src/HOL/RealVector.thy
author wenzelm
Sat, 13 Mar 2010 14:44:47 +0100
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child 36349 39be26d1bc28
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adapted to localized typedef: handle single global interpretation only;
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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 [simp]:
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  "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
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by (auto intro: scale_left_imp_eq)
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lemma scale_cancel_right [simp]:
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  "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
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by (auto intro: scale_right_imp_eq)
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end
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subsection {* Real vector spaces *}
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class scaleR =
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  fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
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begin
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abbreviation
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  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
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where
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  "x /\<^sub>R r == scaleR (inverse r) x"
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end
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class real_vector = scaleR + ab_group_add +
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  assumes scaleR_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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c433e78d4203 define new constant of_real for class real_algebra_1;
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lemma of_real_1 [simp]: "of_real 1 = 1"
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by (simp add: of_real_def)
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lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
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by (simp add: of_real_def scaleR_left_distrib)
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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)
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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
c433e78d4203 define new constant of_real for class real_algebra_1;
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parents: 20551
diff changeset
   278
  show "of_real r = id r"
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   279
    by (simp add: of_real_def)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   280
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   281
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   282
text{*Collapse nested embeddings*}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   283
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   284
by (induct n) auto
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   285
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   286
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   287
by (cases z rule: int_diff_cases, simp)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   288
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   289
lemma of_real_number_of_eq:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   290
  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   291
by (simp add: number_of_eq)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   292
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   293
text{*Every real algebra has characteristic zero*}
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   294
instance real_algebra_1 < ring_char_0
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   295
proof
23282
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   296
  fix m n :: nat
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   297
  have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   298
    by (simp only: of_real_eq_iff of_nat_eq_iff)
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   299
  thus "(of_nat m = (of_nat n::'a)) = (m = n)"
dfc459989d24 add axclass semiring_char_0 for types where of_nat is injective
huffman
parents: 23127
diff changeset
   300
    by (simp only: of_real_of_nat_eq)
22912
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   301
qed
c477862c566d instance real_algebra_1 < ring_char_0
huffman
parents: 22898
diff changeset
   302
27553
d315a513a150 instance real_field < field_char_0;
huffman
parents: 27552
diff changeset
   303
instance real_field < field_char_0 ..
d315a513a150 instance real_field < field_char_0;
huffman
parents: 27552
diff changeset
   304
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   305
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   306
subsection {* The Set of Real Numbers *}
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   307
20772
7a51ed817ec7 tuned definitions/proofs;
wenzelm
parents: 20763
diff changeset
   308
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21210
diff changeset
   309
  Reals :: "'a::real_algebra_1 set" where
30070
76cee7c62782 declare scaleR distrib rules [algebra_simps]; cleaned up
huffman
parents: 30069
diff changeset
   310
  [code del]: "Reals = range of_real"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   311
21210
c17fd2df4e9e renamed 'const_syntax' to 'notation';
wenzelm
parents: 20828
diff changeset
   312
notation (xsymbols)
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   313
  Reals  ("\<real>")
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   314
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   315
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   316
by (simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   317
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   318
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   319
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
20718
4c4869e4ddb7 add lemmas of_int_in_Reals, of_nat_in_Reals
huffman
parents: 20694
diff changeset
   320
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   321
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   322
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   323
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   324
lemma Reals_number_of [simp]:
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   325
  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   326
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
20718
4c4869e4ddb7 add lemmas of_int_in_Reals, of_nat_in_Reals
huffman
parents: 20694
diff changeset
   327
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   328
lemma Reals_0 [simp]: "0 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   329
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   330
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   331
apply (rule of_real_0 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   332
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   333
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   334
lemma Reals_1 [simp]: "1 \<in> Reals"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   335
apply (unfold Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   336
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   337
apply (rule of_real_1 [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   338
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   339
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   340
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
   341
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   342
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   343
apply (rule of_real_add [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   344
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   345
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   346
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
   347
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   348
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   349
apply (rule of_real_minus [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   350
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   351
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   352
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
   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_diff [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
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   358
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
   359
apply (auto simp add: Reals_def)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   360
apply (rule range_eqI)
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   361
apply (rule of_real_mult [symmetric])
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   362
done
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   363
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   364
lemma nonzero_Reals_inverse:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   365
