ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- iSup_split_single
theorem iSup_split_single (f : β → α) (i₀ : β) : ⨆ i, f i = f i₀ ⊔ ⨆ (i) (_ : i ≠ i₀), f i 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : β → α) (i₀ : β)
Goal: ⨆ i, f i = f i₀ ⊔ ⨆ i, ⨆ (_ : i ≠ i₀), f i
-- iInf_split_single
theorem iInf_split_single (f : β → α) (i₀ : β) : ⨅ i, f i = f i₀ ⊓ ⨅ (i) (_ : i ≠ i₀), f i 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : β → α) (i₀ : β)
Goal: ⨅ i, f i = f i₀ ⊓ ⨅ i, ⨅ (_ : i ≠ i₀), f i
-- iSup_le_iSup_of_subset
theorem iSup_le_iSup_of_subset {f : β → α} {s t : Set β} : s ⊆ t → ⨆ x ∈ s, f x ≤ ⨆ x ∈ t, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {s t : Set β}
Goal: s ⊆ t → ⨆ x ∈ s, f x ≤ ⨆ x ∈ t, f x
-- iInf_le_iInf_of_subset
theorem iInf_le_iInf_of_subset {f : β → α} {s t : Set β} : s ⊆ t → ⨅ x ∈ t, f x ≤ ⨅ x ∈ s, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {s t : Set β}
Goal: s ⊆ t → ⨅ x ∈ t, f x ≤ ⨅ x ∈ s, f x
-- iSup_insert
theorem iSup_insert {f : β → α} {s : Set β} {b : β} :
    ⨆ x ∈ insert b s, f x = f b ⊔ ⨆ x ∈ s, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {s : Set β} {b : β}
Goal: ⨆ x ∈ insert b s, f x = f b ⊔ ⨆ x ∈ s, f x
-- iInf_insert
theorem iInf_insert {f : β → α} {s : Set β} {b : β} :
    ⨅ x ∈ insert b s, f x = f b ⊓ ⨅ x ∈ s, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {s : Set β} {b : β}
Goal: ⨅ x ∈ insert b s, f x = f b ⊓ ⨅ x ∈ s, f x
-- iSup_singleton
theorem iSup_singleton {f : β → α} {b : β} : ⨆ x ∈ (singleton b : Set β), f x = f b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {b : β}
Goal: ⨆ x ∈ {b}, f x = f b
-- iInf_singleton
theorem iInf_singleton {f : β → α} {b : β} : ⨅ x ∈ (singleton b : Set β), f x = f b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {b : β}
Goal: ⨅ x ∈ {b}, f x = f b
-- iSup_pair
theorem iSup_pair {f : β → α} {a b : β} : ⨆ x ∈ ({a, b} : Set β), f x = f a ⊔ f b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {a b : β}
Goal: ⨆ x ∈ {a, b}, f x = f a ⊔ f b
-- iInf_pair
theorem iInf_pair {f : β → α} {a b : β} : ⨅ x ∈ ({a, b} : Set β), f x = f a ⊓ f b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α} {a b : β}
Goal: ⨅ x ∈ {a, b}, f x = f a ⊓ f b
-- iSup_image
theorem iSup_image {γ} {f : β → γ} {g : γ → α} {t : Set β} :
    ⨆ c ∈ f '' t, g c = ⨆ b ∈ t, g (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {γ : Type u_9} {f : β → γ} {g : γ → α} {t : Set β}
Goal: ⨆ c ∈ f '' t, g c = ⨆ b ∈ t, g (f b)
-- iInf_image
theorem iInf_image :
    ∀ {γ} {f : β → γ} {g : γ → α} {t : Set β}, ⨅ c ∈ f '' t, g c = ⨅ b ∈ t, g (f b) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {γ : Type u_9} {f : β → γ} {g : γ → α} {t : Set β}
Goal: ⨅ c ∈ f '' t, g c = ⨅ b ∈ t, g (f b)
-- iSup_extend_bot
theorem iSup_extend_bot {e : ι → β} (he : Injective e) (f : ι → α) :
    ⨆ j, extend e f ⊥ j = ⨆ i, f i 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] {e : ι → β} (he : Function.Injective e) (f : ι → α)
