ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- WithTop.coe_iSup
@[norm_cast]
theorem WithTop.coe_iSup [SupSet α] (f : ι → α) (h : BddAbove (Set.range f)) :
    ↑(⨆ i, f i) = (⨆ i, f i : WithTop α) 
Arguments: {α : Type u_1} {ι : Sort u_4} [Preorder α] [SupSet α] (f : ι → α) (h : BddAbove (Set.range f))
Goal: ↑(⨆ i, f i) = ⨆ i, ↑(f i)
-- WithBot.sSup_empty
@[simp]
theorem WithBot.sSup_empty [SupSet α] : sSup (∅ : Set (WithBot α)) = ⊥ 
Arguments: {α : Type u_1} [Preorder α] [SupSet α]
Goal: sSup ∅ = ⊥
-- WithBot.ciSup_empty
@[simp]
theorem WithBot.ciSup_empty [IsEmpty ι] [SupSet α] (f : ι → WithBot α) :
    ⨆ i, f i = ⊥ 
Arguments: {α : Type u_1} {ι : Sort u_4} [Preorder α] [IsEmpty ι] [SupSet α] (f : ι → WithBot α)
Goal: ⨆ i, f i = ⊥
-- WithBot.coe_sSup'
@[norm_cast]
theorem WithBot.coe_sSup' [SupSet α] {s : Set α} (hs : s.Nonempty) (h's : BddAbove s) :
    ↑(sSup s) = (sSup ((fun (a : α) ↦ ↑a) '' s) : WithBot α) 
Arguments: {α : Type u_1} [Preorder α] [SupSet α] {s : Set α} (hs : s.Nonempty) (h's : BddAbove s)
Goal: ↑(sSup s) = sSup ((fun a => ↑a) '' s)
-- WithBot.coe_iSup
@[norm_cast]
theorem WithBot.coe_iSup [Nonempty ι] [SupSet α] {f : ι → α} (hf : BddAbove (range f)) :
    ↑(⨆ i, f i) = (⨆ i, f i : WithBot α) 
Arguments: {α : Type u_1} {ι : Sort u_4} [Preorder α] [Nonempty ι] [SupSet α] {f : ι → α} (hf : BddAbove (Set.range f))
Goal: ↑(⨆ i, f i) = ⨆ i, ↑(f i)
-- WithBot.coe_sInf'
@[norm_cast]
theorem WithBot.coe_sInf' [InfSet α] {s : Set α} (hs : BddBelow s) :
    ↑(sInf s) = (sInf ((fun (a : α) ↦ ↑a) '' s) : WithBot α) 
Arguments: {α : Type u_1} [Preorder α] [InfSet α] {s : Set α} (hs : BddBelow s)
Goal: ↑(sInf s) = sInf ((fun a => ↑a) '' s)
-- WithBot.coe_iInf
@[norm_cast]
theorem WithBot.coe_iInf [InfSet α] (f : ι → α) (h : BddBelow (Set.range f)) :
    ↑(⨅ i, f i) = (⨅ i, f i : WithBot α) 
Arguments: {α : Type u_1} {ι : Sort u_4} [Preorder α] [InfSet α] (f : ι → α) (h : BddBelow (Set.range f))
Goal: ↑(⨅ i, f i) = ⨅ i, ↑(f i)
-- ConditionallyCompleteLattice.csSup_le
class ConditionallyCompleteLattice (α : Type*) extends Lattice α, SupSet α, InfSet α where
    le_csSup : ∀ s a, BddAbove s → a ∈ s → a ≤ sSup s
    csSup_le : ∀ s a, Set.Nonempty s → a ∈ upperBounds s → sSup s ≤ a
    csInf_le : ∀ s a, BddBelow s → a ∈ s → sInf s ≤ a
    le_csInf : ∀ s a, Set.Nonempty s → a ∈ lowerBounds s → a ≤ sInf s
Arguments: {α : Type u_5} [self : ConditionallyCompleteLattice α] (s : Set α) (a : α)
Goal: s.Nonempty → a ∈ upperBounds s → sSup s ≤ a
-- ConditionallyCompleteLattice.le_csInf
class ConditionallyCompleteLattice (α : Type*) extends Lattice α, SupSet α, InfSet α where
    le_csSup : ∀ s a, BddAbove s → a ∈ s → a ≤ sSup s
    csSup_le : ∀ s a, Set.Nonempty s → a ∈ upperBounds s → sSup s ≤ a
    csInf_le : ∀ s a, BddBelow s → a ∈ s → sInf s ≤ a
