feat(library/theories/group_theory): Group and finite group theories
subgroup.lean : general subgroup theories, quotient group using quot
finsubg.lean : finite subgroups (finset and fintype), Lagrange theorem,
finite cosets and lcoset_type, normalizer for finite groups, coset product
and quotient group based on lcoset_type, semidirect product
hom.lean : homomorphism and isomorphism, kernel, first isomorphism theorem
perm.lean : permutation group
cyclic.lean : cyclic subgroup, finite generator, order of generator, sequence and rotation
action.lean : fixed point, action, stabilizer, orbit stabilizer theorem, orbit partition,
Cayley theorem, action on lcoset, cardinality of permutation group
pgroup.lean : subgroup with order of prime power, Cauchy theorem, first Sylow theorem
2015-07-16 03:02:11 +00:00
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/-
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Copyright (c) 2015 Haitao Zhang. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author : Haitao Zhang
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-/
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import algebra.group data data.fintype.function
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2015-12-06 07:27:46 +00:00
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open nat list function
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feat(library/theories/group_theory): Group and finite group theories
subgroup.lean : general subgroup theories, quotient group using quot
finsubg.lean : finite subgroups (finset and fintype), Lagrange theorem,
finite cosets and lcoset_type, normalizer for finite groups, coset product
and quotient group based on lcoset_type, semidirect product
hom.lean : homomorphism and isomorphism, kernel, first isomorphism theorem
perm.lean : permutation group
cyclic.lean : cyclic subgroup, finite generator, order of generator, sequence and rotation
action.lean : fixed point, action, stabilizer, orbit stabilizer theorem, orbit partition,
Cayley theorem, action on lcoset, cardinality of permutation group
pgroup.lean : subgroup with order of prime power, Cauchy theorem, first Sylow theorem
2015-07-16 03:02:11 +00:00
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2015-12-06 07:27:46 +00:00
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namespace group_theory
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feat(library/theories/group_theory): Group and finite group theories
subgroup.lean : general subgroup theories, quotient group using quot
finsubg.lean : finite subgroups (finset and fintype), Lagrange theorem,
finite cosets and lcoset_type, normalizer for finite groups, coset product
and quotient group based on lcoset_type, semidirect product
hom.lean : homomorphism and isomorphism, kernel, first isomorphism theorem
perm.lean : permutation group
cyclic.lean : cyclic subgroup, finite generator, order of generator, sequence and rotation
action.lean : fixed point, action, stabilizer, orbit stabilizer theorem, orbit partition,
Cayley theorem, action on lcoset, cardinality of permutation group
pgroup.lean : subgroup with order of prime power, Cauchy theorem, first Sylow theorem
2015-07-16 03:02:11 +00:00
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open fintype
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section perm
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variable {A : Type}
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variable [finA : fintype A]
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include finA
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variable [deceqA : decidable_eq A]
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include deceqA
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variable {f : A → A}
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lemma perm_surj : injective f → surjective f :=
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surj_of_inj_eq_card (eq.refl (card A))
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variable (perm : injective f)
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definition perm_inv : A → A :=
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right_inv (perm_surj perm)
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lemma perm_inv_right : f ∘ (perm_inv perm) = id :=
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right_inv_of_surj (perm_surj perm)
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lemma perm_inv_left : (perm_inv perm) ∘ f = id :=
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have H : left_inverse f (perm_inv perm), from congr_fun (perm_inv_right perm),
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funext (take x, right_inverse_of_injective_of_left_inverse perm H x)
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lemma perm_inv_inj : injective (perm_inv perm) :=
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injective_of_has_left_inverse (exists.intro f (congr_fun (perm_inv_right perm)))
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end perm
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structure perm (A : Type) [h : fintype A] : Type :=
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(f : A → A) (inj : injective f)
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local attribute perm.f [coercion]
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section perm
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variable {A : Type}
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variable [finA : fintype A]
