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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
is_equivalent.congr_right {u v w : α → β} {l : filter α} (huv : u ~[l] v)
(hvw : v =ᶠ[l] w) : u ~[l] w | (huv.symm.congr_left hvw).symm | lemma | asymptotics.is_equivalent.congr_right | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_zero_iff_eventually_zero : u ~[l] 0 ↔ u =ᶠ[l] 0 | begin
rw [is_equivalent, sub_zero],
exact is_o_zero_right_iff
end | lemma | asymptotics.is_equivalent_zero_iff_eventually_zero | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_zero_iff_is_O_zero : u ~[l] 0 ↔ u =O[l] (0 : α → β) | begin
refine ⟨is_equivalent.is_O, λ h, _⟩,
rw [is_equivalent_zero_iff_eventually_zero, eventually_eq_iff_exists_mem],
exact ⟨{x : α | u x = 0}, is_O_zero_right_iff.mp h, λ x hx, hx⟩,
end | lemma | asymptotics.is_equivalent_zero_iff_is_O_zero | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_const_iff_tendsto {c : β} (h : c ≠ 0) : u ~[l] const _ c ↔ tendsto u l (𝓝 c) | begin
rw [is_equivalent, is_o_const_iff h],
split; intro h;
[ { have := h.sub tendsto_const_nhds, rw zero_sub (-c) at this },
{ have := h.sub tendsto_const_nhds, rw ← sub_self c} ];
convert this; try { ext }; simp
end | lemma | asymptotics.is_equivalent_const_iff_tendsto | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : tendsto u l (𝓝 c) | begin
rcases (em $ c = 0) with ⟨rfl, h⟩,
{ exact (tendsto_congr' $ is_equivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds },
{ exact (is_equivalent_const_iff_tendsto h).mp hu }
end | lemma | asymptotics.is_equivalent.tendsto_const | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"em",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_nhds {c : β} (huv : u ~[l] v) (hu : tendsto u l (𝓝 c)) :
tendsto v l (𝓝 c) | begin
by_cases h : c = 0,
{ subst c, rw ← is_o_one_iff ℝ at hu ⊢,
simpa using (huv.symm.is_o.trans hu).add hu },
{ rw ← is_equivalent_const_iff_tendsto h at hu ⊢,
exact huv.symm.trans hu }
end | lemma | asymptotics.is_equivalent.tendsto_nhds | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_nhds_iff {c : β} (huv : u ~[l] v) :
tendsto u l (𝓝 c) ↔ tendsto v l (𝓝 c) | ⟨huv.tendsto_nhds, huv.symm.tendsto_nhds⟩ | lemma | asymptotics.is_equivalent.tendsto_nhds_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.add_is_o (huv : u ~[l] v) (hwv : w =o[l] v) : (u + w) ~[l] v | by simpa only [is_equivalent, add_sub_right_comm] using huv.add hwv | lemma | asymptotics.is_equivalent.add_is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.sub_is_o (huv : u ~[l] v) (hwv : w =o[l] v) : (u - w) ~[l] v | by simpa only [sub_eq_add_neg] using huv.add_is_o hwv.neg_left | lemma | asymptotics.is_equivalent.sub_is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.add_is_equivalent (hu : u =o[l] w) (hv : v ~[l] w) : (u + v) ~[l] w | add_comm v u ▸ hv.add_is_o hu | lemma | asymptotics.is_o.add_is_equivalent | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.is_equivalent (huv : (u - v) =o[l] v) : u ~[l] v | huv | lemma | asymptotics.is_o.is_equivalent | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.neg (huv : u ~[l] v) : (λ x, - u x) ~[l] (λ x, - v x) | begin
rw is_equivalent,
convert huv.is_o.neg_left.neg_right,
ext,
simp,
end | lemma | asymptotics.is_equivalent.neg | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_iff_exists_eq_mul : u ~[l] v ↔
∃ (φ : α → β) (hφ : tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v | begin
rw [is_equivalent, is_o_iff_exists_eq_mul],
split; rintros ⟨φ, hφ, h⟩; [use (φ + 1), use (φ - 1)]; split,
{ conv in (𝓝 _) { rw ← zero_add (1 : β) },
exact hφ.add (tendsto_const_nhds) },
{ convert h.add (eventually_eq.refl l v); ext; simp [add_mul] },
{ conv in (𝓝 _) { rw ← sub_self (1 : β) },
... | lemma | asymptotics.is_equivalent_iff_exists_eq_mul | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.exists_eq_mul (huv : u ~[l] v) :
∃ (φ : α → β) (hφ : tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v | is_equivalent_iff_exists_eq_mul.mp huv | lemma | asymptotics.is_equivalent.exists_eq_mul | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_of_tendsto_one (hz : ∀ᶠ x in l, v x = 0 → u x = 0)
(huv : tendsto (u/v) l (𝓝 1)) : u ~[l] v | begin
rw is_equivalent_iff_exists_eq_mul,
refine ⟨u/v, huv, hz.mono $ λ x hz', (div_mul_cancel_of_imp hz').symm⟩,
end | lemma | asymptotics.is_equivalent_of_tendsto_one | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"div_mul_cancel_of_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_of_tendsto_one' (hz : ∀ x, v x = 0 → u x = 0) (huv : tendsto (u/v) l (𝓝 1)) :
