ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- intentClosure_empty
@[simp]
theorem intentClosure_empty : intentClosure r ∅ = univ 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop)
Goal: intentClosure r ∅ = Set.univ
-- extentClosure_empty
@[simp]
theorem extentClosure_empty : extentClosure r ∅ = univ 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop)
Goal: extentClosure r ∅ = Set.univ
-- intentClosure_union
@[simp]
theorem intentClosure_union (s₁ s₂ : Set α) :
    intentClosure r (s₁ ∪ s₂) = intentClosure r s₁ ∩ intentClosure r s₂ 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (s₁ s₂ : Set α)
Goal: intentClosure r (s₁ ∪ s₂) = intentClosure r s₁ ∩ intentClosure r s₂
-- extentClosure_union
@[simp]
theorem extentClosure_union (t₁ t₂ : Set β) :
    extentClosure r (t₁ ∪ t₂) = extentClosure r t₁ ∩ extentClosure r t₂ 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (t₁ t₂ : Set β)
Goal: extentClosure r (t₁ ∪ t₂) = extentClosure r t₁ ∩ extentClosure r t₂
-- intentClosure_iUnion
@[simp]
theorem intentClosure_iUnion (f : ι → Set α) :
    intentClosure r (⋃ i, f i) = ⋂ i, intentClosure r (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (f : ι → Set α)
Goal: intentClosure r (⋃ i, f i) = ⋂ i, intentClosure r (f i)
-- extentClosure_iUnion
@[simp]
theorem extentClosure_iUnion (f : ι → Set β) :
    extentClosure r (⋃ i, f i) = ⋂ i, extentClosure r (f i) 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (f : ι → Set β)
Goal: extentClosure r (⋃ i, f i) = ⋂ i, extentClosure r (f i)
-- intentClosure_iUnion₂
theorem intentClosure_iUnion₂ (f : ∀ i, κ i → Set α) :
    intentClosure r (⋃ (i) (j), f i j) = ⋂ (i) (j), intentClosure r (f i j) 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} {κ : ι → Sort u_5} (r : α → β → Prop) (f : (i : ι) → κ i → Set α)
Goal: intentClosure r (⋃ i, ⋃ j, f i j) = ⋂ i, ⋂ j, intentClosure r (f i j)
-- extentClosure_iUnion₂
theorem extentClosure_iUnion₂ (f : ∀ i, κ i → Set β) :
    extentClosure r (⋃ (i) (j), f i j) = ⋂ (i) (j), extentClosure r (f i j) 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} {κ : ι → Sort u_5} (r : α → β → Prop) (f : (i : ι) → κ i → Set β)
Goal: extentClosure r (⋃ i, ⋃ j, f i j) = ⋂ i, ⋂ j, extentClosure r (f i j)
-- subset_extentClosure_intentClosure
theorem subset_extentClosure_intentClosure (s : Set α) :
    s ⊆ extentClosure r (intentClosure r s) 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (s : Set α)
Goal: s ⊆ extentClosure r (intentClosure r s)
-- subset_intentClosure_extentClosure
theorem subset_intentClosure_extentClosure (t : Set β) :
    t ⊆ intentClosure r (extentClosure r t) 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (t : Set β)
Goal: t ⊆ intentClosure r (extentClosure r t)
-- intentClosure_extentClosure_intentClosure
@[simp]
theorem intentClosure_extentClosure_intentClosure (s : Set α) :
    intentClosure r (extentClosure r <| intentClosure r s) = intentClosure r s 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (s : Set α)
Goal: intentClosure r (extentClosure r (intentClosure r s)) = intentClosure r s
-- extentClosure_intentClosure_extentClosure
@[simp]
theorem extentClosure_intentClosure_extentClosure (t : Set β) :
