ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- inf_ind
theorem inf_ind (a b : α) {p : α → Prop} : p a → p b → p (a ⊓ b) 
Arguments: {α : Type u} [LinearOrder α] (a b : α) {p : α → Prop}
Goal: p a → p b → p (a ⊓ b)
-- inf_le_iff
@[simp]
theorem inf_le_iff : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a
-- inf_lt_iff
@[simp]
theorem inf_lt_iff : b ⊓ c < a ↔ b < a ∨ c < a 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: b ⊓ c < a ↔ b < a ∨ c < a
-- lt_inf_iff
@[simp]
theorem lt_inf_iff : a < b ⊓ c ↔ a < b ∧ a < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a < b ⊓ c ↔ a < b ∧ a < c
-- max_max_max_comm
theorem max_max_max_comm : max (max a b) (max c d) = max (max a c) (max b d) 
Arguments: {α : Type u} [LinearOrder α] (a b c d : α)
Goal: max (max a b) (max c d) = max (max a c) (max b d)
-- min_min_min_comm
theorem min_min_min_comm : min (min a b) (min c d) = min (min a c) (min b d) 
Arguments: {α : Type u} [LinearOrder α] (a b c d : α)
Goal: min (min a b) (min c d) = min (min a c) (min b d)
-- sup_eq_maxDefault
theorem sup_eq_maxDefault [SemilatticeSup α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
    [IsTotal α (· ≤ ·)] :
    (· ⊔ ·) = (maxDefault : α → α → α) 
Arguments: {α : Type u} [SemilatticeSup α] [DecidableRel fun x x_1 => x ≤ x_1] [IsTotal α fun x x_1 => x ≤ x_1]
Goal: (fun x x_1 => x ⊔ x_1) = maxDefault
-- inf_eq_minDefault
theorem inf_eq_minDefault [SemilatticeInf α] [DecidableRel ((· ≤ ·) : α → α → Prop)]
    [IsTotal α (· ≤ ·)] :
    (· ⊓ ·) = (minDefault : α → α → α) 
Arguments: {α : Type u} [SemilatticeInf α] [DecidableRel fun x x_1 => x ≤ x_1] [IsTotal α fun x x_1 => x ≤ x_1]
Goal: (fun x x_1 => x ⊓ x_1) = minDefault
-- ofDual_inf
@[simp]
theorem ofDual_inf [Sup α] (a b : αᵒᵈ) : ofDual (a ⊓ b) = ofDual a ⊔ ofDual b 
Arguments: {α : Type u} [Sup α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a ⊓ b) = OrderDual.ofDual a ⊔ OrderDual.ofDual b
-- ofDual_sup
@[simp]
theorem ofDual_sup [Inf α] (a b : αᵒᵈ) : ofDual (a ⊔ b) = ofDual a ⊓ ofDual b 
Arguments: {α : Type u} [Inf α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (a ⊔ b) = OrderDual.ofDual a ⊓ OrderDual.ofDual b
-- toDual_inf
@[simp]
theorem toDual_inf [Inf α] (a b : α) : toDual (a ⊓ b) = toDual a ⊔ toDual b 
Arguments: {α : Type u} [Inf α] (a b : α)
Goal: OrderDual.toDual (a ⊓ b) = OrderDual.toDual a ⊔ OrderDual.toDual b
-- toDual_sup
@[simp]
theorem toDual_sup [Sup α] (a b : α) : toDual (a ⊔ b) = toDual a ⊓ toDual b 
Arguments: {α : Type u} [Sup α] (a b : α)
Goal: OrderDual.toDual (a ⊔ b) = OrderDual.toDual a ⊓ OrderDual.toDual b
-- ofDual_min
@[simp]
theorem ofDual_min (a b : αᵒᵈ) : ofDual (min a b) = max (ofDual a) (ofDual b) 
Arguments: {α : Type u} [LinearOrder α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (min a b) = max (OrderDual.ofDual a) (OrderDual.ofDual b)
-- ofDual_max
@[simp]
theorem ofDual_max (a b : αᵒᵈ) : ofDual (max a b) = min (ofDual a) (ofDual b) 
Arguments: {α : Type u} [LinearOrder α] (a b : αᵒᵈ)
Goal: OrderDual.ofDual (max a b) = min (OrderDual.ofDual a) (OrderDual.ofDual b)
-- toDual_min
@[simp]
theorem toDual_min (a b : α) : toDual (min a b) = max (toDual a) (toDual b) 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: OrderDual.toDual (min a b) = max (OrderDual.toDual a) (OrderDual.toDual b)
