Bundled non-unital subsemirings

We define bundled non-unital subsemirings and some standard constructions: subtype and inclusion ring homomorphisms.

class NonUnitalSubsemiringClass (S : Type u_1) (R : outParam (Type u)) [NonUnitalNonAssocSemiring R] [SetLike S R] extends AddSubmonoidClass S R : Prop

NonUnitalSubsemiringClass S R states that S is a type of subsets s ⊆ R that are both an additive submonoid and also a multiplicative subsemigroup.

• add_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a + b ∈ s
• zero_mem : ∀ (s : S), 0 ∈ s
• mul_mem : ∀ {s : S} {a b : R}, a ∈ s → b ∈ s → a * b ∈ s

@[instance 100]

instance NonUnitalSubsemiringClass.mulMemClass (S : Type u_1) (R : Type u) [NonUnitalNonAssocSemiring R] [SetLike S R] [h : NonUnitalSubsemiringClass S R] : MulMemClass S R

Equations

• ⋯ = ⋯

@[instance 75]

instance NonUnitalSubsemiringClass.toNonUnitalNonAssocSemiring {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) : NonUnitalNonAssocSemiring ↥s

A non-unital subsemiring of a NonUnitalNonAssocSemiring inherits a NonUnitalNonAssocSemiring structure

Equations

• NonUnitalSubsemiringClass.toNonUnitalNonAssocSemiring s = Function.Injective.nonUnitalNonAssocSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯

instance NonUnitalSubsemiringClass.noZeroDivisors {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) [NoZeroDivisors R] : NoZeroDivisors ↥s

Equations

• ⋯ = ⋯

def NonUnitalSubsemiringClass.subtype {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) : ↥s →ₙ+* R

The natural non-unital ring hom from a non-unital subsemiring of a non-unital semiring R to R.

Equations

• NonUnitalSubsemiringClass.subtype s = { toFun := Subtype.val, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem NonUnitalSubsemiringClass.coeSubtype {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] (s : S) : ⇑(NonUnitalSubsemiringClass.subtype s) = Subtype.val

instance NonUnitalSubsemiringClass.toNonUnitalSemiring {S : Type v} (s : S) {R : Type u_1} [NonUnitalSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] : NonUnitalSemiring ↥s

A non-unital subsemiring of a NonUnitalSemiring is a NonUnitalSemiring.

Equations

• NonUnitalSubsemiringClass.toNonUnitalSemiring s = Function.Injective.nonUnitalSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯

instance NonUnitalSubsemiringClass.toNonUnitalCommSemiring {S : Type v} (s : S) {R : Type u_1} [NonUnitalCommSemiring R] [SetLike S R] [NonUnitalSubsemiringClass S R] : NonUnitalCommSemiring ↥s

A non-unital subsemiring of a NonUnitalCommSemiring is a NonUnitalCommSemiring.

Equations

• NonUnitalSubsemiringClass.toNonUnitalCommSemiring s = Function.Injective.nonUnitalCommSemiring Subtype.val ⋯ ⋯ ⋯ ⋯ ⋯

Note: currently, there are no ordered versions of non-unital rings.

structure NonUnitalSubsemiring (R : Type u) [NonUnitalNonAssocSemiring R] extends AddSubmonoid R, Subsemigroup R : Type u

A non-unital subsemiring of a non-unital semiring R is a subset s that is both an additive submonoid and a semigroup.

• carrier : Set R
• add_mem' : ∀ {a b : R}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier
• zero_mem' : 0 ∈ self.carrier
• mul_mem' : ∀ {a b : R}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier

instance NonUnitalSubsemiring.instSetLike {R : Type u} [NonUnitalNonAssocSemiring R] : SetLike (NonUnitalSubsemiring R) R

Equations

• NonUnitalSubsemiring.instSetLike = { coe := fun (s : NonUnitalSubsemiring R) => s.carrier, coe_injective' := ⋯ }

instance NonUnitalSubsemiring.instNonUnitalSubsemiringClass {R : Type u} [NonUnitalNonAssocSemiring R] : NonUnitalSubsemiringClass (NonUnitalSubsemiring R) R

