src/HOL/RealVector.thy
author huffman
Thu May 28 17:00:02 2009 -0700 (2009-05-28)
changeset 31289 847f00f435d4
parent 31285 0a3f9ee4117c
child 31413 729d90a531e4
permissions -rw-r--r--
move dist operation to new metric_space class
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(*  Title:      HOL/RealVector.thy
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    Author:     Brian Huffman
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*)
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header {* Vector Spaces and Algebras over the Reals *}
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theory RealVector
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imports RealPow
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begin
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subsection {* Locale for additive functions *}
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locale additive =
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  fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
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  assumes add: "f (x + y) = f x + f y"
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begin
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lemma zero: "f 0 = 0"
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proof -
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  have "f 0 = f (0 + 0)" by simp
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  also have "\<dots> = f 0 + f 0" by (rule add)
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  finally show "f 0 = 0" by simp
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qed
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lemma minus: "f (- x) = - f x"
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proof -
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  have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
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  also have "\<dots> = - f x + f x" by (simp add: zero)
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  finally show "f (- x) = - f x" by (rule add_right_imp_eq)
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qed
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lemma diff: "f (x - y) = f x - f y"
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by (simp add: diff_def 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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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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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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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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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:
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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:
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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_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
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  and scaleR_left_distrib: "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_right_distrib)
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apply (rule scaleR_left_distrib)
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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_left_diff_distrib = real_vector.scale_left_diff_distrib
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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_right_diff_distrib = real_vector.scale_right_diff_distrib
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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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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_by_zero}"
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  shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
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apply (case_tac "a = 0", simp)
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apply (case_tac "x = 0", simp)
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apply (erule (1) nonzero_inverse_scaleR_distrib)
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done
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subsection {* Embedding of the Reals into any @{text real_algebra_1}:
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@{term of_real} *}
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definition
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  of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
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  "of_real r = scaleR r 1"
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lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
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by (simp add: of_real_def)
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lemma of_real_0 [simp]: "of_real 0 = 0"
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by (simp add: of_real_def)
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lemma of_real_1 [simp]: "of_real 1 = 1"
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by (simp add: of_real_def)
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lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
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by (simp add: of_real_def scaleR_left_distrib)
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lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
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by (simp add: of_real_def)
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lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
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by (simp add: of_real_def scaleR_left_diff_distrib)
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lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
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by (simp add: of_real_def mult_commute)
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lemma nonzero_of_real_inverse:
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  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
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   inverse (of_real x :: 'a::real_div_algebra)"
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by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
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lemma of_real_inverse [simp]:
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  "of_real (inverse x) =
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   inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
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by (simp add: of_real_def inverse_scaleR_distrib)
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lemma nonzero_of_real_divide:
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  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
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   (of_real x / of_real y :: 'a::real_field)"
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by (simp add: divide_inverse nonzero_of_real_inverse)
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lemma of_real_divide [simp]:
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  "of_real (x / y) =
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   (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
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by (simp add: divide_inverse)
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lemma of_real_power [simp]:
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  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
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by (induct n) simp_all
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lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
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by (simp add: of_real_def scaleR_cancel_right)
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lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
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lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
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proof
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  fix r
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  show "of_real r = id r"
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    by (simp add: of_real_def)
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qed
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text{*Collapse nested embeddings*}
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lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
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by (induct n) auto
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lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
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by (cases z rule: int_diff_cases, simp)
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lemma of_real_number_of_eq:
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  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
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by (simp add: number_of_eq)
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text{*Every real algebra has characteristic zero*}
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instance real_algebra_1 < ring_char_0
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proof
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  fix m n :: nat
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  have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
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    by (simp only: of_real_eq_iff of_nat_eq_iff)
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  thus "(of_nat m = (of_nat n::'a)) = (m = n)"
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    by (simp only: of_real_of_nat_eq)
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qed
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instance real_field < field_char_0 ..
