clarified font_domain: strict excludes e.g. space character;
(* Title: HOL/Modules.thy
Author: Amine Chaieb, University of Cambridge
Author: Jose Divasón <jose.divasonm at unirioja.es>
Author: Jesús Aransay <jesus-maria.aransay at unirioja.es>
Author: Johannes Hölzl, VU Amsterdam
Author: Fabian Immler, TUM
*)
section \<open>Modules\<close>
text \<open>Bases of a linear algebra based on modules (i.e. vector spaces of rings). \<close>
theory Modules
imports Hull
begin
subsection \<open>Locale for additive functions\<close>
locale additive =
fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
assumes add: "f (x + y) = f x + f y"
begin
lemma zero: "f 0 = 0"
proof -
have "f 0 = f (0 + 0)" by simp
also have "\<dots> = f 0 + f 0" by (rule add)
finally show "f 0 = 0" by simp
qed
lemma minus: "f (- x) = - f x"
proof -
have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
also have "\<dots> = - f x + f x" by (simp add: zero)
finally show "f (- x) = - f x" by (rule add_right_imp_eq)
qed
lemma diff: "f (x - y) = f x - f y"
using add [of x "- y"] by (simp add: minus)
lemma sum: "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
by (induct A rule: infinite_finite_induct) (simp_all add: zero add)
end
text \<open>Modules form the central spaces in linear algebra. They are a generalization from vector
spaces by replacing the scalar field by a scalar ring.\<close>
locale module =
fixes scale :: "'a::comm_ring_1 \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" (infixr "*s" 75)
assumes scale_right_distrib [algebra_simps]: "a *s (x + y) = a *s x + a *s y"
and scale_left_distrib [algebra_simps]: "(a + b) *s x = a *s x + b *s x"
and scale_scale [simp]: "a *s (b *s x) = (a * b) *s x"
and scale_one [simp]: "1 *s x = x"
begin
lemma scale_left_commute: "a *s (b *s x) = b *s (a *s x)"
by (simp add: mult.commute)
lemma scale_zero_left [simp]: "0 *s x = 0"
and scale_minus_left [simp]: "(- a) *s x = - (a *s x)"
and scale_left_diff_distrib [algebra_simps]: "(a - b) *s x = a *s x - b *s x"
and scale_sum_left: "(sum f A) *s x = (\<Sum>a\<in>A. (f a) *s x)"
proof -
interpret s: additive "\<lambda>a. a *s x"
by standard (rule scale_left_distrib)
show "0 *s x = 0" by (rule s.zero)
show "(- a) *s x = - (a *s x)" by (rule s.minus)
show "(a - b) *s x = a *s x - b *s x" by (rule s.diff)
show "(sum f A) *s x = (\<Sum>a\<in>A. (f a) *s x)" by (rule s.sum)
qed
lemma scale_zero_right [simp]: "a *s 0 = 0"
and scale_minus_right [simp]: "a *s (- x) = - (a *s x)"
and scale_right_diff_distrib [algebra_simps]: "a *s (x - y) = a *s x - a *s y"
and scale_sum_right: "a *s (sum f A) = (\<Sum>x\<in>A. a *s (f x))"
proof -
interpret s: additive "\<lambda>x. a *s x"
by standard (rule scale_right_distrib)
show "a *s 0 = 0" by (rule s.zero)
show "a *s (- x) = - (a *s x)" by (rule s.minus)
show "a *s (x - y) = a *s x - a *s y" by (rule s.diff)
show "a *s (sum f A) = (\<Sum>x\<in>A. a *s (f x))" by (rule s.sum)
qed
lemma sum_constant_scale: "(\<Sum>x\<in>A. y) = scale (of_nat (card A)) y"
by (induct A rule: infinite_finite_induct) (simp_all add: algebra_simps)
section \<open>Subspace\<close>
definition subspace :: "'b set \<Rightarrow> bool"
where "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *s x \<in> S)"
lemma subspaceI:
"0 \<in> S \<Longrightarrow> (\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S) \<Longrightarrow> (\<And>c x. x \<in> S \<Longrightarrow> c *s x \<in> S) \<Longrightarrow> subspace S"
by (auto simp: subspace_def)
lemma subspace_UNIV[simp]: "subspace UNIV"
by (simp add: subspace_def)
lemma subspace_single_0[simp]: "subspace {0}"
by (simp add: subspace_def)
lemma subspace_0: "subspace S \<Longrightarrow> 0 \<in> S"
by (metis subspace_def)
lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x + y \<in> S"
by (metis subspace_def)
lemma subspace_scale: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
by (metis subspace_def)
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
by (metis scale_minus_left scale_one subspace_scale)
lemma subspace_diff: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
by (metis diff_conv_add_uminus subspace_add subspace_neg)
lemma subspace_sum: "subspace A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<in> A) \<Longrightarrow> sum f B \<in> A"
by (induct B rule: infinite_finite_induct) (auto simp add: subspace_add subspace_0)
lemma subspace_Int: "(\<And>i. i \<in> I \<Longrightarrow> subspace (s i)) \<Longrightarrow> subspace (\<Inter>i\<in>I. s i)"
by (auto simp: subspace_def)
lemma subspace_Inter: "\<forall>s \<in> f. subspace s \<Longrightarrow> subspace (\<Inter>f)"
unfolding subspace_def by auto
lemma subspace_inter: "subspace A \<Longrightarrow> subspace B \<Longrightarrow> subspace (A \<inter> B)"
by (simp add: subspace_def)
section \<open>Span: subspace generated by a set\<close>
definition span :: "'b set \<Rightarrow> 'b set"
where span_explicit: "span b = {(\<Sum>a\<in>t. r a *s a) | t r. finite t \<and> t \<subseteq> b}"
lemma span_explicit':
"span b = {(\<Sum>v | f v \<noteq> 0. f v *s v) | f. finite {v. f v \<noteq> 0} \<and> (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> b)}"
unfolding span_explicit
proof safe
fix t r assume "finite t" "t \<subseteq> b"
