ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- inter_preimage_minimals_eq_of_rel_iff_rel_on_of_subset
theorem inter_preimage_minimals_eq_of_rel_iff_rel_on_of_subset {x : Set α} {y : Set β}
    (hf : ∀ ⦃a a'⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (hy : y ⊆ f '' x) :
    x ∩ f ⁻¹' (minimals s y) = minimals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} {y : Set β} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (hy : y ⊆ f '' x)
Goal: x ∩ f ⁻¹' minimals s y = minimals r (x ∩ f ⁻¹' y)
-- RelEmbedding.inter_preimage_minimals_eq
theorem RelEmbedding.inter_preimage_minimals_eq (f : r ↪r s) (x : Set α) (y : Set β) :
    x ∩ f⁻¹' (minimals s ((f '' x) ∩ y)) = minimals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (x : Set α) (y : Set β)
Goal: x ∩ ⇑f ⁻¹' minimals s (⇑f '' x ∩ y) = minimals r (x ∩ ⇑f ⁻¹' y)
-- RelEmbedding.inter_preimage_minimals_eq_of_subset
theorem RelEmbedding.inter_preimage_minimals_eq_of_subset {y : Set β} {x : Set α}
    (f : r ↪r s) (h : y ⊆ f '' x) :
    x ∩ f ⁻¹' (minimals s y) = minimals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {y : Set β} {x : Set α} (f : r ↪r s) (h : y ⊆ ⇑f '' x)
Goal: x ∩ ⇑f ⁻¹' minimals s y = minimals r (x ∩ ⇑f ⁻¹' y)
-- RelEmbedding.minimals_preimage_eq
theorem RelEmbedding.minimals_preimage_eq (f : r ↪r s) (y : Set β) :
    minimals r (f ⁻¹' y) = f ⁻¹' minimals s (y ∩ range f) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (y : Set β)
Goal: minimals r (⇑f ⁻¹' y) = ⇑f ⁻¹' minimals s (y ∩ Set.range ⇑f)
-- RelIso.minimals_preimage_eq
theorem RelIso.minimals_preimage_eq (f : r ≃r s) (y : Set β) :
    minimals r (f ⁻¹' y) = f ⁻¹' (minimals s y) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) (y : Set β)
Goal: minimals r (⇑f ⁻¹' y) = ⇑f ⁻¹' minimals s y
-- RelIso.maximals_preimage_eq
theorem RelIso.maximals_preimage_eq (f : r ≃r s) (y : Set β) :
    maximals r (f ⁻¹' y) = f ⁻¹' (maximals s y) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≃r s) (y : Set β)
Goal: maximals r (⇑f ⁻¹' y) = ⇑f ⁻¹' maximals s y
-- inter_maximals_preimage_inter_eq_of_rel_iff_rel_on
theorem inter_maximals_preimage_inter_eq_of_rel_iff_rel_on {x : Set α}
    (hf : ∀ ⦃a a'⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (y : Set β) :
    x ∩ f ⁻¹' (maximals s ((f '' x) ∩ y)) = maximals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (y : Set β)
Goal: x ∩ f ⁻¹' maximals s (f '' x ∩ y) = maximals r (x ∩ f ⁻¹' y)
-- inter_preimage_maximals_eq_of_rel_iff_rel_on_of_subset
theorem inter_preimage_maximals_eq_of_rel_iff_rel_on_of_subset {y : Set β} {x : Set α}
    (hf : ∀ ⦃a a'⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (hy : y ⊆ f '' x) :
    x ∩ f ⁻¹' (maximals s y) = maximals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {f : α → β} {r : α → α → Prop} {s : β → β → Prop} {y : Set β} {x : Set α} (hf : ∀ ⦃a a' : α⦄, a ∈ x → a' ∈ x → (r a a' ↔ s (f a) (f a'))) (hy : y ⊆ f '' x)
