ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Booleanisation.lift_lt_comp
@[simp] lemma lift_lt_comp : lift a < comp b ↔ Disjoint a b 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] {a b : α}
Goal: Booleanisation.lift a < Booleanisation.comp b ↔ Disjoint a b
-- Booleanisation.not_comp_lt_lift
@[simp] lemma not_comp_lt_lift : ¬ comp a < lift b 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] {a b : α}
Goal: ¬Booleanisation.comp a < Booleanisation.lift b
-- Booleanisation.lift_sup_lift
@[simp] lemma lift_sup_lift (a b : α) : lift a ⊔ lift b = lift (a ⊔ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.lift a ⊔ Booleanisation.lift b = Booleanisation.lift (a ⊔ b)
-- Booleanisation.lift_sup_comp
@[simp] lemma lift_sup_comp (a b : α) : lift a ⊔ comp b = comp (b \ a) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.lift a ⊔ Booleanisation.comp b = Booleanisation.comp (b \ a)
-- Booleanisation.comp_sup_lift
@[simp] lemma comp_sup_lift (a b : α) : comp a ⊔ lift b = comp (a \ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.comp a ⊔ Booleanisation.lift b = Booleanisation.comp (a \ b)
-- Booleanisation.comp_sup_comp
@[simp] lemma comp_sup_comp (a b : α) : comp a ⊔ comp b = comp (a ⊓ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.comp a ⊔ Booleanisation.comp b = Booleanisation.comp (a ⊓ b)
-- Booleanisation.lift_inf_lift
@[simp] lemma lift_inf_lift (a b : α) : lift a ⊓ lift b = lift (a ⊓ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.lift a ⊓ Booleanisation.lift b = Booleanisation.lift (a ⊓ b)
-- Booleanisation.lift_inf_comp
@[simp] lemma lift_inf_comp (a b : α) : lift a ⊓ comp b = lift (a \ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.lift a ⊓ Booleanisation.comp b = Booleanisation.lift (a \ b)
-- Booleanisation.comp_inf_lift
@[simp] lemma comp_inf_lift (a b : α) : comp a ⊓ lift b = lift (b \ a) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.comp a ⊓ Booleanisation.lift b = Booleanisation.lift (b \ a)
-- Booleanisation.comp_inf_comp
@[simp] lemma comp_inf_comp (a b : α) : comp a ⊓ comp b = comp (a ⊔ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.comp a ⊓ Booleanisation.comp b = Booleanisation.comp (a ⊔ b)
-- Booleanisation.lift_bot
@[simp] lemma lift_bot : lift (⊥ : α) = ⊥ 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α]
Goal: Booleanisation.lift ⊥ = ⊥
-- Booleanisation.comp_bot
@[simp] lemma comp_bot : comp (⊥ : α) = ⊤ 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α]
Goal: Booleanisation.comp ⊥ = ⊤
-- Booleanisation.lift_sdiff_lift
@[simp] lemma lift_sdiff_lift (a b : α) : lift a \ lift b = lift (a \ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.lift a \ Booleanisation.lift b = Booleanisation.lift (a \ b)
-- Booleanisation.lift_sdiff_comp
@[simp] lemma lift_sdiff_comp (a b : α) : lift a \ comp b = lift (a ⊓ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.lift a \ Booleanisation.comp b = Booleanisation.lift (a ⊓ b)
-- Booleanisation.comp_sdiff_lift
@[simp] lemma comp_sdiff_lift (a b : α) : comp a \ lift b = comp (a ⊔ b) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.comp a \ Booleanisation.lift b = Booleanisation.comp (a ⊔ b)
-- Booleanisation.comp_sdiff_comp
