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_O_iff_eventually_is_O_with : f =O[l] g' ↔ ∀ᶠ c in at_top, is_O_with c l f g' | is_O_iff_is_O_with.trans
⟨λ ⟨c, hc⟩, mem_at_top_sets.2 ⟨c, λ c' hc', hc.weaken hc'⟩, λ h, h.exists⟩ | lemma | asymptotics.is_O_iff_eventually_is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | `f = O(g)` if and only if `is_O_with c f g` for all sufficiently large `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_iff_eventually : f =O[l] g' ↔ ∀ᶠ c in at_top, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g' x‖ | is_O_iff_eventually_is_O_with.trans $ by simp only [is_O_with] | lemma | asymptotics.is_O_iff_eventually | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | `f = O(g)` if and only if `∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖` for all sufficiently large `c`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O.exists_mem_basis {ι} {p : ι → Prop} {s : ι → set α} (h : f =O[l] g')
(hb : l.has_basis p s) :
∃ (c : ℝ) (hc : 0 < c) (i : ι) (hi : p i), ∀ x ∈ s i, ‖f x‖ ≤ c * ‖g' x‖ | flip Exists₂.imp h.exists_pos $ λ c hc h,
by simpa only [is_O_with_iff, hb.eventually_iff, exists_prop] using h | lemma | asymptotics.is_O.exists_mem_basis | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"Exists₂.imp",
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_inv (hc : 0 < c) : is_O_with c⁻¹ l f g ↔ ∀ᶠ x in l, c * ‖f x‖ ≤ ‖g x‖ | by simp only [is_O_with, ← div_eq_inv_mul, le_div_iff' hc] | lemma | asymptotics.is_O_with_inv | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"div_eq_inv_mul",
"le_div_iff'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_iff_nat_mul_le_aux (h₀ : (∀ x, 0 ≤ ‖f x‖) ∨ ∀ x, 0 ≤ ‖g x‖) :
f =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g x‖ | begin
split,
{ rintro H (_|n),
{ refine (H.def one_pos).mono (λ x h₀', _),
rw [nat.cast_zero, zero_mul],
refine h₀.elim (λ hf, (hf x).trans _) (λ hg, hg x),
rwa one_mul at h₀' },
{ have : (0 : ℝ) < n.succ, from nat.cast_pos.2 n.succ_pos,
exact (is_O_with_inv this).1 (H.def' $ inv_pos... | lemma | asymptotics.is_o_iff_nat_mul_le_aux | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"exists_nat_gt",
"inv_le_of_inv_le",
"mul_le_mul_of_nonneg_right",
"nat.cast_zero",
"nonneg_of_mul_nonneg_right",
"one_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_iff_nat_mul_le : f =o[l] g' ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f x‖ ≤ ‖g' x‖ | is_o_iff_nat_mul_le_aux (or.inr $ λ x, norm_nonneg _) | lemma | asymptotics.is_o_iff_nat_mul_le | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_iff_nat_mul_le' : f' =o[l] g ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖f' x‖ ≤ ‖g x‖ | is_o_iff_nat_mul_le_aux (or.inl $ λ x, norm_nonneg _) | lemma | asymptotics.is_o_iff_nat_mul_le' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_of_subsingleton [subsingleton E'] : f' =o[l] g' | is_o.of_bound $ λ c hc, by simp [subsingleton.elim (f' _) 0, mul_nonneg hc.le] | lemma | asymptotics.is_o_of_subsingleton | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_of_subsingleton [subsingleton E'] : f' =O[l] g' | is_o_of_subsingleton.is_O | lemma | asymptotics.is_O_of_subsingleton | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_congr (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) :
is_O_with c₁ l f₁ g₁ ↔ is_O_with c₂ l f₂ g₂ | begin
unfold is_O_with,
subst c₂,
apply filter.eventually_congr,