Goal: x ∩ f ⁻¹' maximals s y = maximals r (x ∩ f ⁻¹' y)
-- RelEmbedding.inter_preimage_maximals_eq
theorem RelEmbedding.inter_preimage_maximals_eq (f : r ↪r s) (x : Set α) (y : Set β) :
    x ∩ f⁻¹' (maximals s ((f '' x) ∩ y)) = maximals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (x : Set α) (y : Set β)
Goal: x ∩ ⇑f ⁻¹' maximals s (⇑f '' x ∩ y) = maximals r (x ∩ ⇑f ⁻¹' y)
-- RelEmbedding.inter_preimage_maximals_eq_of_subset
theorem RelEmbedding.inter_preimage_maximals_eq_of_subset {y : Set β} {x : Set α}
    (f : r ↪r s) (h : y ⊆ f '' x) : x ∩ f ⁻¹' (maximals s y) = maximals r (x ∩ f ⁻¹' y) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} {y : Set β} {x : Set α} (f : r ↪r s) (h : y ⊆ ⇑f '' x)
Goal: x ∩ ⇑f ⁻¹' maximals s y = maximals r (x ∩ ⇑f ⁻¹' y)
-- RelEmbedding.maximals_preimage_eq
theorem RelEmbedding.maximals_preimage_eq (f : r ↪r s) (y : Set β) :
    maximals r (f ⁻¹' y) = f ⁻¹' maximals s (y ∩ range f) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (y : Set β)
Goal: maximals r (⇑f ⁻¹' y) = ⇑f ⁻¹' maximals s (y ∩ Set.range ⇑f)
-- maximals_Iic
@[simp] theorem maximals_Iic (a : α) : maximals (· ≤ ·) (Iic a) = {a} 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: maximals (fun x x_1 => x ≤ x_1) (Set.Iic a) = {a}
-- minimals_Ici
@[simp] theorem minimals_Ici (a : α) : minimals (· ≤ ·) (Ici a) = {a} 
Arguments: {α : Type u_1} [PartialOrder α] (a : α)
Goal: minimals (fun x x_1 => x ≤ x_1) (Set.Ici a) = {a}
-- maximals_Icc
theorem maximals_Icc (hab : a ≤ b) : maximals (· ≤ ·) (Icc a b) = {b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a ≤ b)
Goal: maximals (fun x x_1 => x ≤ x_1) (Set.Icc a b) = {b}
-- minimals_Icc
theorem minimals_Icc (hab : a ≤ b) : minimals (· ≤ ·) (Icc a b) = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a ≤ b)
Goal: minimals (fun x x_1 => x ≤ x_1) (Set.Icc a b) = {a}
-- maximals_Ioc
theorem maximals_Ioc (hab : a < b) : maximals (· ≤ ·) (Ioc a b) = {b} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a < b)
Goal: maximals (fun x x_1 => x ≤ x_1) (Set.Ioc a b) = {b}
-- minimals_Ico
theorem minimals_Ico (hab : a < b) : minimals (· ≤ ·) (Ico a b) = {a} 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α} (hab : a < b)
Goal: minimals (fun x x_1 => x ≤ x_1) (Set.Ico a b) = {a}
-- supClosed_empty
@[simp] lemma supClosed_empty : SupClosed (∅ : Set α) 
Arguments: {α : Type u_2} [SemilatticeSup α]
Goal: SupClosed ∅
-- supClosed_singleton
@[simp] lemma supClosed_singleton : SupClosed ({a} : Set α) 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: SupClosed {a}
-- supClosed_univ
@[simp] lemma supClosed_univ : SupClosed (univ : Set α) 
Arguments: {α : Type u_2} [SemilatticeSup α]
Goal: SupClosed Set.univ
-- SupClosed.inter
lemma SupClosed.inter (hs : SupClosed s) (ht : SupClosed t) : SupClosed (s ∩ t) 
Arguments: {α : Type u_2} [SemilatticeSup α] {s t : Set α} (hs : SupClosed s) (ht : SupClosed t)
Goal: SupClosed (s ∩ t)
-- supClosed_sInter
