Subsemigroups: definition and CompleteLattice structure

This file defines bundled multiplicative and additive subsemigroups. We also define a CompleteLattice structure on Subsemigroups, and define the closure of a set as the minimal subsemigroup that includes this set.

Main definitions

• Subsemigroup M: the type of bundled subsemigroup of a magma M; the underlying set is given in the carrier field of the structure, and should be accessed through coercion as in (S : Set M).
• AddSubsemigroup M : the type of bundled subsemigroups of an additive magma M.

For each of the following definitions in the Subsemigroup namespace, there is a corresponding definition in the AddSubsemigroup namespace.

• Subsemigroup.copy : copy of a subsemigroup with carrier replaced by a set that is equal but possibly not definitionally equal to the carrier of the original Subsemigroup.
• Subsemigroup.closure : semigroup closure of a set, i.e., the least subsemigroup that includes the set.
• Subsemigroup.gi : closure : Set M → Subsemigroup M and coercion coe : Subsemigroup M → Set M form a GaloisInsertion;

Implementation notes

Subsemigroup inclusion is denoted ≤ rather than ⊆, although ∈ is defined as membership of a subsemigroup's underlying set.

Note that Subsemigroup M does not actually require Semigroup M, instead requiring only the weaker Mul M.

This file is designed to have very few dependencies. In particular, it should not use natural numbers.

Tags

subsemigroup, subsemigroups

class MulMemClass (S : Type u_4) (M : Type u_5) [Mul M] [SetLike S M] : Prop

MulMemClass S M says S is a type of sets s : Set M that are closed under (*)

• mul_mem : ∀ {s : S} {a b : M}, a ∈ s → b ∈ s → a * b ∈ s

  A substructure satisfying MulMemClass is closed under multiplication.

class AddMemClass (S : Type u_4) (M : Type u_5) [Add M] [SetLike S M] : Prop

AddMemClass S M says S is a type of sets s : Set M that are closed under (+)

• add_mem : ∀ {s : S} {a b : M}, a ∈ s → b ∈ s → a + b ∈ s

  A substructure satisfying AddMemClass is closed under addition.

structure Subsemigroup (M : Type u_4) [Mul M] : Type u_4

A subsemigroup of a magma M is a subset closed under multiplication.

• carrier : Set M

  The carrier of a subsemigroup.

• mul_mem' : ∀ {a b : M}, a ∈ self.carrier → b ∈ self.carrier → a * b ∈ self.carrier

  The product of two elements of a subsemigroup belongs to the subsemigroup.

structure AddSubsemigroup (M : Type u_4) [Add M] : Type u_4

An additive subsemigroup of an additive magma M is a subset closed under addition.

• carrier : Set M

  The carrier of an additive subsemigroup.

• add_mem' : ∀ {a b : M}, a ∈ self.carrier → b ∈ self.carrier → a + b ∈ self.carrier

  The sum of two elements of an additive subsemigroup belongs to the subsemigroup.

instance AddSubsemigroup.instSetLikeSubsemigroup {M : Type u_1} [Add M] : SetLike (AddSubsemigroup M) M

Equations

• AddSubsemigroup.instSetLikeSubsemigroup = { coe := AddSubsemigroup.carrier, coe_injective' := ⋯ }

theorem AddSubsemigroup.instSetLikeSubsemigroup.proof_1 {M : Type u_1} [Add M] (p : AddSubsemigroup M) (q : AddSubsemigroup M) (h : p.carrier = q.carrier) : p = q

instance Subsemigroup.instSetLikeSubsemigroup {M : Type u_1} [Mul M] : SetLike (Subsemigroup M) M

Equations

• Subsemigroup.instSetLikeSubsemigroup = { coe := Subsemigroup.carrier, coe_injective' := ⋯ }

instance AddSubsemigroup.instAddMemClassSubsemigroupInstSetLikeSubsemigroup {M : Type u_1} [Add M] : AddMemClass (AddSubsemigroup M) M

Equations

• ⋯ = ⋯

instance Subsemigroup.instMulMemClassSubsemigroupInstSetLikeSubsemigroup {M : Type u_1} [Mul M] : MulMemClass (Subsemigroup M) M

Equations

• ⋯ = ⋯

@[simp]

theorem AddSubsemigroup.mem_carrier {M : Type u_1} [Add M] {s : AddSubsemigroup M} {x : M} : x ∈ s.carrier ↔ x ∈ s

@[simp]

theorem Subsemigroup.mem_carrier {M : Type u_1} [Mul M] {s : Subsemigroup M} {x : M} : x ∈ s.carrier ↔ x ∈ s

@[simp]

theorem AddSubsemigroup.mem_mk {M : Type u_1} [Add M] {s : Set M} {x : M} (h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s) : x ∈ { carrier := s, add_mem' := h_mul } ↔ x ∈ s

