(* Title : RealVector.thy
ID: $Id$
Author : Brian Huffman
*)
header {* Vector Spaces and Algebras over the Reals *}
theory RealVector
imports RealPow
begin
subsection {* Locale for additive functions *}
locale additive =
fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
assumes add: "f (x + y) = f x + f y"
lemma (in additive) zero: "f 0 = 0"
proof -
have "f 0 = f (0 + 0)" by simp
also have "\<dots> = f 0 + f 0" by (rule add)
finally show "f 0 = 0" by simp
qed
lemma (in additive) minus: "f (- x) = - f x"
proof -
have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
also have "\<dots> = - f x + f x" by (simp add: zero)
finally show "f (- x) = - f x" by (rule add_right_imp_eq)
qed
lemma (in additive) diff: "f (x - y) = f x - f y"
by (simp add: diff_def add minus)
subsection {* Real vector spaces *}
axclass scaleR < type
consts
scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a::scaleR" (infixr "*#" 75)
abbreviation
divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a::scaleR" (infixl "'/#" 70) where
"x /# r == scaleR (inverse r) x"
notation (xsymbols)
scaleR (infixr "*\<^sub>R" 75) and
divideR (infixl "'/\<^sub>R" 70)
instance real :: scaleR ..
defs (overloaded)
real_scaleR_def: "scaleR a x \<equiv> a * x"
axclass real_vector < scaleR, ab_group_add
scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
scaleR_one [simp]: "scaleR 1 x = x"
axclass real_algebra < real_vector, ring
mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
axclass real_algebra_1 < real_algebra, ring_1
axclass real_div_algebra < real_algebra_1, division_ring
axclass real_field < real_div_algebra, field
instance real :: real_field
apply (intro_classes, unfold real_scaleR_def)
apply (rule right_distrib)
apply (rule left_distrib)
apply (rule mult_assoc [symmetric])
apply (rule mult_1_left)
apply (rule mult_assoc)
apply (rule mult_left_commute)
done
lemma scaleR_left_commute:
fixes x :: "'a::real_vector"
shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)"
by (simp add: mult_commute)
lemma additive_scaleR_right: "additive (\<lambda>x. scaleR a x::'a::real_vector)"
by (rule additive.intro, rule scaleR_right_distrib)
lemma additive_scaleR_left: "additive (\<lambda>a. scaleR a x::'a::real_vector)"
by (rule additive.intro, rule scaleR_left_distrib)
lemmas scaleR_zero_left [simp] =
additive.zero [OF additive_scaleR_left, standard]
lemmas scaleR_zero_right [simp] =
additive.zero [OF additive_scaleR_right, standard]
lemmas scaleR_minus_left [simp] =
additive.minus [OF additive_scaleR_left, standard]
lemmas scaleR_minus_right [simp] =
additive.minus [OF additive_scaleR_right, standard]
lemmas scaleR_left_diff_distrib =
additive.diff [OF additive_scaleR_left, standard]
lemmas scaleR_right_diff_distrib =
additive.diff [OF additive_scaleR_right, standard]
lemma scaleR_eq_0_iff:
fixes x :: "'a::real_vector"
shows "(scaleR a x = 0) = (a = 0 \<or> x = 0)"
proof cases
assume "a = 0" thus ?thesis by simp
next
assume anz [simp]: "a \<noteq> 0"
{ assume "scaleR a x = 0"
hence "scaleR (inverse a) (scaleR a x) = 0" by simp
hence "x = 0" by simp }
thus ?thesis by force
qed
lemma scaleR_left_imp_eq:
fixes x y :: "'a::real_vector"
shows "\<lbrakk>a \<noteq> 0; scaleR a x = scaleR a y\<rbrakk> \<Longrightarrow> x = y"
proof -
assume nonzero: "a \<noteq> 0"
assume "scaleR a x = scaleR a y"
hence "scaleR a (x - y) = 0"
by (simp add: scaleR_right_diff_distrib)
hence "x - y = 0"
by (simp add: scaleR_eq_0_iff nonzero)
thus "x = y" by simp
qed
lemma scaleR_right_imp_eq:
fixes x y :: "'a::real_vector"
shows "\<lbrakk>x \<noteq> 0; scaleR a x = scaleR b x\<rbrakk> \<Longrightarrow> a = b"
proof -
assume nonzero: "x \<noteq> 0"
assume "scaleR a x = scaleR b x"
hence "scaleR (a - b) x = 0"
by (simp add: scaleR_left_diff_distrib)
hence "a - b = 0"
by (simp add: scaleR_eq_0_iff nonzero)
thus "a = b" by simp
qed
lemma scaleR_cancel_left:
fixes x y :: "'a::real_vector"
shows "(scaleR a x = scaleR a y) = (x = y \<or> a = 0)"
by (auto intro: scaleR_left_imp_eq)
lemma scaleR_cancel_right:
fixes x y :: "'a::real_vector"
shows "(scaleR a x = scaleR b x) = (a = b \<or> x = 0)"
by (auto intro: scaleR_right_imp_eq)
lemma nonzero_inverse_scaleR_distrib:
fixes x :: "'a::real_div_algebra" shows
"\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
by (rule inverse_unique, simp)
lemma inverse_scaleR_distrib:
fixes x :: "'a::{real_div_algebra,division_by_zero}"
shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
apply (case_tac "a = 0", simp)
