ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.image_subtype_val_Iic_Iio
@[simp]
lemma image_subtype_val_Iic_Iio {a : α} (b : Iic a) : Subtype.val '' Iio b = Iio b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iic a))
Goal: Subtype.val '' Set.Iio b = Set.Iio ↑b
-- Set.image_subtype_val_Ioi_Ici
@[simp]
lemma image_subtype_val_Ioi_Ici {a : α} (b : Ioi a) : Subtype.val '' Ici b = Ici b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ioi a))
Goal: Subtype.val '' Set.Ici b = Set.Ici ↑b
-- Set.image_subtype_val_Ioi_Iic
@[simp]
lemma image_subtype_val_Ioi_Iic {a : α} (b : Ioi a) : Subtype.val '' Iic b = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ioi a))
Goal: Subtype.val '' Set.Iic b = Set.Ioc a ↑b
-- Set.image_subtype_val_Ioi_Ioi
@[simp]
lemma image_subtype_val_Ioi_Ioi {a : α} (b : Ioi a) : Subtype.val '' Ioi b = Ioi b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ioi a))
Goal: Subtype.val '' Set.Ioi b = Set.Ioi ↑b
-- Set.image_subtype_val_Ioi_Iio
@[simp]
lemma image_subtype_val_Ioi_Iio {a : α} (b : Ioi a) : Subtype.val '' Iio b = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Ioi a))
Goal: Subtype.val '' Set.Iio b = Set.Ioo a ↑b
-- Set.image_subtype_val_Iio_Ici
@[simp]
lemma image_subtype_val_Iio_Ici {a : α} (b : Iio a) : Subtype.val '' Ici b = Ico b.1 a 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iio a))
Goal: Subtype.val '' Set.Ici b = Set.Ico (↑b) a
-- Set.image_subtype_val_Iio_Iic
@[simp]
lemma image_subtype_val_Iio_Iic {a : α} (b : Iio a) : Subtype.val '' Iic b = Iic b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iio a))
Goal: Subtype.val '' Set.Iic b = Set.Iic ↑b
-- Set.image_subtype_val_Iio_Ioi
@[simp]
lemma image_subtype_val_Iio_Ioi {a : α} (b : Iio a) : Subtype.val '' Ioi b = Ioo b.1 a 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iio a))
Goal: Subtype.val '' Set.Ioi b = Set.Ioo (↑b) a
-- Set.image_subtype_val_Iio_Iio
@[simp]
lemma image_subtype_val_Iio_Iio {a : α} (b : Iio a) : Subtype.val '' Iio b = Iio b.1 
Arguments: {α : Type u_1} [Preorder α] {a : α} (b : ↑(Set.Iio a))
Goal: Subtype.val '' Set.Iio b = Set.Iio ↑b
-- Set.image_subtype_val_Icc_Ici
@[simp]
lemma image_subtype_val_Icc_Ici {a b : α} (c : Icc a b) : Subtype.val '' Ici c = Icc c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Icc a b))
Goal: Subtype.val '' Set.Ici c = Set.Icc (↑c) b
-- Set.image_subtype_val_Icc_Iic
@[simp]
lemma image_subtype_val_Icc_Iic {a b : α} (c : Icc a b) : Subtype.val '' Iic c = Icc a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Icc a b))
Goal: Subtype.val '' Set.Iic c = Set.Icc a ↑c
-- Set.image_subtype_val_Icc_Ioi
@[simp]
lemma image_subtype_val_Icc_Ioi {a b : α} (c : Icc a b) : Subtype.val '' Ioi c = Ioc c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Icc a b))
Goal: Subtype.val '' Set.Ioi c = Set.Ioc (↑c) b
-- Set.image_subtype_val_Icc_Iio
@[simp]
lemma image_subtype_val_Icc_Iio {a b : α} (c : Icc a b) : Subtype.val '' Iio c = Ico a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Icc a b))
Goal: Subtype.val '' Set.Iio c = Set.Ico a ↑c
-- Set.image_subtype_val_Ico_Ici
@[simp]
lemma image_subtype_val_Ico_Ici {a b : α} (c : Ico a b) : Subtype.val '' Ici c = Ico c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ico a b))
Goal: Subtype.val '' Set.Ici c = Set.Ico (↑c) b
