ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Finset.disjoint_sup_left
protected theorem disjoint_sup_left : Disjoint (s.sup f) a ↔ ∀ ⦃i⦄, i ∈ s → Disjoint (f i) a 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [OrderBot α] {s : Finset ι} {f : ι → α} {a : α}
Goal: Disjoint (s.sup f) a ↔ ∀ ⦃i : ι⦄, i ∈ s → Disjoint (f i) a
-- Finset.sup_inf_sup
theorem sup_inf_sup (s : Finset ι) (t : Finset κ) (f : ι → α) (g : κ → α) :
    s.sup f ⊓ t.sup g = (s ×ˢ t).sup fun i => f i.1 ⊓ g i.2 
Arguments: {α : Type u_2} {ι : Type u_5} {κ : Type u_6} [DistribLattice α] [OrderBot α] (s : Finset ι) (t : Finset κ) (f : ι → α) (g : κ → α)
Goal: s.sup f ⊓ t.sup g = (s ×ˢ t).sup fun i => f i.1 ⊓ g i.2
-- Finset.inf_sup_distrib_left
theorem inf_sup_distrib_left (s : Finset ι) (f : ι → α) (a : α) :
    a ⊔ s.inf f = s.inf fun i => a ⊔ f i 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [OrderTop α] (s : Finset ι) (f : ι → α) (a : α)
Goal: a ⊔ s.inf f = s.inf fun i => a ⊔ f i
-- Finset.inf_sup_distrib_right
theorem inf_sup_distrib_right (s : Finset ι) (f : ι → α) (a : α) :
    s.inf f ⊔ a = s.inf fun i => f i ⊔ a 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [OrderTop α] (s : Finset ι) (f : ι → α) (a : α)
Goal: s.inf f ⊔ a = s.inf fun i => f i ⊔ a
-- Finset.codisjoint_inf_right
protected theorem codisjoint_inf_right :
    Codisjoint a (s.inf f) ↔ ∀ ⦃i⦄, i ∈ s → Codisjoint a (f i) 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [OrderTop α] {f : ι → α} {s : Finset ι} {a : α}
Goal: Codisjoint a (s.inf f) ↔ ∀ ⦃i : ι⦄, i ∈ s → Codisjoint a (f i)
-- Finset.codisjoint_inf_left
protected theorem codisjoint_inf_left :
    Codisjoint (s.inf f) a ↔ ∀ ⦃i⦄, i ∈ s → Codisjoint (f i) a 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [OrderTop α] {f : ι → α} {s : Finset ι} {a : α}
Goal: Codisjoint (s.inf f) a ↔ ∀ ⦃i : ι⦄, i ∈ s → Codisjoint (f i) a
-- Finset.inf_sup_inf
theorem inf_sup_inf (s : Finset ι) (t : Finset κ) (f : ι → α) (g : κ → α) :
    s.inf f ⊔ t.inf g = (s ×ˢ t).inf fun i => f i.1 ⊔ g i.2 
Arguments: {α : Type u_2} {ι : Type u_5} {κ : Type u_6} [DistribLattice α] [OrderTop α] (s : Finset ι) (t : Finset κ) (f : ι → α) (g : κ → α)
Goal: s.inf f ⊔ t.inf g = (s ×ˢ t).inf fun i => f i.1 ⊔ g i.2
-- Finset.inf_sup
theorem inf_sup {κ : ι → Type*} (s : Finset ι) (t : ∀ i, Finset (κ i)) (f : ∀ i, κ i → α) :
    (s.inf fun i => (t i).sup (f i)) =
      (s.pi t).sup fun g => s.attach.inf fun i => f _ <| g _ i.2 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [BoundedOrder α] [DecidableEq ι] {κ : ι → Type u_7} (s : Finset ι) (t : (i : ι) → Finset (κ i)) (f : (i : ι) → κ i → α)
Goal: (s.inf fun i => (t i).sup (f i)) = (s.pi t).sup fun g => s.attach.inf fun i => f (↑i) (g ↑i ⋯)
-- Finset.sup_inf
theorem sup_inf {κ : ι → Type*} (s : Finset ι) (t : ∀ i, Finset (κ i)) (f : ∀ i, κ i → α) :
    (s.sup fun i => (t i).inf (f i)) = (s.pi t).inf fun g => s.attach.sup fun i => f _ <| g _ i.2 
