ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.SupIndep.image
theorem SupIndep.image [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g)) :
    (s.image g).SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [Lattice α] [OrderBot α] {f : ι → α} [DecidableEq ι] {s : Finset ι'} {g : ι' → ι} (hs : s.SupIndep (f ∘ g))
Goal: (Finset.image g s).SupIndep f
-- Finset.supIndep_map
theorem supIndep_map {s : Finset ι'} {g : ι' ↪ ι} : (s.map g).SupIndep f ↔ s.SupIndep (f ∘ g) 
Arguments: {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [Lattice α] [OrderBot α] {f : ι → α} {s : Finset ι'} {g : ι' ↪ ι}
Goal: (Finset.map g s).SupIndep f ↔ s.SupIndep (f ∘ ⇑g)
-- Finset.supIndep_pair
@[simp]
theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
    ({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) 
Arguments: {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {f : ι → α} [DecidableEq ι] {i j : ι} (hij : i ≠ j)
Goal: {i, j}.SupIndep f ↔ Disjoint (f i) (f j)
-- Finset.supIndep_univ_bool
theorem supIndep_univ_bool (f : Bool → α) :
    (Finset.univ : Finset Bool).SupIndep f ↔ Disjoint (f false) (f true) 
Arguments: {α : Type u_1} [Lattice α] [OrderBot α] (f : Bool → α)
Goal: Finset.univ.SupIndep f ↔ Disjoint (f false) (f true)
-- Finset.supIndep_univ_fin_two
@[simp]
theorem supIndep_univ_fin_two (f : Fin 2 → α) :
    (Finset.univ : Finset (Fin 2)).SupIndep f ↔ Disjoint (f 0) (f 1) 
Arguments: {α : Type u_1} [Lattice α] [OrderBot α] (f : Fin 2 → α)
Goal: Finset.univ.SupIndep f ↔ Disjoint (f 0) (f 1)
-- Finset.SupIndep.attach
theorem SupIndep.attach (hs : s.SupIndep f) : s.attach.SupIndep fun a => f a 
Arguments: {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α} (hs : s.SupIndep f)
Goal: s.attach.SupIndep fun a => f ↑a
-- Finset.supIndep_attach
@[simp, nolint simpNF]
theorem supIndep_attach : (s.attach.SupIndep fun a => f a) ↔ s.SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} [Lattice α] [OrderBot α] {s : Finset ι} {f : ι → α}
Goal: (s.attach.SupIndep fun a => f ↑a) ↔ s.SupIndep f
-- Finset.supIndep_iff_pairwiseDisjoint
theorem supIndep_iff_pairwiseDisjoint : s.SupIndep f ↔ (s : Set ι).PairwiseDisjoint f 
Arguments: {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α}
Goal: s.SupIndep f ↔ (↑s).PairwiseDisjoint f
-- Finset.SupIndep.sup
theorem SupIndep.sup [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
    (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) :
    (s.sup g).SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f)
Goal: (s.sup g).SupIndep f
-- Finset.SupIndep.biUnion
theorem SupIndep.biUnion [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α}
    (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f) :
    (s.biUnion g).SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] [DecidableEq ι] {s : Finset ι'} {g : ι' → Finset ι} {f : ι → α} (hs : s.SupIndep fun i => (g i).sup f) (hg : ∀ i' ∈ s, (g i').SupIndep f)
Goal: (s.biUnion g).SupIndep f
-- Finset.SupIndep.sigma
theorem SupIndep.sigma {β : ι → Type*} {s : Finset ι} {g : ∀ i, Finset (β i)} {f : Sigma β → α}
    (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩)
    (hg : ∀ i ∈ s, (g i).SupIndep fun b => f ⟨i, b⟩) : (s.sigma g).SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} [DistribLattice α] [OrderBot α] {β : ι → Type u_5} {s : Finset ι} {g : (i : ι) → Finset (β i)} {f : Sigma β → α} (hs : s.SupIndep fun i => (g i).sup fun b => f ⟨i, b⟩) (hg : ∀ i ∈ s, (g i).SupIndep fun b => f ⟨i, b⟩)
