39 lines
1.1 KiB
Text
39 lines
1.1 KiB
Text
import logic
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context
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variable {A : Type}
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variable f : A → A → A
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variable one : A
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variable inv : A → A
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infixl `*`:75 := f
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postfix `^-1`:100 := inv
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definition is_assoc := ∀ a b c, (a*b)*c = a*b*c
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definition is_id := ∀ a, a*one = a
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definition is_inv := ∀ a, a*a^-1 = one
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end
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inductive group_struct [class] (A : Type) : Type :=
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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
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inductive group : Type :=
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mk_group : Π (A : Type), group_struct A → group
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definition carrier (g : group) : Type
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:= group.rec (λ c s, c) g
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definition group_to_struct [instance] (g : group) : group_struct (carrier g)
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:= group.rec (λ (A : Type) (s : group_struct A), s) g
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check group_struct
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definition mul {A : Type} {s : group_struct A} (a b : A) : A
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:= group_struct.rec (λ mul one inv h1 h2 h3, mul) s a b
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infixl `*`:75 := mul
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constant G1 : group.{1}
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constant G2 : group.{1}
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constants a b c : (carrier G2)
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constants d e : (carrier G1)
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check a * b * b
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check d * e
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