--- a/src/HOL/Algebra/Exact_Sequence.thy Wed Apr 03 14:55:30 2019 +0100
+++ b/src/HOL/Algebra/Exact_Sequence.thy Wed Apr 03 15:14:36 2019 +0100
@@ -1,12 +1,13 @@
(* Title: HOL/Algebra/Exact_Sequence.thy
- Author: Martin Baillon
+ Author: Martin Baillon (first part) and LC Paulson (material ported from HOL Light)
*)
+section \<open>Exact Sequences\<close>
+
theory Exact_Sequence
- imports Group Coset Solvable_Groups
+ imports Elementary_Groups Solvable_Groups
begin
-section \<open>Exact Sequences\<close>
subsection \<open>Definitions\<close>
@@ -16,6 +17,9 @@
extension: "\<lbrakk> exact_seq ((G # K # l), (g # q)); group H ; h \<in> hom G H ;
kernel G H h = image g (carrier K) \<rbrakk> \<Longrightarrow> exact_seq (H # G # K # l, h # g # q)"
+inductive_simps exact_seq_end_iff [simp]: "exact_seq ([G,H], (g # q))"
+inductive_simps exact_seq_cons_iff [simp]: "exact_seq ((G # K # H # l), (g # h # q))"
+
abbreviation exact_seq_arrow ::
"('a \<Rightarrow> 'a) \<Rightarrow> 'a monoid list \<times> ('a \<Rightarrow> 'a) list \<Rightarrow> 'a monoid \<Rightarrow> 'a monoid list \<times> ('a \<Rightarrow> 'a) list"
("(3_ / \<longlongrightarrow>\<index> _)" [1000, 60])
@@ -173,4 +177,295 @@
shows "(solvable G1) \<and> (solvable G3) \<longleftrightarrow> solvable G2"
using exact_seq_solvable_recip exact_seq_solvable_imp assms by blast
+
+lemma exact_seq_eq_triviality:
+ assumes "exact_seq ([E,D,C,B,A], [k,h,g,f])"
+ shows "trivial_group C \<longleftrightarrow> f ` carrier A = carrier B \<and> inj_on k (carrier D)" (is "_ = ?rhs")
+proof
+ assume C: "trivial_group C"
+ with assms have "inj_on k (carrier D)"
+ apply (auto simp: group_hom.image_from_trivial_group trivial_group_def hom_one)
+ apply (simp add: group_hom_def group_hom_axioms_def group_hom.inj_iff_trivial_ker)
+ done
+ with assms C show ?rhs
+ apply (auto simp: group_hom.image_from_trivial_group trivial_group_def hom_one)
+ apply (auto simp: group_hom_def group_hom_axioms_def hom_def kernel_def)
+ done
+next
+ assume ?rhs
+ with assms show "trivial_group C"
+ apply (simp add: trivial_group_def)
+ by (metis group_hom.inj_iff_trivial_ker group_hom.trivial_hom_iff group_hom_axioms.intro group_hom_def)
+qed
+
+lemma exact_seq_imp_triviality:
+ "\<lbrakk>exact_seq ([E,D,C,B,A], [k,h,g,f]); f \<in> iso A B; k \<in> iso D E\<rbrakk> \<Longrightarrow> trivial_group C"
+ by (metis (no_types, lifting) Group.iso_def bij_betw_def exact_seq_eq_triviality mem_Collect_eq)
+
+lemma exact_seq_epi_eq_triviality:
+ "exact_seq ([D,C,B,A], [h,g,f]) \<Longrightarrow> (f ` carrier A = carrier B) \<longleftrightarrow> trivial_homomorphism B C g"
+ by (auto simp: trivial_homomorphism_def kernel_def)
+
+lemma exact_seq_mon_eq_triviality:
+ "exact_seq ([D,C,B,A], [h,g,f]) \<Longrightarrow> inj_on h (carrier C) \<longleftrightarrow> trivial_homomorphism B C g"
+ by (auto simp: trivial_homomorphism_def kernel_def group.is_monoid inj_on_one_iff' image_def) blast
+
+lemma exact_sequence_sum_lemma:
+ assumes "comm_group G" and h: "h \<in> iso A C" and k: "k \<in> iso B D"
+ and ex: "exact_seq ([D,G,A], [g,i])" "exact_seq ([C,G,B], [f,j])"
+ and fih: "\<And>x. x \<in> carrier A \<Longrightarrow> f(i x) = h x"
+ and gjk: "\<And>x. x \<in> carrier B \<Longrightarrow> g(j x) = k x"