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subsection {* The Set of Real Numbers *}
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definition
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  Reals :: "'a::real_algebra_1 set" where
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  [code del]: "Reals = range of_real"
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notation (xsymbols)
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  Reals  ("\<real>")
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lemma Reals_of_real [simp]: "of_real r \<in> Reals"
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by (simp add: Reals_def)
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lemma Reals_of_int [simp]: "of_int z \<in> Reals"
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by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
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lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
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by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
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lemma Reals_number_of [simp]:
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  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
huffman@21809
   326
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
huffman@20718
   327
huffman@20554
   328
lemma Reals_0 [simp]: "0 \<in> Reals"
huffman@20554
   329
apply (unfold Reals_def)
huffman@20554
   330
apply (rule range_eqI)
huffman@20554
   331
apply (rule of_real_0 [symmetric])
huffman@20554
   332
done
huffman@20554
   333
huffman@20554
   334
lemma Reals_1 [simp]: "1 \<in> Reals"
huffman@20554
   335
apply (unfold Reals_def)
huffman@20554
   336
apply (rule range_eqI)
huffman@20554
   337
apply (rule of_real_1 [symmetric])
huffman@20554
   338
done
huffman@20554
   339
huffman@20584
   340
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
huffman@20554
   341
apply (auto simp add: Reals_def)
huffman@20554
   342
apply (rule range_eqI)
huffman@20554
   343
apply (rule of_real_add [symmetric])
huffman@20554
   344
done
huffman@20554
   345
huffman@20584
   346
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
huffman@20584
   347
apply (auto simp add: Reals_def)
huffman@20584
   348
apply (rule range_eqI)
huffman@20584
   349
apply (rule of_real_minus [symmetric])
huffman@20584
   350
done
huffman@20584
   351
huffman@20584
   352
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
huffman@20584
   353
apply (auto simp add: Reals_def)
huffman@20584
   354
apply (rule range_eqI)
huffman@20584
   355
apply (rule of_real_diff [symmetric])
huffman@20584
   356
done
huffman@20584
   357
huffman@20584
   358
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
huffman@20554
   359
apply (auto simp add: Reals_def)
huffman@20554
   360
apply (rule range_eqI)
huffman@20554
   361
apply (rule of_real_mult [symmetric])
huffman@20554
   362
done
huffman@20554
   363
huffman@20584
   364
lemma nonzero_Reals_inverse:
huffman@20584
   365
  fixes a :: "'a::real_div_algebra"
huffman@20584
   366
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   367
apply (auto simp add: Reals_def)
huffman@20584
   368
apply (rule range_eqI)
huffman@20584
   369
apply (erule nonzero_of_real_inverse [symmetric])
huffman@20584
   370
done
huffman@20584
   371
huffman@20584
   372
lemma Reals_inverse [simp]:
huffman@20584
   373
  fixes a :: "'a::{real_div_algebra,division_by_zero}"
huffman@20584
   374
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   375
apply (auto simp add: Reals_def)
huffman@20584
   376
apply (rule range_eqI)
huffman@20584
   377
apply (rule of_real_inverse [symmetric])
huffman@20584
   378
done
huffman@20584
   379
huffman@20584
   380
lemma nonzero_Reals_divide:
huffman@20584
   381
  fixes a b :: "'a::real_field"
huffman@20584
   382
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   383
apply (auto simp add: Reals_def)
huffman@20584
   384
apply (rule range_eqI)
huffman@20584
   385
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   386
done
huffman@20584
   387
huffman@20584
   388
lemma Reals_divide [simp]:
huffman@20584
   389
  fixes a b :: "'a::{real_field,division_by_zero}"
huffman@20584
   390
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   391
apply (auto simp add: Reals_def)
huffman@20584
   392
apply (rule range_eqI)
huffman@20584
   393
apply (rule of_real_divide [symmetric])
huffman@20584
   394
done
huffman@20584
   395
huffman@20722
   396
lemma Reals_power [simp]:
haftmann@31017
   397
  fixes a :: "'a::{real_algebra_1}"
huffman@20722
   398
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   399
apply (auto simp add: Reals_def)
huffman@20722
   400
apply (rule range_eqI)
huffman@20722
   401
apply (rule of_real_power [symmetric])
huffman@20722
   402
done
huffman@20722
   403
huffman@20554
   404
lemma Reals_cases [cases set: Reals]:
huffman@20554
   405
  assumes "q \<in> \<real>"
huffman@20554
   406
  obtains (of_real) r where "q = of_real r"
huffman@20554
   407
  unfolding Reals_def
huffman@20554
   408
proof -
huffman@20554
   409
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   410
  then obtain r where "q = of_real r" ..
