ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- UpperSet.not_mem_top
@[simp]
theorem not_mem_top : a ∉ (⊤ : UpperSet α) 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: a ∉ ⊤
-- UpperSet.mem_bot
@[simp]
theorem mem_bot : a ∈ (⊥ : UpperSet α) 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: a ∈ ⊥
-- UpperSet.mem_sup_iff
@[simp]
theorem mem_sup_iff : a ∈ s ⊔ t ↔ a ∈ s ∧ a ∈ t 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α} {a : α}
Goal: a ∈ s ⊔ t ↔ a ∈ s ∧ a ∈ t
-- UpperSet.mem_inf_iff
@[simp]
theorem mem_inf_iff : a ∈ s ⊓ t ↔ a ∈ s ∨ a ∈ t 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α} {a : α}
Goal: a ∈ s ⊓ t ↔ a ∈ s ∨ a ∈ t
-- UpperSet.mem_sSup_iff
@[simp]
theorem mem_sSup_iff : a ∈ sSup S ↔ ∀ s ∈ S, a ∈ s 
Arguments: {α : Type u_1} [LE α] {S : Set (UpperSet α)} {a : α}
Goal: a ∈ sSup S ↔ ∀ s ∈ S, a ∈ s
-- UpperSet.mem_sInf_iff
@[simp]
theorem mem_sInf_iff : a ∈ sInf S ↔ ∃ s ∈ S, a ∈ s 
Arguments: {α : Type u_1} [LE α] {S : Set (UpperSet α)} {a : α}
Goal: a ∈ sInf S ↔ ∃ s ∈ S, a ∈ s
-- UpperSet.mem_iSup_iff
@[simp]
theorem mem_iSup_iff {f : ι → UpperSet α} : (a ∈ ⨆ i, f i) ↔ ∀ i, a ∈ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {a : α} {f : ι → UpperSet α}
Goal: a ∈ ⨆ i, f i ↔ ∀ (i : ι), a ∈ f i
-- UpperSet.mem_iInf_iff
@[simp]
theorem mem_iInf_iff {f : ι → UpperSet α} : (a ∈ ⨅ i, f i) ↔ ∃ i, a ∈ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {a : α} {f : ι → UpperSet α}
Goal: a ∈ ⨅ i, f i ↔ ∃ i, a ∈ f i
-- UpperSet.mem_iSup₂_iff
theorem mem_iSup₂_iff {f : ∀ i, κ i → UpperSet α} : (a ∈ ⨆ (i) (j), f i j) ↔ ∀ i j, a ∈ f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {a : α} {f : (i : ι) → κ i → UpperSet α}
Goal: a ∈ ⨆ i, ⨆ j, f i j ↔ ∀ (i : ι) (j : κ i), a ∈ f i j
-- UpperSet.mem_iInf₂_iff
theorem mem_iInf₂_iff {f : ∀ i, κ i → UpperSet α} : (a ∈ ⨅ (i) (j), f i j) ↔ ∃ i j, a ∈ f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {a : α} {f : (i : ι) → κ i → UpperSet α}
Goal: a ∈ ⨅ i, ⨅ j, f i j ↔ ∃ i j, a ∈ f i j
-- UpperSet.codisjoint_coe
@[simp, norm_cast]
theorem codisjoint_coe : Codisjoint (s : Set α) t ↔ Disjoint s t 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α}
Goal: Codisjoint ↑s ↑t ↔ Disjoint s t
-- LowerSet.coe_subset_coe
@[norm_cast] lemma coe_subset_coe : (s : Set α) ⊆ t ↔ s ≤ t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α}
Goal: ↑s ⊆ ↑t ↔ s ≤ t
-- LowerSet.coe_ssubset_coe
@[norm_cast] lemma coe_ssubset_coe : (s : Set α) ⊂ t ↔ s < t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α}
Goal: ↑s ⊂ ↑t ↔ s < t
-- LowerSet.coe_top
@[simp, norm_cast]
theorem coe_top : ((⊤ : LowerSet α) : Set α) = univ 
Arguments: {α : Type u_1} [LE α]
Goal: ↑⊤ = Set.univ
-- LowerSet.coe_bot
@[simp, norm_cast]
theorem coe_bot : ((⊥ : LowerSet α) : Set α) = ∅ 
Arguments: {α : Type u_1} [LE α]
Goal: ↑⊥ = ∅
-- LowerSet.coe_eq_univ
@[simp, norm_cast]
theorem coe_eq_univ : (s : Set α) = univ ↔ s = ⊤ 
Arguments: {α : Type u_1} [LE α] {s : LowerSet α}
