ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.min'_lt_max'
theorem min'_lt_max' {i j} (H1 : i ∈ s) (H2 : j ∈ s) (H3 : i ≠ j) :
    s.min' ⟨i, H1⟩ < s.max' ⟨i, H1⟩ 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α) {i j : α} (H1 : i ∈ s) (H2 : j ∈ s) (H3 : i ≠ j)
Goal: s.min' ⋯ < s.max' ⋯
-- Finset.min'_lt_max'_of_card
theorem min'_lt_max'_of_card (h₂ : 1 < card s) :
    s.min' (Finset.card_pos.1 <| by omega) < s.max' (Finset.card_pos.1 <| by omega) 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α) (h₂ : 1 < s.card)
Goal: s.min' ⋯ < s.max' ⋯
-- Finset.map_ofDual_min
theorem map_ofDual_min (s : Finset αᵒᵈ) : s.min.map ofDual = (s.image ofDual).max 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset αᵒᵈ)
Goal: WithTop.map (⇑OrderDual.ofDual) s.min = (Finset.image (⇑OrderDual.ofDual) s).max
-- Finset.map_ofDual_max
theorem map_ofDual_max (s : Finset αᵒᵈ) : s.max.map ofDual = (s.image ofDual).min 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset αᵒᵈ)
Goal: WithBot.map (⇑OrderDual.ofDual) s.max = (Finset.image (⇑OrderDual.ofDual) s).min
-- Finset.map_toDual_min
theorem map_toDual_min (s : Finset α) : s.min.map toDual = (s.image toDual).max 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α)
Goal: WithTop.map (⇑OrderDual.toDual) s.min = (Finset.image (⇑OrderDual.toDual) s).max
-- Finset.map_toDual_max
theorem map_toDual_max (s : Finset α) : s.max.map toDual = (s.image toDual).min 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α)
Goal: WithBot.map (⇑OrderDual.toDual) s.max = (Finset.image (⇑OrderDual.toDual) s).min
-- Finset.ofDual_min'
theorem ofDual_min' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
    ofDual (min' s hs) = max' (s.image ofDual) (hs.image _) 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset αᵒᵈ} (hs : s.Nonempty)
Goal: OrderDual.ofDual (s.min' hs) = (Finset.image (⇑OrderDual.ofDual) s).max' ⋯
-- Finset.ofDual_max'
theorem ofDual_max' {s : Finset αᵒᵈ} (hs : s.Nonempty) :
    ofDual (max' s hs) = min' (s.image ofDual) (hs.image _) 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset αᵒᵈ} (hs : s.Nonempty)
Goal: OrderDual.ofDual (s.max' hs) = (Finset.image (⇑OrderDual.ofDual) s).min' ⋯
-- Finset.toDual_min'
theorem toDual_min' {s : Finset α} (hs : s.Nonempty) :
    toDual (min' s hs) = max' (s.image toDual) (hs.image _) 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset α} (hs : s.Nonempty)
Goal: OrderDual.toDual (s.min' hs) = (Finset.image (⇑OrderDual.toDual) s).max' ⋯
-- Finset.toDual_max'
theorem toDual_max' {s : Finset α} (hs : s.Nonempty) :
    toDual (max' s hs) = min' (s.image toDual) (hs.image _) 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset α} (hs : s.Nonempty)
Goal: OrderDual.toDual (s.max' hs) = (Finset.image (⇑OrderDual.toDual) s).min' ⋯
-- Finset.max'_subset
theorem max'_subset {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t) :
    s.max' H ≤ t.max' (H.mono hst) 
Arguments: {α : Type u_2} [LinearOrder α] {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t)
