--- a/src/HOL/Real/HahnBanach/Subspace.thy Thu Aug 22 12:28:41 2002 +0200
+++ b/src/HOL/Real/HahnBanach/Subspace.thy Thu Aug 22 20:49:43 2002 +0200
@@ -16,122 +16,109 @@
and scalar multiplication.
*}
-constdefs
- is_subspace :: "'a::{plus, minus, zero} set \<Rightarrow> 'a set \<Rightarrow> bool"
- "is_subspace U V \<equiv> U \<noteq> {} \<and> U \<subseteq> V
- \<and> (\<forall>x \<in> U. \<forall>y \<in> U. \<forall>a. x + y \<in> U \<and> a \<cdot> x \<in> U)"
+locale subspace = var U + var V +
+ assumes non_empty [iff, intro]: "U \<noteq> {}"
+ and subset [iff]: "U \<subseteq> V"
+ and add_closed [iff]: "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
+ and mult_closed [iff]: "x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
-lemma subspaceI [intro]:
- "0 \<in> U \<Longrightarrow> U \<subseteq> V \<Longrightarrow> \<forall>x \<in> U. \<forall>y \<in> U. (x + y \<in> U) \<Longrightarrow>
- \<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U
- \<Longrightarrow> is_subspace U V"
-proof (unfold is_subspace_def, intro conjI)
- assume "0 \<in> U" thus "U \<noteq> {}" by fast
-qed (simp+)
+syntax (symbols)
+ subspace :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" (infix "\<unlhd>" 50)
-lemma subspace_not_empty [intro?]: "is_subspace U V \<Longrightarrow> U \<noteq> {}"
- by (unfold is_subspace_def) blast
+lemma subspace_subset [elim]: "U \<unlhd> V \<Longrightarrow> U \<subseteq> V"
+ by (rule subspace.subset)
-lemma subspace_subset [intro?]: "is_subspace U V \<Longrightarrow> U \<subseteq> V"
- by (unfold is_subspace_def) blast
+lemma (in subspace) subsetD [iff]: "x \<in> U \<Longrightarrow> x \<in> V"
+ using subset by blast
-lemma subspace_subsetD [simp, intro?]:
- "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
- by (unfold is_subspace_def) blast
+lemma subspaceD [elim]: "U \<unlhd> V \<Longrightarrow> x \<in> U \<Longrightarrow> x \<in> V"
+ by (rule subspace.subsetD)
-lemma subspace_add_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x + y \<in> U"
- by (unfold is_subspace_def) blast
+lemma rev_subspaceD [elim?]: "x \<in> U \<Longrightarrow> U \<unlhd> V \<Longrightarrow> x \<in> V"
+ by (rule subspace.subsetD)
+
-lemma subspace_mult_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> x \<in> U \<Longrightarrow> a \<cdot> x \<in> U"
- by (unfold is_subspace_def) blast
+locale (open) subvectorspace =
+ subspace + vectorspace
-lemma subspace_diff_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> y \<in> U
- \<Longrightarrow> x - y \<in> U"
+lemma (in subvectorspace) diff_closed [iff]:
+ "x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x - y \<in> U"
by (simp add: diff_eq1 negate_eq1)
-text {* Similar as for linear spaces, the existence of the
-zero element in every subspace follows from the non-emptiness
-of the carrier set and by vector space laws.*}
-lemma zero_in_subspace [intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> 0 \<in> U"
+text {*
+ \medskip Similar as for linear spaces, the existence of the zero
+ element in every subspace follows from the non-emptiness of the
+ carrier set and by vector space laws.
+*}
+
+lemma (in subvectorspace) zero [intro]: "0 \<in> U"
proof -
- assume "is_subspace U V" and v: "is_vectorspace V"
- have "U \<noteq> {}" ..
- hence "\<exists>x. x \<in> U" by blast
- thus ?thesis
- proof
- fix x assume u: "x \<in> U"
- hence "x \<in> V" by (simp!)
- with v have "0 = x - x" by (simp!)
- also have "... \<in> U" by (rule subspace_diff_closed)
- finally show ?thesis .