@[simp] lemma comp_sdiff_comp (a b : α) : comp a \ comp b = lift (b \ a) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α] (a b : α)
Goal: Booleanisation.comp a \ Booleanisation.comp b = Booleanisation.lift (b \ a)
-- Booleanisation.liftLatticeHom_injective
lemma liftLatticeHom_injective : Injective (liftLatticeHom (α := α)) 
Arguments: {α : Type u_1} [GeneralizedBooleanAlgebra α]
Goal: Function.Injective ⇑Booleanisation.liftLatticeHom
-- Pi.toLex_apply
@[simp]
theorem toLex_apply (x : ∀ i, β i) (i : ι) : toLex x i = x i 
Arguments: {ι : Type u_1} {β : ι → Type u_2} (x : (i : ι) → β i) (i : ι)
Goal: toLex x i = x i
-- Pi.ofLex_apply
@[simp]
theorem ofLex_apply (x : Lex (∀ i, β i)) (i : ι) : ofLex x i = x i 
Arguments: {ι : Type u_1} {β : ι → Type u_2} (x : Lex ((i : ι) → β i)) (i : ι)
Goal: ofLex x i = x i
-- Pi.lex_lt_of_lt_of_preorder
theorem lex_lt_of_lt_of_preorder [∀ i, Preorder (β i)] {r} (hwf : WellFounded r) {x y : ∀ i, β i}
    (hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [(i : ι) → Preorder (β i)] {r : ι → ι → Prop} (hwf : WellFounded r) {x y : (i : ι) → β i} (hlt : x < y)
Goal: ∃ i, (∀ (j : ι), r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i
-- Pi.lex_lt_of_lt
theorem lex_lt_of_lt [∀ i, PartialOrder (β i)] {r} (hwf : WellFounded r) {x y : ∀ i, β i}
    (hlt : x < y) : Pi.Lex r (@fun i => (· < ·)) x y 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [(i : ι) → PartialOrder (β i)] {r : ι → ι → Prop} (hwf : WellFounded r) {x y : (i : ι) → β i} (hlt : x < y)
Goal: Pi.Lex r (fun i x x_1 => x < x_1) x y
-- Pi.isTrichotomous_lex
theorem isTrichotomous_lex [∀ i, IsTrichotomous (β i) s] (wf : WellFounded r) :
    IsTrichotomous (∀ i, β i) (Pi.Lex r @s) 
Arguments: {ι : Type u_1} {β : ι → Type u_2} (r : ι → ι → Prop) (s : {i : ι} → β i → β i → Prop) [∀ (i : ι), IsTrichotomous (β i) s] (wf : WellFounded r)
Goal: IsTrichotomous ((i : ι) → β i) (Pi.Lex r s)
-- Pi.toLex_monotone
theorem toLex_monotone : Monotone (@toLex (∀ i, β i)) 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)]
Goal: Monotone ⇑toLex
-- Pi.toLex_strictMono
theorem toLex_strictMono : StrictMono (@toLex (∀ i, β i)) 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)]
Goal: StrictMono ⇑toLex
-- Pi.lt_toLex_update_self_iff
@[simp]
theorem lt_toLex_update_self_iff : toLex x < toLex (update x i a) ↔ x i < a 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i}
Goal: toLex x < toLex (Function.update x i a) ↔ x i < a
-- Pi.toLex_update_lt_self_iff
@[simp]
theorem toLex_update_lt_self_iff : toLex (update x i a) < toLex x ↔ a < x i 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i}
Goal: toLex (Function.update x i a) < toLex x ↔ a < x i
-- Pi.le_toLex_update_self_iff
@[simp]
theorem le_toLex_update_self_iff : toLex x ≤ toLex (update x i a) ↔ x i ≤ a 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i}
Goal: toLex x ≤ toLex (Function.update x i a) ↔ x i ≤ a
-- Pi.toLex_update_le_self_iff
@[simp]
theorem toLex_update_le_self_iff : toLex (update x i a) ≤ toLex x ↔ a ≤ x i 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [LinearOrder ι] [IsWellOrder ι fun x x_1 => x < x_1] [(i : ι) → PartialOrder (β i)] {x : (i : ι) → β i} {i : ι} {a : β i}
Goal: toLex (Function.update x i a) ≤ toLex x ↔ a ≤ x i
-- Pi.Lex.noMaxOrder'
