Theory RealVector

Up to index of Isabelle/HOL

theory RealVector
imports RComplete

(*  Title:      HOL/RealVector.thy
Author: Brian Huffman
*)


header {* Vector Spaces and Algebras over the Reals *}

theory RealVector
imports RComplete
begin


subsection {* Locale for additive functions *}

locale additive =
fixes f :: "'a::ab_group_add => 'b::ab_group_add"
assumes add: "f (x + y) = f x + f y"
begin


lemma zero: "f 0 = 0"
proof -
have "f 0 = f (0 + 0)" by simp
also have "… = f 0 + f 0" by (rule add)
finally show "f 0 = 0" by simp
qed

lemma minus: "f (- x) = - f x"
proof -
have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
also have "… = - f x + f x" by (simp add: zero)
finally show "f (- x) = - f x" by (rule add_right_imp_eq)
qed

lemma diff: "f (x - y) = f x - f y"
by (simp add: diff_def add minus)

lemma setsum: "f (setsum g A) = (∑x∈A. f (g x))"
apply (cases "finite A")
apply (induct set: finite)
apply (simp add: zero)
apply (simp add: add)
apply (simp add: zero)
done

end

subsection {* Vector spaces *}

locale vector_space =
fixes scale :: "'a::field => 'b::ab_group_add => 'b"
assumes scale_right_distrib [algebra_simps]:
"scale a (x + y) = scale a x + scale a y"
and scale_left_distrib [algebra_simps]:
"scale (a + b) x = scale a x + scale b x"
and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
and scale_one [simp]: "scale 1 x = x"
begin


lemma scale_left_commute:
"scale a (scale b x) = scale b (scale a x)"

by (simp add: mult_commute)

lemma scale_zero_left [simp]: "scale 0 x = 0"
and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
and scale_left_diff_distrib [algebra_simps]:
"scale (a - b) x = scale a x - scale b x"

proof -
interpret s: additive "λa. scale a x"
proof qed (rule scale_left_distrib)
show "scale 0 x = 0" by (rule s.zero)
show "scale (- a) x = - (scale a x)" by (rule s.minus)
show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
qed

lemma scale_zero_right [simp]: "scale a 0 = 0"
and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
and scale_right_diff_distrib [algebra_simps]:
"scale a (x - y) = scale a x - scale a y"

proof -
interpret s: additive "λx. scale a x"
proof qed (rule scale_right_distrib)
show "scale a 0 = 0" by (rule s.zero)
show "scale a (- x) = - (scale a x)" by (rule s.minus)
show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
qed

lemma scale_eq_0_iff [simp]:
"scale a x = 0 <-> a = 0 ∨ x = 0"

proof cases
assume "a = 0" thus ?thesis by simp
next
assume anz [simp]: "a ≠ 0"
{ assume "scale a x = 0"
hence "scale (inverse a) (scale a x) = 0" by simp
hence "x = 0" by simp }
thus ?thesis by force
qed

lemma scale_left_imp_eq:
"[|a ≠ 0; scale a x = scale a y|] ==> x = y"

proof -
assume nonzero: "a ≠ 0"
assume "scale a x = scale a y"
hence "scale a (x - y) = 0"
by (simp add: scale_right_diff_distrib)
hence "x - y = 0" by (simp add: nonzero)
thus "x = y" by (simp only: right_minus_eq)
qed

lemma scale_right_imp_eq:
"[|x ≠ 0; scale a x = scale b x|] ==> a = b"

proof -
assume nonzero: "x ≠ 0"
assume "scale a x = scale b x"
hence "scale (a - b) x = 0"
by (simp add: scale_left_diff_distrib)
hence "a - b = 0" by (simp add: nonzero)
thus "a = b" by (simp only: right_minus_eq)
qed

lemma scale_cancel_left [simp]:
"scale a x = scale a y <-> x = y ∨ a = 0"

by (auto intro: scale_left_imp_eq)

lemma scale_cancel_right [simp]:
"scale a x = scale b x <-> a = b ∨ x = 0"

by (auto intro: scale_right_imp_eq)

end

subsection {* Real vector spaces *}

class scaleR =
fixes scaleR :: "real => 'a => 'a" (infixr "*R" 75)
begin


abbreviation
divideR :: "'a => real => 'a" (infixl "'/R" 70)
where
"x /R r == scaleR (inverse r) x"