-- toDual_max
@[simp]
theorem toDual_max (a b : α) : toDual (max a b) = min (toDual a) (toDual b) 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: OrderDual.toDual (max a b) = min (OrderDual.toDual a) (OrderDual.toDual b)
-- Pi.sup_apply
@[simp]
theorem sup_apply [∀ i, Sup (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i 
Arguments: {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Sup (α' i)] (f g : (i : ι) → α' i) (i : ι)
Goal: (f ⊔ g) i = f i ⊔ g i
-- Pi.sup_def
theorem sup_def [∀ i, Sup (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i 
Arguments: {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Sup (α' i)] (f g : (i : ι) → α' i)
Goal: f ⊔ g = fun i => f i ⊔ g i
-- Pi.inf_apply
@[simp]
theorem inf_apply [∀ i, Inf (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊓ g) i = f i ⊓ g i 
Arguments: {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Inf (α' i)] (f g : (i : ι) → α' i) (i : ι)
Goal: (f ⊓ g) i = f i ⊓ g i
-- Pi.inf_def
theorem inf_def [∀ i, Inf (α' i)] (f g : ∀ i, α' i) : f ⊓ g = fun i => f i ⊓ g i 
Arguments: {ι : Type u_1} {α' : ι → Type u_2} [(i : ι) → Inf (α' i)] (f g : (i : ι) → α' i)
Goal: f ⊓ g = fun i => f i ⊓ g i
-- Function.update_sup
theorem update_sup [∀ i, SemilatticeSup (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
    update f i (a ⊔ b) = update f i a ⊔ update f i b 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [DecidableEq ι] [(i : ι) → SemilatticeSup (π i)] (f : (i : ι) → π i) (i : ι) (a b : π i)
Goal: Function.update f i (a ⊔ b) = Function.update f i a ⊔ Function.update f i b
-- Function.update_inf
theorem update_inf [∀ i, SemilatticeInf (π i)] (f : ∀ i, π i) (i : ι) (a b : π i) :
    update f i (a ⊓ b) = update f i a ⊓ update f i b 
Arguments: {ι : Type u_1} {π : ι → Type u_2} [DecidableEq ι] [(i : ι) → SemilatticeInf (π i)] (f : (i : ι) → π i) (i : ι) (a b : π i)
Goal: Function.update f i (a ⊓ b) = Function.update f i a ⊓ Function.update f i b
-- Monotone.sup
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f)
    (hg : Monotone g) :
    Monotone (f ⊔ g) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Monotone f) (hg : Monotone g)
Goal: Monotone (f ⊔ g)
-- Monotone.inf
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f)
    (hg : Monotone g) :
    Monotone (f ⊓ g) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Monotone f) (hg : Monotone g)
Goal: Monotone (f ⊓ g)
-- Monotone.max
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
    (hg : Monotone g) :
    Monotone fun x => max (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f) (hg : Monotone g)
Goal: Monotone fun x => max (f x) (g x)
-- Monotone.min
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f)
    (hg : Monotone g) :
    Monotone fun x => min (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} (hf : Monotone f) (hg : Monotone g)
Goal: Monotone fun x => min (f x) (g x)
-- Monotone.le_map_sup
theorem le_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α) :
    f x ⊔ f y ≤ f (x ⊔ y) 
Arguments: {α : Type u} {β : Type v} [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : Monotone f) (x y : α)
Goal: f x ⊔ f y ≤ f (x ⊔ y)
-- Monotone.map_inf_le
theorem map_inf_le [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α) :
