import logic definition Type1 := Type.{1} context variable {A : Type} variable f : A → A → A variable one : A variable inv : A → A infixl `*` := 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 namespace algebra inductive mul_struct [class] (A : Type) : Type := mk : (A → A → A) → mul_struct A inductive add_struct [class] (A : Type) : Type := mk : (A → A → A) → add_struct A definition mul {A : Type} [s : mul_struct A] (a b : A) := mul_struct.rec (fun f, f) s a b infixl `*` := mul definition add {A : Type} [s : add_struct A] (a b : A) := add_struct.rec (fun f, f) s a b infixl `+` := add end algebra open algebra inductive nat : Type := zero : nat, succ : nat → nat namespace nat constant add : nat → nat → nat constant mul : nat → nat → nat definition is_mul_struct [instance] : algebra.mul_struct nat := algebra.mul_struct.mk mul definition is_add_struct [instance] : algebra.add_struct nat := algebra.add_struct.mk add definition to_nat (n : num) : nat := #algebra num.rec nat.zero (λ n, pos_num.rec (succ zero) (λ n r, r + r) (λ n r, r + r + succ zero) n) n end nat namespace algebra namespace semigroup inductive semigroup_struct [class] (A : Type) : Type := mk : Π (mul : A → A → A), is_assoc mul → semigroup_struct A definition mul {A : Type} (s : semigroup_struct A) (a b : A) := semigroup_struct.rec (fun f h, f) s a b definition assoc {A : Type} (s : semigroup_struct A) : is_assoc (mul s) := semigroup_struct.rec (fun f h, h) s definition is_mul_struct [instance] (A : Type) (s : semigroup_struct A) : mul_struct A := mul_struct.mk (mul s) inductive semigroup : Type := mk : Π (A : Type), semigroup_struct A → semigroup definition carrier [coercion] (g : semigroup) := semigroup.rec (fun c s, c) g definition is_semigroup [instance] (g : semigroup) : semigroup_struct (carrier g) := semigroup.rec (fun c s, s) g end semigroup namespace monoid check semigroup.mul inductive monoid_struct [class] (A : Type) : Type := mk_monoid_struct : Π (mul : A → A → A) (id : A), is_assoc mul → is_id mul id → monoid_struct A definition mul {A : Type} (s : monoid_struct A) (a b : A) := monoid_struct.rec (fun mul id a i, mul) s a b definition assoc {A : Type} (s : monoid_struct A) : is_assoc (mul s) := monoid_struct.rec (fun mul id a i, a) s open semigroup definition is_semigroup_struct [instance] (A : Type) (s : monoid_struct A) : semigroup_struct A := semigroup_struct.mk (mul s) (assoc s) inductive monoid : Type := mk_monoid : Π (A : Type), monoid_struct A → monoid definition carrier [coercion] (m : monoid) := monoid.rec (fun c s, c) m definition is_monoid [instance] (m : monoid) : monoid_struct (carrier m) := monoid.rec (fun c s, s) m end monoid end algebra section open algebra algebra.semigroup algebra.monoid variable M : monoid variables a b c : M check a*b*c*a*b*c*a*b*a*b*c*a check a*b end