Goal: s.max' H ≤ t.max' ⋯
-- Finset.min'_subset
theorem min'_subset {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t) :
    t.min' (H.mono hst) ≤ s.min' H 
Arguments: {α : Type u_2} [LinearOrder α] {s t : Finset α} (H : s.Nonempty) (hst : s ⊆ t)
Goal: t.min' ⋯ ≤ s.min' H
-- Finset.max'_insert
theorem max'_insert (a : α) (s : Finset α) (H : s.Nonempty) :
    (insert a s).max' (s.insert_nonempty a) = max (s.max' H) a 
Arguments: {α : Type u_2} [LinearOrder α] (a : α) (s : Finset α) (H : s.Nonempty)
Goal: (insert a s).max' ⋯ = max (s.max' H) a
-- Finset.min'_insert
theorem min'_insert (a : α) (s : Finset α) (H : s.Nonempty) :
    (insert a s).min' (s.insert_nonempty a) = min (s.min' H) a 
Arguments: {α : Type u_2} [LinearOrder α] (a : α) (s : Finset α) (H : s.Nonempty)
Goal: (insert a s).min' ⋯ = min (s.min' H) a
-- Finset.lt_max'_of_mem_erase_max'
theorem lt_max'_of_mem_erase_max' [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.max' H)) :
    a < s.max' H 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α) (H : s.Nonempty) [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.max' H))
Goal: a < s.max' H
-- Finset.min'_lt_of_mem_erase_min'
theorem min'_lt_of_mem_erase_min' [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.min' H)) :
    s.min' H < a 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α) (H : s.Nonempty) [DecidableEq α] {a : α} (ha : a ∈ s.erase (s.min' H))
Goal: s.min' H < a
-- Finset.max'_image
@[simp]
theorem max'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
    (h : (s.image f).Nonempty) : (s.image f).max' h = f (s.max' h.of_image) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α) (h : (Finset.image f s).Nonempty)
Goal: (Finset.image f s).max' h = f (s.max' ⋯)
-- Monotone.map_finset_max'
lemma _root_.Monotone.map_finset_max' [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α}
    (h : s.Nonempty) : f (s.max' h) = (s.image f).max' (h.image f) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α} (h : s.Nonempty)
Goal: f (s.max' h) = (Finset.image f s).max' ⋯
-- Finset.min'_image
@[simp]
theorem min'_image [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α)
    (h : (s.image f).Nonempty) : (s.image f).min' h = f (s.min' h.of_image) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [LinearOrder β] {f : α → β} (hf : Monotone f) (s : Finset α) (h : (Finset.image f s).Nonempty)
Goal: (Finset.image f s).min' h = f (s.min' ⋯)
-- Monotone.map_finset_min'
lemma _root_.Monotone.map_finset_min' [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α}
    (h : s.Nonempty) : f (s.min' h) = (s.image f).min' (h.image f) 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] [LinearOrder β] {f : α → β} (hf : Monotone f) {s : Finset α} (h : s.Nonempty)
Goal: f (s.min' h) = (Finset.image f s).min' ⋯
-- Finset.coe_max'
theorem coe_max' {s : Finset α} (hs : s.Nonempty) : ↑(s.max' hs) = s.max 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset α} (hs : s.Nonempty)
Goal: ↑(s.max' hs) = s.max
