(* Title: HOL/Algebra/Product_Groups.thy Author: LC Paulson (ported from HOL Light)*)section \<open>Product and Sum Groups\<close>theory Product_Groups imports Elementary_Groups "HOL-Library.Equipollence" beginsubsection \<open>Product of a Family of Groups\<close>definition product_group:: "'a set \<Rightarrow> ('a \<Rightarrow> ('b, 'c) monoid_scheme) \<Rightarrow> ('a \<Rightarrow> 'b) monoid" where "product_group I G \<equiv> \<lparr>carrier = (\<Pi>\<^sub>E i\<in>I. carrier (G i)), monoid.mult = (\<lambda>x y. (\<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i)), one = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)\<rparr>"lemma carrier_product_group [simp]: "carrier(product_group I G) = (\<Pi>\<^sub>E i\<in>I. carrier (G i))" by (simp add: product_group_def)lemma one_product_group [simp]: "one(product_group I G) = (\<lambda>i\<in>I. one (G i))" by (simp add: product_group_def)lemma mult_product_group [simp]: "(\<otimes>\<^bsub>product_group I G\<^esub>) = (\<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i)" by (simp add: product_group_def)lemma product_group [simp]: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "group (product_group I G)"proof (rule groupI; simp) show "(\<lambda>i. x i \<otimes>\<^bsub>G i\<^esub> y i) \<in> (\<Pi> i\<in>I. carrier (G i))" if "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" "y \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" for x y using that assms group.subgroup_self subgroup.m_closed by fastforce show "(\<lambda>i. \<one>\<^bsub>G i\<^esub>) \<in> (\<Pi> i\<in>I. carrier (G i))" by (simp add: assms group.is_monoid) show "(\<lambda>i\<in>I. (if i \<in> I then x i \<otimes>\<^bsub>G i\<^esub> y i else undefined) \<otimes>\<^bsub>G i\<^esub> z i) = (\<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> (if i \<in> I then y i \<otimes>\<^bsub>G i\<^esub> z i else undefined))" if "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" "y \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" "z \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" for x y z using that by (auto simp: PiE_iff assms group.is_monoid monoid.m_assoc intro: restrict_ext) show "(\<lambda>i\<in>I. (if i \<in> I then \<one>\<^bsub>G i\<^esub> else undefined) \<otimes>\<^bsub>G i\<^esub> x i) = x" if "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" for x using assms that by (fastforce simp: Group.group_def PiE_iff) show "\<exists>y\<in>\<Pi>\<^sub>E i\<in>I. carrier (G i). (\<lambda>i\<in>I. y i \<otimes>\<^bsub>G i\<^esub> x i) = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)" if "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" for x by (rule_tac x="\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> x i" in bexI) (use assms that in \<open>auto simp: PiE_iff group.l_inv\<close>)qedlemma inv_product_group [simp]: assumes "f \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "inv\<^bsub>product_group I G\<^esub> f = (\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> f i)"proof (rule group.inv_equality) show "Group.group (product_group I G)" by (simp add: assms) show "(\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> f i) \<otimes>\<^bsub>product_group I G\<^esub> f = \<one>\<^bsub>product_group I G\<^esub>" using assms by (auto simp: PiE_iff group.l_inv) show "f \<in> carrier (product_group I G)" using