ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Iio_toDual
theorem Iio_toDual (a : α) : Iio (toDual a) = (Ioi a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : α)
Goal: Finset.Iio (OrderDual.toDual a) = Finset.map OrderDual.toDual.toEmbedding (Finset.Ioi a)
-- Ici_ofDual
theorem Ici_ofDual (a : αᵒᵈ) : Ici (ofDual a) = (Iic a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : αᵒᵈ)
Goal: Finset.Ici (OrderDual.ofDual a) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Iic a)
-- Ioi_ofDual
theorem Ioi_ofDual (a : αᵒᵈ) : Ioi (ofDual a) = (Iio a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : αᵒᵈ)
Goal: Finset.Ioi (OrderDual.ofDual a) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Iio a)
-- Ici_toDual
theorem Ici_toDual (a : α) : Ici (toDual a) = (Iic a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (a : α)
Goal: Finset.Ici (OrderDual.toDual a) = Finset.map OrderDual.toDual.toEmbedding (Finset.Iic a)
-- Ioi_toDual
theorem Ioi_toDual (a : α) : Ioi (toDual a) = (Iio a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (a : α)
Goal: Finset.Ioi (OrderDual.toDual a) = Finset.map OrderDual.toDual.toEmbedding (Finset.Iio a)
-- Iic_ofDual
theorem Iic_ofDual (a : αᵒᵈ) : Iic (ofDual a) = (Ici a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (a : αᵒᵈ)
Goal: Finset.Iic (OrderDual.ofDual a) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ici a)
-- Iio_ofDual
theorem Iio_ofDual (a : αᵒᵈ) : Iio (ofDual a) = (Ioi a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (a : αᵒᵈ)
Goal: Finset.Iio (OrderDual.ofDual a) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ioi a)
-- Prod.Icc_eq
theorem Icc_eq [LocallyFiniteOrder α] [LocallyFiniteOrder β]
    [DecidableRel ((· ≤ ·) : α × β → α × β → Prop)] (p q : α × β) :
    Finset.Icc p q = Finset.Icc p.1 q.1 ×ˢ Finset.Icc p.2 q.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableRel fun x x_1 => x ≤ x_1] (p q : α × β)
Goal: Finset.Icc p q = Finset.Icc p.1 q.1 ×ˢ Finset.Icc p.2 q.2
-- Prod.Icc_mk_mk
@[simp]
theorem Icc_mk_mk [LocallyFiniteOrder α] [LocallyFiniteOrder β]
    [DecidableRel ((· ≤ ·) : α × β → α × β → Prop)] (a₁ a₂ : α) (b₁ b₂ : β) :
    Finset.Icc (a₁, b₁) (a₂, b₂) = Finset.Icc a₁ a₂ ×ˢ Finset.Icc b₁ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableRel fun x x_1 => x ≤ x_1] (a₁ a₂ : α) (b₁ b₂ : β)
Goal: Finset.Icc (a₁, b₁) (a₂, b₂) = Finset.Icc a₁ a₂ ×ˢ Finset.Icc b₁ b₂
-- Prod.card_Icc
theorem card_Icc [LocallyFiniteOrder α] [LocallyFiniteOrder β]
    [DecidableRel ((· ≤ ·) : α × β → α × β → Prop)] (p q : α × β) :
    (Finset.Icc p q).card = (Finset.Icc p.1 q.1).card * (Finset.Icc p.2 q.2).card 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableRel fun x x_1 => x ≤ x_1] (p q : α × β)
Goal: (Finset.Icc p q).card = (Finset.Icc p.1 q.1).card * (Finset.Icc p.2 q.2).card
-- Prod.uIcc_eq
theorem uIcc_eq [LocallyFiniteOrder α] [LocallyFiniteOrder β]
    [DecidableRel ((· ≤ ·) : α × β → α × β → Prop)] (p q : α × β) :
    Finset.uIcc p q = Finset.uIcc p.1 q.1 ×ˢ Finset.uIcc p.2 q.2 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableRel fun x x_1 => x ≤ x_1] (p q : α × β)
Goal: Finset.uIcc p q = Finset.uIcc p.1 q.1 ×ˢ Finset.uIcc p.2 q.2
-- Prod.uIcc_mk_mk
@[simp]
theorem uIcc_mk_mk [LocallyFiniteOrder α] [LocallyFiniteOrder β]
    [DecidableRel ((· ≤ ·) : α × β → α × β → Prop)] (a₁ a₂ : α) (b₁ b₂ : β) :
    Finset.uIcc (a₁, b₁) (a₂, b₂) = Finset.uIcc a₁ a₂ ×ˢ Finset.uIcc b₁ b₂ 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableRel fun x x_1 => x ≤ x_1] (a₁ a₂ : α) (b₁ b₂ : β)
Goal: Finset.uIcc (a₁, b₁) (a₂, b₂) = Finset.uIcc a₁ a₂ ×ˢ Finset.uIcc b₁ b₂
