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theory VectorSpace(* Title: HOL/Real/HahnBanach/VectorSpace.thy ID: $Id: VectorSpace.thy,v 1.28 2007/06/13 22:22:45 wenzelm Exp $ 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 ·} is declared. *} consts prod :: "real => 'a::{plus, minus, zero} => 'a" (infixr "'(*')" 70) notation (xsymbols) prod (infixr "·" 70) notation (HTML output) prod (infixr "·" 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 ≠ {}" and add_closed [iff]: "x ∈ V ==> y ∈ V ==> x + y ∈ V" and mult_closed [iff]: "x ∈ V ==> a · x ∈ V" and add_assoc: "x ∈ V ==> y ∈ V ==> z ∈ V ==> (x + y) + z = x + (y + z)" and add_commute: "x ∈ V ==> y ∈ V ==> x + y = y + x" and diff_self [simp]: "x ∈ V ==> x - x = 0" and add_zero_left [simp]: "x ∈ V ==> 0 + x = x" and add_mult_distrib1: "x ∈ V ==> y ∈ V ==> a · (x + y) = a · x + a · y" and add_mult_distrib2: "x ∈ V ==> (a + b) · x = a · x + b · x" and mult_assoc: "x ∈ V ==> (a * b) · x = a · (b · x)" and mult_1 [simp]: "x ∈ V ==> 1 · x = x" and negate_eq1: "x ∈ V ==> - x = (- 1) · x" and diff_eq1: "x ∈ V ==> y ∈ V ==> x - y = x + - y" lemma (in vectorspace) negate_eq2: "x ∈ V ==> (- 1) · x = - x" by (rule negate_eq1 [symmetric]) lemma (in vectorspace) negate_eq2a: "x ∈ V ==> -1 · x = - x" by (simp add: negate_eq1) lemma (in vectorspace) diff_eq2: "x ∈ V ==> y ∈ V ==> x + - y = x - y" by (rule diff_eq1 [symmetric]) lemma (in vectorspace) diff_closed [iff]: "x ∈ V ==> y ∈ V ==> x - y ∈ V" by (simp add: diff_eq1 negate_eq1) lemma (in vectorspace) neg_closed [iff]: "x ∈ V ==> - x ∈ V" by (simp add: negate_eq1) lemma (in vectorspace) add_left_commute: "x ∈ V ==> y ∈ V ==> z ∈ V ==> x + (y + z) = y + (x + z)" proof - assume xyz: "x ∈ V" "y ∈ V" "z ∈ V" hence "x + (y + z) = (x + y) + z" by (simp only: add_assoc) also from xyz have "... = (y + x) + z" by (simp only: add_commute) also from xyz have "... = 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 ∈ V" proof - from non_empty obtain x where x: "x ∈ V" by blast then have "0 = x - x" by (rule diff_self [symmetric]) also from x x have "... ∈ V" by (rule diff_closed) finally show ?thesis . qed lemma (in vectorspace) add_zero_right [simp]: "x ∈ V ==> x + 0 = x" proof - assume x: "x ∈ V" from this and zero have "x + 0 = 0 + x" by (rule add_commute) also from x have "... = x" by (rule add_zero_left) finally show ?thesis . qed lemma (in vectorspace) mult_assoc2: "x ∈ V ==> a · b · x = (a * b) · x" by (simp only: mult_assoc) lemma (in vectorspace) diff_mult_distrib1: "x ∈ V ==> y ∈ V ==> a · (x - y) = a · x - a · y" by (simp add: diff_eq1 negate_eq1 add_mult_distrib1 mult_assoc2) lemma (in vectorspace) diff_mult_distrib2: "x ∈ V ==> (a - b) · x = a · x - (b · x)" proof - assume x: "x ∈ V" have " (a - b) · x = (a + - b) · x" by (simp add: real_diff_def) also from x have "... = a · x + (- b) · x" by (rule add_mult_distrib2) also from x have "... = a · x + - (b · x)" by (simp add: negate_eq1 mult_assoc2) also from x have "... = a · x - (b · 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 ∈ V ==> 0 · x = 0" proof - assume x: "x ∈ V" have "0 · x = (1 - 1) · x" by simp also have "... = (1 + - 1) · x" by simp also from x have "... = 1 · x + (- 1) · x" by (rule add_mult_distrib2) also from x have "... = x + (- 1) · x" by simp also from x have "... = x + - x" by (simp add: negate_eq2a) also from x have "... = x - x" by (simp add: diff_eq2) also from x have "... = 0" by simp finally show ?thesis . qed lemma (in vectorspace) mult_zero_right [simp]: "a · 0 = (0::'a)" proof - have "a · 0 = a · (0 - (0::'a))" by simp also have "... = a · 0 - a · 0" by (rule diff_mult_distrib1) simp_all also have "... = 0" by simp finally show ?thesis . qed lemma (in vectorspace) minus_mult_cancel [simp]: "x ∈ V ==> (- a) · - x = a · x" by (simp add: negate_eq1 mult_assoc2) lemma (in vectorspace) add_minus_left_eq_diff: "x ∈ V ==> y ∈ V ==> - x + y = y - x" proof - assume xy: "x ∈ V" "y ∈ V" hence "- x + y = y + - x" by (simp add: add_commute) also from xy have "... = y - x" by (simp add: diff_eq1) finally show ?thesis . qed lemma (in vectorspace) add_minus [simp]: "x ∈ V ==> x + - x = 0" by (simp add: diff_eq2) lemma (in vectorspace) add_minus_left [simp]: "x ∈ V ==> - x + x = 0" by (simp add: diff_eq2 add_commute) lemma (in vectorspace) minus_minus [simp]: "x ∈ V ==> - (- 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 ∈ V ==> (- x = 0) = (x = 0)" proof assume x: "x ∈ V" { from x have "x = - (- x)" by (simp add: minus_minus) also assume "- x = 0" also have "- ... = 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 ∈ V ==> y ∈ V ==> x + (- x + y) = y" by (simp add: add_assoc [symmetric] del: add_commute) lemma (in vectorspace) minus_add_cancel [simp]: "x ∈ V ==> y ∈ V ==> - x + (x + y) = y" by (simp add: add_assoc [symmetric] del: add_commute) lemma (in vectorspace) minus_add_distrib [simp]: "x ∈ V ==> y ∈ V ==> - (x + y) = - x + - y" by (simp add: negate_eq1 add_mult_distrib1) lemma (in vectorspace) diff_zero [simp]: "x ∈ V ==> x - 0 = x" by (simp add: diff_eq1) lemma (in vectorspace) diff_zero_right [simp]: "x ∈ V ==> 0 - x = - x" by (simp add: diff_eq1) lemma (in vectorspace) add_left_cancel: "x ∈ V ==> y ∈ V ==> z ∈ V ==> (x + y = x + z) = (y = z)" proof assume x: "x ∈ V" and y: "y ∈ V" and z: "z ∈ V" { from y have "y = 0 + y" by simp also from x y have "... = (- x + x) + y" by simp also from x y have "... = - 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 "... = 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 ∈ V ==> y ∈ V ==> z ∈ V ==> (y + x = z + x) = (y = z)" by (simp only: add_commute add_left_cancel) lemma (in vectorspace) add_assoc_cong: "x ∈ V ==> y ∈ V ==> x' ∈ V ==> y' ∈ V ==> z ∈ V ==> x + y = x' + y' ==> x + (y + z) = x' + (y' + z)" by (simp only: add_assoc [symmetric]) lemma (in vectorspace) mult_left_commute: "x ∈ V ==> a · b · x = b · a · x" by (simp add: real_mult_commute mult_assoc2) lemma (in vectorspace) mult_zero_uniq: "x ∈ V ==> x ≠ 0 ==> a · x = 0 ==> a = 0" proof (rule classical) assume a: "a ≠ 0" assume x: "x ∈ V" "x ≠ 0" and ax: "a · x = 0" from x a have "x = (inverse a * a) · x" by simp also from `x ∈ V` have "... = inverse a · (a · x)" by (rule mult_assoc) also from ax have "... = inverse a · 0" by simp also have "... = 0" by simp finally have "x = 0" . with `x ≠ 0` show "a = 0" by contradiction qed lemma (in vectorspace) mult_left_cancel: "x ∈ V ==> y ∈ V ==> a ≠ 0 ==> (a · x = a · y) = (x = y)" proof assume x: "x ∈ V" and y: "y ∈ V" and a: "a ≠ 0" from x have "x = 1 · x" by simp also from a have "... = (inverse a * a) · x" by simp also from x have "... = inverse a · (a · x)" by (simp only: mult_assoc) also assume "a · x = a · y" also from a y have "inverse a · ... = y" by (simp add: mult_assoc2) finally show "x = y" . next assume "x = y" then show "a · x = a · y" by (simp only:) qed lemma (in vectorspace) mult_right_cancel: "x ∈ V ==> x ≠ 0 ==> (a · x = b · x) = (a = b)" proof assume x: "x ∈ V" and neq: "x ≠ 0" { from x have "(a - b) · x = a · x - b · x" by (simp add: diff_mult_distrib2) also assume "a · x = b · x" with x have "a · x - b · x = 0" by simp finally have "(a - b) · x = 0" . with x neq have "a - b = 0" by (rule mult_zero_uniq) thus "a = b" by simp next assume "a = b" then show "a · x = b · x" by (simp only:) } qed lemma (in vectorspace) eq_diff_eq: "x ∈ V ==> y ∈ V ==> z ∈ V ==> (x = z - y) = (x + y = z)" proof assume x: "x ∈ V" and y: "y ∈ V" and z: "z ∈ V" { assume "x = z - y" hence "x + y = z - y + y" by simp also from y z have "... = z + - y + y" by (simp add: diff_eq1) also have "... = z + (- y + y)" by (rule add_assoc) (simp_all add: y z) also from y z have "... = z + 0" by (simp only: add_minus_left) also from z have "... = z" by (simp only: add_zero_right) finally show "x + y = z" . next assume "x + y = z" hence "z - y = (x + y) - y" by simp also from x y have "... = x + y + - y" by (simp add: diff_eq1) also have "... = x + (y + - y)" by (rule add_assoc) (simp_all add: x y) also from x y have "... = x" by simp finally show "x = z - y" .. } qed lemma (in vectorspace) add_minus_eq_minus: "x ∈ V ==> y ∈ V ==> x + y = 0 ==> x = - y" proof - assume x: "x ∈ V" and y: "y ∈ V" from x y have "x = (- y + y) + x" by simp also from x y have "... = - 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 ∈ V ==> y ∈ V ==> x - y = 0 ==> x = y" proof - assume x: "x ∈ V" and y: "y ∈ 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 ∈ V ==> b ∈ V ==> c ∈ V ==> d ∈ V ==> a + b = c + d ==> a - c = d - b" proof - assume vs: "a ∈ V" "b ∈ V" "c ∈ V" "d ∈ V" and eq: "a + b = c + d" then have "- c + (a + b) = - c + (c + d)" by (simp add: add_left_cancel) also have "... = d" using `c ∈ V` `d ∈ 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 "... = d + - b" by (simp add: add_right_cancel) also from vs have "... = d - b" by (simp add: diff_eq2) finally show "a - c = d - b" . qed lemma (in vectorspace) vs_add_cancel_21: "x ∈ V ==> y ∈ V ==> z ∈ V ==> u ∈ V ==> (x + (y + z) = y + u) = (x + z = u)" proof assume vs: "x ∈ V" "y ∈ V" "z ∈ V" "u ∈ V" { from vs have "x + z = - y + y + (x + z)" by simp also have "... = - 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 ∈ V ==> y ∈ V ==> z ∈ V ==> (x + (y + z) = y) = (x = - z)" proof assume vs: "x ∈ V" "y ∈ V" "z ∈ 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" hence "x + (y + z) = - z + (y + z)" by simp also have "... = y + (- z + z)" by (rule add_left_commute) (simp_all add: vs) also from vs have "... = y" by simp finally show "x + (y + z) = y" . } qed end