assms by simp show "(\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> f i) \<in> carrier (product_group I G)" using PiE_mem assms by fastforceqedlemma trivial_product_group: "trivial_group(product_group I G) \<longleftrightarrow> (\<forall>i \<in> I. trivial_group(G i))" (is "?lhs = ?rhs")proof assume L: ?lhs then have "inv\<^bsub>product_group I G\<^esub> (\<lambda>a\<in>I. \<one>\<^bsub>G a\<^esub>) = \<one>\<^bsub>product_group I G\<^esub>" by (metis group.is_monoid monoid.inv_one one_product_group trivial_group_def) have [simp]: "\<one>\<^bsub>G i\<^esub> \<otimes>\<^bsub>G i\<^esub> \<one>\<^bsub>G i\<^esub> = \<one>\<^bsub>G i\<^esub>" if "i \<in> I" for i unfolding trivial_group_def proof - have 1: "(\<lambda>a\<in>I. \<one>\<^bsub>G a\<^esub>) i = \<one>\<^bsub>G i\<^esub>" by (simp add: that) have "(\<lambda>a\<in>I. \<one>\<^bsub>G a\<^esub>) = (\<lambda>a\<in>I. \<one>\<^bsub>G a\<^esub>) \<otimes>\<^bsub>product_group I G\<^esub> (\<lambda>a\<in>I. \<one>\<^bsub>G a\<^esub>)" by (metis (no_types) L group.is_monoid monoid.l_one one_product_group singletonI trivial_group_def) then show ?thesis using 1 by (simp add: that) qed show ?rhs using L by (auto simp: trivial_group_def product_group_def PiE_eq_singleton intro: groupI)next assume ?rhs then show ?lhs by (simp add: PiE_eq_singleton trivial_group_def)qedlemma PiE_subgroup_product_group: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "subgroup (PiE I H) (product_group I G) \<longleftrightarrow> (\<forall>i \<in> I. subgroup (H i) (G i))" (is "?lhs = ?rhs")proof assume L: ?lhs then have [simp]: "PiE I H \<noteq> {}" using subgroup_nonempty by force show ?rhs proof (clarify; unfold_locales) show sub: "H i \<subseteq> carrier (G i)" if "i \<in> I" for i using that L by (simp add: subgroup_def) (metis (no_types, lifting) L subgroup_nonempty subset_PiE) show "x \<otimes>\<^bsub>G i\<^esub> y \<in> H i" if "i \<in> I" "x \<in> H i" "y \<in> H i" for i x y proof - have *: "\<And>x. x \<in> Pi\<^sub>E I H \<Longrightarrow> (\<forall>y \<in> Pi\<^sub>E I H. \<forall>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i \<in> H i)" using L by (auto simp: subgroup_def Pi_iff) have "\<forall>y\<in>H i. f i \<otimes>\<^bsub>G i\<^esub> y \<in> H i" if f: "f \<in> Pi\<^sub>E I H" and "i \<in> I" for i f using * [OF f] \<open>i \<in> I\<close> by (subst(asm) all_PiE_elements) auto then have "\<forall>f \<in> Pi\<^sub>E I H. \<forall>i \<in> I. \<forall>y\<in>H i. f i \<otimes>\<^bsub>G i\<^esub> y \<in> H i" by blast with that show ?thesis by (subst(asm) all_PiE_elements) auto qed show "\<one>\<^bsub>G i\<^esub> \<in> H i" if "i \<in> I" for i using L subgroup.one_closed that by fastforce show "inv\<^bsub>G i\<^esub> x \<in> H i" if "i \<in> I" and x: "x \<in> H i" for i x proof - have *: "\<forall>y \<in> Pi\<^sub>E I H. \<forall>i\<in>I. inv\<^bsub>G i\<^esub> y i \<in> H i" proof fix y assume y: "y \<in> Pi\<^sub>E I H" then have yc: "y \<in> carrier (product_group I G)" by (metis (no_types) L subgroup_def subsetCE) have "inv\<^bsub>product_group I G\<^esub> y \<in> Pi\<^sub>E I H" by (simp add: y L subgroup.m_inv_closed) moreover have "inv\<^bsub>product_group I G\<^esub> y = (\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> y i)" using yc by (simp add: assms) ultimately show "\<forall>i\<in>I. inv\<^bsub>G i\<^esub> y