# Theory Exact_Sequence

```(*  Title:      HOL/Algebra/Exact_Sequence.thy
Author:     Martin Baillon (first part) and LC Paulson (material ported from HOL Light)
*)

section ‹Exact Sequences›

theory Exact_Sequence
imports Elementary_Groups Solvable_Groups
begin

subsection ‹Definitions›

inductive exact_seq :: "'a monoid list × ('a ⇒ 'a) list ⇒ bool"  where
unity:     " group_hom G1 G2 f ⟹ exact_seq ([G2, G1], [f])" |
extension: "⟦ exact_seq ((G # K # l), (g # q)); group H ; h ∈ hom G H ;
kernel G H h = image g (carrier K) ⟧ ⟹ 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 ⇒ 'a) ⇒ 'a monoid list × ('a ⇒ 'a) list ⇒  'a monoid ⇒ 'a monoid list × ('a ⇒ 'a) list"
("(3_ / ⤏ı _)" [1000, 60])
where "exact_seq_arrow  f t G ≡ (G # (fst t), f # (snd t))"

subsection ‹Basic Properties›

lemma exact_seq_length1: "exact_seq t ⟹ length (fst t) = Suc (length (snd t))"
by (induct t rule: exact_seq.induct) auto

lemma exact_seq_length2: "exact_seq t ⟹ length (snd t) ≥ 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
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 ≠ 0" hence "i ≥ 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 ≥ 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) ⤏⇘g1⇙ G2 ⤏⇘g2⇙ G3"
shows "g1 ` (carrier G1) = kernel G2 G3 g2"
proof-
{ fix t assume "exact_seq t" and "length (fst t) ≥ 3 ∧ length (snd t) ≥ 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) ≥ 2" "length (drop i G) ≥ 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)) ⤏⇘g⇙ 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) ≥ 1" "length (drop i G) ≥ 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 ‹Link Between Exact Sequences and Solvable Conditions›

lemma exact_seq_solvable_imp :
assumes "exact_seq ([G1],[]) ⤏⇘g1⇙ G2 ⤏⇘g2⇙ G3"
and "inj_on g1 (carrier G1)"
and "g2 ` (carrier G2) = carrier G3"
shows "solvable G2 ⟹ (solvable G1) ∧ (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],[]) ⤏⇘g1⇙ G2 ⤏⇘g2⇙ G3"
and "inj_on g1 (carrier G1)"
and "g2 ` (carrier G2) = carrier G3"
shows "(solvable G1) ∧ (solvable G3) ⟹ solvable G2"
proof -
assume "(solvable G1) ∧ (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],[]) ⤏⇘g1⇙ G2 ⤏⇘g2⇙ G3"
and "inj_on g1 (carrier G1)"
and "g2 ` (carrier G2) = carrier G3"
shows "(solvable G1) ∧ (solvable G3) ⟷  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 ⟷ f ` carrier A = carrier B ∧ 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"
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:
"⟦exact_seq ([E,D,C,B,A], [k,h,g,f]); f ∈ iso A B; k ∈ iso D E⟧ ⟹ 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]) ⟹ (f ` carrier A = carrier B) ⟷ 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]) ⟹ inj_on h (carrier C) ⟷ 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 ∈ iso A C" and k: "k ∈ iso B D"
and ex: "exact_seq ([D,G,A], [g,i])" "exact_seq ([C,G,B], [f,j])"
and fih: "⋀x. x ∈ carrier A ⟹ f(i x) = h x"
and gjk: "⋀x. x ∈ carrier B ⟹ g(j x) = k x"
shows "(λ(x, y). i x ⊗⇘G⇙ j y) ∈ Group.iso (A ×× B) G ∧ (λz. (f z, g z)) ∈ Group.iso G (C ×× D)"
(is "?ij ∈ _ ∧ ?gf ∈ _")
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 ∈ hom A G"
using ex by (auto simp: group_hom_def group_hom_axioms_def)
then obtain h' where "h' ∈ hom C A" "(∀x ∈ carrier A. h'(h x) = x)"
and hh': "(∀y ∈ 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 ∈ hom (A ×× 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 ∈ hom G (C ×× D)"
using ex by (simp add: hom_paired)
show "?ij ∈ epi (A ×× B) G"
proof (clarsimp simp add: epi_iff_subset homij)
fix x
assume x: "x ∈ carrier G"
with ‹i ∈ hom A G› ‹h' ∈ hom C A› have "x ⊗⇘G⇙ inv⇘G⇙(i(h'(f x))) ∈ kernel G C f"
by (simp add: kernel_def hom_in_carrier hh' fih)
with kerf obtain y where y: "y ∈ carrier B" "j y = x ⊗⇘G⇙ inv⇘G⇙(i(h'(f x)))"
by auto
have "i (h' (f x)) ⊗⇘G⇙ (x ⊗⇘G⇙ inv⇘G⇙ i (h' (f x))) = x ⊗⇘G⇙ (i (h' (f x)) ⊗⇘G⇙ inv⇘G⇙ i (h' (f x)))"
by (meson ‹h' ∈ hom C A› x f.hom_closed hom_in_carrier i.hom_closed inv_closed m_lcomm)
also have "… = x"
using ‹h' ∈ hom C A› hom_in_carrier x by fastforce
