ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.iInter_subset
theorem iInter_subset : ∀ (s : ι → Set β) (i : ι), ⋂ i, s i ⊆ s i 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β) (i : ι)
Goal: ⋂ i, s i ⊆ s i
-- Set.subset_iUnion₂
theorem subset_iUnion₂ {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : s i j ⊆ ⋃ (i') (j'), s i' j' 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} (i : ι) (j : κ i)
Goal: s i j ⊆ ⋃ i', ⋃ j', s i' j'
-- Set.iInter₂_subset
theorem iInter₂_subset {s : ∀ i, κ i → Set α} (i : ι) (j : κ i) : ⋂ (i) (j), s i j ⊆ s i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} (i : ι) (j : κ i)
Goal: ⋂ i, ⋂ j, s i j ⊆ s i j
-- Set.subset_iUnion_of_subset
theorem subset_iUnion_of_subset {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i) : s ⊆ ⋃ i, t i 
Arguments: {α : Type u_1} {ι : Sort u_4} {s : Set α} {t : ι → Set α} (i : ι) (h : s ⊆ t i)
Goal: s ⊆ ⋃ i, t i
-- Set.iInter_subset_of_subset
theorem iInter_subset_of_subset {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t) :
    ⋂ i, s i ⊆ t 
Arguments: {α : Type u_1} {ι : Sort u_4} {s : ι → Set α} {t : Set α} (i : ι) (h : s i ⊆ t)
Goal: ⋂ i, s i ⊆ t
-- Set.subset_iUnion₂_of_subset
theorem subset_iUnion₂_of_subset {s : Set α} {t : ∀ i, κ i → Set α} (i : ι) (j : κ i)
    (h : s ⊆ t i j) : s ⊆ ⋃ (i) (j), t i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : Set α} {t : (i : ι) → κ i → Set α} (i : ι) (j : κ i) (h : s ⊆ t i j)
Goal: s ⊆ ⋃ i, ⋃ j, t i j
-- Set.iInter₂_subset_of_subset
theorem iInter₂_subset_of_subset {s : ∀ i, κ i → Set α} {t : Set α} (i : ι) (j : κ i)
    (h : s i j ⊆ t) : ⋂ (i) (j), s i j ⊆ t 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s : (i : ι) → κ i → Set α} {t : Set α} (i : ι) (j : κ i) (h : s i j ⊆ t)
Goal: ⋂ i, ⋂ j, s i j ⊆ t
-- Set.iUnion_mono
theorem iUnion_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋃ i, s i ⊆ ⋃ i, t i 
Arguments: {α : Type u_1} {ι : Sort u_4} {s t : ι → Set α} (h : ∀ (i : ι), s i ⊆ t i)
Goal: ⋃ i, s i ⊆ ⋃ i, t i
-- Set.iUnion_mono''
@[gcongr]
theorem iUnion_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iUnion s ⊆ iUnion t 
Arguments: {α : Type u_1} {ι : Sort u_4} {s t : ι → Set α} (h : ∀ (i : ι), s i ⊆ t i)
Goal: Set.iUnion s ⊆ Set.iUnion t
-- Set.iUnion₂_mono
theorem iUnion₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
    ⋃ (i) (j), s i j ⊆ ⋃ (i) (j), t i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s t : (i : ι) → κ i → Set α} (h : ∀ (i : ι) (j : κ i), s i j ⊆ t i j)
Goal: ⋃ i, ⋃ j, s i j ⊆ ⋃ i, ⋃ j, t i j
-- Set.iInter_mono
theorem iInter_mono {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : ⋂ i, s i ⊆ ⋂ i, t i 
Arguments: {α : Type u_1} {ι : Sort u_4} {s t : ι → Set α} (h : ∀ (i : ι), s i ⊆ t i)
Goal: ⋂ i, s i ⊆ ⋂ i, t i
-- Set.iInter_mono''
@[gcongr]
theorem iInter_mono'' {s t : ι → Set α} (h : ∀ i, s i ⊆ t i) : iInter s ⊆ iInter t 
Arguments: {α : Type u_1} {ι : Sort u_4} {s t : ι → Set α} (h : ∀ (i : ι), s i ⊆ t i)
Goal: Set.iInter s ⊆ Set.iInter t
-- Set.iInter₂_mono
