ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ofDual_iInf
@[simp]
theorem ofDual_iInf (f : ι → αᵒᵈ) : ofDual (⨅ i, f i) = ⨆ i, ofDual (f i) 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → αᵒᵈ)
Goal: OrderDual.ofDual (⨅ i, f i) = ⨆ i, OrderDual.ofDual (f i)
-- sInf_le_sSup
theorem sInf_le_sSup (hs : s.Nonempty) : sInf s ≤ sSup s 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} (hs : s.Nonempty)
Goal: sInf s ≤ sSup s
-- sSup_union
theorem sSup_union {s t : Set α} : sSup (s ∪ t) = sSup s ⊔ sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] {s t : Set α}
Goal: sSup (s ∪ t) = sSup s ⊔ sSup t
-- sInf_union
theorem sInf_union {s t : Set α} : sInf (s ∪ t) = sInf s ⊓ sInf t 
Arguments: {α : Type u_1} [CompleteLattice α] {s t : Set α}
Goal: sInf (s ∪ t) = sInf s ⊓ sInf t
-- sSup_inter_le
theorem sSup_inter_le {s t : Set α} : sSup (s ∩ t) ≤ sSup s ⊓ sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] {s t : Set α}
Goal: sSup (s ∩ t) ≤ sSup s ⊓ sSup t
-- le_sInf_inter
theorem le_sInf_inter {s t : Set α} : sInf s ⊔ sInf t ≤ sInf (s ∩ t) 
Arguments: {α : Type u_1} [CompleteLattice α] {s t : Set α}
Goal: sInf s ⊔ sInf t ≤ sInf (s ∩ t)
-- sSup_empty
@[simp]
theorem sSup_empty : sSup ∅ = (⊥ : α) 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: sSup ∅ = ⊥
-- sInf_empty
@[simp]
theorem sInf_empty : sInf ∅ = (⊤ : α) 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: sInf ∅ = ⊤
-- sSup_univ
@[simp]
theorem sSup_univ : sSup univ = (⊤ : α) 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: sSup Set.univ = ⊤
-- sInf_univ
@[simp]
theorem sInf_univ : sInf univ = (⊥ : α) 
Arguments: {α : Type u_1} [CompleteLattice α]
Goal: sInf Set.univ = ⊥
-- sSup_insert
@[simp]
theorem sSup_insert {a : α} {s : Set α} : sSup (insert a s) = a ⊔ sSup s 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} {s : Set α}
Goal: sSup (insert a s) = a ⊔ sSup s
-- sInf_insert
@[simp]
theorem sInf_insert {a : α} {s : Set α} : sInf (insert a s) = a ⊓ sInf s 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} {s : Set α}
Goal: sInf (insert a s) = a ⊓ sInf s
-- sSup_le_sSup_of_subset_insert_bot
theorem sSup_le_sSup_of_subset_insert_bot (h : s ⊆ insert ⊥ t) : sSup s ≤ sSup t 
Arguments: {α : Type u_1} [CompleteLattice α] {s t : Set α} (h : s ⊆ insert ⊥ t)
Goal: sSup s ≤ sSup t
-- sInf_le_sInf_of_subset_insert_top
theorem sInf_le_sInf_of_subset_insert_top (h : s ⊆ insert ⊤ t) : sInf t ≤ sInf s 
Arguments: {α : Type u_1} [CompleteLattice α] {s t : Set α} (h : s ⊆ insert ⊤ t)
Goal: sInf t ≤ sInf s
-- sSup_diff_singleton_bot
@[simp]
theorem sSup_diff_singleton_bot (s : Set α) : sSup (s \ {⊥}) = sSup s 
Arguments: {α : Type u_1} [CompleteLattice α] (s : Set α)
Goal: sSup (s \ {⊥}) = sSup s
-- sInf_diff_singleton_top
@[simp]
theorem sInf_diff_singleton_top (s : Set α) : sInf (s \ {⊤}) = sInf s 