filter_upwards [hf, hg] with _ e₁ e₂,
rw [e₁, e₂],
end | theorem | asymptotics.is_O_with_congr | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter.eventually_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.congr' (h : is_O_with c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : f₁ =ᶠ[l] f₂)
(hg : g₁ =ᶠ[l] g₂) : is_O_with c₂ l f₂ g₂ | (is_O_with_congr hc hf hg).mp h | theorem | asymptotics.is_O_with.congr' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.congr (h : is_O_with c₁ l f₁ g₁) (hc : c₁ = c₂) (hf : ∀ x, f₁ x = f₂ x)
(hg : ∀ x, g₁ x = g₂ x) : is_O_with c₂ l f₂ g₂ | h.congr' hc (univ_mem' hf) (univ_mem' hg) | theorem | asymptotics.is_O_with.congr | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.congr_left (h : is_O_with c l f₁ g) (hf : ∀ x, f₁ x = f₂ x) :
is_O_with c l f₂ g | h.congr rfl hf (λ _, rfl) | theorem | asymptotics.is_O_with.congr_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.congr_right (h : is_O_with c l f g₁) (hg : ∀ x, g₁ x = g₂ x) :
is_O_with c l f g₂ | h.congr rfl (λ _, rfl) hg | theorem | asymptotics.is_O_with.congr_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.congr_const (h : is_O_with c₁ l f g) (hc : c₁ = c₂) : is_O_with c₂ l f g | h.congr hc (λ _, rfl) (λ _, rfl) | theorem | asymptotics.is_O_with.congr_const | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =O[l] g₁ ↔ f₂ =O[l] g₂ | by { unfold is_O, exact exists_congr (λ c, is_O_with_congr rfl hf hg) } | theorem | asymptotics.is_O_congr | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.congr' (h : f₁ =O[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₂ =O[l] g₂ | (is_O_congr hf hg).mp h | theorem | asymptotics.is_O.congr' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.congr (h : f₁ =O[l] g₁) (hf : ∀ x, f₁ x = f₂ x) (hg : ∀ x, g₁ x = g₂ x) :
f₂ =O[l] g₂ | h.congr' (univ_mem' hf) (univ_mem' hg) | theorem | asymptotics.is_O.congr | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.congr_left (h : f₁ =O[l] g) (hf : ∀ x, f₁ x = f₂ x) : f₂ =O[l] g | h.congr hf (λ _, rfl) | theorem | asymptotics.is_O.congr_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.congr_right (h : f =O[l] g₁) (hg : ∀ x, g₁ x = g₂ x) : f =O[l] g₂ | h.congr (λ _, rfl) hg | theorem | asymptotics.is_O.congr_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =o[l] g₁ ↔ f₂ =o[l] g₂ | by { unfold is_o, exact forall₂_congr (λ c hc, is_O_with_congr (eq.refl c) hf hg) } | theorem | asymptotics.is_o_congr | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.congr' (h : f₁ =o[l] g₁) (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₂ =o[l] g₂ | (is_o_congr hf hg).mp h | theorem | asymptotics.is_o.congr' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.congr (h : f₁ =o[l] g₁) (hf : ∀ x, f₁ x = f₂ x) (hg : ∀ x, g₁ x = g₂ x) :
f₂ =o[l] g₂ | h.congr' (univ_mem' hf) (univ_mem' hg) | theorem | asymptotics.is_o.congr | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.congr_left (h : f₁ =o[l] g) (hf : ∀ x, f₁ x = f₂ x) : f₂ =o[l] g | h.congr hf (λ _, rfl) | theorem | asymptotics.is_o.congr_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.congr_right (h : f =o[l] g₁) (hg : ∀ x, g₁ x = g₂ x) : f =o[l] g₂ | h.congr (λ _, rfl) hg | theorem | asymptotics.is_o.congr_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.eventually_eq.trans_is_O {f₁ f₂ : α → E} {g : α → F}
