# Theory Sylow

theory Sylow
imports Coset Exponent
(*  Title:      HOL/Algebra/Sylow.thy
Author:     Florian Kammueller, with new proofs by L C Paulson
*)

theory Sylow
imports Coset Exponent
begin

text ‹
›

text‹The combinatorial argument is in theory Exponent›

lemma le_extend_mult:
fixes c::nat shows "⟦0 < c; a ≤ b⟧ ⟹ a ≤ b * c"
by (metis divisors_zero dvd_triv_left leI less_le_trans nat_dvd_not_less zero_less_iff_neq_zero)

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 ⊆ carrier(G) & card(s) = p^a}"
and "RelM == {(N1,N2). N1 ∈ calM & N2 ∈ calM &
(∃g ∈ carrier(G). N1 = (N2 #> g) )}"
begin

lemma 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 RelM_sym: "sym RelM"
proof (unfold sym_def RelM_def, clarify)
fix y g
assume   "y ∈ calM"
and g: "g ∈ carrier G"
hence "y = y #> g #> (inv g)" by (simp add: coset_mult_assoc calM_def)
thus "∃g'∈carrier G. y = y #> g #> g'" by (blast intro: g)
qed

lemma RelM_trans: "trans RelM"
by (auto simp add: trans_def RelM_def calM_def coset_mult_assoc)

lemma RelM_equiv: "equiv calM RelM"
apply (unfold equiv_def)
apply (blast intro: RelM_refl_on RelM_sym RelM_trans)
done

lemma M_subset_calM_prep: "M' ∈ calM // RelM  ==> M' ⊆ calM"
apply (unfold RelM_def)
apply (blast elim!: quotientE)
done

end

subsection‹Main Part of the Proof›

locale sylow_central = sylow +
fixes H and M1 and M
assumes M_in_quot:  "M ∈ calM // RelM"
and not_dvd_M:  "~(p ^ Suc(multiplicity p m) dvd card(M))"
and M1_in_M:    "M1 ∈ M"
defines "H == {g. g∈carrier G & M1 #> g = M1}"

begin

lemma M_subset_calM: "M ⊆ calM"
by (rule M_in_quot [THEN M_subset_calM_prep])

lemma card_M1: "card(M1) = p^a"
using M1_in_M M_subset_calM calM_def by blast

lemma exists_x_in_M1: "∃x. x ∈ M1"
using prime_p [THEN prime_gt_Suc_0_nat] card_M1
by (metis Suc_lessD card_eq_0_iff empty_subsetI equalityI gr_implies_not0 nat_zero_less_power_iff subsetI)

lemma M1_subset_G [simp]: "M1 ⊆ carrier G"
using M1_in_M  M_subset_calM calM_def mem_Collect_eq subsetCE by blast

lemma M1_inj_H: "∃f ∈ H→M1. inj_on f H"
proof -
from exists_x_in_M1 obtain m1 where m1M: "m1 ∈ M1"..
have m1G: "m1 ∈ carrier G" by (simp add: m1M M1_subset_G [THEN subsetD])
show ?thesis
proof
show "inj_on (λz∈H. m1 ⊗ z) H"
by (simp add: inj_on_def l_cancel [of m1 x y, THEN iffD1] H_def m1G)
show "restrict (op ⊗ m1) H ∈ H → M1"
proof (rule restrictI)
fix z assume zH: "z ∈ H"
show "m1 ⊗ z ∈ M1"
proof -
from zH
have zG: "z ∈ carrier G" and M1zeq: "M1 #> z = M1"
show ?thesis
by (rule subst [OF M1zeq], simp add: m1M zG rcosI)
qed
qed
qed
qed

end

subsection‹Discharging the Assumptions of ‹sylow_central››

context sylow
begin

lemma EmptyNotInEquivSet: "{} ∉ calM // RelM"
by (blast elim!: quotientE dest: RelM_equiv [THEN equiv_class_self])

lemma existsM1inM: "M ∈ calM // RelM ==> ∃M1. M1 ∈ M"
using RelM_equiv equiv_Eps_in by blast

lemma zero_less_o_G: "0 < order(G)"
by (simp add: order_def card_gt_0_iff carrier_not_empty)

lemma zero_less_m: "m > 0"
using zero_less_o_G by (simp add: order_G)

lemma card_calM: "card(calM) = (p^a) * m choose p^a"
by (simp add: calM_def n_subsets order_G [symmetric] order_def)