Goal: (s.sigma g).SupIndep f
-- Finset.SupIndep.product
theorem SupIndep.product {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α}
    (hs : s.SupIndep fun i => t.sup fun i' => f (i, i'))
    (ht : t.SupIndep fun i' => s.sup fun i => f (i, i')) : (s ×ˢ t).SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} (hs : s.SupIndep fun i => t.sup fun i' => f (i, i')) (ht : t.SupIndep fun i' => s.sup fun i => f (i, i'))
Goal: (s ×ˢ t).SupIndep f
-- Finset.supIndep_product_iff
theorem supIndep_product_iff {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α} :
    (s.product t).SupIndep f ↔ (s.SupIndep fun i => t.sup fun i' => f (i, i'))
      ∧ t.SupIndep fun i' => s.sup fun i => f (i, i') 
Arguments: {α : Type u_1} {ι : Type u_3} {ι' : Type u_4} [DistribLattice α] [OrderBot α] {s : Finset ι} {t : Finset ι'} {f : ι × ι' → α}
Goal: (s.product t).SupIndep f ↔
  (s.SupIndep fun i => t.sup fun i' => f (i, i')) ∧ t.SupIndep fun i' => s.sup fun i => f (i, i')
-- CompleteLattice.setIndependent_empty
@[simp]
theorem setIndependent_empty : SetIndependent (∅ : Set α) 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: CompleteLattice.SetIndependent ∅
-- CompleteLattice.SetIndependent.mono
theorem SetIndependent.mono {t : Set α} (hst : t ⊆ s) : SetIndependent t 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : CompleteLattice.SetIndependent s) {t : Set α} (hst : t ⊆ s)
Goal: CompleteLattice.SetIndependent t
-- CompleteLattice.SetIndependent.pairwiseDisjoint
theorem SetIndependent.pairwiseDisjoint : s.PairwiseDisjoint id 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : CompleteLattice.SetIndependent s)
Goal: s.PairwiseDisjoint id
-- CompleteLattice.setIndependent_singleton
theorem setIndependent_singleton (a : α) : SetIndependent ({a} : Set α) 
Arguments: {α : Type u_1} [CompleteLattice α] (a : α)
Goal: CompleteLattice.SetIndependent {a}
-- CompleteLattice.setIndependent_pair
theorem setIndependent_pair {a b : α} (hab : a ≠ b) :
    SetIndependent ({a, b} : Set α) ↔ Disjoint a b 
Arguments: {α : Type u_1} [CompleteLattice α] {a b : α} (hab : a ≠ b)
Goal: CompleteLattice.SetIndependent {a, b} ↔ Disjoint a b
-- CompleteLattice.SetIndependent.disjoint_sSup
theorem SetIndependent.disjoint_sSup {x : α} {y : Set α} (hx : x ∈ s) (hy : y ⊆ s) (hxy : x ∉ y) :
    Disjoint x (sSup y) 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : CompleteLattice.SetIndependent s) {x : α} {y : Set α} (hx : x ∈ s) (hy : y ⊆ s) (hxy : x ∉ y)
Goal: Disjoint x (sSup y)
-- CompleteLattice.setIndependent_iff
theorem setIndependent_iff {α : Type*} [CompleteLattice α] (s : Set α) :
    SetIndependent s ↔ Independent ((↑) : s → α) 
Arguments: {α : Type u_5} [CompleteLattice α] (s : Set α)
Goal: CompleteLattice.SetIndependent s ↔ CompleteLattice.Independent Subtype.val
-- CompleteLattice.independent_def
theorem independent_def : Independent t ↔ ∀ i : ι, Disjoint (t i) (⨆ (j) (_ : j ≠ i), t j) 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α}
Goal: CompleteLattice.Independent t ↔ ∀ (i : ι), Disjoint (t i) (⨆ j, ⨆ (_ : j ≠ i), t j)
-- CompleteLattice.independent_def'
theorem independent_def' : Independent t ↔ ∀ i, Disjoint (t i) (sSup (t '' { j | j ≠ i })) 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α}
