File: Mathlib/Algebra/Order/Group/OrderIso.lean

full_name: OrderIso.neg_symm_apply
code: /-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ -x` as an order-reversing equivalence."]
def OrderIso.inv : α ≃o αᵒᵈ 
goal: ∀ (a : αᵒᵈ), (RelIso.symm (OrderIso.neg α)) a = -OrderDual.ofDual a

full_name: OrderIso.inv_symm_apply
code: /-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ -x` as an order-reversing equivalence."]
def OrderIso.inv : α ≃o αᵒᵈ 
goal: ∀ (a : αᵒᵈ), (RelIso.symm (OrderIso.inv α)) a = (OrderDual.ofDual a)⁻¹

full_name: OrderIso.neg_apply
code: /-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ -x` as an order-reversing equivalence."]
def OrderIso.inv : α ≃o αᵒᵈ 
goal: ∀ (a : α), (OrderIso.neg α) a = OrderDual.toDual (-a)

full_name: OrderIso.neg
code: /-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ -x` as an order-reversing equivalence."]
def OrderIso.inv : α ≃o αᵒᵈ 
goal: α ≃o αᵒᵈ

full_name: OrderIso.inv
code: /-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ -x` as an order-reversing equivalence."]
def OrderIso.inv : α ≃o αᵒᵈ 
goal: α ≃o αᵒᵈ

full_name: OrderIso.inv_apply
code: /-- `x ↦ x⁻¹` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ -x` as an order-reversing equivalence."]
def OrderIso.inv : α ≃o αᵒᵈ 
goal: ∀ (a : α), (OrderIso.inv α) a = OrderDual.toDual a⁻¹

full_name: inv_le'
code: @[to_additive neg_le]
theorem inv_le' : a⁻¹ ≤ b ↔ b⁻¹ ≤ a 
goal: a⁻¹ ≤ b ↔ b⁻¹ ≤ a

full_name: neg_le
code: @[to_additive neg_le]
theorem inv_le' : a⁻¹ ≤ b ↔ b⁻¹ ≤ a 
goal: -a ≤ b ↔ -b ≤ a

full_name: le_neg
code: @[to_additive le_neg]
theorem le_inv' : a ≤ b⁻¹ ↔ b ≤ a⁻¹ 
goal: a ≤ -b ↔ b ≤ -a

full_name: le_inv'
code: @[to_additive le_neg]
theorem le_inv' : a ≤ b⁻¹ ↔ b ≤ a⁻¹ 
goal: a ≤ b⁻¹ ↔ b ≤ a⁻¹

full_name: OrderIso.divLeft_symm_apply
code: /-- `x ↦ a / x` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ a - x` as an order-reversing equivalence."]
def OrderIso.divLeft (a : α) : α ≃o αᵒᵈ 
goal: ∀ (a_1 : αᵒᵈ), (RelIso.symm (OrderIso.divLeft a)) a_1 = (OrderDual.ofDual a_1)⁻¹ * a

full_name: OrderIso.subLeft_symm_apply
code: /-- `x ↦ a / x` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ a - x` as an order-reversing equivalence."]
def OrderIso.divLeft (a : α) : α ≃o αᵒᵈ 
goal: ∀ (a_1 : αᵒᵈ), (RelIso.symm (OrderIso.subLeft a)) a_1 = -OrderDual.ofDual a_1 + a

full_name: OrderIso.divLeft
code: /-- `x ↦ a / x` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ a - x` as an order-reversing equivalence."]
def OrderIso.divLeft (a : α) : α ≃o αᵒᵈ 
goal: α ≃o αᵒᵈ

full_name: OrderIso.subLeft_apply
code: /-- `x ↦ a / x` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ a - x` as an order-reversing equivalence."]
def OrderIso.divLeft (a : α) : α ≃o αᵒᵈ 
goal: ∀ (a_1 : α), (OrderIso.subLeft a) a_1 = OrderDual.toDual (a - a_1)

