207 lines
7.3 KiB
Text
207 lines
7.3 KiB
Text
/-
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Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jeremy Avigad, Leonardo de Moura
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Various multiplicative and additive structures. Partially modeled on Isabelle's library.
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-/
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import algebra.inf_group
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open eq eq.ops -- note: ⁻¹ will be overloaded
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open binary algebra is_trunc
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set_option class.force_new true
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variable {A : Type}
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/- semigroup -/
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namespace algebra
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structure is_set_structure [class] (A : Type) :=
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(is_set_carrier : is_set A)
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attribute is_set_structure.is_set_carrier [instance] [priority 950]
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structure semigroup [class] (A : Type) extends is_set_structure A, inf_semigroup A
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structure comm_semigroup [class] (A : Type) extends semigroup A, comm_inf_semigroup A
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structure left_cancel_semigroup [class] (A : Type) extends semigroup A, left_cancel_inf_semigroup A
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structure right_cancel_semigroup [class] (A : Type) extends semigroup A, right_cancel_inf_semigroup A
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/- additive semigroup -/
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definition add_semigroup [class] : Type → Type := semigroup
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definition add_semigroup.is_set_carrier [instance] [priority 900] (A : Type) [H : add_semigroup A] :
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is_set A :=
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@is_set_structure.is_set_carrier A (@semigroup.to_is_set_structure A H)
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definition add_inf_semigroup_of_add_semigroup [reducible] [trans_instance] (A : Type)
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[H : add_semigroup A] : add_inf_semigroup A :=
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@semigroup.to_inf_semigroup A H
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definition add_comm_semigroup [class] : Type → Type := comm_semigroup
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definition add_semigroup_of_add_comm_semigroup [reducible] [trans_instance] (A : Type)
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[H : add_comm_semigroup A] : add_semigroup A :=
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@comm_semigroup.to_semigroup A H
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definition add_comm_inf_semigroup_of_add_comm_semigroup [reducible] [trans_instance] (A : Type)
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[H : add_comm_semigroup A] : add_comm_inf_semigroup A :=
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@comm_semigroup.to_comm_inf_semigroup A H
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definition add_left_cancel_semigroup [class] : Type → Type := left_cancel_semigroup
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definition add_semigroup_of_add_left_cancel_semigroup [reducible] [trans_instance] (A : Type)
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[H : add_left_cancel_semigroup A] : add_semigroup A :=
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@left_cancel_semigroup.to_semigroup A H
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definition add_left_cancel_inf_semigroup_of_add_left_cancel_semigroup [reducible] [trans_instance]
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(A : Type) [H : add_left_cancel_semigroup A] : add_left_cancel_inf_semigroup A :=
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@left_cancel_semigroup.to_left_cancel_inf_semigroup A H
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definition add_right_cancel_semigroup [class] : Type → Type := right_cancel_semigroup
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definition add_semigroup_of_add_right_cancel_semigroup [reducible] [trans_instance] (A : Type)
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[H : add_right_cancel_semigroup A] : add_semigroup A :=
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@right_cancel_semigroup.to_semigroup A H
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definition add_right_cancel_inf_semigroup_of_add_right_cancel_semigroup [reducible] [trans_instance]
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(A : Type) [H : add_right_cancel_semigroup A] : add_right_cancel_inf_semigroup A :=
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@right_cancel_semigroup.to_right_cancel_inf_semigroup A H
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/- monoid -/
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structure monoid [class] (A : Type) extends semigroup A, inf_monoid A
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structure comm_monoid [class] (A : Type) extends monoid A, comm_semigroup A, comm_inf_monoid A
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/- additive monoid -/
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definition add_monoid [class] : Type → Type := monoid
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definition add_semigroup_of_add_monoid [reducible] [trans_instance] (A : Type)
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[H : add_monoid A] : add_semigroup A :=
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@monoid.to_semigroup A H
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definition add_inf_monoid_of_add_monoid [reducible] [trans_instance] (A : Type)
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[H : add_monoid A] : add_inf_monoid A :=
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@monoid.to_inf_monoid A H
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definition add_comm_monoid [class] : Type → Type := comm_monoid
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definition add_monoid_of_add_comm_monoid [reducible] [trans_instance] (A : Type)
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[H : add_comm_monoid A] : add_monoid A :=
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@comm_monoid.to_monoid A H
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definition add_comm_semigroup_of_add_comm_monoid [reducible] [trans_instance] (A : Type)
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[H : add_comm_monoid A] : add_comm_semigroup A :=
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@comm_monoid.to_comm_semigroup A H
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definition add_comm_inf_monoid_of_add_comm_monoid [reducible] [trans_instance] (A : Type)