Goal: ↑s = Set.univ ↔ s = ⊤
-- LowerSet.coe_eq_empty
@[simp, norm_cast]
theorem coe_eq_empty : (s : Set α) = ∅ ↔ s = ⊥ 
Arguments: {α : Type u_1} [LE α] {s : LowerSet α}
Goal: ↑s = ∅ ↔ s = ⊥
-- LowerSet.coe_nonempty
@[simp, norm_cast] lemma coe_nonempty : (s : Set α).Nonempty ↔ s ≠ ⊥ 
Arguments: {α : Type u_1} [LE α] {s : LowerSet α}
Goal: (↑s).Nonempty ↔ s ≠ ⊥
-- LowerSet.coe_sup
@[simp, norm_cast]
theorem coe_sup (s t : LowerSet α) : (↑(s ⊔ t) : Set α) = (s : Set α) ∪ t 
Arguments: {α : Type u_1} [LE α] (s t : LowerSet α)
Goal: ↑(s ⊔ t) = ↑s ∪ ↑t
-- LowerSet.coe_inf
@[simp, norm_cast]
theorem coe_inf (s t : LowerSet α) : (↑(s ⊓ t) : Set α) = (s : Set α) ∩ t 
Arguments: {α : Type u_1} [LE α] (s t : LowerSet α)
Goal: ↑(s ⊓ t) = ↑s ∩ ↑t
-- LowerSet.coe_sSup
@[simp, norm_cast]
theorem coe_sSup (S : Set (LowerSet α)) : (↑(sSup S) : Set α) = ⋃ s ∈ S, ↑s 
Arguments: {α : Type u_1} [LE α] (S : Set (LowerSet α))
Goal: ↑(sSup S) = ⋃ s ∈ S, ↑s
-- LowerSet.coe_sInf
@[simp, norm_cast]
theorem coe_sInf (S : Set (LowerSet α)) : (↑(sInf S) : Set α) = ⋂ s ∈ S, ↑s 
Arguments: {α : Type u_1} [LE α] (S : Set (LowerSet α))
Goal: ↑(sInf S) = ⋂ s ∈ S, ↑s
-- LowerSet.coe_iSup
@[simp, norm_cast]
theorem coe_iSup (f : ι → LowerSet α) : (↑(⨆ i, f i) : Set α) = ⋃ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → LowerSet α)
Goal: ↑(⨆ i, f i) = ⋃ i, ↑(f i)
-- LowerSet.coe_iInf
@[simp, norm_cast]
theorem coe_iInf (f : ι → LowerSet α) : (↑(⨅ i, f i) : Set α) = ⋂ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → LowerSet α)
Goal: ↑(⨅ i, f i) = ⋂ i, ↑(f i)
-- LowerSet.coe_iSup₂
@[norm_cast] -- Porting note: no longer a `simp`
theorem coe_iSup₂ (f : ∀ i, κ i → LowerSet α) :
    (↑(⨆ (i) (j), f i j) : Set α) = ⋃ (i) (j), f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → LowerSet α)
Goal: ↑(⨆ i, ⨆ j, f i j) = ⋃ i, ⋃ j, ↑(f i j)
-- LowerSet.coe_iInf₂
@[norm_cast] -- Porting note: no longer a `simp`
theorem coe_iInf₂ (f : ∀ i, κ i → LowerSet α) :
    (↑(⨅ (i) (j), f i j) : Set α) = ⋂ (i) (j), f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → LowerSet α)
Goal: ↑(⨅ i, ⨅ j, f i j) = ⋂ i, ⋂ j, ↑(f i j)
-- LowerSet.mem_top
@[simp]
theorem mem_top : a ∈ (⊤ : LowerSet α) 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: a ∈ ⊤
-- LowerSet.not_mem_bot
@[simp]
theorem not_mem_bot : a ∉ (⊥ : LowerSet α) 
Arguments: {α : Type u_1} [LE α] {a : α}
Goal: a ∉ ⊥
-- LowerSet.mem_sup_iff
@[simp]
theorem mem_sup_iff : a ∈ s ⊔ t ↔ a ∈ s ∨ a ∈ t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α} {a : α}
Goal: a ∈ s ⊔ t ↔ a ∈ s ∨ a ∈ t
-- LowerSet.mem_inf_iff
@[simp]
theorem mem_inf_iff : a ∈ s ⊓ t ↔ a ∈ s ∧ a ∈ t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α} {a : α}
Goal: a ∈ s ⊓ t ↔ a ∈ s ∧ a ∈ t
-- LowerSet.mem_sSup_iff
@[simp]
theorem mem_sSup_iff : a ∈ sSup S ↔ ∃ s ∈ S, a ∈ s 
Arguments: {α : Type u_1} [LE α] {S : Set (LowerSet α)} {a : α}
Goal: a ∈ sSup S ↔ ∃ s ∈ S, a ∈ s
-- LowerSet.mem_sInf_iff
@[simp]
theorem mem_sInf_iff : a ∈ sInf S ↔ ∀ s ∈ S, a ∈ s 