    le_csInf : ∀ s a, Set.Nonempty s → a ∈ lowerBounds s → a ≤ sInf s
Arguments: {α : Type u_5} [self : ConditionallyCompleteLattice α] (s : Set α) (a : α)
Goal: s.Nonempty → a ∈ lowerBounds s → a ≤ sInf s
-- ConditionallyCompleteLattice.csInf_le
class ConditionallyCompleteLattice (α : Type*) extends Lattice α, SupSet α, InfSet α where
    le_csSup : ∀ s a, BddAbove s → a ∈ s → a ≤ sSup s
    csSup_le : ∀ s a, Set.Nonempty s → a ∈ upperBounds s → sSup s ≤ a
    csInf_le : ∀ s a, BddBelow s → a ∈ s → sInf s ≤ a
    le_csInf : ∀ s a, Set.Nonempty s → a ∈ lowerBounds s → a ≤ sInf s
Arguments: {α : Type u_5} [self : ConditionallyCompleteLattice α] (s : Set α) (a : α)
Goal: BddBelow s → a ∈ s → sInf s ≤ a
-- ConditionallyCompleteLattice.le_csSup
class ConditionallyCompleteLattice (α : Type*) extends Lattice α, SupSet α, InfSet α where
    le_csSup : ∀ s a, BddAbove s → a ∈ s → a ≤ sSup s
    csSup_le : ∀ s a, Set.Nonempty s → a ∈ upperBounds s → sSup s ≤ a
    csInf_le : ∀ s a, BddBelow s → a ∈ s → sInf s ≤ a
    le_csInf : ∀ s a, Set.Nonempty s → a ∈ lowerBounds s → a ≤ sInf s
Arguments: {α : Type u_5} [self : ConditionallyCompleteLattice α] (s : Set α) (a : α)
Goal: BddAbove s → a ∈ s → a ≤ sSup s
-- ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove
class ConditionallyCompleteLinearOrder (α : Type*) extends ConditionallyCompleteLattice α where
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
    csSup_of_not_bddAbove : ∀ s, ¬BddAbove s → sSup s = sSup (∅ : Set α)
    csInf_of_not_bddBelow : ∀ s, ¬BddBelow s → sInf s = sInf (∅ : Set α)
Arguments: {α : Type u_5} [self : ConditionallyCompleteLinearOrder α] (s : Set α)
Goal: ¬BddAbove s → sSup s = sSup ∅
-- ConditionallyCompleteLinearOrder.csInf_of_not_bddBelow
class ConditionallyCompleteLinearOrder (α : Type*) extends ConditionallyCompleteLattice α where
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
    csSup_of_not_bddAbove : ∀ s, ¬BddAbove s → sSup s = sSup (∅ : Set α)
    csInf_of_not_bddBelow : ∀ s, ¬BddBelow s → sInf s = sInf (∅ : Set α)
Arguments: {α : Type u_5} [self : ConditionallyCompleteLinearOrder α] (s : Set α)
Goal: ¬BddBelow s → sInf s = sInf ∅
-- ConditionallyCompleteLinearOrder.le_total
class ConditionallyCompleteLinearOrder (α : Type*) extends ConditionallyCompleteLattice α where
    le_total (a b : α) : a ≤ b ∨ b ≤ a
    decidableLE : DecidableRel (· ≤ · : α → α → Prop)
    decidableEq : DecidableEq α := @decidableEqOfDecidableLE _ _ decidableLE
    decidableLT : DecidableRel (· < · : α → α → Prop) :=
    @decidableLTOfDecidableLE _ _ decidableLE
    csSup_of_not_bddAbove : ∀ s, ¬BddAbove s → sSup s = sSup (∅ : Set α)
    csInf_of_not_bddBelow : ∀ s, ¬BddBelow s → sInf s = sInf (∅ : Set α)
Arguments: {α : Type u_5} [self : ConditionallyCompleteLinearOrder α] (a b : α)
Goal: a ≤ b ∨ b ≤ a
