ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.Ioc_subset_Ioi_self
theorem Ioc_subset_Ioi_self : Ioc a b ⊆ Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioc a b ⊆ Set.Ioi a
-- Set.Ioo_subset_Ioi_self
theorem Ioo_subset_Ioi_self : Ioo a b ⊆ Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioo a b ⊆ Set.Ioi a
-- Set.Ioi_subset_Ici_self
theorem Ioi_subset_Ici_self : Ioi a ⊆ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Ioi a ⊆ Set.Ici a
-- Set.Iio_subset_Iic_self
theorem Iio_subset_Iic_self : Iio a ⊆ Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Iio a ⊆ Set.Iic a
-- Set.Ico_subset_Ici_self
theorem Ico_subset_Ici_self : Ico a b ⊆ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ico a b ⊆ Set.Ici a
-- Set.Ioi_ssubset_Ici_self
theorem Ioi_ssubset_Ici_self : Ioi a ⊂ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Ioi a ⊂ Set.Ici a
-- Set.Iio_ssubset_Iic_self
theorem Iio_ssubset_Iic_self : Iio a ⊂ Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Iio a ⊂ Set.Iic a
-- Set.Icc_subset_Icc_iff
theorem Icc_subset_Icc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Icc a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂
-- Set.Icc_subset_Ioo_iff
theorem Icc_subset_Ioo_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Ioo a₂ b₂ ↔ a₂ < a₁ ∧ b₁ < b₂
-- Set.Icc_subset_Ico_iff
theorem Icc_subset_Ico_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ < b₂
-- Set.Icc_subset_Ioc_iff
theorem Icc_subset_Ioc_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Ioc a₂ b₂ ↔ a₂ < a₁ ∧ b₁ ≤ b₂
-- Set.Icc_subset_Iio_iff
theorem Icc_subset_Iio_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iio b₂ ↔ b₁ < b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Iio b₂ ↔ b₁ < b₂
-- Set.Icc_subset_Ioi_iff
theorem Icc_subset_Ioi_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ioi a₂ ↔ a₂ < a₁ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Ioi a₂ ↔ a₂ < a₁
-- Set.Icc_subset_Iic_iff
theorem Icc_subset_Iic_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Iic b₂ ↔ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ b₁ b₂ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Iic b₂ ↔ b₁ ≤ b₂
-- Set.Icc_subset_Ici_iff
theorem Icc_subset_Ici_iff (h₁ : a₁ ≤ b₁) : Icc a₁ b₁ ⊆ Ici a₂ ↔ a₂ ≤ a₁ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ : α} (h₁ : a₁ ≤ b₁)
Goal: Set.Icc a₁ b₁ ⊆ Set.Ici a₂ ↔ a₂ ≤ a₁
-- Set.Icc_ssubset_Icc_left
theorem Icc_ssubset_Icc_left (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂) : Icc a₁ b₁ ⊂ Icc a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (hI : a₂ ≤ b₂) (ha : a₂ < a₁) (hb : b₁ ≤ b₂)
Goal: Set.Icc a₁ b₁ ⊂ Set.Icc a₂ b₂
-- Set.Icc_ssubset_Icc_right
theorem Icc_ssubset_Icc_right (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂) :
    Icc a₁ b₁ ⊂ Icc a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (hI : a₂ ≤ b₂) (ha : a₂ ≤ a₁) (hb : b₁ < b₂)
Goal: Set.Icc a₁ b₁ ⊂ Set.Icc a₂ b₂
-- Set.Ioi_subset_Ioi
@[gcongr]
theorem Ioi_subset_Ioi (h : a ≤ b) : Ioi b ⊆ Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Set.Ioi b ⊆ Set.Ioi a
-- Set.Ioi_subset_Ici
theorem Ioi_subset_Ici (h : a ≤ b) : Ioi b ⊆ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Set.Ioi b ⊆ Set.Ici a
-- Set.Iio_subset_Iio
@[gcongr]