Goal: ⨆ j, Function.extend e f ⊥ j = ⨆ i, f i
-- iInf_extend_top
theorem iInf_extend_top {e : ι → β} (he : Injective e) (f : ι → α) :
    ⨅ j, extend e f ⊤ j = iInf f 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] {e : ι → β} (he : Function.Injective e) (f : ι → α)
Goal: ⨅ j, Function.extend e f ⊤ j = iInf f
-- iSup_of_empty'
theorem iSup_of_empty' {α ι} [SupSet α] [IsEmpty ι] (f : ι → α) : iSup f = sSup (∅ : Set α) 
Arguments: {α : Type u_9} {ι : Sort u_10} [SupSet α] [IsEmpty ι] (f : ι → α)
Goal: iSup f = sSup ∅
-- iInf_of_isEmpty
theorem iInf_of_isEmpty {α ι} [InfSet α] [IsEmpty ι] (f : ι → α) : iInf f = sInf (∅ : Set α) 
Arguments: {α : Type u_9} {ι : Sort u_10} [InfSet α] [IsEmpty ι] (f : ι → α)
Goal: iInf f = sInf ∅
-- iSup_of_empty
theorem iSup_of_empty [IsEmpty ι] (f : ι → α) : iSup f = ⊥ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [IsEmpty ι] (f : ι → α)
Goal: iSup f = ⊥
-- iInf_of_empty
theorem iInf_of_empty [IsEmpty ι] (f : ι → α) : iInf f = ⊤ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [IsEmpty ι] (f : ι → α)
Goal: iInf f = ⊤
-- iSup_bool_eq
theorem iSup_bool_eq {f : Bool → α} : ⨆ b : Bool, f b = f true ⊔ f false 
Arguments: {α : Type u_1} [CompleteLattice α] {f : Bool → α}
Goal: ⨆ b, f b = f true ⊔ f false
-- iInf_bool_eq
theorem iInf_bool_eq {f : Bool → α} : ⨅ b : Bool, f b = f true ⊓ f false 
Arguments: {α : Type u_1} [CompleteLattice α] {f : Bool → α}
Goal: ⨅ b, f b = f true ⊓ f false
-- sup_eq_iSup
theorem sup_eq_iSup (x y : α) : x ⊔ y = ⨆ b : Bool, cond b x y 
Arguments: {α : Type u_1} [CompleteLattice α] (x y : α)
Goal: x ⊔ y = ⨆ b, bif b then x else y
-- inf_eq_iInf
theorem inf_eq_iInf (x y : α) : x ⊓ y = ⨅ b : Bool, cond b x y 
Arguments: {α : Type u_1} [CompleteLattice α] (x y : α)
Goal: x ⊓ y = ⨅ b, bif b then x else y
-- isGLB_biInf
theorem isGLB_biInf {s : Set β} {f : β → α} : IsGLB (f '' s) (⨅ x ∈ s, f x) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {s : Set β} {f : β → α}
Goal: IsGLB (f '' s) (⨅ x ∈ s, f x)
-- isLUB_biSup
theorem isLUB_biSup {s : Set β} {f : β → α} : IsLUB (f '' s) (⨆ x ∈ s, f x) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {s : Set β} {f : β → α}
Goal: IsLUB (f '' s) (⨆ x ∈ s, f x)
-- iSup_sigma
theorem iSup_sigma {p : β → Type*} {f : Sigma p → α} : ⨆ x, f x = ⨆ (i) (j), f ⟨i, j⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {p : β → Type u_9} {f : Sigma p → α}
Goal: ⨆ x, f x = ⨆ i, ⨆ j, f ⟨i, j⟩
-- iInf_sigma
theorem iInf_sigma {p : β → Type*} {f : Sigma p → α} : ⨅ x, f x = ⨅ (i) (j), f ⟨i, j⟩ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {p : β → Type u_9} {f : Sigma p → α}
Goal: ⨅ x, f x = ⨅ i, ⨅ j, f ⟨i, j⟩
-- iSup_sigma'
lemma iSup_sigma' {κ : β → Type*} (f : ∀ i, κ i → α) :
    (⨆ i, ⨆ j, f i j) = ⨆ x : Σ i, κ i, f x.1 x.2 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {κ : β → Type u_9} (f : (i : β) → κ i → α)
Goal: ⨆ i, ⨆ j, f i j = ⨆ x, f x.fst x.snd
-- iInf_sigma'