end

class real_vector = scaleR + ab_group_add +
assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
and scaleR_one: "scaleR 1 x = x"


interpretation real_vector:
vector_space "scaleR :: real => 'a => 'a::real_vector"

apply unfold_locales
apply (rule scaleR_right_distrib)
apply (rule scaleR_left_distrib)
apply (rule scaleR_scaleR)
apply (rule scaleR_one)
done

text {* Recover original theorem names *}

lemmas scaleR_left_commute = real_vector.scale_left_commute
lemmas scaleR_zero_left = real_vector.scale_zero_left
lemmas scaleR_minus_left = real_vector.scale_minus_left
lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib
lemmas scaleR_zero_right = real_vector.scale_zero_right
lemmas scaleR_minus_right = real_vector.scale_minus_right
lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib
lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
lemmas scaleR_cancel_left = real_vector.scale_cancel_left
lemmas scaleR_cancel_right = real_vector.scale_cancel_right

lemma scaleR_minus1_left [simp]:
fixes x :: "'a::real_vector"
shows "scaleR (-1) x = - x"

using scaleR_minus_left [of 1 x] by simp

class real_algebra = real_vector + ring +
assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"


class real_algebra_1 = real_algebra + ring_1

class real_div_algebra = real_algebra_1 + division_ring

class real_field = real_div_algebra + field

instantiation real :: real_field
begin


definition
real_scaleR_def [simp]: "scaleR a x = a * x"


instance proof
qed (simp_all add: algebra_simps)

end

interpretation scaleR_left: additive "(λa. scaleR a x::'a::real_vector)"
proof qed (rule scaleR_left_distrib)

interpretation scaleR_right: additive "(λx. scaleR a x::'a::real_vector)"
proof qed (rule scaleR_right_distrib)

lemma nonzero_inverse_scaleR_distrib:
fixes x :: "'a::real_div_algebra" shows
"[|a ≠ 0; x ≠ 0|] ==> 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_ring_inverse_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 => '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 ≠ 0 ==> 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_ring_inverse_zero})"

by (simp add: of_real_def inverse_scaleR_distrib)

lemma nonzero_of_real_divide:
"y ≠ 0 ==> 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, field_inverse_zero})"

by (simp add: divide_inverse)

lemma of_real_power [simp]:
"of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"

by (induct n) simp_all

lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
by (simp add: of_real_def)

lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]

lemma of_real_eq_id [simp]: "of_real = (id :: real => real)"
proof
fix r
show "of_real r = id r"
by (simp add: of_real_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)

text{*Every real algebra has characteristic zero*}
instance real_algebra_1 < ring_char_0
proof
fix m n :: nat
have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
by (simp only: of_real_eq_iff of_nat_eq_iff)
thus "(of_nat m = (of_nat n::'a)) = (m = n)"
by (simp only: of_real_of_nat_eq)
qed

instance real_field < field_char_0 ..


subsection {* The Set of Real Numbers *}

definition
Reals :: "'a::real_algebra_1 set" where
[code del]: "Reals = range of_real"


notation (xsymbols)
Reals ("\<real>")


lemma Reals_of_real [simp]: "of_real r ∈ Reals"
by (simp add: Reals_def)

lemma Reals_of_int [simp]: "of_int z ∈ Reals"
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)

lemma Reals_of_nat [simp]: "of_nat n ∈ 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}) ∈ Reals"

by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)

lemma Reals_0 [simp]: "0 ∈ Reals"
apply (unfold Reals_def)
apply (rule range_eqI)
apply (rule of_real_0 [symmetric])
done

lemma Reals_1 [simp]: "1 ∈ Reals"
apply (unfold Reals_def)
apply (rule range_eqI)
apply (rule of_real_1 [symmetric])
done

lemma Reals_add [simp]: "[|a ∈ Reals; b ∈ Reals|] ==> a + b ∈ Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_add [symmetric])
done

lemma Reals_minus [simp]: "a ∈ Reals ==> - a ∈ Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_minus [symmetric])
done

lemma Reals_diff [simp]: "[|a ∈ Reals; b ∈ Reals|] ==> a - b ∈ Reals"
apply (auto simp add: Reals_def)
apply (rule range_eqI)
apply (rule of_real_diff [symmetric])
done