finally show "x ∈ (λ(x, y). i x ⊗⇘G⇙ j y) ` (carrier A × 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 ‹h' ∈ hom C A› f.hom_closed hom_in_carrier)
qed
show "(λz. (f z, g z)) ∘ (λ(x, y). i x ⊗⇘G⇙ j y) ∈ Group.iso (A ×× B) (C ×× D)"
apply (rule group.iso_eq [where f = "λ(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 ‹Splitting lemmas and Short exact sequences›
text‹Ported from HOL Light by LCP›

definition short_exact_sequence
where "short_exact_sequence A B C f g ≡ ∃T1 T2 e1 e2. exact_seq ([T1,A,B,C,T2], [e1,f,g,e2]) ∧ trivial_group T1 ∧ trivial_group T2"

lemma short_exact_sequenceD:
assumes "short_exact_sequence A B C f g" shows "exact_seq ([A,B,C], [f,g]) ∧ f ∈ epi B A ∧ g ∈ 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 ⟷ exact_seq ([A,B,C], [f,g]) ∧ f ∈ epi B A ∧ g ∈ mon C B"
proof -
have "short_exact_sequence A B C f g"
if "exact_seq ([A, B, C], [f, g])" and "f ∈ epi B A" and "g ∈ mon C B"
proof -
show ?thesis
unfolding short_exact_sequence_def
proof (intro exI conjI)
have "kernel A (singleton_group 𝟭⇘A⇙) (λx. 𝟭⇘A⇙) = f ` carrier B"
using that by (simp add: kernel_def singleton_group_def epi_def)
moreover have "kernel C B g = {𝟭⇘C⇙}"
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)], [λx. 𝟭⇘A⇙, 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 ∈ 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 ∩ K ⊆ {one B}" and eq: "set_mult B H K = carrier B"
shows "g ∈ 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 ∈ hom (subgroup_generated B K) (subgroup_generated C(g ` carrier B))"
by (metis ker_eq ‹subgroup K B› 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 ‹group C›)
have *: "x = 𝟭⇘B⇙"
if x: "x ∈ carrier (subgroup_generated B K)" and "g x = 𝟭⇘C⇙" for x
proof -
have x': "x ∈ carrier B"
using that fAB.H.carrier_subgroup_generated_subset by blast
moreover have "x ∈ 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 ∩ K ⊆ {one B}" and eq: "set_mult B H K = carrier B"
shows "f ∈ iso A (subgroup_generated B H)" (is ?f)
"g ∈ 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" ‹subgroup K B› eq fim splitting_sublemma_gen)
qed

lemma splitting_lemma_left_gen:
assumes ex: "exact_seq ([C,B,A], [g,f])" and f': "f' ∈ hom B A" and iso: "(f' ∘ f) ∈ iso A A"
and injf: "inj_on f (carrier A)" and surj: "g ` carrier B = carrier C"
obtains H K where "H ⊲ B" "K ⊲ B" "H ∩ K ⊆ {one B}" "set_mult B H K = carrier B"
"f ∈ iso A (subgroup_generated B H)" "g ∈ 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 ∩ kernel B A f' = {𝟭⇘B⇙} ∧ f ` carrier A <#>⇘B⇙ 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 ⊲ B"
by (simp add: gBC.normal_kernel flip: kerg)
show "?K ⊲ B"
by (rule f'AB.normal_kernel)
show "?H ∩ ?K ⊆ {𝟭⇘B⇙}" "?H <#>⇘B⇙ ?K = carrier B"
using * by auto
show "f ∈ 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 ∈ 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' ∈ hom B A"
and inv: "(⋀x. x ∈ carrier A ⟹ f'(f x) = x)"
and injf: "inj_on f (carrier A)" and surj: "g ` carrier B = carrier C"
obtains H K where "H ⊲ B" "K ⊲ B" "H ∩ K ⊆ {one B}" "set_mult B H K = carrier B"
"f ∈ iso A (subgroup_generated B H)" "g ∈ 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' ∘ f ∈ Group.iso A A"
using ex by (auto simp: inv intro:  group.iso_eq [OF ‹group A› 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' ∈ hom C B" and iso: "(g ∘ g') ∈ iso C C"
obtains H K where "H ⊲ B" "subgroup K B" "H ∩ K ⊆ {one B}" "set_mult B H K = carrier B"
"f ∈ iso A (subgroup_generated B H)" "g ∈ 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 ∩ g' ` carrier C = {𝟭⇘B⇙}"
"f ` carrier A <#>⇘B⇙ 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 ⊲ B"
show "(kernel B C g) ∩ (g' ` carrier C) ⊆ {𝟭⇘B⇙}" "(kernel B C g) <#>⇘B⇙ (g' ` carrier C) = carrier B"
by (auto simp: *)
show "f ∈ 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 ∈ 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' ∈ hom C B" and gg': "⋀z. z ∈ carrier C ⟹ g(g' z) = z"
obtains H K where "H ⊲ B" "subgroup K B" "H ∩ K ⊆ {one B}" "set_mult B H K = carrier B"
"f ∈ iso A (subgroup_generated B H)" "g ∈ 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]