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

full_name: Function.Injective.orderedAddCommMonoid
code: /-- Pullback an `OrderedCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) "Pullback an `OrderedAddCommMonoid` under an injective map."]
def Function.Injective.orderedCommMonoid [OrderedCommMonoid α] {β : Type*} [One β] [Mul β]
    [Pow β ℕ] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1)
    (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) :
    OrderedCommMonoid β 
goal: OrderedAddCommMonoid β

full_name: Function.Injective.orderedCommMonoid
code: /-- Pullback an `OrderedCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) "Pullback an `OrderedAddCommMonoid` under an injective map."]
def Function.Injective.orderedCommMonoid [OrderedCommMonoid α] {β : Type*} [One β] [Mul β]
    [Pow β ℕ] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1)
    (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) :
    OrderedCommMonoid β 
goal: OrderedCommMonoid β

full_name: Function.Injective.orderedCancelCommMonoid
code: /-- Pullback an `OrderedCancelCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) Function.Injective.orderedCancelAddCommMonoid
    "Pullback an `OrderedCancelAddCommMonoid` under an injective map."]
def Function.Injective.orderedCancelCommMonoid [OrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ℕ]
    (f : β → α) (hf : Injective f) (one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y)
    (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) : OrderedCancelCommMonoid β 
goal: OrderedCancelCommMonoid β

full_name: Function.Injective.orderedCancelAddCommMonoid
code: /-- Pullback an `OrderedCancelCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) Function.Injective.orderedCancelAddCommMonoid
    "Pullback an `OrderedCancelAddCommMonoid` under an injective map."]
def Function.Injective.orderedCancelCommMonoid [OrderedCancelCommMonoid α] [One β] [Mul β] [Pow β ℕ]
    (f : β → α) (hf : Injective f) (one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y)
    (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n) : OrderedCancelCommMonoid β 
goal: OrderedCancelAddCommMonoid β

full_name: Function.Injective.linearOrderedAddCommMonoid
code: /-- Pullback a `LinearOrderedCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) "Pullback an `OrderedAddCommMonoid` under an injective map."]
def Function.Injective.linearOrderedCommMonoid [LinearOrderedCommMonoid α] {β : Type*} [One β]
    [Mul β] [Pow β ℕ] [Sup β] [Inf β] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1)
    (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
    (sup : ∀ x y, f (x ⊔ y) = max (f x) (f y)) (inf : ∀ x y, f (x ⊓ y) = min (f x) (f y)) :
    LinearOrderedCommMonoid β 
goal: LinearOrderedAddCommMonoid β

full_name: Function.Injective.linearOrderedCommMonoid
code: /-- Pullback a `LinearOrderedCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) "Pullback an `OrderedAddCommMonoid` under an injective map."]
def Function.Injective.linearOrderedCommMonoid [LinearOrderedCommMonoid α] {β : Type*} [One β]
    [Mul β] [Pow β ℕ] [Sup β] [Inf β] (f : β → α) (hf : Function.Injective f) (one : f 1 = 1)
    (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
    (sup : ∀ x y, f (x ⊔ y) = max (f x) (f y)) (inf : ∀ x y, f (x ⊓ y) = min (f x) (f y)) :
    LinearOrderedCommMonoid β 
goal: LinearOrderedCommMonoid β

full_name: Function.Injective.linearOrderedCancelAddCommMonoid
code: /-- Pullback a `LinearOrderedCancelCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) Function.Injective.linearOrderedCancelAddCommMonoid
    "Pullback a `LinearOrderedCancelAddCommMonoid` under an injective map."]
def Function.Injective.linearOrderedCancelCommMonoid [LinearOrderedCancelCommMonoid α] [One β]
    [Mul β] [Pow β ℕ] [Sup β] [Inf β] (f : β → α) (hf : Injective f) (one : f 1 = 1)
    (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
    (hsup : ∀ x y, f (x ⊔ y) = max (f x) (f y)) (hinf : ∀ x y, f (x ⊓ y) = min (f x) (f y)) :
    LinearOrderedCancelCommMonoid β 
goal: LinearOrderedCancelAddCommMonoid β