Arguments: {α : Type u_2} {ι : Type u_5} [DistribLattice α] [BoundedOrder α] [DecidableEq ι] {κ : ι → Type u_7} (s : Finset ι) (t : (i : ι) → Finset (κ i)) (f : (i : ι) → κ i → α)
Goal: (s.sup fun i => (t i).inf (f i)) = (s.pi t).inf fun g => s.attach.sup fun i => f (↑i) (g ↑i ⋯)
-- Finset.sup_sdiff_left
theorem sup_sdiff_left (s : Finset ι) (f : ι → α) (a : α) :
    (s.sup fun b => a \ f b) = a \ s.inf f 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] (s : Finset ι) (f : ι → α) (a : α)
Goal: (s.sup fun b => a \ f b) = a \ s.inf f
-- Finset.inf_sdiff_left
theorem inf_sdiff_left (hs : s.Nonempty) (f : ι → α) (a : α) :
    (s.inf fun b => a \ f b) = a \ s.sup f 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] {s : Finset ι} (hs : s.Nonempty) (f : ι → α) (a : α)
Goal: (s.inf fun b => a \ f b) = a \ s.sup f
-- Finset.inf_sdiff_right
theorem inf_sdiff_right (hs : s.Nonempty) (f : ι → α) (a : α) :
    (s.inf fun b => f b \ a) = s.inf f \ a 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] {s : Finset ι} (hs : s.Nonempty) (f : ι → α) (a : α)
Goal: (s.inf fun b => f b \ a) = s.inf f \ a
-- Finset.inf_himp_right
theorem inf_himp_right (s : Finset ι) (f : ι → α) (a : α) :
    (s.inf fun b => f b ⇨ a) = s.sup f ⇨ a 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] (s : Finset ι) (f : ι → α) (a : α)
Goal: (s.inf fun b => f b ⇨ a) = s.sup f ⇨ a
-- Finset.sup_himp_right
theorem sup_himp_right (hs : s.Nonempty) (f : ι → α) (a : α) :
    (s.sup fun b => f b ⇨ a) = s.inf f ⇨ a 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] {s : Finset ι} (hs : s.Nonempty) (f : ι → α) (a : α)
Goal: (s.sup fun b => f b ⇨ a) = s.inf f ⇨ a
-- Finset.sup_himp_left
theorem sup_himp_left (hs : s.Nonempty) (f : ι → α) (a : α) :
    (s.sup fun b => a ⇨ f b) = a ⇨ s.sup f 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] {s : Finset ι} (hs : s.Nonempty) (f : ι → α) (a : α)
Goal: (s.sup fun b => a ⇨ f b) = a ⇨ s.sup f
-- Finset.compl_sup
@[simp]
protected theorem compl_sup (s : Finset ι) (f : ι → α) : (s.sup f)ᶜ = s.inf fun i => (f i)ᶜ 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] (s : Finset ι) (f : ι → α)
Goal: (s.sup f)ᶜ = s.inf fun i => (f i)ᶜ
-- Finset.compl_inf
@[simp]
protected theorem compl_inf (s : Finset ι) (f : ι → α) : (s.inf f)ᶜ = s.sup fun i => (f i)ᶜ 
Arguments: {α : Type u_2} {ι : Type u_5} [BooleanAlgebra α] (s : Finset ι) (f : ι → α)
Goal: (s.inf f)ᶜ = s.sup fun i => (f i)ᶜ
-- Finset.comp_sup_eq_sup_comp_of_is_total
theorem comp_sup_eq_sup_comp_of_is_total [SemilatticeSup β] [OrderBot β] (g : α → β)
    (mono_g : Monotone g) (bot : g ⊥ = ⊥) : g (s.sup f) = s.sup (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Type u_5} [LinearOrder α] [OrderBot α] {s : Finset ι} {f : ι → α} [SemilatticeSup β] [OrderBot β] (g : α → β) (mono_g : Monotone g) (bot : g ⊥ = ⊥)
Goal: g (s.sup f) = s.sup (g ∘ f)
-- Finset.le_sup_iff
@[simp]
protected theorem le_sup_iff (ha : ⊥ < a) : a ≤ s.sup f ↔ ∃ b ∈ s, a ≤ f b 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [OrderBot α] {s : Finset ι} {f : ι → α} {a : α} (ha : ⊥ < a)