    extentClosure r (intentClosure r <| extentClosure r t) = extentClosure r t 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop) (t : Set β)
Goal: extentClosure r (intentClosure r (extentClosure r t)) = extentClosure r t
-- intentClosure_anti
theorem intentClosure_anti : Antitone (intentClosure r) 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop)
Goal: Antitone (intentClosure r)
-- extentClosure_anti
theorem extentClosure_anti : Antitone (extentClosure r) 
Arguments: {α : Type u_2} {β : Type u_3} (r : α → β → Prop)
Goal: Antitone (extentClosure r)
-- Concept.closure_fst
structure Concept extends Set α × Set β where
    closure_fst : intentClosure r fst = snd
    closure_snd : extentClosure r snd = fst
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (self : Concept α β r)
Goal: intentClosure r self.toProd.1 = self.toProd.2
-- Concept.closure_snd
structure Concept extends Set α × Set β where
    closure_fst : intentClosure r fst = snd
    closure_snd : extentClosure r snd = fst
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (self : Concept α β r)
Goal: extentClosure r self.toProd.2 = self.toProd.1
-- Concept.ext
@[ext]
theorem ext (h : c.fst = d.fst) : c = d 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r} (h : c.toProd.1 = d.toProd.1)
Goal: c = d
-- Concept.ext'
theorem ext' (h : c.snd = d.snd) : c = d 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r} (h : c.toProd.2 = d.toProd.2)
Goal: c = d
-- Concept.fst_injective
theorem fst_injective : Injective fun c : Concept α β r => c.fst 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: Function.Injective fun c => c.toProd.1
-- Concept.snd_injective
theorem snd_injective : Injective fun c : Concept α β r => c.snd 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: Function.Injective fun c => c.toProd.2
-- Concept.fst_subset_fst_iff
@[simp]
theorem fst_subset_fst_iff : c.fst ⊆ d.fst ↔ c ≤ d 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r}
Goal: c.toProd.1 ⊆ d.toProd.1 ↔ c ≤ d
-- Concept.fst_ssubset_fst_iff
@[simp]
theorem fst_ssubset_fst_iff : c.fst ⊂ d.fst ↔ c < d 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r}
Goal: c.toProd.1 ⊂ d.toProd.1 ↔ c < d
-- Concept.snd_subset_snd_iff
@[simp]
theorem snd_subset_snd_iff : c.snd ⊆ d.snd ↔ d ≤ c 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r}
Goal: c.toProd.2 ⊆ d.toProd.2 ↔ d ≤ c
-- Concept.snd_ssubset_snd_iff
@[simp]
theorem snd_ssubset_snd_iff : c.snd ⊂ d.snd ↔ d < c 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r}
Goal: c.toProd.2 ⊂ d.toProd.2 ↔ d < c
-- Concept.strictMono_fst
theorem strictMono_fst : StrictMono (Prod.fst ∘ toProd : Concept α β r → Set α) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: StrictMono (Prod.fst ∘ Concept.toProd)
-- Concept.strictAnti_snd
theorem strictAnti_snd : StrictAnti (Prod.snd ∘ toProd : Concept α β r → Set β) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: StrictAnti (Prod.snd ∘ Concept.toProd)
-- Concept.top_fst
@[simp]
theorem top_fst : (⊤ : Concept α β r).fst = univ 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: ⊤.toProd.1 = Set.univ
-- Concept.top_snd
@[simp]
theorem top_snd : (⊤ : Concept α β r).snd = intentClosure r univ 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: ⊤.toProd.2 = intentClosure r Set.univ
-- Concept.bot_fst