lemma iInf_sigma' {κ : β → Type*} (f : ∀ i, κ i → α) :
    (⨅ i, ⨅ j, f i j) = ⨅ x : Σ i, κ i, f x.1 x.2 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {κ : β → Type u_9} (f : (i : β) → κ i → α)
Goal: ⨅ i, ⨅ j, f i j = ⨅ x, f x.fst x.snd
-- iSup_prod
theorem iSup_prod {f : β × γ → α} : ⨆ x, f x = ⨆ (i) (j), f (i, j) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β × γ → α}
Goal: ⨆ x, f x = ⨆ i, ⨆ j, f (i, j)
-- iInf_prod
theorem iInf_prod {f : β × γ → α} : ⨅ x, f x = ⨅ (i) (j), f (i, j) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β × γ → α}
Goal: ⨅ x, f x = ⨅ i, ⨅ j, f (i, j)
-- iSup_prod'
lemma iSup_prod' (f : β → γ → α) : (⨆ i, ⨆ j, f i j) = ⨆ x : β × γ, f x.1 x.2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] (f : β → γ → α)
Goal: ⨆ i, ⨆ j, f i j = ⨆ x, f x.1 x.2
-- iInf_prod'
lemma iInf_prod' (f : β → γ → α) : (⨅ i, ⨅ j, f i j) = ⨅ x : β × γ, f x.1 x.2 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] (f : β → γ → α)
Goal: ⨅ i, ⨅ j, f i j = ⨅ x, f x.1 x.2
-- biSup_prod
theorem biSup_prod {f : β × γ → α} {s : Set β} {t : Set γ} :
    ⨆ x ∈ s ×ˢ t, f x = ⨆ (a ∈ s) (b ∈ t), f (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β × γ → α} {s : Set β} {t : Set γ}
Goal: ⨆ x ∈ s ×ˢ t, f x = ⨆ a ∈ s, ⨆ b ∈ t, f (a, b)
-- biInf_prod
theorem biInf_prod {f : β × γ → α} {s : Set β} {t : Set γ} :
    ⨅ x ∈ s ×ˢ t, f x = ⨅ (a ∈ s) (b ∈ t), f (a, b) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β × γ → α} {s : Set β} {t : Set γ}
Goal: ⨅ x ∈ s ×ˢ t, f x = ⨅ a ∈ s, ⨅ b ∈ t, f (a, b)
-- iSup_sum
theorem iSup_sum {f : β ⊕ γ → α} : ⨆ x, f x = (⨆ i, f (Sum.inl i)) ⊔ ⨆ j, f (Sum.inr j) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β ⊕ γ → α}
Goal: ⨆ x, f x = (⨆ i, f (Sum.inl i)) ⊔ ⨆ j, f (Sum.inr j)
-- iInf_sum
theorem iInf_sum {f : β ⊕ γ → α} : ⨅ x, f x = (⨅ i, f (Sum.inl i)) ⊓ ⨅ j, f (Sum.inr j) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β ⊕ γ → α}
Goal: ⨅ x, f x = (⨅ i, f (Sum.inl i)) ⊓ ⨅ j, f (Sum.inr j)
-- iSup_option
theorem iSup_option (f : Option β → α) : ⨆ o, f o = f none ⊔ ⨆ b, f (Option.some b) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : Option β → α)
Goal: ⨆ o, f o = f none ⊔ ⨆ b, f (some b)
-- iInf_option
theorem iInf_option (f : Option β → α) : ⨅ o, f o = f none ⊓ ⨅ b, f (Option.some b) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : Option β → α)
Goal: ⨅ o, f o = f none ⊓ ⨅ b, f (some b)
-- iSup_option_elim
theorem iSup_option_elim (a : α) (f : β → α) : ⨆ o : Option β, o.elim a f = a ⊔ ⨆ b, f b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (a : α) (f : β → α)
Goal: ⨆ o, o.elim a f = a ⊔ ⨆ b, f b
-- iInf_option_elim
theorem iInf_option_elim (a : α) (f : β → α) : ⨅ o : Option β, o.elim a f = a ⊓ ⨅ b, f b 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (a : α) (f : β → α)
Goal: ⨅ o, o.elim a f = a ⊓ ⨅ b, f b
-- iSup_ne_bot_subtype
@[simp]
theorem iSup_ne_bot_subtype (f : ι → α) : ⨆ i : { i // f i ≠ ⊥ }, f i = ⨆ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → α)
