs_diff_less_iff)
hoelzl@51531
  1486
hoelzl@51531
  1487
  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
hoelzl@51531
  1488
  hence "X N - r/2 \<in> S" by (rule mem_S)
hoelzl@54263
  1489
  hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
hoelzl@51531
  1490
hoelzl@51531
  1491
  from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
hoelzl@54263
  1492
  from bound_isUb[OF this]
hoelzl@54263
  1493
  have 2: "Sup S \<le> X N + r/2"
hoelzl@54263
  1494
    by (intro cSup_least) simp_all
hoelzl@51531
  1495
hoelzl@54263
  1496
  show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
hoelzl@51531
  1497
  proof (intro exI allI impI)
hoelzl@51531
  1498
    fix n assume n: "N \<le> n"
hoelzl@51531
  1499
    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
hoelzl@54263
  1500
    thus "dist (X n) (Sup S) < r" using 1 2
hoelzl@51531
  1501
      by (simp add: abs_diff_less_iff dist_real_def)
hoelzl@51531
  1502
  qed
hoelzl@51531
  1503
  qed
hoelzl@51531
  1504
  then show ?thesis unfolding convergent_def by auto
hoelzl@51531
  1505
qed
hoelzl@51531
  1506
hoelzl@51531
  1507
instance real :: complete_space
hoelzl@51531
  1508
  by intro_classes (rule real_Cauchy_convergent)
hoelzl@51531
  1509
hoelzl@51531
  1510
class banach = real_normed_vector + complete_space
hoelzl@51531
  1511
hoelzl@51531
  1512
instance real :: banach by default
hoelzl@51531
  1513
hoelzl@51531
  1514
lemma tendsto_at_topI_sequentially:
hoelzl@51531
  1515
  fixes f :: "real \<Rightarrow> real"
hoelzl@51531
  1516
  assumes mono: "mono f"
hoelzl@51531
  1517
  assumes limseq: "(\<lambda>n. f (real n)) ----> y"
hoelzl@51531
  1518
  shows "(f ---> y) at_top"
hoelzl@51531
  1519
proof (rule tendstoI)
hoelzl@51531
  1520
  fix e :: real assume "0 < e"
hoelzl@51531
  1521
  with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
hoelzl@51531
  1522
    by (auto simp: LIMSEQ_def dist_real_def)
hoelzl@51531
  1523
  { fix x :: real
wenzelm@53381
  1524
    obtain n where "x \<le> real_of_nat n"
wenzelm@53381
  1525
      using ex_le_of_nat[of x] ..
hoelzl@51531
  1526
    note monoD[OF mono this]
hoelzl@51531
  1527
    also have "f (real_of_nat n) \<le> y"
hoelzl@51531
  1528
      by (rule LIMSEQ_le_const[OF limseq])
hoelzl@51531
  1529
         (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
hoelzl@51531
  1530
    finally have "f x \<le> y" . }
hoelzl@51531
  1531
  note le = this
hoelzl@51531
  1532
  have "eventually (\<lambda>x. real N \<le> x) at_top"
hoelzl@51531
  1533
    by (rule eventually_ge_at_top)
hoelzl@51531
  1534
  then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
hoelzl@51531
  1535
  proof eventually_elim
hoelzl@51531
  1536
    fix x assume N': "real N \<le> x"
hoelzl@51531
  1537
    with N[of N] le have "y - f (real N) < e" by auto
hoelzl@51531
  1538
    moreover note monoD[OF mono N']
hoelzl@51531
  1539
    ultimately show "dist (f x) y < e"
hoelzl@51531
  1540
      using le[of x] by (auto simp: dist_real_def field_simps)
hoelzl@51531
  1541
  qed
hoelzl@51531
  1542
qed
hoelzl@51531
  1543
huffman@20504
  1544
end