Goal: a ≤ s.sup f ↔ ∃ b ∈ s, a ≤ f b
-- Finset.sup_eq_top_iff
protected theorem sup_eq_top_iff {α : Type*} [LinearOrder α] [BoundedOrder α] [Nontrivial α]
    {s : Finset ι} {f : ι → α} : s.sup f = ⊤ ↔ ∃ b ∈ s, f b = ⊤ 
Arguments: {ι : Type u_5} {α : Type u_7} [LinearOrder α] [BoundedOrder α] [Nontrivial α] {s : Finset ι} {f : ι → α}
Goal: s.sup f = ⊤ ↔ ∃ b ∈ s, f b = ⊤
-- Finset.Nonempty.sup_eq_top_iff
protected theorem Nonempty.sup_eq_top_iff {α : Type*} [LinearOrder α] [BoundedOrder α]
    {s : Finset ι} {f : ι → α} (hs : s.Nonempty) : s.sup f = ⊤ ↔ ∃ b ∈ s, f b = ⊤ 
Arguments: {ι : Type u_5} {α : Type u_7} [LinearOrder α] [BoundedOrder α] {s : Finset ι} {f : ι → α} (hs : s.Nonempty)
Goal: s.sup f = ⊤ ↔ ∃ b ∈ s, f b = ⊤
-- Finset.lt_sup_iff
@[simp]
protected theorem lt_sup_iff : a < s.sup f ↔ ∃ b ∈ s, a < f b 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [OrderBot α] {s : Finset ι} {f : ι → α} {a : α}
Goal: a < s.sup f ↔ ∃ b ∈ s, a < f b
-- Finset.sup_lt_iff
@[simp]
protected theorem sup_lt_iff (ha : ⊥ < a) : s.sup f < a ↔ ∀ b ∈ s, f b < a 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [OrderBot α] {s : Finset ι} {f : ι → α} {a : α} (ha : ⊥ < a)
Goal: s.sup f < a ↔ ∀ b ∈ s, f b < a
-- Finset.comp_inf_eq_inf_comp_of_is_total
theorem comp_inf_eq_inf_comp_of_is_total [SemilatticeInf β] [OrderTop β] (g : α → β)
    (mono_g : Monotone g) (top : g ⊤ = ⊤) : g (s.inf f) = s.inf (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {ι : Type u_5} [LinearOrder α] [OrderTop α] {s : Finset ι} {f : ι → α} [SemilatticeInf β] [OrderTop β] (g : α → β) (mono_g : Monotone g) (top : g ⊤ = ⊤)
Goal: g (s.inf f) = s.inf (g ∘ f)
-- Finset.inf_le_iff
@[simp]
protected theorem inf_le_iff (ha : a < ⊤) : s.inf f ≤ a ↔ ∃ b ∈ s, f b ≤ a 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [OrderTop α] {s : Finset ι} {f : ι → α} {a : α} (ha : a < ⊤)
Goal: s.inf f ≤ a ↔ ∃ b ∈ s, f b ≤ a
-- Finset.inf_eq_bot_iff
protected theorem inf_eq_bot_iff {α : Type*} [LinearOrder α] [BoundedOrder α] [Nontrivial α]
    {s : Finset ι} {f : ι → α} : s.inf f = ⊥ ↔ ∃ b ∈ s, f b = ⊥ 
Arguments: {ι : Type u_5} {α : Type u_7} [LinearOrder α] [BoundedOrder α] [Nontrivial α] {s : Finset ι} {f : ι → α}
Goal: s.inf f = ⊥ ↔ ∃ b ∈ s, f b = ⊥
-- Finset.Nonempty.inf_eq_bot_iff
protected theorem Nonempty.inf_eq_bot_iff {α : Type*} [LinearOrder α] [BoundedOrder α]
    {s : Finset ι} {f : ι → α} (h : s.Nonempty) : s.inf f = ⊥ ↔ ∃ b ∈ s, f b = ⊥ 
Arguments: {ι : Type u_5} {α : Type u_7} [LinearOrder α] [BoundedOrder α] {s : Finset ι} {f : ι → α} (h : s.Nonempty)
Goal: s.inf f = ⊥ ↔ ∃ b ∈ s, f b = ⊥
-- Finset.inf_lt_iff
@[simp]
protected theorem inf_lt_iff : s.inf f < a ↔ ∃ b ∈ s, f b < a 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [OrderTop α] {s : Finset ι} {f : ι → α} {a : α}
Goal: s.inf f < a ↔ ∃ b ∈ s, f b < a
-- Finset.lt_inf_iff
@[simp]
