Quelle Exact_Sequence.thy Sprache: Isabelle

    Author    Author:     Martin Baillon (first part)*)
*)

section \<open>Exact Sequences\<close>

theory Exact_Sequence
  importsElementary_Groups
begin

ction \<open>Definitions\<close>

inductive exact_seq   imports Solvable_Groups
unity     " group_hom G2 f\java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
:
               G Hh  g carrier

inductive_simps exact_seq_end_iff [simp]: "exact_seq ([G,H], (g # q))"
inductive_simps exact_seq_cons_iff [simp]: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma
assumes
( G1 f thus
  (\<open>(\<open>indent=3 notation=\<open>mixfix exact_seq\<close>\<close>_ / \<longlongrightarrow>\<index> _)\<close> [1000, 60]).unity
  where " assume " \<noteq> 0" hence "i \<ge> Suc 0" by simp

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         Suc_le_D. by  ? using.IHsimp

lemma dropped_seq_is_exact_seqjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
   "exact_seq (G ) "i:natlength
  shows "exact_seq (drop i G, drop i F)"
proof- :
    " ( # ,)\\<^bsub>g1\<^esub> G2 \\<^bsub>g2\<^esub> G3"
    using that
  proof " ` carrierG1) kernelG2G3"
    case(nity  f  ?java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
by add.)
  next
    case (extension ift and t
     ()
      assume "i = case( G1 G2 f)
        using exact_seq
    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
      assume "i \ 0" hence "i \ Suc 0" by simp
then   "kk
        using Suc_le_D extension.
      thus andSuc
qed

  thus  )
qed

lemma (2 [OF()  auto
  assumes  q
  shows " i =G! G! ( i) #(G S ( i)) "
  using exact_seq_length1[OF assms(1)] dropped_seq_is_exact_seq
exact_seq_length2(]assms(simpdrop_Suc

lemma exact_seq_imp_exact_hom le_imp_less_Sucprod())
    "exact_seq (G1 #,)\\<^bsub>g1\<^esub> G2 \\<^bsub>g2\<^esub> G3"
    "g1 `(carrier G1) =k G2 G3 "
proof exact_seq_imp_exact_hom "G !i" " (Suc i)" "G ! (Suc(uci))" lq by uto
  have "(hd (tl (snd t))) ` (carrier (hd (tl (tl (fst t))))) =
            kernel
    if " t" andjava.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 85
    usingproof()
  proof (    case (unity G1G2)
    case (unity G1 G2 f)
    then show ?  showcasebyauto
  next
    casecase( Gl Hh
     show  group_hom_axioms_def
  qed
  with assms
qedshowthesisusingaux_lemma assms

lemma
   assumes
"
    "(F!( )) carrier(G!( (Suc )) kernel ( !(uci)! )( !i"
proof -
  have    exact_seq F  "i
using2[OF)byjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
   obtain
    where "drop i G = (G ! thenobtainl
a  dropF=F! i # F!Suc#q"
    by (metis Cons_nth_drop_Suc Suc_less_eq  " i F ( i #q"
        le_eq_less_or_eq prodsel)
  thus ?thesis
  using[OFassms, of]assms2)
        exact_seq_imp_exact_hom[of "G ! i" "G ! (Suc i)" "G ! (Suc (thus ?thesis
qed

lemma exact_seq_imp_group_hom :
  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))) usingdropped_seq_is_exact_seq[OF assms(1), of ] (2)
    sing
  proof (induction)
    subsection
    then  ?case  auto
  next
    case (extension G l g
    then ?caseunfolding  group_hom_axioms_def
  qed
  show ?   "exact_seq ([G1][
    by simp
qed    andinj_onG1

a:
assumesexact_seq,F"and "( : )<length
-
proofassumeG2 " G2"
  length i F
    using assms2) [OFassmsauto
  then obtain l q[ (1),  " 0"]
     drop =G! ) # java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
     and     [ assms),of by simp
    by metis Suc_leI exact_seq_length1
        le_eq_less_or_eq le_imp_less_Sucjava.lang.StringIndexOutOfBoundsException: Range [12, 2) out of bounds for length 33
   ?
  using dropped_seq_is_exact_seq " ( G2) carrier G3"
exact_seq_imp_group_hom G  i G Suc) lqFi" bybysimp
qed

