ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Sublattice.coe_iInf
@[simp, norm_cast] lemma coe_iInf (f : ι → Sublattice α) : ⨅ i, f i = ⋂ i, (f i : Set α) 
Arguments: {ι : Sort u_1} {α : Type u_2} [Lattice α] (f : ι → Sublattice α)
Goal: ↑(⨅ i, f i) = ⋂ i, ↑(f i)
-- Sublattice.coe_eq_univ
@[simp, norm_cast] lemma coe_eq_univ : L = (univ : Set α) ↔ L = ⊤ 
Arguments: {α : Type u_2} [Lattice α] {L : Sublattice α}
Goal: ↑L = Set.univ ↔ L = ⊤
-- Sublattice.coe_eq_empty
@[simp, norm_cast] lemma coe_eq_empty : L = (∅ : Set α) ↔ L = ⊥ 
Arguments: {α : Type u_2} [Lattice α] {L : Sublattice α}
Goal: ↑L = ∅ ↔ L = ⊥
-- Sublattice.not_mem_bot
@[simp] lemma not_mem_bot (a : α) : a ∉ (⊥ : Sublattice α) 
Arguments: {α : Type u_2} [Lattice α] (a : α)
Goal: a ∉ ⊥
-- Sublattice.mem_top
@[simp] lemma mem_top (a : α) : a ∈ (⊤ : Sublattice α) 
Arguments: {α : Type u_2} [Lattice α] (a : α)
Goal: a ∈ ⊤
-- Sublattice.mem_inf
@[simp] lemma mem_inf : a ∈ L ⊓ M ↔ a ∈ L ∧ a ∈ M 
Arguments: {α : Type u_2} [Lattice α] {L M : Sublattice α} {a : α}
Goal: a ∈ L ⊓ M ↔ a ∈ L ∧ a ∈ M
-- Sublattice.mem_sInf
@[simp] lemma mem_sInf {S : Set (Sublattice α)} : a ∈ sInf S ↔ ∀ L ∈ S, a ∈ L 
Arguments: {α : Type u_2} [Lattice α] {a : α} {S : Set (Sublattice α)}
Goal: a ∈ sInf S ↔ ∀ L ∈ S, a ∈ L
-- Sublattice.mem_iInf
@[simp] lemma mem_iInf {f : ι → Sublattice α} : a ∈ ⨅ i, f i ↔ ∀ i, a ∈ f i 
Arguments: {ι : Sort u_1} {α : Type u_2} [Lattice α] {a : α} {f : ι → Sublattice α}
Goal: a ∈ ⨅ i, f i ↔ ∀ (i : ι), a ∈ f i
-- Sublattice.subsingleton_iff
lemma subsingleton_iff : Subsingleton (Sublattice α) ↔ IsEmpty α 
Arguments: {α : Type u_2} [Lattice α]
Goal: Subsingleton (Sublattice α) ↔ IsEmpty α
-- Sublattice.coe_comap
@[simp, norm_cast] lemma coe_comap (L : Sublattice β) (f : LatticeHom α β) : L.comap f = f ⁻¹' L 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (L : Sublattice β) (f : LatticeHom α β)
Goal: ↑(Sublattice.comap f L) = ⇑f ⁻¹' ↑L
-- Sublattice.mem_comap
@[simp] lemma mem_comap {L : Sublattice β} : a ∈ L.comap f ↔ f a ∈ L 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {f : LatticeHom α β} {a : α} {L : Sublattice β}
Goal: a ∈ Sublattice.comap f L ↔ f a ∈ L
-- Sublattice.comap_mono
lemma comap_mono : Monotone (comap f) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {f : LatticeHom α β}
Goal: Monotone (Sublattice.comap f)
-- Sublattice.comap_id
@[simp] lemma comap_id (L : Sublattice α) : L.comap (LatticeHom.id _) = L 
Arguments: {α : Type u_2} [Lattice α] (L : Sublattice α)
Goal: Sublattice.comap (LatticeHom.id α) L = L
-- Sublattice.comap_comap
@[simp] lemma comap_comap (L : Sublattice γ) (g : LatticeHom β γ) (f : LatticeHom α β) :
    (L.comap g).comap f = L.comap (g.comp f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Lattice α] [Lattice β] [Lattice γ] (L : Sublattice γ) (g : LatticeHom β γ) (f : LatticeHom α β)
Goal: Sublattice.comap f (Sublattice.comap g L) = Sublattice.comap (g.comp f) L
-- Sublattice.coe_map
@[simp] lemma coe_map (f : LatticeHom α β) (L : Sublattice α) : (L.map f : Set β) = f '' L 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β) (L : Sublattice α)
Goal: ↑(Sublattice.map f L) = ⇑f '' ↑L
-- Sublattice.mem_map
@[simp] lemma mem_map {b : β} : b ∈ L.map f ↔ ∃ a ∈ L, f a = b 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} {f : LatticeHom α β} {b : β}
