ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- le_ciInf
theorem le_ciInf [Nonempty ι] {f : ι → α} {c : α} (H : ∀ x, c ≤ f x) : c ≤ iInf f 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} {c : α} (H : ∀ (x : ι), c ≤ f x)
Goal: c ≤ iInf f
-- ciInf_le
theorem ciInf_le {f : ι → α} (H : BddBelow (range f)) (c : ι) : iInf f ≤ f c 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {f : ι → α} (H : BddBelow (Set.range f)) (c : ι)
Goal: iInf f ≤ f c
-- ciInf_le_of_le
theorem ciInf_le_of_le {f : ι → α} (H : BddBelow (range f)) (c : ι) (h : f c ≤ a) : iInf f ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {a : α} {f : ι → α} (H : BddBelow (Set.range f)) (c : ι) (h : f c ≤ a)
Goal: iInf f ≤ a
-- ciInf_set_le
theorem ciInf_set_le {f : β → α} {s : Set β} (H : BddBelow (f '' s)) {c : β} (hc : c ∈ s) :
    ⨅ i : s, f i ≤ f c 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : β → α} {s : Set β} (H : BddBelow (f '' s)) {c : β} (hc : c ∈ s)
Goal: ⨅ i, f ↑i ≤ f c
-- ciSup_const
@[simp]
theorem ciSup_const [hι : Nonempty ι] {a : α} : ⨆ _ : ι, a = a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [hι : Nonempty ι] {a : α}
Goal: ⨆ x, a = a
-- ciInf_const
@[simp]
theorem ciInf_const [Nonempty ι] {a : α} : ⨅ _ : ι, a = a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {a : α}
Goal: ⨅ x, a = a
-- ciSup_unique
@[simp]
theorem ciSup_unique [Unique ι] {s : ι → α} : ⨆ i, s i = s default 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Unique ι] {s : ι → α}
Goal: ⨆ i, s i = s default
-- ciInf_unique
@[simp]
theorem ciInf_unique [Unique ι] {s : ι → α} : ⨅ i, s i = s default 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Unique ι] {s : ι → α}
Goal: ⨅ i, s i = s default
-- ciSup_subsingleton
theorem ciSup_subsingleton [Subsingleton ι] (i : ι) (s : ι → α) : ⨆ i, s i = s i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Subsingleton ι] (i : ι) (s : ι → α)
Goal: ⨆ i, s i = s i
-- ciInf_subsingleton
theorem ciInf_subsingleton [Subsingleton ι] (i : ι) (s : ι → α) : ⨅ i, s i = s i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Subsingleton ι] (i : ι) (s : ι → α)
Goal: ⨅ i, s i = s i
-- ciSup_pos
@[simp]
theorem ciSup_pos {p : Prop} {f : p → α} (hp : p) : ⨆ h : p, f h = f hp 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {p : Prop} {f : p → α} (hp : p)
Goal: ⨆ (h : p), f h = f hp
-- ciInf_pos
@[simp]
theorem ciInf_pos {p : Prop} {f : p → α} (hp : p) : ⨅ h : p, f h = f hp 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {p : Prop} {f : p → α} (hp : p)
Goal: ⨅ (h : p), f h = f hp
-- ciSup_neg
lemma ciSup_neg {p : Prop} {f : p → α} (hp : ¬ p) :
    ⨆ (h : p), f h = sSup (∅ : Set α) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {p : Prop} {f : p → α} (hp : ¬p)
Goal: ⨆ (h : p), f h = sSup ∅
-- ciInf_neg
lemma ciInf_neg {p : Prop} {f : p → α} (hp : ¬ p) :
    ⨅ (h : p), f h = sInf (∅ : Set α) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {p : Prop} {f : p → α} (hp : ¬p)
Goal: ⨅ (h : p), f h = sInf ∅
-- ciSup_eq_ite
lemma ciSup_eq_ite {p : Prop} [Decidable p] {f : p → α} :
