fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
@[to_additive
/-- If a function is countably sub-additive then it is sub-additive on countable types -/]
rel_iSup_tprod [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s : β → α) :
R (m (⨆ b : β, s b)) (∏' b : β, m (s b)) := by
cases ... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | rel_iSup_tprod | If a function is countably sub-multiplicative then it is sub-multiplicative on countable
types |
@[to_additive /-- If a function is countably sub-additive then it is sub-additive on finite sets -/]
rel_iSup_prod [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s : γ → α) (t : Finset γ) :
R (m (⨆ d ∈ t, s d)) (∏ d ∈ t, m (s d)) := by... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | rel_iSup_prod | If a function is countably sub-multiplicative then it is sub-multiplicative on finite sets |
@[to_additive /-- If a function is countably sub-additive then it is binary sub-additive -/]
rel_sup_mul [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s₁ s₂ : α) :
R (m (s₁ ⊔ s₂)) (m s₁ * m s₂) := by
convert rel_iSup_tprod m m0 R m_... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | rel_sup_mul | If a function is countably sub-multiplicative then it is binary sub-multiplicative |
@[to_additive]
protected Multipliable.prod_mul_tprod_nat_mul'
{f : ℕ → M} {k : ℕ} (h : Multipliable (fun n ↦ f (n + k))) :
((∏ i ∈ range k, f i) * ∏' i, f (i + k)) = ∏' i, f i :=
h.hasProd.prod_range_mul.tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias sum_add_tsum_nat_add' := Summable.sum_add_tsum_n... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.prod_mul_tprod_nat_mul' | null |
tprod_eq_zero_mul'
{f : ℕ → M} (hf : Multipliable (fun n ↦ f (n + 1))) :
∏' b, f b = f 0 * ∏' b, f (b + 1) := by
simpa only [prod_range_one] using hf.prod_mul_tprod_nat_mul'.symm
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_eq_zero_mul' | null |
tprod_even_mul_odd {f : ℕ → M} (he : Multipliable fun k ↦ f (2 * k))
(ho : Multipliable fun k ↦ f (2 * k + 1)) :
(∏' k, f (2 * k)) * ∏' k, f (2 * k + 1) = ∏' k, f k :=
(he.hasProd.even_mul_odd ho.hasProd).tprod_eq.symm | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_even_mul_odd | null |
@[to_additive]
hasProd_nat_add_iff {f : ℕ → G} (k : ℕ) :
HasProd (fun n ↦ f (n + k)) g ↔ HasProd f (g * ∏ i ∈ range k, f i) := by
refine Iff.trans ?_ (range k).hasProd_compl_iff
rw [← (notMemRangeEquiv k).symm.hasProd_iff, Function.comp_def, coe_notMemRangeEquiv_symm]
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | hasProd_nat_add_iff | null |
multipliable_nat_add_iff {f : ℕ → G} (k : ℕ) :
(Multipliable fun n ↦ f (n + k)) ↔ Multipliable f :=
Iff.symm <|
(Equiv.mulRight (∏ i ∈ range k, f i)).surjective.multipliable_iff_of_hasProd_iff
(hasProd_nat_add_iff k).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | multipliable_nat_add_iff | null |
hasProd_nat_add_iff' {f : ℕ → G} (k : ℕ) :
HasProd (fun n ↦ f (n + k)) (g / ∏ i ∈ range k, f i) ↔ HasProd f g := by
simp [hasProd_nat_add_iff]
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | hasProd_nat_add_iff' | null |
protected Multipliable.prod_mul_tprod_nat_add [T2Space G] {f : ℕ → G} (k : ℕ)
(h : Multipliable f) : ((∏ i ∈ range k, f i) * ∏' i, f (i + k)) = ∏' i, f i :=
Multipliable.prod_mul_tprod_nat_mul' <| (multipliable_nat_add_iff k).2 h
@[deprecated (since := "2025-04-12")] alias sum_add_tsum_nat_add :=
Summable.sum_a... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.prod_mul_tprod_nat_add | null |
protected Multipliable.tprod_eq_zero_mul [T2Space G] {f : ℕ → G} (hf : Multipliable f) :
∏' b, f b = f 0 * ∏' b, f (b + 1) :=
tprod_eq_zero_mul' <| (multipliable_nat_add_iff 1).2 hf
@[deprecated (since := "2025-04-12")] alias tsum_eq_zero_add := Summable.tsum_eq_zero_add
@[to_additive existing, deprecated (since ... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.tprod_eq_zero_mul | null |
@[to_additive /-- For `f : ℕ → G`, the sum `∑' k, f (k + i)` tends to zero. This does not require a
summability assumption on `f`, as otherwise all such sums are zero. -/]
tendsto_prod_nat_add [T2Space G] (f : ℕ → G) :
