ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.Ico_subset_Ici_self
theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderTop α]
Goal: Finset.Ico a b ⊆ Finset.Ici a
-- Finset.Ioc_subset_Ioi_self
theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderTop α]
Goal: Finset.Ioc a b ⊆ Finset.Ioi a
-- Finset.Ioo_subset_Ioi_self
theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderTop α]
Goal: Finset.Ioo a b ⊆ Finset.Ioi a
-- Finset.Ioc_subset_Ici_self
theorem Ioc_subset_Ici_self : Ioc a b ⊆ Ici a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderTop α]
Goal: Finset.Ioc a b ⊆ Finset.Ici a
-- Finset.Ioo_subset_Ici_self
theorem Ioo_subset_Ici_self : Ioo a b ⊆ Ici a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderTop α]
Goal: Finset.Ioo a b ⊆ Finset.Ici a
-- Finset.nonempty_Iic
@[simp] lemma nonempty_Iic : (Iic a).Nonempty 
Arguments: {α : Type u_2} [Preorder α] {a : α} [LocallyFiniteOrderBot α]
Goal: (Finset.Iic a).Nonempty
-- Finset.nonempty_Iio
@[simp] lemma nonempty_Iio : (Iio a).Nonempty ↔ ¬ IsMin a 
Arguments: {α : Type u_2} [Preorder α] {a : α} [LocallyFiniteOrderBot α]
Goal: (Finset.Iio a).Nonempty ↔ ¬IsMin a
-- Finset.Icc_subset_Iic_self
theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderBot α]
Goal: Finset.Icc a b ⊆ Finset.Iic b
-- Finset.Ioc_subset_Iic_self
theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderBot α]
Goal: Finset.Ioc a b ⊆ Finset.Iic b
-- Finset.Ico_subset_Iio_self
theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderBot α]
Goal: Finset.Ico a b ⊆ Finset.Iio b
-- Finset.Ioo_subset_Iio_self
theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderBot α]
Goal: Finset.Ioo a b ⊆ Finset.Iio b
-- Finset.Ico_subset_Iic_self
theorem Ico_subset_Iic_self : Ico a b ⊆ Iic b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderBot α]
Goal: Finset.Ico a b ⊆ Finset.Iic b
-- Finset.Ioo_subset_Iic_self
theorem Ioo_subset_Iic_self : Ioo a b ⊆ Iic b 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrder α] {a b : α} [LocallyFiniteOrderBot α]
Goal: Finset.Ioo a b ⊆ Finset.Iic b
-- Finset.Ioi_subset_Ici_self
theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] {a : α}
Goal: Finset.Ioi a ⊆ Finset.Ici a
-- BddBelow.finite
theorem _root_.BddBelow.finite {s : Set α} (hs : BddBelow s) : s.Finite 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] {s : Set α} (hs : BddBelow s)
Goal: s.Finite
-- Set.Infinite.not_bddBelow
theorem _root_.Set.Infinite.not_bddBelow {s : Set α} : s.Infinite → ¬BddBelow s 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] {s : Set α}
Goal: s.Infinite → ¬BddBelow s
-- Finset.filter_lt_eq_Ioi
theorem filter_lt_eq_Ioi [DecidablePred (a < ·)] : univ.filter (a < ·) = Ioi a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] {a : α} [Fintype α] [DecidablePred fun x => a < x]
Goal: Finset.filter (fun x => a < x) Finset.univ = Finset.Ioi a
-- Finset.filter_le_eq_Ici
theorem filter_le_eq_Ici [DecidablePred (a ≤ ·)] : univ.filter (a ≤ ·) = Ici a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] {a : α} [Fintype α] [DecidablePred fun x => a ≤ x]
