src/HOL/RealVector.thy
author haftmann
Mon Jul 19 16:09:44 2010 +0200 (2010-07-19)
changeset 37887 2ae085b07f2f
parent 37767 a2b7a20d6ea3
child 38621 d6cb7e625d75
permissions -rw-r--r--
diff_minus subsumes diff_def
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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 RComplete
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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: add minus diff_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_ring_inverse_zero}"
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  shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
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apply (case_tac "a = 0", simp)
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apply (case_tac "x = 0", simp)
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apply (erule (1) nonzero_inverse_scaleR_distrib)
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done
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subsection {* Embedding of the Reals into any @{text real_algebra_1}:
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@{term of_real} *}
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definition
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  of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
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  "of_real r = scaleR r 1"
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lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
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by (simp add: of_real_def)
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lemma of_real_0 [simp]: "of_real 0 = 0"
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by (simp add: of_real_def)
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lemma of_real_1 [simp]: "of_real 1 = 1"
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by (simp add: of_real_def)
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lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
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by (simp add: of_real_def scaleR_left_distrib)
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lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
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by (simp add: of_real_def)
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lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
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by (simp add: of_real_def scaleR_left_diff_distrib)
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lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
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by (simp add: of_real_def mult_commute)
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lemma nonzero_of_real_inverse:
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  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
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   inverse (of_real x :: 'a::real_div_algebra)"
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by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
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lemma of_real_inverse [simp]:
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  "of_real (inverse x) =
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   inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
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by (simp add: of_real_def inverse_scaleR_distrib)
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lemma nonzero_of_real_divide:
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  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
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   (of_real x / of_real y :: 'a::real_field)"
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by (simp add: divide_inverse nonzero_of_real_inverse)
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lemma of_real_divide [simp]:
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  "of_real (x / y) =
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   (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
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by (simp add: divide_inverse)
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lemma of_real_power [simp]:
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  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
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by (induct n) simp_all
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lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
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by (simp add: of_real_def)
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lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
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lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
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proof
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  fix r
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  show "of_real r = id r"
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    by (simp add: of_real_def)
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qed
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text{*Collapse nested embeddings*}
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lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
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by (induct n) auto
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lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
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by (cases z rule: int_diff_cases, simp)
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lemma of_real_number_of_eq:
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  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
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by (simp add: number_of_eq)
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text{*Every real algebra has characteristic zero*}
