Loogle!

Result

Found 219 declarations mentioning Set.Infinite. Of these, only the first 200 are shown.

Set.Infinite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} (s : Set α) : Prop
Set.finite_or_infinite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} (s : Set α) : s.Finite ∨ s.Infinite
Set.infinite_or_finite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} (s : Set α) : s.Infinite ∨ s.Finite
Set.Finite.not_infinite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} {s : Set α} : s.Finite → ¬s.Infinite
Set.Infinite.not_finite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} {s : Set α} (hs : s.Infinite) : ¬s.Finite
Set.Infinite.to_subtype 📋 Mathlib.Basic.Finite.Defs
{α : Type u} {s : Set α} : s.Infinite → Infinite ↑s
Set.infinite_coe_iff 📋 Mathlib.Basic.Finite.Defs
{α : Type u} {s : Set α} : Infinite ↑s ↔ s.Infinite
Set.not_finite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} {s : Set α} : ¬s.Finite ↔ s.Infinite
Set.not_infinite 📋 Mathlib.Basic.Finite.Defs
{α : Type u} {s : Set α} : ¬s.Infinite ↔ s.Finite
Set.infinite_univ 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} [h : Infinite α] : Set.univ.Infinite
Set.infinite_univ_iff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} : Set.univ.Infinite ↔ Infinite α
Set.Infinite.nonempty 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s : Set α} (h : s.Infinite) : s.Nonempty
Set.Infinite.nontrivial 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s : Set α} (hs : s.Infinite) : s.Nontrivial
Set.Infinite.natEmbedding 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} (s : Set α) (h : s.Infinite) : ℕ ↪ ↑s
Set.infinite_of_finite_compl 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} [Infinite α] {s : Set α} (hs : sᶜ.Finite) : s.Infinite
Set.infinite_range_of_injective 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} [Infinite α] {f : α → β} (hi : Function.Injective f) : (Set.range f).Infinite
Set.Finite.infinite_compl 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} [Infinite α] {s : Set α} (hs : s.Finite) : sᶜ.Infinite
Set.Infinite.of_image 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} (f : α → β) {s : Set α} (hs : (f '' s).Infinite) : s.Infinite
Set.infinite_range_iff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {f : α → β} (hf : Function.Injective f) : (Set.range f).Infinite ↔ Infinite α
Set.Infinite.mono 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s t : Set α} (h : s ⊆ t) : s.Infinite → t.Infinite
Set.Infinite.diff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s t : Set α} (hs : s.Infinite) (ht : t.Finite) : (s \ t).Infinite
Set.Infinite.sdiff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s t : Set α} (hs : s.Infinite) (ht : t.Finite) : (s \ t).Infinite
Set.Infinite.image 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {s : Set α} {f : α → β} (hi : Set.InjOn f s) : s.Infinite → (f '' s).Infinite
Set.infinite_image_iff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {s : Set α} {f : α → β} (hi : Set.InjOn f s) : (f '' s).Infinite ↔ s.Infinite
Set.infinite_union 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s t : Set α} : (s ∪ t).Infinite ↔ s.Infinite ∨ t.Infinite
Set.not_injOn_infinite_finite_image 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {f : α → β} {s : Set α} (h_inf : s.Infinite) (h_fin : (f '' s).Finite) : ¬Set.InjOn f s
Set.infinite_of_injOn_mapsTo 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {s : Set α} {t : Set β} {f : α → β} (hi : Set.InjOn f s) (hm : Set.MapsTo f s t) (hs : s.Infinite) : t.Infinite
Set.infinite_of_injective_forall_mem 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} [Infinite α] {s : Set β} {f : α → β} (hi : Function.Injective f) (hf : ∀ (x : α), f x ∈ s) : s.Infinite
Set.Infinite.preimage' 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {f : α → β} {s : Set β} (hs : (s ∩ Set.range f).Infinite) : (f ⁻¹' s).Infinite
Set.Infinite.inter_of_finite_diff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u_1} {s t : Set α} (hs : s.Infinite) (ht : (s \ t).Finite) : (s ∩ t).Infinite
Set.Infinite.inter_of_finite_sdiff 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u_1} {s t : Set α} (hs : s.Infinite) (ht : (s \ t).Finite) : (s ∩ t).Infinite