-- Set.image_subtype_val_Ico_Iic
@[simp]
lemma image_subtype_val_Ico_Iic {a b : α} (c : Ico a b) : Subtype.val '' Iic c = Icc a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ico a b))
Goal: Subtype.val '' Set.Iic c = Set.Icc a ↑c
-- Set.image_subtype_val_Ico_Ioi
@[simp]
lemma image_subtype_val_Ico_Ioi {a b : α} (c : Ico a b) : Subtype.val '' Ioi c = Ioo c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ico a b))
Goal: Subtype.val '' Set.Ioi c = Set.Ioo (↑c) b
-- Set.image_subtype_val_Ico_Iio
@[simp]
lemma image_subtype_val_Ico_Iio {a b : α} (c : Ico a b) : Subtype.val '' Iio c = Ico a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ico a b))
Goal: Subtype.val '' Set.Iio c = Set.Ico a ↑c
-- Set.image_subtype_val_Ioc_Ici
@[simp]
lemma image_subtype_val_Ioc_Ici {a b : α} (c : Ioc a b) : Subtype.val '' Ici c = Icc c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioc a b))
Goal: Subtype.val '' Set.Ici c = Set.Icc (↑c) b
-- Set.image_subtype_val_Ioc_Iic
@[simp]
lemma image_subtype_val_Ioc_Iic {a b : α} (c : Ioc a b) : Subtype.val '' Iic c = Ioc a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioc a b))
Goal: Subtype.val '' Set.Iic c = Set.Ioc a ↑c
-- Set.image_subtype_val_Ioc_Ioi
@[simp]
lemma image_subtype_val_Ioc_Ioi {a b : α} (c : Ioc a b) : Subtype.val '' Ioi c = Ioc c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioc a b))
Goal: Subtype.val '' Set.Ioi c = Set.Ioc (↑c) b
-- Set.image_subtype_val_Ioc_Iio
@[simp]
lemma image_subtype_val_Ioc_Iio {a b : α} (c : Ioc a b) : Subtype.val '' Iio c = Ioo a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioc a b))
Goal: Subtype.val '' Set.Iio c = Set.Ioo a ↑c
-- Set.image_subtype_val_Ioo_Ici
@[simp]
lemma image_subtype_val_Ioo_Ici {a b : α} (c : Ioo a b) : Subtype.val '' Ici c = Ico c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioo a b))
Goal: Subtype.val '' Set.Ici c = Set.Ico (↑c) b
-- Set.image_subtype_val_Ioo_Iic
@[simp]
lemma image_subtype_val_Ioo_Iic {a b : α} (c : Ioo a b) : Subtype.val '' Iic c = Ioc a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioo a b))
Goal: Subtype.val '' Set.Iic c = Set.Ioc a ↑c
-- Set.image_subtype_val_Ioo_Ioi
@[simp]
lemma image_subtype_val_Ioo_Ioi {a b : α} (c : Ioo a b) : Subtype.val '' Ioi c = Ioo c.1 b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioo a b))
Goal: Subtype.val '' Set.Ioi c = Set.Ioo (↑c) b
-- Set.image_subtype_val_Ioo_Iio
@[simp]
lemma image_subtype_val_Ioo_Iio {a b : α} (c : Ioo a b) : Subtype.val '' Iio c = Ioo a c 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (c : ↑(Set.Ioo a b))
Goal: Subtype.val '' Set.Iio c = Set.Ioo a ↑c
-- directedOn_le_Iic
lemma directedOn_le_Iic (b : α) : DirectedOn (· ≤ ·) (Iic b) 
Arguments: {α : Type u_1} [Preorder α] (b : α)
Goal: DirectedOn (fun x x_1 => x ≤ x_1) (Set.Iic b)
-- directedOn_le_Icc
lemma directedOn_le_Icc (a b : α) : DirectedOn (· ≤ ·) (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: DirectedOn (fun x x_1 => x ≤ x_1) (Set.Icc a b)
-- directedOn_le_Ioc
lemma directedOn_le_Ioc (a b : α) : DirectedOn (· ≤ ·) (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: DirectedOn (fun x x_1 => x ≤ x_1) (Set.Ioc a b)
-- directedOn_ge_Ici
lemma directedOn_ge_Ici (a : α) : DirectedOn (· ≥ ·) (Ici a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: DirectedOn (fun x x_1 => x ≥ x_1) (Set.Ici a)