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instance real_algebra_1 < ring_char_0
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proof
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  fix m n :: nat
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  have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
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    by (simp only: of_real_eq_iff of_nat_eq_iff)
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  thus "(of_nat m = (of_nat n::'a)) = (m = n)"
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    by (simp only: of_real_of_nat_eq)
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qed
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instance real_field < field_char_0 ..
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subsection {* The Set of Real Numbers *}
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definition Reals :: "'a::real_algebra_1 set" where
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  "Reals = range of_real"
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notation (xsymbols)
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  Reals  ("\<real>")
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lemma Reals_of_real [simp]: "of_real r \<in> Reals"
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by (simp add: Reals_def)
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lemma Reals_of_int [simp]: "of_int z \<in> Reals"
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by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
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lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
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by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
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lemma Reals_number_of [simp]:
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  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
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by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
huffman@20718
   326
huffman@20554
   327
lemma Reals_0 [simp]: "0 \<in> Reals"
huffman@20554
   328
apply (unfold Reals_def)
huffman@20554
   329
apply (rule range_eqI)
huffman@20554
   330
apply (rule of_real_0 [symmetric])
huffman@20554
   331
done
huffman@20554
   332
huffman@20554
   333
lemma Reals_1 [simp]: "1 \<in> Reals"
huffman@20554
   334
apply (unfold Reals_def)
huffman@20554
   335
apply (rule range_eqI)
huffman@20554
   336
apply (rule of_real_1 [symmetric])
huffman@20554
   337
done
huffman@20554
   338
huffman@20584
   339
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
huffman@20554
   340
apply (auto simp add: Reals_def)
huffman@20554
   341
apply (rule range_eqI)
huffman@20554
   342
apply (rule of_real_add [symmetric])
huffman@20554
   343
done
huffman@20554
   344
huffman@20584
   345
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
huffman@20584
   346
apply (auto simp add: Reals_def)
huffman@20584
   347
apply (rule range_eqI)
huffman@20584
   348
apply (rule of_real_minus [symmetric])
huffman@20584
   349
done
huffman@20584
   350
huffman@20584
   351
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
huffman@20584
   352
apply (auto simp add: Reals_def)
huffman@20584
   353
apply (rule range_eqI)
huffman@20584
   354
apply (rule of_real_diff [symmetric])
huffman@20584
   355
done
huffman@20584
   356
huffman@20584
   357
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
huffman@20554
   358
apply (auto simp add: Reals_def)
huffman@20554
   359
apply (rule range_eqI)
huffman@20554
   360
apply (rule of_real_mult [symmetric])
huffman@20554
   361
done
huffman@20554
   362
huffman@20584
   363
lemma nonzero_Reals_inverse:
huffman@20584
   364
  fixes a :: "'a::real_div_algebra"
huffman@20584
   365
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   366
apply (auto simp add: Reals_def)
huffman@20584
   367
apply (rule range_eqI)
huffman@20584
   368
apply (erule nonzero_of_real_inverse [symmetric])
huffman@20584
   369
done
huffman@20584
   370
huffman@20584
   371
lemma Reals_inverse [simp]:
haftmann@36409
   372
  fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
huffman@20584
   373
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   374
apply (auto simp add: Reals_def)
huffman@20584
   375
apply (rule range_eqI)
huffman@20584
   376
apply (rule of_real_inverse [symmetric])
huffman@20584
   377
done
huffman@20584
   378
huffman@20584
   379
lemma nonzero_Reals_divide:
huffman@20584
   380
  fixes a b :: "'a::real_field"
huffman@20584
   381
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   382
apply (auto simp add: Reals_def)
huffman@20584
   383
apply (rule range_eqI)
huffman@20584
   384
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   385
done
huffman@20584
   386
huffman@20584
   387
lemma Reals_divide [simp]:
haftmann@36409
   388
  fixes a b :: "'a::{real_field, field_inverse_zero}"
huffman@20584
   389
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   390
apply (auto simp add: Reals_def)
huffman@20584
   391
apply (rule range_eqI)
huffman@20584
   392
apply (rule of_real_divide [symmetric])
huffman@20584
   393
done
huffman@20584
   394
huffman@20722
   395
lemma Reals_power [simp]:
haftmann@31017
   396
  fixes a :: "'a::{real_algebra_1}"
huffman@20722
   397
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   398
apply (auto simp add: Reals_def)
huffman@20722
   399
apply (rule range_eqI)
huffman@20722
   400
apply (rule of_real_power [symmetric])
huffman@20722
   401
done
huffman@20722
   402
huffman@20554
   403
lemma Reals_cases [cases set: Reals]:
huffman@20554
   404
  assumes "q \<in> \<real>"
huffman@20554
   405
  obtains (of_real) r where "q = of_real r"
huffman@20554
   406
  unfolding Reals_def
huffman@20554
   407
proof -
huffman@20554
   408
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   409
  then obtain r where "q = of_real r" ..
huffman@20554
   410
  then show thesis ..