  fixes a :: "'a::real_div_algebra"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   366
  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
   367
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   368
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   369
apply (erule nonzero_of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   370
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   371
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   372
lemma Reals_inverse [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   373
  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
   374
  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
   375
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   376
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   377
apply (rule of_real_inverse [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   378
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   379
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   380
lemma nonzero_Reals_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   381
  fixes a b :: "'a::real_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   382
  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
   383
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   384
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   385
apply (erule nonzero_of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   386
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   387
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   388
lemma Reals_divide [simp]:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   389
  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
   390
  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
   391
apply (auto simp add: Reals_def)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   392
apply (rule range_eqI)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   393
apply (rule of_real_divide [symmetric])
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   394
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   395
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   396
lemma Reals_power [simp]:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   397
  fixes a :: "'a::{real_algebra_1}"
20722
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   398
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   399
apply (auto simp add: Reals_def)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   400
apply (rule range_eqI)
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   401
apply (rule of_real_power [symmetric])
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   402
done
741737aa70b2 add lemmas about of_real and power
huffman
parents: 20718
diff changeset
   403
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   404
lemma Reals_cases [cases set: Reals]:
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   405
  assumes "q \<in> \<real>"
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   406
  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
   407
  unfolding Reals_def
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   408
proof -
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   409
  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
   410
  then obtain r where "q = of_real r" ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   411
  then show thesis ..
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   412
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   413
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   414
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
   415
  "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
   416
  by (rule Reals_cases) auto
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   417
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   418
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   419
subsection {* Topological spaces *}
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   420
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   421
class "open" =
31494
1ba61c7b129f fix type of open
huffman
parents: 31492
diff changeset
   422
  fixes "open" :: "'a set \<Rightarrow> bool"
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   423
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   424
class topological_space = "open" +
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   425
  assumes open_UNIV [simp, intro]: "open UNIV"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   426
  assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   427
  assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
31490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   428
begin
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   429
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   430
definition
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   431
  closed :: "'a set \<Rightarrow> bool" where
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   432
  "closed S \<longleftrightarrow> open (- S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   433
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   434
lemma open_empty [intro, simp]: "open {}"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   435
  using open_Union [of "{}"] by simp
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   436
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   437
lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   438
  using open_Union [of "{S, T}"] by simp
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   439
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   440
lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   441
  unfolding UN_eq by (rule open_Union) auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   442
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   443
lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   444
  by (induct set: finite) auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   445
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   446
lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   447
  unfolding Inter_def by (rule open_INT)
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   448
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   449
lemma closed_empty [intro, simp]:  "closed {}"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   450
  unfolding closed_def by simp
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   451
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   452
lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   453
  unfolding closed_def by auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   454
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   455
lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   456
  unfolding closed_def Inter_def by auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   457
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   458
lemma closed_UNIV [intro, simp]: "closed UNIV"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   459
  unfolding closed_def by simp
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   460
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   461
lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   462
  unfolding closed_def by auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   463
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   464
lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   465
  unfolding closed_def by auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   466
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   467
lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   468
  by (induct set: finite) auto
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   469
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   470
lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   471
  unfolding Union_def by (rule closed_UN)
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   472
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   473
lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   474
  unfolding closed_def by simp
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   475
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   476
lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   477
  unfolding closed_def by simp
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   478
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   479
lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   480
  unfolding closed_open Diff_eq by (rule open_Int)
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   481
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   482
lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   483
  unfolding open_closed Diff_eq by (rule closed_Int)
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   484
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   485
lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   486
  unfolding closed_open .
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   487
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   488
lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   489
  unfolding open_closed .