Goal: b ∈ Sublattice.map f L ↔ ∃ a ∈ L, f a = b
-- Sublattice.mem_map_of_mem
lemma mem_map_of_mem (f : LatticeHom α β) {a : α} : a ∈ L → f a ∈ L.map f 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} (f : LatticeHom α β) {a : α}
Goal: a ∈ L → f a ∈ Sublattice.map f L
-- Sublattice.apply_coe_mem_map
lemma apply_coe_mem_map (f : LatticeHom α β) (a : L) : f a ∈ L.map f 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} (f : LatticeHom α β) (a : ↥L)
Goal: f ↑a ∈ Sublattice.map f L
-- Sublattice.map_mono
lemma map_mono : Monotone (map f) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {f : LatticeHom α β}
Goal: Monotone (Sublattice.map f)
-- Sublattice.map_id
@[simp] lemma map_id : L.map (LatticeHom.id α) = L 
Arguments: {α : Type u_2} [Lattice α] {L : Sublattice α}
Goal: Sublattice.map (LatticeHom.id α) L = L
-- Sublattice.map_map
@[simp] lemma map_map (g : LatticeHom β γ) (f : LatticeHom α β) :
    (L.map f).map g = L.map (g.comp f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [Lattice α] [Lattice β] [Lattice γ] {L : Sublattice α} (g : LatticeHom β γ) (f : LatticeHom α β)
Goal: Sublattice.map g (Sublattice.map f L) = Sublattice.map (g.comp f) L
-- Sublattice.mem_map_equiv
lemma mem_map_equiv {f : α ≃o β} {a : β} : a ∈ L.map f ↔ f.symm a ∈ L 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} {f : α ≃o β} {a : β}
Goal: a ∈ Sublattice.map { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯ } L ↔ f.symm a ∈ L
-- Sublattice.apply_mem_map_iff
lemma apply_mem_map_iff (hf : Injective f) : f a ∈ L.map f ↔ a ∈ L 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} {f : LatticeHom α β} {a : α} (hf : Function.Injective ⇑f)
Goal: f a ∈ Sublattice.map f L ↔ a ∈ L
-- Sublattice.map_equiv_eq_comap_symm
lemma map_equiv_eq_comap_symm (f : α ≃o β) (L : Sublattice α) :
    L.map f = L.comap (f.symm : LatticeHom β α) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : α ≃o β) (L : Sublattice α)
Goal: Sublattice.map { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯ } L =
  Sublattice.comap { toFun := ⇑f.symm, map_sup' := ⋯, map_inf' := ⋯ } L
-- Sublattice.comap_equiv_eq_map_symm
lemma comap_equiv_eq_map_symm (f : β ≃o α) (L : Sublattice α) :
    L.comap f = L.map (f.symm : LatticeHom α β) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : β ≃o α) (L : Sublattice α)
Goal: Sublattice.comap { toFun := ⇑f, map_sup' := ⋯, map_inf' := ⋯ } L =
  Sublattice.map { toFun := ⇑f.symm, map_sup' := ⋯, map_inf' := ⋯ } L
-- Sublattice.map_symm_eq_iff_eq_map
lemma map_symm_eq_iff_eq_map {M : Sublattice β} {e : β ≃o α} :
    L.map ↑e.symm = M ↔ L = M.map ↑e 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} {M : Sublattice β} {e : β ≃o α}
Goal: Sublattice.map { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ } L = M ↔
  L = Sublattice.map { toFun := ⇑e, map_sup' := ⋯, map_inf' := ⋯ } M
-- Sublattice.map_le_iff_le_comap
lemma map_le_iff_le_comap {f : LatticeHom α β} {M : Sublattice β} : L.map f ≤ M ↔ L ≤ M.comap f 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] {L : Sublattice α} {f : LatticeHom α β} {M : Sublattice β}
Goal: Sublattice.map f L ≤ M ↔ L ≤ Sublattice.comap f M
-- Sublattice.gc_map_comap
lemma gc_map_comap (f : LatticeHom α β) : GaloisConnection (map f) (comap f) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: GaloisConnection (Sublattice.map f) (Sublattice.comap f)