full_name: OrderIso.divLeft_apply
code: /-- `x ↦ a / x` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ a - x` as an order-reversing equivalence."]
def OrderIso.divLeft (a : α) : α ≃o αᵒᵈ 
goal: ∀ (a_1 : α), (OrderIso.divLeft a) a_1 = OrderDual.toDual (a / a_1)

full_name: OrderIso.subLeft
code: /-- `x ↦ a / x` as an order-reversing equivalence. -/
@[to_additive (attr := simps!) "`x ↦ a - x` as an order-reversing equivalence."]
def OrderIso.divLeft (a : α) : α ≃o αᵒᵈ 
goal: α ≃o αᵒᵈ

full_name: OrderIso.mulRight
code: /-- `Equiv.mulRight` as an `OrderIso`. See also `OrderEmbedding.mulRight`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addRight` as an `OrderIso`. See also `OrderEmbedding.addRight`."]
def OrderIso.mulRight (a : α) : α ≃o α 
goal: α ≃o α

full_name: OrderIso.mulRight_toEquiv
code: /-- `Equiv.mulRight` as an `OrderIso`. See also `OrderEmbedding.mulRight`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addRight` as an `OrderIso`. See also `OrderEmbedding.addRight`."]
def OrderIso.mulRight (a : α) : α ≃o α 
goal: (OrderIso.mulRight a).toEquiv = Equiv.mulRight a

full_name: OrderIso.addRight
code: /-- `Equiv.mulRight` as an `OrderIso`. See also `OrderEmbedding.mulRight`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addRight` as an `OrderIso`. See also `OrderEmbedding.addRight`."]
def OrderIso.mulRight (a : α) : α ≃o α 
goal: α ≃o α

full_name: OrderIso.addRight_apply
code: /-- `Equiv.mulRight` as an `OrderIso`. See also `OrderEmbedding.mulRight`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addRight` as an `OrderIso`. See also `OrderEmbedding.addRight`."]
def OrderIso.mulRight (a : α) : α ≃o α 
goal: (OrderIso.addRight a) x = x + a

full_name: OrderIso.mulRight_apply
code: /-- `Equiv.mulRight` as an `OrderIso`. See also `OrderEmbedding.mulRight`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addRight` as an `OrderIso`. See also `OrderEmbedding.addRight`."]
def OrderIso.mulRight (a : α) : α ≃o α 
goal: (OrderIso.mulRight a) x = x * a

full_name: OrderIso.addRight_toEquiv
code: /-- `Equiv.mulRight` as an `OrderIso`. See also `OrderEmbedding.mulRight`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addRight` as an `OrderIso`. See also `OrderEmbedding.addRight`."]
def OrderIso.mulRight (a : α) : α ≃o α 
goal: (OrderIso.addRight a).toEquiv = Equiv.addRight a

full_name: OrderIso.mulRight_symm
code: @[to_additive (attr := simp)]
theorem OrderIso.mulRight_symm (a : α) : (OrderIso.mulRight a).symm = OrderIso.mulRight a⁻¹ 
goal: (OrderIso.mulRight a).symm = OrderIso.mulRight a⁻¹

full_name: OrderIso.addRight_symm
code: @[to_additive (attr := simp)]
theorem OrderIso.mulRight_symm (a : α) : (OrderIso.mulRight a).symm = OrderIso.mulRight a⁻¹ 
goal: (OrderIso.addRight a).symm = OrderIso.addRight (-a)

full_name: OrderIso.subRight_symm_apply
code: /-- `x ↦ x / a` as an order isomorphism. -/
@[to_additive (attr := simps!) "`x ↦ x - a` as an order isomorphism."]
def OrderIso.divRight (a : α) : α ≃o α 
goal: (RelIso.symm (OrderIso.subRight a)) b = b + a

full_name: OrderIso.divRight_apply
code: /-- `x ↦ x / a` as an order isomorphism. -/
@[to_additive (attr := simps!) "`x ↦ x - a` as an order isomorphism."]
def OrderIso.divRight (a : α) : α ≃o α 
goal: (OrderIso.divRight a) b = b / a

full_name: OrderIso.subRight
code: /-- `x ↦ x / a` as an order isomorphism. -/
@[to_additive (attr := simps!) "`x ↦ x - a` as an order isomorphism."]
def OrderIso.divRight (a : α) : α ≃o α 
goal: α ≃o α