u ~[l] v | is_equivalent_of_tendsto_one (eventually_of_forall hz) huv | lemma | asymptotics.is_equivalent_of_tendsto_one' | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_iff_tendsto_one (hz : ∀ᶠ x in l, v x ≠ 0) :
u ~[l] v ↔ tendsto (u/v) l (𝓝 1) | begin
split,
{ intro hequiv,
have := hequiv.is_o.tendsto_div_nhds_zero,
simp only [pi.sub_apply, sub_div] at this,
have key : tendsto (λ x, v x / v x) l (𝓝 1),
{ exact (tendsto_congr' $ hz.mono $ λ x hnz, @div_self _ _ (v x) hnz).mpr tendsto_const_nhds },
convert this.add key,
{ ext, simp }... | lemma | asymptotics.is_equivalent_iff_tendsto_one | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"div_self",
"sub_div",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.smul {α E 𝕜 : Type*} [normed_field 𝕜] [normed_add_comm_group E]
[normed_space 𝕜 E] {a b : α → 𝕜} {u v : α → E} {l : filter α} (hab : a ~[l] b) (huv : u ~[l] v) :
(λ x, a x • u x) ~[l] (λ x, b x • v x) | begin
rcases hab.exists_eq_mul with ⟨φ, hφ, habφ⟩,
have : (λ (x : α), a x • u x) - (λ (x : α), b x • v x) =ᶠ[l] λ x, b x • ((φ x • u x) - v x),
{ convert (habφ.comp₂ (•) $ eventually_eq.refl _ u).sub (eventually_eq.refl _ (λ x, b x • v x)),
ext,
rw [pi.mul_apply, mul_comm, mul_smul, ← smul_sub] },
refin... | lemma | asymptotics.is_equivalent.smul | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"div_pos",
"filter",
"metric.tendsto_nhds",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"norm_smul",
"normed_add_comm_group",
"normed_field",
"normed_space",
"pi.mul_apply",
"ring",
"smul_sub",
"sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.mul (htu : t ~[l] u) (hvw : v ~[l] w) : t * v ~[l] u * w | htu.smul hvw | lemma | asymptotics.is_equivalent.mul | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.inv (huv : u ~[l] v) : (λ x, (u x)⁻¹) ~[l] (λ x, (v x)⁻¹) | begin
rw is_equivalent_iff_exists_eq_mul at *,
rcases huv with ⟨φ, hφ, h⟩,
rw ← inv_one,
refine ⟨λ x, (φ x)⁻¹, tendsto.inv₀ hφ (by norm_num) , _⟩,
convert h.inv,
ext,
simp [mul_inv]
end | lemma | asymptotics.is_equivalent.inv | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"inv_one",
"mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.div (htu : t ~[l] u) (hvw : v ~[l] w) :
(λ x, t x / v x) ~[l] (λ x, u x / w x) | by simpa only [div_eq_mul_inv] using htu.mul hvw.inv | lemma | asymptotics.is_equivalent.div | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_at_top [order_topology β] (huv : u ~[l] v) (hu : tendsto u l at_top) :
tendsto v l at_top | let ⟨φ, hφ, h⟩ := huv.symm.exists_eq_mul in
tendsto.congr' h.symm ((mul_comm u φ) ▸ (hu.at_top_mul zero_lt_one hφ)) | lemma | asymptotics.is_equivalent.tendsto_at_top | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"mul_comm",
"order_topology",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_at_top_iff [order_topology β] (huv : u ~[l] v) :
tendsto u l at_top ↔ tendsto v l at_top | ⟨huv.tendsto_at_top, huv.symm.tendsto_at_top⟩ | lemma | asymptotics.is_equivalent.tendsto_at_top_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"order_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_at_bot [order_topology β] (huv : u ~[l] v) (hu : tendsto u l at_bot) :
tendsto v l at_bot | begin
convert tendsto_neg_at_top_at_bot.comp
(huv.neg.tendsto_at_top $ tendsto_neg_at_bot_at_top.comp hu),
ext,
simp
end | lemma | asymptotics.is_equivalent.tendsto_at_bot | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"order_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.tendsto_at_bot_iff [order_topology β] (huv : u ~[l] v) :
tendsto u l at_bot ↔ tendsto v l at_bot | ⟨huv.tendsto_at_bot, huv.symm.tendsto_at_bot⟩ | lemma | asymptotics.is_equivalent.tendsto_at_bot_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"order_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.is_equivalent {u v : α → β} {l : filter α} (h : u =ᶠ[l] v) : u ~[l] v | is_equivalent.congr_right (is_o_refl_left _ _) h | lemma | filter.eventually_eq.is_equivalent | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.is_bounded_under.is_o_sub_self_inv {𝕜 E : Type*} [normed_field 𝕜] [has_norm E]
{a : 𝕜} {f : 𝕜 → E} (h : is_bounded_under (≤) (𝓝[≠] a) (norm ∘ f)) :
f =o[𝓝[≠] a] (λ x, (x - a)⁻¹) | begin
refine (h.is_O_const (one_ne_zero' ℝ)).trans_is_o (is_o_const_left.2 $ or.inr _),
simp only [(∘), norm_inv],
exact (tendsto_norm_sub_self_punctured_nhds a).inv_tendsto_zero