    f (x ⊓ y) ≤ f x ⊓ f y 
Arguments: {α : Type u} {β : Type v} [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : Monotone f) (x y : α)
Goal: f (x ⊓ y) ≤ f x ⊓ f y
-- Monotone.of_map_inf
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β] {f : α → β}
    (h : ∀ x y, f (x ⊓ y) = f x ⊓ f y) : Monotone f 
Arguments: {α : Type u} {β : Type v} [SemilatticeInf α] [SemilatticeInf β] {f : α → β} (h : ∀ (x y : α), f (x ⊓ y) = f x ⊓ f y)
Goal: Monotone f
-- Monotone.of_map_sup
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β] {f : α → β}
    (h : ∀ x y, f (x ⊔ y) = f x ⊔ f y) : Monotone f 
Arguments: {α : Type u} {β : Type v} [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (h : ∀ (x y : α), f (x ⊔ y) = f x ⊔ f y)
Goal: Monotone f
-- Monotone.map_sup
theorem map_sup [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α) :
    f (x ⊔ y) = f x ⊔ f y 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [SemilatticeSup β] {f : α → β} (hf : Monotone f) (x y : α)
Goal: f (x ⊔ y) = f x ⊔ f y
-- Monotone.map_inf
theorem map_inf [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α) :
    f (x ⊓ y) = f x ⊓ f y 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [SemilatticeInf β] {f : α → β} (hf : Monotone f) (x y : α)
Goal: f (x ⊓ y) = f x ⊓ f y
-- MonotoneOn.sup
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
    (hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊔ g) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonotoneOn (f ⊔ g) s
-- MonotoneOn.inf
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
    (hf : MonotoneOn f s) (hg : MonotoneOn g s) : MonotoneOn (f ⊓ g) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonotoneOn (f ⊓ g) s
-- MonotoneOn.max
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
    (hg : MonotoneOn g s) : MonotoneOn (fun x => max (f x) (g x)) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonotoneOn (fun x => max (f x) (g x)) s
-- MonotoneOn.min
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s)
    (hg : MonotoneOn g s) : MonotoneOn (fun x => min (f x) (g x)) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : MonotoneOn f s) (hg : MonotoneOn g s)
Goal: MonotoneOn (fun x => min (f x) (g x)) s
-- MonotoneOn.of_map_inf
theorem of_map_inf [SemilatticeInf α] [SemilatticeInf β]
    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊓ f y) : MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} [SemilatticeInf α] [SemilatticeInf β] (h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊓ f y)
Goal: MonotoneOn f s
-- MonotoneOn.of_map_sup
theorem of_map_sup [SemilatticeSup α] [SemilatticeSup β]
    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊔ f y) : MonotoneOn f s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} [SemilatticeSup α] [SemilatticeSup β] (h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊔ y) = f x ⊔ f y)
Goal: MonotoneOn f s
-- MonotoneOn.map_sup
theorem map_sup [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
    f (x ⊔ y) = f x ⊔ f y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} {x y : α} [LinearOrder α] [SemilatticeSup β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s)
Goal: f (x ⊔ y) = f x ⊔ f y
-- MonotoneOn.map_inf
theorem map_inf [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