@[simp]

theorem Subsemigroup.mem_mk {M : Type u_1} [Mul M] {s : Set M} {x : M} (h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a * b ∈ s) : x ∈ { carrier := s, mul_mem' := h_mul } ↔ x ∈ s

@[simp]

theorem AddSubsemigroup.coe_set_mk {M : Type u_1} [Add M] {s : Set M} (h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s) : ↑{ carrier := s, add_mem' := h_mul } = s

@[simp]

theorem Subsemigroup.coe_set_mk {M : Type u_1} [Mul M] {s : Set M} (h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a * b ∈ s) : ↑{ carrier := s, mul_mem' := h_mul } = s

@[simp]

theorem AddSubsemigroup.mk_le_mk {M : Type u_1} [Add M] {s : Set M} {t : Set M} (h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s) (h_mul' : ∀ {a b : M}, a ∈ t → b ∈ t → a + b ∈ t) : { carrier := s, add_mem' := h_mul } ≤ { carrier := t, add_mem' := h_mul' } ↔ s ⊆ t

@[simp]

theorem Subsemigroup.mk_le_mk {M : Type u_1} [Mul M] {s : Set M} {t : Set M} (h_mul : ∀ {a b : M}, a ∈ s → b ∈ s → a * b ∈ s) (h_mul' : ∀ {a b : M}, a ∈ t → b ∈ t → a * b ∈ t) : { carrier := s, mul_mem' := h_mul } ≤ { carrier := t, mul_mem' := h_mul' } ↔ s ⊆ t

theorem AddSubsemigroup.ext {M : Type u_1} [Add M] {S : AddSubsemigroup M} {T : AddSubsemigroup M} (h : ∀ (x : M), x ∈ S ↔ x ∈ T) : S = T

Two AddSubsemigroups are equal if they have the same elements.

theorem Subsemigroup.ext {M : Type u_1} [Mul M] {S : Subsemigroup M} {T : Subsemigroup M} (h : ∀ (x : M), x ∈ S ↔ x ∈ T) : S = T

Two subsemigroups are equal if they have the same elements.

def AddSubsemigroup.copy {M : Type u_1} [Add M] (S : AddSubsemigroup M) (s : Set M) (hs : s = ↑S) : AddSubsemigroup M

Copy an additive subsemigroup replacing carrier with a set that is equal to it.

Equations

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

theorem AddSubsemigroup.copy.proof_1 {M : Type u_1} [Add M] (S : AddSubsemigroup M) (s : Set M) (hs : s = ↑S) : ∀ {a b : M}, a ∈ s → b ∈ s → a + b ∈ s

def Subsemigroup.copy {M : Type u_1} [Mul M] (S : Subsemigroup M) (s : Set M) (hs : s = ↑S) : Subsemigroup M

Copy a subsemigroup replacing carrier with a set that is equal to it.

Equations

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

@[simp]

theorem AddSubsemigroup.coe_copy {M : Type u_1} [Add M] {S : AddSubsemigroup M} {s : Set M} (hs : s = ↑S) : ↑(AddSubsemigroup.copy S s hs) = s

@[simp]

theorem Subsemigroup.coe_copy {M : Type u_1} [Mul M] {S : Subsemigroup M} {s : Set M} (hs : s = ↑S) : ↑(Subsemigroup.copy S s hs) = s

theorem AddSubsemigroup.copy_eq {M : Type u_1} [Add M] {S : AddSubsemigroup M} {s : Set M} (hs : s = ↑S) : AddSubsemigroup.copy S s hs = S

theorem Subsemigroup.copy_eq {M : Type u_1} [Mul M] {S : Subsemigroup M} {s : Set M} (hs : s = ↑S) : Subsemigroup.copy S s hs = S

theorem AddSubsemigroup.add_mem {M : Type u_1} [Add M] (S : AddSubsemigroup M) {x : M} {y : M} : x ∈ S → y ∈ S → x + y ∈ S

An AddSubsemigroup is closed under addition.

theorem Subsemigroup.mul_mem {M : Type u_1} [Mul M] (S : Subsemigroup M) {x : M} {y : M} : x ∈ S → y ∈ S → x * y ∈ S

A subsemigroup is closed under multiplication.

theorem AddSubsemigroup.instTopSubsemigroup.proof_1 {M : Type u_1} [Add M] : ∀ {a b : M}, a ∈ Set.univ → b ∈ Set.univ → a + b ∈ Set.univ

instance AddSubsemigroup.instTopSubsemigroup {M : Type u_1} [Add M] : Top (AddSubsemigroup M)

The additive subsemigroup M of the magma M.

Equations

• AddSubsemigroup.instTopSubsemigroup = { top := { carrier := Set.univ, add_mem' := ⋯ } }

instance Subsemigroup.instTopSubsemigroup {M : Type u_1} [Mul M] : Top (Subsemigroup M)

The subsemigroup M of the magma M.