end | lemma | filter.is_bounded_under.is_o_sub_self_inv | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"has_norm",
"norm_inv",
"normed_field",
"one_ne_zero'"
] | If `f : 𝕜 → E` is bounded in a punctured neighborhood of `a`, then `f(x) = o((x - a)⁻¹)` as
`x → a`, `x ≠ a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_div_pow_eventually_eq_at_top {p q : ℕ} :
(λ x : 𝕜, x^p / x^q) =ᶠ[at_top] (λ x, x^((p : ℤ) -q)) | begin
apply ((eventually_gt_at_top (0 : 𝕜)).mono (λ x hx, _)),
simp [zpow_sub₀ hx.ne'],
end | lemma | pow_div_pow_eventually_eq_at_top | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"zpow_sub₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_div_pow_eventually_eq_at_bot {p q : ℕ} :
(λ x : 𝕜, x^p / x^q) =ᶠ[at_bot] (λ x, x^((p : ℤ) -q)) | begin
apply ((eventually_lt_at_bot (0 : 𝕜)).mono (λ x hx, _)),
simp [zpow_sub₀ hx.ne],
end | lemma | pow_div_pow_eventually_eq_at_bot | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"zpow_sub₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_zpow_at_top_at_top {n : ℤ}
(hn : 0 < n) : tendsto (λ x : 𝕜, x^n) at_top at_top | begin
lift n to ℕ using hn.le,
simp only [zpow_coe_nat],
exact tendsto_pow_at_top (nat.cast_pos.mp hn).ne'
end | lemma | tendsto_zpow_at_top_at_top | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"lift",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_div_pow_at_top_at_top {p q : ℕ}
(hpq : q < p) : tendsto (λ x : 𝕜, x^p / x^q) at_top at_top | begin
rw tendsto_congr' pow_div_pow_eventually_eq_at_top,
apply tendsto_zpow_at_top_at_top,
linarith
end | lemma | tendsto_pow_div_pow_at_top_at_top | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"pow_div_pow_eventually_eq_at_top",
"tendsto_zpow_at_top_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_div_pow_at_top_zero [topological_space 𝕜] [order_topology 𝕜] {p q : ℕ}
(hpq : p < q) : tendsto (λ x : 𝕜, x^p / x^q) at_top (𝓝 0) | begin
rw tendsto_congr' pow_div_pow_eventually_eq_at_top,
apply tendsto_zpow_at_top_zero,
linarith
end | lemma | tendsto_pow_div_pow_at_top_zero | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"order_topology",
"pow_div_pow_eventually_eq_at_top",
"tendsto_zpow_at_top_zero",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
asymptotics.is_o_pow_pow_at_top_of_lt
[order_topology 𝕜] {p q : ℕ} (hpq : p < q) :
(λ x : 𝕜, x^p) =o[at_top] (λ x, x^q) | begin
refine (is_o_iff_tendsto' _).mpr (tendsto_pow_div_pow_at_top_zero hpq),
exact (eventually_gt_at_top 0).mono (λ x hx hxq, (pow_ne_zero q hx.ne' hxq).elim),
end | lemma | asymptotics.is_o_pow_pow_at_top_of_lt | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"order_topology",
"pow_ne_zero",
"tendsto_pow_div_pow_at_top_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
asymptotics.is_O.trans_tendsto_norm_at_top {α : Type*} {u v : α → 𝕜} {l : filter α}
(huv : u =O[l] v) (hu : tendsto (λ x, ‖u x‖) l at_top) : tendsto (λ x, ‖v x‖) l at_top | begin
rcases huv.exists_pos with ⟨c, hc, hcuv⟩,
rw is_O_with at hcuv,
convert tendsto.at_top_div_const hc (tendsto_at_top_mono' l hcuv hu),
ext x,
rw mul_div_cancel_left _ hc.ne.symm,
end | lemma | asymptotics.is_O.trans_tendsto_norm_at_top | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"filter",
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
asymptotics.is_o.sum_range {α : Type*} [normed_add_comm_group α]
{f : ℕ → α} {g : ℕ → ℝ} (h : f =o[at_top] g)
(hg : 0 ≤ g) (h'g : tendsto (λ n, ∑ i in range n, g i) at_top at_top) :
(λ n, ∑ i in range n, f i) =o[at_top] (λ n, ∑ i in range n, g i) | begin
have A : ∀ i, ‖g i‖ = g i := λ i, real.norm_of_nonneg (hg i),
have B : ∀ n, ‖∑ i in range n, g i‖ = ∑ i in range n, g i,
from λ n, by rwa [real.norm_eq_abs, abs_sum_of_nonneg'],
apply is_o_iff.2 (λ ε εpos, _),
obtain ⟨N, hN⟩ : ∃ (N : ℕ), ∀ (b : ℕ), N ≤ b → ‖f b‖ ≤ ε / 2 * g b,
by simpa only [A, ev... | lemma | asymptotics.is_o.sum_range | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"half_pos",
"le_rfl",
"mul_le_mul_of_nonneg_left",
"normed_add_comm_group",
"real.norm_eq_abs",
"real.norm_of_nonneg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
asymptotics.is_o_sum_range_of_tendsto_zero {α : Type*} [normed_add_comm_group α]
{f : ℕ → α} (h : tendsto f at_top (𝓝 0)) :
(λ n, ∑ i in range n, f i) =o[at_top] (λ n, (n : ℝ)) | begin
have := ((is_o_one_iff ℝ).2 h).sum_range (λ i, zero_le_one),
simp only [sum_const, card_range, nat.smul_one_eq_coe] at this,
exact this tendsto_coe_nat_at_top_at_top
