ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsLeast.dual
theorem IsLeast.dual (h : IsLeast s a) : IsGreatest (ofDual ⁻¹' s) (toDual a) 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLeast s a)
Goal: IsGreatest (⇑OrderDual.ofDual ⁻¹' s) (OrderDual.toDual a)
-- IsGreatest.dual
theorem IsGreatest.dual (h : IsGreatest s a) : IsLeast (ofDual ⁻¹' s) (toDual a) 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGreatest s a)
Goal: IsLeast (⇑OrderDual.ofDual ⁻¹' s) (OrderDual.toDual a)
-- IsLUB.dual
theorem IsLUB.dual (h : IsLUB s a) : IsGLB (ofDual ⁻¹' s) (toDual a) 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLUB s a)
Goal: IsGLB (⇑OrderDual.ofDual ⁻¹' s) (OrderDual.toDual a)
-- IsGLB.dual
theorem IsGLB.dual (h : IsGLB s a) : IsLUB (ofDual ⁻¹' s) (toDual a) 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGLB s a)
Goal: IsLUB (⇑OrderDual.ofDual ⁻¹' s) (OrderDual.toDual a)
-- upperBounds_mono_set
theorem upperBounds_mono_set ⦃s t : Set α⦄ (hst : s ⊆ t) : upperBounds t ⊆ upperBounds s 
Arguments: {α : Type u} [Preorder α] ⦃s t : Set α⦄ (hst : s ⊆ t)
Goal: upperBounds t ⊆ upperBounds s
-- lowerBounds_mono_set
theorem lowerBounds_mono_set ⦃s t : Set α⦄ (hst : s ⊆ t) : lowerBounds t ⊆ lowerBounds s 
Arguments: {α : Type u} [Preorder α] ⦃s t : Set α⦄ (hst : s ⊆ t)
Goal: lowerBounds t ⊆ lowerBounds s
-- upperBounds_mono_mem
theorem upperBounds_mono_mem ⦃a b⦄ (hab : a ≤ b) : a ∈ upperBounds s → b ∈ upperBounds s 
Arguments: {α : Type u} [Preorder α] {s : Set α} ⦃a b : α⦄ (hab : a ≤ b)
Goal: a ∈ upperBounds s → b ∈ upperBounds s
-- lowerBounds_mono_mem
theorem lowerBounds_mono_mem ⦃a b⦄ (hab : a ≤ b) : b ∈ lowerBounds s → a ∈ lowerBounds s 
Arguments: {α : Type u} [Preorder α] {s : Set α} ⦃a b : α⦄ (hab : a ≤ b)
Goal: b ∈ lowerBounds s → a ∈ lowerBounds s
-- upperBounds_mono
theorem upperBounds_mono ⦃s t : Set α⦄ (hst : s ⊆ t) ⦃a b⦄ (hab : a ≤ b) :
    a ∈ upperBounds t → b ∈ upperBounds s 
Arguments: {α : Type u} [Preorder α] ⦃s t : Set α⦄ (hst : s ⊆ t) ⦃a b : α⦄ (hab : a ≤ b)
Goal: a ∈ upperBounds t → b ∈ upperBounds s
-- lowerBounds_mono
theorem lowerBounds_mono ⦃s t : Set α⦄ (hst : s ⊆ t) ⦃a b⦄ (hab : a ≤ b) :
    b ∈ lowerBounds t → a ∈ lowerBounds s 
Arguments: {α : Type u} [Preorder α] ⦃s t : Set α⦄ (hst : s ⊆ t) ⦃a b : α⦄ (hab : a ≤ b)
Goal: b ∈ lowerBounds t → a ∈ lowerBounds s
-- BddAbove.mono
theorem BddAbove.mono ⦃s t : Set α⦄ (h : s ⊆ t) : BddAbove t → BddAbove s 
Arguments: {α : Type u} [Preorder α] ⦃s t : Set α⦄ (h : s ⊆ t)
Goal: BddAbove t → BddAbove s
-- BddBelow.mono
theorem BddBelow.mono ⦃s t : Set α⦄ (h : s ⊆ t) : BddBelow t → BddBelow s 
Arguments: {α : Type u} [Preorder α] ⦃s t : Set α⦄ (h : s ⊆ t)
Goal: BddBelow t → BddBelow s
-- IsLUB.of_subset_of_superset
theorem IsLUB.of_subset_of_superset {s t p : Set α} (hs : IsLUB s a) (hp : IsLUB p a) (hst : s ⊆ t)
    (htp : t ⊆ p) : IsLUB t a 
Arguments: {α : Type u} [Preorder α] {a : α} {s t p : Set α} (hs : IsLUB s a) (hp : IsLUB p a) (hst : s ⊆ t) (htp : t ⊆ p)
Goal: IsLUB t a