lemma supClosed_sInter (hS : ∀ s ∈ S, SupClosed s) : SupClosed (⋂₀ S) 
Arguments: {α : Type u_2} [SemilatticeSup α] {S : Set (Set α)} (hS : ∀ s ∈ S, SupClosed s)
Goal: SupClosed (⋂₀ S)
-- supClosed_iInter
lemma supClosed_iInter (hf : ∀ i, SupClosed (f i)) : SupClosed (⋂ i, f i) 
Arguments: {α : Type u_2} [SemilatticeSup α] {ι : Sort u_4} {f : ι → Set α} (hf : ∀ (i : ι), SupClosed (f i))
Goal: SupClosed (⋂ i, f i)
-- SupClosed.directedOn
lemma SupClosed.directedOn (hs : SupClosed s) : DirectedOn (· ≤ ·) s 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α} (hs : SupClosed s)
Goal: DirectedOn (fun x x_1 => x ≤ x_1) s
-- IsUpperSet.supClosed
lemma IsUpperSet.supClosed (hs : IsUpperSet s) : SupClosed s 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α} (hs : IsUpperSet s)
Goal: SupClosed s
-- SupClosed.preimage
lemma SupClosed.preimage [FunLike F β α] [SupHomClass F β α] (hs : SupClosed s) (f : F) :
    SupClosed (f ⁻¹' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] {s : Set α} [FunLike F β α] [SupHomClass F β α] (hs : SupClosed s) (f : F)
Goal: SupClosed (⇑f ⁻¹' s)
-- SupClosed.image
lemma SupClosed.image [FunLike F α β] [SupHomClass F α β] (hs : SupClosed s) (f : F) :
    SupClosed (f '' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] {s : Set α} [FunLike F α β] [SupHomClass F α β] (hs : SupClosed s) (f : F)
Goal: SupClosed (⇑f '' s)
-- supClosed_range
lemma supClosed_range [FunLike F α β] [SupHomClass F α β] (f : F) : SupClosed (Set.range f) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] [FunLike F α β] [SupHomClass F α β] (f : F)
Goal: SupClosed (Set.range ⇑f)
-- SupClosed.prod
lemma SupClosed.prod {t : Set β} (hs : SupClosed s) (ht : SupClosed t) : SupClosed (s ×ˢ t) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] {s : Set α} {t : Set β} (hs : SupClosed s) (ht : SupClosed t)
Goal: SupClosed (s ×ˢ t)
-- supClosed_pi
lemma supClosed_pi {ι : Type*} {α : ι → Type*} [∀ i, SemilatticeSup (α i)] {s : Set ι}
    {t : ∀ i, Set (α i)} (ht : ∀ i ∈ s, SupClosed (t i)) : SupClosed (s.pi t) 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [(i : ι) → SemilatticeSup (α i)] {s : Set ι} {t : (i : ι) → Set (α i)} (ht : ∀ i ∈ s, SupClosed (t i))
Goal: SupClosed (s.pi t)
-- SupClosed.finsetSup'_mem
lemma SupClosed.finsetSup'_mem (hs : SupClosed s) (ht : t.Nonempty) :
    (∀ i ∈ t, f i ∈ s) → t.sup' ht f ∈ s 
Arguments: {α : Type u_2} [SemilatticeSup α] {ι : Type u_4} {f : ι → α} {s : Set α} {t : Finset ι} (hs : SupClosed s) (ht : t.Nonempty)
Goal: (∀ i ∈ t, f i ∈ s) → t.sup' ht f ∈ s
-- SupClosed.finsetSup_mem
lemma SupClosed.finsetSup_mem [OrderBot α] (hs : SupClosed s) (ht : t.Nonempty) :
    (∀ i ∈ t, f i ∈ s) → t.sup f ∈ s 
Arguments: {α : Type u_2} [SemilatticeSup α] {ι : Type u_4} {f : ι → α} {s : Set α} {t : Finset ι} [OrderBot α] (hs : SupClosed s) (ht : t.Nonempty)
Goal: (∀ i ∈ t, f i ∈ s) → t.sup f ∈ s
-- infClosed_empty
@[simp] lemma infClosed_empty : InfClosed (∅ : Set α) 
