ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrderHom.withBotMap_coe
@[simps (config := .asFn)]
protected def withBotMap (f : α →o β) : WithBot α →o WithBot β 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α →o β)
Goal: ⇑f.withBotMap = WithBot.map ⇑f
-- OrderHom.withTopMap_coe
@[simps (config := .asFn)]
protected def withTopMap (f : α →o β) : WithTop α →o WithTop β 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α →o β)
Goal: ⇑f.withTopMap = WithTop.map ⇑f
-- RelEmbedding.orderEmbeddingOfLTEmbedding_apply
@[simp]
theorem RelEmbedding.orderEmbeddingOfLTEmbedding_apply [PartialOrder α] [PartialOrder β]
    {f : ((· < ·) : α → α → Prop) ↪r ((· < ·) : β → β → Prop)} {x : α} :
    RelEmbedding.orderEmbeddingOfLTEmbedding f x = f x 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] {f : (fun x x_1 => x < x_1) ↪r fun x x_1 => x < x_1} {x : α}
Goal: f.orderEmbeddingOfLTEmbedding x = f x
-- OrderEmbedding.ltEmbedding_apply
@[simp]
theorem ltEmbedding_apply (x : α) : f.ltEmbedding x = f x 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β) (x : α)
Goal: f.ltEmbedding x = f x
-- OrderEmbedding.le_iff_le
@[simp]
theorem le_iff_le {a b} : f a ≤ f b ↔ a ≤ b 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β) {a b : α}
Goal: f a ≤ f b ↔ a ≤ b
-- OrderEmbedding.lt_iff_lt
@[simp]
theorem lt_iff_lt {a b} : f a < f b ↔ a < b 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β) {a b : α}
Goal: f a < f b ↔ a < b
-- OrderEmbedding.eq_iff_eq
theorem eq_iff_eq {a b} : f a = f b ↔ a = b 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β) {a b : α}
Goal: f a = f b ↔ a = b
-- OrderEmbedding.monotone
protected theorem monotone : Monotone f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β)
Goal: Monotone ⇑f
-- OrderEmbedding.strictMono
protected theorem strictMono : StrictMono f 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β)
Goal: StrictMono ⇑f
-- OrderEmbedding.acc
protected theorem acc (a : α) : Acc (· < ·) (f a) → Acc (· < ·) a 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β) (a : α)
Goal: Acc (fun x x_1 => x < x_1) (f a) → Acc (fun x x_1 => x < x_1) a
-- OrderEmbedding.wellFounded
protected theorem wellFounded (f : α ↪o β) :
    WellFounded ((· < ·) : β → β → Prop) → WellFounded ((· < ·) : α → α → Prop) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β)
Goal: (WellFounded fun x x_1 => x < x_1) → WellFounded fun x x_1 => x < x_1
-- OrderEmbedding.isWellOrder
protected theorem isWellOrder [IsWellOrder β (· < ·)] (f : α ↪o β) : IsWellOrder α (· < ·) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [IsWellOrder β fun x x_1 => x < x_1] (f : α ↪o β)
Goal: IsWellOrder α fun x x_1 => x < x_1
-- OrderEmbedding.wellFoundedLT
protected theorem wellFoundedLT [WellFoundedLT β] (f : α ↪o β) : WellFoundedLT α 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [WellFoundedLT β] (f : α ↪o β)
Goal: WellFoundedLT α
-- OrderEmbedding.wellFoundedGT
