statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
summable.has_sum_iff (h : summable f) : has_sum f a ↔ ∑'b, f b = a | iff.intro has_sum.tsum_eq (assume eq, eq ▸ h.has_sum) | lemma | summable.has_sum_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum.tsum_eq",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_empty [is_empty β] : ∑'b, f b = 0 | has_sum_empty.tsum_eq | lemma | tsum_empty | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"is_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_sum {f : β → α} {s : finset β} (hf : ∀b∉s, f b = 0) :
∑' b, f b = ∑ b in s, f b | (has_sum_sum_of_ne_finset_zero hf).tsum_eq | lemma | tsum_eq_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum_sum_of_ne_finset_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_eq_tsum_indicator (f : β → α) (s : finset β) :
∑ x in s, f x = ∑' x, set.indicator ↑s f x | have ∀ x ∉ s, set.indicator ↑s f x = 0,
from λ x hx, set.indicator_apply_eq_zero.2 (λ hx', (hx $ finset.mem_coe.1 hx').elim),
(finset.sum_congr rfl (λ x hx, (set.indicator_apply_eq_self.2 $
λ hx', (hx' $ finset.mem_coe.2 hx).elim).symm)).trans (tsum_eq_sum this).symm | lemma | sum_eq_tsum_indicator | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"set.indicator",
"tsum_eq_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_congr {α β : Type*} [add_comm_monoid α] [topological_space α]
{f g : β → α} (hfg : ∀ b, f b = g b) : ∑' b, f b = ∑' b, g b | congr_arg tsum (funext hfg) | lemma | tsum_congr | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"topological_space",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_fintype [fintype β] (f : β → α) : ∑'b, f b = ∑ b, f b | (has_sum_fintype f).tsum_eq | lemma | tsum_fintype | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"fintype",
"has_sum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_bool (f : bool → α) : ∑' i : bool, f i = f false + f true | by { rw [tsum_fintype, finset.sum_eq_add]; simp } | lemma | tsum_bool | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"tsum_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_single {f : β → α} (b : β) (hf : ∀b' ≠ b, f b' = 0) :
∑'b, f b = f b | (has_sum_single b hf).tsum_eq | lemma | tsum_eq_single | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_tsum_eq_single (f : β → γ → α) (b : β) (c : γ) (hfb : ∀ b' ≠ b, f b' c = 0)
(hfc : ∀ (b' : β) (c' : γ), c' ≠ c → f b' c' = 0) :
∑' b' c', f b' c' = f b c | calc ∑' b' c', f b' c' = ∑' b', f b' c : tsum_congr $ λ b', tsum_eq_single _ (hfc b')
... = f b c : tsum_eq_single _ hfb | lemma | tsum_tsum_eq_single | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"tsum_congr",
"tsum_eq_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_ite_eq (b : β) [decidable_pred (= b)] (a : α) :
∑' b', (if b' = b then a else 0) = a | (has_sum_ite_eq b a).tsum_eq | lemma | tsum_ite_eq | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_ite_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_pi_single [decidable_eq β] (b : β) (a : α) :
∑' b', pi.single b a b' = a | (has_sum_pi_single b a).tsum_eq | lemma | tsum_pi_single | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_pi_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_dite_right (P : Prop) [decidable P] (x : β → ¬ P → α) :
∑' (b : β), (if h : P then (0 : α) else x b h) = if h : P then (0 : α) else ∑' (b : β), x b h | by by_cases hP : P; simp [hP] | lemma | tsum_dite_right | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_dite_left (P : Prop) [decidable P] (x : β → P → α) :
∑' (b : β), (if h : P then x b h else 0) = if h : P then (∑' (b : β), x b h) else 0 | by by_cases hP : P; simp [hP] | lemma | tsum_dite_left | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.surjective.tsum_eq_tsum_of_has_sum_iff_has_sum {α' : Type*} [add_comm_monoid α']
[topological_space α'] {e : α' → α} (hes : function.surjective e) (h0 : e 0 = 0)
{f : β → α} {g : γ → α'}
(h : ∀ {a}, has_sum f (e a) ↔ has_sum g a) :
∑' b, f b = e (∑' c, g c) | by_cases
(assume : summable g, (h.mpr this.has_sum).tsum_eq)
(assume hg : ¬ summable g,
have hf : ¬ summable f, from mt (hes.summable_iff_of_has_sum_iff @h).1 hg,
by simp [tsum, hf, hg, h0]) | lemma | function.surjective.tsum_eq_tsum_of_has_sum_iff_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"add_comm_monoid",
"has_sum",
"summable",
"topological_space",
"tsum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_tsum_of_has_sum_iff_has_sum {f : β → α} {g : γ → α}
(h : ∀{a}, has_sum f a ↔ has_sum g a) :