Arguments: {α : Type u_1} [CompleteLattice α] (s : Set α)
Goal: sInf (s \ {⊤}) = sInf s
-- sSup_pair
theorem sSup_pair {a b : α} : sSup {a, b} = a ⊔ b 
Arguments: {α : Type u_1} [CompleteLattice α] {a b : α}
Goal: sSup {a, b} = a ⊔ b
-- sInf_pair
theorem sInf_pair {a b : α} : sInf {a, b} = a ⊓ b 
Arguments: {α : Type u_1} [CompleteLattice α] {a b : α}
Goal: sInf {a, b} = a ⊓ b
-- sSup_eq_bot
@[simp]
theorem sSup_eq_bot : sSup s = ⊥ ↔ ∀ a ∈ s, a = ⊥ 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: sSup s = ⊥ ↔ ∀ a ∈ s, a = ⊥
-- sInf_eq_top
@[simp]
theorem sInf_eq_top : sInf s = ⊤ ↔ ∀ a ∈ s, a = ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: sInf s = ⊤ ↔ ∀ a ∈ s, a = ⊤
-- sSup_eq_bot'
lemma sSup_eq_bot' {s : Set α} : sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥} 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: sSup s = ⊥ ↔ s = ∅ ∨ s = {⊥}
-- eq_singleton_bot_of_sSup_eq_bot_of_nonempty
theorem eq_singleton_bot_of_sSup_eq_bot_of_nonempty {s : Set α} (h_sup : sSup s = ⊥)
    (hne : s.Nonempty) : s = {⊥} 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} (h_sup : sSup s = ⊥) (hne : s.Nonempty)
Goal: s = {⊥}
-- eq_singleton_top_of_sInf_eq_top_of_nonempty
theorem eq_singleton_top_of_sInf_eq_top_of_nonempty : sInf s = ⊤ → s.Nonempty → s = {⊤} 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: sInf s = ⊤ → s.Nonempty → s = {⊤}
-- sSup_eq_of_forall_le_of_forall_lt_exists_gt
theorem sSup_eq_of_forall_le_of_forall_lt_exists_gt (h₁ : ∀ a ∈ s, a ≤ b)
    (h₂ : ∀ w, w < b → ∃ a ∈ s, w < a) : sSup s = b 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} {b : α} (h₁ : ∀ a ∈ s, a ≤ b) (h₂ : ∀ w < b, ∃ a ∈ s, w < a)
Goal: sSup s = b
-- sInf_eq_of_forall_ge_of_forall_gt_exists_lt
theorem sInf_eq_of_forall_ge_of_forall_gt_exists_lt :
    (∀ a ∈ s, b ≤ a) → (∀ w, b < w → ∃ a ∈ s, a < w) → sInf s = b 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α} {b : α}
Goal: (∀ a ∈ s, b ≤ a) → (∀ (w : α), b < w → ∃ a ∈ s, a < w) → sInf s = b
-- lt_sSup_iff
theorem lt_sSup_iff : b < sSup s ↔ ∃ a ∈ s, b < a 
Arguments: {α : Type u_1} [CompleteLinearOrder α] {s : Set α} {b : α}
Goal: b < sSup s ↔ ∃ a ∈ s, b < a
-- sInf_lt_iff
theorem sInf_lt_iff : sInf s < b ↔ ∃ a ∈ s, a < b 
Arguments: {α : Type u_1} [CompleteLinearOrder α] {s : Set α} {b : α}
Goal: sInf s < b ↔ ∃ a ∈ s, a < b
-- sSup_eq_top
theorem sSup_eq_top : sSup s = ⊤ ↔ ∀ b < ⊤, ∃ a ∈ s, b < a 
Arguments: {α : Type u_1} [CompleteLinearOrder α] {s : Set α}
Goal: sSup s = ⊤ ↔ ∀ b < ⊤, ∃ a ∈ s, b < a
-- sInf_eq_bot
theorem sInf_eq_bot : sInf s = ⊥ ↔ ∀ b > ⊥, ∃ a ∈ s, a < b 
Arguments: {α : Type u_1} [CompleteLinearOrder α] {s : Set α}
Goal: sInf s = ⊥ ↔ ∀ b > ⊥, ∃ a ∈ s, a < b
-- lt_iSup_iff
theorem lt_iSup_iff {f : ι → α} : a < iSup f ↔ ∃ i, a < f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLinearOrder α] {a : α} {f : ι → α}
