Datasets:
File size: 2,396 Bytes
bb40bb0 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 | theory linear_map
imports Main
begin
locale linear_map_setup =
fixes zeroV :: "'v"
and addV :: "'v \<Rightarrow> 'v \<Rightarrow> 'v" (infixl "+V" 65)
and smul :: "'r \<Rightarrow> 'v \<Rightarrow> 'v" (infixr "\<cdot>V" 70)
and zeroW :: "'w"
and addW :: "'w \<Rightarrow> 'w \<Rightarrow> 'w" (infixl "+W" 65)
and smulW :: "'r \<Rightarrow> 'w \<Rightarrow> 'w" (infixr "\<cdot>W" 70)
and toFun :: "'v \<Rightarrow> 'w"
assumes addV_comm : "\<And>u v. u +V v = v +V u"
and addV_assoc : "\<And>u v w. (u +V v) +V w = u +V (v +V w)"
and addV_zero : "\<And>u. u +V zeroV = u"
and smul_zeroV : "\<And>a. a \<cdot>V zeroV = zeroV"
and addW_comm : "\<And>u v. u +W v = v +W u"
and addW_assoc : "\<And>u v w. (u +W v) +W w = u +W (v +W w)"
and addW_zero : "\<And>u. u +W zeroW = u"
and smulW_zero : "\<And>a. a \<cdot>W zeroW = zeroW"
and map_add : "\<And>u v. toFun (u +V v) = toFun u +W toFun v"
and map_smul : "\<And>a u. toFun (a \<cdot>V u) = a \<cdot>W toFun u"
begin
definition ker :: "'v \<Rightarrow> bool"
where "ker x \<equiv> toFun x = zeroW"
definition im :: "'w \<Rightarrow> bool"
where "im y \<equiv> \<exists>x. toFun x = y"
lemma ker_add:
assumes "ker x" "ker y" shows "ker (x +V y)"
proof -
have "toFun (x +V y) = toFun x +W toFun y" by (rule map_add)
also have "\<dots> = zeroW +W zeroW"
using assms unfolding ker_def by simp
also have "\<dots> = zeroW"
by (simp add: addW_comm addW_zero)
finally show ?thesis unfolding ker_def .
qed
lemma ker_smul:
assumes "ker x" shows "ker (a \<cdot>V x)"
unfolding ker_def
using assms unfolding ker_def
by (simp add: map_smul smulW_zero)
lemma im_add:
assumes "im y" "im z" shows "im (y +W z)"
proof -
from assms(1) obtain x where hx : "toFun x = y" unfolding im_def by blast
from assms(2) obtain x' where hx' : "toFun x' = z" unfolding im_def by blast
have "toFun (x +V x') = toFun x +W toFun x'" by (rule map_add)
with hx hx' show ?thesis unfolding im_def by blast
qed
lemma im_smul:
assumes "im y" shows "im (a \<cdot>W y)"
proof -
from assms obtain x where hx : "toFun x = y" unfolding im_def by blast
have "toFun (a \<cdot>V x) = a \<cdot>W toFun x" by (rule map_smul)
with hx show ?thesis unfolding im_def by blast
qed
end
end
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