    f (x ⊓ y) = f x ⊓ f y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} {x y : α} [LinearOrder α] [SemilatticeInf β] (hf : MonotoneOn f s) (hx : x ∈ s) (hy : y ∈ s)
Goal: f (x ⊓ y) = f x ⊓ f y
-- Antitone.sup
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f)
    (hg : Antitone g) :
    Antitone (f ⊔ g) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeSup β] {f g : α → β} (hf : Antitone f) (hg : Antitone g)
Goal: Antitone (f ⊔ g)
-- Antitone.inf
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f)
    (hg : Antitone g) :
    Antitone (f ⊓ g) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeInf β] {f g : α → β} (hf : Antitone f) (hg : Antitone g)
Goal: Antitone (f ⊓ g)
-- Antitone.max
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
    (hg : Antitone g) :
    Antitone fun x => max (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f) (hg : Antitone g)
Goal: Antitone fun x => max (f x) (g x)
-- Antitone.min
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f)
    (hg : Antitone g) :
    Antitone fun x => min (f x) (g x) 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} (hf : Antitone f) (hg : Antitone g)
Goal: Antitone fun x => min (f x) (g x)
-- Antitone.map_sup_le
theorem map_sup_le [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α) :
    f (x ⊔ y) ≤ f x ⊓ f y 
Arguments: {α : Type u} {β : Type v} [SemilatticeSup α] [SemilatticeInf β] {f : α → β} (h : Antitone f) (x y : α)
Goal: f (x ⊔ y) ≤ f x ⊓ f y
-- Antitone.le_map_inf
theorem le_map_inf [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α) :
    f x ⊔ f y ≤ f (x ⊓ y) 
Arguments: {α : Type u} {β : Type v} [SemilatticeInf α] [SemilatticeSup β] {f : α → β} (h : Antitone f) (x y : α)
Goal: f x ⊔ f y ≤ f (x ⊓ y)
-- Antitone.map_sup
theorem map_sup [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α) :
    f (x ⊔ y) = f x ⊓ f y 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [SemilatticeInf β] {f : α → β} (hf : Antitone f) (x y : α)
Goal: f (x ⊔ y) = f x ⊓ f y
-- Antitone.map_inf
theorem map_inf [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α) :
    f (x ⊓ y) = f x ⊔ f y 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [SemilatticeSup β] {f : α → β} (hf : Antitone f) (x y : α)
Goal: f (x ⊓ y) = f x ⊔ f y
-- AntitoneOn.sup
protected theorem sup [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α}
    (hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊔ g) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeSup β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: AntitoneOn (f ⊔ g) s
-- AntitoneOn.inf
protected theorem inf [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α}
    (hf : AntitoneOn f s) (hg : AntitoneOn g s) : AntitoneOn (f ⊓ g) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [SemilatticeInf β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: AntitoneOn (f ⊓ g) s
-- AntitoneOn.max
protected theorem max [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
    (hg : AntitoneOn g s) : AntitoneOn (fun x => max (f x) (g x)) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: AntitoneOn (fun x => max (f x) (g x)) s
-- AntitoneOn.min
protected theorem min [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s)
    (hg : AntitoneOn g s) : AntitoneOn (fun x => min (f x) (g x)) s 
