(*  Title:      HOL/Real/HahnBanach/VectorSpace.thy

     ID:         $Id$

     Author:     Gertrud Bauer, TU Munich

 *)

 header {* Vector spaces *}

 theory VectorSpace

 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 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