Documentation

Mathlib.Data.Set.Semiring

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".

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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 instOrderBotSetSemiring (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.

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

The identity function SetSemiring α → Set α.

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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 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 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 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 t : SetSemiring α} :
SetSemiring.down s ⊂ SetSemiring.down t ↔ s < t
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instance SetSemiring.instZero {α : Type u_1} :
Zero (SetSemiring α)
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  • SetSemiring.instZero = { zero := Set.up ∅ }
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instance SetSemiring.instAdd {α : Type u_1} :
Add (SetSemiring α)
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  • SetSemiring.instAdd = { add := fun (s t : SetSemiring α) => Set.up (SetSemiring.down s ∪ SetSemiring.down t) }
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instance SetSemiring.instAddCommMonoid {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 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 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 t : Set α) :
Set.up (s ∪ t) = Set.up s + Set.up t
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instance SetSemiring.addLeftMono {α : Type u_1} :
AddLeftMono (SetSemiring α)
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  • ⋯ = ⋯
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instance SetSemiring.instNonUnitalNonAssocSemiring {α : Type u_1} [Mul α] :
NonUnitalNonAssocSemiring (SetSemiring α)
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theorem SetSemiring.mul_def {α : Type u_1} [Mul α] (s 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 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 t : Set α) :
Set.up (s * t) = Set.up s * Set.up t
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instance SetSemiring.instNoZeroDivisors {α : Type u_1} [Mul α] :
NoZeroDivisors (SetSemiring α)
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  • ⋯ = ⋯
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instance SetSemiring.mulLeftMono {α : Type u_1} [Mul α] :
MulLeftMono (SetSemiring α)
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  • ⋯ = ⋯
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instance SetSemiring.mulRightMono {α : Type u_1} [Mul α] :
MulRightMono (SetSemiring α)
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  • ⋯ = ⋯
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instance SetSemiring.instOne {α : Type u_1} [One α] :
One (SetSemiring α)
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  • SetSemiring.instOne = { one := Set.up 1 }
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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.instNonAssocSemiringOfMulOneClass {α : Type u_1} [MulOneClass α] :
NonAssocSemiring (SetSemiring α)
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instance SetSemiring.instNonUnitalSemiringOfSemigroup {α : Type u_1} [Semigroup α] :
NonUnitalSemiring (SetSemiring α)
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instance SetSemiring.instIdemSemiringOfMonoid {α : Type u_1} [Monoid α] :
IdemSemiring (SetSemiring α)
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instance SetSemiring.instNonUnitalCommSemiringOfCommSemigroup {α : Type u_1} [CommSemigroup α] :
NonUnitalCommSemiring (SetSemiring α)
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instance SetSemiring.instIdemCommSemiringOfCommMonoid {α : Type u_1} [CommMonoid α] :
IdemCommSemiring (SetSemiring α)
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instance SetSemiring.instCommMonoid {α : Type u_1} [CommMonoid α] :
CommMonoid (SetSemiring α)
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instance SetSemiring.instCanonicallyOrderedCommSemiringOfCommMonoid {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.

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  • SetSemiring.imageHom f = { toFun := fun (s : SetSemiring α) => Set.up (⇑f '' SetSemiring.down s), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }
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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)