-- Prod.card_uIcc
theorem card_uIcc [LocallyFiniteOrder α] [LocallyFiniteOrder β]
    [DecidableRel ((· ≤ ·) : α × β → α × β → Prop)] (p q : α × β) :
    (Finset.uIcc p q).card = (Finset.uIcc p.1 q.1).card * (Finset.uIcc p.2 q.2).card 
Arguments: {α : Type u_1} {β : Type u_2} [Lattice α] [Lattice β] [LocallyFiniteOrder α] [LocallyFiniteOrder β] [DecidableRel fun x x_1 => x ≤ x_1] (p q : α × β)
Goal: (Finset.uIcc p q).card = (Finset.uIcc p.1 q.1).card * (Finset.uIcc p.2 q.2).card
-- _private.Mathlib.Order.Interval.Finset.Defs.0.WithTop.aux
private lemma aux (x : α) (p : α → Prop) :
    (∃ a : α, p a ∧ WithTop.some a = WithTop.some x) ↔ p x 
Arguments: (α : Type u_1) (x : α) (p : α → Prop)
Goal: (∃ a, p a ∧ ↑a = ↑x) ↔ p x
-- WithTop.Icc_coe_top
theorem Icc_coe_top : Icc (a : WithTop α) ⊤ = insertNone (Ici a) 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Icc ↑a ⊤ = Finset.insertNone (Finset.Ici a)
-- WithTop.Icc_coe_coe
theorem Icc_coe_coe : Icc (a : WithTop α) b = (Icc a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Icc ↑a ↑b = Finset.map Function.Embedding.some (Finset.Icc a b)
-- WithTop.Ico_coe_top
theorem Ico_coe_top : Ico (a : WithTop α) ⊤ = (Ici a).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Ico ↑a ⊤ = Finset.map Function.Embedding.some (Finset.Ici a)
-- WithTop.Ico_coe_coe
theorem Ico_coe_coe : Ico (a : WithTop α) b = (Ico a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ico ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ico a b)
-- WithTop.Ioc_coe_top
theorem Ioc_coe_top : Ioc (a : WithTop α) ⊤ = insertNone (Ioi a) 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Ioc ↑a ⊤ = Finset.insertNone (Finset.Ioi a)
-- WithTop.Ioc_coe_coe
theorem Ioc_coe_coe : Ioc (a : WithTop α) b = (Ioc a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ioc ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ioc a b)
-- WithTop.Ioo_coe_top
theorem Ioo_coe_top : Ioo (a : WithTop α) ⊤ = (Ioi a).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Ioo ↑a ⊤ = Finset.map Function.Embedding.some (Finset.Ioi a)
-- WithTop.Ioo_coe_coe
theorem Ioo_coe_coe : Ioo (a : WithTop α) b = (Ioo a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderTop α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ioo ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ioo a b)
-- WithBot.Icc_bot_coe
theorem Icc_bot_coe : Icc (⊥ : WithBot α) b = insertNone (Iic b) 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (b : α)
Goal: Finset.Icc ⊥ ↑b = Finset.insertNone (Finset.Iic b)
-- WithBot.Icc_coe_coe
theorem Icc_coe_coe : Icc (a : WithBot α) b = (Icc a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Icc ↑a ↑b = Finset.map Function.Embedding.some (Finset.Icc a b)
-- WithBot.Ico_bot_coe
theorem Ico_bot_coe : Ico (⊥ : WithBot α) b = insertNone (Iio b) 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (b : α)
Goal: Finset.Ico ⊥ ↑b = Finset.insertNone (Finset.Iio b)
-- WithBot.Ico_coe_coe
theorem Ico_coe_coe : Ico (a : WithBot α) b = (Ico a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ico ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ico a b)
-- WithBot.Ioc_bot_coe
theorem Ioc_bot_coe : Ioc (⊥ : WithBot α) b = (Iic b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (b : α)
Goal: Finset.Ioc ⊥ ↑b = Finset.map Function.Embedding.some (Finset.Iic b)
-- WithBot.Ioc_coe_coe
theorem Ioc_coe_coe : Ioc (a : WithBot α) b = (Ioc a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ioc ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ioc a b)
-- WithBot.Ioo_bot_coe