    (⨆ h : p, f h) = if h : p then f h else sSup (∅ : Set α) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {p : Prop} [Decidable p] {f : p → α}
Goal: ⨆ (h : p), f h = if h : p then f h else sSup ∅
-- ciInf_eq_ite
lemma ciInf_eq_ite {p : Prop} [Decidable p] {f : p → α} :
    (⨅ h : p, f h) = if h : p then f h else sInf (∅ : Set α) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {p : Prop} [Decidable p] {f : p → α}
Goal: ⨅ (h : p), f h = if h : p then f h else sInf ∅
-- cbiSup_eq_of_forall
theorem cbiSup_eq_of_forall {p : ι → Prop} {f : Subtype p → α} (hp : ∀ i, p i) :
    ⨆ (i) (h : p i), f ⟨i, h⟩ = iSup f 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {p : ι → Prop} {f : Subtype p → α} (hp : ∀ (i : ι), p i)
Goal: ⨆ i, ⨆ (h : p i), f ⟨i, h⟩ = iSup f
-- cbiInf_eq_of_forall
theorem cbiInf_eq_of_forall {p : ι → Prop} {f : Subtype p → α} (hp : ∀ i, p i) :
    ⨅ (i) (h : p i), f ⟨i, h⟩ = iInf f 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {p : ι → Prop} {f : Subtype p → α} (hp : ∀ (i : ι), p i)
Goal: ⨅ i, ⨅ (h : p i), f ⟨i, h⟩ = iInf f
-- ciSup_eq_of_forall_le_of_forall_lt_exists_gt
theorem ciSup_eq_of_forall_le_of_forall_lt_exists_gt [Nonempty ι] {f : ι → α} (h₁ : ∀ i, f i ≤ b)
    (h₂ : ∀ w, w < b → ∃ i, w < f i) : ⨆ i : ι, f i = b 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {b : α} [Nonempty ι] {f : ι → α} (h₁ : ∀ (i : ι), f i ≤ b) (h₂ : ∀ w < b, ∃ i, w < f i)
Goal: ⨆ i, f i = b
-- ciInf_eq_of_forall_ge_of_forall_gt_exists_lt
theorem ciInf_eq_of_forall_ge_of_forall_gt_exists_lt [Nonempty ι] {f : ι → α} (h₁ : ∀ i, b ≤ f i)
    (h₂ : ∀ w, b < w → ∃ i, f i < w) : ⨅ i : ι, f i = b 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {b : α} [Nonempty ι] {f : ι → α} (h₁ : ∀ (i : ι), b ≤ f i) (h₂ : ∀ (w : α), b < w → ∃ i, f i < w)
Goal: ⨅ i, f i = b
-- Monotone.ciSup_mem_iInter_Icc_of_antitone
theorem Monotone.ciSup_mem_iInter_Icc_of_antitone [SemilatticeSup β] {f g : β → α} (hf : Monotone f)
    (hg : Antitone g) (h : f ≤ g) : (⨆ n, f n) ∈ ⋂ n, Icc (f n) (g n) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [SemilatticeSup β] {f g : β → α} (hf : Monotone f) (hg : Antitone g) (h : f ≤ g)
Goal: ⨆ n, f n ∈ ⋂ n, Set.Icc (f n) (g n)
-- ciSup_mem_iInter_Icc_of_antitone_Icc
theorem ciSup_mem_iInter_Icc_of_antitone_Icc [SemilatticeSup β] {f g : β → α}
    (h : Antitone fun n => Icc (f n) (g n)) (h' : ∀ n, f n ≤ g n) :
    (⨆ n, f n) ∈ ⋂ n, Icc (f n) (g n) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [SemilatticeSup β] {f g : β → α} (h : Antitone fun n => Set.Icc (f n) (g n)) (h' : ∀ (n : β), f n ≤ g n)
Goal: ⨆ n, f n ∈ ⋂ n, Set.Icc (f n) (g n)
-- csSup_eq_of_is_forall_le_of_forall_le_imp_ge
theorem csSup_eq_of_is_forall_le_of_forall_le_imp_ge (hs : s.Nonempty) (h_is_ub : ∀ a ∈ s, a ≤ b)
    (h_b_le_ub : ∀ ub, (∀ a ∈ s, a ≤ ub) → b ≤ ub) : sSup s = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {b : α} (hs : s.Nonempty) (h_is_ub : ∀ a ∈ s, a ≤ b) (h_b_le_ub : ∀ (ub : α), (∀ a ∈ s, a ≤ ub) → b ≤ ub)
Goal: sSup s = b
-- Set.Iic_ciInf
lemma Set.Iic_ciInf [Nonempty ι] {f : ι → α} (hf : BddBelow (range f)) :
    Iic (⨅ i, f i) = ⋂ i, Iic (f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} (hf : BddBelow (Set.range f))