-- IsGLB.of_subset_of_superset
theorem IsGLB.of_subset_of_superset {s t p : Set α} (hs : IsGLB s a) (hp : IsGLB p a) (hst : s ⊆ t)
    (htp : t ⊆ p) : IsGLB t a 
Arguments: {α : Type u} [Preorder α] {a : α} {s t p : Set α} (hs : IsGLB s a) (hp : IsGLB p a) (hst : s ⊆ t) (htp : t ⊆ p)
Goal: IsGLB t a
-- IsLeast.mono
theorem IsLeast.mono (ha : IsLeast s a) (hb : IsLeast t b) (hst : s ⊆ t) : b ≤ a 
Arguments: {α : Type u} [Preorder α] {s t : Set α} {a b : α} (ha : IsLeast s a) (hb : IsLeast t b) (hst : s ⊆ t)
Goal: b ≤ a
-- IsGreatest.mono
theorem IsGreatest.mono (ha : IsGreatest s a) (hb : IsGreatest t b) (hst : s ⊆ t) : a ≤ b 
Arguments: {α : Type u} [Preorder α] {s t : Set α} {a b : α} (ha : IsGreatest s a) (hb : IsGreatest t b) (hst : s ⊆ t)
Goal: a ≤ b
-- IsLUB.mono
theorem IsLUB.mono (ha : IsLUB s a) (hb : IsLUB t b) (hst : s ⊆ t) : a ≤ b 
Arguments: {α : Type u} [Preorder α] {s t : Set α} {a b : α} (ha : IsLUB s a) (hb : IsLUB t b) (hst : s ⊆ t)
Goal: a ≤ b
-- IsGLB.mono
theorem IsGLB.mono (ha : IsGLB s a) (hb : IsGLB t b) (hst : s ⊆ t) : b ≤ a 
Arguments: {α : Type u} [Preorder α] {s t : Set α} {a b : α} (ha : IsGLB s a) (hb : IsGLB t b) (hst : s ⊆ t)
Goal: b ≤ a
-- subset_lowerBounds_upperBounds
theorem subset_lowerBounds_upperBounds (s : Set α) : s ⊆ lowerBounds (upperBounds s) 
Arguments: {α : Type u} [Preorder α] (s : Set α)
Goal: s ⊆ lowerBounds (upperBounds s)
-- subset_upperBounds_lowerBounds
theorem subset_upperBounds_lowerBounds (s : Set α) : s ⊆ upperBounds (lowerBounds s) 
Arguments: {α : Type u} [Preorder α] (s : Set α)
Goal: s ⊆ upperBounds (lowerBounds s)
-- Set.Nonempty.bddAbove_lowerBounds
theorem Set.Nonempty.bddAbove_lowerBounds (hs : s.Nonempty) : BddAbove (lowerBounds s) 
Arguments: {α : Type u} [Preorder α] {s : Set α} (hs : s.Nonempty)
Goal: BddAbove (lowerBounds s)
-- Set.Nonempty.bddBelow_upperBounds
theorem Set.Nonempty.bddBelow_upperBounds (hs : s.Nonempty) : BddBelow (upperBounds s) 
Arguments: {α : Type u} [Preorder α] {s : Set α} (hs : s.Nonempty)
Goal: BddBelow (upperBounds s)
-- IsLeast.isGLB
theorem IsLeast.isGLB (h : IsLeast s a) : IsGLB s a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLeast s a)
Goal: IsGLB s a
-- IsGreatest.isLUB
theorem IsGreatest.isLUB (h : IsGreatest s a) : IsLUB s a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGreatest s a)
Goal: IsLUB s a
-- IsLUB.upperBounds_eq
theorem IsLUB.upperBounds_eq (h : IsLUB s a) : upperBounds s = Ici a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLUB s a)
Goal: upperBounds s = Set.Ici a
-- IsGLB.lowerBounds_eq
theorem IsGLB.lowerBounds_eq (h : IsGLB s a) : lowerBounds s = Iic a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGLB s a)
Goal: lowerBounds s = Set.Iic a
-- IsLeast.lowerBounds_eq
theorem IsLeast.lowerBounds_eq (h : IsLeast s a) : lowerBounds s = Iic a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLeast s a)
Goal: lowerBounds s = Set.Iic a
-- IsGreatest.upperBounds_eq
theorem IsGreatest.upperBounds_eq (h : IsGreatest s a) : upperBounds s = Ici a 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGreatest s a)
