(*  Title:      HOL/Hahn_Banach/Vector_Space.thy
    Author:     Gertrud Bauer, TU Munich
*)

section \<open>Vector spaces\<close>

theory Vector_Space
imports Complex_Main Bounds
begin

subsection \<open>Signature\<close>

text \<open>
  For the definition of real vector spaces a type @{typ 'a} of the
  sort @{text "{plus, minus, zero}"} is considered, on which a real
  scalar multiplication @{text \<cdot>} is declared.
\<close>

consts
  prod :: "real \<Rightarrow> 'a::{plus,minus,zero} \<Rightarrow> 'a"  (infixr "\<cdot>" 70)


subsection \<open>Vector space laws\<close>

text \<open>
  A \emph{vector space} is a non-empty set @{text V} of elements from
  @{typ 'a} with the following vector space laws: The set @{text V} is
  closed under addition and scalar multiplication, addition is
  associative and commutative; @{text "- x"} is the inverse of @{text
  x} w.~r.~t.~addition and @{text 0} is the neutral element of
  addition.  Addition and multiplication are distributive; scalar
  multiplication is associative and the real number @{text "1"} is
  the neutral element of scalar multiplication.
\<close>

locale vectorspace =
  fixes V
  assumes non_empty [iff, intro?]: "V \<noteq> {}"
    and add_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y \<in> V"
    and mult_closed [iff]: "x \<in> V \<Longrightarrow> a \<cdot> x \<in> V"
    and add_assoc: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y) + z = x + (y + z)"
    and add_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = y + x"
    and diff_self [simp]: "x \<in> V \<Longrightarrow> x - x = 0"
    and add_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 + x = x"
    and add_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
    and add_mult_distrib2: "x \<in> V \<Longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x"
    and mult_assoc: "x \<in> V \<Longrightarrow> (a * b) \<cdot> x = a \<cdot> (b \<cdot> x)"
    and mult_1 [simp]: "x \<in> V \<Longrightarrow> 1 \<cdot> x = x"
    and negate_eq1: "x \<in> V \<Longrightarrow> - x = (- 1) \<cdot> x"
    and diff_eq1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = x + - y"
begin

lemma negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
  by (rule negate_eq1 [symmetric])

lemma negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
  by (simp add: negate_eq1)

lemma diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
  by (rule diff_eq1 [symmetric])

lemma diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
  by (simp add: diff_eq1 negate_eq1)

lemma neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
  by (simp add: negate_eq1)

lemma add_left_commute: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> x + (y + z) = y + (x + z)"
proof -
  assume xyz: "x \<in> V"  "y \<in> V"  "z \<in> V"
  then have "x + (y + z) = (x + y) + z"
    by (simp only: add_assoc)
  also from xyz have "\<dots> = (y + x) + z" by (simp only: add_commute)
  also from xyz have "\<dots> = y + (x + z)" by (simp only: add_assoc)
  finally show ?thesis .
qed

theorems add_ac = add_assoc add_commute add_left_commute


text \<open>The existence of the zero element of a vector space
  follows from the non-emptiness of carrier set.\<close>

lemma zero [iff]: "0 \<in> V"
proof -
  from non_empty obtain x where x: "x \<in> V" by blast
  then have "0 = x - x" by (rule diff_self [symmetric])
  also from x x have "\<dots> \<in> V" by (rule diff_closed)
  finally show ?thesis .
qed

lemma add_zero_right [simp]: "x \<in> V \<Longrightarrow>  x + 0 = x"
proof -
  assume x: "x \<in> V"
  from this and zero have "x + 0 = 0 + x" by (rule add_commute)
  also from x have "\<dots> = x" by (rule add_zero_left)
  finally show ?thesis .
qed

lemma mult_assoc2: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
  by (simp only: mult_assoc)

lemma diff_mult_distrib1: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<cdot> (x - y) = a \<cdot> x - a \<cdot> y"
  by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2)

lemma diff_mult_distrib2: "x \<in> V \<Longrightarrow> (a - b) \<cdot> x = a \<cdot> x - (b \<cdot> x)"
proof -
  assume x: "x \<in> V"
  have " (a - b) \<cdot> x = (a + - b) \<cdot> x"
    by simp
  also from x have "\<dots> = a \<cdot> x + (- b) \<cdot> x"
    by (rule add_mult_distrib2)
  also from x have "\<dots> = a \<cdot> x + - (b \<cdot> x)"
    by (simp add: negate_eq1 mult_assoc2)
  also from x have "\<dots> = a \<cdot> x - (b \<cdot> x)"
    by (simp add: diff_eq1)
  finally show ?thesis .
qed

lemmas distrib =
  add_mult_distrib1 add_mult_distrib2
  diff_mult_distrib1 diff_mult_distrib2