apply (case_tac "x = 0", simp)
apply (erule (1) nonzero_inverse_scaleR_distrib)
done
subsection {* Embedding of the Reals into any @{text real_algebra_1}:
@{term of_real} *}
definition
of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
"of_real r = scaleR r 1"
lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
by (simp add: of_real_def)
lemma of_real_0 [simp]: "of_real 0 = 0"
by (simp add: of_real_def)
lemma of_real_1 [simp]: "of_real 1 = 1"
by (simp add: of_real_def)
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
by (simp add: of_real_def scaleR_left_distrib)
lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
by (simp add: of_real_def)
lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
by (simp add: of_real_def scaleR_left_diff_distrib)
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
by (simp add: of_real_def mult_commute)
lemma nonzero_of_real_inverse:
"x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
inverse (of_real x :: 'a::real_div_algebra)"
by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
lemma of_real_inverse [simp]:
"of_real (inverse x) =
inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
by (simp add: of_real_def inverse_scaleR_distrib)
lemma nonzero_of_real_divide:
"y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
(of_real x / of_real y :: 'a::real_field)"
by (simp add: divide_inverse nonzero_of_real_inverse)
lemma of_real_divide [simp]:
"of_real (x / y) =
(of_real x / of_real y :: 'a::{real_field,division_by_zero})"
by (simp add: divide_inverse)
lemma of_real_power [simp]:
"of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
by (induct n) (simp_all add: power_Suc)
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
by (simp add: of_real_def scaleR_cancel_right)
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
proof
fix r
show "of_real r = id r"
by (simp add: of_real_def real_scaleR_def)
qed
text{*Collapse nested embeddings*}
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
by (induct n) auto
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
by (cases z rule: int_diff_cases, simp)
lemma of_real_number_of_eq:
"of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
by (simp add: number_of_eq)
subsection {* The Set of Real Numbers *}
definition
Reals :: "'a::real_algebra_1 set" where
"Reals \<equiv> range of_real"
notation (xsymbols)
Reals ("\<real>")
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
by (simp add: Reals_def)
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
lemma Reals_number_of [simp]:
"(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
lemma Reals_0 [simp]: "0 \<in> Reals"
apply (unfold Reals_def)
apply (rule range_eqI)
apply (rule of_real_0 [symmetric])
done
lemma Reals_1 [simp]: "1 \<in> Reals"
apply (unfold Reals_def)
apply (rule range_eqI)
apply (rule of_real_1 [symmetric])
done
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_add [symmetric])
done
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_minus [symmetric])
done
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_diff [symmetric])
done
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_mult [symmetric])
done
lemma nonzero_Reals_inverse:
fixes a :: "'a::real_div_algebra"
shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (erule nonzero_of_real_inverse [symmetric])
done
lemma Reals_inverse [simp]:
fixes a :: "'a::{real_div_algebra,division_by_zero}"
shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_inverse [symmetric])
done
lemma nonzero_Reals_divide:
fixes a b :: "'a::real_field"
shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (erule nonzero_of_real_divide [symmetric])
done
lemma Reals_divide [simp]:
fixes a b :: "'a::{real_field,division_by_zero}"
shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_divide [symmetric])
done
lemma Reals_power [simp]:
fixes a :: "'a::{real_algebra_1,recpower}"
shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_power [symmetric])
done
lemma Reals_cases [cases set: Reals]:
assumes "q \<in> \<real>"
obtains (of_real) r where "q = of_real r"
unfolding Reals_def
proof -
from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
then obtain r where "q = of_real r" ..
then show thesis ..
qed
lemma Reals_induct [case_names of_real, induct set: Reals]:
"q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
by (rule Reals_cases) auto
subsection {* Real normed vector spaces *}
axclass norm < type
consts norm :: "'a::norm \<Rightarrow> real"
instance real :: norm ..