Set.Infinite.preimage 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {f : α → β} {s : Set β} (hs : s.Infinite) (hf : s ⊆ Set.range f) : (f ⁻¹' s).Infinite
Set.Infinite.exists_notMem_finite 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s t : Set α} (hs : s.Infinite) (ht : t.Finite) : ∃ a ∈ s, a ∉ t
Set.Infinite.exists_subset_card_eq 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s : Set α} (hs : s.Infinite) (n : ℕ) : ∃ t, ↑t ⊆ s ∧ t.card = n
Set.Infinite.exists_notMem_finset 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {s : Set α} (hs : s.Infinite) (t : Finset α) : ∃ a ∈ s, a ∉ t
Set.Infinite.exists_ne_map_eq_of_mapsTo 📋 Mathlib.Data.Set.Finite.Basic
{α : Type u} {β : Type v} {s : Set α} {t : Set β} {f : α → β} (hs : s.Infinite) (hf : Set.MapsTo f s t) (ht : t.Finite) : ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ f x = f y
Set.Infinite.prod_left 📋 Mathlib.Basic.Finite.Prod
{α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} (hs : s.Infinite) (ht : t.Nonempty) : (s ×ˢ t).Infinite
Set.Infinite.prod_right 📋 Mathlib.Basic.Finite.Prod
{α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} (ht : t.Infinite) (hs : s.Nonempty) : (s ×ˢ t).Infinite
Set.Infinite.image2_right 📋 Mathlib.Basic.Finite.Prod
{α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {s : Set α} {t : Set β} {a : α} (ht : t.Infinite) (ha : a ∈ s) (hf : Set.InjOn (f a) t) : (Set.image2 f s t).Infinite
Set.Infinite.image2_left 📋 Mathlib.Basic.Finite.Prod
{α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {s : Set α} {t : Set β} {b : β} (hs : s.Infinite) (hb : b ∈ t) (hf : Set.InjOn (fun a => f a b) s) : (Set.image2 f s t).Infinite
Set.infinite_prod 📋 Mathlib.Basic.Finite.Prod
{α : Type u_1} {β : Type u_2} {s : Set α} {t : Set β} : (s ×ˢ t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty
Set.infinite_image2 📋 Mathlib.Basic.Finite.Prod
{α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → β → γ} {s : Set α} {t : Set β} (hfs : ∀ b ∈ t, Set.InjOn (fun a => f a b) s) (hft : ∀ a ∈ s, Set.InjOn (f a) t) : (Set.image2 f s t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty
Set.infinite_inv 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} [InvolutiveInv α] {s : Set α} : s⁻¹.Infinite ↔ s.Infinite
Set.infinite_neg 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} [InvolutiveNeg α] {s : Set α} : (-s).Infinite ↔ s.Infinite
Set.Infinite.of_smul_set 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} {β : Type u_2} [SMul α β] {s : Set β} {a : α} : (a • s).Infinite → s.Infinite
Set.Infinite.of_vadd_set 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} {β : Type u_2} [VAdd α β] {s : Set β} {a : α} : (a +ᵥ s).Infinite → s.Infinite
Set.infinite_div 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} [Group α] {s t : Set α} : (s / t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty
Set.infinite_sub 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} [AddGroup α] {s t : Set α} : (s - t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty
Set.infinite_add 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} [Add α] [IsLeftCancelAdd α] [IsRightCancelAdd α] {s t : Set α} : (s + t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty
Set.infinite_mul 📋 Mathlib.Algebra.Group.Pointwise.Set.Finite
{α : Type u_1} [Mul α] [IsLeftCancelMul α] [IsRightCancelMul α] {s t : Set α} : (s * t).Infinite ↔ s.Infinite ∧ t.Nonempty ∨ t.Infinite ∧ s.Nonempty
Set.Infinite.sUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {s : Set (Set α)} (hs : s.Infinite) : (⋃₀ s).Infinite
Set.Infinite.iUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {ι : Sort u_1} {s : ι → Set α} (i : ι) (hi : (s i).Infinite) : (⋃ i, s i).Infinite
Set.infinite_iUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {ι : Type u_1} [Infinite ι] {s : ι → Set α} (hs : Function.Injective s) : (⋃ i, s i).Infinite
Set.infinite_of_not_bddAbove 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} [Preorder α] [IsDirectedOrder α] [Nonempty α] {s : Set α} : ¬BddAbove s → s.Infinite
Set.infinite_of_not_bddBelow 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} [Preorder α] [IsCodirectedOrder α] [Nonempty α] {s : Set α} : ¬BddBelow s → s.Infinite