theorem Iio_subset_Iio (h : a ≤ b) : Iio a ⊆ Iio b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Set.Iio a ⊆ Set.Iio b
-- Set.Iio_subset_Iic
theorem Iio_subset_Iic (h : a ≤ b) : Iio a ⊆ Iic b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Set.Iio a ⊆ Set.Iic b
-- Set.Ici_inter_Iic
theorem Ici_inter_Iic : Ici a ∩ Iic b = Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ici a ∩ Set.Iic b = Set.Icc a b
-- Set.Ici_inter_Iio
theorem Ici_inter_Iio : Ici a ∩ Iio b = Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ici a ∩ Set.Iio b = Set.Ico a b
-- Set.Ioi_inter_Iic
theorem Ioi_inter_Iic : Ioi a ∩ Iic b = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioi a ∩ Set.Iic b = Set.Ioc a b
-- Set.Ioi_inter_Iio
theorem Ioi_inter_Iio : Ioi a ∩ Iio b = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioi a ∩ Set.Iio b = Set.Ioo a b
-- Set.Iic_inter_Ici
theorem Iic_inter_Ici : Iic a ∩ Ici b = Icc b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Iic a ∩ Set.Ici b = Set.Icc b a
-- Set.Iio_inter_Ici
theorem Iio_inter_Ici : Iio a ∩ Ici b = Ico b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Iio a ∩ Set.Ici b = Set.Ico b a
-- Set.Iic_inter_Ioi
theorem Iic_inter_Ioi : Iic a ∩ Ioi b = Ioc b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Iic a ∩ Set.Ioi b = Set.Ioc b a
-- Set.Iio_inter_Ioi
theorem Iio_inter_Ioi : Iio a ∩ Ioi b = Ioo b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Iio a ∩ Set.Ioi b = Set.Ioo b a
-- Set.mem_Icc_of_Ioo
theorem mem_Icc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α} (h : x ∈ Set.Ioo a b)
Goal: x ∈ Set.Icc a b
-- Set.mem_Ico_of_Ioo
theorem mem_Ico_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α} (h : x ∈ Set.Ioo a b)
Goal: x ∈ Set.Ico a b
-- Set.mem_Ioc_of_Ioo
theorem mem_Ioc_of_Ioo (h : x ∈ Ioo a b) : x ∈ Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α} (h : x ∈ Set.Ioo a b)
Goal: x ∈ Set.Ioc a b
-- Set.mem_Icc_of_Ico
theorem mem_Icc_of_Ico (h : x ∈ Ico a b) : x ∈ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α} (h : x ∈ Set.Ico a b)
Goal: x ∈ Set.Icc a b
-- Set.mem_Icc_of_Ioc
theorem mem_Icc_of_Ioc (h : x ∈ Ioc a b) : x ∈ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α} (h : x ∈ Set.Ioc a b)
Goal: x ∈ Set.Icc a b
-- Set.mem_Ici_of_Ioi
theorem mem_Ici_of_Ioi (h : x ∈ Ioi a) : x ∈ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a x : α} (h : x ∈ Set.Ioi a)
Goal: x ∈ Set.Ici a
-- Set.mem_Iic_of_Iio
theorem mem_Iic_of_Iio (h : x ∈ Iio a) : x ∈ Iic a 
Arguments: {α : Type u_1} [Preorder α] {a x : α} (h : x ∈ Set.Iio a)
Goal: x ∈ Set.Iic a
-- Set.Icc_eq_empty_iff
theorem Icc_eq_empty_iff : Icc a b = ∅ ↔ ¬a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Icc a b = ∅ ↔ ¬a ≤ b
-- Set.Ico_eq_empty_iff
theorem Ico_eq_empty_iff : Ico a b = ∅ ↔ ¬a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ico a b = ∅ ↔ ¬a < b
-- Set.Ioc_eq_empty_iff
theorem Ioc_eq_empty_iff : Ioc a b = ∅ ↔ ¬a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioc a b = ∅ ↔ ¬a < b
-- Set.Ioo_eq_empty_iff
theorem Ioo_eq_empty_iff [DenselyOrdered α] : Ioo a b = ∅ ↔ ¬a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} [DenselyOrdered α]
Goal: Set.Ioo a b = ∅ ↔ ¬a < b
-- IsTop.Iic_eq
theorem _root_.IsTop.Iic_eq (h : IsTop a) : Iic a = univ 