+ shows "(\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) \<in> Group.iso (A \<times>\<times> B) G \<and> (\<lambda>z. (f z, g z)) \<in> Group.iso G (C \<times>\<times> D)"
+ (is "?ij \<in> _ \<and> ?gf \<in> _")
+proof (rule epi_iso_compose_rev)
+ interpret comm_group G
+ by (rule assms)
+ interpret f: group_hom G C f
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ interpret g: group_hom G D g
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ interpret i: group_hom A G i
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ interpret j: group_hom B G j
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ have kerf: "kernel G C f = j ` carrier B" and "group A" "group B" "i \<in> hom A G"
+ using ex by (auto simp: group_hom_def group_hom_axioms_def)
+ then obtain h' where "h' \<in> hom C A" "(\<forall>x \<in> carrier A. h'(h x) = x)"
+ and hh': "(\<forall>y \<in> carrier C. h(h' y) = y)" and "group_isomorphisms A C h h'"
+ using h by (auto simp: group.iso_iff_group_isomorphisms group_isomorphisms_def)
+ have homij: "?ij \<in> hom (A \<times>\<times> B) G"
+ unfolding case_prod_unfold
+ apply (rule hom_group_mult)
+ using ex by (simp_all add: group_hom_def hom_of_fst [unfolded o_def] hom_of_snd [unfolded o_def])
+ show homgf: "?gf \<in> hom G (C \<times>\<times> D)"
+ using ex by (simp add: hom_paired)
+ show "?ij \<in> epi (A \<times>\<times> B) G"
+ proof (clarsimp simp add: epi_iff_subset homij)
+ fix x
+ assume x: "x \<in> carrier G"
+ with \<open>i \<in> hom A G\<close> \<open>h' \<in> hom C A\<close> have "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub>(i(h'(f x))) \<in> kernel G C f"
+ by (simp add: kernel_def hom_in_carrier hh' fih)
+ with kerf obtain y where y: "y \<in> carrier B" "j y = x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub>(i(h'(f x)))"
+ by auto
+ have "i (h' (f x)) \<otimes>\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> i (h' (f x))) = x \<otimes>\<^bsub>G\<^esub> (i (h' (f x)) \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> i (h' (f x)))"
+ by (meson \<open>h' \<in> hom C A\<close> x f.hom_closed hom_in_carrier i.hom_closed inv_closed m_lcomm)
+ also have "\<dots> = x"
+ using \<open>h' \<in> hom C A\<close> hom_in_carrier x by fastforce
+ finally show "x \<in> (\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) ` (carrier A \<times> carrier B)"
+ using x y apply (clarsimp simp: image_def)
+ apply (rule_tac x="h'(f x)" in bexI)
+ apply (rule_tac x=y in bexI, auto)
+ by (meson \<open>h' \<in> hom C A\<close> f.hom_closed hom_in_carrier)
+ qed
+ show "(\<lambda>z. (f z, g z)) \<circ> (\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) \<in> Group.iso (A \<times>\<times> B) (C \<times>\<times> D)"
+ apply (rule group.iso_eq [where f = "\<lambda>(x,y). (h x,k y)"])
+ using ex
+ apply (auto simp: group_hom_def group_hom_axioms_def DirProd_group iso_paired2 h k fih gjk kernel_def set_eq_iff)
+ apply (metis f.hom_closed f.r_one fih imageI)
+ apply (metis g.hom_closed g.l_one gjk imageI)
+ done
+qed
+
+subsection \<open>Splitting lemmas and Short exact sequences\<close>
+text\<open>Ported from HOL Light by LCP\<close>
+
+definition short_exact_sequence
+ where "short_exact_sequence A B C f g \<equiv> \<exists>T1 T2 e1 e2. exact_seq ([T1,A,B,C,T2], [e1,f,g,e2]) \<and> trivial_group T1 \<and> trivial_group T2"