theorem Lex.noMaxOrder' [Preorder ι] [∀ i, LT (β i)] (i : ι) [NoMaxOrder (β i)] :
    NoMaxOrder (Lex (∀ i, β i)) 
Arguments: {ι : Type u_1} {β : ι → Type u_2} [Preorder ι] [(i : ι) → LT (β i)] (i : ι) [NoMaxOrder (β i)]
Goal: NoMaxOrder (Lex ((i : ι) → β i))
-- Pi.lex_desc
theorem lex_desc {α} [Preorder ι] [DecidableEq ι] [Preorder α] {f : ι → α} {i j : ι} (h₁ : i ≤ j)
    (h₂ : f j < f i) : toLex (f ∘ Equiv.swap i j) < toLex f 
Arguments: {ι : Type u_1} {α : Type u_3} [Preorder ι] [DecidableEq ι] [Preorder α] {f : ι → α} {i j : ι} (h₁ : i ≤ j) (h₂ : f j < f i)
Goal: toLex (f ∘ ⇑(Equiv.swap i j)) < toLex f
-- NoMaxOrder.infinite
theorem NoMaxOrder.infinite [Nonempty α] [NoMaxOrder α] : Infinite α 
Arguments: {α : Type u_1} [Preorder α] [Nonempty α] [NoMaxOrder α]
Goal: Infinite α
-- NoMinOrder.infinite
theorem NoMinOrder.infinite [Nonempty α] [NoMinOrder α] : Infinite α 
Arguments: {α : Type u_1} [Preorder α] [Nonempty α] [NoMinOrder α]
Goal: Infinite α
-- Set.Ioo.infinite
theorem Ioo.infinite : Infinite (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: Infinite ↑(Set.Ioo a b)
-- Set.Ioo_infinite
theorem Ioo_infinite : (Ioo a b).Infinite 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: (Set.Ioo a b).Infinite
-- Set.Ico_infinite
theorem Ico_infinite : (Ico a b).Infinite 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: (Set.Ico a b).Infinite
-- Set.Ico.infinite
theorem Ico.infinite : Infinite (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: Infinite ↑(Set.Ico a b)
-- Set.Ioc_infinite
theorem Ioc_infinite : (Ioc a b).Infinite 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: (Set.Ioc a b).Infinite
-- Set.Ioc.infinite
theorem Ioc.infinite : Infinite (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: Infinite ↑(Set.Ioc a b)
-- Set.Icc_infinite
theorem Icc_infinite : (Icc a b).Infinite 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: (Set.Icc a b).Infinite
-- Set.Icc.infinite
theorem Icc.infinite : Infinite (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b)
Goal: Infinite ↑(Set.Icc a b)
-- Set.Iio_infinite
theorem Iio_infinite [NoMinOrder α] (a : α) : (Iio a).Infinite 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] (a : α)
Goal: (Set.Iio a).Infinite
-- Set.Iic_infinite
theorem Iic_infinite [NoMinOrder α] (a : α) : (Iic a).Infinite 
Arguments: {α : Type u_1} [Preorder α] [NoMinOrder α] (a : α)
Goal: (Set.Iic a).Infinite
-- Set.Ioi_infinite
theorem Ioi_infinite [NoMaxOrder α] (a : α) : (Ioi a).Infinite 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α)
Goal: (Set.Ioi a).Infinite
-- Set.Ici_infinite
theorem Ici_infinite [NoMaxOrder α] (a : α) : (Ici a).Infinite 
Arguments: {α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α)
Goal: (Set.Ici a).Infinite
-- subset_sSup_def
@[simp]
theorem subset_sSup_def [Inhabited s] :
    @sSup s _ = fun t =>
      if ht : t.Nonempty ∧ BddAbove t ∧ sSup ((↑) '' t : Set α) ∈ s
      then ⟨sSup ((↑) '' t : Set α), ht.2.2⟩
      else default 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [SupSet α] [Inhabited ↑s]
Goal: sSup = fun t =>
  if ht : t.Nonempty ∧ BddAbove t ∧ sSup (Subtype.val '' t) ∈ s then ⟨sSup (Subtype.val '' t), ⋯⟩ else default
-- subset_sSup_of_within
theorem subset_sSup_of_within [Inhabited s] {t : Set s}