Arguments: {α : Type u_2} [SemilatticeInf α]
Goal: InfClosed ∅
-- infClosed_singleton
@[simp] lemma infClosed_singleton : InfClosed ({a} : Set α) 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: InfClosed {a}
-- infClosed_univ
@[simp] lemma infClosed_univ : InfClosed (univ : Set α) 
Arguments: {α : Type u_2} [SemilatticeInf α]
Goal: InfClosed Set.univ
-- InfClosed.inter
lemma InfClosed.inter (hs : InfClosed s) (ht : InfClosed t) : InfClosed (s ∩ t) 
Arguments: {α : Type u_2} [SemilatticeInf α] {s t : Set α} (hs : InfClosed s) (ht : InfClosed t)
Goal: InfClosed (s ∩ t)
-- infClosed_sInter
lemma infClosed_sInter (hS : ∀ s ∈ S, InfClosed s) : InfClosed (⋂₀ S) 
Arguments: {α : Type u_2} [SemilatticeInf α] {S : Set (Set α)} (hS : ∀ s ∈ S, InfClosed s)
Goal: InfClosed (⋂₀ S)
-- infClosed_iInter
lemma infClosed_iInter (hf : ∀ i, InfClosed (f i)) : InfClosed (⋂ i, f i) 
Arguments: {α : Type u_2} [SemilatticeInf α] {ι : Sort u_4} {f : ι → Set α} (hf : ∀ (i : ι), InfClosed (f i))
Goal: InfClosed (⋂ i, f i)
-- InfClosed.codirectedOn
lemma InfClosed.codirectedOn (hs : InfClosed s) : DirectedOn (· ≥ ·) s 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α} (hs : InfClosed s)
Goal: DirectedOn (fun x x_1 => x ≥ x_1) s
-- IsLowerSet.infClosed
lemma IsLowerSet.infClosed (hs : IsLowerSet s) : InfClosed s 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α} (hs : IsLowerSet s)
Goal: InfClosed s
-- InfClosed.preimage
lemma InfClosed.preimage [FunLike F β α] [InfHomClass F β α] (hs : InfClosed s) (f : F) :
    InfClosed (f ⁻¹' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] {s : Set α} [FunLike F β α] [InfHomClass F β α] (hs : InfClosed s) (f : F)
Goal: InfClosed (⇑f ⁻¹' s)
-- InfClosed.image
lemma InfClosed.image [FunLike F α β] [InfHomClass F α β] (hs : InfClosed s) (f : F) :
    InfClosed (f '' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] {s : Set α} [FunLike F α β] [InfHomClass F α β] (hs : InfClosed s) (f : F)
Goal: InfClosed (⇑f '' s)
-- infClosed_range
lemma infClosed_range [FunLike F α β] [InfHomClass F α β] (f : F) : InfClosed (Set.range f) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] [FunLike F α β] [InfHomClass F α β] (f : F)
Goal: InfClosed (Set.range ⇑f)
-- InfClosed.prod
lemma InfClosed.prod {t : Set β} (hs : InfClosed s) (ht : InfClosed t) : InfClosed (s ×ˢ t) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] {s : Set α} {t : Set β} (hs : InfClosed s) (ht : InfClosed t)
Goal: InfClosed (s ×ˢ t)
-- infClosed_pi
lemma infClosed_pi {ι : Type*} {α : ι → Type*} [∀ i, SemilatticeInf (α i)] {s : Set ι}
    {t : ∀ i, Set (α i)} (ht : ∀ i ∈ s, InfClosed (t i)) : InfClosed (s.pi t) 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [(i : ι) → SemilatticeInf (α i)] {s : Set ι} {t : (i : ι) → Set (α i)} (ht : ∀ i ∈ s, InfClosed (t i))
Goal: InfClosed (s.pi t)
-- InfClosed.finsetInf'_mem
lemma InfClosed.finsetInf'_mem (hs : InfClosed s) (ht : t.Nonempty) :
