src/HOL/Algebra/Sylow.thy
author paulson <lp15@cam.ac.uk>
Sun Jun 24 11:41:32 2018 +0100 (14 months ago)
changeset 68488 dfbd80c3d180
parent 68484 59793df7f853
child 68561 5e85cda58af6
permissions -rw-r--r--
more modernisaton and de-applying
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(*  Title:      HOL/Algebra/Sylow.thy
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    Author:     Florian Kammueller, with new proofs by L C Paulson
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*)
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theory Sylow
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  imports Coset Exponent
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begin
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text \<open>See also @{cite "Kammueller-Paulson:1999"}.\<close>
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text \<open>The combinatorial argument is in theory @{theory "HOL-Algebra.Exponent"}.\<close>
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lemma le_extend_mult: "\<lbrakk>0 < c; a \<le> b\<rbrakk> \<Longrightarrow> a \<le> b * c"
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  for c :: nat
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  by (metis divisors_zero dvd_triv_left leI less_le_trans nat_dvd_not_less zero_less_iff_neq_zero)
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locale sylow = group +
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  fixes p and a and m and calM and RelM
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  assumes prime_p: "prime p"
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    and order_G: "order G = (p^a) * m"
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    and finite_G[iff]: "finite (carrier G)"
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  defines "calM \<equiv> {s. s \<subseteq> carrier G \<and> card s = p^a}"
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    and "RelM \<equiv> {(N1, N2). N1 \<in> calM \<and> N2 \<in> calM \<and> (\<exists>g \<in> carrier G. N1 = N2 #> g)}"
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begin
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lemma RelM_refl_on: "refl_on calM RelM"
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  by (auto simp: refl_on_def RelM_def calM_def) (blast intro!: coset_mult_one [symmetric])
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lemma RelM_sym: "sym RelM"
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proof (unfold sym_def RelM_def, clarify)
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  fix y g
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  assume "y \<in> calM"
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    and g: "g \<in> carrier G"
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  then have "y = y #> g #> (inv g)"
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    by (simp add: coset_mult_assoc calM_def)
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  then show "\<exists>g'\<in>carrier G. y = y #> g #> g'"
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    by (blast intro: g)
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qed
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lemma RelM_trans: "trans RelM"
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  by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc)
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lemma RelM_equiv: "equiv calM RelM"
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  unfolding equiv_def by (blast intro: RelM_refl_on RelM_sym RelM_trans)
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lemma M_subset_calM_prep: "M' \<in> calM // RelM  \<Longrightarrow> M' \<subseteq> calM"
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  unfolding RelM_def by (blast elim!: quotientE)
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end
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subsection \<open>Main Part of the Proof\<close>
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locale sylow_central = sylow +
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  fixes H and M1 and M
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  assumes M_in_quot: "M \<in> calM // RelM"
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    and not_dvd_M: "\<not> (p ^ Suc (multiplicity p m) dvd card M)"
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    and M1_in_M: "M1 \<in> M"
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  defines "H \<equiv> {g. g \<in> carrier G \<and> M1 #> g = M1}"
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begin
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lemma M_subset_calM: "M \<subseteq> calM"
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  by (rule M_in_quot [THEN M_subset_calM_prep])
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lemma card_M1: "card M1 = p^a"
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  using M1_in_M M_subset_calM calM_def by blast
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lemma exists_x_in_M1: "\<exists>x. x \<in> M1"
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  using prime_p [THEN prime_gt_Suc_0_nat] card_M1
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  by (metis Suc_lessD card_eq_0_iff empty_subsetI equalityI gr_implies_not0 nat_zero_less_power_iff subsetI)
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lemma M1_subset_G [simp]: "M1 \<subseteq> carrier G"
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  using M1_in_M M_subset_calM calM_def mem_Collect_eq subsetCE by blast
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lemma M1_inj_H: "\<exists>f \<in> H\<rightarrow>M1. inj_on f H"
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proof -
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  from exists_x_in_M1 obtain m1 where m1M: "m1 \<in> M1"..