-- directedOn_ge_Icc
lemma directedOn_ge_Icc (a b : α) : DirectedOn (· ≥ ·) (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: DirectedOn (fun x x_1 => x ≥ x_1) (Set.Icc a b)
-- directedOn_ge_Ico
lemma directedOn_ge_Ico (a b : α) : DirectedOn (· ≥ ·) (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: DirectedOn (fun x x_1 => x ≥ x_1) (Set.Ico a b)
-- Set.coe_projIci
@[norm_cast]
theorem coe_projIci (a x : α) : (projIci a x : α) = max a x 
Arguments: {α : Type u_1} [LinearOrder α] (a x : α)
Goal: ↑(Set.projIci a x) = max a x
-- Set.coe_projIic
@[norm_cast]
theorem coe_projIic (b x : α) : (projIic b x : α) = min b x 
Arguments: {α : Type u_1} [LinearOrder α] (b x : α)
Goal: ↑(Set.projIic b x) = min b x
-- Set.coe_projIcc
@[norm_cast]
theorem coe_projIcc (a b : α) (h : a ≤ b) (x : α) : (projIcc a b h x : α) = max a (min b x) 
Arguments: {α : Type u_1} [LinearOrder α] (a b : α) (h : a ≤ b) (x : α)
Goal: ↑(Set.projIcc a b h x) = max a (min b x)
-- Set.projIci_of_le
theorem projIci_of_le (hx : x ≤ a) : projIci a x = ⟨a, le_rfl⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a x : α} (hx : x ≤ a)
Goal: Set.projIci a x = ⟨a, ⋯⟩
-- Set.projIic_of_le
theorem projIic_of_le (hx : b ≤ x) : projIic b x = ⟨b, le_rfl⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {b x : α} (hx : b ≤ x)
Goal: Set.projIic b x = ⟨b, ⋯⟩
-- Set.projIcc_of_le_left
theorem projIcc_of_le_left (hx : x ≤ a) : projIcc a b h x = ⟨a, left_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b) {x : α} (hx : x ≤ a)
Goal: Set.projIcc a b h x = ⟨a, ⋯⟩
-- Set.projIcc_of_right_le
theorem projIcc_of_right_le (hx : b ≤ x) : projIcc a b h x = ⟨b, right_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b) {x : α} (hx : b ≤ x)
Goal: Set.projIcc a b h x = ⟨b, ⋯⟩
-- Set.projIci_self
@[simp]
theorem projIci_self (a : α) : projIci a a = ⟨a, le_rfl⟩ 
Arguments: {α : Type u_1} [LinearOrder α] (a : α)
Goal: Set.projIci a a = ⟨a, ⋯⟩
-- Set.projIic_self
@[simp]
theorem projIic_self (b : α) : projIic b b = ⟨b, le_rfl⟩ 
Arguments: {α : Type u_1} [LinearOrder α] (b : α)
Goal: Set.projIic b b = ⟨b, ⋯⟩
-- Set.projIcc_left
@[simp]
theorem projIcc_left : projIcc a b h a = ⟨a, left_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.projIcc a b h a = ⟨a, ⋯⟩
-- Set.projIcc_right
@[simp]
theorem projIcc_right : projIcc a b h b = ⟨b, right_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.projIcc a b h b = ⟨b, ⋯⟩
-- Set.projIci_eq_self
theorem projIci_eq_self : projIci a x = ⟨a, le_rfl⟩ ↔ x ≤ a 
Arguments: {α : Type u_1} [LinearOrder α] {a x : α}
Goal: Set.projIci a x = ⟨a, ⋯⟩ ↔ x ≤ a
-- Set.projIic_eq_self
theorem projIic_eq_self : projIic b x = ⟨b, le_rfl⟩ ↔ b ≤ x 
Arguments: {α : Type u_1} [LinearOrder α] {b x : α}
Goal: Set.projIic b x = ⟨b, ⋯⟩ ↔ b ≤ x
-- Set.projIcc_eq_left
theorem projIcc_eq_left (h : a < b) : projIcc a b h.le x = ⟨a, left_mem_Icc.mpr h.le⟩ ↔ x ≤ a 
Arguments: {α : Type u_1} [LinearOrder α] {a b x : α} (h : a < b)
Goal: Set.projIcc a b ⋯ x = ⟨a, ⋯⟩ ↔ x ≤ a
-- Set.projIcc_eq_right
theorem projIcc_eq_right (h : a < b) : projIcc a b h.le x = ⟨b, right_mem_Icc.2 h.le⟩ ↔ b ≤ x 
Arguments: {α : Type u_1} [LinearOrder α] {a b x : α} (h : a < b)
Goal: Set.projIcc a b ⋯ x = ⟨b, ⋯⟩ ↔ b ≤ x
-- Set.projIci_of_mem