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   490
c350f3ad6b0d move definitions of open, closed to RealVector.thy
huffman
parents: 31446
diff changeset
   491
end
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   492
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   493
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   494
subsection {* Metric spaces *}
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   495
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   496
class dist =
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   497
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   498
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   499
class open_dist = "open" + dist +
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   500
  assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   501
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   502
class metric_space = open_dist +
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   503
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   504
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   505
begin
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   506
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   507
lemma dist_self [simp]: "dist x x = 0"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   508
by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   509
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   510
lemma zero_le_dist [simp]: "0 \<le> dist x y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   511
using dist_triangle2 [of x x y] by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   512
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   513
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   514
by (simp add: less_le)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   515
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   516
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   517
by (simp add: not_less)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   518
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   519
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   520
by (simp add: le_less)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   521
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   522
lemma dist_commute: "dist x y = dist y x"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   523
proof (rule order_antisym)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   524
  show "dist x y \<le> dist y x"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   525
    using dist_triangle2 [of x y x] by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   526
  show "dist y x \<le> dist x y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   527
    using dist_triangle2 [of y x y] by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   528
qed
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   529
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   530
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   531
using dist_triangle2 [of x z y] by (simp add: dist_commute)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   532
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31564
diff changeset
   533
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31564
diff changeset
   534
using dist_triangle2 [of x y a] by (simp add: dist_commute)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31564
diff changeset
   535
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   536
subclass topological_space
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   537
proof
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   538
  have "\<exists>e::real. 0 < e"
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   539
    by (fast intro: zero_less_one)
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   540
  then show "open UNIV"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   541
    unfolding open_dist by simp
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   542
next
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   543
  fix S T assume "open S" "open T"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   544
  then show "open (S \<inter> T)"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   545
    unfolding open_dist
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   546
    apply clarify
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   547
    apply (drule (1) bspec)+
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   548
    apply (clarify, rename_tac r s)
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   549
    apply (rule_tac x="min r s" in exI, simp)
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   550
    done
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   551
next
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   552
  fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   553
    unfolding open_dist by fast
31413
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   554
qed
729d90a531e4 introduce class topological_space as a superclass of metric_space
huffman
parents: 31289
diff changeset
   555
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   556
end
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   557
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   558
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   559
subsection {* Real normed vector spaces *}
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   560
29608
564ea783ace8 no base sort in class import
haftmann
parents: 29252
diff changeset
   561
class norm =
22636
c40465deaf20 new class syntax for scaleR and norm classes
huffman
parents: 22625
diff changeset
   562
  fixes norm :: "'a \<Rightarrow> real"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   563
24520
40b220403257 fix sgn_div_norm class
huffman
parents: 24513
diff changeset
   564
class sgn_div_norm = scaleR + norm + sgn +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   565
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   566
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   567
class dist_norm = dist + norm + minus +
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   568
  assumes dist_norm: "dist x y = norm (x - y)"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   569
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   570
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   571
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   572
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   573
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
   574
  and norm_scaleR [simp]: "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
   575
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   576
class real_normed_algebra = real_algebra + real_normed_vector +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   577
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   578
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   579
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   580
  assumes norm_one [simp]: "norm 1 = 1"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   581
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   582
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
25062
af5ef0d4d655 global class syntax
haftmann
parents: 24901
diff changeset
   583
  assumes norm_mult: "norm (x * y) = norm x * norm y"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   584
24588
ed9a1254d674 introduced classes
haftmann
parents: 24520
diff changeset
   585
class real_normed_field = real_field + real_normed_div_algebra
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   586
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   587
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
   588
proof
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   589
  fix x y :: 'a
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   590
  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
   591
    by (simp add: norm_mult)
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   592
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   593
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   594
    by (rule norm_mult)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   595
  thus "norm (1::'a) = 1" by simp
20554
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   596
qed
c433e78d4203 define new constant of_real for class real_algebra_1;
huffman
parents: 20551
diff changeset
   597
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   598
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
   599
by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   600
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   601
lemma zero_less_norm_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   602
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   603