i \<in> H i" by auto qed then have "\<forall>i\<in>I. \<forall>x\<in>H i. inv\<^bsub>G i\<^esub> x \<in> H i" by (subst(asm) all_PiE_elements) auto then show ?thesis using that(1) x by blast qed qednext assume R: ?rhs show ?lhs proof show "Pi\<^sub>E I H \<subseteq> carrier (product_group I G)" using R by (force simp: subgroup_def) show "x \<otimes>\<^bsub>product_group I G\<^esub> y \<in> Pi\<^sub>E I H" if "x \<in> Pi\<^sub>E I H" "y \<in> Pi\<^sub>E I H" for x y using R that by (auto simp: PiE_iff subgroup_def) show "\<one>\<^bsub>product_group I G\<^esub> \<in> Pi\<^sub>E I H" using R by (force simp: subgroup_def) show "inv\<^bsub>product_group I G\<^esub> x \<in> Pi\<^sub>E I H" if "x \<in> Pi\<^sub>E I H" for x proof - have x: "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" using R that by (force simp: subgroup_def) show ?thesis using assms R that by (fastforce simp: x assms subgroup_def) qed qedqedlemma product_group_subgroup_generated: assumes "\<And>i. i \<in> I \<Longrightarrow> subgroup (H i) (G i)" and gp: "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "product_group I (\<lambda>i. subgroup_generated (G i) (H i)) = subgroup_generated (product_group I G) (PiE I H)"proof (rule monoid.equality) have [simp]: "\<And>i. i \<in> I \<Longrightarrow> carrier (G i) \<inter> H i = H i" "(\<Pi>\<^sub>E i\<in>I. carrier (G i)) \<inter> Pi\<^sub>E I H = Pi\<^sub>E I H" using assms by (force simp: subgroup_def)+ have "(\<Pi>\<^sub>E i\<in>I. generate (G i) (H i)) = generate (product_group I G) (Pi\<^sub>E I H)" proof (rule group.generateI) show "Group.group (product_group I G)" using assms by simp show "subgroup (\<Pi>\<^sub>E i\<in>I. generate (G i) (H i)) (product_group I G)" using assms by (simp add: PiE_subgroup_product_group group.generate_is_subgroup subgroup.subset) show "Pi\<^sub>E I H \<subseteq> (\<Pi>\<^sub>E i\<in>I. generate (G i) (H i))" using assms by (auto simp: PiE_iff generate.incl) show "(\<Pi>\<^sub>E i\<in>I. generate (G i) (H i)) \<subseteq> K" if "subgroup K (product_group I G)" "Pi\<^sub>E I H \<subseteq> K" for K using assms that group.generate_subgroup_incl by fastforce qed with assms show "carrier (product_group I (\<lambda>i. subgroup_generated (G i) (H i))) = carrier (subgroup_generated (product_group I G) (Pi\<^sub>E I H))" by (simp add: carrier_subgroup_generated cong: PiE_cong)qed autolemma finite_product_group: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "finite (carrier (product_group I G)) \<longleftrightarrow> finite {i. i \<in> I \<and> ~ trivial_group(G i)} \<and> (\<forall>i \<in> I. finite(carrier(G i)))"proof - have [simp]: "\<And>i. i \<in> I \<Longrightarrow> carrier (G i) \<noteq> {}" using assms group.is_monoid by blast show ?thesis by (auto simp: finite_PiE_iff PiE_eq_empty_iff group.trivial_group_alt [OF assms] cong: Collect_cong conj_cong)qedsubsection \<open>Sum of a Family of Groups\<close>definition sum_group :: "'a set \<Rightarrow> ('a \<Rightarrow> ('b, 'c) monoid_scheme) \<Rightarrow> ('a \<Rightarrow> 'b) monoid" where "sum_group I G \<equiv> subgroup_generated (product_group I G) {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"lemma subgroup_sum_group: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "subgroup {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}} (product_group I G)"proof unfold_locales fix