Goal: upperBounds s = Set.Ici a
-- IsGreatest.lt_iff
theorem IsGreatest.lt_iff (h : IsGreatest s a) : a < b ↔ ∀ x ∈ s, x < b 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a b : α} (h : IsGreatest s a)
Goal: a < b ↔ ∀ x ∈ s, x < b
-- IsLeast.lt_iff
theorem IsLeast.lt_iff (h : IsLeast s a) : b < a ↔ ∀ x ∈ s, b < x 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a b : α} (h : IsLeast s a)
Goal: b < a ↔ ∀ x ∈ s, b < x
-- isLUB_le_iff
theorem isLUB_le_iff (h : IsLUB s a) : a ≤ b ↔ b ∈ upperBounds s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a b : α} (h : IsLUB s a)
Goal: a ≤ b ↔ b ∈ upperBounds s
-- le_isGLB_iff
theorem le_isGLB_iff (h : IsGLB s a) : b ≤ a ↔ b ∈ lowerBounds s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a b : α} (h : IsGLB s a)
Goal: b ≤ a ↔ b ∈ lowerBounds s
-- isLUB_iff_le_iff
theorem isLUB_iff_le_iff : IsLUB s a ↔ ∀ b, a ≤ b ↔ b ∈ upperBounds s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α}
Goal: IsLUB s a ↔ ∀ (b : α), a ≤ b ↔ b ∈ upperBounds s
-- isGLB_iff_le_iff
theorem isGLB_iff_le_iff : IsGLB s a ↔ ∀ b, b ≤ a ↔ b ∈ lowerBounds s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α}
Goal: IsGLB s a ↔ ∀ (b : α), b ≤ a ↔ b ∈ lowerBounds s
-- IsLUB.bddAbove
theorem IsLUB.bddAbove (h : IsLUB s a) : BddAbove s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLUB s a)
Goal: BddAbove s
-- IsGLB.bddBelow
theorem IsGLB.bddBelow (h : IsGLB s a) : BddBelow s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGLB s a)
Goal: BddBelow s
-- IsGreatest.bddAbove
theorem IsGreatest.bddAbove (h : IsGreatest s a) : BddAbove s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGreatest s a)
Goal: BddAbove s
-- IsLeast.bddBelow
theorem IsLeast.bddBelow (h : IsLeast s a) : BddBelow s 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLeast s a)
Goal: BddBelow s
-- IsLeast.nonempty
theorem IsLeast.nonempty (h : IsLeast s a) : s.Nonempty 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsLeast s a)
Goal: s.Nonempty
-- IsGreatest.nonempty
theorem IsGreatest.nonempty (h : IsGreatest s a) : s.Nonempty 
Arguments: {α : Type u} [Preorder α] {s : Set α} {a : α} (h : IsGreatest s a)
Goal: s.Nonempty
-- upperBounds_union
@[simp]
theorem upperBounds_union : upperBounds (s ∪ t) = upperBounds s ∩ upperBounds t 
Arguments: {α : Type u} [Preorder α] {s t : Set α}
Goal: upperBounds (s ∪ t) = upperBounds s ∩ upperBounds t
-- lowerBounds_union
@[simp]
theorem lowerBounds_union : lowerBounds (s ∪ t) = lowerBounds s ∩ lowerBounds t 
Arguments: {α : Type u} [Preorder α] {s t : Set α}
Goal: lowerBounds (s ∪ t) = lowerBounds s ∩ lowerBounds t
-- union_upperBounds_subset_upperBounds_inter
theorem union_upperBounds_subset_upperBounds_inter :
    upperBounds s ∪ upperBounds t ⊆ upperBounds (s ∩ t) 
Arguments: {α : Type u} [Preorder α] {s t : Set α}
Goal: upperBounds s ∪ upperBounds t ⊆ upperBounds (s ∩ t)
-- union_lowerBounds_subset_lowerBounds_inter
theorem union_lowerBounds_subset_lowerBounds_inter :
