trying to show that quotient groups of logically equivalent subgroups are isomorphic
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@ -6,9 +6,9 @@ Authors: Floris van Doorn, Egbert Rijke
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Constructions with groups
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Constructions with groups
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-/
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-/
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import hit.set_quotient .subgroup ..move_to_lib
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import hit.set_quotient .subgroup ..move_to_lib types.equiv
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod trunc function equiv
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open eq algebra is_trunc set_quotient relation sigma sigma.ops prod trunc function equiv is_equiv
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namespace group
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namespace group
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@ -269,6 +269,31 @@ namespace group
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exact H
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exact H
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end
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end
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definition quotient_group_functor_contr {K L : subgroup_rel A} (H : Π (a : A), K a → L a) :
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is_contr ((Σ(g : quotient_ab_group K →g quotient_ab_group L), g ∘g ab_qg_map K ~ ab_qg_map L) ) :=
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begin
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fapply ab_qg_universal_property,
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intro a p,
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fapply qg_map_eq_one,
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exact H a p,
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end
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definition quotient_group_iso {K L : subgroup_rel A} (H1 : Π (a : A), K a → L a) (H2 : Π (a : A), L a → K a) :
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is_contr (Σ gh : (Σ (g : quotient_ab_group K →g quotient_ab_group L), g ∘g ab_qg_map K ~ ab_qg_map L),
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is_equiv (group_fun (pr1 gh))) :=
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begin
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fapply is_trunc_sigma,
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exact quotient_group_functor_contr H1,
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intro a, induction a with g h,
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fapply is_contr_of_inhabited_prop,
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fapply adjointify,
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rexact group_fun (pr1 (center' (@quotient_group_functor_contr A L K H2))),
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have KK : is_contr ((Σ(g' : quotient_ab_group K →g quotient_ab_group K), g' ∘g ab_qg_map K ~ ab_qg_map K) ), from
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quotient_group_functor_contr (λ a, (H2 a) ∘ (H1 a)),
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-- have KK_path : ⟨g, h⟩ = ⟨id, λ a, refl (ab_qg_map K a)⟩, from eq_of_is_contr ⟨g, h⟩ ⟨id, λ a, refl (ab_qg_map K a)⟩,
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repeat exact sorry
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end
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definition quotient_group_functor [constructor] (φ : G →g G') (h : Πg, N g → N' (φ g)) :
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definition quotient_group_functor [constructor] (φ : G →g G') (h : Πg, N g → N' (φ g)) :
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quotient_group N →g quotient_group N' :=
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quotient_group N →g quotient_group N' :=
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begin
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begin
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