-- ConditionallyCompleteLinearOrderBot.csSup_empty
class ConditionallyCompleteLinearOrderBot (α : Type*) extends ConditionallyCompleteLinearOrder α,
  Bot α where
    bot_le : ∀ x : α, ⊥ ≤ x
    csSup_empty : sSup ∅ = ⊥
Arguments: {α : Type u_5} [self : ConditionallyCompleteLinearOrderBot α]
Goal: sSup ∅ = ⊥
-- ConditionallyCompleteLinearOrderBot.bot_le
class ConditionallyCompleteLinearOrderBot (α : Type*) extends ConditionallyCompleteLinearOrder α,
  Bot α where
    bot_le : ∀ x : α, ⊥ ≤ x
    csSup_empty : sSup ∅ = ⊥
Arguments: {α : Type u_5} [self : ConditionallyCompleteLinearOrderBot α] (x : α)
Goal: ⊥ ≤ x
-- le_csSup
theorem le_csSup (h₁ : BddAbove s) (h₂ : a ∈ s) : a ≤ sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (h₁ : BddAbove s) (h₂ : a ∈ s)
Goal: a ≤ sSup s
-- csSup_le
theorem csSup_le (h₁ : s.Nonempty) (h₂ : ∀ b ∈ s, b ≤ a) : sSup s ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (h₁ : s.Nonempty) (h₂ : ∀ b ∈ s, b ≤ a)
Goal: sSup s ≤ a
-- csInf_le
theorem csInf_le (h₁ : BddBelow s) (h₂ : a ∈ s) : sInf s ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (h₁ : BddBelow s) (h₂ : a ∈ s)
Goal: sInf s ≤ a
-- le_csInf
theorem le_csInf (h₁ : s.Nonempty) (h₂ : ∀ b ∈ s, a ≤ b) : a ≤ sInf s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (h₁ : s.Nonempty) (h₂ : ∀ b ∈ s, a ≤ b)
Goal: a ≤ sInf s
-- le_csSup_of_le
theorem le_csSup_of_le (hs : BddAbove s) (hb : b ∈ s) (h : a ≤ b) : a ≤ sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a b : α} (hs : BddAbove s) (hb : b ∈ s) (h : a ≤ b)
Goal: a ≤ sSup s
-- csInf_le_of_le
theorem csInf_le_of_le (hs : BddBelow s) (hb : b ∈ s) (h : b ≤ a) : sInf s ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a b : α} (hs : BddBelow s) (hb : b ∈ s) (h : b ≤ a)
Goal: sInf s ≤ a
-- csSup_le_csSup
theorem csSup_le_csSup (ht : BddAbove t) (hs : s.Nonempty) (h : s ⊆ t) : sSup s ≤ sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α} (ht : BddAbove t) (hs : s.Nonempty) (h : s ⊆ t)
Goal: sSup s ≤ sSup t
-- csInf_le_csInf
theorem csInf_le_csInf (ht : BddBelow t) (hs : s.Nonempty) (h : s ⊆ t) : sInf t ≤ sInf s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α} (ht : BddBelow t) (hs : s.Nonempty) (h : s ⊆ t)
Goal: sInf t ≤ sInf s
-- le_csSup_iff
theorem le_csSup_iff (h : BddAbove s) (hs : s.Nonempty) :
    a ≤ sSup s ↔ ∀ b, b ∈ upperBounds s → a ≤ b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (h : BddAbove s) (hs : s.Nonempty)
Goal: a ≤ sSup s ↔ ∀ b ∈ upperBounds s, a ≤ b
-- csInf_le_iff
theorem csInf_le_iff (h : BddBelow s) (hs : s.Nonempty) : sInf s ≤ a ↔ ∀ b ∈ lowerBounds s, b ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (h : BddBelow s) (hs : s.Nonempty)
Goal: sInf s ≤ a ↔ ∀ b ∈ lowerBounds s, b ≤ a
-- isLUB_csSup
