ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.Iio_subset_Iio_iff
@[simp]
theorem Iio_subset_Iio_iff : 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.Iio_subset_Iic_iff
@[simp]
theorem Iio_subset_Iic_iff [DenselyOrdered α] : Iio a ⊆ Iic b ↔ a ≤ b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} [DenselyOrdered α]
Goal: Set.Iio a ⊆ Set.Iic b ↔ a ≤ b
-- Set.Iic_union_Ioi_of_le
theorem Iic_union_Ioi_of_le (h : a ≤ b) : Iic b ∪ Ioi a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iic b ∪ Set.Ioi a = Set.univ
-- Set.Iio_union_Ici_of_le
theorem Iio_union_Ici_of_le (h : a ≤ b) : Iio b ∪ Ici a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iio b ∪ Set.Ici a = Set.univ
-- Set.Iic_union_Ici_of_le
theorem Iic_union_Ici_of_le (h : a ≤ b) : Iic b ∪ Ici a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iic b ∪ Set.Ici a = Set.univ
-- Set.Iio_union_Ioi_of_lt
theorem Iio_union_Ioi_of_lt (h : a < b) : Iio b ∪ Ioi a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a < b)
Goal: Set.Iio b ∪ Set.Ioi a = Set.univ
-- Set.Iic_union_Ici
@[simp]
theorem Iic_union_Ici : Iic a ∪ Ici a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.Iic a ∪ Set.Ici a = Set.univ
-- Set.Iio_union_Ici
@[simp]
theorem Iio_union_Ici : Iio a ∪ Ici a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.Iio a ∪ Set.Ici a = Set.univ
-- Set.Iic_union_Ioi
@[simp]
theorem Iic_union_Ioi : Iic a ∪ Ioi a = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.Iic a ∪ Set.Ioi a = Set.univ
-- Set.Iio_union_Ioi
@[simp]
theorem Iio_union_Ioi : Iio a ∪ Ioi a = {a}ᶜ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.Iio a ∪ Set.Ioi a = {a}ᶜ
-- Set.Ioo_union_Ioi'
theorem Ioo_union_Ioi' (h₁ : c < b) : Ioo a b ∪ Ioi c = Ioi (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : c < b)
Goal: Set.Ioo a b ∪ Set.Ioi c = Set.Ioi (min a c)
-- Set.Ioo_union_Ioi
theorem Ioo_union_Ioi (h : c < max a b) : Ioo a b ∪ Ioi c = Ioi (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h : c < max a b)
Goal: Set.Ioo a b ∪ Set.Ioi c = Set.Ioi (min a c)
-- Set.Ioi_subset_Ioo_union_Ici
theorem Ioi_subset_Ioo_union_Ici : Ioi a ⊆ Ioo a b ∪ Ici b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a ⊆ Set.Ioo a b ∪ Set.Ici b
-- Set.Ioo_union_Ici_eq_Ioi
@[simp]
theorem Ioo_union_Ici_eq_Ioi (h : a < b) : Ioo a b ∪ Ici b = Ioi a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a < b)
Goal: Set.Ioo a b ∪ Set.Ici b = Set.Ioi a
-- Set.Ici_subset_Ico_union_Ici
theorem Ici_subset_Ico_union_Ici : Ici a ⊆ Ico a b ∪ Ici b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ici a ⊆ Set.Ico a b ∪ Set.Ici b
-- Set.Ico_union_Ici_eq_Ici
@[simp]
theorem Ico_union_Ici_eq_Ici (h : a ≤ b) : Ico a b ∪ Ici b = Ici a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Ico a b ∪ Set.Ici b = Set.Ici a
-- Set.Ico_union_Ici'