∑'b, f b = ∑'c, g c | surjective_id.tsum_eq_tsum_of_has_sum_iff_has_sum rfl @h | lemma | tsum_eq_tsum_of_has_sum_iff_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.tsum_eq (j : γ ≃ β) (f : β → α) : ∑'c, f (j c) = ∑'b, f b | tsum_eq_tsum_of_has_sum_iff_has_sum $ λ a, j.has_sum_iff | lemma | equiv.tsum_eq | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"tsum_eq_tsum_of_has_sum_iff_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv.tsum_eq_tsum_of_support {f : β → α} {g : γ → α} (e : support f ≃ support g)
(he : ∀ x, g (e x) = f x) :
(∑' x, f x) = ∑' y, g y | tsum_eq_tsum_of_has_sum_iff_has_sum $ λ _, e.has_sum_iff_of_support he | lemma | equiv.tsum_eq_tsum_of_support | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"tsum_eq_tsum_of_has_sum_iff_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_tsum_of_ne_zero_bij {g : γ → α} (i : support g → β)
(hi : ∀ ⦃x y⦄, i x = i y → (x : γ) = y)
(hf : support f ⊆ set.range i) (hfg : ∀ x, f (i x) = g x) :
∑' x, f x = ∑' y, g y | tsum_eq_tsum_of_has_sum_iff_has_sum $ λ _, has_sum_iff_has_sum_of_ne_zero_bij i hi hf hfg | lemma | tsum_eq_tsum_of_ne_zero_bij | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_iff_has_sum_of_ne_zero_bij",
"set.range",
"tsum_eq_tsum_of_has_sum_iff_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.tsum_subtype (s : finset β) (f : β → α) :
∑' x : {x // x ∈ s}, f x = ∑ x in s, f x | (s.has_sum f).tsum_eq | lemma | finset.tsum_subtype | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.tsum_subtype' (s : finset β) (f : β → α) :
∑' x : (s : set β), f x = ∑ x in s, f x | s.tsum_subtype f | lemma | finset.tsum_subtype' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_subtype (s : set β) (f : β → α) :
∑' x : s, f x = ∑' x, s.indicator f x | tsum_eq_tsum_of_has_sum_iff_has_sum $ λ _, has_sum_subtype_iff_indicator | lemma | tsum_subtype | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_subtype_iff_indicator",
"tsum_eq_tsum_of_has_sum_iff_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_subtype_eq_of_support_subset {f : β → α} {s : set β} (hs : support f ⊆ s) :
∑' x : s, f x = ∑' x, f x | tsum_eq_tsum_of_has_sum_iff_has_sum $ λ x, has_sum_subtype_iff_of_support_subset hs | lemma | tsum_subtype_eq_of_support_subset | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_subtype_iff_of_support_subset",
"tsum_eq_tsum_of_has_sum_iff_has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_univ (f : β → α) : ∑' x : (set.univ : set β), f x = ∑' x, f x | tsum_subtype_eq_of_support_subset $ set.subset_univ _ | lemma | tsum_univ | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"set.subset_univ",
"tsum_subtype_eq_of_support_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_singleton (b : β) (f : β → α) :
∑' x : ({b} : set β), f x = f b | begin
rw [tsum_subtype, tsum_eq_single b],
{ simp },
{ intros b' hb',
rw set.indicator_of_not_mem,
rwa set.mem_singleton_iff },
{ apply_instance }
end | lemma | tsum_singleton | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"set.mem_singleton_iff",
"tsum_eq_single",
"tsum_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_image {g : γ → β} (f : β → α) {s : set γ} (hg : set.inj_on g s) :
∑' x : g '' s, f x = ∑' x : s, f (g x) | ((equiv.set.image_of_inj_on _ _ hg).tsum_eq (λ x, f x)).symm | lemma | tsum_image | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.set.image_of_inj_on",
"set.inj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_range {g : γ → β} (f : β → α) (hg : injective g) :
∑' x : set.range g, f x = ∑' x, f (g x) | by rw [← set.image_univ, tsum_image f (hg.inj_on _), tsum_univ (f ∘ g)] | lemma | tsum_range | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"set.image_univ",
"set.range",
"tsum_image",
"tsum_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_add (hf : summable f) (hg : summable g) : ∑'b, (f b + g b) = (∑'b, f b) + (∑'b, g b) | (hf.has_sum.add hg.has_sum).tsum_eq | lemma | tsum_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_sum {f : γ → β → α} {s : finset γ} (hf : ∀i∈s, summable (f i)) :
∑'b, ∑ i in s, f i b = ∑ i in s, ∑'b, f i b | (has_sum_sum $ assume i hi, (hf i hi).has_sum).tsum_eq | lemma | tsum_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum",
"has_sum_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_add_tsum_ite' {f : β → α} (b : β) (hf : summable (f.update b 0)) :
∑' x, f x = f b + ∑' x, ite (x = b) 0 (f x) | calc ∑' x, f x = ∑' x, ((ite (x = b) (f x) 0) + (f.update b 0 x)) :
tsum_congr (λ n, by split_ifs; simp [function.update_apply, h])
... = ∑' x, ite (x = b) (f x) 0 + ∑' x, f.update b 0 x :
tsum_add ⟨ite (b = b) (f b) 0, has_sum_single b (λ b hb, if_neg hb)⟩ (hf)
... = (ite (b = b) (f b) 0) + ∑' x, f.update ... | lemma | tsum_eq_add_tsum_ite' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"dite_eq_ite",
"eq_rec_constant",
"has_sum_single",
"summable",
"tsum_add",
"tsum_congr",
"tsum_eq_single"
] | Version of `tsum_eq_add_tsum_ite` for `add_comm_monoid` rather than `add_comm_group`.
