lean2/tests/lean/run/algebra3.lean

import logic

infixl `*`   := has_mul.mul
postfix `⁻¹` := has_inv.inv
notation 1   := has_one.one

structure semigroup [class] (A : Type) extends has_mul A :=
(assoc : ∀ a b c, mul (mul a b) c = mul a (mul b c))

structure comm_semigroup [class] (A : Type) extends semigroup A renaming mul→add:=
(comm : ∀a b, add a b = add b a)

infixl `+` := comm_semigroup.add

structure monoid [class] (A : Type) extends semigroup A, has_one A :=
(right_id : ∀a, mul a one = a) (left_id : ∀a, mul one a = a)

-- We can suppress := and :: when we are not declaring any new field.
structure comm_monoid [class] (A : Type) extends monoid A renaming mul→add, comm_semigroup A

print fields comm_monoid

structure group [class] (A : Type) extends monoid A, has_inv A :=
(is_inv : ∀ a, mul a (inv a) = one)

structure abelian_group [class] (A : Type) extends group A renaming mul→add, comm_monoid A

structure ring [class] (A : Type)
  extends abelian_group A renaming
    assoc→add.assoc
    comm→add.comm
    one→zero
    right_id→add_zero
    left_id→add.left_id
    inv→uminus
    is_inv→uminus_is_inv,
  monoid A renaming
    assoc→mul.assoc
    right_id→mul.right_id
    left_id→mul.left_id
:=
(dist_left  : ∀ a b c, mul a (add b c) = add (mul a b) (mul a c))
(dist_right : ∀ a b c, mul (add a b) c = add (mul a c) (mul b c))

print fields ring

variable A : Type₁
variables a b c d : A
variable R : ring A

check a + b * c

set_option pp.implicit true
set_option pp.notation false
set_option pp.coercions true

check a + b * c