then show "\<exists>f. (\<Sum>a\<in>t. r a *s a) = (\<Sum>v | f v \<noteq> 0. f v *s v) \<and> finite {v. f v \<noteq> 0} \<and> (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> b)"
by (intro exI[of _ "\<lambda>v. if v \<in> t then r v else 0"]) (auto intro!: sum.mono_neutral_cong_right)
next
fix f :: "'b \<Rightarrow> 'a" assume "finite {v. f v \<noteq> 0}" "(\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> b)"
then show "\<exists>t r. (\<Sum>v | f v \<noteq> 0. f v *s v) = (\<Sum>a\<in>t. r a *s a) \<and> finite t \<and> t \<subseteq> b"
by (intro exI[of _ "{v. f v \<noteq> 0}"] exI[of _ f]) auto
qed
lemma span_alt:
"span B = {(\<Sum>x | f x \<noteq> 0. f x *s x) | f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}"
unfolding span_explicit' by auto
lemma span_finite:
assumes fS: "finite S"
shows "span S = range (\<lambda>u. \<Sum>v\<in>S. u v *s v)"
unfolding span_explicit
proof safe
fix t r assume "t \<subseteq> S" then show "(\<Sum>a\<in>t. r a *s a) \<in> range (\<lambda>u. \<Sum>v\<in>S. u v *s v)"
by (intro image_eqI[of _ _ "\<lambda>a. if a \<in> t then r a else 0"])
(auto simp: if_distrib[of "\<lambda>r. r *s a" for a] sum.If_cases fS Int_absorb1)
next
show "\<exists>t r. (\<Sum>v\<in>S. u v *s v) = (\<Sum>a\<in>t. r a *s a) \<and> finite t \<and> t \<subseteq> S" for u
by (intro exI[of _ u] exI[of _ S]) (auto intro: fS)
qed
lemma span_induct_alt [consumes 1, case_names base step, induct set: span]:
assumes x: "x \<in> span S"
assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *s x + y)"
shows "h x"
using x unfolding span_explicit
proof safe
fix t r assume "finite t" "t \<subseteq> S" then show "h (\<Sum>a\<in>t. r a *s a)"
by (induction t) (auto intro!: hS h0)
qed
lemma span_mono: "A \<subseteq> B \<Longrightarrow> span A \<subseteq> span B"
by (auto simp: span_explicit)
lemma span_base: "a \<in> S \<Longrightarrow> a \<in> span S"
by (auto simp: span_explicit intro!: exI[of _ "{a}"] exI[of _ "\<lambda>_. 1"])
lemma span_superset: "S \<subseteq> span S"
by (auto simp: span_base)
lemma span_zero: "0 \<in> span S"
by (auto simp: span_explicit intro!: exI[of _ "{}"])
lemma span_UNIV[simp]: "span UNIV = UNIV"
by (auto intro: span_base)
lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x + y \<in> span S"
unfolding span_explicit
proof safe
fix tx ty rx ry assume *: "finite tx" "finite ty" "tx \<subseteq> S" "ty \<subseteq> S"
have [simp]: "(tx \<union> ty) \<inter> tx = tx" "(tx \<union> ty) \<inter> ty = ty"
by auto
show "\<exists>t r. (\<Sum>a\<in>tx. rx a *s a) + (\<Sum>a\<in>ty. ry a *s a) = (\<Sum>a\<in>t. r a *s a) \<and> finite t \<and> t \<subseteq> S"
apply (intro exI[of _ "tx \<union> ty"])
apply (intro exI[of _ "\<lambda>a. (if a \<in> tx then rx a else 0) + (if a \<in> ty then ry a else 0)"])
apply (auto simp: * scale_left_distrib sum.distrib if_distrib[of "\<lambda>r. r *s a" for a] sum.If_cases)
done
qed
lemma span_scale: "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
unfolding span_explicit
proof safe
fix t r assume *: "finite t" "t \<subseteq> S"
show "\<exists>t' r'. c *s (\<Sum>a\<in>t. r a *s a) = (\<Sum>a\<in>t'. r' a *s a) \<and> finite t' \<and> t' \<subseteq> S"
by (intro exI[of _ t] exI[of _ "\<lambda>a. c * r a"]) (auto simp: * scale_sum_right)
qed
lemma subspace_span [iff]: "subspace (span S)"
by (auto simp: subspace_def span_zero span_add span_scale)
lemma span_neg: "x \<in> span S \<Longrightarrow> - x \<in> span S"
by (metis subspace_neg subspace_span)
lemma span_diff: "x \<in> span S \<Longrightarrow> y \<in> span S \<Longrightarrow> x - y \<in> span S"
by (metis subspace_span subspace_diff)
lemma span_sum: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> span S) \<Longrightarrow> sum f A \<in> span S"
by (rule subspace_sum, rule subspace_span)
lemma span_minimal: "S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> span S \<subseteq> T"
by (auto simp: span_explicit intro!: subspace_sum subspace_scale)
lemma span_def: "span S = subspace hull S"
by (intro hull_unique[symmetric] span_superset subspace_span span_minimal)
lemma span_unique:
"S \<subseteq> T \<Longrightarrow> subspace T \<Longrightarrow> (\<And>T'. S \<subseteq> T' \<Longrightarrow> subspace T' \<Longrightarrow> T \<subseteq> T') \<Longrightarrow> span S = T"
unfolding span_def by (rule hull_unique)
lemma span_subspace_induct[consumes 2]:
assumes x: "x \<in> span S"
and P: "subspace P"
and SP: "\<And>x. x \<in> S \<Longrightarrow> x \<in> P"
shows "x \<in> P"
proof -
from SP have SP': "S \<subseteq> P"
by (simp add: subset_eq)
from x hull_minimal[where S=subspace, OF SP' P, unfolded span_def[symmetric]]
show "x \<in> P"
by (metis subset_eq)
qed
lemma (in module) span_induct[consumes 1, case_names base step, induct set: span]:
assumes x: "x \<in> span S"
and P: "subspace (Collect P)"
and SP: "\<And>x. x \<in> S \<Longrightarrow> P x"
shows "P x"
using P SP span_subspace_induct x by fastforce
lemma span_empty[simp]: "span {} = {0}"
by (rule span_unique) (auto simp add: subspace_def)
lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<subseteq> span A \<Longrightarrow> subspace B \<Longrightarrow> span A = B"
by (metis order_antisym span_def hull_minimal)
lemma span_span: "span (span A) = span A"
unfolding span_def hull_hull ..