-- Sublattice.map_bot
@[simp] lemma map_bot (f : LatticeHom α β) : (⊥ : Sublattice α).map f = ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: Sublattice.map f ⊥ = ⊥
-- Sublattice.map_sup
lemma map_sup (f : LatticeHom α β) (L M : Sublattice α) : (L ⊔ M).map f = L.map f ⊔ M.map f 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β) (L M : Sublattice α)
Goal: Sublattice.map f (L ⊔ M) = Sublattice.map f L ⊔ Sublattice.map f M
-- Sublattice.map_iSup
lemma map_iSup (f : LatticeHom α β) (L : ι → Sublattice α) : (⨆ i, L i).map f = ⨆ i, (L i).map f 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β) (L : ι → Sublattice α)
Goal: Sublattice.map f (⨆ i, L i) = ⨆ i, Sublattice.map f (L i)
-- Sublattice.comap_top
@[simp] lemma comap_top (f : LatticeHom α β) : (⊤ : Sublattice β).comap f = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β)
Goal: Sublattice.comap f ⊤ = ⊤
-- Sublattice.comap_inf
lemma comap_inf (L M : Sublattice β) (f : LatticeHom α β) :
    (L ⊓ M).comap f = L.comap f ⊓ M.comap f 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (L M : Sublattice β) (f : LatticeHom α β)
Goal: Sublattice.comap f (L ⊓ M) = Sublattice.comap f L ⊓ Sublattice.comap f M
-- Sublattice.comap_iInf
lemma comap_iInf (f : LatticeHom α β) (s : ι → Sublattice β) :
    (iInf s).comap f = ⨅ i, (s i).comap f 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β) (s : ι → Sublattice β)
Goal: Sublattice.comap f (iInf s) = ⨅ i, Sublattice.comap f (s i)
-- Sublattice.map_inf_le
lemma map_inf_le (L M : Sublattice α) (f : LatticeHom α β) : map f (L ⊓ M) ≤ map f L ⊓ map f M 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (L M : Sublattice α) (f : LatticeHom α β)
Goal: Sublattice.map f (L ⊓ M) ≤ Sublattice.map f L ⊓ Sublattice.map f M
-- Sublattice.le_comap_sup
lemma le_comap_sup (L M : Sublattice β) (f : LatticeHom α β) :
    comap f L ⊔ comap f M ≤ comap f (L ⊔ M) 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (L M : Sublattice β) (f : LatticeHom α β)
Goal: Sublattice.comap f L ⊔ Sublattice.comap f M ≤ Sublattice.comap f (L ⊔ M)
-- Sublattice.le_comap_iSup
lemma le_comap_iSup (f : LatticeHom α β) (L : ι → Sublattice β) :
    ⨆ i, (L i).comap f ≤ (⨆ i, L i).comap f 
Arguments: {ι : Sort u_1} {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β) (L : ι → Sublattice β)
Goal: ⨆ i, Sublattice.comap f (L i) ≤ Sublattice.comap f (⨆ i, L i)
-- Sublattice.map_inf
lemma map_inf (L M : Sublattice α) (f : LatticeHom α β) (hf : Injective f) :
    map f (L ⊓ M) = map f L ⊓ map f M 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (L M : Sublattice α) (f : LatticeHom α β) (hf : Function.Injective ⇑f)
Goal: Sublattice.map f (L ⊓ M) = Sublattice.map f L ⊓ Sublattice.map f M
-- Sublattice.map_top
lemma map_top (f : LatticeHom α β) (h : Surjective f) : Sublattice.map f ⊤ = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [Lattice α] [Lattice β] (f : LatticeHom α β) (h : Function.Surjective ⇑f)
Goal: Sublattice.map f ⊤ = ⊤
-- le_min_iff
@[simp]
theorem le_min_iff : c ≤ min a b ↔ c ≤ a ∧ c ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: c ≤ min a b ↔ c ≤ a ∧ c ≤ b
-- le_max_iff
@[simp]
theorem le_max_iff : a ≤ max b c ↔ a ≤ b ∨ a ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a ≤ max b c ↔ a ≤ b ∨ a ≤ c
-- min_le_iff
@[simp]
theorem min_le_iff : min a b ≤ c ↔ a ≤ c ∨ b ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: min a b ≤ c ↔ a ≤ c ∨ b ≤ c