@[simp]
theorem bot_fst : (⊥ : Concept α β r).fst = extentClosure r univ 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: ⊥.toProd.1 = extentClosure r Set.univ
-- Concept.bot_snd
@[simp]
theorem bot_snd : (⊥ : Concept α β r).snd = univ 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: ⊥.toProd.2 = Set.univ
-- Concept.sup_fst
@[simp]
theorem sup_fst (c d : Concept α β r) : (c ⊔ d).fst = extentClosure r (c.snd ∩ d.snd) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (c d : Concept α β r)
Goal: (c ⊔ d).toProd.1 = extentClosure r (c.toProd.2 ∩ d.toProd.2)
-- Concept.sup_snd
@[simp]
theorem sup_snd (c d : Concept α β r) : (c ⊔ d).snd = c.snd ∩ d.snd 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (c d : Concept α β r)
Goal: (c ⊔ d).toProd.2 = c.toProd.2 ∩ d.toProd.2
-- Concept.inf_fst
@[simp]
theorem inf_fst (c d : Concept α β r) : (c ⊓ d).fst = c.fst ∩ d.fst 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (c d : Concept α β r)
Goal: (c ⊓ d).toProd.1 = c.toProd.1 ∩ d.toProd.1
-- Concept.inf_snd
@[simp]
theorem inf_snd (c d : Concept α β r) : (c ⊓ d).snd = intentClosure r (c.fst ∩ d.fst) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (c d : Concept α β r)
Goal: (c ⊓ d).toProd.2 = intentClosure r (c.toProd.1 ∩ d.toProd.1)
-- Concept.sSup_fst
@[simp]
theorem sSup_fst (S : Set (Concept α β r)) :
    (sSup S).fst = extentClosure r (⋂ c ∈ S, (c : Concept _ _ _).snd) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (S : Set (Concept α β r))
Goal: (sSup S).toProd.1 = extentClosure r (⋂ c ∈ S, c.toProd.2)
-- Concept.sSup_snd
@[simp]
theorem sSup_snd (S : Set (Concept α β r)) : (sSup S).snd = ⋂ c ∈ S, (c : Concept _ _ _).snd 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (S : Set (Concept α β r))
Goal: (sSup S).toProd.2 = ⋂ c ∈ S, c.toProd.2
-- Concept.sInf_fst
@[simp]
theorem sInf_fst (S : Set (Concept α β r)) : (sInf S).fst = ⋂ c ∈ S, (c : Concept _ _ _).fst 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (S : Set (Concept α β r))
Goal: (sInf S).toProd.1 = ⋂ c ∈ S, c.toProd.1
-- Concept.sInf_snd
@[simp]
theorem sInf_snd (S : Set (Concept α β r)) :
    (sInf S).snd = intentClosure r (⋂ c ∈ S, (c : Concept _ _ _).fst) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (S : Set (Concept α β r))
Goal: (sInf S).toProd.2 = intentClosure r (⋂ c ∈ S, c.toProd.1)
-- Concept.swap_toProd
@[simps]
def swap (c : Concept α β r) : Concept β α (swap r) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (c : Concept α β r)
Goal: c.swap.toProd = c.swap
-- Concept.swap_swap
@[simp]
theorem swap_swap (c : Concept α β r) : c.swap.swap = c 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} (c : Concept α β r)
Goal: c.swap.swap = c
-- Concept.swap_le_swap_iff
@[simp]
theorem swap_le_swap_iff : c.swap ≤ d.swap ↔ d ≤ c 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r}
Goal: c.swap ≤ d.swap ↔ d ≤ c
-- Concept.swap_lt_swap_iff
@[simp]
theorem swap_lt_swap_iff : c.swap < d.swap ↔ d < c 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop} {c d : Concept α β r}
Goal: c.swap < d.swap ↔ d < c
-- Concept.swapEquiv_symm_apply
@[simps]
def swapEquiv : (Concept α β r)ᵒᵈ ≃o Concept β α (Function.swap r) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: ∀ (a : Concept β α (Function.swap r)), (RelIso.symm Concept.swapEquiv) a = (⇑OrderDual.toDual ∘ Concept.swap) a