Equations

• ⋯ = ⋯

theorem NonUnitalSubsemiring.mem_carrier {R : Type u} [NonUnitalNonAssocSemiring R] {s : NonUnitalSubsemiring R} {x : R} : x ∈ s.carrier ↔ x ∈ s

theorem NonUnitalSubsemiring.ext {R : Type u} [NonUnitalNonAssocSemiring R] {S T : NonUnitalSubsemiring R} (h : ∀ (x : R), x ∈ S ↔ x ∈ T) : S = T

Two non-unital subsemirings are equal if they have the same elements.

def NonUnitalSubsemiring.copy {R : Type u} [NonUnitalNonAssocSemiring R] (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) : NonUnitalSubsemiring R

Copy of a non-unital subsemiring with a new carrier equal to the old one. Useful to fix definitional equalities.

Equations

• S.copy s hs = { carrier := s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

theorem NonUnitalSubsemiring.coe_copy {R : Type u} [NonUnitalNonAssocSemiring R] (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) : ↑(S.copy s hs) = s

theorem NonUnitalSubsemiring.copy_eq {R : Type u} [NonUnitalNonAssocSemiring R] (S : NonUnitalSubsemiring R) (s : Set R) (hs : s = ↑S) : S.copy s hs = S

theorem NonUnitalSubsemiring.toSubsemigroup_injective {R : Type u} [NonUnitalNonAssocSemiring R] : Function.Injective NonUnitalSubsemiring.toSubsemigroup

theorem NonUnitalSubsemiring.toAddSubmonoid_injective {R : Type u} [NonUnitalNonAssocSemiring R] : Function.Injective NonUnitalSubsemiring.toAddSubmonoid

def NonUnitalSubsemiring.mk' {R : Type u} [NonUnitalNonAssocSemiring R] (s : Set R) (sg : Subsemigroup R) (hg : ↑sg = s) (sa : AddSubmonoid R) (ha : ↑sa = s) : NonUnitalSubsemiring R

Construct a NonUnitalSubsemiring R from a set s, a subsemigroup sg, and an additive submonoid sa such that x ∈ s ↔ x ∈ sg ↔ x ∈ sa.

Equations

• NonUnitalSubsemiring.mk' s sg hg sa ha = { carrier := s, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

@[simp]

theorem NonUnitalSubsemiring.coe_mk' {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : ↑(NonUnitalSubsemiring.mk' s sg hg sa ha) = s

@[simp]

theorem NonUnitalSubsemiring.mem_mk' {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) {x : R} : x ∈ NonUnitalSubsemiring.mk' s sg hg sa ha ↔ x ∈ s

@[simp]

theorem NonUnitalSubsemiring.mk'_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : (NonUnitalSubsemiring.mk' s sg hg sa ha).toSubsemigroup = sg

@[simp]

theorem NonUnitalSubsemiring.mk'_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] {s : Set R} {sg : Subsemigroup R} (hg : ↑sg = s) {sa : AddSubmonoid R} (ha : ↑sa = s) : (NonUnitalSubsemiring.mk' s sg hg sa ha).toAddSubmonoid = sa

@[simp]

theorem NonUnitalSubsemiring.coe_zero {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : ↑0 = 0

@[simp]

theorem NonUnitalSubsemiring.coe_add {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) (x y : ↥s) : ↑(x + y) = ↑x + ↑y

@[simp]

theorem NonUnitalSubsemiring.coe_mul {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) (x y : ↥s) : ↑(x * y) = ↑x * ↑y

Note: currently, there are no ordered versions of non-unital rings.