    lowerBounds s ∪ lowerBounds t ⊆ lowerBounds (s ∩ t) 
Arguments: {α : Type u} [Preorder α] {s t : Set α}
Goal: lowerBounds s ∪ lowerBounds t ⊆ lowerBounds (s ∩ t)
-- isLeast_union_iff
theorem isLeast_union_iff {a : α} {s t : Set α} :
    IsLeast (s ∪ t) a ↔ IsLeast s a ∧ a ∈ lowerBounds t ∨ a ∈ lowerBounds s ∧ IsLeast t a 
Arguments: {α : Type u} [Preorder α] {a : α} {s t : Set α}
Goal: IsLeast (s ∪ t) a ↔ IsLeast s a ∧ a ∈ lowerBounds t ∨ a ∈ lowerBounds s ∧ IsLeast t a
-- isGreatest_union_iff
theorem isGreatest_union_iff :
    IsGreatest (s ∪ t) a ↔
      IsGreatest s a ∧ a ∈ upperBounds t ∨ a ∈ upperBounds s ∧ IsGreatest t a 
Arguments: {α : Type u} [Preorder α] {s t : Set α} {a : α}
Goal: IsGreatest (s ∪ t) a ↔ IsGreatest s a ∧ a ∈ upperBounds t ∨ a ∈ upperBounds s ∧ IsGreatest t a
-- BddAbove.inter_of_left
theorem BddAbove.inter_of_left (h : BddAbove s) : BddAbove (s ∩ t) 
Arguments: {α : Type u} [Preorder α] {s t : Set α} (h : BddAbove s)
Goal: BddAbove (s ∩ t)
-- BddAbove.inter_of_right
theorem BddAbove.inter_of_right (h : BddAbove t) : BddAbove (s ∩ t) 
Arguments: {α : Type u} [Preorder α] {s t : Set α} (h : BddAbove t)
Goal: BddAbove (s ∩ t)
-- BddBelow.inter_of_left
theorem BddBelow.inter_of_left (h : BddBelow s) : BddBelow (s ∩ t) 
Arguments: {α : Type u} [Preorder α] {s t : Set α} (h : BddBelow s)
Goal: BddBelow (s ∩ t)
-- BddBelow.inter_of_right
theorem BddBelow.inter_of_right (h : BddBelow t) : BddBelow (s ∩ t) 
Arguments: {α : Type u} [Preorder α] {s t : Set α} (h : BddBelow t)
Goal: BddBelow (s ∩ t)
-- BddAbove.union
theorem BddAbove.union [IsDirected α (· ≤ ·)] {s t : Set α} :
    BddAbove s → BddAbove t → BddAbove (s ∪ t) 
Arguments: {α : Type u} [Preorder α] [IsDirected α fun x x_1 => x ≤ x_1] {s t : Set α}
Goal: BddAbove s → BddAbove t → BddAbove (s ∪ t)
-- bddAbove_union
theorem bddAbove_union [IsDirected α (· ≤ ·)] {s t : Set α} :
    BddAbove (s ∪ t) ↔ BddAbove s ∧ BddAbove t 
Arguments: {α : Type u} [Preorder α] [IsDirected α fun x x_1 => x ≤ x_1] {s t : Set α}
Goal: BddAbove (s ∪ t) ↔ BddAbove s ∧ BddAbove t
-- BddBelow.union
theorem BddBelow.union [IsDirected α (· ≥ ·)] {s t : Set α} :
    BddBelow s → BddBelow t → BddBelow (s ∪ t) 
Arguments: {α : Type u} [Preorder α] [IsDirected α fun x x_1 => x ≥ x_1] {s t : Set α}
Goal: BddBelow s → BddBelow t → BddBelow (s ∪ t)
-- bddBelow_union
theorem bddBelow_union [IsDirected α (· ≥ ·)] {s t : Set α} :
    BddBelow (s ∪ t) ↔ BddBelow s ∧ BddBelow t 
Arguments: {α : Type u} [Preorder α] [IsDirected α fun x x_1 => x ≥ x_1] {s t : Set α}
Goal: BddBelow (s ∪ t) ↔ BddBelow s ∧ BddBelow t
-- IsLUB.union
theorem IsLUB.union [SemilatticeSup γ] {a b : γ} {s t : Set γ} (hs : IsLUB s a) (ht : IsLUB t b) :
    IsLUB (s ∪ t) (a ⊔ b) 
Arguments: {γ : Type w} [SemilatticeSup γ] {a b : γ} {s t : Set γ} (hs : IsLUB s a) (ht : IsLUB t b)
Goal: IsLUB (s ∪ t) (a ⊔ b)
-- IsGLB.union
theorem IsGLB.union [SemilatticeInf γ] {a₁ a₂ : γ} {s t : Set γ} (hs : IsGLB s a₁)
    (ht : IsGLB t a₂) : IsGLB (s ∪ t) (a₁ ⊓ a₂) 