Equations

• Subsemigroup.instTopSubsemigroup = { top := { carrier := Set.univ, mul_mem' := ⋯ } }

theorem AddSubsemigroup.instBotSubsemigroup.proof_1 {M : Type u_1} [Add M] : ∀ {a b : M}, False → b ∈ ∅ → a + b ∈ ∅

instance AddSubsemigroup.instBotSubsemigroup {M : Type u_1} [Add M] : Bot (AddSubsemigroup M)

The trivial AddSubsemigroup ∅ of an additive magma M.

Equations

• AddSubsemigroup.instBotSubsemigroup = { bot := { carrier := ∅, add_mem' := ⋯ } }

instance Subsemigroup.instBotSubsemigroup {M : Type u_1} [Mul M] : Bot (Subsemigroup M)

The trivial subsemigroup ∅ of a magma M.

Equations

• Subsemigroup.instBotSubsemigroup = { bot := { carrier := ∅, mul_mem' := ⋯ } }

instance AddSubsemigroup.instInhabitedSubsemigroup {M : Type u_1} [Add M] : Inhabited (AddSubsemigroup M)

Equations

• AddSubsemigroup.instInhabitedSubsemigroup = { default := ⊥ }

instance Subsemigroup.instInhabitedSubsemigroup {M : Type u_1} [Mul M] : Inhabited (Subsemigroup M)

Equations

• Subsemigroup.instInhabitedSubsemigroup = { default := ⊥ }

theorem AddSubsemigroup.not_mem_bot {M : Type u_1} [Add M] {x : M} : x ∉ ⊥

theorem Subsemigroup.not_mem_bot {M : Type u_1} [Mul M] {x : M} : x ∉ ⊥

@[simp]

theorem AddSubsemigroup.mem_top {M : Type u_1} [Add M] (x : M) : x ∈ ⊤

@[simp]

theorem Subsemigroup.mem_top {M : Type u_1} [Mul M] (x : M) : x ∈ ⊤

@[simp]

theorem AddSubsemigroup.coe_top {M : Type u_1} [Add M] : ↑⊤ = Set.univ

@[simp]

theorem Subsemigroup.coe_top {M : Type u_1} [Mul M] : ↑⊤ = Set.univ

@[simp]

theorem AddSubsemigroup.coe_bot {M : Type u_1} [Add M] : ↑⊥ = ∅

@[simp]

theorem Subsemigroup.coe_bot {M : Type u_1} [Mul M] : ↑⊥ = ∅

abbrev AddSubsemigroup.instInfSubsemigroup.match_1 {M : Type u_1} [Add M] (S₁ : AddSubsemigroup M) (S₂ : AddSubsemigroup M) : ∀ {b : M} (motive : b ∈ ↑S₁ ∩ ↑S₂ → Prop) (x : b ∈ ↑S₁ ∩ ↑S₂), (∀ (hy : b ∈ ↑S₁) (hy' : b ∈ ↑S₂), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem AddSubsemigroup.instInfSubsemigroup.proof_1 {M : Type u_1} [Add M] (S₁ : AddSubsemigroup M) (S₂ : AddSubsemigroup M) : ∀ {a b : M}, a ∈ ↑S₁ ∩ ↑S₂ → b ∈ ↑S₁ ∩ ↑S₂ → a + b ∈ ↑S₁ ∩ ↑S₂

instance AddSubsemigroup.instInfSubsemigroup {M : Type u_1} [Add M] : Inf (AddSubsemigroup M)

The inf of two AddSubsemigroups is their intersection.

Equations

• AddSubsemigroup.instInfSubsemigroup = { inf := fun (S₁ S₂ : AddSubsemigroup M) => { carrier := ↑S₁ ∩ ↑S₂, add_mem' := ⋯ } }

instance Subsemigroup.instInfSubsemigroup {M : Type u_1} [Mul M] : Inf (Subsemigroup M)

The inf of two subsemigroups is their intersection.

Equations

• Subsemigroup.instInfSubsemigroup = { inf := fun (S₁ S₂ : Subsemigroup M) => { carrier := ↑S₁ ∩ ↑S₂, mul_mem' := ⋯ } }

@[simp]

theorem AddSubsemigroup.coe_inf {M : Type u_1} [Add M] (p : AddSubsemigroup M) (p' : AddSubsemigroup M) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem Subsemigroup.coe_inf {M : Type u_1} [Mul M] (p : Subsemigroup M) (p' : Subsemigroup M) : ↑(p ⊓ p') = ↑p ∩ ↑p'

@[simp]

theorem AddSubsemigroup.mem_inf {M : Type u_1} [Add M] {p : AddSubsemigroup M} {p' : AddSubsemigroup M} {x : M} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

@[simp]

theorem Subsemigroup.mem_inf {M : Type u_1} [Mul M] {p : Subsemigroup M} {p' : Subsemigroup M} {x : M} : x ∈ p ⊓ p' ↔ x ∈ p ∧ x ∈ p'

instance AddSubsemigroup.instInfSetSubsemigroup {M : Type u_1} [Add M] : InfSet (AddSubsemigroup M)

