src/HOL/Real/RealVector.thy
author huffman
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(*  Title       : RealVector.thy
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    ID:         $Id$
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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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lemma (in additive) 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 (in additive) 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 (in additive) diff: "f (x - y) = f x - f y"
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by (simp add: diff_def add minus)
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lemma (in additive) 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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subsection {* Real vector spaces *}
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class scaleR = type +
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  fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"
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notation
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  scaleR (infixr "*#" 75)
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abbreviation
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  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a::scaleR" (infixl "'/#" 70) where
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  "x /# r == scaleR (inverse r) x"
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notation (xsymbols)
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  scaleR (infixr "*\<^sub>R" 75) and
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  divideR (infixl "'/\<^sub>R" 70)
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instance real :: scaleR
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  real_scaleR_def: "scaleR a x \<equiv> a * x" ..
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axclass real_vector < scaleR, ab_group_add
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  scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
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  scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
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  scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
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  scaleR_one [simp]: "scaleR 1 x = x"
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axclass real_algebra < real_vector, ring
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  mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
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  mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
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axclass real_algebra_1 < real_algebra, ring_1
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axclass real_div_algebra < real_algebra_1, division_ring
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axclass real_field < real_div_algebra, field
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instance real :: real_field
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apply (intro_classes, unfold real_scaleR_def)
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apply (rule right_distrib)
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apply (rule left_distrib)
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apply (rule mult_assoc [symmetric])
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apply (rule mult_1_left)
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apply (rule mult_assoc)
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apply (rule mult_left_commute)
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done
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lemma scaleR_left_commute:
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  fixes x :: "'a::real_vector"
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  shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)"
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by (simp add: mult_commute)
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lemma additive_scaleR_right: "additive (\<lambda>x. scaleR a x::'a::real_vector)"
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by (rule additive.intro, rule scaleR_right_distrib)
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lemma additive_scaleR_left: "additive (\<lambda>a. scaleR a x::'a::real_vector)"
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by (rule additive.intro, rule scaleR_left_distrib)
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lemmas scaleR_zero_left [simp] =
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  additive.zero [OF additive_scaleR_left, standard]
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lemmas scaleR_zero_right [simp] =
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  additive.zero [OF additive_scaleR_right, standard]
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lemmas scaleR_minus_left [simp] =
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  additive.minus [OF additive_scaleR_left, standard]
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lemmas scaleR_minus_right [simp] =
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  additive.minus [OF additive_scaleR_right, standard]
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lemmas scaleR_left_diff_distrib =
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  additive.diff [OF additive_scaleR_left, standard]
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lemmas scaleR_right_diff_distrib =
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  additive.diff [OF additive_scaleR_right, standard]
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lemma scaleR_eq_0_iff:
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  fixes x :: "'a::real_vector"
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  shows "(scaleR a x = 0) = (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 "scaleR a x = 0"
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    hence "scaleR (inverse a) (scaleR 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 scaleR_left_imp_eq:
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  fixes x y :: "'a::real_vector"
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  shows "\<lbrakk>a \<noteq> 0; scaleR a x = scaleR 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 "scaleR a x = scaleR a y"
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  hence "scaleR a (x - y) = 0"
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     by (simp add: scaleR_right_diff_distrib)
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  hence "x - y = 0"
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     by (simp add: scaleR_eq_0_iff nonzero)
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  thus "x = y" by simp
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qed
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lemma scaleR_right_imp_eq:
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  fixes x y :: "'a::real_vector"
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  shows "\<lbrakk>x \<noteq> 0; scaleR a x = scaleR 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 "scaleR a x = scaleR b x"
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  hence "scaleR (a - b) x = 0"
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     by (simp add: scaleR_left_diff_distrib)
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  hence "a - b = 0"
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     by (simp add: scaleR_eq_0_iff nonzero)
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  thus "a = b" by simp
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qed
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c433e78d4203 define new constant of_real for class real_algebra_1;
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lemma scaleR_cancel_left:
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  fixes x y :: "'a::real_vector"
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  shows "(scaleR a x = scaleR a y) = (x = y \<or> a = 0)"
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by (auto intro: scaleR_left_imp_eq)
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c433e78d4203 define new constant of_real for class real_algebra_1;
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lemma scaleR_cancel_right:
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  fixes x y :: "'a::real_vector"
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  shows "(scaleR a x = scaleR b x) = (a = b \<or> x = 0)"
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by (auto intro: scaleR_right_imp_eq)
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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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c433e78d4203 define new constant of_real for class real_algebra_1;
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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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   193
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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   196
c433e78d4203 define new constant of_real for class real_algebra_1;
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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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   199
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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   202
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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   204
by (simp add: of_real_def scaleR_left_diff_distrib)
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   205
c433e78d4203 define new constant of_real for class real_algebra_1;
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lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
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   207
by (simp add: of_real_def mult_commute)
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   208
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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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   213