protected theorem wellFoundedGT [WellFoundedGT β] (f : α ↪o β) : WellFoundedGT α 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [WellFoundedGT β] (f : α ↪o β)
Goal: WellFoundedGT α
-- OrderEmbedding.withBotMap_apply
@[simps (config := .asFn)]
protected def withBotMap (f : α ↪o β) : WithBot α ↪o WithBot β 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β)
Goal: ⇑f.withBotMap = WithBot.map ⇑f
-- OrderEmbedding.withTopMap_apply
@[simps (config := .asFn)]
protected def withTopMap (f : α ↪o β) : WithTop α ↪o WithTop β 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (f : α ↪o β)
Goal: ⇑f.withTopMap = WithTop.map ⇑f
-- OrderEmbedding.coe_ofMapLEIff
@[simp]
theorem coe_ofMapLEIff {α β} [PartialOrder α] [Preorder β] {f : α → β} (h) :
    ⇑(ofMapLEIff f h) = f 
Arguments: {α : Type u_6} {β : Type u_7} [PartialOrder α] [Preorder β] {f : α → β} (h : ∀ (a b : α), f a ≤ f b ↔ a ≤ b)
Goal: ⇑(OrderEmbedding.ofMapLEIff f h) = f
-- OrderEmbedding.coe_ofStrictMono
@[simp]
theorem coe_ofStrictMono {α β} [LinearOrder α] [Preorder β] {f : α → β} (h : StrictMono f) :
    ⇑(ofStrictMono f h) = f 
Arguments: {α : Type u_6} {β : Type u_7} [LinearOrder α] [Preorder β] {f : α → β} (h : StrictMono f)
Goal: ⇑(OrderEmbedding.ofStrictMono f h) = f
-- OrderEmbedding.subtype_apply
@[simps! (config := .asFn)]
def subtype (p : α → Prop) : Subtype p ↪o α 
Arguments: {α : Type u_2} [Preorder α] (p : α → Prop)
Goal: ⇑(OrderEmbedding.subtype p) = Subtype.val
-- OrderEmbedding.toOrderHom_coe
@[simps (config := .asFn)]
def toOrderHom {X Y : Type*} [Preorder X] [Preorder Y] (f : X ↪o Y) : X →o Y 
Arguments: {X : Type u_6} {Y : Type u_7} [Preorder X] [Preorder Y] (f : X ↪o Y)
Goal: ⇑f.toOrderHom = ⇑f
-- OrderEmbedding.ofIsEmpty_apply
@[simps] def ofIsEmpty [IsEmpty α] : α ↪o β 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [IsEmpty α] (a : α)
Goal: OrderEmbedding.ofIsEmpty a = isEmptyElim a
-- OrderEmbedding.coe_ofIsEmpty
@[simp, norm_cast]
lemma coe_ofIsEmpty [IsEmpty α] : (ofIsEmpty : α ↪o β) = (isEmptyElim : α → β) 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] [IsEmpty α]
Goal: ⇑OrderEmbedding.ofIsEmpty = fun a => isEmptyElim a
-- Disjoint.of_orderEmbedding
lemma Disjoint.of_orderEmbedding [OrderBot α] [OrderBot β] {a₁ a₂ : α} :
    Disjoint (f a₁) (f a₂) → Disjoint a₁ a₂ 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] (f : α ↪o β) [OrderBot α] [OrderBot β] {a₁ a₂ : α}
Goal: Disjoint (f a₁) (f a₂) → Disjoint a₁ a₂
-- Codisjoint.of_orderEmbedding
lemma Codisjoint.of_orderEmbedding [OrderTop α] [OrderTop β] {a₁ a₂ : α} :
    Codisjoint (f a₁) (f a₂) → Codisjoint a₁ a₂ 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] (f : α ↪o β) [OrderTop α] [OrderTop β] {a₁ a₂ : α}
Goal: Codisjoint (f a₁) (f a₂) → Codisjoint a₁ a₂
-- IsCompl.of_orderEmbedding
lemma IsCompl.of_orderEmbedding [BoundedOrder α] [BoundedOrder β] {a₁ a₂ : α} :