@[simp]

theorem NonUnitalSubsemiring.mem_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] {s : NonUnitalSubsemiring R} {x : R} : x ∈ s.toSubsemigroup ↔ x ∈ s

@[simp]

theorem NonUnitalSubsemiring.coe_toSubsemigroup {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : ↑s.toSubsemigroup = ↑s

@[simp]

theorem NonUnitalSubsemiring.mem_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] {s : NonUnitalSubsemiring R} {x : R} : x ∈ s.toAddSubmonoid ↔ x ∈ s

@[simp]

theorem NonUnitalSubsemiring.coe_toAddSubmonoid {R : Type u} [NonUnitalNonAssocSemiring R] (s : NonUnitalSubsemiring R) : ↑s.toAddSubmonoid = ↑s

instance NonUnitalSubsemiring.instTop {R : Type u} [NonUnitalNonAssocSemiring R] : Top (NonUnitalSubsemiring R)

The non-unital subsemiring R of the non-unital semiring R.

Equations

• NonUnitalSubsemiring.instTop = { top := let __src := ⊤; let __src_1 := ⊤; { carrier := __src.carrier, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ } }

@[simp]

theorem NonUnitalSubsemiring.mem_top {R : Type u} [NonUnitalNonAssocSemiring R] (x : R) : x ∈ ⊤

@[simp]

theorem NonUnitalSubsemiring.coe_top {R : Type u} [NonUnitalNonAssocSemiring R] : ↑⊤ = Set.univ

instance NonUnitalSubsemiring.instBot {R : Type u} [NonUnitalNonAssocSemiring R] : Bot (NonUnitalSubsemiring R)

Equations

• NonUnitalSubsemiring.instBot = { bot := { carrier := {0}, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ } }

instance NonUnitalSubsemiring.instInhabited {R : Type u} [NonUnitalNonAssocSemiring R] : Inhabited (NonUnitalSubsemiring R)

Equations

• NonUnitalSubsemiring.instInhabited = { default := ⊥ }

theorem NonUnitalSubsemiring.coe_bot {R : Type u} [NonUnitalNonAssocSemiring R] : ↑⊥ = {0}

theorem NonUnitalSubsemiring.mem_bot {R : Type u} [NonUnitalNonAssocSemiring R] {x : R} : x ∈ ⊥ ↔ x = 0

instance NonUnitalSubsemiring.instMin {R : Type u} [NonUnitalNonAssocSemiring R] : Min (NonUnitalSubsemiring R)

The inf of two non-unital subsemirings is their intersection.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem NonUnitalSubsemiring.coe_inf {R : Type u} [NonUnitalNonAssocSemiring R] (p p' : NonUnitalSubsemiring R) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem NonUnitalSubsemiring.mem_inf {R : Type u} [NonUnitalNonAssocSemiring R] {p p' : NonUnitalSubsemiring R} {x : R} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

def NonUnitalRingHom.codRestrict {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] {S' : Type u_2} [SetLike S' S] [NonUnitalSubsemiringClass S' S] (f : F) (s : S') (h : ∀ (x : R), f x ∈ s) : R →ₙ+* ↥s

Restriction of a non-unital ring homomorphism to a non-unital subsemiring of the codomain.

Equations

• NonUnitalRingHom.codRestrict f s h = { toFun := fun (n : R) => ⟨f n, ⋯⟩, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

def NonUnitalRingHom.eqSlocus {R : Type u} {S : Type v} [NonUnitalNonAssocSemiring R] {F : Type u_1} [FunLike F R S] [NonUnitalNonAssocSemiring S] [NonUnitalRingHomClass F R S] (f g : F) : NonUnitalSubsemiring R

The non-unital subsemiring of elements x : R such that f x = g x

Equations

• NonUnitalRingHom.eqSlocus f g = { carrier := {x : R | f x = g x}, add_mem' := ⋯, zero_mem' := ⋯, mul_mem' := ⋯ }

def NonUnitalSubsemiring.inclusion {R : Type u} [NonUnitalNonAssocSemiring R] {S T : NonUnitalSubsemiring R} (h : S ≤ T) : ↥S →ₙ+* ↥T

The non-unital ring homomorphism associated to an inclusion of non-unital subsemirings.

Equations

• NonUnitalSubsemiring.inclusion h = NonUnitalRingHom.codRestrict (NonUnitalSubsemiringClass.subtype S) T ⋯