huffman@20554
   411
  then show thesis ..
huffman@20554
   412
qed
huffman@20554
   413
huffman@20554
   414
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   415
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   416
  by (rule Reals_cases) auto
huffman@20554
   417
huffman@20504
   418
huffman@31289
   419
subsection {* Metric spaces *}
huffman@31289
   420
huffman@31289
   421
class dist =
huffman@31289
   422
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
huffman@31289
   423
huffman@31289
   424
class metric_space = dist +
huffman@31289
   425
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   426
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
huffman@31289
   427
begin
huffman@31289
   428
huffman@31289
   429
lemma dist_self [simp]: "dist x x = 0"
huffman@31289
   430
by simp
huffman@31289
   431
huffman@31289
   432
lemma zero_le_dist [simp]: "0 \<le> dist x y"
huffman@31289
   433
using dist_triangle2 [of x x y] by simp
huffman@31289
   434
huffman@31289
   435
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
huffman@31289
   436
by (simp add: less_le)
huffman@31289
   437
huffman@31289
   438
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
huffman@31289
   439
by (simp add: not_less)
huffman@31289
   440
huffman@31289
   441
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
huffman@31289
   442
by (simp add: le_less)
huffman@31289
   443
huffman@31289
   444
lemma dist_commute: "dist x y = dist y x"
huffman@31289
   445
proof (rule order_antisym)
huffman@31289
   446
  show "dist x y \<le> dist y x"
huffman@31289
   447
    using dist_triangle2 [of x y x] by simp
huffman@31289
   448
  show "dist y x \<le> dist x y"
huffman@31289
   449
    using dist_triangle2 [of y x y] by simp
huffman@31289
   450
qed
huffman@31289
   451
huffman@31289
   452
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
huffman@31289
   453
using dist_triangle2 [of x z y] by (simp add: dist_commute)
huffman@31289
   454
huffman@31289
   455
end
huffman@31289
   456
huffman@31289
   457
huffman@20504
   458
subsection {* Real normed vector spaces *}
huffman@20504
   459
haftmann@29608
   460
class norm =
huffman@22636
   461
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   462
huffman@24520
   463
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   464
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   465
huffman@31289
   466
class dist_norm = dist + norm + minus +
huffman@31289
   467
  assumes dist_norm: "dist x y = norm (x - y)"
huffman@31289
   468
huffman@31289
   469
class real_normed_vector = real_vector + sgn_div_norm + dist_norm +
haftmann@24588
   470
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
haftmann@25062
   471
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   472
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
haftmann@24588
   473
  and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@20504
   474
haftmann@24588
   475
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   476
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   477
haftmann@24588
   478
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   479
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   480
haftmann@24588
   481
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   482
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   483
haftmann@24588
   484
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   485
huffman@22852
   486
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   487
proof
huffman@20554
   488
  fix x y :: 'a
huffman@20554
   489
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   490
    by (simp add: norm_mult)
huffman@22852
   491
next
huffman@22852
   492
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   493
    by (rule norm_mult)
huffman@22852
   494
  thus "norm (1::'a) = 1" by simp
huffman@20554
   495
qed
huffman@20554
   496
huffman@30069
   497
instantiation real :: real_normed_field
huffman@30069
   498
begin
huffman@30069
   499
huffman@30069
   500
definition
huffman@30069
   501
  real_norm_def [simp]: "norm r = \<bar>r\<bar>"
huffman@30069
   502
huffman@31289
   503
definition
huffman@31289
   504
  dist_real_def: "dist x y = \<bar>x - y\<bar>"
huffman@31289
   505
huffman@30069
   506
instance
huffman@22852
   507
apply (intro_classes, unfold real_norm_def real_scaleR_def)