huffman@20554
   411
qed
huffman@20554
   412
huffman@20554
   413
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   414
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   415
  by (rule Reals_cases) auto
huffman@20554
   416
huffman@20504
   417
huffman@31413
   418
subsection {* Topological spaces *}
huffman@31413
   419
huffman@31492
   420
class "open" =
huffman@31494
   421
  fixes "open" :: "'a set \<Rightarrow> bool"
huffman@31490
   422
huffman@31492
   423
class topological_space = "open" +
huffman@31492
   424
  assumes open_UNIV [simp, intro]: "open UNIV"
huffman@31492
   425
  assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
huffman@31492
   426
  assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
huffman@31490
   427
begin
huffman@31490
   428
huffman@31490
   429
definition
huffman@31490
   430
  closed :: "'a set \<Rightarrow> bool" where
huffman@31490
   431
  "closed S \<longleftrightarrow> open (- S)"
huffman@31490
   432
huffman@31490
   433
lemma open_empty [intro, simp]: "open {}"
huffman@31490
   434
  using open_Union [of "{}"] by simp
huffman@31490
   435
huffman@31490
   436
lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
huffman@31490
   437
  using open_Union [of "{S, T}"] by simp
huffman@31490
   438
huffman@31490
   439
lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
huffman@31490
   440
  unfolding UN_eq by (rule open_Union) auto
huffman@31490
   441
huffman@31490
   442
lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
huffman@31490
   443
  by (induct set: finite) auto
huffman@31490
   444
huffman@31490
   445
lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
huffman@31490
   446
  unfolding Inter_def by (rule open_INT)
huffman@31490
   447
huffman@31490
   448
lemma closed_empty [intro, simp]:  "closed {}"
huffman@31490
   449
  unfolding closed_def by simp
huffman@31490
   450
huffman@31490
   451
lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
huffman@31490
   452
  unfolding closed_def by auto
huffman@31490
   453
huffman@31490
   454
lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
huffman@31490
   455
  unfolding closed_def Inter_def by auto
huffman@31490
   456
huffman@31490
   457
lemma closed_UNIV [intro, simp]: "closed UNIV"
huffman@31490
   458
  unfolding closed_def by simp
huffman@31490
   459
huffman@31490
   460
lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
huffman@31490
   461
  unfolding closed_def by auto
huffman@31490
   462
huffman@31490
   463
lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
huffman@31490
   464
  unfolding closed_def by auto
huffman@31490
   465
huffman@31490
   466
lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
huffman@31490
   467
  by (induct set: finite) auto
huffman@31490
   468
huffman@31490
   469
lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
huffman@31490
   470
  unfolding Union_def by (rule closed_UN)
huffman@31490
   471
huffman@31490
   472
lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
huffman@31490
   473
  unfolding closed_def by simp
huffman@31490
   474
huffman@31490
   475
lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
huffman@31490
   476
  unfolding closed_def by simp
huffman@31490
   477
huffman@31490
   478
lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
huffman@31490
   479
  unfolding closed_open Diff_eq by (rule open_Int)
huffman@31490
   480
huffman@31490
   481
lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
huffman@31490
   482
  unfolding open_closed Diff_eq by (rule closed_Int)
huffman@31490
   483
huffman@31490
   484
lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
huffman@31490
   485
  unfolding closed_open .
huffman@31490
   486
huffman@31490
   487
lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
huffman@31490
   488
  unfolding open_closed .
huffman@31490
   489
huffman@31490
   490
end
huffman@31413
   491
huffman@31413
   492
huffman@31289
   493
subsection {* Metric spaces *}
huffman@31289
   494
huffman@31289
   495
class dist =
huffman@31289
   496
  fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
huffman@31289
   497
huffman@31492
   498
class open_dist = "open" + dist +
huffman@31492
   499
  assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31413
   500
huffman@31492
   501
class metric_space = open_dist +
huffman@31289
   502
  assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   503
  assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
huffman@31289
   504
begin
huffman@31289
   505
huffman@31289
   506
lemma dist_self [simp]: "dist x x = 0"
huffman@31289
   507
by simp
huffman@31289
   508
huffman@31289
   509
lemma zero_le_dist [simp]: "0 \<le> dist x y"
huffman@31289
   510
using dist_triangle2 [of x x y] by simp
huffman@31289
   511
huffman@31289
   512
lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
huffman@31289
   513
by (simp add: less_le)
huffman@31289
   514
huffman@31289
   515
lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
huffman@31289
   516
by (simp add: not_less)
huffman@31289
   517
huffman@31289
   518
lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
huffman@31289
   519
by (simp add: le_less)
huffman@31289
   520
huffman@31289
   521
lemma dist_commute: "dist x y = dist y x"
huffman@31289
   522
proof (rule order_antisym)
huffman@31289
   523
  show "dist x y \<le> dist y x"
huffman@31289
   524
    using dist_triangle2 [of x y x] by simp
huffman@31289
   525
  show "dist y x \<le> dist x y"
huffman@31289
   526
    using dist_triangle2 [of y x y] by simp
huffman@31289
   527
qed
huffman@31289
   528
huffman@31289
   529
lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
huffman@31289
   530
using dist_triangle2 [of x z y] by (simp add: dist_commute)
huffman@31289
   531
huffman@31565
   532
lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
huffman@31565
   533
using dist_triangle2 [of x y a] by (simp add: dist_commute)
huffman@31565
   534
huffman@31413
   535
subclass topological_space
huffman@31413
   536
proof
huffman@31413
   537
  have "\<exists>e::real. 0 < e"
huffman@31413
   538
    by (fast intro: zero_less_one)
huffman@31492
   539
  then show "open UNIV"
huffman@31492
   540
    unfolding open_dist by simp
huffman@31413
   541
next
huffman@31492
   542
  fix S T assume "open S" "open T"
huffman@31492
   543