lemma Reals_mult [simp]: "[|a ∈ Reals; b ∈ Reals|] ==> a * b ∈ 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 "[|a ∈ Reals; a ≠ 0|] ==> inverse a ∈ 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_ring_inverse_zero}"
shows "a ∈ Reals ==> inverse a ∈ 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 "[|a ∈ Reals; b ∈ Reals; b ≠ 0|] ==> a / b ∈ 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, field_inverse_zero}"
shows "[|a ∈ Reals; b ∈ Reals|] ==> a / b ∈ 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}"
shows "a ∈ Reals ==> a ^ n ∈ 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 ∈ \<real>"
obtains (of_real) r where "q = of_real r"

unfolding Reals_def
proof -
from `q ∈ \<real>` have "q ∈ 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 ∈ \<real> ==> (!!r. P (of_real r)) ==> P q"

by (rule Reals_cases) auto


subsection {* Topological spaces *}

class "open" =
fixes "open" :: "'a set => bool"


class topological_space = "open" +
assumes open_UNIV [simp, intro]: "open UNIV"
assumes open_Int [intro]: "open S ==> open T ==> open (S ∩ T)"
assumes open_Union [intro]: "∀S∈K. open S ==> open (\<Union> K)"
begin


definition
closed :: "'a set => bool" where
"closed S <-> open (- S)"


lemma open_empty [intro, simp]: "open {}"
using open_Union [of "{}"] by simp

lemma open_Un [intro]: "open S ==> open T ==> open (S ∪ T)"
using open_Union [of "{S, T}"] by simp

lemma open_UN [intro]: "∀x∈A. open (B x) ==> open (\<Union>x∈A. B x)"
unfolding UN_eq by (rule open_Union) auto

lemma open_INT [intro]: "finite A ==> ∀x∈A. open (B x) ==> open (\<Inter>x∈A. B x)"
by (induct set: finite) auto

lemma open_Inter [intro]: "finite S ==> ∀T∈S. open T ==> open (\<Inter>S)"
unfolding Inter_def by (rule open_INT)

lemma closed_empty [intro, simp]: "closed {}"
unfolding closed_def by simp

lemma closed_Un [intro]: "closed S ==> closed T ==> closed (S ∪ T)"
unfolding closed_def by auto

lemma closed_Inter [intro]: "∀S∈K. closed S ==> closed (\<Inter> K)"
unfolding closed_def Inter_def by auto

lemma closed_UNIV [intro, simp]: "closed UNIV"
unfolding closed_def by simp

lemma closed_Int [intro]: "closed S ==> closed T ==> closed (S ∩ T)"
unfolding closed_def by auto

lemma closed_INT [intro]: "∀x∈A. closed (B x) ==> closed (\<Inter>x∈A. B x)"
unfolding closed_def by auto

lemma closed_UN [intro]: "finite A ==> ∀x∈A. closed (B x) ==> closed (\<Union>x∈A. B x)"
by (induct set: finite) auto

lemma closed_Union [intro]: "finite S ==> ∀T∈S. closed T ==> closed (\<Union>S)"
unfolding Union_def by (rule closed_UN)

lemma open_closed: "open S <-> closed (- S)"
unfolding closed_def by simp

lemma closed_open: "closed S <-> open (- S)"
unfolding closed_def by simp

lemma open_Diff [intro]: "open S ==> closed T ==> open (S - T)"
unfolding closed_open Diff_eq by (rule open_Int)

lemma closed_Diff [intro]: "closed S ==> open T ==> closed (S - T)"
unfolding open_closed Diff_eq by (rule closed_Int)

lemma open_Compl [intro]: "closed S ==> open (- S)"
unfolding closed_open .

lemma closed_Compl [intro]: "open S ==> closed (- S)"
unfolding open_closed .