Goal: Finset.filter (fun x => a ≤ x) Finset.univ = Finset.Ici a
-- Finset.Iio_subset_Iic_self
theorem Iio_subset_Iic_self : Iio a ⊆ Iic a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderBot α] {a : α}
Goal: Finset.Iio a ⊆ Finset.Iic a
-- BddAbove.finite
theorem _root_.BddAbove.finite {s : Set α} (hs : BddAbove s) : s.Finite 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderBot α] {s : Set α} (hs : BddAbove s)
Goal: s.Finite
-- Set.Infinite.not_bddAbove
theorem _root_.Set.Infinite.not_bddAbove {s : Set α} : s.Infinite → ¬BddAbove s 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderBot α] {s : Set α}
Goal: s.Infinite → ¬BddAbove s
-- Finset.filter_gt_eq_Iio
theorem filter_gt_eq_Iio [DecidablePred (· < a)] : univ.filter (· < a) = Iio a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderBot α] {a : α} [Fintype α] [DecidablePred fun x => x < a]
Goal: Finset.filter (fun x => x < a) Finset.univ = Finset.Iio a
-- Finset.filter_ge_eq_Iic
theorem filter_ge_eq_Iic [DecidablePred (· ≤ a)] : univ.filter (· ≤ a) = Iic a 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderBot α] {a : α} [Fintype α] [DecidablePred fun x => x ≤ a]
Goal: Finset.filter (fun x => x ≤ a) Finset.univ = Finset.Iic a
-- Finset.disjoint_Ioi_Iio
theorem disjoint_Ioi_Iio (a : α) : Disjoint (Ioi a) (Iio a) 
Arguments: {α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] [LocallyFiniteOrderBot α] (a : α)
Goal: Disjoint (Finset.Ioi a) (Finset.Iio a)
-- Finset.Icc_self
@[simp]
theorem Icc_self (a : α) : Icc a a = {a} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a : α)
Goal: Finset.Icc a a = {a}
-- Finset.Icc_eq_singleton_iff
@[simp]
theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b c : α}
Goal: Finset.Icc a b = {c} ↔ a = c ∧ b = c
-- Finset.Ico_disjoint_Ico_consecutive
theorem Ico_disjoint_Ico_consecutive (a b c : α) : Disjoint (Ico a b) (Ico b c) 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Disjoint (Finset.Ico a b) (Finset.Ico b c)
-- Finset.Icc_erase_left
@[simp]
theorem Icc_erase_left (a b : α) : (Icc a b).erase a = Ioc a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] (a b : α)
Goal: (Finset.Icc a b).erase a = Finset.Ioc a b
-- Finset.Icc_erase_right
@[simp]
theorem Icc_erase_right (a b : α) : (Icc a b).erase b = Ico a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] (a b : α)
Goal: (Finset.Icc a b).erase b = Finset.Ico a b
-- Finset.Ico_erase_left
@[simp]
theorem Ico_erase_left (a b : α) : (Ico a b).erase a = Ioo a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] (a b : α)
Goal: (Finset.Ico a b).erase a = Finset.Ioo a b
-- Finset.Ioc_erase_right
@[simp]
theorem Ioc_erase_right (a b : α) : (Ioc a b).erase b = Ioo a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] (a b : α)
Goal: (Finset.Ioc a b).erase b = Finset.Ioo a b
-- Finset.Icc_diff_both
@[simp]
theorem Icc_diff_both (a b : α) : Icc a b \ {a, b} = Ioo a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] (a b : α)
Goal: Finset.Icc a b \ {a, b} = Finset.Ioo a b
-- Finset.Ico_insert_right
@[simp]
theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a ≤ b)
