(*  Title:      HOL/Algebra/Sylow.thy

     Author:     Florian Kammueller, with new proofs by L C Paulson

 *)

 theory Sylow

 imports Coset Exponent

 begin

 text {*

   See also \cite{Kammueller-Paulson:1999}.

 *}

 text{*The combinatorial argument is in theory Exponent*}

 locale sylow = group +

   fixes p and a and m and calM and RelM

   assumes prime_p:   "prime p"

       and order_G:   "order(G) = (p^a) * m"

       and finite_G [iff]:  "finite (carrier G)"

   defines "calM == {s. s \<subseteq> carrier(G) & card(s) = p^a}"

       and "RelM == {(N1,N2). N1 \<in> calM & N2 \<in> calM &

                              (\<exists>g \<in> carrier(G). N1 = (N2 #> g) )}"

 lemma (in sylow) RelM_refl_on: "refl_on calM RelM"

 apply (auto simp add: refl_on_def RelM_def calM_def)

 apply (blast intro!: coset_mult_one [symmetric])

 done

 lemma (in sylow) RelM_sym: "sym RelM"

 proof (unfold sym_def RelM_def, clarify)

   fix y g

   assume   "y \<in> calM"

     and g: "g \<in> carrier G"

   hence "y = y #> g #> (inv g)" by (simp add: coset_mult_assoc calM_def)

   thus "\<exists>g'\<in>carrier G. y = y #> g #> g'" by (blast intro: g)

 qed

 lemma (in sylow) RelM_trans: "trans RelM"

 by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc)

 lemma (in sylow) RelM_equiv: "equiv calM RelM"

 apply (unfold equiv_def)

 apply (blast intro: RelM_refl_on RelM_sym RelM_trans)

 done

 lemma (in sylow) M_subset_calM_prep: "M' \<in> calM // RelM  ==> M' \<subseteq> calM"

 apply (unfold RelM_def)

 apply (blast elim!: quotientE)

 done

 subsection{*Main Part of the Proof*}

 locale sylow_central = sylow +

   fixes H and M1 and M

   assumes M_in_quot:  "M \<in> calM // RelM"

       and not_dvd_M:  "~(p ^ Suc(exponent p m) dvd card(M))"

       and M1_in_M:    "M1 \<in> M"

   defines "H == {g. g\<in>carrier G & M1 #> g = M1}"

 lemma (in sylow_central) M_subset_calM: "M \<subseteq> calM"

 by (rule M_in_quot [THEN M_subset_calM_prep])

 lemma (in sylow_central) card_M1: "card(M1) = p^a"

 apply (cut_tac M_subset_calM M1_in_M)

 apply (simp add: calM_def, blast)

 done

 lemma card_nonempty: "0 < card(S) ==> S \<noteq> {}"

 by force

 lemma (in sylow_central) exists_x_in_M1: "\<exists>x. x\<in>M1"

 apply (subgoal_tac "0 < card M1")

  apply (blast dest: card_nonempty)

 apply (cut_tac prime_p [THEN prime_imp_one_less])

 apply (simp (no_asm_simp) add: card_M1)

 done

 lemma (in sylow_central) M1_subset_G [simp]: "M1 \<subseteq> carrier G"

 apply (rule subsetD [THEN PowD])

 apply (rule_tac [2] M1_in_M)

 apply (rule M_subset_calM [THEN subset_trans])

 apply (auto simp add: calM_def)

 done

 lemma (in sylow_central) M1_inj_H: "\<exists>f \<in> H\<rightarrow>M1. inj_on f H"

   proof -

     from exists_x_in_M1 obtain m1 where m1M: "m1 \<in> M1"..

