ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OrderIso.map_iSup
theorem OrderIso.map_iSup [CompleteLattice β] (f : α ≃o β) (x : ι → α) :
    f (⨆ i, x i) = ⨆ i, f (x i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (x : ι → α)
Goal: f (⨆ i, x i) = ⨆ i, f (x i)
-- OrderIso.map_iInf
theorem OrderIso.map_iInf [CompleteLattice β] (f : α ≃o β) (x : ι → α) :
    f (⨅ i, x i) = ⨅ i, f (x i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (x : ι → α)
Goal: f (⨅ i, x i) = ⨅ i, f (x i)
-- OrderIso.map_sSup
theorem OrderIso.map_sSup [CompleteLattice β] (f : α ≃o β) (s : Set α) :
    f (sSup s) = ⨆ a ∈ s, f a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (s : Set α)
Goal: f (sSup s) = ⨆ a ∈ s, f a
-- OrderIso.map_sInf
theorem OrderIso.map_sInf [CompleteLattice β] (f : α ≃o β) (s : Set α) :
    f (sInf s) = ⨅ a ∈ s, f a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (s : Set α)
Goal: f (sInf s) = ⨅ a ∈ s, f a
-- iSup_comp_le
theorem iSup_comp_le {ι' : Sort*} (f : ι' → α) (g : ι → ι') : ⨆ x, f (g x) ≤ ⨆ y, f y 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {ι' : Sort u_9} (f : ι' → α) (g : ι → ι')
Goal: ⨆ x, f (g x) ≤ ⨆ y, f y
-- le_iInf_comp
theorem le_iInf_comp {ι' : Sort*} (f : ι' → α) (g : ι → ι') : ⨅ y, f y ≤ ⨅ x, f (g x) 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {ι' : Sort u_9} (f : ι' → α) (g : ι → ι')
Goal: ⨅ y, f y ≤ ⨅ x, f (g x)
-- Monotone.iSup_comp_eq
theorem Monotone.iSup_comp_eq [Preorder β] {f : β → α} (hf : Monotone f) {s : ι → β}
    (hs : ∀ x, ∃ i, x ≤ s i) : ⨆ x, f (s x) = ⨆ y, f y 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] [Preorder β] {f : β → α} (hf : Monotone f) {s : ι → β} (hs : ∀ (x : β), ∃ i, x ≤ s i)
Goal: ⨆ x, f (s x) = ⨆ y, f y
-- Monotone.iInf_comp_eq
theorem Monotone.iInf_comp_eq [Preorder β] {f : β → α} (hf : Monotone f) {s : ι → β}
    (hs : ∀ x, ∃ i, s i ≤ x) : ⨅ x, f (s x) = ⨅ y, f y 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] [Preorder β] {f : β → α} (hf : Monotone f) {s : ι → β} (hs : ∀ (x : β), ∃ i, s i ≤ x)
Goal: ⨅ x, f (s x) = ⨅ y, f y
-- Antitone.map_iSup_le
theorem Antitone.map_iSup_le [CompleteLattice β] {f : α → β} (hf : Antitone f) :
    f (iSup s) ≤ ⨅ i, f (s i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] {s : ι → α} [CompleteLattice β] {f : α → β} (hf : Antitone f)
Goal: f (iSup s) ≤ ⨅ i, f (s i)
-- Monotone.map_iInf_le
theorem Monotone.map_iInf_le [CompleteLattice β] {f : α → β} (hf : Monotone f) :
    f (iInf s) ≤ ⨅ i, f (s i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] {s : ι → α} [CompleteLattice β] {f : α → β} (hf : Monotone f)
Goal: f (iInf s) ≤ ⨅ i, f (s i)
-- Antitone.map_iSup₂_le
theorem Antitone.map_iSup₂_le [CompleteLattice β] {f : α → β} (hf : Antitone f) (s : ∀ i, κ i → α) :
