(* linear_map — isabelle *)

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