(hf : f₁ =ᶠ[l] f₂) (h : f₂ =O[l] g) : f₁ =O[l] g | h.congr' hf.symm eventually_eq.rfl | theorem | filter.eventually_eq.trans_is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.eventually_eq.trans_is_o {f₁ f₂ : α → E} {g : α → F}
(hf : f₁ =ᶠ[l] f₂) (h : f₂ =o[l] g) : f₁ =o[l] g | h.congr' hf.symm eventually_eq.rfl | theorem | filter.eventually_eq.trans_is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.trans_eventually_eq {f : α → E} {g₁ g₂ : α → F}
(h : f =O[l] g₁) (hg : g₁ =ᶠ[l] g₂) : f =O[l] g₂ | h.congr' eventually_eq.rfl hg | theorem | asymptotics.is_O.trans_eventually_eq | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.trans_eventually_eq {f : α → E} {g₁ g₂ : α → F}
(h : f =o[l] g₁) (hg : g₁ =ᶠ[l] g₂) : f =o[l] g₂ | h.congr' eventually_eq.rfl hg | theorem | asymptotics.is_o.trans_eventually_eq | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.comp_tendsto (hcfg : is_O_with c l f g)
{k : β → α} {l' : filter β} (hk : tendsto k l' l):
is_O_with c l' (f ∘ k) (g ∘ k) | is_O_with.of_bound $ hk hcfg.bound | theorem | asymptotics.is_O_with.comp_tendsto | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.comp_tendsto (hfg : f =O[l] g) {k : β → α} {l' : filter β} (hk : tendsto k l' l) :
(f ∘ k) =O[l'] (g ∘ k) | is_O_iff_is_O_with.2 $ hfg.is_O_with.imp (λ c h, h.comp_tendsto hk) | theorem | asymptotics.is_O.comp_tendsto | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.comp_tendsto (hfg : f =o[l] g) {k : β → α} {l' : filter β} (hk : tendsto k l' l) :
(f ∘ k) =o[l'] (g ∘ k) | is_o.of_is_O_with $ λ c cpos, (hfg.forall_is_O_with cpos).comp_tendsto hk | theorem | asymptotics.is_o.comp_tendsto | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_map {k : β → α} {l : filter β} :
is_O_with c (map k l) f g ↔ is_O_with c l (f ∘ k) (g ∘ k) | by { unfold is_O_with, exact eventually_map } | theorem | asymptotics.is_O_with_map | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_map {k : β → α} {l : filter β} : f =O[map k l] g ↔ (f ∘ k) =O[l] (g ∘ k) | by simp only [is_O, is_O_with_map] | theorem | asymptotics.is_O_map | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_map {k : β → α} {l : filter β} : f =o[map k l] g ↔ (f ∘ k) =o[l] (g ∘ k) | by simp only [is_o, is_O_with_map] | theorem | asymptotics.is_o_map | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.mono (h : is_O_with c l' f g) (hl : l ≤ l') : is_O_with c l f g | is_O_with.of_bound $ hl h.bound | theorem | asymptotics.is_O_with.mono | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.mono (h : f =O[l'] g) (hl : l ≤ l') : f =O[l] g | is_O_iff_is_O_with.2 $ h.is_O_with.imp (λ c h, h.mono hl) | theorem | asymptotics.is_O.mono | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.mono (h : f =o[l'] g) (hl : l ≤ l') : f =o[l] g | is_o.of_is_O_with $ λ c cpos, (h.forall_is_O_with cpos).mono hl | theorem | asymptotics.is_o.mono | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.trans (hfg : is_O_with c l f g) (hgk : is_O_with c' l g k) (hc : 0 ≤ c) :
is_O_with (c * c') l f k | begin
unfold is_O_with at *,
filter_upwards [hfg, hgk] with x hx hx',
calc ‖f x‖ ≤ c * ‖g x‖ : hx
... ≤ c * (c' * ‖k x‖) : mul_le_mul_of_nonneg_left hx' hc
... = c * c' * ‖k x‖ : (mul_assoc _ _ _).symm
end | theorem | asymptotics.is_O_with.trans | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"mul_assoc",
"mul_le_mul_of_nonneg_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.trans {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g)