(* TODO: proof generally for subspace: *)
lemma span_add_eq: assumes x: "x \<in> span S" shows "x + y \<in> span S \<longleftrightarrow> y \<in> span S"
proof
assume *: "x + y \<in> span S"
have "(x + y) - x \<in> span S" using * x by (rule span_diff)
then show "y \<in> span S" by simp
qed (intro span_add x)
lemma span_add_eq2: assumes y: "y \<in> span S" shows "x + y \<in> span S \<longleftrightarrow> x \<in> span S"
using span_add_eq[of y S x] y by (auto simp: ac_simps)
lemma span_singleton: "span {x} = range (\<lambda>k. k *s x)"
by (auto simp: span_finite)
lemma span_Un: "span (S \<union> T) = {x + y | x y. x \<in> span S \<and> y \<in> span T}"
proof safe
fix x assume "x \<in> span (S \<union> T)"
then obtain t r where t: "finite t" "t \<subseteq> S \<union> T" and x: "x = (\<Sum>a\<in>t. r a *s a)"
by (auto simp: span_explicit)
moreover have "t \<inter> S \<union> (t - S) = t" by auto
ultimately show "\<exists>xa y. x = xa + y \<and> xa \<in> span S \<and> y \<in> span T"
unfolding x
apply (rule_tac exI[of _ "\<Sum>a\<in>t \<inter> S. r a *s a"])
apply (rule_tac exI[of _ "\<Sum>a\<in>t - S. r a *s a"])
apply (subst sum.union_inter_neutral[symmetric])
apply (auto intro!: span_sum span_scale intro: span_base)
done
next
fix x y assume"x \<in> span S" "y \<in> span T" then show "x + y \<in> span (S \<union> T)"
using span_mono[of S "S \<union> T"] span_mono[of T "S \<union> T"]
by (auto intro!: span_add)
qed
lemma span_insert: "span (insert a S) = {x. \<exists>k. (x - k *s a) \<in> span S}"
proof -
have "span ({a} \<union> S) = {x. \<exists>k. (x - k *s a) \<in> span S}"
unfolding span_Un span_singleton
apply (auto simp add: set_eq_iff)
subgoal for y k by (auto intro!: exI[of _ "k"])
subgoal for y k by (rule exI[of _ "k *s a"], rule exI[of _ "y - k *s a"]) auto
done
then show ?thesis by simp
qed
lemma span_breakdown:
assumes bS: "b \<in> S"
and aS: "a \<in> span S"
shows "\<exists>k. a - k *s b \<in> span (S - {b})"
using assms span_insert [of b "S - {b}"]
by (simp add: insert_absorb)
lemma span_breakdown_eq: "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. x - k *s a \<in> span S)"
by (simp add: span_insert)
lemmas span_clauses = span_base span_zero span_add span_scale
lemma span_eq_iff[simp]: "span s = s \<longleftrightarrow> subspace s"
unfolding span_def by (rule hull_eq) (rule subspace_Inter)
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
by (metis span_minimal span_subspace span_superset subspace_span)
lemma eq_span_insert_eq:
assumes "(x - y) \<in> span S"
shows "span(insert x S) = span(insert y S)"
proof -
have *: "span(insert x S) \<subseteq> span(insert y S)" if "(x - y) \<in> span S" for x y
proof -
have 1: "(r *s x - r *s y) \<in> span S" for r
by (metis scale_right_diff_distrib span_scale that)
have 2: "(z - k *s y) - k *s (x - y) = z - k *s x" for z k
by (simp add: scale_right_diff_distrib)
show ?thesis
apply (clarsimp simp add: span_breakdown_eq)
by (metis 1 2 diff_add_cancel scale_right_diff_distrib span_add_eq)
qed
show ?thesis
apply (intro subset_antisym * assms)
using assms subspace_neg subspace_span minus_diff_eq by force
qed
section \<open>Dependent and independent sets\<close>
definition dependent :: "'b set \<Rightarrow> bool"
where dependent_explicit: "dependent s \<longleftrightarrow> (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *s v) = 0 \<and> (\<exists>v\<in>t. u v \<noteq> 0))"
abbreviation "independent s \<equiv> \<not> dependent s"
lemma dependent_mono: "dependent B \<Longrightarrow> B \<subseteq> A \<Longrightarrow> dependent A"
by (auto simp add: dependent_explicit)
lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> independent B"
by (auto intro: dependent_mono)
lemma dependent_zero: "0 \<in> A \<Longrightarrow> dependent A"
by (auto simp: dependent_explicit intro!: exI[of _ "\<lambda>i. 1"] exI[of _ "{0}"])
lemma independent_empty[intro]: "independent {}"
by (simp add: dependent_explicit)
lemma independent_explicit_module:
"independent s \<longleftrightarrow> (\<forall>t u v. finite t \<longrightarrow> t \<subseteq> s \<longrightarrow> (\<Sum>v\<in>t. u v *s v) = 0 \<longrightarrow> v \<in> t \<longrightarrow> u v = 0)"
unfolding dependent_explicit by auto
lemma independentD: "independent s \<Longrightarrow> finite t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (\<Sum>v\<in>t. u v *s v) = 0 \<Longrightarrow> v \<in> t \<Longrightarrow> u v = 0"
by (simp add: independent_explicit_module)
lemma independent_Union_directed:
assumes directed: "\<And>c d. c \<in> C \<Longrightarrow> d \<in> C \<Longrightarrow> c \<subseteq> d \<or> d \<subseteq> c"