  then show "open (S \<inter> T)"
huffman@31492
   544
    unfolding open_dist
huffman@31413
   545
    apply clarify
huffman@31413
   546
    apply (drule (1) bspec)+
huffman@31413
   547
    apply (clarify, rename_tac r s)
huffman@31413
   548
    apply (rule_tac x="min r s" in exI, simp)
huffman@31413
   549
    done
huffman@31413
   550
next
huffman@31492
   551
  fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
huffman@31492
   552
    unfolding open_dist by fast
huffman@31413
   553
qed
huffman@31413
   554
huffman@31289
   555
end
huffman@31289
   556
huffman@31289
   557
huffman@20504
   558
subsection {* Real normed vector spaces *}
huffman@20504
   559
haftmann@29608
   560
class norm =
huffman@22636
   561
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   562
huffman@24520
   563
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   564
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   565
huffman@31289
   566
class dist_norm = dist + norm + minus +
huffman@31289
   567
  assumes dist_norm: "dist x y = norm (x - y)"
huffman@31289
   568
huffman@31492
   569
class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
haftmann@24588
   570
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
haftmann@25062
   571
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   572
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@31586
   573
  and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@20504
   574
haftmann@24588
   575
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   576
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   577
haftmann@24588
   578
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   579
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   580
haftmann@24588
   581
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   582
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   583
haftmann@24588
   584
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   585
huffman@22852
   586
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   587
proof
huffman@20554
   588
  fix x y :: 'a
huffman@20554
   589
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   590
    by (simp add: norm_mult)
huffman@22852
   591
next
huffman@22852
   592
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   593
    by (rule norm_mult)
huffman@22852
   594
  thus "norm (1::'a) = 1" by simp
huffman@20554
   595
qed
huffman@20554
   596
huffman@22852
   597
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   598
by simp
huffman@20504
   599
huffman@22852
   600
lemma zero_less_norm_iff [simp]:
huffman@22852
   601
  fixes x :: "'a::real_normed_vector"
huffman@22852
   602
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   603
by (simp add: order_less_le)
huffman@20504
   604
huffman@22852
   605
lemma norm_not_less_zero [simp]:
huffman@22852
   606
  fixes x :: "'a::real_normed_vector"
huffman@22852
   607
  shows "\<not> norm x < 0"
huffman@20828
   608
by (simp add: linorder_not_less)
huffman@20828
   609
huffman@22852
   610
lemma norm_le_zero_iff [simp]:
huffman@22852
   611
  fixes x :: "'a::real_normed_vector"
huffman@22852
   612
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   613
by (simp add: order_le_less)
huffman@20828
   614
huffman@20504
   615
lemma norm_minus_cancel [simp]:
huffman@20584
   616
  fixes x :: "'a::real_normed_vector"
huffman@20584
   617
  shows "norm (- x) = norm x"
huffman@20504
   618
proof -
huffman@21809
   619
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   620
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   621
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   622
    by (rule norm_scaleR)
huffman@20504
   623
  finally show ?thesis by simp
huffman@20504
   624
qed
huffman@20504
   625
huffman@20504
   626
lemma norm_minus_commute:
huffman@20584
   627
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   628
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   629
proof -
huffman@22898
   630
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   631
    by (rule norm_minus_cancel)
huffman@22898
   632
  thus ?thesis by simp
huffman@20504
   633
qed
huffman@20504
   634
huffman@20504
   635
lemma norm_triangle_ineq2:
huffman@20584
   636
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   637
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   638
proof -
huffman@20533
   639
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   640
    by (rule norm_triangle_ineq)
huffman@22898
   641
  thus ?thesis by simp
huffman@20504
   642
qed
huffman@20504
   643
huffman@20584
   644
lemma norm_triangle_ineq3:
huffman@20584
   645
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   646
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   647
apply (subst abs_le_iff)
huffman@20584
   648
apply auto
huffman@20584
   649
apply (rule norm_triangle_ineq2)
huffman@20584
   650
apply (subst norm_minus_commute)
huffman@20584
   651
apply (rule norm_triangle_ineq2)
huffman@20584
   652
done
huffman@20584
   653
huffman@20504
   654
lemma norm_triangle_ineq4:
huffman@20584
   655
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   656
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   657
proof -
huffman@22898
   658
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   659
    by (rule norm_triangle_ineq)
huffman@22898
   660
  thus ?thesis
huffman@22898
   661
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   662
qed
huffman@22898
   663
huffman@22898
   664
lemma norm_diff_ineq:
huffman@22898
   665
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   666
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   667
proof -
huffman@22898
   668
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   669
    by (rule norm_triangle_ineq2)
huffman@22898
   670
  thus ?thesis by simp
huffman@20504
   671
qed
huffman@20504
   672
huffman@20551
   673
lemma norm_diff_triangle_ineq:
huffman@20551
   674
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   675
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   676
proof -
huffman@20551
   677
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   678
    by (simp add: diff_minus add_ac)
huffman@20551
   679
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   680
    by (rule norm_triangle_ineq)
huffman@20551
   681
  finally show ?thesis .
