Quelle  Exact_Sequence.thy   Sprache: Isabelle

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

section

theoryExact_Sequenceimports Elementary_Groups  Solvable_Groupssubse
 Elementary_Groups
begin



subsection:G1  <Longrightarrow> exact_seq ([G2, G1], [f])" |

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) \<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 \ 'a) \ 'a monoid list \ ('a \ 'a) list \ 'a monoid \ 'a monoid list \ ('a \ 'a) list"
  (\<open>(\<open>indent=3 notation=\<open>mixfix exact_seq\<close>\<close>_ / \<longlongrightarrow>\<index> _)\<close> [1000, 60])
  where "exact_seq_arrow f textension "\<lbrakk> exact_seq ((G # K # l), (g # q)); group H ; h \<in> hom G H ;kernel  =image( K) \<rbrakk> \<Longrightarrow> exact_seq (H # G # K # l, h # g # q)"


subsection \<open>Basic Properties\<close>

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:
   "exact_seq (G, F)" and "(i :: nat) < length F"
  shows "exact_seq (drop i G, drop i F)"
proof-
  have "exact_seq (drop i (fst t), drop i (snd t))" if "exact_seq t" "i < length (snd t)" for t i
    using that
  proof (induction arbitrary: i)
    case(unity G2) ?case
      by (simp add: exact_seq)
  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
assumei 
      then obtain k where "k < length (snd (G # K java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using extensionprems auto
      thusthesis extensionIH by  
    qed
  qed
  thus ?thesis using assms
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  assumesexact_seqG,F"and " : nat) < length F"
        exact_seq_length2[proof-

lemmaexact_seq_imp_exact_hom
assumesexact_seqG1 lq \<longlongrightarrow>\<^bsub>g1\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
   showsg1 ( G1=    g2
proof (nity G1G2)thuscase
  have "(hd (tl (snd t))) ` (carrier (hd (tl (tl (fst t))))) =
            kernel (hd (tl (fst t))      by (simpadd: exact_sequnity
    if "exact_seq " "length (fst tt) \ 3 \ length (snd t) \ 2" for t
    using that
  proofinduction
     unityG2)
    then show ?case by auto
  next
    case (extension G lnext
    then show ?case by auto
  qed
  with assms show ?thesis      then obtainkwhere < sndG#K# ,#q)" i = Suc k"
ed

lemma exact_seq_imp_exact_hom_arbitrary:
   assumes "exact_seq (G, F)"
     and "Suc i < length F"
       qed
proof  qed
  have"length (drop i F \ 2" "length (drop i G) \ 3"
    usingassms)exact_seq_length1OF assms1)by
  thenobtain l  
    wheredropG  ( !i)#( ! Suc))# G!(uc (Suc) # ljava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 50
    by        exact_seq_length2[OF assms1) assms(2) by (simp add: drop_Suc)
        le_eq_less_or_eqle_imp_less_Suc .sel2))
  thusassumesexact_seqG1#lq 
  usingshows carrier ernelg2
        [of! G!(Suc" ! ( ( )"  ]ajava.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89
qed

lemma exact_seq_imp_group_hom :exact_seqand "length (fst t) \ 3 \ length (snd t) \ 2" for t
  assumes "exact_seq ((G # l, q)) \\<^bsub>g\<^esub> H"
  shows "group_hom G H g"
proof-
  have aux_lemma: "group_hom (hd (tl (fst t))) (hd (fst t)) (hd(snd t))" if "exact_seq t" for t
    using that
   (induction
caseunity  fjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
   then ?  java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  next
    case extensionG  gq  h)
    thenshow ?case unfoldinggroup_hom_def by auto
  qed
   ?thesis using [OF ]
    by simp
qed

lemma      and "Suc i < length Fshows  Suci) `( G  Suci)) =kernelG S ) (G ! i (F! )
assumes" (G,)"and(:: nat)   "
  shows "group_hom (G ! (Suc i)) (G ! i) (F ! i)"
proof -
  have      assms() exact_seq_length1[OF assms(1]  auto
    using assms(2) exact_seq_length1[OF assms(1)] by autothen l q
    l q
    where "drop i nd " i F  (  )  (F !( i)) #"
     anddrop=F!i) "
    by (metis le_imp_less_Sucprod.(2))
        le_eq_less_or_eq dropped_seq_is_exact_seq (1), i (java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
 java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length 14
   dropped_seq_is_exact_seq,iassms
        exact_seq_imp_group_hom[of "using that
qed