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
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   214
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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   217
by (simp add: of_real_def inverse_scaleR_distrib)
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   218
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
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   219
lemma nonzero_of_real_divide:
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  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
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   221
   (of_real x / of_real y :: 'a::real_field)"
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   222
by (simp add: divide_inverse nonzero_of_real_inverse)
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   223
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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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   227
by (simp add: divide_inverse)
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   228
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lemma of_real_power [simp]:
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  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
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   231
by (induct n) (simp_all add: power_Suc)
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   232
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lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
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   234
by (simp add: of_real_def scaleR_cancel_right)
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   235
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   236
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
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   237
c433e78d4203 define new constant of_real for class real_algebra_1;
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   238
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
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   239
proof
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   240
  fix r
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   241
  show "of_real r = id r"
c433e78d4203 define new constant of_real for class real_algebra_1;
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   242
    by (simp add: of_real_def real_scaleR_def)
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   243
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
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   244
c433e78d4203 define new constant of_real for class real_algebra_1;
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   245
text{*Collapse nested embeddings*}
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   246
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
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by (induct n) auto
20554
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   248
c433e78d4203 define new constant of_real for class real_algebra_1;
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   249
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
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   250
by (cases z rule: int_diff_cases, simp)
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   251
c433e78d4203 define new constant of_real for class real_algebra_1;
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   252
lemma of_real_number_of_eq:
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   253
  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
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   254
by (simp add: number_of_eq)
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   255
22912
c477862c566d instance real_algebra_1 < ring_char_0
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   256
text{*Every real algebra has characteristic zero*}
c477862c566d instance real_algebra_1 < ring_char_0
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   257
instance real_algebra_1 < ring_char_0
c477862c566d instance real_algebra_1 < ring_char_0
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   258
proof
c477862c566d instance real_algebra_1 < ring_char_0
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   259
  fix w z :: int
c477862c566d instance real_algebra_1 < ring_char_0
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   260
  assume "of_int w = (of_int z::'a)"
c477862c566d instance real_algebra_1 < ring_char_0
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   261
  hence "of_real (of_int w) = (of_real (of_int z)::'a)"
c477862c566d instance real_algebra_1 < ring_char_0
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   262
    by (simp only: of_real_of_int_eq)
c477862c566d instance real_algebra_1 < ring_char_0
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   263
  thus "w = z"
c477862c566d instance real_algebra_1 < ring_char_0
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diff changeset
   264
    by (simp only: of_real_eq_iff of_int_eq_iff)
c477862c566d instance real_algebra_1 < ring_char_0
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   265
qed
c477862c566d instance real_algebra_1 < ring_char_0
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diff changeset
   266
20554
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   267
c433e78d4203 define new constant of_real for class real_algebra_1;
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   268
subsection {* The Set of Real Numbers *}
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   269
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definition
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  Reals :: "'a::real_algebra_1 set" where
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  "Reals \<equiv> range of_real"
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   273
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   274
notation (xsymbols)
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   275
  Reals  ("\<real>")
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   276
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   277
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
20554
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   278
by (simp add: Reals_def)
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   279
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   280
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
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   281
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
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   282
21809
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   283
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
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   284
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
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diff changeset
   285
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   286
lemma Reals_number_of [simp]:
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   287
  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
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diff changeset
   288
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
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diff changeset
   289
20554
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   290
lemma Reals_0 [simp]: "0 \<in> Reals"
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   291
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   292
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   293
apply (rule of_real_0 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   294
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   295
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   296
lemma Reals_1 [simp]: "1 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   297
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   298
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   299
apply (rule of_real_1 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   300
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   301
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   302
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
   303
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   304
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   305
apply (rule of_real_add [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   306
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   307
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   308
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
   309
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   310
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   311
apply (rule of_real_minus [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   312
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   313
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   314
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
   315
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   316
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   317
apply (rule of_real_diff [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   318
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   319
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   320
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
   321
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   322
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   323
apply (rule of_real_mult [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   324
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   325
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   326
lemma nonzero_Reals_inverse:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   327
  fixes a :: "'a::real_div_algebra"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   328
  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
   329
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   330
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   331