huffman@31289
   508
apply (rule dist_real_def)
nipkow@24506
   509
apply (simp add: real_sgn_def)
huffman@20554
   510
apply (rule abs_ge_zero)
huffman@20554
   511
apply (rule abs_eq_0)
huffman@20554
   512
apply (rule abs_triangle_ineq)
huffman@22852
   513
apply (rule abs_mult)
huffman@20554
   514
apply (rule abs_mult)
huffman@20554
   515
done
huffman@20504
   516
huffman@30069
   517
end
huffman@30069
   518
huffman@22852
   519
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   520
by simp
huffman@20504
   521
huffman@22852
   522
lemma zero_less_norm_iff [simp]:
huffman@22852
   523
  fixes x :: "'a::real_normed_vector"
huffman@22852
   524
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   525
by (simp add: order_less_le)
huffman@20504
   526
huffman@22852
   527
lemma norm_not_less_zero [simp]:
huffman@22852
   528
  fixes x :: "'a::real_normed_vector"
huffman@22852
   529
  shows "\<not> norm x < 0"
huffman@20828
   530
by (simp add: linorder_not_less)
huffman@20828
   531
huffman@22852
   532
lemma norm_le_zero_iff [simp]:
huffman@22852
   533
  fixes x :: "'a::real_normed_vector"
huffman@22852
   534
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   535
by (simp add: order_le_less)
huffman@20828
   536
huffman@20504
   537
lemma norm_minus_cancel [simp]:
huffman@20584
   538
  fixes x :: "'a::real_normed_vector"
huffman@20584
   539
  shows "norm (- x) = norm x"
huffman@20504
   540
proof -
huffman@21809
   541
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   542
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   543
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   544
    by (rule norm_scaleR)
huffman@20504
   545
  finally show ?thesis by simp
huffman@20504
   546
qed
huffman@20504
   547
huffman@20504
   548
lemma norm_minus_commute:
huffman@20584
   549
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   550
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   551
proof -
huffman@22898
   552
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   553
    by (rule norm_minus_cancel)
huffman@22898
   554
  thus ?thesis by simp
huffman@20504
   555
qed
huffman@20504
   556
huffman@20504
   557
lemma norm_triangle_ineq2:
huffman@20584
   558
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   559
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   560
proof -
huffman@20533
   561
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   562
    by (rule norm_triangle_ineq)
huffman@22898
   563
  thus ?thesis by simp
huffman@20504
   564
qed
huffman@20504
   565
huffman@20584
   566
lemma norm_triangle_ineq3:
huffman@20584
   567
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   568
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   569
apply (subst abs_le_iff)
huffman@20584
   570
apply auto
huffman@20584
   571
apply (rule norm_triangle_ineq2)
huffman@20584
   572
apply (subst norm_minus_commute)
huffman@20584
   573
apply (rule norm_triangle_ineq2)
huffman@20584
   574
done
huffman@20584
   575
huffman@20504
   576
lemma norm_triangle_ineq4:
huffman@20584
   577
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   578
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   579
proof -
huffman@22898
   580
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   581
    by (rule norm_triangle_ineq)
huffman@22898
   582
  thus ?thesis
huffman@22898
   583
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   584
qed
huffman@22898
   585
huffman@22898
   586
lemma norm_diff_ineq:
huffman@22898
   587
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   588
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   589
proof -
huffman@22898
   590
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   591
    by (rule norm_triangle_ineq2)
huffman@22898
   592
  thus ?thesis by simp
huffman@20504
   593
qed
huffman@20504
   594
huffman@20551
   595
lemma norm_diff_triangle_ineq:
huffman@20551
   596
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   597
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   598
proof -
huffman@20551
   599
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   600
    by (simp add: diff_minus add_ac)
huffman@20551
   601
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   602
    by (rule norm_triangle_ineq)
huffman@20551
   603
  finally show ?thesis .