- qed
+ have "U \<noteq> {}" by (rule U_V.non_empty)
+ then obtain x where x: "x \<in> U" by blast
+ hence "x \<in> V" .. hence "0 = x - x" by simp
+ also have "... \<in> U" by (rule U_V.diff_closed)
+ finally show ?thesis .
qed
-lemma subspace_neg_closed [simp, intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> x \<in> U \<Longrightarrow> - x \<in> U"
+lemma (in subvectorspace) neg_closed [iff]: "x \<in> U \<Longrightarrow> - x \<in> U"
by (simp add: negate_eq1)
+
text {* \medskip Further derived laws: every subspace is a vector space. *}
-lemma subspace_vs [intro?]:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_vectorspace U"
-proof -
- assume "is_subspace U V" "is_vectorspace V"
- show ?thesis
- proof
- show "0 \<in> U" ..
- show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U" by (simp!)
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U" by (simp!)
- show "\<forall>x \<in> U. - x = - 1 \<cdot> x" by (simp! add: negate_eq1)
- show "\<forall>x \<in> U. \<forall>y \<in> U. x - y = x + - y"
- by (simp! add: diff_eq1)
- qed (simp! add: vs_add_mult_distrib1 vs_add_mult_distrib2)+
+lemma (in subvectorspace) vectorspace [iff]:
+ "vectorspace U"
+proof
+ show "U \<noteq> {}" ..
+ fix x y z assume x: "x \<in> U" and y: "y \<in> U" and z: "z \<in> U"
+ fix a b :: real
+ from x y show "x + y \<in> U" by simp
+ from x show "a \<cdot> x \<in> U" by simp
+ from x y z show "(x + y) + z = x + (y + z)" by (simp add: add_ac)
+ from x y show "x + y = y + x" by (simp add: add_ac)
+ from x show "x - x = 0" by simp
+ from x show "0 + x = x" by simp
+ from x y show "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y" by (simp add: distrib)
+ from x show "(a + b) \<cdot> x = a \<cdot> x + b \<cdot> x" by (simp add: distrib)
+ from x show "(a * b) \<cdot> x = a \<cdot> b \<cdot> x" by (simp add: mult_assoc)
+ from x show "1 \<cdot> x = x" by simp
+ from x show "- x = - 1 \<cdot> x" by (simp add: negate_eq1)
+ from x y show "x - y = x + - y" by (simp add: diff_eq1)
qed
+
text {* The subspace relation is reflexive. *}
-lemma subspace_refl [intro]: "is_vectorspace V \<Longrightarrow> is_subspace V V"
+lemma (in vectorspace) subspace_refl [intro]: "V \<unlhd> V"
proof
- assume "is_vectorspace V"
- show "0 \<in> V" ..
+ show "V \<noteq> {}" ..
show "V \<subseteq> V" ..
- show "\<forall>x \<in> V. \<forall>y \<in> V. x + y \<in> V" by (simp!)
- show "\<forall>x \<in> V. \<forall>a. a \<cdot> x \<in> V" by (simp!)
+ fix x y assume x: "x \<in> V" and y: "y \<in> V"
+ fix a :: real
+ from x y show "x + y \<in> V" by simp
+ from x show "a \<cdot> x \<in> V" by simp
qed
text {* The subspace relation is transitive. *}
-lemma subspace_trans:
- "is_subspace U V \<Longrightarrow> is_vectorspace V \<Longrightarrow> is_subspace V W
- \<Longrightarrow> is_subspace U W"
+lemma (in vectorspace) subspace_trans [trans]:
+ "U \<unlhd> V \<Longrightarrow> V \<unlhd> W \<Longrightarrow> U \<unlhd> W"
proof
- assume "is_subspace U V" "is_subspace V W" "is_vectorspace V"
- show "0 \<in> U" ..
-
- have "U \<subseteq> V" ..
- also have "V \<subseteq> W" ..
- finally show "U \<subseteq> W" .
-
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
- proof (intro ballI)
- fix x y assume "x \<in> U" "y \<in> U"
- show "x + y \<in> U" by (simp!)
+ assume uv: "U \<unlhd> V" and vw: "V \<unlhd> W"
+ from uv show "U \<noteq> {}" by (rule subspace.non_empty)
+ show "U \<subseteq> W"
+ proof -
+ from uv have "U \<subseteq> V" by (rule subspace.subset)
+ also from vw have "V \<subseteq> W" by (rule subspace.subset)
+ finally show ?thesis .
qed
-
- show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
- proof (intro ballI allI)
- fix x a assume "x \<in> U"
- show "a \<cdot> x \<in> U" by (simp!)