theorem isLUB_csSup (ne : s.Nonempty) (H : BddAbove s) : IsLUB s (sSup s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (ne : s.Nonempty) (H : BddAbove s)
Goal: IsLUB s (sSup s)
-- isLUB_ciSup
theorem isLUB_ciSup [Nonempty ι] {f : ι → α} (H : BddAbove (range f)) :
    IsLUB (range f) (⨆ i, f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} (H : BddAbove (Set.range f))
Goal: IsLUB (Set.range f) (⨆ i, f i)
-- isLUB_ciSup_set
theorem isLUB_ciSup_set {f : β → α} {s : Set β} (H : BddAbove (f '' s)) (Hne : s.Nonempty) :
    IsLUB (f '' s) (⨆ i : s, f i) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : β → α} {s : Set β} (H : BddAbove (f '' s)) (Hne : s.Nonempty)
Goal: IsLUB (f '' s) (⨆ i, f ↑i)
-- isGLB_csInf
theorem isGLB_csInf (ne : s.Nonempty) (H : BddBelow s) : IsGLB s (sInf s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (ne : s.Nonempty) (H : BddBelow s)
Goal: IsGLB s (sInf s)
-- isGLB_ciInf
theorem isGLB_ciInf [Nonempty ι] {f : ι → α} (H : BddBelow (range f)) :
    IsGLB (range f) (⨅ i, f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} (H : BddBelow (Set.range f))
Goal: IsGLB (Set.range f) (⨅ i, f i)
-- isGLB_ciInf_set
theorem isGLB_ciInf_set {f : β → α} {s : Set β} (H : BddBelow (f '' s)) (Hne : s.Nonempty) :
    IsGLB (f '' s) (⨅ i : s, f i) 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : β → α} {s : Set β} (H : BddBelow (f '' s)) (Hne : s.Nonempty)
Goal: IsGLB (f '' s) (⨅ i, f ↑i)
-- ciSup_le_iff
theorem ciSup_le_iff [Nonempty ι] {f : ι → α} {a : α} (hf : BddAbove (range f)) :
    iSup f ≤ a ↔ ∀ i, f i ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} {a : α} (hf : BddAbove (Set.range f))
Goal: iSup f ≤ a ↔ ∀ (i : ι), f i ≤ a
-- le_ciInf_iff
theorem le_ciInf_iff [Nonempty ι] {f : ι → α} {a : α} (hf : BddBelow (range f)) :
    a ≤ iInf f ↔ ∀ i, a ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} {a : α} (hf : BddBelow (Set.range f))
Goal: a ≤ iInf f ↔ ∀ (i : ι), a ≤ f i
-- ciSup_set_le_iff
theorem ciSup_set_le_iff {ι : Type*} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty)
    (hf : BddAbove (f '' s)) : ⨆ i : s, f i ≤ a ↔ ∀ i ∈ s, f i ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {ι : Type u_5} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty) (hf : BddAbove (f '' s))
Goal: ⨆ i, f ↑i ≤ a ↔ ∀ i ∈ s, f i ≤ a
-- le_ciInf_set_iff
theorem le_ciInf_set_iff {ι : Type*} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty)
    (hf : BddBelow (f '' s)) : (a ≤ ⨅ i : s, f i) ↔ ∀ i ∈ s, a ≤ f i 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {ι : Type u_5} {s : Set ι} {f : ι → α} {a : α} (hs : s.Nonempty) (hf : BddBelow (f '' s))
Goal: a ≤ ⨅ i, f ↑i ↔ ∀ i ∈ s, a ≤ f i
-- IsLUB.csSup_eq
theorem IsLUB.csSup_eq (H : IsLUB s a) (ne : s.Nonempty) : sSup s = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (H : IsLUB s a) (ne : s.Nonempty)
Goal: sSup s = a