huffman@20551
   604
qed
huffman@20551
   605
huffman@22857
   606
lemma abs_norm_cancel [simp]:
huffman@22857
   607
  fixes a :: "'a::real_normed_vector"
huffman@22857
   608
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   609
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   610
huffman@22880
   611
lemma norm_add_less:
huffman@22880
   612
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   613
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   614
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   615
huffman@22880
   616
lemma norm_mult_less:
huffman@22880
   617
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   618
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   619
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   620
apply (simp add: mult_strict_mono')
huffman@22880
   621
done
huffman@22880
   622
huffman@22857
   623
lemma norm_of_real [simp]:
huffman@22857
   624
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@22852
   625
unfolding of_real_def by (simp add: norm_scaleR)
huffman@20560
   626
huffman@22876
   627
lemma norm_number_of [simp]:
huffman@22876
   628
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
huffman@22876
   629
    = \<bar>number_of w\<bar>"
huffman@22876
   630
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
huffman@22876
   631
huffman@22876
   632
lemma norm_of_int [simp]:
huffman@22876
   633
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   634
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   635
huffman@22876
   636
lemma norm_of_nat [simp]:
huffman@22876
   637
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   638
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   639
apply (subst norm_of_real, simp)
huffman@22876
   640
done
huffman@22876
   641
huffman@20504
   642
lemma nonzero_norm_inverse:
huffman@20504
   643
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   644
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   645
apply (rule inverse_unique [symmetric])
huffman@20504
   646
apply (simp add: norm_mult [symmetric])
huffman@20504
   647
done
huffman@20504
   648
huffman@20504
   649
lemma norm_inverse:
huffman@20504
   650
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
huffman@20533
   651
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   652
apply (case_tac "a = 0", simp)
huffman@20504
   653
apply (erule nonzero_norm_inverse)
huffman@20504
   654
done
huffman@20504
   655
huffman@20584
   656
lemma nonzero_norm_divide:
huffman@20584
   657
  fixes a b :: "'a::real_normed_field"
huffman@20584
   658
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   659
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   660
huffman@20584
   661
lemma norm_divide:
huffman@20584
   662
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
huffman@20584
   663
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   664
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   665
huffman@22852
   666
lemma norm_power_ineq:
haftmann@31017
   667
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@22852
   668
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   669
proof (induct n)
huffman@22852
   670
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   671
next
huffman@22852
   672
  case (Suc n)
huffman@22852
   673
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   674
    by (rule norm_mult_ineq)
huffman@22852
   675
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   676
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   677
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@30273
   678
    by simp
huffman@22852
   679
qed
huffman@22852
   680
huffman@20684
   681
lemma norm_power:
haftmann@31017
   682
  fixes x :: "'a::{real_normed_div_algebra}"
huffman@20684
   683
  shows "norm (x ^ n) = norm x ^ n"
huffman@30273
   684
by (induct n) (simp_all add: norm_mult)
huffman@20684
   685
huffman@31289
   686
text {* Every normed vector space is a metric space. *}
huffman@31285
   687
huffman@31289
   688
instance real_normed_vector < metric_space
huffman@31289
   689
proof
huffman@31289
   690
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   691
    unfolding dist_norm by simp
huffman@31289
   692
next
huffman@31289
   693
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
huffman@31289
   694
    unfolding dist_norm
huffman@31289
   695
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
huffman@31289
   696
qed
huffman@31285
   697
huffman@31285
   698
huffman@22972
   699
subsection {* Sign function *}