    (h' : t.Nonempty) (h'' : BddAbove t)  (h : sSup ((↑) '' t : Set α) ∈ s) :
    sSup ((↑) '' t : Set α) = (@sSup s _ t : α) 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [SupSet α] [Inhabited ↑s] {t : Set ↑s} (h' : t.Nonempty) (h'' : BddAbove t) (h : sSup (Subtype.val '' t) ∈ s)
Goal: sSup (Subtype.val '' t) = ↑(sSup t)
-- subset_sSup_emptyset
theorem subset_sSup_emptyset [Inhabited s] :
    sSup (∅ : Set s) = default 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [SupSet α] [Inhabited ↑s]
Goal: sSup ∅ = default
-- subset_sSup_of_not_bddAbove
theorem subset_sSup_of_not_bddAbove [Inhabited s] {t : Set s} (ht : ¬BddAbove t) :
    sSup t = default 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [SupSet α] [Inhabited ↑s] {t : Set ↑s} (ht : ¬BddAbove t)
Goal: sSup t = default
-- subset_sInf_def
@[simp]
theorem subset_sInf_def [Inhabited s] :
    @sInf s _ = fun t =>
      if ht : t.Nonempty ∧ BddBelow t ∧ sInf ((↑) '' t : Set α) ∈ s
      then ⟨sInf ((↑) '' t : Set α), ht.2.2⟩ else
      default 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [InfSet α] [Inhabited ↑s]
Goal: sInf = fun t =>
  if ht : t.Nonempty ∧ BddBelow t ∧ sInf (Subtype.val '' t) ∈ s then ⟨sInf (Subtype.val '' t), ⋯⟩ else default
-- subset_sInf_of_within
theorem subset_sInf_of_within [Inhabited s] {t : Set s}
    (h' : t.Nonempty) (h'' : BddBelow t) (h : sInf ((↑) '' t : Set α) ∈ s) :
    sInf ((↑) '' t : Set α) = (@sInf s _ t : α) 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [InfSet α] [Inhabited ↑s] {t : Set ↑s} (h' : t.Nonempty) (h'' : BddBelow t) (h : sInf (Subtype.val '' t) ∈ s)
Goal: sInf (Subtype.val '' t) = ↑(sInf t)
-- subset_sInf_emptyset
theorem subset_sInf_emptyset [Inhabited s] :
    sInf (∅ : Set s) = default 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [InfSet α] [Inhabited ↑s]
Goal: sInf ∅ = default
-- subset_sInf_of_not_bddBelow
theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
    sInf t = default 
Arguments: {α : Type u_2} (s : Set α) [Preorder α] [InfSet α] [Inhabited ↑s] {t : Set ↑s} (ht : ¬BddBelow t)
Goal: sInf t = default
-- sSup_within_of_ordConnected
theorem sSup_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
    (h_bdd : BddAbove t) : sSup ((↑) '' t : Set α) ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} [hs : s.OrdConnected] ⦃t : Set ↑s⦄ (ht : t.Nonempty) (h_bdd : BddAbove t)
Goal: sSup (Subtype.val '' t) ∈ s
-- sInf_within_of_ordConnected
theorem sInf_within_of_ordConnected {s : Set α} [hs : OrdConnected s] ⦃t : Set s⦄ (ht : t.Nonempty)
    (h_bdd : BddBelow t) : sInf ((↑) '' t : Set α) ∈ s 
Arguments: {α : Type u_2} [ConditionallyCompleteLinearOrder α] {s : Set α} [hs : s.OrdConnected] ⦃t : Set ↑s⦄ (ht : t.Nonempty) (h_bdd : BddBelow t)
Goal: sInf (Subtype.val '' t) ∈ s
-- Set.Icc.coe_sSup
lemma Set.Icc.coe_sSup [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
    {S : Set (Set.Icc a b)} (hS : S.Nonempty) : letI := Set.Icc.completeLattice h
    ↑(sSup S) = sSup ((↑) '' S : Set α) 
Arguments: {α : Type u_2} [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b) {S : Set ↑(Set.Icc a b)} (hS : S.Nonempty)
Goal: ↑(sSup S) = sSup (Subtype.val '' S)
-- Set.Icc.coe_sInf
lemma Set.Icc.coe_sInf [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