    f (⨆ (i) (j), s i j) ≤ ⨅ (i) (j), f (s i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] [CompleteLattice β] {f : α → β} (hf : Antitone f) (s : (i : ι) → κ i → α)
Goal: f (⨆ i, ⨆ j, s i j) ≤ ⨅ i, ⨅ j, f (s i j)
-- Monotone.map_iInf₂_le
theorem Monotone.map_iInf₂_le [CompleteLattice β] {f : α → β} (hf : Monotone f) (s : ∀ i, κ i → α) :
    f (⨅ (i) (j), s i j) ≤ ⨅ (i) (j), f (s i j) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] [CompleteLattice β] {f : α → β} (hf : Monotone f) (s : (i : ι) → κ i → α)
Goal: f (⨅ i, ⨅ j, s i j) ≤ ⨅ i, ⨅ j, f (s i j)
-- Antitone.map_sSup_le
theorem Antitone.map_sSup_le [CompleteLattice β] {s : Set α} {f : α → β} (hf : Antitone f) :
    f (sSup s) ≤ ⨅ a ∈ s, f a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] {s : Set α} {f : α → β} (hf : Antitone f)
Goal: f (sSup s) ≤ ⨅ a ∈ s, f a
-- Monotone.map_sInf_le
theorem Monotone.map_sInf_le [CompleteLattice β] {s : Set α} {f : α → β} (hf : Monotone f) :
    f (sInf s) ≤ ⨅ a ∈ s, f a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] {s : Set α} {f : α → β} (hf : Monotone f)
Goal: f (sInf s) ≤ ⨅ a ∈ s, f a
-- iSup_const_le
theorem iSup_const_le : ⨆ _ : ι, a ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {a : α}
Goal: ⨆ x, a ≤ a
-- le_iInf_const
theorem le_iInf_const : a ≤ ⨅ _ : ι, a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {a : α}
Goal: a ≤ ⨅ x, a
-- iSup_const
theorem iSup_const [Nonempty ι] : ⨆ _ : ι, a = a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {a : α} [Nonempty ι]
Goal: ⨆ x, a = a
-- iInf_const
theorem iInf_const [Nonempty ι] : ⨅ _ : ι, a = a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {a : α} [Nonempty ι]
Goal: ⨅ x, a = a
-- iSup_unique
lemma iSup_unique [Unique ι] (f : ι → α) : ⨆ i, f i = f default 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [Unique ι] (f : ι → α)
Goal: ⨆ i, f i = f default
-- iInf_unique
lemma iInf_unique [Unique ι] (f : ι → α) : ⨅ i, f i = f default 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [Unique ι] (f : ι → α)
Goal: ⨅ i, f i = f default
-- iSup_bot
@[simp]
theorem iSup_bot : (⨆ _ : ι, ⊥ : α) = ⊥ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α]
Goal: ⨆ x, ⊥ = ⊥
-- iInf_top
@[simp]
theorem iInf_top : (⨅ _ : ι, ⊤ : α) = ⊤ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α]
Goal: ⨅ x, ⊤ = ⊤
-- iSup_eq_bot
@[simp]
theorem iSup_eq_bot : iSup s = ⊥ ↔ ∀ i, s i = ⊥ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {s : ι → α}
Goal: iSup s = ⊥ ↔ ∀ (i : ι), s i = ⊥
-- iInf_eq_top
@[simp]
theorem iInf_eq_top : iInf s = ⊤ ↔ ∀ i, s i = ⊤ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {s : ι → α}
Goal: iInf s = ⊤ ↔ ∀ (i : ι), s i = ⊤
-- iSup₂_eq_bot
theorem iSup₂_eq_bot {f : ∀ i, κ i → α} : ⨆ (i) (j), f i j = ⊥ ↔ ∀ i j, f i j = ⊥ 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {f : (i : ι) → κ i → α}