end | lemma | asymptotics.is_o_sum_range_of_tendsto_zero | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"nat.smul_one_eq_coe",
"normed_add_comm_group",
"tendsto_coe_nat_at_top_at_top",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.cesaro_smul {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
{u : ℕ → E} {l : E} (h : tendsto u at_top (𝓝 l)) :
tendsto (λ (n : ℕ), (n ⁻¹ : ℝ) • (∑ i in range n, u i)) at_top (𝓝 l) | begin
rw [← tendsto_sub_nhds_zero_iff, ← is_o_one_iff ℝ],
have := asymptotics.is_o_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 h),
apply ((is_O_refl (λ (n : ℕ), (n : ℝ) ⁻¹) at_top).smul_is_o this).congr' _ _,
{ filter_upwards [Ici_mem_at_top 1] with n npos,
have nposℝ : (0 : ℝ) < n := nat.cast_po... | lemma | filter.tendsto.cesaro_smul | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [
"algebra.id.smul_eq_mul",
"asymptotics.is_o_sum_range_of_tendsto_zero",
"inv_mul_cancel",
"normed_add_comm_group",
"normed_space",
"nsmul_eq_smul_cast",
"one_smul",
"smul_smul",
"smul_sub"
] | The Cesaro average of a converging sequence converges to the same limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.cesaro {u : ℕ → ℝ} {l : ℝ} (h : tendsto u at_top (𝓝 l)) :
tendsto (λ (n : ℕ), (n ⁻¹ : ℝ) * (∑ i in range n, u i)) at_top (𝓝 l) | h.cesaro_smul | lemma | filter.tendsto.cesaro | analysis.asymptotics | src/analysis/asymptotics/specific_asymptotics.lean | [
"analysis.normed.order.basic",
"analysis.asymptotics.asymptotics"
] | [] | The Cesaro average of a converging sequence converges to the same limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
superpolynomial_decay {α β : Type*} [topological_space β] [comm_semiring β]
(l : filter α) (k : α → β) (f : α → β) | ∀ (n : ℕ), tendsto (λ (a : α), (k a) ^ n * f a) l (𝓝 0) | def | asymptotics.superpolynomial_decay | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"comm_semiring",
"filter",
"topological_space"
] | `f` has superpolynomial decay in parameter `k` along filter `l` if
`k ^ n * f` tends to zero at `l` for all naturals `n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
superpolynomial_decay.congr' (hf : superpolynomial_decay l k f)
(hfg : f =ᶠ[l] g) : superpolynomial_decay l k g | λ z, (hf z).congr' (eventually_eq.mul (eventually_eq.refl l _) hfg) | lemma | asymptotics.superpolynomial_decay.congr' | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.congr (hf : superpolynomial_decay l k f)
(hfg : ∀ x, f x = g x) : superpolynomial_decay l k g | λ z, (hf z).congr (λ x, congr_arg (λ a, k x ^ z * a) $ hfg x) | lemma | asymptotics.superpolynomial_decay.congr | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_zero (l : filter α) (k : α → β) :
superpolynomial_decay l k 0 | λ z, by simpa only [pi.zero_apply, mul_zero] using tendsto_const_nhds | lemma | asymptotics.superpolynomial_decay_zero | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"filter",
"mul_zero",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.add [has_continuous_add β] (hf : superpolynomial_decay l k f)
(hg : superpolynomial_decay l k g) : superpolynomial_decay l k (f + g) | λ z, by simpa only [mul_add, add_zero, pi.add_apply] using (hf z).add (hg z) | lemma | asymptotics.superpolynomial_decay.add | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.mul [has_continuous_mul β] (hf : superpolynomial_decay l k f)
(hg : superpolynomial_decay l k g) : superpolynomial_decay l k (f * g) | λ z, by simpa only [mul_assoc, one_mul, mul_zero, pow_zero] using (hf z).mul (hg 0) | lemma | asymptotics.superpolynomial_decay.mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_mul",
"mul_assoc",
"mul_zero",
"one_mul",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.mul_const [has_continuous_mul β] (hf : superpolynomial_decay l k f)
(c : β) : superpolynomial_decay l k (λ n, f n * c) | λ z, by simpa only [←mul_assoc, zero_mul] using tendsto.mul_const c (hf z) | lemma | asymptotics.superpolynomial_decay.mul_const | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.const_mul [has_continuous_mul β] (hf : superpolynomial_decay l k f)
(c : β) : superpolynomial_decay l k (λ n, c * f n) | (hf.mul_const c).congr (λ _, mul_comm _ _) | lemma | asymptotics.superpolynomial_decay.const_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.param_mul (hf : superpolynomial_decay l k f) :
superpolynomial_decay l k (k * f) | λ z, tendsto_nhds.2 (λ s hs hs0, l.sets_of_superset ((tendsto_nhds.1 (hf $ z + 1)) s hs hs0)