Arguments: {α : Type u} {β : Type v} [Preorder α] [LinearOrder β] {f g : α → β} {s : Set α} (hf : AntitoneOn f s) (hg : AntitoneOn g s)
Goal: AntitoneOn (fun x => min (f x) (g x)) s
-- AntitoneOn.of_map_inf
theorem of_map_inf [SemilatticeInf α] [SemilatticeSup β]
    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊓ y) = f x ⊔ f y) : AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} [SemilatticeInf α] [SemilatticeSup β] (h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊓ y) = f x ⊔ f y)
Goal: AntitoneOn f s
-- AntitoneOn.of_map_sup
theorem of_map_sup [SemilatticeSup α] [SemilatticeInf β]
    (h : ∀ x ∈ s, ∀ y ∈ s, f (x ⊔ y) = f x ⊓ f y) : AntitoneOn f s 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} [SemilatticeSup α] [SemilatticeInf β] (h : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → f (x ⊔ y) = f x ⊓ f y)
Goal: AntitoneOn f s
-- AntitoneOn.map_sup
theorem map_sup [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
    f (x ⊔ y) = f x ⊓ f y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} {x y : α} [LinearOrder α] [SemilatticeInf β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s)
Goal: f (x ⊔ y) = f x ⊓ f y
-- AntitoneOn.map_inf
theorem map_inf [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s) :
    f (x ⊓ y) = f x ⊔ f y 
Arguments: {α : Type u} {β : Type v} {f : α → β} {s : Set α} {x y : α} [LinearOrder α] [SemilatticeSup β] (hf : AntitoneOn f s) (hx : x ∈ s) (hy : y ∈ s)
Goal: f (x ⊓ y) = f x ⊔ f y
-- Prod.mk_sup_mk
@[simp]
theorem mk_sup_mk [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β) :
    (a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂) 
Arguments: (α : Type u) (β : Type v) [Sup α] [Sup β] (a₁ a₂ : α) (b₁ b₂ : β)
Goal: (a₁, b₁) ⊔ (a₂, b₂) = (a₁ ⊔ a₂, b₁ ⊔ b₂)
-- Prod.mk_inf_mk
@[simp]
theorem mk_inf_mk [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β) :
    (a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂) 
Arguments: (α : Type u) (β : Type v) [Inf α] [Inf β] (a₁ a₂ : α) (b₁ b₂ : β)
Goal: (a₁, b₁) ⊓ (a₂, b₂) = (a₁ ⊓ a₂, b₁ ⊓ b₂)
-- Prod.fst_sup
@[simp]
theorem fst_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).fst = p.fst ⊔ q.fst 
Arguments: (α : Type u) (β : Type v) [Sup α] [Sup β] (p q : α × β)
Goal: (p ⊔ q).1 = p.1 ⊔ q.1
-- Prod.fst_inf
@[simp]
theorem fst_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).fst = p.fst ⊓ q.fst 
Arguments: (α : Type u) (β : Type v) [Inf α] [Inf β] (p q : α × β)
Goal: (p ⊓ q).1 = p.1 ⊓ q.1
-- Prod.snd_sup
@[simp]
theorem snd_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).snd = p.snd ⊔ q.snd 
Arguments: (α : Type u) (β : Type v) [Sup α] [Sup β] (p q : α × β)
Goal: (p ⊔ q).2 = p.2 ⊔ q.2
-- Prod.snd_inf
@[simp]
theorem snd_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).snd = p.snd ⊓ q.snd 
Arguments: (α : Type u) (β : Type v) [Inf α] [Inf β] (p q : α × β)
Goal: (p ⊓ q).2 = p.2 ⊓ q.2
-- Prod.swap_sup
@[simp]
theorem swap_sup [Sup α] [Sup β] (p q : α × β) : (p ⊔ q).swap = p.swap ⊔ q.swap 
Arguments: (α : Type u) (β : Type v) [Sup α] [Sup β] (p q : α × β)
Goal: (p ⊔ q).swap = p.swap ⊔ q.swap
-- Prod.swap_inf
@[simp]
theorem swap_inf [Inf α] [Inf β] (p q : α × β) : (p ⊓ q).swap = p.swap ⊓ q.swap 
Arguments: (α : Type u) (β : Type v) [Inf α] [Inf β] (p q : α × β)