lemma zero_less_card_calM: "card calM > 0"
by (simp add: card_calM zero_less_binomial le_extend_mult zero_less_m)

lemma max_p_div_calM:
"~ (p ^ Suc(multiplicity p m) dvd card(calM))"
proof
assume "p ^ Suc (multiplicity p m) dvd card calM"
with zero_less_card_calM prime_p
have "Suc (multiplicity p m) ≤ multiplicity p (card calM)"
by (intro multiplicity_geI) auto
hence "multiplicity p m < multiplicity p (card calM)" by simp
also have "multiplicity p m = multiplicity p (card calM)"
by (simp add: const_p_fac prime_p zero_less_m card_calM)
finally show False by simp
qed

lemma finite_calM: "finite calM"
unfolding calM_def
by (rule_tac B = "Pow (carrier G) " in finite_subset) auto

lemma lemma_A1:
"∃M ∈ calM // RelM. ~ (p ^ Suc(multiplicity p m) dvd card(M))"
using RelM_equiv equiv_imp_dvd_card finite_calM max_p_div_calM by blast

end

subsubsection‹Introduction and Destruct Rules for @{term H}›

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

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

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

lemma (in sylow_central) H_m_closed: "[| x∈H; y∈H|] ==> x ⊗ y ∈ H"
apply (unfold H_def)
done

lemma (in sylow_central) H_not_empty: "H ≠ {}"
apply (rule exI [of _ 𝟭], 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 (erule_tac P = "%z. lhs(z) = M1" for lhs in subst)
apply (blast intro: H_m_closed)
done

lemma (in sylow_central) rcosetGM1g_subset_G:
"[| g ∈ carrier G; x ∈ M1 #>  g |] ==> x ∈ 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∈carrier G ==> finite (M1 #> g)"
using rcosetGM1g_subset_G finite_G M1_subset_G cosets_finite rcosetsI by blast

lemma (in sylow_central) M1_cardeq_rcosetGM1g:
"g ∈ 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 ∈ carrier G ==> (M1, M1 #> g) ∈ RelM"
apply (simp add: RelM_def calM_def card_M1)
apply (rule conjI)
apply (blast intro: rcosetGM1g_subset_G)
apply (metis M1_subset_G coset_mult_assoc coset_mult_one r_inv_ex)
done

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

text‹Injections between @{term M} and @{term "rcosetsG H"} show that
their cardinalities are equal.›

lemma ElemClassEquiv:
"[| equiv A r; C ∈ A // r |] ==> ∀x ∈ C. ∀y ∈ C. (x,y)∈r"
by (unfold equiv_def quotient_def sym_def trans_def, blast)

lemma (in sylow_central) M_elem_map:
"M2 ∈ M ==> ∃g. g ∈ carrier G & M1 #> g = M2"
apply (cut_tac M1_in_M M_in_quot [THEN RelM_equiv [THEN ElemClassEquiv]])
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 ∈ carrier G & M1 #> g = x)) ∈ M → rcosets H"
by (metis (lifting) H_is_subgroup M_elem_map_carrier rcosetsI restrictI subgroup_imp_subset)

lemma (in sylow_central) inj_M_GmodH: "∃f ∈ M → 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 ∈ rcosets H ==> ∃g. g ∈ carrier G & H #> g = H1"

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 (in sylow_central) rcosets_H_funcset_M:
"(λC ∈ rcosets H. M1 #> (@g. g ∈ carrier G ∧ H #> g = C)) ∈ rcosets H → M"
apply (fast intro: someI2
intro!: M1_in_M in_quotient_imp_closed [OF RelM_equiv M_in_quot _  M1_RelM_rcosetGM1g])
done

text‹close to a duplicate of ‹inj_M_GmodH››
lemma (in sylow_central) inj_GmodH_M:
"∃g ∈ rcosets H→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 ⊆ Pow(carrier G)"

lemma (in sylow_central) finite_M: "finite M"
by (metis M_subset_calM finite_calM rev_finite_subset)

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 ≤ card(H)"
apply (rule dvd_imp_le)
apply (rule div_combine [OF prime_imp_prime_elem[OF prime_p] not_dvd_M])
prefer 2 apply (blast intro: subgroup.finite_imp_card_positive H_is_subgroup)
zero_less_m)
done

lemma (in sylow_central) lemma_leq2: "card(H) ≤ 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: "∃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)"