Goal: a < iSup f ↔ ∃ i, a < f i
-- iInf_lt_iff
theorem iInf_lt_iff {f : ι → α} : iInf f < a ↔ ∃ i, f i < a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLinearOrder α] {a : α} {f : ι → α}
Goal: iInf f < a ↔ ∃ i, f i < a
-- sSup_range
theorem sSup_range : sSup (range f) = iSup f 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] {f : ι → α}
Goal: sSup (Set.range f) = iSup f
-- sSup_eq_iSup'
theorem sSup_eq_iSup' (s : Set α) : sSup s = ⨆ a : s, (a : α) 
Arguments: {α : Type u_1} [SupSet α] (s : Set α)
Goal: sSup s = ⨆ a, ↑a
-- iSup_congr
theorem iSup_congr (h : ∀ i, f i = g i) : ⨆ i, f i = ⨆ i, g i 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] {f g : ι → α} (h : ∀ (i : ι), f i = g i)
Goal: ⨆ i, f i = ⨆ i, g i
-- biSup_congr
theorem biSup_congr {p : ι → Prop} (h : ∀ i, p i → f i = g i) :
    ⨆ (i) (_ : p i), f i = ⨆ (i) (_ : p i), g i 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] {f g : ι → α} {p : ι → Prop} (h : ∀ (i : ι), p i → f i = g i)
Goal: ⨆ i, ⨆ (_ : p i), f i = ⨆ i, ⨆ (_ : p i), g i
-- biSup_congr'
theorem biSup_congr' {p : ι → Prop} {f g : (i : ι) → p i → α}
    (h : ∀ i (hi : p i), f i hi = g i hi) :
    ⨆ i, ⨆ (hi : p i), f i hi = ⨆ i, ⨆ (hi : p i), g i hi 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] {p : ι → Prop} {f g : (i : ι) → p i → α} (h : ∀ (i : ι) (hi : p i), f i hi = g i hi)
Goal: ⨆ i, ⨆ (hi : p i), f i hi = ⨆ i, ⨆ (hi : p i), g i hi
-- Function.Surjective.iSup_comp
theorem Function.Surjective.iSup_comp {f : ι → ι'} (hf : Surjective f) (g : ι' → α) :
    ⨆ x, g (f x) = ⨆ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [SupSet α] {f : ι → ι'} (hf : Function.Surjective f) (g : ι' → α)
Goal: ⨆ x, g (f x) = ⨆ y, g y
-- Equiv.iSup_comp
theorem Equiv.iSup_comp {g : ι' → α} (e : ι ≃ ι') : ⨆ x, g (e x) = ⨆ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [SupSet α] {g : ι' → α} (e : ι ≃ ι')
Goal: ⨆ x, g (e x) = ⨆ y, g y
-- Function.Surjective.iSup_congr
protected theorem Function.Surjective.iSup_congr {g : ι' → α} (h : ι → ι') (h1 : Surjective h)
    (h2 : ∀ x, g (h x) = f x) : ⨆ x, f x = ⨆ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [SupSet α] {f : ι → α} {g : ι' → α} (h : ι → ι') (h1 : Function.Surjective h) (h2 : ∀ (x : ι), g (h x) = f x)
Goal: ⨆ x, f x = ⨆ y, g y
-- Equiv.iSup_congr
protected theorem Equiv.iSup_congr {g : ι' → α} (e : ι ≃ ι') (h : ∀ x, g (e x) = f x) :
    ⨆ x, f x = ⨆ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [SupSet α] {f : ι → α} {g : ι' → α} (e : ι ≃ ι') (h : ∀ (x : ι), g (e x) = f x)
Goal: ⨆ x, f x = ⨆ y, g y
-- iSup_congr_Prop
@[congr]
theorem iSup_congr_Prop {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q)
    (f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iSup f₁ = iSup f₂ 