-- max_le_iff
@[simp]
theorem max_le_iff : max a b ≤ c ↔ a ≤ c ∧ b ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: max a b ≤ c ↔ a ≤ c ∧ b ≤ c
-- lt_min_iff
@[simp]
theorem lt_min_iff : a < min b c ↔ a < b ∧ a < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a < min b c ↔ a < b ∧ a < c
-- lt_max_iff
@[simp]
theorem lt_max_iff : a < max b c ↔ a < b ∨ a < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a < max b c ↔ a < b ∨ a < c
-- min_lt_iff
@[simp]
theorem min_lt_iff : min a b < c ↔ a < c ∨ b < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: min a b < c ↔ a < c ∨ b < c
-- max_lt_iff
@[simp]
theorem max_lt_iff : max a b < c ↔ a < c ∧ b < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: max a b < c ↔ a < c ∧ b < c
-- max_le_max
@[gcongr]
theorem max_le_max : a ≤ c → b ≤ d → max a b ≤ max c d 
Arguments: {α : Type u} [LinearOrder α] {a b c d : α}
Goal: a ≤ c → b ≤ d → max a b ≤ max c d
-- max_le_max_left
@[gcongr] theorem max_le_max_left (c) (h : a ≤ b) : max c a ≤ max c b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (c : α) (h : a ≤ b)
Goal: max c a ≤ max c b
-- max_le_max_right
@[gcongr] theorem max_le_max_right (c) (h : a ≤ b) : max a c ≤ max b c 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (c : α) (h : a ≤ b)
Goal: max a c ≤ max b c
-- min_le_min
@[gcongr]
theorem min_le_min : a ≤ c → b ≤ d → min a b ≤ min c d 
Arguments: {α : Type u} [LinearOrder α] {a b c d : α}
Goal: a ≤ c → b ≤ d → min a b ≤ min c d
-- min_le_min_left
@[gcongr] theorem min_le_min_left (c) (h : a ≤ b) : min c a ≤ min c b 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (c : α) (h : a ≤ b)
Goal: min c a ≤ min c b
-- min_le_min_right
@[gcongr] theorem min_le_min_right (c) (h : a ≤ b) : min a c ≤ min b c 
Arguments: {α : Type u} [LinearOrder α] {a b : α} (c : α) (h : a ≤ b)
Goal: min a c ≤ min b c
-- le_max_of_le_left
theorem le_max_of_le_left : a ≤ b → a ≤ max b c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a ≤ b → a ≤ max b c
-- le_max_of_le_right
theorem le_max_of_le_right : a ≤ c → a ≤ max b c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a ≤ c → a ≤ max b c
-- lt_max_of_lt_left
theorem lt_max_of_lt_left (h : a < b) : a < max b c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h : a < b)
Goal: a < max b c
-- lt_max_of_lt_right
theorem lt_max_of_lt_right (h : a < c) : a < max b c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h : a < c)
Goal: a < max b c
-- min_le_of_left_le
theorem min_le_of_left_le : a ≤ c → min a b ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: a ≤ c → min a b ≤ c
-- min_le_of_right_le
theorem min_le_of_right_le : b ≤ c → min a b ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: b ≤ c → min a b ≤ c
-- min_lt_of_left_lt
theorem min_lt_of_left_lt (h : a < c) : min a b < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h : a < c)
Goal: min a b < c
-- min_lt_of_right_lt
theorem min_lt_of_right_lt (h : b < c) : min a b < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h : b < c)
Goal: min a b < c
-- max_min_distrib_left
lemma max_min_distrib_left (a b c : α) : max a (min b c) = min (max a b) (max a c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: max a (min b c) = min (max a b) (max a c)
-- max_min_distrib_right