theorem iInter₂_mono {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j ⊆ t i j) :
    ⋂ (i) (j), s i j ⊆ ⋂ (i) (j), t i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s t : (i : ι) → κ i → Set α} (h : ∀ (i : ι) (j : κ i), s i j ⊆ t i j)
Goal: ⋂ i, ⋂ j, s i j ⊆ ⋂ i, ⋂ j, t i j
-- Set.iUnion_mono'
theorem iUnion_mono' {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ i, ∃ j, s i ⊆ t j) :
    ⋃ i, s i ⊆ ⋃ i, t i 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι₂ : Sort u_6} {s : ι → Set α} {t : ι₂ → Set α} (h : ∀ (i : ι), ∃ j, s i ⊆ t j)
Goal: ⋃ i, s i ⊆ ⋃ i, t i
-- Set.iUnion₂_mono'
theorem iUnion₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
    (h : ∀ i j, ∃ i' j', s i j ⊆ t i' j') : ⋃ (i) (j), s i j ⊆ ⋃ (i') (j'), t i' j' 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {κ : ι → Sort u_7} {κ' : ι' → Sort u_10} {s : (i : ι) → κ i → Set α} {t : (i' : ι') → κ' i' → Set α} (h : ∀ (i : ι) (j : κ i), ∃ i' j', s i j ⊆ t i' j')
Goal: ⋃ i, ⋃ j, s i j ⊆ ⋃ i', ⋃ j', t i' j'
-- Set.iInter_mono'
theorem iInter_mono' {s : ι → Set α} {t : ι' → Set α} (h : ∀ j, ∃ i, s i ⊆ t j) :
    ⋂ i, s i ⊆ ⋂ j, t j 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {s : ι → Set α} {t : ι' → Set α} (h : ∀ (j : ι'), ∃ i, s i ⊆ t j)
Goal: ⋂ i, s i ⊆ ⋂ j, t j
-- Set.iInter₂_mono'
theorem iInter₂_mono' {s : ∀ i, κ i → Set α} {t : ∀ i', κ' i' → Set α}
    (h : ∀ i' j', ∃ i j, s i j ⊆ t i' j') : ⋂ (i) (j), s i j ⊆ ⋂ (i') (j'), t i' j' 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {κ : ι → Sort u_7} {κ' : ι' → Sort u_10} {s : (i : ι) → κ i → Set α} {t : (i' : ι') → κ' i' → Set α} (h : ∀ (i' : ι') (j' : κ' i'), ∃ i j, s i j ⊆ t i' j')
Goal: ⋂ i, ⋂ j, s i j ⊆ ⋂ i', ⋂ j', t i' j'
-- Set.iUnion₂_subset_iUnion
theorem iUnion₂_subset_iUnion (κ : ι → Sort*) (s : ι → Set α) :
    ⋃ (i) (_ : κ i), s i ⊆ ⋃ i, s i 
Arguments: {α : Type u_1} {ι : Sort u_4} (κ : ι → Sort u_11) (s : ι → Set α)
Goal: ⋃ i, ⋃ x, s i ⊆ ⋃ i, s i
-- Set.iInter_subset_iInter₂
theorem iInter_subset_iInter₂ (κ : ι → Sort*) (s : ι → Set α) :
    ⋂ i, s i ⊆ ⋂ (i) (_ : κ i), s i 
Arguments: {α : Type u_1} {ι : Sort u_4} (κ : ι → Sort u_11) (s : ι → Set α)
Goal: ⋂ i, s i ⊆ ⋂ i, ⋂ x, s i
-- Set.iUnion_setOf
theorem iUnion_setOf (P : ι → α → Prop) : ⋃ i, { x : α | P i x } = { x : α | ∃ i, P i x } 
Arguments: {α : Type u_1} {ι : Sort u_4} (P : ι → α → Prop)
Goal: ⋃ i, {x | P i x} = {x | ∃ i, P i x}
-- Set.iInter_setOf
theorem iInter_setOf (P : ι → α → Prop) : ⋂ i, { x : α | P i x } = { x : α | ∀ i, P i x } 
Arguments: {α : Type u_1} {ι : Sort u_4} (P : ι → α → Prop)
Goal: ⋂ i, {x | P i x} = {x | ∀ (i : ι), P i x}
-- Set.iUnion_congr_of_surjective
theorem iUnion_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
    (h2 : ∀ x, g (h x) = f x) : ⋃ x, f x = ⋃ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι₂ : Sort u_6} {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Function.Surjective h) (h2 : ∀ (x : ι), g (h x) = f x)
Goal: ⋃ x, f x = ⋃ y, g y
-- Set.iInter_congr_of_surjective
theorem iInter_congr_of_surjective {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Surjective h)