(λ x hx, by simpa only [set.mem_preimage, pi.mul_apply, ← mul_assoc, ← pow_succ'] using hx)) | lemma | asymptotics.superpolynomial_decay.param_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_assoc",
"pi.mul_apply",
"pow_succ'",
"set.mem_preimage"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.mul_param (hf : superpolynomial_decay l k f) :
superpolynomial_decay l k (f * k) | (hf.param_mul).congr (λ _, mul_comm _ _) | lemma | asymptotics.superpolynomial_decay.mul_param | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.param_pow_mul (hf : superpolynomial_decay l k f)
(n : ℕ) : superpolynomial_decay l k (k ^ n * f) | begin
induction n with n hn,
{ simpa only [one_mul, pow_zero] using hf },
{ simpa only [pow_succ, mul_assoc] using hn.param_mul }
end | lemma | asymptotics.superpolynomial_decay.param_pow_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_assoc",
"one_mul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.mul_param_pow (hf : superpolynomial_decay l k f)
(n : ℕ) : superpolynomial_decay l k (f * k ^ n) | (hf.param_pow_mul n).congr (λ _, mul_comm _ _) | lemma | asymptotics.superpolynomial_decay.mul_param_pow | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.polynomial_mul [has_continuous_add β] [has_continuous_mul β]
(hf : superpolynomial_decay l k f) (p : β[X]) :
superpolynomial_decay l k (λ x, (p.eval $ k x) * f x) | polynomial.induction_on' p (λ p q hp hq, by simpa [add_mul] using hp.add hq)
(λ n c, by simpa [mul_assoc] using (hf.param_pow_mul n).const_mul c) | lemma | asymptotics.superpolynomial_decay.polynomial_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_add",
"has_continuous_mul",
"mul_assoc",
"polynomial.induction_on'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.mul_polynomial [has_continuous_add β] [has_continuous_mul β]
(hf : superpolynomial_decay l k f) (p : β[X]) :
superpolynomial_decay l k (λ x, f x * (p.eval $ k x)) | (hf.polynomial_mul p).congr (λ _, mul_comm _ _) | lemma | asymptotics.superpolynomial_decay.mul_polynomial | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_add",
"has_continuous_mul",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.trans_eventually_le (hk : 0 ≤ᶠ[l] k)
(hg : superpolynomial_decay l k g) (hg' : superpolynomial_decay l k g')
(hfg : g ≤ᶠ[l] f) (hfg' : f ≤ᶠ[l] g') : superpolynomial_decay l k f | λ z, tendsto_of_tendsto_of_tendsto_of_le_of_le' (hg z) (hg' z)
(hfg.mp (hk.mono $ λ x hx hx', mul_le_mul_of_nonneg_left hx' (pow_nonneg hx z)))
(hfg'.mp (hk.mono $ λ x hx hx', mul_le_mul_of_nonneg_left hx' (pow_nonneg hx z))) | lemma | asymptotics.superpolynomial_decay.trans_eventually_le | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_le_mul_of_nonneg_left",
"pow_nonneg",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_abs_tendsto_zero :
superpolynomial_decay l k f ↔ ∀ (n : ℕ), tendsto (λ (a : α), |(k a) ^ n * f a|) l (𝓝 0) | ⟨λ h z, (tendsto_zero_iff_abs_tendsto_zero _).1 (h z),
λ h z, (tendsto_zero_iff_abs_tendsto_zero _).2 (h z)⟩ | lemma | asymptotics.superpolynomial_decay_iff_abs_tendsto_zero | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"tendsto_zero_iff_abs_tendsto_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_superpolynomial_decay_abs :
superpolynomial_decay l k f ↔ superpolynomial_decay l (λ a, |k a|) (λ a, |f a|) | (superpolynomial_decay_iff_abs_tendsto_zero l k f).trans
(by simp_rw [superpolynomial_decay, abs_mul, abs_pow]) | lemma | asymptotics.superpolynomial_decay_iff_superpolynomial_decay_abs | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"abs_mul",
"abs_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.trans_eventually_abs_le (hf : superpolynomial_decay l k f)
(hfg : abs ∘ g ≤ᶠ[l] abs ∘ f) : superpolynomial_decay l k g | begin
rw superpolynomial_decay_iff_abs_tendsto_zero at hf ⊢,
refine λ z, tendsto_of_tendsto_of_tendsto_of_le_of_le' (tendsto_const_nhds) (hf z)
(eventually_of_forall $ λ x, abs_nonneg _) (hfg.mono $ λ x hx, _),
calc |k x ^ z * g x| = |k x ^ z| * |g x| : abs_mul (k x ^ z) (g x)
... ≤ |k x ^ z| * |f x| : mu... | lemma | asymptotics.superpolynomial_decay.trans_eventually_abs_le | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"abs_mul",
"abs_nonneg",
"le_rfl",
"mul_le_mul",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.trans_abs_le (hf : superpolynomial_decay l k f)
(hfg : ∀ x, |g x| ≤ |f x|) : superpolynomial_decay l k g | hf.trans_eventually_abs_le (eventually_of_forall hfg) | lemma | asymptotics.superpolynomial_decay.trans_abs_le | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_mul_const_iff [has_continuous_mul β] {c : β} (hc0 : c ≠ 0) :