Goal: insert b (Finset.Ico a b) = Finset.Icc a b
-- Finset.Ioc_insert_left
@[simp]
theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a ≤ b)
Goal: insert a (Finset.Ioc a b) = Finset.Icc a b
-- Finset.Ioo_insert_left
@[simp]
theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a < b)
Goal: insert a (Finset.Ioo a b) = Finset.Ico a b
-- Finset.Ioo_insert_right
@[simp]
theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a < b)
Goal: insert b (Finset.Ioo a b) = Finset.Ioc a b
-- Finset.Icc_diff_Ico_self
@[simp]
theorem Icc_diff_Ico_self (h : a ≤ b) : Icc a b \ Ico a b = {b} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a ≤ b)
Goal: Finset.Icc a b \ Finset.Ico a b = {b}
-- Finset.Icc_diff_Ioc_self
@[simp]
theorem Icc_diff_Ioc_self (h : a ≤ b) : Icc a b \ Ioc a b = {a} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a ≤ b)
Goal: Finset.Icc a b \ Finset.Ioc a b = {a}
-- Finset.Icc_diff_Ioo_self
@[simp]
theorem Icc_diff_Ioo_self (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a ≤ b)
Goal: Finset.Icc a b \ Finset.Ioo a b = {a, b}
-- Finset.Ico_diff_Ioo_self
@[simp]
theorem Ico_diff_Ioo_self (h : a < b) : Ico a b \ Ioo a b = {a} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a < b)
Goal: Finset.Ico a b \ Finset.Ioo a b = {a}
-- Finset.Ioc_diff_Ioo_self
@[simp]
theorem Ioc_diff_Ioo_self (h : a < b) : Ioc a b \ Ioo a b = {b} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidableEq α] (h : a < b)
Goal: Finset.Ioc a b \ Finset.Ioo a b = {b}
-- Finset.Ico_inter_Ico_consecutive
@[simp]
theorem Ico_inter_Ico_consecutive (a b c : α) : Ico a b ∩ Ico b c = ∅ 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.Ico a b ∩ Finset.Ico b c = ∅
-- Finset.Icc_eq_cons_Ico
theorem Icc_eq_cons_Ico (h : a ≤ b) : Icc a b = (Ico a b).cons b right_not_mem_Ico 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a ≤ b)
Goal: Finset.Icc a b = Finset.cons b (Finset.Ico a b) ⋯
-- Finset.Icc_eq_cons_Ioc
theorem Icc_eq_cons_Ioc (h : a ≤ b) : Icc a b = (Ioc a b).cons a left_not_mem_Ioc 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a ≤ b)
Goal: Finset.Icc a b = Finset.cons a (Finset.Ioc a b) ⋯
-- Finset.Ioc_eq_cons_Ioo
theorem Ioc_eq_cons_Ioo (h : a < b) : Ioc a b = (Ioo a b).cons b right_not_mem_Ioo 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a < b)
Goal: Finset.Ioc a b = Finset.cons b (Finset.Ioo a b) ⋯
-- Finset.Ico_eq_cons_Ioo
theorem Ico_eq_cons_Ioo (h : a < b) : Ico a b = (Ioo a b).cons a left_not_mem_Ioo 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} (h : a < b)
Goal: Finset.Ico a b = Finset.cons a (Finset.Ioo a b) ⋯
-- Finset.Ico_filter_le_left
theorem Ico_filter_le_left {a b : α} [DecidablePred (· ≤ a)] (hab : a < b) :
    ((Ico a b).filter fun x => x ≤ a) = {a} 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] {a b : α} [DecidablePred fun x => x ≤ a] (hab : a < b)
Goal: Finset.filter (fun x => x ≤ a) (Finset.Ico a b) = {a}
-- Finset.card_Ico_eq_card_Icc_sub_one
theorem card_Ico_eq_card_Icc_sub_one (a b : α) : (Ico a b).card = (Icc a b).card - 1 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Finset.Ico a b).card = (Finset.Icc a b).card - 1
