File: Mathlib/GroupTheory/MonoidLocalization/Order.lean

full_name: AddLocalization.le
code: @[to_additive]
instance le : LE (Localization s) 
goal: LE (AddLocalization s)

full_name: Localization.le
code: @[to_additive]
instance le : LE (Localization s) 
goal: LE (Localization s)

full_name: AddLocalization.lt
code: @[to_additive]
instance lt : LT (Localization s) 
goal: LT (AddLocalization s)

full_name: Localization.lt
code: @[to_additive]
instance lt : LT (Localization s) 
goal: LT (Localization s)

full_name: Localization.mk_le_mk
code: @[to_additive]
theorem mk_le_mk : mk a₁ a₂ ≤ mk b₁ b₂ ↔ ↑b₂ * a₁ ≤ a₂ * b₁ 
goal: Localization.mk a₁ a₂ ≤ Localization.mk b₁ b₂ ↔ ↑b₂ * a₁ ≤ ↑a₂ * b₁

full_name: AddLocalization.mk_le_mk
code: @[to_additive]
theorem mk_le_mk : mk a₁ a₂ ≤ mk b₁ b₂ ↔ ↑b₂ * a₁ ≤ a₂ * b₁ 
goal: AddLocalization.mk a₁ a₂ ≤ AddLocalization.mk b₁ b₂ ↔ ↑b₂ + a₁ ≤ ↑a₂ + b₁

full_name: AddLocalization.mk_lt_mk
code: @[to_additive]
theorem mk_lt_mk : mk a₁ a₂ < mk b₁ b₂ ↔ ↑b₂ * a₁ < a₂ * b₁ 
goal: AddLocalization.mk a₁ a₂ < AddLocalization.mk b₁ b₂ ↔ ↑b₂ + a₁ < ↑a₂ + b₁

full_name: Localization.mk_lt_mk
code: @[to_additive]
theorem mk_lt_mk : mk a₁ a₂ < mk b₁ b₂ ↔ ↑b₂ * a₁ < a₂ * b₁ 
goal: Localization.mk a₁ a₂ < Localization.mk b₁ b₂ ↔ ↑b₂ * a₁ < ↑a₂ * b₁

full_name: AddLocalization.partialOrder
code: @[to_additive]
instance partialOrder : PartialOrder (Localization s) 
goal: PartialOrder (AddLocalization s)

full_name: Localization.partialOrder
code: @[to_additive]
instance partialOrder : PartialOrder (Localization s) 
goal: PartialOrder (Localization s)

full_name: Localization.orderedCancelCommMonoid
code: @[to_additive]
instance orderedCancelCommMonoid : OrderedCancelCommMonoid (Localization s) 
goal: OrderedCancelCommMonoid (Localization s)

full_name: AddLocalization.orderedAddCancelCommMonoid
code: @[to_additive]
instance orderedCancelCommMonoid : OrderedCancelCommMonoid (Localization s) 
goal: OrderedCancelAddCommMonoid (AddLocalization s)

full_name: AddLocalization.decidableLE
code: @[to_additive]
instance decidableLE [DecidableRel ((· ≤ ·) : α → α → Prop)] :
    DecidableRel ((· ≤ ·) : Localization s → Localization s → Prop) 
goal: DecidableRel fun x x_1 => x ≤ x_1

full_name: Localization.decidableLE
code: @[to_additive]
instance decidableLE [DecidableRel ((· ≤ ·) : α → α → Prop)] :
    DecidableRel ((· ≤ ·) : Localization s → Localization s → Prop) 
goal: DecidableRel fun x x_1 => x ≤ x_1

full_name: AddLocalization.decidableLT
code: @[to_additive]
instance decidableLT [DecidableRel ((· < ·) : α → α → Prop)] :
    DecidableRel ((· < ·) : Localization s → Localization s → Prop) 
goal: DecidableRel fun x x_1 => x < x_1

full_name: Localization.decidableLT
code: @[to_additive]
instance decidableLT [DecidableRel ((· < ·) : α → α → Prop)] :
    DecidableRel ((· < ·) : Localization s → Localization s → Prop) 
goal: DecidableRel fun x x_1 => x < x_1

full_name: AddLocalization.mkOrderEmbedding_apply
code: /-- An ordered cancellative monoid injects into its localization by sending `a` to `a / b`. -/
@[to_additive (attr := simps!) "An ordered cancellative monoid injects into its localization by
sending `a` to `a - b`."]
def mkOrderEmbedding (b : s) : α ↪o Localization s 
goal: (AddLocalization.mkOrderEmbedding b) a = (AddLocalization.addMonoidOf s).mk' a b

full_name: AddLocalization.mkOrderEmbedding
code: /-- An ordered cancellative monoid injects into its localization by sending `a` to `a / b`. -/
@[to_additive (attr := simps!) "An ordered cancellative monoid injects into its localization by
sending `a` to `a - b`."]
def mkOrderEmbedding (b : s) : α ↪o Localization s 
goal: α ↪o AddLocalization s

full_name: Localization.mkOrderEmbedding_apply
code: /-- An ordered cancellative monoid injects into its localization by sending `a` to `a / b`. -/
@[to_additive (attr := simps!) "An ordered cancellative monoid injects into its localization by
sending `a` to `a - b`."]
def mkOrderEmbedding (b : s) : α ↪o Localization s 
goal: (Localization.mkOrderEmbedding b) a = (Localization.monoidOf s).mk' a b

full_name: Localization.mkOrderEmbedding
code: /-- An ordered cancellative monoid injects into its localization by sending `a` to `a / b`. -/
@[to_additive (attr := simps!) "An ordered cancellative monoid injects into its localization by
sending `a` to `a - b`."]
def mkOrderEmbedding (b : s) : α ↪o Localization s 
goal: α ↪o Localization s

full_name: Localization.instLinearOrderedCancelCommMonoid
code: @[to_additive]
instance [LinearOrderedCancelCommMonoid α] {s : Submonoid α} :
    LinearOrderedCancelCommMonoid (Localization s) 
goal: LinearOrderedCancelCommMonoid (Localization s)

full_name: AddLocalization.instLinearOrderedAddCancelCommMonoid
code: @[to_additive]
instance [LinearOrderedCancelCommMonoid α] {s : Submonoid α} :
    LinearOrderedCancelCommMonoid (Localization s) 
goal: LinearOrderedCancelAddCommMonoid (AddLocalization s)