lemma max_min_distrib_right (a b c : α) : max (min a b) c = min (max a c) (max b c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: max (min a b) c = min (max a c) (max b c)
-- min_max_distrib_left
lemma min_max_distrib_left (a b c : α) : min a (max b c) = max (min a b) (min a c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: min a (max b c) = max (min a b) (min a c)
-- min_max_distrib_right
lemma min_max_distrib_right (a b c : α) : min (max a b) c = max (min a c) (min b c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: min (max a b) c = max (min a c) (min b c)
-- min_le_max
theorem min_le_max : min a b ≤ max a b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: min a b ≤ max a b
-- min_eq_left_iff
@[simp]
theorem min_eq_left_iff : min a b = a ↔ a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: min a b = a ↔ a ≤ b
-- min_eq_right_iff
@[simp]
theorem min_eq_right_iff : min a b = b ↔ b ≤ a 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: min a b = b ↔ b ≤ a
-- max_eq_left_iff
@[simp]
theorem max_eq_left_iff : max a b = a ↔ b ≤ a 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: max a b = a ↔ b ≤ a
-- max_eq_right_iff
@[simp]
theorem max_eq_right_iff : max a b = b ↔ a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: max a b = b ↔ a ≤ b
-- min_cases
theorem min_cases (a b : α) : min a b = a ∧ a ≤ b ∨ min a b = b ∧ b < a 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: min a b = a ∧ a ≤ b ∨ min a b = b ∧ b < a
-- max_cases
theorem max_cases (a b : α) : max a b = a ∧ b ≤ a ∨ max a b = b ∧ a < b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: max a b = a ∧ b ≤ a ∨ max a b = b ∧ a < b
-- min_eq_iff
theorem min_eq_iff : min a b = c ↔ a = c ∧ a ≤ b ∨ b = c ∧ b ≤ a 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: min a b = c ↔ a = c ∧ a ≤ b ∨ b = c ∧ b ≤ a
-- max_eq_iff
theorem max_eq_iff : max a b = c ↔ a = c ∧ b ≤ a ∨ b = c ∧ a ≤ b 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: max a b = c ↔ a = c ∧ b ≤ a ∨ b = c ∧ a ≤ b
-- min_lt_min_left_iff
theorem min_lt_min_left_iff : min a c < min b c ↔ a < b ∧ a < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: min a c < min b c ↔ a < b ∧ a < c
-- min_lt_min_right_iff
theorem min_lt_min_right_iff : min a b < min a c ↔ b < c ∧ b < a 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: min a b < min a c ↔ b < c ∧ b < a
-- max_lt_max_left_iff
theorem max_lt_max_left_iff : max a c < max b c ↔ a < b ∧ c < b 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: max a c < max b c ↔ a < b ∧ c < b
-- max_lt_max_right_iff
theorem max_lt_max_right_iff : max a b < max a c ↔ b < c ∧ a < c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α}
Goal: max a b < max a c ↔ b < c ∧ a < c
-- min_lt_max
theorem min_lt_max : min a b < max a b ↔ a ≠ b 
Arguments: {α : Type u} [LinearOrder α] {a b : α}
Goal: min a b < max a b ↔ a ≠ b
-- max_lt_max
theorem max_lt_max (h₁ : a < c) (h₂ : b < d) : max a b < max c d 
Arguments: {α : Type u} [LinearOrder α] {a b c d : α} (h₁ : a < c) (h₂ : b < d)
Goal: max a b < max c d
-- min_lt_min
theorem min_lt_min (h₁ : a < c) (h₂ : b < d) : min a b < min c d 
Arguments: {α : Type u} [LinearOrder α] {a b c d : α} (h₁ : a < c) (h₂ : b < d)
Goal: min a b < min c d
-- min_right_comm
theorem min_right_comm (a b c : α) : min (min a b) c = min (min a c) b 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: min (min a b) c = min (min a c) b