text \<open>\medskip Further derived laws:\<close>

lemma mult_zero_left [simp]: "x \<in> V \<Longrightarrow> 0 \<cdot> x = 0"
proof -
  assume x: "x \<in> V"
  have "0 \<cdot> x = (1 - 1) \<cdot> x" by simp
  also have "\<dots> = (1 + - 1) \<cdot> x" by simp
  also from x have "\<dots> =  1 \<cdot> x + (- 1) \<cdot> x"
    by (rule add_mult_distrib2)
  also from x have "\<dots> = x + (- 1) \<cdot> x" by simp
  also from x have "\<dots> = x + - x" by (simp add: negate_eq2a)
  also from x have "\<dots> = x - x" by (simp add: diff_eq2)
  also from x have "\<dots> = 0" by simp
  finally show ?thesis .
qed

lemma mult_zero_right [simp]: "a \<cdot> 0 = (0::'a)"
proof -
  have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))" by simp
  also have "\<dots> =  a \<cdot> 0 - a \<cdot> 0"
    by (rule diff_mult_distrib1) simp_all
  also have "\<dots> = 0" by simp
  finally show ?thesis .
qed

lemma minus_mult_cancel [simp]: "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
  by (simp add: negate_eq1 mult_assoc2)

lemma add_minus_left_eq_diff: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + y = y - x"
proof -
  assume xy: "x \<in> V"  "y \<in> V"
  then have "- x + y = y + - x" by (simp add: add_commute)
  also from xy have "\<dots> = y - x" by (simp add: diff_eq1)
  finally show ?thesis .
qed

lemma add_minus [simp]: "x \<in> V \<Longrightarrow> x + - x = 0"
  by (simp add: diff_eq2)

lemma add_minus_left [simp]: "x \<in> V \<Longrightarrow> - x + x = 0"
  by (simp add: diff_eq2 add_commute)

lemma minus_minus [simp]: "x \<in> V \<Longrightarrow> - (- x) = x"
  by (simp add: negate_eq1 mult_assoc2)

lemma minus_zero [simp]: "- (0::'a) = 0"
  by (simp add: negate_eq1)

lemma minus_zero_iff [simp]:
  assumes x: "x \<in> V"
  shows "(- x = 0) = (x = 0)"
proof
  from x have "x = - (- x)" by simp
  also assume "- x = 0"
  also have "- \<dots> = 0" by (rule minus_zero)
  finally show "x = 0" .
next
  assume "x = 0"
  then show "- x = 0" by simp
qed

lemma add_minus_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
  by (simp add: add_assoc [symmetric])

lemma minus_add_cancel [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
  by (simp add: add_assoc [symmetric])

lemma minus_add_distrib [simp]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - (x + y) = - x + - y"
  by (simp add: negate_eq1 add_mult_distrib1)

lemma diff_zero [simp]: "x \<in> V \<Longrightarrow> x - 0 = x"
  by (simp add: diff_eq1)

lemma diff_zero_right [simp]: "x \<in> V \<Longrightarrow> 0 - x = - x"
  by (simp add: diff_eq1)

lemma add_left_cancel:
  assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
  shows "(x + y = x + z) = (y = z)"
proof
  from y have "y = 0 + y" by simp
  also from x y have "\<dots> = (- x + x) + y" by simp
  also from x y have "\<dots> = - x + (x + y)" by (simp add: add.assoc)
  also assume "x + y = x + z"
  also from x z have "- x + (x + z) = - x + x + z" by (simp add: add.assoc)
  also from x z have "\<dots> = z" by simp
  finally show "y = z" .
next
  assume "y = z"
  then show "x + y = x + z" by (simp only:)
qed

lemma add_right_cancel: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (y + x = z + x) = (y = z)"
  by (simp only: add_commute add_left_cancel)

lemma add_assoc_cong:
  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x' \<in> V \<Longrightarrow> y' \<in> V \<Longrightarrow> z \<in> V
    \<Longrightarrow> x + y = x' + y' \<Longrightarrow> x + (y + z) = x' + (y' + z)"
  by (simp only: add_assoc [symmetric])

lemma mult_left_commute: "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
  by (simp add: mult.commute mult_assoc2)

lemma mult_zero_uniq:
  assumes x: "x \<in> V"  "x \<noteq> 0" and ax: "a \<cdot> x = 0"
  shows "a = 0"
proof (rule classical)
  assume a: "a \<noteq> 0"
  from x a have "x = (inverse a * a) \<cdot> x" by simp
  also from \<open>x \<in> V\<close> have "\<dots> = inverse a \<cdot> (a \<cdot> x)" by (rule mult_assoc)
  also from ax have "\<dots> = inverse a \<cdot> 0" by simp
  also have "\<dots> = 0" by simp
  finally have "x = 0" .
  with \<open>x \<noteq> 0\<close> show "a = 0" by contradiction
qed

lemma mult_left_cancel:
  assumes x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
  shows "(a \<cdot> x = a \<cdot> y) = (x = y)"
proof
  from x have "x = 1 \<cdot> x" by simp
  also from a have "\<dots> = (inverse a * a) \<cdot> x" by simp
  also from x have "\<dots> = inverse a \<cdot> (a \<cdot> x)"
    by (simp only: mult_assoc)
  also assume "a \<cdot> x = a \<cdot> y"
  also from a y have "inverse a \<cdot> \<dots> = y"
    by (simp add: mult_assoc2)
  finally show "x = y" .
next
  assume "x = y"
  then show "a \<cdot> x = a \<cdot> y" by (simp only:)
qed