theorem projIci_of_mem (hx : x ∈ Ici a) : projIci a x = ⟨x, hx⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a x : α} (hx : x ∈ Set.Ici a)
Goal: Set.projIci a x = ⟨x, hx⟩
-- Set.projIic_of_mem
theorem projIic_of_mem (hx : x ∈ Iic b) : projIic b x = ⟨x, hx⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {b x : α} (hx : x ∈ Set.Iic b)
Goal: Set.projIic b x = ⟨x, hx⟩
-- Set.projIcc_of_mem
theorem projIcc_of_mem (hx : x ∈ Icc a b) : projIcc a b h x = ⟨x, hx⟩ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b) {x : α} (hx : x ∈ Set.Icc a b)
Goal: Set.projIcc a b h x = ⟨x, hx⟩
-- Set.projIci_coe
@[simp]
theorem projIci_coe (x : Ici a) : projIci a x = x 
Arguments: {α : Type u_1} [LinearOrder α] {a : α} (x : ↑(Set.Ici a))
Goal: Set.projIci a ↑x = x
-- Set.projIic_coe
@[simp]
theorem projIic_coe (x : Iic b) : projIic b x = x 
Arguments: {α : Type u_1} [LinearOrder α] {b : α} (x : ↑(Set.Iic b))
Goal: Set.projIic b ↑x = x
-- Set.projIcc_val
@[simp]
theorem projIcc_val (x : Icc a b) : projIcc a b h x = x 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b) (x : ↑(Set.Icc a b))
Goal: Set.projIcc a b h ↑x = x
-- Set.projIci_surjOn
theorem projIci_surjOn : SurjOn (projIci a) (Ici a) univ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.SurjOn (Set.projIci a) (Set.Ici a) Set.univ
-- Set.projIic_surjOn
theorem projIic_surjOn : SurjOn (projIic b) (Iic b) univ 
Arguments: {α : Type u_1} [LinearOrder α] {b : α}
Goal: Set.SurjOn (Set.projIic b) (Set.Iic b) Set.univ
-- Set.projIcc_surjOn
theorem projIcc_surjOn : SurjOn (projIcc a b h) (Icc a b) univ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.SurjOn (Set.projIcc a b h) (Set.Icc a b) Set.univ
-- Set.projIci_surjective
theorem projIci_surjective : Surjective (projIci a) 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Function.Surjective (Set.projIci a)
-- Set.projIic_surjective
theorem projIic_surjective : Surjective (projIic b) 
Arguments: {α : Type u_1} [LinearOrder α] {b : α}
Goal: Function.Surjective (Set.projIic b)
-- Set.projIcc_surjective
theorem projIcc_surjective : Surjective (projIcc a b h) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Function.Surjective (Set.projIcc a b h)
-- Set.range_projIci
@[simp]
theorem range_projIci : range (projIci a) = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.range (Set.projIci a) = Set.univ
-- Set.range_projIic
@[simp]
theorem range_projIic : range (projIic a) = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Set.range (Set.projIic a) = Set.univ
-- Set.range_projIcc
@[simp]
theorem range_projIcc : range (projIcc a b h) = univ 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Set.range (Set.projIcc a b h) = Set.univ
-- Set.monotone_projIci
theorem monotone_projIci : Monotone (projIci a) 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Monotone (Set.projIci a)
-- Set.monotone_projIic
theorem monotone_projIic : Monotone (projIic a) 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: Monotone (Set.projIic a)
-- Set.monotone_projIcc
theorem monotone_projIcc : Monotone (projIcc a b h) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: Monotone (Set.projIcc a b h)
-- Set.strictMonoOn_projIci
theorem strictMonoOn_projIci : StrictMonoOn (projIci a) (Ici a) 
Arguments: {α : Type u_1} [LinearOrder α] {a : α}
Goal: StrictMonoOn (Set.projIci a) (Set.Ici a)
-- Set.strictMonoOn_projIic