     have m1G: "m1 \<in> carrier G" by (simp add: m1M M1_subset_G [THEN subsetD])

     show ?thesis

     proof

       show "inj_on (\<lambda>z\<in>H. m1 \<otimes> z) H"

         by (simp add: inj_on_def l_cancel [of m1 x y, THEN iffD1] H_def m1G)

       show "restrict (op \<otimes> m1) H \<in> H \<rightarrow> M1"

       proof (rule restrictI)

         fix z assume zH: "z \<in> H"

         show "m1 \<otimes> z \<in> M1"

         proof -

           from zH

           have zG: "z \<in> carrier G" and M1zeq: "M1 #> z = M1"

             by (auto simp add: H_def)

           show ?thesis

             by (rule subst [OF M1zeq], simp add: m1M zG rcosI)

         qed

       qed

     qed

   qed

 subsection{*Discharging the Assumptions of @{text sylow_central}*}

 lemma (in sylow) EmptyNotInEquivSet: "{} \<notin> calM // RelM"

 by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self])

 lemma (in sylow) existsM1inM: "M \<in> calM // RelM ==> \<exists>M1. M1 \<in> M"

 apply (subgoal_tac "M \<noteq> {}")

  apply blast

 apply (cut_tac EmptyNotInEquivSet, blast)

 done

 lemma (in sylow) zero_less_o_G: "0 < order(G)"

 apply (unfold order_def)

 apply (blast intro: zero_less_card_empty)

 done

 lemma (in sylow) zero_less_m: "m > 0"

 apply (cut_tac zero_less_o_G)

 apply (simp add: order_G)

 done

 lemma (in sylow) card_calM: "card(calM) = (p^a) * m choose p^a"

 by (simp add: calM_def n_subsets order_G [symmetric] order_def)

 lemma (in sylow) zero_less_card_calM: "card calM > 0"

 by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m)

 lemma (in sylow) max_p_div_calM:

      "~ (p ^ Suc(exponent p m) dvd card(calM))"

 apply (subgoal_tac "exponent p m = exponent p (card calM) ")

  apply (cut_tac zero_less_card_calM prime_p)

  apply (force dest: power_Suc_exponent_Not_dvd)

 apply (simp add: card_calM zero_less_m [THEN const_p_fac])

 done

 lemma (in sylow) finite_calM: "finite calM"

 apply (unfold calM_def)

 apply (rule_tac B = "Pow (carrier G) " in finite_subset)

 apply auto

 done

 lemma (in sylow) lemma_A1:

      "\<exists>M \<in> calM // RelM. ~ (p ^ Suc(exponent p m) dvd card(M))"

 apply (rule max_p_div_calM [THEN contrapos_np])

 apply (simp add: finite_calM equiv_imp_dvd_card [OF _ RelM_equiv])

 done

 subsubsection{*Introduction and Destruct Rules for @{term H}*}

 lemma (in sylow_central) H_I: "[|g \<in> carrier G; M1 #> g = M1|] ==> g \<in> H"

 by (simp add: H_def)

 lemma (in sylow_central) H_into_carrier_G: "x \<in> H ==> x \<in> carrier G"

 by (simp add: H_def)

 lemma (in sylow_central) in_H_imp_eq: "g : H ==> M1 #> g = M1"

 by (simp add: H_def)

 lemma (in sylow_central) H_m_closed: "[| x\<in>H; y\<in>H|] ==> x \<otimes> y \<in> H"

 apply (unfold H_def)

 apply (simp add: coset_mult_assoc [symmetric])

 done

 lemma (in sylow_central) H_not_empty: "H \<noteq> {}"

 apply (simp add: H_def)

 apply (rule exI [of _ \<one>], simp)

 done

 lemma (in sylow_central) H_is_subgroup: "subgroup H G"

 apply (rule subgroupI)

 apply (rule subsetI)

 apply (erule H_into_carrier_G)

 apply (rule H_not_empty)

 apply (simp add: H_def, clarify)

 apply (erule_tac P = "%z. ?lhs(z) = M1" in subst)

 apply (simp add: coset_mult_assoc )

 apply (blast intro: H_m_closed)

 done

 lemma (in sylow_central) rcosetGM1g_subset_G:

      "[| g \<in> carrier G; x \<in> M1 #>  g |] ==> x \<in> carrier G"

 by (blast intro: M1_subset_G [THEN r_coset_subset_G, THEN subsetD])

 lemma (in sylow_central) finite_M1: "finite M1"

 by (rule finite_subset [OF M1_subset_G finite_G])

 lemma (in sylow_central) finite_rcosetGM1g: "g\<in>carrier G ==> finite (M1 #> g)"

 apply (rule finite_subset)

 apply (rule subsetI)

 apply (erule rcosetGM1g_subset_G, assumption)

 apply (rule finite_G)

 done

 lemma (in sylow_central) M1_cardeq_rcosetGM1g:

      "g \<in> carrier G ==> card(M1 #> g) = card(M1)"

 by (simp (no_asm_simp) add: card_cosets_equal rcosetsI)

 lemma (in sylow_central) M1_RelM_rcosetGM1g:

      "g \<in> carrier G ==> (M1, M1 #> g) \<in> RelM"

 apply (simp (no_asm) add: RelM_def calM_def card_M1)

 apply (rule conjI)

  apply (blast intro: rcosetGM1g_subset_G)

 apply (simp (no_asm_simp) add: card_M1 M1_cardeq_rcosetGM1g)

 apply (rule bexI [of _ "inv g"])

 apply (simp_all add: coset_mult_assoc)

 done

 subsection{*Equal Cardinalities of @{term M} and the Set of Cosets*}

 text{*Injections between @{term M} and @{term "rcosets\<^bsub>G\<^esub> H"} show that

  their cardinalities are equal.*}

 lemma ElemClassEquiv:

      "[| equiv A r; C \<in> A // r |] ==> \<forall>x \<in> C. \<forall>y \<in> C. (x,y)\<in>r"

 by (unfold equiv_def quotient_def sym_def trans_def, blast)

 lemma (in sylow_central) M_elem_map:

      "M2 \<in> M ==> \<exists>g. g \<in> carrier G & M1 #> g = M2"

 apply (cut_tac M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]])

 apply (simp add: RelM_def)

 apply (blast dest!: bspec)

 done

 lemmas (in sylow_central) M_elem_map_carrier =

         M_elem_map [THEN someI_ex, THEN conjunct1]

 lemmas (in sylow_central) M_elem_map_eq =

         M_elem_map [THEN someI_ex, THEN conjunct2]

 lemma (in sylow_central) M_funcset_rcosets_H:

      "(%x:M. H #> (SOME g. g \<in> carrier G & M1 #> g = x)) \<in> M \<rightarrow> rcosets H"

 apply (rule rcosetsI [THEN restrictI])

 apply (rule H_is_subgroup [THEN subgroup.subset])

 apply (erule M_elem_map_carrier)

 done

 lemma (in sylow_central) inj_M_GmodH: "\<exists>f \<in> M\<rightarrow>rcosets H. inj_on f M"

 apply (rule bexI)

 apply (rule_tac [2] M_funcset_rcosets_H)

 apply (rule inj_onI, simp)

 apply (rule trans [OF _ M_elem_map_eq])

 prefer 2 apply assumption

 apply (rule M_elem_map_eq [symmetric, THEN trans], assumption)

 apply (rule coset_mult_inv1)

 apply (erule_tac [2] M_elem_map_carrier)+

 apply (rule_tac [2] M1_subset_G)

 apply (rule coset_join1 [THEN in_H_imp_eq])

 apply (rule_tac [3] H_is_subgroup)

 prefer 2 apply (blast intro: M_elem_map_carrier)

 apply (simp add: coset_mult_inv2 H_def M_elem_map_carrier subset_eq)

 done

 subsubsection{*The Opposite Injection*}

 lemma (in sylow_central) H_elem_map:

      "H1 \<in> rcosets H ==> \<exists>g. g \<in> carrier G & H #> g = H1"

 by (auto simp add: RCOSETS_def)

 lemmas (in sylow_central) H_elem_map_carrier =

         H_elem_map [THEN someI_ex, THEN conjunct1]

 lemmas (in sylow_central) H_elem_map_eq =

         H_elem_map [THEN someI_ex, THEN conjunct2]

 lemma EquivElemClass:

      "[|equiv A r; M \<in> A//r; M1\<in>M; (M1,M2) \<in> r |] ==> M2 \<in> M"

 by (unfold equiv_def quotient_def sym_def trans_def, blast)

 lemma (in sylow_central) rcosets_H_funcset_M:

   "(\<lambda>C \<in> rcosets H. M1 #> (@g. g \<in> carrier G \<and> H #> g = C)) \<in> rcosets H \<rightarrow> M"

 apply (simp add: RCOSETS_def)

 apply (fast intro: someI2

             intro!: M1_in_M EquivElemClass [OF RelM_equiv M_in_quot _  M1_RelM_rcosetGM1g])

 done

 text{*close to a duplicate of @{text inj_M_GmodH}*}

 lemma (in sylow_central) inj_GmodH_M:

      "\<exists>g \<in> rcosets H\<rightarrow>M. inj_on g (rcosets H)"

 apply (rule bexI)

 apply (rule_tac [2] rcosets_H_funcset_M)

 apply (rule inj_onI)

 apply (simp)

 apply (rule trans [OF _ H_elem_map_eq])

 prefer 2 apply assumption

 apply (rule H_elem_map_eq [symmetric, THEN trans], assumption)

 apply (rule coset_mult_inv1)

 apply (erule_tac [2] H_elem_map_carrier)+

 apply (rule_tac [2] H_is_subgroup [THEN subgroup.subset])

 apply (rule coset_join2)

 apply (blast intro: H_elem_map_carrier)

 apply (rule H_is_subgroup)

 apply (simp add: H_I coset_mult_inv2 H_elem_map_carrier)

 done

 lemma (in sylow_central) calM_subset_PowG: "calM \<subseteq> Pow(carrier G)"

 by (auto simp add: calM_def)

 lemma (in sylow_central) finite_M: "finite M"

 apply (rule finite_subset)

 apply (rule M_subset_calM [THEN subset_trans])

 apply (rule calM_subset_PowG, blast)

 done

 lemma (in sylow_central) cardMeqIndexH: "card(M) = card(rcosets H)"

 apply (insert inj_M_GmodH inj_GmodH_M)

 apply (blast intro: card_bij finite_M H_is_subgroup

              rcosets_subset_PowG [THEN finite_subset]

              finite_Pow_iff [THEN iffD2])

 done

 lemma (in sylow_central) index_lem: "card(M) * card(H) = order(G)"

 by (simp add: cardMeqIndexH lagrange H_is_subgroup)

 lemma (in sylow_central) lemma_leq1: "p^a \<le> card(H)"

 apply (rule dvd_imp_le)

  apply (rule div_combine [OF prime_p not_dvd_M])

  prefer 2 apply (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)

 apply (simp add: index_lem order_G power_add mult_dvd_mono power_exponent_dvd

                  zero_less_m)

 done

 lemma (in sylow_central) lemma_leq2: "card(H) \<le> p^a"

 apply (subst card_M1 [symmetric])

 apply (cut_tac M1_inj_H)

 apply (blast intro!: M1_subset_G intro:

              card_inj H_into_carrier_G finite_subset [OF _ finite_G])

 done

 lemma (in sylow_central) card_H_eq: "card(H) = p^a"

 by (blast intro: le_antisym lemma_leq1 lemma_leq2)

 lemma (in sylow) sylow_thm: "\<exists>H. subgroup H G & card(H) = p^a"

 apply (cut_tac lemma_A1, clarify)

 apply (frule existsM1inM, clarify)

 apply (subgoal_tac "sylow_central G p a m M1 M")

  apply (blast dest:  sylow_central.H_is_subgroup sylow_central.card_H_eq)

 apply (simp add: sylow_central_def sylow_central_axioms_def sylow_axioms calM_def RelM_def)

 done

 text{*Needed because the locale's automatic definition refers to

    @{term "semigroup G"} and @{term "group_axioms G"} rather than

   simply to @{term "group G"}.*}

 lemma sylow_eq: "sylow G p a m = (group G & sylow_axioms G p a m)"

 by (simp add: sylow_def group_def)

 subsection {* Sylow's Theorem *}

 theorem sylow_thm:

      "[| prime p;  group(G);  order(G) = (p^a) * m; finite (carrier G)|]

       ==> \<exists>H. subgroup H G & card(H) = p^a"

 apply (rule sylow.sylow_thm [of G p a m])

 apply (simp add: sylow_eq sylow_axioms_def)

 done

 end