    IsCompl (f a₁) (f a₂) → IsCompl a₁ a₂ 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [PartialOrder β] (f : α ↪o β) [BoundedOrder α] [BoundedOrder β] {a₁ a₂ : α}
Goal: IsCompl (f a₁) (f a₂) → IsCompl a₁ a₂
-- RelHom.toOrderHom_coe
@[simps (config := .asFn)]
def toOrderHom : α →o β 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [Preorder β] (f : (fun x x_1 => x < x_1) →r fun x x_1 => x < x_1)
Goal: ⇑f.toOrderHom = ⇑f
-- RelEmbedding.toOrderHom_injective
theorem RelEmbedding.toOrderHom_injective
    (f : ((· < ·) : α → α → Prop) ↪r ((· < ·) : β → β → Prop)) :
    Function.Injective (f : ((· < ·) : α → α → Prop) →r ((· < ·) : β → β → Prop)).toOrderHom 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] [Preorder β] (f : (fun x x_1 => x < x_1) ↪r fun x x_1 => x < x_1)
Goal: Function.Injective ⇑f.toRelHom.toOrderHom
-- OrderIso.toFun_eq_coe
@[simp]
theorem toFun_eq_coe {f : α ≃o β} : f.toFun = f 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] {f : α ≃o β}
Goal: f.toFun = ⇑f
-- OrderIso.ext
@[ext]
theorem ext {f g : α ≃o β} (h : (f : α → β) = g) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] {f g : α ≃o β} (h : ⇑f = ⇑g)
Goal: f = g
-- OrderIso.coe_toOrderEmbedding
@[simp]
theorem coe_toOrderEmbedding (e : α ≃o β) : ⇑e.toOrderEmbedding = e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: ⇑e.toOrderEmbedding = ⇑e
-- OrderIso.bijective
protected theorem bijective (e : α ≃o β) : Function.Bijective e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: Function.Bijective ⇑e
-- OrderIso.injective
protected theorem injective (e : α ≃o β) : Function.Injective e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: Function.Injective ⇑e
-- OrderIso.surjective
protected theorem surjective (e : α ≃o β) : Function.Surjective e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: Function.Surjective ⇑e
-- OrderIso.apply_eq_iff_eq
theorem apply_eq_iff_eq (e : α ≃o β) {x y : α} : e x = e y ↔ x = y 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) {x y : α}
Goal: e x = e y ↔ x = y
-- OrderIso.coe_refl
@[simp]
theorem coe_refl : ⇑(refl α) = id 
Arguments: {α : Type u_2} [LE α]
Goal: ⇑(OrderIso.refl α) = id
-- OrderIso.refl_apply
@[simp]
theorem refl_apply (x : α) : refl α x = x 
Arguments: {α : Type u_2} [LE α] (x : α)
Goal: (OrderIso.refl α) x = x
-- OrderIso.refl_toEquiv
@[simp]
theorem refl_toEquiv : (refl α).toEquiv = Equiv.refl α 
Arguments: {α : Type u_2} [LE α]
Goal: (OrderIso.refl α).toEquiv = Equiv.refl α
-- OrderIso.apply_symm_apply
@[simp]
theorem apply_symm_apply (e : α ≃o β) (x : β) : e (e.symm x) = x 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) (x : β)
Goal: e (e.symm x) = x
-- OrderIso.symm_apply_apply
@[simp]
theorem symm_apply_apply (e : α ≃o β) (x : α) : e.symm (e x) = x 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) (x : α)
Goal: e.symm (e x) = x
-- OrderIso.symm_refl
@[simp]
theorem symm_refl (α : Type*) [LE α] : (refl α).symm = refl α 