theorem Ico_union_Ici' (h₁ : c ≤ b) : Ico a b ∪ Ici c = Ici (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : c ≤ b)
Goal: Set.Ico a b ∪ Set.Ici c = Set.Ici (min a c)
-- Set.Ico_union_Ici
theorem Ico_union_Ici (h : c ≤ max a b) : Ico a b ∪ Ici c = Ici (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h : c ≤ max a b)
Goal: Set.Ico a b ∪ Set.Ici c = Set.Ici (min a c)
-- Set.Ioi_subset_Ioc_union_Ioi
theorem Ioi_subset_Ioc_union_Ioi : Ioi a ⊆ Ioc a b ∪ Ioi b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a ⊆ Set.Ioc a b ∪ Set.Ioi b
-- Set.Ioc_union_Ioi_eq_Ioi
@[simp]
theorem Ioc_union_Ioi_eq_Ioi (h : a ≤ b) : Ioc a b ∪ Ioi b = Ioi a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Ioc a b ∪ Set.Ioi b = Set.Ioi a
-- Set.Ioc_union_Ioi'
theorem Ioc_union_Ioi' (h₁ : c ≤ b) : Ioc a b ∪ Ioi c = Ioi (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : c ≤ b)
Goal: Set.Ioc a b ∪ Set.Ioi c = Set.Ioi (min a c)
-- Set.Ioc_union_Ioi
theorem Ioc_union_Ioi (h : c ≤ max a b) : Ioc a b ∪ Ioi c = Ioi (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h : c ≤ max a b)
Goal: Set.Ioc a b ∪ Set.Ioi c = Set.Ioi (min a c)
-- Set.Ici_subset_Icc_union_Ioi
theorem Ici_subset_Icc_union_Ioi : Ici a ⊆ Icc a b ∪ Ioi b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ici a ⊆ Set.Icc a b ∪ Set.Ioi b
-- Set.Icc_union_Ioi_eq_Ici
@[simp]
theorem Icc_union_Ioi_eq_Ici (h : a ≤ b) : Icc a b ∪ Ioi b = Ici a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Icc a b ∪ Set.Ioi b = Set.Ici a
-- Set.Ioi_subset_Ioc_union_Ici
theorem Ioi_subset_Ioc_union_Ici : Ioi a ⊆ Ioc a b ∪ Ici b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a ⊆ Set.Ioc a b ∪ Set.Ici b
-- Set.Ioc_union_Ici_eq_Ioi
@[simp]
theorem Ioc_union_Ici_eq_Ioi (h : a < b) : Ioc a b ∪ Ici b = Ioi a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a < b)
Goal: Set.Ioc a b ∪ Set.Ici b = Set.Ioi a
-- Set.Ici_subset_Icc_union_Ici
theorem Ici_subset_Icc_union_Ici : Ici a ⊆ Icc a b ∪ Ici b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ici a ⊆ Set.Icc a b ∪ Set.Ici b
-- Set.Icc_union_Ici_eq_Ici
@[simp]
theorem Icc_union_Ici_eq_Ici (h : a ≤ b) : Icc a b ∪ Ici b = Ici a 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Icc a b ∪ Set.Ici b = Set.Ici a
-- Set.Icc_union_Ici'
theorem Icc_union_Ici' (h₁ : c ≤ b) : Icc a b ∪ Ici c = Ici (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : c ≤ b)
Goal: Set.Icc a b ∪ Set.Ici c = Set.Ici (min a c)
-- Set.Icc_union_Ici
theorem Icc_union_Ici (h : c ≤ max a b) : Icc a b ∪ Ici c = Ici (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h : c ≤ max a b)
Goal: Set.Icc a b ∪ Set.Ici c = Set.Ici (min a c)
-- Set.Iic_subset_Iio_union_Icc
theorem Iic_subset_Iio_union_Icc : 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_union_Icc_eq_Iic
@[simp]