    {S : Set (Set.Icc a b)} (hS : S.Nonempty) : letI := Set.Icc.completeLattice h
    ↑(sInf S) = sInf ((↑) '' S : Set α) 
Arguments: {α : Type u_2} [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b) {S : Set ↑(Set.Icc a b)} (hS : S.Nonempty)
Goal: ↑(sInf S) = sInf (Subtype.val '' S)
-- Set.Icc.coe_iSup
lemma Set.Icc.coe_iSup [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
    [Nonempty ι] {S : ι → Set.Icc a b} : letI := Set.Icc.completeLattice h
    ↑(iSup S) = (⨆ i, S i : α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b) [Nonempty ι] {S : ι → ↑(Set.Icc a b)}
Goal: ↑(iSup S) = ⨆ i, ↑(S i)
-- Set.Icc.coe_iInf
lemma Set.Icc.coe_iInf [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
    [Nonempty ι] {S : ι → Set.Icc a b} : letI := Set.Icc.completeLattice h
    ↑(iInf S) = (⨅ i, S i : α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b) [Nonempty ι] {S : ι → ↑(Set.Icc a b)}
Goal: ↑(iInf S) = ⨅ i, ↑(S i)
-- Set.Iic.coe_sSup
@[simp] theorem coe_sSup : (↑(sSup S) : α) = sSup ((↑) '' S) 
Arguments: {α : Type u_2} [CompleteLattice α] {a : α} (S : Set ↑(Set.Iic a))
Goal: ↑(sSup S) = sSup (Subtype.val '' S)
-- Set.Iic.coe_iSup
@[simp] theorem coe_iSup : (↑(⨆ i, f i) : α) = ⨆ i, (f i : α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [CompleteLattice α] {a : α} (f : ι → ↑(Set.Iic a))
Goal: ↑(⨆ i, f i) = ⨆ i, ↑(f i)
-- Set.Iic.coe_biSup
theorem coe_biSup : (↑(⨆ i, ⨆ (_ : p i), f i) : α) = ⨆ i, ⨆ (_ : p i), (f i : α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [CompleteLattice α] {a : α} (f : ι → ↑(Set.Iic a)) (p : ι → Prop)
Goal: ↑(⨆ i, ⨆ (_ : p i), f i) = ⨆ i, ⨆ (_ : p i), ↑(f i)
-- Set.Iic.coe_sInf
@[simp] theorem coe_sInf : (↑(sInf S) : α) = a ⊓ sInf ((↑) '' S) 
Arguments: {α : Type u_2} [CompleteLattice α] {a : α} (S : Set ↑(Set.Iic a))
Goal: ↑(sInf S) = a ⊓ sInf (Subtype.val '' S)
-- Set.Iic.coe_iInf
@[simp] theorem coe_iInf : (↑(⨅ i, f i) : α) = a ⊓ ⨅ i, (f i : α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [CompleteLattice α] {a : α} (f : ι → ↑(Set.Iic a))
Goal: ↑(⨅ i, f i) = a ⊓ ⨅ i, ↑(f i)
-- Set.Iic.coe_biInf
theorem coe_biInf : (↑(⨅ i, ⨅ (_ : p i), f i) : α) = a ⊓ ⨅ i, ⨅ (_ : p i), (f i : α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [CompleteLattice α] {a : α} (f : ι → ↑(Set.Iic a)) (p : ι → Prop)
Goal: ↑(⨅ i, ⨅ (_ : p i), f i) = a ⊓ ⨅ i, ⨅ (_ : p i), ↑(f i)
-- CompleteLattice.isCompactElement_iff
theorem isCompactElement_iff.{u} {α : Type u} [CompleteLattice α] (k : α) :
    CompleteLattice.IsCompactElement k ↔
      ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t : Finset ι, k ≤ t.sup s 
Arguments: {α : Type u} [CompleteLattice α] (k : α)
Goal: CompleteLattice.IsCompactElement k ↔ ∀ (ι : Type u) (s : ι → α), k ≤ iSup s → ∃ t, k ≤ t.sup s
-- CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le
theorem isCompactElement_iff_le_of_directed_sSup_le (k : α) :
    IsCompactElement k ↔
      ∀ s : Set α, s.Nonempty → DirectedOn (· ≤ ·) s → k ≤ sSup s → ∃ x : α, x ∈ s ∧ k ≤ x 
Arguments: (α : Type u_2) [CompleteLattice α] (k : α)
Goal: CompleteLattice.IsCompactElement k ↔
  ∀ (s : Set α), s.Nonempty → DirectedOn (fun x x_1 => x ≤ x_1) s → k ≤ sSup s → ∃ x ∈ s, k ≤ x
-- CompleteLattice.IsCompactElement.exists_finset_of_le_iSup