Arguments: {α : Type u_1} [Preorder α] {a : α} (h : IsTop a)
Goal: Set.Iic a = Set.univ
-- IsBot.Ici_eq
theorem _root_.IsBot.Ici_eq (h : IsBot a) : Ici a = univ 
Arguments: {α : Type u_1} [Preorder α] {a : α} (h : IsBot a)
Goal: Set.Ici a = Set.univ
-- IsMax.Ioi_eq
theorem _root_.IsMax.Ioi_eq (h : IsMax a) : Ioi a = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a : α} (h : IsMax a)
Goal: Set.Ioi a = ∅
-- IsMin.Iio_eq
theorem _root_.IsMin.Iio_eq (h : IsMin a) : Iio a = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a : α} (h : IsMin a)
Goal: Set.Iio a = ∅
-- Set.Iic_inter_Ioc_of_le
theorem Iic_inter_Ioc_of_le (h : a ≤ c) : Iic a ∩ Ioc b c = Ioc b a 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (h : a ≤ c)
Goal: Set.Iic a ∩ Set.Ioc b c = Set.Ioc b a
-- Set.not_mem_Icc_of_lt
theorem not_mem_Icc_of_lt (ha : c < a) : c ∉ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (ha : c < a)
Goal: c ∉ Set.Icc a b
-- Set.not_mem_Icc_of_gt
theorem not_mem_Icc_of_gt (hb : b < c) : c ∉ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hb : b < c)
Goal: c ∉ Set.Icc a b
-- Set.not_mem_Ico_of_lt
theorem not_mem_Ico_of_lt (ha : c < a) : c ∉ Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (ha : c < a)
Goal: c ∉ Set.Ico a b
-- Set.not_mem_Ioc_of_gt
theorem not_mem_Ioc_of_gt (hb : b < c) : c ∉ Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hb : b < c)
Goal: c ∉ Set.Ioc a b
-- Set.not_mem_Ioi_self
theorem not_mem_Ioi_self : a ∉ Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: a ∉ Set.Ioi a
-- Set.not_mem_Iio_self
theorem not_mem_Iio_self : b ∉ Iio b 
Arguments: {α : Type u_1} [Preorder α] {b : α}
Goal: b ∉ Set.Iio b
-- Set.not_mem_Ioc_of_le
theorem not_mem_Ioc_of_le (ha : c ≤ a) : c ∉ Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (ha : c ≤ a)
Goal: c ∉ Set.Ioc a b
-- Set.not_mem_Ico_of_ge
theorem not_mem_Ico_of_ge (hb : b ≤ c) : c ∉ Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hb : b ≤ c)
Goal: c ∉ Set.Ico a b
-- Set.not_mem_Ioo_of_le
theorem not_mem_Ioo_of_le (ha : c ≤ a) : c ∉ Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (ha : c ≤ a)
Goal: c ∉ Set.Ioo a b
-- Set.not_mem_Ioo_of_ge
theorem not_mem_Ioo_of_ge (hb : b ≤ c) : c ∉ Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {a b c : α} (hb : b ≤ c)
Goal: c ∉ Set.Ioo a b
-- Set.Icc_self
@[simp]
theorem Icc_self (a : α) : Icc a a = {a} 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: Set.Icc a a = {a}
-- Set.Icc_eq_singleton_iff
@[simp]
theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c 
Arguments: {α : Type u_1} [PartialOrder α] {a b c : α}
Goal: Set.Icc a b = {c} ↔ a = c ∧ b = c
-- Set.subsingleton_Icc_of_ge
lemma subsingleton_Icc_of_ge (hba : b ≤ a) : Set.Subsingleton (Icc a b) 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hba : b ≤ a)
Goal: (Set.Icc a b).Subsingleton
-- Set.subsingleton_Icc_iff
@[simp] lemma subsingleton_Icc_iff {α : Type*} [LinearOrder α] {a b : α} :
    Set.Subsingleton (Icc a b) ↔ b ≤ a 
Arguments: {α : Type u_3} [LinearOrder α] {a b : α}
Goal: (Set.Icc a b).Subsingleton ↔ b ≤ a
-- Set.Icc_diff_left
@[simp]
theorem Icc_diff_left : Icc a b \ {a} = Ioc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Icc a b \ {a} = Set.Ioc a b
-- Set.Icc_diff_right
@[simp]