(hgk : g =O[l] k) : f =O[l] k | let ⟨c, cnonneg, hc⟩ := hfg.exists_nonneg, ⟨c', hc'⟩ := hgk.is_O_with in
(hc.trans hc' cnonneg).is_O | theorem | asymptotics.is_O.trans | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.trans_is_O_with (hfg : f =o[l] g) (hgk : is_O_with c l g k) (hc : 0 < c) :
f =o[l] k | begin
unfold is_o at *,
intros c' c'pos,
have : 0 < c' / c, from div_pos c'pos hc,
exact ((hfg this).trans hgk this.le).congr_const (div_mul_cancel _ hc.ne')
end | theorem | asymptotics.is_o.trans_is_O_with | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"div_mul_cancel",
"div_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.trans_is_O {f : α → E} {g : α → F} {k : α → G'} (hfg : f =o[l] g)
(hgk : g =O[l] k) :
f =o[l] k | let ⟨c, cpos, hc⟩ := hgk.exists_pos in hfg.trans_is_O_with hc cpos | theorem | asymptotics.is_o.trans_is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.trans_is_o (hfg : is_O_with c l f g) (hgk : g =o[l] k) (hc : 0 < c) :
f =o[l] k | begin
unfold is_o at *,
intros c' c'pos,
have : 0 < c' / c, from div_pos c'pos hc,
exact (hfg.trans (hgk this) hc.le).congr_const (mul_div_cancel' _ hc.ne')
end | theorem | asymptotics.is_O_with.trans_is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"div_pos",
"mul_div_cancel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.trans_is_o {f : α → E} {g : α → F'} {k : α → G} (hfg : f =O[l] g)
(hgk : g =o[l] k) :
f =o[l] k | let ⟨c, cpos, hc⟩ := hfg.exists_pos in hc.trans_is_o hgk cpos | theorem | asymptotics.is_O.trans_is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.trans {f : α → E} {g : α → F} {k : α → G} (hfg : f =o[l] g)
(hgk : g =o[l] k) : f =o[l] k | hfg.trans_is_O_with hgk.is_O_with one_pos | theorem | asymptotics.is_o.trans | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.eventually.trans_is_O {f : α → E} {g : α → F'} {k : α → G}
(hfg : ∀ᶠ x in l, ‖f x‖ ≤ ‖g x‖) (hgk : g =O[l] k) : f =O[l] k | (is_O.of_bound' hfg).trans hgk | lemma | filter.eventually.trans_is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.eventually.is_O {f : α → E} {g : α → ℝ} {l : filter α}
(hfg : ∀ᶠ x in l, ‖f x‖ ≤ g x) : f =O[l] g | is_O.of_bound' $ hfg.mono $ λ x hx, hx.trans $ real.le_norm_self _ | lemma | filter.eventually.is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter",
"real.le_norm_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_of_le' (hfg : ∀ x, ‖f x‖ ≤ c * ‖g x‖) : is_O_with c l f g | is_O_with.of_bound $ univ_mem' hfg | theorem | asymptotics.is_O_with_of_le' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_of_le (hfg : ∀ x, ‖f x‖ ≤ ‖g x‖) : is_O_with 1 l f g | is_O_with_of_le' l $ λ x, by { rw one_mul, exact hfg x } | theorem | asymptotics.is_O_with_of_le | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_of_le' (hfg : ∀ x, ‖f x‖ ≤ c * ‖g x‖) : f =O[l] g | (is_O_with_of_le' l hfg).is_O | theorem | asymptotics.is_O_of_le' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_of_le (hfg : ∀ x, ‖f x‖ ≤ ‖g x‖) : f =O[l] g | (is_O_with_of_le l hfg).is_O | theorem | asymptotics.is_O_of_le | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_refl (f : α → E) (l : filter α) : is_O_with 1 l f f | is_O_with_of_le l $ λ _, le_rfl | theorem | asymptotics.is_O_with_refl | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_refl (f : α → E) (l : filter α) : f =O[l] f | (is_O_with_refl f l).is_O | theorem | asymptotics.is_O_refl | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.trans_le (hfg : is_O_with c l f g) (hgk : ∀ x, ‖g x‖ ≤ ‖k x‖) (hc : 0 ≤ c) :