theorem Ioo_bot_coe : Ioo (⊥ : WithBot α) b = (Iio b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (b : α)
Goal: Finset.Ioo ⊥ ↑b = Finset.map Function.Embedding.some (Finset.Iio b)
-- WithBot.Ioo_coe_coe
theorem Ioo_coe_coe : Ioo (a : WithBot α) b = (Ioo a b).map Embedding.some 
Arguments: (α : Type u_1) [PartialOrder α] [OrderBot α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ioo ↑a ↑b = Finset.map Function.Embedding.some (Finset.Ioo a b)
-- Finset.subtype_Icc_eq
theorem subtype_Icc_eq : Icc a b = (Icc (a : α) b).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p)
Goal: Finset.Icc a b = Finset.subtype p (Finset.Icc ↑a ↑b)
-- Finset.subtype_Ico_eq
theorem subtype_Ico_eq : Ico a b = (Ico (a : α) b).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p)
Goal: Finset.Ico a b = Finset.subtype p (Finset.Ico ↑a ↑b)
-- Finset.subtype_Ioc_eq
theorem subtype_Ioc_eq : Ioc a b = (Ioc (a : α) b).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p)
Goal: Finset.Ioc a b = Finset.subtype p (Finset.Ioc ↑a ↑b)
-- Finset.subtype_Ioo_eq
theorem subtype_Ioo_eq : Ioo a b = (Ioo (a : α) b).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p)
Goal: Finset.Ioo a b = Finset.subtype p (Finset.Ioo ↑a ↑b)
-- Finset.map_subtype_embedding_Icc
theorem map_subtype_embedding_Icc (hp : ∀ ⦃a b x⦄, a ≤ x → x ≤ b → p a → p b → p x):
    (Icc a b).map (Embedding.subtype p) = (Icc a b : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p) (hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Icc a b) = Finset.Icc ↑a ↑b
-- Finset.map_subtype_embedding_Ico
theorem map_subtype_embedding_Ico (hp : ∀ ⦃a b x⦄, a ≤ x → x ≤ b → p a → p b → p x):
    (Ico a b).map (Embedding.subtype p) = (Ico a b : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p) (hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Ico a b) = Finset.Ico ↑a ↑b
-- Finset.map_subtype_embedding_Ioc
theorem map_subtype_embedding_Ioc (hp : ∀ ⦃a b x⦄, a ≤ x → x ≤ b → p a → p b → p x):
    (Ioc a b).map (Embedding.subtype p) = (Ioc a b : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p) (hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Ioc a b) = Finset.Ioc ↑a ↑b
-- Finset.map_subtype_embedding_Ioo
theorem map_subtype_embedding_Ioo (hp : ∀ ⦃a b x⦄, a ≤ x → x ≤ b → p a → p b → p x):
    (Ioo a b).map (Embedding.subtype p) = (Ioo a b : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrder α] (a b : Subtype p) (hp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Ioo a b) = Finset.Ioo ↑a ↑b
-- Finset.subtype_Ici_eq
theorem subtype_Ici_eq : Ici a = (Ici (a : α)).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderTop α] (a : Subtype p)
Goal: Finset.Ici a = Finset.subtype p (Finset.Ici ↑a)
-- Finset.subtype_Ioi_eq
theorem subtype_Ioi_eq : Ioi a = (Ioi (a : α)).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderTop α] (a : Subtype p)
Goal: Finset.Ioi a = Finset.subtype p (Finset.Ioi ↑a)
-- Finset.map_subtype_embedding_Ici
theorem map_subtype_embedding_Ici : (Ici a).map (Embedding.subtype p) = (Ici a : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderTop α] (a : Subtype p) (hp : ∀ ⦃a x : α⦄, a ≤ x → p a → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Ici a) = Finset.Ici ↑a
-- Finset.map_subtype_embedding_Ioi
theorem map_subtype_embedding_Ioi : (Ioi a).map (Embedding.subtype p) = (Ioi a : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderTop α] (a : Subtype p) (hp : ∀ ⦃a x : α⦄, a ≤ x → p a → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Ioi a) = Finset.Ioi ↑a
-- Finset.subtype_Iic_eq
theorem subtype_Iic_eq : Iic a = (Iic (a : α)).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderBot α] (a : Subtype p)