assumes indep: "\<And>c. c \<in> C \<Longrightarrow> independent c"
shows "independent (\<Union>C)"
proof
assume "dependent (\<Union>C)"
then obtain u v S where S: "finite S" "S \<subseteq> \<Union>C" "v \<in> S" "u v \<noteq> 0" "(\<Sum>v\<in>S. u v *s v) = 0"
by (auto simp: dependent_explicit)
have "S \<noteq> {}"
using \<open>v \<in> S\<close> by auto
have "\<exists>c\<in>C. S \<subseteq> c"
using \<open>finite S\<close> \<open>S \<noteq> {}\<close> \<open>S \<subseteq> \<Union>C\<close>
proof (induction rule: finite_ne_induct)
case (insert i I)
then obtain c d where cd: "c \<in> C" "d \<in> C" and iI: "I \<subseteq> c" "i \<in> d"
by blast
from directed[OF cd] cd have "c \<union> d \<in> C"
by (auto simp: sup.absorb1 sup.absorb2)
with iI show ?case
by (intro bexI[of _ "c \<union> d"]) auto
qed auto
then obtain c where "c \<in> C" "S \<subseteq> c"
by auto
have "dependent c"
unfolding dependent_explicit
by (intro exI[of _ S] exI[of _ u] bexI[of _ v] conjI) fact+
with indep[OF \<open>c \<in> C\<close>] show False
by auto
qed
lemma dependent_finite:
assumes "finite S"
shows "dependent S \<longleftrightarrow> (\<exists>u. (\<exists>v \<in> S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *s v) = 0)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain T u v
where "finite T" "T \<subseteq> S" "v\<in>T" "u v \<noteq> 0" "(\<Sum>v\<in>T. u v *s v) = 0"
by (force simp: dependent_explicit)
with assms show ?rhs
apply (rule_tac x="\<lambda>v. if v \<in> T then u v else 0" in exI)
apply (auto simp: sum.mono_neutral_right)
done
next
assume ?rhs with assms show ?lhs
by (fastforce simp add: dependent_explicit)
qed
lemma dependent_alt:
"dependent B \<longleftrightarrow>
(\<exists>X. finite {x. X x \<noteq> 0} \<and> {x. X x \<noteq> 0} \<subseteq> B \<and> (\<Sum>x|X x \<noteq> 0. X x *s x) = 0 \<and> (\<exists>x. X x \<noteq> 0))"
unfolding dependent_explicit
apply safe
subgoal for S u v
apply (intro exI[of _ "\<lambda>x. if x \<in> S then u x else 0"])
apply (subst sum.mono_neutral_cong_left[where T=S])
apply (auto intro!: sum.mono_neutral_cong_right cong: rev_conj_cong)
done
apply auto
done
lemma independent_alt:
"independent B \<longleftrightarrow>
(\<forall>X. finite {x. X x \<noteq> 0} \<longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *s x) = 0 \<longrightarrow> (\<forall>x. X x = 0))"
unfolding dependent_alt by auto
lemma independentD_alt:
"independent B \<Longrightarrow> finite {x. X x \<noteq> 0} \<Longrightarrow> {x. X x \<noteq> 0} \<subseteq> B \<Longrightarrow> (\<Sum>x|X x \<noteq> 0. X x *s x) = 0 \<Longrightarrow> X x = 0"
unfolding independent_alt by blast
lemma independentD_unique:
assumes B: "independent B"
and X: "finite {x. X x \<noteq> 0}" "{x. X x \<noteq> 0} \<subseteq> B"
and Y: "finite {x. Y x \<noteq> 0}" "{x. Y x \<noteq> 0} \<subseteq> B"
and "(\<Sum>x | X x \<noteq> 0. X x *s x) = (\<Sum>x| Y x \<noteq> 0. Y x *s x)"
shows "X = Y"
proof -
have "X x - Y x = 0" for x
using B
proof (rule independentD_alt)
have "{x. X x - Y x \<noteq> 0} \<subseteq> {x. X x \<noteq> 0} \<union> {x. Y x \<noteq> 0}"
by auto
then show "finite {x. X x - Y x \<noteq> 0}" "{x. X x - Y x \<noteq> 0} \<subseteq> B"
using X Y by (auto dest: finite_subset)
then have "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *s x) = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. (X v - Y v) *s v)"
using X Y by (intro sum.mono_neutral_cong_left) auto
also have "\<dots> = (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *s v) - (\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *s v)"
by (simp add: scale_left_diff_distrib sum_subtractf assms)
also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. X v *s v) = (\<Sum>v\<in>{S. X S \<noteq> 0}. X v *s v)"
using X Y by (intro sum.mono_neutral_cong_right) auto
also have "(\<Sum>v\<in>{S. X S \<noteq> 0} \<union> {S. Y S \<noteq> 0}. Y v *s v) = (\<Sum>v\<in>{S. Y S \<noteq> 0}. Y v *s v)"
using X Y by (intro sum.mono_neutral_cong_right) auto
finally show "(\<Sum>x | X x - Y x \<noteq> 0. (X x - Y x) *s x) = 0"
using assms by simp
qed
then show ?thesis
by auto
qed
section \<open>Representation of a vector on a specific basis\<close>
definition representation :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"
where "representation basis v =
(if independent basis \<and> v \<in> span basis then
SOME f. (\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> 0} \<and> (\<Sum>v\<in>{v. f v \<noteq> 0}. f v *s v) = v
else (\<lambda>b. 0))"
lemma unique_representation:
assumes basis: "independent basis"