defs (overloaded)
real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>"
axclass normed < plus, zero, norm
norm_ge_zero [simp]: "0 \<le> norm x"
norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
axclass real_normed_vector < real_vector, normed
norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
axclass real_normed_algebra < real_algebra, real_normed_vector
norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
axclass real_normed_div_algebra < real_div_algebra, normed
norm_of_real: "norm (of_real r) = abs r"
norm_mult: "norm (x * y) = norm x * norm y"
axclass real_normed_field < real_field, real_normed_div_algebra
instance real_normed_div_algebra < real_normed_algebra
proof
fix a :: real and x :: 'a
have "norm (scaleR a x) = norm (of_real a * x)"
by (simp add: of_real_def)
also have "\<dots> = abs a * norm x"
by (simp add: norm_mult norm_of_real)
finally show "norm (scaleR a x) = abs a * norm x" .
next
fix x y :: 'a
show "norm (x * y) \<le> norm x * norm y"
by (simp add: norm_mult)
qed
instance real :: real_normed_field
apply (intro_classes, unfold real_norm_def)
apply (rule abs_ge_zero)
apply (rule abs_eq_0)
apply (rule abs_triangle_ineq)
apply simp
apply (rule abs_mult)
done
lemma norm_zero [simp]: "norm (0::'a::normed) = 0"
by simp
lemma zero_less_norm_iff [simp]: "(0 < norm x) = (x \<noteq> (0::'a::normed))"
by (simp add: order_less_le)
lemma norm_not_less_zero [simp]: "\<not> norm (x::'a::normed) < 0"
by (simp add: linorder_not_less)
lemma norm_le_zero_iff [simp]: "(norm x \<le> 0) = (x = (0::'a::normed))"
by (simp add: order_le_less)
lemma norm_minus_cancel [simp]:
fixes x :: "'a::real_normed_vector"
shows "norm (- x) = norm x"
proof -
have "norm (- x) = norm (scaleR (- 1) x)"
by (simp only: scaleR_minus_left scaleR_one)
also have "\<dots> = \<bar>- 1\<bar> * norm x"
by (rule norm_scaleR)
finally show ?thesis by simp
qed
lemma norm_minus_commute:
fixes a b :: "'a::real_normed_vector"
shows "norm (a - b) = norm (b - a)"
proof -
have "norm (a - b) = norm (- (a - b))"
by (simp only: norm_minus_cancel)
also have "\<dots> = norm (b - a)" by simp
finally show ?thesis .
qed
lemma norm_triangle_ineq2:
fixes a b :: "'a::real_normed_vector"
shows "norm a - norm b \<le> norm (a - b)"
proof -
have "norm (a - b + b) \<le> norm (a - b) + norm b"
by (rule norm_triangle_ineq)
also have "(a - b + b) = a"
by simp
finally show ?thesis
by (simp add: compare_rls)
qed
lemma norm_triangle_ineq3:
fixes a b :: "'a::real_normed_vector"
shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
apply (subst abs_le_iff)
apply auto
apply (rule norm_triangle_ineq2)
apply (subst norm_minus_commute)
apply (rule norm_triangle_ineq2)
done
lemma norm_triangle_ineq4:
fixes a b :: "'a::real_normed_vector"
shows "norm (a - b) \<le> norm a + norm b"
proof -
have "norm (a - b) = norm (a + - b)"
by (simp only: diff_minus)
also have "\<dots> \<le> norm a + norm (- b)"
by (rule norm_triangle_ineq)
finally show ?thesis
by simp
qed
lemma norm_diff_triangle_ineq:
fixes a b c d :: "'a::real_normed_vector"
shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
proof -
have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
by (simp add: diff_minus add_ac)
also have "\<dots> \<le> norm (a - c) + norm (b - d)"
by (rule norm_triangle_ineq)
finally show ?thesis .
qed
lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1"
proof -
have "norm (of_real 1 :: 'a) = abs 1"
by (rule norm_of_real)
thus ?thesis by simp
qed
lemma nonzero_norm_inverse:
fixes a :: "'a::real_normed_div_algebra"
shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
apply (rule inverse_unique [symmetric])
apply (simp add: norm_mult [symmetric])
done
lemma norm_inverse:
fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
shows "norm (inverse a) = inverse (norm a)"
apply (case_tac "a = 0", simp)
apply (erule nonzero_norm_inverse)
done
lemma nonzero_norm_divide:
fixes a b :: "'a::real_normed_field"
shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
lemma norm_divide:
fixes a b :: "'a::{real_normed_field,division_by_zero}"
shows "norm (a / b) = norm a / norm b"
by (simp add: divide_inverse norm_mult norm_inverse)
lemma norm_power:
fixes x :: "'a::{real_normed_div_algebra,recpower}"
shows "norm (x ^ n) = norm x ^ n"
by (induct n) (simp_all add: power_Suc norm_mult)
end