end


subsection {* Metric spaces *}

class dist =
fixes dist :: "'a => 'a => real"


class open_dist = "open" + dist +
assumes open_dist: "open S <-> (∀x∈S. ∃e>0. ∀y. dist y x < e --> y ∈ S)"


class metric_space = open_dist +
assumes dist_eq_0_iff [simp]: "dist x y = 0 <-> x = y"
assumes dist_triangle2: "dist x y ≤ dist x z + dist y z"
begin


lemma dist_self [simp]: "dist x x = 0"
by simp

lemma zero_le_dist [simp]: "0 ≤ dist x y"
using dist_triangle2 [of x x y] by simp

lemma zero_less_dist_iff: "0 < dist x y <-> x ≠ y"
by (simp add: less_le)

lemma dist_not_less_zero [simp]: "¬ dist x y < 0"
by (simp add: not_less)

lemma dist_le_zero_iff [simp]: "dist x y ≤ 0 <-> x = y"
by (simp add: le_less)

lemma dist_commute: "dist x y = dist y x"
proof (rule order_antisym)
show "dist x y ≤ dist y x"
using dist_triangle2 [of x y x] by simp
show "dist y x ≤ dist x y"
using dist_triangle2 [of y x y] by simp
qed

lemma dist_triangle: "dist x z ≤ dist x y + dist y z"
using dist_triangle2 [of x z y] by (simp add: dist_commute)

lemma dist_triangle3: "dist x y ≤ dist a x + dist a y"
using dist_triangle2 [of x y a] by (simp add: dist_commute)

subclass topological_space
proof
have "∃e::real. 0 < e"
by (fast intro: zero_less_one)
then show "open UNIV"
unfolding open_dist by simp
next
fix S T assume "open S" "open T"
then show "open (S ∩ T)"
unfolding open_dist
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac r s)
apply (rule_tac x="min r s" in exI, simp)
done
next
fix K assume "∀S∈K. open S" thus "open (\<Union>K)"
unfolding open_dist by fast
qed

end


subsection {* Real normed vector spaces *}

class norm =
fixes norm :: "'a => real"


class sgn_div_norm = scaleR + norm + sgn +
assumes sgn_div_norm: "sgn x = x /R norm x"


class dist_norm = dist + norm + minus +
assumes dist_norm: "dist x y = norm (x - y)"


class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
assumes norm_ge_zero [simp]: "0 ≤ norm x"
and norm_eq_zero [simp]: "norm x = 0 <-> x = 0"
and norm_triangle_ineq: "norm (x + y) ≤ norm x + norm y"
and norm_scaleR [simp]: "norm (scaleR a x) = ¦a¦ * norm x"


class real_normed_algebra = real_algebra + real_normed_vector +
assumes norm_mult_ineq: "norm (x * y) ≤ norm x * norm y"


class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
assumes norm_one [simp]: "norm 1 = 1"


class real_normed_div_algebra = real_div_algebra + real_normed_vector +
assumes norm_mult: "norm (x * y) = norm x * norm y"


class real_normed_field = real_field + real_normed_div_algebra

instance real_normed_div_algebra < real_normed_algebra_1
proof
fix x y :: 'a
show "norm (x * y) ≤ norm x * norm y"
by (simp add: norm_mult)
next
have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
by (rule norm_mult)
thus "norm (1::'a) = 1" by simp
qed

lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
by simp

lemma zero_less_norm_iff [simp]:
fixes x :: "'a::real_normed_vector"
shows "(0 < norm x) = (x ≠ 0)"

by (simp add: order_less_le)

lemma norm_not_less_zero [simp]:
fixes x :: "'a::real_normed_vector"
shows "¬ norm x < 0"

by (simp add: linorder_not_less)

lemma norm_le_zero_iff [simp]:
fixes x :: "'a::real_normed_vector"
shows "(norm x ≤ 0) = (x = 0)"

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 "… = ¦- 1¦ * 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 (- (b - a)) = norm (b - a)"
by (rule norm_minus_cancel)
thus ?thesis by simp
qed

lemma norm_triangle_ineq2:
fixes a b :: "'a::real_normed_vector"
shows "norm a - norm b ≤ norm (a - b)"

proof -
have "norm (a - b + b) ≤ norm (a - b) + norm b"
by (rule norm_triangle_ineq)
thus ?thesis by simp
qed

lemma norm_triangle_ineq3:
fixes a b :: "'a::real_normed_vector"
shows "¦norm a - norm b¦ ≤ 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) ≤ norm a + norm b"