Requires a different convergence assumption involving `function.update`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_sigma' {γ : β → Type*} {f : (Σb:β, γ b) → δ} (h₁ : ∀b, summable (λc, f ⟨b, c⟩))
(h₂ : summable f) : ∑'p, f p = ∑'b c, f ⟨b, c⟩ | (h₂.has_sum.sigma (assume b, (h₁ b).has_sum)).tsum_eq.symm | lemma | tsum_sigma' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_prod' {f : β × γ → δ} (h : summable f) (h₁ : ∀b, summable (λc, f (b, c))) :
∑'p, f p = ∑'b c, f (b, c) | (h.has_sum.prod_fiberwise (assume b, (h₁ b).has_sum)).tsum_eq.symm | lemma | tsum_prod' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_comm' {f : β → γ → δ} (h : summable (function.uncurry f)) (h₁ : ∀b, summable (f b))
(h₂ : ∀ c, summable (λ b, f b c)) :
∑' c b, f b c = ∑' b c, f b c | begin
erw [← tsum_prod' h h₁, ← tsum_prod' h.prod_symm h₂, ← (equiv.prod_comm γ β).tsum_eq (uncurry f)],
refl
end | lemma | tsum_comm' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"equiv.prod_comm",
"summable",
"tsum_prod'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_supr_decode₂ [complete_lattice β] (m : β → α) (m0 : m ⊥ = 0)
(s : γ → β) : ∑' i : ℕ, m (⨆ b ∈ decode₂ γ i, s b) = ∑' b : γ, m (s b) | begin
have H : ∀ n, m (⨆ b ∈ decode₂ γ n, s b) ≠ 0 → (decode₂ γ n).is_some,
{ intros n h,
cases decode₂ γ n with b,
{ refine (h $ by simp [m0]).elim },
{ exact rfl } },
symmetry, refine tsum_eq_tsum_of_ne_zero_bij (λ a, option.get (H a.1 a.2)) _ _ _,
{ rintros ⟨m, hm⟩ ⟨n, hn⟩ e,
have := mem_deco... | theorem | tsum_supr_decode₂ | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"complete_lattice",
"option.get_mem",
"option.get_of_mem",
"option.some_get",
"set.ext_iff",
"subtype.coe_mk",
"tsum_eq_tsum_of_ne_zero_bij"
] | You can compute a sum over an encodably type by summing over the natural numbers and
taking a supremum. This is useful for outer measures. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_Union_decode₂ (m : set β → α) (m0 : m ∅ = 0)
(s : γ → set β) : ∑' i, m (⋃ b ∈ decode₂ γ i, s b) = ∑' b, m (s b) | tsum_supr_decode₂ m m0 s | theorem | tsum_Union_decode₂ | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"tsum_supr_decode₂"
] | `tsum_supr_decode₂` specialized to the complete lattice of sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_supr_tsum [complete_lattice β] (m : β → α) (m0 : m ⊥ = 0)
(R : α → α → Prop) (m_supr : ∀(s : ℕ → β), R (m (⨆ i, s i)) ∑' i, m (s i))
(s : γ → β) : R (m (⨆ b : γ, s b)) ∑' b : γ, m (s b) | by { casesI nonempty_encodable γ, rw [←supr_decode₂, ←tsum_supr_decode₂ _ m0 s], exact m_supr _ } | theorem | rel_supr_tsum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"complete_lattice",
"nonempty_encodable"
] | If a function is countably sub-additive then it is sub-additive on countable types | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_supr_sum [complete_lattice β] (m : β → α) (m0 : m ⊥ = 0)
(R : α → α → Prop) (m_supr : ∀(s : ℕ → β), R (m (⨆ i, s i)) (∑' i, m (s i)))
(s : δ → β) (t : finset δ) :
R (m (⨆ d ∈ t, s d)) (∑ d in t, m (s d)) | by { rw [supr_subtype', ←finset.tsum_subtype], exact rel_supr_tsum m m0 R m_supr _ } | theorem | rel_supr_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"complete_lattice",
"finset",
"rel_supr_tsum",
"supr_subtype'"
] | If a function is countably sub-additive then it is sub-additive on finite sets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rel_sup_add [complete_lattice β] (m : β → α) (m0 : m ⊥ = 0)
(R : α → α → Prop) (m_supr : ∀(s : ℕ → β), R (m (⨆ i, s i)) (∑' i, m (s i)))
(s₁ s₂ : β) : R (m (s₁ ⊔ s₂)) (m s₁ + m s₂) | begin
convert rel_supr_tsum m m0 R m_supr (λ b, cond b s₁ s₂),
{ simp only [supr_bool_eq, cond] },
{ rw [tsum_fintype, fintype.sum_bool, cond, cond] }
end | theorem | rel_sup_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"complete_lattice",
"rel_supr_tsum",
"supr_bool_eq",
"tsum_fintype"
] | If a function is countably sub-additive then it is binary sub-additive | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_add_tsum_compl {s : set β} (hs : summable (f ∘ coe : s → α))
(hsc : summable (f ∘ coe : sᶜ → α)) :
(∑' x : s, f x) + (∑' x : sᶜ, f x) = ∑' x, f x | (hs.has_sum.add_compl hsc.has_sum).tsum_eq.symm | lemma | tsum_add_tsum_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_union_disjoint {s t : set β} (hd : disjoint s t)
(hs : summable (f ∘ coe : s → α)) (ht : summable (f ∘ coe : t → α)) :
(∑' x : s ∪ t, f x) = (∑' x : s, f x) + (∑' x : t, f x) | (hs.has_sum.add_disjoint hd ht.has_sum).tsum_eq | lemma | tsum_union_disjoint | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"disjoint",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_finset_bUnion_disjoint {ι} {s : finset ι} {t : ι → set β}
(hd : (s : set ι).pairwise (disjoint on t))