+
+lemma short_exact_sequenceD:
+ assumes "short_exact_sequence A B C f g" shows "exact_seq ([A,B,C], [f,g]) \<and> f \<in> epi B A \<and> g \<in> mon C B"
+ using assms
+ apply (auto simp: short_exact_sequence_def group_hom_def group_hom_axioms_def)
+ apply (simp add: epi_iff_subset group_hom.intro group_hom.kernel_to_trivial_group group_hom_axioms.intro)
+ by (metis (no_types, lifting) group_hom.inj_iff_trivial_ker group_hom.intro group_hom_axioms.intro
+ hom_one image_empty image_insert mem_Collect_eq mon_def trivial_group_def)
+
+lemma short_exact_sequence_iff:
+ "short_exact_sequence A B C f g \<longleftrightarrow> exact_seq ([A,B,C], [f,g]) \<and> f \<in> epi B A \<and> g \<in> mon C B"
+proof -
+ have "short_exact_sequence A B C f g"
+ if "exact_seq ([A, B, C], [f, g])" and "f \<in> epi B A" and "g \<in> mon C B"
+ proof -
+ show ?thesis
+ unfolding short_exact_sequence_def
+ proof (intro exI conjI)
+ have "kernel A (singleton_group \<one>\<^bsub>A\<^esub>) (\<lambda>x. \<one>\<^bsub>A\<^esub>) = f ` carrier B"
+ using that by (simp add: kernel_def singleton_group_def epi_def)
+ moreover have "kernel C B g = {\<one>\<^bsub>C\<^esub>}"
+ using that group_hom.inj_iff_trivial_ker mon_def by fastforce
+ ultimately show "exact_seq ([singleton_group (one A), A, B, C, singleton_group (one C)], [\<lambda>x. \<one>\<^bsub>A\<^esub>, f, g, id])"
+ using that
+ by (simp add: group_hom_def group_hom_axioms_def group.id_hom_singleton)
+ qed auto
+qed
+ then show ?thesis
+ using short_exact_sequenceD by blast
+qed
+
+lemma very_short_exact_sequence:
+ assumes "exact_seq ([D,C,B,A], [h,g,f])" "trivial_group A" "trivial_group D"
+ shows "g \<in> iso B C"
+ using assms
+ apply simp
+ by (metis (no_types, lifting) group_hom.image_from_trivial_group group_hom.iso_iff
+ group_hom.kernel_to_trivial_group group_hom.trivial_ker_imp_inj group_hom_axioms.intro group_hom_def hom_carrier inj_on_one_iff')
+
+lemma splitting_sublemma_gen:
+ assumes ex: "exact_seq ([C,B,A], [g,f])" and fim: "f ` carrier A = H"
+ and "subgroup K B" and 1: "H \<inter> K \<subseteq> {one B}" and eq: "set_mult B H K = carrier B"
+ shows "g \<in> iso (subgroup_generated B K) (subgroup_generated C(g ` carrier B))"
+proof -
+ interpret KB: subgroup K B
+ by (rule assms)
+ interpret fAB: group_hom A B f
+ using ex by simp
+ interpret gBC: group_hom B C g
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ have "group A" "group B" "group C" and kerg: "kernel B C g = f ` carrier A"
+ using ex by (auto simp: group_hom_def group_hom_axioms_def)
+ have ker_eq: "kernel B C g = H"
+ using ex by (simp add: fim)
+ then have "subgroup H B"
+ using ex by (simp add: group_hom.img_is_subgroup)
+ show ?thesis
+ unfolding iso_iff
+ proof (intro conjI)
+ show "g \<in> hom (subgroup_generated B K) (subgroup_generated C(g ` carrier B))"
+ by (metis ker_eq \<open>subgroup K B\<close> eq gBC.hom_between_subgroups gBC.set_mult_ker_hom(2) order_refl subgroup.subset)
+ show "g ` carrier (subgroup_generated B K) = carrier (subgroup_generated C(g ` carrier B))"
+ by (metis assms(3) eq fAB.H.subgroupE(1) gBC.img_is_subgroup gBC.set_mult_ker_hom(2) ker_eq subgroup.carrier_subgroup_generated_subgroup)
+ interpret gKBC: group_hom "subgroup_generated B K" C g