lemma mult_right_cancel:
  assumes x: "x \<in> V" and neq: "x \<noteq> 0"
  shows "(a \<cdot> x = b \<cdot> x) = (a = b)"
proof
  from x have "(a - b) \<cdot> x = a \<cdot> x - b \<cdot> x"
    by (simp add: diff_mult_distrib2)
  also assume "a \<cdot> x = b \<cdot> x"
  with x have "a \<cdot> x - b \<cdot> x = 0" by simp
  finally have "(a - b) \<cdot> x = 0" .
  with x neq have "a - b = 0" by (rule mult_zero_uniq)
  then show "a = b" by simp
next
  assume "a = b"
  then show "a \<cdot> x = b \<cdot> x" by (simp only:)
qed

lemma eq_diff_eq:
  assumes x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
  shows "(x = z - y) = (x + y = z)"
proof
  assume "x = z - y"
  then have "x + y = z - y + y" by simp
  also from y z have "\<dots> = z + - y + y"
    by (simp add: diff_eq1)
  also have "\<dots> = z + (- y + y)"
    by (rule add_assoc) (simp_all add: y z)
  also from y z have "\<dots> = z + 0"
    by (simp only: add_minus_left)
  also from z have "\<dots> = z"
    by (simp only: add_zero_right)
  finally show "x + y = z" .
next
  assume "x + y = z"
  then have "z - y = (x + y) - y" by simp
  also from x y have "\<dots> = x + y + - y"
    by (simp add: diff_eq1)
  also have "\<dots> = x + (y + - y)"
    by (rule add_assoc) (simp_all add: x y)
  also from x y have "\<dots> = x" by simp
  finally show "x = z - y" ..
qed

lemma add_minus_eq_minus:
  assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x + y = 0"
  shows "x = - y"
proof -
  from x y have "x = (- y + y) + x" by simp
  also from x y have "\<dots> = - y + (x + y)" by (simp add: add_ac)
  also note xy
  also from y have "- y + 0 = - y" by simp
  finally show "x = - y" .
qed

lemma add_minus_eq:
  assumes x: "x \<in> V" and y: "y \<in> V" and xy: "x - y = 0"
  shows "x = y"
proof -
  from x y xy have eq: "x + - y = 0" by (simp add: diff_eq1)
  with _ _ have "x = - (- y)"
    by (rule add_minus_eq_minus) (simp_all add: x y)
  with x y show "x = y" by simp
qed

lemma add_diff_swap:
  assumes vs: "a \<in> V"  "b \<in> V"  "c \<in> V"  "d \<in> V"
    and eq: "a + b = c + d"
  shows "a - c = d - b"
proof -
  from assms have "- c + (a + b) = - c + (c + d)"
    by (simp add: add_left_cancel)
  also have "\<dots> = d" using \<open>c \<in> V\<close> \<open>d \<in> V\<close> by (rule minus_add_cancel)
  finally have eq: "- c + (a + b) = d" .
  from vs have "a - c = (- c + (a + b)) + - b"
    by (simp add: add_ac diff_eq1)
  also from vs eq have "\<dots>  = d + - b"
    by (simp add: add_right_cancel)
  also from vs have "\<dots> = d - b" by (simp add: diff_eq2)
  finally show "a - c = d - b" .
qed

lemma vs_add_cancel_21:
  assumes vs: "x \<in> V"  "y \<in> V"  "z \<in> V"  "u \<in> V"
  shows "(x + (y + z) = y + u) = (x + z = u)"
proof
  from vs have "x + z = - y + y + (x + z)" by simp
  also have "\<dots> = - y + (y + (x + z))"
    by (rule add_assoc) (simp_all add: vs)
  also from vs have "y + (x + z) = x + (y + z)"
    by (simp add: add_ac)
  also assume "x + (y + z) = y + u"
  also from vs have "- y + (y + u) = u" by simp
  finally show "x + z = u" .
next
  assume "x + z = u"
  with vs show "x + (y + z) = y + u"
    by (simp only: add_left_commute [of x])
qed

lemma add_cancel_end:
  assumes vs: "x \<in> V"  "y \<in> V"  "z \<in> V"
  shows "(x + (y + z) = y) = (x = - z)"
proof
  assume "x + (y + z) = y"
  with vs have "(x + z) + y = 0 + y" by (simp add: add_ac)
  with vs have "x + z = 0" by (simp only: add_right_cancel add_closed zero)
  with vs show "x = - z" by (simp add: add_minus_eq_minus)
next
  assume eq: "x = - z"
  then have "x + (y + z) = - z + (y + z)" by simp
  also have "\<dots> = y + (- z + z)" by (rule add_left_commute) (simp_all add: vs)
  also from vs have "\<dots> = y"  by simp
  finally show "x + (y + z) = y" .
qed

end

end