(hf : ∀ i ∈ s, summable (f ∘ coe : t i → α)) :
(∑' x : (⋃ i ∈ s, t i), f x) = ∑ i in s, ∑' x : t i, f x | (has_sum_sum_disjoint _ hd (λ i hi, (hf i hi).has_sum)).tsum_eq | lemma | tsum_finset_bUnion_disjoint | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"disjoint",
"finset",
"has_sum",
"has_sum_sum_disjoint",
"pairwise",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_even_add_odd {f : ℕ → α} (he : summable (λ k, f (2 * k)))
(ho : summable (λ k, f (2 * k + 1))) :
(∑' k, f (2 * k)) + (∑' k, f (2 * k + 1)) = ∑' k, f k | (he.has_sum.even_add_odd ho.has_sum).tsum_eq.symm | lemma | tsum_even_add_odd | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.neg (h : has_sum f a) : has_sum (λb, - f b) (- a) | by simpa only using h.map (-add_monoid_hom.id α) continuous_neg | lemma | has_sum.neg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.neg (hf : summable f) : summable (λb, - f b) | hf.has_sum.neg.summable | lemma | summable.neg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.of_neg (hf : summable (λb, - f b)) : summable f | by simpa only [neg_neg] using hf.neg | lemma | summable.of_neg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_neg_iff : summable (λ b, - f b) ↔ summable f | ⟨summable.of_neg, summable.neg⟩ | lemma | summable_neg_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.sub (hf : has_sum f a₁) (hg : has_sum g a₂) : has_sum (λb, f b - g b) (a₁ - a₂) | by { simp only [sub_eq_add_neg], exact hf.add hg.neg } | lemma | has_sum.sub | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.sub (hf : summable f) (hg : summable g) : summable (λb, f b - g b) | (hf.has_sum.sub hg.has_sum).summable | lemma | summable.sub | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.trans_sub (hg : summable g) (hfg : summable (λb, f b - g b)) :
summable f | by simpa only [sub_add_cancel] using hfg.add hg | lemma | summable.trans_sub | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_iff_of_summable_sub (hfg : summable (λb, f b - g b)) :
summable f ↔ summable g | ⟨λ hf, hf.trans_sub $ by simpa only [neg_sub] using hfg.neg, λ hg, hg.trans_sub hfg⟩ | lemma | summable_iff_of_summable_sub | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.update (hf : has_sum f a₁) (b : β) [decidable_eq β] (a : α) :
has_sum (update f b a) (a - f b + a₁) | begin
convert ((has_sum_ite_eq b _).add hf),
ext b',
by_cases h : b' = b,
{ rw [h, update_same],
simp only [eq_self_iff_true, if_true, sub_add_cancel] },
simp only [h, update_noteq, if_false, ne.def, zero_add, not_false_iff],
end | lemma | has_sum.update | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_ite_eq",
"update",
"update_noteq",
"update_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.update (hf : summable f) (b : β) [decidable_eq β] (a : α) :
summable (update f b a) | (hf.has_sum.update b a).summable | lemma | summable.update | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.has_sum_compl_iff {s : set β} (hf : has_sum (f ∘ coe : s → α) a₁) :
has_sum (f ∘ coe : sᶜ → α) a₂ ↔ has_sum f (a₁ + a₂) | begin
refine ⟨λ h, hf.add_compl h, λ h, _⟩,
rw [has_sum_subtype_iff_indicator] at hf ⊢,
rw [set.indicator_compl],
simpa only [add_sub_cancel'] using h.sub hf
end | lemma | has_sum.has_sum_compl_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_subtype_iff_indicator",
"set.indicator_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.has_sum_iff_compl {s : set β} (hf : has_sum (f ∘ coe : s → α) a₁) :
has_sum f a₂ ↔ has_sum (f ∘ coe : sᶜ → α) (a₂ - a₁) | iff.symm $ hf.has_sum_compl_iff.trans $ by rw [add_sub_cancel'_right] | lemma | has_sum.has_sum_iff_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.summable_compl_iff {s : set β} (hf : summable (f ∘ coe : s → α)) :
summable (f ∘ coe : sᶜ → α) ↔ summable f | ⟨λ ⟨a, ha⟩, (hf.has_sum.has_sum_compl_iff.1 ha).summable,
λ ⟨a, ha⟩, (hf.has_sum.has_sum_iff_compl.1 ha).summable⟩ | lemma | summable.summable_compl_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.has_sum_compl_iff (s : finset β) :
has_sum (λ x : {x // x ∉ s}, f x) a ↔ has_sum f (a + ∑ i in s, f i) | (s.has_sum f).has_sum_compl_iff.trans $ by rw [add_comm] | lemma | finset.has_sum_compl_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.has_sum_iff_compl (s : finset β) :
has_sum f a ↔ has_sum (λ x : {x // x ∉ s}, f x) (a - ∑ i in s, f i) | (s.has_sum f).has_sum_iff_compl | lemma | finset.has_sum_iff_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.summable_compl_iff (s : finset β) :
summable (λ x : {x // x ∉ s}, f x) ↔ summable f | (s.summable f).summable_compl_iff | lemma | finset.summable_compl_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.finite.summable_compl_iff {s : set β} (hs : s.finite) :