Goal: (p ⊓ q).swap = p.swap ⊓ q.swap
-- Prod.sup_def
theorem sup_def [Sup α] [Sup β] (p q : α × β) : p ⊔ q = (p.fst ⊔ q.fst, p.snd ⊔ q.snd) 
Arguments: (α : Type u) (β : Type v) [Sup α] [Sup β] (p q : α × β)
Goal: p ⊔ q = (p.1 ⊔ q.1, p.2 ⊔ q.2)
-- Prod.inf_def
theorem inf_def [Inf α] [Inf β] (p q : α × β) : p ⊓ q = (p.fst ⊓ q.fst, p.snd ⊓ q.snd) 
Arguments: (α : Type u) (β : Type v) [Inf α] [Inf β] (p q : α × β)
Goal: p ⊓ q = (p.1 ⊓ q.1, p.2 ⊓ q.2)
-- Subtype.coe_sup
@[simp, norm_cast]
theorem coe_sup [SemilatticeSup α] {P : α → Prop}
    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P) :
    (haveI := Subtype.semilatticeSup Psup; (x ⊔ y : Subtype P) : α) = (x ⊔ y : α) 
Arguments: {α : Type u} [SemilatticeSup α] {P : α → Prop} (Psup : ∀ ⦃x y : α⦄, P x → P y → P (x ⊔ y)) (x y : Subtype P)
Goal: ↑(x ⊔ y) = ↑x ⊔ ↑y
-- Subtype.coe_inf
@[simp, norm_cast]
theorem coe_inf [SemilatticeInf α] {P : α → Prop}
    (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P) :
    (haveI := Subtype.semilatticeInf Pinf; (x ⊓ y : Subtype P) : α) = (x ⊓ y : α) 
Arguments: {α : Type u} [SemilatticeInf α] {P : α → Prop} (Pinf : ∀ ⦃x y : α⦄, P x → P y → P (x ⊓ y)) (x y : Subtype P)
Goal: ↑(x ⊓ y) = ↑x ⊓ ↑y
-- Subtype.mk_sup_mk
@[simp]
theorem mk_sup_mk [SemilatticeSup α] {P : α → Prop}
    (Psup : ∀ ⦃x y⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y) :
    (haveI := Subtype.semilatticeSup Psup; (⟨x, hx⟩ ⊔ ⟨y, hy⟩ : Subtype P)) =
      ⟨x ⊔ y, Psup hx hy⟩ 
Arguments: {α : Type u} [SemilatticeSup α] {P : α → Prop} (Psup : ∀ ⦃x y : α⦄, P x → P y → P (x ⊔ y)) {x y : α} (hx : P x) (hy : P y)
Goal: ⟨x, hx⟩ ⊔ ⟨y, hy⟩ = ⟨x ⊔ y, ⋯⟩
-- Subtype.mk_inf_mk
@[simp]
theorem mk_inf_mk [SemilatticeInf α] {P : α → Prop}
    (Pinf : ∀ ⦃x y⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y) :
    (haveI := Subtype.semilatticeInf Pinf; (⟨x, hx⟩ ⊓ ⟨y, hy⟩ : Subtype P)) =
      ⟨x ⊓ y, Pinf hx hy⟩ 
Arguments: {α : Type u} [SemilatticeInf α] {P : α → Prop} (Pinf : ∀ ⦃x y : α⦄, P x → P y → P (x ⊓ y)) {x y : α} (hx : P x) (hy : P y)
Goal: ⟨x, hx⟩ ⊓ ⟨y, hy⟩ = ⟨x ⊓ y, ⋯⟩
-- Filter.isBounded_iff
theorem isBounded_iff : f.IsBounded r ↔ ∃ s ∈ f.sets, ∃ b, s ⊆ { x | r x b } 
Arguments: {α : Type u_1} {r : α → α → Prop} {f : Filter α}
Goal: Filter.IsBounded r f ↔ ∃ s ∈ f.sets, ∃ b, s ⊆ {x | r x b}
-- Filter.isBoundedUnder_of
theorem isBoundedUnder_of {f : Filter β} {u : β → α} : (∃ b, ∀ x, r (u x) b) → f.IsBoundedUnder r u
  
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : Filter β} {u : β → α}
Goal: (∃ b, ∀ (x : β), r (u x) b) → Filter.IsBoundedUnder r f u
-- Filter.isBounded_bot
theorem isBounded_bot : IsBounded r ⊥ ↔ Nonempty α 
Arguments: {α : Type u_1} {r : α → α → Prop}
Goal: Filter.IsBounded r ⊥ ↔ Nonempty α
-- Filter.isBounded_top
theorem isBounded_top : IsBounded r ⊤ ↔ ∃ t, ∀ x, r x t 
Arguments: {α : Type u_1} {r : α → α → Prop}
Goal: Filter.IsBounded r ⊤ ↔ ∃ t, ∀ (x : α), r x t
-- Filter.isBounded_principal
theorem isBounded_principal (s : Set α) : IsBounded r (𝓟 s) ↔ ∃ t, ∀ x ∈ s, r x t 
Arguments: {α : Type u_1} {r : α → α → Prop} (s : Set α)
Goal: Filter.IsBounded r (𝓟 s) ↔ ∃ t, ∀ x ∈ s, r x t
-- Filter.isBounded_sup
theorem isBounded_sup [IsTrans α r] [IsDirected α r] :
    IsBounded r f → IsBounded r g → IsBounded r (f ⊔ g)
  