theorem strictMonoOn_projIic : StrictMonoOn (projIic b) (Iic b) 
Arguments: {α : Type u_1} [LinearOrder α] {b : α}
Goal: StrictMonoOn (Set.projIic b) (Set.Iic b)
-- Set.strictMonoOn_projIcc
theorem strictMonoOn_projIcc : StrictMonoOn (projIcc a b h) (Icc a b) 
Arguments: {α : Type u_1} [LinearOrder α] {a b : α} (h : a ≤ b)
Goal: StrictMonoOn (Set.projIcc a b h) (Set.Icc a b)
-- Set.IciExtend_apply
theorem IciExtend_apply (f : Ici a → β) (x : α) : IciExtend f x = f ⟨max a x, le_max_left _ _⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a : α} (f : ↑(Set.Ici a) → β) (x : α)
Goal: Set.IciExtend f x = f ⟨max a x, ⋯⟩
-- Set.IicExtend_apply
theorem IicExtend_apply (f : Iic b → β) (x : α) : IicExtend f x = f ⟨min b x, min_le_left _ _⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {b : α} (f : ↑(Set.Iic b) → β) (x : α)
Goal: Set.IicExtend f x = f ⟨min b x, ⋯⟩
-- Set.IccExtend_apply
theorem IccExtend_apply (h : a ≤ b) (f : Icc a b → β) (x : α) :
    IccExtend h f x = f ⟨max a (min b x), le_max_left _ _, max_le h (min_le_left _ _)⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) (x : α)
Goal: Set.IccExtend h f x = f ⟨max a (min b x), ⋯⟩
-- Set.range_IciExtend
@[simp]
theorem range_IciExtend (f : Ici a → β) : range (IciExtend f) = range f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a : α} (f : ↑(Set.Ici a) → β)
Goal: Set.range (Set.IciExtend f) = Set.range f
-- Set.range_IicExtend
@[simp]
theorem range_IicExtend (f : Iic b → β) : range (IicExtend f) = range f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {b : α} (f : ↑(Set.Iic b) → β)
Goal: Set.range (Set.IicExtend f) = Set.range f
-- Set.IccExtend_range
@[simp]
theorem IccExtend_range (f : Icc a b → β) : range (IccExtend h f) = range f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β)
Goal: Set.range (Set.IccExtend h f) = Set.range f
-- Set.IciExtend_of_le
theorem IciExtend_of_le (f : Ici a → β) (hx : x ≤ a) : IciExtend f x = f ⟨a, le_rfl⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a x : α} (f : ↑(Set.Ici a) → β) (hx : x ≤ a)
Goal: Set.IciExtend f x = f ⟨a, ⋯⟩
-- Set.IicExtend_of_le
theorem IicExtend_of_le (f : Iic b → β) (hx : b ≤ x) : IicExtend f x = f ⟨b, le_rfl⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {b x : α} (f : ↑(Set.Iic b) → β) (hx : b ≤ x)
Goal: Set.IicExtend f x = f ⟨b, ⋯⟩
-- Set.IccExtend_of_le_left
theorem IccExtend_of_le_left (f : Icc a b → β) (hx : x ≤ a) :
    IccExtend h f x = f ⟨a, left_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) {x : α} (f : ↑(Set.Icc a b) → β) (hx : x ≤ a)
Goal: Set.IccExtend h f x = f ⟨a, ⋯⟩
-- Set.IccExtend_of_right_le
theorem IccExtend_of_right_le (f : Icc a b → β) (hx : b ≤ x) :
    IccExtend h f x = f ⟨b, right_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) {x : α} (f : ↑(Set.Icc a b) → β) (hx : b ≤ x)
Goal: Set.IccExtend h f x = f ⟨b, ⋯⟩
-- Set.IciExtend_self
@[simp]
theorem IciExtend_self (f : Ici a → β) : IciExtend f a = f ⟨a, le_rfl⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a : α} (f : ↑(Set.Ici a) → β)
Goal: Set.IciExtend f a = f ⟨a, ⋯⟩
-- Set.IicExtend_self
@[simp]
theorem IicExtend_self (f : Iic b → β) : IicExtend f b = f ⟨b, le_rfl⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {b : α} (f : ↑(Set.Iic b) → β)
Goal: Set.IicExtend f b = f ⟨b, ⋯⟩
-- Set.IccExtend_left