- qed
+ fix x y assume x: "x \<in> U" and y: "y \<in> U"
+ from uv and x y show "x + y \<in> U" by (rule subspace.add_closed)
+ from uv and x show "\<And>a. a \<cdot> x \<in> U" by (rule subspace.mult_closed)
qed
-
subsection {* Linear closure *}
text {*
@@ -140,73 +127,75 @@
*}
constdefs
- lin :: "('a::{minus,plus,zero}) \<Rightarrow> 'a set"
+ lin :: "('a::{minus, plus, zero}) \<Rightarrow> 'a set"
"lin x \<equiv> {a \<cdot> x | a. True}"
-lemma linD: "(x \<in> lin v) = (\<exists>a::real. x = a \<cdot> v)"
- by (unfold lin_def) fast
+lemma linI [intro]: "y = a \<cdot> x \<Longrightarrow> y \<in> lin x"
+ by (unfold lin_def) blast
-lemma linI [intro?]: "a \<cdot> x0 \<in> lin x0"
- by (unfold lin_def) fast
+lemma linI' [iff]: "a \<cdot> x \<in> lin x"
+ by (unfold lin_def) blast
+
+lemma linE [elim]:
+ "x \<in> lin v \<Longrightarrow> (\<And>a::real. x = a \<cdot> v \<Longrightarrow> C) \<Longrightarrow> C"
+ by (unfold lin_def) blast
+
text {* Every vector is contained in its linear closure. *}
-lemma x_lin_x: "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> x \<in> lin x"
-proof (unfold lin_def, intro CollectI exI conjI)
- assume "is_vectorspace V" "x \<in> V"
- show "x = 1 \<cdot> x" by (simp!)
-qed simp
+lemma (in vectorspace) x_lin_x [iff]: "x \<in> V \<Longrightarrow> x \<in> lin x"
+proof -
+ assume "x \<in> V"
+ hence "x = 1 \<cdot> x" by simp
+ also have "\<dots> \<in> lin x" ..
+ finally show ?thesis .
+qed
+
+lemma (in vectorspace) "0_lin_x" [iff]: "x \<in> V \<Longrightarrow> 0 \<in> lin x"
+proof
+ assume "x \<in> V"
+ thus "0 = 0 \<cdot> x" by simp
+qed
text {* Any linear closure is a subspace. *}
-lemma lin_subspace [intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_subspace (lin x) V"
+lemma (in vectorspace) lin_subspace [intro]:
+ "x \<in> V \<Longrightarrow> lin x \<unlhd> V"
proof
- assume "is_vectorspace V" "x \<in> V"
- show "0 \<in> lin x"
- proof (unfold lin_def, intro CollectI exI conjI)
- show "0 = (0::real) \<cdot> x" by (simp!)
- qed simp
-
+ assume x: "x \<in> V"
+ thus "lin x \<noteq> {}" by (auto simp add: x_lin_x)
show "lin x \<subseteq> V"
- proof (unfold lin_def, intro subsetI, elim CollectE exE conjE)
- fix xa a assume "xa = a \<cdot> x"
- show "xa \<in> V" by (simp!)
+ proof
+ fix x' assume "x' \<in> lin x"
+ then obtain a where "x' = a \<cdot> x" ..
+ with x show "x' \<in> V" by simp
qed
-
- show "\<forall>x1 \<in> lin x. \<forall>x2 \<in> lin x. x1 + x2 \<in> lin x"
- proof (intro ballI)
- fix x1 x2 assume "x1 \<in> lin x" "x2 \<in> lin x"
- thus "x1 + x2 \<in> lin x"
- proof (unfold lin_def, elim CollectE exE conjE,
- intro CollectI exI conjI)
- fix a1 a2 assume "x1 = a1 \<cdot> x" "x2 = a2 \<cdot> x"
- show "x1 + x2 = (a1 + a2) \<cdot> x"
- by (simp! add: vs_add_mult_distrib2)
- qed simp
+ fix x' x'' assume x': "x' \<in> lin x" and x'': "x'' \<in> lin x"
+ show "x' + x'' \<in> lin x"
+ proof -
+ from x' obtain a' where "x' = a' \<cdot> x" ..