and in_basis: "\<And>v. f v \<noteq> 0 \<Longrightarrow> v \<in> basis" "\<And>v. g v \<noteq> 0 \<Longrightarrow> v \<in> basis"
and [simp]: "finite {v. f v \<noteq> 0}" "finite {v. g v \<noteq> 0}"
and eq: "(\<Sum>v\<in>{v. f v \<noteq> 0}. f v *s v) = (\<Sum>v\<in>{v. g v \<noteq> 0}. g v *s v)"
shows "f = g"
proof (rule ext, rule ccontr)
fix v assume ne: "f v \<noteq> g v"
have "dependent basis"
unfolding dependent_explicit
proof (intro exI conjI)
have *: "{v. f v - g v \<noteq> 0} \<subseteq> {v. f v \<noteq> 0} \<union> {v. g v \<noteq> 0}"
by auto
show "finite {v. f v - g v \<noteq> 0}"
by (rule finite_subset[OF *]) simp
show "\<exists>v\<in>{v. f v - g v \<noteq> 0}. f v - g v \<noteq> 0"
by (rule bexI[of _ v]) (auto simp: ne)
have "(\<Sum>v | f v - g v \<noteq> 0. (f v - g v) *s v) =
(\<Sum>v\<in>{v. f v \<noteq> 0} \<union> {v. g v \<noteq> 0}. (f v - g v) *s v)"
by (intro sum.mono_neutral_cong_left *) auto
also have "... =
(\<Sum>v\<in>{v. f v \<noteq> 0} \<union> {v. g v \<noteq> 0}. f v *s v) - (\<Sum>v\<in>{v. f v \<noteq> 0} \<union> {v. g v \<noteq> 0}. g v *s v)"
by (simp add: algebra_simps sum_subtractf)
also have "... = (\<Sum>v | f v \<noteq> 0. f v *s v) - (\<Sum>v | g v \<noteq> 0. g v *s v)"
by (intro arg_cong2[where f= "(-)"] sum.mono_neutral_cong_right) auto
finally show "(\<Sum>v | f v - g v \<noteq> 0. (f v - g v) *s v) = 0"
by (simp add: eq)
show "{v. f v - g v \<noteq> 0} \<subseteq> basis"
using in_basis * by auto
qed
with basis show False by auto
qed
lemma
shows representation_ne_zero: "\<And>b. representation basis v b \<noteq> 0 \<Longrightarrow> b \<in> basis"
and finite_representation: "finite {b. representation basis v b \<noteq> 0}"
and sum_nonzero_representation_eq:
"independent basis \<Longrightarrow> v \<in> span basis \<Longrightarrow> (\<Sum>b | representation basis v b \<noteq> 0. representation basis v b *s b) = v"
proof -
{ assume basis: "independent basis" and v: "v \<in> span basis"
define p where "p f \<longleftrightarrow>
(\<forall>v. f v \<noteq> 0 \<longrightarrow> v \<in> basis) \<and> finite {v. f v \<noteq> 0} \<and> (\<Sum>v\<in>{v. f v \<noteq> 0}. f v *s v) = v" for f
obtain t r where *: "finite t" "t \<subseteq> basis" "(\<Sum>b\<in>t. r b *s b) = v"
using \<open>v \<in> span basis\<close> by (auto simp: span_explicit)
define f where "f b = (if b \<in> t then r b else 0)" for b
have "p f"
using * by (auto simp: p_def f_def intro!: sum.mono_neutral_cong_left)
have *: "representation basis v = Eps p" by (simp add: p_def[abs_def] representation_def basis v)
from someI[of p f, OF \<open>p f\<close>] have "p (representation basis v)"
unfolding * . }
note * = this
show "representation basis v b \<noteq> 0 \<Longrightarrow> b \<in> basis" for b
using * by (cases "independent basis \<and> v \<in> span basis") (auto simp: representation_def)
show "finite {b. representation basis v b \<noteq> 0}"
using * by (cases "independent basis \<and> v \<in> span basis") (auto simp: representation_def)
show "independent basis \<Longrightarrow> v \<in> span basis \<Longrightarrow> (\<Sum>b | representation basis v b \<noteq> 0. representation basis v b *s b) = v"
using * by auto
qed
lemma sum_representation_eq:
"(\<Sum>b\<in>B. representation basis v b *s b) = v"
if "independent basis" "v \<in> span basis" "finite B" "basis \<subseteq> B"
proof -
have "(\<Sum>b\<in>B. representation basis v b *s b) =
(\<Sum>b | representation basis v b \<noteq> 0. representation basis v b *s b)"
apply (rule sum.mono_neutral_cong)
apply (rule finite_representation)
apply fact
subgoal for b
using that representation_ne_zero[of basis v b]
by auto
subgoal by auto
subgoal by simp
done
also have "\<dots> = v"
by (rule sum_nonzero_representation_eq; fact)
finally show ?thesis .
qed
lemma representation_eqI:
assumes basis: "independent basis" and b: "v \<in> span basis"
and ne_zero: "\<And>b. f b \<noteq> 0 \<Longrightarrow> b \<in> basis"
and finite: "finite {b. f b \<noteq> 0}"
and eq: "(\<Sum>b | f b \<noteq> 0. f b *s b) = v"
shows "representation basis v = f"
by (rule unique_representation[OF basis])
(auto simp: representation_ne_zero finite_representation
sum_nonzero_representation_eq[OF basis b] ne_zero finite eq)
lemma representation_basis:
assumes basis: "independent basis" and b: "b \<in> basis"
shows "representation basis b = (\<lambda>v. if v = b then 1 else 0)"
proof (rule unique_representation[OF basis])
show "representation basis b v \<noteq> 0 \<Longrightarrow> v \<in> basis" for v
using representation_ne_zero .
show "finite {v. representation basis b v \<noteq> 0}"
using finite_representation .