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  have m1: "m1 \<in> carrier G"
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    by (simp add: m1M M1_subset_G [THEN subsetD])
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  show ?thesis
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  proof
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    show "inj_on (\<lambda>z\<in>H. m1 \<otimes> z) H"
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      by (simp add: H_def inj_on_def m1)
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    show "restrict ((\<otimes>) m1) H \<in> H \<rightarrow> M1"
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    proof (rule restrictI)
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      fix z
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      assume zH: "z \<in> H"
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      show "m1 \<otimes> z \<in> M1"
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      proof -
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        from zH
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        have zG: "z \<in> carrier G" and M1zeq: "M1 #> z = M1"
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          by (auto simp add: H_def)
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        show ?thesis
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          by (rule subst [OF M1zeq]) (simp add: m1M zG rcosI)
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      qed
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    qed
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  qed
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qed
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end
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subsection \<open>Discharging the Assumptions of \<open>sylow_central\<close>\<close>
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context sylow
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begin
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lemma EmptyNotInEquivSet: "{} \<notin> calM // RelM"
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  by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self])
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lemma existsM1inM: "M \<in> calM // RelM \<Longrightarrow> \<exists>M1. M1 \<in> M"
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  using RelM_equiv equiv_Eps_in by blast
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lemma zero_less_o_G: "0 < order G"
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  by (simp add: order_def card_gt_0_iff carrier_not_empty)
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lemma zero_less_m: "m > 0"
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  using zero_less_o_G by (simp add: order_G)
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lemma card_calM: "card calM = (p^a) * m choose p^a"
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  by (simp add: calM_def n_subsets order_G [symmetric] order_def)
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lemma zero_less_card_calM: "card calM > 0"
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  by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m)
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lemma max_p_div_calM: "\<not> (p ^ Suc (multiplicity p m) dvd card calM)"
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proof
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  assume "p ^ Suc (multiplicity p m) dvd card calM"
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  with zero_less_card_calM prime_p
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  have "Suc (multiplicity p m) \<le> multiplicity p (card calM)"
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    by (intro multiplicity_geI) auto
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  then have "multiplicity p m < multiplicity p (card calM)" by simp
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  also have "multiplicity p m = multiplicity p (card calM)"
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    by (simp add: const_p_fac prime_p zero_less_m card_calM)
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  finally show False by simp
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qed
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lemma finite_calM: "finite calM"
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  unfolding calM_def by (rule finite_subset [where B = "Pow (carrier G)"]) auto
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lemma lemma_A1: "\<exists>M \<in> calM // RelM. \<not> (p ^ Suc (multiplicity p m) dvd card M)"
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  using RelM_equiv equiv_imp_dvd_card finite_calM max_p_div_calM by blast
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end
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subsubsection \<open>Introduction and Destruct Rules for \<open>H\<close>\<close>
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context sylow_central
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begin
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lemma H_I: "\<lbrakk>g \<in> carrier G; M1 #> g = M1\<rbrakk> \<Longrightarrow> g \<in> H"
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  by (simp add: H_def)
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lemma H_into_carrier_G: "x \<in> H \<Longrightarrow> x \<in> carrier G"
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  by (simp add: H_def)
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lemma in_H_imp_eq: "g \<in> H \<Longrightarrow> M1 #> g = M1"
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  by (simp add: H_def)
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lemma H_m_closed: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"
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  by (simp add: H_def coset_mult_assoc [symmetric])
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lemma H_not_empty: "H \<noteq> {}"
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  by (force simp add: H_def intro: exI [of _ \<one>])
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lemma H_is_subgroup: "subgroup H G"
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proof (rule subgroupI)
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  show "H \<subseteq> carrier G"