    (∀ i ∈ t, f i ∈ s) → t.inf' ht f ∈ s 
Arguments: {α : Type u_2} [SemilatticeInf α] {ι : Type u_4} {f : ι → α} {s : Set α} {t : Finset ι} (hs : InfClosed s) (ht : t.Nonempty)
Goal: (∀ i ∈ t, f i ∈ s) → t.inf' ht f ∈ s
-- InfClosed.finsetInf_mem
lemma InfClosed.finsetInf_mem [OrderTop α] (hs : InfClosed s) (ht : t.Nonempty) :
    (∀ i ∈ t, f i ∈ s) → t.inf f ∈ s 
Arguments: {α : Type u_2} [SemilatticeInf α] {ι : Type u_4} {f : ι → α} {s : Set α} {t : Finset ι} [OrderTop α] (hs : InfClosed s) (ht : t.Nonempty)
Goal: (∀ i ∈ t, f i ∈ s) → t.inf f ∈ s
-- IsSublattice.infClosed
structure IsSublattice (s : Set α) : Prop where
  supClosed : SupClosed s
  infClosed : InfClosed s
Arguments: {α : Type u_2} [Lattice α] {s : Set α} (self : IsSublattice s)
Goal: InfClosed s
-- IsSublattice.supClosed
structure IsSublattice (s : Set α) : Prop where
  supClosed : SupClosed s
  infClosed : InfClosed s
Arguments: {α : Type u_2} [Lattice α] {s : Set α} (self : IsSublattice s)
Goal: SupClosed s
-- isSublattice_empty
@[simp] lemma isSublattice_empty : IsSublattice (∅ : Set α) 
Arguments: {α : Type u_2} [Lattice α]
Goal: IsSublattice ∅
-- isSublattice_singleton
@[simp] lemma isSublattice_singleton : IsSublattice ({a} : Set α) 
Arguments: {α : Type u_2} [Lattice α] {a : α}
Goal: IsSublattice {a}
-- isSublattice_univ
@[simp] lemma isSublattice_univ : IsSublattice (Set.univ : Set α) 
Arguments: {α : Type u_2} [Lattice α]
Goal: IsSublattice Set.univ
-- IsSublattice.inter
lemma IsSublattice.inter (hs : IsSublattice s) (ht : IsSublattice t) : IsSublattice (s ∩ t) 
Arguments: {α : Type u_2} [Lattice α] {s t : Set α} (hs : IsSublattice s) (ht : IsSublattice t)
Goal: IsSublattice (s ∩ t)
-- isSublattice_sInter
lemma isSublattice_sInter (hS : ∀ s ∈ S, IsSublattice s) : IsSublattice (⋂₀ S) 
Arguments: {α : Type u_2} [Lattice α] {S : Set (Set α)} (hS : ∀ s ∈ S, IsSublattice s)
Goal: IsSublattice (⋂₀ S)
-- isSublattice_iInter
lemma isSublattice_iInter (hf : ∀ i, IsSublattice (f i)) : IsSublattice (⋂ i, f i) 
Arguments: {α : Type u_2} {ι : Sort u_4} [Lattice α] {f : ι → Set α} (hf : ∀ (i : ι), IsSublattice (f i))
Goal: IsSublattice (⋂ i, f i)
-- IsSublattice.preimage
lemma IsSublattice.preimage [FunLike F β α] [LatticeHomClass F β α]
    (hs : IsSublattice s) (f : F) :
    IsSublattice (f ⁻¹' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {s : Set α} [FunLike F β α] [LatticeHomClass F β α] (hs : IsSublattice s) (f : F)
Goal: IsSublattice (⇑f ⁻¹' s)
-- IsSublattice.image
lemma IsSublattice.image [FunLike F α β] [LatticeHomClass F α β] (hs : IsSublattice s) (f : F) :
    IsSublattice (f '' s) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {s : Set α} [FunLike F α β] [LatticeHomClass F α β] (hs : IsSublattice s) (f : F)
Goal: IsSublattice (⇑f '' s)
-- IsSublattice_range
lemma IsSublattice_range [FunLike F α β] [LatticeHomClass F α β] (f : F) :