-- Concept.swapEquiv_apply
@[simps]
def swapEquiv : (Concept α β r)ᵒᵈ ≃o Concept β α (Function.swap r) 
Arguments: {α : Type u_2} {β : Type u_3} {r : α → β → Prop}
Goal: ∀ (a : (Concept α β r)ᵒᵈ), Concept.swapEquiv a = (Concept.swap ∘ ⇑OrderDual.ofDual) a
-- Filter.mem_cocardinal
@[simp]
theorem mem_cocardinal {s : Set α} :
    s ∈ cocardinal α hreg ↔ Cardinal.mk (sᶜ : Set α) < c 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {s : Set α}
Goal: s ∈ Filter.cocardinal α hreg ↔ #↑sᶜ < c
-- Filter.cocardinal_aleph0_eq_cofinite
@[simp] lemma cocardinal_aleph0_eq_cofinite :
    cocardinal (α := α) isRegular_aleph0 = cofinite 
Arguments: {α : Type u}
Goal: Filter.cocardinal α Cardinal.isRegular_aleph0 = Filter.cofinite
-- Filter.eventually_cocardinal
@[simp]
theorem eventually_cocardinal {p : α → Prop} :
    (∀ᶠ x in cocardinal α hreg, p x) ↔ #{ x | ¬p x } < c 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {p : α → Prop}
Goal: (∀ᶠ (x : α) in Filter.cocardinal α hreg, p x) ↔ #↑{x | ¬p x} < c
-- Filter.hasBasis_cocardinal
theorem hasBasis_cocardinal : HasBasis (cocardinal α hreg) {s : Set α | #s < c} compl 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular}
Goal: (Filter.cocardinal α hreg).HasBasis {s | #↑s < c} compl
-- Filter.frequently_cocardinal
theorem frequently_cocardinal {p : α → Prop} :
    (∃ᶠ x in cocardinal α hreg, p x) ↔ c ≤ # { x | p x } 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {p : α → Prop}
Goal: (∃ᶠ (x : α) in Filter.cocardinal α hreg, p x) ↔ c ≤ #↑{x | p x}
-- Filter.frequently_cocardinal_mem
lemma frequently_cocardinal_mem {s : Set α} :
    (∃ᶠ x in cocardinal α hreg, x ∈ s) ↔ c ≤ #s 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {s : Set α}
Goal: (∃ᶠ (x : α) in Filter.cocardinal α hreg, x ∈ s) ↔ c ≤ #↑s
-- Filter.cocardinal_inf_principal_neBot_iff
@[simp]
lemma cocardinal_inf_principal_neBot_iff {s : Set α} :
    (cocardinal α hreg ⊓ 𝓟 s).NeBot ↔ c ≤ #s 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {s : Set α}
Goal: (Filter.cocardinal α hreg ⊓ 𝓟 s).NeBot ↔ c ≤ #↑s
-- Filter.compl_mem_cocardinal_of_card_lt
theorem compl_mem_cocardinal_of_card_lt {s : Set α} (hs : #s < c) :
    sᶜ ∈ cocardinal α hreg 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {s : Set α} (hs : #↑s < c)
Goal: sᶜ ∈ Filter.cocardinal α hreg
-- Set.Finite.compl_mem_cocardinal
theorem _root_.Set.Finite.compl_mem_cocardinal {s : Set α} (hs : s.Finite) :
    sᶜ ∈ cocardinal α hreg 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {s : Set α} (hs : s.Finite)
Goal: sᶜ ∈ Filter.cocardinal α hreg
-- Filter.eventually_cocardinal_nmem_of_card_lt
theorem eventually_cocardinal_nmem_of_card_lt  {s : Set α} (hs : #s < c) :
    ∀ᶠ x in cocardinal α hreg, x ∉ s 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} {s : Set α} (hs : #↑s < c)
Goal: ∀ᶠ (x : α) in Filter.cocardinal α hreg, x ∉ s
-- Finset.eventually_cocardinal_nmem
theorem _root_.Finset.eventually_cocardinal_nmem (s : Finset α) :
    ∀ᶠ x in cocardinal α hreg, x ∉ s 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} (s : Finset α)
Goal: ∀ᶠ (x : α) in Filter.cocardinal α hreg, x ∉ s
-- Filter.eventually_cocardinal_ne