proof -
have "norm (a + - b) ≤ norm a + norm (- b)"
by (rule norm_triangle_ineq)
thus ?thesis
by (simp only: diff_minus norm_minus_cancel)
qed

lemma norm_diff_ineq:
fixes a b :: "'a::real_normed_vector"
shows "norm a - norm b ≤ norm (a + b)"

proof -
have "norm a - norm (- b) ≤ norm (a - - b)"
by (rule norm_triangle_ineq2)
thus ?thesis by simp
qed

lemma norm_diff_triangle_ineq:
fixes a b c d :: "'a::real_normed_vector"
shows "norm ((a + b) - (c + d)) ≤ 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 "… ≤ norm (a - c) + norm (b - d)"
by (rule norm_triangle_ineq)
finally show ?thesis .
qed

lemma abs_norm_cancel [simp]:
fixes a :: "'a::real_normed_vector"
shows "¦norm a¦ = norm a"

by (rule abs_of_nonneg [OF norm_ge_zero])

lemma norm_add_less:
fixes x y :: "'a::real_normed_vector"
shows "[|norm x < r; norm y < s|] ==> norm (x + y) < r + s"

by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])

lemma norm_mult_less:
fixes x y :: "'a::real_normed_algebra"
shows "[|norm x < r; norm y < s|] ==> norm (x * y) < r * s"

apply (rule order_le_less_trans [OF norm_mult_ineq])
apply (simp add: mult_strict_mono')
done

lemma norm_of_real [simp]:
"norm (of_real r :: 'a::real_normed_algebra_1) = ¦r¦"

unfolding of_real_def by simp

lemma norm_number_of [simp]:
"norm (number_of w::'a::{number_ring,real_normed_algebra_1})
= ¦number_of w¦"

by (subst of_real_number_of_eq [symmetric], rule norm_of_real)

lemma norm_of_int [simp]:
"norm (of_int z::'a::real_normed_algebra_1) = ¦of_int z¦"

by (subst of_real_of_int_eq [symmetric], rule norm_of_real)

lemma norm_of_nat [simp]:
"norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"

apply (subst of_real_of_nat_eq [symmetric])
apply (subst norm_of_real, simp)
done

lemma nonzero_norm_inverse:
fixes a :: "'a::real_normed_div_algebra"
shows "a ≠ 0 ==> 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_ring_inverse_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 ≠ 0 ==> 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, field_inverse_zero}"
shows "norm (a / b) = norm a / norm b"

by (simp add: divide_inverse norm_mult norm_inverse)

lemma norm_power_ineq:
fixes x :: "'a::{real_normed_algebra_1}"
shows "norm (x ^ n) ≤ norm x ^ n"

proof (induct n)
case 0 show "norm (x ^ 0) ≤ norm x ^ 0" by simp
next
case (Suc n)
have "norm (x * x ^ n) ≤ norm x * norm (x ^ n)"
by (rule norm_mult_ineq)
also from Suc have "… ≤ norm x * norm x ^ n"
using norm_ge_zero by (rule mult_left_mono)
finally show "norm (x ^ Suc n) ≤ norm x ^ Suc n"
by simp
qed

lemma norm_power:
fixes x :: "'a::{real_normed_div_algebra}"
shows "norm (x ^ n) = norm x ^ n"

by (induct n) (simp_all add: norm_mult)

text {* Every normed vector space is a metric space. *}

instance real_normed_vector < metric_space
proof
fix x y :: 'a show "dist x y = 0 <-> x = y"
unfolding dist_norm by simp
next
fix x y z :: 'a show "dist x y ≤ dist x z + dist y z"
unfolding dist_norm
using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
qed


subsection {* Class instances for real numbers *}

instantiation real :: real_normed_field
begin


definition real_norm_def [simp]:
"norm r = ¦r¦"


definition dist_real_def:
"dist x y = ¦x - y¦"


definition open_real_def [code del]:
"open (S :: real set) <-> (∀x∈S. ∃e>0. ∀y. dist y x < e --> y ∈ S)"


instance
apply (intro_classes, unfold real_norm_def real_scaleR_def)
apply (rule dist_real_def)
apply (rule open_real_def)
apply (simp add: sgn_real_def)
apply (rule abs_ge_zero)
apply (rule abs_eq_0)
apply (rule abs_triangle_ineq)
apply (rule abs_mult)
apply (rule abs_mult)
done