Arguments: (α : Type u_6) [LE α]
Goal: (OrderIso.refl α).symm = OrderIso.refl α
-- OrderIso.apply_eq_iff_eq_symm_apply
theorem apply_eq_iff_eq_symm_apply (e : α ≃o β) (x : α) (y : β) : e x = y ↔ x = e.symm y 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) (x : α) (y : β)
Goal: e x = y ↔ x = e.symm y
-- OrderIso.symm_apply_eq
theorem symm_apply_eq (e : α ≃o β) {x : α} {y : β} : e.symm y = x ↔ y = e x 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) {x : α} {y : β}
Goal: e.symm y = x ↔ y = e x
-- OrderIso.symm_symm
@[simp]
theorem symm_symm (e : α ≃o β) : e.symm.symm = e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: e.symm.symm = e
-- OrderIso.symm_bijective
theorem symm_bijective : Function.Bijective (OrderIso.symm : (α ≃o β) → β ≃o α) 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β]
Goal: Function.Bijective OrderIso.symm
-- OrderIso.symm_injective
theorem symm_injective : Function.Injective (symm : α ≃o β → β ≃o α) 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β]
Goal: Function.Injective OrderIso.symm
-- OrderIso.toEquiv_symm
@[simp]
theorem toEquiv_symm (e : α ≃o β) : e.toEquiv.symm = e.symm.toEquiv 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: e.symm = e.symm.toEquiv
-- OrderIso.coe_trans
@[simp]
theorem coe_trans (e : α ≃o β) (e' : β ≃o γ) : ⇑(e.trans e') = e' ∘ e 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [LE β] [LE γ] (e : α ≃o β) (e' : β ≃o γ)
Goal: ⇑(e.trans e') = ⇑e' ∘ ⇑e
-- OrderIso.trans_apply
@[simp]
theorem trans_apply (e : α ≃o β) (e' : β ≃o γ) (x : α) : e.trans e' x = e' (e x) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [LE β] [LE γ] (e : α ≃o β) (e' : β ≃o γ) (x : α)
Goal: (e.trans e') x = e' (e x)
-- OrderIso.refl_trans
@[simp]
theorem refl_trans (e : α ≃o β) : (refl α).trans e = e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: (OrderIso.refl α).trans e = e
-- OrderIso.trans_refl
@[simp]
theorem trans_refl (e : α ≃o β) : e.trans (refl β) = e 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: e.trans (OrderIso.refl β) = e
-- OrderIso.symm_trans_apply
@[simp]
theorem symm_trans_apply (e₁ : α ≃o β) (e₂ : β ≃o γ) (c : γ) :
    (e₁.trans e₂).symm c = e₁.symm (e₂.symm c) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [LE β] [LE γ] (e₁ : α ≃o β) (e₂ : β ≃o γ) (c : γ)
Goal: (e₁.trans e₂).symm c = e₁.symm (e₂.symm c)
-- OrderIso.symm_trans
theorem symm_trans (e₁ : α ≃o β) (e₂ : β ≃o γ) : (e₁.trans e₂).symm = e₂.symm.trans e₁.symm 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [LE α] [LE β] [LE γ] (e₁ : α ≃o β) (e₂ : β ≃o γ)
Goal: (e₁.trans e₂).symm = e₂.symm.trans e₁.symm
-- OrderIso.self_trans_symm
@[simp]
theorem self_trans_symm (e : α ≃o β) : e.trans e.symm = OrderIso.refl α 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: e.trans e.symm = OrderIso.refl α
-- OrderIso.symm_trans_self
@[simp]