theorem eventually_cocardinal_ne (x : α) : ∀ᶠ a in cocardinal α hreg, a ≠ x 
Arguments: {α : Type u} {c : Cardinal.{u}} {hreg : c.IsRegular} (x : α)
Goal: ∀ᶠ (a : α) in Filter.cocardinal α hreg, a ≠ x
-- Filter.mem_cocountable
theorem mem_cocountable {s : Set α} :
    s ∈ cocountable ↔ (sᶜ : Set α).Countable 
Arguments: {α : Type u} {s : Set α}
Goal: s ∈ Filter.cocountable ↔ sᶜ.Countable
-- Set.Bounded.mono
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounded r s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} (hst : s ⊆ t) (hs : Set.Bounded r t)
Goal: Set.Bounded r s
-- Set.Unbounded.mono
theorem Unbounded.mono (hst : s ⊆ t) (hs : Unbounded r s) : Unbounded r t 
Arguments: {α : Type u_1} {r : α → α → Prop} {s t : Set α} (hst : s ⊆ t) (hs : Set.Unbounded r s)
Goal: Set.Unbounded r t
-- Set.unbounded_le_of_forall_exists_lt
theorem unbounded_le_of_forall_exists_lt [Preorder α] (h : ∀ a, ∃ b ∈ s, a < b) :
    Unbounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : ∀ (a : α), ∃ b ∈ s, a < b)
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) s
-- Set.unbounded_le_iff
theorem unbounded_le_iff [LinearOrder α] : Unbounded (· ≤ ·) s ↔ ∀ a, ∃ b ∈ s, a < b 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α]
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) s ↔ ∀ (a : α), ∃ b ∈ s, a < b
-- Set.unbounded_lt_of_forall_exists_le
theorem unbounded_lt_of_forall_exists_le [Preorder α] (h : ∀ a, ∃ b ∈ s, a ≤ b) :
    Unbounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : ∀ (a : α), ∃ b ∈ s, a ≤ b)
Goal: Set.Unbounded (fun x x_1 => x < x_1) s
-- Set.unbounded_lt_iff
theorem unbounded_lt_iff [LinearOrder α] : Unbounded (· < ·) s ↔ ∀ a, ∃ b ∈ s, a ≤ b 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α]
Goal: Set.Unbounded (fun x x_1 => x < x_1) s ↔ ∀ (a : α), ∃ b ∈ s, a ≤ b
-- Set.unbounded_ge_of_forall_exists_gt
theorem unbounded_ge_of_forall_exists_gt [Preorder α] (h : ∀ a, ∃ b ∈ s, b < a) :
    Unbounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : ∀ (a : α), ∃ b ∈ s, b < a)
Goal: Set.Unbounded (fun x x_1 => x ≥ x_1) s
-- Set.unbounded_ge_iff
theorem unbounded_ge_iff [LinearOrder α] : Unbounded (· ≥ ·) s ↔ ∀ a, ∃ b ∈ s, b < a 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α]
Goal: Set.Unbounded (fun x x_1 => x ≥ x_1) s ↔ ∀ (a : α), ∃ b ∈ s, b < a
-- Set.unbounded_gt_of_forall_exists_ge
theorem unbounded_gt_of_forall_exists_ge [Preorder α] (h : ∀ a, ∃ b ∈ s, b ≤ a) :
    Unbounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : ∀ (a : α), ∃ b ∈ s, b ≤ a)
Goal: Set.Unbounded (fun x x_1 => x > x_1) s
-- Set.unbounded_gt_iff
theorem unbounded_gt_iff [LinearOrder α] : Unbounded (· > ·) s ↔ ∀ a, ∃ b ∈ s, b ≤ a 
Arguments: {α : Type u_1} {s : Set α} [LinearOrder α]
Goal: Set.Unbounded (fun x x_1 => x > x_1) s ↔ ∀ (a : α), ∃ b ∈ s, b ≤ a
-- Set.Bounded.rel_mono
theorem Bounded.rel_mono {r' : α → α → Prop} (h : Bounded r s) (hrr' : r ≤ r') : Bounded r' s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {r' : α → α → Prop} (h : Set.Bounded r s) (hrr' : r ≤ r')
Goal: Set.Bounded r' s
-- Set.bounded_le_of_bounded_lt
theorem bounded_le_of_bounded_lt [Preorder α] (h : Bounded (· < ·) s) : Bounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : Set.Bounded (fun x x_1 => x < x_1) s)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) s