summable (f ∘ coe : sᶜ → α) ↔ summable f | (hs.summable f).summable_compl_iff | lemma | set.finite.summable_compl_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_ite_sub_has_sum [decidable_eq β] (hf : has_sum f a) (b : β) :
has_sum (λ n, ite (n = b) 0 (f n)) (a - f b) | begin
convert hf.update b 0 using 1,
{ ext n, rw function.update_apply, },
{ rw [sub_add_eq_add_sub, zero_add], },
end | lemma | has_sum_ite_sub_has_sum | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_neg : ∑'b, - f b = - ∑'b, f b | begin
by_cases hf : summable f,
{ exact hf.has_sum.neg.tsum_eq, },
{ simp [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable (mt summable.of_neg hf)] },
end | lemma | tsum_neg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"summable.of_neg",
"tsum_eq_zero_of_not_summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_sub (hf : summable f) (hg : summable g) : ∑'b, (f b - g b) = ∑'b, f b - ∑'b, g b | (hf.has_sum.sub hg.has_sum).tsum_eq | lemma | tsum_sub | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add_tsum_compl {s : finset β} (hf : summable f) :
(∑ x in s, f x) + (∑' x : (↑s : set β)ᶜ, f x) = ∑' x, f x | ((s.has_sum f).add_compl (s.summable_compl_iff.2 hf).has_sum).tsum_eq.symm | lemma | sum_add_tsum_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"has_sum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_add_tsum_ite [decidable_eq β] (hf : summable f) (b : β) :
∑' n, f n = f b + ∑' n, ite (n = b) 0 (f n) | begin
rw (has_sum_ite_sub_has_sum hf.has_sum b).tsum_eq,
exact (add_sub_cancel'_right _ _).symm,
end | lemma | tsum_eq_add_tsum_ite | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_ite_sub_has_sum",
"summable"
] | Let `f : β → α` be a sequence with summable series and let `b ∈ β` be an index.
Lemma `tsum_eq_add_tsum_ite` writes `Σ f n` as the sum of `f b` plus the series of the
remaining terms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_nat_add_iff {f : ℕ → α} (k : ℕ) {a : α} :
has_sum (λ n, f (n + k)) a ↔ has_sum f (a + ∑ i in range k, f i) | begin
refine iff.trans _ ((range k).has_sum_compl_iff),
rw [← (not_mem_range_equiv k).symm.has_sum_iff],
refl
end | lemma | has_sum_nat_add_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"not_mem_range_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_nat_add_iff {f : ℕ → α} (k : ℕ) : summable (λ n, f (n + k)) ↔ summable f | iff.symm $ (equiv.add_right (∑ i in range k, f i)).surjective.summable_iff_of_has_sum_iff $
λ a, (has_sum_nat_add_iff k).symm | lemma | summable_nat_add_iff | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum_nat_add_iff",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_nat_add_iff' {f : ℕ → α} (k : ℕ) {a : α} :
has_sum (λ n, f (n + k)) (a - ∑ i in range k, f i) ↔ has_sum f a | by simp [has_sum_nat_add_iff] | lemma | has_sum_nat_add_iff' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_nat_add_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add_tsum_nat_add [t2_space α] {f : ℕ → α} (k : ℕ) (h : summable f) :
(∑ i in range k, f i) + (∑' i, f (i + k)) = ∑' i, f i | by simpa only [add_comm] using
((has_sum_nat_add_iff k).1 ((summable_nat_add_iff k).2 h).has_sum).unique h.has_sum | lemma | sum_add_tsum_nat_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_nat_add_iff",
"summable",
"summable_nat_add_iff",
"t2_space",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_eq_zero_add [t2_space α] {f : ℕ → α} (hf : summable f) :
∑'b, f b = f 0 + ∑'b, f (b + 1) | by simpa only [sum_range_one] using (sum_add_tsum_nat_add 1 hf).symm | lemma | tsum_eq_zero_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"sum_add_tsum_nat_add",
"summable",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sum_nat_add [t2_space α] (f : ℕ → α) : tendsto (λ i, ∑' k, f (k + i)) at_top (𝓝 0) | begin
by_cases hf : summable f,
{ have h₀ : (λ i, (∑' i, f i) - ∑ j in range i, f j) = λ i, ∑' (k : ℕ), f (k + i),
{ ext1 i,
rw [sub_eq_iff_eq_add, add_comm, sum_add_tsum_nat_add i hf] },
have h₁ : tendsto (λ i : ℕ, ∑' i, f i) at_top (𝓝 (∑' i, f i)) := tendsto_const_nhds,
simpa only [h₀, sub_self... | lemma | tendsto_sum_nat_add | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"sum_add_tsum_nat_add",
"summable",
"summable_nat_add_iff",
"t2_space",
"tendsto_const_nhds",
"tsum_eq_zero_of_not_summable"
] | For `f : ℕ → α`, then `∑' k, f (k + i)` tends to zero. This does not require a summability
assumption on `f`, as otherwise all sums are zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.int_rec {b : α} {f g : ℕ → α} (hf : has_sum f a) (hg : has_sum g b) :
@has_sum α _ _ _ (@int.rec (λ _, α) f g : ℤ → α) (a + b) | begin
-- note this proof works for any two-case inductive
have h₁ : injective (coe : ℕ → ℤ) := @int.of_nat.inj,
have h₂ : injective int.neg_succ_of_nat := @int.neg_succ_of_nat.inj,