theorem symm_trans_self (e : α ≃o β) : e.symm.trans e = OrderIso.refl β 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β)
Goal: e.symm.trans e = OrderIso.refl β
-- OrderIso.arrowCongr_symm_apply
@[simps apply symm_apply]
def arrowCongr {α β γ δ} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ]
    (f : α ≃o γ) (g : β ≃o δ) : (α →o β) ≃o (γ →o δ) 
Arguments: {α : Type u_6} {β : Type u_7} {γ : Type u_8} {δ : Type u_9} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] (f : α ≃o γ) (g : β ≃o δ) (p : γ →o δ)
Goal: (RelIso.symm (f.arrowCongr g)) p = (↑g.symm).comp (p.comp ↑f)
-- OrderIso.arrowCongr_apply
@[simps apply symm_apply]
def arrowCongr {α β γ δ} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ]
    (f : α ≃o γ) (g : β ≃o δ) : (α →o β) ≃o (γ →o δ) 
Arguments: {α : Type u_6} {β : Type u_7} {γ : Type u_8} {δ : Type u_9} [Preorder α] [Preorder β] [Preorder γ] [Preorder δ] (f : α ≃o γ) (g : β ≃o δ) (p : α →o β)
Goal: (f.arrowCongr g) p = (↑g).comp (p.comp ↑f.symm)
-- OrderIso.conj_apply
@[simps! apply symm_apply]
def conj {α β} [Preorder α] [Preorder β] (f : α ≃o β) : (α →o α) ≃ (β →o β) 
Arguments: {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] (f : α ≃o β) (a : α →o α)
Goal: f.conj a = (↑f).comp (a.comp ↑f.symm)
-- OrderIso.conj_symm_apply
@[simps! apply symm_apply]
def conj {α β} [Preorder α] [Preorder β] (f : α ≃o β) : (α →o α) ≃ (β →o β) 
Arguments: {α : Type u_6} {β : Type u_7} [Preorder α] [Preorder β] (f : α ≃o β)
Goal: ∀ (a : β →o β), f.conj.symm a = EquivLike.inv (f.arrowCongr f) a
-- OrderIso.coe_prodComm
@[simp]
theorem coe_prodComm : ⇑(prodComm : α × β ≃o β × α) = Prod.swap 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β]
Goal: ⇑OrderIso.prodComm = Prod.swap
-- OrderIso.prodComm_symm
@[simp]
theorem prodComm_symm : (prodComm : α × β ≃o β × α).symm = prodComm 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β]
Goal: OrderIso.prodComm.symm = OrderIso.prodComm
-- OrderIso.coe_dualDual
@[simp]
theorem coe_dualDual : ⇑(dualDual α) = toDual ∘ toDual 
Arguments: (α : Type u_2) [LE α]
Goal: ⇑(OrderIso.dualDual α) = ⇑OrderDual.toDual ∘ ⇑OrderDual.toDual
-- OrderIso.coe_dualDual_symm
@[simp]
theorem coe_dualDual_symm : ⇑(dualDual α).symm = ofDual ∘ ofDual 
Arguments: (α : Type u_2) [LE α]
Goal: ⇑(OrderIso.dualDual α).symm = ⇑OrderDual.ofDual ∘ ⇑OrderDual.ofDual
-- OrderIso.dualDual_apply
@[simp]
theorem dualDual_apply (a : α) : dualDual α a = toDual (toDual a) 
Arguments: {α : Type u_2} [LE α] (a : α)
Goal: (OrderIso.dualDual α) a = OrderDual.toDual (OrderDual.toDual a)
-- OrderIso.dualDual_symm_apply
@[simp]
theorem dualDual_symm_apply (a : αᵒᵈᵒᵈ) : (dualDual α).symm a = ofDual (ofDual a) 
Arguments: {α : Type u_2} [LE α] (a : αᵒᵈᵒᵈ)
Goal: (OrderIso.dualDual α).symm a = OrderDual.ofDual (OrderDual.ofDual a)
-- OrderIso.le_iff_le
theorem le_iff_le (e : α ≃o β) {x y : α} : e x ≤ e y ↔ x ≤ y 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) {x y : α}
Goal: e x ≤ e y ↔ x ≤ y