huffman@22972
   700
nipkow@24506
   701
lemma norm_sgn:
nipkow@24506
   702
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
nipkow@24506
   703
by (simp add: sgn_div_norm norm_scaleR)
huffman@22972
   704
nipkow@24506
   705
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   706
by (simp add: sgn_div_norm)
huffman@22972
   707
nipkow@24506
   708
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   709
by (simp add: sgn_div_norm)
huffman@22972
   710
nipkow@24506
   711
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   712
by (simp add: sgn_div_norm)
huffman@22972
   713
nipkow@24506
   714
lemma sgn_scaleR:
nipkow@24506
   715
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
nipkow@24506
   716
by (simp add: sgn_div_norm norm_scaleR mult_ac)
huffman@22973
   717
huffman@22972
   718
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   719
by (simp add: sgn_div_norm)
huffman@22972
   720
huffman@22972
   721
lemma sgn_of_real:
huffman@22972
   722
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   723
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   724
huffman@22973
   725
lemma sgn_mult:
huffman@22973
   726
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   727
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   728
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   729
huffman@22972
   730
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   731
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   732
huffman@22972
   733
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   734
unfolding real_sgn_eq by simp
huffman@22972
   735
huffman@22972
   736
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   737
unfolding real_sgn_eq by simp
huffman@22972
   738
huffman@22972
   739
huffman@22442
   740
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   741
huffman@22442
   742
locale bounded_linear = additive +
huffman@22442
   743
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@22442
   744
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   745
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
   746
begin
huffman@22442
   747
huffman@27443
   748
lemma pos_bounded:
huffman@22442
   749
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   750
proof -
huffman@22442
   751
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   752
    using bounded by fast
huffman@22442
   753
  show ?thesis
huffman@22442
   754
  proof (intro exI impI conjI allI)
huffman@22442
   755
    show "0 < max 1 K"
huffman@22442
   756
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   757
  next
huffman@22442
   758
    fix x
huffman@22442
   759
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   760
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   761
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   762
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   763
  qed
huffman@22442
   764
qed
huffman@22442
   765
huffman@27443
   766
lemma nonneg_bounded:
huffman@22442
   767
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   768
proof -
huffman@22442
   769
  from pos_bounded
huffman@22442
   770
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   771
qed
huffman@22442
   772
huffman@27443
   773
end
huffman@27443
   774
huffman@22442
   775
locale bounded_bilinear =
huffman@22442
   776
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   777
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   778
    (infixl "**" 70)
huffman@22442
   779
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   780
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   781
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   782
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   783
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
   784
begin
huffman@22442
   785
huffman@27443
   786
lemma pos_bounded:
huffman@22442
   787
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   788
apply (cut_tac bounded, erule exE)
huffman@22442
   789
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   790
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   791
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   792
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   793
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   794
done
huffman@22442
   795
huffman@27443
   796
lemma nonneg_bounded:
huffman@22442
   797
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   798
proof -
huffman@22442
   799
  from pos_bounded
huffman@22442
   800
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   801