x y have *: "{i. (i \<in> I \<longrightarrow> x i \<otimes>\<^bsub>G i\<^esub> y i \<noteq> \<one>\<^bsub>G i\<^esub>) \<and> i \<in> I} \<subseteq> {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>} \<union> {i \<in> I. y i \<noteq> \<one>\<^bsub>G i\<^esub>}" by (auto simp: Group.group_def dest: assms) assume "x \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" "y \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" then show "x \<otimes>\<^bsub>product_group I G\<^esub> y \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" using assms apply (auto simp: Group.group_def monoid.m_closed PiE_iff) apply (rule finite_subset [OF *]) by blastnext fix x assume "x \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" then show "inv\<^bsub>product_group I G\<^esub> x \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" using assms by (auto simp: PiE_iff assms group.inv_eq_1_iff [OF assms] conj_commute cong: rev_conj_cong)qed (use assms [unfolded Group.group_def] in auto)lemma carrier_sum_group: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "carrier(sum_group I G) = {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"proof - interpret SG: subgroup "{x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" "(product_group I G)" by (simp add: assms subgroup_sum_group) show ?thesis by (simp add: sum_group_def subgroup_sum_group carrier_subgroup_generated_alt)qedlemma one_sum_group [simp]: "\<one>\<^bsub>sum_group I G\<^esub> = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)" by (simp add: sum_group_def)lemma mult_sum_group [simp]: "(\<otimes>\<^bsub>sum_group I G\<^esub>) = (\<lambda>x y. (\<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i))" by (auto simp: sum_group_def)lemma sum_group [simp]: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" shows "group (sum_group I G)"proof (rule groupI) note group.is_monoid [OF assms, simp] show "x \<otimes>\<^bsub>sum_group I G\<^esub> y \<in> carrier (sum_group I G)" if "x \<in> carrier (sum_group I G)" and "y \<in> carrier (sum_group I G)" for x y proof - have *: "{i \<in> I. x i \<otimes>\<^bsub>G i\<^esub> y i \<noteq> \<one>\<^bsub>G i\<^esub>} \<subseteq> {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>} \<union> {i \<in> I. y i \<noteq> \<one>\<^bsub>G i\<^esub>}" by auto show ?thesis using that apply (simp add: assms carrier_sum_group PiE_iff monoid.m_closed conj_commute cong: rev_conj_cong) apply (blast intro: finite_subset [OF *]) done qed show "\<one>\<^bsub>sum_group I G\<^esub> \<otimes>\<^bsub>sum_group I G\<^esub> x = x" if "x \<in> carrier (sum_group I G)" for x using that by (auto simp: assms carrier_sum_group PiE_iff extensional_def) show "\<exists>y\<in>carrier (sum_group I G). y \<otimes>\<^bsub>sum_group I G\<^esub> x = \<one>\<^bsub>sum_group I G\<^esub>" if "x \<in> carrier (sum_group I G)" for x proof let ?y = "\<lambda>i\<in>I. m_inv (G i) (x i)" show "?y \<otimes>\<^bsub>sum_group I G\<^esub> x = \<one>\<^bsub>sum_group I G\<^esub>" using that assms by (auto simp: carrier_sum_group PiE_iff group.l_inv) show "?y \<in> carrier (sum_group I G)" using that assms by (auto simp: carrier_sum_group PiE_iff group.inv_eq_1_iff group.l_inv cong: conj_cong) qedqed (auto simp: assms carrier_sum_group PiE_iff group.is_monoid monoid.m_assoc)lemma inv_sum_group [simp]: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" and