src/HOL/Real_Vector_Spaces.thy
author haftmann
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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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c433e78d4203 define new constant of_real for class real_algebra_1;
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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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c433e78d4203 define new constant of_real for class real_algebra_1;
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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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c433e78d4203 define new constant of_real for class real_algebra_1;
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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 of_real_setsum[simp]: "of_real (setsum f s) = (\<Sum>x\<in>s. of_real (f x))"
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  by (induct s rule: infinite_finite_induct) auto
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lemma of_real_setprod[simp]: "of_real (setprod f s) = (\<Prod>x\<in>s. of_real (f x))"
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  by (induct s rule: infinite_finite_induct) auto
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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]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   272
  "of_real (inverse x) =
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   273
   inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   274
by (simp add: of_real_def inverse_scaleR_distrib)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   275
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   276
lemma nonzero_of_real_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   277
  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   278
   (of_real x / of_real y :: 'a::real_field)"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   279
by (simp add: divide_inverse nonzero_of_real_inverse)
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   280
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   281
lemma of_real_divide [simp]:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   282
  "of_real (x / y) =
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   283
   (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   284
by (simp add: divide_inverse)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   285
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   286
lemma of_real_power [simp]:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   287
  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   288
by (induct n) simp_all
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   289
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   290
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
35216
7641e8d831d2 get rid of many duplicate simp rule warnings
huffman
parents: 31586
diff changeset
   291
by (simp add: of_real_def)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   292
38621
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   293
lemma inj_of_real:
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   294
  "inj of_real"
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   295
  by (auto intro: injI)
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   296
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   297
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   298
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   299
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   300
proof
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   301
  fix r
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   302
  show "of_real r = id r"
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   303
    by (simp add: of_real_def)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   304
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   305
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   306
text{*Collapse nested embeddings*}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   307
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   308
by (induct n) auto
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   309
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   310
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   311
by (cases z rule: int_diff_cases, simp)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   312
56889
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
   313
lemma of_real_real_of_nat_eq [simp]: "of_real (real n) = of_nat n"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
   314
  by (simp add: real_of_nat_def)
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
   315
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
   316
lemma of_real_real_of_int_eq [simp]: "of_real (real z) = of_int z"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
   317
  by (simp add: real_of_int_def)
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
   318
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   319
lemma of_real_numeral: "of_real (numeral w) = numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   320
using of_real_of_int_eq [of "numeral w"] by simp
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   321
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54263
diff changeset
   322
lemma of_real_neg_numeral: "of_real (- numeral w) = - numeral w"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54263
diff changeset
   323
using of_real_of_int_eq [of "- numeral w"] by simp
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   324
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   325
text{*Every real algebra has characteristic zero*}
38621
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   326
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   327
instance real_algebra_1 < ring_char_0
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   328
proof
38621
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   329
  from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
d6cb7e625d75 more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents: 37887
diff changeset
   330
  then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   331
qed
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   332
27553
d315a513a150 instance real_field < field_char_0;
huffman
parents: 27552
diff changeset
   333
instance real_field < field_char_0 ..
d315a513a150 instance real_field < field_char_0;
huffman
parents: 27552
diff changeset
   334
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   335
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   336
subsection {* The Set of Real Numbers *}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   337
37767
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36839
diff changeset
   338
definition Reals :: "'a::real_algebra_1 set" where
a2b7a20d6ea3 dropped superfluous [code del]s
haftmann
parents: 36839
diff changeset
   339
  "Reals = range of_real"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   340
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20828
diff changeset
   341
notation (xsymbols)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   342
  Reals  ("\<real>")
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   343
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   344
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   345
by (simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   346
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   347
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   348
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
20718
4c4869e4ddb7 add lemmas of_int_in_Reals, of_nat_in_Reals
huffman
parents: 20694
diff changeset
   349
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   350
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   351
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   352
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   353
lemma Reals_numeral [simp]: "numeral w \<in> Reals"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   354
by (subst of_real_numeral [symmetric], rule Reals_of_real)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   355
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   356
lemma Reals_0 [simp]: "0 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   357
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   358
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   359
apply (rule of_real_0 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   360
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   361
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   362
lemma Reals_1 [simp]: "1 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   363
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   364
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   365
apply (rule of_real_1 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   366
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   367
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   368
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   369
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   370
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   371
apply (rule of_real_add [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   372
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   373
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   374
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   375
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   376
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   377
apply (rule of_real_minus [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   378
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   379
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   380
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   381
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   382
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   383
apply (rule of_real_diff [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   384
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   385
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   386
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   387
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   388
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   389
apply (rule of_real_mult [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   390
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   391
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   392
lemma nonzero_Reals_inverse:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   393
  fixes a :: "'a::real_div_algebra"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   394
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   395
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   396
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   397
apply (erule nonzero_of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   398
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   399
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   400
lemma Reals_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   401
  fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   402
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   403
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   404
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   405
apply (rule of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   406
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   407
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   408
lemma Reals_inverse_iff [simp]: 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   409
  fixes x:: "'a :: {real_div_algebra, division_ring_inverse_zero}"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   410
  shows "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   411
by (metis Reals_inverse inverse_inverse_eq)
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   412
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   413
lemma nonzero_Reals_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   414
  fixes a b :: "'a::real_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   415
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   416
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   417
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   418
apply (erule nonzero_of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   419
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   420
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   421
lemma Reals_divide [simp]:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   422
  fixes a b :: "'a::{real_field, field_inverse_zero}"
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   423
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   424
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   425
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   426
apply (rule of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   427
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   428
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   429
lemma Reals_power [simp]:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   430
  fixes a :: "'a::{real_algebra_1}"
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   431
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   432
apply (auto simp add: Reals_def)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   433
apply (rule range_eqI)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   434
apply (rule of_real_power [symmetric])
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   435
done
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   436
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   437
lemma Reals_cases [cases set: Reals]:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   438
  assumes "q \<in> \<real>"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   439
  obtains (of_real) r where "q = of_real r"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   440
  unfolding Reals_def
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   441
proof -
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   442
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   443
  then obtain r where "q = of_real r" ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   444
  then show thesis ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   445
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   446
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   447
lemma setsum_in_Reals: assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setsum f s \<in> \<real>"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   448
proof (cases "finite s")
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   449
  case True then show ?thesis using assms
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   450
    by (induct s rule: finite_induct) auto
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   451
next
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   452
  case False then show ?thesis using assms
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57276
diff changeset
   453
    by (metis Reals_0 setsum.infinite)
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   454
qed
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   455
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   456
lemma setprod_in_Reals: assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setprod f s \<in> \<real>"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   457
proof (cases "finite s")
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   458
  case True then show ?thesis using assms
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   459
    by (induct s rule: finite_induct) auto
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   460
next
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   461
  case False then show ?thesis using assms
57418
6ab1c7cb0b8d fact consolidation
haftmann
parents: 57276
diff changeset
   462
    by (metis Reals_1 setprod.infinite)
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   463
qed
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   464
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   465
lemma Reals_induct [case_names of_real, induct set: Reals]:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   466
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   467
  by (rule Reals_cases) auto
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   468
54778
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   469
subsection {* Ordered real vector spaces *}
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   470
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   471
class ordered_real_vector = real_vector + ordered_ab_group_add +
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   472
  assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   473
  assumes scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   474
begin
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   475
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   476
lemma scaleR_mono:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   477
  "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"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   478
apply (erule scaleR_right_mono [THEN order_trans], assumption)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   479
apply (erule scaleR_left_mono, assumption)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   480