Goal: Finset.Iic a = Finset.subtype p (Finset.Iic ↑a)
-- Finset.subtype_Iio_eq
theorem subtype_Iio_eq : Iio a = (Iio (a : α)).subtype p 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderBot α] (a : Subtype p)
Goal: Finset.Iio a = Finset.subtype p (Finset.Iio ↑a)
-- Finset.map_subtype_embedding_Iic
theorem map_subtype_embedding_Iic : (Iic a).map (Embedding.subtype p) = (Iic a : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderBot α] (a : Subtype p) (hp : ∀ ⦃a x : α⦄, x ≤ a → p a → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Iic a) = Finset.Iic ↑a
-- Finset.map_subtype_embedding_Iio
theorem map_subtype_embedding_Iio : (Iio a).map (Embedding.subtype p) = (Iio a : Finset α) 
Arguments: {α : Type u_1} [Preorder α] (p : α → Prop) [DecidablePred p] [LocallyFiniteOrderBot α] (a : Subtype p) (hp : ∀ ⦃a x : α⦄, x ≤ a → p a → p x)
Goal: Finset.map (Function.Embedding.subtype p) (Finset.Iio a) = Finset.Iio ↑a
-- BddBelow.finite_of_bddAbove
theorem BddBelow.finite_of_bddAbove [Preorder α] [LocallyFiniteOrder α]
    {s : Set α} (h₀ : BddBelow s) (h₁ : BddAbove s) :
    s.Finite 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrder α] {s : Set α} (h₀ : BddBelow s) (h₁ : BddAbove s)
Goal: s.Finite
-- Set.finite_iff_bddAbove
theorem Set.finite_iff_bddAbove [SemilatticeSup α] [LocallyFiniteOrder α] [OrderBot α] :
    s.Finite ↔ BddAbove s 
Arguments: {α : Type u_3} {s : Set α} [SemilatticeSup α] [LocallyFiniteOrder α] [OrderBot α]
Goal: s.Finite ↔ BddAbove s
-- Set.finite_iff_bddBelow
theorem Set.finite_iff_bddBelow [SemilatticeInf α] [LocallyFiniteOrder α] [OrderTop α] :
    s.Finite ↔ BddBelow s 
Arguments: {α : Type u_3} {s : Set α} [SemilatticeInf α] [LocallyFiniteOrder α] [OrderTop α]
Goal: s.Finite ↔ BddBelow s
-- Set.finite_iff_bddBelow_bddAbove
theorem Set.finite_iff_bddBelow_bddAbove [Nonempty α] [Lattice α] [LocallyFiniteOrder α] :
    s.Finite ↔ BddBelow s ∧ BddAbove s 
Arguments: {α : Type u_3} {s : Set α} [Nonempty α] [Lattice α] [LocallyFiniteOrder α]
Goal: s.Finite ↔ BddBelow s ∧ BddAbove s
-- Set.toFinset_Icc
@[simp] lemma toFinset_Icc (a b : α) [Fintype (Icc a b)] : (Icc a b).toFinset = Finset.Icc a b 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype ↑(Set.Icc a b)]
Goal: (Set.Icc a b).toFinset = Finset.Icc a b
-- Set.toFinset_Ico
@[simp] lemma toFinset_Ico (a b : α) [Fintype (Ico a b)] : (Ico a b).toFinset = Finset.Ico a b 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype ↑(Set.Ico a b)]
Goal: (Set.Ico a b).toFinset = Finset.Ico a b
-- Set.toFinset_Ioc
@[simp] lemma toFinset_Ioc (a b : α) [Fintype (Ioc a b)] : (Ioc a b).toFinset = Finset.Ioc a b 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype ↑(Set.Ioc a b)]
Goal: (Set.Ioc a b).toFinset = Finset.Ioc a b
-- Set.toFinset_Ioo
@[simp] lemma toFinset_Ioo (a b : α) [Fintype (Ioo a b)] : (Ioo a b).toFinset = Finset.Ioo a b 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrder α] (a b : α) [Fintype ↑(Set.Ioo a b)]
Goal: (Set.Ioo a b).toFinset = Finset.Ioo a b
-- Set.toFinset_Ici
@[simp]
lemma toFinset_Ici (a : α) [Fintype (Ici a)] : (Ici a).toFinset = Finset.Ici a 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrderTop α] (a : α) [Fintype ↑(Set.Ici a)]
Goal: (Set.Ici a).toFinset = Finset.Ici a
-- Set.toFinset_Ioi
@[simp]
lemma toFinset_Ioi (a : α) [Fintype (Ioi a)] : (Ioi a).toFinset = Finset.Ioi a 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrderTop α] (a : α) [Fintype ↑(Set.Ioi a)]
Goal: (Set.Ioi a).toFinset = Finset.Ioi a
-- Set.toFinset_Iic
@[simp]