theorem Icc_diff_right : Icc a b \ {b} = Ico a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Icc a b \ {b} = Set.Ico a b
-- Set.Ico_diff_left
@[simp]
theorem Ico_diff_left : Ico a b \ {a} = Ioo a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Ico a b \ {a} = Set.Ioo a b
-- Set.Ioc_diff_right
@[simp]
theorem Ioc_diff_right : Ioc a b \ {b} = Ioo a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Ioc a b \ {b} = Set.Ioo a b
-- Set.Icc_diff_both
@[simp]
theorem Icc_diff_both : Icc a b \ {a, b} = Ioo a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Icc a b \ {a, b} = Set.Ioo a b
-- Set.Ici_diff_left
@[simp]
theorem Ici_diff_left : Ici a \ {a} = Ioi a 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: Set.Ici a \ {a} = Set.Ioi a
-- Set.Iic_diff_right
@[simp]
theorem Iic_diff_right : Iic a \ {a} = Iio a 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: Set.Iic a \ {a} = Set.Iio a
-- Set.Ico_diff_Ioo_same
@[simp]
theorem Ico_diff_Ioo_same (h : a < b) : Ico a b \ Ioo a b = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a < b)
Goal: Set.Ico a b \ Set.Ioo a b = {a}
-- Set.Ioc_diff_Ioo_same
@[simp]
theorem Ioc_diff_Ioo_same (h : a < b) : Ioc a b \ Ioo a b = {b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a < b)
Goal: Set.Ioc a b \ Set.Ioo a b = {b}
-- Set.Icc_diff_Ico_same
@[simp]
theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Icc a b \ Set.Ico a b = {b}
-- Set.Icc_diff_Ioc_same
@[simp]
theorem Icc_diff_Ioc_same (h : a ≤ b) : Icc a b \ Ioc a b = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Icc a b \ Set.Ioc a b = {a}
-- Set.Icc_diff_Ioo_same
@[simp]
theorem Icc_diff_Ioo_same (h : a ≤ b) : Icc a b \ Ioo a b = {a, b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Icc a b \ Set.Ioo a b = {a, b}
-- Set.Ici_diff_Ioi_same
@[simp]
theorem Ici_diff_Ioi_same : Ici a \ Ioi a = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: Set.Ici a \ Set.Ioi a = {a}
-- Set.Iic_diff_Iio_same
@[simp]
theorem Iic_diff_Iio_same : Iic a \ Iio a = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: Set.Iic a \ Set.Iio a = {a}
-- Set.Ioi_union_left
theorem Ioi_union_left : Ioi a ∪ {a} = Ici a 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: Set.Ioi a ∪ {a} = Set.Ici a
-- Set.Iio_union_right
theorem Iio_union_right : Iio a ∪ {a} = Iic a 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: Set.Iio a ∪ {a} = Set.Iic a
-- Set.Ioo_union_left
theorem Ioo_union_left (hab : a < b) : Ioo a b ∪ {a} = Ico a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a < b)
Goal: Set.Ioo a b ∪ {a} = Set.Ico a b
-- Set.Ioo_union_right
theorem Ioo_union_right (hab : a < b) : Ioo a b ∪ {b} = Ioc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a < b)
Goal: Set.Ioo a b ∪ {b} = Set.Ioc a b
-- Set.Ioo_union_both
theorem Ioo_union_both (h : a ≤ b) : Ioo a b ∪ {a, b} = Icc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Ioo a b ∪ {a, b} = Set.Icc a b
-- Set.Ioc_union_left
theorem Ioc_union_left (hab : a ≤ b) : Ioc a b ∪ {a} = Icc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a ≤ b)
Goal: Set.Ioc a b ∪ {a} = Set.Icc a b
-- Set.Ico_union_right
theorem Ico_union_right (hab : a ≤ b) : Ico a b ∪ {b} = Icc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a ≤ b)
Goal: Set.Ico a b ∪ {b} = Set.Icc a b
-- Set.Ico_insert_right
@[simp]