  shows "(0 < norm x) = (x \<noteq> 0)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   604
by (simp add: order_less_le)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   605
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   606
lemma norm_not_less_zero [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   607
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   608
  shows "\<not> norm x < 0"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   609
by (simp add: linorder_not_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   610
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   611
lemma norm_le_zero_iff [simp]:
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   612
  fixes x :: "'a::real_normed_vector"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   613
  shows "(norm x \<le> 0) = (x = 0)"
20828
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   614
by (simp add: order_le_less)
68ed2e514ca0 add lemmas norm_not_less_zero, norm_le_zero_iff
huffman
parents: 20772
diff changeset
   615
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   616
lemma norm_minus_cancel [simp]:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   617
  fixes x :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   618
  shows "norm (- x) = norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   619
proof -
21809
4b93e949ac33 remove uses of scaleR infix syntax; add lemma Reals_number_of
huffman
parents: 21404
diff changeset
   620
  have "norm (- x) = norm (scaleR (- 1) x)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   621
    by (simp only: scaleR_minus_left scaleR_one)
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   622
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   623
    by (rule norm_scaleR)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   624
  finally show ?thesis by simp
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   625
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   626
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   627
lemma norm_minus_commute:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   628
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   629
  shows "norm (a - b) = norm (b - a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   630
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   631
  have "norm (- (b - a)) = norm (b - a)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   632
    by (rule norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   633
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   634
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   635
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   636
lemma norm_triangle_ineq2:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   637
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   638
  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
   639
proof -
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   640
  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
   641
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   642
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   643
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   644
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   645
lemma norm_triangle_ineq3:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   646
  fixes a b :: "'a::real_normed_vector"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   647
  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
   648
apply (subst abs_le_iff)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   649
apply auto
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   650
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   651
apply (subst norm_minus_commute)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   652
apply (rule norm_triangle_ineq2)
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   653
done
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   654
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   655
lemma norm_triangle_ineq4:
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   656
  fixes a b :: "'a::real_normed_vector"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   657
  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
   658
proof -
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   659
  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
   660
    by (rule norm_triangle_ineq)
22898
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   661
  thus ?thesis
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   662
    by (simp only: diff_minus norm_minus_cancel)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   663
qed
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   664
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   665
lemma norm_diff_ineq:
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   666
  fixes a b :: "'a::real_normed_vector"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   667
  shows "norm a - norm b \<le> norm (a + b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   668
proof -
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   669
  have "norm a - norm (- b) \<le> norm (a - - b)"
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   670
    by (rule norm_triangle_ineq2)
38ae2815989f add lemma norm_diff_ineq; shorten other proofs
huffman
parents: 22880
diff changeset
   671
  thus ?thesis by simp
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   672
qed
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   673
20551
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   674
lemma norm_diff_triangle_ineq:
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   675
  fixes a b c d :: "'a::real_normed_vector"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   676
  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
   677
proof -
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   678
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   679
    by (simp add: diff_minus add_ac)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   680
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   681
    by (rule norm_triangle_ineq)
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   682
  finally show ?thesis .
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   683
qed
ba543692bfa1 add theorem norm_diff_triangle_ineq
huffman
parents: 20533
diff changeset
   684
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   685
lemma abs_norm_cancel [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   686
  fixes a :: "'a::real_normed_vector"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   687
  shows "\<bar>norm a\<bar> = norm a"
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   688
by (rule abs_of_nonneg [OF norm_ge_zero])
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   689
22880
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   690
lemma norm_add_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   691
  fixes x y :: "'a::real_normed_vector"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   692
  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
   693
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
   694
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   695
lemma norm_mult_less:
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   696
  fixes x y :: "'a::real_normed_algebra"
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   697
  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
   698
apply (rule order_le_less_trans [OF norm_mult_ineq])
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   699
apply (simp add: mult_strict_mono')
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   700
done
424d6fb67513 add lemmas norm_add_less, norm_mult_less
huffman
parents: 22876
diff changeset
   701
22857
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   702
lemma norm_of_real [simp]:
cb84e886cc90 add lemma abs_norm_cancel
huffman
parents: 22852
diff changeset
   703
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
   704
unfolding of_real_def by simp
20560
49996715bc6e norm_one is now proved from other class axioms
huffman
parents: 20554
diff changeset
   705
22876
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   706
lemma norm_number_of [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   707
  "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
   708
    = \<bar>number_of w\<bar>"
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   709
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
   710
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   711
lemma norm_of_int [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   712
  "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
   713
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
   714
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   715
lemma norm_of_nat [simp]:
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   716