    (h2 : ∀ x, g (h x) = f x) : ⋂ x, f x = ⋂ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι₂ : Sort u_6} {f : ι → Set α} {g : ι₂ → Set α} (h : ι → ι₂) (h1 : Function.Surjective h) (h2 : ∀ (x : ι), g (h x) = f x)
Goal: ⋂ x, f x = ⋂ y, g y
-- Set.iUnion_congr
lemma iUnion_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋃ i, s i = ⋃ i, t i 
Arguments: {α : Type u_1} {ι : Sort u_4} {s t : ι → Set α} (h : ∀ (i : ι), s i = t i)
Goal: ⋃ i, s i = ⋃ i, t i
-- Set.iInter_congr
lemma iInter_congr {s t : ι → Set α} (h : ∀ i, s i = t i) : ⋂ i, s i = ⋂ i, t i 
Arguments: {α : Type u_1} {ι : Sort u_4} {s t : ι → Set α} (h : ∀ (i : ι), s i = t i)
Goal: ⋂ i, s i = ⋂ i, t i
-- Set.iUnion₂_congr
lemma iUnion₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
    ⋃ (i) (j), s i j = ⋃ (i) (j), t i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s t : (i : ι) → κ i → Set α} (h : ∀ (i : ι) (j : κ i), s i j = t i j)
Goal: ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, t i j
-- Set.iInter₂_congr
lemma iInter₂_congr {s t : ∀ i, κ i → Set α} (h : ∀ i j, s i j = t i j) :
    ⋂ (i) (j), s i j = ⋂ (i) (j), t i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} {s t : (i : ι) → κ i → Set α} (h : ∀ (i : ι) (j : κ i), s i j = t i j)
Goal: ⋂ i, ⋂ j, s i j = ⋂ i, ⋂ j, t i j
-- Set.iUnion_const
lemma iUnion_const (s : Set β) : ⋃ _ : ι, s = s 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β)
Goal: ⋃ x, s = s
-- Set.iInter_const
lemma iInter_const (s : Set β) : ⋂ _ : ι, s = s 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β)
Goal: ⋂ x, s = s
-- Set.iUnion_eq_const
lemma iUnion_eq_const (hf : ∀ i, f i = s) : ⋃ i, f i = s 
Arguments: {α : Type u_1} {ι : Sort u_4} [Nonempty ι] {f : ι → Set α} {s : Set α} (hf : ∀ (i : ι), f i = s)
Goal: ⋃ i, f i = s
-- Set.iInter_eq_const
lemma iInter_eq_const (hf : ∀ i, f i = s) : ⋂ i, f i = s 
Arguments: {α : Type u_1} {ι : Sort u_4} [Nonempty ι] {f : ι → Set α} {s : Set α} (hf : ∀ (i : ι), f i = s)
Goal: ⋂ i, f i = s
-- Set.compl_iUnion
@[simp]
theorem compl_iUnion (s : ι → Set β) : (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β)
Goal: (⋃ i, s i)ᶜ = ⋂ i, (s i)ᶜ
-- Set.compl_iUnion₂
theorem compl_iUnion₂ (s : ∀ i, κ i → Set α) : (⋃ (i) (j), s i j)ᶜ = ⋂ (i) (j), (s i j)ᶜ 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} (s : (i : ι) → κ i → Set α)
Goal: (⋃ i, ⋃ j, s i j)ᶜ = ⋂ i, ⋂ j, (s i j)ᶜ
-- Set.compl_iInter
@[simp]
theorem compl_iInter (s : ι → Set β) : (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β)
Goal: (⋂ i, s i)ᶜ = ⋃ i, (s i)ᶜ
-- Set.compl_iInter₂
theorem compl_iInter₂ (s : ∀ i, κ i → Set α) : (⋂ (i) (j), s i j)ᶜ = ⋃ (i) (j), (s i j)ᶜ 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ : ι → Sort u_7} (s : (i : ι) → κ i → Set α)
Goal: (⋂ i, ⋂ j, s i j)ᶜ = ⋃ i, ⋃ j, (s i j)ᶜ
-- Set.iUnion_eq_compl_iInter_compl
theorem iUnion_eq_compl_iInter_compl (s : ι → Set β) : ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β)
Goal: ⋃ i, s i = (⋂ i, (s i)ᶜ)ᶜ
-- Set.iInter_eq_compl_iUnion_compl