 \<open>Link Between Exact Sequences and Solvable Conditions\<close>showcaseby

lemma exact_seq_solvable_imp show unfoldinggroup_hom_def byauto
assumesexact_seq,])\<longlongrightarrow>\<^bsub>g1\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
     "inj_on g1 (carrier G1)"
     exact_seq_imp_group_hom_arbitrary
   "exact_seq (G F)" "i : nat < F"
proof
   G2:solvable
  have "have" (dropF)\<ge> 1" "length (drop i G) \<ge> 2"using(2 exact_seq_length1 (1)] by auto
    using exact_seq_imp_group_hom_arbitraryOFassms ofSuc]bysimp
  hence "solvable where"drop i G  ( i) # (G ! (Suc i)) # l"
    using group_hom.inj_hom_imp_solvable[of G1 G2 g1] assms(2) G2 by simp
  moreover have "group_hom G2 G3 g2"
usingexact_seq_imp_group_hom_arbitraryOF(1,  0] by simp
  hence "solvable G3"
    usingby ( Cons_nth_drop_Suc assms fst_conv
  ultimately show ?thesis by simp
qed

lemma exact_seq_solvable_recip :
  assumes "exact_seq ([G1],[]) \\<^bsub>g1\<^esub> G2 \\<^bsub>g2\<^esub> G3"
    and  thusthesis
    andg2 `(carrier =carrier
  shows        [of" !"" !( i"   " ! ] by simp
proof -
  assume "(solvable G1) \ (solvable G3)"
  hence G1: " G1" and G3 "solvableG3 by
  have g1: "group_hom G1 G2 g1" and g2: "group_hom G2 G3 g2"
    using " ([G1,] <^bsub>g1\<^esub> G2 \\<^bsub>g2\<^esub> G3"
          exact_seq_imp_group_hom_arbitrary[OF assms(1), of 0] by auto "inj_on g1 (carrier G1)"
  show ?thesis "solvable G2 \ (solvable G1) \ (solvable G3)"
    using solvable_condition g1g2 (3)]
          xact_seq_imp_exact_homassms]G1byauto
qed

proposition exact_seq_solvable_iff :
  assumes "exact_seq ([G1],[]) \\<^bsub>g1\<^esub> G2 \\<^bsub>g2\<^esub> G3"
    andinj_onG1
    and "g2 ` (carrier have group_hom G2G3g2"
   "solvableG1)
  using exact_seq_solvable_recip exact_seq_solvable_imp assms 


lemma exact_seq_eq_triviality: group_hom[of G3] assms(3) G2 by simp
   "exact_seq ([E,D,C,B,A], [k,h,g,f])"
  shows java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
proof
  assume C: "trivial_group C"
  with assms have "inj_on k (carrier D)"
applyauto simp.image_from_trivial_group trivial_group_def )
    apply (simphave: "group_hom G2 g1" and:"group_hom G2 G3G3 g2"
    done
  with assms C show ?rhs
    apply (auto    usingexact_seq_imp_group_hom_arbitraryOFassms1), ofSuc]
      (auto: group_hom_defgroup_hom_axioms_def kernel_def)
    done
next
  assume ?rhsusing solvable_conditionOFg1 assms(3)]
  with assms show "trivial_group C"
    apply (simp add: trivial_group_def          exact_seq_imp_exact_hom assms()] G1 G3by auto
    by (metis
qed

lemmaexact_seq_imp_triviality
   " "g2 ` (carrier G2) = carrier G3"
  y (metisno_types) Groupiso_def exact_seq_eq_triviality mem_Collect_eq

lemma exact_seq_epi_eq_triviality
   lemmaexact_seq_eq_triviality
  by assumesexact_seq,,CBA, k,h,,]"