  "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
   717
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
   718
apply (subst norm_of_real, simp)
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   719
done
2b4c831ceca7 add lemmas norm_number_of, norm_of_int, norm_of_nat
huffman
parents: 22857
diff changeset
   720
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   721
lemma nonzero_norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   722
  fixes a :: "'a::real_normed_div_algebra"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   723
  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
   724
apply (rule inverse_unique [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   725
apply (simp add: norm_mult [symmetric])
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   726
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   727
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   728
lemma norm_inverse:
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   729
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
20533
49442b3024bb remove conflicting norm syntax
huffman
parents: 20504
diff changeset
   730
  shows "norm (inverse a) = inverse (norm a)"
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   731
apply (case_tac "a = 0", simp)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   732
apply (erule nonzero_norm_inverse)
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   733
done
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
   734
20584
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   735
lemma nonzero_norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   736
  fixes a b :: "'a::real_normed_field"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   737
  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
   738
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
   739
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   740
lemma norm_divide:
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   741
  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
   742
  shows "norm (a / b) = norm a / norm b"
60b1d52a455d added classes real_div_algebra and real_field; added lemmas
huffman
parents: 20560
diff changeset
   743
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
   744
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   745
lemma norm_power_ineq:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   746
  fixes x :: "'a::{real_normed_algebra_1}"
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   747
  shows "norm (x ^ n) \<le> norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   748
proof (induct n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   749
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   750
next
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   751
  case (Suc n)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   752
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   753
    by (rule norm_mult_ineq)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   754
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   755
    using norm_ge_zero by (rule mult_left_mono)
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   756
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   757
    by simp
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   758
qed
2490d4b4671a clean up RealVector classes
huffman
parents: 22636
diff changeset
   759
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   760
lemma norm_power:
31017
2c227493ea56 stripped class recpower further
haftmann
parents: 30729
diff changeset
   761
  fixes x :: "'a::{real_normed_div_algebra}"
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   762
  shows "norm (x ^ n) = norm x ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 30242
diff changeset
   763
by (induct n) (simp_all add: norm_mult)
20684
74e8b46abb97 add lemma norm_power
huffman
parents: 20584
diff changeset
   764
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   765
text {* Every normed vector space is a metric space. *}
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31017
diff changeset
   766
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   767
instance real_normed_vector < metric_space
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   768
proof
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   769
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   770
    unfolding dist_norm by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   771
next
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   772
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   773
    unfolding dist_norm
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   774
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   775
qed
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31017
diff changeset
   776
31564
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   777
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   778
subsection {* Class instances for real numbers *}
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   779
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   780
instantiation real :: real_normed_field
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   781
begin
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   782
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   783
definition real_norm_def [simp]:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   784
  "norm r = \<bar>r\<bar>"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   785
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   786
definition dist_real_def:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   787
  "dist x y = \<bar>x - y\<bar>"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   788
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   789
definition open_real_def [code del]:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   790
  "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   791
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   792
instance
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   793
apply (intro_classes, unfold real_norm_def real_scaleR_def)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   794
apply (rule dist_real_def)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   795
apply (rule open_real_def)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   796
apply (simp add: real_sgn_def)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   797
apply (rule abs_ge_zero)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   798
apply (rule abs_eq_0)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   799
apply (rule abs_triangle_ineq)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   800
apply (rule abs_mult)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   801
apply (rule abs_mult)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   802
done
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   803
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   804
end
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   805
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   806
lemma open_real_lessThan [simp]:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   807
  fixes a :: real shows "open {..<a}"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   808
unfolding open_real_def dist_real_def
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   809
proof (clarify)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   810
  fix x assume "x < a"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   811
  hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   812
  thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   813
qed
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   814
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   815
lemma open_real_greaterThan [simp]:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   816
  fixes a :: real shows "open {a<..}"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   817
unfolding open_real_def dist_real_def
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   818
proof (clarify)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   819
  fix x assume "a < x"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   820
  hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   821
  thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   822
qed
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   823
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   824
lemma open_real_greaterThanLessThan [simp]:
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   825
  fixes a b :: real shows "open {a<..<b}"
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   826
proof -
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   827
  have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   828
  thus "open {a<..<b}" by (simp add: open_Int)