Arguments: {α : Type u_1} [LE α] {S : Set (LowerSet α)} {a : α}
Goal: a ∈ sInf S ↔ ∀ s ∈ S, a ∈ s
-- LowerSet.mem_iSup_iff
@[simp]
theorem mem_iSup_iff {f : ι → LowerSet α} : (a ∈ ⨆ i, f i) ↔ ∃ i, a ∈ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {a : α} {f : ι → LowerSet α}
Goal: a ∈ ⨆ i, f i ↔ ∃ i, a ∈ f i
-- LowerSet.mem_iInf_iff
@[simp]
theorem mem_iInf_iff {f : ι → LowerSet α} : (a ∈ ⨅ i, f i) ↔ ∀ i, a ∈ f i 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] {a : α} {f : ι → LowerSet α}
Goal: a ∈ ⨅ i, f i ↔ ∀ (i : ι), a ∈ f i
-- LowerSet.mem_iSup₂_iff
theorem mem_iSup₂_iff {f : ∀ i, κ i → LowerSet α} : (a ∈ ⨆ (i) (j), f i j) ↔ ∃ i j, a ∈ f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {a : α} {f : (i : ι) → κ i → LowerSet α}
Goal: a ∈ ⨆ i, ⨆ j, f i j ↔ ∃ i j, a ∈ f i j
-- LowerSet.mem_iInf₂_iff
theorem mem_iInf₂_iff {f : ∀ i, κ i → LowerSet α} : (a ∈ ⨅ (i) (j), f i j) ↔ ∀ i j, a ∈ f i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] {a : α} {f : (i : ι) → κ i → LowerSet α}
Goal: a ∈ ⨅ i, ⨅ j, f i j ↔ ∀ (i : ι) (j : κ i), a ∈ f i j
-- LowerSet.disjoint_coe
@[simp, norm_cast]
theorem disjoint_coe : Disjoint (s : Set α) t ↔ Disjoint s t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α}
Goal: Disjoint ↑s ↑t ↔ Disjoint s t
-- UpperSet.coe_compl
@[simp]
theorem coe_compl (s : UpperSet α) : (s.compl : Set α) = (↑s)ᶜ 
Arguments: {α : Type u_1} [LE α] (s : UpperSet α)
Goal: ↑s.compl = (↑s)ᶜ
-- UpperSet.mem_compl_iff
@[simp]
theorem mem_compl_iff : a ∈ s.compl ↔ a ∉ s 
Arguments: {α : Type u_1} [LE α] {s : UpperSet α} {a : α}
Goal: a ∈ s.compl ↔ a ∉ s
-- UpperSet.compl_compl
@[simp]
nonrec theorem compl_compl (s : UpperSet α) : s.compl.compl = s 
Arguments: {α : Type u_1} [LE α] (s : UpperSet α)
Goal: s.compl.compl = s
-- UpperSet.compl_le_compl
@[simp]
theorem compl_le_compl : s.compl ≤ t.compl ↔ s ≤ t 
Arguments: {α : Type u_1} [LE α] {s t : UpperSet α}
Goal: s.compl ≤ t.compl ↔ s ≤ t
-- UpperSet.compl_sup
@[simp]
protected theorem compl_sup (s t : UpperSet α) : (s ⊔ t).compl = s.compl ⊔ t.compl 
Arguments: {α : Type u_1} [LE α] (s t : UpperSet α)
Goal: (s ⊔ t).compl = s.compl ⊔ t.compl
-- UpperSet.compl_inf
@[simp]
protected theorem compl_inf (s t : UpperSet α) : (s ⊓ t).compl = s.compl ⊓ t.compl 
Arguments: {α : Type u_1} [LE α] (s t : UpperSet α)
Goal: (s ⊓ t).compl = s.compl ⊓ t.compl
-- UpperSet.compl_top
@[simp]
protected theorem compl_top : (⊤ : UpperSet α).compl = ⊤ 
Arguments: {α : Type u_1} [LE α]
Goal: ⊤.compl = ⊤
-- UpperSet.compl_bot
@[simp]
protected theorem compl_bot : (⊥ : UpperSet α).compl = ⊥ 
Arguments: {α : Type u_1} [LE α]
Goal: ⊥.compl = ⊥
-- UpperSet.compl_sSup
@[simp]
protected theorem compl_sSup (S : Set (UpperSet α)) : (sSup S).compl = ⨆ s ∈ S, UpperSet.compl s 
Arguments: {α : Type u_1} [LE α] (S : Set (UpperSet α))
Goal: (sSup S).compl = ⨆ s ∈ S, s.compl
-- UpperSet.compl_sInf
@[simp]
protected theorem compl_sInf (S : Set (UpperSet α)) : (sInf S).compl = ⨅ s ∈ S, UpperSet.compl s 