Goal: Set.Iic (⨅ i, f i) = ⋂ i, Set.Iic (f i)
-- Set.Ici_ciSup
lemma Set.Ici_ciSup [Nonempty ι] {f : ι → α} (hf : BddAbove (range f)) :
    Ici (⨆ i, f i) = ⋂ i, Ici (f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} (hf : BddAbove (Set.range f))
Goal: Set.Ici (⨆ i, f i) = ⋂ i, Set.Ici (f i)
-- sup_eq_top_of_top_mem
lemma sup_eq_top_of_top_mem [OrderTop α] (h : ⊤ ∈ s) : sSup s = ⊤ 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} [OrderTop α] (h : ⊤ ∈ s)
Goal: sSup s = ⊤
-- inf_eq_bot_of_bot_mem
lemma inf_eq_bot_of_bot_mem [OrderBot α] (h : ⊥ ∈ s) : sInf s = ⊥ 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} [OrderBot α] (h : ⊥ ∈ s)
Goal: sInf s = ⊥
-- exists_lt_of_lt_csSup
theorem exists_lt_of_lt_csSup (hs : s.Nonempty) (hb : b < sSup s) : ∃ a ∈ s, b < a 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} {b : α} (hs : s.Nonempty) (hb : b < sSup s)
Goal: ∃ a ∈ s, b < a
-- exists_lt_of_lt_ciSup
theorem exists_lt_of_lt_ciSup [Nonempty ι] {f : ι → α} (h : b < iSup f) : ∃ i, b < f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] {b : α} [Nonempty ι] {f : ι → α} (h : b < iSup f)
Goal: ∃ i, b < f i
-- exists_lt_of_csInf_lt
theorem exists_lt_of_csInf_lt (hs : s.Nonempty) (hb : sInf s < b) : ∃ a ∈ s, a < b 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} {b : α} (hs : s.Nonempty) (hb : sInf s < b)
Goal: ∃ a ∈ s, a < b
-- exists_lt_of_ciInf_lt
theorem exists_lt_of_ciInf_lt [Nonempty ι] {f : ι → α} (h : iInf f < a) : ∃ i, f i < a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] {a : α} [Nonempty ι] {f : ι → α} (h : iInf f < a)
Goal: ∃ i, f i < a
-- csSup_of_not_bddAbove
theorem csSup_of_not_bddAbove {s : Set α} (hs : ¬BddAbove s) : sSup s = sSup ∅ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} (hs : ¬BddAbove s)
Goal: sSup s = sSup ∅
-- csSup_eq_univ_of_not_bddAbove
theorem csSup_eq_univ_of_not_bddAbove {s : Set α} (hs : ¬BddAbove s) : sSup s = sSup univ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} (hs : ¬BddAbove s)
Goal: sSup s = sSup Set.univ
-- csInf_of_not_bddBelow
theorem csInf_of_not_bddBelow {s : Set α} (hs : ¬BddBelow s) : sInf s = sInf ∅ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} (hs : ¬BddBelow s)
Goal: sInf s = sInf ∅
-- csInf_eq_univ_of_not_bddBelow
theorem csInf_eq_univ_of_not_bddBelow {s : Set α} (hs : ¬BddBelow s) : sInf s = sInf univ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} (hs : ¬BddBelow s)
Goal: sInf s = sInf Set.univ
-- csSup_eq_csSup_of_forall_exists_le
theorem csSup_eq_csSup_of_forall_exists_le {s t : Set α}
    (hs : ∀ x ∈ s, ∃ y ∈ t, x ≤ y) (ht : ∀ y ∈ t, ∃ x ∈ s, y ≤ x) :
    sSup s = sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s t : Set α} (hs : ∀ x ∈ s, ∃ y ∈ t, x ≤ y) (ht : ∀ y ∈ t, ∃ x ∈ s, y ≤ x)
Goal: sSup s = sSup t
-- csInf_eq_csInf_of_forall_exists_le
theorem csInf_eq_csInf_of_forall_exists_le {s t : Set α}
    (hs : ∀ x ∈ s, ∃ y ∈ t, y ≤ x) (ht : ∀ y ∈ t, ∃ x ∈ s, x ≤ y) :
    sInf s = sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s t : Set α} (hs : ∀ x ∈ s, ∃ y ∈ t, y ≤ x) (ht : ∀ y ∈ t, ∃ x ∈ s, x ≤ y)
Goal: sInf s = sInf t
-- sSup_iUnion_Iic
