| theory linear_map |
| imports Main |
| begin |
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| 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" |
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| 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" |
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| 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" |
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| 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 |
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| definition ker :: "'v \<Rightarrow> bool" |
| where "ker x \<equiv> toFun x = zeroW" |
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| definition im :: "'w \<Rightarrow> bool" |
| where "im y \<equiv> \<exists>x. toFun x = y" |
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| 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 |
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| 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) |
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| 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 |
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| 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 |
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| end |
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| end |
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