Arguments: {α : Type u_1} [LE α] (S : Set (UpperSet α))
Goal: (sInf S).compl = ⨅ s ∈ S, s.compl
-- UpperSet.compl_iSup
@[simp]
protected theorem compl_iSup (f : ι → UpperSet α) : (⨆ i, f i).compl = ⨆ i, (f i).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → UpperSet α)
Goal: (⨆ i, f i).compl = ⨆ i, (f i).compl
-- UpperSet.compl_iInf
@[simp]
protected theorem compl_iInf (f : ι → UpperSet α) : (⨅ i, f i).compl = ⨅ i, (f i).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → UpperSet α)
Goal: (⨅ i, f i).compl = ⨅ i, (f i).compl
-- UpperSet.compl_iSup₂
theorem compl_iSup₂ (f : ∀ i, κ i → UpperSet α) :
    (⨆ (i) (j), f i j).compl = ⨆ (i) (j), (f i j).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → UpperSet α)
Goal: (⨆ i, ⨆ j, f i j).compl = ⨆ i, ⨆ j, (f i j).compl
-- UpperSet.compl_iInf₂
theorem compl_iInf₂ (f : ∀ i, κ i → UpperSet α) :
    (⨅ (i) (j), f i j).compl = ⨅ (i) (j), (f i j).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → UpperSet α)
Goal: (⨅ i, ⨅ j, f i j).compl = ⨅ i, ⨅ j, (f i j).compl
-- LowerSet.coe_compl
@[simp]
theorem coe_compl (s : LowerSet α) : (s.compl : Set α) = (↑s)ᶜ 
Arguments: {α : Type u_1} [LE α] (s : LowerSet α)
Goal: ↑s.compl = (↑s)ᶜ
-- LowerSet.mem_compl_iff
@[simp]
theorem mem_compl_iff : a ∈ s.compl ↔ a ∉ s 
Arguments: {α : Type u_1} [LE α] {s : LowerSet α} {a : α}
Goal: a ∈ s.compl ↔ a ∉ s
-- LowerSet.compl_compl
@[simp]
nonrec theorem compl_compl (s : LowerSet α) : s.compl.compl = s 
Arguments: {α : Type u_1} [LE α] (s : LowerSet α)
Goal: s.compl.compl = s
-- LowerSet.compl_le_compl
@[simp]
theorem compl_le_compl : s.compl ≤ t.compl ↔ s ≤ t 
Arguments: {α : Type u_1} [LE α] {s t : LowerSet α}
Goal: s.compl ≤ t.compl ↔ s ≤ t
-- LowerSet.compl_sup
protected theorem compl_sup (s t : LowerSet α) : (s ⊔ t).compl = s.compl ⊔ t.compl 
Arguments: {α : Type u_1} [LE α] (s t : LowerSet α)
Goal: (s ⊔ t).compl = s.compl ⊔ t.compl
-- LowerSet.compl_inf
protected theorem compl_inf (s t : LowerSet α) : (s ⊓ t).compl = s.compl ⊓ t.compl 
Arguments: {α : Type u_1} [LE α] (s t : LowerSet α)
Goal: (s ⊓ t).compl = s.compl ⊓ t.compl
-- LowerSet.compl_top
protected theorem compl_top : (⊤ : LowerSet α).compl = ⊤ 
Arguments: {α : Type u_1} [LE α]
Goal: ⊤.compl = ⊤
-- LowerSet.compl_bot
protected theorem compl_bot : (⊥ : LowerSet α).compl = ⊥ 
Arguments: {α : Type u_1} [LE α]
Goal: ⊥.compl = ⊥
-- LowerSet.compl_sSup
protected theorem compl_sSup (S : Set (LowerSet α)) : (sSup S).compl = ⨆ s ∈ S, LowerSet.compl s 
Arguments: {α : Type u_1} [LE α] (S : Set (LowerSet α))
Goal: (sSup S).compl = ⨆ s ∈ S, s.compl
-- LowerSet.compl_sInf
protected theorem compl_sInf (S : Set (LowerSet α)) : (sInf S).compl = ⨅ s ∈ S, LowerSet.compl s 
Arguments: {α : Type u_1} [LE α] (S : Set (LowerSet α))
Goal: (sInf S).compl = ⨅ s ∈ S, s.compl
-- LowerSet.compl_iSup
protected theorem compl_iSup (f : ι → LowerSet α) : (⨆ i, f i).compl = ⨆ i, (f i).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → LowerSet α)