Set.Infinite.iUnion₂ 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {ι : Sort u_1} {κ : ι → Sort u_2} {s : (i : ι) → κ i → Set α} (i : ι) (j : κ i) (hij : (s i j).Infinite) : (⋃ i, ⋃ j, s i j).Infinite
Set.Infinite.biUnion 📋 Mathlib.Data.Set.Finite.Lattice
{α : Type u} {ι : Type u_1} {s : ι → Set α} {a : Set ι} (ha : a.Infinite) (hs : Set.InjOn s a) : (⋃ i ∈ a, s i).Infinite
Set.infinite_of_forall_exists_gt 📋 Mathlib.Order.Preorder.Finite
{α : Type u_2} [Preorder α] {s : Set α} [Nonempty α] (h : ∀ (a : α), ∃ b ∈ s, a < b) : s.Infinite
Set.infinite_of_forall_exists_lt 📋 Mathlib.Order.Preorder.Finite
{α : Type u_2} [Preorder α] {s : Set α} [Nonempty α] (h : ∀ (a : α), ∃ b ∈ s, b < a) : s.Infinite
Set.Infinite.exists_lt_map_eq_of_mapsTo 📋 Mathlib.Order.Preorder.Finite
{α : Type u_2} {β : Type u_3} [LinearOrder α] {s : Set α} {t : Set β} {f : α → β} (hs : s.Infinite) (hf : Set.MapsTo f s t) (ht : t.Finite) : ∃ x ∈ s, ∃ y ∈ s, x < y ∧ f x = f y
Set.Infinite.not_bddAbove 📋 Mathlib.Order.Interval.Finset.Basic
{α : Type u_2} [Preorder α] [LocallyFiniteOrderBot α] {s : Set α} : s.Infinite → ¬BddAbove s
Set.Infinite.not_bddBelow 📋 Mathlib.Order.Interval.Finset.Basic
{α : Type u_2} [Preorder α] [LocallyFiniteOrderTop α] {s : Set α} : s.Infinite → ¬BddBelow s
Set.Infinite.exists_gt 📋 Mathlib.Order.Interval.Finset.Basic
{α : Type u_2} [LinearOrder α] [LocallyFiniteOrderBot α] {s : Set α} (hs : s.Infinite) (a : α) : ∃ b ∈ s, a < b
Set.Infinite.exists_lt 📋 Mathlib.Order.Interval.Finset.Basic
{α : Type u_2} [LinearOrder α] [LocallyFiniteOrderTop α] {s : Set α} (hs : s.Infinite) (a : α) : ∃ b ∈ s, b < a
Set.infinite_iff_exists_gt 📋 Mathlib.Order.Interval.Finset.Basic
{α : Type u_2} [LinearOrder α] [LocallyFiniteOrderBot α] {s : Set α} [Nonempty α] : s.Infinite ↔ ∀ (a : α), ∃ b ∈ s, a < b
Set.infinite_iff_exists_lt 📋 Mathlib.Order.Interval.Finset.Basic
{α : Type u_2} [LinearOrder α] [LocallyFiniteOrderTop α] {s : Set α} [Nonempty α] : s.Infinite ↔ ∀ (a : α), ∃ b ∈ s, b < a
Nat.decreasing_induction_of_infinite 📋 Mathlib.Order.Interval.Finset.Nat
{P : ℕ → Prop} (h : ∀ (n : ℕ), P (n + 1) → P n) (hP : {x | P x}.Infinite) (n : ℕ) : P n
Set.Infinite.Nat.sSup_eq_zero 📋 Mathlib.Order.Lattice.Nat
{s : Set ℕ} (h : s.Infinite) : sSup s = 0
Set.Infinite.exists_subset_countable_infinite 📋 Mathlib.Data.Set.Countable
{α : Type u} {s : Set α} (hs : s.Infinite) : ∃ t ⊆ s, t.Countable ∧ t.Infinite
Cardinal.aleph0_le_mk_set 📋 Mathlib.SetTheory.Cardinal.Basic
{α : Type u} {s : Set α} : Cardinal.aleph0 ≤ Cardinal.mk ↑s ↔ s.Infinite
Set.countable_infinite_iff_nonempty_denumerable 📋 Mathlib.SetTheory.Cardinal.Basic
{α : Type u_1} {s : Set α} : s.Countable ∧ s.Infinite ↔ Nonempty (Denumerable ↑s)
finprod_of_infinite_mulSupport 📋 Mathlib.Algebra.BigOperators.Finprod
{α : Type u_1} {M : Type u_5} [CommMonoid M] {f : α → M} (hf : (Function.mulSupport f).Infinite) : ∏ᶠ (i : α), f i = 1
finsum_of_infinite_support 📋 Mathlib.Algebra.BigOperators.Finprod
{α : Type u_1} {M : Type u_5} [AddCommMonoid M] {f : α → M} (hf : (Function.support f).Infinite) : ∑ᶠ (i : α), f i = 0
finprod_mem_eq_one_of_infinite 📋 Mathlib.Algebra.BigOperators.Finprod
{α : Type u_1} {M : Type u_5} [CommMonoid M] {f : α → M} {s : Set α} (hs : (s ∩ Function.mulSupport f).Infinite) : ∏ᶠ (i : α) (_ : i ∈ s), f i = 1
finsum_mem_eq_zero_of_infinite 📋 Mathlib.Algebra.BigOperators.Finprod
{α : Type u_1} {M : Type u_5} [AddCommMonoid M] {f : α → M} {s : Set α} (hs : (s ∩ Function.support f).Infinite) : ∑ᶠ (i : α) (_ : i ∈ s), f i = 0
Set.Infinite.card_eq_zero 📋 Mathlib.SetTheory.Cardinal.Finite
{α : Type u_1} {s : Set α} (hs : s.Infinite) : Nat.card ↑s = 0
Set.encard_eq_top 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s : Set α} : s.Infinite → s.encard = ⊤
Set.Infinite.encard_eq 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s : Set α} (h : s.Infinite) : s.encard = ⊤
Set.encard_eq_top_iff 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s : Set α} : s.encard = ⊤ ↔ s.Infinite
Set.Infinite.ncard 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s : Set α} (hs : s.Infinite) : s.ncard = 0