theorem Ico_insert_right (h : a ≤ b) : insert b (Ico a b) = Icc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: insert b (Set.Ico a b) = Set.Icc a b
-- Set.Ioc_insert_left
@[simp]
theorem Ioc_insert_left (h : a ≤ b) : insert a (Ioc a b) = Icc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: insert a (Set.Ioc a b) = Set.Icc a b
-- Set.Ioo_insert_left
@[simp]
theorem Ioo_insert_left (h : a < b) : insert a (Ioo a b) = Ico a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a < b)
Goal: insert a (Set.Ioo a b) = Set.Ico a b
-- Set.Ioo_insert_right
@[simp]
theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (h : a < b)
Goal: insert b (Set.Ioo a b) = Set.Ioc a b
-- Set.Iio_insert
@[simp]
theorem Iio_insert : insert a (Iio a) = Iic a 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: insert a (Set.Iio a) = Set.Iic a
-- Set.Ioi_insert
@[simp]
theorem Ioi_insert : insert a (Ioi a) = Ici a 
Arguments: {α : Type u_1} [PartialOrder α] {a : α}
Goal: insert a (Set.Ioi a) = Set.Ici a
-- Set.mem_Ici_Ioi_of_subset_of_subset
theorem mem_Ici_Ioi_of_subset_of_subset {s : Set α} (ho : Ioi a ⊆ s) (hc : s ⊆ Ici a) :
    s ∈ ({Ici a, Ioi a} : Set (Set α)) 
Arguments: {α : Type u_1} [PartialOrder α] {a : α} {s : Set α} (ho : Set.Ioi a ⊆ s) (hc : s ⊆ Set.Ici a)
Goal: s ∈ {Set.Ici a, Set.Ioi a}
-- Set.mem_Iic_Iio_of_subset_of_subset
theorem mem_Iic_Iio_of_subset_of_subset {s : Set α} (ho : Iio a ⊆ s) (hc : s ⊆ Iic a) :
    s ∈ ({Iic a, Iio a} : Set (Set α)) 
Arguments: {α : Type u_1} [PartialOrder α] {a : α} {s : Set α} (ho : Set.Iio a ⊆ s) (hc : s ⊆ Set.Iic a)
Goal: s ∈ {Set.Iic a, Set.Iio a}
-- Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset
theorem mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset {s : Set α} (ho : Ioo a b ⊆ s) (hc : s ⊆ Icc a b) :
    s ∈ ({Icc a b, Ico a b, Ioc a b, Ioo a b} : Set (Set α)) 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} {s : Set α} (ho : Set.Ioo a b ⊆ s) (hc : s ⊆ Set.Icc a b)
Goal: s ∈ {Set.Icc a b, Set.Ico a b, Set.Ioc a b, Set.Ioo a b}
-- Set.eq_left_or_mem_Ioo_of_mem_Ico
theorem eq_left_or_mem_Ioo_of_mem_Ico {x : α} (hmem : x ∈ Ico a b) : x = a ∨ x ∈ Ioo a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b x : α} (hmem : x ∈ Set.Ico a b)
Goal: x = a ∨ x ∈ Set.Ioo a b
-- Set.eq_right_or_mem_Ioo_of_mem_Ioc
theorem eq_right_or_mem_Ioo_of_mem_Ioc {x : α} (hmem : x ∈ Ioc a b) : x = b ∨ x ∈ Ioo a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b x : α} (hmem : x ∈ Set.Ioc a b)
Goal: x = b ∨ x ∈ Set.Ioo a b
-- Set.eq_endpoints_or_mem_Ioo_of_mem_Icc
theorem eq_endpoints_or_mem_Ioo_of_mem_Icc {x : α} (hmem : x ∈ Icc a b) :
    x = a ∨ x = b ∨ x ∈ Ioo a b 
Arguments: {α : Type u_1} [PartialOrder α] {a b x : α} (hmem : x ∈ Set.Icc a b)
Goal: x = a ∨ x = b ∨ x ∈ Set.Ioo a b
-- IsMax.Ici_eq
theorem _root_.IsMax.Ici_eq (h : IsMax a) : Ici a = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a : α} (h : IsMax a)
Goal: Set.Ici a = {a}
-- IsMin.Iic_eq
theorem _root_.IsMin.Iic_eq (h : IsMin a) : Iic a = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a : α} (h : IsMin a)
Goal: Set.Iic a = {a}
-- Set.Ici_injective
theorem Ici_injective : Injective (Ici : α → Set α) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: Function.Injective Set.Ici