-- Set.Unbounded.rel_mono
theorem Unbounded.rel_mono {r' : α → α → Prop} (hr : r' ≤ r) (h : Unbounded r s) : Unbounded r' s 
Arguments: {α : Type u_1} {r : α → α → Prop} {s : Set α} {r' : α → α → Prop} (hr : r' ≤ r) (h : Set.Unbounded r s)
Goal: Set.Unbounded r' s
-- Set.unbounded_lt_of_unbounded_le
theorem unbounded_lt_of_unbounded_le [Preorder α] (h : Unbounded (· ≤ ·) s) : Unbounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : Set.Unbounded (fun x x_1 => x ≤ x_1) s)
Goal: Set.Unbounded (fun x x_1 => x < x_1) s
-- Set.bounded_le_iff_bounded_lt
theorem bounded_le_iff_bounded_lt [Preorder α] [NoMaxOrder α] :
    Bounded (· ≤ ·) s ↔ Bounded (· < ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] [NoMaxOrder α]
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) s ↔ Set.Bounded (fun x x_1 => x < x_1) s
-- Set.unbounded_lt_iff_unbounded_le
theorem unbounded_lt_iff_unbounded_le [Preorder α] [NoMaxOrder α] :
    Unbounded (· < ·) s ↔ Unbounded (· ≤ ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] [NoMaxOrder α]
Goal: Set.Unbounded (fun x x_1 => x < x_1) s ↔ Set.Unbounded (fun x x_1 => x ≤ x_1) s
-- Set.bounded_ge_of_bounded_gt
theorem bounded_ge_of_bounded_gt [Preorder α] (h : Bounded (· > ·) s) : Bounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : Set.Bounded (fun x x_1 => x > x_1) s)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) s
-- Set.unbounded_gt_of_unbounded_ge
theorem unbounded_gt_of_unbounded_ge [Preorder α] (h : Unbounded (· ≥ ·) s) : Unbounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] (h : Set.Unbounded (fun x x_1 => x ≥ x_1) s)
Goal: Set.Unbounded (fun x x_1 => x > x_1) s
-- Set.bounded_ge_iff_bounded_gt
theorem bounded_ge_iff_bounded_gt [Preorder α] [NoMinOrder α] :
    Bounded (· ≥ ·) s ↔ Bounded (· > ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] [NoMinOrder α]
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) s ↔ Set.Bounded (fun x x_1 => x > x_1) s
-- Set.unbounded_gt_iff_unbounded_ge
theorem unbounded_gt_iff_unbounded_ge [Preorder α] [NoMinOrder α] :
    Unbounded (· > ·) s ↔ Unbounded (· ≥ ·) s 
Arguments: {α : Type u_1} {s : Set α} [Preorder α] [NoMinOrder α]
Goal: Set.Unbounded (fun x x_1 => x > x_1) s ↔ Set.Unbounded (fun x x_1 => x ≥ x_1) s
-- Set.unbounded_le_univ
theorem unbounded_le_univ [LE α] [NoTopOrder α] : Unbounded (· ≤ ·) (@Set.univ α) 
Arguments: {α : Type u_1} [LE α] [NoTopOrder α]
Goal: Set.Unbounded (fun x x_1 => x ≤ x_1) Set.univ
-- Set.unbounded_lt_univ
theorem unbounded_lt_univ [Preorder α] [NoTopOrder α] : Unbounded (· < ·) (@Set.univ α) 
Arguments: {α : Type u_1} [Preorder α] [NoTopOrder α]
Goal: Set.Unbounded (fun x x_1 => x < x_1) Set.univ
-- Set.unbounded_ge_univ
theorem unbounded_ge_univ [LE α] [NoBotOrder α] : Unbounded (· ≥ ·) (@Set.univ α) 
Arguments: {α : Type u_1} [LE α] [NoBotOrder α]
Goal: Set.Unbounded (fun x x_1 => x ≥ x_1) Set.univ
-- Set.unbounded_gt_univ
theorem unbounded_gt_univ [Preorder α] [NoBotOrder α] : Unbounded (· > ·) (@Set.univ α) 
Arguments: {α : Type u_1} [Preorder α] [NoBotOrder α]
Goal: Set.Unbounded (fun x x_1 => x > x_1) Set.univ
-- Set.bounded_self
theorem bounded_self (a : α) : Bounded r { b | r b a } 