lemma toFinset_Iic (a : α) [Fintype (Iic a)] : (Iic a).toFinset = Finset.Iic a 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrderBot α] (a : α) [Fintype ↑(Set.Iic a)]
Goal: (Set.Iic a).toFinset = Finset.Iic a
-- Set.toFinset_Iio
@[simp]
lemma toFinset_Iio (a : α) [Fintype (Iio a)] : (Iio a).toFinset = Finset.Iio a 
Arguments: {α : Type u_3} [Preorder α] [LocallyFiniteOrderBot α] (a : α) [Fintype ↑(Set.Iio a)]
Goal: (Set.Iio a).toFinset = Finset.Iio a
-- min_def
theorem min_def (a b : α) : min a b = if a ≤ b then a else b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: min a b = if a ≤ b then a else b
-- max_def
theorem max_def (a b : α) : max a b = if a ≤ b then b else a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: max a b = if a ≤ b then b else a
-- min_le_left
theorem min_le_left (a b : α) : min a b ≤ a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: min a b ≤ a
-- min_le_right
theorem min_le_right (a b : α) : min a b ≤ b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: min a b ≤ b
-- le_min
theorem le_min {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) : c ≤ min a b 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b)
Goal: c ≤ min a b
-- le_max_left
theorem le_max_left (a b : α) : a ≤ max a b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: a ≤ max a b
-- le_max_right
theorem le_max_right (a b : α) : b ≤ max a b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: b ≤ max a b
-- max_le
theorem max_le {a b c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) : max a b ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h₁ : a ≤ c) (h₂ : b ≤ c)
Goal: max a b ≤ c
-- eq_min
theorem eq_min {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d}, d ≤ a → d ≤ b → d ≤ c) :
    c = min a b 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h₁ : c ≤ a) (h₂ : c ≤ b) (h₃ : ∀ {d : α}, d ≤ a → d ≤ b → d ≤ c)
Goal: c = min a b
-- min_comm
theorem min_comm (a b : α) : min a b = min b a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: min a b = min b a
-- min_assoc
theorem min_assoc (a b c : α) : min (min a b) c = min a (min b c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: min (min a b) c = min a (min b c)
-- min_left_comm
theorem min_left_comm : ∀ a b c : α, min a (min b c) = min b (min a c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: min a (min b c) = min b (min a c)
-- min_self
@[simp]
theorem min_self (a : α) : min a a = a 
Arguments: {α : Type u} [LinearOrder α] (a : α)
Goal: min a a = a
-- min_eq_left
theorem min_eq_left {a b : α} (h : a ≤ b) : min a b = a 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: min a b = a
-- min_eq_right
theorem min_eq_right {a b : α} (h : b ≤ a) : min a b = b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : b ≤ a)
Goal: min a b = b
-- eq_max
theorem eq_max {a b c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) (h₃ : ∀ {d}, a ≤ d → b ≤ d → c ≤ d) :
    c = max a b 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h₁ : a ≤ c) (h₂ : b ≤ c) (h₃ : ∀ {d : α}, a ≤ d → b ≤ d → c ≤ d)
Goal: c = max a b
-- max_comm
theorem max_comm (a b : α) : max a b = max b a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: max a b = max b a
-- max_assoc
theorem max_assoc (a b c : α) : max (max a b) c = max a (max b c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: max (max a b) c = max a (max b c)
-- max_left_comm
theorem max_left_comm : ∀ a b c : α, max a (max b c) = max b (max a c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: max a (max b c) = max b (max a c)
-- max_self
@[simp]
theorem max_self (a : α) : max a a = a 
