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+(*  Title:      HOL/Hahn_Banach/Vector_Space.thy
+    Author:     Gertrud Bauer, TU Munich
+*)
+
+header {* Vector spaces *}
+
+theory Vector_Space
+imports Real Bounds Zorn
+begin
+
+subsection {* Signature *}
+
+text {*
+  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.
+*}
+
+consts
+  prod  :: "real \<Rightarrow> 'a::{plus, minus, zero} \<Rightarrow> 'a"     (infixr "'(*')" 70)
+
+notation (xsymbols)
+  prod  (infixr "\<cdot>" 70)
+notation (HTML output)
+  prod  (infixr "\<cdot>" 70)
+
+
+subsection {* Vector space laws *}
+
+text {*
+  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.
+*}
+
+locale var_V = fixes V
+
+locale vectorspace = var_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"
+
+lemma (in vectorspace) negate_eq2: "x \<in> V \<Longrightarrow> (- 1) \<cdot> x = - x"
+  by (rule negate_eq1 [symmetric])
+
+lemma (in vectorspace) negate_eq2a: "x \<in> V \<Longrightarrow> -1 \<cdot> x = - x"
+  by (simp add: negate_eq1)
+
+lemma (in vectorspace) diff_eq2: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + - y = x - y"
+  by (rule diff_eq1 [symmetric])
+
+lemma (in vectorspace) diff_closed [iff]: "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y \<in> V"
+  by (simp add: diff_eq1 negate_eq1)
+
+lemma (in vectorspace) neg_closed [iff]: "x \<in> V \<Longrightarrow> - x \<in> V"
+  by (simp add: negate_eq1)
+
+lemma (in vectorspace) 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 (in vectorspace) add_ac =
+  add_assoc add_commute add_left_commute
+
+
+text {* The existence of the zero element of a vector space
+  follows from the non-emptiness of carrier set. *}
+
+lemma (in vectorspace) 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 (in vectorspace) 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 (in vectorspace) mult_assoc2:
+    "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
+  by (simp only: mult_assoc)
+
+lemma (in vectorspace) 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 (in vectorspace) 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 add: real_diff_def)
+  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 (in vectorspace) distrib =
+  add_mult_distrib1 add_mult_distrib2
+  diff_mult_distrib1 diff_mult_distrib2
+
+
+text {* \medskip Further derived laws: *}
+
+lemma (in vectorspace) 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 (in vectorspace) 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 (in vectorspace) minus_mult_cancel [simp]:
+    "x \<in> V \<Longrightarrow> (- a) \<cdot> - x = a \<cdot> x"
+  by (simp add: negate_eq1 mult_assoc2)
+
+lemma (in vectorspace) 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 (in vectorspace) add_minus [simp]:
+    "x \<in> V \<Longrightarrow> x + - x = 0"
+  by (simp add: diff_eq2)
+
+lemma (in vectorspace) add_minus_left [simp]:
+    "x \<in> V \<Longrightarrow> - x + x = 0"
+  by (simp add: diff_eq2 add_commute)
+
+lemma (in vectorspace) minus_minus [simp]:
+    "x \<in> V \<Longrightarrow> - (- x) = x"
+  by (simp add: negate_eq1 mult_assoc2)
+
+lemma (in vectorspace) minus_zero [simp]:
+    "- (0::'a) = 0"
+  by (simp add: negate_eq1)
+
+lemma (in vectorspace) minus_zero_iff [simp]:
+  "x \<in> V \<Longrightarrow> (- x = 0) = (x = 0)"
+proof
+  assume x: "x \<in> V"
+  {
+    from x have "x = - (- x)" by (simp add: minus_minus)
+    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 (in vectorspace) add_minus_cancel [simp]:
+    "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + (- x + y) = y"
+  by (simp add: add_assoc [symmetric] del: add_commute)
+
+lemma (in vectorspace) minus_add_cancel [simp]:
+    "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> - x + (x + y) = y"
+  by (simp add: add_assoc [symmetric] del: add_commute)
+
+lemma (in vectorspace) 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 (in vectorspace) diff_zero [simp]:
+    "x \<in> V \<Longrightarrow> x - 0 = x"
+  by (simp add: diff_eq1)
+
+lemma (in vectorspace) diff_zero_right [simp]:
+    "x \<in> V \<Longrightarrow> 0 - x = - x"
+  by (simp add: diff_eq1)
+
+lemma (in vectorspace) add_left_cancel:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + y = x + z) = (y = z)"
+proof
+  assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
+  {
+    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 neg_closed)
+    also assume "x + y = x + z"
+    also from x z have "- x + (x + z) = - x + x + z"
+      by (simp add: add_assoc [symmetric] neg_closed)
+    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 (in vectorspace) 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 (in vectorspace) 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 (in vectorspace) mult_left_commute:
+    "x \<in> V \<Longrightarrow> a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
+  by (simp add: real_mult_commute mult_assoc2)
+
+lemma (in vectorspace) mult_zero_uniq:
+  "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> a \<cdot> x = 0 \<Longrightarrow> a = 0"
+proof (rule classical)
+  assume a: "a \<noteq> 0"
+  assume x: "x \<in> V"  "x \<noteq> 0" and ax: "a \<cdot> x = 0"
+  from x a have "x = (inverse a * a) \<cdot> x" by simp
+  also from `x \<in> V` 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 `x \<noteq> 0` show "a = 0" by contradiction
+qed
+
+lemma (in vectorspace) mult_left_cancel:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (a \<cdot> x = a \<cdot> y) = (x = y)"
+proof
+  assume x: "x \<in> V" and y: "y \<in> V" and a: "a \<noteq> 0"
+  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 (in vectorspace) mult_right_cancel:
+  "x \<in> V \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> (a \<cdot> x = b \<cdot> x) = (a = b)"
+proof
+  assume x: "x \<in> V" and neq: "x \<noteq> 0"
+  {
+    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 (in vectorspace) eq_diff_eq:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x = z - y) = (x + y = z)"
+proof
+  assume x: "x \<in> V" and y: "y \<in> V" and z: "z \<in> V"
+  {
+    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 (in vectorspace) add_minus_eq_minus:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x + y = 0 \<Longrightarrow> x = - y"
+proof -
+  assume x: "x \<in> V" and y: "y \<in> V"
+  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 assume "x + y = 0"
+  also from y have "- y + 0 = - y" by simp
+  finally show "x = - y" .
+qed
+
+lemma (in vectorspace) add_minus_eq:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> x - y = 0 \<Longrightarrow> x = y"
+proof -
+  assume x: "x \<in> V" and y: "y \<in> V"
+  assume "x - y = 0"
+  with x y 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 (in vectorspace) add_diff_swap:
+  "a \<in> V \<Longrightarrow> b \<in> V \<Longrightarrow> c \<in> V \<Longrightarrow> d \<in> V \<Longrightarrow> a + b = c + d
+    \<Longrightarrow> a - c = d - b"
+proof -
+  assume vs: "a \<in> V"  "b \<in> V"  "c \<in> V"  "d \<in> V"
+    and eq: "a + b = c + d"
+  then have "- c + (a + b) = - c + (c + d)"
+    by (simp add: add_left_cancel)
+  also have "\<dots> = d" using `c \<in> V` `d \<in> V` 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 (in vectorspace) vs_add_cancel_21:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> u \<in> V
+    \<Longrightarrow> (x + (y + z) = y + u) = (x + z = u)"
+proof
+  assume vs: "x \<in> V"  "y \<in> V"  "z \<in> V"  "u \<in> V"
+  {
+    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 (in vectorspace) add_cancel_end:
+  "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> z \<in> V \<Longrightarrow> (x + (y + z) = y) = (x = - z)"
+proof
+  assume vs: "x \<in> V"  "y \<in> V"  "z \<in> V"
+  {
+    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