full_name: Function.Injective.linearOrderedCancelCommMonoid
code: /-- Pullback a `LinearOrderedCancelCommMonoid` under an injective map.
See note [reducible non-instances]. -/
@[to_additive (attr := reducible) Function.Injective.linearOrderedCancelAddCommMonoid
    "Pullback a `LinearOrderedCancelAddCommMonoid` under an injective map."]
def Function.Injective.linearOrderedCancelCommMonoid [LinearOrderedCancelCommMonoid α] [One β]
    [Mul β] [Pow β ℕ] [Sup β] [Inf β] (f : β → α) (hf : Injective f) (one : f 1 = 1)
    (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ (x) (n : ℕ), f (x ^ n) = f x ^ n)
    (hsup : ∀ x y, f (x ⊔ y) = max (f x) (f y)) (hinf : ∀ x y, f (x ⊓ y) = min (f x) (f y)) :
    LinearOrderedCancelCommMonoid β 
goal: LinearOrderedCancelCommMonoid β

full_name: OrderEmbedding.addLeft_apply
code: /-- The order embedding sending `b` to `a * b`, for some fixed `a`.
See also `OrderIso.mulLeft` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `a + b`, for some fixed `a`.
       See also `OrderIso.addLeft` when working in an additive ordered group."]
def OrderEmbedding.mulLeft {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· < ·)] (m : α) : α ↪o α 
goal: (OrderEmbedding.addLeft m) n = m + n

full_name: OrderEmbedding.mulLeft
code: /-- The order embedding sending `b` to `a * b`, for some fixed `a`.
See also `OrderIso.mulLeft` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `a + b`, for some fixed `a`.
       See also `OrderIso.addLeft` when working in an additive ordered group."]
def OrderEmbedding.mulLeft {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· < ·)] (m : α) : α ↪o α 
goal: α ↪o α

full_name: OrderEmbedding.addLeft
code: /-- The order embedding sending `b` to `a * b`, for some fixed `a`.
See also `OrderIso.mulLeft` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `a + b`, for some fixed `a`.
       See also `OrderIso.addLeft` when working in an additive ordered group."]
def OrderEmbedding.mulLeft {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· < ·)] (m : α) : α ↪o α 
goal: α ↪o α

full_name: OrderEmbedding.mulLeft_apply
code: /-- The order embedding sending `b` to `a * b`, for some fixed `a`.
See also `OrderIso.mulLeft` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `a + b`, for some fixed `a`.
       See also `OrderIso.addLeft` when working in an additive ordered group."]
def OrderEmbedding.mulLeft {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (· * ·) (· < ·)] (m : α) : α ↪o α 
goal: (OrderEmbedding.mulLeft m) n = m * n

full_name: OrderEmbedding.addRight_apply
code: /-- The order embedding sending `b` to `b * a`, for some fixed `a`.
See also `OrderIso.mulRight` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `b + a`, for some fixed `a`.
       See also `OrderIso.addRight` when working in an additive ordered group."]
def OrderEmbedding.mulRight {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (m : α) : α ↪o α 
goal: (OrderEmbedding.addRight m) n = n + m

full_name: OrderEmbedding.mulRight
code: /-- The order embedding sending `b` to `b * a`, for some fixed `a`.
See also `OrderIso.mulRight` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `b + a`, for some fixed `a`.
       See also `OrderIso.addRight` when working in an additive ordered group."]
def OrderEmbedding.mulRight {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (m : α) : α ↪o α 
goal: α ↪o α

full_name: OrderEmbedding.addRight
code: /-- The order embedding sending `b` to `b * a`, for some fixed `a`.
See also `OrderIso.mulRight` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `b + a`, for some fixed `a`.
       See also `OrderIso.addRight` when working in an additive ordered group."]
def OrderEmbedding.mulRight {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (m : α) : α ↪o α 
goal: α ↪o α

full_name: OrderEmbedding.mulRight_apply
code: /-- The order embedding sending `b` to `b * a`, for some fixed `a`.
See also `OrderIso.mulRight` when working in an ordered group. -/
@[to_additive (attr := simps!)
      "The order embedding sending `b` to `b + a`, for some fixed `a`.
       See also `OrderIso.addRight` when working in an additive ordered group."]
def OrderEmbedding.mulRight {α : Type*} [Mul α] [LinearOrder α]
    [CovariantClass α α (swap (· * ·)) (· < ·)] (m : α) : α ↪o α 
goal: (OrderEmbedding.mulRight m) n = n * m