have : is_compl (set.range (coe : ℕ → ℤ)) (set.range int.neg_succ_of_nat),
{ split,
{ rw disjoint_iff_inf_le,
rintros _... | lemma | has_sum.int_rec | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"codisjoint_iff_le_sup",
"disjoint_iff_inf_le",
"has_sum",
"has_sum.add_is_compl",
"is_compl",
"set.range"
] | If `f₀, f₁, f₂, ...` and `g₀, g₁, g₂, ...` are both convergent then so is the `ℤ`-indexed
sequence: `..., g₂, g₁, g₀, f₀, f₁, f₂, ...`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.nonneg_add_neg {b : α} {f : ℤ → α}
(hnonneg : has_sum (λ n : ℕ, f n) a) (hneg : has_sum (λ (n : ℕ), f (-n.succ)) b) :
has_sum f (a + b) | begin
simp_rw ← int.neg_succ_of_nat_coe at hneg,
convert hnonneg.int_rec hneg using 1,
ext (i | j); refl,
end | lemma | has_sum.nonneg_add_neg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.pos_add_zero_add_neg {b : α} {f : ℤ → α}
(hpos : has_sum (λ n:ℕ, f(n + 1)) a) (hneg : has_sum (λ (n : ℕ), f (-n.succ)) b) :
has_sum f (a + f 0 + b) | begin
have : ∀ g : ℕ → α, has_sum (λ k, g (k + 1)) a → has_sum g (a + g 0),
{ intros g hg, simpa using (has_sum_nat_add_iff _).mp hg },
exact (this (λ n, f n) hpos).nonneg_add_neg hneg,
end | lemma | has_sum.pos_add_zero_add_neg | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum_nat_add_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_int_of_summable_nat {f : ℤ → α}
(hp : summable (λ n:ℕ, f n)) (hn : summable (λ n:ℕ, f (-n))) : summable f | (has_sum.nonneg_add_neg hp.has_sum $ summable.has_sum $ (summable_nat_add_iff 1).mpr hn).summable | lemma | summable_int_of_summable_nat | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum.nonneg_add_neg",
"summable",
"summable.has_sum",
"summable_nat_add_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.sum_nat_of_sum_int {α : Type*} [add_comm_monoid α] [topological_space α]
[has_continuous_add α] {a : α} {f : ℤ → α} (hf : has_sum f a) :
has_sum (λ n:ℕ, f n + f (-n)) (a + f 0) | begin
apply (hf.add (has_sum_ite_eq (0 : ℤ) (f 0))).has_sum_of_sum_eq (λ u, _),
refine ⟨u.image int.nat_abs, λ v' hv', _⟩,
let u1 := v'.image (λ (x : ℕ), (x : ℤ)),
let u2 := v'.image (λ (x : ℕ), - (x : ℤ)),
have A : u ⊆ u1 ∪ u2,
{ assume x hx,
simp only [mem_union, mem_image, exists_prop],
rcases le... | lemma | has_sum.sum_nat_of_sum_int | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"abs_eq_self",
"abs_of_nonpos",
"add_comm_monoid",
"exists_prop",
"has_continuous_add",
"has_sum",
"has_sum_ite_eq",
"imp_self",
"int.coe_nat_abs",
"nat.cast_inj",
"nat.cast_nonneg",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_iff_cauchy_seq_finset [complete_space α] {f : β → α} :
summable f ↔ cauchy_seq (λ (s : finset β), ∑ b in s, f b) | cauchy_map_iff_exists_tendsto.symm | lemma | summable_iff_cauchy_seq_finset | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"cauchy_seq",
"complete_space",
"finset",
"summable"
] | The **Cauchy criterion** for infinite sums, also known as the **Cauchy convergence test** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_finset_iff_vanishing :
cauchy_seq (λ (s : finset β), ∑ b in s, f b)
↔ ∀ e ∈ 𝓝 (0:α), (∃s:finset β, ∀t, disjoint t s → ∑ b in t, f b ∈ e) | begin
simp only [cauchy_seq, cauchy_map_iff, and_iff_right at_top_ne_bot,
prod_at_top_at_top_eq, uniformity_eq_comap_nhds_zero α, tendsto_comap_iff, (∘)],
rw [tendsto_at_top'],
split,
{ assume h e he,
rcases h e he with ⟨⟨s₁, s₂⟩, h⟩,
use [s₁ ∪ s₂],
assume t ht,
specialize h (s₁ ∪ s₂, (s₁ ∪ ... | lemma | cauchy_seq_finset_iff_vanishing | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"cauchy_map_iff",
"cauchy_seq",
"disjoint",
"finset",
"finset.sdiff_disjoint",
"le_sup_of_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_tsum_compl_at_top_zero (f : β → α) :
tendsto (λ (s : finset β), ∑' b : {x // x ∉ s}, f b) at_top (𝓝 0) | begin
by_cases H : summable f,
{ assume e he,
rcases exists_mem_nhds_is_closed_subset he with ⟨o, ho, o_closed, oe⟩,
simp only [le_eq_subset, set.mem_preimage, mem_at_top_sets, filter.mem_map, ge_iff_le],
obtain ⟨s, hs⟩ : ∃ (s : finset β), ∀ (t : finset β), disjoint t s → ∑ (b : β) in t, f b ∈ o :=
... | lemma | tendsto_tsum_compl_at_top_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"disjoint",
"exists_mem_nhds_is_closed_subset",
"filter.mem_map",
"finset",
"finset.image",
"finset.summable_compl_iff",
"ge_iff_le",
"is_closed.mem_of_tendsto",
"set.mem_preimage",
"subtype.ext",
"summable",
"tendsto_const_nhds",
"tsum_eq_zero_of_not_summable"
] | The sum over the complement of a finset tends to `0` when the finset grows to cover the whole
space. This does not need a summability assumption, as otherwise all sums are zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_iff_vanishing :