Equations

• AddSubsemigroup.instInfSetSubsemigroup = { sInf := fun (s : Set (AddSubsemigroup M)) => { carrier := ⋂ t ∈ s, ↑t, add_mem' := ⋯ } }

theorem AddSubsemigroup.instInfSetSubsemigroup.proof_1 {M : Type u_1} [Add M] (s : Set (AddSubsemigroup M)) : ∀ {a b : M}, a ∈ ⋂ t ∈ s, ↑t → b ∈ ⋂ t ∈ s, ↑t → a + b ∈ ⋂ x ∈ s, ↑x

instance Subsemigroup.instInfSetSubsemigroup {M : Type u_1} [Mul M] : InfSet (Subsemigroup M)

Equations

• Subsemigroup.instInfSetSubsemigroup = { sInf := fun (s : Set (Subsemigroup M)) => { carrier := ⋂ t ∈ s, ↑t, mul_mem' := ⋯ } }

@[simp]

theorem AddSubsemigroup.coe_sInf {M : Type u_1} [Add M] (S : Set (AddSubsemigroup M)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

@[simp]

theorem Subsemigroup.coe_sInf {M : Type u_1} [Mul M] (S : Set (Subsemigroup M)) : ↑(sInf S) = ⋂ s ∈ S, ↑s

theorem AddSubsemigroup.mem_sInf {M : Type u_1} [Add M] {S : Set (AddSubsemigroup M)} {x : M} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

theorem Subsemigroup.mem_sInf {M : Type u_1} [Mul M] {S : Set (Subsemigroup M)} {x : M} : x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p

theorem AddSubsemigroup.mem_iInf {M : Type u_1} [Add M] {ι : Sort u_4} {S : ι → AddSubsemigroup M} {x : M} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

theorem Subsemigroup.mem_iInf {M : Type u_1} [Mul M] {ι : Sort u_4} {S : ι → Subsemigroup M} {x : M} : x ∈ ⨅ (i : ι), S i ↔ ∀ (i : ι), x ∈ S i

@[simp]

theorem AddSubsemigroup.coe_iInf {M : Type u_1} [Add M] {ι : Sort u_4} {S : ι → AddSubsemigroup M} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

@[simp]

theorem Subsemigroup.coe_iInf {M : Type u_1} [Mul M] {ι : Sort u_4} {S : ι → Subsemigroup M} : ↑(⨅ (i : ι), S i) = ⋂ (i : ι), ↑(S i)

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_9 {M : Type u_1} [Add M] : ∀ (x x_1 : AddSubsemigroup M) (x_2 : M), x_2 ∈ ↑x ∧ x_2 ∈ ↑x_1 → x_2 ∈ ↑x

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_4 {M : Type u_1} [Add M] (a : AddSubsemigroup M) (b : AddSubsemigroup M) : a < b ↔ a ≤ b ∧ ¬b ≤ a

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_11 {M : Type u_1} [Add M] : ∀ (x x_1 x_2 : AddSubsemigroup M), x ≤ x_1 → x ≤ x_2 → ∀ x_3 ∈ x, x_3 ∈ ↑x_1 ∧ x_3 ∈ ↑x_2

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_2 {M : Type u_1} [Add M] (a : AddSubsemigroup M) : a ≤ a

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_16 {M : Type u_1} [Add M] : ∀ (x : AddSubsemigroup M), ∀ x_1 ∈ ⊥, x_1 ∈ x

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_15 {M : Type u_1} [Add M] (s : Set (AddSubsemigroup M)) (a : AddSubsemigroup M) : (∀ b ∈ s, a ≤ b) → a ≤ sInf s

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_14 {M : Type u_1} [Add M] (s : Set (AddSubsemigroup M)) (a : AddSubsemigroup M) : a ∈ s → sInf s ≤ a

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_12 {M : Type u_1} [Add M] (s : Set (AddSubsemigroup M)) (a : AddSubsemigroup M) : a ∈ s → a ≤ sSup s

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_7 {M : Type u_1} [Add M] (a : AddSubsemigroup M) (b : AddSubsemigroup M) : b ≤ a ⊔ b

instance AddSubsemigroup.instCompleteLatticeSubsemigroup {M : Type u_1} [Add M] : CompleteLattice (AddSubsemigroup M)

The AddSubsemigroups of an AddMonoid form a complete lattice.