Arguments: {γ : Type w} [SemilatticeInf γ] {a₁ a₂ : γ} {s t : Set γ} (hs : IsGLB s a₁) (ht : IsGLB t a₂)
Goal: IsGLB (s ∪ t) (a₁ ⊓ a₂)
-- IsLeast.union
theorem IsLeast.union [LinearOrder γ] {a b : γ} {s t : Set γ} (ha : IsLeast s a)
    (hb : IsLeast t b) : IsLeast (s ∪ t) (min a b) 
Arguments: {γ : Type w} [LinearOrder γ] {a b : γ} {s t : Set γ} (ha : IsLeast s a) (hb : IsLeast t b)
Goal: IsLeast (s ∪ t) (min a b)
-- IsGreatest.union
theorem IsGreatest.union [LinearOrder γ] {a b : γ} {s t : Set γ} (ha : IsGreatest s a)
    (hb : IsGreatest t b) : IsGreatest (s ∪ t) (max a b) 
Arguments: {γ : Type w} [LinearOrder γ] {a b : γ} {s t : Set γ} (ha : IsGreatest s a) (hb : IsGreatest t b)
Goal: IsGreatest (s ∪ t) (max a b)
-- IsLUB.inter_Ici_of_mem
theorem IsLUB.inter_Ici_of_mem [LinearOrder γ] {s : Set γ} {a b : γ} (ha : IsLUB s a) (hb : b ∈ s) :
    IsLUB (s ∩ Ici b) a 
Arguments: {γ : Type w} [LinearOrder γ] {s : Set γ} {a b : γ} (ha : IsLUB s a) (hb : b ∈ s)
Goal: IsLUB (s ∩ Set.Ici b) a
-- IsGLB.inter_Iic_of_mem
theorem IsGLB.inter_Iic_of_mem [LinearOrder γ] {s : Set γ} {a b : γ} (ha : IsGLB s a) (hb : b ∈ s) :
    IsGLB (s ∩ Iic b) a 
Arguments: {γ : Type w} [LinearOrder γ] {s : Set γ} {a b : γ} (ha : IsGLB s a) (hb : b ∈ s)
Goal: IsGLB (s ∩ Set.Iic b) a
-- bddAbove_iff_exists_ge
theorem bddAbove_iff_exists_ge [SemilatticeSup γ] {s : Set γ} (x₀ : γ) :
    BddAbove s ↔ ∃ x, x₀ ≤ x ∧ ∀ y ∈ s, y ≤ x 
Arguments: {γ : Type w} [SemilatticeSup γ] {s : Set γ} (x₀ : γ)
Goal: BddAbove s ↔ ∃ x, x₀ ≤ x ∧ ∀ y ∈ s, y ≤ x
-- bddBelow_iff_exists_le
theorem bddBelow_iff_exists_le [SemilatticeInf γ] {s : Set γ} (x₀ : γ) :
    BddBelow s ↔ ∃ x, x ≤ x₀ ∧ ∀ y ∈ s, x ≤ y 
Arguments: {γ : Type w} [SemilatticeInf γ] {s : Set γ} (x₀ : γ)
Goal: BddBelow s ↔ ∃ x ≤ x₀, ∀ y ∈ s, x ≤ y
-- BddAbove.exists_ge
theorem BddAbove.exists_ge [SemilatticeSup γ] {s : Set γ} (hs : BddAbove s) (x₀ : γ) :
    ∃ x, x₀ ≤ x ∧ ∀ y ∈ s, y ≤ x 
Arguments: {γ : Type w} [SemilatticeSup γ] {s : Set γ} (hs : BddAbove s) (x₀ : γ)
Goal: ∃ x, x₀ ≤ x ∧ ∀ y ∈ s, y ≤ x
-- BddBelow.exists_le
theorem BddBelow.exists_le [SemilatticeInf γ] {s : Set γ} (hs : BddBelow s) (x₀ : γ) :
    ∃ x, x ≤ x₀ ∧ ∀ y ∈ s, x ≤ y 
Arguments: {γ : Type w} [SemilatticeInf γ] {s : Set γ} (hs : BddBelow s) (x₀ : γ)
Goal: ∃ x ≤ x₀, ∀ y ∈ s, x ≤ y
-- isLeast_Ici
theorem isLeast_Ici : IsLeast (Ici a) a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsLeast (Set.Ici a) a
-- isGreatest_Iic
theorem isGreatest_Iic : IsGreatest (Iic a) a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsGreatest (Set.Iic a) a
-- isLUB_Iic
theorem isLUB_Iic : IsLUB (Iic a) a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsLUB (Set.Iic a) a
-- isGLB_Ici
theorem isGLB_Ici : IsGLB (Ici a) a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsGLB (Set.Ici a) a
-- upperBounds_Iic
theorem upperBounds_Iic : upperBounds (Iic a) = Ici a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: upperBounds (Set.Iic a) = Set.Ici a
-- lowerBounds_Ici