Goal: ⨆ i, f ↑i = ⨆ i, f i
-- iInf_ne_top_subtype
theorem iInf_ne_top_subtype (f : ι → α) : ⨅ i : { i // f i ≠ ⊤ }, f i = ⨅ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → α)
Goal: ⨅ i, f ↑i = ⨅ i, f i
-- sSup_image2
theorem sSup_image2 {f : β → γ → α} {s : Set β} {t : Set γ} :
    sSup (image2 f s t) = ⨆ (a ∈ s) (b ∈ t), f a b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β → γ → α} {s : Set β} {t : Set γ}
Goal: sSup (Set.image2 f s t) = ⨆ a ∈ s, ⨆ b ∈ t, f a b
-- sInf_image2
theorem sInf_image2 {f : β → γ → α} {s : Set β} {t : Set γ} :
    sInf (image2 f s t) = ⨅ (a ∈ s) (b ∈ t), f a b 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_4} [CompleteLattice α] {f : β → γ → α} {s : Set β} {t : Set γ}
Goal: sInf (Set.image2 f s t) = ⨅ a ∈ s, ⨅ b ∈ t, f a b
-- iSup_ge_eq_iSup_nat_add
theorem iSup_ge_eq_iSup_nat_add (u : ℕ → α) (n : ℕ) : ⨆ i ≥ n, u i = ⨆ i, u (i + n) 
Arguments: {α : Type u_1} [CompleteLattice α] (u : ℕ → α) (n : ℕ)
Goal: ⨆ i, ⨆ (_ : i ≥ n), u i = ⨆ i, u (i + n)
-- iInf_ge_eq_iInf_nat_add
theorem iInf_ge_eq_iInf_nat_add (u : ℕ → α) (n : ℕ) : ⨅ i ≥ n, u i = ⨅ i, u (i + n) 
Arguments: {α : Type u_1} [CompleteLattice α] (u : ℕ → α) (n : ℕ)
Goal: ⨅ i, ⨅ (_ : i ≥ n), u i = ⨅ i, u (i + n)
-- Monotone.iSup_nat_add
theorem Monotone.iSup_nat_add {f : ℕ → α} (hf : Monotone f) (k : ℕ) : ⨆ n, f (n + k) = ⨆ n, f n 
Arguments: {α : Type u_1} [CompleteLattice α] {f : ℕ → α} (hf : Monotone f) (k : ℕ)
Goal: ⨆ n, f (n + k) = ⨆ n, f n
-- Antitone.iInf_nat_add
theorem Antitone.iInf_nat_add {f : ℕ → α} (hf : Antitone f) (k : ℕ) : ⨅ n, f (n + k) = ⨅ n, f n 
Arguments: {α : Type u_1} [CompleteLattice α] {f : ℕ → α} (hf : Antitone f) (k : ℕ)
Goal: ⨅ n, f (n + k) = ⨅ n, f n
-- iSup_iInf_ge_nat_add
theorem iSup_iInf_ge_nat_add (f : ℕ → α) (k : ℕ) :
    ⨆ n, ⨅ i ≥ n, f (i + k) = ⨆ n, ⨅ i ≥ n, f i 
Arguments: {α : Type u_1} [CompleteLattice α] (f : ℕ → α) (k : ℕ)
Goal: ⨆ n, ⨅ i, ⨅ (_ : i ≥ n), f (i + k) = ⨆ n, ⨅ i, ⨅ (_ : i ≥ n), f i
-- iInf_iSup_ge_nat_add
theorem iInf_iSup_ge_nat_add :
    ∀ (f : ℕ → α) (k : ℕ), ⨅ n, ⨆ i ≥ n, f (i + k) = ⨅ n, ⨆ i ≥ n, f i 
Arguments: {α : Type u_1} [CompleteLattice α] (f : ℕ → α) (k : ℕ)
Goal: ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), f (i + k) = ⨅ n, ⨆ i, ⨆ (_ : i ≥ n), f i
-- sup_iSup_nat_succ
theorem sup_iSup_nat_succ (u : ℕ → α) : (u 0 ⊔ ⨆ i, u (i + 1)) = ⨆ i, u i 
Arguments: {α : Type u_1} [CompleteLattice α] (u : ℕ → α)
Goal: u 0 ⊔ ⨆ i, u (i + 1) = ⨆ i, u i
-- inf_iInf_nat_succ
theorem inf_iInf_nat_succ (u : ℕ → α) : (u 0 ⊓ ⨅ i, u (i + 1)) = ⨅ i, u i 
Arguments: {α : Type u_1} [CompleteLattice α] (u : ℕ → α)
Goal: u 0 ⊓ ⨅ i, u (i + 1) = ⨅ i, u i
-- iInf_nat_gt_zero_eq
theorem iInf_nat_gt_zero_eq (f : ℕ → α) : ⨅ i > 0, f i = ⨅ i, f (i + 1) 
Arguments: {α : Type u_1} [CompleteLattice α] (f : ℕ → α)
Goal: ⨅ i, ⨅ (_ : i > 0), f i = ⨅ i, f (i + 1)
-- iSup_nat_gt_zero_eq
theorem iSup_nat_gt_zero_eq (f : ℕ → α) : ⨆ i > 0, f i = ⨆ i, f (i + 1) 