-- Finset.card_Ioc_eq_card_Icc_sub_one
theorem card_Ioc_eq_card_Icc_sub_one (a b : α) : (Ioc a b).card = (Icc a b).card - 1 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Finset.Ioc a b).card = (Finset.Icc a b).card - 1
-- Finset.card_Ioo_eq_card_Ico_sub_one
theorem card_Ioo_eq_card_Ico_sub_one (a b : α) : (Ioo a b).card = (Ico a b).card - 1 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Finset.Ioo a b).card = (Finset.Ico a b).card - 1
-- Finset.card_Ioo_eq_card_Ioc_sub_one
theorem card_Ioo_eq_card_Ioc_sub_one (a b : α) : (Ioo a b).card = (Ioc a b).card - 1 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Finset.Ioo a b).card = (Finset.Ioc a b).card - 1
-- Finset.card_Ioo_eq_card_Icc_sub_two
theorem card_Ioo_eq_card_Icc_sub_two (a b : α) : (Ioo a b).card = (Icc a b).card - 2 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Finset.Ioo a b).card = (Finset.Icc a b).card - 2
-- Finset.Ici_erase
@[simp]
theorem Ici_erase [DecidableEq α] (a : α) : (Ici a).erase a = Ioi a 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderTop α] [DecidableEq α] (a : α)
Goal: (Finset.Ici a).erase a = Finset.Ioi a
-- Finset.Ioi_insert
@[simp]
theorem Ioi_insert [DecidableEq α] (a : α) : insert a (Ioi a) = Ici a 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderTop α] [DecidableEq α] (a : α)
Goal: insert a (Finset.Ioi a) = Finset.Ici a
-- Finset.not_mem_Ioi_self
theorem not_mem_Ioi_self {b : α} : b ∉ Ioi b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderTop α] {b : α}
Goal: b ∉ Finset.Ioi b
-- Finset.Ici_eq_cons_Ioi
theorem Ici_eq_cons_Ioi (a : α) : Ici a = (Ioi a).cons a not_mem_Ioi_self 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderTop α] (a : α)
Goal: Finset.Ici a = Finset.cons a (Finset.Ioi a) ⋯
-- Finset.card_Ioi_eq_card_Ici_sub_one
theorem card_Ioi_eq_card_Ici_sub_one (a : α) : (Ioi a).card = (Ici a).card - 1 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderTop α] (a : α)
Goal: (Finset.Ioi a).card = (Finset.Ici a).card - 1
-- Finset.Iic_erase
@[simp]
theorem Iic_erase [DecidableEq α] (b : α) : (Iic b).erase b = Iio b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderBot α] [DecidableEq α] (b : α)
Goal: (Finset.Iic b).erase b = Finset.Iio b
-- Finset.Iio_insert
@[simp]
theorem Iio_insert [DecidableEq α] (b : α) : insert b (Iio b) = Iic b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderBot α] [DecidableEq α] (b : α)
Goal: insert b (Finset.Iio b) = Finset.Iic b
-- Finset.not_mem_Iio_self
theorem not_mem_Iio_self {b : α} : b ∉ Iio b 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderBot α] {b : α}
Goal: b ∉ Finset.Iio b
-- Finset.Iic_eq_cons_Iio
theorem Iic_eq_cons_Iio (b : α) : Iic b = (Iio b).cons b not_mem_Iio_self 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderBot α] (b : α)
Goal: Finset.Iic b = Finset.cons b (Finset.Iio b) ⋯
-- Finset.card_Iio_eq_card_Iic_sub_one
theorem card_Iio_eq_card_Iic_sub_one (a : α) : (Iio a).card = (Iic a).card - 1 
Arguments: {α : Type u_2} [PartialOrder α] [LocallyFiniteOrderBot α] (a : α)