Goal: ⨆ i, ⨆ j, f i j = ⊥ ↔ ∀ (i : ι) (j : κ i), f i j = ⊥
-- iInf₂_eq_top
theorem iInf₂_eq_top {f : ∀ i, κ i → α} : ⨅ (i) (j), f i j = ⊤ ↔ ∀ i j, f i j = ⊤ 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {f : (i : ι) → κ i → α}
Goal: ⨅ i, ⨅ j, f i j = ⊤ ↔ ∀ (i : ι) (j : κ i), f i j = ⊤
-- iSup_pos
@[simp]
theorem iSup_pos {p : Prop} {f : p → α} (hp : p) : ⨆ h : p, f h = f hp 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} {f : p → α} (hp : p)
Goal: ⨆ (h : p), f h = f hp
-- iInf_pos
@[simp]
theorem iInf_pos {p : Prop} {f : p → α} (hp : p) : ⨅ h : p, f h = f hp 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} {f : p → α} (hp : p)
Goal: ⨅ (h : p), f h = f hp
-- iSup_neg
@[simp]
theorem iSup_neg {p : Prop} {f : p → α} (hp : ¬p) : ⨆ h : p, f h = ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} {f : p → α} (hp : ¬p)
Goal: ⨆ (h : p), f h = ⊥
-- iInf_neg
@[simp]
theorem iInf_neg {p : Prop} {f : p → α} (hp : ¬p) : ⨅ h : p, f h = ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} {f : p → α} (hp : ¬p)
Goal: ⨅ (h : p), f h = ⊤
-- iSup_eq_of_forall_le_of_forall_lt_exists_gt
theorem iSup_eq_of_forall_le_of_forall_lt_exists_gt {f : ι → α} (h₁ : ∀ i, f i ≤ b)
    (h₂ : ∀ w, w < b → ∃ i, w < f i) : ⨆ i : ι, f i = b 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {b : α} {f : ι → α} (h₁ : ∀ (i : ι), f i ≤ b) (h₂ : ∀ w < b, ∃ i, w < f i)
Goal: ⨆ i, f i = b
-- iInf_eq_of_forall_ge_of_forall_gt_exists_lt
theorem iInf_eq_of_forall_ge_of_forall_gt_exists_lt :
    (∀ i, b ≤ f i) → (∀ w, b < w → ∃ i, f i < w) → ⨅ i, f i = b 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {b : α}
Goal: (∀ (i : ι), b ≤ f i) → (∀ (w : α), b < w → ∃ i, f i < w) → ⨅ i, f i = b
-- iSup_eq_dif
theorem iSup_eq_dif {p : Prop} [Decidable p] (a : p → α) :
    ⨆ h : p, a h = if h : p then a h else ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} [Decidable p] (a : p → α)
Goal: ⨆ (h : p), a h = if h : p then a h else ⊥
-- iSup_eq_if
theorem iSup_eq_if {p : Prop} [Decidable p] (a : α) : ⨆ _ : p, a = if p then a else ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} [Decidable p] (a : α)
Goal: ⨆ (_ : p), a = if p then a else ⊥
-- iInf_eq_dif
theorem iInf_eq_dif {p : Prop} [Decidable p] (a : p → α) :
    ⨅ h : p, a h = if h : p then a h else ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} [Decidable p] (a : p → α)
Goal: ⨅ (h : p), a h = if h : p then a h else ⊤
-- iInf_eq_if
theorem iInf_eq_if {p : Prop} [Decidable p] (a : α) : ⨅ _ : p, a = if p then a else ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α] {p : Prop} [Decidable p] (a : α)
Goal: ⨅ (_ : p), a = if p then a else ⊤
-- iSup_comm
theorem iSup_comm {f : ι → ι' → α} : ⨆ (i) (j), f i j = ⨆ (j) (i), f i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [CompleteLattice α] {f : ι → ι' → α}
Goal: ⨆ i, ⨆ j, f i j = ⨆ j, ⨆ i, f i j
