Sets as a semiring under union

This file defines SetSemiring α, an alias of Set α, which we endow with ∪ as addition and pointwise * as multiplication. If α is a (commutative) monoid, SetSemiring α is a (commutative) semiring.

def SetSemiring (α : Type u_3) : Type u_3

An alias for Set α, which has a semiring structure given by ∪ as "addition" and pointwise multiplication * as "multiplication".

@[implicit_reducible]

instance instInhabitedSetSemiring (α : Type u_1) : Inhabited (SetSemiring α)

@[implicit_reducible]

instance instPartialOrderSetSemiring (α : Type u_1) : PartialOrder (SetSemiring α)

@[implicit_reducible]

instance instOrderBotSetSemiring (α : Type u_1) : OrderBot (SetSemiring α)

def Set.up {α : Type u_1} : Set α ≃ SetSemiring α

The identity function Set α → SetSemiring α.

def SetSemiring.down {α : Type u_1} : SetSemiring α ≃ Set α

The identity function SetSemiring α → Set α.

@[simp]

theorem SetSemiring.down_up {α : Type u_1} (s : Set α) : SetSemiring.down (Set.up s) = s

@[simp]

theorem SetSemiring.up_down {α : Type u_1} (s : SetSemiring α) : Set.up (SetSemiring.down s) = s

theorem SetSemiring.up_le_up {α : Type u_1} {s t : Set α} : Set.up s ≤ Set.up t ↔ s ⊆ t

theorem SetSemiring.up_lt_up {α : Type u_1} {s t : Set α} : Set.up s < Set.up t ↔ s ⊂ t

@[simp]

theorem SetSemiring.down_subset_down {α : Type u_1} {s t : SetSemiring α} : SetSemiring.down s ⊆ SetSemiring.down t ↔ s ≤ t

@[simp]

theorem SetSemiring.down_ssubset_down {α : Type u_1} {s t : SetSemiring α} : SetSemiring.down s ⊂ SetSemiring.down t ↔ s < t

@[implicit_reducible]

instance SetSemiring.instZero {α : Type u_1} : Zero (SetSemiring α)

@[implicit_reducible]

instance SetSemiring.instAdd {α : Type u_1} : Add (SetSemiring α)

@[implicit_reducible]

instance SetSemiring.instAddCommMonoid {α : Type u_1} : AddCommMonoid (SetSemiring α)

theorem SetSemiring.zero_def {α : Type u_1} : 0 = Set.up ∅

@[simp]

theorem SetSemiring.down_zero {α : Type u_1} : SetSemiring.down 0 = ∅

@[simp]

theorem Set.up_empty {α : Type u_1} : Set.up ∅ = 0

theorem SetSemiring.add_def {α : Type u_1} (s t : SetSemiring α) : s + t = Set.up (SetSemiring.down s ∪ SetSemiring.down t)

@[simp]

theorem SetSemiring.down_add {α : Type u_1} (s t : SetSemiring α) : SetSemiring.down (s + t) = SetSemiring.down s ∪ SetSemiring.down t

@[simp]

theorem Set.up_union {α : Type u_1} (s t : Set α) : Set.up (s ∪ t) = Set.up s + Set.up t

instance SetSemiring.addLeftMono {α : Type u_1} : AddLeftMono (SetSemiring α)

@[implicit_reducible]

instance SetSemiring.instNonUnitalNonAssocSemiring {α : Type u_1} [Mul α] : NonUnitalNonAssocSemiring (SetSemiring α)

theorem SetSemiring.mul_def {α : Type u_1} [Mul α] (s t : SetSemiring α) : s * t = Set.up (SetSemiring.down s * SetSemiring.down t)

@[simp]

theorem SetSemiring.down_mul {α : Type u_1} [Mul α] (s t : SetSemiring α) : SetSemiring.down (s * t) = SetSemiring.down s * SetSemiring.down t

@[simp]

theorem Set.up_mul {α : Type u_1} [Mul α] (s t : Set α) : Set.up (s * t) = Set.up s * Set.up t

instance SetSemiring.instNoZeroDivisors {α : Type u_1} [Mul α] : NoZeroDivisors (SetSemiring α)

instance SetSemiring.mulLeftMono {α : Type u_1} [Mul α] : MulLeftMono (SetSemiring α)

instance SetSemiring.mulRightMono {α : Type u_1} [Mul α] : MulRightMono (SetSemiring α)

@[implicit_reducible]

instance SetSemiring.instOne {α : Type u_1} [One α] : One (SetSemiring α)

theorem SetSemiring.one_def {α : Type u_1} [One α] : 1 = Set.up 1

@[simp]

theorem SetSemiring.down_one {α : Type u_1} [One α] : SetSemiring.down 1 = 1

@[simp]

theorem Set.up_one {α : Type u_1} [One α] : Set.up 1 = 1

@[implicit_reducible]

noncomputable instance SetSemiring.instNonAssocSemiringOfMulOneClass {α : Type u_1} [MulOneClass α] : NonAssocSemiring (SetSemiring α)

@[implicit_reducible]

instance SetSemiring.instNonUnitalSemiringOfSemigroup {α : Type u_1} [Semigroup α] : NonUnitalSemiring (SetSemiring α)

@[implicit_reducible]

noncomputable instance SetSemiring.instIdemSemiringOfMonoid {α : Type u_1} [Monoid α] : IdemSemiring (SetSemiring α)

@[implicit_reducible]

instance SetSemiring.instNonUnitalCommSemiringOfCommSemigroup {α : Type u_1} [CommSemigroup α] : NonUnitalCommSemiring (SetSemiring α)

@[implicit_reducible]

noncomputable instance SetSemiring.instIdemCommSemiringOfCommMonoid {α : Type u_1} [CommMonoid α] : IdemCommSemiring (SetSemiring α)

@[implicit_reducible]

noncomputable instance SetSemiring.instCommMonoid {α : Type u_1} [CommMonoid α] : CommMonoid (SetSemiring α)

instance SetSemiring.instCanonicallyOrderedAdd {α : Type u_1} : CanonicallyOrderedAdd (SetSemiring α)

instance SetSemiring.instIsOrderedRing {α : Type u_1} [CommMonoid α] : IsOrderedRing (SetSemiring α)

noncomputable def SetSemiring.singletonMonoidHom {α : Type u_1} [Monoid α] : α →* SetSemiring α

If α is a monoid, the map that sends a : α to the singleton set {a} is a monoid homomorphism.

noncomputable def SetSemiring.imageHom {α : Type u_1} {β : Type u_2} [MulOneClass α] [MulOneClass β] (f : α →* β) : SetSemiring α →+* SetSemiring β

The image of a set under a multiplicative homomorphism is a ring homomorphism with respect to the pointwise operations on sets.

theorem SetSemiring.imageHom_def {α : Type u_1} {β : Type u_2} [MulOneClass α] [MulOneClass β] (f : α →* β) (s : SetSemiring α) : (imageHom f) s = Set.up (⇑f '' SetSemiring.down s)

@[simp]

theorem SetSemiring.down_imageHom {α : Type u_1} {β : Type u_2} [MulOneClass α] [MulOneClass β] (f : α →* β) (s : SetSemiring α) : SetSemiring.down ((imageHom f) s) = ⇑f '' SetSemiring.down s

@[simp]

theorem Set.up_image {α : Type u_1} {β : Type u_2} [MulOneClass α] [MulOneClass β] (f : α →* β) (s : Set α) : Set.up (⇑f '' s) = (SetSemiring.imageHom f) (Set.up s)