Goal: (Finset.Iio a).card = (Finset.Iic a).card - 1
-- Finset.sup'_Iic
lemma sup'_Iic (a : α) : (Iic a).sup' nonempty_Iic id = a 
Arguments: {α : Type u_2} [SemilatticeSup α] [LocallyFiniteOrderBot α] (a : α)
Goal: (Finset.Iic a).sup' ⋯ id = a
-- Finset.sup_Iic
@[simp] lemma sup_Iic [OrderBot α] (a : α) : (Iic a).sup id = a 
Arguments: {α : Type u_2} [SemilatticeSup α] [LocallyFiniteOrderBot α] [OrderBot α] (a : α)
Goal: (Finset.Iic a).sup id = a
-- Finset.inf'_Ici
lemma inf'_Ici (a : α) : (Ici a).inf' nonempty_Ici id = a 
Arguments: {α : Type u_2} [SemilatticeInf α] [LocallyFiniteOrderTop α] (a : α)
Goal: (Finset.Ici a).inf' ⋯ id = a
-- Finset.inf_Ici
@[simp] lemma inf_Ici [OrderTop α] (a : α) : (Ici a).inf id = a 
Arguments: {α : Type u_2} [SemilatticeInf α] [LocallyFiniteOrderTop α] [OrderTop α] (a : α)
Goal: (Finset.Ici a).inf id = a
-- Finset.Ico_subset_Ico_iff
theorem Ico_subset_Ico_iff {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁) :
    Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a₁ b₁ a₂ b₂ : α} (h : a₁ < b₁)
Goal: Finset.Ico a₁ b₁ ⊆ Finset.Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂
-- Finset.Ico_union_Ico_eq_Ico
theorem Ico_union_Ico_eq_Ico {a b c : α} (hab : a ≤ b) (hbc : b ≤ c) :
    Ico a b ∪ Ico b c = Ico a c 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a b c : α} (hab : a ≤ b) (hbc : b ≤ c)
Goal: Finset.Ico a b ∪ Finset.Ico b c = Finset.Ico a c
-- Finset.Ioc_union_Ioc_eq_Ioc
@[simp]
theorem Ioc_union_Ioc_eq_Ioc {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c) :
    Ioc a b ∪ Ioc b c = Ioc a c 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c)
Goal: Finset.Ioc a b ∪ Finset.Ioc b c = Finset.Ioc a c
-- Finset.Ico_subset_Ico_union_Ico
theorem Ico_subset_Ico_union_Ico {a b c : α} : Ico a c ⊆ Ico a b ∪ Ico b c 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a b c : α}
Goal: Finset.Ico a c ⊆ Finset.Ico a b ∪ Finset.Ico b c
-- Finset.Ico_union_Ico'
theorem Ico_union_Ico' {a b c d : α} (hcb : c ≤ b) (had : a ≤ d) :
    Ico a b ∪ Ico c d = Ico (min a c) (max b d) 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a b c d : α} (hcb : c ≤ b) (had : a ≤ d)
Goal: Finset.Ico a b ∪ Finset.Ico c d = Finset.Ico (min a c) (max b d)
-- Finset.Ico_union_Ico
theorem Ico_union_Ico {a b c d : α} (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b) :
    Ico a b ∪ Ico c d = Ico (min a c) (max b d) 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a b c d : α} (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b)
Goal: Finset.Ico a b ∪ Finset.Ico c d = Finset.Ico (min a c) (max b d)
-- Finset.Ico_inter_Ico
theorem Ico_inter_Ico {a b c d : α} : Ico a b ∩ Ico c d = Ico (max a c) (min b d) 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] {a b c d : α}
Goal: Finset.Ico a b ∩ Finset.Ico c d = Finset.Ico (max a c) (min b d)
-- Finset.Ico_filter_lt
@[simp]
theorem Ico_filter_lt (a b c : α) : ((Ico a b).filter fun x => x < c) = Ico a (min b c) 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.filter (fun x => x < c) (Finset.Ico a b) = Finset.Ico a (min b c)
-- Finset.Ico_filter_le