-- Max.left_comm
theorem Max.left_comm (a b c : α) : max a (max b c) = max b (max a c) 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: max a (max b c) = max b (max a c)
-- Max.right_comm
theorem Max.right_comm (a b c : α) : max (max a b) c = max (max a c) b 
Arguments: {α : Type u} [LinearOrder α] (a b c : α)
Goal: max (max a b) c = max (max a c) b
-- MonotoneOn.map_max
theorem MonotoneOn.map_max (hf : MonotoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (max a b) =
    max (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Set α} {a b : α} (hf : MonotoneOn f s) (ha : a ∈ s) (hb : b ∈ s)
Goal: f (max a b) = max (f a) (f b)
-- MonotoneOn.map_min
theorem MonotoneOn.map_min (hf : MonotoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (min a b) =
    min (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Set α} {a b : α} (hf : MonotoneOn f s) (ha : a ∈ s) (hb : b ∈ s)
Goal: f (min a b) = min (f a) (f b)
-- AntitoneOn.map_max
theorem AntitoneOn.map_max (hf : AntitoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (max a b) =
    min (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Set α} {a b : α} (hf : AntitoneOn f s) (ha : a ∈ s) (hb : b ∈ s)
Goal: f (max a b) = min (f a) (f b)
-- AntitoneOn.map_min
theorem AntitoneOn.map_min (hf : AntitoneOn f s) (ha : a ∈ s) (hb : b ∈ s) : f (min a b) =
    max (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Set α} {a b : α} (hf : AntitoneOn f s) (ha : a ∈ s) (hb : b ∈ s)
Goal: f (min a b) = max (f a) (f b)
-- Monotone.map_max
theorem Monotone.map_max (hf : Monotone f) : f (max a b) = max (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {a b : α} (hf : Monotone f)
Goal: f (max a b) = max (f a) (f b)
-- Monotone.map_min
theorem Monotone.map_min (hf : Monotone f) : f (min a b) = min (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {a b : α} (hf : Monotone f)
Goal: f (min a b) = min (f a) (f b)
-- Antitone.map_max
theorem Antitone.map_max (hf : Antitone f) : f (max a b) = min (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {a b : α} (hf : Antitone f)
Goal: f (max a b) = min (f a) (f b)
-- Antitone.map_min
theorem Antitone.map_min (hf : Antitone f) : f (min a b) = max (f a) (f b) 
Arguments: {α : Type u} {β : Type v} [LinearOrder α] [LinearOrder β] {f : α → β} {a b : α} (hf : Antitone f)
Goal: f (min a b) = max (f a) (f b)
-- min_choice
theorem min_choice (a b : α) : min a b = a ∨ min a b = b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: min a b = a ∨ min a b = b
-- max_choice
theorem max_choice (a b : α) : max a b = a ∨ max a b = b 
Arguments: {α : Type u} [LinearOrder α] (a b : α)
Goal: max a b = a ∨ max a b = b
-- le_of_max_le_left
theorem le_of_max_le_left {a b c : α} (h : max a b ≤ c) : a ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h : max a b ≤ c)
Goal: a ≤ c
-- le_of_max_le_right
theorem le_of_max_le_right {a b c : α} (h : max a b ≤ c) : b ≤ c 
Arguments: {α : Type u} [LinearOrder α] {a b c : α} (h : max a b ≤ c)
Goal: b ≤ c
-- max_commutative
theorem max_commutative : Commutative (max : α → α → α) 
Arguments: {α : Type u} [LinearOrder α]
Goal: Commutative max
-- max_associative
theorem max_associative : Associative (max : α → α → α) 
Arguments: {α : Type u} [LinearOrder α]
Goal: Associative max
-- max_left_commutative
theorem max_left_commutative : LeftCommutative (max : α → α → α) 