+ moreover from x'' obtain a'' where "x'' = a'' \<cdot> x" ..
+ ultimately have "x' + x'' = (a' + a'') \<cdot> x"
+ using x by (simp add: distrib)
+ also have "\<dots> \<in> lin x" ..
+ finally show ?thesis .
qed
-
- show "\<forall>xa \<in> lin x. \<forall>a. a \<cdot> xa \<in> lin x"
- proof (intro ballI allI)
- fix x1 a assume "x1 \<in> lin x"
- thus "a \<cdot> x1 \<in> lin x"
- proof (unfold lin_def, elim CollectE exE conjE,
- intro CollectI exI conjI)
- fix a1 assume "x1 = a1 \<cdot> x"
- show "a \<cdot> x1 = (a * a1) \<cdot> x" by (simp!)
- qed simp
+ fix a :: real
+ show "a \<cdot> x' \<in> lin x"
+ proof -
+ from x' obtain a' where "x' = a' \<cdot> x" ..
+ with x have "a \<cdot> x' = (a * a') \<cdot> x" by (simp add: mult_assoc)
+ also have "\<dots> \<in> lin x" ..
+ finally show ?thesis .
qed
qed
+
text {* Any linear closure is a vector space. *}
-lemma lin_vs [intro?]:
- "is_vectorspace V \<Longrightarrow> x \<in> V \<Longrightarrow> is_vectorspace (lin x)"
-proof (rule subspace_vs)
- assume "is_vectorspace V" "x \<in> V"
- show "is_subspace (lin x) V" ..
-qed
-
+lemma (in vectorspace) lin_vectorspace [intro]:
+ "x \<in> V \<Longrightarrow> vectorspace (lin x)"
+ by (rule subvectorspace.vectorspace) (rule lin_subspace)
subsection {* Sum of two vectorspaces *}
@@ -219,101 +208,92 @@
instance set :: (plus) plus ..
defs (overloaded)
- vs_sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}"
+ sum_def: "U + V \<equiv> {u + v | u v. u \<in> U \<and> v \<in> V}"
-lemma vs_sumD:
- "(x \<in> U + V) = (\<exists>u \<in> U. \<exists>v \<in> V. x = u + v)"
- by (unfold vs_sum_def) fast
+lemma sumE [elim]:
+ "x \<in> U + V \<Longrightarrow> (\<And>u v. x = u + v \<Longrightarrow> u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> C) \<Longrightarrow> C"
+ by (unfold sum_def) blast
-lemmas vs_sumE = vs_sumD [THEN iffD1, elim_format, standard]
+lemma sumI [intro]:
+ "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> x = u + v \<Longrightarrow> x \<in> U + V"
+ by (unfold sum_def) blast
-lemma vs_sumI [intro?]:
- "x \<in> U \<Longrightarrow> y \<in> V \<Longrightarrow> t = x + y \<Longrightarrow> t \<in> U + V"
- by (unfold vs_sum_def) fast
+lemma sumI' [intro]:
+ "u \<in> U \<Longrightarrow> v \<in> V \<Longrightarrow> u + v \<in> U + V"
+ by (unfold sum_def) blast
text {* @{text U} is a subspace of @{text "U + V"}. *}
-lemma subspace_vs_sum1 [intro?]:
- "is_vectorspace U \<Longrightarrow> is_vectorspace V
- \<Longrightarrow> is_subspace U (U + V)"
+lemma subspace_sum1 [iff]:
+ includes vectorspace U + vectorspace V
+ shows "U \<unlhd> U + V"
proof
- assume "is_vectorspace U" "is_vectorspace V"
- show "0 \<in> U" ..
+ show "U \<noteq> {}" ..
show "U \<subseteq> U + V"
- proof (intro subsetI vs_sumI)
- fix x assume "x \<in> U"
- show "x = x + 0" by (simp!)
- show "0 \<in> V" by (simp!)
+ proof
+ fix x assume x: "x \<in> U"
+ moreover have "0 \<in> V" ..
+ ultimately have "x + 0 \<in> U + V" ..