Tendsto (fun i ↦ ∏' k, f (k + i)) atTop (𝓝 1) := by
by_cases hf : Multipliable f
· have h₀ :... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tendsto_prod_nat_add | For `f : ℕ → G`, the product `∏' k, f (k + i)` tends to one. This does not require a
multipliability assumption on `f`, as otherwise all such products are one. |
@[to_additive]
cauchySeq_finset_iff_nat_tprod_vanishing {f : ℕ → G} :
(CauchySeq fun s : Finset ℕ ↦ ∏ n ∈ s, f n) ↔
∀ e ∈ 𝓝 (1 : G), ∃ N : ℕ, ∀ t ⊆ {n | N ≤ n}, (∏' n : t, f n) ∈ e := by
refine cauchySeq_finset_iff_tprod_vanishing.trans ⟨fun vanish e he ↦ ?_, fun vanish e he ↦ ?_⟩
· obtain ⟨s, hs⟩ := van... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | cauchySeq_finset_iff_nat_tprod_vanishing | null |
multipliable_iff_nat_tprod_vanishing {f : ℕ → G} : Multipliable f ↔
∀ e ∈ 𝓝 1, ∃ N : ℕ, ∀ t ⊆ {n | N ≤ n}, (∏' n : t, f n) ∈ e := by
rw [multipliable_iff_cauchySeq_finset, cauchySeq_finset_iff_nat_tprod_vanishing] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | multipliable_iff_nat_tprod_vanishing | null |
@[to_additive]
Multipliable.nat_tprod_vanishing {f : ℕ → G} (hf : Multipliable f) ⦃e : Set G⦄
(he : e ∈ 𝓝 1) : ∃ N : ℕ, ∀ t ⊆ {n | N ≤ n}, (∏' n : t, f n) ∈ e :=
letI : UniformSpace G := IsTopologicalGroup.toUniformSpace G
have : IsUniformGroup G := isUniformGroup_of_commGroup
cauchySeq_finset_iff_nat_tprod_... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.nat_tprod_vanishing | null |
Multipliable.tendsto_atTop_one {f : ℕ → G} (hf : Multipliable f) :
Tendsto f atTop (𝓝 1) := by
rw [← Nat.cofinite_eq_atTop]
exact hf.tendsto_cofinite_one | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.tendsto_atTop_one | null |
@[to_additive HasSum.nat_add_neg_add_one]
HasProd.nat_mul_neg_add_one {f : ℤ → M} (hf : HasProd f m) :
HasProd (fun n : ℕ ↦ f n * f (-(n + 1))) m := by
change HasProd (fun n : ℕ ↦ f n * f (Int.negSucc n)) m
have : Injective Int.negSucc := @Int.negSucc.inj
refine hf.hasProd_of_prod_eq fun u ↦ ?_
refine ⟨u.pr... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.nat_mul_neg_add_one | null |
Multipliable.nat_mul_neg_add_one {f : ℤ → M} (hf : Multipliable f) :
Multipliable (fun n : ℕ ↦ f n * f (-(n + 1))) :=
hf.hasProd.nat_mul_neg_add_one.multipliable
@[to_additive tsum_nat_add_neg_add_one] | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.nat_mul_neg_add_one | null |
tprod_nat_mul_neg_add_one [T2Space M] {f : ℤ → M} (hf : Multipliable f) :
∏' (n : ℕ), (f n * f (-(n + 1))) = ∏' (n : ℤ), f n :=
hf.hasProd.nat_mul_neg_add_one.tprod_eq | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_nat_mul_neg_add_one | null |
@[to_additive HasSum.of_nat_of_neg_add_one]
HasProd.of_nat_of_neg_add_one {f : ℤ → M}
(hf₁ : HasProd (fun n : ℕ ↦ f n) m) (hf₂ : HasProd (fun n : ℕ ↦ f (-(n + 1))) m') :
HasProd f (m * m') := by
have hi₂ : Injective Int.negSucc := @Int.negSucc.inj
have : IsCompl (Set.range ((↑) : ℕ → ℤ)) (Set.range Int.negS... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.of_nat_of_neg_add_one | null |
Multipliable.of_nat_of_neg_add_one {f : ℤ → M}
(hf₁ : Multipliable fun n : ℕ ↦ f n) (hf₂ : Multipliable fun n : ℕ ↦ f (-(n + 1))) :
Multipliable f :=
(hf₁.hasProd.of_nat_of_neg_add_one hf₂.hasProd).multipliable
@[to_additive tsum_of_nat_of_neg_add_one] | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.of_nat_of_neg_add_one | null |
tprod_of_nat_of_neg_add_one [T2Space M] {f : ℤ → M}
(hf₁ : Multipliable fun n : ℕ ↦ f n) (hf₂ : Multipliable fun n : ℕ ↦ f (-(n + 1))) :
∏' n : ℤ, f n = (∏' n : ℕ, f n) * ∏' n : ℕ, f (-(n + 1)) :=
(hf₁.hasProd.of_nat_of_neg_add_one hf₂.hasProd).tprod_eq | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_of_nat_of_neg_add_one | null |
@[to_additive /-- If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` have sums `a`, `b` respectively, then
the `ℤ`-indexed sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...` (with `f₀` at the `0`-th position) has
sum `a + b`. -/]
HasProd.int_rec {f g : ℕ → M} (hf : HasProd f m) (hg : HasProd g m') :
HasProd (Int.rec f g) (m * m'... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.int_rec | If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` have products `a`, `b` respectively, then