end

lemma open_real_lessThan [simp]:
fixes a :: real shows "open {..<a}"

unfolding open_real_def dist_real_def
proof (clarify)
fix x assume "x < a"
hence "0 < a - x ∧ (∀y. ¦y - x¦ < a - x --> y ∈ {..<a})" by auto
thus "∃e>0. ∀y. ¦y - x¦ < e --> y ∈ {..<a}" ..
qed

lemma open_real_greaterThan [simp]:
fixes a :: real shows "open {a<..}"

unfolding open_real_def dist_real_def
proof (clarify)
fix x assume "a < x"
hence "0 < x - a ∧ (∀y. ¦y - x¦ < x - a --> y ∈ {a<..})" by auto
thus "∃e>0. ∀y. ¦y - x¦ < e --> y ∈ {a<..}" ..
qed

lemma open_real_greaterThanLessThan [simp]:
fixes a b :: real shows "open {a<..<b}"

proof -
have "{a<..<b} = {a<..} ∩ {..<b}" by auto
thus "open {a<..<b}" by (simp add: open_Int)
qed

lemma closed_real_atMost [simp]:
fixes a :: real shows "closed {..a}"

unfolding closed_open by simp

lemma closed_real_atLeast [simp]:
fixes a :: real shows "closed {a..}"

unfolding closed_open by simp

lemma closed_real_atLeastAtMost [simp]:
fixes a b :: real shows "closed {a..b}"

proof -
have "{a..b} = {a..} ∩ {..b}" by auto
thus "closed {a..b}" by (simp add: closed_Int)
qed


subsection {* Extra type constraints *}

text {* Only allow @{term "open"} in class @{text topological_space}. *}

setup {* Sign.add_const_constraint
(@{const_name "open"}, SOME @{typ "'a::topological_space set => bool"}) *}


text {* Only allow @{term dist} in class @{text metric_space}. *}

setup {* Sign.add_const_constraint
(@{const_name dist}, SOME @{typ "'a::metric_space => 'a => real"}) *}


text {* Only allow @{term norm} in class @{text real_normed_vector}. *}

setup {* Sign.add_const_constraint
(@{const_name norm}, SOME @{typ "'a::real_normed_vector => real"}) *}



subsection {* Sign function *}

lemma norm_sgn:
"norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"

by (simp add: sgn_div_norm)

lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
by (simp add: sgn_div_norm)

lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
by (simp add: sgn_div_norm)

lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
by (simp add: sgn_div_norm)

lemma sgn_scaleR:
"sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"

by (simp add: sgn_div_norm mult_ac)

lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
by (simp add: sgn_div_norm)

lemma sgn_of_real:
"sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"

unfolding of_real_def by (simp only: sgn_scaleR sgn_one)

lemma sgn_mult:
fixes x y :: "'a::real_normed_div_algebra"
shows "sgn (x * y) = sgn x * sgn y"

by (simp add: sgn_div_norm norm_mult mult_commute)

lemma real_sgn_eq: "sgn (x::real) = x / ¦x¦"
by (simp add: sgn_div_norm divide_inverse)

lemma real_sgn_pos: "0 < (x::real) ==> sgn x = 1"
unfolding real_sgn_eq by simp

lemma real_sgn_neg: "(x::real) < 0 ==> sgn x = -1"
unfolding real_sgn_eq by simp


subsection {* Bounded Linear and Bilinear Operators *}

locale bounded_linear = additive +
constrains f :: "'a::real_normed_vector => 'b::real_normed_vector"
assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
assumes bounded: "∃K. ∀x. norm (f x) ≤ norm x * K"
begin


lemma pos_bounded:
"∃K>0. ∀x. norm (f x) ≤ norm x * K"

proof -
obtain K where K: "!!x. norm (f x) ≤ norm x * K"
using bounded by fast
show ?thesis
proof (intro exI impI conjI allI)
show "0 < max 1 K"
by (rule order_less_le_trans [OF zero_less_one le_maxI1])
next
fix x
have "norm (f x) ≤ norm x * K" using K .
also have "… ≤ norm x * max 1 K"
by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
finally show "norm (f x) ≤ norm x * max 1 K" .
qed
qed

lemma nonneg_bounded:
"∃K≥0. ∀x. norm (f x) ≤ norm x * K"

proof -
from pos_bounded
show ?thesis by (auto intro: order_less_imp_le)
qed

end

locale bounded_bilinear =
fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
=> 'c::real_normed_vector"