qed
huffman@22442
   802
huffman@27443
   803
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
   804
by (rule additive.intro, rule add_right)
huffman@22442
   805
huffman@27443
   806
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
   807
by (rule additive.intro, rule add_left)
huffman@22442
   808
huffman@27443
   809
lemma zero_left: "prod 0 b = 0"
huffman@22442
   810
by (rule additive.zero [OF additive_left])
huffman@22442
   811
huffman@27443
   812
lemma zero_right: "prod a 0 = 0"
huffman@22442
   813
by (rule additive.zero [OF additive_right])
huffman@22442
   814
huffman@27443
   815
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
   816
by (rule additive.minus [OF additive_left])
huffman@22442
   817
huffman@27443
   818
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
   819
by (rule additive.minus [OF additive_right])
huffman@22442
   820
huffman@27443
   821
lemma diff_left:
huffman@22442
   822
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
   823
by (rule additive.diff [OF additive_left])
huffman@22442
   824
huffman@27443
   825
lemma diff_right:
huffman@22442
   826
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
   827
by (rule additive.diff [OF additive_right])
huffman@22442
   828
huffman@27443
   829
lemma bounded_linear_left:
huffman@22442
   830
  "bounded_linear (\<lambda>a. a ** b)"
huffman@22442
   831
apply (unfold_locales)
huffman@22442
   832
apply (rule add_left)
huffman@22442
   833
apply (rule scaleR_left)
huffman@22442
   834
apply (cut_tac bounded, safe)
huffman@22442
   835
apply (rule_tac x="norm b * K" in exI)
huffman@22442
   836
apply (simp add: mult_ac)
huffman@22442
   837
done
huffman@22442
   838
huffman@27443
   839
lemma bounded_linear_right:
huffman@22442
   840
  "bounded_linear (\<lambda>b. a ** b)"
huffman@22442
   841
apply (unfold_locales)
huffman@22442
   842
apply (rule add_right)
huffman@22442
   843
apply (rule scaleR_right)
huffman@22442
   844
apply (cut_tac bounded, safe)
huffman@22442
   845
apply (rule_tac x="norm a * K" in exI)
huffman@22442
   846
apply (simp add: mult_ac)
huffman@22442
   847
done
huffman@22442
   848
huffman@27443
   849
lemma prod_diff_prod:
huffman@22442
   850
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
   851
by (simp add: diff_left diff_right)
huffman@22442
   852
huffman@27443
   853
end
huffman@27443
   854
wenzelm@30729
   855
interpretation mult:
ballarin@29229
   856
  bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"
huffman@22442
   857
apply (rule bounded_bilinear.intro)
huffman@22442
   858
apply (rule left_distrib)
huffman@22442
   859
apply (rule right_distrib)
huffman@22442
   860
apply (rule mult_scaleR_left)
huffman@22442
   861
apply (rule mult_scaleR_right)
huffman@22442
   862
apply (rule_tac x="1" in exI)
huffman@22442
   863
apply (simp add: norm_mult_ineq)
huffman@22442
   864
done
huffman@22442
   865
wenzelm@30729
   866
interpretation mult_left:
ballarin@29229
   867
  bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)"
huffman@23127
   868
by (rule mult.bounded_linear_left)
huffman@22442
   869
wenzelm@30729
   870
interpretation mult_right:
ballarin@29229
   871
  bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)"
huffman@23127
   872
by (rule mult.bounded_linear_right)
huffman@23127
   873
wenzelm@30729
   874
interpretation divide:
ballarin@29229
   875
  bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)"
huffman@23127
   876
unfolding divide_inverse by (rule mult.bounded_linear_left)
huffman@23120
   877
wenzelm@30729
   878
interpretation scaleR: bounded_bilinear "scaleR"
huffman@22442
   879
apply (rule bounded_bilinear.intro)
huffman@22442
   880
apply (rule scaleR_left_distrib)
huffman@22442
   881
apply (rule scaleR_right_distrib)
huffman@22973
   882
apply simp
huffman@22442
   883
apply (rule scaleR_left_commute)
huffman@22442
   884
apply (rule_tac x="1" in exI)
huffman@22442
   885
apply (simp add: norm_scaleR)
huffman@22442
   886
done
huffman@22442
   887
wenzelm@30729
   888
interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x"
huffman@23127
   889
by (rule scaleR.bounded_linear_left)
huffman@23127
   890
wenzelm@30729
   891
interpretation scaleR_right: bounded_linear "\<lambda>x. scaleR r x"
huffman@23127
   892
by (rule scaleR.bounded_linear_right)
huffman@23127
   893
wenzelm@30729
   894
interpretation of_real: bounded_linear "\<lambda>r. of_real r"
huffman@23127
   895
unfolding of_real_def by (rule scaleR.bounded_linear_left)
huffman@22625
   896
huffman@20504
   897
end