lemma negate_eq2:
x ∈ V ==> - 1 · x = - x
lemma negate_eq2a:
x ∈ V ==> -1 · x = - x
lemma diff_eq2:
[| x ∈ V; y ∈ V |] ==> x + - y = x - y
lemma diff_closed:
[| x ∈ V; y ∈ V |] ==> x - y ∈ V
lemma neg_closed:
x ∈ V ==> - x ∈ V
lemma add_left_commute:
[| x ∈ V; y ∈ V; z ∈ V |] ==> x + (y + z) = y + (x + z)
theorem add_ac:
[| x ∈ V; y ∈ V; z ∈ V |] ==> x + y + z = x + (y + z)
[| x ∈ V; y ∈ V |] ==> x + y = y + x
[| x ∈ V; y ∈ V; z ∈ V |] ==> x + (y + z) = y + (x + z)
lemma zero:
(0::'a) ∈ V
lemma add_zero_right:
x ∈ V ==> x + (0::'a) = x
lemma mult_assoc2:
x ∈ V ==> a · b · x = (a * b) · x
lemma diff_mult_distrib1:
[| x ∈ V; y ∈ V |] ==> a · (x - y) = a · x - a · y
lemma diff_mult_distrib2:
x ∈ V ==> (a - b) · x = a · x - b · x
lemma distrib:
[| x ∈ V; y ∈ V |] ==> a · (x + y) = a · x + a · y
x ∈ V ==> (a + b) · x = a · x + b · x
[| x ∈ V; y ∈ V |] ==> a · (x - y) = a · x - a · y
x ∈ V ==> (a - b) · x = a · x - b · x
lemma mult_zero_left:
x ∈ V ==> 0 · x = (0::'a)
lemma mult_zero_right:
a · (0::'a) = (0::'a)
lemma minus_mult_cancel:
x ∈ V ==> - a · - x = a · x
lemma add_minus_left_eq_diff:
[| x ∈ V; y ∈ V |] ==> - x + y = y - x
lemma add_minus:
x ∈ V ==> x + - x = (0::'a)
lemma add_minus_left:
x ∈ V ==> - x + x = (0::'a)
lemma minus_minus:
x ∈ V ==> - (- x) = x
lemma minus_zero:
- (0::'a) = (0::'a)
lemma minus_zero_iff:
x ∈ V ==> (- x = (0::'a)) = (x = (0::'a))
lemma add_minus_cancel:
[| x ∈ V; y ∈ V |] ==> x + (- x + y) = y
lemma minus_add_cancel:
[| x ∈ V; y ∈ V |] ==> - x + (x + y) = y
lemma minus_add_distrib:
[| x ∈ V; y ∈ V |] ==> - (x + y) = - x + - y
lemma diff_zero:
x ∈ V ==> x - (0::'a) = x
lemma diff_zero_right:
x ∈ V ==> (0::'a) - x = - x
lemma add_left_cancel:
[| x ∈ V; y ∈ V; z ∈ V |] ==> (x + y = x + z) = (y = z)
lemma add_right_cancel:
[| x ∈ V; y ∈ V; z ∈ V |] ==> (y + x = z + x) = (y = z)
lemma add_assoc_cong:
[| x ∈ V; y ∈ V; x' ∈ V; y' ∈ V; z ∈ V; x + y = x' + y' |]
==> x + (y + z) = x' + (y' + z)
lemma mult_left_commute:
x ∈ V ==> a · b · x = b · a · x
lemma mult_zero_uniq:
[| x ∈ V; x ≠ (0::'a); a · x = (0::'a) |] ==> a = 0
lemma mult_left_cancel:
[| x ∈ V; y ∈ V; a ≠ 0 |] ==> (a · x = a · y) = (x = y)
lemma mult_right_cancel:
[| x ∈ V; x ≠ (0::'a) |] ==> (a · x = b · x) = (a = b)
lemma eq_diff_eq:
[| x ∈ V; y ∈ V; z ∈ V |] ==> (x = z - y) = (x + y = z)
lemma add_minus_eq_minus:
[| x ∈ V; y ∈ V; x + y = (0::'a) |] ==> x = - y
lemma add_minus_eq:
[| x ∈ V; y ∈ V; x - y = (0::'a) |] ==> x = y
lemma add_diff_swap:
[| a ∈ V; b ∈ V; c ∈ V; d ∈ V; a + b = c + d |] ==> a - c = d - b
lemma vs_add_cancel_21:
[| x ∈ V; y ∈ V; z ∈ V; u ∈ V |] ==> (x + (y + z) = y + u) = (x + z = u)
lemma add_cancel_end:
[| x ∈ V; y ∈ V; z ∈ V |] ==> (x + (y + z) = y) = (x = - z)