huffman@20551
   682
qed
huffman@20551
   683
huffman@22857
   684
lemma abs_norm_cancel [simp]:
huffman@22857
   685
  fixes a :: "'a::real_normed_vector"
huffman@22857
   686
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   687
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   688
huffman@22880
   689
lemma norm_add_less:
huffman@22880
   690
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   691
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   692
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   693
huffman@22880
   694
lemma norm_mult_less:
huffman@22880
   695
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   696
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   697
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   698
apply (simp add: mult_strict_mono')
huffman@22880
   699
done
huffman@22880
   700
huffman@22857
   701
lemma norm_of_real [simp]:
huffman@22857
   702
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@31586
   703
unfolding of_real_def by simp
huffman@20560
   704
huffman@22876
   705
lemma norm_number_of [simp]:
huffman@22876
   706
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
huffman@22876
   707
    = \<bar>number_of w\<bar>"
huffman@22876
   708
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
huffman@22876
   709
huffman@22876
   710
lemma norm_of_int [simp]:
huffman@22876
   711
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   712
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   713
huffman@22876
   714
lemma norm_of_nat [simp]:
huffman@22876
   715
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   716
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   717
apply (subst norm_of_real, simp)
huffman@22876
   718
done
huffman@22876
   719
huffman@20504
   720
lemma nonzero_norm_inverse:
huffman@20504
   721
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   722
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   723
apply (rule inverse_unique [symmetric])
huffman@20504
   724
apply (simp add: norm_mult [symmetric])
huffman@20504
   725
done
huffman@20504
   726
huffman@20504
   727
lemma norm_inverse:
haftmann@36409
   728
  fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
huffman@20533
   729
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   730
apply (case_tac "a = 0", simp)
huffman@20504
   731
apply (erule nonzero_norm_inverse)
huffman@20504
   732
done
huffman@20504
   733
huffman@20584
   734
lemma nonzero_norm_divide:
huffman@20584
   735
  fixes a b :: "'a::real_normed_field"
huffman@20584
   736
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   737
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   738
huffman@20584
   739
lemma norm_divide:
haftmann@36409
   740
  fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
huffman@20584
   741
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   742
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   743
huffman@22852
   744
lemma norm_power_ineq:
haftmann@31017
   745
  fixes x :: "'a::{real_normed_algebra_1}"
huffman@22852
   746
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   747
proof (induct n)
huffman@22852
   748
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   749
next
huffman@22852
   750
  case (Suc n)
huffman@22852
   751
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   752
    by (rule norm_mult_ineq)
huffman@22852
   753
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   754
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   755
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@30273
   756
    by simp
huffman@22852
   757
qed
huffman@22852
   758
huffman@20684
   759
lemma norm_power:
haftmann@31017
   760
  fixes x :: "'a::{real_normed_div_algebra}"
huffman@20684
   761
  shows "norm (x ^ n) = norm x ^ n"
huffman@30273
   762
by (induct n) (simp_all add: norm_mult)
huffman@20684
   763
huffman@31289
   764
text {* Every normed vector space is a metric space. *}
huffman@31285
   765
huffman@31289
   766
instance real_normed_vector < metric_space
huffman@31289
   767
proof
huffman@31289
   768
  fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
huffman@31289
   769
    unfolding dist_norm by simp
huffman@31289
   770
next
huffman@31289
   771
  fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
huffman@31289
   772
    unfolding dist_norm
huffman@31289
   773