Arguments: {α : Type u_1} [SupSet α] {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q) (f : ∀ (x : q), f₁ ⋯ = f₂ x)
Goal: iSup f₁ = iSup f₂
-- iSup_plift_up
theorem iSup_plift_up (f : PLift ι → α) : ⨆ i, f (PLift.up i) = ⨆ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] (f : PLift ι → α)
Goal: ⨆ i, f { down := i } = ⨆ i, f i
-- iSup_plift_down
theorem iSup_plift_down (f : ι → α) : ⨆ i, f (PLift.down i) = ⨆ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] (f : ι → α)
Goal: ⨆ i, f i.down = ⨆ i, f 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} [SupSet α] (g : β → α) (f : ι → β)
Goal: ⨆ b, 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} [SupSet α] {s : Set β} {f : β → α}
Goal: sSup (f '' s) = ⨆ a, f ↑a
-- sInf_range
theorem sInf_range : sInf (range f) = iInf f 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] {f : ι → α}
Goal: sInf (Set.range f) = iInf f
-- sInf_eq_iInf'
theorem sInf_eq_iInf' (s : Set α) : sInf s = ⨅ a : s, (a : α) 
Arguments: {α : Type u_1} [InfSet α] (s : Set α)
Goal: sInf s = ⨅ a, ↑a
-- iInf_congr
theorem iInf_congr (h : ∀ i, f i = g i) : ⨅ i, f i = ⨅ i, g i 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] {f g : ι → α} (h : ∀ (i : ι), f i = g i)
Goal: ⨅ i, f i = ⨅ i, g i
-- biInf_congr
theorem biInf_congr {p : ι → Prop} (h : ∀ i, p i → f i = g i) :
    ⨅ (i) (_ : p i), f i = ⨅ (i) (_ : p i), g i 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] {f g : ι → α} {p : ι → Prop} (h : ∀ (i : ι), p i → f i = g i)
Goal: ⨅ i, ⨅ (_ : p i), f i = ⨅ i, ⨅ (_ : p i), g i
-- biInf_congr'
theorem biInf_congr' {p : ι → Prop} {f g : (i : ι) → p i → α}
    (h : ∀ i (hi : p i), f i hi = g i hi) :
    ⨅ i, ⨅ (hi : p i), f i hi = ⨅ i, ⨅ (hi : p i), g i hi 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] {p : ι → Prop} {f g : (i : ι) → p i → α} (h : ∀ (i : ι) (hi : p i), f i hi = g i hi)
Goal: ⨅ i, ⨅ (hi : p i), f i hi = ⨅ i, ⨅ (hi : p i), g i hi
-- Function.Surjective.iInf_comp
theorem Function.Surjective.iInf_comp {f : ι → ι'} (hf : Surjective f) (g : ι' → α) :
    ⨅ x, g (f x) = ⨅ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [InfSet α] {f : ι → ι'} (hf : Function.Surjective f) (g : ι' → α)
Goal: ⨅ x, g (f x) = ⨅ y, g y
-- Equiv.iInf_comp
theorem Equiv.iInf_comp {g : ι' → α} (e : ι ≃ ι') : ⨅ x, g (e x) = ⨅ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [InfSet α] {g : ι' → α} (e : ι ≃ ι')
Goal: ⨅ x, g (e x) = ⨅ y, g y
-- Function.Surjective.iInf_congr
protected theorem Function.Surjective.iInf_congr {g : ι' → α} (h : ι → ι') (h1 : Surjective h)
    (h2 : ∀ x, g (h x) = f x) : ⨅ x, f x = ⨅ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [InfSet α] {f : ι → α} {g : ι' → α} (h : ι → ι') (h1 : Function.Surjective h) (h2 : ∀ (x : ι), g (h x) = f x)
Goal: ⨅ x, f x = ⨅ y, g y
-- Equiv.iInf_congr