summable f ↔ ∀ e ∈ 𝓝 (0:α), (∃s:finset β, ∀t, disjoint t s → ∑ b in t, f b ∈ e) | by rw [summable_iff_cauchy_seq_finset, cauchy_seq_finset_iff_vanishing] | lemma | summable_iff_vanishing | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"cauchy_seq_finset_iff_vanishing",
"disjoint",
"finset",
"summable",
"summable_iff_cauchy_seq_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.summable_of_eq_zero_or_self (hf : summable f) (h : ∀b, g b = 0 ∨ g b = f b) :
summable g | summable_iff_vanishing.2 $
assume e he,
let ⟨s, hs⟩ := summable_iff_vanishing.1 hf e he in
⟨s, assume t ht,
have eq : ∑ b in t.filter (λb, g b = f b), f b = ∑ b in t, g b :=
calc ∑ b in t.filter (λb, g b = f b), f b = ∑ b in t.filter (λb, g b = f b), g b :
finset.sum_congr rfl (assume b hb, (f... | lemma | summable.summable_of_eq_zero_or_self | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset.disjoint_of_subset_left",
"finset.filter_subset",
"finset.mem_filter",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.indicator (hf : summable f) (s : set β) :
summable (s.indicator f) | hf.summable_of_eq_zero_or_self $ set.indicator_eq_zero_or_self _ _ | lemma | summable.indicator | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.comp_injective {i : γ → β} (hf : summable f) (hi : injective i) :
summable (f ∘ i) | begin
simpa only [set.indicator_range_comp]
using (hi.summable_iff _).2 (hf.indicator (set.range i)),
exact λ x hx, set.indicator_of_not_mem hx _
end | lemma | summable.comp_injective | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"set.range",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.subtype (hf : summable f) (s : set β) : summable (f ∘ coe : s → α) | hf.comp_injective subtype.coe_injective | lemma | summable.subtype | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"subtype.coe_injective",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_subtype_and_compl {s : set β} :
summable (λ x : s, f x) ∧ summable (λ x : sᶜ, f x) ↔ summable f | ⟨and_imp.2 summable.add_compl, λ h, ⟨h.subtype s, h.subtype sᶜ⟩⟩ | lemma | summable_subtype_and_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"summable.add_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.sigma_factor {γ : β → Type*} {f : (Σb:β, γ b) → α}
(ha : summable f) (b : β) : summable (λc, f ⟨b, c⟩) | ha.comp_injective sigma_mk_injective | lemma | summable.sigma_factor | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"sigma_mk_injective",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.sigma {γ : β → Type*} {f : (Σb:β, γ b) → α}
(ha : summable f) : summable (λb, ∑'c, f ⟨b, c⟩) | ha.sigma' (λ b, ha.sigma_factor b) | lemma | summable.sigma | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.prod_factor {f : β × γ → α} (h : summable f) (b : β) :
summable (λ c, f (b, c)) | h.comp_injective $ λ c₁ c₂ h, (prod.ext_iff.1 h).2 | lemma | summable.prod_factor | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_sigma [t0_space α] {γ : β → Type*} {f : (Σb:β, γ b) → α}
(ha : summable f) : ∑'p, f p = ∑'b c, f ⟨b, c⟩ | tsum_sigma' (λ b, ha.sigma_factor b) ha | lemma | tsum_sigma | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"t0_space",
"tsum_sigma'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_prod [t0_space α] {f : β × γ → α} (h : summable f) :
∑'p, f p = ∑'b c, f ⟨b, c⟩ | tsum_prod' h h.prod_factor | lemma | tsum_prod | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"t0_space",
"tsum_prod'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_comm [t0_space α] {f : β → γ → α} (h : summable (function.uncurry f)) :
∑' c b, f b c = ∑' b c, f b c | tsum_comm' h h.prod_factor h.prod_symm.prod_factor | lemma | tsum_comm | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"t0_space",
"tsum_comm'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_subtype_add_tsum_subtype_compl [t2_space α] {f : β → α} (hf : summable f) (s : set β) :
∑' x : s, f x + ∑' x : sᶜ, f x = ∑' x, f x | ((hf.subtype s).has_sum.add_compl (hf.subtype {x | x ∉ s}).has_sum).unique hf.has_sum | lemma | tsum_subtype_add_tsum_subtype_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"has_sum",
"has_sum.add_compl",
"summable",
"t2_space",
"unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_add_tsum_subtype_compl [t2_space α] {f : β → α} (hf : summable f) (s : finset β) :
∑ x in s, f x + ∑' x : {x // x ∉ s}, f x = ∑' x, f x | begin
rw ← tsum_subtype_add_tsum_subtype_compl hf s,
simp only [finset.tsum_subtype', add_right_inj],
refl,
end | lemma | sum_add_tsum_subtype_compl | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"finset",
"finset.tsum_subtype'",
"summable",
"t2_space",
"tsum_subtype_add_tsum_subtype_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.vanishing (hf : summable f) ⦃e : set G⦄ (he : e ∈ 𝓝 (0 : G)) :