theorem iInter_eq_compl_iUnion_compl (s : ι → Set β) : ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β)
Goal: ⋂ i, s i = (⋃ i, (s i)ᶜ)ᶜ
-- Set.inter_iUnion
theorem inter_iUnion (s : Set β) (t : ι → Set β) : (s ∩ ⋃ i, t i) = ⋃ i, s ∩ t i 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : Set β) (t : ι → Set β)
Goal: s ∩ ⋃ i, t i = ⋃ i, s ∩ t i
-- Set.iUnion_inter
theorem iUnion_inter (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : Set β) (t : ι → Set β)
Goal: (⋃ i, t i) ∩ s = ⋃ i, t i ∩ s
-- Set.iUnion_union_distrib
theorem iUnion_union_distrib (s : ι → Set β) (t : ι → Set β) :
    ⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i 
Arguments: {β : Type u_2} {ι : Sort u_4} (s t : ι → Set β)
Goal: ⋃ i, s i ∪ t i = (⋃ i, s i) ∪ ⋃ i, t i
-- Set.iInter_inter_distrib
theorem iInter_inter_distrib (s : ι → Set β) (t : ι → Set β) :
    ⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i 
Arguments: {β : Type u_2} {ι : Sort u_4} (s t : ι → Set β)
Goal: ⋂ i, s i ∩ t i = (⋂ i, s i) ∩ ⋂ i, t i
-- Set.union_iUnion
theorem union_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∪ ⋃ i, t i) = ⋃ i, s ∪ t i 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β) (t : ι → Set β)
Goal: s ∪ ⋃ i, t i = ⋃ i, s ∪ t i
-- Set.iUnion_union
theorem iUnion_union [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β) (t : ι → Set β)
Goal: (⋃ i, t i) ∪ s = ⋃ i, t i ∪ s
-- Set.inter_iInter
theorem inter_iInter [Nonempty ι] (s : Set β) (t : ι → Set β) : (s ∩ ⋂ i, t i) = ⋂ i, s ∩ t i 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β) (t : ι → Set β)
Goal: s ∩ ⋂ i, t i = ⋂ i, s ∩ t i
-- Set.iInter_inter
theorem iInter_inter [Nonempty ι] (s : Set β) (t : ι → Set β) : (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β) (t : ι → Set β)
Goal: (⋂ i, t i) ∩ s = ⋂ i, t i ∩ s
-- Set.union_iInter
theorem union_iInter (s : Set β) (t : ι → Set β) : (s ∪ ⋂ i, t i) = ⋂ i, s ∪ t i 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : Set β) (t : ι → Set β)
Goal: s ∪ ⋂ i, t i = ⋂ i, s ∪ t i
-- Set.iInter_union
theorem iInter_union (s : ι → Set β) (t : Set β) : (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : ι → Set β) (t : Set β)
Goal: (⋂ i, s i) ∪ t = ⋂ i, s i ∪ t
-- Set.iUnion_diff
theorem iUnion_diff (s : Set β) (t : ι → Set β) : (⋃ i, t i) \ s = ⋃ i, t i \ s 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : Set β) (t : ι → Set β)
Goal: (⋃ i, t i) \ s = ⋃ i, t i \ s
-- Set.diff_iUnion
theorem diff_iUnion [Nonempty ι] (s : Set β) (t : ι → Set β) : (s \ ⋃ i, t i) = ⋂ i, s \ t i 
Arguments: {β : Type u_2} {ι : Sort u_4} [Nonempty ι] (s : Set β) (t : ι → Set β)
Goal: s \ ⋃ i, t i = ⋂ i, s \ t i
-- Set.diff_iInter
theorem diff_iInter (s : Set β) (t : ι → Set β) : (s \ ⋂ i, t i) = ⋃ i, s \ t i 
Arguments: {β : Type u_2} {ι : Sort u_4} (s : Set β) (t : ι → Set β)
Goal: s \ ⋂ i, t i = ⋃ i, s \ t i
-- Set.iUnion_inter_subset
theorem iUnion_inter_subset {ι α} {s t : ι → Set α} : ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i 
Arguments: {ι : Sort u_11} {α : Type u_12} {s t : ι → Set α}