lemma sSup_iUnion_Iic (f : ι → α) : sSup (⋃ (i : ι), Iic (f i)) = ⨆ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] (f : ι → α)
Goal: sSup (⋃ i, Set.Iic (f i)) = ⨆ i, f i
-- sInf_iUnion_Ici
lemma sInf_iUnion_Ici (f : ι → α) : sInf (⋃ (i : ι), Ici (f i)) = ⨅ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] (f : ι → α)
Goal: sInf (⋃ i, Set.Ici (f i)) = ⨅ i, f i
-- cbiSup_eq_of_not_forall
theorem cbiSup_eq_of_not_forall {p : ι → Prop} {f : Subtype p → α} (hp : ¬ (∀ i, p i)) :
    ⨆ (i) (h : p i), f ⟨i, h⟩ = iSup f ⊔ sSup ∅ 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] {p : ι → Prop} {f : Subtype p → α} (hp : ¬∀ (i : ι), p i)
Goal: ⨆ i, ⨆ (h : p i), f ⟨i, h⟩ = iSup f ⊔ sSup ∅
-- cbiInf_eq_of_not_forall
theorem cbiInf_eq_of_not_forall {p : ι → Prop} {f : Subtype p → α} (hp : ¬ (∀ i, p i)) :
    ⨅ (i) (h : p i), f ⟨i, h⟩ = iInf f ⊓ sInf ∅ 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] {p : ι → Prop} {f : Subtype p → α} (hp : ¬∀ (i : ι), p i)
Goal: ⨅ i, ⨅ (h : p i), f ⟨i, h⟩ = iInf f ⊓ sInf ∅
-- sInf_eq_argmin_on
theorem sInf_eq_argmin_on (hs : s.Nonempty) : sInf s = argminOn id wellFounded_lt s hs 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} [IsWellOrder α fun x x_1 => x < x_1] (hs : s.Nonempty)
Goal: sInf s = Function.argminOn id ⋯ s hs
-- isLeast_csInf
theorem isLeast_csInf (hs : s.Nonempty) : IsLeast s (sInf s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} [IsWellOrder α fun x x_1 => x < x_1] (hs : s.Nonempty)
Goal: IsLeast s (sInf s)
-- le_csInf_iff'
theorem le_csInf_iff' (hs : s.Nonempty) : b ≤ sInf s ↔ b ∈ lowerBounds s 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} {b : α} [IsWellOrder α fun x x_1 => x < x_1] (hs : s.Nonempty)
Goal: b ≤ sInf s ↔ b ∈ lowerBounds s
-- csInf_mem
theorem csInf_mem (hs : s.Nonempty) : sInf s ∈ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} [IsWellOrder α fun x x_1 => x < x_1] (hs : s.Nonempty)
Goal: sInf s ∈ s
-- ciInf_mem
theorem ciInf_mem [Nonempty ι] (f : ι → α) : iInf f ∈ range f 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrder α] [IsWellOrder α fun x x_1 => x < x_1] [Nonempty ι] (f : ι → α)
Goal: iInf f ∈ Set.range f
-- MonotoneOn.map_csInf
theorem MonotoneOn.map_csInf {β : Type*} [ConditionallyCompleteLattice β] {f : α → β}
    (hf : MonotoneOn f s) (hs : s.Nonempty) : f (sInf s) = sInf (f '' s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} [IsWellOrder α fun x x_1 => x < x_1] {β : Type u_5} [ConditionallyCompleteLattice β] {f : α → β} (hf : MonotoneOn f s) (hs : s.Nonempty)
Goal: f (sInf s) = sInf (f '' s)
-- Monotone.map_csInf
theorem Monotone.map_csInf {β : Type*} [ConditionallyCompleteLattice β] {f : α → β}
    (hf : Monotone f) (hs : s.Nonempty) : f (sInf s) = sInf (f '' s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] {s : Set α} [IsWellOrder α fun x x_1 => x < x_1] {β : Type u_5} [ConditionallyCompleteLattice β] {f : α → β} (hf : Monotone f) (hs : s.Nonempty)
Goal: f (sInf s) = sInf (f '' s)
-- csInf_univ
@[simp]
theorem csInf_univ [ConditionallyCompleteLinearOrder α] [OrderBot α] : sInf (univ : Set α) = ⊥ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrder α] [OrderBot α]