Goal: (⨆ i, f i).compl = ⨆ i, (f i).compl
-- LowerSet.compl_iInf
protected theorem compl_iInf (f : ι → LowerSet α) : (⨅ i, f i).compl = ⨅ i, (f i).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} [LE α] (f : ι → LowerSet α)
Goal: (⨅ i, f i).compl = ⨅ i, (f i).compl
-- LowerSet.compl_iSup₂
@[simp]
theorem compl_iSup₂ (f : ∀ i, κ i → LowerSet α) :
    (⨆ (i) (j), f i j).compl = ⨆ (i) (j), (f i j).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → LowerSet α)
Goal: (⨆ i, ⨆ j, f i j).compl = ⨆ i, ⨆ j, (f i j).compl
-- LowerSet.compl_iInf₂
@[simp]
theorem compl_iInf₂ (f : ∀ i, κ i → LowerSet α) :
    (⨅ (i) (j), f i j).compl = ⨅ (i) (j), (f i j).compl 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [LE α] (f : (i : ι) → κ i → LowerSet α)
Goal: (⨅ i, ⨅ j, f i j).compl = ⨅ i, ⨅ j, (f i j).compl
-- upperSetIsoLowerSet_symm_apply
@[simps]
def upperSetIsoLowerSet : UpperSet α ≃o LowerSet α 
Arguments: {α : Type u_1} [LE α] (s : LowerSet α)
Goal: (RelIso.symm upperSetIsoLowerSet) s = s.compl
-- upperSetIsoLowerSet_apply
@[simps]
def upperSetIsoLowerSet : UpperSet α ≃o LowerSet α 
Arguments: {α : Type u_1} [LE α] (s : UpperSet α)
Goal: upperSetIsoLowerSet s = s.compl
-- UpperSet.symm_map
@[simp]
theorem symm_map (f : α ≃o β) : (map f).symm = map f.symm 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β)
Goal: (UpperSet.map f).symm = UpperSet.map f.symm
-- UpperSet.mem_map
@[simp]
theorem mem_map : b ∈ map f s ↔ f.symm b ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {f : α ≃o β} {s : UpperSet α} {b : β}
Goal: b ∈ (UpperSet.map f) s ↔ f.symm b ∈ s
-- UpperSet.map_refl
@[simp]
theorem map_refl : map (OrderIso.refl α) = OrderIso.refl _ 
Arguments: {α : Type u_1} [Preorder α]
Goal: UpperSet.map (OrderIso.refl α) = OrderIso.refl (UpperSet α)
-- UpperSet.map_map
@[simp]
theorem map_map (g : β ≃o γ) (f : α ≃o β) : map g (map f s) = map (f.trans g) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] {s : UpperSet α} (g : β ≃o γ) (f : α ≃o β)
Goal: (UpperSet.map g) ((UpperSet.map f) s) = (UpperSet.map (f.trans g)) s
-- UpperSet.coe_map
@[simp, norm_cast]
theorem coe_map : (map f s : Set β) = f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : UpperSet α)
Goal: ↑((UpperSet.map f) s) = ⇑f '' ↑s
-- LowerSet.symm_map
@[simp]
theorem symm_map (f : α ≃o β) : (map f).symm = map f.symm 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β)
Goal: (LowerSet.map f).symm = LowerSet.map f.symm
-- LowerSet.mem_map
@[simp]
theorem mem_map {f : α ≃o β} {b : β} : b ∈ map f s ↔ f.symm b ∈ s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] {s : LowerSet α} {f : α ≃o β} {b : β}
Goal: b ∈ (LowerSet.map f) s ↔ f.symm b ∈ s
-- LowerSet.map_refl
@[simp]
theorem map_refl : map (OrderIso.refl α) = OrderIso.refl _ 
Arguments: {α : Type u_1} [Preorder α]
Goal: LowerSet.map (OrderIso.refl α) = OrderIso.refl (LowerSet α)
-- LowerSet.map_map
@[simp]