-- iInf_comm
theorem iInf_comm {f : ι → ι' → α} : ⨅ (i) (j), f i j = ⨅ (j) (i), f i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [CompleteLattice α] {f : ι → ι' → α}
Goal: ⨅ i, ⨅ j, f i j = ⨅ j, ⨅ i, f i j
-- iSup₂_comm
theorem iSup₂_comm {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}
    (f : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → α) :
    ⨆ (i₁) (j₁) (i₂) (j₂), f i₁ j₁ i₂ j₂ = ⨆ (i₂) (j₂) (i₁) (j₁), f i₁ j₁ i₂ j₂ 
Arguments: {α : Type u_1} [CompleteLattice α] {ι₁ : Sort u_9} {ι₂ : Sort u_10} {κ₁ : ι₁ → Sort u_11} {κ₂ : ι₂ → Sort u_12} (f : (i₁ : ι₁) → κ₁ i₁ → (i₂ : ι₂) → κ₂ i₂ → α)
Goal: ⨆ i₁, ⨆ j₁, ⨆ i₂, ⨆ j₂, f i₁ j₁ i₂ j₂ = ⨆ i₂, ⨆ j₂, ⨆ i₁, ⨆ j₁, f i₁ j₁ i₂ j₂
-- iInf₂_comm
theorem iInf₂_comm {ι₁ ι₂ : Sort*} {κ₁ : ι₁ → Sort*} {κ₂ : ι₂ → Sort*}
    (f : ∀ i₁, κ₁ i₁ → ∀ i₂, κ₂ i₂ → α) :
    ⨅ (i₁) (j₁) (i₂) (j₂), f i₁ j₁ i₂ j₂ = ⨅ (i₂) (j₂) (i₁) (j₁), f i₁ j₁ i₂ j₂ 
Arguments: {α : Type u_1} [CompleteLattice α] {ι₁ : Sort u_9} {ι₂ : Sort u_10} {κ₁ : ι₁ → Sort u_11} {κ₂ : ι₂ → Sort u_12} (f : (i₁ : ι₁) → κ₁ i₁ → (i₂ : ι₂) → κ₂ i₂ → α)
Goal: ⨅ i₁, ⨅ j₁, ⨅ i₂, ⨅ j₂, f i₁ j₁ i₂ j₂ = ⨅ i₂, ⨅ j₂, ⨅ i₁, ⨅ j₁, f i₁ j₁ i₂ j₂
-- iSup_iSup_eq_left
@[simp]
theorem iSup_iSup_eq_left {b : β} {f : ∀ x : β, x = b → α} : ⨆ x, ⨆ h : x = b, f x h = f b rfl 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {b : β} {f : (x : β) → x = b → α}
Goal: ⨆ x, ⨆ (h : x = b), f x h = f b ⋯
-- iInf_iInf_eq_left
@[simp]
theorem iInf_iInf_eq_left {b : β} {f : ∀ x : β, x = b → α} : ⨅ x, ⨅ h : x = b, f x h = f b rfl 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {b : β} {f : (x : β) → x = b → α}
Goal: ⨅ x, ⨅ (h : x = b), f x h = f b ⋯
-- iSup_iSup_eq_right
@[simp]
theorem iSup_iSup_eq_right {b : β} {f : ∀ x : β, b = x → α} : ⨆ x, ⨆ h : b = x, f x h = f b rfl 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {b : β} {f : (x : β) → b = x → α}
Goal: ⨆ x, ⨆ (h : b = x), f x h = f b ⋯
-- iInf_iInf_eq_right
@[simp]
theorem iInf_iInf_eq_right {b : β} {f : ∀ x : β, b = x → α} : ⨅ x, ⨅ h : b = x, f x h = f b rfl 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {b : β} {f : (x : β) → b = x → α}
Goal: ⨅ x, ⨅ (h : b = x), f x h = f b ⋯
-- iSup_subtype
theorem iSup_subtype {p : ι → Prop} {f : Subtype p → α} : iSup f = ⨆ (i) (h : p i), f ⟨i, h⟩ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : Subtype p → α}
Goal: iSup f = ⨆ i, ⨆ (h : p i), f ⟨i, h⟩
-- iInf_subtype
theorem iInf_subtype : ∀ {p : ι → Prop} {f : Subtype p → α}, iInf f = ⨅ (i) (h : p i), f ⟨i, h⟩ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : Subtype p → α}
Goal: iInf f = ⨅ i, ⨅ (h : p i), f ⟨i, h⟩
-- iSup_subtype'
theorem iSup_subtype' {p : ι → Prop} {f : ∀ i, p i → α} :
    ⨆ (i) (h), f i h = ⨆ x : Subtype p, f x x.property 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α}