+ with x show "x \<in> U + V" by simp
qed
- show "\<forall>x \<in> U. \<forall>y \<in> U. x + y \<in> U"
- proof (intro ballI)
- fix x y assume "x \<in> U" "y \<in> U" show "x + y \<in> U" by (simp!)
- qed
- show "\<forall>x \<in> U. \<forall>a. a \<cdot> x \<in> U"
- proof (intro ballI allI)
- fix x a assume "x \<in> U" show "a \<cdot> x \<in> U" by (simp!)
- qed
+ fix x y assume x: "x \<in> U" and "y \<in> U"
+ thus "x + y \<in> U" by simp
+ from x show "\<And>a. a \<cdot> x \<in> U" by simp
qed
-text{* The sum of two subspaces is again a subspace.*}
+text {* The sum of two subspaces is again a subspace. *}
-lemma vs_sum_subspace [intro?]:
- "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_subspace (U + V) E"
+lemma sum_subspace [intro?]:
+ includes subvectorspace U E + subvectorspace V E
+ shows "U + V \<unlhd> E"
proof
- assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
- show "0 \<in> U + V"
- proof (intro vs_sumI)
+ have "0 \<in> U + V"
+ proof
show "0 \<in> U" ..
show "0 \<in> V" ..
- show "(0::'a) = 0 + 0" by (simp!)
+ show "(0::'a) = 0 + 0" by simp
qed
-
+ thus "U + V \<noteq> {}" by blast
show "U + V \<subseteq> E"
- proof (intro subsetI, elim vs_sumE bexE)
- fix x u v assume "u \<in> U" "v \<in> V" "x = u + v"
- show "x \<in> E" by (simp!)
+ proof
+ fix x assume "x \<in> U + V"
+ then obtain u v where x: "x = u + v" and
+ u: "u \<in> U" and v: "v \<in> V" ..
+ have "U \<unlhd> E" . with u have "u \<in> E" ..
+ moreover have "V \<unlhd> E" . with v have "v \<in> E" ..
+ ultimately show "x \<in> E" using x by simp
qed
-
- show "\<forall>x \<in> U + V. \<forall>y \<in> U + V. x + y \<in> U + V"
- proof (intro ballI)
- fix x y assume "x \<in> U + V" "y \<in> U + V"
- thus "x + y \<in> U + V"
- proof (elim vs_sumE bexE, intro vs_sumI)
- fix ux vx uy vy
- assume "ux \<in> U" "vx \<in> V" "x = ux + vx"
- and "uy \<in> U" "vy \<in> V" "y = uy + vy"
- show "x + y = (ux + uy) + (vx + vy)" by (simp!)
- qed (simp_all!)
+ fix x y assume x: "x \<in> U + V" and y: "y \<in> U + V"
+ show "x + y \<in> U + V"
+ proof -
+ from x obtain ux vx where "x = ux + vx" and "ux \<in> U" and "vx \<in> V" ..
+ moreover
+ from y obtain uy vy where "y = uy + vy" and "uy \<in> U" and "vy \<in> V" ..
+ ultimately
+ have "ux + uy \<in> U"
+ and "vx + vy \<in> V"
+ and "x + y = (ux + uy) + (vx + vy)"
+ using x y by (simp_all add: add_ac)
+ thus ?thesis ..
qed
-
- show "\<forall>x \<in> U + V. \<forall>a. a \<cdot> x \<in> U + V"
- proof (intro ballI allI)
- fix x a assume "x \<in> U + V"
- thus "a \<cdot> x \<in> U + V"
- proof (elim vs_sumE bexE, intro vs_sumI)
- fix a x u v assume "u \<in> U" "v \<in> V" "x = u + v"
- show "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)"
- by (simp! add: vs_add_mult_distrib1)
- qed (simp_all!)
+ fix a show "a \<cdot> x \<in> U + V"
+ proof -
+ from x obtain u v where "x = u + v" and "u \<in> U" and "v \<in> V" ..
+ hence "a \<cdot> u \<in> U" and "a \<cdot> v \<in> V"
+ and "a \<cdot> x = (a \<cdot> u) + (a \<cdot> v)" by (simp_all add: distrib)
+ thus ?thesis ..
qed
qed
text{* The sum of two subspaces is a vectorspace. *}
-lemma vs_sum_vs [intro?]:
- "is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow> is_vectorspace E
- \<Longrightarrow> is_vectorspace (U + V)"
-proof (rule subspace_vs)
- assume "is_subspace U E" "is_subspace V E" "is_vectorspace E"
- show "is_subspace (U + V) E" ..
