(* Title: HOL/Algebra/Exact_Sequence.thy
Author: Martin Baillon (first part) and LC Paulson (material ported from HOL Light)
*)
section \<open>Exact Sequences\<close>
theory Exact_Sequence
imports Elementary_Groups Solvable_Groups
begin
subsection \<open>Definitions\<close>
inductive exact_seq :: "'a monoid list \<times> ('a \<Rightarrow> 'a) list \<Rightarrow> bool" where
unity: " group_hom G1 G2 f \<Longrightarrow> exact_seq ([G2, G1], [f])" |
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])
where "exact_seq_arrow f t G \<equiv> (G # (fst t), f # (snd t))"
subsection \<open>Basic Properties\<close>
lemma exact_seq_length1: "exact_seq t \<Longrightarrow> length (fst t) = Suc (length (snd t))"
by (induct t rule: exact_seq.induct) auto
lemma exact_seq_length2: "exact_seq t \<Longrightarrow> length (snd t) \<ge> Suc 0"
by (induct t rule: exact_seq.induct) auto
lemma dropped_seq_is_exact_seq:
assumes "exact_seq (G, F)" and "(i :: nat) < length F"
shows "exact_seq (drop i G, drop i F)"
proof-
{ fix t i assume "exact_seq t" "i < length (snd t)"
hence "exact_seq (drop i (fst t), drop i (snd t))"
proof (induction arbitrary: i)
case (unity G1 G2 f) thus ?case
by (simp add: exact_seq.unity)
next
case (extension G K l g q H h) show ?case
proof (cases)
assume "i = 0" thus ?case
using exact_seq.extension[OF extension.hyps] by simp
next
assume "i \<noteq> 0" hence "i \<ge> Suc 0" by simp
then obtain k where "k < length (snd (G # K # l, g # q))" "i = Suc k"
using Suc_le_D extension.prems by auto
thus ?thesis using extension.IH by simp
qed
qed }
thus ?thesis using assms by auto
qed
lemma truncated_seq_is_exact_seq:
assumes "exact_seq (l, q)" and "length l \<ge> 3"
shows "exact_seq (tl l, tl q)"
using exact_seq_length1[OF assms(1)] dropped_seq_is_exact_seq[OF assms(1), of "Suc 0"]
exact_seq_length2[OF assms(1)] assms(2) by (simp add: drop_Suc)
lemma exact_seq_imp_exact_hom:
assumes "exact_seq (G1 # l,q) \<longlongrightarrow>\<^bsub>g1\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
shows "g1 ` (carrier G1) = kernel G2 G3 g2"
proof-
{ fix t assume "exact_seq t" and "length (fst t) \<ge> 3 \<and> length (snd t) \<ge> 2"
hence "(hd (tl (snd t))) ` (carrier (hd (tl (tl (fst t))))) =
kernel (hd (tl (fst t))) (hd (fst t)) (hd (snd t))"
proof (induction)
case (unity G1 G2 f)
then show ?case by auto
next
case (extension G l g q H h)
then show ?case by auto
qed }
thus ?thesis using assms by fastforce
qed
lemma exact_seq_imp_exact_hom_arbitrary:
assumes "exact_seq (G, F)"
and "Suc i < length F"
shows "(F ! (Suc i)) ` (carrier (G ! (Suc (Suc i)))) = kernel (G ! (Suc i)) (G ! i) (F ! i)"
proof -
have "length (drop i F) \<ge> 2" "length (drop i G) \<ge> 3"
using assms(2) exact_seq_length1[OF assms(1)] by auto
then obtain l q
where "drop i G = (G ! i) # (G ! (Suc i)) # (G ! (Suc (Suc i))) # l"
and "drop i F = (F ! i) # (F ! (Suc i)) # q"
by (metis Cons_nth_drop_Suc Suc_less_eq assms exact_seq_length1 fst_conv
le_eq_less_or_eq le_imp_less_Suc prod.sel(2))
thus ?thesis