Goal: ⨆ i, ⨆ (h : p i), f i h = ⨆ x, f ↑x ⋯
-- iInf_subtype'
theorem iInf_subtype' {p : ι → Prop} {f : ∀ i, p i → α} :
    ⨅ (i) (h : p i), f i h = ⨅ x : Subtype p, f x x.property 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α}
Goal: ⨅ i, ⨅ (h : p i), f i h = ⨅ x, f ↑x ⋯
-- iSup_subtype''
theorem iSup_subtype'' {ι} (s : Set ι) (f : ι → α) : ⨆ i : s, f i = ⨆ (t : ι) (_ : t ∈ s), f t 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} (s : Set ι) (f : ι → α)
Goal: ⨆ i, f ↑i = ⨆ t ∈ s, f t
-- iInf_subtype''
theorem iInf_subtype'' {ι} (s : Set ι) (f : ι → α) : ⨅ i : s, f i = ⨅ (t : ι) (_ : t ∈ s), f t 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} (s : Set ι) (f : ι → α)
Goal: ⨅ i, f ↑i = ⨅ t ∈ s, f t
-- biSup_const
theorem biSup_const {ι : Sort _} {a : α} {s : Set ι} (hs : s.Nonempty) : ⨆ i ∈ s, a = a 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} {a : α} {s : Set ι} (hs : s.Nonempty)
Goal: ⨆ i ∈ s, a = a
-- biInf_const
theorem biInf_const {ι : Sort _} {a : α} {s : Set ι} (hs : s.Nonempty) : ⨅ i ∈ s, a = a 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} {a : α} {s : Set ι} (hs : s.Nonempty)
Goal: ⨅ i ∈ s, a = a
-- iSup_sup_eq
theorem iSup_sup_eq : ⨆ x, f x ⊔ g x = (⨆ x, f x) ⊔ ⨆ x, g x 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f g : ι → α}
Goal: ⨆ x, f x ⊔ g x = (⨆ x, f x) ⊔ ⨆ x, g x
-- iInf_inf_eq
theorem iInf_inf_eq : ⨅ x, f x ⊓ g x = (⨅ x, f x) ⊓ ⨅ x, g x 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f g : ι → α}
Goal: ⨅ x, f x ⊓ g x = (⨅ x, f x) ⊓ ⨅ x, g x
-- Equiv.biSup_comp
lemma Equiv.biSup_comp {ι ι' : Type*} {g : ι' → α} (e : ι ≃ ι') (s : Set ι') :
    ⨆ i ∈ e.symm '' s, g (e i) = ⨆ i ∈ s, g i 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} {ι' : Type u_10} {g : ι' → α} (e : ι ≃ ι') (s : Set ι')
Goal: ⨆ i ∈ ⇑e.symm '' s, g (e i) = ⨆ i ∈ s, g i
-- Equiv.biInf_comp
lemma Equiv.biInf_comp {ι ι' : Type*} {g : ι' → α} (e : ι ≃ ι') (s : Set ι') :
    ⨅ i ∈ e.symm '' s, g (e i) = ⨅ i ∈ s, g i 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} {ι' : Type u_10} {g : ι' → α} (e : ι ≃ ι') (s : Set ι')
Goal: ⨅ i ∈ ⇑e.symm '' s, g (e i) = ⨅ i ∈ s, g i
-- biInf_le
lemma biInf_le {ι : Type*} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s) :
    ⨅ i ∈ s, f i ≤ f i 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s)
Goal: ⨅ i ∈ s, f i ≤ f i
-- le_biSup
lemma le_biSup {ι : Type*} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s) :
    f i ≤ ⨆ i ∈ s, f i 
Arguments: {α : Type u_1} [CompleteLattice α] {ι : Type u_9} {s : Set ι} (f : ι → α) {i : ι} (hi : i ∈ s)
Goal: f i ≤ ⨆ i ∈ s, f i
-- iSup_sup
theorem iSup_sup [Nonempty ι] {f : ι → α} {a : α} : (⨆ x, f x) ⊔ a = ⨆ x, f x ⊔ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [Nonempty ι] {f : ι → α} {a : α}