-- Finset.coe_min'
theorem coe_min' {s : Finset α} (hs : s.Nonempty) : ↑(s.min' hs) = s.min 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset α} (hs : s.Nonempty)
Goal: ↑(s.min' hs) = s.min
-- Finset.max_mem_image_coe
theorem max_mem_image_coe {s : Finset α} (hs : s.Nonempty) :
    s.max ∈ (s.image (↑) : Finset (WithBot α)) 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset α} (hs : s.Nonempty)
Goal: s.max ∈ Finset.image WithBot.some s
-- Finset.min_mem_image_coe
theorem min_mem_image_coe {s : Finset α} (hs : s.Nonempty) :
    s.min ∈ (s.image (↑) : Finset (WithTop α)) 
Arguments: {α : Type u_2} [LinearOrder α] {s : Finset α} (hs : s.Nonempty)
Goal: s.min ∈ Finset.image WithTop.some s
-- Finset.max_mem_insert_bot_image_coe
theorem max_mem_insert_bot_image_coe (s : Finset α) :
    s.max ∈ (insert ⊥ (s.image (↑)) : Finset (WithBot α)) 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α)
Goal: s.max ∈ insert ⊥ (Finset.image WithBot.some s)
-- Finset.min_mem_insert_top_image_coe
theorem min_mem_insert_top_image_coe (s : Finset α) :
    s.min ∈ (insert ⊤ (s.image (↑)) : Finset (WithTop α)) 
Arguments: {α : Type u_2} [LinearOrder α] (s : Finset α)
Goal: s.min ∈ insert ⊤ (Finset.image WithTop.some s)
-- Finset.max'_erase_ne_self
theorem max'_erase_ne_self {s : Finset α} (s0 : (s.erase x).Nonempty) : (s.erase x).max' s0 ≠ x 
Arguments: {α : Type u_2} [LinearOrder α] {x : α} {s : Finset α} (s0 : (s.erase x).Nonempty)
Goal: (s.erase x).max' s0 ≠ x
-- Finset.min'_erase_ne_self
theorem min'_erase_ne_self {s : Finset α} (s0 : (s.erase x).Nonempty) : (s.erase x).min' s0 ≠ x 
Arguments: {α : Type u_2} [LinearOrder α] {x : α} {s : Finset α} (s0 : (s.erase x).Nonempty)
Goal: (s.erase x).min' s0 ≠ x
-- Finset.max_erase_ne_self
theorem max_erase_ne_self {s : Finset α} : (s.erase x).max ≠ x 
Arguments: {α : Type u_2} [LinearOrder α] {x : α} {s : Finset α}
Goal: (s.erase x).max ≠ ↑x
-- Finset.min_erase_ne_self
theorem min_erase_ne_self {s : Finset α} : (s.erase x).min ≠ x 
Arguments: {α : Type u_2} [LinearOrder α] {x : α} {s : Finset α}
Goal: (s.erase x).min ≠ ↑x
-- Finset.exists_next_right
theorem exists_next_right {x : α} {s : Finset α} (h : ∃ y ∈ s, x < y) :
    ∃ y ∈ s, x < y ∧ ∀ z ∈ s, x < z → y ≤ z 
Arguments: {α : Type u_2} [LinearOrder α] {x : α} {s : Finset α} (h : ∃ y ∈ s, x < y)
Goal: ∃ y ∈ s, x < y ∧ ∀ z ∈ s, x < z → y ≤ z
-- Finset.exists_next_left
theorem exists_next_left {x : α} {s : Finset α} (h : ∃ y ∈ s, y < x) :
    ∃ y ∈ s, y < x ∧ ∀ z ∈ s, z < x → z ≤ y 
Arguments: {α : Type u_2} [LinearOrder α] {x : α} {s : Finset α} (h : ∃ y ∈ s, y < x)
Goal: ∃ y ∈ s, y < x ∧ ∀ z ∈ s, z < x → z ≤ y
-- Finset.card_le_of_interleaved
theorem card_le_of_interleaved {s t : Finset α}
    (h : ∀ᵉ (x ∈ s) (y ∈ s),
        x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
    s.card ≤ t.card + 1 
Arguments: {α : Type u_2} [LinearOrder α] {s t : Finset α} (h : ∀ x ∈ s, ∀ y ∈ s, x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y)
Goal: s.card ≤ t.card + 1