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[H : add_comm_monoid A] : add_comm_inf_monoid A :=
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@comm_monoid.to_comm_inf_monoid A H
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definition add_monoid.to_monoid {A : Type} [s : add_monoid A] : monoid A := s
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definition add_comm_monoid.to_comm_monoid {A : Type} [s : add_comm_monoid A] : comm_monoid A := s
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definition monoid.to_add_monoid {A : Type} [s : monoid A] : add_monoid A := s
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definition comm_monoid.to_add_comm_monoid {A : Type} [s : comm_monoid A] : add_comm_monoid A := s
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/- group -/
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structure group [class] (A : Type) extends monoid A, inf_group A
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definition group_of_inf_group (A : Type) [s : inf_group A] [is_set A] : group A :=
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⦃group, s, is_set_carrier := _⦄
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section group
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variable [s : group A]
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include s
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definition group.to_left_cancel_semigroup [trans_instance] : left_cancel_semigroup A :=
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⦃ left_cancel_semigroup, s,
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mul_left_cancel := @mul_left_cancel A _ ⦄
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definition group.to_right_cancel_semigroup [trans_instance] : right_cancel_semigroup A :=
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⦃ right_cancel_semigroup, s,
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mul_right_cancel := @mul_right_cancel A _ ⦄
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end group
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structure ab_group [class] (A : Type) extends group A, comm_monoid A, ab_inf_group A
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definition ab_group_of_ab_inf_group (A : Type) [s : ab_inf_group A] [is_set A] : ab_group A :=
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⦃ab_group, s, is_set_carrier := _⦄
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/- additive group -/
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definition add_group [class] : Type → Type := group
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definition add_semigroup_of_add_group [reducible] [trans_instance] (A : Type)
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[H : add_group A] : add_monoid A :=
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@group.to_monoid A H
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definition add_inf_group_of_add_group [reducible] [trans_instance] (A : Type)
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[H : add_group A] : add_inf_group A :=
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@group.to_inf_group A H
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definition add_group.to_group {A : Type} [s : add_group A] : group A := s
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definition group.to_add_group {A : Type} [s : group A] : add_group A := s
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definition add_group_of_add_inf_group (A : Type) [s : add_inf_group A] [is_set A] :
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add_group A :=
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⦃group, s, is_set_carrier := _⦄
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section add_group
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variables [s : add_group A]
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include s
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definition add_group.to_add_left_cancel_semigroup [reducible] [trans_instance] :
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add_left_cancel_semigroup A :=
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@group.to_left_cancel_semigroup A s
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definition add_group.to_add_right_cancel_semigroup [reducible] [trans_instance] :
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add_right_cancel_semigroup A :=
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@group.to_right_cancel_semigroup A s
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end add_group
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definition add_ab_group [class] : Type → Type := ab_group
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definition add_group_of_add_ab_group [reducible] [trans_instance] (A : Type)
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[H : add_ab_group A] : add_group A :=
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@ab_group.to_group A H
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definition add_comm_monoid_of_add_ab_group [reducible] [trans_instance] (A : Type)
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[H : add_ab_group A] : add_comm_monoid A :=
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@ab_group.to_comm_monoid A H
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definition add_ab_inf_group_of_add_ab_group [reducible] [trans_instance] (A : Type)
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[H : add_ab_group A] : add_ab_inf_group A :=
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@ab_group.to_ab_inf_group A H
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definition add_ab_group.to_ab_group {A : Type} [s : add_ab_group A] : ab_group A := s
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definition ab_group.to_add_ab_group {A : Type} [s : ab_group A] : add_ab_group A := s
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definition add_ab_group_of_add_ab_inf_group (A : Type) [s : add_ab_inf_group A] [is_set A] :
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add_ab_group A :=
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⦃ab_group, s, is_set_carrier := _⦄
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definition group_of_add_group (A : Type) [G : add_group A] : group A :=
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⦃group,
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mul := has_add.add,
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mul_assoc := add.assoc,
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one := !has_zero.zero,
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one_mul := zero_add,
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mul_one := add_zero,
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inv := has_neg.neg,
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mul_left_inv := add.left_inv,
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is_set_carrier := _⦄
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end algebra
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open algebra
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