40 lines
1.1 KiB
Text
40 lines
1.1 KiB
Text
import logic
|
|
using num
|
|
|
|
section
|
|
parameter {A : Type}
|
|
parameter f : A → A → A
|
|
parameter one : A
|
|
parameter inv : A → A
|
|
infixl `*`:75 := f
|
|
postfix `^-1`:100 := inv
|
|
definition is_assoc := ∀ a b c, (a*b)*c = a*b*c
|
|
definition is_id := ∀ a, a*one = a
|
|
definition is_inv := ∀ a, a*a^-1 = one
|
|
end
|
|
|
|
inductive group_struct (A : Type) : Type :=
|
|
mk_group_struct : Π (mul : A → A → A) (one : A) (inv : A → A), is_assoc mul → is_id mul one → is_inv mul one inv → group_struct A
|
|
|
|
inductive group : Type :=
|
|
mk_group : Π (A : Type), group_struct A → group
|
|
|
|
definition carrier (g : group) : Type
|
|
:= group_rec (λ c s, c) g
|
|
|
|
definition group_to_struct [instance] (g : group) : group_struct (carrier g)
|
|
:= group_rec (λ (A : Type) (s : group_struct A), s) g
|
|
|
|
check group_struct
|
|
|
|
definition mul [inline] {A : Type} {s : group_struct A} (a b : A) : A
|
|
:= group_struct_rec (λ mul one inv h1 h2 h3, mul) s a b
|
|
|
|
infixl `*`:75 := mul
|
|
|
|
variable G1 : group.{1}
|
|
variable G2 : group.{1}
|
|
variables a b c : (carrier G2)
|
|
variables d e : (carrier G1)
|
|
check a * b * b
|
|
check d * e
|