done
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   481
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   482
lemma scaleR_mono':
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   483
  "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"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   484
  by (rule scaleR_mono) (auto intro: order.trans)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   485
54785
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   486
lemma pos_le_divideRI:
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   487
  assumes "0 < c"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   488
  assumes "c *\<^sub>R a \<le> b"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   489
  shows "a \<le> b /\<^sub>R c"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   490
proof -
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   491
  from scaleR_left_mono[OF assms(2)] assms(1)
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   492
  have "c *\<^sub>R a /\<^sub>R c \<le> b /\<^sub>R c"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   493
    by simp
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   494
  with assms show ?thesis
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   495
    by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   496
qed
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   497
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   498
lemma pos_le_divideR_eq:
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   499
  assumes "0 < c"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   500
  shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   501
proof rule
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   502
  assume "a \<le> b /\<^sub>R c"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   503
  from scaleR_left_mono[OF this] assms
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   504
  have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   505
    by simp
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   506
  with assms show "c *\<^sub>R a \<le> b"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   507
    by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   508
qed (rule pos_le_divideRI[OF assms])
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   509
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   510
lemma scaleR_image_atLeastAtMost:
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   511
  "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   512
  apply (auto intro!: scaleR_left_mono)
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   513
  apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI)
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   514
  apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   515
  done
4876fb408c0d lemmas about divideR and scaleR
immler
parents: 54778
diff changeset
   516
54778
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   517
end
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   518
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   519
lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> 0 \<le> a *\<^sub>R x"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   520
  using scaleR_left_mono [of 0 x a]
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   521
  by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   522
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   523
lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> (x::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   524
  using scaleR_left_mono [of x 0 a] by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   525
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   526
lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> a *\<^sub>R x \<le> 0"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   527
  using scaleR_right_mono [of a 0 x] by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   528
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   529
lemma split_scaleR_neg_le: "(0 \<le> a & x \<le> 0) | (a \<le> 0 & 0 \<le> x) \<Longrightarrow>
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   530
  a *\<^sub>R (x::'a::ordered_real_vector) \<le> 0"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   531
  by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   532
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   533
lemma le_add_iff1:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   534
  fixes c d e::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   535
  shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   536
  by (simp add: algebra_simps)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   537
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   538
lemma le_add_iff2:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   539
  fixes c d e::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   540
  shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   541
  by (simp add: algebra_simps)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   542
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   543
lemma scaleR_left_mono_neg:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   544
  fixes a b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   545
  shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   546
  apply (drule scaleR_left_mono [of _ _ "- c"])
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   547
  apply simp_all
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   548
  done
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   549
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   550
lemma scaleR_right_mono_neg:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   551
  fixes c::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   552
  shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   553
  apply (drule scaleR_right_mono [of _ _ "- c"])
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   554
  apply simp_all
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   555
  done
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   556
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   557
lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> (b::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   558
using scaleR_right_mono_neg [of a 0 b] by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   559
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   560
lemma split_scaleR_pos_le:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   561
  fixes b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   562
  shows "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   563
  by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   564
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   565
lemma zero_le_scaleR_iff:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   566
  fixes b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   567
  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")
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   568
proof cases
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   569
  assume "a \<noteq> 0"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   570
  show ?thesis
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   571
  proof
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   572
    assume lhs: ?lhs
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   573
    {
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   574
      assume "0 < a"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   575
      with lhs have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   576
        by (intro scaleR_mono) auto
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   577
      hence ?rhs using `0 < a`
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   578
        by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   579
    } moreover {
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   580
      assume "0 > a"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   581
      with lhs have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   582
        by (intro scaleR_mono) auto
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   583
      hence ?rhs using `0 > a`
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   584
        by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   585
    } ultimately show ?rhs using `a \<noteq> 0` by arith
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   586
  qed (auto simp: not_le `a \<noteq> 0` intro!: split_scaleR_pos_le)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   587
qed simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   588
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   589
lemma scaleR_le_0_iff:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   590
  fixes b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   591
  shows "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   592
  by (insert zero_le_scaleR_iff [of "-a" b]) force
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   593
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   594
lemma scaleR_le_cancel_left:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   595
  fixes b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   596
  shows "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   597
  by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   598
    dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   599
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   600
lemma scaleR_le_cancel_left_pos:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   601
  fixes b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   602
  shows "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   603
  by (auto simp: scaleR_le_cancel_left)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   604
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   605
lemma scaleR_le_cancel_left_neg:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   606
  fixes b::"'a::ordered_real_vector"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   607
  shows "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   608
  by (auto simp: scaleR_le_cancel_left)
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   609
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   610
lemma scaleR_left_le_one_le:
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   611
  fixes x::"'a::ordered_real_vector" and a::real
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   612
  shows "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   613
  using scaleR_right_mono[of a 1 x] by simp
13f08c876899 introduced ordered real vector spaces
immler
parents: 54703
diff changeset
   614
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   615
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   616
subsection {* Real normed vector spaces *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   617
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   618
class dist =
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   619
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   620
29608
564ea783ace8 no base sort in class import
haftmann
parents: 29252
diff changeset
   621
class norm =
22636
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   622
  fixes norm :: "'a \<Rightarrow> real"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   623
24520
40b220403257 fix sgn_div_norm class
huffman
parents: 24513
diff changeset
   624
class sgn_div_norm = scaleR + norm + sgn +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   625
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   626
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   627
class dist_norm = dist + norm + minus +
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   628
  assumes dist_norm: "dist x y = norm (x - y)"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   629
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   630
class open_dist = "open" + dist +
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   631
  assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   632
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   633
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
51002
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   634
  assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   635
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
   636
  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
51002
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   637
begin
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   638
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   639
lemma norm_ge_zero [simp]: "0 \<le> norm x"
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   640
proof -
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   641
  have "0 = norm (x + -1 *\<^sub>R x)" 
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   642
    using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   643
  also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   644
  finally show ?thesis by simp
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   645
qed
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   646
496013a6eb38 remove unnecessary assumption from real_normed_vector
hoelzl
parents: 50999
diff changeset
   647
end
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   648
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   649
class real_normed_algebra = real_algebra + real_normed_vector +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   650
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   651
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   652
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   653
  assumes norm_one [simp]: "norm 1 = 1"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   654
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   655
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   656
  assumes norm_mult: "norm (x * y) = norm x * norm y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   657
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   658
class real_normed_field = real_field + real_normed_div_algebra
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   659
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   660
instance real_normed_div_algebra < real_normed_algebra_1
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   661
proof
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   662
  fix x y :: 'a
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   663
  show "norm (x * y) \<le> norm x * norm y"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   664
    by (simp add: norm_mult)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   665
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   666
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   667
    by (rule norm_mult)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   668
  thus "norm (1::'a) = 1" by simp
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   669
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   670
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   671
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   672
by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   673
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   674
lemma zero_less_norm_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   675
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   676
  shows "(0 < norm x) = (x \<noteq> 0)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   677
by (simp add: order_less_le)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   678
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   679
lemma norm_not_less_zero [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   680
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   681
  shows "\<not> norm x < 0"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   682
by (simp add: linorder_not_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   683
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   684
lemma norm_le_zero_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   685
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   686
  shows "(norm x \<le> 0) = (x = 0)"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   687
by (simp add: order_le_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   688
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   689
lemma norm_minus_cancel [simp]:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   690
  fixes x :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   691
  shows "norm (- x) = norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   692
proof -
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   693
  have "norm (- x) = norm (scaleR (- 1) x)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   694
    by (simp only: scaleR_minus_left scaleR_one)
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   695
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   696
    by (rule norm_scaleR)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   697
  finally show ?thesis by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   698
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   699
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   700
lemma norm_minus_commute:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   701
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   702
  shows "norm (a - b) = norm (b - a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   703