∃ s : finset α, ∀ t, disjoint t s → ∑ k in t, f k ∈ e | begin
letI : uniform_space G := topological_add_group.to_uniform_space G,
letI : uniform_add_group G := topological_add_comm_group_is_uniform,
rcases hf with ⟨y, hy⟩,
exact cauchy_seq_finset_iff_vanishing.1 hy.cauchy_seq e he
end | lemma | summable.vanishing | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"disjoint",
"finset",
"summable",
"uniform_add_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.tendsto_cofinite_zero (hf : summable f) : tendsto f cofinite (𝓝 0) | begin
intros e he,
rw [filter.mem_map],
rcases hf.vanishing he with ⟨s, hs⟩,
refine s.eventually_cofinite_nmem.mono (λ x hx, _),
by simpa using hs {x} (disjoint_singleton_left.2 hx)
end | lemma | summable.tendsto_cofinite_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"filter.mem_map",
"summable"
] | Series divergence test: if `f` is a convergent series, then `f x` tends to zero along
`cofinite`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable.tendsto_at_top_zero {f : ℕ → G} (hf : summable f) : tendsto f at_top (𝓝 0) | by { rw ←nat.cofinite_eq_at_top, exact hf.tendsto_cofinite_zero } | lemma | summable.tendsto_at_top_zero | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum.const_smul {a : α} (b : γ) (hf : has_sum f a) : has_sum (λ i, b • f i) (b • a) | hf.map (distrib_mul_action.to_add_monoid_hom α _) $ continuous_const_smul _ | lemma | has_sum.const_smul | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"distrib_mul_action.to_add_monoid_hom",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable.const_smul (b : γ) (hf : summable f) : summable (λ i, b • f i) | (hf.has_sum.const_smul _).summable | lemma | summable.const_smul | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_const_smul [t2_space α] (b : γ) (hf : summable f) : ∑' i, b • f i = b • ∑' i, f i | (hf.has_sum.const_smul _).tsum_eq | lemma | tsum_const_smul | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"summable",
"t2_space"
] | Infinite sums commute with scalar multiplication. Version for scalars living in a `monoid`, but
requiring a summability hypothesis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_const_smul' {γ : Type*} [group γ] [distrib_mul_action γ α]
[has_continuous_const_smul γ α] [t2_space α] (g : γ) :
∑' (i : β), g • f i = g • ∑' (i : β), f i | begin
by_cases hf : summable f,
{ exact tsum_const_smul _ hf, },
rw tsum_eq_zero_of_not_summable hf,
simp only [smul_zero],
let mul_g : α ≃+ α := distrib_mul_action.to_add_equiv α g,
apply tsum_eq_zero_of_not_summable,
change ¬ summable (mul_g ∘ f),
rwa summable.map_iff_of_equiv mul_g; apply continuous_... | lemma | tsum_const_smul' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"distrib_mul_action",
"distrib_mul_action.to_add_equiv",
"group",
"has_continuous_const_smul",
"smul_zero",
"summable",
"summable.map_iff_of_equiv",
"t2_space",
"tsum_const_smul",
"tsum_eq_zero_of_not_summable"
] | Infinite sums commute with scalar multiplication. Version for scalars living in a `group`, but
not requiring any summability hypothesis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_const_smul'' {γ : Type*} [division_ring γ] [module γ α] [has_continuous_const_smul γ α]
[t2_space α] (g : γ) :
∑' (i : β), g • f i = g • ∑' (i : β), f i | begin
by_cases hf : summable f,
{ exact tsum_const_smul _ hf, },
rw tsum_eq_zero_of_not_summable hf,
simp only [smul_zero],
by_cases hg : g = 0,
{ simp [hg], },
let mul_g : α ≃+ α := distrib_mul_action.to_add_equiv₀ α g hg,
apply tsum_eq_zero_of_not_summable,
change ¬ summable (mul_g ∘ f),
rwa summa... | lemma | tsum_const_smul'' | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"distrib_mul_action.to_add_equiv₀",
"division_ring",
"has_continuous_const_smul",
"module",
"smul_zero",
"summable",
"summable.map_iff_of_equiv",
"t2_space",
"tsum_const_smul",
"tsum_eq_zero_of_not_summable"
] | Infinite sums commute with scalar multiplication. Version for scalars living in a
`division_ring`; no summability hypothesis. This could be made to work for a
`[group_with_zero γ]` if there was such a thing as `distrib_mul_action_with_zero`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.prod_mk {f : β → α} {g : β → γ} {a : α} {b : γ}
(hf : has_sum f a) (hg : has_sum g b) :
has_sum (λ x, (⟨f x, g x⟩ : α × γ)) ⟨a, b⟩ | by simp [has_sum, ← prod_mk_sum, filter.tendsto.prod_mk_nhds hf hg] | lemma | has_sum.prod_mk | topology.algebra.infinite_sum | src/topology/algebra/infinite_sum/basic.lean | [
"data.nat.parity",
"logic.encodable.lattice",
"topology.algebra.uniform_group",
"topology.algebra.star"
] | [
"filter.tendsto.prod_mk_nhds",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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