Goal: (⨆ x, f x) ⊔ a = ⨆ x, f x ⊔ a
-- iInf_inf
theorem iInf_inf [Nonempty ι] {f : ι → α} {a : α} : (⨅ x, f x) ⊓ a = ⨅ x, f x ⊓ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [Nonempty ι] {f : ι → α} {a : α}
Goal: (⨅ x, f x) ⊓ a = ⨅ x, f x ⊓ a
-- sup_iSup
theorem sup_iSup [Nonempty ι] {f : ι → α} {a : α} : (a ⊔ ⨆ x, f x) = ⨆ x, a ⊔ f x 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [Nonempty ι] {f : ι → α} {a : α}
Goal: a ⊔ ⨆ x, f x = ⨆ x, a ⊔ f x
-- inf_iInf
theorem inf_iInf [Nonempty ι] {f : ι → α} {a : α} : (a ⊓ ⨅ x, f x) = ⨅ x, a ⊓ f x 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] [Nonempty ι] {f : ι → α} {a : α}
Goal: a ⊓ ⨅ x, f x = ⨅ x, a ⊓ f x
-- biSup_sup
theorem biSup_sup {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
    (⨆ (i) (h : p i), f i h) ⊔ a = ⨆ (i) (h : p i), f i h ⊔ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α} {a : α} (h : ∃ i, p i)
Goal: (⨆ i, ⨆ (h : p i), f i h) ⊔ a = ⨆ i, ⨆ (h : p i), f i h ⊔ a
-- sup_biSup
theorem sup_biSup {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
    (a ⊔ ⨆ (i) (h : p i), f i h) = ⨆ (i) (h : p i), a ⊔ f i h 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α} {a : α} (h : ∃ i, p i)
Goal: a ⊔ ⨆ i, ⨆ (h : p i), f i h = ⨆ i, ⨆ (h : p i), a ⊔ f i h
-- biInf_inf
theorem biInf_inf {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
    (⨅ (i) (h : p i), f i h) ⊓ a = ⨅ (i) (h : p i), f i h ⊓ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α} {a : α} (h : ∃ i, p i)
Goal: (⨅ i, ⨅ (h : p i), f i h) ⊓ a = ⨅ i, ⨅ (h : p i), f i h ⊓ a
-- inf_biInf
theorem inf_biInf {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
    (a ⊓ ⨅ (i) (h : p i), f i h) = ⨅ (i) (h : p i), a ⊓ f i h 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : (i : ι) → p i → α} {a : α} (h : ∃ i, p i)
Goal: a ⊓ ⨅ i, ⨅ (h : p i), f i h = ⨅ i, ⨅ (h : p i), a ⊓ f i h
-- iSup_false
theorem iSup_false {s : False → α} : iSup s = ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α] {s : False → α}
Goal: iSup s = ⊥
-- iInf_false
theorem iInf_false {s : False → α} : iInf s = ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α] {s : False → α}
Goal: iInf s = ⊤
-- iSup_true
theorem iSup_true {s : True → α} : iSup s = s trivial 
Arguments: {α : Type u_1} [CompleteLattice α] {s : True → α}
Goal: iSup s = s trivial
-- iInf_true
theorem iInf_true {s : True → α} : iInf s = s trivial 
Arguments: {α : Type u_1} [CompleteLattice α] {s : True → α}
Goal: iInf s = s trivial
-- iSup_exists
@[simp]
theorem iSup_exists {p : ι → Prop} {f : Exists p → α} : ⨆ x, f x = ⨆ (i) (h), f ⟨i, h⟩ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : Exists p → α}
Goal: ⨆ (x : Exists p), f x = ⨆ i, ⨆ (h : p i), f ⋯
-- iInf_exists
@[simp]
theorem iInf_exists {p : ι → Prop} {f : Exists p → α} : ⨅ x, f x = ⨅ (i) (h), f ⟨i, h⟩ 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {p : ι → Prop} {f : Exists p → α}