Goal: ⋃ i, s i ∩ t i ⊆ (⋃ i, s i) ∩ ⋃ i, t i
-- Set.iUnion_inter_of_monotone
theorem iUnion_inter_of_monotone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
    (hs : Monotone s) (ht : Monotone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i 
Arguments: {ι : Type u_11} {α : Type u_12} [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {s t : ι → Set α} (hs : Monotone s) (ht : Monotone t)
Goal: ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i
-- Set.iUnion_inter_of_antitone
theorem iUnion_inter_of_antitone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
    (hs : Antitone s) (ht : Antitone t) : ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i 
Arguments: {ι : Type u_11} {α : Type u_12} [Preorder ι] [IsDirected ι (Function.swap fun x x_1 => x ≤ x_1)] {s t : ι → Set α} (hs : Antitone s) (ht : Antitone t)
Goal: ⋃ i, s i ∩ t i = (⋃ i, s i) ∩ ⋃ i, t i
-- Set.iInter_union_of_monotone
theorem iInter_union_of_monotone {ι α} [Preorder ι] [IsDirected ι (swap (· ≤ ·))] {s t : ι → Set α}
    (hs : Monotone s) (ht : Monotone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i 
Arguments: {ι : Type u_11} {α : Type u_12} [Preorder ι] [IsDirected ι (Function.swap fun x x_1 => x ≤ x_1)] {s t : ι → Set α} (hs : Monotone s) (ht : Monotone t)
Goal: ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i
-- Set.iInter_union_of_antitone
theorem iInter_union_of_antitone {ι α} [Preorder ι] [IsDirected ι (· ≤ ·)] {s t : ι → Set α}
    (hs : Antitone s) (ht : Antitone t) : ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i 
Arguments: {ι : Type u_11} {α : Type u_12} [Preorder ι] [IsDirected ι fun x x_1 => x ≤ x_1] {s t : ι → Set α} (hs : Antitone s) (ht : Antitone t)
Goal: ⋂ i, s i ∪ t i = (⋂ i, s i) ∪ ⋂ i, t i
-- Set.iUnion_iInter_subset
theorem iUnion_iInter_subset {s : ι → ι' → Set α} : (⋃ j, ⋂ i, s i j) ⊆ ⋂ i, ⋃ j, s i j 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} {s : ι → ι' → Set α}
Goal: ⋃ j, ⋂ i, s i j ⊆ ⋂ i, ⋃ j, s i j
-- Set.iUnion_option
theorem iUnion_option {ι} (s : Option ι → Set α) : ⋃ o, s o = s none ∪ ⋃ i, s (some i) 
Arguments: {α : Type u_1} {ι : Type u_11} (s : Option ι → Set α)
Goal: ⋃ o, s o = s none ∪ ⋃ i, s (some i)
-- Set.iInter_option
theorem iInter_option {ι} (s : Option ι → Set α) : ⋂ o, s o = s none ∩ ⋂ i, s (some i) 
Arguments: {α : Type u_1} {ι : Type u_11} (s : Option ι → Set α)
Goal: ⋂ o, s o = s none ∩ ⋂ i, s (some i)
-- Set.iUnion_dite
theorem iUnion_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set α) :
    ⋃ i, (if h : p i then f i h else g i h) = (⋃ (i) (h : p i), f i h) ∪ ⋃ (i) (h : ¬p i), g i h 
Arguments: {α : Type u_1} {ι : Sort u_4} (p : ι → Prop) [DecidablePred p] (f : (i : ι) → p i → Set α) (g : (i : ι) → ¬p i → Set α)
Goal: (⋃ i, if h : p i then f i h else g i h) = (⋃ i, ⋃ (h : p i), f i h) ∪ ⋃ i, ⋃ (h : ¬p i), g i h
-- Set.iUnion_ite
theorem iUnion_ite (f g : ι → Set α) :
    ⋃ i, (if p i then f i else g i) = (⋃ (i) (_ : p i), f i) ∪ ⋃ (i) (_ : ¬p i), g i 