+ apply (auto simp: group_hom_def group_hom_axioms_def \<open>group C\<close>)
+ by (simp add: fAB.H.hom_from_subgroup_generated gBC.homh)
+ have *: "x = \<one>\<^bsub>B\<^esub>"
+ if x: "x \<in> carrier (subgroup_generated B K)" and "g x = \<one>\<^bsub>C\<^esub>" for x
+ proof -
+ have x': "x \<in> carrier B"
+ using that fAB.H.carrier_subgroup_generated_subset by blast
+ moreover have "x \<in> H"
+ using kerg fim x' that by (auto simp: kernel_def set_eq_iff)
+ ultimately show ?thesis
+ by (metis "1" x Int_iff singletonD KB.carrier_subgroup_generated_subgroup subsetCE)
+ qed
+ show "inj_on g (carrier (subgroup_generated B K))"
+ using "*" gKBC.inj_on_one_iff by auto
+ qed
+qed
+
+lemma splitting_sublemma:
+ assumes ex: "short_exact_sequence C B A g f" and fim: "f ` carrier A = H"
+ and "subgroup K B" and 1: "H \<inter> K \<subseteq> {one B}" and eq: "set_mult B H K = carrier B"
+ shows "f \<in> iso A (subgroup_generated B H)" (is ?f)
+ "g \<in> iso (subgroup_generated B K) C" (is ?g)
+proof -
+ show ?f
+ using short_exact_sequenceD [OF ex]
+ apply (clarsimp simp add: group_hom_def group.iso_onto_image)
+ using fim group.iso_onto_image by blast
+ have "C = subgroup_generated C(g ` carrier B)"
+ using short_exact_sequenceD [OF ex]
+ apply simp
+ by (metis epi_iff_subset group.subgroup_generated_group_carrier hom_carrier subset_antisym)
+ then show ?g
+ using short_exact_sequenceD [OF ex]
+ by (metis "1" \<open>subgroup K B\<close> eq fim splitting_sublemma_gen)
+qed
+
+lemma splitting_lemma_left_gen:
+ assumes ex: "exact_seq ([C,B,A], [g,f])" and f': "f' \<in> hom B A" and iso: "(f' \<circ> f) \<in> iso A A"
+ and injf: "inj_on f (carrier A)" and surj: "g ` carrier B = carrier C"
+ obtains H K where "H \<lhd> B" "K \<lhd> B" "H \<inter> K \<subseteq> {one B}" "set_mult B H K = carrier B"
+ "f \<in> iso A (subgroup_generated B H)" "g \<in> iso (subgroup_generated B K) C"
+proof -
+ interpret gBC: group_hom B C g
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ have "group A" "group B" "group C" and kerg: "kernel B C g = f ` carrier A"
+ using ex by (auto simp: group_hom_def group_hom_axioms_def)
+ then have *: "f ` carrier A \<inter> kernel B A f' = {\<one>\<^bsub>B\<^esub>} \<and> f ` carrier A <#>\<^bsub>B\<^esub> kernel B A f' = carrier B"
+ using group_semidirect_sum_image_ker [of f A B f' A] assms by auto
+ interpret f'AB: group_hom B A f'
+ using assms by (auto simp: group_hom_def group_hom_axioms_def)
+ let ?H = "f ` carrier A"
+ let ?K = "kernel B A f'"
+ show thesis
+ proof
+ show "?H \<lhd> B"
+ by (simp add: gBC.normal_kernel flip: kerg)
+ show "?K \<lhd> B"
+ by (rule f'AB.normal_kernel)
+ show "?H \<inter> ?K \<subseteq> {\<one>\<^bsub>B\<^esub>}" "?H <#>\<^bsub>B\<^esub> ?K = carrier B"
+ using * by auto
+ show "f \<in> Group.iso A (subgroup_generated B ?H)"
+ using ex by (simp add: injf iso_onto_image group_hom_def group_hom_axioms_def)
+ have C: "C = subgroup_generated C(g ` carrier B)"
+ using surj by (simp add: gBC.subgroup_generated_group_carrier)
+ show "g \<in> Group.iso (subgroup_generated B ?K) C"
+ apply (subst C)
+ apply (rule splitting_sublemma_gen [OF ex refl])
+ using * by (auto simp: f'AB.subgroup_kernel)
+ qed
+qed
+
+lemma splitting_lemma_left:
+ assumes ex: "exact_seq ([C,B,A], [g,f])" and f': "f' \<in> hom B A"