Goal: CompleteLattice.Independent t ↔ ∀ (i : ι), Disjoint (t i) (sSup (t '' {j | j ≠ i}))
-- CompleteLattice.independent_def''
theorem independent_def'' :
    Independent t ↔ ∀ i, Disjoint (t i) (sSup { a | ∃ j ≠ i, t j = a }) 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α}
Goal: CompleteLattice.Independent t ↔ ∀ (i : ι), Disjoint (t i) (sSup {a | ∃ j, j ≠ i ∧ t j = a})
-- CompleteLattice.independent_empty
@[simp]
theorem independent_empty (t : Empty → α) : Independent t 
Arguments: {α : Type u_1} [CompleteLattice α] (t : Empty → α)
Goal: CompleteLattice.Independent t
-- CompleteLattice.independent_pempty
@[simp]
theorem independent_pempty (t : PEmpty → α) : Independent t 
Arguments: {α : Type u_1} [CompleteLattice α] (t : PEmpty.{u_5} → α)
Goal: CompleteLattice.Independent t
-- CompleteLattice.Independent.pairwiseDisjoint
theorem Independent.pairwiseDisjoint : Pairwise (Disjoint on t) 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : CompleteLattice.Independent t)
Goal: Pairwise (Disjoint on t)
-- CompleteLattice.Independent.mono
theorem Independent.mono {s t : ι → α} (hs : Independent s) (hst : t ≤ s) : Independent t 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {s t : ι → α} (hs : CompleteLattice.Independent s) (hst : t ≤ s)
Goal: CompleteLattice.Independent t
-- CompleteLattice.Independent.comp
theorem Independent.comp {ι ι' : Sort*} {t : ι → α} {f : ι' → ι} (ht : Independent t)
    (hf : Injective f) : Independent (t ∘ f) 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Sort u_5} {ι' : Sort u_6} {t : ι → α} {f : ι' → ι} (ht : CompleteLattice.Independent t) (hf : Function.Injective f)
Goal: CompleteLattice.Independent (t ∘ f)
-- CompleteLattice.Independent.comp'
theorem Independent.comp' {ι ι' : Sort*} {t : ι → α} {f : ι' → ι} (ht : Independent <| t ∘ f)
    (hf : Surjective f) : Independent t 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Sort u_5} {ι' : Sort u_6} {t : ι → α} {f : ι' → ι} (ht : CompleteLattice.Independent (t ∘ f)) (hf : Function.Surjective f)
Goal: CompleteLattice.Independent t
-- CompleteLattice.Independent.setIndependent_range
theorem Independent.setIndependent_range (ht : Independent t) : SetIndependent <| range t 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : CompleteLattice.Independent t)
Goal: CompleteLattice.SetIndependent (Set.range t)
-- CompleteLattice.independent_ne_bot_iff_independent
@[simp]
theorem independent_ne_bot_iff_independent :
    Independent (fun i : {i // t i ≠ ⊥} ↦ t i) ↔ Independent t 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α}
Goal: (CompleteLattice.Independent fun i => t ↑i) ↔ CompleteLattice.Independent t
-- CompleteLattice.Independent.injOn
theorem Independent.injOn (ht : Independent t) : InjOn t {i | t i ≠ ⊥} 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : CompleteLattice.Independent t)
Goal: Set.InjOn t {i | t i ≠ ⊥}
-- CompleteLattice.Independent.injective
theorem Independent.injective (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Injective t 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} (ht : CompleteLattice.Independent t) (h_ne_bot : ∀ (i : ι), t i ≠ ⊥)
Goal: Function.Injective t
-- CompleteLattice.independent_pair
theorem independent_pair {i j : ι} (hij : i ≠ j) (huniv : ∀ k, k = i ∨ k = j) :