Arguments: {α : Type u_1} {ι : Sort u_4} (p : ι → Prop) [DecidablePred p] (f g : ι → Set α)
Goal: (⋃ i, if p i then f i else g i) = (⋃ i, ⋃ (_ : p i), f i) ∪ ⋃ i, ⋃ (_ : ¬p i), g i
-- Set.iInter_dite
theorem iInter_dite (f : ∀ i, p i → Set α) (g : ∀ i, ¬p i → Set α) :
    ⋂ i, (if h : p i then f i h else g i h) = (⋂ (i) (h : p i), f i h) ∩ ⋂ (i) (h : ¬p i), g i h 
Arguments: {α : Type u_1} {ι : Sort u_4} (p : ι → Prop) [DecidablePred p] (f : (i : ι) → p i → Set α) (g : (i : ι) → ¬p i → Set α)
Goal: (⋂ i, if h : p i then f i h else g i h) = (⋂ i, ⋂ (h : p i), f i h) ∩ ⋂ i, ⋂ (h : ¬p i), g i h
-- Set.iInter_ite
theorem iInter_ite (f g : ι → Set α) :
    ⋂ i, (if p i then f i else g i) = (⋂ (i) (_ : p i), f i) ∩ ⋂ (i) (_ : ¬p i), g i 
Arguments: {α : Type u_1} {ι : Sort u_4} (p : ι → Prop) [DecidablePred p] (f g : ι → Set α)
Goal: (⋂ i, if p i then f i else g i) = (⋂ i, ⋂ (_ : p i), f i) ∩ ⋂ i, ⋂ (_ : ¬p i), g i
-- Set.image_projection_prod
theorem image_projection_prod {ι : Type*} {α : ι → Type*} {v : ∀ i : ι, Set (α i)}
    (hv : (pi univ v).Nonempty) (i : ι) :
    ((fun x : ∀ i : ι, α i => x i) '' ⋂ k, (fun x : ∀ j : ι, α j => x k) ⁻¹' v k) = v i 
Arguments: {ι : Type u_11} {α : ι → Type u_12} {v : (i : ι) → Set (α i)} (hv : (Set.univ.pi v).Nonempty) (i : ι)
Goal: (fun x => x i) '' ⋂ k, (fun x => x k) ⁻¹' v k = v i
-- Set.iInter_false
theorem iInter_false {s : False → Set α} : iInter s = univ 
Arguments: {α : Type u_1} {s : False → Set α}
Goal: Set.iInter s = Set.univ
-- Set.iUnion_false
theorem iUnion_false {s : False → Set α} : iUnion s = ∅ 
Arguments: {α : Type u_1} {s : False → Set α}
Goal: Set.iUnion s = ∅
-- Set.iInter_true
@[simp]
theorem iInter_true {s : True → Set α} : iInter s = s trivial 
Arguments: {α : Type u_1} {s : True → Set α}
Goal: Set.iInter s = s trivial
-- Set.iUnion_true
@[simp]
theorem iUnion_true {s : True → Set α} : iUnion s = s trivial 
Arguments: {α : Type u_1} {s : True → Set α}
Goal: Set.iUnion s = s trivial
-- Set.iInter_exists
@[simp]
theorem iInter_exists {p : ι → Prop} {f : Exists p → Set α} :
    ⋂ x, f x = ⋂ (i) (h : p i), f ⟨i, h⟩ 
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {f : Exists p → Set α}
Goal: ⋂ (x : Exists p), f x = ⋂ i, ⋂ (h : p i), f ⋯
-- Set.iUnion_exists
@[simp]
theorem iUnion_exists {p : ι → Prop} {f : Exists p → Set α} :
    ⋃ x, f x = ⋃ (i) (h : p i), f ⟨i, h⟩ 
Arguments: {α : Type u_1} {ι : Sort u_4} {p : ι → Prop} {f : Exists p → Set α}
Goal: ⋃ (x : Exists p), f x = ⋃ i, ⋃ (h : p i), f ⋯
-- Set.iUnion_empty
@[simp]
theorem iUnion_empty : (⋃ _ : ι, ∅ : Set α) = ∅ 
Arguments: {α : Type u_1} {ι : Sort u_4}
Goal: ⋃ x, ∅ = ∅
-- Set.iInter_univ
@[simp]
theorem iInter_univ : (⋂ _ : ι, univ : Set α) = univ 
Arguments: {α : Type u_1} {ι : Sort u_4}
Goal: ⋂ x, Set.univ = Set.univ
-- Set.iUnion_eq_empty
@[simp]
theorem iUnion_eq_empty : ⋃ i, s i = ∅ ↔ ∀ i, s i = ∅ 
Arguments: {α : Type u_1} {ι : Sort u_4} {s : ι → Set α}
Goal: ⋃ i, s i = ∅ ↔ ∀ (i : ι), s i = ∅
-- Set.iInter_eq_univ
@[simp]
theorem iInter_eq_univ : ⋂ i, s i = univ ↔ ∀ i, s i = univ 