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   704
  have "norm (- (b - a)) = norm (b - a)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   705
    by (rule norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   706
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   707
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   708
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   709
lemma norm_triangle_ineq2:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   710
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   711
  shows "norm a - norm b \<le> norm (a - b)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   712
proof -
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   713
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   714
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   715
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   716
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   717
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   718
lemma norm_triangle_ineq3:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   719
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   720
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   721
apply (subst abs_le_iff)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   722
apply auto
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   723
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   724
apply (subst norm_minus_commute)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   725
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   726
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   727
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   728
lemma norm_triangle_ineq4:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   729
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   730
  shows "norm (a - b) \<le> norm a + norm b"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   731
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   732
  have "norm (a + - b) \<le> norm a + norm (- b)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   733
    by (rule norm_triangle_ineq)
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53600
diff changeset
   734
  then show ?thesis by simp
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   735
qed
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   736
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   737
lemma norm_diff_ineq:
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   738
  fixes a b :: "'a::real_normed_vector"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   739
  shows "norm a - norm b \<le> norm (a + b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   740
proof -
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   741
  have "norm a - norm (- b) \<le> norm (a - - b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   742
    by (rule norm_triangle_ineq2)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   743
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   744
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   745
20551
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   746
lemma norm_diff_triangle_ineq:
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   747
  fixes a b c d :: "'a::real_normed_vector"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   748
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   749
proof -
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   750
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
54230
b1d955791529 more simplification rules on unary and binary minus
haftmann
parents: 53600
diff changeset
   751
    by (simp add: algebra_simps)
20551
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   752
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   753
    by (rule norm_triangle_ineq)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   754
  finally show ?thesis .
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   755
qed
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   756
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   757
lemma norm_triangle_mono: 
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   758
  fixes a b :: "'a::real_normed_vector"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   759
  shows "\<lbrakk>norm a \<le> r; norm b \<le> s\<rbrakk> \<Longrightarrow> norm (a + b) \<le> r + s"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   760
by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   761
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 55719
diff changeset
   762
lemma norm_setsum:
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 55719
diff changeset
   763
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 55719
diff changeset
   764
  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 55719
diff changeset
   765
  by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono)
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 55719
diff changeset
   766
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
   767
lemma setsum_norm_le:
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
   768
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
   769
  assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
   770
  shows "norm (setsum f S) \<le> setsum g S"
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
   771
  by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
   772
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   773
lemma abs_norm_cancel [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   774
  fixes a :: "'a::real_normed_vector"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   775
  shows "\<bar>norm a\<bar> = norm a"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   776
by (rule abs_of_nonneg [OF norm_ge_zero])
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   777
22880
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   778
lemma norm_add_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   779
  fixes x y :: "'a::real_normed_vector"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   780
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   781
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   782
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   783
lemma norm_mult_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   784
  fixes x y :: "'a::real_normed_algebra"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   785
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   786
apply (rule order_le_less_trans [OF norm_mult_ineq])
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   787
apply (simp add: mult_strict_mono')
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   788
done
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   789
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   790
lemma norm_of_real [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   791
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
   792
unfolding of_real_def by simp
20560
49996715bc6e norm_one is now proved from other class axioms
huffman
parents: 20554
diff changeset
   793
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   794
lemma norm_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   795
  "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   796
by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   797
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   798
lemma norm_neg_numeral [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54263
diff changeset
   799
  "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46868
diff changeset
   800
by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
22876
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   801
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   802
lemma norm_of_int [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   803
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   804
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   805
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   806
lemma norm_of_nat [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   807
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   808
apply (subst of_real_of_nat_eq [symmetric])
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   809
apply (subst norm_of_real, simp)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   810
done
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   811
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   812
lemma nonzero_norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   813
  fixes a :: "'a::real_normed_div_algebra"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   814
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   815
apply (rule inverse_unique [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   816
apply (simp add: norm_mult [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   817
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   818
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   819
lemma norm_inverse:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   820
  fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   821
  shows "norm (inverse a) = inverse (norm a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   822
apply (case_tac "a = 0", simp)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   823
apply (erule nonzero_norm_inverse)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   824
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   825
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   826
lemma nonzero_norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   827
  fixes a b :: "'a::real_normed_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   828
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   829
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   830
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   831
lemma norm_divide:
36409
d323e7773aa8 use new classes (linordered_)field_inverse_zero
haftmann
parents: 36349
diff changeset
   832
  fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   833
  shows "norm (a / b) = norm a / norm b"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   834
by (simp add: divide_inverse norm_mult norm_inverse)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   835
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   836
lemma norm_power_ineq:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   837
  fixes x :: "'a::{real_normed_algebra_1}"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   838
  shows "norm (x ^ n) \<le> norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   839
proof (induct n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   840
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   841
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   842
  case (Suc n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   843
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   844
    by (rule norm_mult_ineq)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   845
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   846
    using norm_ge_zero by (rule mult_left_mono)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   847
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   848
    by simp
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   849
qed
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   850
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   851
lemma norm_power:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   852
  fixes x :: "'a::{real_normed_div_algebra}"
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   853
  shows "norm (x ^ n) = norm x ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   854
by (induct n) (simp_all add: norm_mult)
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   855
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   856
lemma setprod_norm:
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   857
  fixes f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}"
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   858
  shows "setprod (\<lambda>x. norm(f x)) A = norm (setprod f A)"
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   859
  by (induct A rule: infinite_finite_induct) (auto simp: norm_mult)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   860
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   861
lemma norm_setprod_le: 
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   862
  "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   863
proof (induction A rule: infinite_finite_induct)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   864
  case (insert a A)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   865
  then have "norm (setprod f (insert a A)) \<le> norm (f a) * norm (setprod f A)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   866
    by (simp add: norm_mult_ineq)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   867
  also have "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a))"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   868
    by (rule insert)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   869
  finally show ?case
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   870
    by (simp add: insert mult_left_mono)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   871
qed simp_all
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   872
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   873
lemma norm_setprod_diff:
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   874
  fixes z w :: "'i \<Rightarrow> 'a::{real_normed_algebra_1, comm_monoid_mult}"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   875
  shows "(\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1) \<Longrightarrow>
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   876
    norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))" 
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   877
proof (induction I rule: infinite_finite_induct)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   878
  case (insert i I)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   879
  note insert.hyps[simp]
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   880
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   881
  have "norm ((\<Prod>i\<in>insert i I. z i) - (\<Prod>i\<in>insert i I. w i)) =
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   882
    norm ((\<Prod>i\<in>I. z i) * (z i - w i) + ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * w i)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   883
    (is "_ = norm (?t1 + ?t2)")
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   884
    by (auto simp add: field_simps)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   885
  also have "... \<le> norm ?t1 + norm ?t2"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   886
    by (rule norm_triangle_ineq)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   887
  also have "norm ?t1 \<le> norm (\<Prod>i\<in>I. z i) * norm (z i - w i)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   888
    by (rule norm_mult_ineq)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   889
  also have "\<dots> \<le> (\<Prod>i\<in>I. norm (z i)) * norm(z i - w i)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   890
    by (rule mult_right_mono) (auto intro: norm_setprod_le)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   891
  also have "(\<Prod>i\<in>I. norm (z i)) \<le> (\<Prod>i\<in>I. 1)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   892
    by (intro setprod_mono) (auto intro!: insert)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   893
  also have "norm ?t2 \<le> norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * norm (w i)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   894
    by (rule norm_mult_ineq)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   895
  also have "norm (w i) \<le> 1"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   896
    by (auto intro: insert)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   897
  also have "norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   898
    using insert by auto
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   899
  finally show ?case
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   900
    by (auto simp add: add_ac mult_right_mono mult_left_mono)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   901
qed simp_all
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   902
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   903
lemma norm_power_diff: 
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   904
  fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   905
  assumes "norm z \<le> 1" "norm w \<le> 1"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   906
  shows "norm (z^m - w^m) \<le> m * norm (z - w)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   907
proof -
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   908
  have "norm (z^m - w^m) = norm ((\<Prod> i < m. z) - (\<Prod> i < m. w))"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   909
    by (simp add: setprod_constant)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   910
  also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   911
    by (intro norm_setprod_diff) (auto simp add: assms)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   912
  also have "\<dots> = m * norm (z - w)"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   913
    by (simp add: real_of_nat_def)
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
   914
  finally show ?thesis .