Arguments: {α : Type u} [LinearOrder α] (a : α)
Goal: max a a = a
-- max_eq_left
theorem max_eq_left {a b : α} (h : b ≤ a) : max a b = a 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : b ≤ a)
Goal: max a b = a
-- max_eq_right
theorem max_eq_right {a b : α} (h : a ≤ b) : max a b = b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: max a b = b
-- min_eq_left_of_lt
theorem min_eq_left_of_lt {a b : α} (h : a < b) : min a b = a 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : a < b)
Goal: min a b = a
-- min_eq_right_of_lt
theorem min_eq_right_of_lt {a b : α} (h : b < a) : min a b = b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : b < a)
Goal: min a b = b
-- max_eq_left_of_lt
theorem max_eq_left_of_lt {a b : α} (h : b < a) : max a b = a 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : b < a)
Goal: max a b = a
-- max_eq_right_of_lt
theorem max_eq_right_of_lt {a b : α} (h : a < b) : max a b = b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (h : a < b)
Goal: max a b = b
-- lt_min
theorem lt_min {a b c : α} (h₁ : a < b) (h₂ : a < c) : a < min b c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h₁ : a < b) (h₂ : a < c)
Goal: a < min b c
-- max_lt
theorem max_lt {a b c : α} (h₁ : a < c) (h₂ : b < c) : max a b < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h₁ : a < c) (h₂ : b < c)
Goal: max a b < c
-- monotone_inclusion_le_le_of_le
lemma monotone_inclusion_le_le_of_le [Preorder α] {k j : α} (hkj : k ≤ j) :
    Monotone (fun ⟨i, hi⟩ => ⟨i, hi.trans hkj⟩ : { i // i ≤ k } → { i // i ≤ j}) 
Arguments: {α : Type u} [Preorder α] {k j : α} (hkj : k ≤ j)
Goal: Monotone fun x =>
  match x with
  | ⟨i, hi⟩ => ⟨i, ⋯⟩
-- monotone_inclusion_lt_le_of_le
lemma monotone_inclusion_lt_le_of_le [Preorder α] {k j : α} (hkj : k ≤ j) :
    Monotone (fun ⟨i, hi⟩ => ⟨i, hi.le.trans hkj⟩ : { i // i < k } → { i // i ≤ j}) 
Arguments: {α : Type u} [Preorder α] {k j : α} (hkj : k ≤ j)
Goal: Monotone fun x =>
  match x with
  | ⟨i, hi⟩ => ⟨i, ⋯⟩
-- monotone_inclusion_lt_lt_of_le
lemma monotone_inclusion_lt_lt_of_le [Preorder α] {k j : α} (hkj : k ≤ j) :
    Monotone (fun ⟨i, hi⟩ => ⟨i, lt_of_lt_of_le hi hkj⟩ : { i // i < k } → { i // i < j}) 
Arguments: {α : Type u} [Preorder α] {k j : α} (hkj : k ≤ j)
Goal: Monotone fun x =>
  match x with
  | ⟨i, hi⟩ => ⟨i, ⋯⟩
-- monotone_comp_ofDual_iff
@[simp]
theorem monotone_comp_ofDual_iff : Monotone (f ∘ ofDual) ↔ Antitone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: Monotone (f ∘ ⇑OrderDual.ofDual) ↔ Antitone f
-- antitone_comp_ofDual_iff
@[simp]
theorem antitone_comp_ofDual_iff : Antitone (f ∘ ofDual) ↔ Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: Antitone (f ∘ ⇑OrderDual.ofDual) ↔ Monotone f
-- monotone_toDual_comp_iff
@[simp]
theorem monotone_toDual_comp_iff : Monotone (toDual ∘ f : α → βᵒᵈ) ↔ Antitone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: Monotone (⇑OrderDual.toDual ∘ f) ↔ Antitone f
-- antitone_toDual_comp_iff
@[simp]
theorem antitone_toDual_comp_iff : Antitone (toDual ∘ f : α → βᵒᵈ) ↔ Monotone f 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β}
Goal: Antitone (⇑OrderDual.toDual ∘ f) ↔ Monotone f
-- monotoneOn_comp_ofDual_iff
@[simp]
theorem monotoneOn_comp_ofDual_iff : MonotoneOn (f ∘ ofDual) s ↔ AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: MonotoneOn (f ∘ ⇑OrderDual.ofDual) s ↔ AntitoneOn f s
-- antitoneOn_comp_ofDual_iff
@[simp]
theorem antitoneOn_comp_ofDual_iff : AntitoneOn (f ∘ ofDual) s ↔ MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [Preorder β] {f : α → β} {s : Set α}
Goal: AntitoneOn (f ∘ ⇑OrderDual.ofDual) s ↔ MonotoneOn f s