Equations

• AddSubsemigroup.instCompleteLatticeSubsemigroup = let __src := completeLatticeOfInf (AddSubsemigroup M) ⋯; CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_1 {M : Type u_1} [Add M] : ∀ (x : Set (AddSubsemigroup M)), IsGLB x (sInf x)

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_10 {M : Type u_1} [Add M] : ∀ (x x_1 : AddSubsemigroup M) (x_2 : M), x_2 ∈ ↑x ∧ x_2 ∈ ↑x_1 → x_2 ∈ ↑x_1

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_8 {M : Type u_1} [Add M] (a : AddSubsemigroup M) (b : AddSubsemigroup M) (c : AddSubsemigroup M) : a ≤ c → b ≤ c → a ⊔ b ≤ c

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_13 {M : Type u_1} [Add M] (s : Set (AddSubsemigroup M)) (a : AddSubsemigroup M) : (∀ b ∈ s, b ≤ a) → sSup s ≤ a

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_5 {M : Type u_1} [Add M] (a : AddSubsemigroup M) (b : AddSubsemigroup M) : a ≤ b → b ≤ a → a = b

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_6 {M : Type u_1} [Add M] (a : AddSubsemigroup M) (b : AddSubsemigroup M) : a ≤ a ⊔ b

theorem AddSubsemigroup.instCompleteLatticeSubsemigroup.proof_3 {M : Type u_1} [Add M] (a : AddSubsemigroup M) (b : AddSubsemigroup M) (c : AddSubsemigroup M) : a ≤ b → b ≤ c → a ≤ c

instance Subsemigroup.instCompleteLatticeSubsemigroup {M : Type u_1} [Mul M] : CompleteLattice (Subsemigroup M)

subsemigroups of a monoid form a complete lattice.

Equations

• Subsemigroup.instCompleteLatticeSubsemigroup = let __src := completeLatticeOfInf (Subsemigroup M) ⋯; CompleteLattice.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem AddSubsemigroup.subsingleton_of_subsingleton {M : Type u_1} [Add M] [Subsingleton (AddSubsemigroup M)] : Subsingleton M

theorem Subsemigroup.subsingleton_of_subsingleton {M : Type u_1} [Mul M] [Subsingleton (Subsemigroup M)] : Subsingleton M

instance AddSubsemigroup.instNontrivialSubsemigroup {M : Type u_1} [Add M] [hn : Nonempty M] : Nontrivial (AddSubsemigroup M)

Equations

• ⋯ = ⋯

instance Subsemigroup.instNontrivialSubsemigroup {M : Type u_1} [Mul M] [hn : Nonempty M] : Nontrivial (Subsemigroup M)

Equations

• ⋯ = ⋯

def AddSubsemigroup.closure {M : Type u_1} [Add M] (s : Set M) : AddSubsemigroup M

The AddSubsemigroup generated by a set

Equations

• AddSubsemigroup.closure s = sInf {S : AddSubsemigroup M | s ⊆ ↑S}

def Subsemigroup.closure {M : Type u_1} [Mul M] (s : Set M) : Subsemigroup M

The Subsemigroup generated by a set.

Equations

• Subsemigroup.closure s = sInf {S : Subsemigroup M | s ⊆ ↑S}

theorem AddSubsemigroup.mem_closure {M : Type u_1} [Add M] {s : Set M} {x : M} : x ∈ AddSubsemigroup.closure s ↔ ∀ (S : AddSubsemigroup M), s ⊆ ↑S → x ∈ S

theorem Subsemigroup.mem_closure {M : Type u_1} [Mul M] {s : Set M} {x : M} : x ∈ Subsemigroup.closure s ↔ ∀ (S : Subsemigroup M), s ⊆ ↑S → x ∈ S

@[simp]

theorem AddSubsemigroup.subset_closure {M : Type u_1} [Add M] {s : Set M} : s ⊆ ↑(AddSubsemigroup.closure s)

The AddSubsemigroup generated by a set includes the set.

@[simp]

theorem Subsemigroup.subset_closure {M : Type u_1} [Mul M] {s : Set M} : s ⊆ ↑(Subsemigroup.closure s)

The subsemigroup generated by a set includes the set.

theorem AddSubsemigroup.not_mem_of_not_mem_closure {M : Type u_1} [Add M] {s : Set M} {P : M} (hP : P ∉ AddSubsemigroup.closure s) : P ∉ s

theorem Subsemigroup.not_mem_of_not_mem_closure {M : Type u_1} [Mul M] {s : Set M} {P : M} (hP : P ∉ Subsemigroup.closure s) : P ∉ s

@[simp]

theorem AddSubsemigroup.closure_le {M : Type u_1} [Add M] {s : Set M} {S : AddSubsemigroup M} : AddSubsemigroup.closure s ≤ S ↔ s ⊆ ↑S

An additive subsemigroup S includes closure s if and only if it includes s

@[simp]

theorem Subsemigroup.closure_le {M : Type u_1} [Mul M] {s : Set M} {S : Subsemigroup M} : Subsemigroup.closure s ≤ S ↔ s ⊆ ↑S

A subsemigroup S includes closure s if and only if it includes s.

theorem AddSubsemigroup.closure_mono {M : Type u_1} [Add M] ⦃s : Set M⦄ ⦃t : Set M⦄ (h : s ⊆ t) : AddSubsemigroup.closure s ≤ AddSubsemigroup.closure t