apply (erule nonzero_of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   332
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   333
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   334
lemma Reals_inverse [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   335
  fixes a :: "'a::{real_div_algebra,division_by_zero}"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   336
  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
   337
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   338
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   339
apply (rule of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   340
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   341
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   342
lemma nonzero_Reals_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   343
  fixes a b :: "'a::real_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   344
  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
   345
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   346
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   347
apply (erule nonzero_of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   348
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   349
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   350
lemma Reals_divide [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   351
  fixes a b :: "'a::{real_field,division_by_zero}"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   352
  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
   353
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   354
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   355
apply (rule of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   356
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   357
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   358
lemma Reals_power [simp]:
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   359
  fixes a :: "'a::{real_algebra_1,recpower}"
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   360
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   361
apply (auto simp add: Reals_def)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   362
apply (rule range_eqI)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   363
apply (rule of_real_power [symmetric])
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   364
done
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   365
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   366
lemma Reals_cases [cases set: Reals]:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   367
  assumes "q \<in> \<real>"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   368
  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
   369
  unfolding Reals_def
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   370
proof -
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   371
  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
   372
  then obtain r where "q = of_real r" ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   373
  then show thesis ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   374
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   375
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   376
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
   377
  "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
   378
  by (rule Reals_cases) auto
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   379
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   380
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   381
subsection {* Real normed vector spaces *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   382
22636
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   383
class norm = type +
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   384
  fixes norm :: "'a \<Rightarrow> real"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   385
22636
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   386
instance real :: norm
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   387
  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>" ..
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   388
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   389
axclass real_normed_vector < real_vector, norm
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   390
  norm_ge_zero [simp]: "0 \<le> norm x"
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   391
  norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   392
  norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   393
  norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   394
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   395
axclass real_normed_algebra < real_algebra, real_normed_vector
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   396
  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
   397
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   398
axclass real_normed_algebra_1 < real_algebra_1, real_normed_algebra
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   399
  norm_one [simp]: "norm 1 = 1"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   400
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   401
axclass real_normed_div_algebra < real_div_algebra, real_normed_vector
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   402
  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
   403
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   404
axclass real_normed_field < real_field, real_normed_div_algebra
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   405
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   406
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
   407
proof
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   408
  fix x y :: 'a
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   409
  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
   410
    by (simp add: norm_mult)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   411
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   412
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   413
    by (rule norm_mult)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   414
  thus "norm (1::'a) = 1" by simp
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   415
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   416
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   417
instance real :: real_normed_field
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   418
apply (intro_classes, unfold real_norm_def real_scaleR_def)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   419
apply (rule abs_ge_zero)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   420
apply (rule abs_eq_0)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   421
apply (rule abs_triangle_ineq)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   422
apply (rule abs_mult)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   423
apply (rule abs_mult)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   424
done
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   425
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   426
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
   427
by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   428
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   429
lemma zero_less_norm_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   430
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   431
  shows "(0 < norm x) = (x \<noteq> 0)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   432
by (simp add: order_less_le)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   433
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   434
lemma norm_not_less_zero [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   435
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   436
  shows "\<not> norm x < 0"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   437
by (simp add: linorder_not_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   438
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   439
lemma norm_le_zero_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   440
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   441
  shows "(norm x \<le> 0) = (x = 0)"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   442
by (simp add: order_le_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   443
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   444
lemma norm_minus_cancel [simp]:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   445
  fixes x :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   446
  shows "norm (- x) = norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   447
proof -
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   448
  have "norm (- x) = norm (scaleR (- 1) x)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   449
    by (simp only: scaleR_minus_left scaleR_one)
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   450
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   451
    by (rule norm_scaleR)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   452
  finally show ?thesis by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   453
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   454
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   455
lemma norm_minus_commute:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   456
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   457
  shows "norm (a - b) = norm (b - a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   458
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   459
  have "norm (- (b - a)) = norm (b - a)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   460
    by (rule norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   461
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   462
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   463
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   464
lemma norm_triangle_ineq2:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   465
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   466
  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