full_name: OrderIso.subRight_apply
code: /-- `x ↦ x / a` as an order isomorphism. -/
@[to_additive (attr := simps!) "`x ↦ x - a` as an order isomorphism."]
def OrderIso.divRight (a : α) : α ≃o α 
goal: (OrderIso.subRight a) b = b - a

full_name: OrderIso.divRight
code: /-- `x ↦ x / a` as an order isomorphism. -/
@[to_additive (attr := simps!) "`x ↦ x - a` as an order isomorphism."]
def OrderIso.divRight (a : α) : α ≃o α 
goal: α ≃o α

full_name: OrderIso.divRight_symm_apply
code: /-- `x ↦ x / a` as an order isomorphism. -/
@[to_additive (attr := simps!) "`x ↦ x - a` as an order isomorphism."]
def OrderIso.divRight (a : α) : α ≃o α 
goal: (RelIso.symm (OrderIso.divRight a)) b = b * a

full_name: OrderIso.addLeft_toEquiv
code: /-- `Equiv.mulLeft` as an `OrderIso`. See also `OrderEmbedding.mulLeft`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addLeft` as an `OrderIso`. See also `OrderEmbedding.addLeft`."]
def OrderIso.mulLeft (a : α) : α ≃o α 
goal: (OrderIso.addLeft a).toEquiv = Equiv.addLeft a

full_name: OrderIso.addLeft
code: /-- `Equiv.mulLeft` as an `OrderIso`. See also `OrderEmbedding.mulLeft`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addLeft` as an `OrderIso`. See also `OrderEmbedding.addLeft`."]
def OrderIso.mulLeft (a : α) : α ≃o α 
goal: α ≃o α

full_name: OrderIso.mulLeft_toEquiv
code: /-- `Equiv.mulLeft` as an `OrderIso`. See also `OrderEmbedding.mulLeft`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addLeft` as an `OrderIso`. See also `OrderEmbedding.addLeft`."]
def OrderIso.mulLeft (a : α) : α ≃o α 
goal: (OrderIso.mulLeft a).toEquiv = Equiv.mulLeft a

full_name: OrderIso.addLeft_apply
code: /-- `Equiv.mulLeft` as an `OrderIso`. See also `OrderEmbedding.mulLeft`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addLeft` as an `OrderIso`. See also `OrderEmbedding.addLeft`."]
def OrderIso.mulLeft (a : α) : α ≃o α 
goal: (OrderIso.addLeft a) x = a + x

full_name: OrderIso.mulLeft
code: /-- `Equiv.mulLeft` as an `OrderIso`. See also `OrderEmbedding.mulLeft`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addLeft` as an `OrderIso`. See also `OrderEmbedding.addLeft`."]
def OrderIso.mulLeft (a : α) : α ≃o α 
goal: α ≃o α

full_name: OrderIso.mulLeft_apply
code: /-- `Equiv.mulLeft` as an `OrderIso`. See also `OrderEmbedding.mulLeft`. -/
@[to_additive (attr := simps! (config := { simpRhs := true }) toEquiv apply)
  "`Equiv.addLeft` as an `OrderIso`. See also `OrderEmbedding.addLeft`."]
def OrderIso.mulLeft (a : α) : α ≃o α 
goal: (OrderIso.mulLeft a) x = a * x

full_name: OrderIso.addLeft_symm
code: @[to_additive (attr := simp)]
theorem OrderIso.mulLeft_symm (a : α) : (OrderIso.mulLeft a).symm = OrderIso.mulLeft a⁻¹ 
goal: (OrderIso.addLeft a).symm = OrderIso.addLeft (-a)

full_name: OrderIso.mulLeft_symm
code: @[to_additive (attr := simp)]
theorem OrderIso.mulLeft_symm (a : α) : (OrderIso.mulLeft a).symm = OrderIso.mulLeft a⁻¹ 
goal: (OrderIso.mulLeft a).symm = OrderIso.mulLeft a⁻¹