protected theorem Equiv.iInf_congr {g : ι' → α} (e : ι ≃ ι') (h : ∀ x, g (e x) = f x) :
    ⨅ x, f x = ⨅ y, g y 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [InfSet α] {f : ι → α} {g : ι' → α} (e : ι ≃ ι') (h : ∀ (x : ι), g (e x) = f x)
Goal: ⨅ x, f x = ⨅ y, g y
-- iInf_congr_Prop
@[congr]
theorem iInf_congr_Prop {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q)
    (f : ∀ x, f₁ (pq.mpr x) = f₂ x) : iInf f₁ = iInf f₂ 
Arguments: {α : Type u_1} [InfSet α] {p q : Prop} {f₁ : p → α} {f₂ : q → α} (pq : p ↔ q) (f : ∀ (x : q), f₁ ⋯ = f₂ x)
Goal: iInf f₁ = iInf f₂
-- iInf_plift_up
theorem iInf_plift_up (f : PLift ι → α) : ⨅ i, f (PLift.up i) = ⨅ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] (f : PLift ι → α)
Goal: ⨅ i, f { down := i } = ⨅ i, f i
-- iInf_plift_down
theorem iInf_plift_down (f : ι → α) : ⨅ i, f (PLift.down i) = ⨅ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] (f : ι → α)
Goal: ⨅ i, f i.down = ⨅ i, 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} [InfSet α] (g : β → α) (f : ι → β)
Goal: ⨅ b, g ↑b = ⨅ i, g (f i)
-- 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} [InfSet α] {s : Set β} {f : β → α}
Goal: sInf (f '' s) = ⨅ a, f ↑a
-- le_iSup
theorem le_iSup (f : ι → α) (i : ι) : f i ≤ iSup f 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → α) (i : ι)
Goal: f i ≤ iSup f
-- iInf_le
theorem iInf_le (f : ι → α) (i : ι) : iInf f ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → α) (i : ι)
Goal: iInf f ≤ f i
-- le_iSup'
theorem le_iSup' (f : ι → α) (i : ι) : f i ≤ iSup f 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → α) (i : ι)
Goal: f i ≤ iSup f
-- iInf_le'
theorem iInf_le' (f : ι → α) (i : ι) : iInf f ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (f : ι → α) (i : ι)
Goal: iInf f ≤ f i
-- isLUB_iSup
theorem isLUB_iSup : IsLUB (range f) (⨆ j, f j) 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α}
Goal: IsLUB (Set.range f) (⨆ j, f j)
-- isGLB_iInf
theorem isGLB_iInf : IsGLB (range f) (⨅ j, f j) 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α}
Goal: IsGLB (Set.range f) (⨅ j, f j)
-- IsLUB.iSup_eq
theorem IsLUB.iSup_eq (h : IsLUB (range f) a) : ⨆ j, f j = a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α} (h : IsLUB (Set.range f) a)
Goal: ⨆ j, f j = a
-- IsGLB.iInf_eq
theorem IsGLB.iInf_eq (h : IsGLB (range f) a) : ⨅ j, f j = a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α} (h : IsGLB (Set.range f) a)
Goal: ⨅ j, f j = a
-- le_iSup_of_le
theorem le_iSup_of_le (i : ι) (h : a ≤ f i) : a ≤ iSup f 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α} (i : ι) (h : a ≤ f i)
Goal: a ≤ iSup f
-- iInf_le_of_le
theorem iInf_le_of_le (i : ι) (h : f i ≤ a) : iInf f ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α} (i : ι) (h : f i ≤ a)
Goal: iInf f ≤ a
-- le_iSup₂