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: "exact_seq ([D,G,A], [g,i])" "exact_seq ([C,G,B], [f,j])"
    and fih  withassms have "inj_on k (carrier D)"
    and gjk: "\x. x \ carrier B \ g(j x) = k x"
  shows"\(, y. ix\\<^bsub>G\<^esub> j y) \ Group.iso (A \\ B) G \ (\z. (f z, g z)) \ Group.iso G (C \\ D)"
    (is? <in> _ \<and> ?gf \<in> _")
proof (rule epi_iso_compose_rev)
  interpret comm_group G
    by (rule assms)
  interpret f:group_hom  f
    usingwith assms  ?rhs
  interpretapply (auto simp: group_hom trivial_group_def hom_one
    using ex (simp: group_hom_def)
  interpret i: group_hom A Gjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 8
    using ex (simp add trivial_group_def)
      by metis.inj_iff_trivial_ker group_hom group_hom_axioms.intro)
    using exjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  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)
  thenobtainh' where "' \<in> hom C A" "(\<forall>x \<in> carrier A. h'(h x) = x)"
> carrier C h(h'y)= yy" and group_isomorphismsAC h '"
    using h by (auto simp: group.iso_iff_group_isomorphisms group_isomorphisms_def)
  have lemma exact_seq_epi_eq_triviality
    unfolding
    apply (rule b ( simp:  kernel_def)
    using xact_seq_mon_eq_triviality
  showhomgf "gf \ hom G (C \\ D)"
    sing by (simp: hom_paired)
  show "? \ epi (A \\ B) G"
  proof (clarsimp add: epi_iff_subset)
    fixx
    assume x: "x \ carrier G"
    with
      by (simp add: kernel_def hom_in_carrierhhfih
    with kerf obtain y where y: "y \ carrier B" "j y = x \\<^bsub>G\<^esub> inv\<^bsub>G\<^esub>(i(h'(f x)))"
      by auto
java.lang.StringIndexOutOfBoundsException: Index 222 out of bounds for length 222
     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 "proof (rule epi_iso_compose_rev)
usingjava.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
finallyshowx \<in> (\<lambda>(x, y). i x \<otimes>\<^bsub>G\<^esub> j y) ` (carrier A \<times> carrier B)"
       x y  (clarsimp: image_def
      apply (rule_tac x="h'(finterpretg: group_hom G D g
       apply(rule_tac x=y inbexI, auto
      by( \<open>h' \<in> hom C A\<close> f.hom_closed hom_in_carrier)
  qed
  show "(java.lang.StringIndexOutOfBoundsException: Range [0, 156) out of bounds for length 30
     rule.iso_eqwhere="(x,y). (h x,k y)"])
    using ex
    apply (auto simp kerf " G C f = j` carrier B" and" A" "group B" " \ hom A G"
     apply ( f.hom_closedfr_one imageI
    apply (metis g.hom_closed obtain h' where "' <in> hom C A" "(\<forall>x \<in> carrier A. h'(h x) = x)"
    done
qed

subsection \<open>Splitting lemmas and Short exact sequences\<close>
text<> from HOL Light by LCP\<close>

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 (rulehom_group_mult)
  assumes "short_exact_sequence A B C f g" shows "exact_seq ([A,B,C], [f,g by(simp_all add: group_hom_def hom_of_fst [unfolded o_def] hom_of_snd [unfolded o_def])
  using assms
  apply (autousingexby (simpadd: hom_paired)
   ( addepi_iff_subset. group_homkernel_to_trivial_group)
  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)

mmashort_exact_sequence_iff:
  "short_exact_sequencefix x
proof -
  have "short_exact_sequence A B C f g"
    if    assume x: " \ carrier G"
  proof -
    show ?thesis
      unfolding short_exact_sequence_def
    proof(ntro conjI)
      have       by (simp add: kernel_def hom_in_carrier hh' fih)
        using that by (simpadd: kernel_def epi_def)
      moreoverhave "ernelCBg = {\\<^bsub>C\<^esub>}"
        using that     "i(h (fx) \<^bsub>G\<^esub> (x \\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> i (h' (f x))) = x \\<^bsub>G\<^esub> (i (h' (f x)) \\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> i (h' (f x)))"
      ultimately show "exact_seq ([singleton_group (one A), A, B, C, singleton_group (one C)], [\x. \\<^bsub>A\<^esub>, f, g, id])"
        using that
        by (simpadd: group_hom_defgroup_hom_axioms_def group.id_hom_singleton
    qed auto
qed
  then show ?thesis
    using short_exact_sequenceD by blast
qed

lemmaapply (rule_tac x="h'(f x)" in bexI)
  assumes "exact_seq ([D,C,B,A], [h,g,f])" " apply (rule_tac x=y in bexI, auto)
  shows       meson
  using assms
  applysjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
  by (metis (no_types) group_homimage_from_trivial_group group_homiso_iff
      group_hom.kernel_to_trivial_group group_hom.trivial_ker_imp_inj group_hom_axioms.intro group_hom_def ex