Arguments: {α : Type u} [LinearOrder α]
Goal: LeftCommutative max
-- min_commutative
theorem min_commutative : Commutative (min : α → α → α) 
Arguments: {α : Type u} [LinearOrder α]
Goal: Commutative min
-- min_associative
theorem min_associative : Associative (α := α) min 
Arguments: {α : Type u} [LinearOrder α]
Goal: Associative min
-- min_left_commutative
theorem min_left_commutative : LeftCommutative (min : α → α → α) 
Arguments: {α : Type u} [LinearOrder α]
Goal: LeftCommutative min
-- Nat.comap_cast_atTop
@[simp]
theorem Nat.comap_cast_atTop [StrictOrderedSemiring R] [Archimedean R] :
    comap ((↑) : ℕ → R) atTop = atTop 
Arguments: {R : Type u_2} [StrictOrderedSemiring R] [Archimedean R]
Goal: Filter.comap Nat.cast Filter.atTop = Filter.atTop
-- tendsto_natCast_atTop_iff
theorem tendsto_natCast_atTop_iff [StrictOrderedSemiring R] [Archimedean R] {f : α → ℕ}
    {l : Filter α} : Tendsto (fun n => (f n : R)) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_1} {R : Type u_2} [StrictOrderedSemiring R] [Archimedean R] {f : α → ℕ} {l : Filter α}
Goal: Filter.Tendsto (fun n => ↑(f n)) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- tendsto_natCast_atTop_atTop
theorem tendsto_natCast_atTop_atTop [OrderedSemiring R] [Archimedean R] :
    Tendsto ((↑) : ℕ → R) atTop atTop 
Arguments: {R : Type u_2} [OrderedSemiring R] [Archimedean R]
Goal: Filter.Tendsto Nat.cast Filter.atTop Filter.atTop
-- Filter.Eventually.natCast_atTop
theorem Filter.Eventually.natCast_atTop [OrderedSemiring R] [Archimedean R] {p : R → Prop}
    (h : ∀ᶠ (x : R) in atTop, p x) : ∀ᶠ (n : ℕ) in atTop, p n 
Arguments: {R : Type u_2} [OrderedSemiring R] [Archimedean R] {p : R → Prop} (h : ∀ᶠ (x : R) in Filter.atTop, p x)
Goal: ∀ᶠ (n : ℕ) in Filter.atTop, p ↑n
-- Int.comap_cast_atTop
@[simp] theorem Int.comap_cast_atTop [StrictOrderedRing R] [Archimedean R] :
    comap ((↑) : ℤ → R) atTop = atTop 
Arguments: {R : Type u_2} [StrictOrderedRing R] [Archimedean R]
Goal: Filter.comap Int.cast Filter.atTop = Filter.atTop
-- Int.comap_cast_atBot
@[simp]
theorem Int.comap_cast_atBot [StrictOrderedRing R] [Archimedean R] :
    comap ((↑) : ℤ → R) atBot = atBot 
Arguments: {R : Type u_2} [StrictOrderedRing R] [Archimedean R]
Goal: Filter.comap Int.cast Filter.atBot = Filter.atBot
-- tendsto_intCast_atTop_iff
theorem tendsto_intCast_atTop_iff [StrictOrderedRing R] [Archimedean R] {f : α → ℤ}
    {l : Filter α} : Tendsto (fun n => (f n : R)) l atTop ↔ Tendsto f l atTop 
Arguments: {α : Type u_1} {R : Type u_2} [StrictOrderedRing R] [Archimedean R] {f : α → ℤ} {l : Filter α}
Goal: Filter.Tendsto (fun n => ↑(f n)) l Filter.atTop ↔ Filter.Tendsto f l Filter.atTop
-- tendsto_intCast_atBot_iff
theorem tendsto_intCast_atBot_iff [StrictOrderedRing R] [Archimedean R] {f : α → ℤ}
    {l : Filter α} : Tendsto (fun n => (f n : R)) l atBot ↔ Tendsto f l atBot 
Arguments: {α : Type u_1} {R : Type u_2} [StrictOrderedRing R] [Archimedean R] {f : α → ℤ} {l : Filter α}
Goal: Filter.Tendsto (fun n => ↑(f n)) l Filter.atBot ↔ Filter.Tendsto f l Filter.atBot
-- tendsto_intCast_atTop_atTop
theorem tendsto_intCast_atTop_atTop [StrictOrderedRing R] [Archimedean R] :
    Tendsto ((↑) : ℤ → R) atTop atTop 
Arguments: {R : Type u_2} [StrictOrderedRing R] [Archimedean R]
Goal: Filter.Tendsto Int.cast Filter.atTop Filter.atTop