show "(if v = b then 1 else 0) \<noteq> 0 \<Longrightarrow> v \<in> basis" for v
by (cases "v = b") (auto simp: b)
have *: "{v. (if v = b then 1 else 0 :: 'a) \<noteq> 0} = {b}"
by auto
show "finite {v. (if v = b then 1 else 0) \<noteq> 0}" unfolding * by auto
show "(\<Sum>v | representation basis b v \<noteq> 0. representation basis b v *s v) =
(\<Sum>v | (if v = b then 1 else 0::'a) \<noteq> 0. (if v = b then 1 else 0) *s v)"
unfolding * sum_nonzero_representation_eq[OF basis span_base[OF b]] by auto
qed
lemma representation_zero: "representation basis 0 = (\<lambda>b. 0)"
proof cases
assume basis: "independent basis" show ?thesis
by (rule representation_eqI[OF basis span_zero]) auto
qed (simp add: representation_def)
lemma representation_diff:
assumes basis: "independent basis" and v: "v \<in> span basis" and u: "u \<in> span basis"
shows "representation basis (u - v) = (\<lambda>b. representation basis u b - representation basis v b)"
proof (rule representation_eqI[OF basis span_diff[OF u v]])
let ?R = "representation basis"
note finite_representation[simp] u[simp] v[simp]
have *: "{b. ?R u b - ?R v b \<noteq> 0} \<subseteq> {b. ?R u b \<noteq> 0} \<union> {b. ?R v b \<noteq> 0}"
by auto
then show "?R u b - ?R v b \<noteq> 0 \<Longrightarrow> b \<in> basis" for b
by (auto dest: representation_ne_zero)
show "finite {b. ?R u b - ?R v b \<noteq> 0}"
by (intro finite_subset[OF *]) simp_all
have "(\<Sum>b | ?R u b - ?R v b \<noteq> 0. (?R u b - ?R v b) *s b) =
(\<Sum>b\<in>{b. ?R u b \<noteq> 0} \<union> {b. ?R v b \<noteq> 0}. (?R u b - ?R v b) *s b)"
by (intro sum.mono_neutral_cong_left *) auto
also have "... =
(\<Sum>b\<in>{b. ?R u b \<noteq> 0} \<union> {b. ?R v b \<noteq> 0}. ?R u b *s b) - (\<Sum>b\<in>{b. ?R u b \<noteq> 0} \<union> {b. ?R v b \<noteq> 0}. ?R v b *s b)"
by (simp add: algebra_simps sum_subtractf)
also have "... = (\<Sum>b | ?R u b \<noteq> 0. ?R u b *s b) - (\<Sum>b | ?R v b \<noteq> 0. ?R v b *s b)"
by (intro arg_cong2[where f= "(-)"] sum.mono_neutral_cong_right) auto
finally show "(\<Sum>b | ?R u b - ?R v b \<noteq> 0. (?R u b - ?R v b) *s b) = u - v"
by (simp add: sum_nonzero_representation_eq[OF basis])
qed
lemma representation_neg:
"independent basis \<Longrightarrow> v \<in> span basis \<Longrightarrow> representation basis (- v) = (\<lambda>b. - representation basis v b)"
using representation_diff[of basis v 0] by (simp add: representation_zero span_zero)
lemma representation_add:
"independent basis \<Longrightarrow> v \<in> span basis \<Longrightarrow> u \<in> span basis \<Longrightarrow>
representation basis (u + v) = (\<lambda>b. representation basis u b + representation basis v b)"
using representation_diff[of basis "-v" u] by (simp add: representation_neg representation_diff span_neg)
lemma representation_sum:
"independent basis \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> v i \<in> span basis) \<Longrightarrow>
representation basis (sum v I) = (\<lambda>b. \<Sum>i\<in>I. representation basis (v i) b)"
by (induction I rule: infinite_finite_induct)
(auto simp: representation_zero representation_add span_sum)
lemma representation_scale:
assumes basis: "independent basis" and v: "v \<in> span basis"
shows "representation basis (r *s v) = (\<lambda>b. r * representation basis v b)"
proof (rule representation_eqI[OF basis span_scale[OF v]])
let ?R = "representation basis"
note finite_representation[simp] v[simp]
have *: "{b. r * ?R v b \<noteq> 0} \<subseteq> {b. ?R v b \<noteq> 0}"
by auto
then show "r * representation basis v b \<noteq> 0 \<Longrightarrow> b \<in> basis" for b
using representation_ne_zero by auto
show "finite {b. r * ?R v b \<noteq> 0}"
by (intro finite_subset[OF *]) simp_all
have "(\<Sum>b | r * ?R v b \<noteq> 0. (r * ?R v b) *s b) = (\<Sum>b\<in>{b. ?R v b \<noteq> 0}. (r * ?R v b) *s b)"
by (intro sum.mono_neutral_cong_left *) auto
also have "... = r *s (\<Sum>b | ?R v b \<noteq> 0. ?R v b *s b)"
by (simp add: scale_scale[symmetric] scale_sum_right del: scale_scale)
finally show "(\<Sum>b | r * ?R v b \<noteq> 0. (r * ?R v b) *s b) = r *s v"
by (simp add: sum_nonzero_representation_eq[OF basis])
qed
lemma representation_extend:
assumes basis: "independent basis" and v: "v \<in> span basis'" and basis': "basis' \<subseteq> basis"
shows "representation basis v = representation basis' v"
proof (rule representation_eqI[OF basis])
show v': "v \<in> span basis" using span_mono[OF basis'] v by auto
have *: "independent basis'" using basis' basis by (auto intro: dependent_mono)
show "representation basis' v b \<noteq> 0 \<Longrightarrow> b \<in> basis" for b
using representation_ne_zero basis' by auto
show "finite {b. representation basis' v b \<noteq> 0}"
using finite_representation .
show "(\<Sum>b | representation basis' v b \<noteq> 0. representation basis' v b *s b) = v"
using sum_nonzero_representation_eq[OF * v] .
qed
text \<open>The set \<open>B\<close> is the maximal independent set for \<open>span B\<close>, or \<open>A\<close> is the minimal spanning set\<close>
lemma spanning_subset_independent:
assumes BA: "B \<subseteq> A"
and iA: "independent A"
and AsB: "A \<subseteq> span B"
shows "A = B"
proof (intro antisym[OF _ BA] subsetI)
have iB: "independent B" using independent_mono [OF iA BA] .
fix v assume "v \<in> A"
with AsB have "v \<in> span B" by auto
let ?RB = "representation B v" and ?RA = "representation A v"
have "?RB v = 1"
unfolding representation_extend[OF iA \<open>v \<in> span B\<close> BA, symmetric] representation_basis[OF iA \<open>v \<in> A\<close>] by simp
then show "v \<in> B"
using representation_ne_zero[of B v v] by auto
qed
end
(* We need to introduce more specific modules, where the ring structure gets more and more finer,
i.e. Bezout rings & domains, division rings, fields *)
text \<open>A linear function is a mapping between two modules over the same ring.\<close>
locale module_hom = m1: module s1 + m2: module s2
for s1 :: "'a::comm_ring_1 \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b" (infixr "*a" 75)
and s2 :: "'a::comm_ring_1 \<Rightarrow> 'c::ab_group_add \<Rightarrow> 'c" (infixr "*b" 75) +
fixes f :: "'b \<Rightarrow> 'c"
assumes add: "f (b1 + b2) = f b1 + f b2"
and scale: "f (r *a b) = r *b f b"
begin
lemma zero[simp]: "f 0 = 0"
using scale[of 0 0] by simp
lemma neg: "f (- x) = - f x"
using scale [where r="-1"] by (metis add add_eq_0_iff zero)
lemma diff: "f (x - y) = f x - f y"
by (metis diff_conv_add_uminus add neg)
lemma sum: "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
proof (induct S rule: infinite_finite_induct)
case (insert x F)
have "f (sum g (insert x F)) = f (g x + sum g F)"
using insert.hyps by simp
also have "\<dots> = f (g x) + f (sum g F)"
using add by simp
also have "\<dots> = (\<Sum>a\<in>insert x F. f (g a))"
using insert.hyps by simp
finally show ?case .