Goal: ⨅ (x : Exists p), f x = ⨅ i, ⨅ (h : p i), f ⋯
-- iSup_and
theorem iSup_and {p q : Prop} {s : p ∧ q → α} : iSup s = ⨆ (h₁) (h₂), s ⟨h₁, h₂⟩ 
Arguments: {α : Type u_1} [CompleteLattice α] {p q : Prop} {s : p ∧ q → α}
Goal: iSup s = ⨆ (h₁ : p), ⨆ (h₂ : q), s ⋯
-- iInf_and
theorem iInf_and {p q : Prop} {s : p ∧ q → α} : iInf s = ⨅ (h₁) (h₂), s ⟨h₁, h₂⟩ 
Arguments: {α : Type u_1} [CompleteLattice α] {p q : Prop} {s : p ∧ q → α}
Goal: iInf s = ⨅ (h₁ : p), ⨅ (h₂ : q), s ⋯
-- iSup_and'
theorem iSup_and' {p q : Prop} {s : p → q → α} :
    ⨆ (h₁ : p) (h₂ : q), s h₁ h₂ = ⨆ h : p ∧ q, s h.1 h.2 
Arguments: {α : Type u_1} [CompleteLattice α] {p q : Prop} {s : p → q → α}
Goal: ⨆ (h₁ : p), ⨆ (h₂ : q), s h₁ h₂ = ⨆ (h : p ∧ q), s ⋯ ⋯
-- iInf_and'
theorem iInf_and' {p q : Prop} {s : p → q → α} :
    ⨅ (h₁ : p) (h₂ : q), s h₁ h₂ = ⨅ h : p ∧ q, s h.1 h.2 
Arguments: {α : Type u_1} [CompleteLattice α] {p q : Prop} {s : p → q → α}
Goal: ⨅ (h₁ : p), ⨅ (h₂ : q), s h₁ h₂ = ⨅ (h : p ∧ q), s ⋯ ⋯
-- iSup_or
theorem iSup_or {p q : Prop} {s : p ∨ q → α} :
    ⨆ x, s x = (⨆ i, s (Or.inl i)) ⊔ ⨆ j, s (Or.inr j) 
Arguments: {α : Type u_1} [CompleteLattice α] {p q : Prop} {s : p ∨ q → α}
Goal: ⨆ (x : p ∨ q), s x = (⨆ (i : p), s ⋯) ⊔ ⨆ (j : q), s ⋯
-- iInf_or
theorem iInf_or {p q : Prop} {s : p ∨ q → α} :
    ⨅ x, s x = (⨅ i, s (Or.inl i)) ⊓ ⨅ j, s (Or.inr j) 
Arguments: {α : Type u_1} [CompleteLattice α] {p q : Prop} {s : p ∨ q → α}
Goal: ⨅ (x : p ∨ q), s x = (⨅ (i : p), s ⋯) ⊓ ⨅ (j : q), s ⋯
-- iSup_dite
theorem iSup_dite (f : ∀ i, p i → α) (g : ∀ i, ¬p i → 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_5} [CompleteLattice α] (p : ι → Prop) [DecidablePred p] (f : (i : ι) → p i → α) (g : (i : ι) → ¬p i → 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
-- iInf_dite
theorem iInf_dite (f : ∀ i, p i → α) (g : ∀ i, ¬p i → 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_5} [CompleteLattice α] (p : ι → Prop) [DecidablePred p] (f : (i : ι) → p i → α) (g : (i : ι) → ¬p i → 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
-- iSup_ite
theorem iSup_ite (f g : ι → 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_5} [CompleteLattice α] (p : ι → Prop) [DecidablePred p] (f g : ι → α)
Goal: (⨆ i, if p i then f i else g i) = (⨆ i, ⨆ (_ : p i), f i) ⊔ ⨆ i, ⨆ (_ : ¬p i), g i
-- iInf_ite
theorem iInf_ite (f g : ι → 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_5} [CompleteLattice α] (p : ι → Prop) [DecidablePred p] (f g : ι → α)
Goal: (⨅ i, if p i then f i else g i) = (⨅ i, ⨅ (_ : p i), f i) ⊓ ⨅ i, ⨅ (_ : ¬p i), g i
-- iSup_range
theorem iSup_range {g : β → α} {f : ι → β} : ⨆ b ∈ range f, g b = ⨆ i, g (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] {g : β → α} {f : ι → β}