@[simp]
theorem Ico_filter_le (a b c : α) : ((Ico a b).filter fun x => c ≤ x) = Ico (max a c) b 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.filter (fun x => c ≤ x) (Finset.Ico a b) = Finset.Ico (max a c) b
-- Finset.Ioo_filter_lt
@[simp]
theorem Ioo_filter_lt (a b c : α) : (Ioo a b).filter (· < c) = Ioo a (min b c) 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.filter (fun x => x < c) (Finset.Ioo a b) = Finset.Ioo a (min b c)
-- Finset.Iio_filter_lt
@[simp]
theorem Iio_filter_lt {α} [LinearOrder α] [LocallyFiniteOrderBot α] (a b : α) :
    (Iio a).filter (· < b) = Iio (min a b) 
Arguments: {α : Type u_3} [LinearOrder α] [LocallyFiniteOrderBot α] (a b : α)
Goal: Finset.filter (fun x => x < b) (Finset.Iio a) = Finset.Iio (min a b)
-- Finset.Ico_diff_Ico_left
@[simp]
theorem Ico_diff_Ico_left (a b c : α) : Ico a b \ Ico a c = Ico (max a c) b 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.Ico a b \ Finset.Ico a c = Finset.Ico (max a c) b
-- Finset.Ico_diff_Ico_right
@[simp]
theorem Ico_diff_Ico_right (a b c : α) : Ico a b \ Ico c b = Ico a (min b c) 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.Ico a b \ Finset.Ico c b = Finset.Ico a (min b c)
-- Set.Infinite.exists_gt
theorem _root_.Set.Infinite.exists_gt (hs : s.Infinite) : ∀ a, ∃ b ∈ s, a < b 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrderBot α] {s : Set α} (hs : s.Infinite) (a : α)
Goal: ∃ b ∈ s, a < b
-- Set.infinite_iff_exists_gt
theorem _root_.Set.infinite_iff_exists_gt [Nonempty α] : s.Infinite ↔ ∀ a, ∃ b ∈ s, a < b 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrderBot α] {s : Set α} [Nonempty α]
Goal: s.Infinite ↔ ∀ (a : α), ∃ b ∈ s, a < b
-- Set.Infinite.exists_lt
theorem _root_.Set.Infinite.exists_lt (hs : s.Infinite) : ∀ a, ∃ b ∈ s, b < a 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrderTop α] {s : Set α} (hs : s.Infinite) (a : α)
Goal: ∃ b ∈ s, b < a
-- Set.infinite_iff_exists_lt
theorem _root_.Set.infinite_iff_exists_lt [Nonempty α] : s.Infinite ↔ ∀ a, ∃ b ∈ s, b < a 
Arguments: {α : Type u_2} [LinearOrder α] [LocallyFiniteOrderTop α] {s : Set α} [Nonempty α]
Goal: s.Infinite ↔ ∀ (a : α), ∃ b ∈ s, b < a
-- Finset.Ioi_disjUnion_Iio
theorem Ioi_disjUnion_Iio (a : α) :
    (Ioi a).disjUnion (Iio a) (disjoint_Ioi_Iio a) = ({a} : Finset α)ᶜ 
Arguments: {α : Type u_2} [LinearOrder α] [Fintype α] [LocallyFiniteOrderTop α] [LocallyFiniteOrderBot α] (a : α)
Goal: (Finset.Ioi a).disjUnion (Finset.Iio a) ⋯ = {a}ᶜ
-- Finset.uIcc_toDual
theorem uIcc_toDual (a b : α) : [[toDual a, toDual b]] = [[a, b]].map toDual.toEmbedding 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] (a b : α)
Goal: [[OrderDual.toDual a, OrderDual.toDual b]] = Finset.map OrderDual.toDual.toEmbedding [[a, b]]
-- Finset.uIcc_of_le
@[simp]
theorem uIcc_of_le (h : a ≤ b) : [[a, b]] = Icc a b 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α} (h : a ≤ b)
Goal: [[a, b]] = Finset.Icc a b
-- Finset.uIcc_of_ge
@[simp]
theorem uIcc_of_ge (h : b ≤ a) : [[a, b]] = Icc b a 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α} (h : b ≤ a)
Goal: [[a, b]] = Finset.Icc b a