-qed
-
+lemma sum_vs [intro?]:
+ "U \<unlhd> E \<Longrightarrow> V \<unlhd> E \<Longrightarrow> vectorspace E \<Longrightarrow> vectorspace (U + V)"
+ by (rule subvectorspace.vectorspace) (rule sum_subspace)
subsection {* Direct sums *}
-
text {*
The sum of @{text U} and @{text V} is called \emph{direct}, iff the
zero element is the only common element of @{text U} and @{text
@@ -323,31 +303,35 @@
*}
lemma decomp:
- "is_vectorspace E \<Longrightarrow> is_subspace U E \<Longrightarrow> is_subspace V E \<Longrightarrow>
- U \<inter> V = {0} \<Longrightarrow> u1 \<in> U \<Longrightarrow> u2 \<in> U \<Longrightarrow> v1 \<in> V \<Longrightarrow> v2 \<in> V \<Longrightarrow>
- u1 + v1 = u2 + v2 \<Longrightarrow> u1 = u2 \<and> v1 = v2"
+ includes vectorspace E + subspace U E + subspace V E
+ assumes direct: "U \<inter> V = {0}"
+ and u1: "u1 \<in> U" and u2: "u2 \<in> U"
+ and v1: "v1 \<in> V" and v2: "v2 \<in> V"
+ and sum: "u1 + v1 = u2 + v2"
+ shows "u1 = u2 \<and> v1 = v2"
proof
- assume "is_vectorspace E" "is_subspace U E" "is_subspace V E"
- "U \<inter> V = {0}" "u1 \<in> U" "u2 \<in> U" "v1 \<in> V" "v2 \<in> V"
- "u1 + v1 = u2 + v2"
- have eq: "u1 - u2 = v2 - v1" by (simp! add: vs_add_diff_swap)
- have u: "u1 - u2 \<in> U" by (simp!)
- with eq have v': "v2 - v1 \<in> U" by simp
- have v: "v2 - v1 \<in> V" by (simp!)
- with eq have u': "u1 - u2 \<in> V" by simp
+ have U: "vectorspace U" by (rule subvectorspace.vectorspace)
+ have V: "vectorspace V" by (rule subvectorspace.vectorspace)
+ from u1 u2 v1 v2 and sum have eq: "u1 - u2 = v2 - v1"
+ by (simp add: add_diff_swap)
+ from u1 u2 have u: "u1 - u2 \<in> U"
+ by (rule vectorspace.diff_closed [OF U])
+ with eq have v': "v2 - v1 \<in> U" by (simp only:)
+ from v2 v1 have v: "v2 - v1 \<in> V"
+ by (rule vectorspace.diff_closed [OF V])
+ with eq have u': " u1 - u2 \<in> V" by (simp only:)
show "u1 = u2"
- proof (rule vs_add_minus_eq)
- show "u1 - u2 = 0" by (rule Int_singletonD [OF _ u u'])
+ proof (rule add_minus_eq)
show "u1 \<in> E" ..
show "u2 \<in> E" ..
+ from u u' and direct show "u1 - u2 = 0" by blast
qed
-
show "v1 = v2"
- proof (rule vs_add_minus_eq [symmetric])
- show "v2 - v1 = 0" by (rule Int_singletonD [OF _ v' v])
+ proof (rule add_minus_eq [symmetric])
show "v1 \<in> E" ..
show "v2 \<in> E" ..