is_O_with c l f k | (hfg.trans (is_O_with_of_le l hgk) hc).congr_const $ mul_one c | theorem | asymptotics.is_O_with.trans_le | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.trans_le (hfg : f =O[l] g') (hgk : ∀ x, ‖g' x‖ ≤ ‖k x‖) : f =O[l] k | hfg.trans (is_O_of_le l hgk) | theorem | asymptotics.is_O.trans_le | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.trans_le (hfg : f =o[l] g) (hgk : ∀ x, ‖g x‖ ≤ ‖k x‖) : f =o[l] k | hfg.trans_is_O_with (is_O_with_of_le _ hgk) zero_lt_one | theorem | asymptotics.is_o.trans_le | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_irrefl' (h : ∃ᶠ x in l, ‖f' x‖ ≠ 0) : ¬f' =o[l] f' | begin
intro ho,
rcases ((ho.bound one_half_pos).and_frequently h).exists with ⟨x, hle, hne⟩,
rw [one_div, ← div_eq_inv_mul] at hle,
exact (half_lt_self (lt_of_le_of_ne (norm_nonneg _) hne.symm)).not_le hle
end | theorem | asymptotics.is_o_irrefl' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"div_eq_inv_mul",
"one_div",
"one_half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_irrefl (h : ∃ᶠ x in l, f'' x ≠ 0) : ¬f'' =o[l] f'' | is_o_irrefl' $ h.mono $ λ x, norm_ne_zero_iff.mpr | theorem | asymptotics.is_o_irrefl | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.not_is_o (h : f'' =O[l] g') (hf : ∃ᶠ x in l, f'' x ≠ 0) : ¬g' =o[l] f'' | λ h', is_o_irrefl hf (h.trans_is_o h') | theorem | asymptotics.is_O.not_is_o | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.not_is_O (h : f'' =o[l] g') (hf : ∃ᶠ x in l, f'' x ≠ 0) : ¬g' =O[l] f'' | λ h', is_o_irrefl hf (h.trans_is_O h') | theorem | asymptotics.is_o.not_is_O | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_bot : is_O_with c ⊥ f g | is_O_with.of_bound $ trivial | theorem | asymptotics.is_O_with_bot | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_bot : f =O[⊥] g | (is_O_with_bot 1 f g).is_O | theorem | asymptotics.is_O_bot | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_bot : f =o[⊥] g | is_o.of_is_O_with $ λ c _, is_O_with_bot c f g | theorem | asymptotics.is_o_bot | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_pure {x} : is_O_with c (pure x) f g ↔ ‖f x‖ ≤ c * ‖g x‖ | is_O_with_iff | theorem | asymptotics.is_O_with_pure | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.sup (h : is_O_with c l f g) (h' : is_O_with c l' f g) :
is_O_with c (l ⊔ l') f g | is_O_with.of_bound $ mem_sup.2 ⟨h.bound, h'.bound⟩ | theorem | asymptotics.is_O_with.sup | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.sup' (h : is_O_with c l f g') (h' : is_O_with c' l' f g') :
is_O_with (max c c') (l ⊔ l') f g' | is_O_with.of_bound $
mem_sup.2 ⟨(h.weaken $ le_max_left c c').bound, (h'.weaken $ le_max_right c c').bound⟩ | theorem | asymptotics.is_O_with.sup' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"bound"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.sup (h : f =O[l] g') (h' : f =O[l'] g') : f =O[l ⊔ l'] g' | let ⟨c, hc⟩ := h.is_O_with, ⟨c', hc'⟩ := h'.is_O_with in (hc.sup' hc').is_O | theorem | asymptotics.is_O.sup | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.sup (h : f =o[l] g) (h' : f =o[l'] g) : f =o[l ⊔ l'] g | is_o.of_is_O_with $ λ c cpos, (h.forall_is_O_with cpos).sup (h'.forall_is_O_with cpos) | theorem | asymptotics.is_o.sup | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_sup : f =O[l ⊔ l'] g' ↔ f =O[l] g' ∧ f =O[l'] g' | ⟨λ h, ⟨h.mono le_sup_left, h.mono le_sup_right⟩, λ h, h.1.sup h.2⟩ | lemma | asymptotics.is_O_sup | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_sup : f =o[l ⊔ l'] g ↔ f =o[l] g ∧ f =o[l'] g | ⟨λ h, ⟨h.mono le_sup_left, h.mono le_sup_right⟩, λ h, h.1.sup h.2⟩ | lemma | asymptotics.is_o_sup | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"le_sup_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_insert [topological_space α] {x : α} {s : set α} {C : ℝ} {g : α → E} {g' : α → F}