theorem le_iSup₂ {f : ∀ i, κ i → α} (i : ι) (j : κ i) : 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} (i : ι) (j : κ i)
Goal: f i j ≤ ⨆ i, ⨆ j, f i j
-- iInf₂_le
theorem iInf₂_le {f : ∀ i, κ i → α} (i : ι) (j : κ i) : ⨅ (i) (j), f 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} (i : ι) (j : κ i)
Goal: ⨅ i, ⨅ j, f i j ≤ f i j
-- le_iSup₂_of_le
theorem le_iSup₂_of_le {f : ∀ i, κ i → α} (i : ι) (j : κ i) (h : a ≤ f i j) :
    a ≤ ⨆ (i) (j), f i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {a : α} {f : (i : ι) → κ i → α} (i : ι) (j : κ i) (h : a ≤ f i j)
Goal: a ≤ ⨆ i, ⨆ j, f i j
-- iInf₂_le_of_le
theorem iInf₂_le_of_le {f : ∀ i, κ i → α} (i : ι) (j : κ i) (h : f i j ≤ a) :
    ⨅ (i) (j), f i j ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {a : α} {f : (i : ι) → κ i → α} (i : ι) (j : κ i) (h : f i j ≤ a)
Goal: ⨅ i, ⨅ j, f i j ≤ a
-- iSup_le
theorem iSup_le (h : ∀ i, f i ≤ a) : iSup f ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α} (h : ∀ (i : ι), f i ≤ a)
Goal: iSup f ≤ a
-- le_iInf
theorem le_iInf (h : ∀ i, a ≤ f i) : a ≤ iInf f 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α} (h : ∀ (i : ι), a ≤ f i)
Goal: a ≤ iInf f
-- iSup₂_le
theorem iSup₂_le {f : ∀ i, κ i → α} (h : ∀ i j, f i j ≤ a) : ⨆ (i) (j), f i j ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {a : α} {f : (i : ι) → κ i → α} (h : ∀ (i : ι) (j : κ i), f i j ≤ a)
Goal: ⨆ i, ⨆ j, f i j ≤ a
-- le_iInf₂
theorem le_iInf₂ {f : ∀ i, κ i → α} (h : ∀ i j, a ≤ f i j) : a ≤ ⨅ (i) (j), f i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {a : α} {f : (i : ι) → κ i → α} (h : ∀ (i : ι) (j : κ i), a ≤ f i j)
Goal: a ≤ ⨅ i, ⨅ j, f i j
-- iSup₂_le_iSup
theorem iSup₂_le_iSup (κ : ι → Sort*) (f : ι → α) : ⨆ (i) (_ : κ i), f i ≤ ⨆ i, f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (κ : ι → Sort u_9) (f : ι → α)
Goal: ⨆ i, ⨆ x, f i ≤ ⨆ i, f i
-- iInf_le_iInf₂
theorem iInf_le_iInf₂ (κ : ι → Sort*) (f : ι → α) : ⨅ i, f i ≤ ⨅ (i) (_ : κ i), f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] (κ : ι → Sort u_9) (f : ι → α)
Goal: ⨅ i, f i ≤ ⨅ i, ⨅ x, f i
-- iSup_mono
@[gcongr]
theorem iSup_mono (h : ∀ i, f i ≤ g i) : iSup f ≤ iSup g 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f g : ι → α} (h : ∀ (i : ι), f i ≤ g i)
Goal: iSup f ≤ iSup g
-- iInf_mono
@[gcongr]
theorem iInf_mono (h : ∀ i, f i ≤ g i) : iInf f ≤ iInf g 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f g : ι → α} (h : ∀ (i : ι), f i ≤ g i)
Goal: iInf f ≤ iInf g
-- iSup₂_mono
theorem iSup₂_mono {f g : ∀ i, κ i → α} (h : ∀ i j, f i j ≤ g i j) :
    ⨆ (i) (j), f i j ≤ ⨆ (i) (j), g i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {f g : (i : ι) → κ i → α} (h : ∀ (i : ι) (j : κ i), f i j ≤ g i j)
Goal: ⨆ i, ⨆ j, f i j ≤ ⨆ i, ⨆ j, g i j
-- iInf₂_mono
theorem iInf₂_mono {f g : ∀ i, κ i → α} (h : ∀ i j, f i j ≤ g i j) :