    Independent t ↔ Disjoint (t i) (t j) 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {t : ι → α} {i j : ι} (hij : i ≠ j) (huniv : ∀ (k : ι), k = i ∨ k = j)
Goal: CompleteLattice.Independent t ↔ Disjoint (t i) (t j)
-- CompleteLattice.Independent.map_orderIso
theorem Independent.map_orderIso {ι : Sort*} {α β : Type*} [CompleteLattice α]
    [CompleteLattice β] (f : α ≃o β) {a : ι → α} (ha : Independent a) : Independent (f ∘ a) 
Arguments: {ι : Sort u_5} {α : Type u_6} {β : Type u_7} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) {a : ι → α} (ha : CompleteLattice.Independent a)
Goal: CompleteLattice.Independent (⇑f ∘ a)
-- CompleteLattice.independent_map_orderIso_iff
@[simp]
theorem independent_map_orderIso_iff {ι : Sort*} {α β : Type*} [CompleteLattice α]
    [CompleteLattice β] (f : α ≃o β) {a : ι → α} : Independent (f ∘ a) ↔ Independent a 
Arguments: {ι : Sort u_5} {α : Type u_6} {β : Type u_7} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) {a : ι → α}
Goal: CompleteLattice.Independent (⇑f ∘ a) ↔ CompleteLattice.Independent a
-- CompleteLattice.Independent.disjoint_biSup
theorem Independent.disjoint_biSup {ι : Type*} {α : Type*} [CompleteLattice α] {t : ι → α}
    (ht : Independent t) {x : ι} {y : Set ι} (hx : x ∉ y) : Disjoint (t x) (⨆ i ∈ y, t i) 
Arguments: {ι : Type u_5} {α : Type u_6} [CompleteLattice α] {t : ι → α} (ht : CompleteLattice.Independent t) {x : ι} {y : Set ι} (hx : x ∉ y)
Goal: Disjoint (t x) (⨆ i ∈ y, t i)
-- CompleteLattice.independent_of_independent_coe_Iic_comp
lemma independent_of_independent_coe_Iic_comp {ι : Sort*} {a : α} {t : ι → Set.Iic a}
    (ht : Independent ((↑) ∘ t : ι → α)) : Independent t 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Sort u_5} {a : α} {t : ι → ↑(Set.Iic a)} (ht : CompleteLattice.Independent (Subtype.val ∘ t))
Goal: CompleteLattice.Independent t
-- CompleteLattice.independent_iff_supIndep
theorem CompleteLattice.independent_iff_supIndep [CompleteLattice α] {s : Finset ι} {f : ι → α} :
    CompleteLattice.Independent (f ∘ ((↑) : s → ι)) ↔ s.SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {s : Finset ι} {f : ι → α}
Goal: CompleteLattice.Independent (f ∘ Subtype.val) ↔ s.SupIndep f
-- CompleteLattice.Independent.supIndep'
theorem CompleteLattice.Independent.supIndep' [CompleteLattice α] {f : ι → α} (s : Finset ι)
    (h : CompleteLattice.Independent f) : s.SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] {f : ι → α} (s : Finset ι) (h : CompleteLattice.Independent f)
Goal: s.SupIndep f
-- CompleteLattice.independent_iff_supIndep_univ
theorem CompleteLattice.independent_iff_supIndep_univ [CompleteLattice α] [Fintype ι] {f : ι → α} :
    CompleteLattice.Independent f ↔ Finset.univ.SupIndep f 
Arguments: {α : Type u_1} {ι : Type u_3} [CompleteLattice α] [Fintype ι] {f : ι → α}
Goal: CompleteLattice.Independent f ↔ Finset.univ.SupIndep f
-- CompleteLattice.setIndependent_iff_pairwiseDisjoint
theorem setIndependent_iff_pairwiseDisjoint {s : Set α} :
    SetIndependent s ↔ s.PairwiseDisjoint id 
Arguments: {α : Type u_1} [Order.Frame α] {s : Set α}
Goal: CompleteLattice.SetIndependent s ↔ s.PairwiseDisjoint id
-- CompleteLattice.independent_iff_pairwiseDisjoint
theorem independent_iff_pairwiseDisjoint {f : ι → α} : Independent f ↔ Pairwise (Disjoint on f) 
Arguments: {α : Type u_1} {ι : Type u_3} [Order.Frame α] {f : ι → α}