theorem Iio_union_Icc_eq_Iic (h : a ≤ b) : Iio a ∪ Icc a b = Iic b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iio a ∪ Set.Icc a b = Set.Iic b
-- Set.Iio_subset_Iio_union_Ico
theorem Iio_subset_Iio_union_Ico : 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.Iio_union_Ico_eq_Iio
@[simp]
theorem Iio_union_Ico_eq_Iio (h : a ≤ b) : Iio a ∪ Ico a b = Iio b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iio a ∪ Set.Ico a b = Set.Iio b
-- Set.Iio_union_Ico'
theorem Iio_union_Ico' (h₁ : c ≤ b) : Iio b ∪ Ico c d = Iio (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h₁ : c ≤ b)
Goal: Set.Iio b ∪ Set.Ico c d = Set.Iio (max b d)
-- Set.Iio_union_Ico
theorem Iio_union_Ico (h : min c d ≤ b) : Iio b ∪ Ico c d = Iio (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h : min c d ≤ b)
Goal: Set.Iio b ∪ Set.Ico c d = Set.Iio (max b d)
-- Set.Iic_subset_Iic_union_Ioc
theorem Iic_subset_Iic_union_Ioc : 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.Iic_union_Ioc_eq_Iic
@[simp]
theorem Iic_union_Ioc_eq_Iic (h : a ≤ b) : Iic a ∪ Ioc a b = Iic b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iic a ∪ Set.Ioc a b = Set.Iic b
-- Set.Iic_union_Ioc'
theorem Iic_union_Ioc' (h₁ : c < b) : Iic b ∪ Ioc c d = Iic (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h₁ : c < b)
Goal: Set.Iic b ∪ Set.Ioc c d = Set.Iic (max b d)
-- Set.Iic_union_Ioc
theorem Iic_union_Ioc (h : min c d < b) : Iic b ∪ Ioc c d = Iic (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h : min c d < b)
Goal: Set.Iic b ∪ Set.Ioc c d = Set.Iic (max b d)
-- Set.Iio_subset_Iic_union_Ioo
theorem Iio_subset_Iic_union_Ioo : 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_union_Ioo_eq_Iio
@[simp]
theorem Iic_union_Ioo_eq_Iio (h : a < b) : Iic a ∪ Ioo a b = Iio b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a < b)
Goal: Set.Iic a ∪ Set.Ioo a b = Set.Iio b
-- Set.Iio_union_Ioo'
theorem Iio_union_Ioo' (h₁ : c < b) : Iio b ∪ Ioo c d = Iio (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h₁ : c < b)
Goal: Set.Iio b ∪ Set.Ioo c d = Set.Iio (max b d)
-- Set.Iio_union_Ioo
theorem Iio_union_Ioo (h : min c d < b) : Iio b ∪ Ioo c d = Iio (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h : min c d < b)
Goal: Set.Iio b ∪ Set.Ioo c d = Set.Iio (max b d)
-- Set.Iic_subset_Iic_union_Icc
theorem Iic_subset_Iic_union_Icc : Iic b ⊆ Iic a ∪ Icc a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iic b ⊆ Set.Iic a ∪ Set.Icc a b
-- Set.Iic_union_Icc_eq_Iic
@[simp]
theorem Iic_union_Icc_eq_Iic (h : a ≤ b) : Iic a ∪ Icc a b = Iic b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.Iic a ∪ Set.Icc a b = Set.Iic b
-- Set.Iic_union_Icc'
theorem Iic_union_Icc' (h₁ : c ≤ b) : Iic b ∪ Icc c d = Iic (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h₁ : c ≤ b)
Goal: Set.Iic b ∪ Set.Icc c d = Set.Iic (max b d)
-- Set.Iic_union_Icc