theorem IsCompactElement.exists_finset_of_le_iSup {k : α} (hk : IsCompactElement k) {ι : Type*}
    (f : ι → α) (h : k ≤ ⨆ i, f i) : ∃ s : Finset ι, k ≤ ⨆ i ∈ s, f i 
Arguments: (α : Type u_2) [CompleteLattice α] {k : α} (hk : CompleteLattice.IsCompactElement k) {ι : Type u_3} (f : ι → α) (h : k ≤ ⨆ i, f i)
Goal: ∃ s, k ≤ ⨆ i ∈ s, f i
-- CompleteLattice.IsCompactElement.directed_sSup_lt_of_lt
theorem IsCompactElement.directed_sSup_lt_of_lt {α : Type*} [CompleteLattice α] {k : α}
    (hk : IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (· ≤ ·) s)
    (hbelow : ∀ x ∈ s, x < k) : sSup s < k 
Arguments: {α : Type u_3} [CompleteLattice α] {k : α} (hk : CompleteLattice.IsCompactElement k) {s : Set α} (hemp : s.Nonempty) (hdir : DirectedOn (fun x x_1 => x ≤ x_1) s) (hbelow : ∀ x ∈ s, x < k)
Goal: sSup s < k
-- CompleteLattice.isCompactElement_finsetSup
theorem isCompactElement_finsetSup {α β : Type*} [CompleteLattice α] {f : β → α} (s : Finset β)
    (h : ∀ x ∈ s, IsCompactElement (f x)) : IsCompactElement (s.sup f) 
Arguments: {α : Type u_3} {β : Type u_4} [CompleteLattice α] {f : β → α} (s : Finset β) (h : ∀ x ∈ s, CompleteLattice.IsCompactElement (f x))
Goal: CompleteLattice.IsCompactElement (s.sup f)
-- CompleteLattice.WellFounded.isSupFiniteCompact
theorem WellFounded.isSupFiniteCompact (h : WellFounded ((· > ·) : α → α → Prop)) :
    IsSupFiniteCompact α 
Arguments: (α : Type u_2) [CompleteLattice α] (h : WellFounded fun x x_1 => x > x_1)
Goal: CompleteLattice.IsSupFiniteCompact α
-- CompleteLattice.IsSupFiniteCompact.isSupClosedCompact
theorem IsSupFiniteCompact.isSupClosedCompact (h : IsSupFiniteCompact α) :
    IsSupClosedCompact α 
Arguments: (α : Type u_2) [CompleteLattice α] (h : CompleteLattice.IsSupFiniteCompact α)
Goal: CompleteLattice.IsSupClosedCompact α
-- CompleteLattice.IsSupClosedCompact.wellFounded
theorem IsSupClosedCompact.wellFounded (h : IsSupClosedCompact α) :
    WellFounded ((· > ·) : α → α → Prop) 
Arguments: (α : Type u_2) [CompleteLattice α] (h : CompleteLattice.IsSupClosedCompact α)
Goal: WellFounded fun x x_1 => x > x_1
-- CompleteLattice.isSupFiniteCompact_iff_all_elements_compact
theorem isSupFiniteCompact_iff_all_elements_compact :
    IsSupFiniteCompact α ↔ ∀ k : α, IsCompactElement k 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: CompleteLattice.IsSupFiniteCompact α ↔ ∀ (k : α), CompleteLattice.IsCompactElement k
-- CompleteLattice.wellFounded_characterisations
theorem wellFounded_characterisations : List.TFAE
    [WellFounded ((· > ·) : α → α → Prop),
      IsSupFiniteCompact α, IsSupClosedCompact α, ∀ k : α, IsCompactElement k] 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: [WellFounded fun x x_1 => x > x_1, CompleteLattice.IsSupFiniteCompact α, CompleteLattice.IsSupClosedCompact α,
    ∀ (k : α), CompleteLattice.IsCompactElement k].TFAE
-- CompleteLattice.wellFounded_iff_isSupFiniteCompact
theorem wellFounded_iff_isSupFiniteCompact :
    WellFounded ((· > ·) : α → α → Prop) ↔ IsSupFiniteCompact α 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: (WellFounded fun x x_1 => x > x_1) ↔ CompleteLattice.IsSupFiniteCompact α
-- CompleteLattice.isSupFiniteCompact_iff_isSupClosedCompact