protected theorem lt_inf_iff (ha : a < ⊤) : a < s.inf f ↔ ∀ b ∈ s, a < f b 
Arguments: {α : Type u_2} {ι : Type u_5} [LinearOrder α] [OrderTop α] {s : Finset ι} {f : ι → α} {a : α} (ha : a < ⊤)
Goal: a < s.inf f ↔ ∀ b ∈ s, a < f b
-- Finset.inf_eq_iInf
theorem inf_eq_iInf [CompleteLattice β] (s : Finset α) (f : α → β) : s.inf f = ⨅ a ∈ s, f a 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] (s : Finset α) (f : α → β)
Goal: s.inf f = ⨅ a ∈ s, f a
-- Finset.inf_id_eq_sInf
theorem inf_id_eq_sInf [CompleteLattice α] (s : Finset α) : s.inf id = sInf s 
Arguments: {α : Type u_2} [CompleteLattice α] (s : Finset α)
Goal: s.inf id = sInf ↑s
-- Finset.inf_id_set_eq_sInter
theorem inf_id_set_eq_sInter (s : Finset (Set α)) : s.inf id = ⋂₀ ↑s 
Arguments: {α : Type u_2} (s : Finset (Set α))
Goal: s.inf id = ⋂₀ ↑s
-- Finset.inf_set_eq_iInter
@[simp]
theorem inf_set_eq_iInter (s : Finset α) (f : α → Set β) : s.inf f = ⋂ x ∈ s, f x 
Arguments: {α : Type u_2} {β : Type u_3} (s : Finset α) (f : α → Set β)
Goal: s.inf f = ⋂ x ∈ s, f x
-- Finset.inf_eq_sInf_image
theorem inf_eq_sInf_image [CompleteLattice β] (s : Finset α) (f : α → β) :
    s.inf f = sInf (f '' s) 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice β] (s : Finset α) (f : α → β)
Goal: s.inf f = sInf (f '' ↑s)
-- Finset.sup_of_mem
theorem sup_of_mem {s : Finset β} (f : β → α) {b : β} (h : b ∈ s) :
    ∃ a : α, s.sup ((↑) ∘ f : β → WithBot α) = ↑a 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (f : β → α) {b : β} (h : b ∈ s)
Goal: ∃ a, s.sup (WithBot.some ∘ f) = ↑a
-- Finset.coe_sup'
@[simp]
theorem coe_sup' : ((s.sup' H f : α) : WithBot α) = s.sup ((↑) ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) (f : β → α)
Goal: ↑(s.sup' H f) = s.sup (WithBot.some ∘ f)
-- Finset.sup'_cons
@[simp]
theorem sup'_cons {b : β} {hb : b ∉ s} :
    (cons b s hb).sup' (nonempty_cons hb) f = f b ⊔ s.sup' H f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) (f : β → α) {b : β} {hb : b ∉ s}
Goal: (Finset.cons b s hb).sup' ⋯ f = f b ⊔ s.sup' H f
-- Finset.sup'_insert
@[simp]
theorem sup'_insert [DecidableEq β] {b : β} :
    (insert b s).sup' (insert_nonempty _ _) f = f b ⊔ s.sup' H f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) (f : β → α) [DecidableEq β] {b : β}
Goal: (insert b s).sup' ⋯ f = f b ⊔ s.sup' H f
-- Finset.sup'_singleton
@[simp]
theorem sup'_singleton {b : β} : ({b} : Finset β).sup' (singleton_nonempty _) f = f b 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] (f : β → α) {b : β}
Goal: {b}.sup' ⋯ f = f b
-- Finset.sup'_le_iff
@[simp]
theorem sup'_le_iff {a : α} : s.sup' H f ≤ a ↔ ∀ b ∈ s, f b ≤ a 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) (f : β → α) {a : α}
Goal: s.sup' H f ≤ a ↔ ∀ b ∈ s, f b ≤ a
-- Finset.le_sup'
theorem le_sup' {b : β} (h : b ∈ s) : f b ≤ s.sup' ⟨b, h⟩ f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (f : β → α) {b : β} (h : b ∈ s)
Goal: f b ≤ s.sup' ⋯ f
-- Finset.le_sup'_of_le