Additive subsemigroup closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t

theorem Subsemigroup.closure_mono {M : Type u_1} [Mul M] ⦃s : Set M⦄ ⦃t : Set M⦄ (h : s ⊆ t) : Subsemigroup.closure s ≤ Subsemigroup.closure t

subsemigroup closure of a set is monotone in its argument: if s ⊆ t, then closure s ≤ closure t.

theorem AddSubsemigroup.closure_eq_of_le {M : Type u_1} [Add M] {s : Set M} {S : AddSubsemigroup M} (h₁ : s ⊆ ↑S) (h₂ : S ≤ AddSubsemigroup.closure s) : AddSubsemigroup.closure s = S

theorem Subsemigroup.closure_eq_of_le {M : Type u_1} [Mul M] {s : Set M} {S : Subsemigroup M} (h₁ : s ⊆ ↑S) (h₂ : S ≤ Subsemigroup.closure s) : Subsemigroup.closure s = S

theorem AddSubsemigroup.closure_induction {M : Type u_1} [Add M] {s : Set M} {p : M → Prop} {x : M} (h : x ∈ AddSubsemigroup.closure s) (mem : ∀ x ∈ s, p x) (mul : ∀ (x y : M), p x → p y → p (x + y)) : p x

An induction principle for additive closure membership. If p holds for all elements of s, and is preserved under addition, then p holds for all elements of the additive closure of s.

theorem Subsemigroup.closure_induction {M : Type u_1} [Mul M] {s : Set M} {p : M → Prop} {x : M} (h : x ∈ Subsemigroup.closure s) (mem : ∀ x ∈ s, p x) (mul : ∀ (x y : M), p x → p y → p (x * y)) : p x

An induction principle for closure membership. If p holds for all elements of s, and is preserved under multiplication, then p holds for all elements of the closure of s.

theorem AddSubsemigroup.closure_induction' {M : Type u_1} [Add M] (s : Set M) {p : (x : M) → x ∈ AddSubsemigroup.closure s → Prop} (mem : ∀ (x : M) (h : x ∈ s), p x ⋯) (mul : ∀ (x : M) (hx : x ∈ AddSubsemigroup.closure s) (y : M) (hy : y ∈ AddSubsemigroup.closure s), p x hx → p y hy → p (x + y) ⋯) {x : M} (hx : x ∈ AddSubsemigroup.closure s) : p x hx

A dependent version of AddSubsemigroup.closure_induction.

abbrev AddSubsemigroup.closure_induction'.match_1 {M : Type u_1} [Add M] (s : Set M) {p : (x : M) → x ∈ AddSubsemigroup.closure s → Prop} (y : M) (motive : (∃ (x : y ∈ AddSubsemigroup.closure s), p y x) → Prop) : ∀ (x : ∃ (x : y ∈ AddSubsemigroup.closure s), p y x), (∀ (hy' : y ∈ AddSubsemigroup.closure s) (hy : p y hy'), motive ⋯) → motive x

Equations

• ⋯ = ⋯

theorem Subsemigroup.closure_induction' {M : Type u_1} [Mul M] (s : Set M) {p : (x : M) → x ∈ Subsemigroup.closure s → Prop} (mem : ∀ (x : M) (h : x ∈ s), p x ⋯) (mul : ∀ (x : M) (hx : x ∈ Subsemigroup.closure s) (y : M) (hy : y ∈ Subsemigroup.closure s), p x hx → p y hy → p (x * y) ⋯) {x : M} (hx : x ∈ Subsemigroup.closure s) : p x hx

A dependent version of Subsemigroup.closure_induction.

theorem AddSubsemigroup.closure_induction₂ {M : Type u_1} [Add M] {s : Set M} {p : M → M → Prop} {x : M} {y : M} (hx : x ∈ AddSubsemigroup.closure s) (hy : y ∈ AddSubsemigroup.closure s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (Hmul_left : ∀ (x y z : M), p x z → p y z → p (x + y) z) (Hmul_right : ∀ (x y z : M), p z x → p z y → p z (x + y)) : p x y

An induction principle for additive closure membership for predicates with two arguments.

theorem Subsemigroup.closure_induction₂ {M : Type u_1} [Mul M] {s : Set M} {p : M → M → Prop} {x : M} {y : M} (hx : x ∈ Subsemigroup.closure s) (hy : y ∈ Subsemigroup.closure s) (Hs : ∀ x ∈ s, ∀ y ∈ s, p x y) (Hmul_left : ∀ (x y z : M), p x z → p y z → p (x * y) z) (Hmul_right : ∀ (x y z : M), p z x → p z y → p z (x * y)) : p x y

An induction principle for closure membership for predicates with two arguments.

theorem AddSubsemigroup.dense_induction {M : Type u_1} [Add M] {p : M → Prop} (x : M) {s : Set M} (hs : AddSubsemigroup.closure s = ⊤) (mem : ∀ x ∈ s, p x) (mul : ∀ (x y : M), p x → p y → p (x + y)) : p x