    using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
huffman@31289
   774
qed
huffman@31285
   775
huffman@31564
   776
huffman@31564
   777
subsection {* Class instances for real numbers *}
huffman@31564
   778
huffman@31564
   779
instantiation real :: real_normed_field
huffman@31564
   780
begin
huffman@31564
   781
huffman@31564
   782
definition real_norm_def [simp]:
huffman@31564
   783
  "norm r = \<bar>r\<bar>"
huffman@31564
   784
huffman@31564
   785
definition dist_real_def:
huffman@31564
   786
  "dist x y = \<bar>x - y\<bar>"
huffman@31564
   787
haftmann@37767
   788
definition open_real_def:
huffman@31564
   789
  "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
huffman@31564
   790
huffman@31564
   791
instance
huffman@31564
   792
apply (intro_classes, unfold real_norm_def real_scaleR_def)
huffman@31564
   793
apply (rule dist_real_def)
huffman@31564
   794
apply (rule open_real_def)
huffman@36795
   795
apply (simp add: sgn_real_def)
huffman@31564
   796
apply (rule abs_ge_zero)
huffman@31564
   797
apply (rule abs_eq_0)
huffman@31564
   798
apply (rule abs_triangle_ineq)
huffman@31564
   799
apply (rule abs_mult)
huffman@31564
   800
apply (rule abs_mult)
huffman@31564
   801
done
huffman@31564
   802
huffman@31564
   803
end
huffman@31564
   804
huffman@31564
   805
lemma open_real_lessThan [simp]:
huffman@31564
   806
  fixes a :: real shows "open {..<a}"
huffman@31564
   807
unfolding open_real_def dist_real_def
huffman@31564
   808
proof (clarify)
huffman@31564
   809
  fix x assume "x < a"
huffman@31564
   810
  hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
huffman@31564
   811
  thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
huffman@31564
   812
qed
huffman@31564
   813
huffman@31564
   814
lemma open_real_greaterThan [simp]:
huffman@31564
   815
  fixes a :: real shows "open {a<..}"
huffman@31564
   816
unfolding open_real_def dist_real_def
huffman@31564
   817
proof (clarify)
huffman@31564
   818
  fix x assume "a < x"
huffman@31564
   819
  hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
huffman@31564
   820
  thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
huffman@31564
   821
qed
huffman@31564
   822
huffman@31564
   823
lemma open_real_greaterThanLessThan [simp]:
huffman@31564
   824
  fixes a b :: real shows "open {a<..<b}"
huffman@31564
   825
proof -
huffman@31564
   826
  have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
huffman@31564
   827
  thus "open {a<..<b}" by (simp add: open_Int)
huffman@31564
   828
qed
huffman@31564
   829
huffman@31567
   830
lemma closed_real_atMost [simp]: 
huffman@31567
   831
  fixes a :: real shows "closed {..a}"
huffman@31567
   832
unfolding closed_open by simp
huffman@31567
   833
huffman@31567
   834
lemma closed_real_atLeast [simp]:
huffman@31567
   835
  fixes a :: real shows "closed {a..}"
huffman@31567
   836
unfolding closed_open by simp
huffman@31567
   837
huffman@31567
   838
lemma closed_real_atLeastAtMost [simp]:
huffman@31567
   839
  fixes a b :: real shows "closed {a..b}"
huffman@31567
   840
proof -
huffman@31567
   841
  have "{a..b} = {a..} \<inter> {..b}" by auto
huffman@31567
   842
  thus "closed {a..b}" by (simp add: closed_Int)
huffman@31567
   843
qed
huffman@31567
   844
huffman@31564
   845
huffman@31446
   846
subsection {* Extra type constraints *}
huffman@31446
   847
huffman@31492
   848
text {* Only allow @{term "open"} in class @{text topological_space}. *}
huffman@31492
   849
huffman@31492
   850
setup {* Sign.add_const_constraint
huffman@31492
   851
  (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
huffman@31492
   852
huffman@31446
   853
text {* Only allow @{term dist} in class @{text metric_space}. *}
huffman@31446
   854
huffman@31446
   855
setup {* Sign.add_const_constraint
huffman@31446
   856
  (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
huffman@31446
   857
huffman@31446
   858
text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
huffman@31446
   859
huffman@31446
   860
setup {* Sign.add_const_constraint
huffman@31446
   861
  (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
huffman@31446
   862
huffman@31285
   863
huffman@22972
   864
subsection {* Sign function *}
huffman@22972
   865
nipkow@24506
   866
lemma norm_sgn:
nipkow@24506
   867
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
huffman@31586
   868
by (simp add: sgn_div_norm)
huffman@22972
   869
nipkow@24506