Goal: CompleteLattice.Independent f ↔ Pairwise (Disjoint on f)
-- csSup_mem_of_not_isSuccLimit
lemma csSup_mem_of_not_isSuccLimit
    (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬ IsSuccLimit (sSup s)) :
    sSup s ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s) (hlim : ¬Order.IsSuccLimit (sSup s))
Goal: sSup s ∈ s
-- csInf_mem_of_not_isPredLimit
lemma csInf_mem_of_not_isPredLimit
    (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬ IsPredLimit (sInf s)) :
    sInf s ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} (hne : s.Nonempty) (hbdd : BddBelow s) (hlim : ¬Order.IsPredLimit (sInf s))
Goal: sInf s ∈ s
-- exists_eq_ciSup_of_not_isSuccLimit
lemma exists_eq_ciSup_of_not_isSuccLimit
    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i 
Arguments: {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} (hf : BddAbove (Set.range f)) (hf' : ¬Order.IsSuccLimit (⨆ i, f i))
Goal: ∃ i, f i = ⨆ i, f i
-- exists_eq_ciInf_of_not_isPredLimit
lemma exists_eq_ciInf_of_not_isPredLimit
    (hf : BddBelow (Set.range f)) (hf' : ¬ IsPredLimit (⨅ i, f i)) :
    ∃ i, f i = ⨅ i, f i 
Arguments: {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} (hf : BddBelow (Set.range f)) (hf' : ¬Order.IsPredLimit (⨅ i, f i))
Goal: ∃ i, f i = ⨅ i, f i
-- IsLUB.mem_of_nonempty_of_not_isSuccLimit
lemma IsLUB.mem_of_nonempty_of_not_isSuccLimit
    (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬ IsSuccLimit x) : x ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} {x : α} (hs : IsLUB s x) (hne : s.Nonempty) (hx : ¬Order.IsSuccLimit x)
Goal: x ∈ s
-- IsGLB.mem_of_nonempty_of_not_isPredLimit
lemma IsGLB.mem_of_nonempty_of_not_isPredLimit
    (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬ IsPredLimit x) : x ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} {x : α} (hs : IsGLB s x) (hne : s.Nonempty) (hx : ¬Order.IsPredLimit x)
Goal: x ∈ s
-- IsLUB.exists_of_nonempty_of_not_isSuccLimit
lemma IsLUB.exists_of_nonempty_of_not_isSuccLimit
    (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccLimit x) :
    ∃ i, f i = x 
Arguments: {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} {x : α} (hf : IsLUB (Set.range f) x) (hx : ¬Order.IsSuccLimit x)
Goal: ∃ i, f i = x
-- IsGLB.exists_of_nonempty_of_not_isPredLimit
lemma IsGLB.exists_of_nonempty_of_not_isPredLimit
    (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredLimit x) :
    ∃ i, f i = x 
Arguments: {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrder α] [Nonempty ι] {f : ι → α} {x : α} (hf : IsGLB (Set.range f) x) (hx : ¬Order.IsPredLimit x)
Goal: ∃ i, f i = x
-- csSup_mem_of_not_isSuccLimit'
lemma csSup_mem_of_not_isSuccLimit'
    (hbdd : BddAbove s) (hlim : ¬ IsSuccLimit (sSup s)) :
    sSup s ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hbdd : BddAbove s) (hlim : ¬Order.IsSuccLimit (sSup s))
Goal: sSup s ∈ s
-- exists_eq_ciSup_of_not_isSuccLimit'
lemma exists_eq_ciSup_of_not_isSuccLimit'
    (hf : BddAbove (Set.range f)) (hf' : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i 
Arguments: {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} (hf : BddAbove (Set.range f)) (hf' : ¬Order.IsSuccLimit (⨆ i, f i))
Goal: ∃ i, f i = ⨆ i, f i
-- IsLUB.mem_of_not_isSuccLimit
lemma IsLUB.mem_of_not_isSuccLimit (hs : IsLUB s x) (hx : ¬ IsSuccLimit x) :