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    using H_into_carrier_G by blast
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  show "\<And>a. a \<in> H \<Longrightarrow> inv a \<in> H"
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    by (metis H_I H_into_carrier_G H_m_closed M1_subset_G Units_eq Units_inv_closed Units_inv_inv coset_mult_inv1 in_H_imp_eq)
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  show "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"
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    by (blast intro: H_m_closed)
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qed (use H_not_empty in auto)
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lemma rcosetGM1g_subset_G: "\<lbrakk>g \<in> carrier G; x \<in> M1 #> g\<rbrakk> \<Longrightarrow> x \<in> carrier G"
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  by (blast intro: M1_subset_G [THEN r_coset_subset_G, THEN subsetD])
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lemma finite_M1: "finite M1"
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  by (rule finite_subset [OF M1_subset_G finite_G])
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lemma finite_rcosetGM1g: "g \<in> carrier G \<Longrightarrow> finite (M1 #> g)"
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  using rcosetGM1g_subset_G finite_G M1_subset_G cosets_finite rcosetsI by blast
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lemma M1_cardeq_rcosetGM1g: "g \<in> carrier G \<Longrightarrow> card (M1 #> g) = card M1"
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  by (metis M1_subset_G card_rcosets_equal rcosetsI)
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lemma M1_RelM_rcosetGM1g: 
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  assumes "g \<in> carrier G"
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  shows "(M1, M1 #> g) \<in> RelM"
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proof -
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  have "M1 #> g \<subseteq> carrier G"
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    by (simp add: assms r_coset_subset_G)
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  moreover have "card (M1 #> g) = p ^ a"
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    using assms by (simp add: card_M1 M1_cardeq_rcosetGM1g)
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  moreover have "\<exists>h\<in>carrier G. M1 = M1 #> g #> h"
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    by (metis assms M1_subset_G coset_mult_assoc coset_mult_one r_inv_ex)
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  ultimately show ?thesis
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    by (simp add: RelM_def calM_def card_M1)
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qed
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end
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subsection \<open>Equal Cardinalities of \<open>M\<close> and the Set of Cosets\<close>
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text \<open>Injections between @{term M} and @{term "rcosets\<^bsub>G\<^esub> H"} show that
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 their cardinalities are equal.\<close>
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lemma ElemClassEquiv: "\<lbrakk>equiv A r; C \<in> A // r\<rbrakk> \<Longrightarrow> \<forall>x \<in> C. \<forall>y \<in> C. (x, y) \<in> r"
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  unfolding equiv_def quotient_def sym_def trans_def by blast
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context sylow_central
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begin
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lemma M_elem_map: "M2 \<in> M \<Longrightarrow> \<exists>g. g \<in> carrier G \<and> M1 #> g = M2"
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  using M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]]
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  by (simp add: RelM_def) (blast dest!: bspec)
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lemmas M_elem_map_carrier = M_elem_map [THEN someI_ex, THEN conjunct1]
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lemmas M_elem_map_eq = M_elem_map [THEN someI_ex, THEN conjunct2]
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lemma M_funcset_rcosets_H:
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  "(\<lambda>x\<in>M. H #> (SOME g. g \<in> carrier G \<and> M1 #> g = x)) \<in> M \<rightarrow> rcosets H"
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  by (metis (lifting) H_is_subgroup M_elem_map_carrier rcosetsI restrictI subgroup.subset)
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lemma inj_M_GmodH: "\<exists>f \<in> M \<rightarrow> rcosets H. inj_on f M"
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proof
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  let ?inv = "\<lambda>x. SOME g. g \<in> carrier G \<and> M1 #> g = x"
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  show "inj_on (\<lambda>x\<in>M. H #> ?inv x) M"
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  proof (rule inj_onI, simp)
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    fix x y
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    assume eq: "H #> ?inv x = H #> ?inv y" and xy: "x \<in> M" "y \<in> M"
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    have "x = M1 #> ?inv x"
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      by (simp add: M_elem_map_eq \<open>x \<in> M\<close>)
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    also have "... = M1 #> ?inv y"
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    proof (rule coset_mult_inv1 [OF in_H_imp_eq [OF coset_join1]])
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      show "H #> ?inv x \<otimes> inv (?inv y) = H"
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        by (simp add: H_into_carrier_G M_elem_map_carrier xy coset_mult_inv2 eq subsetI)
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    qed (simp_all add: H_is_subgroup M_elem_map_carrier xy)
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    also have "... = y"
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      using M_elem_map_eq \<open>y \<in> M\<close> by simp
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    finally show "x=y" .