theorem le_sup'_of_le {a : α} {b : β} (hb : b ∈ s) (h : a ≤ f b) : a ≤ s.sup' ⟨b, hb⟩ f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (f : β → α) {a : α} {b : β} (hb : b ∈ s) (h : a ≤ f b)
Goal: a ≤ s.sup' ⋯ f
-- Finset.sup'_const
@[simp]
theorem sup'_const (a : α) : s.sup' H (fun _ => a) = a 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) (a : α)
Goal: (s.sup' H fun x => a) = a
-- Finset.sup'_union
theorem sup'_union [DecidableEq β] {s₁ s₂ : Finset β} (h₁ : s₁.Nonempty) (h₂ : s₂.Nonempty)
    (f : β → α) :
    (s₁ ∪ s₂).sup' (h₁.mono subset_union_left) f = s₁.sup' h₁ f ⊔ s₂.sup' h₂ f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [DecidableEq β] {s₁ s₂ : Finset β} (h₁ : s₁.Nonempty) (h₂ : s₂.Nonempty) (f : β → α)
Goal: (s₁ ∪ s₂).sup' ⋯ f = s₁.sup' h₁ f ⊔ s₂.sup' h₂ f
-- Finset.sup'_biUnion
theorem sup'_biUnion [DecidableEq β] {s : Finset γ} (Hs : s.Nonempty) {t : γ → Finset β}
    (Ht : ∀ b, (t b).Nonempty) :
    (s.biUnion t).sup' (Hs.biUnion fun b _ => Ht b) f = s.sup' Hs (fun b => (t b).sup' (Ht b) f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] (f : β → α) [DecidableEq β] {s : Finset γ} (Hs : s.Nonempty) {t : γ → Finset β} (Ht : ∀ (b : γ), (t b).Nonempty)
Goal: (s.biUnion t).sup' ⋯ f = s.sup' Hs fun b => (t b).sup' ⋯ f
-- Finset.sup'_comm
protected theorem sup'_comm {t : Finset γ} (hs : s.Nonempty) (ht : t.Nonempty) (f : β → γ → α) :
    (s.sup' hs fun b => t.sup' ht (f b)) = t.sup' ht fun c => s.sup' hs fun b => f b c 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] {s : Finset β} {t : Finset γ} (hs : s.Nonempty) (ht : t.Nonempty) (f : β → γ → α)
Goal: (s.sup' hs fun b => t.sup' ht (f b)) = t.sup' ht fun c => s.sup' hs fun b => f b c
-- Finset.sup'_product_left
theorem sup'_product_left {t : Finset γ} (h : (s ×ˢ t).Nonempty) (f : β × γ → α) :
    (s ×ˢ t).sup' h f = s.sup' h.fst fun i => t.sup' h.snd fun i' => f ⟨i, i'⟩ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] {s : Finset β} {t : Finset γ} (h : (s ×ˢ t).Nonempty) (f : β × γ → α)
Goal: (s ×ˢ t).sup' h f = s.sup' ⋯ fun i => t.sup' ⋯ fun i' => f (i, i')
-- Finset.sup'_product_right
theorem sup'_product_right {t : Finset γ} (h : (s ×ˢ t).Nonempty) (f : β × γ → α) :
    (s ×ˢ t).sup' h f = t.sup' h.snd fun i' => s.sup' h.fst fun i => f ⟨i, i'⟩ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] {s : Finset β} {t : Finset γ} (h : (s ×ˢ t).Nonempty) (f : β × γ → α)
Goal: (s ×ˢ t).sup' h f = t.sup' ⋯ fun i' => s.sup' ⋯ fun i => f (i, i')
-- Finset.prodMk_sup'_sup'
lemma prodMk_sup'_sup' (hs : s.Nonempty) (ht : t.Nonempty) (f : ι → α) (g : κ → β) :
    (sup' s hs f, sup' t ht g) = sup' (s ×ˢ t) (hs.product ht) (Prod.map f g) 
Arguments: {ι : Type u_7} {κ : Type u_8} {α : Type u_9} {β : Type u_10} [SemilatticeSup α] [SemilatticeSup β] {s : Finset ι} {t : Finset κ} (hs : s.Nonempty) (ht : t.Nonempty) (f : ι → α) (g : κ → β)
Goal: (s.sup' hs f, t.sup' ht g) = (s ×ˢ t).sup' ⋯ (Prod.map f g)
-- Finset.sup'_prodMap
-- @[simp] -- TODO: Why does `Prod.map_apply` simplify the LHS?