the `ℤ`-indexed sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...` (with `f₀` at the `0`-th position) has
product `a + b`. |
@[to_additive /-- If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` are both summable then so is the
`ℤ`-indexed sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...` (with `f₀` at the `0`-th position). -/]
Multipliable.int_rec {f g : ℕ → M} (hf : Multipliable f) (hg : Multipliable g) :
Multipliable (Int.rec f g) :=
.of_nat_of_n... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.int_rec | If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` are both multipliable then so is the
`ℤ`-indexed sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...` (with `f₀` at the `0`-th position). |
@[to_additive /-- If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` are both summable, then the sum of the
`ℤ`-indexed sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...` (with `f₀` at the `0`-th position) is
`∑' n, f n + ∑' n, g n`. -/]
tprod_int_rec [T2Space M] {f g : ℕ → M} (hf : Multipliable f) (hg : Multipliable g) :
∏' n :... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_int_rec | If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` are both multipliable, then the product of the
`ℤ`-indexed sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...` (with `f₀` at the `0`-th position) is
`(∏' n, f n) * ∏' n, g n`. |
HasProd.nat_mul_neg {f : ℤ → M} (hf : HasProd f m) :
HasProd (fun n : ℕ ↦ f n * f (-n)) (m * f 0) := by
apply (hf.mul (hasProd_ite_eq (0 : ℤ) (f 0))).hasProd_of_prod_eq fun u ↦ ?_
refine ⟨u.image Int.natAbs, fun v' hv' ↦ ?_⟩
let u1 := v'.image fun x : ℕ ↦ (x : ℤ)
let u2 := v'.image fun x : ℕ ↦ -(x : ℤ)
ha... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.nat_mul_neg | null |
Multipliable.nat_mul_neg {f : ℤ → M} (hf : Multipliable f) :
Multipliable fun n : ℕ ↦ f n * f (-n) :=
hf.hasProd.nat_mul_neg.multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.nat_mul_neg | null |
tprod_nat_mul_neg [T2Space M] {f : ℤ → M} (hf : Multipliable f) :
∏' n : ℕ, (f n * f (-n)) = (∏' n : ℤ, f n) * f 0 :=
hf.hasProd.nat_mul_neg.tprod_eq
@[to_additive HasSum.of_add_one_of_neg_add_one] | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_nat_mul_neg | null |
HasProd.of_add_one_of_neg_add_one {f : ℤ → M}
(hf₁ : HasProd (fun n : ℕ ↦ f (n + 1)) m) (hf₂ : HasProd (fun n : ℕ ↦ f (-(n + 1))) m') :
HasProd f (m * f 0 * m') :=
HasProd.of_nat_of_neg_add_one (mul_comm _ m ▸ HasProd.zero_mul hf₁) hf₂
@[to_additive Summable.of_add_one_of_neg_add_one] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.of_add_one_of_neg_add_one | null |
Multipliable.of_add_one_of_neg_add_one {f : ℤ → M}
(hf₁ : Multipliable fun n : ℕ ↦ f (n + 1)) (hf₂ : Multipliable fun n : ℕ ↦ f (-(n + 1))) :
Multipliable f :=
(hf₁.hasProd.of_add_one_of_neg_add_one hf₂.hasProd).multipliable
@[to_additive tsum_of_add_one_of_neg_add_one] | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.of_add_one_of_neg_add_one | null |
tprod_of_add_one_of_neg_add_one [T2Space M] {f : ℤ → M}
(hf₁ : Multipliable fun n : ℕ ↦ f (n + 1)) (hf₂ : Multipliable fun n : ℕ ↦ f (-(n + 1))) :
∏' n : ℤ, f n = (∏' n : ℕ, f (n + 1)) * f 0 * ∏' n : ℕ, f (-(n + 1)) :=
(hf₁.hasProd.of_add_one_of_neg_add_one hf₂.hasProd).tprod_eq | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_of_add_one_of_neg_add_one | null |
@[to_additive]
HasProd.of_nat_of_neg {f : ℤ → G} (hf₁ : HasProd (fun n : ℕ ↦ f n) g)
(hf₂ : HasProd (fun n : ℕ ↦ f (-n)) g') : HasProd f (g * g' / f 0) := by
refine mul_div_assoc' g .. ▸ hf₁.of_nat_of_neg_add_one (m' := g' / f 0) ?_
rwa [← hasProd_nat_add_iff' 1, prod_range_one, Nat.cast_zero, neg_zero] at hf₂
... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | HasProd.of_nat_of_neg | null |
Multipliable.of_nat_of_neg {f : ℤ → G} (hf₁ : Multipliable fun n : ℕ ↦ f n)
(hf₂ : Multipliable fun n : ℕ ↦ f (-n)) : Multipliable f :=
(hf₁.hasProd.of_nat_of_neg hf₂.hasProd).multipliable
@[to_additive] | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.of_nat_of_neg | null |
protected Multipliable.tprod_of_nat_of_neg [T2Space G] {f : ℤ → G}