qed simp_all
lemma inj_on_iff_eq_0:
assumes s: "m1.subspace s"
shows "inj_on f s \<longleftrightarrow> (\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0)"
proof -
have "inj_on f s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. f x - f y = 0 \<longrightarrow> x - y = 0)"
by (simp add: inj_on_def)
also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. f (x - y) = 0 \<longrightarrow> x - y = 0)"
by (simp add: diff)
also have "\<dots> \<longleftrightarrow> (\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0)" (is "?l = ?r")(* TODO: sledgehammer! *)
proof safe
fix x assume ?l assume "x \<in> s" "f x = 0" with \<open>?l\<close>[rule_format, of x 0] s show "x = 0"
by (auto simp: m1.subspace_0)
next
fix x y assume ?r assume "x \<in> s" "y \<in> s" "f (x - y) = 0"
with \<open>?r\<close>[rule_format, of "x - y"] s
show "x - y = 0"
by (auto simp: m1.subspace_diff)
qed
finally show ?thesis
by auto
qed
lemma inj_iff_eq_0: "inj f = (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
by (rule inj_on_iff_eq_0[OF m1.subspace_UNIV, unfolded ball_UNIV])
lemma subspace_image: assumes S: "m1.subspace S" shows "m2.subspace (f ` S)"
unfolding m2.subspace_def
proof safe
show "0 \<in> f ` S"
by (rule image_eqI[of _ _ 0]) (auto simp: S m1.subspace_0)
show "x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> f x + f y \<in> f ` S" for x y
by (rule image_eqI[of _ _ "x + y"]) (auto simp: S m1.subspace_add add)
show "x \<in> S \<Longrightarrow> r *b f x \<in> f ` S" for r x
by (rule image_eqI[of _ _ "r *a x"]) (auto simp: S m1.subspace_scale scale)
qed
lemma subspace_vimage: "m2.subspace S \<Longrightarrow> m1.subspace (f -` S)"
by (simp add: vimage_def add scale m1.subspace_def m2.subspace_0 m2.subspace_add m2.subspace_scale)
lemma subspace_kernel: "m1.subspace {x. f x = 0}"
using subspace_vimage[OF m2.subspace_single_0] by (simp add: vimage_def)
lemma span_image: "m2.span (f ` S) = f ` (m1.span S)"
proof (rule m2.span_unique)
show "f ` S \<subseteq> f ` m1.span S"
by (rule image_mono, rule m1.span_superset)
show "m2.subspace (f ` m1.span S)"
using m1.subspace_span by (rule subspace_image)
next
fix T assume "f ` S \<subseteq> T" and "m2.subspace T" then show "f ` m1.span S \<subseteq> T"
unfolding image_subset_iff_subset_vimage by (metis subspace_vimage m1.span_minimal)
qed
lemma dependent_inj_imageD:
assumes d: "m2.dependent (f ` s)" and i: "inj_on f (m1.span s)"
shows "m1.dependent s"
proof -
have [intro]: "inj_on f s"
using \<open>inj_on f (m1.span s)\<close> m1.span_superset by (rule inj_on_subset)
from d obtain s' r v where *: "finite s'" "s' \<subseteq> s" "(\<Sum>v\<in>f ` s'. r v *b v) = 0" "v \<in> s'" "r (f v) \<noteq> 0"
by (auto simp: m2.dependent_explicit subset_image_iff dest!: finite_imageD intro: inj_on_subset)
have "f (\<Sum>v\<in>s'. r (f v) *a v) = (\<Sum>v\<in>s'. r (f v) *b f v)"
by (simp add: sum scale)
also have "... = (\<Sum>v\<in>f ` s'. r v *b v)"
using \<open>s' \<subseteq> s\<close> by (subst sum.reindex) (auto dest!: finite_imageD intro: inj_on_subset)
finally have "f (\<Sum>v\<in>s'. r (f v) *a v) = 0"
by (simp add: *)
with \<open>s' \<subseteq> s\<close> have "(\<Sum>v\<in>s'. r (f v) *a v) = 0"
by (intro inj_onD[OF i] m1.span_zero m1.span_sum m1.span_scale) (auto intro: m1.span_base)
then show "m1.dependent s"
using \<open>finite s'\<close> \<open>s' \<subseteq> s\<close> \<open>v \<in> s'\<close> \<open>r (f v) \<noteq> 0\<close> by (force simp add: m1.dependent_explicit)
qed
lemma eq_0_on_span:
assumes f0: "\<And>x. x \<in> b \<Longrightarrow> f x = 0" and x: "x \<in> m1.span b" shows "f x = 0"
using m1.span_induct[OF x subspace_kernel] f0 by simp
lemma independent_injective_image: "m1.independent s \<Longrightarrow> inj_on f (m1.span s) \<Longrightarrow> m2.independent (f ` s)"
using dependent_inj_imageD[of s] by auto
lemma inj_on_span_independent_image:
assumes ifB: "m2.independent (f ` B)" and f: "inj_on f B" shows "inj_on f (m1.span B)"
unfolding inj_on_iff_eq_0[OF m1.subspace_span] unfolding m1.span_explicit'
proof safe
fix r assume fr: "finite {v. r v \<noteq> 0}" and r: "\<forall>v. r v \<noteq> 0 \<longrightarrow> v \<in> B"
and eq0: "f (\<Sum>v | r v \<noteq> 0. r v *a v) = 0"
have "0 = (\<Sum>v | r v \<noteq> 0. r v *b f v)"
using eq0 by (simp add: sum scale)
also have "... = (\<Sum>v\<in>f ` {v. r v \<noteq> 0}. r (the_inv_into B f v) *b v)"
using r by (subst sum.reindex) (auto simp: the_inv_into_f_f[OF f] intro!: inj_on_subset[OF f] sum.cong)
finally have "r v \<noteq> 0 \<Longrightarrow> r (the_inv_into B f (f v)) = 0" for v
using fr r ifB[unfolded m2.independent_explicit_module, rule_format,
of "f ` {v. r v \<noteq> 0}" "\<lambda>v. r (the_inv_into B f v)"]
by auto
then have "r v = 0" for v
using the_inv_into_f_f[OF f] r by auto
then show "(\<Sum>v | r v \<noteq> 0. r v *a v) = 0" by auto
qed
lemma inj_on_span_iff_independent_image: "m2.independent (f ` B) \<Longrightarrow> inj_on f (m1.span B) \<longleftrightarrow> inj_on f B"
using inj_on_span_independent_image[of B] inj_on_subset[OF _ m1.span_superset, of f B] by auto
lemma subspace_linear_preimage: "m2.subspace S \<Longrightarrow> m1.subspace {x. f x \<in> S}"
by (simp add: add scale m1.subspace_def m2.subspace_def)
lemma spans_image: "V \<subseteq> m1.span B \<Longrightarrow> f ` V \<subseteq> m2.span (f ` B)"
by (metis image_mono span_image)
text \<open>Relation between bases and injectivity/surjectivity of map.\<close>
lemma spanning_surjective_image:
assumes us: "UNIV \<subseteq> m1.span S"
and sf: "surj f"
shows "UNIV \<subseteq> m2.span (f ` S)"
proof -
have "UNIV \<subseteq> f ` UNIV"
using sf by (auto simp add: surj_def)
also have " \<dots> \<subseteq> m2.span (f ` S)"
using spans_image[OF us] .