+ from v v' and direct show "v2 - v1 = 0" by blast
qed
qed
@@ -361,58 +345,48 @@
*}
lemma decomp_H':
- "is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow> y1 \<in> H \<Longrightarrow> y2 \<in> H \<Longrightarrow>
- x' \<notin> H \<Longrightarrow> x' \<in> E \<Longrightarrow> x' \<noteq> 0 \<Longrightarrow> y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'
- \<Longrightarrow> y1 = y2 \<and> a1 = a2"
+ includes vectorspace E + subvectorspace H E
+ assumes y1: "y1 \<in> H" and y2: "y2 \<in> H"
+ and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ and eq: "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
+ shows "y1 = y2 \<and> a1 = a2"
proof
- assume "is_vectorspace E" and h: "is_subspace H E"
- and "y1 \<in> H" "y2 \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- "y1 + a1 \<cdot> x' = y2 + a2 \<cdot> x'"
-
have c: "y1 = y2 \<and> a1 \<cdot> x' = a2 \<cdot> x'"
proof (rule decomp)
show "a1 \<cdot> x' \<in> lin x'" ..
show "a2 \<cdot> x' \<in> lin x'" ..
- show "H \<inter> (lin x') = {0}"
+ show "H \<inter> lin x' = {0}"
proof
show "H \<inter> lin x' \<subseteq> {0}"
- proof (intro subsetI, elim IntE, rule singleton_iff [THEN iffD2])
- fix x assume "x \<in> H" "x \<in> lin x'"
- thus "x = 0"
- proof (unfold lin_def, elim CollectE exE conjE)
- fix a assume "x = a \<cdot> x'"
- show ?thesis
- proof cases
- assume "a = (0::real)" show ?thesis by (simp!)
- next
- assume "a \<noteq> (0::real)"
- from h have "inverse a \<cdot> a \<cdot> x' \<in> H"
- by (rule subspace_mult_closed) (simp!)
- also have "inverse a \<cdot> a \<cdot> x' = x'" by (simp!)
- finally have "x' \<in> H" .
- thus ?thesis by contradiction
- qed
- qed
+ proof
+ fix x assume x: "x \<in> H \<inter> lin x'"
+ then obtain a where xx': "x = a \<cdot> x'"
+ by blast
+ have "x = 0"
+ proof cases
+ assume "a = 0"
+ with xx' and x' show ?thesis by simp
+ next
+ assume a: "a \<noteq> 0"
+ from x have "x \<in> H" ..
+ with xx' have "inverse a \<cdot> a \<cdot> x' \<in> H" by simp
+ with a and x' have "x' \<in> H" by (simp add: mult_assoc2)
+ thus ?thesis by contradiction
+ qed
+ thus "x \<in> {0}" ..
qed
show "{0} \<subseteq> H \<inter> lin x'"
proof -
- have "0 \<in> H \<inter> lin x'"
- proof (rule IntI)
- show "0 \<in> H" ..
- from lin_vs show "0 \<in> lin x'" ..
- qed
- thus ?thesis by simp
+ have "0 \<in> H" ..
+ moreover have "0 \<in> lin x'" ..
+ ultimately show ?thesis by blast
qed
qed
- show "is_subspace (lin x') E" ..
+ show "lin x' \<unlhd> E" ..
qed
-
- from c show "y1 = y2" by simp
-
- show "a1 = a2"
- proof (rule vs_mult_right_cancel [THEN iffD1])
- from c show "a1 \<cdot> x' = a2 \<cdot> x'" by simp
- qed
+ thus "y1 = y2" ..
+ from c have "a1 \<cdot> x' = a2 \<cdot> x'" ..
+ with x' show "a1 = a2" by (simp add: mult_right_cancel)
qed
text {*
@@ -423,18 +397,20 @@
*}
lemma decomp_H'_H:
- "is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow> t \<in> H \<Longrightarrow> x' \<notin> H \<Longrightarrow> x' \<in> E
- \<Longrightarrow> x' \<noteq> 0
- \<Longrightarrow> (SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, (0::real))"
-proof (rule, unfold split_tupled_all)
- assume "is_vectorspace E" "is_subspace H E" "t \<in> H" "x' \<notin> H" "x' \<in> E"
- "x' \<noteq> 0"
- have h: "is_vectorspace H" ..
- fix y a presume t1: "t = y + a \<cdot> x'" and "y \<in> H"
- have "y = t \<and> a = (0::real)"
- by (rule decomp_H') (auto!)
- thus "(y, a) = (t, (0::real))" by (simp!)
-qed (simp_all!)