Goal: sInf Set.univ = ⊥
-- csSup_empty
@[simp]
theorem csSup_empty : (sSup ∅ : α) = ⊥ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α]
Goal: sSup ∅ = ⊥
-- ciSup_of_empty
@[simp]
theorem ciSup_of_empty [IsEmpty ι] (f : ι → α) : ⨆ i, f i = ⊥ 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] [IsEmpty ι] (f : ι → α)
Goal: ⨆ i, f i = ⊥
-- ciSup_false
theorem ciSup_false (f : False → α) : ⨆ i, f i = ⊥ 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] (f : False → α)
Goal: ⨆ (i : False), f i = ⊥
-- isLUB_csSup'
theorem isLUB_csSup' {s : Set α} (hs : BddAbove s) : IsLUB s (sSup s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hs : BddAbove s)
Goal: IsLUB s (sSup s)
-- csSup_le_iff'
theorem csSup_le_iff' {s : Set α} (hs : BddAbove s) {a : α} : sSup s ≤ a ↔ ∀ x ∈ s, x ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hs : BddAbove s) {a : α}
Goal: sSup s ≤ a ↔ ∀ x ∈ s, x ≤ a
-- csSup_le'
theorem csSup_le' {s : Set α} {a : α} (h : a ∈ upperBounds s) : sSup s ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {a : α} (h : a ∈ upperBounds s)
Goal: sSup s ≤ a
-- le_csSup_iff'
theorem le_csSup_iff' {s : Set α} {a : α} (h : BddAbove s) :
    a ≤ sSup s ↔ ∀ b, b ∈ upperBounds s → a ≤ b 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {a : α} (h : BddAbove s)
Goal: a ≤ sSup s ↔ ∀ b ∈ upperBounds s, a ≤ b
-- le_ciSup_iff'
theorem le_ciSup_iff' {s : ι → α} {a : α} (h : BddAbove (range s)) :
    a ≤ iSup s ↔ ∀ b, (∀ i, s i ≤ b) → a ≤ b 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] {s : ι → α} {a : α} (h : BddAbove (Set.range s))
Goal: a ≤ iSup s ↔ ∀ (b : α), (∀ (i : ι), s i ≤ b) → a ≤ b
-- le_csInf_iff''
theorem le_csInf_iff'' {s : Set α} {a : α} (ne : s.Nonempty) :
    a ≤ sInf s ↔ ∀ b : α, b ∈ s → a ≤ b 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {a : α} (ne : s.Nonempty)
Goal: a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b
-- le_ciInf_iff'
theorem le_ciInf_iff' [Nonempty ι] {f : ι → α} {a : α} : a ≤ iInf f ↔ ∀ i, a ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] [Nonempty ι] {f : ι → α} {a : α}
Goal: a ≤ iInf f ↔ ∀ (i : ι), a ≤ f i
-- csInf_le'
theorem csInf_le' (h : a ∈ s) : sInf s ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {a : α} (h : a ∈ s)
Goal: sInf s ≤ a
-- ciInf_le'
theorem ciInf_le' (f : ι → α) (i : ι) : iInf f ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] (f : ι → α) (i : ι)
Goal: iInf f ≤ f i
-- ciInf_le_of_le'
lemma ciInf_le_of_le' (c : ι) : f c ≤ a → iInf f ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {a : α} (c : ι)
Goal: f c ≤ a → iInf f ≤ a
-- exists_lt_of_lt_csSup'
theorem exists_lt_of_lt_csSup' {s : Set α} {a : α} (h : a < sSup s) : ∃ b ∈ s, a < b 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} {a : α} (h : a < sSup s)
Goal: ∃ b ∈ s, a < b
-- ciSup_le_iff'
theorem ciSup_le_iff' {f : ι → α} (h : BddAbove (range f)) {a : α} :
    ⨆ i, f i ≤ a ↔ ∀ i, f i ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} (h : BddAbove (Set.range f)) {a : α}
Goal: ⨆ i, f i ≤ a ↔ ∀ (i : ι), f i ≤ a
-- ciSup_le'