superpolynomial_decay l k (λ n, f n * c) ↔ superpolynomial_decay l k f | ⟨λ h, (h.mul_const c⁻¹).congr (λ x, by simp [mul_assoc, mul_inv_cancel hc0]), λ h, h.mul_const c⟩ | lemma | asymptotics.superpolynomial_decay_mul_const_iff | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_mul",
"mul_assoc",
"mul_inv_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_const_mul_iff [has_continuous_mul β] {c : β} (hc0 : c ≠ 0) :
superpolynomial_decay l k (λ n, c * f n) ↔ superpolynomial_decay l k f | ⟨λ h, (h.const_mul c⁻¹).congr (λ x, by simp [← mul_assoc, inv_mul_cancel hc0]), λ h, h.const_mul c⟩ | lemma | asymptotics.superpolynomial_decay_const_mul_iff | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"has_continuous_mul",
"inv_mul_cancel",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_abs_is_bounded_under (hk : tendsto k l at_top) :
superpolynomial_decay l k f ↔ ∀ (z : ℕ), is_bounded_under (≤) l (λ (a : α), |(k a) ^ z * f a|) | begin
refine ⟨λ h z, tendsto.is_bounded_under_le (tendsto.abs (h z)),
λ h, (superpolynomial_decay_iff_abs_tendsto_zero l k f).2 (λ z, _)⟩,
obtain ⟨m, hm⟩ := h (z + 1),
have h1 : tendsto (λ (a : α), (0 : β)) l (𝓝 0) := tendsto_const_nhds,
have h2 : tendsto (λ (a : α), |(k a)⁻¹| * m) l (𝓝 0) := (zero_mul m)... | lemma | asymptotics.superpolynomial_decay_iff_abs_is_bounded_under | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"abs_mul",
"abs_nonneg",
"inv_mul_cancel",
"mul_assoc",
"mul_le_mul_of_nonneg_left",
"one_mul",
"pow_succ",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le'",
"tendsto_zero_iff_abs_tendsto_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_zpow_tendsto_zero (hk : tendsto k l at_top) :
superpolynomial_decay l k f ↔ ∀ (z : ℤ), tendsto (λ (a : α), (k a) ^ z * f a) l (𝓝 0) | begin
refine ⟨λ h z, _, λ h n, by simpa only [zpow_coe_nat] using h (n : ℤ)⟩,
by_cases hz : 0 ≤ z,
{ lift z to ℕ using hz,
simpa using h z },
{ have : tendsto (λ a, (k a) ^ z) l (𝓝 0) :=
tendsto.comp (tendsto_zpow_at_top_zero (not_le.1 hz)) hk,
have h : tendsto f l (𝓝 0) := by simpa using h 0,
... | lemma | asymptotics.superpolynomial_decay_iff_zpow_tendsto_zero | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"lift",
"tendsto_zpow_at_top_zero",
"zero_mul",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.param_zpow_mul (hk : tendsto k l at_top)
(hf : superpolynomial_decay l k f) (z : ℤ) : superpolynomial_decay l k (λ a, k a ^ z * f a) | begin
rw superpolynomial_decay_iff_zpow_tendsto_zero _ hk at hf ⊢,
refine λ z', (hf $ z' + z).congr' ((hk.eventually_ne_at_top 0).mono (λ x hx, _)),
simp [zpow_add₀ hx, mul_assoc, pi.mul_apply],
end | lemma | asymptotics.superpolynomial_decay.param_zpow_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_assoc",
"pi.mul_apply",
"zpow_add₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.mul_param_zpow (hk : tendsto k l at_top)
(hf : superpolynomial_decay l k f) (z : ℤ) : superpolynomial_decay l k (λ a, f a * k a ^ z) | (hf.param_zpow_mul hk z).congr (λ _, mul_comm _ _) | lemma | asymptotics.superpolynomial_decay.mul_param_zpow | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.inv_param_mul (hk : tendsto k l at_top)
(hf : superpolynomial_decay l k f) : superpolynomial_decay l k (k⁻¹ * f) | by simpa using (hf.param_zpow_mul hk (-1)) | lemma | asymptotics.superpolynomial_decay.inv_param_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay.param_inv_mul (hk : tendsto k l at_top)
(hf : superpolynomial_decay l k f) : superpolynomial_decay l k (f * k⁻¹) | (hf.inv_param_mul hk).congr (λ _, mul_comm _ _) | lemma | asymptotics.superpolynomial_decay.param_inv_mul | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_param_mul_iff (hk : tendsto k l at_top) :
superpolynomial_decay l k (k * f) ↔ superpolynomial_decay l k f | ⟨λ h, (h.inv_param_mul hk).congr' ((hk.eventually_ne_at_top 0).mono
(λ x hx, by simp [← mul_assoc, inv_mul_cancel hx])), λ h, h.param_mul⟩ | lemma | asymptotics.superpolynomial_decay_param_mul_iff | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"inv_mul_cancel",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_mul_param_iff (hk : tendsto k l at_top) :
superpolynomial_decay l k (f * k) ↔ superpolynomial_decay l k f | by simpa [mul_comm k] using superpolynomial_decay_param_mul_iff f hk | lemma | asymptotics.superpolynomial_decay_mul_param_iff | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_param_pow_mul_iff (hk : tendsto k l at_top) (n : ℕ) :
