File: Mathlib/Algebra/Order/Monoid/Submonoid.lean

full_name: SubmonoidClass.toOrderedCommMonoid
code: /-- A submonoid of an `OrderedCommMonoid` is an `OrderedCommMonoid`. -/
@[to_additive "An `AddSubmonoid` of an `OrderedAddCommMonoid` is an `OrderedAddCommMonoid`."]
instance (priority := 75) toOrderedCommMonoid [OrderedCommMonoid M]
    [SubmonoidClass S M] (s : S) : OrderedCommMonoid s 
goal: OrderedCommMonoid ↥s

full_name: AddSubmonoidClass.toOrderedAddCommMonoid
code: /-- A submonoid of an `OrderedCommMonoid` is an `OrderedCommMonoid`. -/
@[to_additive "An `AddSubmonoid` of an `OrderedAddCommMonoid` is an `OrderedAddCommMonoid`."]
instance (priority := 75) toOrderedCommMonoid [OrderedCommMonoid M]
    [SubmonoidClass S M] (s : S) : OrderedCommMonoid s 
goal: OrderedAddCommMonoid ↥s

full_name: SubmonoidClass.toLinearOrderedCommMonoid
code: /-- A submonoid of a `LinearOrderedCommMonoid` is a `LinearOrderedCommMonoid`. -/
@[to_additive
      "An `AddSubmonoid` of a `LinearOrderedAddCommMonoid` is a `LinearOrderedAddCommMonoid`."]
instance (priority := 75) toLinearOrderedCommMonoid [LinearOrderedCommMonoid M]
    [SubmonoidClass S M] (s : S) : LinearOrderedCommMonoid s 
goal: LinearOrderedCommMonoid ↥s

full_name: AddSubmonoidClass.toLinearOrderedAddCommMonoid
code: /-- A submonoid of a `LinearOrderedCommMonoid` is a `LinearOrderedCommMonoid`. -/
@[to_additive
      "An `AddSubmonoid` of a `LinearOrderedAddCommMonoid` is a `LinearOrderedAddCommMonoid`."]
instance (priority := 75) toLinearOrderedCommMonoid [LinearOrderedCommMonoid M]
    [SubmonoidClass S M] (s : S) : LinearOrderedCommMonoid s 
goal: LinearOrderedAddCommMonoid ↥s

full_name: AddSubmonoidClass.toOrderedCancelAddCommMonoid
code: /-- A submonoid of an `OrderedCancelCommMonoid` is an `OrderedCancelCommMonoid`. -/
@[to_additive AddSubmonoidClass.toOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of an `OrderedCancelAddCommMonoid` is an `OrderedCancelAddCommMonoid`."]
instance (priority := 75) toOrderedCancelCommMonoid [OrderedCancelCommMonoid M]
    [SubmonoidClass S M] (s : S) : OrderedCancelCommMonoid s 
goal: OrderedCancelAddCommMonoid ↥s

full_name: SubmonoidClass.toOrderedCancelCommMonoid
code: /-- A submonoid of an `OrderedCancelCommMonoid` is an `OrderedCancelCommMonoid`. -/
@[to_additive AddSubmonoidClass.toOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of an `OrderedCancelAddCommMonoid` is an `OrderedCancelAddCommMonoid`."]
instance (priority := 75) toOrderedCancelCommMonoid [OrderedCancelCommMonoid M]
    [SubmonoidClass S M] (s : S) : OrderedCancelCommMonoid s 
goal: OrderedCancelCommMonoid ↥s

full_name: SubmonoidClass.toLinearOrderedCancelCommMonoid
code: /-- A submonoid of a `LinearOrderedCancelCommMonoid` is a `LinearOrderedCancelCommMonoid`.
-/
@[to_additive AddSubmonoidClass.toLinearOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of a `LinearOrderedCancelAddCommMonoid` is
      a `LinearOrderedCancelAddCommMonoid`."]
instance (priority := 75) toLinearOrderedCancelCommMonoid [LinearOrderedCancelCommMonoid M]
    [SubmonoidClass S M] (s : S) : LinearOrderedCancelCommMonoid s 
goal: LinearOrderedCancelCommMonoid ↥s

full_name: AddSubmonoidClass.toLinearOrderedCancelAddCommMonoid
code: /-- A submonoid of a `LinearOrderedCancelCommMonoid` is a `LinearOrderedCancelCommMonoid`.
-/
@[to_additive AddSubmonoidClass.toLinearOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of a `LinearOrderedCancelAddCommMonoid` is
      a `LinearOrderedCancelAddCommMonoid`."]
instance (priority := 75) toLinearOrderedCancelCommMonoid [LinearOrderedCancelCommMonoid M]
    [SubmonoidClass S M] (s : S) : LinearOrderedCancelCommMonoid s 
goal: LinearOrderedCancelAddCommMonoid ↥s

full_name: Submonoid.toOrderedCommMonoid
code: /-- A submonoid of an `OrderedCommMonoid` is an `OrderedCommMonoid`. -/
@[to_additive "An `AddSubmonoid` of an `OrderedAddCommMonoid` is an `OrderedAddCommMonoid`."]
instance toOrderedCommMonoid [OrderedCommMonoid M] (S : Submonoid M) : OrderedCommMonoid S 
goal: OrderedCommMonoid ↥S

full_name: AddSubmonoid.toOrderedAddCommMonoid
code: /-- A submonoid of an `OrderedCommMonoid` is an `OrderedCommMonoid`. -/
@[to_additive "An `AddSubmonoid` of an `OrderedAddCommMonoid` is an `OrderedAddCommMonoid`."]
instance toOrderedCommMonoid [OrderedCommMonoid M] (S : Submonoid M) : OrderedCommMonoid S 
goal: OrderedAddCommMonoid ↥S