Arguments: {α : Type u_1} {r : α → α → Prop} {f g : Filter α} [IsTrans α r] [IsDirected α r]
Goal: Filter.IsBounded r f → Filter.IsBounded r g → Filter.IsBounded r (f ⊔ g)
-- Filter.IsBounded.mono
theorem IsBounded.mono (h : f ≤ g) : IsBounded r g → IsBounded r f
  
Arguments: {α : Type u_1} {r : α → α → Prop} {f g : Filter α} (h : f ≤ g)
Goal: Filter.IsBounded r g → Filter.IsBounded r f
-- Filter.IsBoundedUnder.mono
theorem IsBoundedUnder.mono {f g : Filter β} {u : β → α} (h : f ≤ g) :
    g.IsBoundedUnder r u → f.IsBoundedUnder r u 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f g : Filter β} {u : β → α} (h : f ≤ g)
Goal: Filter.IsBoundedUnder r g u → Filter.IsBoundedUnder r f u
-- Filter.IsBoundedUnder.mono_le
theorem IsBoundedUnder.mono_le [Preorder β] {l : Filter α} {u v : α → β}
    (hu : IsBoundedUnder (· ≤ ·) l u) (hv : v ≤ᶠ[l] u) : IsBoundedUnder (· ≤ ·) l v 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder β] {l : Filter α} {u v : α → β} (hu : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l u) (hv : v ≤ᶠ[l] u)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) l v
-- Filter.IsBoundedUnder.mono_ge
theorem IsBoundedUnder.mono_ge [Preorder β] {l : Filter α} {u v : α → β}
    (hu : IsBoundedUnder (· ≥ ·) l u) (hv : u ≤ᶠ[l] v) : IsBoundedUnder (· ≥ ·) l v 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder β] {l : Filter α} {u v : α → β} (hu : Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l u) (hv : u ≤ᶠ[l] v)
Goal: Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) l v
-- Filter.isBoundedUnder_const
theorem isBoundedUnder_const [IsRefl α r] {l : Filter β} {a : α} : IsBoundedUnder r l fun _ => a 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} [IsRefl α r] {l : Filter β} {a : α}
Goal: Filter.IsBoundedUnder r l fun x => a
-- Filter.IsBounded.isBoundedUnder
theorem IsBounded.isBoundedUnder {q : β → β → Prop} {u : α → β}
    (hu : ∀ a₀ a₁, r a₀ a₁ → q (u a₀) (u a₁)) : f.IsBounded r → f.IsBoundedUnder q u
  
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {f : Filter α} {q : β → β → Prop} {u : α → β} (hu : ∀ (a₀ a₁ : α), r a₀ a₁ → q (u a₀) (u a₁))
Goal: Filter.IsBounded r f → Filter.IsBoundedUnder q f u
-- Filter.IsBoundedUnder.comp
theorem IsBoundedUnder.comp {l : Filter γ} {q : β → β → Prop} {u : γ → α} {v : α → β}
    (hv : ∀ a₀ a₁, r a₀ a₁ → q (v a₀) (v a₁)) : l.IsBoundedUnder r u → l.IsBoundedUnder q (v ∘ u)
  
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {l : Filter γ} {q : β → β → Prop} {u : γ → α} {v : α → β} (hv : ∀ (a₀ a₁ : α), r a₀ a₁ → q (v a₀) (v a₁))
Goal: Filter.IsBoundedUnder r l u → Filter.IsBoundedUnder q l (v ∘ u)
-- BddAbove.isBoundedUnder
lemma _root_.BddAbove.isBoundedUnder [Preorder α] {f : Filter β} {u : β → α} :
    BddAbove (Set.range u) → f.IsBoundedUnder (· ≤ ·) u
  
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] {f : Filter β} {u : β → α}
Goal: BddAbove (Set.range u) → Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) f u
-- BddBelow.isBoundedUnder
lemma _root_.BddBelow.isBoundedUnder [Preorder α] {f : Filter β} {u : β → α} :
    BddBelow (Set.range u) → f.IsBoundedUnder (· ≥ ·) u
  
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] {f : Filter β} {u : β → α}
Goal: BddBelow (Set.range u) → Filter.IsBoundedUnder (fun x x_1 => x ≥ x_1) f u