theorem ciSup_le' {f : ι → α} {a : α} (h : ∀ i, f i ≤ a) : ⨆ i, f i ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {a : α} (h : ∀ (i : ι), f i ≤ a)
Goal: ⨆ i, f i ≤ a
-- exists_lt_of_lt_ciSup'
theorem exists_lt_of_lt_ciSup' {f : ι → α} {a : α} (h : a < ⨆ i, f i) : ∃ i, a < f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {a : α} (h : a < ⨆ i, f i)
Goal: ∃ i, a < f i
-- ciSup_mono'
theorem ciSup_mono' {ι'} {f : ι → α} {g : ι' → α} (hg : BddAbove (range g))
    (h : ∀ i, ∃ i', f i ≤ g i') : iSup f ≤ iSup g 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLinearOrderBot α] {ι' : Sort u_5} {f : ι → α} {g : ι' → α} (hg : BddAbove (Set.range g)) (h : ∀ (i : ι), ∃ i', f i ≤ g i')
Goal: iSup f ≤ iSup g
-- csInf_le_csInf'
theorem csInf_le_csInf' {s t : Set α} (h₁ : t.Nonempty) (h₂ : t ⊆ s) : sInf s ≤ sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s t : Set α} (h₁ : t.Nonempty) (h₂ : t ⊆ s)
Goal: sInf s ≤ sInf t
-- WithTop.isLUB_sSup'
theorem isLUB_sSup' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)}
    (hs : s.Nonempty) : IsLUB s (sSup s) 
Arguments: {β : Type u_5} [ConditionallyCompleteLattice β] {s : Set (WithTop β)} (hs : s.Nonempty)
Goal: IsLUB s (sSup s)
-- WithTop.isLUB_sSup
theorem isLUB_sSup (s : Set (WithTop α)) : IsLUB s (sSup s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] (s : Set (WithTop α))
Goal: IsLUB s (sSup s)
-- WithTop.isGLB_sInf'
theorem isGLB_sInf' {β : Type*} [ConditionallyCompleteLattice β] {s : Set (WithTop β)}
    (hs : BddBelow s) : IsGLB s (sInf s) 
Arguments: {β : Type u_5} [ConditionallyCompleteLattice β] {s : Set (WithTop β)} (hs : BddBelow s)
Goal: IsGLB s (sInf s)
-- WithTop.isGLB_sInf
theorem isGLB_sInf (s : Set (WithTop α)) : IsGLB s (sInf s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] (s : Set (WithTop α))
Goal: IsGLB s (sInf s)
-- WithTop.coe_sSup
@[norm_cast]
theorem coe_sSup {s : Set α} (hb : BddAbove s) : ↑(sSup s) = (⨆ a ∈ s, ↑a : WithTop α) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hb : BddAbove s)
Goal: ↑(sSup s) = ⨆ a ∈ s, ↑a
-- WithTop.coe_sInf
@[norm_cast]
theorem coe_sInf {s : Set α} (hs : s.Nonempty) (h's : BddBelow s) :
    ↑(sInf s) = (⨅ a ∈ s, ↑a : WithTop α) 
Arguments: {α : Type u_1} [ConditionallyCompleteLinearOrderBot α] {s : Set α} (hs : s.Nonempty) (h's : BddBelow s)
Goal: ↑(sInf s) = ⨅ a ∈ s, ↑a
-- Monotone.le_csSup_image
theorem le_csSup_image {s : Set α} {c : α} (hcs : c ∈ s) (h_bdd : BddAbove s) :
    f c ≤ sSup (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [ConditionallyCompleteLattice β] {f : α → β} (h_mono : Monotone f) {s : Set α} {c : α} (hcs : c ∈ s) (h_bdd : BddAbove s)
Goal: f c ≤ sSup (f '' s)
-- Monotone.csSup_image_le
theorem csSup_image_le {s : Set α} (hs : s.Nonempty) {B : α} (hB : B ∈ upperBounds s) :
    sSup (f '' s) ≤ f B 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [ConditionallyCompleteLattice β] {f : α → β} (h_mono : Monotone f) {s : Set α} (hs : s.Nonempty) {B : α} (hB : B ∈ upperBounds s)
Goal: sSup (f '' s) ≤ f B
-- Monotone.csInf_image_le
theorem csInf_image_le {s : Set α} {c : α} (hcs : c ∈ s) (h_bdd : BddBelow s) :