theorem Iic_union_Icc (h : min c d ≤ b) : Iic b ∪ Icc c d = Iic (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {b c d : α} (h : min c d ≤ b)
Goal: Set.Iic b ∪ Set.Icc c d = Set.Iic (max b d)
-- Set.Iio_subset_Iic_union_Ico
theorem Iio_subset_Iic_union_Ico : Iio b ⊆ Iic a ∪ Ico a b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iio b ⊆ Set.Iic a ∪ Set.Ico a b
-- Set.Iic_union_Ico_eq_Iio
@[simp]
theorem Iic_union_Ico_eq_Iio (h : a < b) : Iic a ∪ Ico a b = Iio b 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a < b)
Goal: Set.Iic a ∪ Set.Ico a b = Set.Iio b
-- Set.Ioo_subset_Ioo_union_Ico
theorem Ioo_subset_Ioo_union_Ico : Ioo a c ⊆ Ioo a b ∪ Ico b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioo a c ⊆ Set.Ioo a b ∪ Set.Ico b c
-- Set.Ioo_union_Ico_eq_Ioo
@[simp]
theorem Ioo_union_Ico_eq_Ioo (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Ico b c = Ioo a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a < b) (h₂ : b ≤ c)
Goal: Set.Ioo a b ∪ Set.Ico b c = Set.Ioo a c
-- Set.Ico_subset_Ico_union_Ico
theorem Ico_subset_Ico_union_Ico : Ico a c ⊆ Ico a b ∪ Ico b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ico a c ⊆ Set.Ico a b ∪ Set.Ico b c
-- Set.Ico_union_Ico_eq_Ico
@[simp]
theorem Ico_union_Ico_eq_Ico (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Ico b c = Ico a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c)
Goal: Set.Ico a b ∪ Set.Ico b c = Set.Ico a c
-- Set.Ico_union_Ico'
theorem Ico_union_Ico' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ico a b ∪ Ico c d = Ico (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : c ≤ b) (h₂ : a ≤ d)
Goal: Set.Ico a b ∪ Set.Ico c d = Set.Ico (min a c) (max b d)
-- Set.Ico_union_Ico
theorem Ico_union_Ico (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_1} [LinearOrder α] {a b c d : α} (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b)
Goal: Set.Ico a b ∪ Set.Ico c d = Set.Ico (min a c) (max b d)
-- Set.Icc_subset_Ico_union_Icc
theorem Icc_subset_Ico_union_Icc : Icc a c ⊆ Ico a b ∪ Icc b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Icc a c ⊆ Set.Ico a b ∪ Set.Icc b c
-- Set.Ico_union_Icc_eq_Icc
@[simp]
theorem Ico_union_Icc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Ico a b ∪ Icc b c = Icc a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c)
Goal: Set.Ico a b ∪ Set.Icc b c = Set.Icc a c
-- Set.Ioc_subset_Ioo_union_Icc
theorem Ioc_subset_Ioo_union_Icc : Ioc a c ⊆ Ioo a b ∪ Icc b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a c ⊆ Set.Ioo a b ∪ Set.Icc b c
-- Set.Ioo_union_Icc_eq_Ioc
@[simp]
theorem Ioo_union_Icc_eq_Ioc (h₁ : a < b) (h₂ : b ≤ c) : Ioo a b ∪ Icc b c = Ioc a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a < b) (h₂ : b ≤ c)
Goal: Set.Ioo a b ∪ Set.Icc b c = Set.Ioc a c
-- Set.Ioo_subset_Ioc_union_Ioo