+ and inv: "(\<And>x. x \<in> carrier A \<Longrightarrow> f'(f x) = x)"
+ and injf: "inj_on f (carrier A)" and surj: "g ` carrier B = carrier C"
+ obtains H K where "H \<lhd> B" "K \<lhd> B" "H \<inter> K \<subseteq> {one B}" "set_mult B H K = carrier B"
+ "f \<in> iso A (subgroup_generated B H)" "g \<in> iso (subgroup_generated B K) C"
+proof -
+ interpret fAB: group_hom A B f
+ using ex by simp
+ interpret gBC: group_hom B C g
+ using ex by (simp add: group_hom_def group_hom_axioms_def)
+ have "group A" "group B" "group C" and kerg: "kernel B C g = f ` carrier A"
+ using ex by (auto simp: group_hom_def group_hom_axioms_def)
+ have iso: "f' \<circ> f \<in> Group.iso A A"
+ using ex by (auto simp: inv intro: group.iso_eq [OF \<open>group A\<close> id_iso])
+ show thesis
+ by (metis that splitting_lemma_left_gen [OF ex f' iso injf surj])
+qed
+
+lemma splitting_lemma_right_gen:
+ assumes ex: "short_exact_sequence C B A g f" and g': "g' \<in> hom C B" and iso: "(g \<circ> g') \<in> iso C C"
+ obtains H K where "H \<lhd> B" "subgroup K B" "H \<inter> K \<subseteq> {one B}" "set_mult B H K = carrier B"
+ "f \<in> iso A (subgroup_generated B H)" "g \<in> iso (subgroup_generated B K) C"
+proof
+ interpret fAB: group_hom A B f
+ using short_exact_sequenceD [OF ex] by (simp add: group_hom_def group_hom_axioms_def)
+ interpret gBC: group_hom B C g
+ using short_exact_sequenceD [OF ex] by (simp add: group_hom_def group_hom_axioms_def)
+ have *: "f ` carrier A \<inter> g' ` carrier C = {\<one>\<^bsub>B\<^esub>}"
+ "f ` carrier A <#>\<^bsub>B\<^esub> g' ` carrier C = carrier B"
+ "group A" "group B" "group C"
+ "kernel B C g = f ` carrier A"
+ using group_semidirect_sum_ker_image [of g g' C C B] short_exact_sequenceD [OF ex]
+ by (simp_all add: g' iso group_hom_def)
+ show "kernel B C g \<lhd> B"
+ by (simp add: gBC.normal_kernel)
+ show "(kernel B C g) \<inter> (g' ` carrier C) \<subseteq> {\<one>\<^bsub>B\<^esub>}" "(kernel B C g) <#>\<^bsub>B\<^esub> (g' ` carrier C) = carrier B"
+ by (auto simp: *)
+ show "f \<in> Group.iso A (subgroup_generated B (kernel B C g))"
+ by (metis "*"(6) fAB.group_hom_axioms group.iso_onto_image group_hom_def short_exact_sequenceD [OF ex])
+ show "subgroup (g' ` carrier C) B"
+ using splitting_sublemma
+ by (simp add: fAB.H.is_group g' gBC.is_group group_hom.img_is_subgroup group_hom_axioms_def group_hom_def)
+ then show "g \<in> Group.iso (subgroup_generated B (g' ` carrier C)) C"
+ by (metis (no_types, lifting) iso_iff fAB.H.hom_from_subgroup_generated gBC.homh image_comp inj_on_imageI iso subgroup.carrier_subgroup_generated_subgroup)
+qed
+
+lemma splitting_lemma_right:
+ assumes ex: "short_exact_sequence C B A g f" and g': "g' \<in> hom C B" and gg': "\<And>z. z \<in> carrier C \<Longrightarrow> g(g' z) = z"
+ obtains H K where "H \<lhd> B" "subgroup K B" "H \<inter> K \<subseteq> {one B}" "set_mult B H K = carrier B"
+ "f \<in> iso A (subgroup_generated B H)" "g \<in> iso (subgroup_generated B K) C"
+proof -
+ have *: "group A" "group B" "group C"
+ using group_semidirect_sum_ker_image [of g g' C C B] short_exact_sequenceD [OF ex]
+ by (simp_all add: g' group_hom_def)
+ show thesis
+ apply (rule splitting_lemma_right_gen [OF ex g' group.iso_eq [OF _ id_iso]])
+ using * apply (auto simp: gg' intro: that)
+ done
+qed
+
+
end