Arguments: {α : Type u_1} [CompleteLattice α] (f : ℕ → α)
Goal: ⨆ i, ⨆ (_ : i > 0), f i = ⨆ i, f (i + 1)
-- iSup_eq_top
theorem iSup_eq_top (f : ι → α) : iSup f = ⊤ ↔ ∀ b < ⊤, ∃ i, b < f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLinearOrder α] (f : ι → α)
Goal: iSup f = ⊤ ↔ ∀ b < ⊤, ∃ i, b < f i
-- iInf_eq_bot
theorem iInf_eq_bot (f : ι → α) : iInf f = ⊥ ↔ ∀ b > ⊥, ∃ i, f i < b 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLinearOrder α] (f : ι → α)
Goal: iInf f = ⊥ ↔ ∀ b > ⊥, ∃ i, f i < b
-- sSup_Prop_eq
@[simp]
theorem sSup_Prop_eq {s : Set Prop} : sSup s = ∃ p ∈ s, p 
Arguments: {s : Set Prop}
Goal: sSup s = ∃ p ∈ s, p
-- sInf_Prop_eq
@[simp]
theorem sInf_Prop_eq {s : Set Prop} : sInf s = ∀ p ∈ s, p 
Arguments: {s : Set Prop}
Goal: sInf s = ∀ p ∈ s, p
-- iSup_Prop_eq
@[simp]
theorem iSup_Prop_eq {p : ι → Prop} : ⨆ i, p i = ∃ i, p i 
Arguments: {ι : Sort u_5} {p : ι → Prop}
Goal: ⨆ i, p i = ∃ i, p i
-- iInf_Prop_eq
@[simp]
theorem iInf_Prop_eq {p : ι → Prop} : ⨅ i, p i = ∀ i, p i 
Arguments: {ι : Sort u_5} {p : ι → Prop}
Goal: ⨅ i, p i = ∀ (i : ι), p i
-- sSup_apply
@[simp]
theorem sSup_apply {α : Type*} {β : α → Type*} [∀ i, SupSet (β i)] {s : Set (∀ a, β a)} {a : α} :
    (sSup s) a = ⨆ f : s, (f : ∀ a, β a) a 
Arguments: {α : Type u_9} {β : α → Type u_10} [(i : α) → SupSet (β i)] {s : Set ((a : α) → β a)} {a : α}
Goal: sSup s a = ⨆ f, ↑f a
-- sInf_apply
@[simp]
theorem sInf_apply {α : Type*} {β : α → Type*} [∀ i, InfSet (β i)] {s : Set (∀ a, β a)} {a : α} :
    sInf s a = ⨅ f : s, (f : ∀ a, β a) a 
Arguments: {α : Type u_9} {β : α → Type u_10} [(i : α) → InfSet (β i)] {s : Set ((a : α) → β a)} {a : α}
Goal: sInf s a = ⨅ f, ↑f a
-- iSup_apply
@[simp]
theorem iSup_apply {α : Type*} {β : α → Type*} {ι : Sort*} [∀ i, SupSet (β i)] {f : ι → ∀ a, β a}
    {a : α} : (⨆ i, f i) a = ⨆ i, f i a 
Arguments: {α : Type u_9} {β : α → Type u_10} {ι : Sort u_11} [(i : α) → SupSet (β i)] {f : ι → (a : α) → β a} {a : α}
Goal: (⨆ i, f i) a = ⨆ i, f i a
-- iInf_apply
@[simp]
theorem iInf_apply {α : Type*} {β : α → Type*} {ι : Sort*} [∀ i, InfSet (β i)] {f : ι → ∀ a, β a}
    {a : α} : (⨅ i, f i) a = ⨅ i, f i a 
Arguments: {α : Type u_9} {β : α → Type u_10} {ι : Sort u_11} [(i : α) → InfSet (β i)] {f : ι → (a : α) → β a} {a : α}
Goal: (⨅ i, f i) a = ⨅ i, f i a
-- unary_relation_sSup_iff
theorem unary_relation_sSup_iff {α : Type*} (s : Set (α → Prop)) {a : α} :
    sSup s a ↔ ∃ r : α → Prop, r ∈ s ∧ r a 
Arguments: {α : Type u_9} (s : Set (α → Prop)) {a : α}
Goal: sSup s a ↔ ∃ r ∈ s, r a
-- unary_relation_sInf_iff
theorem unary_relation_sInf_iff {α : Type*} (s : Set (α → Prop)) {a : α} :
    sInf s a ↔ ∀ r : α → Prop, r ∈ s → r a 
Arguments: {α : Type u_9} (s : Set (α → Prop)) {a : α}
Goal: sInf s a ↔ ∀ r ∈ s, r a
-- binary_relation_sSup_iff
theorem binary_relation_sSup_iff {α β : Type*} (s : Set (α → β → Prop)) {a : α} {b : β} :
    sSup s a b ↔ ∃ r : α → β → Prop, r ∈ s ∧ r a b 