theorem Ioo_subset_Ioc_union_Ioo : Ioo a c ⊆ Ioc a b ∪ Ioo b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioo a c ⊆ Set.Ioc a b ∪ Set.Ioo b c
-- Set.Ioc_union_Ioo_eq_Ioo
@[simp]
theorem Ioc_union_Ioo_eq_Ioo (h₁ : a ≤ b) (h₂ : b < c) : Ioc a b ∪ Ioo b c = Ioo a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b < c)
Goal: Set.Ioc a b ∪ Set.Ioo b c = Set.Ioo a c
-- Set.Ico_subset_Icc_union_Ioo
theorem Ico_subset_Icc_union_Ioo : Ico a c ⊆ Icc a b ∪ Ioo b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ico a c ⊆ Set.Icc a b ∪ Set.Ioo b c
-- Set.Icc_union_Ioo_eq_Ico
@[simp]
theorem Icc_union_Ioo_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ioo b c = Ico a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b < c)
Goal: Set.Icc a b ∪ Set.Ioo b c = Set.Ico a c
-- Set.Icc_subset_Icc_union_Ioc
theorem Icc_subset_Icc_union_Ioc : Icc a c ⊆ Icc a b ∪ Ioc b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Icc a c ⊆ Set.Icc a b ∪ Set.Ioc b c
-- Set.Icc_union_Ioc_eq_Icc
@[simp]
theorem Icc_union_Ioc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Ioc b c = Icc a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c)
Goal: Set.Icc a b ∪ Set.Ioc b c = Set.Icc a c
-- Set.Ioc_subset_Ioc_union_Ioc
theorem Ioc_subset_Ioc_union_Ioc : Ioc a c ⊆ Ioc a b ∪ Ioc b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a c ⊆ Set.Ioc a b ∪ Set.Ioc b c
-- Set.Ioc_union_Ioc_eq_Ioc
@[simp]
theorem Ioc_union_Ioc_eq_Ioc (h₁ : a ≤ b) (h₂ : b ≤ c) : Ioc a b ∪ Ioc b c = Ioc a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c)
Goal: Set.Ioc a b ∪ Set.Ioc b c = Set.Ioc a c
-- Set.Ioc_union_Ioc'
theorem Ioc_union_Ioc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : c ≤ b) (h₂ : a ≤ d)
Goal: Set.Ioc a b ∪ Set.Ioc c d = Set.Ioc (min a c) (max b d)
-- Set.Ioc_union_Ioc
theorem Ioc_union_Ioc (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b) :
    Ioc a b ∪ Ioc c d = Ioc (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : min a b ≤ max c d) (h₂ : min c d ≤ max a b)
Goal: Set.Ioc a b ∪ Set.Ioc c d = Set.Ioc (min a c) (max b d)
-- Set.Ioo_subset_Ioc_union_Ico
theorem Ioo_subset_Ioc_union_Ico : Ioo a c ⊆ Ioc a b ∪ Ico b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioo a c ⊆ Set.Ioc a b ∪ Set.Ico b c
-- Set.Ioc_union_Ico_eq_Ioo
@[simp]
theorem Ioc_union_Ico_eq_Ioo (h₁ : a < b) (h₂ : b < c) : Ioc a b ∪ Ico b c = Ioo a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a < b) (h₂ : b < c)
Goal: Set.Ioc a b ∪ Set.Ico b c = Set.Ioo a c
-- Set.Ico_subset_Icc_union_Ico
theorem Ico_subset_Icc_union_Ico : Ico a c ⊆ Icc a b ∪ Ico b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ico a c ⊆ Set.Icc a b ∪ Set.Ico b c
-- Set.Icc_union_Ico_eq_Ico
@[simp]
theorem Icc_union_Ico_eq_Ico (h₁ : a ≤ b) (h₂ : b < c) : Icc a b ∪ Ico b c = Ico a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b < c)
Goal: Set.Icc a b ∪ Set.Ico b c = Set.Ico a c