@[simp]
theorem IccExtend_left (f : Icc a b → β) : IccExtend h f a = f ⟨a, left_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β)
Goal: Set.IccExtend h f a = f ⟨a, ⋯⟩
-- Set.IccExtend_right
@[simp]
theorem IccExtend_right (f : Icc a b → β) : IccExtend h f b = f ⟨b, right_mem_Icc.2 h⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β)
Goal: Set.IccExtend h f b = f ⟨b, ⋯⟩
-- Set.IciExtend_of_mem
theorem IciExtend_of_mem (f : Ici a → β) (hx : x ∈ Ici a) : IciExtend f x = f ⟨x, hx⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a x : α} (f : ↑(Set.Ici a) → β) (hx : x ∈ Set.Ici a)
Goal: Set.IciExtend f x = f ⟨x, hx⟩
-- Set.IicExtend_of_mem
theorem IicExtend_of_mem (f : Iic b → β) (hx : x ∈ Iic b) : IicExtend f x = f ⟨x, hx⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {b x : α} (f : ↑(Set.Iic b) → β) (hx : x ∈ Set.Iic b)
Goal: Set.IicExtend f x = f ⟨x, hx⟩
-- Set.IccExtend_of_mem
theorem IccExtend_of_mem (f : Icc a b → β) (hx : x ∈ Icc a b) : IccExtend h f x = f ⟨x, hx⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) {x : α} (f : ↑(Set.Icc a b) → β) (hx : x ∈ Set.Icc a b)
Goal: Set.IccExtend h f x = f ⟨x, hx⟩
-- Set.IciExtend_coe
@[simp]
theorem IciExtend_coe (f : Ici a → β) (x : Ici a) : IciExtend f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a : α} (f : ↑(Set.Ici a) → β) (x : ↑(Set.Ici a))
Goal: Set.IciExtend f ↑x = f x
-- Set.IicExtend_coe
@[simp]
theorem IicExtend_coe (f : Iic b → β) (x : Iic b) : IicExtend f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {b : α} (f : ↑(Set.Iic b) → β) (x : ↑(Set.Iic b))
Goal: Set.IicExtend f ↑x = f x
-- Set.IccExtend_val
@[simp]
theorem IccExtend_val (f : Icc a b → β) (x : Icc a b) : IccExtend h f x = f x 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (f : ↑(Set.Icc a b) → β) (x : ↑(Set.Icc a b))
Goal: Set.IccExtend h f ↑x = f x
-- Set.IccExtend_eq_self
theorem IccExtend_eq_self (f : α → β) (ha : ∀ x < a, f x = f a) (hb : ∀ x, b < x → f x = f b) :
    IccExtend h (f ∘ (↑)) = f 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (f : α → β) (ha : ∀ x < a, f x = f a) (hb : ∀ (x : α), b < x → f x = f b)
Goal: Set.IccExtend h (f ∘ Subtype.val) = f
-- Monotone.IciExtend
protected theorem Monotone.IciExtend {f : Ici a → β} (hf : Monotone f) : Monotone (IciExtend f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a : α} {f : ↑(Set.Ici a) → β} (hf : Monotone f)
Goal: Monotone (Set.IciExtend f)
-- Monotone.IicExtend
protected theorem Monotone.IicExtend {f : Iic b → β} (hf : Monotone f) : Monotone (IicExtend f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {b : α} {f : ↑(Set.Iic b) → β} (hf : Monotone f)
Goal: Monotone (Set.IicExtend f)
-- Monotone.IccExtend
protected theorem Monotone.IccExtend (hf : Monotone f) : Monotone (IccExtend h f) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a b : α} (h : a ≤ b) {f : ↑(Set.Icc a b) → β} (hf : Monotone f)
Goal: Monotone (Set.IccExtend h f)
-- StrictMono.strictMonoOn_IciExtend
theorem StrictMono.strictMonoOn_IciExtend {f : Ici a → β} (hf : StrictMono f) :
    StrictMonoOn (IciExtend f) (Ici a) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a : α} {f : ↑(Set.Ici a) → β} (hf : StrictMono f)
Goal: StrictMonoOn (Set.IciExtend f) (Set.Ici a)
-- StrictMono.strictMonoOn_IicExtend