subsectionsolvableand :" "byauto

lemma exact_seq_solvable_imp :
exact_seq][)\<\^bsub\<^esub> G2 \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
    andg1carrier
    and "g2 ` (carrier G2) = carrier G3"
  showsG2
proof -
  assume G2 [OF g2 assms
  have "group_hom G1 G2exact_seq_imp_exact_hom[OF assms(1) G3 auto
    using java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  hence "solvable G1"
    using "inj_on g1 (carrier )"
moreover" G2G2 G3 g2"
    using exact_seq_imp_group_hom_arbitrary[OF assms(1), ofshows( G1\<and> (solvable G3) \<longleftrightarrow>  solvable G2"
java.lang.StringIndexOutOfBoundsException: Range [62, 21) out of bounds for length 21
    using.surj_hom_imp_solvable G2 g2assmsG2 simp
  ultimately assumesCB]
qed

lemma exact_seq_solvable_recip :
  assumes "exact_seq ([G1],[]) \\<^bsub>g1\<^esub> G2 \\<^bsub>g2\<^esub> G3"
java.lang.StringIndexOutOfBoundsException: Range [29, 2) out of bounds for length 29
    and "g2 ` (carrier G2) = carrier G3"
  shows "(solvable G1) \ (solvable G3) \ solvable G2"
proof -
  assume "(solvable G1) \ (solvable G3)"
  hence      (auto : group_homimage_from_trivial_grouptrivial_group_def hom_one
   g1G1" g2 group_hom java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 60
     [ ()  " 0"
apply( simp  hom_def)
  showdone
    using[  g2()java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
          [OF1 G1 by
qed

proposition
exact_seq_imp_triviality:
    and "inj_on g1 (carrier G1)"
    and
  b (, lifting. bij_betw_def )
  using exact_seq_solvable_recip

:
   "exact_seq ([ED,,,],[k,gf)java.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof
  assume C: ex
assms inj_on
    apply (x,)
    apply (simp "ij\java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
    done f  GCf
  with Cshow
     auto.image_from_trivial_group)
     apply (auto simp by add group_hom_axioms_def
    done
next
  assume ?rhs
  with assms show "trivial_group C"
    applysimp:trivial_group_def
    by( group_hom.trivial_hom_iff group_hom_def
qed

lemma exact_seq_imp_triviality:
       'where"h
  by (metis (no_types, lifting .hh   ) " hjava.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 90

lemma:
   "exact_seq ([D case_prod_unfold
  y(autotrivial_homomorphism_def

lemmae:
   "exact_seq homgf: ? \ hom G (C \\ D)"
u ex add

lemma exact_sequence_sum_lemmaij
  assumes( simp  homij
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
    and \<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" ' )
    and gjk: "\x. x \ carrier B \ g(j x) = k x"
  shows "( have "i (h' (f x)) \\<^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)))"
    (is " by java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
proofrule
  interpret comm_group G
    by (rule       <open>h' \<in> hom C A\<close> hom_in_carrier x by fastforce
  interpret      " \ (\(x, y). i x \\<^bsub>G\<^esub> j y) ` (carrier A \ carrier B)"
    using ex by (simp addusingapply( simp)
   : G g
    using ex rule_tac bexI)
  interpret i:  meson
    using ex by (simp add
  interpret j: group_hom B G j
    using ex by (simp apply( group [ f  java.lang.NullPointerException
  have: kernel  carrier group "i
    using ex by (auto simp metis . fih)
  then'h\java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85
    and java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    using h by (\openPorted
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    unfolding case_prod_unfold
    apply hom_group_mult
    using ex simp_allunfolded)
  show homgf: "?gf \ hom G (C \\ D)"
      by add
  show "applysimp: epi_iff_subset group_homintrogroup_hom.kernel_to_trivial_group group_hom_axioms.introjava.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
   short_exact_sequence_iff
     java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
assumex xjava.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
     ( exI

    withthat add singleton_group_def
      by have" {
have '( )\
      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 "\ = x"
      using \<open>h' \<in> hom C A\<close> hom_in_carrier x by fastforce add  group_hom_axioms_def)
    finally java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
      using x y apply (clarsimp
      java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
applyrule_tac java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
by ( \<open>h' \<in> hom C A\<close> f.hom_closed hom_in_carrier)
  apply imp
  show "(\z. (f z, g z)) \ (\(x, y). i x \\<^bsub>G\<^esub> j y) \ Group.iso (A \\ B) (C \\ D)"
    apply (rule group.iso_eq (, lifting.image_from_trivial_group.java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
    using
    apply (auto simp:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     pply . f. fih)
    applyandKB  1:"H\java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
done
qed