Arguments: {α : Type u_9} {β : Type u_10} (s : Set (α → β → Prop)) {a : α} {b : β}
Goal: sSup s a b ↔ ∃ r ∈ s, r a b
-- binary_relation_sInf_iff
theorem binary_relation_sInf_iff {α β : Type*} (s : Set (α → β → Prop)) {a : α} {b : β} :
    sInf s a b ↔ ∀ r : α → β → Prop, r ∈ s → r a b 
Arguments: {α : Type u_9} {β : Type u_10} (s : Set (α → β → Prop)) {a : α} {b : β}
Goal: sInf s a b ↔ ∀ r ∈ s, r a b
-- Monotone.sSup
protected lemma Monotone.sSup (hs : ∀ f ∈ s, Monotone f) : Monotone (sSup s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] {s : Set (α → β)} (hs : ∀ f ∈ s, Monotone f)
Goal: Monotone (sSup s)
-- Monotone.sInf
protected lemma Monotone.sInf (hs : ∀ f ∈ s, Monotone f) : Monotone (sInf s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] {s : Set (α → β)} (hs : ∀ f ∈ s, Monotone f)
Goal: Monotone (sInf s)
-- Antitone.sSup
protected lemma Antitone.sSup (hs : ∀ f ∈ s, Antitone f) : Antitone (sSup s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] {s : Set (α → β)} (hs : ∀ f ∈ s, Antitone f)
Goal: Antitone (sSup s)
-- Antitone.sInf
protected lemma Antitone.sInf (hs : ∀ f ∈ s, Antitone f) : Antitone (sInf s) 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [CompleteLattice β] {s : Set (α → β)} (hs : ∀ f ∈ s, Antitone f)
Goal: Antitone (sInf s)
-- Monotone.iSup
protected lemma Monotone.iSup (hf : ∀ i, Monotone (f i)) : Monotone (⨆ i, f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_9} [Preorder α] [CompleteLattice β] {f : ι → α → β} (hf : ∀ (i : ι), Monotone (f i))
Goal: Monotone (⨆ i, f i)
-- Monotone.iInf
protected lemma Monotone.iInf (hf : ∀ i, Monotone (f i)) : Monotone (⨅ i, f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_9} [Preorder α] [CompleteLattice β] {f : ι → α → β} (hf : ∀ (i : ι), Monotone (f i))
Goal: Monotone (⨅ i, f i)
-- Antitone.iSup
protected lemma Antitone.iSup (hf : ∀ i, Antitone (f i)) : Antitone (⨆ i, f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_9} [Preorder α] [CompleteLattice β] {f : ι → α → β} (hf : ∀ (i : ι), Antitone (f i))
Goal: Antitone (⨆ i, f i)
-- Antitone.iInf
protected lemma Antitone.iInf (hf : ∀ i, Antitone (f i)) : Antitone (⨅ i, f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_9} [Preorder α] [CompleteLattice β] {f : ι → α → β} (hf : ∀ (i : ι), Antitone (f i))
Goal: Antitone (⨅ i, f i)
-- Prod.fst_sInf
theorem fst_sInf [InfSet α] [InfSet β] (s : Set (α × β)) : (sInf s).fst = sInf (Prod.fst '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [InfSet α] [InfSet β] (s : Set (α × β))
Goal: (sInf s).1 = sInf (Prod.fst '' s)
-- Prod.snd_sInf
theorem snd_sInf [InfSet α] [InfSet β] (s : Set (α × β)) : (sInf s).snd = sInf (Prod.snd '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [InfSet α] [InfSet β] (s : Set (α × β))
Goal: (sInf s).2 = sInf (Prod.snd '' s)
-- Prod.swap_sInf
theorem swap_sInf [InfSet α] [InfSet β] (s : Set (α × β)) : (sInf s).swap = sInf (Prod.swap '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [InfSet α] [InfSet β] (s : Set (α × β))