55719
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   915
qed
cdddd073bff8 Lemmas about Reals, norm, etc., and cleaner variants of existing ones
paulson <lp15@cam.ac.uk>
parents: 54890
diff changeset
   916
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   917
subsection {* Metric spaces *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   918
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   919
class metric_space = open_dist +
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   920
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   921
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   922
begin
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   923
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   924
lemma dist_self [simp]: "dist x x = 0"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   925
by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   926
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   927
lemma zero_le_dist [simp]: "0 \<le> dist x y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   928
using dist_triangle2 [of x x y] by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   929
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   930
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   931
by (simp add: less_le)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   932
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   933
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   934
by (simp add: not_less)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   935
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   936
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   937
by (simp add: le_less)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   938
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   939
lemma dist_commute: "dist x y = dist y x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   940
proof (rule order_antisym)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   941
  show "dist x y \<le> dist y x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   942
    using dist_triangle2 [of x y x] by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   943
  show "dist y x \<le> dist x y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   944
    using dist_triangle2 [of y x y] by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   945
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   946
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   947
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   948
using dist_triangle2 [of x z y] by (simp add: dist_commute)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   949
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   950
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   951
using dist_triangle2 [of x y a] by (simp add: dist_commute)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   952
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   953
lemma dist_triangle_alt:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   954
  shows "dist y z <= dist x y + dist x z"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   955
by (rule dist_triangle3)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   956
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   957
lemma dist_pos_lt:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   958
  shows "x \<noteq> y ==> 0 < dist x y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   959
by (simp add: zero_less_dist_iff)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   960
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   961
lemma dist_nz:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   962
  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   963
by (simp add: zero_less_dist_iff)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   964
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   965
lemma dist_triangle_le:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   966
  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   967
by (rule order_trans [OF dist_triangle2])
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   968
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   969
lemma dist_triangle_lt:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   970
  shows "dist x z + dist y z < e ==> dist x y < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   971
by (rule le_less_trans [OF dist_triangle2])
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   972
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   973
lemma dist_triangle_half_l:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   974
  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   975
by (rule dist_triangle_lt [where z=y], simp)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   976
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   977
lemma dist_triangle_half_r:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   978
  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   979
by (rule dist_triangle_half_l, simp_all add: dist_commute)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   980
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   981
subclass topological_space
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   982
proof
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   983
  have "\<exists>e::real. 0 < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   984
    by (fast intro: zero_less_one)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   985
  then show "open UNIV"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   986
    unfolding open_dist by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   987
next
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   988
  fix S T assume "open S" "open T"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   989
  then show "open (S \<inter> T)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   990
    unfolding open_dist
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   991
    apply clarify
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   992
    apply (drule (1) bspec)+
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   993
    apply (clarify, rename_tac r s)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   994
    apply (rule_tac x="min r s" in exI, simp)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   995
    done
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   996
next
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   997
  fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   998
    unfolding open_dist by fast
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
   999
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1000
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1001
lemma open_ball: "open {y. dist x y < d}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1002
proof (unfold open_dist, intro ballI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1003
  fix y assume *: "y \<in> {y. dist x y < d}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1004
  then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1005
    by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1006
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1007
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1008
subclass first_countable_topology
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1009
proof
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1010
  fix x 
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1011
  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))"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1012
  proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1013
    fix S assume "open S" "x \<in> S"
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 52381
diff changeset
  1014
    then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1015
      by (auto simp: open_dist subset_eq dist_commute)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1016
    moreover
53374
a14d2a854c02 tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents: 52381
diff changeset
  1017
    from e obtain i where "inverse (Suc i) < e"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1018
      by (auto dest!: reals_Archimedean)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1019
    then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1020
      by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1021
    ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1022
      by blast
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1023
  qed (auto intro: open_ball)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1024
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1025
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1026
end
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1027
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1028
instance metric_space \<subseteq> t2_space
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1029
proof
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1030
  fix x y :: "'a::metric_space"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1031
  assume xy: "x \<noteq> y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1032
  let ?U = "{y'. dist x y' < dist x y / 2}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1033
  let ?V = "{x'. dist y x' < dist x y / 2}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1034
  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
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1035
               \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1036
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1037
    using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1038
    using open_ball[of _ "dist x y / 2"] by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1039
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1040
    by blast
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1041
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1042
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1043
text {* Every normed vector space is a metric space. *}
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31017
diff changeset
  1044
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1045
instance real_normed_vector < metric_space
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1046
proof
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1047
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1048
    unfolding dist_norm by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1049
next
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1050
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1051
    unfolding dist_norm
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1052
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1053
qed
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31017
diff changeset
  1054
31564
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1055
subsection {* Class instances for real numbers *}
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1056
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1057
instantiation real :: real_normed_field
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1058
begin
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1059
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1060
definition dist_real_def:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1061
  "dist x y = \<bar>x - y\<bar>"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1062
52381
63eec9cea2c7 pragmatic executability for instance real :: open
haftmann
parents: 51775
diff changeset
  1063
definition open_real_def [code del]:
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1064
  "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1065
31564
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1066
definition real_norm_def [simp]:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1067
  "norm r = \<bar>r\<bar>"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1068
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1069
instance
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1070
apply (intro_classes, unfold real_norm_def real_scaleR_def)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1071
apply (rule dist_real_def)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1072
apply (rule open_real_def)
36795
e05e1283c550 new construction of real numbers using Cauchy sequences
huffman
parents: 36409
diff changeset
  1073
apply (simp add: sgn_real_def)
31564
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1074
apply (rule abs_eq_0)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1075
apply (rule abs_triangle_ineq)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1076
apply (rule abs_mult)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1077
apply (rule abs_mult)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1078
done
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1079
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1080
end
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
  1081
54890
cb892d835803 fundamental treatment of undefined vs. universally partial replaces code_abort
haftmann
parents: 54863
diff changeset
  1082
declare [[code abort: "open :: real set \<Rightarrow> bool"]]
52381
63eec9cea2c7 pragmatic executability for instance real :: open
haftmann
parents: 51775
diff changeset
  1083
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1084
instance real :: linorder_topology
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1085
proof
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1086
  show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1087
  proof (rule ext, safe)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1088
    fix S :: "real set" assume "open S"
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1089
    then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1090
      unfolding open_real_def bchoice_iff ..