(h : ‖g x‖ ≤ C * ‖g' x‖) :
is_O_with C (𝓝[insert x s] x) g g' ↔ is_O_with C (𝓝[s] x) g g' | by simp_rw [is_O_with, nhds_within_insert, eventually_sup, eventually_pure, h, true_and] | lemma | asymptotics.is_O_with_insert | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"nhds_within_insert",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.insert [topological_space α] {x : α} {s : set α} {C : ℝ} {g : α → E} {g' : α → F}
(h1 : is_O_with C (𝓝[s] x) g g') (h2 : ‖g x‖ ≤ C * ‖g' x‖) :
is_O_with C (𝓝[insert x s] x) g g' | (is_O_with_insert h2).mpr h1 | lemma | asymptotics.is_O_with.insert | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_insert [topological_space α] {x : α} {s : set α} {g : α → E'} {g' : α → F'}
(h : g x = 0) : g =o[𝓝[insert x s] x] g' ↔ g =o[𝓝[s] x] g' | begin
simp_rw [is_o],
refine forall_congr (λ c, forall_congr (λ hc, _)),
rw [is_O_with_insert],
rw [h, norm_zero],
exact mul_nonneg hc.le (norm_nonneg _)
end | lemma | asymptotics.is_o_insert | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.insert [topological_space α] {x : α} {s : set α} {g : α → E'} {g' : α → F'}
(h1 : g =o[𝓝[s] x] g') (h2 : g x = 0) : g =o[𝓝[insert x s] x] g' | (is_o_insert h2).mpr h1 | lemma | asymptotics.is_o.insert | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_norm_right : is_O_with c l f (λ x, ‖g' x‖) ↔ is_O_with c l f g' | by simp only [is_O_with, norm_norm] | theorem | asymptotics.is_O_with_norm_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_abs_right : is_O_with c l f (λ x, |u x|) ↔ is_O_with c l f u | @is_O_with_norm_right _ _ _ _ _ _ f u l | theorem | asymptotics.is_O_with_abs_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_norm_right : f =O[l] (λ x, ‖g' x‖) ↔ f =O[l] g' | by { unfold is_O, exact exists_congr (λ _, is_O_with_norm_right) } | theorem | asymptotics.is_O_norm_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_abs_right : f =O[l] (λ x, |u x|) ↔ f =O[l] u | @is_O_norm_right _ _ ℝ _ _ _ _ _ | theorem | asymptotics.is_O_abs_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_norm_right : f =o[l] (λ x, ‖g' x‖) ↔ f =o[l] g' | by { unfold is_o, exact forall₂_congr (λ _ _, is_O_with_norm_right) } | theorem | asymptotics.is_o_norm_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_abs_right : f =o[l] (λ x, |u x|) ↔ f =o[l] u | @is_o_norm_right _ _ ℝ _ _ _ _ _ | theorem | asymptotics.is_o_abs_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_norm_left : is_O_with c l (λ x, ‖f' x‖) g ↔ is_O_with c l f' g | by simp only [is_O_with, norm_norm] | theorem | asymptotics.is_O_with_norm_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_abs_left : is_O_with c l (λ x, |u x|) g ↔ is_O_with c l u g | @is_O_with_norm_left _ _ _ _ _ _ g u l | theorem | asymptotics.is_O_with_abs_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_norm_left : (λ x, ‖f' x‖) =O[l] g ↔ f' =O[l] g | by { unfold is_O, exact exists_congr (λ _, is_O_with_norm_left) } | theorem | asymptotics.is_O_norm_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_abs_left : (λ x, |u x|) =O[l] g ↔ u =O[l] g | @is_O_norm_left _ _ _ _ _ g u l | theorem | asymptotics.is_O_abs_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_norm_left : (λ x, ‖f' x‖) =o[l] g ↔ f' =o[l] g | by { unfold is_o, exact forall₂_congr (λ _ _, is_O_with_norm_left) } | theorem | asymptotics.is_o_norm_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_abs_left : (λ x, |u x|) =o[l] g ↔ u =o[l] g | @is_o_norm_left _ _ _ _ _ g u l | theorem | asymptotics.is_o_abs_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_norm_norm : is_O_with c l (λ x, ‖f' x‖) (λ x, ‖g' x‖) ↔ is_O_with c l f' g' | is_O_with_norm_left.trans is_O_with_norm_right | theorem | asymptotics.is_O_with_norm_norm | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_abs_abs : is_O_with c l (λ x, |u x|) (λ x, |v x|) ↔ is_O_with c l u v | is_O_with_abs_left.trans is_O_with_abs_right | theorem | asymptotics.is_O_with_abs_abs | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_norm_norm : (λ x, ‖f' x‖) =O[l] (λ x, ‖g' x‖) ↔ f' =O[l] g' | is_O_norm_left.trans is_O_norm_right | theorem | asymptotics.is_O_norm_norm | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_abs_abs : (λ x, |u x|) =O[l] (λ x, |v x|) ↔ u =O[l] v | is_O_abs_left.trans is_O_abs_right | theorem | asymptotics.is_O_abs_abs | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_norm_norm : (λ x, ‖f' x‖) =o[l] (λ x, ‖g' x‖) ↔ f' =o[l] g' | is_o_norm_left.trans is_o_norm_right | theorem | asymptotics.is_o_norm_norm | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_abs_abs : (λ x, |u x|) =o[l] (λ x, |v x|) ↔ u =o[l] v | is_o_abs_left.trans is_o_abs_right | theorem | asymptotics.is_o_abs_abs | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_neg_right : is_O_with c l f (λ x, -(g' x)) ↔ is_O_with c l f g' | by simp only [is_O_with, norm_neg] | theorem | asymptotics.is_O_with_neg_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_neg_right : f =O[l] (λ x, -(g' x)) ↔ f =O[l] g' | by { unfold is_O, exact exists_congr (λ _, is_O_with_neg_right) } | theorem | asymptotics.is_O_neg_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_neg_right : f =o[l] (λ x, -(g' x)) ↔ f =o[l] g' | by { unfold is_o, exact forall₂_congr (λ _ _, is_O_with_neg_right) } | theorem | asymptotics.is_o_neg_right | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_neg_left : is_O_with c l (λ x, -(f' x)) g ↔ is_O_with c l f' g | by simp only [is_O_with, norm_neg] | theorem | asymptotics.is_O_with_neg_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_neg_left : (λ x, -(f' x)) =O[l] g ↔ f' =O[l] g | by { unfold is_O, exact exists_congr (λ _, is_O_with_neg_left) } | theorem | asymptotics.is_O_neg_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_neg_left : (λ x, -(f' x)) =o[l] g ↔ f' =o[l] g | by { unfold is_o, exact forall₂_congr (λ _ _, is_O_with_neg_left) } | theorem | asymptotics.is_o_neg_left | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_fst_prod : is_O_with 1 l f' (λ x, (f' x, g' x)) | is_O_with_of_le l $ λ x, le_max_left _ _ | lemma | asymptotics.is_O_with_fst_prod | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_snd_prod : is_O_with 1 l g' (λ x, (f' x, g' x)) | is_O_with_of_le l $ λ x, le_max_right _ _ | lemma | asymptotics.is_O_with_snd_prod | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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