theorem lowerBounds_Ici : lowerBounds (Ici a) = Iic a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: lowerBounds (Set.Ici a) = Set.Iic a
-- bddAbove_Iic
theorem bddAbove_Iic : BddAbove (Iic a) 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: BddAbove (Set.Iic a)
-- bddBelow_Ici
theorem bddBelow_Ici : BddBelow (Ici a) 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: BddBelow (Set.Ici a)
-- bddAbove_Iio
theorem bddAbove_Iio : BddAbove (Iio a) 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: BddAbove (Set.Iio a)
-- bddBelow_Ioi
theorem bddBelow_Ioi : BddBelow (Ioi a) 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: BddBelow (Set.Ioi a)
-- lub_Iio_le
theorem lub_Iio_le (a : α) (hb : IsLUB (Iio a) b) : b ≤ a 
Arguments: {α : Type u} [Preorder α] {b : α} (a : α) (hb : IsLUB (Set.Iio a) b)
Goal: b ≤ a
-- le_glb_Ioi
theorem le_glb_Ioi (a : α) (hb : IsGLB (Ioi a) b) : a ≤ b 
Arguments: {α : Type u} [Preorder α] {b : α} (a : α) (hb : IsGLB (Set.Ioi a) b)
Goal: a ≤ b
-- lub_Iio_eq_self_or_Iio_eq_Iic
theorem lub_Iio_eq_self_or_Iio_eq_Iic [PartialOrder γ] {j : γ} (i : γ) (hj : IsLUB (Iio i) j) :
    j = i ∨ Iio i = Iic j 
Arguments: {γ : Type w} [PartialOrder γ] {j : γ} (i : γ) (hj : IsLUB (Set.Iio i) j)
Goal: j = i ∨ Set.Iio i = Set.Iic j
-- glb_Ioi_eq_self_or_Ioi_eq_Ici
theorem glb_Ioi_eq_self_or_Ioi_eq_Ici [PartialOrder γ] {j : γ} (i : γ) (hj : IsGLB (Ioi i) j) :
    j = i ∨ Ioi i = Ici j 
Arguments: {γ : Type w} [PartialOrder γ] {j : γ} (i : γ) (hj : IsGLB (Set.Ioi i) j)
Goal: j = i ∨ Set.Ioi i = Set.Ici j
-- exists_lub_Iio
theorem exists_lub_Iio (i : γ) : ∃ j, IsLUB (Iio i) j 
Arguments: {γ : Type w} [LinearOrder γ] (i : γ)
Goal: ∃ j, IsLUB (Set.Iio i) j
-- exists_glb_Ioi
theorem exists_glb_Ioi (i : γ) : ∃ j, IsGLB (Ioi i) j 
Arguments: {γ : Type w} [LinearOrder γ] (i : γ)
Goal: ∃ j, IsGLB (Set.Ioi i) j
-- isLUB_Iio
theorem isLUB_Iio {a : γ} : IsLUB (Iio a) a 
Arguments: {γ : Type w} [LinearOrder γ] [DenselyOrdered γ] {a : γ}
Goal: IsLUB (Set.Iio a) a
-- isGLB_Ioi
theorem isGLB_Ioi {a : γ} : IsGLB (Ioi a) a 
Arguments: {γ : Type w} [LinearOrder γ] [DenselyOrdered γ] {a : γ}
Goal: IsGLB (Set.Ioi a) a
-- upperBounds_Iio
theorem upperBounds_Iio {a : γ} : upperBounds (Iio a) = Ici a 
Arguments: {γ : Type w} [LinearOrder γ] [DenselyOrdered γ] {a : γ}
Goal: upperBounds (Set.Iio a) = Set.Ici a
-- lowerBounds_Ioi
theorem lowerBounds_Ioi {a : γ} : lowerBounds (Ioi a) = Iic a 
Arguments: {γ : Type w} [LinearOrder γ] [DenselyOrdered γ] {a : γ}
Goal: lowerBounds (Set.Ioi a) = Set.Iic a
-- isGreatest_singleton
@[simp] theorem isGreatest_singleton : IsGreatest {a} a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsGreatest {a} a
-- isLeast_singleton
@[simp] theorem isLeast_singleton : IsLeast {a} a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsLeast {a} a
-- isLUB_singleton
@[simp] theorem isLUB_singleton : IsLUB {a} a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsLUB {a} a
-- isGLB_singleton
@[simp] theorem isGLB_singleton : IsGLB {a} a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: IsGLB {a} a