lemma sup'_prodMap (hst : (s ×ˢ t).Nonempty) (f : ι → α) (g : κ → β) :
    sup' (s ×ˢ t) hst (Prod.map f g) = (sup' s hst.fst f, sup' t hst.snd g) 
Arguments: {ι : Type u_7} {κ : Type u_8} {α : Type u_9} {β : Type u_10} [SemilatticeSup α] [SemilatticeSup β] {s : Finset ι} {t : Finset κ} (hst : (s ×ˢ t).Nonempty) (f : ι → α) (g : κ → β)
Goal: (s ×ˢ t).sup' hst (Prod.map f g) = (s.sup' ⋯ f, t.sup' ⋯ g)
-- Finset.sup'_induction
theorem sup'_induction {p : α → Prop} (hp : ∀ a₁, p a₁ → ∀ a₂, p a₂ → p (a₁ ⊔ a₂))
    (hs : ∀ b ∈ s, p (f b)) : p (s.sup' H f) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) (f : β → α) {p : α → Prop} (hp : ∀ (a₁ : α), p a₁ → ∀ (a₂ : α), p a₂ → p (a₁ ⊔ a₂)) (hs : ∀ b ∈ s, p (f b))
Goal: p (s.sup' H f)
-- Finset.sup'_mem
theorem sup'_mem (s : Set α) (w : ∀ᵉ (x ∈ s) (y ∈ s), x ⊔ y ∈ s) {ι : Type*}
    (t : Finset ι) (H : t.Nonempty) (p : ι → α) (h : ∀ i ∈ t, p i ∈ s) : t.sup' H p ∈ s 
Arguments: {α : Type u_2} [SemilatticeSup α] (s : Set α) (w : ∀ x ∈ s, ∀ y ∈ s, x ⊔ y ∈ s) {ι : Type u_7} (t : Finset ι) (H : t.Nonempty) (p : ι → α) (h : ∀ i ∈ t, p i ∈ s)
Goal: t.sup' H p ∈ s
-- Finset.sup'_congr
@[congr]
theorem sup'_congr {t : Finset β} {f g : β → α} (h₁ : s = t) (h₂ : ∀ x ∈ s, f x = g x) :
    s.sup' H f = t.sup' (h₁ ▸ H) g 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] {s : Finset β} (H : s.Nonempty) {t : Finset β} {f g : β → α} (h₁ : s = t) (h₂ : ∀ x ∈ s, f x = g x)
Goal: s.sup' H f = t.sup' ⋯ g
-- Finset.comp_sup'_eq_sup'_comp
theorem comp_sup'_eq_sup'_comp [SemilatticeSup γ] {s : Finset β} (H : s.Nonempty) {f : β → α}
    (g : α → γ) (g_sup : ∀ x y, g (x ⊔ y) = g x ⊔ g y) : g (s.sup' H f) = s.sup' H (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] [SemilatticeSup γ] {s : Finset β} (H : s.Nonempty) {f : β → α} (g : α → γ) (g_sup : ∀ (x y : α), g (x ⊔ y) = g x ⊔ g y)
Goal: g (s.sup' H f) = s.sup' H (g ∘ f)
-- map_finset_sup'
@[simp]
theorem _root_.map_finset_sup' [SemilatticeSup β] [FunLike F α β] [SupHomClass F α β]
    (f : F) {s : Finset ι} (hs) (g : ι → α) :
    f (s.sup' hs g) = s.sup' hs (f ∘ g) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Type u_5} [SemilatticeSup α] [SemilatticeSup β] [FunLike F α β] [SupHomClass F α β] (f : F) {s : Finset ι} (hs : s.Nonempty) (g : ι → α)
Goal: f (s.sup' hs g) = s.sup' hs (⇑f ∘ g)
-- Finset.nsmul_sup'
lemma nsmul_sup' {α'} [LinearOrderedAddCommMonoid β] {s : Finset α'}
    (hs : s.Nonempty) (f : α' → β) (n : ℕ) :
    s.sup' hs (fun a => n • f a) = n • s.sup' hs f 
Arguments: {β : Type u_3} {α' : Type u_7} [LinearOrderedAddCommMonoid β] {s : Finset α'} (hs : s.Nonempty) (f : α' → β) (n : ℕ)
Goal: (s.sup' hs fun a => n • f a) = n • s.sup' hs f
-- Finset.sup'_image
@[simp]
theorem sup'_image [DecidableEq β] {s : Finset γ} {f : γ → β} (hs : (s.image f).Nonempty)
    (g : β → α) :