Set.infinite_iff_infinite_of_encard_eq_encard 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s t : Set α} (h : s.encard = t.encard) : s.Infinite ↔ t.Infinite
Set.Infinite.exists_subset_ncard_eq 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s : Set α} (hs : s.Infinite) (k : ℕ) : ∃ t ⊆ s, t.Finite ∧ t.ncard = k
Set.Infinite.exists_superset_ncard_eq 📋 Mathlib.Data.Set.Card
{α : Type u_1} {s t : Set α} (ht : t.Infinite) (hst : s ⊆ t) (hs : s.Finite) {k : ℕ} (hsk : s.ncard ≤ k) : ∃ s', s ⊆ s' ∧ s' ⊆ t ∧ s'.ncard = k
Cardinal.mk_sdiff_eq_left_of_finite 📋 Mathlib.SetTheory.Cardinal.Arithmetic
{α : Type u_1} {s t : Set α} (hs : s.Infinite) (ht : t.Finite) : Cardinal.mk ↑(s \ t) = Cardinal.mk ↑s
Cardinal.mk_sdiff_eq_left_of_finite' 📋 Mathlib.SetTheory.Cardinal.Arithmetic
{α : Type u_1} {s t : Set α} (hs : s.Infinite) (hst : (s ∩ t).Finite) : Cardinal.mk ↑(s \ t) = Cardinal.mk ↑s
Cardinal.mk_sdiff_eq_left 📋 Mathlib.SetTheory.Cardinal.Arithmetic
{α : Type u_1} {s t : Set α} (hs : s.Infinite) (hts : Cardinal.mk ↑t < Cardinal.mk ↑s) : Cardinal.mk ↑(s \ t) = Cardinal.mk ↑s
Cardinal.mk_sdiff_eq_left' 📋 Mathlib.SetTheory.Cardinal.Arithmetic
{α : Type u_1} {s t : Set α} (hs : s.Infinite) (hst : Cardinal.mk ↑(s ∩ t) < Cardinal.mk ↑s) : Cardinal.mk ↑(s \ t) = Cardinal.mk ↑s
ENat.sSup_eq_top_of_infinite 📋 Mathlib.Data.ENat.Lattice
{s : Set ℕ∞} (h : s.Infinite) : sSup s = ⊤
Nat.infinite_setOfPred_prime 📋 Mathlib.Data.Nat.PrimeFin
: {p | Nat.Prime p}.Infinite
Nat.infinite_setOf_prime 📋 Mathlib.Data.Nat.PrimeFin
: {p | Nat.Prime p}.Infinite
Set.Ici_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α) : (Set.Ici a).Infinite
Set.Iic_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [NoMinOrder α] (a : α) : (Set.Iic a).Infinite
Set.Iio_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [NoMinOrder α] (a : α) : (Set.Iio a).Infinite
Set.Ioi_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [NoMaxOrder α] (a : α) : (Set.Ioi a).Infinite
Set.Icc_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) : (Set.Icc a b).Infinite
Set.Ico_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) : (Set.Ico a b).Infinite
Set.Ioc_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) : (Set.Ioc a b).Infinite
Set.Ioo_infinite 📋 Mathlib.Order.Interval.Set.Infinite
{α : Type u_1} [Preorder α] [DenselyOrdered α] {a b : α} (h : a < b) : (Set.Ioo a b).Infinite
infinite_not_isOfFinAddOrder 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [AddLeftCancelMonoid G] {x : G} (h : ¬IsOfFinAddOrder x) : {y | ¬IsOfFinAddOrder y}.Infinite
infinite_not_isOfFinOrder 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [LeftCancelMonoid G] {x : G} (h : ¬IsOfFinOrder x) : {y | ¬IsOfFinOrder y}.Infinite
AddRightCancelMonoid.infinite_not_isOfFinAddOrder 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [AddRightCancelMonoid G] {x : G} (h : ¬IsOfFinAddOrder x) : {y | ¬IsOfFinAddOrder y}.Infinite
RightCancelMonoid.infinite_not_isOfFinOrder 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [RightCancelMonoid G] {x : G} (h : ¬IsOfFinOrder x) : {y | ¬IsOfFinOrder y}.Infinite
infinite_multiples 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [AddLeftCancelMonoid G] {a : G} : (↑(AddSubmonoid.multiples a)).Infinite ↔ ¬IsOfFinAddOrder a
infinite_powers 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [LeftCancelMonoid G] {a : G} : (↑(Submonoid.powers a)).Infinite ↔ ¬IsOfFinOrder a
AddRightCancelMonoid.infinite_multiples 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [AddRightCancelMonoid G] {a : G} : (↑(AddSubmonoid.multiples a)).Infinite ↔ ¬IsOfFinAddOrder a
RightCancelMonoid.infinite_powers 📋 Mathlib.GroupTheory.OrderOfElement
{G : Type u_1} [RightCancelMonoid G] {a : G} : (↑(Submonoid.powers a)).Infinite ↔ ¬IsOfFinOrder a
Filter.frequently_cofinite_iff_infinite 📋 Mathlib.Order.Filter.Cofinite
{α : Type u_2} {p : α → Prop} : (∃ᶠ (x : α) in Filter.cofinite, p x) ↔ {x | p x}.Infinite
Nat.frequently_atTop_iff_infinite 📋 Mathlib.Order.Filter.Cofinite
{p : ℕ → Prop} : (∃ᶠ (n : ℕ) in Filter.atTop, p n) ↔ {n | p n}.Infinite
Set.Infinite.cofinite_inf_principal_neBot 📋 Mathlib.Order.Filter.Cofinite
{α : Type u_2} {s : Set α} : s.Infinite → (Filter.cofinite ⊓ Filter.principal s).NeBot
Filter.cofinite_inf_principal_neBot_iff 📋 Mathlib.Order.Filter.Cofinite