-- IsLUB.ciSup_eq
theorem IsLUB.ciSup_eq [Nonempty ι] {f : ι → α} (H : IsLUB (range f) a) : ⨆ i, f i = a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {a : α} [Nonempty ι] {f : ι → α} (H : IsLUB (Set.range f) a)
Goal: ⨆ i, f i = a
-- IsLUB.ciSup_set_eq
theorem IsLUB.ciSup_set_eq {s : Set β} {f : β → α} (H : IsLUB (f '' s) a) (Hne : s.Nonempty) :
    ⨆ i : s, f i = a 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {a : α} {s : Set β} {f : β → α} (H : IsLUB (f '' s) a) (Hne : s.Nonempty)
Goal: ⨆ i, f ↑i = a
-- IsGreatest.csSup_eq
theorem IsGreatest.csSup_eq (H : IsGreatest s a) : sSup s = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (H : IsGreatest s a)
Goal: sSup s = a
-- IsGreatest.csSup_mem
theorem IsGreatest.csSup_mem (H : IsGreatest s a) : sSup s ∈ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (H : IsGreatest s a)
Goal: sSup s ∈ s
-- IsGLB.csInf_eq
theorem IsGLB.csInf_eq (H : IsGLB s a) (ne : s.Nonempty) : sInf s = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (H : IsGLB s a) (ne : s.Nonempty)
Goal: sInf s = a
-- IsGLB.ciInf_eq
theorem IsGLB.ciInf_eq [Nonempty ι] {f : ι → α} (H : IsGLB (range f) a) : ⨅ i, f i = a 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {a : α} [Nonempty ι] {f : ι → α} (H : IsGLB (Set.range f) a)
Goal: ⨅ i, f i = a
-- IsGLB.ciInf_set_eq
theorem IsGLB.ciInf_set_eq {s : Set β} {f : β → α} (H : IsGLB (f '' s) a) (Hne : s.Nonempty) :
    ⨅ i : s, f i = a 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {a : α} {s : Set β} {f : β → α} (H : IsGLB (f '' s) a) (Hne : s.Nonempty)
Goal: ⨅ i, f ↑i = a
-- IsLeast.csInf_eq
theorem IsLeast.csInf_eq (H : IsLeast s a) : sInf s = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (H : IsLeast s a)
Goal: sInf s = a
-- IsLeast.csInf_mem
theorem IsLeast.csInf_mem (H : IsLeast s a) : sInf s ∈ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (H : IsLeast s a)
Goal: sInf s ∈ s
-- subset_Icc_csInf_csSup
theorem subset_Icc_csInf_csSup (hb : BddBelow s) (ha : BddAbove s) : s ⊆ Icc (sInf s) (sSup s) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (hb : BddBelow s) (ha : BddAbove s)
Goal: s ⊆ Set.Icc (sInf s) (sSup s)
-- csSup_le_iff
theorem csSup_le_iff (hb : BddAbove s) (hs : s.Nonempty) : sSup s ≤ a ↔ ∀ b ∈ s, b ≤ a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (hb : BddAbove s) (hs : s.Nonempty)
Goal: sSup s ≤ a ↔ ∀ b ∈ s, b ≤ a
-- le_csInf_iff
theorem le_csInf_iff (hb : BddBelow s) (hs : s.Nonempty) : a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (hb : BddBelow s) (hs : s.Nonempty)
Goal: a ≤ sInf s ↔ ∀ b ∈ s, a ≤ b
-- csSup_lower_bounds_eq_csInf
theorem csSup_lower_bounds_eq_csInf {s : Set α} (h : BddBelow s) (hs : s.Nonempty) :
    sSup (lowerBounds s) = sInf s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (h : BddBelow s) (hs : s.Nonempty)