finally show ?thesis .
qed
lemmas independent_inj_on_image = independent_injective_image
lemma independent_inj_image:
"m1.independent S \<Longrightarrow> inj f \<Longrightarrow> m2.independent (f ` S)"
using independent_inj_on_image[of S] by (auto simp: subset_inj_on)
end
lemma module_hom_iff:
"module_hom s1 s2 f \<longleftrightarrow>
module s1 \<and> module s2 \<and>
(\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (s1 c x) = s2 c (f x))"
by (simp add: module_hom_def module_hom_axioms_def)
locale module_pair = m1: module s1 + m2: module s2
for s1 :: "'a :: comm_ring_1 \<Rightarrow> 'b \<Rightarrow> 'b :: ab_group_add"
and s2 :: "'a :: comm_ring_1 \<Rightarrow> 'c \<Rightarrow> 'c :: ab_group_add"
begin
lemma module_hom_zero: "module_hom s1 s2 (\<lambda>x. 0)"
by (simp add: module_hom_iff m1.module_axioms m2.module_axioms)
lemma module_hom_add: "module_hom s1 s2 f \<Longrightarrow> module_hom s1 s2 g \<Longrightarrow> module_hom s1 s2 (\<lambda>x. f x + g x)"
by (simp add: module_hom_iff module.scale_right_distrib)
lemma module_hom_sub: "module_hom s1 s2 f \<Longrightarrow> module_hom s1 s2 g \<Longrightarrow> module_hom s1 s2 (\<lambda>x. f x - g x)"
by (simp add: module_hom_iff module.scale_right_diff_distrib)
lemma module_hom_neg: "module_hom s1 s2 f \<Longrightarrow> module_hom s1 s2 (\<lambda>x. - f x)"
by (simp add: module_hom_iff module.scale_minus_right)
lemma module_hom_scale: "module_hom s1 s2 f \<Longrightarrow> module_hom s1 s2 (\<lambda>x. s2 c (f x))"
by (simp add: module_hom_iff module.scale_scale module.scale_right_distrib ac_simps)
lemma module_hom_compose_scale:
"module_hom s1 s2 (\<lambda>x. s2 (f x) (c))"
if "module_hom s1 (*) f"
proof -
interpret mh: module_hom s1 "(*)" f by fact
show ?thesis
by unfold_locales (simp_all add: mh.add mh.scale m2.scale_left_distrib)
qed
lemma bij_module_hom_imp_inv_module_hom: "module_hom scale1 scale2 f \<Longrightarrow> bij f \<Longrightarrow>
module_hom scale2 scale1 (inv f)"
by (auto simp: module_hom_iff bij_is_surj bij_is_inj surj_f_inv_f
intro!: Hilbert_Choice.inv_f_eq)
lemma module_hom_sum: "(\<And>i. i \<in> I \<Longrightarrow> module_hom s1 s2 (f i)) \<Longrightarrow> (I = {} \<Longrightarrow> module s1 \<and> module s2) \<Longrightarrow> module_hom s1 s2 (\<lambda>x. \<Sum>i\<in>I. f i x)"
apply (induction I rule: infinite_finite_induct)
apply (auto intro!: module_hom_zero module_hom_add)
using m1.module_axioms m2.module_axioms by blast
lemma module_hom_eq_on_span: "f x = g x"
if "module_hom s1 s2 f" "module_hom s1 s2 g"
and "(\<And>x. x \<in> B \<Longrightarrow> f x = g x)" "x \<in> m1.span B"
proof -
interpret module_hom s1 s2 "\<lambda>x. f x - g x"
by (rule module_hom_sub that)+
from eq_0_on_span[OF _ that(4)] that(3) show ?thesis by auto
qed
end
context module begin
lemma module_hom_scale_self[simp]:
"module_hom scale scale (\<lambda>x. scale c x)"
using module_axioms module_hom_iff scale_left_commute scale_right_distrib by blast
lemma module_hom_scale_left[simp]:
"module_hom (*) scale (\<lambda>r. scale r x)"
by unfold_locales (auto simp: algebra_simps)
lemma module_hom_id: "module_hom scale scale id"
by (simp add: module_hom_iff module_axioms)
lemma module_hom_ident: "module_hom scale scale (\<lambda>x. x)"
by (simp add: module_hom_iff module_axioms)
lemma module_hom_uminus: "module_hom scale scale uminus"
by (simp add: module_hom_iff module_axioms)
end
lemma module_hom_compose: "module_hom s1 s2 f \<Longrightarrow> module_hom s2 s3 g \<Longrightarrow> module_hom s1 s3 (g o f)"
by (auto simp: module_hom_iff)
end