-- OrderIso.le_symm_apply
theorem le_symm_apply (e : α ≃o β) {x : α} {y : β} : x ≤ e.symm y ↔ e x ≤ y 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) {x : α} {y : β}
Goal: x ≤ e.symm y ↔ e x ≤ y
-- OrderIso.symm_apply_le
theorem symm_apply_le (e : α ≃o β) {x : α} {y : β} : e.symm y ≤ x ↔ y ≤ e x 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [LE β] (e : α ≃o β) {x : α} {y : β}
Goal: e.symm y ≤ x ↔ y ≤ e x
-- OrderIso.monotone
protected theorem monotone (e : α ≃o β) : Monotone e 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β)
Goal: Monotone ⇑e
-- OrderIso.strictMono
protected theorem strictMono (e : α ≃o β) : StrictMono e 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β)
Goal: StrictMono ⇑e
-- OrderIso.lt_iff_lt
@[simp]
theorem lt_iff_lt (e : α ≃o β) {x y : α} : e x < e y ↔ x < y 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β) {x y : α}
Goal: e x < e y ↔ x < y
-- OrderIso.toRelIsoLT_apply
@[simp]
theorem toRelIsoLT_apply (e : α ≃o β) (x : α) : e.toRelIsoLT x = e x 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β) (x : α)
Goal: e.toRelIsoLT x = e x
-- OrderIso.toRelIsoLT_symm
@[simp]
theorem toRelIsoLT_symm (e : α ≃o β) : e.toRelIsoLT.symm = e.symm.toRelIsoLT 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃o β)
Goal: e.toRelIsoLT.symm = e.symm.toRelIsoLT
-- OrderIso.ofRelIsoLT_apply
@[simp]
theorem ofRelIsoLT_apply {α β} [PartialOrder α] [PartialOrder β]
    (e : ((· < ·) : α → α → Prop) ≃r ((· < ·) : β → β → Prop)) (x : α) : ofRelIsoLT e x = e x 
Arguments: {α : Type u_6} {β : Type u_7} [PartialOrder α] [PartialOrder β] (e : (fun x x_1 => x < x_1) ≃r fun x x_1 => x < x_1) (x : α)
Goal: (OrderIso.ofRelIsoLT e) x = e x
-- OrderIso.ofRelIsoLT_symm
@[simp]
theorem ofRelIsoLT_symm {α β} [PartialOrder α] [PartialOrder β]
    (e : ((· < ·) : α → α → Prop) ≃r ((· < ·) : β → β → Prop)) :
    (ofRelIsoLT e).symm = ofRelIsoLT e.symm 
Arguments: {α : Type u_6} {β : Type u_7} [PartialOrder α] [PartialOrder β] (e : (fun x x_1 => x < x_1) ≃r fun x x_1 => x < x_1)
Goal: (OrderIso.ofRelIsoLT e).symm = OrderIso.ofRelIsoLT e.symm
-- OrderIso.ofRelIsoLT_toRelIsoLT
@[simp]
theorem ofRelIsoLT_toRelIsoLT {α β} [PartialOrder α] [PartialOrder β] (e : α ≃o β) :
    ofRelIsoLT (toRelIsoLT e) = e 
Arguments: {α : Type u_6} {β : Type u_7} [PartialOrder α] [PartialOrder β] (e : α ≃o β)
Goal: OrderIso.ofRelIsoLT e.toRelIsoLT = e
-- OrderIso.toRelIsoLT_ofRelIsoLT
@[simp]
theorem toRelIsoLT_ofRelIsoLT {α β} [PartialOrder α] [PartialOrder β]
    (e : ((· < ·) : α → α → Prop) ≃r ((· < ·) : β → β → Prop)) : toRelIsoLT (ofRelIsoLT e) = e 
Arguments: {α : Type u_6} {β : Type u_7} [PartialOrder α] [PartialOrder β] (e : (fun x x_1 => x < x_1) ≃r fun x x_1 => x < x_1)
Goal: (OrderIso.ofRelIsoLT e).toRelIsoLT = e
-- OrderIso.funUnique_toEquiv
@[simps! toEquiv apply]
def funUnique (α β : Type*) [Unique α] [Preorder β] : (α → β) ≃o β 