-- Set.Icc_subset_Icc_union_Icc
theorem Icc_subset_Icc_union_Icc : Icc a c ⊆ Icc a b ∪ Icc b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Icc a c ⊆ Set.Icc a b ∪ Set.Icc b c
-- Set.Icc_union_Icc_eq_Icc
@[simp]
theorem Icc_union_Icc_eq_Icc (h₁ : a ≤ b) (h₂ : b ≤ c) : Icc a b ∪ Icc b c = Icc a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a ≤ b) (h₂ : b ≤ c)
Goal: Set.Icc a b ∪ Set.Icc b c = Set.Icc a c
-- Set.Icc_union_Icc'
theorem Icc_union_Icc' (h₁ : c ≤ b) (h₂ : a ≤ d) : Icc a b ∪ Icc c d = Icc (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : c ≤ b) (h₂ : a ≤ d)
Goal: Set.Icc a b ∪ Set.Icc c d = Set.Icc (min a c) (max b d)
-- Set.Icc_union_Icc
theorem Icc_union_Icc (h₁ : min a b < max c d) (h₂ : min c d < max a b) :
    Icc a b ∪ Icc c d = Icc (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : min a b < max c d) (h₂ : min c d < max a b)
Goal: Set.Icc a b ∪ Set.Icc c d = Set.Icc (min a c) (max b d)
-- Set.Ioc_subset_Ioc_union_Icc
theorem Ioc_subset_Ioc_union_Icc : Ioc a c ⊆ Ioc a b ∪ Icc b c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a c ⊆ Set.Ioc a b ∪ Set.Icc b c
-- Set.Ioc_union_Icc_eq_Ioc
@[simp]
theorem Ioc_union_Icc_eq_Ioc (h₁ : a < b) (h₂ : b ≤ c) : Ioc a b ∪ Icc b c = Ioc a c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α} (h₁ : a < b) (h₂ : b ≤ c)
Goal: Set.Ioc a b ∪ Set.Icc b c = Set.Ioc a c
-- Set.Ioo_union_Ioo'
theorem Ioo_union_Ioo' (h₁ : c < b) (h₂ : a < d) : Ioo a b ∪ Ioo c d = Ioo (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : c < b) (h₂ : a < d)
Goal: Set.Ioo a b ∪ Set.Ioo c d = Set.Ioo (min a c) (max b d)
-- Set.Ioo_union_Ioo
theorem Ioo_union_Ioo (h₁ : min a b < max c d) (h₂ : min c d < max a b) :
    Ioo a b ∪ Ioo c d = Ioo (min a c) (max b d) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c d : α} (h₁ : min a b < max c d) (h₂ : min c d < max a b)
Goal: Set.Ioo a b ∪ Set.Ioo c d = Set.Ioo (min a c) (max b d)
-- Set.Iic_inter_Iic
@[simp]
theorem Iic_inter_Iic {a b : α} : Iic a ∩ Iic b = Iic (a ⊓ b) 
Arguments: {α : Type u_1} [SemilatticeInf α] {a b : α}
Goal: Set.Iic a ∩ Set.Iic b = Set.Iic (a ⊓ b)
-- Set.Ioc_inter_Iic
@[simp]
theorem Ioc_inter_Iic (a b c : α) : Ioc a b ∩ Iic c = Ioc a (b ⊓ c) 
Arguments: {α : Type u_1} [SemilatticeInf α] (a b c : α)
Goal: Set.Ioc a b ∩ Set.Iic c = Set.Ioc a (b ⊓ c)
-- Set.Ici_inter_Ici
@[simp]
theorem Ici_inter_Ici {a b : α} : Ici a ∩ Ici b = Ici (a ⊔ b) 
Arguments: {α : Type u_1} [SemilatticeSup α] {a b : α}
Goal: Set.Ici a ∩ Set.Ici b = Set.Ici (a ⊔ b)
-- Set.Ico_inter_Ici
@[simp]
theorem Ico_inter_Ici (a b c : α) : Ico a b ∩ Ici c = Ico (a ⊔ c) b 
Arguments: {α : Type u_1} [SemilatticeSup α] (a b c : α)
Goal: Set.Ico a b ∩ Set.Ici c = Set.Ico (a ⊔ c) b
-- Set.Icc_inter_Icc
theorem Icc_inter_Icc : Icc a₁ b₁ ∩ Icc a₂ b₂ = Icc (a₁ ⊔ a₂) (b₁ ⊓ b₂) 