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   829
qed
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   830
31567
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   831
lemma closed_real_atMost [simp]: 
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   832
  fixes a :: real shows "closed {..a}"
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   833
unfolding closed_open by simp
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   834
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   835
lemma closed_real_atLeast [simp]:
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   836
  fixes a :: real shows "closed {a..}"
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   837
unfolding closed_open by simp
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   838
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   839
lemma closed_real_atLeastAtMost [simp]:
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   840
  fixes a b :: real shows "closed {a..b}"
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   841
proof -
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   842
  have "{a..b} = {a..} \<inter> {..b}" by auto
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   843
  thus "closed {a..b}" by (simp add: closed_Int)
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   844
qed
0fb78b3a9145 new lemmas
huffman
parents: 31565
diff changeset
   845
31564
d2abf6f6f619 subsection for real instances; new lemmas for open sets of reals
huffman
parents: 31494
diff changeset
   846
31446
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   847
subsection {* Extra type constraints *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   848
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   849
text {* Only allow @{term "open"} in class @{text topological_space}. *}
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   850
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   851
setup {* Sign.add_const_constraint
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   852
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31490
diff changeset
   853
31446
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   854
text {* Only allow @{term dist} in class @{text metric_space}. *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   855
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   856
setup {* Sign.add_const_constraint
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   857
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   858
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   859
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   860
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   861
setup {* Sign.add_const_constraint
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   862
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
2d91b2416de8 add extra type constraints for dist, norm
huffman
parents: 31419
diff changeset
   863
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31017
diff changeset
   864
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   865
subsection {* Sign function *}
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   866
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   867
lemma norm_sgn:
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   868
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
   869
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   870
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   871
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   872
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   873
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   874
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   875
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   876
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   877
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   878
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   879
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   880
lemma sgn_scaleR:
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   881
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
   882
by (simp add: sgn_div_norm mult_ac)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   883
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   884
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   885
by (simp add: sgn_div_norm)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   886
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   887
lemma sgn_of_real:
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   888
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   889
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   890
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   891
lemma sgn_mult:
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   892
  fixes x y :: "'a::real_normed_div_algebra"
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   893
  shows "sgn (x * y) = sgn x * sgn y"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   894
by (simp add: sgn_div_norm norm_mult mult_commute)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
   895
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   896
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
24506
020db6ec334a final(?) iteration of sgn saga.
nipkow
parents: 23282
diff changeset
   897
by (simp add: sgn_div_norm divide_inverse)
22972
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   898
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   899
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   900
unfolding real_sgn_eq by simp
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   901
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   902
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   903
unfolding real_sgn_eq by simp
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   904
3e96b98d37c6 generalized sgn function to work on any real normed vector space
huffman
parents: 22942
diff changeset
   905
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   906
subsection {* Bounded Linear and Bilinear Operators *}
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   907
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   908
locale bounded_linear = additive +
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   909
  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
   910
  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
   911
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   912
begin
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   913
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   914
lemma pos_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   915
  "\<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
   916
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   917
  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
   918
    using bounded by fast
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   919
  show ?thesis
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   920
  proof (intro exI impI conjI allI)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   921
    show "0 < max 1 K"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   922
      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
   923
  next
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   924
    fix x
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   925
    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
   926
    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
   927
      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
   928
    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
   929
  qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   930
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   931
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   932
lemma nonneg_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   933
  "\<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
   934
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   935
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   936
  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
   937
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   938
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   939
end
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   940
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   941
locale bounded_bilinear =
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   942
  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
   943
                 \<Rightarrow> 'c::real_normed_vector"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   944
    (infixl "**" 70)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   945
  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
   946
  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
   947