superpolynomial_decay l k (k ^ n * f) ↔ superpolynomial_decay l k f | begin
induction n with n hn,
{ simp },
{ simpa [pow_succ, ← mul_comm k, mul_assoc,
superpolynomial_decay_param_mul_iff (k ^ n * f) hk] using hn }
end | lemma | asymptotics.superpolynomial_decay_param_pow_mul_iff | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_assoc",
"mul_comm",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_mul_param_pow_iff (hk : tendsto k l at_top) (n : ℕ) :
superpolynomial_decay l k (f * k ^ n) ↔ superpolynomial_decay l k f | by simpa [mul_comm f] using superpolynomial_decay_param_pow_mul_iff f hk n | lemma | asymptotics.superpolynomial_decay_mul_param_pow_iff | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_norm_tendsto_zero :
superpolynomial_decay l k f ↔ ∀ (n : ℕ), tendsto (λ (a : α), ‖(k a) ^ n * f a‖) l (𝓝 0) | ⟨λ h z, tendsto_zero_iff_norm_tendsto_zero.1 (h z),
λ h z, tendsto_zero_iff_norm_tendsto_zero.2 (h z)⟩ | lemma | asymptotics.superpolynomial_decay_iff_norm_tendsto_zero | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_superpolynomial_decay_norm :
superpolynomial_decay l k f ↔ superpolynomial_decay l (λ a, ‖k a‖) (λ a, ‖f a‖) | (superpolynomial_decay_iff_norm_tendsto_zero l k f).trans (by simp [superpolynomial_decay]) | lemma | asymptotics.superpolynomial_decay_iff_superpolynomial_decay_norm | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_is_O (hk : tendsto k l at_top) :
superpolynomial_decay l k f ↔ ∀ (z : ℤ), f =O[l] (λ (a : α), (k a) ^ z) | begin
refine (superpolynomial_decay_iff_zpow_tendsto_zero f hk).trans _,
have hk0 : ∀ᶠ x in l, k x ≠ 0 := hk.eventually_ne_at_top 0,
refine ⟨λ h z, _, λ h z, _⟩,
{ refine is_O_of_div_tendsto_nhds (hk0.mono (λ x hx hxz, absurd (zpow_eq_zero hxz) hx)) 0 _,
have : (λ (a : α), k a ^ z)⁻¹ = (λ (a : α), k a ^ (- ... | lemma | asymptotics.superpolynomial_decay_iff_is_O | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"div_eq_mul_inv",
"mul_assoc",
"mul_comm",
"mul_inv_cancel",
"one_mul",
"zpow_add₀",
"zpow_eq_zero",
"zpow_ne_zero",
"zpow_neg",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
superpolynomial_decay_iff_is_o (hk : tendsto k l at_top) :
superpolynomial_decay l k f ↔ ∀ (z : ℤ), f =o[l] (λ (a : α), (k a) ^ z) | begin
refine ⟨λ h z, _, λ h, (superpolynomial_decay_iff_is_O f hk).2 (λ z, (h z).is_O)⟩,
have hk0 : ∀ᶠ x in l, k x ≠ 0 := hk.eventually_ne_at_top 0,
have : (λ (x : α), (1 : β)) =o[l] k := is_o_of_tendsto'
(hk0.mono (λ x hkx hkx', absurd hkx' hkx)) (by simpa using hk.inv_tendsto_at_top),
have : f =o[l] (λ (x... | lemma | asymptotics.superpolynomial_decay_iff_is_o | analysis.asymptotics | src/analysis/asymptotics/superpolynomial_decay.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic",
"data.polynomial.eval",
"topology.algebra.order.liminf_limsup"
] | [
"inv_mul_cancel",
"mul_assoc",
"mul_comm",
"mul_one",
"one_mul",
"zpow_sub_one₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta (l : filter α) (f : α → E) (g : α → F) : Prop | is_O l f g ∧ is_O l g f | def | asymptotics.is_Theta | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"filter"
] | We say that `f` is `Θ(g)` along a filter `l` (notation: `f =Θ[l] g`) if `f =O[l] g` and
`g =O[l] f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O.antisymm (h₁ : f =O[l] g) (h₂ : g =O[l] f) : f =Θ[l] g | ⟨h₁, h₂⟩ | lemma | asymptotics.is_O.antisymm | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_refl (f : α → E) (l : filter α) : f =Θ[l] f | ⟨is_O_refl _ _, is_O_refl _ _⟩ | lemma | asymptotics.is_Theta_refl | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_rfl : f =Θ[l] f | is_Theta_refl _ _ | lemma | asymptotics.is_Theta_rfl | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.symm (h : f =Θ[l] g) : g =Θ[l] f | h.symm | lemma | asymptotics.is_Theta.symm | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_comm : f =Θ[l] g ↔ g =Θ[l] f | ⟨λ h, h.symm, λ h, h.symm⟩ | lemma | asymptotics.is_Theta_comm | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.trans {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
(h₂ : g =Θ[l] k) : f =Θ[l] k | ⟨h₁.1.trans h₂.1, h₂.2.trans h₁.2⟩ | lemma | asymptotics.is_Theta.trans | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.trans_is_Theta {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =O[l] g)