    ⨅ (i) (j), f i j ≤ ⨅ (i) (j), g i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {f g : (i : ι) → κ i → α} (h : ∀ (i : ι) (j : κ i), f i j ≤ g i j)
Goal: ⨅ i, ⨅ j, f i j ≤ ⨅ i, ⨅ j, g i j
-- iSup_mono'
theorem iSup_mono' {g : ι' → α} (h : ∀ i, ∃ i', f i ≤ g i') : iSup f ≤ iSup g 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [CompleteLattice α] {f : ι → α} {g : ι' → α} (h : ∀ (i : ι), ∃ i', f i ≤ g i')
Goal: iSup f ≤ iSup g
-- iInf_mono'
theorem iInf_mono' {g : ι' → α} (h : ∀ i', ∃ i, f i ≤ g i') : iInf f ≤ iInf g 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [CompleteLattice α] {f : ι → α} {g : ι' → α} (h : ∀ (i' : ι'), ∃ i, f i ≤ g i')
Goal: iInf f ≤ iInf g
-- iSup₂_mono'
theorem iSup₂_mono' {f : ∀ i, κ i → α} {g : ∀ i', κ' i' → α} (h : ∀ i j, ∃ i' j', f i j ≤ g i' j') :
    ⨆ (i) (j), f i j ≤ ⨆ (i) (j), g i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} {κ : ι → Sort u_7} {κ' : ι' → Sort u_8} [CompleteLattice α] {f : (i : ι) → κ i → α} {g : (i' : ι') → κ' i' → α} (h : ∀ (i : ι) (j : κ i), ∃ i' j', f i j ≤ g i' j')
Goal: ⨆ i, ⨆ j, f i j ≤ ⨆ i, ⨆ j, g i j
-- iInf₂_mono'
theorem iInf₂_mono' {f : ∀ i, κ i → α} {g : ∀ i', κ' i' → α} (h : ∀ i j, ∃ i' j', f i' j' ≤ g i j) :
    ⨅ (i) (j), f i j ≤ ⨅ (i) (j), g i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} {κ : ι → Sort u_7} {κ' : ι' → Sort u_8} [CompleteLattice α] {f : (i : ι) → κ i → α} {g : (i' : ι') → κ' i' → α} (h : ∀ (i : ι') (j : κ' i), ∃ i' j', f i' j' ≤ g i j)
Goal: ⨅ i, ⨅ j, f i j ≤ ⨅ i, ⨅ j, g i j
-- iSup_const_mono
theorem iSup_const_mono (h : ι → ι') : ⨆ _ : ι, a ≤ ⨆ _ : ι', a 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [CompleteLattice α] {a : α} (h : ι → ι')
Goal: ⨆ x, a ≤ ⨆ x, a
-- iInf_const_mono
theorem iInf_const_mono (h : ι' → ι) : ⨅ _ : ι, a ≤ ⨅ _ : ι', a 
Arguments: {α : Type u_1} {ι : Sort u_5} {ι' : Sort u_6} [CompleteLattice α] {a : α} (h : ι' → ι)
Goal: ⨅ x, a ≤ ⨅ x, a
-- iSup_iInf_le_iInf_iSup
theorem iSup_iInf_le_iInf_iSup (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
-- biSup_mono
theorem biSup_mono {p q : ι → Prop} (hpq : ∀ i, p i → q i) :
    ⨆ (i) (_ : p i), f i ≤ ⨆ (i) (_ : q i), f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {p q : ι → Prop} (hpq : ∀ (i : ι), p i → q i)
Goal: ⨆ i, ⨆ (_ : p i), f i ≤ ⨆ i, ⨆ (_ : q i), f i
-- biInf_mono
theorem biInf_mono {p q : ι → Prop} (hpq : ∀ i, p i → q i) :
    ⨅ (i) (_ : q i), f i ≤ ⨅ (i) (_ : p i), f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {p q : ι → Prop} (hpq : ∀ (i : ι), p i → q i)
Goal: ⨅ i, ⨅ (_ : q i), f i ≤ ⨅ i, ⨅ (_ : p i), f i
-- iSup_le_iff
@[simp]
theorem iSup_le_iff : iSup f ≤ a ↔ ∀ i, f i ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α}
Goal: iSup f ≤ a ↔ ∀ (i : ι), f i ≤ a
-- le_iInf_iff
@[simp]
theorem le_iInf_iff : a ≤ iInf f ↔ ∀ i, a ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α}