{α : Type u_2} {s : Set α} : (Filter.cofinite ⊓ Filter.principal s).NeBot ↔ s.Infinite
Set.Infinite.frequently_cofinite 📋 Mathlib.Order.Filter.Cofinite
{α : Type u_2} {s : Set α} : s.Infinite → ∃ᶠ (x : α) in Filter.cofinite, x ∈ s
Filter.frequently_cofinite_mem_iff_infinite 📋 Mathlib.Order.Filter.Cofinite
{α : Type u_2} {s : Set α} : (∃ᶠ (x : α) in Filter.cofinite, x ∈ s) ↔ s.Infinite
Set.infinite_iff_frequently_cofinite 📋 Mathlib.Order.Filter.Cofinite
{α : Type u_2} {s : Set α} : s.Infinite ↔ ∃ᶠ (x : α) in Filter.cofinite, x ∈ s
Polynomial.eq_of_infinite_eval_eq 📋 Mathlib.Algebra.Polynomial.Roots
{R : Type u} [CommRing R] [IsDomain R] (p q : Polynomial R) (h : {x | Polynomial.eval x p = Polynomial.eval x q}.Infinite) : p = q
Polynomial.eq_zero_of_infinite_isRoot 📋 Mathlib.Algebra.Polynomial.Roots
{R : Type u} [CommRing R] [IsDomain R] (p : Polynomial R) (h : {x | p.IsRoot x}.Infinite) : p = 0
Algebraic.infinite_of_charZero 📋 Mathlib.Algebra.AlgebraicCard
(R : Type u_1) (A : Type u_2) [CommRing R] [Ring A] [Algebra R A] [CharZero A] : {x | IsAlgebraic R x}.Infinite
infinite_zmultiples 📋 Mathlib.Data.ZMod.QuotientGroup
{α : Type u_2} [AddGroup α] {a : α} : (↑(AddSubgroup.zmultiples a)).Infinite ↔ ¬IsOfFinAddOrder a
infinite_zpowers 📋 Mathlib.Data.ZMod.QuotientGroup
{α : Type u_2} [Group α] {a : α} : (↑(Subgroup.zpowers a)).Infinite ↔ ¬IsOfFinOrder a
Set.Infinite.exists_accPt_principal 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {s : Set X} [CompactSpace X] (hs : s.Infinite) : ∃ x, AccPt x (Filter.principal s)
Set.Infinite.exists_accPt_cofinite_inf_principal 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {s : Set X} [CompactSpace X] (hs : s.Infinite) : ∃ x, AccPt x (Filter.cofinite ⊓ Filter.principal s)
Set.Infinite.exists_accPt_of_subset_isCompact 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {s K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K) : ∃ x ∈ K, AccPt x (Filter.principal s)
Set.Infinite.exists_accPt_cofinite_inf_principal_of_subset_isCompact 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {s K : Set X} (hs : s.Infinite) (hK : IsCompact K) (hsub : s ⊆ K) : ∃ x ∈ K, AccPt x (Filter.cofinite ⊓ Filter.principal s)
exists_nhds_ne_inf_principal_neBot 📋 Mathlib.Topology.Compactness.Compact
{X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) (hs' : s.Infinite) : ∃ z ∈ s, (nhdsWithin z {z}ᶜ ⊓ Filter.principal s).NeBot
infinite_of_mem_nhds 📋 Mathlib.Topology.Separation.Basic
{X : Type u_3} [TopologicalSpace X] [T1Space X] (x : X) [hx : (nhdsWithin x {x}ᶜ).NeBot] {s : Set X} (hs : s ∈ nhds x) : s.Infinite
Set.Infinite.of_accPt 📋 Mathlib.Topology.DiscreteSubset
{X : Type u_1} [TopologicalSpace X] [T1Space X] {S : Set X} {x : X} (h : AccPt x (Filter.principal S)) : S.Infinite
AddMonoid.exponent_eq_zero_iff_range_addOrderOf_infinite 📋 Mathlib.GroupTheory.Exponent
{G : Type u} [AddMonoid G] (h : ∀ (g : G), 0 < addOrderOf g) : AddMonoid.exponent G = 0 ↔ (Set.range addOrderOf).Infinite
Monoid.exponent_eq_zero_iff_range_orderOf_infinite 📋 Mathlib.GroupTheory.Exponent
{G : Type u} [Monoid G] (h : ∀ (g : G), 0 < orderOf g) : Monoid.exponent G = 0 ↔ (Set.range orderOf).Infinite
exists_covby_infinite_Ici_of_infinite_Ici 📋 Mathlib.Order.Atoms.Finite
{α : Type u_1} [PartialOrder α] {a : α} [IsStronglyAtomic α] (ha : (Set.Ici a).Infinite) (hfin : {x | a ⋖ x}.Finite) : ∃ b, a ⋖ b ∧ (Set.Ici b).Infinite
exists_covby_infinite_Iic_of_infinite_Iic 📋 Mathlib.Order.Atoms.Finite
{α : Type u_1} [PartialOrder α] {a : α} [IsStronglyCoatomic α] (ha : (Set.Iic a).Infinite) (hfin : {x | x ⋖ a}.Finite) : ∃ b, b ⋖ a ∧ (Set.Iic b).Infinite
infinite_range_add_nsmul_iff 📋 Mathlib.Algebra.Module.Torsion.Basic
{M : Type u_2} [AddCommGroup M] [IsAddTorsionFree M] (x y : M) : (Set.range fun n => x + n • y).Infinite ↔ y ≠ 0
infinite_range_add_smul_iff 📋 Mathlib.Algebra.Module.Torsion.Basic
{R : Type u_1} {M : Type u_2} [Ring R] [IsDomain R] [Infinite R] [AddCommGroup M] [Module R M] [Module.IsTorsionFree R M] (x y : M) : (Set.range fun r => x + r • y).Infinite ↔ y ≠ 0
Set.Infinite.smul_set 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Finite
{G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a : G} {s : Set α} : s.Infinite → (a • s).Infinite