x: "x \<in> carrier (sum_group I G)" shows "m_inv (sum_group I G) x = (\<lambda>i\<in>I. m_inv (G i) (x i))"proof (rule group.inv_equality) show "(\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> x i) \<otimes>\<^bsub>sum_group I G\<^esub> x = \<one>\<^bsub>sum_group I G\<^esub>" using x by (auto simp: carrier_sum_group PiE_iff group.l_inv assms intro: restrict_ext) show "(\<lambda>i\<in>I. inv\<^bsub>G i\<^esub> x i) \<in> carrier (sum_group I G)" using x by (simp add: carrier_sum_group PiE_iff group.inv_eq_1_iff assms conj_commute cong: rev_conj_cong)qed (auto simp: assms)thm group.subgroups_Inter (*REPLACE*)theorem subgroup_Inter: assumes subgr: "(\<And>H. H \<in> A \<Longrightarrow> subgroup H G)" and not_empty: "A \<noteq> {}" shows "subgroup (\<Inter>A) G"proof show "\<Inter> A \<subseteq> carrier G" by (simp add: Inf_less_eq not_empty subgr subgroup.subset)qed (auto simp: subgr subgroup.m_closed subgroup.one_closed subgroup.m_inv_closed)thm group.subgroups_Inter_pair (*REPLACE*)lemma subgroup_Int: assumes "subgroup I G" "subgroup J G" shows "subgroup (I \<inter> J) G" using subgroup_Inter[ where ?A = "{I,J}"] assms by autolemma sum_group_subgroup_generated: assumes "\<And>i. i \<in> I \<Longrightarrow> group (G i)" and sg: "\<And>i. i \<in> I \<Longrightarrow> subgroup (H i) (G i)" shows "sum_group I (\<lambda>i. subgroup_generated (G i) (H i)) = subgroup_generated (sum_group I G) (PiE I H)"proof (rule monoid.equality) have "subgroup (carrier (sum_group I G) \<inter> Pi\<^sub>E I H) (product_group I G)" by (rule subgroup_Int) (auto simp: assms carrier_sum_group subgroup_sum_group PiE_subgroup_product_group) moreover have "carrier (sum_group I G) \<inter> Pi\<^sub>E I H \<subseteq> carrier (subgroup_generated (product_group I G) {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}})" by (simp add: assms subgroup_sum_group subgroup.carrier_subgroup_generated_subgroup carrier_sum_group) ultimately have "subgroup (carrier (sum_group I G) \<inter> Pi\<^sub>E I H) (sum_group I G)" by (simp add: assms sum_group_def group.subgroup_subgroup_generated_iff) then have *: "{f \<in> \<Pi>\<^sub>E i\<in>I. carrier (subgroup_generated (G i) (H i)). finite {i \<in> I. f i \<noteq> \<one>\<^bsub>G i\<^esub>}} = carrier (subgroup_generated (sum_group I G) (carrier (sum_group I G) \<inter> Pi\<^sub>E I H))" apply (simp only: subgroup.carrier_subgroup_generated_subgroup) using subgroup.subset [OF sg] apply (auto simp: set_eq_iff PiE_def Pi_def assms carrier_sum_group subgroup.carrier_subgroup_generated_subgroup) done then show "carrier (sum_group I (\<lambda>i. subgroup_generated (G i) (H i))) = carrier (subgroup_generated (sum_group I G) (Pi\<^sub>E I H))" by simp (simp add: assms group.subgroupE(1) group.group_subgroup_generated carrier_sum_group)qed (auto simp: sum_group_def subgroup_generated_def)lemma iso_product_groupI: assumes iso: "\<And>i. i \<in> I \<Longrightarrow> G i \<cong> H i" and G: "\<And>i. i \<in> I \<Longrightarrow> group (G i)" and H: "\<And>i. i \<in> I \<Longrightarrow> group (H i)" shows "product_group I G \<cong> product_group I H" (is "?IG \<cong> ?IH")proof - have "\<And>i. i \<in> I \<Longrightarrow> \<exists>h. h \<in> iso (G i) (H i)" using iso by (auto simp: is_iso_def) then obtain f where f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> iso (G i) (H i)" by metis define h where "h \<equiv> \<lambda>x. (\<lambda>i\<in>I. f i (x i))" have hom: "h \<in> iso ?IG ?IH" proof (rule isoI) show hom: "h \<in> hom ?IG ?IH" proof (rule homI) fix x assume "x \<in> carrier ?IG" with f show "h x \<in> carrier ?IH" using PiE by (fastforce simp add: h_def PiE_def iso_def hom_def) next fix x y assume "x \<in> carrier ?IG" "y \<in> carrier ?IG" with f show "h (x \<otimes>\<^bsub>?IG\<^esub> y) = h x \<otimes>\<^bsub>?IH\<^esub> h y" apply (simp add: h_def PiE_def iso_def hom_def) using PiE by (fastforce simp add: h_def PiE_def iso_def hom_def intro: restrict_ext) qed with G H interpret GH : group_hom "?IG" "?IH" h by (simp add: group_hom_def group_hom_axioms_def) show "bij_betw h (carrier ?IG) (carrier ?IH)" unfolding bij_betw_def proof (intro conjI subset_antisym) have "\<gamma> i = \<one>\<^bsub>G i\<^esub>" if \<gamma>: "\<gamma> \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" and eq: "(\<lambda>i\<in>I. f i (\<gamma> i)) = (\<lambda>i\<in>I. \<one>\<^bsub>H i\<^esub>)" and "i \<in> I" for \<gamma> i proof - have "inj_on (f i) (carrier (G i))" "f i \<in> hom (G i) (H i)" using \<open>i \<in> I\<close> f by (auto simp: iso_def bij_betw_def) then have *: "\<And>x. \<lbrakk>f i x = \<one>\<^bsub>H i\<^esub>; x \<in> carrier (G i)\<rbrakk> \<Longrightarrow> x = \<one>\<^bsub>G i\<^esub>" by (metis G Group.group_def H hom_one inj_onD monoid.one_closed \<open>i \<in> I\<close>) show ?thesis using eq \<open>i \<in> I\<close> * \<gamma> by (simp add: fun_eq_iff) (meson PiE_iff) qed then show "inj_on h (carrier ?IG)" apply (simp add: iso_def bij_betw_def GH.inj_on_one_iff flip: carrier_product_group) apply (force simp: h_def) done next show "h ` carrier ?IG \<subseteq> carrier ?IH" unfolding h_def using f by (force simp: PiE_def Pi_def Group.iso_def dest!: bij_betwE) next show "carrier ?IH \<subseteq> h ` carrier ?IG" unfolding h_def proof (clarsimp simp: iso_def bij_betw_def) fix x assume "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (H i))" with f have x: "x \<in> (\<Pi>\<^sub>E i\<in>I. f i ` carrier (G i))" unfolding h_def by (auto simp: iso_def bij_betw_def) have "\<And>i. i \<in> I \<Longrightarrow> inj_on (f i) (carrier (G i))" using f by (auto simp: iso_def bij_betw_def) let ?g = "\<lambda>i\<in>I. inv_into (carrier (G i)) (f i) (x i)" show "x \<in> (\<lambda>g. \<lambda>i\<in>I. f i (g i)) ` (\<Pi>\<^sub>E i\<in>I. carrier (G i))" proof show "x = (\<lambda>i\<in>I. f i (?g i))" using x by (auto simp: PiE_iff fun_eq_iff extensional_def f_inv_into_f) show "?g \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" using x by (auto simp: PiE_iff inv_into_into) qed qed qed qed then show ?thesis using is_iso_def by autoqedlemma iso_sum_groupI: assumes iso: "\<And>i. i \<in> I \<Longrightarrow> G i \<cong> H i" and G: "\<And>i. i \<in> I \<Longrightarrow> group (G i)" and H: "\<And>i. i \<in> I \<Longrightarrow> group (H i)" shows "sum_group I G \<cong> sum_group I H" (is "?IG \<cong> ?IH")proof - have "\<And>i. i \<in> I \<Longrightarrow> \<exists>h. h \<in> iso (G i) (H i)" using iso by (auto simp: is_iso_def) then obtain f where f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> iso (G i) (H i)" by metis then have injf: "inj_on (f i) (carrier (G i))" and homf: "f i \<in> hom (G i) (H i)" if "i \<in> I" for i using \<open>i \<in> I\<close> f by (auto simp: iso_def bij_betw_def) then have one: "\<And>x. \<lbrakk>f i x = \<one>\<^bsub>H i\<^esub>; x \<in> carrier (G i)\<rbrakk> \<Longrightarrow> x = \<one>\<^bsub>G i\<^esub>" if "i \<in> I" for i by (metis G H group.subgroup_self hom_one inj_on_eq_iff subgroup.one_closed that) have fin1: "finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>} \<Longrightarrow> finite {i \<in> I. f i (x i) \<noteq> \<one>\<^bsub>H i\<^esub>}" for x using homf by (auto simp: G H hom_one elim!: rev_finite_subset) define h where "h \<equiv> \<lambda>x. (\<lambda>i\<in>I. f i (x i))" have hom: "h \<in> iso ?IG ?IH" proof (rule isoI) show hom: "h \<in> hom ?IG ?IH" proof (rule homI) fix x assume "x \<in> carrier ?IG" with f fin1 show "h x \<in> carrier ?IH" by (force simp: h_def PiE_def iso_def hom_def carrier_sum_group assms conj_commute cong: conj_cong) next fix x y assume "x \<in> carrier ?IG" "y \<in> carrier ?IG" with homf show "h (x \<otimes>\<^bsub>?IG\<^esub> y) = h x \<otimes>\<^bsub>?IH\<^esub> h y" by (fastforce simp add: h_def PiE_def hom_def carrier_sum_group assms intro: restrict_ext) qed with G H interpret GH : group_hom "?IG" "?IH" h by (simp add: group_hom_def group_hom_axioms_def) show "bij_betw h (carrier ?IG) (carrier ?IH)" unfolding bij_betw_def proof (intro conjI subset_antisym) have \<gamma>: "\<gamma> i = \<one>\<^bsub>G i\<^esub>" if "\<gamma> \<in> (\<Pi>\<^sub>E i\<in>I. carrier (G i))" and eq: "(\<lambda>i\<in>I. f i (\<gamma> i)) = (\<lambda>i\<in>I. \<one>\<^bsub>H i\<^esub>)" and "i \<in> I" for \<gamma> i using \<open>i \<in> I\<close> one that by (simp add: fun_eq_iff) (meson PiE_iff) show "inj_on h (carrier ?IG)" apply (simp add: iso_def bij_betw_def GH.inj_on_one_iff assms one flip: carrier_sum_group) apply (auto simp: h_def fun_eq_iff carrier_sum_group assms PiE_def Pi_def extensional_def one) done next show "h ` carrier ?IG \<subseteq> carrier ?IH" using homf GH.hom_closed by (fastforce simp: h_def PiE_def Pi_def dest!: bij_betwE) next show "carrier ?IH \<subseteq> h ` carrier ?IG" unfolding h_def proof (clarsimp simp: iso_def bij_betw_def carrier_sum_group assms) fix x assume x: "x \<in> (\<Pi>\<^sub>E i\<in>I. carrier (H i))" and fin: "finite {i \<in> I. x i \<noteq> \<one>\<^bsub>H i\<^esub>}" with f have xf: "x \<in> (\<Pi>\<^sub>E i\<in>I. f i ` carrier (G i))" unfolding h_def by (auto simp: iso_def bij_betw_def) have "\<And>i. i \<in> I \<Longrightarrow> inj_on (f i) (carrier (G i))" using f by (auto simp: iso_def bij_betw_def) let ?g = "\<lambda>i\<in>I. inv_into (carrier (G i)) (f i) (x i)" show "x \<in> (\<lambda>g. \<lambda>i\<in>I. f i (g i)) ` {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" proof show xeq: "x = (\<lambda>i\<in>I. f i (?g i))" using x by (clarsimp simp: PiE_iff fun_eq_iff extensional_def) (metis iso_iff f_inv_into_f f) have "finite {i \<in> I. inv_into (carrier (G i)) (f i) (x i) \<noteq> \<one>\<^bsub>G i\<^esub>}" apply (rule finite_subset [OF _ fin]) using G H group.subgroup_self hom_one homf injf inv_into_f_eq subgroup.one_closed by fastforce with x show "?g \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" apply (auto simp: PiE_iff inv_into_into conj_commute cong: conj_cong) by (metis (no_types, opaque_lifting) iso_iff f inv_into_into) qed qed qed qed then show ?thesis using is_iso_def by autoqedend