(infixl "**" 70)
assumes add_left: "prod (a + a') b = prod a b + prod a' b"
assumes add_right: "prod a (b + b') = prod a b + prod a b'"
assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
assumes bounded: "∃K. ∀a b. norm (prod a b) ≤ norm a * norm b * K"
begin


lemma pos_bounded:
"∃K>0. ∀a b. norm (a ** b) ≤ norm a * norm b * K"

apply (cut_tac bounded, erule exE)
apply (rule_tac x="max 1 K" in exI, safe)
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
apply (drule spec, drule spec, erule order_trans)
apply (rule mult_left_mono [OF le_maxI2])
apply (intro mult_nonneg_nonneg norm_ge_zero)
done

lemma nonneg_bounded:
"∃K≥0. ∀a b. norm (a ** b) ≤ norm a * norm b * K"

proof -
from pos_bounded
show ?thesis by (auto intro: order_less_imp_le)
qed

lemma additive_right: "additive (λb. prod a b)"
by (rule additive.intro, rule add_right)

lemma additive_left: "additive (λa. prod a b)"
by (rule additive.intro, rule add_left)

lemma zero_left: "prod 0 b = 0"
by (rule additive.zero [OF additive_left])

lemma zero_right: "prod a 0 = 0"
by (rule additive.zero [OF additive_right])

lemma minus_left: "prod (- a) b = - prod a b"
by (rule additive.minus [OF additive_left])

lemma minus_right: "prod a (- b) = - prod a b"
by (rule additive.minus [OF additive_right])

lemma diff_left:
"prod (a - a') b = prod a b - prod a' b"

by (rule additive.diff [OF additive_left])

lemma diff_right:
"prod a (b - b') = prod a b - prod a b'"

by (rule additive.diff [OF additive_right])

lemma bounded_linear_left:
"bounded_linear (λa. a ** b)"

apply (unfold_locales)
apply (rule add_left)
apply (rule scaleR_left)
apply (cut_tac bounded, safe)
apply (rule_tac x="norm b * K" in exI)
apply (simp add: mult_ac)
done

lemma bounded_linear_right:
"bounded_linear (λb. a ** b)"

apply (unfold_locales)
apply (rule add_right)
apply (rule scaleR_right)
apply (cut_tac bounded, safe)
apply (rule_tac x="norm a * K" in exI)
apply (simp add: mult_ac)
done

lemma prod_diff_prod:
"(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"

by (simp add: diff_left diff_right)

end

interpretation mult:
bounded_bilinear "op * :: 'a => 'a => 'a::real_normed_algebra"

apply (rule bounded_bilinear.intro)
apply (rule left_distrib)
apply (rule right_distrib)
apply (rule mult_scaleR_left)
apply (rule mult_scaleR_right)
apply (rule_tac x="1" in exI)
apply (simp add: norm_mult_ineq)
done

interpretation mult_left:
bounded_linear "(λx::'a::real_normed_algebra. x * y)"

by (rule mult.bounded_linear_left)

interpretation mult_right:
bounded_linear "(λy::'a::real_normed_algebra. x * y)"

by (rule mult.bounded_linear_right)

interpretation divide:
bounded_linear "(λx::'a::real_normed_field. x / y)"

unfolding divide_inverse by (rule mult.bounded_linear_left)

interpretation scaleR: bounded_bilinear "scaleR"
apply (rule bounded_bilinear.intro)
apply (rule scaleR_left_distrib)
apply (rule scaleR_right_distrib)
apply simp
apply (rule scaleR_left_commute)
apply (rule_tac x="1" in exI, simp)
done

interpretation scaleR_left: bounded_linear "λr. scaleR r x"
by (rule scaleR.bounded_linear_left)

interpretation scaleR_right: bounded_linear "λx. scaleR r x"
by (rule scaleR.bounded_linear_right)

interpretation of_real: bounded_linear "λr. of_real r"
unfolding of_real_def by (rule scaleR.bounded_linear_left)

end