Set.Infinite.vadd_set 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Finite
{G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a : G} {s : Set α} : s.Infinite → (a +ᵥ s).Infinite
Set.infinite_smul_set 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Finite
{G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] {a : G} {s : Set α} : (a • s).Infinite ↔ s.Infinite
Set.infinite_vadd_set 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Finite
{G : Type u_1} {α : Type u_2} [AddGroup G] [AddAction G α] {a : G} {s : Set α} : (a +ᵥ s).Infinite ↔ s.Infinite
Module.infinite_range_reflection_reflection_iterate_iff 📋 Mathlib.LinearAlgebra.Reflection
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] {x : M} {f : Module.Dual R M} {y : M} {g : Module.Dual R M} [IsAddTorsionFree M] (hfx : f x = 2) (hgy : g y = 2) (hgxfy : f y * g x = 4) : (Set.range fun n => (⇑(Module.reflection hgy ≪≫ₗ Module.reflection hfx))^[n] y).Infinite ↔ f y • x ≠ 2 • y
Matroid.IsBase.infinite 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} {B : Set α} [M.RankInfinite] (hB : M.IsBase B) : B.Infinite
Matroid.IsBase.rankInfinite_of_infinite 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} {B : Set α} (hB : M.IsBase B) (h : B.Infinite) : M.RankInfinite
Matroid.RankInfinite.exists_infinite_isBase 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} [self : M.RankInfinite] : ∃ B, M.IsBase B ∧ B.Infinite
Matroid.RankInfinite.mk 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} (exists_infinite_isBase : ∃ B, M.IsBase B ∧ B.Infinite) : M.RankInfinite
Matroid.rankInfinite_iff 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} (M : Matroid α) : M.RankInfinite ↔ ∃ B, M.IsBase B ∧ B.Infinite
Matroid.IsBase.infinite_of_infinite 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} {B B₁ : Set α} (hB : M.IsBase B) (h : B.Infinite) (hB₁ : M.IsBase B₁) : B₁.Infinite
Matroid.IsBase.diff_infinite_comm 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} {B₁ B₂ : Set α} (hB₁ : M.IsBase B₁) (hB₂ : M.IsBase B₂) : (B₁ \ B₂).Infinite ↔ (B₂ \ B₁).Infinite
Matroid.IsBase.sdiff_infinite_comm 📋 Mathlib.Combinatorics.Matroid.Basic
{α : Type u_1} {M : Matroid α} {B₁ B₂ : Set α} (hB₁ : M.IsBase B₁) (hB₂ : M.IsBase B₂) : (B₁ \ B₂).Infinite ↔ (B₂ \ B₁).Infinite
Filter.cofinite.limsup_set_eq 📋 Mathlib.Order.LiminfLimsup
{α : Type u_1} {ι : Type u_4} {s : ι → Set α} : Filter.limsup s Filter.cofinite = {x | {n | x ∈ s n}.Infinite}
Filter.cofinite.blimsup_set_eq 📋 Mathlib.Order.LiminfLimsup
{α : Type u_1} {ι : Type u_4} {p : ι → Prop} {s : ι → Set α} : Filter.blimsup s Filter.cofinite p = {x | {n | p n ∧ x ∈ s n}.Infinite}
MeasurableSet.setOfPred_infinite 📋 Mathlib.MeasureTheory.MeasurableSpace.Constructions
{α : Type u_1} [Countable α] : MeasurableSet {s | s.Infinite}
MeasurableSet.setOf_infinite 📋 Mathlib.MeasureTheory.MeasurableSpace.Constructions
{α : Type u_1} [Countable α] : MeasurableSet {s | s.Infinite}
MeasurableSet.sep_infinite 📋 Mathlib.MeasureTheory.MeasurableSpace.Constructions
{α : Type u_1} [Countable α] {S : Set (Set α)} (hS : MeasurableSet S) : MeasurableSet {s | s ∈ S ∧ s.Infinite}
Set.Infinite.meas_eq_top 📋 Mathlib.MeasureTheory.Measure.Typeclasses.SFinite
{α : Type u_1} {m0 : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasurableSingletonClass α] {s : Set α} (hs : s.Infinite) (h' : ∃ ε, ε ≠ 0 ∧ ∀ x ∈ s, ε ≤ μ {x}) : μ s = ⊤
MeasureTheory.Measure.count_apply_infinite 📋 Mathlib.MeasureTheory.Measure.Count
{α : Type u_1} [MeasurableSpace α] {s : Set α} (hs : s.Infinite) : MeasureTheory.Measure.count s = ⊤
MeasureTheory.Measure.count_apply_eq_top 📋 Mathlib.MeasureTheory.Measure.Count
{α : Type u_1} [MeasurableSpace α] {s : Set α} [MeasurableSingletonClass α] : MeasureTheory.Measure.count s = ⊤ ↔ s.Infinite
MeasureTheory.Measure.count_apply_eq_top' 📋 Mathlib.MeasureTheory.Measure.Count
{α : Type u_1} [MeasurableSpace α] {s : Set α} (s_mble : MeasurableSet s) : MeasureTheory.Measure.count s = ⊤ ↔ s.Infinite
Real.range_cos_infinite 📋 Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
: (Set.range Real.cos).Infinite
Real.range_sin_infinite 📋 Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