   870
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   871
by (simp add: sgn_div_norm)
huffman@22972
   872
nipkow@24506
   873
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   874
by (simp add: sgn_div_norm)
huffman@22972
   875
nipkow@24506
   876
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   877
by (simp add: sgn_div_norm)
huffman@22972
   878
nipkow@24506
   879
lemma sgn_scaleR:
nipkow@24506
   880
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
huffman@31586
   881
by (simp add: sgn_div_norm mult_ac)
huffman@22973
   882
huffman@22972
   883
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   884
by (simp add: sgn_div_norm)
huffman@22972
   885
huffman@22972
   886
lemma sgn_of_real:
huffman@22972
   887
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   888
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   889
huffman@22973
   890
lemma sgn_mult:
huffman@22973
   891
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   892
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   893
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   894
huffman@22972
   895
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   896
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   897
huffman@22972
   898
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   899
unfolding real_sgn_eq by simp
huffman@22972
   900
huffman@22972
   901
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   902
unfolding real_sgn_eq by simp
huffman@22972
   903
huffman@22972
   904
huffman@22442
   905
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   906
huffman@22442
   907
locale bounded_linear = additive +
huffman@22442
   908
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@22442
   909
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   910
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
   911
begin
huffman@22442
   912
huffman@27443
   913
lemma pos_bounded:
huffman@22442
   914
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   915
proof -
huffman@22442
   916
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   917
    using bounded by fast
huffman@22442
   918
  show ?thesis
huffman@22442
   919
  proof (intro exI impI conjI allI)
huffman@22442
   920
    show "0 < max 1 K"
huffman@22442
   921
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   922
  next
huffman@22442
   923
    fix x
huffman@22442
   924
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   925
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   926
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   927
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   928
  qed
huffman@22442
   929
qed
huffman@22442
   930
huffman@27443
   931
lemma nonneg_bounded:
huffman@22442
   932
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   933
proof -
huffman@22442
   934
  from pos_bounded
huffman@22442
   935
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   936
qed
huffman@22442
   937
huffman@27443
   938
end
huffman@27443
   939
huffman@22442
   940
locale bounded_bilinear =
huffman@22442
   941
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   942
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   943
    (infixl "**" 70)
huffman@22442
   944
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   945
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   946
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   947
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   948
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
   949
begin
huffman@22442
   950
huffman@27443
   951
lemma pos_bounded:
huffman@22442
   952
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   953
apply (cut_tac bounded, erule exE)
huffman@22442
   954
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   955
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   956
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   957
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   958
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   959
done
huffman@22442
   960
huffman@27443
   961
lemma nonneg_bounded:
huffman@22442
   962
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   963
proof -
huffman@22442
   964
  from pos_bounded
huffman@22442
   965
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   966
qed
huffman@22442
   967
huffman@27443
   968
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