-- Finset.card_le_diff_of_interleaved
theorem card_le_diff_of_interleaved {s t : Finset α}
    (h :
      ∀ᵉ (x ∈ s) (y ∈ s),
        x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y) :
    s.card ≤ (t \ s).card + 1 
Arguments: {α : Type u_2} [LinearOrder α] {s t : Finset α} (h : ∀ x ∈ s, ∀ y ∈ s, x < y → (∀ z ∈ s, z ∉ Set.Ioo x y) → ∃ z ∈ t, x < z ∧ z < y)
Goal: s.card ≤ (t \ s).card + 1
-- Finset.induction_on_max
@[elab_as_elim]
theorem induction_on_max [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅)
    (step : ∀ a s, (∀ x ∈ s, x < a) → p s → p (insert a s)) : p s 
Arguments: {α : Type u_2} [LinearOrder α] [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅) (step : ∀ (a : α) (s : Finset α), (∀ x ∈ s, x < a) → p s → p (insert a s))
Goal: p s
-- Finset.induction_on_min
@[elab_as_elim]
theorem induction_on_min [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅)
    (step : ∀ a s, (∀ x ∈ s, a < x) → p s → p (insert a s)) : p s 
Arguments: {α : Type u_2} [LinearOrder α] [DecidableEq α] {p : Finset α → Prop} (s : Finset α) (h0 : p ∅) (step : ∀ (a : α) (s : Finset α), (∀ x ∈ s, a < x) → p s → p (insert a s))
Goal: p s
-- Finset.induction_on_max_value
@[elab_as_elim]
theorem induction_on_max_value [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι)
    (h0 : p ∅) (step : ∀ a s, a ∉ s → (∀ x ∈ s, f x ≤ f a) → p s → p (insert a s)) : p s 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι) (h0 : p ∅) (step : ∀ (a : ι) (s : Finset ι), a ∉ s → (∀ x ∈ s, f x ≤ f a) → p s → p (insert a s))
Goal: p s
-- Finset.induction_on_min_value
@[elab_as_elim]
theorem induction_on_min_value [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι)
    (h0 : p ∅) (step : ∀ a s, a ∉ s → (∀ x ∈ s, f a ≤ f x) → p s → p (insert a s)) : p s 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [DecidableEq ι] (f : ι → α) {p : Finset ι → Prop} (s : Finset ι) (h0 : p ∅) (step : ∀ (a : ι) (s : Finset ι), a ∉ s → (∀ x ∈ s, f a ≤ f x) → p s → p (insert a s))
Goal: p s
-- Finset.exists_max_image
theorem exists_max_image (s : Finset β) (f : β → α) (h : s.Nonempty) :
    ∃ x ∈ s, ∀ x' ∈ s, f x' ≤ f x 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] (s : Finset β) (f : β → α) (h : s.Nonempty)
Goal: ∃ x ∈ s, ∀ x' ∈ s, f x' ≤ f x
-- Finset.exists_min_image
theorem exists_min_image (s : Finset β) (f : β → α) (h : s.Nonempty) :
    ∃ x ∈ s, ∀ x' ∈ s, f x ≤ f x' 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrder α] (s : Finset β) (f : β → α) (h : s.Nonempty)
Goal: ∃ x ∈ s, ∀ x' ∈ s, f x ≤ f x'
-- Finset.isGLB_iff_isLeast
theorem isGLB_iff_isLeast [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty) :
    IsGLB (s : Set α) i ↔ IsLeast (↑s) i 
Arguments: {α : Type u_2} [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty)
Goal: IsGLB (↑s) i ↔ IsLeast (↑s) i
-- Finset.isLUB_iff_isGreatest
theorem isLUB_iff_isGreatest [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty) :
    IsLUB (s : Set α) i ↔ IsGreatest (↑s) i 