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

definition short_exact_sequence
  whereshort_exact_sequence Cfg\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:
   " A BCf g " ([,,C,[,])
  using assms
  apply (auto simp: short_exact_sequence_def group_hom_def group_hom_axioms_def)
  apply( addepi_iff_subset group_hom group_hom. group_hom_axioms)
  by (metis (no_types, liftinghavegroup" B" groupC and:" BCg ` by (auto simp: group_hom_axioms_def)
hom_one  mem_Collect_eq trivial_group_def

lemma short_exact_sequence_iff have" H B
"java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 129

bmetis" ( B ) ( C(`carrier B)"
    if "exact_seq (metisassms3) fABHsubgroupE1) BCimg_is_subgroup .set_mult_ker_hom(2) ker_eq subgroup.carrier_subgroup_generated_subgroup)
-
    show ?thesis simp group_hom_axioms_def
      unfolding short_exact_sequence_def
    proof exIjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
      have "kernel A (singleton_group \\<^bsub>A\<^esub>) (\x. \\<^bsub>A\<^esub>) = f ` carrier B"
        using that by (simp that.Hcarrier_subgroup_generated_subset
moreover CBg  \one
        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)], [\x. \\<^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 by blast
qed

lemma:
  assumesqed
  showsg
  using assms"" gKBC byauto
  applysimp
  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:
a exshort_exact_sequence "and:"`carrier
subgroup 1H
  shows "g \ iso (subgroup_generated B K) (subgroup_generated C(g ` carrier B))"
proof -
  interpret KB: subgroup K B
    proof
  interpretusing sh [ex
    using  java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
  interpreth C= C(  arrier
    ex add )
  have "group A" "group B" "group applyjava.lang.StringIndexOutOfBoundsException: Range [14, 10) out of bounds for length 14
      using  short_exact_sequenceD ex
: kernel
    using ex byjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  then assumes  [,,A,gf) and f:"f' \ hom B A" and iso: "(f' \ f) \ iso A A"
    using by simp: group_hom)
  show ?thesis
    unfoldingiso_iff
  proof (intro conjI)
    showg\<in> hom (subgroup_generated B K) (subgroup_generated C(g ` carrier B))"
by( 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_generatedgroup_hom_axioms_def)
      by (metis assms(3) eq fAB.H   "group A" " B"" : " B Cg=f  A"
    interpret gKBC: group_hom "subgroup_generated have :"f  Ajava.lang.StringIndexOutOfBoundsException: Index 149 out of bounds for length 149
      apply (auto simp: group_hom_def group_hom_axioms_def  interpret ': B Af'
      by (simp add:     using assms by (auto: group_hom_def)
x  \<one>\<^bsub>B\<^esub>"
       x:" carrier (subgroup_generated B K)" and "g x = \\<^bsub>C\<^esub>" for xin (subgroup_generated B ) "g x =x = \<one>\<^bsub>C\<^esub>" for x
    proof -java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
      have:"x
        using thatjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
      moreover have "x \ H"
          fimauto
      ultimately C
by 1 Int_iff KB subsetCE
    qed
    showusing  ( simp'.subgroup_kernel)
      using "*" gKBC
  qed
qed

lemma splitting_sublemma:
    assumes " ([C, g,)" and f' "f \ hom B A"
      and"subgroupK B"and1 " K \ {one B}" and eq: "set_mult B H K = carrier B"
    shows "f \ iso A (subgroup_generated B H)" (is ?f)
          g
proof -
  show ?f
    using [OF]
    apply (clarsimp simp add"f \ iso A (subgroup_generated B H)" "g \ iso (subgroup_generated B K) C"
    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 gBCgroup_homBCg
      using by ( addgroup_hom_defgroup_hom_axioms_def
    using [OF
    by (metis "1" \<open>subgroup K B\<close> eq fim splitting_sublemma_gen)using by(auto: group_hom_def)
qedusing by (uto simp  introgroup [OF

lemma  that [OF 'iso surj)
  assumes ex: "exact_seq ([C,B,A], [g,f]java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    and injf: "inj_on f (carrier A)" and surj K java.lang.NullPointerException
obtains
"\ iso A (subgroup_generated B H)" "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

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