Arguments: {α : Type u_1} {r : α → α → Prop} (a : α)
Goal: Set.Bounded r {b | r b a}
-- Set.bounded_lt_Iio
theorem bounded_lt_Iio [Preorder α] (a : α) : Bounded (· < ·) (Iio a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Iio a)
-- Set.bounded_le_Iio
theorem bounded_le_Iio [Preorder α] (a : α) : Bounded (· ≤ ·) (Iio a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Iio a)
-- Set.bounded_le_Iic
theorem bounded_le_Iic [Preorder α] (a : α) : Bounded (· ≤ ·) (Iic a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Iic a)
-- Set.bounded_lt_Iic
theorem bounded_lt_Iic [Preorder α] [NoMaxOrder α] (a : α) : Bounded (· < ·) (Iic a) 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Iic a)
-- Set.bounded_gt_Ioi
theorem bounded_gt_Ioi [Preorder α] (a : α) : Bounded (· > ·) (Ioi a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ioi a)
-- Set.bounded_ge_Ioi
theorem bounded_ge_Ioi [Preorder α] (a : α) : Bounded (· ≥ ·) (Ioi a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ioi a)
-- Set.bounded_ge_Ici
theorem bounded_ge_Ici [Preorder α] (a : α) : Bounded (· ≥ ·) (Ici a) 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x ≥ x_1) (Set.Ici a)
-- Set.bounded_gt_Ici
theorem bounded_gt_Ici [Preorder α] [NoMinOrder α] (a : α) : Bounded (· > ·) (Ici a) 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] (a : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ici a)
-- Set.bounded_lt_Ioo
theorem bounded_lt_Ioo [Preorder α] (a b : α) : Bounded (· < ·) (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Ioo a b)
-- Set.bounded_lt_Ico
theorem bounded_lt_Ico [Preorder α] (a b : α) : Bounded (· < ·) (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Ico a b)
-- Set.bounded_lt_Ioc
theorem bounded_lt_Ioc [Preorder α] [NoMaxOrder α] (a b : α) : Bounded (· < ·) (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Ioc a b)
-- Set.bounded_lt_Icc
theorem bounded_lt_Icc [Preorder α] [NoMaxOrder α] (a b : α) : Bounded (· < ·) (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x < x_1) (Set.Icc a b)
-- Set.bounded_le_Ioo
theorem bounded_le_Ioo [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Ioo a b)
-- Set.bounded_le_Ico
theorem bounded_le_Ico [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Ico a b)
-- Set.bounded_le_Ioc
theorem bounded_le_Ioc [Preorder α] (a b : α) : Bounded (· ≤ ·) (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Ioc a b)
-- Set.bounded_le_Icc
theorem bounded_le_Icc [Preorder α] (a b : α) : Bounded (· ≤ ·) (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x ≤ x_1) (Set.Icc a b)
-- Set.bounded_gt_Ioo
theorem bounded_gt_Ioo [Preorder α] (a b : α) : Bounded (· > ·) (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ioo a b)
-- Set.bounded_gt_Ioc
theorem bounded_gt_Ioc [Preorder α] (a b : α) : Bounded (· > ·) (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ioc a b)
-- Set.bounded_gt_Ico
theorem bounded_gt_Ico [Preorder α] [NoMinOrder α] (a b : α) : Bounded (· > ·) (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Ico a b)
-- Set.bounded_gt_Icc
theorem bounded_gt_Icc [Preorder α] [NoMinOrder α] (a b : α) : Bounded (· > ·) (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] (a b : α)
Goal: Set.Bounded (fun x x_1 => x > x_1) (Set.Icc a b)