(hf₁ : Multipliable fun n : ℕ ↦ f n) (hf₂ : Multipliable fun n : ℕ ↦ f (-n)) :
∏' n : ℤ, f n = (∏' n : ℕ, f n) * (∏' n : ℕ, f (-n)) / f 0 :=
(hf₁.hasProd.of_nat_of_neg hf₂.hasProd).tprod_eq
@[deprecated (since := "2025-04-12")] alias tsum_of_n... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Multipliable.tprod_of_nat_of_neg | null |
@[to_additive /-- "iff" version of `Summable.of_nat_of_neg_add_one`. -/]
multipliable_int_iff_multipliable_nat_and_neg_add_one {f : ℤ → G} : Multipliable f ↔
(Multipliable fun n : ℕ ↦ f n) ∧ (Multipliable fun n : ℕ ↦ f (-(n + 1))) := by
refine ⟨fun p ↦ ⟨?_, ?_⟩, fun ⟨hf₁, hf₂⟩ ↦ Multipliable.of_nat_of_neg_add_one... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | multipliable_int_iff_multipliable_nat_and_neg_add_one | "iff" version of `Multipliable.of_nat_of_neg_add_one`. |
@[to_additive /-- "iff" version of `Summable.of_nat_of_neg`. -/]
multipliable_int_iff_multipliable_nat_and_neg {f : ℤ → G} :
Multipliable f ↔ (Multipliable fun n : ℕ ↦ f n) ∧ (Multipliable fun n : ℕ ↦ f (-n)) := by
refine ⟨fun p ↦ ⟨?_, ?_⟩, fun ⟨hf₁, hf₂⟩ ↦ Multipliable.of_nat_of_neg hf₁ hf₂⟩ <;>
apply p.comp_i... | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | multipliable_int_iff_multipliable_nat_and_neg | "iff" version of `Multipliable.of_nat_of_neg`. |
Summable.alternating {α} [Ring α]
[UniformSpace α] [IsUniformAddGroup α] [CompleteSpace α] {f : ℕ → α} (hf : Summable f) :
Summable (fun n => (-1) ^ n * f n) := by
apply Summable.even_add_odd
· simp only [even_two, Even.mul_right, Even.neg_pow, one_pow, one_mul]
exact hf.comp_injective (mul_right_inject... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | Summable.alternating | null |
@[to_additive]
multipliable_pnat_iff_multipliable_succ {f : ℕ → M} :
Multipliable (fun x : ℕ+ ↦ f x) ↔ Multipliable fun x ↦ f (x + 1) :=
Equiv.pnatEquivNat.symm.multipliable_iff.symm
@[deprecated (since := "2025-09-31")]
alias pnat_multipliable_iff_multipliable_succ := multipliable_pnat_iff_multipliable_succ
@[to... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | multipliable_pnat_iff_multipliable_succ | null |
tprod_pnat_eq_tprod_succ {f : ℕ → M} : ∏' n : ℕ+, f n = ∏' n, f (n + 1) :=
(Equiv.pnatEquivNat.symm.tprod_eq _).symm
@[to_additive] | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_pnat_eq_tprod_succ | null |
tprod_zero_pnat_eq_tprod_nat [TopologicalSpace G] [IsTopologicalGroup G] [T2Space G]
{f : ℕ → G} (hf : Multipliable f) :
f 0 * ∏' n : ℕ+, f ↑n = ∏' n, f n := by
simpa [hf.tprod_eq_zero_mul] using tprod_pnat_eq_tprod_succ
@[to_additive tsum_int_eq_zero_add_two_mul_tsum_pnat] | lemma | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_zero_pnat_eq_tprod_nat | null |
tprod_int_eq_zero_mul_tprod_pnat_sq [UniformSpace G] [IsUniformGroup G] [CompleteSpace G]
[T2Space G] {f : ℤ → G} (hf : ∀ n : ℤ, f (-n) = f n) (hf2 : Multipliable f) :
∏' n, f n = f 0 * (∏' n : ℕ+, f n) ^ 2 := by
have hf3 : Multipliable fun n : ℕ ↦ f n :=
(multipliable_int_iff_multipliable_nat_and_neg.mp ... | theorem | Topology | [
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Order.Filter.AtTopBot.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.Group"
] | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | tprod_int_eq_zero_mul_tprod_pnat_sq | null |
@[to_additive /-- Let `G` be a nonarchimedean additive abelian group, and let `f : α → G` be a
function that tends to zero on the filter of cofinite sets. For each finite subset of `α`, consider
the partial sum of `f` on that subset. These partial sums form a Cauchy filter. -/]
cauchySeq_prod_of_tendsto_cofinite_one {f... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | cauchySeq_prod_of_tendsto_cofinite_one | Let `G` be a nonarchimedean multiplicative abelian group, and let `f : α → G` be a function that
tends to one on the filter of cofinite sets. For each finite subset of `α`, consider the partial
product of `f` on that subset. These partial products form a Cauchy filter. |
@[to_additive /-- Let `G` be a nonarchimedean additive abelian group, and let `f : ℕ → G` be a
function such that the differences `f (n + 1) - f n` tend to zero.
