(* Title: HOL/Hahn_Banach/Vector_Space.thy Author: Gertrud Bauer, TU Munich *) section ‹Vector spaces› theory Vector_Space imports Complex_Main Bounds begin subsection ‹Signature› text ‹ For the definition of real vector spaces a type @{typ 'a} of the sort ‹{plus, minus, zero}› is considered, on which a real scalar multiplication ‹⋅› is declared. › consts prod :: "real ⇒ 'a::{plus,minus,zero} ⇒ 'a" (infixr "⋅" 70) subsection ‹Vector space laws› text ‹ A ∗‹vector space› is a non-empty set ‹V› of elements from @{typ 'a} with the following vector space laws: The set ‹V› is closed under addition and scalar multiplication, addition is associative and commutative; ‹- x› is the inverse of ‹x› wrt.\ addition and ‹0› is the neutral element of addition. Addition and multiplication are distributive; scalar multiplication is associative and the real number ‹1› is the neutral element of scalar multiplication. › locale vectorspace = fixes 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" begin lemma negate_eq2: "x ∈ V ⟹ (- 1) ⋅ x = - x" by (rule negate_eq1 [symmetric]) lemma negate_eq2a: "x ∈ V ⟹ -1 ⋅ x = - x" by (simp add: negate_eq1) lemma diff_eq2: "x ∈ V ⟹ y ∈ V ⟹ x + - y = x - y" by (rule diff_eq1 [symmetric]) lemma diff_closed [iff]: "x ∈ V ⟹ y ∈ V ⟹ x - y ∈ V" by (simp add: diff_eq1 negate_eq1) lemma neg_closed [iff]: "x ∈ V ⟹ - x ∈ V" by (simp add: negate_eq1) lemma add_left_commute: "x ∈ V ⟹ y ∈ V ⟹ z ∈ V ⟹ x + (y + z) = y + (x + z)" proof - assume xyz: "x ∈ V" "y ∈ V" "z ∈ V" then have "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 lemmas 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 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 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 mult_assoc2: "x ∈ V ⟹ a ⋅ b ⋅ x = (a * b) ⋅ x" by (simp only: mult_assoc) lemma 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 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 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 distrib = add_mult_distrib1 add_mult_distrib2 diff_mult_distrib1 diff_mult_distrib2 text ‹┉ Further derived laws:› lemma 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 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 minus_mult_cancel [simp]: "x ∈ V ⟹ (- a) ⋅ - x = a ⋅ x" by (simp add: negate_eq1 mult_assoc2) lemma add_minus_left_eq_diff: "x ∈ V ⟹ y ∈ V ⟹ - x + y = y - x" proof - assume xy: "x ∈ V" "y ∈ V" then have "- 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 add_minus [simp]: "x ∈ V ⟹ x + - x = 0" by (simp add: diff_eq2) lemma add_minus_left [simp]: "x ∈ V ⟹ - x + x = 0" by (simp add: diff_eq2 add_commute) lemma minus_minus [simp]: "x ∈ V ⟹ - (- 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 ∈ V" shows "(- x = 0) = (x = 0)" proof from x have "x = - (- x)" by simp 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 add_minus_cancel [simp]: "x ∈ V ⟹ y ∈ V ⟹ x + (- x + y) = y" by (simp add: add_assoc [symmetric]) lemma minus_add_cancel [simp]: "x ∈ V ⟹ y ∈ V ⟹ - x + (x + y) = y" by (simp add: add_assoc [symmetric]) lemma minus_add_distrib [simp]: "x ∈ V ⟹ y ∈ V ⟹ - (x + y) = - x + - y" by (simp add: negate_eq1 add_mult_distrib1) lemma diff_zero [simp]: "x ∈ V ⟹ x - 0 = x" by (simp add: diff_eq1) lemma diff_zero_right [simp]: "x ∈ V ⟹ 0 - x = - x" by (simp add: diff_eq1) lemma add_left_cancel: assumes x: "x ∈ V" and y: "y ∈ V" and z: "z ∈ V" shows "(x + y = x + z) = (y = z)" proof 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) 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 "… = 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 ∈ V ⟹ y ∈ V ⟹ z ∈ V ⟹ (y + x = z + x) = (y = z)" by (simp only: add_commute add_left_cancel) lemma 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 mult_left_commute: "x ∈ V ⟹ a ⋅ b ⋅ x = b ⋅ a ⋅ x" by (simp add: mult.commute mult_assoc2) lemma mult_zero_uniq: assumes x: "x ∈ V" "x ≠ 0" and ax: "a ⋅ x = 0" shows "a = 0" proof (rule classical) assume a: "a ≠ 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 mult_left_cancel: assumes x: "x ∈ V" and y: "y ∈ V" and a: "a ≠ 0" shows "(a ⋅ x = a ⋅ y) = (x = y)" proof 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 mult_right_cancel: assumes x: "x ∈ V" and neq: "x ≠ 0" shows "(a ⋅ x = b ⋅ x) = (a = b)" proof 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) then show "a = b" by simp next assume "a = b" then show "a ⋅ x = b ⋅ x" by (simp only:) qed lemma eq_diff_eq: assumes x: "x ∈ V" and y: "y ∈ V" and z: "z ∈ 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 "… = 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" then have "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 add_minus_eq_minus: assumes x: "x ∈ V" and y: "y ∈ 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 "… = - 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 ∈ V" and y: "y ∈ 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 ∈ V" "b ∈ V" "c ∈ V" "d ∈ 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 "… = 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 vs_add_cancel_21: assumes vs: "x ∈ V" "y ∈ V" "z ∈ V" "u ∈ V" shows "(x + (y + z) = y + u) = (x + z = u)" proof 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 add_cancel_end: assumes vs: "x ∈ V" "y ∈ V" "z ∈ 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 "… = 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 end