Arguments: {α : Type u_1} [Lattice α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.Icc a₁ b₁ ∩ Set.Icc a₂ b₂ = Set.Icc (a₁ ⊔ a₂) (b₁ ⊓ b₂)
-- Set.Icc_inter_Icc_eq_singleton
@[simp]
theorem Icc_inter_Icc_eq_singleton (hab : a ≤ b) (hbc : b ≤ c) : Icc a b ∩ Icc b c = {b} 
Arguments: {α : Type u_1} [Lattice α] {a b c : α} (hab : a ≤ b) (hbc : b ≤ c)
Goal: Set.Icc a b ∩ Set.Icc b c = {b}
-- Set.Ioi_inter_Ioi
@[simp]
theorem Ioi_inter_Ioi : Ioi a ∩ Ioi b = Ioi (a ⊔ b) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioi a ∩ Set.Ioi b = Set.Ioi (a ⊔ b)
-- Set.Iio_inter_Iio
@[simp]
theorem Iio_inter_Iio : Iio a ∩ Iio b = Iio (a ⊓ b) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Iio a ∩ Set.Iio b = Set.Iio (a ⊓ b)
-- Set.Ico_inter_Ico
theorem Ico_inter_Ico : Ico a₁ b₁ ∩ Ico a₂ b₂ = Ico (a₁ ⊔ a₂) (b₁ ⊓ b₂) 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.Ico a₁ b₁ ∩ Set.Ico a₂ b₂ = Set.Ico (a₁ ⊔ a₂) (b₁ ⊓ b₂)
-- Set.Ioc_inter_Ioc
theorem Ioc_inter_Ioc : Ioc a₁ b₁ ∩ Ioc a₂ b₂ = Ioc (a₁ ⊔ a₂) (b₁ ⊓ b₂) 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.Ioc a₁ b₁ ∩ Set.Ioc a₂ b₂ = Set.Ioc (a₁ ⊔ a₂) (b₁ ⊓ b₂)
-- Set.Ioo_inter_Ioo
theorem Ioo_inter_Ioo : Ioo a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (a₁ ⊔ a₂) (b₁ ⊓ b₂) 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α}
Goal: Set.Ioo a₁ b₁ ∩ Set.Ioo a₂ b₂ = Set.Ioo (a₁ ⊔ a₂) (b₁ ⊓ b₂)
-- Set.Ioo_inter_Iio
theorem Ioo_inter_Iio : Ioo a b ∩ Iio c = Ioo a (min b c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioo a b ∩ Set.Iio c = Set.Ioo a (min b c)
-- Set.Iio_inter_Ioo
theorem Iio_inter_Ioo : Iio a ∩ Ioo b c = Ioo b (min a c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Iio a ∩ Set.Ioo b c = Set.Ioo b (min a c)
-- Set.Ioo_inter_Ioi
theorem Ioo_inter_Ioi : Ioo a b ∩ Ioi c = Ioo (max a c) b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioo a b ∩ Set.Ioi c = Set.Ioo (max a c) b
-- Set.Ioi_inter_Ioo
theorem Ioi_inter_Ioo : Set.Ioi a ∩ Set.Ioo b c = Set.Ioo (max a b) c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioi a ∩ Set.Ioo b c = Set.Ioo (max a b) c
-- Set.Ioc_inter_Ioo_of_left_lt
theorem Ioc_inter_Ioo_of_left_lt (h : b₁ < b₂) : Ioc a₁ b₁ ∩ Ioo a₂ b₂ = Ioc (max a₁ a₂) b₁ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h : b₁ < b₂)
Goal: Set.Ioc a₁ b₁ ∩ Set.Ioo a₂ b₂ = Set.Ioc (max a₁ a₂) b₁
-- Set.Ioc_inter_Ioo_of_right_le
theorem Ioc_inter_Ioo_of_right_le (h : b₂ ≤ b₁) : Ioc a₁ b₁ ∩ Ioo a₂ b₂ = Ioo (max a₁ a₂) b₂ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h : b₂ ≤ b₁)
Goal: Set.Ioc a₁ b₁ ∩ Set.Ioo a₂ b₂ = Set.Ioo (max a₁ a₂) b₂
-- Set.Ioo_inter_Ioc_of_left_le
theorem Ioo_inter_Ioc_of_left_le (h : b₁ ≤ b₂) : Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioo (max a₁ a₂) b₁ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Set.Ioo a₁ b₁ ∩ Set.Ioc a₂ b₂ = Set.Ioo (max a₁ a₂) b₁