Then the function is a Cauchy sequence. -/]
cauchySeq_of_tendsto_div_nhds_one {f : ℕ → G}
(hf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)) :
Cauc... | lemma | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | cauchySeq_of_tendsto_div_nhds_one | Let `G` be a nonarchimedean abelian group, and let `f : ℕ → G` be a function
such that the quotients `f (n + 1) / f n` tend to one. Then the function is a Cauchy sequence. |
@[to_additive /-- Let `G` be a complete nonarchimedean additive abelian group, and let `f : α → G`
be a function that tends to zero on the filter of cofinite sets. Then `f` is unconditionally
summable. -/]
multipliable_of_tendsto_cofinite_one [CompleteSpace G] {f : α → G}
(hf : Tendsto f cofinite (𝓝 1)) : Multipli... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | multipliable_of_tendsto_cofinite_one | Let `G` be a complete nonarchimedean multiplicative abelian group, and let `f : α → G` be a
function that tends to one on the filter of cofinite sets. Then `f` is unconditionally
multipliable. |
@[to_additive /-- Let `G` be a complete nonarchimedean additive abelian group. Then a function
`f : α → G` is unconditionally summable if and only if it tends to zero on the filter of cofinite
sets. -/]
multipliable_iff_tendsto_cofinite_one [CompleteSpace G] (f : α → G) :
Multipliable f ↔ Tendsto f cofinite (𝓝 1) ... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | multipliable_iff_tendsto_cofinite_one | Let `G` be a complete nonarchimedean multiplicative abelian group. Then a function `f : α → G`
is unconditionally multipliable if and only if it tends to one on the filter of cofinite sets. |
private Summable.mul_of_complete_nonarchimedean [CompleteSpace R] {f : α → R} {g : β → R}
(hf : Summable f) (hg : Summable g) : Summable (fun i : α × β ↦ f i.1 * g i.2) := by
rw [NonarchimedeanAddGroup.summable_iff_tendsto_cofinite_zero] at *
exact tendsto_mul_cofinite_nhds_zero hf hg | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | Summable.mul_of_complete_nonarchimedean | null |
HasSum.mul_of_nonarchimedean {f : α → R} {g : β → R} {a b : R} (hf : HasSum f a)
(hg : HasSum g b) : HasSum (fun i : α × β ↦ f i.1 * g i.2) (a * b) := by
rw [← hasSum_iff_hasSum_compl] at *
simp only [Function.comp_def, UniformSpace.Completion.toCompl_apply,
UniformSpace.Completion.coe_mul]
exact (hf.mul ... | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | HasSum.mul_of_nonarchimedean | Let `R` be a nonarchimedean ring, let `f : α → R` be a function that sums to `a : R`,
and let `g : β → R` be a function that sums to `b : R`. Then `fun i : α × β ↦ f i.1 * g i.2`
sums to `a * b`. |
Summable.mul_of_nonarchimedean {f : α → R} {g : β → R} (hf : Summable f)
(hg : Summable g) : Summable (fun i : α × β ↦ f i.1 * g i.2) :=
(hf.hasSum.mul_of_nonarchimedean hg.hasSum).summable | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | Summable.mul_of_nonarchimedean | Let `R` be a nonarchimedean ring. If functions `f : α → R` and `g : β → R` are summable, then
so is `fun i : α × β ↦ f i.1 * g i.2`. |
tsum_mul_tsum_of_nonarchimedean [T0Space R] {f : α → R} {g : β → R} (hf : Summable f)
(hg : Summable g) : (∑' i, f i) * (∑' i, g i) = ∑' i : α × β, f i.1 * g i.2 :=
(hf.hasSum.mul_of_nonarchimedean hg.hasSum).tsum_eq.symm | theorem | Topology | [
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Algebra.Nonarchimedean.Completion"
] | Mathlib/Topology/Algebra/InfiniteSum/Nonarchimedean.lean | tsum_mul_tsum_of_nonarchimedean | null |
@[to_additive]
hasProd_le_of_prod_le [ClosedIicTopology α]
(hf : HasProd f a) (h : ∀ s, ∏ i ∈ s, f i ≤ c) : a ≤ c :=
le_of_tendsto' hf h
@[to_additive] | lemma | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_le_of_prod_le | null |
le_hasProd_of_le_prod [ClosedIciTopology α]
(hf : HasProd f a) (h : ∀ s, c ≤ ∏ i ∈ s, f i) : c ≤ a :=
ge_of_tendsto' hf h
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | le_hasProd_of_le_prod | null |
protected Multipliable.tprod_le_of_prod_range_le [ClosedIicTopology α] {f : ℕ → α}
(hf : Multipliable f) (h : ∀ n, ∏ i ∈ range n, f i ≤ c) : ∏' n, f n ≤ c :=
le_of_tendsto' hf.hasProd.tendsto_prod_nat h
@[deprecated (since := "2025-04-12")] alias tsum_le_of_sum_range_le :=
Summable.tsum_le_of_sum_range_le
@[to_... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_le_of_prod_range_le | null |
@[to_additive]
hasProd_le (h : ∀ i, f i ≤ g i) (hf : HasProd f a₁) (hg : HasProd g a₂) : a₁ ≤ a₂ :=
le_of_tendsto_of_tendsto' hf hg fun _ ↦ prod_le_prod' fun i _ ↦ h i
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_le | null |
hasProd_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f ≤ g) : a₁ ≤ a₂ :=
hasProd_le h hf hg
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_mono | null |
hasProd_le_inj {g : κ → α} (e : ι → κ) (he : Injective e)
(hs : ∀ c, c ∉ Set.range e → 1 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : HasProd f a₁)
(hg : HasProd g a₂) : a₁ ≤ a₂ := by
rw [← hasProd_extend_one he] at hf
refine hasProd_le (fun c ↦ ?_) hf hg
obtain ⟨i, rfl⟩ | h := em (c ∈ Set.range e)