-- bddAbove_singleton
@[simp] lemma bddAbove_singleton : BddAbove ({a} : Set α) 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: BddAbove {a}
-- bddBelow_singleton
@[simp] lemma bddBelow_singleton : BddBelow ({a} : Set α) 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: BddBelow {a}
-- upperBounds_singleton
@[simp]
theorem upperBounds_singleton : upperBounds {a} = Ici a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: upperBounds {a} = Set.Ici a
-- lowerBounds_singleton
@[simp]
theorem lowerBounds_singleton : lowerBounds {a} = Iic a 
Arguments: {α : Type u} [Preorder α] {a : α}
Goal: lowerBounds {a} = Set.Iic a
-- bddAbove_Icc
theorem bddAbove_Icc : BddAbove (Icc a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddAbove (Set.Icc a b)
-- bddBelow_Icc
theorem bddBelow_Icc : BddBelow (Icc a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddBelow (Set.Icc a b)
-- bddAbove_Ico
theorem bddAbove_Ico : BddAbove (Ico a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddAbove (Set.Ico a b)
-- bddBelow_Ico
theorem bddBelow_Ico : BddBelow (Ico a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddBelow (Set.Ico a b)
-- bddAbove_Ioc
theorem bddAbove_Ioc : BddAbove (Ioc a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddAbove (Set.Ioc a b)
-- bddBelow_Ioc
theorem bddBelow_Ioc : BddBelow (Ioc a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddBelow (Set.Ioc a b)
-- bddAbove_Ioo
theorem bddAbove_Ioo : BddAbove (Ioo a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddAbove (Set.Ioo a b)
-- bddBelow_Ioo
theorem bddBelow_Ioo : BddBelow (Ioo a b) 
Arguments: {α : Type u} [Preorder α] {a b : α}
Goal: BddBelow (Set.Ioo a b)
-- isGreatest_Icc
theorem isGreatest_Icc (h : a ≤ b) : IsGreatest (Icc a b) b 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a ≤ b)
Goal: IsGreatest (Set.Icc a b) b
-- isLUB_Icc
theorem isLUB_Icc (h : a ≤ b) : IsLUB (Icc a b) b 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a ≤ b)
Goal: IsLUB (Set.Icc a b) b
-- upperBounds_Icc
theorem upperBounds_Icc (h : a ≤ b) : upperBounds (Icc a b) = Ici b 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a ≤ b)
Goal: upperBounds (Set.Icc a b) = Set.Ici b
-- isLeast_Icc
theorem isLeast_Icc (h : a ≤ b) : IsLeast (Icc a b) a 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a ≤ b)
Goal: IsLeast (Set.Icc a b) a
-- isGLB_Icc
theorem isGLB_Icc (h : a ≤ b) : IsGLB (Icc a b) a 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a ≤ b)
Goal: IsGLB (Set.Icc a b) a
-- lowerBounds_Icc
theorem lowerBounds_Icc (h : a ≤ b) : lowerBounds (Icc a b) = Iic a 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a ≤ b)
Goal: lowerBounds (Set.Icc a b) = Set.Iic a
-- isGreatest_Ioc
theorem isGreatest_Ioc (h : a < b) : IsGreatest (Ioc a b) b 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a < b)
Goal: IsGreatest (Set.Ioc a b) b
-- isLUB_Ioc
theorem isLUB_Ioc (h : a < b) : IsLUB (Ioc a b) b 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a < b)