theorem StrictMono.strictMonoOn_IicExtend {f : Iic b → β} (hf : StrictMono f) :
    StrictMonoOn (IicExtend f) (Iic b) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {b : α} {f : ↑(Set.Iic b) → β} (hf : StrictMono f)
Goal: StrictMonoOn (Set.IicExtend f) (Set.Iic b)
-- StrictMono.strictMonoOn_IccExtend
theorem StrictMono.strictMonoOn_IccExtend (hf : StrictMono f) :
    StrictMonoOn (IccExtend h f) (Icc a b) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Preorder β] {a b : α} (h : a ≤ b) {f : ↑(Set.Icc a b) → β} (hf : StrictMono f)
Goal: StrictMonoOn (Set.IccExtend h f) (Set.Icc a b)
-- Set.OrdConnected.IciExtend
protected theorem Set.OrdConnected.IciExtend {s : Set (Ici a)} (hs : s.OrdConnected) :
    {x | IciExtend (· ∈ s) x}.OrdConnected 
Arguments: {α : Type u_1} [LinearOrder α] {a : α} {s : Set ↑(Set.Ici a)} (hs : s.OrdConnected)
Goal: {x | Set.IciExtend (fun x => x ∈ s) x}.OrdConnected
-- Set.OrdConnected.IicExtend
protected theorem Set.OrdConnected.IicExtend {s : Set (Iic b)} (hs : s.OrdConnected) :
    {x | IicExtend (· ∈ s) x}.OrdConnected 
Arguments: {α : Type u_1} [LinearOrder α] {b : α} {s : Set ↑(Set.Iic b)} (hs : s.OrdConnected)
Goal: {x | Set.IicExtend (fun x => x ∈ s) x}.OrdConnected
-- Set.OrdConnected.restrict
protected theorem Set.OrdConnected.restrict (hs : s.OrdConnected) :
    {x | restrict t (· ∈ s) x}.OrdConnected 
Arguments: {α : Type u_1} [LinearOrder α] {s t : Set α} (hs : s.OrdConnected)
Goal: {x | t.restrict (fun x => x ∈ s) x}.OrdConnected
-- Fin.coe_sup
@[simp, norm_cast]
theorem coe_sup : ↑(a ⊔ b) = (a ⊔ b : ℕ) 
Arguments: {n : ℕ} (a b : Fin n)
Goal: ↑(a ⊔ b) = ↑a ⊔ ↑b
-- Fin.coe_inf
@[simp, norm_cast]
theorem coe_inf : ↑(a ⊓ b) = (a ⊓ b : ℕ) 
Arguments: {n : ℕ} (a b : Fin n)
Goal: ↑(a ⊓ b) = ↑a ⊓ ↑b
-- Fin.coe_max
@[simp, norm_cast]
theorem coe_max : ↑(max a b) = (max a b : ℕ) 
Arguments: {n : ℕ} (a b : Fin n)
Goal: ↑(max a b) = max ↑a ↑b
-- Fin.coe_min
@[simp, norm_cast]
theorem coe_min : ↑(min a b) = (min a b : ℕ) 
Arguments: {n : ℕ} (a b : Fin n)
Goal: ↑(min a b) = min ↑a ↑b
-- Fin.Icc_eq_finset_subtype
theorem Icc_eq_finset_subtype : Icc a b = (Icc (a : ℕ) b).fin n 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.Icc a b = Finset.fin n (Finset.Icc ↑a ↑b)
-- Fin.Ico_eq_finset_subtype
theorem Ico_eq_finset_subtype : Ico a b = (Ico (a : ℕ) b).fin n 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.Ico a b = Finset.fin n (Finset.Ico ↑a ↑b)
-- Fin.Ioc_eq_finset_subtype
theorem Ioc_eq_finset_subtype : Ioc a b = (Ioc (a : ℕ) b).fin n 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.Ioc a b = Finset.fin n (Finset.Ioc ↑a ↑b)
-- Fin.Ioo_eq_finset_subtype
theorem Ioo_eq_finset_subtype : Ioo a b = (Ioo (a : ℕ) b).fin n 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.Ioo a b = Finset.fin n (Finset.Ioo ↑a ↑b)
-- Fin.uIcc_eq_finset_subtype
theorem uIcc_eq_finset_subtype : uIcc a b = (uIcc (a : ℕ) b).fin n 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.uIcc a b = Finset.fin n (Finset.uIcc ↑a ↑b)
-- Fin.map_valEmbedding_Icc
@[simp]
theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.map Fin.valEmbedding (Finset.Icc a b) = Finset.Icc ↑a ↑b
-- Fin.map_valEmbedding_Ico
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico ↑a ↑b 
Arguments: {n : ℕ} (a b : Fin n)
Goal: Finset.map Fin.valEmbedding (Finset.Ico a b) = Finset.Ico ↑a ↑b