· rw [he.ex... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_le_inj | null |
protected Multipliable.tprod_le_tprod_of_inj {g : κ → α} (e : ι → κ) (he : Injective e)
(hs : ∀ c, c ∉ Set.range e → 1 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : Multipliable f)
(hg : Multipliable g) : tprod f ≤ tprod g :=
hasProd_le_inj _ he hs h hf.hasProd hg.hasProd
@[deprecated (since := "2025-04-12")] alias t... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_le_tprod_of_inj | null |
protected Multipliable.tprod_subtype_le {κ γ : Type*} [CommGroup γ] [PartialOrder γ]
[IsOrderedMonoid γ] [UniformSpace γ] [IsUniformGroup γ] [OrderClosedTopology γ]
[CompleteSpace γ] (f : κ → γ) (β : Set κ) (h : ∀ a : κ, 1 ≤ f a) (hf : Multipliable f) :
(∏' (b : β), f b) ≤ (∏' (a : κ), f a) := by
apply Mu... | lemma | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_subtype_le | null |
prod_le_hasProd (s : Finset ι) (hs : ∀ i, i ∉ s → 1 ≤ f i) (hf : HasProd f a) :
∏ i ∈ s, f i ≤ a :=
ge_of_tendsto hf (eventually_atTop.2
⟨s, fun _t hst ↦ prod_le_prod_of_subset_of_one_le' hst fun i _ hbs ↦ hs i hbs⟩)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | prod_le_hasProd | null |
isLUB_hasProd (h : ∀ i, 1 ≤ f i) (hf : HasProd f a) :
IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) a := by
classical
exact isLUB_of_tendsto_atTop (Finset.prod_mono_set_of_one_le' h) hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | isLUB_hasProd | null |
le_hasProd (hf : HasProd f a) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) : f i ≤ a :=
calc
f i = ∏ i ∈ {i}, f i := by rw [prod_singleton]
_ ≤ a := prod_le_hasProd _ (by simpa) hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | le_hasProd | null |
lt_hasProd [MulRightStrictMono α] (hf : HasProd f a) (i : ι)
(hi : ∀ (j : ι), j ≠ i → 1 ≤ f j) (j : ι) (hij : j ≠ i) (hj : 1 < f j) :
f i < a := by
classical
calc
f i < f j * f i := lt_mul_of_one_lt_left' (f i) hj
_ = ∏ k ∈ {j, i}, f k := by rw [Finset.prod_pair hij]
_ ≤ a := prod_le_hasProd _ (... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | lt_hasProd | null |
protected Multipliable.prod_le_tprod {f : ι → α} (s : Finset ι) (hs : ∀ i, i ∉ s → 1 ≤ f i)
(hf : Multipliable f) : ∏ i ∈ s, f i ≤ ∏' i, f i :=
prod_le_hasProd s hs hf.hasProd
@[deprecated (since := "2025-04-12")] alias sum_le_tsum := Summable.sum_le_tsum
@[to_additive existing, deprecated (since := "2025-04-12")... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.prod_le_tprod | null |
protected Multipliable.le_tprod (hf : Multipliable f) (i : ι) (hb : ∀ j, j ≠ i → 1 ≤ f j) :
f i ≤ ∏' i, f i :=
le_hasProd hf.hasProd i hb
@[deprecated (since := "2025-04-12")] alias le_tsum := Summable.le_tsum
@[to_additive existing, deprecated (since := "2025-04-12")] alias le_tprod := Multipliable.le_tprod
@[to... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.le_tprod | null |
protected Multipliable.tprod_le_tprod (h : ∀ i, f i ≤ g i) (hf : Multipliable f)
(hg : Multipliable g) : ∏' i, f i ≤ ∏' i, g i :=
hasProd_le h hf.hasProd hg.hasProd
@[deprecated (since := "2025-04-12")] alias tsum_le_tsum := Summable.tsum_le_tsum
@[to_additive existing, deprecated (since := "2025-04-12")] alias t... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_le_tprod | null |
protected Multipliable.tprod_mono (hf : Multipliable f) (hg : Multipliable g) (h : f ≤ g) :
∏' n, f n ≤ ∏' n, g n :=
hf.tprod_le_tprod h hg
@[deprecated (since := "2025-04-12")] alias tsum_mono := Summable.tsum_mono
@[to_additive existing (attr := mono), deprecated (since := "2025-04-12")] alias tprod_mono :=
M... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_mono | null |
protected Multipliable.tprod_le_of_prod_le (hf : Multipliable f)
(h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ :=
hasProd_le_of_prod_le hf.hasProd h
@[deprecated (since := "2025-04-12")] alias tsum_le_of_sum_le := Summable.tsum_le_of_sum_le
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_l... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_le_of_prod_le | null |
tprod_le_of_prod_le' (ha₂ : 1 ≤ a₂) (h : ∀ s, ∏ i ∈ s, f i ≤ a₂) : ∏' i, f i ≤ a₂ := by
by_cases hf : Multipliable f
· exact hf.tprod_le_of_prod_le h
· rw [tprod_eq_one_of_not_multipliable hf]
exact ha₂
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | tprod_le_of_prod_le' | null |
HasProd.one_le (h : ∀ i, 1 ≤ g i) (ha : HasProd g a) : 1 ≤ a :=
hasProd_le h hasProd_one ha
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | HasProd.one_le | null |
HasProd.le_one (h : ∀ i, g i ≤ 1) (ha : HasProd g a) : a ≤ 1 :=
hasProd_le h ha hasProd_one
@[to_additive tsum_nonneg] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | HasProd.le_one | null |
one_le_tprod (h : ∀ i, 1 ≤ g i) : 1 ≤ ∏' i, g i := by
by_cases hg : Multipliable g
· exact hg.hasProd.one_le h
· rw [tprod_eq_one_of_not_multipliable hg]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | one_le_tprod | null |
tprod_le_one (h : ∀ i, f i ≤ 1) : ∏' i, f i ≤ 1 := by
by_cases hf : Multipliable f
· exact hf.hasProd.le_one h
