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.

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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".

Instances For

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noncomputable instance instInhabitedSetSemiring (α : Type u_3) :

Inhabited (SetSemiring α)

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instance instPartialOrderSetSemiring (α : Type u_3) :

PartialOrder (SetSemiring α)

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instance instOrderBotSetSemiringToLEToPreorderInstPartialOrderSetSemiring (α : Type u_3) :

OrderBot (SetSemiring α)

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def Set.up {α : Type u_1} :

Set α ≃ SetSemiring α

The identity function Set α → SetSemiring α.

Instances For

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def SetSemiring.down {α : Type u_1} :

SetSemiring α ≃ Set α

The identity function SetSemiring α → Set α.

Instances For

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@[simp]

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

↑SetSemiring.down (↑Set.up s) = s

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@[simp]

theorem SetSemiring.up_down {α : Type u_1} (s : SetSemiring α) :

↑Set.up (↑SetSemiring.down s) = s

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theorem SetSemiring.up_le_up {α : Type u_1} {s : Set α} {t : Set α} :

↑Set.up s ≤ ↑Set.up t ↔ s ⊆ t

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theorem SetSemiring.up_lt_up {α : Type u_1} {s : Set α} {t : Set α} :

↑Set.up s < ↑Set.up t ↔ s ⊂ t

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@[simp]

theorem SetSemiring.down_subset_down {α : Type u_1} {s : SetSemiring α} {t : SetSemiring α} :

↑SetSemiring.down s ⊆ ↑SetSemiring.down t ↔ s ≤ t

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@[simp]

theorem SetSemiring.down_ssubset_down {α : Type u_1} {s : SetSemiring α} {t : SetSemiring α} :

↑SetSemiring.down s ⊂ ↑SetSemiring.down t ↔ s < t

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instance SetSemiring.instAddCommMonoidSetSemiring {α : Type u_1} :

AddCommMonoid (SetSemiring α)

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theorem SetSemiring.zero_def {α : Type u_1} :

0 = ↑Set.up ∅

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@[simp]

theorem SetSemiring.down_zero {α : Type u_1} :

↑SetSemiring.down 0 = ∅

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@[simp]

theorem Set.up_empty {α : Type u_1} :

↑Set.up ∅ = 0

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theorem SetSemiring.add_def {α : Type u_1} (s : SetSemiring α) (t : SetSemiring α) :

s + t = ↑Set.up (↑SetSemiring.down s ∪ ↑SetSemiring.down t)

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@[simp]

theorem SetSemiring.down_add {α : Type u_1} (s : SetSemiring α) (t : SetSemiring α) :

↑SetSemiring.down (s + t) = ↑SetSemiring.down s ∪ ↑SetSemiring.down t

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@[simp]

theorem Set.up_union {α : Type u_1} (s : Set α) (t : Set α) :

↑Set.up (s ∪ t) = ↑Set.up s + ↑Set.up t

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instance SetSemiring.covariantClass_add {α : Type u_1} :

CovariantClass (SetSemiring α) (SetSemiring α) (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

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instance SetSemiring.instNonUnitalNonAssocSemiringSetSemiring {α : Type u_1} [Mul α] :

NonUnitalNonAssocSemiring (SetSemiring α)

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theorem SetSemiring.mul_def {α : Type u_1} [Mul α] (s : SetSemiring α) (t : SetSemiring α) :

s * t = ↑Set.up (↑SetSemiring.down s * ↑SetSemiring.down t)

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@[simp]

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

↑SetSemiring.down (s * t) = ↑SetSemiring.down s * ↑SetSemiring.down t

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@[simp]

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

↑Set.up (s * t) = ↑Set.up s * ↑Set.up t

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instance SetSemiring.instNoZeroDivisorsSetSemiringToMulInstNonUnitalNonAssocSemiringSetSemiringToZeroToMulZeroClass {α : Type u_1} [Mul α] :

NoZeroDivisors (SetSemiring α)

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instance SetSemiring.covariantClass_mul_left {α : Type u_1} [Mul α] :

CovariantClass (SetSemiring α) (SetSemiring α) (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

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instance SetSemiring.covariantClass_mul_right {α : Type u_1} [Mul α] :

CovariantClass (SetSemiring α) (SetSemiring α) (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

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instance SetSemiring.instOneSetSemiring {α : Type u_1} [One α] :

One (SetSemiring α)

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theorem SetSemiring.one_def {α : Type u_1} [One α] :

1 = ↑Set.up 1

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@[simp]

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

↑SetSemiring.down 1 = 1

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@[simp]

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

↑Set.up 1 = 1

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instance SetSemiring.instNonAssocSemiringSetSemiring {α : Type u_1} [MulOneClass α] :

NonAssocSemiring (SetSemiring α)

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instance SetSemiring.instNonUnitalSemiringSetSemiring {α : Type u_1} [Semigroup α] :

NonUnitalSemiring (SetSemiring α)

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instance SetSemiring.instIdemSemiringSetSemiring {α : Type u_1} [Monoid α] :

IdemSemiring (SetSemiring α)

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instance SetSemiring.instNonUnitalCommSemiringSetSemiring {α : Type u_1} [CommSemigroup α] :

NonUnitalCommSemiring (SetSemiring α)

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instance SetSemiring.instIdemCommSemiringSetSemiring {α : Type u_1} [CommMonoid α] :

IdemCommSemiring (SetSemiring α)

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instance SetSemiring.instCommMonoidSetSemiring {α : Type u_1} [CommMonoid α] :

CommMonoid (SetSemiring α)

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instance SetSemiring.instCanonicallyOrderedCommSemiringSetSemiring {α : Type u_1} [CommMonoid α] :

CanonicallyOrderedCommSemiring (SetSemiring α)

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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.

Instances For

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theorem SetSemiring.imageHom_def {α : Type u_1} {β : Type u_2} [MulOneClass α] [MulOneClass β] (f : α →* β) (s : SetSemiring α) :

↑(SetSemiring.imageHom f) s = ↑Set.up (↑f '' ↑SetSemiring.down s)

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@[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 (↑(SetSemiring.imageHom f) s) = ↑f '' ↑SetSemiring.down s

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@[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)