   467
proof -
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   468
  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
   469
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   470
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   471
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   472
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   473
lemma norm_triangle_ineq3:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   474
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   475
  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
   476
apply (subst abs_le_iff)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   477
apply auto
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   478
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   479
apply (subst norm_minus_commute)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   480
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   481
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   482
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   483
lemma norm_triangle_ineq4:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   484
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   485
  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
   486
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   487
  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
   488
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   489
  thus ?thesis
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   490
    by (simp only: diff_minus norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   491
qed
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   492
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   493
lemma norm_diff_ineq:
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   494
  fixes a b :: "'a::real_normed_vector"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   495
  shows "norm a - norm b \<le> norm (a + b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   496
proof -
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   497
  have "norm a - norm (- b) \<le> norm (a - - b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   498
    by (rule norm_triangle_ineq2)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   499
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   500
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   501
20551
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   502
lemma norm_diff_triangle_ineq:
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   503
  fixes a b c d :: "'a::real_normed_vector"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   504
  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
   505
proof -
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   506
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   507
    by (simp add: diff_minus add_ac)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   508
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   509
    by (rule norm_triangle_ineq)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   510
  finally show ?thesis .
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   511
qed
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   512
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   513
lemma abs_norm_cancel [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   514
  fixes a :: "'a::real_normed_vector"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   515
  shows "\<bar>norm a\<bar> = norm a"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   516
by (rule abs_of_nonneg [OF norm_ge_zero])
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   517
22880
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   518
lemma norm_add_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   519
  fixes x y :: "'a::real_normed_vector"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   520
  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
   521
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
   522
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   523
lemma norm_mult_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   524
  fixes x y :: "'a::real_normed_algebra"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   525
  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
   526
apply (rule order_le_less_trans [OF norm_mult_ineq])
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   527
apply (simp add: mult_strict_mono')
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   528
done
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   529
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   530
lemma norm_of_real [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   531
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   532
unfolding of_real_def by (simp add: norm_scaleR)
20560
49996715bc6e norm_one is now proved from other class axioms
huffman
parents: 20554
diff changeset
   533
22876
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   534
lemma norm_number_of [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   535
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   536
    = \<bar>number_of w\<bar>"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   537
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   538
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   539
lemma norm_of_int [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   540
  "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
   541
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
   542
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   543
lemma norm_of_nat [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   544
  "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
   545
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
   546
apply (subst norm_of_real, simp)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   547
done
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   548
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   549
lemma nonzero_norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   550
  fixes a :: "'a::real_normed_div_algebra"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   551
  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
   552
apply (rule inverse_unique [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   553
apply (simp add: norm_mult [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   554
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   555
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   556
lemma norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   557
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   558
  shows "norm (inverse a) = inverse (norm a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   559
apply (case_tac "a = 0", simp)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   560
apply (erule nonzero_norm_inverse)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   561
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   562
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   563
lemma nonzero_norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   564
  fixes a b :: "'a::real_normed_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   565
  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
   566
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
   567
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   568
lemma norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   569
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   570
  shows "norm (a / b) = norm a / norm b"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   571
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
   572
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   573
lemma norm_power_ineq:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   574
  fixes x :: "'a::{real_normed_algebra_1,recpower}"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   575
  shows "norm (x ^ n) \<le> norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   576
proof (induct n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   577
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   578
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   579
  case (Suc n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   580
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   581
    by (rule norm_mult_ineq)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   582
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   583
    using norm_ge_zero by (rule mult_left_mono)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   584
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   585
    by (simp add: power_Suc)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   586
qed
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   587
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   588
lemma norm_power:
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   589
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   590
  shows "norm (x ^ n) = norm x ^ n"
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   591
by (induct n) (simp_all add: power_Suc norm_mult)
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   592
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   593
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   594
subsection {* Bounded Linear and Bilinear Operators *}
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   595
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   596
locale bounded_linear = additive +
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   597
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   598
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   599
  assumes bounded: "\<exists>K. \<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
   600
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   601
lemma (in bounded_linear) pos_bounded:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   602
  "\<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
   603
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   604
  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