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1091
    then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1092
      by (fastforce simp: dist_real_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1093
    show "generate_topology (range lessThan \<union> range greaterThan) S"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1094
      apply (subst *)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1095
      apply (intro generate_topology_Union generate_topology.Int)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1096
      apply (auto intro: generate_topology.Basis)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1097
      done
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1098
  next
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1099
    fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1100
    moreover have "\<And>a::real. open {..<a}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1101
      unfolding open_real_def dist_real_def
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1102
    proof clarify
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1103
      fix x a :: real assume "x < a"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1104
      hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1105
      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1106
    qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1107
    moreover have "\<And>a::real. open {a <..}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1108
      unfolding open_real_def dist_real_def
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1109
    proof clarify
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1110
      fix x a :: real assume "a < x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1111
      hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1112
      thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1113
    qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1114
    ultimately show "open S"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1115
      by induct auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1116
  qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1117
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1118
51775
408d937c9486 revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents: 51774
diff changeset
  1119
instance real :: linear_continuum_topology ..
51518
6a56b7088a6a separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents: 51481
diff changeset
  1120
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1121
lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1122
lemmas open_real_lessThan = open_lessThan[where 'a=real]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1123
lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1124
lemmas closed_real_atMost = closed_atMost[where 'a=real]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1125
lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1126
lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1127
31446
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1128
subsection {* Extra type constraints *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1129
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1130
text {* Only allow @{term "open"} in class @{text topological_space}. *}
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1131
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1132
setup {* Sign.add_const_constraint
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1133
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
  1134
31446
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1135
text {* Only allow @{term dist} in class @{text metric_space}. *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1136
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1137
setup {* Sign.add_const_constraint
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1138
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1139
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1140
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1141
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1142
setup {* Sign.add_const_constraint
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1143
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
  1144
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1145
subsection {* Sign function *}
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1146
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1147
lemma norm_sgn:
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1148
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
  1149
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1150
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1151
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1152
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1153
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1154
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1155
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1156
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1157
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1158
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1159
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1160
lemma sgn_scaleR:
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1161
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
  1162
by (simp add: sgn_div_norm mult_ac)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1163
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1164
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1165
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1166
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1167
lemma sgn_of_real:
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1168
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1169
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1170
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1171
lemma sgn_mult:
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1172
  fixes x y :: "'a::real_normed_div_algebra"
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1173
  shows "sgn (x * y) = sgn x * sgn y"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57448
diff changeset
  1174
by (simp add: sgn_div_norm norm_mult mult.commute)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1175
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1176
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
  1177
by (simp add: sgn_div_norm divide_inverse)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1178
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1179
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56409
diff changeset
  1180
unfolding real_sgn_eq by simp
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1181
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1182
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
56479
91958d4b30f7 revert c1bbd3e22226, a14831ac3023, and 36489d77c484: divide_minus_left/right are again simp rules
hoelzl
parents: 56409
diff changeset
  1183
unfolding real_sgn_eq by simp
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1184
56889
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1185
lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> (x::real)"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1186
  by (cases "0::real" x rule: linorder_cases) simp_all
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1187
  
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1188
lemma zero_less_sgn_iff [simp]: "0 < sgn x \<longleftrightarrow> 0 < (x::real)"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1189
  by (cases "0::real" x rule: linorder_cases) simp_all
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1190
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1191
lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> (x::real) \<le> 0"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1192
  by (cases "0::real" x rule: linorder_cases) simp_all
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1193
  
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1194
lemma sgn_less_0_iff [simp]: "sgn x < 0 \<longleftrightarrow> (x::real) < 0"
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1195
  by (cases "0::real" x rule: linorder_cases) simp_all
48a745e1bde7 avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents: 56479
diff changeset
  1196
51474
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
  1197
lemma norm_conv_dist: "norm x = dist x 0"
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51472
diff changeset
  1198
  unfolding dist_norm by simp
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
  1199
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1200
subsection {* Bounded Linear and Bilinear Operators *}
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1201
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1202
locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1203
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
53600
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1204
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1205
lemma linearI:
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1206
  assumes "\<And>x y. f (x + y) = f x + f y"
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1207
  assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1208
  shows "linear f"
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1209
  by default (rule assms)+
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1210
8fda7ad57466 make 'linear' into a sublocale of 'bounded_linear';
huffman
parents: 53381
diff changeset
  1211
locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1212
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1213
begin
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1214
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1215
lemma pos_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1216
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1217
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1218
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1219
    using bounded by fast
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1220
  show ?thesis
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1221
  proof (intro exI impI conjI allI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1222
    show "0 < max 1 K"
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54785
diff changeset
  1223
      by (rule order_less_le_trans [OF zero_less_one max.cobounded1])
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1224
  next
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1225
    fix x
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1226
    have "norm (f x) \<le> norm x * K" using K .
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1227
    also have "\<dots> \<le> norm x * max 1 K"
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54785
diff changeset
  1228
      by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero])
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1229
    finally show "norm (f x) \<le> norm x * max 1 K" .
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1230
  qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1231
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1232
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1233
lemma nonneg_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1234
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1235
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1236
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1237
  show ?thesis by (auto intro: order_less_imp_le)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1238
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1239
56369
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
  1240
lemma linear: "linear f" ..