   969
by (rule additive.intro, rule add_right)
huffman@22442
   970
huffman@27443
   971
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
   972
by (rule additive.intro, rule add_left)
huffman@22442
   973
huffman@27443
   974
lemma zero_left: "prod 0 b = 0"
huffman@22442
   975
by (rule additive.zero [OF additive_left])
huffman@22442
   976
huffman@27443
   977
lemma zero_right: "prod a 0 = 0"
huffman@22442
   978
by (rule additive.zero [OF additive_right])
huffman@22442
   979
huffman@27443
   980
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
   981
by (rule additive.minus [OF additive_left])
huffman@22442
   982
huffman@27443
   983
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
   984
by (rule additive.minus [OF additive_right])
huffman@22442
   985
huffman@27443
   986
lemma diff_left:
huffman@22442
   987
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
   988
by (rule additive.diff [OF additive_left])
huffman@22442
   989
huffman@27443
   990
lemma diff_right:
huffman@22442
   991
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
   992
by (rule additive.diff [OF additive_right])
huffman@22442
   993
huffman@27443
   994
lemma bounded_linear_left:
huffman@22442
   995
  "bounded_linear (\<lambda>a. a ** b)"
huffman@22442
   996
apply (unfold_locales)
huffman@22442
   997
apply (rule add_left)
huffman@22442
   998
apply (rule scaleR_left)
huffman@22442
   999
apply (cut_tac bounded, safe)
huffman@22442
  1000
apply (rule_tac x="norm b * K" in exI)
huffman@22442
  1001
apply (simp add: mult_ac)
huffman@22442
  1002
done
huffman@22442
  1003
huffman@27443
  1004
lemma bounded_linear_right:
huffman@22442
  1005
  "bounded_linear (\<lambda>b. a ** b)"
huffman@22442
  1006
apply (unfold_locales)
huffman@22442
  1007
apply (rule add_right)
huffman@22442
  1008
apply (rule scaleR_right)
huffman@22442
  1009
apply (cut_tac bounded, safe)
huffman@22442
  1010
apply (rule_tac x="norm a * K" in exI)
huffman@22442
  1011
apply (simp add: mult_ac)
huffman@22442
  1012
done
huffman@22442
  1013
huffman@27443
  1014
lemma prod_diff_prod:
huffman@22442
  1015
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
  1016
by (simp add: diff_left diff_right)
huffman@22442
  1017
huffman@27443
  1018
end
huffman@27443
  1019
wenzelm@30729
  1020
interpretation mult:
ballarin@29229
  1021
  bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"
huffman@22442
  1022
apply (rule bounded_bilinear.intro)
huffman@22442
  1023
apply (rule left_distrib)
huffman@22442
  1024
apply (rule right_distrib)
huffman@22442
  1025
apply (rule mult_scaleR_left)
huffman@22442
  1026
apply (rule mult_scaleR_right)
huffman@22442
  1027
apply (rule_tac x="1" in exI)
huffman@22442
  1028
apply (simp add: norm_mult_ineq)
huffman@22442
  1029
done
huffman@22442
  1030
wenzelm@30729
  1031
interpretation mult_left:
ballarin@29229
  1032
  bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)"
huffman@23127
  1033
by (rule mult.bounded_linear_left)
huffman@22442
  1034
wenzelm@30729
  1035
interpretation mult_right:
ballarin@29229
  1036
  bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)"
huffman@23127
  1037
by (rule mult.bounded_linear_right)
huffman@23127
  1038
wenzelm@30729
  1039
interpretation divide:
ballarin@29229
  1040
  bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)"
huffman@23127
  1041
unfolding divide_inverse by (rule mult.bounded_linear_left)
huffman@23120
  1042
wenzelm@30729
  1043
interpretation scaleR: bounded_bilinear "scaleR"
huffman@22442
  1044
apply (rule bounded_bilinear.intro)
huffman@22442
  1045
apply (rule scaleR_left_distrib)
huffman@22442
  1046
apply (rule scaleR_right_distrib)
huffman@22973
  1047
apply simp
huffman@22442
  1048
apply (rule scaleR_left_commute)
huffman@31586
  1049
apply (rule_tac x="1" in exI, simp)
huffman@22442
  1050
done
huffman@22442
  1051
wenzelm@30729
  1052
interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x"
huffman@23127
  1053
by (rule scaleR.bounded_linear_left)
huffman@23127
  1054
wenzelm@30729
  1055
interpretation scaleR_right: bounded_linear "\<lambda>x. scaleR r x"
huffman@23127
  1056
by (rule scaleR.bounded_linear_right)
huffman@23127
  1057
wenzelm@30729
  1058
interpretation of_real: bounded_linear "\<lambda>r. of_real r"
huffman@23127
  1059
unfolding of_real_def by (rule scaleR.bounded_linear_left)
huffman@22625
  1060
huffman@20504
  1061
end