-- Finset.uIcc_comm
theorem uIcc_comm (a b : α) : [[a, b]] = [[b, a]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] (a b : α)
Goal: [[a, b]] = [[b, a]]
-- Finset.uIcc_self
theorem uIcc_self : [[a, a]] = {a} 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a : α}
Goal: [[a, a]] = {a}
-- Finset.nonempty_uIcc
@[simp]
theorem nonempty_uIcc : Finset.Nonempty [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α}
Goal: [[a, b]].Nonempty
-- Finset.Icc_subset_uIcc
theorem Icc_subset_uIcc : Icc a b ⊆ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Icc a b ⊆ [[a, b]]
-- Finset.Icc_subset_uIcc'
theorem Icc_subset_uIcc' : Icc b a ⊆ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α}
Goal: Finset.Icc b a ⊆ [[a, b]]
-- Finset.left_mem_uIcc
theorem left_mem_uIcc : a ∈ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α}
Goal: a ∈ [[a, b]]
-- Finset.right_mem_uIcc
theorem right_mem_uIcc : b ∈ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b : α}
Goal: b ∈ [[a, b]]
-- Finset.mem_uIcc_of_le
theorem mem_uIcc_of_le (ha : a ≤ x) (hb : x ≤ b) : x ∈ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b x : α} (ha : a ≤ x) (hb : x ≤ b)
Goal: x ∈ [[a, b]]
-- Finset.mem_uIcc_of_ge
theorem mem_uIcc_of_ge (hb : b ≤ x) (ha : x ≤ a) : x ∈ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b x : α} (hb : b ≤ x) (ha : x ≤ a)
Goal: x ∈ [[a, b]]
-- Finset.uIcc_subset_uIcc
theorem uIcc_subset_uIcc (h₁ : a₁ ∈ [[a₂, b₂]]) (h₂ : b₁ ∈ [[a₂, b₂]]) :
    [[a₁, b₁]] ⊆ [[a₂, b₂]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ∈ [[a₂, b₂]]) (h₂ : b₁ ∈ [[a₂, b₂]])
Goal: [[a₁, b₁]] ⊆ [[a₂, b₂]]
-- Finset.uIcc_subset_Icc
theorem uIcc_subset_Icc (ha : a₁ ∈ Icc a₂ b₂) (hb : b₁ ∈ Icc a₂ b₂) : [[a₁, b₁]] ⊆ Icc a₂ b₂ 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ∈ Finset.Icc a₂ b₂) (hb : b₁ ∈ Finset.Icc a₂ b₂)
Goal: [[a₁, b₁]] ⊆ Finset.Icc a₂ b₂
-- Finset.uIcc_subset_uIcc_iff_mem
theorem uIcc_subset_uIcc_iff_mem : [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ a₁ ∈ [[a₂, b₂]] ∧ b₁ ∈ [[a₂, b₂]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ a₁ ∈ [[a₂, b₂]] ∧ b₁ ∈ [[a₂, b₂]]
-- Finset.uIcc_subset_uIcc_iff_le'
theorem uIcc_subset_uIcc_iff_le' :
    [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ a₂ ⊓ b₂ ≤ a₁ ⊓ b₁ ∧ a₁ ⊔ b₁ ≤ a₂ ⊔ b₂ 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: [[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ a₂ ⊓ b₂ ≤ a₁ ⊓ b₁ ∧ a₁ ⊔ b₁ ≤ a₂ ⊔ b₂
-- Finset.uIcc_subset_uIcc_right
theorem uIcc_subset_uIcc_right (h : x ∈ [[a, b]]) : [[x, b]] ⊆ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b x : α} (h : x ∈ [[a, b]])
Goal: [[x, b]] ⊆ [[a, b]]
-- Finset.uIcc_subset_uIcc_left
theorem uIcc_subset_uIcc_left (h : x ∈ [[a, b]]) : [[a, x]] ⊆ [[a, b]] 
Arguments: {α : Type u_2} [Lattice α] [LocallyFiniteOrder α] {a b x : α} (h : x ∈ [[a, b]])
Goal: [[a, x]] ⊆ [[a, b]]
-- Finset.eq_of_mem_uIcc_of_mem_uIcc
theorem eq_of_mem_uIcc_of_mem_uIcc : a ∈ [[b, c]] → b ∈ [[a, c]] → a = b 
Arguments: {α : Type u_2} [DistribLattice α] [LocallyFiniteOrder α] {a b c : α}
Goal: a ∈ [[b, c]] → b ∈ [[a, c]] → a = b