theorem isSupFiniteCompact_iff_isSupClosedCompact : IsSupFiniteCompact α ↔ IsSupClosedCompact α 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: CompleteLattice.IsSupFiniteCompact α ↔ CompleteLattice.IsSupClosedCompact α
-- CompleteLattice.isSupClosedCompact_iff_wellFounded
theorem isSupClosedCompact_iff_wellFounded :
    IsSupClosedCompact α ↔ WellFounded ((· > ·) : α → α → Prop) 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: CompleteLattice.IsSupClosedCompact α ↔ WellFounded fun x x_1 => x > x_1
-- CompleteLattice.WellFounded.finite_of_setIndependent
theorem WellFounded.finite_of_setIndependent (h : WellFounded ((· > ·) : α → α → Prop)) {s : Set α}
    (hs : SetIndependent s) : s.Finite 
Arguments: {α : Type u_2} [CompleteLattice α] (h : WellFounded fun x x_1 => x > x_1) {s : Set α} (hs : CompleteLattice.SetIndependent s)
Goal: s.Finite
-- CompleteLattice.WellFounded.finite_ne_bot_of_independent
theorem WellFounded.finite_ne_bot_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop))
    {ι : Type*} {t : ι → α} (ht : Independent t) : Set.Finite {i | t i ≠ ⊥} 
Arguments: {α : Type u_2} [CompleteLattice α] (hwf : WellFounded fun x x_1 => x > x_1) {ι : Type u_3} {t : ι → α} (ht : CompleteLattice.Independent t)
Goal: {i | t i ≠ ⊥}.Finite
-- CompleteLattice.WellFounded.finite_of_independent
theorem WellFounded.finite_of_independent (hwf : WellFounded ((· > ·) : α → α → Prop)) {ι : Type*}
    {t : ι → α} (ht : Independent t) (h_ne_bot : ∀ i, t i ≠ ⊥) : Finite ι 
Arguments: {α : Type u_2} [CompleteLattice α] (hwf : WellFounded fun x x_1 => x > x_1) {ι : Type u_3} {t : ι → α} (ht : CompleteLattice.Independent t) (h_ne_bot : ∀ (i : ι), t i ≠ ⊥)
Goal: Finite ι
-- IsCompactlyGenerated.exists_sSup_eq
class IsCompactlyGenerated (α : Type*) [CompleteLattice α] : Prop where
    exists_sSup_eq : ∀ x : α, ∃ s : Set α, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ sSup s = x
Arguments: {α : Type u_3} [CompleteLattice α] [self : IsCompactlyGenerated α] (x : α)
Goal: ∃ s, (∀ x ∈ s, CompleteLattice.IsCompactElement x) ∧ sSup s = x
-- sSup_compact_le_eq
@[simp]
theorem sSup_compact_le_eq (b) :
    sSup { c : α | CompleteLattice.IsCompactElement c ∧ c ≤ b } = b 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] (b : α)
Goal: sSup {c | CompleteLattice.IsCompactElement c ∧ c ≤ b} = b
-- sSup_compact_eq_top
@[simp]
theorem sSup_compact_eq_top : sSup { a : α | CompleteLattice.IsCompactElement a } = ⊤ 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α]
Goal: sSup {a | CompleteLattice.IsCompactElement a} = ⊤
-- le_iff_compact_le_imp
theorem le_iff_compact_le_imp {a b : α} :
    a ≤ b ↔ ∀ c : α, CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {a b : α}
Goal: a ≤ b ↔ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a → c ≤ b
-- DirectedOn.inf_sSup_eq
theorem DirectedOn.inf_sSup_eq (h : DirectedOn (· ≤ ·) s) : a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} {s : Set α} (h : DirectedOn (fun x x_1 => x ≤ x_1) s)
Goal: a ⊓ sSup s = ⨆ b ∈ s, a ⊓ b
-- DirectedOn.sSup_inf_eq
protected theorem DirectedOn.sSup_inf_eq (h : DirectedOn (· ≤ ·) s) :
    sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a 
Arguments: {α : Type u_2} [CompleteLattice α] [IsCompactlyGenerated α] {a : α} {s : Set α} (h : DirectedOn (fun x x_1 => x ≤ x_1) s)
Goal: sSup s ⊓ a = ⨆ b ∈ s, b ⊓ a