2704ca85be98 moved generic theorems from Complex_Analysis_Basic; fixed some theorem names
hoelzl
parents: 56194
diff changeset
  1241
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1242
end
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1243
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1244
lemma bounded_linear_intro:
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1245
  assumes "\<And>x y. f (x + y) = f x + f y"
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1246
  assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1247
  assumes "\<And>x. norm (f x) \<le> norm x * K"
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1248
  shows "bounded_linear f"
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1249
  by default (fast intro: assms)+
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1250
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1251
locale bounded_bilinear =
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1252
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1253
                 \<Rightarrow> 'c::real_normed_vector"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1254
    (infixl "**" 70)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1255
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1256
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1257
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1258
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1259
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1260
begin
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1261
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1262
lemma pos_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1263
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1264
apply (cut_tac bounded, erule exE)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1265
apply (rule_tac x="max 1 K" in exI, safe)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54785
diff changeset
  1266
apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1267
apply (drule spec, drule spec, erule order_trans)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54785
diff changeset
  1268
apply (rule mult_left_mono [OF max.cobounded2])
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1269
apply (intro mult_nonneg_nonneg norm_ge_zero)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1270
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1271
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1272
lemma nonneg_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1273
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1274
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1275
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1276
  show ?thesis by (auto intro: order_less_imp_le)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1277
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1278
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1279
lemma additive_right: "additive (\<lambda>b. prod a b)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1280
by (rule additive.intro, rule add_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1281
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1282
lemma additive_left: "additive (\<lambda>a. prod a b)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1283
by (rule additive.intro, rule add_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1284
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1285
lemma zero_left: "prod 0 b = 0"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1286
by (rule additive.zero [OF additive_left])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1287
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1288
lemma zero_right: "prod a 0 = 0"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1289
by (rule additive.zero [OF additive_right])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1290
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1291
lemma minus_left: "prod (- a) b = - prod a b"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1292
by (rule additive.minus [OF additive_left])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1293
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1294
lemma minus_right: "prod a (- b) = - prod a b"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1295
by (rule additive.minus [OF additive_right])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1296
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1297
lemma diff_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1298
  "prod (a - a') b = prod a b - prod a' b"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1299
by (rule additive.diff [OF additive_left])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1300
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1301
lemma diff_right:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1302
  "prod a (b - b') = prod a b - prod a b'"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1303
by (rule additive.diff [OF additive_right])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1304
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1305
lemma bounded_linear_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1306
  "bounded_linear (\<lambda>a. a ** b)"
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1307
apply (cut_tac bounded, safe)
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1308
apply (rule_tac K="norm b * K" in bounded_linear_intro)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1309
apply (rule add_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1310
apply (rule scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1311
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1312
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1313
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1314
lemma bounded_linear_right:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1315
  "bounded_linear (\<lambda>b. a ** b)"
44127
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1316
apply (cut_tac bounded, safe)
7b57b9295d98 lemma bounded_linear_intro
huffman
parents: 41969
diff changeset
  1317
apply (rule_tac K="norm a * K" in bounded_linear_intro)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1318
apply (rule add_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1319
apply (rule scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1320
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1321
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1322
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1323
lemma prod_diff_prod:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1324
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1325
by (simp add: diff_left diff_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1326
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1327
end
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1328
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1329
lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1330
  by default (auto intro!: exI[of _ 1])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1331
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1332
lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1333
  by default (auto intro!: exI[of _ 1])
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1334
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1335
lemma bounded_linear_add:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1336
  assumes "bounded_linear f"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1337
  assumes "bounded_linear g"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1338
  shows "bounded_linear (\<lambda>x. f x + g x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1339
proof -
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1340
  interpret f: bounded_linear f by fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1341
  interpret g: bounded_linear g by fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1342
  show ?thesis
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1343
  proof
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1344
    from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1345
    from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1346
    show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1347
      using add_mono[OF Kf Kg]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1348
      by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1349
  qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1350
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1351
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1352
lemma bounded_linear_minus:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1353
  assumes "bounded_linear f"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1354
  shows "bounded_linear (\<lambda>x. - f x)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1355
proof -
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1356
  interpret f: bounded_linear f by fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1357
  show ?thesis apply (unfold_locales)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1358
    apply (simp add: f.add)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1359
    apply (simp add: f.scaleR)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1360
    apply (simp add: f.bounded)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1361
    done
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1362
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1363
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1364
lemma bounded_linear_compose:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1365
  assumes "bounded_linear f"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1366
  assumes "bounded_linear g"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1367
  shows "bounded_linear (\<lambda>x. f (g x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1368
proof -
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1369
  interpret f: bounded_linear f by fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1370
  interpret g: bounded_linear g by fact
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1371
  show ?thesis proof (unfold_locales)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1372
    fix x y show "f (g (x + y)) = f (g x) + f (g y)"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1373
      by (simp only: f.add g.add)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1374
  next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1375
    fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1376
      by (simp only: f.scaleR g.scaleR)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1377
  next
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1378
    from f.pos_bounded
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1379
    obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1380
    from g.pos_bounded
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1381
    obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1382
    show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1383
    proof (intro exI allI)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1384
      fix x
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1385
      have "norm (f (g x)) \<le> norm (g x) * Kf"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1386
        using f .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1387
      also have "\<dots> \<le> (norm x * Kg) * Kf"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1388
        using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1389
      also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57448
diff changeset
  1390
        by (rule mult.assoc)
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1391
      finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1392
    qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1393
  qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1394
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1395
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1396
lemma bounded_bilinear_mult:
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1397
  "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1398
apply (rule bounded_bilinear.intro)
49962
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47108
diff changeset
  1399
apply (rule distrib_right)
a8cc904a6820 Renamed {left,right}_distrib to distrib_{right,left}.
webertj
parents: 47108
diff changeset
  1400
apply (rule distrib_left)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1401
apply (rule mult_scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1402
apply (rule mult_scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1403
apply (rule_tac x="1" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1404
apply (simp add: norm_mult_ineq)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1405
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1406
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1407
lemma bounded_linear_mult_left:
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1408
  "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1409
  using bounded_bilinear_mult
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1410
  by (rule bounded_bilinear.bounded_linear_left)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1411
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1412
lemma bounded_linear_mult_right:
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1413
  "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1414
  using bounded_bilinear_mult
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1415
  by (rule bounded_bilinear.bounded_linear_right)
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1416
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1417
lemmas bounded_linear_mult_const =
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1418
  bounded_linear_mult_left [THEN bounded_linear_compose]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1419
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1420
lemmas bounded_linear_const_mult =
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1421
  bounded_linear_mult_right [THEN bounded_linear_compose]
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1422
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1423
lemma bounded_linear_divide:
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1424
  "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1425
  unfolding divide_inverse by (rule bounded_linear_mult_left)
23120
a34f204e9c88 interpretation bounded_linear_divide
huffman
parents: 23113
diff changeset
  1426
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1427
lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1428
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1429
apply (rule scaleR_left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1430
apply (rule scaleR_right_distrib)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1431
apply simp
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1432
apply (rule scaleR_left_commute)
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
  1433
apply (rule_tac x="1" in exI, simp)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1434
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1435
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1436
lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1437
  using bounded_bilinear_scaleR
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1438
  by (rule bounded_bilinear.bounded_linear_left)
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1439
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1440
lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1441
  using bounded_bilinear_scaleR
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1442
  by (rule bounded_bilinear.bounded_linear_right)
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1443
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1444
lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44127
diff changeset
  1445
  unfolding of_real_def by (rule bounded_linear_scaleR_left)
22625
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
  1446
51642
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1447
lemma real_bounded_linear:
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1448
  fixes f :: "real \<Rightarrow> real"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1449
  shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1450
proof -
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1451
  { fix x assume "bounded_linear f"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1452
    then interpret bounded_linear f .