If s is a dense set in an additive monoid M, AddSubsemigroup.closure s = ⊤, then in order to prove that some predicate p holds for all x : M it suffices to verify p x for x ∈ s, and verify that p x and p y imply p (x + y).

theorem Subsemigroup.dense_induction {M : Type u_1} [Mul M] {p : M → Prop} (x : M) {s : Set M} (hs : Subsemigroup.closure s = ⊤) (mem : ∀ x ∈ s, p x) (mul : ∀ (x y : M), p x → p y → p (x * y)) : p x

If s is a dense set in a magma M, Subsemigroup.closure s = ⊤, then in order to prove that some predicate p holds for all x : M it suffices to verify p x for x ∈ s, and verify that p x and p y imply p (x * y).

theorem AddSubsemigroup.gi.proof_1 (M : Type u_1) [Add M] : ∀ (x : AddSubsemigroup M), ↑x ⊆ ↑(AddSubsemigroup.closure ↑x)

def AddSubsemigroup.gi (M : Type u_1) [Add M] : GaloisInsertion AddSubsemigroup.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• AddSubsemigroup.gi M = GaloisConnection.toGaloisInsertion ⋯ ⋯

def Subsemigroup.gi (M : Type u_1) [Mul M] : GaloisInsertion Subsemigroup.closure SetLike.coe

closure forms a Galois insertion with the coercion to set.

Equations

• Subsemigroup.gi M = GaloisConnection.toGaloisInsertion ⋯ ⋯

@[simp]

theorem AddSubsemigroup.closure_eq {M : Type u_1} [Add M] (S : AddSubsemigroup M) : AddSubsemigroup.closure ↑S = S

Additive closure of an additive subsemigroup S equals S

@[simp]

theorem Subsemigroup.closure_eq {M : Type u_1} [Mul M] (S : Subsemigroup M) : Subsemigroup.closure ↑S = S

Closure of a subsemigroup S equals S.

@[simp]

theorem AddSubsemigroup.closure_empty {M : Type u_1} [Add M] : AddSubsemigroup.closure ∅ = ⊥

@[simp]

theorem Subsemigroup.closure_empty {M : Type u_1} [Mul M] : Subsemigroup.closure ∅ = ⊥

@[simp]

theorem AddSubsemigroup.closure_univ {M : Type u_1} [Add M] : AddSubsemigroup.closure Set.univ = ⊤

@[simp]

theorem Subsemigroup.closure_univ {M : Type u_1} [Mul M] : Subsemigroup.closure Set.univ = ⊤

theorem AddSubsemigroup.closure_union {M : Type u_1} [Add M] (s : Set M) (t : Set M) : AddSubsemigroup.closure (s ∪ t) = AddSubsemigroup.closure s ⊔ AddSubsemigroup.closure t

theorem Subsemigroup.closure_union {M : Type u_1} [Mul M] (s : Set M) (t : Set M) : Subsemigroup.closure (s ∪ t) = Subsemigroup.closure s ⊔ Subsemigroup.closure t

theorem AddSubsemigroup.closure_iUnion {M : Type u_1} [Add M] {ι : Sort u_4} (s : ι → Set M) : AddSubsemigroup.closure (⋃ (i : ι), s i) = ⨆ (i : ι), AddSubsemigroup.closure (s i)

theorem Subsemigroup.closure_iUnion {M : Type u_1} [Mul M] {ι : Sort u_4} (s : ι → Set M) : Subsemigroup.closure (⋃ (i : ι), s i) = ⨆ (i : ι), Subsemigroup.closure (s i)

theorem AddSubsemigroup.closure_singleton_le_iff_mem {M : Type u_1} [Add M] (m : M) (p : AddSubsemigroup M) : AddSubsemigroup.closure {m} ≤ p ↔ m ∈ p

theorem Subsemigroup.closure_singleton_le_iff_mem {M : Type u_1} [Mul M] (m : M) (p : Subsemigroup M) : Subsemigroup.closure {m} ≤ p ↔ m ∈ p

theorem AddSubsemigroup.mem_iSup {M : Type u_1} [Add M] {ι : Sort u_4} (p : ι → AddSubsemigroup M) {m : M} : m ∈ ⨆ (i : ι), p i ↔ ∀ (N : AddSubsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N

theorem Subsemigroup.mem_iSup {M : Type u_1} [Mul M] {ι : Sort u_4} (p : ι → Subsemigroup M) {m : M} : m ∈ ⨆ (i : ι), p i ↔ ∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N

theorem AddSubsemigroup.iSup_eq_closure {M : Type u_1} [Add M] {ι : Sort u_4} (p : ι → AddSubsemigroup M) : ⨆ (i : ι), p i = AddSubsemigroup.closure (⋃ (i : ι), ↑(p i))

theorem Subsemigroup.iSup_eq_closure {M : Type u_1} [Mul M] {ι : Sort u_4} (p : ι → Subsemigroup M) : ⨆ (i : ι), p i = Subsemigroup.closure (⋃ (i : ι), ↑(p i))

def AddHom.eqLocus {M : Type u_1} {N : Type u_2} [Add M] [Add N] (f : AddHom M N) (g : AddHom M N) : AddSubsemigroup M