    x ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {x : α} (hs : IsLUB s x) (hx : ¬Order.IsSuccLimit x)
Goal: x ∈ s
-- IsLUB.exists_of_not_isSuccLimit
lemma IsLUB.exists_of_not_isSuccLimit (hf : IsLUB (Set.range f) x) (hx : ¬ IsSuccLimit x) :
    ∃ i, f i = x 
Arguments: {ι : Type u_1} {α : Type u_2} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {x : α} (hf : IsLUB (Set.range f) x) (hx : ¬Order.IsSuccLimit x)
Goal: ∃ i, f i = x
-- sSup_mem_of_not_isSuccLimit
lemma sSup_mem_of_not_isSuccLimit (hlim : ¬ IsSuccLimit (sSup s)) :
    sSup s ∈ s 
Arguments: {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsSuccLimit (sSup s))
Goal: sSup s ∈ s
-- sInf_mem_of_not_isPredLimit
lemma sInf_mem_of_not_isPredLimit (hlim : ¬ IsPredLimit (sInf s)) :
    sInf s ∈ s 
Arguments: {α : Type u_2} [CompleteLinearOrder α] {s : Set α} (hlim : ¬Order.IsPredLimit (sInf s))
Goal: sInf s ∈ s
-- exists_eq_iSup_of_not_isSuccLimit
lemma exists_eq_iSup_of_not_isSuccLimit (hf : ¬ IsSuccLimit (⨆ i, f i)) :
    ∃ i, f i = ⨆ i, f i 
Arguments: {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} (hf : ¬Order.IsSuccLimit (⨆ i, f i))
Goal: ∃ i, f i = ⨆ i, f i
-- exists_eq_iInf_of_not_isPredLimit
lemma exists_eq_iInf_of_not_isPredLimit (hf : ¬ IsPredLimit (⨅ i, f i)) :
    ∃ i, f i = ⨅ i, f i 
Arguments: {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} (hf : ¬Order.IsPredLimit (⨅ i, f i))
Goal: ∃ i, f i = ⨅ i, f i
-- IsGLB.mem_of_not_isPredLimit
lemma IsGLB.mem_of_not_isPredLimit (hs : IsGLB s x) (hx : ¬ IsPredLimit x) :
    x ∈ s 
Arguments: {α : Type u_2} [CompleteLinearOrder α] {s : Set α} {x : α} (hs : IsGLB s x) (hx : ¬Order.IsPredLimit x)
Goal: x ∈ s
-- IsGLB.exists_of_not_isPredLimit
lemma IsGLB.exists_of_not_isPredLimit (hf : IsGLB (Set.range f) x) (hx : ¬ IsPredLimit x) :
    ∃ i, f i = x 
Arguments: {ι : Type u_1} {α : Type u_2} [CompleteLinearOrder α] {f : ι → α} {x : α} (hf : IsGLB (Set.range f) x) (hx : ¬Order.IsPredLimit x)
Goal: ∃ i, f i = x
-- SemilatSupCat.coe_of
@[simp]
theorem coe_of (α : Type*) [SemilatticeSup α] [OrderBot α] : ↥(of α) = α 
Arguments: (α : Type u_1) [SemilatticeSup α] [OrderBot α]
Goal: (SemilatSupCat.of α).X = α
-- SemilatSupCat.coe_forget_to_partOrd
@[simp]
theorem coe_forget_to_partOrd (X : SemilatSupCat) :
    ↥((forget₂ SemilatSupCat PartOrd).obj X) = ↥X 
Arguments: (X : SemilatSupCat)
Goal: ↑((CategoryTheory.forget₂ SemilatSupCat PartOrd).obj X) = X.X
-- SemilatInfCat.coe_of
@[simp]
theorem coe_of (α : Type*) [SemilatticeInf α] [OrderTop α] : ↥(of α) = α 
Arguments: (α : Type u_1) [SemilatticeInf α] [OrderTop α]
Goal: (SemilatInfCat.of α).X = α
-- SemilatInfCat.coe_forget_to_partOrd
@[simp]
theorem coe_forget_to_partOrd (X : SemilatInfCat) :
    ↥((forget₂ SemilatInfCat PartOrd).obj X) = ↥X 
Arguments: (X : SemilatInfCat)
Goal: ↑((CategoryTheory.forget₂ SemilatInfCat PartOrd).obj X) = X.X
-- SemilatSupCat.Iso.mk_inv_toSupHom_toFun
@[simps]
def Iso.mk {α β : SemilatSupCat.{u}} (e : α ≃o β) : α ≅ β 
Arguments: {α β : SemilatSupCat} (e : α.X ≃o β.X) (a : β.X)
Goal: (SemilatSupCat.Iso.mk e).inv.toSupHom a = e.symm a
-- SemilatSupCat.Iso.mk_hom_toSupHom_toFun
@[simps]
def Iso.mk {α β : SemilatSupCat.{u}} (e : α ≃o β) : α ≅ β 
Arguments: {α β : SemilatSupCat} (e : α.X ≃o β.X) (a : α.X)
Goal: (SemilatSupCat.Iso.mk e).hom.toSupHom a = e a
-- SemilatSupCat.dual_obj
@[simps]
def dual : SemilatSupCat ⥤ SemilatInfCat 