    (s.image f).sup' hs g = s.sup' hs.of_image (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] [DecidableEq β] {s : Finset γ} {f : γ → β} (hs : (Finset.image f s).Nonempty) (g : β → α)
Goal: (Finset.image f s).sup' hs g = s.sup' ⋯ (g ∘ f)
-- Finset.sup'_comp_eq_image
lemma sup'_comp_eq_image [DecidableEq β] {s : Finset γ} {f : γ → β} (hs : s.Nonempty) (g : β → α) :
    s.sup' hs (g ∘ f) = (s.image f).sup' (hs.image f) g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] [DecidableEq β] {s : Finset γ} {f : γ → β} (hs : s.Nonempty) (g : β → α)
Goal: s.sup' hs (g ∘ f) = (Finset.image f s).sup' ⋯ g
-- Finset.sup'_map
@[simp]
theorem sup'_map {s : Finset γ} {f : γ ↪ β} (g : β → α) (hs : (s.map f).Nonempty) :
    (s.map f).sup' hs g = s.sup' (map_nonempty.1 hs) (g ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] {s : Finset γ} {f : γ ↪ β} (g : β → α) (hs : (Finset.map f s).Nonempty)
Goal: (Finset.map f s).sup' hs g = s.sup' ⋯ (g ∘ ⇑f)
-- Finset.sup'_comp_eq_map
lemma sup'_comp_eq_map {s : Finset γ} {f : γ ↪ β} (g : β → α) (hs : s.Nonempty) :
    s.sup' hs (g ∘ f) = (s.map f).sup' (map_nonempty.2 hs) g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeSup α] {s : Finset γ} {f : γ ↪ β} (g : β → α) (hs : s.Nonempty)
Goal: s.sup' hs (g ∘ ⇑f) = (Finset.map f s).sup' ⋯ g
-- Finset.sup'_mono
theorem sup'_mono {s₁ s₂ : Finset β} (h : s₁ ⊆ s₂) (h₁ : s₁.Nonempty) :
    s₁.sup' h₁ f ≤ s₂.sup' (h₁.mono h) f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] (f : β → α) {s₁ s₂ : Finset β} (h : s₁ ⊆ s₂) (h₁ : s₁.Nonempty)
Goal: s₁.sup' h₁ f ≤ s₂.sup' ⋯ f
-- GCongr.finset_sup'_le
@[gcongr]
lemma _root_.GCongr.finset_sup'_le {s₁ s₂ : Finset β} (h : s₁ ⊆ s₂)
    {h₁ : s₁.Nonempty} {h₂ : s₂.Nonempty} : s₁.sup' h₁ f ≤ s₂.sup' h₂ f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeSup α] (f : β → α) {s₁ s₂ : Finset β} (h : s₁ ⊆ s₂) {h₁ : s₁.Nonempty} {h₂ : s₂.Nonempty}
Goal: s₁.sup' h₁ f ≤ s₂.sup' h₂ f
-- Finset.inf_of_mem
theorem inf_of_mem {s : Finset β} (f : β → α) {b : β} (h : b ∈ s) :
    ∃ a : α, s.inf ((↑) ∘ f : β → WithTop α) = ↑a 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (f : β → α) {b : β} (h : b ∈ s)
Goal: ∃ a, s.inf (WithTop.some ∘ f) = ↑a
-- Finset.coe_inf'
@[simp]
theorem coe_inf' : ((s.inf' H f : α) : WithTop α) = s.inf ((↑) ∘ f) 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (H : s.Nonempty) (f : β → α)
Goal: ↑(s.inf' H f) = s.inf (WithTop.some ∘ f)
-- Finset.inf'_cons
@[simp]
theorem inf'_cons {b : β} {hb : b ∉ s} :
    (cons b s hb).inf' (nonempty_cons hb) f = f b ⊓ s.inf' H f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (H : s.Nonempty) (f : β → α) {b : β} {hb : b ∉ s}
Goal: (Finset.cons b s hb).inf' ⋯ f = f b ⊓ s.inf' H f
-- Finset.inf'_insert
@[simp]
theorem inf'_insert [DecidableEq β] {b : β} :