· rw [tprod_eq_one_of_not_multipliable hf]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | tprod_le_one | null |
hasProd_one_iff_of_one_le (hf : ∀ i, 1 ≤ f i) : HasProd f 1 ↔ f = 1 := by
refine ⟨fun hf' ↦ ?_, ?_⟩
· ext i
exact (hf i).antisymm' (le_hasProd hf' _ fun j _ ↦ hf j)
· rintro rfl
exact hasProd_one | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_one_iff_of_one_le | null |
@[to_additive]
hasProd_lt (h : f ≤ g) (hi : f i < g i) (hf : HasProd f a₁) (hg : HasProd g a₂) :
a₁ < a₂ := by
classical
have : update f i 1 ≤ update g i 1 := update_le_update_iff.mpr ⟨rfl.le, fun i _ ↦ h i⟩
have : 1 / f i * a₁ ≤ 1 / g i * a₂ := hasProd_le this (hf.update i 1) (hg.update i 1)
simpa only [on... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_lt | null |
hasProd_strict_mono (hf : HasProd f a₁) (hg : HasProd g a₂) (h : f < g) : a₁ < a₂ :=
let ⟨hle, _i, hi⟩ := Pi.lt_def.mp h
hasProd_lt hle hi hf hg
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_strict_mono | null |
protected Multipliable.tprod_lt_tprod (h : f ≤ g) (hi : f i < g i) (hf : Multipliable f)
(hg : Multipliable g) : ∏' n, f n < ∏' n, g n :=
hasProd_lt h hi hf.hasProd hg.hasProd
@[deprecated (since := "2025-04-12")] alias tsum_lt_tsum := Summable.tsum_lt_tsum
@[to_additive existing, deprecated (since := "2025-04-12... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_lt_tprod | null |
protected Multipliable.tprod_strict_mono (hf : Multipliable f) (hg : Multipliable g)
(h : f < g) : ∏' n, f n < ∏' n, g n :=
let ⟨hle, _i, hi⟩ := Pi.lt_def.mp h
hf.tprod_lt_tprod hle hi hg
@[deprecated (since := "2025-04-12")] alias tsum_strict_mono := Summable.tsum_strict_mono
@[to_additive existing (attr := mo... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_strict_mono | null |
protected Multipliable.one_lt_tprod (hsum : Multipliable g) (hg : ∀ i, 1 ≤ g i) (i : ι)
(hi : 1 < g i) : 1 < ∏' i, g i := by
rw [← tprod_one]
exact multipliable_one.tprod_lt_tprod hg hi hsum
@[deprecated (since := "2025-04-12")] alias tsum_pos := Summable.tsum_pos
@[to_additive existing tsum_pos, deprecated (si... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.one_lt_tprod | null |
@[to_additive]
le_hasProd' (hf : HasProd f a) (i : ι) : f i ≤ a :=
le_hasProd hf i fun _ _ ↦ one_le _
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | le_hasProd' | null |
protected Multipliable.le_tprod' (hf : Multipliable f) (i : ι) : f i ≤ ∏' i, f i :=
hf.le_tprod i fun _ _ ↦ one_le _
@[deprecated (since := "2025-04-12")] alias le_tsum' := Summable.le_tsum'
@[to_additive existing, deprecated (since := "2025-04-12")] alias le_tprod' :=
Multipliable.le_tprod'
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.le_tprod' | null |
hasProd_one_iff : HasProd f 1 ↔ ∀ x, f x = 1 :=
(hasProd_one_iff_of_one_le fun _ ↦ one_le _).trans funext_iff
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_one_iff | null |
protected Multipliable.tprod_eq_one_iff (hf : Multipliable f) :
∏' i, f i = 1 ↔ ∀ x, f x = 1 := by
rw [← hasProd_one_iff, hf.hasProd_iff]
@[deprecated (since := "2025-04-12")] alias tsum_eq_zero_iff := Summable.tsum_eq_zero_iff
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_eq_one_iff :=
... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_eq_one_iff | null |
protected Multipliable.tprod_ne_one_iff (hf : Multipliable f) :
∏' i, f i ≠ 1 ↔ ∃ x, f x ≠ 1 := by
rw [Ne, hf.tprod_eq_one_iff, not_forall]
@[deprecated (since := "2025-04-12")] alias tsum_ne_zero_iff := Summable.tsum_ne_zero_iff
@[to_additive existing, deprecated (since := "2025-04-12")] alias tprod_ne_one_iff :... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.tprod_ne_one_iff | null |
isLUB_hasProd' (hf : HasProd f a) : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) a := by
classical
exact isLUB_of_tendsto_atTop (Finset.prod_mono_set' f) hf | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | isLUB_hasProd' | null |
@[to_additive]
hasProd_of_isLUB_of_one_le [CommMonoid α] [LinearOrder α] [IsOrderedMonoid α]
[TopologicalSpace α]
[OrderTopology α] {f : ι → α} (i : α) (h : ∀ i, 1 ≤ f i)
(hf : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) i) : HasProd f i :=
tendsto_atTop_isLUB (Finset.prod_mono_set_of_one_le' h) hf
@[to_additi... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_of_isLUB_of_one_le | null |
hasProd_of_isLUB [CommMonoid α] [LinearOrder α]
[CanonicallyOrderedMul α] [TopologicalSpace α]
[OrderTopology α] {f : ι → α} (b : α) (hf : IsLUB (Set.range fun s ↦ ∏ i ∈ s, f i) b) :
HasProd f b :=
tendsto_atTop_isLUB (Finset.prod_mono_set' f) hf
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | hasProd_of_isLUB | null |
multipliable_mabs_iff [CommGroup α] [LinearOrder α] [IsOrderedMonoid α]
[UniformSpace α] [IsUniformGroup α]
[CompleteSpace α] {f : ι → α} : (Multipliable fun x ↦ mabs (f x)) ↔ Multipliable f :=
let s := { x | 1 ≤ f x }
have h1 : ∀ x : s, mabs (f x) = f x := fun x ↦ mabs_of_one_le x.2
have h2 : ∀ x : ↑sᶜ, ... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | multipliable_mabs_iff | null |