Arguments: (α : Type u_6) (β : Type u_7) [Unique α] [Preorder β]
Goal: (OrderIso.funUnique α β).toEquiv = Equiv.funUnique α β
-- OrderIso.funUnique_apply
@[simps! toEquiv apply]
def funUnique (α β : Type*) [Unique α] [Preorder β] : (α → β) ≃o β 
Arguments: (α : Type u_6) (β : Type u_7) [Unique α] [Preorder β] (f : (i : α) → (fun a => β) i)
Goal: (OrderIso.funUnique α β) f = f default
-- OrderIso.funUnique_symm_apply
@[simp]
theorem funUnique_symm_apply {α β : Type*} [Unique α] [Preorder β] :
    ((funUnique α β).symm : β → α → β) = Function.const α 
Arguments: {α : Type u_6} {β : Type u_7} [Unique α] [Preorder β]
Goal: ⇑(OrderIso.funUnique α β).symm = Function.const α
-- Equiv.coe_toOrderIso
@[simp]
theorem coe_toOrderIso (e : α ≃ β) (h₁ : Monotone e) (h₂ : Monotone e.symm) :
    ⇑(e.toOrderIso h₁ h₂) = e 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃ β) (h₁ : Monotone ⇑e) (h₂ : Monotone ⇑e.symm)
Goal: ⇑(e.toOrderIso h₁ h₂) = ⇑e
-- Equiv.toOrderIso_toEquiv
@[simp]
theorem toOrderIso_toEquiv (e : α ≃ β) (h₁ : Monotone e) (h₂ : Monotone e.symm) :
    (e.toOrderIso h₁ h₂).toEquiv = e 
Arguments: {α : Type u_2} {β : Type u_3} [Preorder α] [Preorder β] (e : α ≃ β) (h₁ : Monotone ⇑e) (h₂ : Monotone ⇑e.symm)
Goal: (e.toOrderIso h₁ h₂).toEquiv = e
-- StrictMono.orderIsoOfRightInverse_apply
@[simps (config := .asFn)]
def orderIsoOfRightInverse (g : β → α) (hg : Function.RightInverse g f) : α ≃o β 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [Preorder β] (f : α → β) (h_mono : StrictMono f) (g : β → α) (hg : Function.RightInverse g f)
Goal: ⇑(StrictMono.orderIsoOfRightInverse f h_mono g hg) = f
-- StrictMono.orderIsoOfRightInverse_symm_apply
@[simps (config := .asFn)]
def orderIsoOfRightInverse (g : β → α) (hg : Function.RightInverse g f) : α ≃o β 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [Preorder β] (f : α → β) (h_mono : StrictMono f) (g : β → α) (hg : Function.RightInverse g f)
Goal: ⇑(RelIso.symm (StrictMono.orderIsoOfRightInverse f h_mono g hg)) = g
-- OrderIso.map_bot'
theorem OrderIso.map_bot' [LE α] [PartialOrder β] (f : α ≃o β) {x : α} {y : β} (hx : ∀ x', x ≤ x')
    (hy : ∀ y', y ≤ y') : f x = y 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [PartialOrder β] (f : α ≃o β) {x : α} {y : β} (hx : ∀ (x' : α), x ≤ x') (hy : ∀ (y' : β), y ≤ y')
Goal: f x = y
-- OrderIso.map_bot
theorem OrderIso.map_bot [LE α] [PartialOrder β] [OrderBot α] [OrderBot β] (f : α ≃o β) : f ⊥ = ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [PartialOrder β] [OrderBot α] [OrderBot β] (f : α ≃o β)
Goal: f ⊥ = ⊥
-- OrderIso.map_top'
theorem OrderIso.map_top' [LE α] [PartialOrder β] (f : α ≃o β) {x : α} {y : β} (hx : ∀ x', x' ≤ x)
    (hy : ∀ y', y' ≤ y) : f x = y 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [PartialOrder β] (f : α ≃o β) {x : α} {y : β} (hx : ∀ (x' : α), x' ≤ x) (hy : ∀ (y' : β), y' ≤ y)
Goal: f x = y
-- OrderIso.map_top