-- Set.Iic_injective
theorem Iic_injective : Injective (Iic : α → Set α) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: Function.Injective Set.Iic
-- Set.Ici_inj
theorem Ici_inj : Ici a = Ici b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Ici a = Set.Ici b ↔ a = b
-- Set.Iic_inj
theorem Iic_inj : Iic a = Iic b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: Set.Iic a = Set.Iic b ↔ a = b
-- Set.Ici_top
@[simp]
theorem Ici_top [PartialOrder α] [OrderTop α] : Ici (⊤ : α) = {⊤} 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α]
Goal: Set.Ici ⊤ = {⊤}
-- Set.Ioi_top
@[simp]
theorem Ioi_top : Ioi (⊤ : α) = ∅ 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α]
Goal: Set.Ioi ⊤ = ∅
-- Set.Iic_top
@[simp]
theorem Iic_top : Iic (⊤ : α) = univ 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α]
Goal: Set.Iic ⊤ = Set.univ
-- Set.Icc_top
@[simp]
theorem Icc_top : Icc a ⊤ = Ici a 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α] {a : α}
Goal: Set.Icc a ⊤ = Set.Ici a
-- Set.Ioc_top
@[simp]
theorem Ioc_top : Ioc a ⊤ = Ioi a 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α] {a : α}
Goal: Set.Ioc a ⊤ = Set.Ioi a
-- Set.Iic_bot
@[simp]
theorem Iic_bot [PartialOrder α] [OrderBot α] : Iic (⊥ : α) = {⊥} 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α]
Goal: Set.Iic ⊥ = {⊥}
-- Set.Iio_bot
@[simp]
theorem Iio_bot : Iio (⊥ : α) = ∅ 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α]
Goal: Set.Iio ⊥ = ∅
-- Set.Ici_bot
@[simp]
theorem Ici_bot : Ici (⊥ : α) = univ 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α]
Goal: Set.Ici ⊥ = Set.univ
-- Set.Icc_bot
@[simp]
theorem Icc_bot : Icc ⊥ a = Iic a 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] {a : α}
Goal: Set.Icc ⊥ a = Set.Iic a
-- Set.Ico_bot
@[simp]
theorem Ico_bot : Ico ⊥ a = Iio a 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] {a : α}
Goal: Set.Ico ⊥ a = Set.Iio a
-- Set.Icc_bot_top
theorem Icc_bot_top [PartialOrder α] [BoundedOrder α] : Icc (⊥ : α) ⊤ = univ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α]
Goal: Set.Icc ⊥ ⊤ = Set.univ
-- Set.not_mem_Ici
theorem not_mem_Ici : c ∉ Ici a ↔ c < a 
Arguments: {α : Type u_1} [LinearOrder α] {a c : α}
Goal: c ∉ Set.Ici a ↔ c < a
-- Set.not_mem_Iic
theorem not_mem_Iic : c ∉ Iic b ↔ b < c 
Arguments: {α : Type u_1} [LinearOrder α] {b c : α}
Goal: c ∉ Set.Iic b ↔ b < c
-- Set.not_mem_Ioi
theorem not_mem_Ioi : c ∉ Ioi a ↔ c ≤ a 
Arguments: {α : Type u_1} [LinearOrder α] {a c : α}
Goal: c ∉ Set.Ioi a ↔ c ≤ a
-- Set.not_mem_Iio
theorem not_mem_Iio : c ∉ Iio b ↔ b ≤ c 
Arguments: {α : Type u_1} [LinearOrder α] {b c : α}
Goal: c ∉ Set.Iio b ↔ b ≤ c
-- Set.compl_Iic
@[simp]
theorem compl_Iic : (Iic a)ᶜ = Ioi a 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: (Set.Iic a)ᶜ = Set.Ioi a
-- Set.compl_Ici
@[simp]
theorem compl_Ici : (Ici a)ᶜ = Iio a 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: (Set.Ici a)ᶜ = Set.Iio a
-- Set.compl_Iio
@[simp]
theorem compl_Iio : (Iio a)ᶜ = Ici a 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: (Set.Iio a)ᶜ = Set.Ici a
-- Set.compl_Ioi
@[simp]
theorem compl_Ioi : (Ioi a)ᶜ = Iic a 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: (Set.Ioi a)ᶜ = Set.Iic a
-- Set.Ici_diff_Ici
@[simp]