   605
    using bounded by fast
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   606
  show ?thesis
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   607
  proof (intro exI impI conjI allI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   608
    show "0 < max 1 K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   609
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   610
  next
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   611
    fix x
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   612
    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
   613
    also have "\<dots> \<le> norm x * max 1 K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   614
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   615
    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
   616
  qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   617
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   618
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   619
lemma (in bounded_linear) nonneg_bounded:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   620
  "\<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
   621
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   622
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   623
  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
   624
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   625
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   626
locale bounded_bilinear =
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   627
  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
   628
                 \<Rightarrow> 'c::real_normed_vector"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   629
    (infixl "**" 70)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   630
  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
   631
  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
   632
  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
   633
  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
   634
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod 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
   635
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   636
lemma (in bounded_bilinear) pos_bounded:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   637
  "\<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
   638
apply (cut_tac bounded, erule exE)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   639
apply (rule_tac x="max 1 K" in exI, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   640
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   641
apply (drule spec, drule spec, erule order_trans)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   642
apply (rule mult_left_mono [OF le_maxI2])
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   643
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
   644
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   645
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   646
lemma (in bounded_bilinear) nonneg_bounded:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   647
  "\<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
   648
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   649
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   650
  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
   651
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   652
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   653
lemma (in bounded_bilinear) additive_right: "additive (\<lambda>b. prod a b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   654
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
   655
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   656
lemma (in bounded_bilinear) additive_left: "additive (\<lambda>a. prod a b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   657
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
   658
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   659
lemma (in bounded_bilinear) zero_left: "prod 0 b = 0"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   660
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
   661
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   662
lemma (in bounded_bilinear) zero_right: "prod a 0 = 0"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   663
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
   664
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   665
lemma (in bounded_bilinear) minus_left: "prod (- a) b = - prod a b"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   666
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
   667
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   668
lemma (in bounded_bilinear) minus_right: "prod a (- b) = - prod a b"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   669
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
   670
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   671
lemma (in bounded_bilinear) diff_left:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   672
  "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
   673
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
   674
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   675
lemma (in bounded_bilinear) diff_right:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   676
  "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
   677
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
   678
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   679
lemma (in bounded_bilinear) bounded_linear_left:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   680
  "bounded_linear (\<lambda>a. a ** b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   681
apply (unfold_locales)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   682
apply (rule add_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   683
apply (rule scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   684
apply (cut_tac bounded, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   685
apply (rule_tac x="norm b * K" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   686
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   687
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   688
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   689
lemma (in bounded_bilinear) bounded_linear_right:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   690
  "bounded_linear (\<lambda>b. a ** b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   691
apply (unfold_locales)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   692
apply (rule add_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   693
apply (rule scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   694
apply (cut_tac bounded, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   695
apply (rule_tac x="norm a * K" in exI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   696
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   697
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   698
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   699
lemma (in bounded_bilinear) prod_diff_prod:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   700
  "(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
   701
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
   702
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   703
interpretation bounded_bilinear_mult:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   704
  bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   705
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   706
apply (rule left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   707
apply (rule right_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   708
apply (rule mult_scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   709
apply (rule mult_scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   710
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
   711
apply (simp add: norm_mult_ineq)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   712
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   713
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   714
interpretation bounded_linear_mult_left:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   715
  bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   716
by (rule bounded_bilinear_mult.bounded_linear_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   717
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   718
interpretation bounded_linear_mult_right:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   719
  bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   720
by (rule bounded_bilinear_mult.bounded_linear_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   721
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   722
interpretation bounded_bilinear_scaleR:
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   723
  bounded_bilinear ["scaleR"]
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   724
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   725
apply (rule scaleR_left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   726
apply (rule scaleR_right_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   727
apply (simp add: real_scaleR_def)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   728
apply (rule scaleR_left_commute)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   729
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
   730
apply (simp add: norm_scaleR)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   731
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   732
22625
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   733
interpretation bounded_linear_of_real:
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   734
  bounded_linear ["\<lambda>r. of_real r"]
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   735
apply (unfold of_real_def)
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   736
apply (rule bounded_bilinear_scaleR.bounded_linear_left)
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   737
done
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
   738
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   739
end