Arguments: {α : Type u_2} [LinearOrder α] (i : α) (s : Finset α) (hs : s.Nonempty)
Goal: IsLUB (↑s) i ↔ IsGreatest (↑s) i
-- Finset.isGLB_mem
theorem isGLB_mem [LinearOrder α] {i : α} (s : Finset α) (his : IsGLB (s : Set α) i)
    (hs : s.Nonempty) : i ∈ s 
Arguments: {α : Type u_2} [LinearOrder α] {i : α} (s : Finset α) (his : IsGLB (↑s) i) (hs : s.Nonempty)
Goal: i ∈ s
-- Finset.isLUB_mem
theorem isLUB_mem [LinearOrder α] {i : α} (s : Finset α) (his : IsLUB (s : Set α) i)
    (hs : s.Nonempty) : i ∈ s 
Arguments: {α : Type u_2} [LinearOrder α] {i : α} (s : Finset α) (his : IsLUB (↑s) i) (hs : s.Nonempty)
Goal: i ∈ s
-- Multiset.map_finset_sup
theorem map_finset_sup [DecidableEq α] [DecidableEq β] (s : Finset γ) (f : γ → Multiset β)
    (g : β → α) (hg : Function.Injective g) : map g (s.sup f) = s.sup (map g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [DecidableEq α] [DecidableEq β] (s : Finset γ) (f : γ → Multiset β) (g : β → α) (hg : Function.Injective g)
Goal: Multiset.map g (s.sup f) = s.sup (Multiset.map g ∘ f)
-- Multiset.count_finset_sup
theorem count_finset_sup [DecidableEq β] (s : Finset α) (f : α → Multiset β) (b : β) :
    count b (s.sup f) = s.sup fun a => count b (f a) 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq β] (s : Finset α) (f : α → Multiset β) (b : β)
Goal: Multiset.count b (s.sup f) = s.sup fun a => Multiset.count b (f a)
-- Multiset.mem_sup
theorem mem_sup {α β} [DecidableEq β] {s : Finset α} {f : α → Multiset β} {x : β} :
    x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v 
Arguments: {α : Type u_7} {β : Type u_8} [DecidableEq β] {s : Finset α} {f : α → Multiset β} {x : β}
Goal: x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v
-- Finset.mem_sup
theorem mem_sup {α β} [DecidableEq β] {s : Finset α} {f : α → Finset β} {x : β} :
    x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v 
Arguments: {α : Type u_7} {β : Type u_8} [DecidableEq β] {s : Finset α} {f : α → Finset β} {x : β}
Goal: x ∈ s.sup f ↔ ∃ v ∈ s, x ∈ f v
-- Finset.sup_eq_biUnion
theorem sup_eq_biUnion {α β} [DecidableEq β] (s : Finset α) (t : α → Finset β) :
    s.sup t = s.biUnion t 
Arguments: {α : Type u_7} {β : Type u_8} [DecidableEq β] (s : Finset α) (t : α → Finset β)
Goal: s.sup t = s.biUnion t
-- Finset.sup_singleton''
@[simp]
theorem sup_singleton'' [DecidableEq α] (s : Finset β) (f : β → α) :
    (s.sup fun b => {f b}) = s.image f 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq α] (s : Finset β) (f : β → α)
Goal: (s.sup fun b => {f b}) = Finset.image f s
-- Finset.sup_singleton'
@[simp]
theorem sup_singleton' [DecidableEq α] (s : Finset α) : s.sup singleton = s 
Arguments: {α : Type u_2} [DecidableEq α] (s : Finset α)
Goal: s.sup singleton = s
-- iSup_eq_iSup_finset
theorem iSup_eq_iSup_finset (s : ι → α) : ⨆ i, s i = ⨆ t : Finset ι, ⨆ i ∈ t, s i 
Arguments: {α : Type u_2} {ι : Type u_5} [CompleteLattice α] (s : ι → α)
Goal: ⨆ i, s i = ⨆ t, ⨆ i ∈ t, s i
-- iSup_eq_iSup_finset'
theorem iSup_eq_iSup_finset' (s : ι' → α) :