    sInf (f '' s) ≤ f c 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [ConditionallyCompleteLattice β] {f : α → β} (h_mono : Monotone f) {s : Set α} {c : α} (hcs : c ∈ s) (h_bdd : BddBelow s)
Goal: sInf (f '' s) ≤ f c
-- Monotone.le_csInf_image
theorem le_csInf_image {s : Set α} (hs : s.Nonempty) {B : α} (hB : B ∈ lowerBounds s) :
    f B ≤ sInf (f '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [ConditionallyCompleteLattice β] {f : α → β} (h_mono : Monotone f) {s : Set α} (hs : s.Nonempty) {B : α} (hB : B ∈ lowerBounds s)
Goal: f B ≤ sInf (f '' s)
-- GaloisConnection.l_csSup
theorem l_csSup (gc : GaloisConnection l u) {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s) :
    l (sSup s) = ⨆ x : s, l x 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s)
Goal: l (sSup s) = ⨆ x, l ↑x
-- GaloisConnection.l_csSup'
theorem l_csSup' (gc : GaloisConnection l u) {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s) :
    l (sSup s) = sSup (l '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set α} (hne : s.Nonempty) (hbdd : BddAbove s)
Goal: l (sSup s) = sSup (l '' s)
-- GaloisConnection.l_ciSup
theorem l_ciSup (gc : GaloisConnection l u) {f : ι → α} (hf : BddAbove (range f)) :
    l (⨆ i, f i) = ⨆ i, l (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] [Nonempty ι] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {f : ι → α} (hf : BddAbove (Set.range f))
Goal: l (⨆ i, f i) = ⨆ i, l (f i)
-- GaloisConnection.l_ciSup_set
theorem l_ciSup_set (gc : GaloisConnection l u) {s : Set γ} {f : γ → α} (hf : BddAbove (f '' s))
    (hne : s.Nonempty) : l (⨆ i : s, f i) = ⨆ i : s, l (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set γ} {f : γ → α} (hf : BddAbove (f '' s)) (hne : s.Nonempty)
Goal: l (⨆ i, f ↑i) = ⨆ i, l (f ↑i)
-- GaloisConnection.u_csInf
theorem u_csInf (gc : GaloisConnection l u) {s : Set β} (hne : s.Nonempty) (hbdd : BddBelow s) :
    u (sInf s) = ⨅ x : s, u x 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set β} (hne : s.Nonempty) (hbdd : BddBelow s)
Goal: u (sInf s) = ⨅ x, u ↑x
-- GaloisConnection.u_csInf'
theorem u_csInf' (gc : GaloisConnection l u) {s : Set β} (hne : s.Nonempty) (hbdd : BddBelow s) :
    u (sInf s) = sInf (u '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set β} (hne : s.Nonempty) (hbdd : BddBelow s)
Goal: u (sInf s) = sInf (u '' s)
-- GaloisConnection.u_ciInf
theorem u_ciInf (gc : GaloisConnection l u) {f : ι → β} (hf : BddBelow (range f)) :
    u (⨅ i, f i) = ⨅ i, u (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_4} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] [Nonempty ι] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {f : ι → β} (hf : BddBelow (Set.range f))
Goal: u (⨅ i, f i) = ⨅ i, u (f i)
-- GaloisConnection.u_ciInf_set
theorem u_ciInf_set (gc : GaloisConnection l u) {s : Set γ} {f : γ → β} (hf : BddBelow (f '' s))
    (hne : s.Nonempty) : u (⨅ i : s, f i) = ⨅ i : s, u (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [ConditionallyCompleteLattice α] [ConditionallyCompleteLattice β] {l : α → β} {u : β → α} (gc : GaloisConnection l u) {s : Set γ} {f : γ → β} (hf : BddBelow (f '' s)) (hne : s.Nonempty)
Goal: u (⨅ i, f ↑i) = ⨅ i, u (f ↑i)