lean2/library/data/set/comm_semiring.lean

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/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
(set A) is an instance of a commutative semiring
-/
import data.set.basic algebra.ring
open algebra set
definition set_comm_semiring [instance] (A : Type) : comm_semiring (set A) :=
⦃ comm_semiring,
add := union,
mul := inter,
zero := empty,
one := univ,
add_assoc := union.assoc,
add_comm := union.comm,
zero_add := empty_union,
add_zero := union_empty,
mul_assoc := inter.assoc,
mul_comm := inter.comm,
zero_mul := empty_inter,
mul_zero := inter_empty,
one_mul := univ_inter,
mul_one := inter_univ,
left_distrib := inter.distrib_left,
right_distrib := inter.distrib_right