: (Set.range Real.sin).Infinite
ProbabilityTheory.uniformOn_eq_zero 📋 Mathlib.Probability.UniformOn
{Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω} [MeasurableSingletonClass Ω] : ProbabilityTheory.uniformOn s = 0 ↔ s.Infinite ∨ s = ∅
ProbabilityTheory.uniformOn_eq_zero' 📋 Mathlib.Probability.UniformOn
{Ω : Type u_1} [MeasurableSpace Ω] {s : Set Ω} (hs : MeasurableSet s) : ProbabilityTheory.uniformOn s = 0 ↔ s.Infinite ∨ s = ∅
MvPolynomial.funext_set 📋 Mathlib.Algebra.MvPolynomial.Funext
{R : Type u_1} [CommRing R] [IsDomain R] {σ : Type u_2} {p q : MvPolynomial σ R} (s : σ → Set R) (hs : ∀ (i : σ), (s i).Infinite) (h : ∀ x ∈ Set.univ.pi s, (MvPolynomial.eval x) p = (MvPolynomial.eval x) q) : p = q
MvPolynomial.funext_set_iff 📋 Mathlib.Algebra.MvPolynomial.Funext
{R : Type u_1} [CommRing R] [IsDomain R] {σ : Type u_2} {p q : MvPolynomial σ R} (s : σ → Set R) (hs : ∀ (i : σ), (s i).Infinite) : p = q ↔ ∀ x ∈ Set.univ.pi s, (MvPolynomial.eval x) p = (MvPolynomial.eval x) q
Set.infinite_iff_tendsto_sum_indicator_atTop 📋 Mathlib.Algebra.Order.Archimedean.IndicatorCard
{R : Type u_1} [AddCommMonoid R] [PartialOrder R] [IsOrderedAddMonoid R] [AddLeftStrictMono R] [Archimedean R] {r : R} (h : 0 < r) {s : Set ℕ} : s.Infinite ↔ Filter.Tendsto (fun n => ∑ k ∈ Finset.range n, s.indicator (fun x => r) k) Filter.atTop Filter.atTop
Polynomial.dvd_of_infinite_eval_dvd_eval 📋 Mathlib.Analysis.Polynomial.Basic
{P Q : Polynomial ℤ} (mQ : Q.Monic) (h : {a | Polynomial.eval a Q ∣ Polynomial.eval a P}.Infinite) : Q ∣ P
Nat.exists_lt_modEq_of_infinite 📋 Mathlib.Combinatorics.Pigeonhole
{s : Set ℕ} (hs : s.Infinite) {k : ℕ} (hk : 0 < k) : ∃ m ∈ s, ∃ n ∈ s, m < n ∧ m ≡ n [MOD k]
SimpleGraph.ComponentCompl.hom_infinite 📋 Mathlib.Combinatorics.SimpleGraph.Ends.Defs
{V : Type u} {G : SimpleGraph V} {K L : Set V} (C : G.ComponentCompl L) (h : K ⊆ L) (Cinf : (↑C).Infinite) : (↑(SimpleGraph.ComponentCompl.hom h C)).Infinite
SimpleGraph.ComponentCompl.infinite_iff_in_all_ranges 📋 Mathlib.Combinatorics.SimpleGraph.Ends.Defs
{V : Type u} {G : SimpleGraph V} {K : Finset V} (C : G.ComponentCompl ↑K) : C.supp.Infinite ↔ ∀ (L : Finset V) (h : K ⊆ L), ∃ D, SimpleGraph.ComponentCompl.hom h D = C
SimpleGraph.infinite_iff_in_eventualRange 📋 Mathlib.Combinatorics.SimpleGraph.Ends.Defs
{V : Type u} (G : SimpleGraph V) {K : (Finset V)ᵒᵖ} (C : G.componentComplFunctor.obj K) : (SimpleGraph.ComponentCompl.supp C).Infinite ↔ C ∈ G.componentComplFunctor.eventualRange K
SimpleGraph.end_componentCompl_infinite 📋 Mathlib.Combinatorics.SimpleGraph.Ends.Properties
{V : Type} (G : SimpleGraph V) (e : ↑G.end) (K : (Finset V)ᵒᵖ) : (SimpleGraph.ComponentCompl.supp (↑e K)).Infinite
Set.Infinite.exists_union_disjoint_cardinal_eq_of_infinite 📋 Mathlib.Data.Set.Card.Arithmetic
{α : Type u_1} {s : Set α} (h : s.Infinite) : ∃ t u, t ∪ u = s ∧ Disjoint t u ∧ Cardinal.mk ↑t = Cardinal.mk ↑u
Nat.nth_injective 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) : Function.Injective (Nat.nth p)
Nat.nth_mem_of_infinite 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) (n : ℕ) : p (Nat.nth p n)
Nat.nth_monotone 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) : Monotone (Nat.nth p)
Nat.nth_strictMono 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) : StrictMono (Nat.nth p)
Nat.range_nth_of_infinite 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) : Set.range (Nat.nth p) = Set.ofPred p
Nat.surjective_count_of_infinite_setOf 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (h : {n | p n}.Infinite) : Function.Surjective (Nat.count p)
Nat.surjective_count_of_infinite_setOfPred 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (h : {n | p n}.Infinite) : Function.Surjective (Nat.count p)
Nat.count_nth_of_infinite 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) (n : ℕ) : Nat.count p (Nat.nth p n) = n
Nat.gc_count_nth 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) : GaloisConnection (Nat.count p) (Nat.nth p)
Nat.giCountNth 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) : GaloisInsertion (Nat.count p) (Nat.nth p)
Nat.le_nth_count 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) (n : ℕ) : n ≤ Nat.nth p (Nat.count p n)
