import logic section variable {A : Type} variable f : A → A → A variable one : A variable inv : A → A local infixl `*` := f local 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 [class] (A : Type) : Type := mk : 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 : Π (A : Type), group_struct A → group definition carrier (g : group) : Type := group.rec (λ c s, c) g attribute carrier [coercion] 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 {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 `*` := mul section variable G1 : group variable G2 : group variables a b c : G2 variables d e : G1 check a * b * b check d * e end constant G : group.{1} constants a b : G definition val : G := a*b check val constant pos_real : Type.{1} constant rmul : pos_real → pos_real → pos_real constant rone : pos_real constant rinv : pos_real → pos_real axiom H1 : is_assoc rmul axiom H2 : is_id rmul rone axiom H3 : is_inv rmul rone rinv definition real_group_struct [instance] : group_struct pos_real := group_struct.mk rmul rone rinv H1 H2 H3 constants x y : pos_real check x * y set_option pp.implicit true print "---------------" theorem T (a b : pos_real): (rmul a b) = a*b := eq.refl (rmul a b)