Goal: sSup (lowerBounds s) = sInf s
-- csInf_upper_bounds_eq_csSup
theorem csInf_upper_bounds_eq_csSup {s : Set α} (h : BddAbove s) (hs : s.Nonempty) :
    sInf (upperBounds s) = sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (h : BddAbove s) (hs : s.Nonempty)
Goal: sInf (upperBounds s) = sSup s
-- not_mem_of_lt_csInf
theorem not_mem_of_lt_csInf {x : α} {s : Set α} (h : x < sInf s) (hs : BddBelow s) : x ∉ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {x : α} {s : Set α} (h : x < sInf s) (hs : BddBelow s)
Goal: x ∉ s
-- not_mem_of_csSup_lt
theorem not_mem_of_csSup_lt {x : α} {s : Set α} (h : sSup s < x) (hs : BddAbove s) : x ∉ s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {x : α} {s : Set α} (h : sSup s < x) (hs : BddAbove s)
Goal: x ∉ s
-- csSup_eq_of_forall_le_of_forall_lt_exists_gt
theorem csSup_eq_of_forall_le_of_forall_lt_exists_gt (hs : s.Nonempty) (H : ∀ a ∈ s, a ≤ b)
    (H' : ∀ w, w < b → ∃ a ∈ s, w < a) : sSup s = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {b : α} (hs : s.Nonempty) (H : ∀ a ∈ s, a ≤ b) (H' : ∀ w < b, ∃ a ∈ s, w < a)
Goal: sSup s = b
-- csInf_eq_of_forall_ge_of_forall_gt_exists_lt
theorem csInf_eq_of_forall_ge_of_forall_gt_exists_lt :
    s.Nonempty → (∀ a ∈ s, b ≤ a) → (∀ w, b < w → ∃ a ∈ s, a < w) → sInf s = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {b : α}
Goal: s.Nonempty → (∀ a ∈ s, b ≤ a) → (∀ (w : α), b < w → ∃ a ∈ s, a < w) → sInf s = b
-- lt_csSup_of_lt
theorem lt_csSup_of_lt (hs : BddAbove s) (ha : a ∈ s) (h : b < a) : b < sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a b : α} (hs : BddAbove s) (ha : a ∈ s) (h : b < a)
Goal: b < sSup s
-- csInf_lt_of_lt
theorem csInf_lt_of_lt : BddBelow s → a ∈ s → a < b → sInf s < b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a b : α}
Goal: BddBelow s → a ∈ s → a < b → sInf s < b
-- exists_between_of_forall_le
theorem exists_between_of_forall_le (sne : s.Nonempty) (tne : t.Nonempty)
    (hst : ∀ x ∈ s, ∀ y ∈ t, x ≤ y) : (upperBounds s ∩ lowerBounds t).Nonempty 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α} (sne : s.Nonempty) (tne : t.Nonempty) (hst : ∀ x ∈ s, ∀ y ∈ t, x ≤ y)
Goal: (upperBounds s ∩ lowerBounds t).Nonempty
-- csSup_singleton
@[simp]
theorem csSup_singleton (a : α) : sSup {a} = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] (a : α)
Goal: sSup {a} = a
-- csInf_singleton
@[simp]
theorem csInf_singleton (a : α) : sInf {a} = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] (a : α)
Goal: sInf {a} = a
-- csSup_pair
@[simp]
theorem csSup_pair (a b : α) : sSup {a, b} = a ⊔ b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] (a b : α)
Goal: sSup {a, b} = a ⊔ b
-- csInf_pair
@[simp]
theorem csInf_pair (a b : α) : sInf {a, b} = a ⊓ b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] (a b : α)