The additive subsemigroup of elements x : M such that f x = g x

Equations

• AddHom.eqLocus f g = { carrier := {x : M | f x = g x}, add_mem' := ⋯ }

theorem AddHom.eqLocus.proof_1 {M : Type u_2} {N : Type u_1} [Add M] [Add N] (f : AddHom M N) (g : AddHom M N) : ∀ {a b : M}, f a = g a → f b = g b → f (a + b) = g (a + b)

def MulHom.eqLocus {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] (f : M →ₙ* N) (g : M →ₙ* N) : Subsemigroup M

The subsemigroup of elements x : M such that f x = g x

Equations

• MulHom.eqLocus f g = { carrier := {x : M | f x = g x}, mul_mem' := ⋯ }

theorem AddHom.eqOn_closure {M : Type u_1} {N : Type u_2} [Add M] [Add N] {f : AddHom M N} {g : AddHom M N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(AddSubsemigroup.closure s)

If two add homomorphisms are equal on a set, then they are equal on its additive subsemigroup closure.

theorem MulHom.eqOn_closure {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] {f : M →ₙ* N} {g : M →ₙ* N} {s : Set M} (h : Set.EqOn (⇑f) (⇑g) s) : Set.EqOn ⇑f ⇑g ↑(Subsemigroup.closure s)

If two mul homomorphisms are equal on a set, then they are equal on its subsemigroup closure.

theorem AddHom.eq_of_eqOn_top {M : Type u_1} {N : Type u_2} [Add M] [Add N] {f : AddHom M N} {g : AddHom M N} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g

theorem MulHom.eq_of_eqOn_top {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] {f : M →ₙ* N} {g : M →ₙ* N} (h : Set.EqOn ⇑f ⇑g ↑⊤) : f = g

theorem AddHom.eq_of_eqOn_dense {M : Type u_1} {N : Type u_2} [Add M] [Add N] {s : Set M} (hs : AddSubsemigroup.closure s = ⊤) {f : AddHom M N} {g : AddHom M N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem MulHom.eq_of_eqOn_dense {M : Type u_1} {N : Type u_2} [Mul M] [Mul N] {s : Set M} (hs : Subsemigroup.closure s = ⊤) {f : M →ₙ* N} {g : M →ₙ* N} (h : Set.EqOn (⇑f) (⇑g) s) : f = g

theorem AddHom.ofDense.proof_1 {M : Type u_2} {N : Type u_1} [AddSemigroup M] [AddSemigroup N] {s : Set M} (f : M → N) (hs : AddSubsemigroup.closure s = ⊤) (hmul : ∀ (x y : M), y ∈ s → f (x + y) = f x + f y) (x : M) (y : M) : f (x + y) = f x + f y

def AddHom.ofDense {M : Type u_4} {N : Type u_5} [AddSemigroup M] [AddSemigroup N] {s : Set M} (f : M → N) (hs : AddSubsemigroup.closure s = ⊤) (hmul : ∀ (x y : M), y ∈ s → f (x + y) = f x + f y) : AddHom M N

Let s be a subset of an additive semigroup M such that the closure of s is the whole semigroup. Then AddHom.ofDense defines an additive homomorphism from M asking for a proof of f (x + y) = f x + f y only for y ∈ s.

Equations

• AddHom.ofDense f hs hmul = { toFun := f, map_add' := ⋯ }

def MulHom.ofDense {M : Type u_4} {N : Type u_5} [Semigroup M] [Semigroup N] {s : Set M} (f : M → N) (hs : Subsemigroup.closure s = ⊤) (hmul : ∀ (x y : M), y ∈ s → f (x * y) = f x * f y) : M →ₙ* N

Let s be a subset of a semigroup M such that the closure of s is the whole semigroup. Then MulHom.ofDense defines a mul homomorphism from M asking for a proof of f (x * y) = f x * f y only for y ∈ s.

Equations

• MulHom.ofDense f hs hmul = { toFun := f, map_mul' := ⋯ }

@[simp]

theorem AddHom.coe_ofDense {M : Type u_1} {N : Type u_2} [AddSemigroup M] [AddSemigroup N] {s : Set M} (f : M → N) (hs : AddSubsemigroup.closure s = ⊤) (hmul : ∀ (x y : M), y ∈ s → f (x + y) = f x + f y) : ⇑(AddHom.ofDense f hs hmul) = f

@[simp]

theorem MulHom.coe_ofDense {M : Type u_1} {N : Type u_2} [Semigroup M] [Semigroup N] {s : Set M} (f : M → N) (hs : Subsemigroup.closure s = ⊤) (hmul : ∀ (x y : M), y ∈ s → f (x * y) = f x * f y) : ⇑(MulHom.ofDense f hs hmul) = f