    IsSublattice (Set.range f) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] [FunLike F α β] [LatticeHomClass F α β] (f : F)
Goal: IsSublattice (Set.range ⇑f)
-- IsSublattice.prod
lemma IsSublattice.prod {t : Set β} (hs : IsSublattice s) (ht : IsSublattice t) :
    IsSublattice (s ×ˢ t) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {s : Set α} {t : Set β} (hs : IsSublattice s) (ht : IsSublattice t)
Goal: IsSublattice (s ×ˢ t)
-- isSublattice_pi
lemma isSublattice_pi {ι : Type*} {α : ι → Type*} [∀ i, Lattice (α i)] {s : Set ι}
    {t : ∀ i, Set (α i)} (ht : ∀ i ∈ s, IsSublattice (t i)) : IsSublattice (s.pi t) 
Arguments: {ι : Type u_5} {α : ι → Type u_6} [(i : ι) → Lattice (α i)] {s : Set ι} {t : (i : ι) → Set (α i)} (ht : ∀ i ∈ s, IsSublattice (t i))
Goal: IsSublattice (s.pi t)
-- supClosed_preimage_toDual
@[simp] lemma supClosed_preimage_toDual {s : Set αᵒᵈ} :
    SupClosed (toDual ⁻¹' s) ↔ InfClosed s 
Arguments: {α : Type u_2} [Lattice α] {s : Set αᵒᵈ}
Goal: SupClosed (⇑OrderDual.toDual ⁻¹' s) ↔ InfClosed s
-- infClosed_preimage_toDual
@[simp] lemma infClosed_preimage_toDual {s : Set αᵒᵈ} :
    InfClosed (toDual ⁻¹' s) ↔ SupClosed s 
Arguments: {α : Type u_2} [Lattice α] {s : Set αᵒᵈ}
Goal: InfClosed (⇑OrderDual.toDual ⁻¹' s) ↔ SupClosed s
-- supClosed_preimage_ofDual
@[simp] lemma supClosed_preimage_ofDual {s : Set α} :
    SupClosed (ofDual ⁻¹' s) ↔ InfClosed s 
Arguments: {α : Type u_2} [Lattice α] {s : Set α}
Goal: SupClosed (⇑OrderDual.ofDual ⁻¹' s) ↔ InfClosed s
-- infClosed_preimage_ofDual
@[simp] lemma infClosed_preimage_ofDual {s : Set α} :
    InfClosed (ofDual ⁻¹' s) ↔ SupClosed s 
Arguments: {α : Type u_2} [Lattice α] {s : Set α}
Goal: InfClosed (⇑OrderDual.ofDual ⁻¹' s) ↔ SupClosed s
-- isSublattice_preimage_toDual
@[simp] lemma isSublattice_preimage_toDual {s : Set αᵒᵈ} :
    IsSublattice (toDual ⁻¹' s) ↔ IsSublattice s 
Arguments: {α : Type u_2} [Lattice α] {s : Set αᵒᵈ}
Goal: IsSublattice (⇑OrderDual.toDual ⁻¹' s) ↔ IsSublattice s
-- isSublattice_preimage_ofDual
@[simp] lemma isSublattice_preimage_ofDual :
    IsSublattice (ofDual ⁻¹' s) ↔ IsSublattice s 
Arguments: {α : Type u_2} [Lattice α] {s : Set α}
Goal: IsSublattice (⇑OrderDual.ofDual ⁻¹' s) ↔ IsSublattice s
-- LinearOrder.supClosed
@[simp] protected lemma LinearOrder.supClosed (s : Set α) : SupClosed s 
Arguments: {α : Type u_2} [LinearOrder α] (s : Set α)
Goal: SupClosed s
-- LinearOrder.infClosed
@[simp] protected lemma LinearOrder.infClosed (s : Set α) : InfClosed s 
Arguments: {α : Type u_2} [LinearOrder α] (s : Set α)
Goal: InfClosed s
-- LinearOrder.isSublattice
@[simp] protected lemma LinearOrder.isSublattice (s : Set α) : IsSublattice s 
Arguments: {α : Type u_2} [LinearOrder α] (s : Set α)
Goal: IsSublattice s
-- supClosure_isClosed
@[simps! isClosed]
def supClosure : ClosureOperator (Set α) 
Arguments: {α : Type u_2} [SemilatticeSup α] (s : Set α)
Goal: supClosure.IsClosed s = SupClosed s
-- subset_supClosure