theorem map_map (g : β ≃o γ) (f : α ≃o β) : map g (map f s) = map (f.trans g) s 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} [Preorder α] [Preorder β] [Preorder γ] {s : LowerSet α} (g : β ≃o γ) (f : α ≃o β)
Goal: (LowerSet.map g) ((LowerSet.map f) s) = (LowerSet.map (f.trans g)) s
-- LowerSet.coe_map
@[simp, norm_cast]
theorem coe_map : (map f s : Set β) = f '' s 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : LowerSet α)
Goal: ↑((LowerSet.map f) s) = ⇑f '' ↑s
-- UpperSet.compl_map
@[simp]
theorem compl_map (f : α ≃o β) (s : UpperSet α) : (map f s).compl = LowerSet.map f s.compl 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : UpperSet α)
Goal: ((UpperSet.map f) s).compl = (LowerSet.map f) s.compl
-- LowerSet.compl_map
@[simp]
theorem compl_map (f : α ≃o β) (s : LowerSet α) : (map f s).compl = UpperSet.map f s.compl 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (s : LowerSet α)
Goal: ((LowerSet.map f) s).compl = (UpperSet.map f) s.compl
-- UpperSet.coe_Ici
@[simp]
theorem coe_Ici (a : α) : ↑(Ici a) = Set.Ici a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: ↑(UpperSet.Ici a) = Set.Ici a
-- UpperSet.coe_Ioi
@[simp]
theorem coe_Ioi (a : α) : ↑(Ioi a) = Set.Ioi a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: ↑(UpperSet.Ioi a) = Set.Ioi a
-- UpperSet.mem_Ici_iff
@[simp]
theorem mem_Ici_iff : b ∈ Ici a ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ UpperSet.Ici a ↔ a ≤ b
-- UpperSet.mem_Ioi_iff
@[simp]
theorem mem_Ioi_iff : b ∈ Ioi a ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ UpperSet.Ioi a ↔ a < b
-- UpperSet.map_Ici
@[simp]
theorem map_Ici (f : α ≃o β) (a : α) : map f (Ici a) = Ici (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α)
Goal: (UpperSet.map f) (UpperSet.Ici a) = UpperSet.Ici (f a)
-- UpperSet.map_Ioi
@[simp]
theorem map_Ioi (f : α ≃o β) (a : α) : map f (Ioi a) = Ioi (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α)
Goal: (UpperSet.map f) (UpperSet.Ioi a) = UpperSet.Ioi (f a)
-- UpperSet.Ici_le_Ioi
theorem Ici_le_Ioi (a : α) : Ici a ≤ Ioi a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: UpperSet.Ici a ≤ UpperSet.Ioi a
-- UpperSet.Ici_bot
@[simp]
nonrec theorem Ici_bot [OrderBot α] : Ici (⊥ : α) = ⊥ 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α]
Goal: UpperSet.Ici ⊥ = ⊥
-- UpperSet.Ioi_top
@[simp]
nonrec theorem Ioi_top [OrderTop α] : Ioi (⊤ : α) = ⊤ 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α]
Goal: UpperSet.Ioi ⊤ = ⊤
-- UpperSet.Ici_ne_top
@[simp] lemma Ici_ne_top : Ici a ≠ ⊤ 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: UpperSet.Ici a ≠ ⊤
-- UpperSet.Ici_lt_top
@[simp] lemma Ici_lt_top : Ici a < ⊤ 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: UpperSet.Ici a < ⊤
-- UpperSet.le_Ici
@[simp] lemma le_Ici : s ≤ Ici a ↔ a ∈ s 
Arguments: {α : Type u_1} [Preorder α] {s : UpperSet α} {a : α}
Goal: s ≤ UpperSet.Ici a ↔ a ∈ s
-- UpperSet.Ici_injective
nonrec lemma Ici_injective : Injective (Ici : α → UpperSet α) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: Function.Injective UpperSet.Ici