Goal: sInf {a, b} = a ⊓ b
-- csInf_le_csSup
theorem csInf_le_csSup (hb : BddBelow s) (ha : BddAbove s) (ne : s.Nonempty) : sInf s ≤ sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} (hb : BddBelow s) (ha : BddAbove s) (ne : s.Nonempty)
Goal: sInf s ≤ sSup s
-- csSup_union
theorem csSup_union (hs : BddAbove s) (sne : s.Nonempty) (ht : BddAbove t) (tne : t.Nonempty) :
    sSup (s ∪ t) = sSup s ⊔ sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α} (hs : BddAbove s) (sne : s.Nonempty) (ht : BddAbove t) (tne : t.Nonempty)
Goal: sSup (s ∪ t) = sSup s ⊔ sSup t
-- csInf_union
theorem csInf_union (hs : BddBelow s) (sne : s.Nonempty) (ht : BddBelow t) (tne : t.Nonempty) :
    sInf (s ∪ t) = sInf s ⊓ sInf t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α} (hs : BddBelow s) (sne : s.Nonempty) (ht : BddBelow t) (tne : t.Nonempty)
Goal: sInf (s ∪ t) = sInf s ⊓ sInf t
-- csSup_inter_le
theorem csSup_inter_le (hs : BddAbove s) (ht : BddAbove t) (hst : (s ∩ t).Nonempty) :
    sSup (s ∩ t) ≤ sSup s ⊓ sSup t 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α} (hs : BddAbove s) (ht : BddAbove t) (hst : (s ∩ t).Nonempty)
Goal: sSup (s ∩ t) ≤ sSup s ⊓ sSup t
-- le_csInf_inter
theorem le_csInf_inter :
    BddBelow s → BddBelow t → (s ∩ t).Nonempty → sInf s ⊔ sInf t ≤ sInf (s ∩ t) 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s t : Set α}
Goal: BddBelow s → BddBelow t → (s ∩ t).Nonempty → sInf s ⊔ sInf t ≤ sInf (s ∩ t)
-- csSup_insert
theorem csSup_insert (hs : BddAbove s) (sne : s.Nonempty) : sSup (insert a s) = a ⊔ sSup s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (hs : BddAbove s) (sne : s.Nonempty)
Goal: sSup (insert a s) = a ⊔ sSup s
-- csInf_insert
theorem csInf_insert (hs : BddBelow s) (sne : s.Nonempty) : sInf (insert a s) = a ⊓ sInf s 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {s : Set α} {a : α} (hs : BddBelow s) (sne : s.Nonempty)
Goal: sInf (insert a s) = a ⊓ sInf s
-- csInf_Icc
@[simp]
theorem csInf_Icc (h : a ≤ b) : sInf (Icc a b) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
Goal: sInf (Set.Icc a b) = a
-- csInf_Ici
@[simp]
theorem csInf_Ici : sInf (Ici a) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a : α}
Goal: sInf (Set.Ici a) = a
-- csInf_Ico
@[simp]
theorem csInf_Ico (h : a < b) : sInf (Ico a b) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} (h : a < b)
Goal: sInf (Set.Ico a b) = a
-- csInf_Ioc
@[simp]
theorem csInf_Ioc [DenselyOrdered α] (h : a < b) : sInf (Ioc a b) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} [DenselyOrdered α] (h : a < b)
Goal: sInf (Set.Ioc a b) = a
-- csInf_Ioi
@[simp]
theorem csInf_Ioi [NoMaxOrder α] [DenselyOrdered α] : sInf (Ioi a) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a : α} [NoMaxOrder α] [DenselyOrdered α]
Goal: sInf (Set.Ioi a) = a
-- csInf_Ioo
@[simp]