@[simp] lemma subset_supClosure {s : Set α} : s ⊆ supClosure s 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α}
Goal: s ⊆ supClosure s
-- supClosed_supClosure
@[simp] lemma supClosed_supClosure : SupClosed (supClosure s) 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α}
Goal: SupClosed (supClosure s)
-- supClosure_mono
lemma supClosure_mono : Monotone (supClosure : Set α → Set α) 
Arguments: {α : Type u_2} [SemilatticeSup α]
Goal: Monotone ⇑supClosure
-- supClosure_eq_self
@[simp] lemma supClosure_eq_self : supClosure s = s ↔ SupClosed s 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α}
Goal: supClosure s = s ↔ SupClosed s
-- supClosure_idem
lemma supClosure_idem (s : Set α) : supClosure (supClosure s) = supClosure s 
Arguments: {α : Type u_2} [SemilatticeSup α] (s : Set α)
Goal: supClosure (supClosure s) = supClosure s
-- supClosure_empty
@[simp] lemma supClosure_empty : supClosure (∅ : Set α) = ∅ 
Arguments: {α : Type u_2} [SemilatticeSup α]
Goal: supClosure ∅ = ∅
-- supClosure_singleton
@[simp] lemma supClosure_singleton : supClosure {a} = {a} 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: supClosure {a} = {a}
-- supClosure_univ
@[simp] lemma supClosure_univ : supClosure (Set.univ : Set α) = Set.univ 
Arguments: {α : Type u_2} [SemilatticeSup α]
Goal: supClosure Set.univ = Set.univ
-- upperBounds_supClosure
@[simp] lemma upperBounds_supClosure (s : Set α) : upperBounds (supClosure s) = upperBounds s 
Arguments: {α : Type u_2} [SemilatticeSup α] (s : Set α)
Goal: upperBounds (supClosure s) = upperBounds s
-- isLUB_supClosure
@[simp] lemma isLUB_supClosure : IsLUB (supClosure s) a ↔ IsLUB s a 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α} {a : α}
Goal: IsLUB (supClosure s) a ↔ IsLUB s a
-- sup_mem_supClosure
lemma sup_mem_supClosure (ha : a ∈ s) (hb : b ∈ s) : a ⊔ b ∈ supClosure s 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α} {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: a ⊔ b ∈ supClosure s
-- finsetSup'_mem_supClosure
lemma finsetSup'_mem_supClosure {ι : Type*} {t : Finset ι} (ht : t.Nonempty) {f : ι → α}
    (hf : ∀ i ∈ t, f i ∈ s) : t.sup' ht f ∈ supClosure s 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α} {ι : Type u_4} {t : Finset ι} (ht : t.Nonempty) {f : ι → α} (hf : ∀ i ∈ t, f i ∈ s)
Goal: t.sup' ht f ∈ supClosure s
-- supClosure_min
lemma supClosure_min : s ⊆ t → SupClosed t → supClosure s ⊆ t 
Arguments: {α : Type u_2} [SemilatticeSup α] {s t : Set α}
Goal: s ⊆ t → SupClosed t → supClosure s ⊆ t
-- Set.Finite.supClosure
protected lemma Set.Finite.supClosure (hs : s.Finite) : (supClosure s).Finite 
Arguments: {α : Type u_2} [SemilatticeSup α] {s : Set α} (hs : s.Finite)
Goal: (supClosure s).Finite
-- supClosure_prod
@[simp] lemma supClosure_prod (s : Set α) (t : Set β) :
    supClosure (s ×ˢ t) = supClosure s ×ˢ supClosure t 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] (s : Set α) (t : Set β)
Goal: supClosure (s ×ˢ t) = supClosure s ×ˢ supClosure t