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1453
    from scaleR[of x 1] have "f x = x * f 1"
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1454
      by simp }
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1455
  then show ?thesis
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1456
    by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1457
qed
400ec5ae7f8f move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
hoelzl
parents: 51641
diff changeset
  1458
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1459
instance real_normed_algebra_1 \<subseteq> perfect_space
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1460
proof
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1461
  fix x::'a
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1462
  show "\<not> open {x}"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1463
    unfolding open_dist dist_norm
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1464
    by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1465
qed
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
  1466
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1467
subsection {* Filters and Limits on Metric Space *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1468
57448
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1469
lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1470
  unfolding nhds_def
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1471
proof (safe intro!: INF_eq)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1472
  fix S assume "open S" "x \<in> S"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1473
  then obtain e where "{y. dist y x < e} \<subseteq> S" "0 < e"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1474
    by (auto simp: open_dist subset_eq)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1475
  then show "\<exists>e\<in>{0<..}. principal {y. dist y x < e} \<le> principal S"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1476
    by auto
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1477
qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1478
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1479
lemma (in metric_space) tendsto_iff:
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1480
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1481
  unfolding nhds_metric filterlim_INF filterlim_principal by auto
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1482
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1483
lemma (in metric_space) tendstoI: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f ---> l) F"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1484
  by (auto simp: tendsto_iff)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1485
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1486
lemma (in metric_space) tendstoD: "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1487
  by (auto simp: tendsto_iff)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1488
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1489
lemma (in metric_space) eventually_nhds_metric:
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1490
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1491
  unfolding nhds_metric
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1492
  by (subst eventually_INF_base)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1493
     (auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b])
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1494
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1495
lemma eventually_at:
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1496
  fixes a :: "'a :: metric_space"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1497
  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)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1498
  unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1499
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1500
lemma eventually_at_le:
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1501
  fixes a :: "'a::metric_space"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1502
  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)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1503
  unfolding eventually_at_filter eventually_nhds_metric
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1504
  apply auto
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1505
  apply (rule_tac x="d / 2" in exI)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1506
  apply auto
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1507
  done
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1508
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1509
lemma metric_tendsto_imp_tendsto:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1510
  fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1511
  assumes f: "(f ---> a) F"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1512
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1513
  shows "(g ---> b) F"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1514
proof (rule tendstoI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1515
  fix e :: real assume "0 < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1516
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1517
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1518
    using le_less_trans by (rule eventually_elim2)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1519
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1520
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1521
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1522
  unfolding filterlim_at_top
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1523
  apply (intro allI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1524
  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1525
  apply (auto simp: natceiling_le_eq)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1526
  done
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1527
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1528
subsubsection {* Limits of Sequences *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1529
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1530
lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1531
  unfolding tendsto_iff eventually_sequentially ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1532
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1533
lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1534
  unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1535
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1536
lemma metric_LIMSEQ_I:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1537
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1538
by (simp add: LIMSEQ_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1539
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1540
lemma metric_LIMSEQ_D:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1541
  "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1542
by (simp add: LIMSEQ_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1543
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1544
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1545
subsubsection {* Limits of Functions *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1546
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1547
lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1548
     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1549
        --> dist (f x) L < r)"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1550
  unfolding tendsto_iff eventually_at by simp
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1551
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1552
lemma metric_LIM_I:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1553
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1554
    \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1555
by (simp add: LIM_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1556
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1557
lemma metric_LIM_D:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1558
  "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1559
    \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1560
by (simp add: LIM_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1561
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1562
lemma metric_LIM_imp_LIM:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1563
  assumes f: "f -- a --> (l::'a::metric_space)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1564
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1565
  shows "g -- a --> (m::'b::metric_space)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1566
  by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1567
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1568
lemma metric_LIM_equal2:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1569
  assumes 1: "0 < R"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1570
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1571
  shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1572
apply (rule topological_tendstoI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1573
apply (drule (2) topological_tendstoD)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1574
apply (simp add: eventually_at, safe)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1575
apply (rule_tac x="min d R" in exI, safe)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1576
apply (simp add: 1)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1577
apply (simp add: 2)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1578
done
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1579
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1580
lemma metric_LIM_compose2:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1581
  assumes f: "f -- (a::'a::metric_space) --> b"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1582
  assumes g: "g -- b --> c"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1583
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1584
  shows "(\<lambda>x. g (f x)) -- a --> c"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1585
  using inj
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51531
diff changeset
  1586
  by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1587
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1588
lemma metric_isCont_LIM_compose2:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1589
  fixes f :: "'a :: metric_space \<Rightarrow> _"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1590
  assumes f [unfolded isCont_def]: "isCont f a"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1591
  assumes g: "g -- f a --> l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1592
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1593
  shows "(\<lambda>x. g (f x)) -- a --> l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1594
by (rule metric_LIM_compose2 [OF f g inj])
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1595
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1596
subsection {* Complete metric spaces *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1597
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1598
subsection {* Cauchy sequences *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1599
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1600
definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1601
  "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1602
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1603
subsection {* Cauchy Sequences *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1604
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1605
lemma metric_CauchyI:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1606
  "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1607
  by (simp add: Cauchy_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1608
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1609
lemma metric_CauchyD:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1610
  "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1611
  by (simp add: Cauchy_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1612
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1613
lemma metric_Cauchy_iff2:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1614
  "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1615
apply (simp add: Cauchy_def, auto)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1616
apply (drule reals_Archimedean, safe)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1617
apply (drule_tac x = n in spec, auto)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1618
apply (rule_tac x = M in exI, auto)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1619
apply (drule_tac x = m in spec, simp)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1620
apply (drule_tac x = na in spec, auto)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1621
done
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1622
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1623
lemma Cauchy_iff2:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1624
  "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1625
  unfolding metric_Cauchy_iff2 dist_real_def ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1626
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1627
lemma Cauchy_subseq_Cauchy:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1628
  "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1629
apply (auto simp add: Cauchy_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1630
apply (drule_tac x=e in spec, clarify)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1631
apply (rule_tac x=M in exI, clarify)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1632
apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1633
done
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1634
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1635
theorem LIMSEQ_imp_Cauchy:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1636
  assumes X: "X ----> a" shows "Cauchy X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1637
proof (rule metric_CauchyI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1638
  fix e::real assume "0 < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1639
  hence "0 < e/2" by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1640
  with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1641
  then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1642
  show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1643
  proof (intro exI allI impI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1644
    fix m assume "N \<le> m"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1645
    hence m: "dist (X m) a < e/2" using N by fast
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1646
    fix n assume "N \<le> n"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1647
    hence n: "dist (X n) a < e/2" using N by fast
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1648
    have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1649
      by (rule dist_triangle2)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1650
    also from m n have "\<dots> < e" by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1651
    finally show "dist (X m) (X n) < e" .