-- UpperSet.Ici_inj
@[simp] lemma Ici_inj : Ici a = Ici b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: UpperSet.Ici a = UpperSet.Ici b ↔ a = b
-- UpperSet.Ici_ne_Ici
lemma Ici_ne_Ici : Ici a ≠ Ici b ↔ a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: UpperSet.Ici a ≠ UpperSet.Ici b ↔ a ≠ b
-- UpperSet.Ici_sup
@[simp]
theorem Ici_sup [SemilatticeSup α] (a b : α) : Ici (a ⊔ b) = Ici a ⊔ Ici b 
Arguments: {α : Type u_1} [SemilatticeSup α] (a b : α)
Goal: UpperSet.Ici (a ⊔ b) = UpperSet.Ici a ⊔ UpperSet.Ici b
-- UpperSet.Ici_sSup
@[simp]
theorem Ici_sSup (S : Set α) : Ici (sSup S) = ⨆ a ∈ S, Ici a 
Arguments: {α : Type u_1} [CompleteLattice α] (S : Set α)
Goal: UpperSet.Ici (sSup S) = ⨆ a ∈ S, UpperSet.Ici a
-- UpperSet.Ici_iSup
@[simp]
theorem Ici_iSup (f : ι → α) : Ici (⨆ i, f i) = ⨆ i, Ici (f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [CompleteLattice α] (f : ι → α)
Goal: UpperSet.Ici (⨆ i, f i) = ⨆ i, UpperSet.Ici (f i)
-- UpperSet.Ici_iSup₂
theorem Ici_iSup₂ (f : ∀ i, κ i → α) : Ici (⨆ (i) (j), f i j) = ⨆ (i) (j), Ici (f i j) 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [CompleteLattice α] (f : (i : ι) → κ i → α)
Goal: UpperSet.Ici (⨆ i, ⨆ j, f i j) = ⨆ i, ⨆ j, UpperSet.Ici (f i j)
-- LowerSet.coe_Iic
@[simp]
theorem coe_Iic (a : α) : ↑(Iic a) = Set.Iic a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: ↑(LowerSet.Iic a) = Set.Iic a
-- LowerSet.coe_Iio
@[simp]
theorem coe_Iio (a : α) : ↑(Iio a) = Set.Iio a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: ↑(LowerSet.Iio a) = Set.Iio a
-- LowerSet.mem_Iic_iff
@[simp]
theorem mem_Iic_iff : b ∈ Iic a ↔ b ≤ a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ LowerSet.Iic a ↔ b ≤ a
-- LowerSet.mem_Iio_iff
@[simp]
theorem mem_Iio_iff : b ∈ Iio a ↔ b < a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ LowerSet.Iio a ↔ b < a
-- LowerSet.map_Iic
@[simp]
theorem map_Iic (f : α ≃o β) (a : α) : map f (Iic a) = Iic (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α)
Goal: (LowerSet.map f) (LowerSet.Iic a) = LowerSet.Iic (f a)
-- LowerSet.map_Iio
@[simp]
theorem map_Iio (f : α ≃o β) (a : α) : map f (Iio a) = Iio (f a) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β) (a : α)
Goal: (LowerSet.map f) (LowerSet.Iio a) = LowerSet.Iio (f a)
-- LowerSet.Ioi_le_Ici
theorem Ioi_le_Ici (a : α) : Ioi a ≤ Ici a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Ioi a ≤ Set.Ici a
-- LowerSet.Iic_top
@[simp]
nonrec theorem Iic_top [OrderTop α] : Iic (⊤ : α) = ⊤ 
Arguments: {α : Type u_1} [Preorder α] [OrderTop α]
Goal: LowerSet.Iic ⊤ = ⊤
-- LowerSet.Iio_bot
@[simp]
nonrec theorem Iio_bot [OrderBot α] : Iio (⊥ : α) = ⊥ 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α]
Goal: LowerSet.Iio ⊥ = ⊥
-- LowerSet.Iic_ne_bot
@[simp] lemma Iic_ne_bot : Iic a ≠ ⊥ 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: LowerSet.Iic a ≠ ⊥
-- LowerSet.bot_lt_Iic