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include finA
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lemma eq_of_feq : ∀ {p₁ p₂ : perm A}, (perm.f p₁) = p₂ → p₁ = p₂
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| (perm.mk f₁ P₁) (perm.mk f₂ P₂) := assume (feq : f₁ = f₂), by congruence; assumption
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lemma feq_of_eq : ∀ {p₁ p₂ : perm A}, p₁ = p₂ → (perm.f p₁) = p₂
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| (perm.mk f₁ P₁) (perm.mk f₂ P₂) := assume Peq,
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have feq : f₁ = f₂, from perm.no_confusion Peq (λ Pe Pqe, Pe), feq
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lemma eq_iff_feq {p₁ p₂ : perm A} : (perm.f p₁) = p₂ ↔ p₁ = p₂ :=
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iff.intro eq_of_feq feq_of_eq
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lemma perm.f_mk {f : A → A} {Pinj : injective f} : perm.f (perm.mk f Pinj) = f := rfl
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definition move_by [reducible] (a : A) (f : perm A) : A := f a
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variable [deceqA : decidable_eq A]
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include deceqA
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lemma perm.has_decidable_eq [instance] : decidable_eq (perm A) :=
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take f g,
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perm.destruct f (λ ff finj, perm.destruct g (λ gf ginj,
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decidable.rec_on (decidable_eq_fun ff gf)
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(λ Peq, decidable.inl (by substvars))
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(λ Pne, decidable.inr begin intro P, injection P, contradiction end)))
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lemma dinj_perm_mk : dinj (@injective A A) perm.mk :=
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take a₁ a₂ Pa₁ Pa₂ Pmkeq, perm.no_confusion Pmkeq (λ Pe Pqe, Pe)
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definition all_perms : list (perm A) :=
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dmap injective perm.mk (all_injs A)
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lemma nodup_all_perms : nodup (@all_perms A _ _) :=
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dmap_nodup_of_dinj dinj_perm_mk nodup_all_injs
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lemma all_perms_complete : ∀ p : perm A, p ∈ all_perms :=
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take p, perm.destruct p (take f Pinj,
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assert Pin : f ∈ all_injs A, from all_injs_complete Pinj,
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mem_dmap Pinj Pin)
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definition perm_is_fintype [instance] : fintype (perm A) :=
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fintype.mk all_perms nodup_all_perms all_perms_complete
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definition perm.mul (f g : perm A) :=
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perm.mk (f∘g) (injective_compose (perm.inj f) (perm.inj g))
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definition perm.one [reducible] : perm A := perm.mk id injective_id
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definition perm.inv (f : perm A) := let inj := perm.inj f in
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perm.mk (perm_inv inj) (perm_inv_inj inj)
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local infix `^` := perm.mul
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lemma perm.assoc (f g h : perm A) : f ^ g ^ h = f ^ (g ^ h) := rfl
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lemma perm.one_mul (p : perm A) : perm.one ^ p = p :=
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perm.cases_on p (λ f inj, rfl)
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lemma perm.mul_one (p : perm A) : p ^ perm.one = p :=
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perm.cases_on p (λ f inj, rfl)
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lemma perm.left_inv (p : perm A) : (perm.inv p) ^ p = perm.one :=
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begin rewrite [↑perm.one], generalize @injective_id A,
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rewrite [-perm_inv_left (perm.inj p)], intros, exact rfl
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end
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lemma perm.right_inv (p : perm A) : p ^ (perm.inv p) = perm.one :=
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begin rewrite [↑perm.one], generalize @injective_id A,
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rewrite [-perm_inv_right (perm.inj p)], intros, exact rfl
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end
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definition perm_group [instance] : group (perm A) :=
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group.mk perm.mul perm.assoc perm.one perm.one_mul perm.mul_one perm.inv perm.left_inv
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lemma perm_one : (1 : perm A) = perm.one := rfl
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end perm
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2015-12-06 07:27:46 +00:00
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end group_theory
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