    (insert b s).inf' (insert_nonempty _ _) f = f b ⊓ s.inf' H f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (H : s.Nonempty) (f : β → α) [DecidableEq β] {b : β}
Goal: (insert b s).inf' ⋯ f = f b ⊓ s.inf' H f
-- Finset.inf'_singleton
@[simp]
theorem inf'_singleton {b : β} : ({b} : Finset β).inf' (singleton_nonempty _) f = f b 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] (f : β → α) {b : β}
Goal: {b}.inf' ⋯ f = f b
-- Finset.le_inf'_iff
@[simp]
theorem le_inf'_iff {a : α} : a ≤ s.inf' H f ↔ ∀ b ∈ s, a ≤ f b 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (H : s.Nonempty) (f : β → α) {a : α}
Goal: a ≤ s.inf' H f ↔ ∀ b ∈ s, a ≤ f b
-- Finset.le_inf'
theorem le_inf' {a : α} (hs : ∀ b ∈ s, a ≤ f b) : a ≤ s.inf' H f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (H : s.Nonempty) (f : β → α) {a : α} (hs : ∀ b ∈ s, a ≤ f b)
Goal: a ≤ s.inf' H f
-- Finset.inf'_le
theorem inf'_le {b : β} (h : b ∈ s) : s.inf' ⟨b, h⟩ f ≤ f b 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (f : β → α) {b : β} (h : b ∈ s)
Goal: s.inf' ⋯ f ≤ f b
-- Finset.inf'_le_of_le
theorem inf'_le_of_le {a : α} {b : β} (hb : b ∈ s) (h : f b ≤ a) :
    s.inf' ⟨b, hb⟩ f ≤ a 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (f : β → α) {a : α} {b : β} (hb : b ∈ s) (h : f b ≤ a)
Goal: s.inf' ⋯ f ≤ a
-- Finset.inf'_const
@[simp]
theorem inf'_const (a : α) : (s.inf' H fun _ => a) = a 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] {s : Finset β} (H : s.Nonempty) (a : α)
Goal: (s.inf' H fun x => a) = a
-- Finset.inf'_union
theorem inf'_union [DecidableEq β] {s₁ s₂ : Finset β} (h₁ : s₁.Nonempty) (h₂ : s₂.Nonempty)
    (f : β → α) :
    (s₁ ∪ s₂).inf' (h₁.mono subset_union_left) f = s₁.inf' h₁ f ⊓ s₂.inf' h₂ f 
Arguments: {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [DecidableEq β] {s₁ s₂ : Finset β} (h₁ : s₁.Nonempty) (h₂ : s₂.Nonempty) (f : β → α)
Goal: (s₁ ∪ s₂).inf' ⋯ f = s₁.inf' h₁ f ⊓ s₂.inf' h₂ f
-- Finset.inf'_biUnion
theorem inf'_biUnion [DecidableEq β] {s : Finset γ} (Hs : s.Nonempty) {t : γ → Finset β}
    (Ht : ∀ b, (t b).Nonempty) :
    (s.biUnion t).inf' (Hs.biUnion fun b _ => Ht b) f = s.inf' Hs (fun b => (t b).inf' (Ht b) f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeInf α] (f : β → α) [DecidableEq β] {s : Finset γ} (Hs : s.Nonempty) {t : γ → Finset β} (Ht : ∀ (b : γ), (t b).Nonempty)
Goal: (s.biUnion t).inf' ⋯ f = s.inf' Hs fun b => (t b).inf' ⋯ f
-- Finset.inf'_comm
protected theorem inf'_comm {t : Finset γ} (hs : s.Nonempty) (ht : t.Nonempty) (f : β → γ → α) :
    (s.inf' hs fun b => t.inf' ht (f b)) = t.inf' ht fun c => s.inf' hs fun b => f b c 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SemilatticeInf α] {s : Finset β} {t : Finset γ} (hs : s.Nonempty) (ht : t.Nonempty) (f : β → γ → α)
Goal: (s.inf' hs fun b => t.inf' ht (f b)) = t.inf' ht fun c => s.inf' hs fun b => f b c