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  qed
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  show "(\<lambda>x\<in>M. H #> ?inv x) \<in> M \<rightarrow> rcosets H"
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    by (rule M_funcset_rcosets_H)
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qed
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end
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subsubsection \<open>The Opposite Injection\<close>
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context sylow_central
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begin
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lemma H_elem_map: "H1 \<in> rcosets H \<Longrightarrow> \<exists>g. g \<in> carrier G \<and> H #> g = H1"
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  by (auto simp: RCOSETS_def)
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lemmas H_elem_map_carrier = H_elem_map [THEN someI_ex, THEN conjunct1]
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lemmas H_elem_map_eq = H_elem_map [THEN someI_ex, THEN conjunct2]
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lemma rcosets_H_funcset_M:
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  "(\<lambda>C \<in> rcosets H. M1 #> (SOME g. g \<in> carrier G \<and> H #> g = C)) \<in> rcosets H \<rightarrow> M"
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  using in_quotient_imp_closed [OF RelM_equiv M_in_quot _  M1_RelM_rcosetGM1g]
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  by (simp add: M1_in_M H_elem_map_carrier RCOSETS_def)
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lemma inj_GmodH_M: "\<exists>g \<in> rcosets H\<rightarrow>M. inj_on g (rcosets H)"
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proof
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  let ?inv = "\<lambda>x. SOME g. g \<in> carrier G \<and> H #> g = x"
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  show "inj_on (\<lambda>C\<in>rcosets H. M1 #> ?inv C) (rcosets H)"
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  proof (rule inj_onI, simp)
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    fix x y
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    assume eq: "M1 #> ?inv x = M1 #> ?inv y" and xy: "x \<in> rcosets H" "y \<in> rcosets H"
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    have "x = H #> ?inv x"
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      by (simp add: H_elem_map_eq \<open>x \<in> rcosets H\<close>)
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    also have "... = H #> ?inv y"
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    proof (rule coset_mult_inv1 [OF coset_join2])
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      show "?inv x \<otimes> inv (?inv y) \<in> carrier G"
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        by (simp add: H_elem_map_carrier \<open>x \<in> rcosets H\<close> \<open>y \<in> rcosets H\<close>)
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      then show "(?inv x) \<otimes> inv (?inv y) \<in> H"
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        by (simp add: H_I H_elem_map_carrier xy coset_mult_inv2 eq)
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      show "H \<subseteq> carrier G"
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        by (simp add: H_is_subgroup subgroup.subset)
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    qed (simp_all add: H_is_subgroup H_elem_map_carrier xy)
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    also have "... = y"
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      by (simp add: H_elem_map_eq \<open>y \<in> rcosets H\<close>)
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    finally show "x=y" .
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  qed
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  show "(\<lambda>C\<in>rcosets H. M1 #> ?inv C) \<in> rcosets H \<rightarrow> M"
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    using rcosets_H_funcset_M by blast
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qed
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lemma calM_subset_PowG: "calM \<subseteq> Pow (carrier G)"
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  by (auto simp: calM_def)
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lemma finite_M: "finite M"
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  by (metis M_subset_calM finite_calM rev_finite_subset)
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lemma cardMeqIndexH: "card M = card (rcosets H)"
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  using inj_M_GmodH inj_GmodH_M
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  by (blast intro: card_bij finite_M H_is_subgroup rcosets_subset_PowG [THEN finite_subset])
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lemma index_lem: "card M * card H = order G"
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  by (simp add: cardMeqIndexH lagrange H_is_subgroup)
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lemma card_H_eq: "card H = p^a"
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proof (rule antisym)
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  show "p^a \<le> card H"
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  proof (rule dvd_imp_le)
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    show "p ^ a dvd card H"
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      apply (rule div_combine [OF prime_imp_prime_elem[OF prime_p] not_dvd_M])
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      by (simp add: index_lem multiplicity_dvd order_G power_add)
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    show "0 < card H"
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      by (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)
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  qed
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next
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  show "card H \<le> p^a"
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    using M1_inj_H card_M1 card_inj finite_M1 by fastforce
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qed
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end
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lemma (in sylow) sylow_thm: "\<exists>H. subgroup H G \<and> card H = p^a"
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proof -
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  obtain M where M: "M \<in> calM // RelM" "\<not> (p ^ Suc (multiplicity p m) dvd card M)"
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    using lemma_A1 by blast
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  then obtain M1 where "M1 \<in> M"
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    by (metis existsM1inM) 
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  define H where "H \<equiv> {g. g \<in> carrier G \<and> M1 #> g = M1}"
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  with M \<open>M1 \<in> M\<close>
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  interpret sylow_central G p a m calM RelM H M1 M
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    by unfold_locales (auto simp add: H_def calM_def RelM_def)
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  show ?thesis
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    using H_is_subgroup card_H_eq by blast
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qed
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text \<open>Needed because the locale's automatic definition refers to
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  @{term "semigroup G"} and @{term "group_axioms G"} rather than
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  simply to @{term "group G"}.\<close>
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lemma sylow_eq: "sylow G p a m \<longleftrightarrow> group G \<and> sylow_axioms G p a m"
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  by (simp add: sylow_def group_def)
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subsection \<open>Sylow's Theorem\<close>
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theorem sylow_thm:
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  "\<lbrakk>prime p; group G; order G = (p^a) * m; finite (carrier G)\<rbrakk>
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    \<Longrightarrow> \<exists>H. subgroup H G \<and> card H = p^a"
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  by (rule sylow.sylow_thm [of G p a m]) (simp add: sylow_eq sylow_axioms_def)
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end