full_name: Submonoid.toLinearOrderedCommMonoid
code: /-- A submonoid of a `LinearOrderedCommMonoid` is a `LinearOrderedCommMonoid`. -/
@[to_additive
      "An `AddSubmonoid` of a `LinearOrderedAddCommMonoid` is a `LinearOrderedAddCommMonoid`."]
instance toLinearOrderedCommMonoid [LinearOrderedCommMonoid M] (S : Submonoid M) :
    LinearOrderedCommMonoid S 
goal: LinearOrderedCommMonoid ↥S

full_name: AddSubmonoid.toLinearOrderedAddCommMonoid
code: /-- A submonoid of a `LinearOrderedCommMonoid` is a `LinearOrderedCommMonoid`. -/
@[to_additive
      "An `AddSubmonoid` of a `LinearOrderedAddCommMonoid` is a `LinearOrderedAddCommMonoid`."]
instance toLinearOrderedCommMonoid [LinearOrderedCommMonoid M] (S : Submonoid M) :
    LinearOrderedCommMonoid S 
goal: LinearOrderedAddCommMonoid ↥S

full_name: AddSubmonoid.toOrderedCancelAddCommMonoid
code: /-- A submonoid of an `OrderedCancelCommMonoid` is an `OrderedCancelCommMonoid`. -/
@[to_additive AddSubmonoid.toOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of an `OrderedCancelAddCommMonoid` is an `OrderedCancelAddCommMonoid`."]
instance toOrderedCancelCommMonoid [OrderedCancelCommMonoid M] (S : Submonoid M) :
    OrderedCancelCommMonoid S 
goal: OrderedCancelAddCommMonoid ↥S

full_name: Submonoid.toOrderedCancelCommMonoid
code: /-- A submonoid of an `OrderedCancelCommMonoid` is an `OrderedCancelCommMonoid`. -/
@[to_additive AddSubmonoid.toOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of an `OrderedCancelAddCommMonoid` is an `OrderedCancelAddCommMonoid`."]
instance toOrderedCancelCommMonoid [OrderedCancelCommMonoid M] (S : Submonoid M) :
    OrderedCancelCommMonoid S 
goal: OrderedCancelCommMonoid ↥S

full_name: Submonoid.toLinearOrderedCancelCommMonoid
code: /-- A submonoid of a `LinearOrderedCancelCommMonoid` is a `LinearOrderedCancelCommMonoid`.
-/
@[to_additive AddSubmonoid.toLinearOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of a `LinearOrderedCancelAddCommMonoid` is
      a `LinearOrderedCancelAddCommMonoid`."]
instance toLinearOrderedCancelCommMonoid [LinearOrderedCancelCommMonoid M] (S : Submonoid M) :
    LinearOrderedCancelCommMonoid S 
goal: LinearOrderedCancelCommMonoid ↥S

full_name: AddSubmonoid.toLinearOrderedCancelAddCommMonoid
code: /-- A submonoid of a `LinearOrderedCancelCommMonoid` is a `LinearOrderedCancelCommMonoid`.
-/
@[to_additive AddSubmonoid.toLinearOrderedCancelAddCommMonoid
      "An `AddSubmonoid` of a `LinearOrderedCancelAddCommMonoid` is
      a `LinearOrderedCancelAddCommMonoid`."]
instance toLinearOrderedCancelCommMonoid [LinearOrderedCancelCommMonoid M] (S : Submonoid M) :
    LinearOrderedCancelCommMonoid S 
goal: LinearOrderedCancelAddCommMonoid ↥S

full_name: Submonoid.oneLE
code: /-- The submonoid of elements greater than `1`. -/
@[to_additive (attr := simps) nonneg "The submonoid of nonnegative elements."]
def oneLE : Submonoid M 
goal: Submonoid M

full_name: Submonoid.oneLE_coe
code: /-- The submonoid of elements greater than `1`. -/
@[to_additive (attr := simps) nonneg "The submonoid of nonnegative elements."]
def oneLE : Submonoid M 
goal: ↑(Submonoid.oneLE M) = Set.Ici 1

full_name: AddSubmonoid.nonneg
code: /-- The submonoid of elements greater than `1`. -/
@[to_additive (attr := simps) nonneg "The submonoid of nonnegative elements."]
def oneLE : Submonoid M 
goal: AddSubmonoid M

full_name: AddSubmonoid.nonneg_coe
code: /-- The submonoid of elements greater than `1`. -/
@[to_additive (attr := simps) nonneg "The submonoid of nonnegative elements."]
def oneLE : Submonoid M 
goal: ↑(AddSubmonoid.nonneg M) = Set.Ici 0

full_name: Submonoid.mem_oneLE
code: @[to_additive (attr := simp) mem_nonneg] lemma mem_oneLE : a ∈ oneLE M ↔ 1 ≤ a 
goal: a ∈ Submonoid.oneLE M ↔ 1 ≤ a

full_name: AddSubmonoid.mem_nonneg
code: @[to_additive (attr := simp) mem_nonneg] lemma mem_oneLE : a ∈ oneLE M ↔ 1 ≤ a 
goal: a ∈ AddSubmonoid.nonneg M ↔ 0 ≤ a

full_name: Submonoid.pos
code: /-- The submonoid of positive elements. -/
@[simps] def pos : Submonoid α 
goal: Submonoid α

full_name: Submonoid.pos_coe
code: /-- The submonoid of positive elements. -/
@[simps] def pos : Submonoid α 
goal: ↑(Submonoid.pos α) = Set.Ioi 0

full_name: Submonoid.mem_pos
code: @[simp] lemma mem_pos : a ∈ pos α ↔ 0 < a 
goal: a ∈ Submonoid.pos α ↔ 0 < a