theorem OrderIso.map_top [LE α] [PartialOrder β] [OrderTop α] [OrderTop β] (f : α ≃o β) : f ⊤ = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [LE α] [PartialOrder β] [OrderTop α] [OrderTop β] (f : α ≃o β)
Goal: f ⊤ = ⊤
-- OrderEmbedding.map_inf_le
theorem OrderEmbedding.map_inf_le [SemilatticeInf α] [SemilatticeInf β] (f : α ↪o β) (x y : α) :
    f (x ⊓ y) ≤ f x ⊓ f y 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] (f : α ↪o β) (x y : α)
Goal: f (x ⊓ y) ≤ f x ⊓ f y
-- OrderEmbedding.le_map_sup
theorem OrderEmbedding.le_map_sup [SemilatticeSup α] [SemilatticeSup β] (f : α ↪o β) (x y : α) :
    f x ⊔ f y ≤ f (x ⊔ y) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] (f : α ↪o β) (x y : α)
Goal: f x ⊔ f y ≤ f (x ⊔ y)
-- OrderIso.map_inf
theorem OrderIso.map_inf [SemilatticeInf α] [SemilatticeInf β] (f : α ≃o β) (x y : α) :
    f (x ⊓ y) = f x ⊓ f y 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] (f : α ≃o β) (x y : α)
Goal: f (x ⊓ y) = f x ⊓ f y
-- OrderIso.map_sup
theorem OrderIso.map_sup [SemilatticeSup α] [SemilatticeSup β] (f : α ≃o β) (x y : α) :
    f (x ⊔ y) = f x ⊔ f y 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] (f : α ≃o β) (x y : α)
Goal: f (x ⊔ y) = f x ⊔ f y
-- Disjoint.map_orderIso
theorem Disjoint.map_orderIso [SemilatticeInf α] [OrderBot α] [SemilatticeInf β] [OrderBot β]
    {a b : α} (f : α ≃o β) (ha : Disjoint a b) : Disjoint (f a) (f b) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [OrderBot α] [SemilatticeInf β] [OrderBot β] {a b : α} (f : α ≃o β) (ha : Disjoint a b)
Goal: Disjoint (f a) (f b)
-- Codisjoint.map_orderIso
theorem Codisjoint.map_orderIso [SemilatticeSup α] [OrderTop α] [SemilatticeSup β] [OrderTop β]
    {a b : α} (f : α ≃o β) (ha : Codisjoint a b) : Codisjoint (f a) (f b) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [OrderTop α] [SemilatticeSup β] [OrderTop β] {a b : α} (f : α ≃o β) (ha : Codisjoint a b)
Goal: Codisjoint (f a) (f b)
-- disjoint_map_orderIso_iff
@[simp]
theorem disjoint_map_orderIso_iff [SemilatticeInf α] [OrderBot α] [SemilatticeInf β] [OrderBot β]
    {a b : α} (f : α ≃o β) : Disjoint (f a) (f b) ↔ Disjoint a b 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [OrderBot α] [SemilatticeInf β] [OrderBot β] {a b : α} (f : α ≃o β)
Goal: Disjoint (f a) (f b) ↔ Disjoint a b
-- codisjoint_map_orderIso_iff
@[simp]
theorem codisjoint_map_orderIso_iff [SemilatticeSup α] [OrderTop α] [SemilatticeSup β] [OrderTop β]
    {a b : α} (f : α ≃o β) : Codisjoint (f a) (f b) ↔ Codisjoint a b 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [OrderTop α] [SemilatticeSup β] [OrderTop β] {a b : α} (f : α ≃o β)
Goal: Codisjoint (f a) (f b) ↔ Codisjoint a b
-- WithBot.toDualTopEquiv_coe
@[simp]
theorem toDualTopEquiv_coe [LE α] (a : α) :
    WithBot.toDualTopEquiv ↑(toDual a) = toDual (a : WithTop α) 
Arguments: {α : Type u_2} [LE α] (a : α)
Goal: WithBot.toDualTopEquiv ↑(OrderDual.toDual a) = OrderDual.toDual ↑a