Nat.nth_le_nth 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) {k n : ℕ} : Nat.nth p k ≤ Nat.nth p n ↔ k ≤ n
Nat.nth_lt_nth 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) {k n : ℕ} : Nat.nth p k < Nat.nth p n ↔ k < n
Nat.count_le_iff_le_nth 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) {a b : ℕ} : Nat.count p a ≤ b ↔ a ≤ Nat.nth p b
Nat.lt_nth_iff_count_lt 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) {a b : ℕ} : a < Nat.count p b ↔ Nat.nth p a < b
Nat.isLeast_nth_of_infinite 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) (n : ℕ) : IsLeast {i | p i ∧ ∀ k < n, Nat.nth p k < i} (Nat.nth p n)
Nat.nth_add_one_le_iff 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hp : (Set.ofPred p).Infinite) {n q : ℕ} (hq : p q) : Nat.nth p (n + 1) ≤ q ↔ Nat.nth p n < q
Nat.count_nth_succ_of_infinite 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) (n : ℕ) : Nat.count p (Nat.nth p n + 1) = n + 1
Nat.filter_range_nth_eq_insert_of_infinite 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} [DecidablePred p] (hp : (Set.ofPred p).Infinite) (k : ℕ) : {n ∈ Finset.range (Nat.nth p (k + 1)) | p n} = insert (Nat.nth p k) ({n ∈ Finset.range (Nat.nth p k) | p n})
Nat.nth_apply_eq_orderIsoOfNat 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) (n : ℕ) : Nat.nth p n = ↑((Nat.Subtype.orderIsoOfNat (Set.ofPred p)) n)
Nat.nth_eq_orderIsoOfNat 📋 Mathlib.Data.Nat.Nth
{p : ℕ → Prop} (hf : (Set.ofPred p).Infinite) : Nat.nth p = Subtype.val ∘ ⇑(Nat.Subtype.orderIsoOfNat (Set.ofPred p))
FirstOrder.Field.ACF_zero_realize_iff_infinite_ACF_prime_realize 📋 Mathlib.ModelTheory.Algebra.Field.IsAlgClosed
{φ : FirstOrder.Language.ring.Sentence} : FirstOrder.Language.Theory.ACF 0 ⊨ᵇ φ ↔ {p | FirstOrder.Language.Theory.ACF ↑p ⊨ᵇ φ}.Infinite
IsPreconnected.infinite_of_nontrivial 📋 Mathlib.Topology.Separation.Connected
{X : Type u_1} [TopologicalSpace X] [T1Space X] {s : Set X} (h : IsPreconnected s) (hs : s.Nontrivial) : s.Infinite
Finset.nsmul_right_strictMono 📋 Mathlib.Geometry.Group.Growth.LinearLowerBound
{G : Type u_1} [AddGroup G] [DecidableEq G] {X : Finset G} (hX₁ : 0 ∈ X) (hXclosure : (↑(AddSubgroup.closure ↑X)).Infinite) : StrictMono fun n => n • X
Finset.pow_right_strictMono 📋 Mathlib.Geometry.Group.Growth.LinearLowerBound
{G : Type u_1} [Group G] [DecidableEq G] {X : Finset G} (hX₁ : 1 ∈ X) (hXclosure : (↑(Subgroup.closure ↑X)).Infinite) : StrictMono fun n => X ^ n
Finset.add_nonneg_card_nsmul 📋 Mathlib.Geometry.Group.Growth.LinearLowerBound
{G : Type u_1} [AddGroup G] [DecidableEq G] {X : Finset G} (hX₁ : 0 ∈ X) (hXclosure : (↑(AddSubgroup.closure ↑X)).Infinite) (n : ℕ) : n + 1 ≤ (n • X).card
Finset.add_one_le_card_pow 📋 Mathlib.Geometry.Group.Growth.LinearLowerBound
{G : Type u_1} [Group G] [DecidableEq G] {X : Finset G} (hX₁ : 1 ∈ X) (hXclosure : (↑(Subgroup.closure ↑X)).Infinite) (n : ℕ) : n + 1 ≤ (X ^ n).card
bergelson' 📋 Mathlib.MeasureTheory.Function.Intersectivity
{α : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {r : ENNReal} {s : ℕ → Set α} (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hr₀ : r ≠ 0) (hr : ∀ (n : ℕ), r ≤ μ (s n)) : ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)
bergelson 📋 Mathlib.MeasureTheory.Function.Intersectivity
{ι : Type u_1} {α : Type u_2} [MeasurableSpace α] {μ : MeasureTheory.Measure α} [MeasureTheory.IsFiniteMeasure μ] {r : ENNReal} [Infinite ι] {s : ι → Set α} (hs : ∀ (i : ι), MeasurableSet (s i)) (hr₀ : r ≠ 0) (hr : ∀ (i : ι), r ≤ μ (s i)) : ∃ t, t.Infinite ∧ ∀ ⦃u : Set ι⦄, u ⊆ t → u.Finite → 0 < μ (⋂ i ∈ u, s i)
Nat.infinite_setOfPred_pseudoprimes 📋 Mathlib.NumberTheory.FermatPsp
{b : ℕ} (h : 1 ≤ b) : {n | n.FermatPsp b}.Infinite
Nat.infinite_setOf_pseudoprimes 📋 Mathlib.NumberTheory.FermatPsp
{b : ℕ} (h : 1 ≤ b) : {n | n.FermatPsp b}.Infinite
Real.infinite_rat_abs_sub_lt_one_div_den_sq_of_irrational 📋 Mathlib.NumberTheory.DiophantineApproximation.Basic
{ξ : ℝ} (hξ : Irrational ξ) : {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}.Infinite
Real.infinite_rat_abs_sub_lt_one_div_den_sq_iff_irrational 📋 Mathlib.NumberTheory.DiophantineApproximation.Basic
(ξ : ℝ) : {q | |ξ - ↑q| < 1 / ↑q.den ^ 2}.Infinite ↔ Irrational ξ

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c