    ⨆ i, s i = ⨆ t : Finset (PLift ι'), ⨆ i ∈ t, s (PLift.down i) 
Arguments: {α : Type u_2} {ι' : Sort u_7} [CompleteLattice α] (s : ι' → α)
Goal: ⨆ i, s i = ⨆ t, ⨆ i ∈ t, s i.down
-- iInf_eq_iInf_finset
theorem iInf_eq_iInf_finset (s : ι → α) : ⨅ i, s i = ⨅ (t : Finset ι) (i ∈ t), s i 
Arguments: {α : Type u_2} {ι : Type u_5} [CompleteLattice α] (s : ι → α)
Goal: ⨅ i, s i = ⨅ t, ⨅ i ∈ t, s i
-- iInf_eq_iInf_finset'
theorem iInf_eq_iInf_finset' (s : ι' → α) :
    ⨅ i, s i = ⨅ t : Finset (PLift ι'), ⨅ i ∈ t, s (PLift.down i) 
Arguments: {α : Type u_2} {ι' : Sort u_7} [CompleteLattice α] (s : ι' → α)
Goal: ⨅ i, s i = ⨅ t, ⨅ i ∈ t, s i.down
-- Set.iUnion_eq_iUnion_finset
theorem iUnion_eq_iUnion_finset (s : ι → Set α) : ⋃ i, s i = ⋃ t : Finset ι, ⋃ i ∈ t, s i 
Arguments: {α : Type u_2} {ι : Type u_5} (s : ι → Set α)
Goal: ⋃ i, s i = ⋃ t, ⋃ i ∈ t, s i
-- Set.iUnion_eq_iUnion_finset'
theorem iUnion_eq_iUnion_finset' (s : ι' → Set α) :
    ⋃ i, s i = ⋃ t : Finset (PLift ι'), ⋃ i ∈ t, s (PLift.down i) 
Arguments: {α : Type u_2} {ι' : Sort u_7} (s : ι' → Set α)
Goal: ⋃ i, s i = ⋃ t, ⋃ i ∈ t, s i.down
-- Set.iInter_eq_iInter_finset
theorem iInter_eq_iInter_finset (s : ι → Set α) : ⋂ i, s i = ⋂ t : Finset ι, ⋂ i ∈ t, s i 
Arguments: {α : Type u_2} {ι : Type u_5} (s : ι → Set α)
Goal: ⋂ i, s i = ⋂ t, ⋂ i ∈ t, s i
-- Set.iInter_eq_iInter_finset'
theorem iInter_eq_iInter_finset' (s : ι' → Set α) :
    ⋂ i, s i = ⋂ t : Finset (PLift ι'), ⋂ i ∈ t, s (PLift.down i) 
Arguments: {α : Type u_2} {ι' : Sort u_7} (s : ι' → Set α)
Goal: ⋂ i, s i = ⋂ t, ⋂ i ∈ t, s i.down
-- Finset.mem_maximals_iff_forall_insert
theorem mem_maximals_iff_forall_insert (hP : ∀ ⦃s t⦄, P t → s ⊆ t → P s) :
    s ∈ maximals (· ⊆ ·) {t | P t} ↔ P s ∧ ∀ x ∉ s, ¬ P (insert x s) 
Arguments: {α : Type u_2} [DecidableEq α] {P : Finset α → Prop} {s : Finset α} (hP : ∀ ⦃s t : Finset α⦄, P t → s ⊆ t → P s)
Goal: s ∈ maximals (fun x x_1 => x ⊆ x_1) {t | P t} ↔ P s ∧ ∀ x ∉ s, ¬P (insert x s)
-- Finset.mem_minimals_iff_forall_erase
theorem mem_minimals_iff_forall_erase (hP : ∀ ⦃s t⦄, P s → s ⊆ t → P t) :
    s ∈ minimals (· ⊆ ·) {t | P t} ↔ P s ∧ ∀ x ∈ s, ¬ P (s.erase x) 
Arguments: {α : Type u_2} [DecidableEq α] {P : Finset α → Prop} {s : Finset α} (hP : ∀ ⦃s t : Finset α⦄, P s → s ⊆ t → P t)
Goal: s ∈ minimals (fun x x_1 => x ⊆ x_1) {t | P t} ↔ P s ∧ ∀ x ∈ s, ¬P (s.erase x)
-- Finset.iSup_coe
theorem iSup_coe [SupSet β] (f : α → β) (s : Finset α) : ⨆ x ∈ (↑s : Set α), f x = ⨆ x ∈ s, f x 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet β] (f : α → β) (s : Finset α)
Goal: ⨆ x ∈ ↑s, f x = ⨆ x ∈ s, f x
-- Finset.iInf_coe
theorem iInf_coe [InfSet β] (f : α → β) (s : Finset α) : ⨅ x ∈ (↑s : Set α), f x = ⨅ x ∈ s, f x 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet β] (f : α → β) (s : Finset α)
Goal: ⨅ x ∈ ↑s, f x = ⨅ x ∈ s, f x
-- Finset.iSup_singleton
theorem iSup_singleton (a : α) (s : α → β) : ⨆ x ∈ ({a} : Finset α), s x = s a 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] (a : α) (s : α → β)