using dropped_seq_is_exact_seq[OF assms(1), of i] assms(2)
exact_seq_imp_exact_hom[of "G ! i" "G ! (Suc i)" "G ! (Suc (Suc i))" l q] by auto
qed
lemma exact_seq_imp_group_hom :
assumes "exact_seq ((G # l, q)) \<longlongrightarrow>\<^bsub>g\<^esub> H"
shows "group_hom G H g"
proof-
{ fix t assume "exact_seq t"
hence "group_hom (hd (tl (fst t))) (hd (fst t)) (hd(snd t))"
proof (induction)
case (unity G1 G2 f)
then show ?case by auto
next
case (extension G l g q H h)
then show ?case unfolding group_hom_def group_hom_axioms_def by auto
qed }
note aux_lemma = this
show ?thesis using aux_lemma[OF assms]
by simp
qed
lemma exact_seq_imp_group_hom_arbitrary:
assumes "exact_seq (G, F)" and "(i :: nat) < length F"
shows "group_hom (G ! (Suc i)) (G ! i) (F ! i)"
proof -
have "length (drop i F) \<ge> 1" "length (drop i G) \<ge> 2"
using assms(2) exact_seq_length1[OF assms(1)] by auto
then obtain l q
where "drop i G = (G ! i) # (G ! (Suc i)) # l"
and "drop i F = (F ! i) # q"
by (metis Cons_nth_drop_Suc Suc_leI assms exact_seq_length1 fst_conv
le_eq_less_or_eq le_imp_less_Suc prod.sel(2))
thus ?thesis
using dropped_seq_is_exact_seq[OF assms(1), of i] assms(2)
exact_seq_imp_group_hom[of "G ! i" "G ! (Suc i)" l q "F ! i"] by simp
qed
subsection \<open>Link Between Exact Sequences and Solvable Conditions\<close>
lemma exact_seq_solvable_imp :
assumes "exact_seq ([G1],[]) \<longlongrightarrow>\<^bsub>g1\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
and "inj_on g1 (carrier G1)"
and "g2 ` (carrier G2) = carrier G3"
shows "solvable G2 \<Longrightarrow> (solvable G1) \<and> (solvable G3)"
proof -
assume G2: "solvable G2"
have "group_hom G1 G2 g1"
using exact_seq_imp_group_hom_arbitrary[OF assms(1), of "Suc 0"] by simp
hence "solvable G1"
using group_hom.inj_hom_imp_solvable[of G1 G2 g1] assms(2) G2 by simp
moreover have "group_hom G2 G3 g2"
using exact_seq_imp_group_hom_arbitrary[OF assms(1), of 0] by simp
hence "solvable G3"
using group_hom.surj_hom_imp_solvable[of G2 G3 g2] assms(3) G2 by simp
ultimately show ?thesis by simp
qed
lemma exact_seq_solvable_recip :
assumes "exact_seq ([G1],[]) \<longlongrightarrow>\<^bsub>g1\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
and "inj_on g1 (carrier G1)"
and "g2 ` (carrier G2) = carrier G3"
shows "(solvable G1) \<and> (solvable G3) \<Longrightarrow> solvable G2"
proof -
assume "(solvable G1) \<and> (solvable G3)"
hence G1: "solvable G1" and G3: "solvable G3" by auto
have g1: "group_hom G1 G2 g1" and g2: "group_hom G2 G3 g2"
using exact_seq_imp_group_hom_arbitrary[OF assms(1), of "Suc 0"]
exact_seq_imp_group_hom_arbitrary[OF assms(1), of 0] by auto
show ?thesis
using solvable_condition[OF g1 g2 assms(3)]
exact_seq_imp_exact_hom[OF assms(1)] G1 G3 by auto
qed
proposition exact_seq_solvable_iff :
assumes "exact_seq ([G1],[]) \<longlongrightarrow>\<^bsub>g1\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
and "inj_on g1 (carrier G1)"
and "g2 ` (carrier G2) = carrier G3"
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