@[simp] lemma bot_lt_Iic : ⊥ < Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: ⊥ < LowerSet.Iic a
-- LowerSet.Iic_le
@[simp] lemma Iic_le : Iic a ≤ s ↔ a ∈ s 
Arguments: {α : Type u_1} [Preorder α] {s : LowerSet α} {a : α}
Goal: LowerSet.Iic a ≤ s ↔ a ∈ s
-- LowerSet.Iic_injective
nonrec lemma Iic_injective : Injective (Iic : α → LowerSet α) 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: Function.Injective LowerSet.Iic
-- LowerSet.Iic_inj
@[simp] lemma Iic_inj : Iic a = Iic b ↔ a = b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: LowerSet.Iic a = LowerSet.Iic b ↔ a = b
-- LowerSet.Iic_ne_Iic
lemma Iic_ne_Iic : Iic a ≠ Iic b ↔ a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] {a b : α}
Goal: LowerSet.Iic a ≠ LowerSet.Iic b ↔ a ≠ b
-- LowerSet.Iic_inf
@[simp]
theorem Iic_inf [SemilatticeInf α] (a b : α) : Iic (a ⊓ b) = Iic a ⊓ Iic b 
Arguments: {α : Type u_1} [SemilatticeInf α] (a b : α)
Goal: LowerSet.Iic (a ⊓ b) = LowerSet.Iic a ⊓ LowerSet.Iic b
-- LowerSet.Iic_sInf
@[simp]
theorem Iic_sInf (S : Set α) : Iic (sInf S) = ⨅ a ∈ S, Iic a 
Arguments: {α : Type u_1} [CompleteLattice α] (S : Set α)
Goal: LowerSet.Iic (sInf S) = ⨅ a ∈ S, LowerSet.Iic a
-- LowerSet.Iic_iInf
@[simp]
theorem Iic_iInf (f : ι → α) : Iic (⨅ i, f i) = ⨅ i, Iic (f i) 
Arguments: {α : Type u_1} {ι : Sort u_4} [CompleteLattice α] (f : ι → α)
Goal: LowerSet.Iic (⨅ i, f i) = ⨅ i, LowerSet.Iic (f i)
-- LowerSet.Iic_iInf₂
theorem Iic_iInf₂ (f : ∀ i, κ i → α) : Iic (⨅ (i) (j), f i j) = ⨅ (i) (j), Iic (f i j) 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_5} [CompleteLattice α] (f : (i : ι) → κ i → α)
Goal: LowerSet.Iic (⨅ i, ⨅ j, f i j) = ⨅ i, ⨅ j, LowerSet.Iic (f i j)
-- mem_upperClosure
@[simp]
theorem mem_upperClosure : x ∈ upperClosure s ↔ ∃ a ∈ s, a ≤ x 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} {x : α}
Goal: x ∈ upperClosure s ↔ ∃ a ∈ s, a ≤ x
-- mem_lowerClosure
@[simp]
theorem mem_lowerClosure : x ∈ lowerClosure s ↔ ∃ a ∈ s, x ≤ a 
Arguments: {α : Type u_1} [Preorder α] {s : Set α} {x : α}
Goal: x ∈ lowerClosure s ↔ ∃ a ∈ s, x ≤ a
-- coe_upperClosure
@[norm_cast]
theorem coe_upperClosure (s : Set α) : ↑(upperClosure s) = ⋃ a ∈ s, Ici a 
Arguments: {α : Type u_1} [Preorder α] (s : Set α)
Goal: ↑(upperClosure s) = ⋃ a ∈ s, Set.Ici a
-- coe_lowerClosure
@[norm_cast]
theorem coe_lowerClosure (s : Set α) : ↑(lowerClosure s) = ⋃ a ∈ s, Iic a 
Arguments: {α : Type u_1} [Preorder α] (s : Set α)
Goal: ↑(lowerClosure s) = ⋃ a ∈ s, Set.Iic a
-- subset_upperClosure
theorem subset_upperClosure : s ⊆ upperClosure s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: s ⊆ ↑(upperClosure s)
-- subset_lowerClosure
theorem subset_lowerClosure : s ⊆ lowerClosure s 
Arguments: {α : Type u_1} [Preorder α] {s : Set α}
Goal: s ⊆ ↑(lowerClosure s)
-- upperClosure_min
theorem upperClosure_min (h : s ⊆ t) (ht : IsUpperSet t) : ↑(upperClosure s) ⊆ t 
Arguments: {α : Type u_1} [Preorder α] {s t : Set α} (h : s ⊆ t) (ht : IsUpperSet t)
Goal: ↑(upperClosure s) ⊆ t