Goal: ⨆ x ∈ {a}, s x = s a
-- Finset.iInf_singleton
theorem iInf_singleton (a : α) (s : α → β) : ⨅ x ∈ ({a} : Finset α), s x = s a 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] (a : α) (s : α → β)
Goal: ⨅ x ∈ {a}, s x = s a
-- Finset.iSup_option_toFinset
theorem iSup_option_toFinset (o : Option α) (f : α → β) : ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] (o : Option α) (f : α → β)
Goal: ⨆ x ∈ o.toFinset, f x = ⨆ x ∈ o, f x
-- Finset.iInf_option_toFinset
theorem iInf_option_toFinset (o : Option α) (f : α → β) : ⨅ x ∈ o.toFinset, f x = ⨅ x ∈ o, f x 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] (o : Option α) (f : α → β)
Goal: ⨅ x ∈ o.toFinset, f x = ⨅ x ∈ o, f x
-- Finset.iSup_union
theorem iSup_union {f : α → β} {s t : Finset α} :
    ⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] [DecidableEq α] {f : α → β} {s t : Finset α}
Goal: ⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x
-- Finset.iInf_union
theorem iInf_union {f : α → β} {s t : Finset α} :
    ⨅ x ∈ s ∪ t, f x = (⨅ x ∈ s, f x) ⊓ ⨅ x ∈ t, f x 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] [DecidableEq α] {f : α → β} {s t : Finset α}
Goal: ⨅ x ∈ s ∪ t, f x = (⨅ x ∈ s, f x) ⊓ ⨅ x ∈ t, f x
-- Finset.iSup_insert
theorem iSup_insert (a : α) (s : Finset α) (t : α → β) :
    ⨆ x ∈ insert a s, t x = t a ⊔ ⨆ x ∈ s, t x 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] [DecidableEq α] (a : α) (s : Finset α) (t : α → β)
Goal: ⨆ x ∈ insert a s, t x = t a ⊔ ⨆ x ∈ s, t x
-- Finset.iInf_insert
theorem iInf_insert (a : α) (s : Finset α) (t : α → β) :
    ⨅ x ∈ insert a s, t x = t a ⊓ ⨅ x ∈ s, t x 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] [DecidableEq α] (a : α) (s : Finset α) (t : α → β)
Goal: ⨅ x ∈ insert a s, t x = t a ⊓ ⨅ x ∈ s, t x
-- Finset.iSup_finset_image
theorem iSup_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
    ⨆ x ∈ s.image f, g x = ⨆ y ∈ s, g (f y) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice β] [DecidableEq α] {f : γ → α} {g : α → β} {s : Finset γ}
Goal: ⨆ x ∈ Finset.image f s, g x = ⨆ y ∈ s, g (f y)
-- Finset.iInf_finset_image
theorem iInf_finset_image {f : γ → α} {g : α → β} {s : Finset γ} :
    ⨅ x ∈ s.image f, g x = ⨅ y ∈ s, g (f y) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice β] [DecidableEq α] {f : γ → α} {g : α → β} {s : Finset γ}
Goal: ⨅ x ∈ Finset.image f s, g x = ⨅ y ∈ s, g (f y)
-- Finset.iSup_insert_update
theorem iSup_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
    ⨆ i ∈ insert x t, Function.update f x s i = s ⊔ ⨆ i ∈ t, f i 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] [DecidableEq α] {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t)
Goal: ⨆ i ∈ insert x t, Function.update f x s i = s ⊔ ⨆ i ∈ t, f i
-- Finset.iInf_insert_update
theorem iInf_insert_update {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t) :
    ⨅ i ∈ insert x t, update f x s i = s ⊓ ⨅ i ∈ t, f i 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] [DecidableEq α] {x : α} {t : Finset α} (f : α → β) {s : β} (hx : x ∉ t)