Arguments: {α : Type u_1} {ι : Sort u_4} {s : ι → Set α}
Goal: ⋂ i, s i = Set.univ ↔ ∀ (i : ι), s i = Set.univ
-- Set.nonempty_iUnion
@[simp]
theorem nonempty_iUnion : (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty 
Arguments: {α : Type u_1} {ι : Sort u_4} {s : ι → Set α}
Goal: (⋃ i, s i).Nonempty ↔ ∃ i, (s i).Nonempty
-- Set.nonempty_biUnion
theorem nonempty_biUnion {t : Set α} {s : α → Set β} :
    (⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty 
Arguments: {α : Type u_1} {β : Type u_2} {t : Set α} {s : α → Set β}
Goal: (⋃ i ∈ t, s i).Nonempty ↔ ∃ i ∈ t, (s i).Nonempty
-- Set.iUnion_nonempty_index
theorem iUnion_nonempty_index (s : Set α) (t : s.Nonempty → Set β) :
    ⋃ h, t h = ⋃ x ∈ s, t ⟨x, ‹_›⟩ 
Arguments: {α : Type u_1} {β : Type u_2} (s : Set α) (t : s.Nonempty → Set β)
Goal: ⋃ (h : s.Nonempty), t h = ⋃ x, ⋃ (h : x ∈ s), t ⋯
-- Set.iInter_iInter_eq_left
@[simp]
theorem iInter_iInter_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
    ⋂ (x) (h : x = b), s x h = s b rfl 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {s : (x : β) → x = b → Set α}
Goal: ⋂ x, ⋂ (h : x = b), s x h = s b ⋯
-- Set.iInter_iInter_eq_right
@[simp]
theorem iInter_iInter_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
    ⋂ (x) (h : b = x), s x h = s b rfl 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {s : (x : β) → b = x → Set α}
Goal: ⋂ x, ⋂ (h : b = x), s x h = s b ⋯
-- Set.iUnion_iUnion_eq_left
@[simp]
theorem iUnion_iUnion_eq_left {b : β} {s : ∀ x : β, x = b → Set α} :
    ⋃ (x) (h : x = b), s x h = s b rfl 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {s : (x : β) → x = b → Set α}
Goal: ⋃ x, ⋃ (h : x = b), s x h = s b ⋯
-- Set.iUnion_iUnion_eq_right
@[simp]
theorem iUnion_iUnion_eq_right {b : β} {s : ∀ x : β, b = x → Set α} :
    ⋃ (x) (h : b = x), s x h = s b rfl 
Arguments: {α : Type u_1} {β : Type u_2} {b : β} {s : (x : β) → b = x → Set α}
Goal: ⋃ x, ⋃ (h : b = x), s x h = s b ⋯
-- Set.iInter_or
theorem iInter_or {p q : Prop} (s : p ∨ q → Set α) :
    ⋂ h, s h = (⋂ h : p, s (Or.inl h)) ∩ ⋂ h : q, s (Or.inr h) 
Arguments: {α : Type u_1} {p q : Prop} (s : p ∨ q → Set α)
Goal: ⋂ (h : p ∨ q), s h = (⋂ (h : p), s ⋯) ∩ ⋂ (h : q), s ⋯
-- Set.iUnion_or
theorem iUnion_or {p q : Prop} (s : p ∨ q → Set α) :
    ⋃ h, s h = (⋃ i, s (Or.inl i)) ∪ ⋃ j, s (Or.inr j) 
Arguments: {α : Type u_1} {p q : Prop} (s : p ∨ q → Set α)
Goal: ⋃ (h : p ∨ q), s h = (⋃ (i : p), s ⋯) ∪ ⋃ (j : q), s ⋯
-- Set.iUnion_and
theorem iUnion_and {p q : Prop} (s : p ∧ q → Set α) : ⋃ h, s h = ⋃ (hp) (hq), s ⟨hp, hq⟩ 
Arguments: {α : Type u_1} {p q : Prop} (s : p ∧ q → Set α)
Goal: ⋃ (h : p ∧ q), s h = ⋃ (hp : p), ⋃ (hq : q), s ⋯
-- Set.iInter_and
theorem iInter_and {p q : Prop} (s : p ∧ q → Set α) : ⋂ h, s h = ⋂ (hp) (hq), s ⟨hp, hq⟩ 
Arguments: {α : Type u_1} {p q : Prop} (s : p ∧ q → Set α)
Goal: ⋂ (h : p ∧ q), s h = ⋂ (hp : p), ⋂ (hq : q), s ⋯
-- Set.iUnion_comm
theorem iUnion_comm (s : ι → ι' → Set α) : ⋃ (i) (i'), s i i' = ⋃ (i') (i), s i i' 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} (s : ι → ι' → Set α)