  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
   948
  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
   949
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   950
begin
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   951
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   952
lemma pos_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   953
  "\<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
   954
apply (cut_tac bounded, erule exE)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   955
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
   956
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
   957
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
   958
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
   959
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
   960
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   961
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   962
lemma nonneg_bounded:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   963
  "\<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
   964
proof -
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   965
  from pos_bounded
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   966
  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
   967
qed
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   968
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   969
lemma additive_right: "additive (\<lambda>b. prod a b)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   970
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
   971
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   972
lemma additive_left: "additive (\<lambda>a. prod a b)"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   973
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
   974
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   975
lemma zero_left: "prod 0 b = 0"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   976
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
   977
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   978
lemma zero_right: "prod a 0 = 0"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   979
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
   980
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   981
lemma minus_left: "prod (- a) b = - prod a b"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   982
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
   983
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   984
lemma minus_right: "prod a (- b) = - prod a b"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   985
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
   986
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   987
lemma diff_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   988
  "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
   989
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
   990
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   991
lemma diff_right:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   992
  "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
   993
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
   994
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
   995
lemma bounded_linear_left:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   996
  "bounded_linear (\<lambda>a. a ** b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   997
apply (unfold_locales)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   998
apply (rule add_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
   999
apply (rule scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1000
apply (cut_tac bounded, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1001
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
  1002
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1003
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1004
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1005
lemma bounded_linear_right:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1006
  "bounded_linear (\<lambda>b. a ** b)"
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1007
apply (unfold_locales)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1008
apply (rule add_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1009
apply (rule scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1010
apply (cut_tac bounded, safe)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1011
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
  1012
apply (simp add: mult_ac)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1013
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1014
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1015
lemma prod_diff_prod:
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1016
  "(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
  1017
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
  1018
27443
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1019
end
22b6281d6719 use begin and end for proofs in locales
huffman
parents: 27435
diff changeset
  1020
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1021
interpretation mult:
29229
6f6262027054 Porting to new locales.
ballarin
parents: 28952
diff changeset
  1022
  bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1023
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1024
apply (rule left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1025
apply (rule right_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1026
apply (rule mult_scaleR_left)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1027
apply (rule mult_scaleR_right)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1028
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
  1029
apply (simp add: norm_mult_ineq)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1030
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1031
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1032
interpretation mult_left:
29229
6f6262027054 Porting to new locales.
ballarin
parents: 28952
diff changeset
  1033
  bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1034
by (rule mult.bounded_linear_left)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1035
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1036
interpretation mult_right:
29229
6f6262027054 Porting to new locales.
ballarin
parents: 28952
diff changeset
  1037
  bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1038
by (rule mult.bounded_linear_right)
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1039
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1040
interpretation divide:
29229
6f6262027054 Porting to new locales.
ballarin
parents: 28952
diff changeset
  1041
  bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1042
unfolding divide_inverse by (rule mult.bounded_linear_left)
23120
a34f204e9c88 interpretation bounded_linear_divide
huffman
parents: 23113
diff changeset
  1043
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1044
interpretation scaleR: bounded_bilinear "scaleR"
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1045
apply (rule bounded_bilinear.intro)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1046
apply (rule scaleR_left_distrib)
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1047
apply (rule scaleR_right_distrib)
22973
64d300e16370 add lemma sgn_mult; declare real_scaleR_def and scaleR_eq_0_iff as simp rules
huffman
parents: 22972
diff changeset
  1048
apply simp
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1049
apply (rule scaleR_left_commute)
31586
d4707b99e631 declare norm_scaleR [simp]; declare scaleR_cancel lemmas [simp]
huffman
parents: 31567
diff changeset
  1050
apply (rule_tac x="1" in exI, simp)
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1051
done
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21809
diff changeset
  1052
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1053
interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1054
by (rule scaleR.bounded_linear_left)
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1055
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1056
interpretation scaleR_right: bounded_linear "\<lambda>x. scaleR r x"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1057
by (rule scaleR.bounded_linear_right)
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1058
30729
461ee3e49ad3 interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents: 30273
diff changeset
  1059
interpretation of_real: bounded_linear "\<lambda>r. of_real r"
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23120
diff changeset
  1060
unfolding of_real_def by (rule scaleR.bounded_linear_left)
22625
a2967023d674 interpretation bounded_linear_of_real
huffman
parents: 22442
diff changeset
  1061
20504
6342e872e71d formalization of vector spaces and algebras over the real numbers
huffman
parents:
diff changeset
  1062
end