Goal: a ≤ iInf f ↔ ∀ (i : ι), a ≤ f i
-- iSup₂_le_iff
theorem iSup₂_le_iff {f : ∀ i, κ i → α} : ⨆ (i) (j), f i j ≤ a ↔ ∀ i j, f i j ≤ a 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {a : α} {f : (i : ι) → κ i → α}
Goal: ⨆ i, ⨆ j, f i j ≤ a ↔ ∀ (i : ι) (j : κ i), f i j ≤ a
-- le_iInf₂_iff
theorem le_iInf₂_iff {f : ∀ i, κ i → α} : (a ≤ ⨅ (i) (j), f i j) ↔ ∀ i j, a ≤ f i j 
Arguments: {α : Type u_1} {ι : Sort u_5} {κ : ι → Sort u_7} [CompleteLattice α] {a : α} {f : (i : ι) → κ i → α}
Goal: a ≤ ⨅ i, ⨅ j, f i j ↔ ∀ (i : ι) (j : κ i), a ≤ f i j
-- iSup_lt_iff
theorem iSup_lt_iff : iSup f < a ↔ ∃ b, b < a ∧ ∀ i, f i ≤ b 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α}
Goal: iSup f < a ↔ ∃ b < a, ∀ (i : ι), f i ≤ b
-- lt_iInf_iff
theorem lt_iInf_iff : a < iInf f ↔ ∃ b, a < b ∧ ∀ i, b ≤ f i 
Arguments: {α : Type u_1} {ι : Sort u_5} [CompleteLattice α] {f : ι → α} {a : α}
Goal: a < iInf f ↔ ∃ b, a < b ∧ ∀ (i : ι), b ≤ f i
-- sSup_eq_iSup
theorem sSup_eq_iSup {s : Set α} : sSup s = ⨆ a ∈ s, a 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: sSup s = ⨆ a ∈ s, a
-- sInf_eq_iInf
theorem sInf_eq_iInf {s : Set α} : sInf s = ⨅ a ∈ s, a 
Arguments: {α : Type u_1} [CompleteLattice α] {s : Set α}
Goal: sInf s = ⨅ a ∈ s, a
-- Monotone.le_map_iSup
theorem Monotone.le_map_iSup [CompleteLattice β] {f : α → β} (hf : Monotone f) :
    ⨆ i, f (s i) ≤ f (iSup s) 
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: ⨆ i, f (s i) ≤ f (iSup s)
-- Antitone.le_map_iInf
theorem Antitone.le_map_iInf [CompleteLattice β] {f : α → β} (hf : Antitone f) :
    ⨆ i, f (s i) ≤ f (iInf s) 
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: ⨆ i, f (s i) ≤ f (iInf s)
-- Monotone.le_map_iSup₂
theorem Monotone.le_map_iSup₂ [CompleteLattice β] {f : α → β} (hf : Monotone f) (s : ∀ i, κ i → α) :
    ⨆ (i) (j), f (s i j) ≤ f (⨆ (i) (j), 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: ⨆ i, ⨆ j, f (s i j) ≤ f (⨆ i, ⨆ j, s i j)
-- Antitone.le_map_iInf₂
theorem Antitone.le_map_iInf₂ [CompleteLattice β] {f : α → β} (hf : Antitone f) (s : ∀ i, κ i → α) :
    ⨆ (i) (j), f (s i j) ≤ f (⨅ (i) (j), 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: ⨆ i, ⨆ j, f (s i j) ≤ f (⨅ i, ⨅ j, s i j)
-- Monotone.le_map_sSup
theorem Monotone.le_map_sSup [CompleteLattice β] {s : Set α} {f : α → β} (hf : Monotone f) :
    ⨆ a ∈ s, f a ≤ f (sSup s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] {s : Set α} {f : α → β} (hf : Monotone f)
Goal: ⨆ a ∈ s, f a ≤ f (sSup s)
-- Antitone.le_map_sInf
theorem Antitone.le_map_sInf [CompleteLattice β] {s : Set α} {f : α → β} (hf : Antitone f) :
    ⨆ a ∈ s, f a ≤ f (sInf s) 
Arguments: {α : Type u_1} {β : Type u_2} [CompleteLattice α] [CompleteLattice β] {s : Set α} {f : α → β} (hf : Antitone f)
Goal: ⨆ a ∈ s, f a ≤ f (sInf s)