Goal: ⋃ i, ⋃ i', s i i' = ⋃ i', ⋃ i, s i i'
-- Set.iInter_comm
theorem iInter_comm (s : ι → ι' → Set α) : ⋂ (i) (i'), s i i' = ⋂ (i') (i), s i i' 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} (s : ι → ι' → Set α)
Goal: ⋂ i, ⋂ i', s i i' = ⋂ i', ⋂ i, s i i'
-- Set.iUnion_sigma
theorem iUnion_sigma {γ : α → Type*} (s : Sigma γ → Set β) : ⋃ ia, s ia = ⋃ i, ⋃ a, s ⟨i, a⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : α → Type u_11} (s : Sigma γ → Set β)
Goal: ⋃ ia, s ia = ⋃ i, ⋃ a, s ⟨i, a⟩
-- Set.iUnion_sigma'
theorem iUnion_sigma' {γ : α → Type*} (s : ∀ i, γ i → Set β) :
    ⋃ i, ⋃ a, s i a = ⋃ ia : Sigma γ, s ia.1 ia.2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : α → Type u_11} (s : (i : α) → γ i → Set β)
Goal: ⋃ i, ⋃ a, s i a = ⋃ ia, s ia.fst ia.snd
-- Set.iInter_sigma
theorem iInter_sigma {γ : α → Type*} (s : Sigma γ → Set β) : ⋂ ia, s ia = ⋂ i, ⋂ a, s ⟨i, a⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {γ : α → Type u_11} (s : Sigma γ → Set β)
Goal: ⋂ ia, s ia = ⋂ i, ⋂ a, s ⟨i, a⟩
-- Set.iInter_sigma'
theorem iInter_sigma' {γ : α → Type*} (s : ∀ i, γ i → Set β) :
    ⋂ i, ⋂ a, s i a = ⋂ ia : Sigma γ, s ia.1 ia.2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : α → Type u_11} (s : (i : α) → γ i → Set β)
Goal: ⋂ i, ⋂ a, s i a = ⋂ ia, s ia.fst ia.snd
-- Set.iUnion₂_comm
theorem iUnion₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
    ⋃ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋃ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ₁ : ι → Sort u_8} {κ₂ : ι → Sort u_9} (s : (i₁ : ι) → κ₁ i₁ → (i₂ : ι) → κ₂ i₂ → Set α)
Goal: ⋃ i₁, ⋃ j₁, ⋃ i₂, ⋃ j₂, s i₁ j₁ i₂ j₂ = ⋃ i₂, ⋃ j₂, ⋃ i₁, ⋃ j₁, s i₁ j₁ i₂ j₂
-- Set.iInter₂_comm
theorem iInter₂_comm (s : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → Set α) :
    ⋂ (i₁) (j₁) (i₂) (j₂), s i₁ j₁ i₂ j₂ = ⋂ (i₂) (j₂) (i₁) (j₁), s i₁ j₁ i₂ j₂ 
Arguments: {α : Type u_1} {ι : Sort u_4} {κ₁ : ι → Sort u_8} {κ₂ : ι → Sort u_9} (s : (i₁ : ι) → κ₁ i₁ → (i₂ : ι) → κ₂ i₂ → Set α)
Goal: ⋂ i₁, ⋂ j₁, ⋂ i₂, ⋂ j₂, s i₁ j₁ i₂ j₂ = ⋂ i₂, ⋂ j₂, ⋂ i₁, ⋂ j₁, s i₁ j₁ i₂ j₂
-- Set.biUnion_and
@[simp]
theorem biUnion_and (p : ι → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p x ∧ q x y → Set α) :
    ⋃ (x : ι) (y : ι') (h : p x ∧ q x y), s x y h =
      ⋃ (x : ι) (hx : p x) (y : ι') (hy : q x y), s x y ⟨hx, hy⟩ 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} (p : ι → Prop) (q : ι → ι' → Prop) (s : (x : ι) → (y : ι') → p x ∧ q x y → Set α)
Goal: ⋃ x, ⋃ y, ⋃ (h : p x ∧ q x y), s x y h = ⋃ x, ⋃ (hx : p x), ⋃ y, ⋃ (hy : q x y), s x y ⋯
-- Set.biUnion_and'
@[simp]
theorem biUnion_and' (p : ι' → Prop) (q : ι → ι' → Prop) (s : ∀ x y, p y ∧ q x y → Set α) :
    ⋃ (x : ι) (y : ι') (h : p y ∧ q x y), s x y h =
      ⋃ (y : ι') (hy : p y) (x : ι) (hx : q x y), s x y ⟨hy, hx⟩ 
Arguments: {α : Type u_1} {ι : Sort u_4} {ι' : Sort u_5} (p : ι' → Prop) (q : ι → ι' → Prop) (s : (x : ι) → (y : ι') → p y ∧ q x y → Set α)
Goal: ⋃ x, ⋃ y, ⋃ (h : p y ∧ q x y), s x y h = ⋃ y, ⋃ (hy : p y), ⋃ x, ⋃ (hx : q x y), s x y ⋯