Goal: (sInf s).swap = sInf (Prod.swap '' s)
-- Prod.fst_sSup
theorem fst_sSup [SupSet α] [SupSet β] (s : Set (α × β)) : (sSup s).fst = sSup (Prod.fst '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [SupSet α] [SupSet β] (s : Set (α × β))
Goal: (sSup s).1 = sSup (Prod.fst '' s)
-- Prod.snd_sSup
theorem snd_sSup [SupSet α] [SupSet β] (s : Set (α × β)) : (sSup s).snd = sSup (Prod.snd '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [SupSet α] [SupSet β] (s : Set (α × β))
Goal: (sSup s).2 = sSup (Prod.snd '' s)
-- Prod.swap_sSup
theorem swap_sSup [SupSet α] [SupSet β] (s : Set (α × β)) : (sSup s).swap = sSup (Prod.swap '' s) 
Arguments: {α : Type u_1} {β : Type u_2} [SupSet α] [SupSet β] (s : Set (α × β))
Goal: (sSup s).swap = sSup (Prod.swap '' s)
-- Prod.fst_iInf
theorem fst_iInf [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).fst = ⨅ i, (f i).fst 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [InfSet α] [InfSet β] (f : ι → α × β)
Goal: (iInf f).1 = ⨅ i, (f i).1
-- Prod.snd_iInf
theorem snd_iInf [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).snd = ⨅ i, (f i).snd 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [InfSet α] [InfSet β] (f : ι → α × β)
Goal: (iInf f).2 = ⨅ i, (f i).2
-- Prod.swap_iInf
theorem swap_iInf [InfSet α] [InfSet β] (f : ι → α × β) : (iInf f).swap = ⨅ i, (f i).swap 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [InfSet α] [InfSet β] (f : ι → α × β)
Goal: (iInf f).swap = ⨅ i, (f i).swap
-- Prod.iInf_mk
theorem iInf_mk [InfSet α] [InfSet β] (f : ι → α) (g : ι → β) :
    ⨅ i, (f i, g i) = (⨅ i, f i, ⨅ i, g i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [InfSet α] [InfSet β] (f : ι → α) (g : ι → β)
Goal: ⨅ i, (f i, g i) = (⨅ i, f i, ⨅ i, g i)
-- Prod.fst_iSup
theorem fst_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).fst = ⨆ i, (f i).fst 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [SupSet α] [SupSet β] (f : ι → α × β)
Goal: (iSup f).1 = ⨆ i, (f i).1
-- Prod.snd_iSup
theorem snd_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).snd = ⨆ i, (f i).snd 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [SupSet α] [SupSet β] (f : ι → α × β)
Goal: (iSup f).2 = ⨆ i, (f i).2
-- Prod.swap_iSup
theorem swap_iSup [SupSet α] [SupSet β] (f : ι → α × β) : (iSup f).swap = ⨆ i, (f i).swap 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [SupSet α] [SupSet β] (f : ι → α × β)
Goal: (iSup f).swap = ⨆ i, (f i).swap
-- Prod.iSup_mk
theorem iSup_mk [SupSet α] [SupSet β] (f : ι → α) (g : ι → β) :
    ⨆ i, (f i, g i) = (⨆ i, f i, ⨆ i, g i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [SupSet α] [SupSet β] (f : ι → α) (g : ι → β)
Goal: ⨆ i, (f i, g i) = (⨆ i, f i, ⨆ i, g i)
-- sInf_prod
lemma sInf_prod [InfSet α] [InfSet β] {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty) :
    sInf (s ×ˢ t) = (sInf s, sInf t) 