+ includes vectorspace E + subvectorspace H E
+ assumes t: "t \<in> H"
+ and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ shows "(SOME (y, a). t = y + a \<cdot> x' \<and> y \<in> H) = (t, 0)"
+proof (rule, simp_all only: split_paired_all split_conv)
+ from t x' show "t = t + 0 \<cdot> x' \<and> t \<in> H" by simp
+ fix y and a assume ya: "t = y + a \<cdot> x' \<and> y \<in> H"
+ have "y = t \<and> a = 0"
+ proof (rule decomp_H')
+ from ya x' show "y + a \<cdot> x' = t + 0 \<cdot> x'" by simp
+ from ya show "y \<in> H" ..
+ qed
+ with t x' show "(y, a) = (y + a \<cdot> x', 0)" by simp
+qed
text {*
The components @{text "y \<in> H"} and @{text a} in @{text "y + a \<cdot> x'"}
@@ -443,42 +419,41 @@
*}
lemma h'_definite:
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
- in (h y) + a * xi) \<Longrightarrow>
- x = y + a \<cdot> x' \<Longrightarrow> is_vectorspace E \<Longrightarrow> is_subspace H E \<Longrightarrow>
- y \<in> H \<Longrightarrow> x' \<notin> H \<Longrightarrow> x' \<in> E \<Longrightarrow> x' \<noteq> 0
- \<Longrightarrow> h' x = h y + a * xi"
+ includes var H
+ assumes h'_def:
+ "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
+ in (h y) + a * xi)"
+ and x: "x = y + a \<cdot> x'"
+ includes vectorspace E + subvectorspace H E
+ assumes y: "y \<in> H"
+ and x': "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
+ shows "h' x = h y + a * xi"
proof -
- assume
- "h' \<equiv> (\<lambda>x. let (y, a) = SOME (y, a). (x = y + a \<cdot> x' \<and> y \<in> H)
- in (h y) + a * xi)"
- "x = y + a \<cdot> x'" "is_vectorspace E" "is_subspace H E"
- "y \<in> H" "x' \<notin> H" "x' \<in> E" "x' \<noteq> 0"
- hence "x \<in> H + (lin x')"
- by (auto simp add: vs_sum_def lin_def)
- have "\<exists>! xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
+ from x y x' have "x \<in> H + lin x'" by auto
+ have "\<exists>!p. (\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) p" (is "\<exists>!p. ?P p")
proof
- show "\<exists>xa. ((\<lambda>(y, a). x = y + a \<cdot> x' \<and> y \<in> H) xa)"
- by (blast!)
- next
- fix xa ya
- assume "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) xa"
- "(\<lambda>(y,a). x = y + a \<cdot> x' \<and> y \<in> H) ya"
- show "xa = ya"
+ from x y show "\<exists>p. ?P p" by blast
+ fix p q assume p: "?P p" and q: "?P q"
+ show "p = q"
proof -
- show "fst xa = fst ya \<and> snd xa = snd ya \<Longrightarrow> xa = ya"
- by (simp add: Pair_fst_snd_eq)
- have x: "x = fst xa + snd xa \<cdot> x' \<and> fst xa \<in> H"
- by (auto!)
- have y: "x = fst ya + snd ya \<cdot> x' \<and> fst ya \<in> H"
- by (auto!)
- from x y show "fst xa = fst ya \<and> snd xa = snd ya"
- by (elim conjE) (rule decomp_H', (simp!)+)
+ from p have xp: "x = fst p + snd p \<cdot> x' \<and> fst p \<in> H"
+ by (cases p) simp
+ from q have xq: "x = fst q + snd q \<cdot> x' \<and> fst q \<in> H"
+ by (cases q) simp
+ have "fst p = fst q \<and> snd p = snd q"
+ proof (rule decomp_H')
+ from xp show "fst p \<in> H" ..
+ from xq show "fst q \<in> H" ..
+ from xp and xq show "fst p + snd p \<cdot> x' = fst q + snd q \<cdot> x'"
+ by simp
+ apply_end assumption+
+ qed
+ thus ?thesis by (cases p, cases q) simp
qed
qed
hence eq: "(SOME (y, a). x = y + a \<cdot> x' \<and> y \<in> H) = (y, a)"
- by (rule some1_equality) (blast!)
- thus "h' x = h y + a * xi" by (simp! add: Let_def)
+ by (rule some1_equality) (simp add: x y)
+ with h'_def show "h' x = h y + a * xi" by (simp add: Let_def)
qed
end