lemma splitting_sublemma_gen:
  a (metisfhom_closed.r_one imageI
       "subgroup "and "H K \ {one B}" and eq: "set_mult B H K = carrier B"
  shows    done
proof -
  interpret KB subgroup java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
    by (rule " A B C T1 T2 e1 e2. exact_seq ([T1,A,B,C,T2], [e1,f,g,e2]) \ trivial_group T1 \ trivial_group T2"
  interpret fAB: group_hom A B f
    using ex by simp
  interpret gBC: group_homassumesshort_exact_sequence  f"shows"xact_seq(AB] fg)\<and> f \<in> epi B A \<and> g \<in> mon C B"
    using ex by (simp simp: epi_iff_subset.introgroup_homkernel_to_trivial_group.intro
   "group A" group""group "and kerg kernel C =f` carrier A"
      using ex autogroup_hom_def
  have ker_eq: "kernel B C g = H"
    using ex       image_emptyimage_insert mon_def)
  then subgroup BB""g \ iso (subgroup_generated B K) C"
proof -
  interpret gBC: group_hom B C g
    using ex by      short_exact_sequenceD[ ex by(simp :  group_hom_axioms_def
  have "group A"   [OF] by( add group_hom_axioms_def
usingexsimp group_hom_axioms_def)
  then have " \bsub>B\<^esub> g' ` carrier C = carrier B"
    using" A"groupgroup
  interpretB  =  carrier"
    using assms by (autousinggroup_semidirect_sum_ker_image[f g g CCB]short_exact_sequenceDOF ex]
  let ?H = "f ` carrier A"
  letK  kernel  f
  show thesis
  proof
    show "?H \ B"
      by (simp add: gBC.normal_kernel flip "kernel B C \ (g' ` carrier C) \ {\\<^bsub>B\<^esub>}" "(kernel B C g) <#>\<^bsub>B\<^esub> (g' ` carrier C) = carrier B"
    show "?K \ B"
      by (rule f'by (auto simp: *)
show? <> ?K\<>java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
       * byjava.lang.StringIndexOutOfBoundsException: Range [21, 22) out of bounds for length 21
    show "f \ Group.iso A (subgroup_generated B ?H)"
      using ex by (simp add: by( add .His_group gBCgroup_homgroup_hom_axioms_def
    have C: "C = subgroup_generated C(g ` carrier B)"
       surj simp: gBC)
    show "g \ Group.iso (subgroup_generated B ?K) C"
      apply (subst C)
       splitting_sublemma_gen[ exrefl]
      using * by (auto ex short_exact_sequenceB f  g' g hom C B" and gg': "\z. z \ carrier C \ g(g' z) = z"in CBandgg java.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 141
  qed
qed

lemma -
  assumes ex: " *: " A"" B"groupC"
    and inv: "(\x. x \ carrier A \ f'(f x) = x)"
    and injf: "inj_on f (carrier A)" and    by simp_all'java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
 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 \<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 \ 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 = {\\<^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 \ B"
    by (simp add: gBC.normal_kernel)
  show "(kernel B C g) \ (g' ` carrier C) \ {\\<^bsub>B\<^esub>}" "(kernel B C g) <#>\<^bsub>B\<^esub> (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' \<in> hom C B" and gg': "\<And>z. z \<in> carrier C \<Longrightarrow> 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]
    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

quality100%

B H K = carrier B"
                   "f \ iso A (subgroup_generated B H)" "g \ iso (subgroup_generated B K) C"
proof
  interpret fAB: group_hom                   f java.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
using OF] simp addgroup_hom_def)
  interpret gBC: group_hom B C g
    usingshort_exact_sequenceD ex  (simp: group_hom_def)
  have *: "f ` using by (auto simp: group_hom_defjava.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 63
          f `carrierA<#><^bsub\<^esub> g' ` carrier C = carrier B"
          "groupA " B" " C"
          "kernel B Cg f `carrier A"
      [ 'C C ] short_exact_sequenceD [OF ex
    by (simp_all add: g' iso group_hom_def)
  show " let ?="kernel BA '"
    byjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  show(   g) \<inter> (g' ` carrier C) \<subseteq> {\<one>\<^bsub>B\<^esub>}" "(kernel B C g) <#>\<^bsub>B\<^esub> (g' ` carrier C) = carrier B"
    by( simp
  show " "H\<inter subseteq {<one>\<^bsub>B\<^esub>}" "?H <#>\<^bsub>B\<^esub> ?K = carrier B"
    by (metis       using auto
  show "subgroup (g' ` carrier C) B"
    using splitting_sublemma
     simp:fAB. 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, liftingusing by( add.subgroup_generated_group_carrier
qed

lemma apply (rule OF refl])
  assumes:" C Ag "and:"'\ hom " and ':"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 qed
proof-
have group group " java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    using group_semidirect_sum_ker_image [of g g' C C B] short_exact_sequenceD [OF ex]
by( 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

quality100%

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