theorem Ici_diff_Ici : Ici a \ Ici b = Ico a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ici a \ Set.Ici b = Set.Ico a b
-- Set.Ici_diff_Ioi
@[simp]
theorem Ici_diff_Ioi : Ici a \ Ioi b = Icc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ici a \ Set.Ioi b = Set.Icc a b
-- Set.Ioi_diff_Ioi
@[simp]
theorem Ioi_diff_Ioi : Ioi a \ Ioi b = Ioc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a \ Set.Ioi b = Set.Ioc a b
-- Set.Ioi_diff_Ici
@[simp]
theorem Ioi_diff_Ici : Ioi a \ Ici b = Ioo a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a \ Set.Ici b = Set.Ioo a b
-- Set.Iic_diff_Iic
@[simp]
theorem Iic_diff_Iic : Iic b \ Iic a = Ioc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iic b \ Set.Iic a = Set.Ioc a b
-- Set.Iio_diff_Iic
@[simp]
theorem Iio_diff_Iic : Iio b \ Iic a = Ioo a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iio b \ Set.Iic a = Set.Ioo a b
-- Set.Iic_diff_Iio
@[simp]
theorem Iic_diff_Iio : Iic b \ Iio a = Icc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iic b \ Set.Iio a = Set.Icc a b
-- Set.Iio_diff_Iio
@[simp]
theorem Iio_diff_Iio : Iio b \ Iio a = Ico a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iio b \ Set.Iio a = Set.Ico a b
-- Set.Ioi_injective
theorem Ioi_injective : Injective (Ioi : α → Set α) 
Arguments: {α : Type u_1} [LinearOrder α]
Goal: Function.Injective Set.Ioi
-- Set.Iio_injective
theorem Iio_injective : Injective (Iio : α → Set α) 
Arguments: {α : Type u_1} [LinearOrder α]
Goal: Function.Injective Set.Iio
-- Set.Ioi_inj
theorem Ioi_inj : Ioi a = Ioi b ↔ a = b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a = Set.Ioi b ↔ a = b
-- Set.Iio_inj
theorem Iio_inj : Iio a = Iio b ↔ a = b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iio a = Set.Iio b ↔ a = b
-- Set.Ico_subset_Ico_iff
theorem Ico_subset_Ico_iff (h₁ : a₁ < b₁) : Ico a₁ b₁ ⊆ Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ < b₁)
Goal: Set.Ico a₁ b₁ ⊆ Set.Ico a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂
-- Set.Ioc_subset_Ioc_iff
theorem Ioc_subset_Ioc_iff (h₁ : a₁ < b₁) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ ↔ b₁ ≤ b₂ ∧ a₂ ≤ a₁ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₁ < b₁)
Goal: Set.Ioc a₁ b₁ ⊆ Set.Ioc a₂ b₂ ↔ b₁ ≤ b₂ ∧ a₂ ≤ a₁
-- Set.Ioo_subset_Ioo_iff
theorem Ioo_subset_Ioo_iff [DenselyOrdered α] (h₁ : a₁ < b₁) :
    Ioo a₁ b₁ ⊆ Ioo a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} [DenselyOrdered α] (h₁ : a₁ < b₁)
Goal: Set.Ioo a₁ b₁ ⊆ Set.Ioo a₂ b₂ ↔ a₂ ≤ a₁ ∧ b₁ ≤ b₂
-- Set.Ico_eq_Ico_iff
theorem Ico_eq_Ico_iff (h : a₁ < b₁ ∨ a₂ < b₂) : Ico a₁ b₁ = Ico a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h : a₁ < b₁ ∨ a₂ < b₂)
Goal: Set.Ico a₁ b₁ = Set.Ico a₂ b₂ ↔ a₁ = a₂ ∧ b₁ = b₂
-- Set.Ici_eq_singleton_iff_isTop
lemma Ici_eq_singleton_iff_isTop {x : α} : (Ici x = {x}) ↔ IsTop x 
Arguments: {α : Type u_1} [LinearOrder α] {x : α}
Goal: Set.Ici x = {x} ↔ IsTop x
-- Set.Ioi_subset_Ioi_iff
@[simp]
theorem Ioi_subset_Ioi_iff : Ioi b ⊆ Ioi a ↔ a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi b ⊆ Set.Ioi a ↔ a ≤ b
-- Set.Ioi_subset_Ici_iff
@[simp]
theorem Ioi_subset_Ici_iff [DenselyOrdered α] : Ioi b ⊆ Ici a ↔ a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} [DenselyOrdered α]
Goal: Set.Ioi b ⊆ Set.Ici a ↔ a ≤ b