-- infClosure_isClosed
@[simps! isClosed]
def infClosure : ClosureOperator (Set α) 
Arguments: {α : Type u_2} [SemilatticeInf α] (s : Set α)
Goal: infClosure.IsClosed s = InfClosed s
-- subset_infClosure
@[simp] lemma subset_infClosure {s : Set α} : s ⊆ infClosure s 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α}
Goal: s ⊆ infClosure s
-- infClosed_infClosure
@[simp] lemma infClosed_infClosure : InfClosed (infClosure s) 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α}
Goal: InfClosed (infClosure s)
-- infClosure_mono
lemma infClosure_mono : Monotone (infClosure : Set α → Set α) 
Arguments: {α : Type u_2} [SemilatticeInf α]
Goal: Monotone ⇑infClosure
-- infClosure_eq_self
@[simp] lemma infClosure_eq_self : infClosure s = s ↔ InfClosed s 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α}
Goal: infClosure s = s ↔ InfClosed s
-- infClosure_idem
lemma infClosure_idem (s : Set α) : infClosure (infClosure s) = infClosure s 
Arguments: {α : Type u_2} [SemilatticeInf α] (s : Set α)
Goal: infClosure (infClosure s) = infClosure s
-- infClosure_empty
@[simp] lemma infClosure_empty : infClosure (∅ : Set α) = ∅ 
Arguments: {α : Type u_2} [SemilatticeInf α]
Goal: infClosure ∅ = ∅
-- infClosure_singleton
@[simp] lemma infClosure_singleton : infClosure {a} = {a} 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: infClosure {a} = {a}
-- infClosure_univ
@[simp] lemma infClosure_univ : infClosure (Set.univ : Set α) = Set.univ 
Arguments: {α : Type u_2} [SemilatticeInf α]
Goal: infClosure Set.univ = Set.univ
-- lowerBounds_infClosure
@[simp] lemma lowerBounds_infClosure (s : Set α) : lowerBounds (infClosure s) = lowerBounds s 
Arguments: {α : Type u_2} [SemilatticeInf α] (s : Set α)
Goal: lowerBounds (infClosure s) = lowerBounds s
-- isGLB_infClosure
@[simp] lemma isGLB_infClosure : IsGLB (infClosure s) a ↔ IsGLB s a 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α} {a : α}
Goal: IsGLB (infClosure s) a ↔ IsGLB s a
-- inf_mem_infClosure
lemma inf_mem_infClosure (ha : a ∈ s) (hb : b ∈ s) : a ⊓ b ∈ infClosure s 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α} {a b : α} (ha : a ∈ s) (hb : b ∈ s)
Goal: a ⊓ b ∈ infClosure s
-- finsetInf'_mem_infClosure
lemma finsetInf'_mem_infClosure {ι : Type*} {t : Finset ι} (ht : t.Nonempty) {f : ι → α}
    (hf : ∀ i ∈ t, f i ∈ s) : t.inf' ht f ∈ infClosure s 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α} {ι : Type u_4} {t : Finset ι} (ht : t.Nonempty) {f : ι → α} (hf : ∀ i ∈ t, f i ∈ s)
Goal: t.inf' ht f ∈ infClosure s
-- infClosure_min
lemma infClosure_min : s ⊆ t → InfClosed t → infClosure s ⊆ t 
Arguments: {α : Type u_2} [SemilatticeInf α] {s t : Set α}
Goal: s ⊆ t → InfClosed t → infClosure s ⊆ t
-- Set.Finite.infClosure
protected lemma Set.Finite.infClosure (hs : s.Finite) : (infClosure s).Finite 
Arguments: {α : Type u_2} [SemilatticeInf α] {s : Set α} (hs : s.Finite)
Goal: (infClosure s).Finite