Goal: ⨅ i ∈ insert x t, Function.update f x s i = s ⊓ ⨅ i ∈ t, f i
-- Finset.iSup_biUnion
theorem iSup_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → β) :
    ⨆ y ∈ s.biUnion t, f y = ⨆ (x ∈ s) (y ∈ t x), f y 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice β] [DecidableEq α] (s : Finset γ) (t : γ → Finset α) (f : α → β)
Goal: ⨆ y ∈ s.biUnion t, f y = ⨆ x ∈ s, ⨆ y ∈ t x, f y
-- Finset.iInf_biUnion
theorem iInf_biUnion (s : Finset γ) (t : γ → Finset α) (f : α → β) :
    ⨅ y ∈ s.biUnion t, f y = ⨅ (x ∈ s) (y ∈ t x), f y 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice β] [DecidableEq α] (s : Finset γ) (t : γ → Finset α) (f : α → β)
Goal: ⨅ y ∈ s.biUnion t, f y = ⨅ x ∈ s, ⨅ y ∈ t x, f y
-- Finset.set_biUnion_coe
theorem set_biUnion_coe (s : Finset α) (t : α → Set β) : ⋃ x ∈ (↑s : Set α), t x = ⋃ x ∈ s, t x 
Arguments: {α : Type u_2} {β : Type u_3} (s : Finset α) (t : α → Set β)
Goal: ⋃ x ∈ ↑s, t x = ⋃ x ∈ s, t x
-- Finset.set_biInter_coe
theorem set_biInter_coe (s : Finset α) (t : α → Set β) : ⋂ x ∈ (↑s : Set α), t x = ⋂ x ∈ s, t x 
Arguments: {α : Type u_2} {β : Type u_3} (s : Finset α) (t : α → Set β)
Goal: ⋂ x ∈ ↑s, t x = ⋂ x ∈ s, t x
-- Finset.set_biUnion_singleton
theorem set_biUnion_singleton (a : α) (s : α → Set β) : ⋃ x ∈ ({a} : Finset α), s x = s a 
Arguments: {α : Type u_2} {β : Type u_3} (a : α) (s : α → Set β)
Goal: ⋃ x ∈ {a}, s x = s a
-- Finset.set_biInter_singleton
theorem set_biInter_singleton (a : α) (s : α → Set β) : ⋂ x ∈ ({a} : Finset α), s x = s a 
Arguments: {α : Type u_2} {β : Type u_3} (a : α) (s : α → Set β)
Goal: ⋂ x ∈ {a}, s x = s a
-- Finset.set_biUnion_preimage_singleton
@[simp]
theorem set_biUnion_preimage_singleton (f : α → β) (s : Finset β) :
    ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' s 
Arguments: {α : Type u_2} {β : Type u_3} (f : α → β) (s : Finset β)
Goal: ⋃ y ∈ s, f ⁻¹' {y} = f ⁻¹' ↑s
-- Finset.set_biUnion_option_toFinset
theorem set_biUnion_option_toFinset (o : Option α) (f : α → Set β) :
    ⋃ x ∈ o.toFinset, f x = ⋃ x ∈ o, f x 
Arguments: {α : Type u_2} {β : Type u_3} (o : Option α) (f : α → Set β)
Goal: ⋃ x ∈ o.toFinset, f x = ⋃ x ∈ o, f x
-- Finset.set_biInter_option_toFinset
theorem set_biInter_option_toFinset (o : Option α) (f : α → Set β) :
    ⋂ x ∈ o.toFinset, f x = ⋂ x ∈ o, f x 
Arguments: {α : Type u_2} {β : Type u_3} (o : Option α) (f : α → Set β)
Goal: ⋂ x ∈ o.toFinset, f x = ⋂ x ∈ o, f x
-- Finset.subset_set_biUnion_of_mem
theorem subset_set_biUnion_of_mem {s : Finset α} {f : α → Set β} {x : α} (h : x ∈ s) :
    f x ⊆ ⋃ y ∈ s, f y 
Arguments: {α : Type u_2} {β : Type u_3} {s : Finset α} {f : α → Set β} {x : α} (h : x ∈ s)
Goal: f x ⊆ ⋃ y ∈ s, f y
-- Finset.set_biUnion_union
theorem set_biUnion_union (s t : Finset α) (u : α → Set β) :
    ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x 
Arguments: {α : Type u_2} {β : Type u_3} [DecidableEq α] (s t : Finset α) (u : α → Set β)
Goal: ⋃ x ∈ s ∪ t, u x = (⋃ x ∈ s, u x) ∪ ⋃ x ∈ t, u x