-- Set.Ioo_inter_Ioc_of_right_lt
theorem Ioo_inter_Ioc_of_right_lt (h : b₂ < b₁) : Ioo a₁ b₁ ∩ Ioc a₂ b₂ = Ioc (max a₁ a₂) b₂ 
Arguments: {α : Type u_1} [LinearOrder α] {a₁ a₂ b₁ b₂ : α} (h : b₂ < b₁)
Goal: Set.Ioo a₁ b₁ ∩ Set.Ioc a₂ b₂ = Set.Ioc (max a₁ a₂) b₂
-- Set.Ico_diff_Iio
@[simp]
theorem Ico_diff_Iio : Ico a b \ Iio c = Ico (max a c) b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ico a b \ Set.Iio c = Set.Ico (max a c) b
-- Set.Ioc_diff_Ioi
@[simp]
theorem Ioc_diff_Ioi : Ioc a b \ Ioi c = Ioc a (min b c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a b \ Set.Ioi c = Set.Ioc a (min b c)
-- Set.Ioc_inter_Ioi
@[simp]
theorem Ioc_inter_Ioi : Ioc a b ∩ Ioi c = Ioc (a ⊔ c) b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a b ∩ Set.Ioi c = Set.Ioc (a ⊔ c) b
-- Set.Ico_inter_Iio
@[simp]
theorem Ico_inter_Iio : Ico a b ∩ Iio c = Ico a (min b c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ico a b ∩ Set.Iio c = Set.Ico a (min b c)
-- Set.Ioc_diff_Iic
@[simp]
theorem Ioc_diff_Iic : Ioc a b \ Iic c = Ioc (max a c) b 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a b \ Set.Iic c = Set.Ioc (max a c) b
-- Set.Ioc_union_Ioc_right
@[simp]
theorem Ioc_union_Ioc_right : Ioc a b ∪ Ioc a c = Ioc a (max b c) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a b ∪ Set.Ioc a c = Set.Ioc a (max b c)
-- Set.Ioc_union_Ioc_left
@[simp]
theorem Ioc_union_Ioc_left : Ioc a c ∪ Ioc b c = Ioc (min a b) c 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a c ∪ Set.Ioc b c = Set.Ioc (min a b) c
-- Set.Ioc_union_Ioc_symm
@[simp]
theorem Ioc_union_Ioc_symm : Ioc a b ∪ Ioc b a = Ioc (min a b) (max a b) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α}
Goal: Set.Ioc a b ∪ Set.Ioc b a = Set.Ioc (min a b) (max a b)
-- Set.Ioc_union_Ioc_union_Ioc_cycle
@[simp]
theorem Ioc_union_Ioc_union_Ioc_cycle :
    Ioc a b ∪ Ioc b c ∪ Ioc c a = Ioc (min a (min b c)) (max a (max b c)) 
Arguments: {α : Type u_1} [LinearOrder α] {a b c : α}
Goal: Set.Ioc a b ∪ Set.Ioc b c ∪ Set.Ioc c a = Set.Ioc (min a (min b c)) (max a (max b c))
-- Set.Iic_prod_Iic
@[simp]
theorem Iic_prod_Iic (a : α) (b : β) : Iic a ×ˢ Iic b = Iic (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (a : α) (b : β)
Goal: Set.Iic a ×ˢ Set.Iic b = Set.Iic (a, b)
-- Set.Ici_prod_Ici
@[simp]
theorem Ici_prod_Ici (a : α) (b : β) : Ici a ×ˢ Ici b = Ici (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (a : α) (b : β)
Goal: Set.Ici a ×ˢ Set.Ici b = Set.Ici (a, b)
-- Set.Ici_prod_eq
theorem Ici_prod_eq (a : α × β) : Ici a = Ici a.1 ×ˢ Ici a.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (a : α × β)
Goal: Set.Ici a = Set.Ici a.1 ×ˢ Set.Ici a.2
-- Set.Iic_prod_eq
theorem Iic_prod_eq (a : α × β) : Iic a = Iic a.1 ×ˢ Iic a.2 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (a : α × β)
Goal: Set.Iic a = Set.Iic a.1 ×ˢ Set.Iic a.2