Goal: ⨆ b ∈ Set.range f, g b = ⨆ i, g (f i)
-- iInf_range
theorem iInf_range : ∀ {g : β → α} {f : ι → β}, ⨅ b ∈ range f, g b = ⨅ i, g (f i) 
Arguments: {α : Type u_1} {β : Type u_2} {ι : Sort u_5} [CompleteLattice α] {g : β → α} {f : ι → β}
Goal: ⨅ b ∈ Set.range f, g b = ⨅ i, g (f i)
-- sSup_image
theorem sSup_image {s : Set β} {f : β → α} : sSup (f '' s) = ⨆ a ∈ s, f a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {s : Set β} {f : β → α}
Goal: sSup (f '' s) = ⨆ a ∈ s, f a
-- sInf_image
theorem sInf_image {s : Set β} {f : β → α} : sInf (f '' s) = ⨅ a ∈ s, f a 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {s : Set β} {f : β → α}
Goal: sInf (f '' s) = ⨅ a ∈ s, f a
-- OrderIso.map_sSup_eq_sSup_symm_preimage
theorem OrderIso.map_sSup_eq_sSup_symm_preimage [CompleteLattice β] (f : α ≃o β) (s : Set α) :
    f (sSup s) = sSup (f.symm ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (s : Set α)
Goal: f (sSup s) = sSup (⇑f.symm ⁻¹' s)
-- OrderIso.map_sInf_eq_sInf_symm_preimage
theorem OrderIso.map_sInf_eq_sInf_symm_preimage [CompleteLattice β] (f : α ≃o β) (s : Set α) :
    f (sInf s) = sInf (f.symm ⁻¹' s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (s : Set α)
Goal: f (sInf s) = sInf (⇑f.symm ⁻¹' s)
-- iSup_emptyset
theorem iSup_emptyset {f : β → α} : ⨆ x ∈ (∅ : Set β), f x = ⊥ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α}
Goal: ⨆ x ∈ ∅, f x = ⊥
-- iInf_emptyset
theorem iInf_emptyset {f : β → α} : ⨅ x ∈ (∅ : Set β), f x = ⊤ 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α}
Goal: ⨅ x ∈ ∅, f x = ⊤
-- iSup_univ
theorem iSup_univ {f : β → α} : ⨆ x ∈ (univ : Set β), f x = ⨆ x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α}
Goal: ⨆ x ∈ Set.univ, f x = ⨆ x, f x
-- iInf_univ
theorem iInf_univ {f : β → α} : ⨅ x ∈ (univ : Set β), f x = ⨅ x, f x 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] {f : β → α}
Goal: ⨅ x ∈ Set.univ, f x = ⨅ x, f x
-- iSup_union
theorem iSup_union {f : β → α} {s t : Set β} :
    ⨆ x ∈ s ∪ t, f x = (⨆ 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: ⨆ x ∈ s ∪ t, f x = (⨆ x ∈ s, f x) ⊔ ⨆ x ∈ t, f x
-- iInf_union
theorem iInf_union {f : β → α} {s t : Set β} : ⨅ x ∈ s ∪ t, f x = (⨅ 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: ⨅ x ∈ s ∪ t, f x = (⨅ x ∈ s, f x) ⊓ ⨅ x ∈ t, f x
-- iSup_split
theorem iSup_split (f : β → α) (p : β → Prop) :
    ⨆ i, f i = (⨆ (i) (_ : p i), f i) ⊔ ⨆ (i) (_ : ¬p i), f i 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : β → α) (p : β → Prop)
Goal: ⨆ i, f i = (⨆ i, ⨆ (_ : p i), f i) ⊔ ⨆ i, ⨆ (_ : ¬p i), f i
-- iInf_split
theorem iInf_split :
    ∀ (f : β → α) (p : β → Prop), ⨅ i, f i = (⨅ (i) (_ : p i), f i) ⊓ ⨅ (i) (_ : ¬p i), f i 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] (f : β → α) (p : β → Prop)
Goal: ⨅ i, f i = (⨅ i, ⨅ (_ : p i), f i) ⊓ ⨅ i, ⨅ (_ : ¬p i), f i