Arguments: (X : SemilatSupCat)
Goal: SemilatSupCat.dual.obj X = SemilatInfCat.of X.Xᵒᵈ
-- SemilatSupCat.dual_map
@[simps]
def dual : SemilatSupCat ⥤ SemilatInfCat 
Arguments: {X Y : SemilatSupCat} (a : SupBotHom X.X Y.X)
Goal: SemilatSupCat.dual.map a = SupBotHom.dual a
-- SemilatInfCat.Iso.mk_hom_toInfHom_toFun
@[simps]
def Iso.mk {α β : SemilatInfCat.{u}} (e : α ≃o β) : α ≅ β 
Arguments: {α β : SemilatInfCat} (e : α.X ≃o β.X) (a : α.X)
Goal: (SemilatInfCat.Iso.mk e).hom.toInfHom a = e a
-- SemilatInfCat.Iso.mk_inv_toInfHom_toFun
@[simps]
def Iso.mk {α β : SemilatInfCat.{u}} (e : α ≃o β) : α ≅ β 
Arguments: {α β : SemilatInfCat} (e : α.X ≃o β.X) (a : β.X)
Goal: (SemilatInfCat.Iso.mk e).inv.toInfHom a = e.symm a
-- SemilatInfCat.dual_map
@[simps]
def dual : SemilatInfCat ⥤ SemilatSupCat 
Arguments: {X Y : SemilatInfCat} (a : InfTopHom X.X Y.X)
Goal: SemilatInfCat.dual.map a = InfTopHom.dual a
-- SemilatInfCat.dual_obj
@[simps]
def dual : SemilatInfCat ⥤ SemilatSupCat 
Arguments: (X : SemilatInfCat)
Goal: SemilatInfCat.dual.obj X = SemilatSupCat.of X.Xᵒᵈ
-- SemilatSupCatEquivSemilatInfCat_inverse
@[simps functor inverse]
def SemilatSupCatEquivSemilatInfCat : SemilatSupCat ≌ SemilatInfCat 
Goal: SemilatSupCatEquivSemilatInfCat.inverse = SemilatInfCat.dual
-- SemilatSupCatEquivSemilatInfCat_functor
@[simps functor inverse]
def SemilatSupCatEquivSemilatInfCat : SemilatSupCat ≌ SemilatInfCat 
Goal: SemilatSupCatEquivSemilatInfCat.functor = SemilatSupCat.dual
-- SemilatSupCat_dual_comp_forget_to_partOrd
theorem SemilatSupCat_dual_comp_forget_to_partOrd :
    SemilatSupCat.dual ⋙ forget₂ SemilatInfCat PartOrd =
      forget₂ SemilatSupCat PartOrd ⋙ PartOrd.dual 
Goal: SemilatSupCat.dual ⋙ CategoryTheory.forget₂ SemilatInfCat PartOrd =
  CategoryTheory.forget₂ SemilatSupCat PartOrd ⋙ PartOrd.dual
-- SemilatInfCat_dual_comp_forget_to_partOrd
theorem SemilatInfCat_dual_comp_forget_to_partOrd :
    SemilatInfCat.dual ⋙ forget₂ SemilatSupCat PartOrd =
      forget₂ SemilatInfCat PartOrd ⋙ PartOrd.dual 
Goal: SemilatInfCat.dual ⋙ CategoryTheory.forget₂ SemilatSupCat PartOrd =
  CategoryTheory.forget₂ SemilatInfCat PartOrd ⋙ PartOrd.dual
-- IsCoatomic.of_isChain_bounded
theorem IsCoatomic.of_isChain_bounded {α : Type*} [PartialOrder α] [OrderTop α]
    (h :
      ∀ c : Set α,
        IsChain (· ≤ ·) c → c.Nonempty → ⊤ ∉ c → ∃ x ≠ ⊤, x ∈ upperBounds c) :
    IsCoatomic α 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] (h : ∀ (c : Set α), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ⊤ ∉ c → ∃ x, x ≠ ⊤ ∧ x ∈ upperBounds c)
Goal: IsCoatomic α
-- IsAtomic.of_isChain_bounded
theorem IsAtomic.of_isChain_bounded {α : Type*} [PartialOrder α] [OrderBot α]
    (h :
      ∀ c : Set α,
        IsChain (· ≤ ·) c → c.Nonempty → ⊥ ∉ c → ∃ x ≠ ⊥, x ∈ lowerBounds c) :
    IsAtomic α 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] (h : ∀ (c : Set α), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → ⊥ ∉ c → ∃ x, x ≠ ⊥ ∧ x ∈ lowerBounds c)
Goal: IsAtomic α
-- ULift.up_le
@[simp] theorem up_le [LE α] {a b : α} : up a ≤ up b ↔ a ≤ b 
Arguments: {α : Type u} [LE α] {a b : α}
Goal: { down := a } ≤ { down := b } ↔ a ≤ b
-- ULift.down_le
@[simp] theorem down_le [LE α] {a b : ULift α} : down a ≤ down b ↔ a ≤ b 
Arguments: {α : Type u} [LE α] {a b : ULift.{u_1, u} α}
Goal: a.down ≤ b.down ↔ a ≤ b