(h₂ : g =Θ[l] k) : f =O[l] k | h₁.trans h₂.1 | lemma | asymptotics.is_O.trans_is_Theta | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.trans_is_O {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
(h₂ : g =O[l] k) : f =O[l] k | h₁.1.trans h₂ | lemma | asymptotics.is_Theta.trans_is_O | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.trans_is_Theta {f : α → E} {g : α → F} {k : α → G'} (h₁ : f =o[l] g)
(h₂ : g =Θ[l] k) : f =o[l] k | h₁.trans_is_O h₂.1 | lemma | asymptotics.is_o.trans_is_Theta | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.trans_is_o {f : α → E} {g : α → F'} {k : α → G} (h₁ : f =Θ[l] g)
(h₂ : g =o[l] k) : f =o[l] k | h₁.1.trans_is_o h₂ | lemma | asymptotics.is_Theta.trans_is_o | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.trans_eventually_eq {f : α → E} {g₁ g₂ : α → F} (h : f =Θ[l] g₁)
(hg : g₁ =ᶠ[l] g₂) : f =Θ[l] g₂ | ⟨h.1.trans_eventually_eq hg, hg.symm.trans_is_O h.2⟩ | lemma | asymptotics.is_Theta.trans_eventually_eq | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.eventually_eq.trans_is_Theta {f₁ f₂ : α → E} {g : α → F}
(hf : f₁ =ᶠ[l] f₂) (h : f₂ =Θ[l] g) : f₁ =Θ[l] g | ⟨hf.trans_is_O h.1, h.2.trans_eventually_eq hf.symm⟩ | lemma | filter.eventually_eq.trans_is_Theta | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_norm_left : (λ x, ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g | by simp [is_Theta] | lemma | asymptotics.is_Theta_norm_left | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_norm_right : f =Θ[l] (λ x, ‖g' x‖) ↔ f =Θ[l] g' | by simp [is_Theta] | lemma | asymptotics.is_Theta_norm_right | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_of_norm_eventually_eq (h : (λ x, ‖f x‖) =ᶠ[l] (λ x, ‖g x‖)) : f =Θ[l] g | ⟨is_O.of_bound 1 $ by simpa only [one_mul] using h.le,
is_O.of_bound 1 $ by simpa only [one_mul] using h.symm.le⟩ | lemma | asymptotics.is_Theta_of_norm_eventually_eq | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_of_norm_eventually_eq' {g : α → ℝ} (h : (λ x, ‖f' x‖) =ᶠ[l] g) : f' =Θ[l] g | is_Theta_of_norm_eventually_eq $ h.mono $ λ x hx, by simp only [← hx, norm_norm] | lemma | asymptotics.is_Theta_of_norm_eventually_eq' | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.is_o_congr_left (h : f' =Θ[l] g') : f' =o[l] k ↔ g' =o[l] k | ⟨h.symm.trans_is_o, h.trans_is_o⟩ | lemma | asymptotics.is_Theta.is_o_congr_left | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.is_o_congr_right (h : g' =Θ[l] k') : f =o[l] g' ↔ f =o[l] k' | ⟨λ H, H.trans_is_Theta h, λ H, H.trans_is_Theta h.symm⟩ | lemma | asymptotics.is_Theta.is_o_congr_right | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.is_O_congr_left (h : f' =Θ[l] g') : f' =O[l] k ↔ g' =O[l] k | ⟨h.symm.trans_is_O, h.trans_is_O⟩ | lemma | asymptotics.is_Theta.is_O_congr_left | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.is_O_congr_right (h : g' =Θ[l] k') : f =O[l] g' ↔ f =O[l] k' | ⟨λ H, H.trans_is_Theta h, λ H, H.trans_is_Theta h.symm⟩ | lemma | asymptotics.is_Theta.is_O_congr_right | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.mono (h : f =Θ[l] g) (hl : l' ≤ l) : f =Θ[l'] g | ⟨h.1.mono hl, h.2.mono hl⟩ | lemma | asymptotics.is_Theta.mono | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.sup (h : f' =Θ[l] g') (h' : f' =Θ[l'] g') : f' =Θ[l ⊔ l'] g' | ⟨h.1.sup h'.1, h.2.sup h'.2⟩ | lemma | asymptotics.is_Theta.sup | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta_sup : f' =Θ[l ⊔ l'] g' ↔ f' =Θ[l] g' ∧ f' =Θ[l'] g' | ⟨λ h, ⟨h.mono le_sup_left, h.mono le_sup_right⟩, λ h, h.1.sup h.2⟩ | lemma | asymptotics.is_Theta_sup | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.eq_zero_iff (h : f'' =Θ[l] g'') : ∀ᶠ x in l, f'' x = 0 ↔ g'' x = 0 | h.1.eq_zero_imp.mp $ h.2.eq_zero_imp.mono $ λ x, iff.intro | lemma | asymptotics.is_Theta.eq_zero_iff | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.tendsto_zero_iff (h : f'' =Θ[l] g'') : tendsto f'' l (𝓝 0) ↔ tendsto g'' l (𝓝 0) | by simp only [← is_o_one_iff ℝ, h.is_o_congr_left] | lemma | asymptotics.is_Theta.tendsto_zero_iff | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_Theta.tendsto_norm_at_top_iff (h : f' =Θ[l] g') :
tendsto (norm ∘ f') l at_top ↔ tendsto (norm ∘ g') l at_top | by simp only [← is_o_const_left_of_ne (one_ne_zero' ℝ), h.is_o_congr_right] | lemma | asymptotics.is_Theta.tendsto_norm_at_top_iff | analysis.asymptotics | src/analysis/asymptotics/theta.lean | [
"analysis.asymptotics.asymptotics"
] | [
"one_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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