theorem csInf_Ioo [DenselyOrdered α] (h : a < b) : sInf (Ioo a b) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} [DenselyOrdered α] (h : a < b)
Goal: sInf (Set.Ioo a b) = a
-- csSup_Icc
@[simp]
theorem csSup_Icc (h : a ≤ b) : sSup (Icc a b) = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} (h : a ≤ b)
Goal: sSup (Set.Icc a b) = b
-- csSup_Ico
@[simp]
theorem csSup_Ico [DenselyOrdered α] (h : a < b) : sSup (Ico a b) = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} [DenselyOrdered α] (h : a < b)
Goal: sSup (Set.Ico a b) = b
-- csSup_Iic
@[simp]
theorem csSup_Iic : sSup (Iic a) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a : α}
Goal: sSup (Set.Iic a) = a
-- csSup_Iio
@[simp]
theorem csSup_Iio [NoMinOrder α] [DenselyOrdered α] : sSup (Iio a) = a 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a : α} [NoMinOrder α] [DenselyOrdered α]
Goal: sSup (Set.Iio a) = a
-- csSup_Ioc
@[simp]
theorem csSup_Ioc (h : a < b) : sSup (Ioc a b) = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} (h : a < b)
Goal: sSup (Set.Ioc a b) = b
-- csSup_Ioo
@[simp]
theorem csSup_Ioo [DenselyOrdered α] (h : a < b) : sSup (Ioo a b) = b 
Arguments: {α : Type u_1} [ConditionallyCompleteLattice α] {a b : α} [DenselyOrdered α] (h : a < b)
Goal: sSup (Set.Ioo a b) = b
-- ciSup_le
theorem ciSup_le [Nonempty ι] {f : ι → α} {c : α} (H : ∀ x, f x ≤ c) : iSup f ≤ c 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] [Nonempty ι] {f : ι → α} {c : α} (H : ∀ (x : ι), f x ≤ c)
Goal: iSup f ≤ c
-- le_ciSup
theorem le_ciSup {f : ι → α} (H : BddAbove (range f)) (c : ι) : f c ≤ iSup f 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {f : ι → α} (H : BddAbove (Set.range f)) (c : ι)
Goal: f c ≤ iSup f
-- le_ciSup_of_le
theorem le_ciSup_of_le {f : ι → α} (H : BddAbove (range f)) (c : ι) (h : a ≤ f c) : a ≤ iSup f 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {a : α} {f : ι → α} (H : BddAbove (Set.range f)) (c : ι) (h : a ≤ f c)
Goal: a ≤ iSup f
-- ciSup_mono
theorem ciSup_mono {f g : ι → α} (B : BddAbove (range g)) (H : ∀ x, f x ≤ g x) :
    iSup f ≤ iSup g 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {f g : ι → α} (B : BddAbove (Set.range g)) (H : ∀ (x : ι), f x ≤ g x)
Goal: iSup f ≤ iSup g
-- le_ciSup_set
theorem le_ciSup_set {f : β → α} {s : Set β} (H : BddAbove (f '' s)) {c : β} (hc : c ∈ s) :
    f c ≤ ⨆ i : s, f i 
Arguments: {α : Type u_1} {β : Type u_2} [ConditionallyCompleteLattice α] {f : β → α} {s : Set β} (H : BddAbove (f '' s)) {c : β} (hc : c ∈ s)
Goal: f c ≤ ⨆ i, f ↑i
-- ciInf_mono
theorem ciInf_mono {f g : ι → α} (B : BddBelow (range f)) (H : ∀ x, f x ≤ g x) : iInf f ≤ iInf g 
Arguments: {α : Type u_1} {ι : Sort u_4} [ConditionallyCompleteLattice α] {f g : ι → α} (B : BddBelow (Set.range f)) (H : ∀ (x : ι), f x ≤ g x)
Goal: iInf f ≤ iInf g