Arguments: {α : Type u_1} {β : Type u_2} [InfSet α] [InfSet β] {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty)
Goal: sInf (s ×ˢ t) = (sInf s, sInf t)
-- sSup_prod
lemma sSup_prod [SupSet α] [SupSet β] {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty) :
    sSup (s ×ˢ t) = (sSup s, sSup t) 
Arguments: {α : Type u_1} {β : Type u_2} [SupSet α] [SupSet β] {s : Set α} {t : Set β} (hs : s.Nonempty) (ht : t.Nonempty)
Goal: sSup (s ×ˢ t) = (sSup s, sSup t)
-- sup_sInf_le_iInf_sup
theorem sup_sInf_le_iInf_sup : a ⊔ sInf s ≤ ⨅ b ∈ s, a ⊔ b 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} {s : Set α}
Goal: a ⊔ sInf s ≤ ⨅ b ∈ s, a ⊔ b
-- iSup_inf_le_inf_sSup
theorem iSup_inf_le_inf_sSup : ⨆ b ∈ s, a ⊓ b ≤ a ⊓ sSup s 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} {s : Set α}
Goal: ⨆ b ∈ s, a ⊓ b ≤ a ⊓ sSup s
-- sInf_sup_le_iInf_sup
theorem sInf_sup_le_iInf_sup : sInf s ⊔ a ≤ ⨅ b ∈ s, b ⊔ a 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} {s : Set α}
Goal: sInf s ⊔ a ≤ ⨅ b ∈ s, b ⊔ a
-- iSup_inf_le_sSup_inf
theorem iSup_inf_le_sSup_inf : ⨆ b ∈ s, b ⊓ a ≤ sSup s ⊓ a 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} {s : Set α}
Goal: ⨆ b ∈ s, b ⊓ a ≤ sSup s ⊓ a
-- le_iSup_inf_iSup
theorem le_iSup_inf_iSup (f g : ι → α) : ⨆ i, f i ⊓ g i ≤ (⨆ i, f i) ⊓ ⨆ i, g i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f g : ι → α)
Goal: ⨆ i, f i ⊓ g i ≤ (⨆ i, f i) ⊓ ⨆ i, g i
-- iInf_sup_iInf_le
theorem iInf_sup_iInf_le (f g : ι → α) : (⨅ i, f i) ⊔ ⨅ i, g i ≤ ⨅ i, f i ⊔ g i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f g : ι → α)
Goal: (⨅ i, f i) ⊔ ⨅ i, g i ≤ ⨅ i, f i ⊔ g i
-- disjoint_sSup_left
theorem disjoint_sSup_left {a : Set α} {b : α} (d : Disjoint (sSup a) b) {i} (hi : i ∈ a) :
    Disjoint i b 
Arguments: {α : Type u_1} [CompleteLattice α] {a : Set α} {b : α} (d : Disjoint (sSup a) b) {i : α} (hi : i ∈ a)
Goal: Disjoint i b
-- disjoint_sSup_right
theorem disjoint_sSup_right {a : Set α} {b : α} (d : Disjoint b (sSup a)) {i} (hi : i ∈ a) :
    Disjoint b i 
Arguments: {α : Type u_1} [CompleteLattice α] {a : Set α} {b : α} (d : Disjoint b (sSup a)) {i : α} (hi : i ∈ a)
Goal: Disjoint b i
-- ULift.down_sSup
theorem down_sSup [SupSet α] (s : Set (ULift.{v} α)) : (sSup s).down = sSup (ULift.up ⁻¹' s) 
Arguments: {α : Type u_1} [SupSet α] (s : Set (ULift.{v, u_1} α))
Goal: (sSup s).down = sSup (ULift.up ⁻¹' s)
-- ULift.up_sSup
theorem up_sSup [SupSet α] (s : Set α) : up (sSup s) = sSup (ULift.down ⁻¹' s) 
Arguments: {α : Type u_1} [SupSet α] (s : Set α)
Goal: { down := sSup s } = sSup (ULift.down ⁻¹' s)
-- ULift.down_sInf
theorem down_sInf [InfSet α] (s : Set (ULift.{v} α)) : (sInf s).down = sInf (ULift.up ⁻¹' s) 
Arguments: {α : Type u_1} [InfSet α] (s : Set (ULift.{v, u_1} α))
Goal: (sInf s).down = sInf (ULift.up ⁻¹' s)
-- ULift.up_sInf
theorem up_sInf [InfSet α] (s : Set α) : up (sInf s) = sInf (ULift.down ⁻¹' s) 
Arguments: {α : Type u_1} [InfSet α] (s : Set α)
Goal: { down := sInf s } = sInf (ULift.down ⁻¹' s)