Finite.of_summable_const [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α]
[TopologicalSpace α] [Archimedean α]
[OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun _ : ι ↦ b) :
Finite ι := by
have H : ∀ s : Finset ι, #s • b ≤ ∑' _ : ι, b := fun s ↦ by
simpa using sum_le_hasSum s (fun... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Finite.of_summable_const | null |
Set.Finite.of_summable_const [AddCommGroup α] [LinearOrder α] [IsOrderedAddMonoid α]
[TopologicalSpace α]
[Archimedean α] [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun _ : ι ↦ b) :
(Set.univ : Set ι).Finite :=
finite_univ_iff.2 <| .of_summable_const hb hf | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Set.Finite.of_summable_const | null |
Multipliable.abs (hf : Multipliable f) : Multipliable (|f ·|) :=
let ⟨x, hx⟩ := hf; ⟨|x|, hx.abs⟩ | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.abs | null |
protected Multipliable.abs_tprod (hf : Multipliable f) : |∏' i, f i| = ∏' i, |f i| :=
hf.hasProd.abs.tprod_eq.symm
@[deprecated (since := "2025-04-12")] alias abs_tprod := Multipliable.abs_tprod | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Multipliable.abs_tprod | null |
Summable.tendsto_atTop_of_pos [Field α] [LinearOrder α] [IsStrictOrderedRing α]
[TopologicalSpace α] [OrderTopology α]
{f : ℕ → α} (hf : Summable f⁻¹) (hf' : ∀ n, 0 < f n) : Tendsto f atTop atTop :=
inv_inv f ▸ Filter.Tendsto.inv_tendsto_nhdsGT_zero <|
tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_with... | theorem | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | Summable.tendsto_atTop_of_pos | null |
@[positivity tsum _]
evalTsum : PositivityExt where eval {u α} zα pα e := do
match e with
| ~q(@tsum _ $instCommMonoid $instTopSpace $ι $f) =>
lambdaBoundedTelescope f 1 fun args (body : Q($α)) => do
let #[(i : Q($ι))] := args | failure
let rbody ← core zα pα body
let pbody ← rbody.toNonneg
... | def | Topology | [
"Mathlib.Algebra.Order.Archimedean.Basic",
"Mathlib.Algebra.Order.BigOperators.Ring.Finset",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence"
] | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | evalTsum | Positivity extension for infinite sums.
This extension only proves non-negativity, strict positivity is more delicate for infinite sums and
requires more assumptions. |
cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
(hd : Summable d) : CauchySeq f := by
lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
apply cauchySeq_of_edist_le_of_summable d (α := α) (f := f)
· exact_mod_cast hf
· exact_mod_cast hd | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | cauchySeq_of_dist_le_of_summable | If the distance between consecutive points of a sequence is estimated by a summable series,
then the original sequence is a Cauchy sequence. |
cauchySeq_of_summable_dist (h : Summable fun n ↦ dist (f n) (f n.succ)) : CauchySeq f :=
cauchySeq_of_dist_le_of_summable _ (fun _ ↦ le_rfl) h | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | cauchySeq_of_summable_dist | null |
dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
(hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
dist (f n) a ≤ ∑' m, d (n + m) := by
refine le_of_tendsto (tendsto_const_nhds.dist ha) (eventually_atTop.2 ⟨n, fun m hnm ↦ ?_⟩)
refine le_trans (dist_le_Ic... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | dist_le_tsum_of_dist_le_of_tendsto | null |
dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
(hd : Summable d) (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ tsum d := by
simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0 | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | dist_le_tsum_of_dist_le_of_tendsto₀ | null |
dist_le_tsum_dist_of_tendsto (h : Summable fun n ↦ dist (f n) (f n.succ))
(ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) :=
show dist (f n) a ≤ ∑' m, (fun x ↦ dist (f x) (f x.succ)) (n + m) from
dist_le_tsum_of_dist_le_of_tendsto (fun n ↦ dist (f n) (f n.succ)) (fun... | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | dist_le_tsum_dist_of_tendsto | null |
dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n ↦ dist (f n) (f n.succ))
(ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by
simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0 | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | dist_le_tsum_dist_of_tendsto₀ | null |
not_summable_iff_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by
lift f to ℕ → ℝ≥0 using hf
simpa using mod_cast NNReal.not_summable_iff_tendsto_nat_atTop | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | not_summable_iff_tendsto_nat_atTop_of_nonneg | null |
summable_iff_not_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop atTop := by
rw [← not_iff_not, Classical.not_not, not_summable_iff_tendsto_nat_atTop_of_nonneg hf] | theorem | Topology | [
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Topology.Instances.ENNReal.Lemmas"
] | Mathlib/Topology/Algebra/InfiniteSum/Real.lean | summable_iff_not_tendsto_nat_atTop_of_nonneg | null |
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