Goal: IsLUB (Set.Ioc a b) b
-- upperBounds_Ioc
theorem upperBounds_Ioc (h : a < b) : upperBounds (Ioc a b) = Ici b 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a < b)
Goal: upperBounds (Set.Ioc a b) = Set.Ici b
-- isLeast_Ico
theorem isLeast_Ico (h : a < b) : IsLeast (Ico a b) a 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a < b)
Goal: IsLeast (Set.Ico a b) a
-- isGLB_Ico
theorem isGLB_Ico (h : a < b) : IsGLB (Ico a b) a 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a < b)
Goal: IsGLB (Set.Ico a b) a
-- lowerBounds_Ico
theorem lowerBounds_Ico (h : a < b) : lowerBounds (Ico a b) = Iic a 
Arguments: {α : Type u} [Preorder α] {a b : α} (h : a < b)
Goal: lowerBounds (Set.Ico a b) = Set.Iic a
-- isGLB_Ioo
theorem isGLB_Ioo {a b : γ} (h : a < b) : IsGLB (Ioo a b) a 
Arguments: {γ : Type w} [SemilatticeSup γ] [DenselyOrdered γ] {a b : γ} (h : a < b)
Goal: IsGLB (Set.Ioo a b) a
-- lowerBounds_Ioo
theorem lowerBounds_Ioo {a b : γ} (hab : a < b) : lowerBounds (Ioo a b) = Iic a 
Arguments: {γ : Type w} [SemilatticeSup γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: lowerBounds (Set.Ioo a b) = Set.Iic a
-- isGLB_Ioc
theorem isGLB_Ioc {a b : γ} (hab : a < b) : IsGLB (Ioc a b) a 
Arguments: {γ : Type w} [SemilatticeSup γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: IsGLB (Set.Ioc a b) a
-- lowerBounds_Ioc
theorem lowerBounds_Ioc {a b : γ} (hab : a < b) : lowerBounds (Ioc a b) = Iic a 
Arguments: {γ : Type w} [SemilatticeSup γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: lowerBounds (Set.Ioc a b) = Set.Iic a
-- isLUB_Ioo
theorem isLUB_Ioo {a b : γ} (hab : a < b) : IsLUB (Ioo a b) b 
Arguments: {γ : Type w} [SemilatticeInf γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: IsLUB (Set.Ioo a b) b
-- upperBounds_Ioo
theorem upperBounds_Ioo {a b : γ} (hab : a < b) : upperBounds (Ioo a b) = Ici b 
Arguments: {γ : Type w} [SemilatticeInf γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: upperBounds (Set.Ioo a b) = Set.Ici b
-- isLUB_Ico
theorem isLUB_Ico {a b : γ} (hab : a < b) : IsLUB (Ico a b) b 
Arguments: {γ : Type w} [SemilatticeInf γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: IsLUB (Set.Ico a b) b
-- upperBounds_Ico
theorem upperBounds_Ico {a b : γ} (hab : a < b) : upperBounds (Ico a b) = Ici b 
Arguments: {γ : Type w} [SemilatticeInf γ] [DenselyOrdered γ] {a b : γ} (hab : a < b)
Goal: upperBounds (Set.Ico a b) = Set.Ici b
-- bddBelow_iff_subset_Ici
theorem bddBelow_iff_subset_Ici : BddBelow s ↔ ∃ a, s ⊆ Ici a 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddBelow s ↔ ∃ a, s ⊆ Set.Ici a
-- bddAbove_iff_subset_Iic
theorem bddAbove_iff_subset_Iic : BddAbove s ↔ ∃ a, s ⊆ Iic a 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddAbove s ↔ ∃ a, s ⊆ Set.Iic a
-- bddBelow_bddAbove_iff_subset_Icc
theorem bddBelow_bddAbove_iff_subset_Icc : BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ Icc a b 
Arguments: {α : Type u} [Preorder α] {s : Set α}
Goal: BddBelow s ∧ BddAbove s ↔ ∃ a b, s ⊆ Set.Icc a b