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1652
  qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1653
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1654
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1655
lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1656
unfolding convergent_def
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1657
by (erule exE, erule LIMSEQ_imp_Cauchy)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1658
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1659
subsubsection {* Cauchy Sequences are Convergent *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1660
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1661
class complete_space = metric_space +
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1662
  assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1663
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1664
lemma Cauchy_convergent_iff:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1665
  fixes X :: "nat \<Rightarrow> 'a::complete_space"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1666
  shows "Cauchy X = convergent X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1667
by (fast intro: Cauchy_convergent convergent_Cauchy)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1668
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1669
subsection {* The set of real numbers is a complete metric space *}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1670
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1671
text {*
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1672
Proof that Cauchy sequences converge based on the one from
54703
499f92dc6e45 more antiquotations;
wenzelm
parents: 54489
diff changeset
  1673
@{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1674
*}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1675
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1676
text {*
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1677
  If sequence @{term "X"} is Cauchy, then its limit is the lub of
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1678
  @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1679
*}
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1680
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1681
lemma increasing_LIMSEQ:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1682
  fixes f :: "nat \<Rightarrow> real"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1683
  assumes inc: "\<And>n. f n \<le> f (Suc n)"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1684
      and bdd: "\<And>n. f n \<le> l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1685
      and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1686
  shows "f ----> l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1687
proof (rule increasing_tendsto)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1688
  fix x assume "x < l"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1689
  with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1690
    by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1691
  from en[OF `0 < e`] obtain n where "l - e \<le> f n"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1692
    by (auto simp: field_simps)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1693
  with `e < l - x` `0 < e` have "x < f n" by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1694
  with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1695
    by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1696
qed (insert bdd, auto)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1697
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1698
lemma real_Cauchy_convergent:
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1699
  fixes X :: "nat \<Rightarrow> real"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1700
  assumes X: "Cauchy X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1701
  shows "convergent X"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1702
proof -
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1703
  def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1704
  then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1705
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1706
  { fix N x assume N: "\<forall>n\<ge>N. X n < x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1707
  fix y::real assume "y \<in> S"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1708
  hence "\<exists>M. \<forall>n\<ge>M. y < X n"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1709
    by (simp add: S_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1710
  then obtain M where "\<forall>n\<ge>M. y < X n" ..
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1711
  hence "y < X (max M N)" by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1712
  also have "\<dots> < x" using N by simp
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1713
  finally have "y \<le> x"
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1714
    by (rule order_less_imp_le) }
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1715
  note bound_isUb = this 
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1716
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1717
  obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1718
    using X[THEN metric_CauchyD, OF zero_less_one] by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1719
  hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1720
  have [simp]: "S \<noteq> {}"
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1721
  proof (intro exI ex_in_conv[THEN iffD1])
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1722
    from N have "\<forall>n\<ge>N. X N - 1 < X n"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1723
      by (simp add: abs_diff_less_iff dist_real_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1724
    thus "X N - 1 \<in> S" by (rule mem_S)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1725
  qed
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1726
  have [simp]: "bdd_above S"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1727
  proof
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1728
    from N have "\<forall>n\<ge>N. X n < X N + 1"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1729
      by (simp add: abs_diff_less_iff dist_real_def)
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1730
    thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1731
      by (rule bound_isUb)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1732
  qed
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1733
  have "X ----> Sup S"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1734
  proof (rule metric_LIMSEQ_I)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1735
  fix r::real assume "0 < r"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1736
  hence r: "0 < r/2" by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1737
  obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1738
    using metric_CauchyD [OF X r] by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1739
  hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1740
  hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1741
    by (simp only: dist_real_def abs_diff_less_iff)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1742
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1743
  from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1744
  hence "X N - r/2 \<in> S" by (rule mem_S)
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1745
  hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1746
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1747
  from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1748
  from bound_isUb[OF this]
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1749
  have 2: "Sup S \<le> X N + r/2"
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1750
    by (intro cSup_least) simp_all
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1751
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1752
  show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1753
  proof (intro exI allI impI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1754
    fix n assume n: "N \<le> n"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1755
    from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
54263
c4159fe6fa46 move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
hoelzl
parents: 54230
diff changeset
  1756
    thus "dist (X n) (Sup S) < r" using 1 2
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1757
      by (simp add: abs_diff_less_iff dist_real_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1758
  qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1759
  qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1760
  then show ?thesis unfolding convergent_def by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1761
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1762
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1763
instance real :: complete_space
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1764
  by intro_classes (rule real_Cauchy_convergent)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1765
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1766
class banach = real_normed_vector + complete_space
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1767
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1768
instance real :: banach by default
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1769
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1770
lemma tendsto_at_topI_sequentially:
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1771
  fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1772
  assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) ----> y"
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1773
  shows "(f ---> y) at_top"
57448
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1774
proof -
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1775
  from nhds_countable[of y] guess A . note A = this
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1776
57448
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1777
  have "\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1778
  proof (rule ccontr)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1779
    assume "\<not> (\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m)"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1780
    then obtain m where "\<And>k. \<exists>x\<ge>k. f x \<notin> A m"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1781
      by auto
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1782
    then have "\<exists>X. \<forall>n. (f (X n) \<notin> A m) \<and> max n (X n) + 1 \<le> X (Suc n)"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1783
      by (intro dependent_nat_choice) (auto simp del: max.bounded_iff)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1784
    then obtain X where X: "\<And>n. f (X n) \<notin> A m" "\<And>n. max n (X n) + 1 \<le> X (Suc n)"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1785
      by auto
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1786
    { fix n have "1 \<le> n \<longrightarrow> real n \<le> X n"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1787
        using X[of "n - 1"] by auto }
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1788
    then have "filterlim X at_top sequentially"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1789
      by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially]
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1790
                simp: eventually_sequentially)
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1791
    from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1792
      by auto
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1793
  qed
57448
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1794
  then obtain k where "\<And>m x. k m \<le> x \<Longrightarrow> f x \<in> A m"
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1795
    by metis
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1796
  then show ?thesis
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1797
    unfolding at_top_def A
159e45728ceb more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents: 57418
diff changeset
  1798
    by (intro filterlim_base[where i=k]) auto
57275
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1799
qed
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1800
0ddb5b755cdc moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents: 56889
diff changeset
  1801
lemma tendsto_at_topI_sequentially_real:
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1802
  fixes f :: "real \<Rightarrow> real"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1803
  assumes mono: "mono f"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1804
  assumes limseq: "(\<lambda>n. f (real n)) ----> y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1805
  shows "(f ---> y) at_top"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1806
proof (rule tendstoI)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1807
  fix e :: real assume "0 < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1808
  with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1809
    by (auto simp: LIMSEQ_def dist_real_def)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1810
  { fix x :: real
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1811
    obtain n where "x \<le> real_of_nat n"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
  1812
      using ex_le_of_nat[of x] ..
51531
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1813
    note monoD[OF mono this]
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1814
    also have "f (real_of_nat n) \<le> y"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1815
      by (rule LIMSEQ_le_const[OF limseq])
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1816
         (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1817
    finally have "f x \<le> y" . }
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1818
  note le = this
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1819
  have "eventually (\<lambda>x. real N \<le> x) at_top"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1820
    by (rule eventually_ge_at_top)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1821
  then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1822
  proof eventually_elim
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1823
    fix x assume N': "real N \<le> x"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1824
    with N[of N] le have "y - f (real N) < e" by auto
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1825
    moreover note monoD[OF mono N']
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1826
    ultimately show "dist (f x) y < e"
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1827
      using le[of x] by (auto simp: dist_real_def field_simps)
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1828
  qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1829
qed
f415febf4234 remove Metric_Spaces and move its content into Limits and Real_Vector_Spaces
hoelzl
parents: 51524
diff changeset
  1830
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
  1831
end
57276
49c51eeaa623 filters are easier to define with INF on filters.
hoelzl
parents: 57275
diff changeset
  1832