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_with.const_mul_right' {g : α → R} {u : Rˣ} {c' : ℝ} (hc' : 0 ≤ c')
(h : is_O_with c' l f g) :
is_O_with (c' * ‖(↑u⁻¹:R)‖) l f (λ x, ↑u * g x) | h.trans (is_O_with_self_const_mul' _ _ _) hc' | theorem | asymptotics.is_O_with.const_mul_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.const_mul_right {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0)
{c' : ℝ} (hc' : 0 ≤ c') (h : is_O_with c' l f g) :
is_O_with (c' * ‖c‖⁻¹) l f (λ x, c * g x) | h.trans (is_O_with_self_const_mul c hc g l) hc' | theorem | asymptotics.is_O_with.const_mul_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.const_mul_right' {g : α → R} {c : R} (hc : is_unit c) (h : f =O[l] g) :
f =O[l] (λ x, c * g x) | h.trans (is_O_self_const_mul' hc g l) | theorem | asymptotics.is_O.const_mul_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"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.const_mul_right {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) (h : f =O[l] g) :
f =O[l] (λ x, c * g x) | h.const_mul_right' $ is_unit.mk0 c hc | theorem | asymptotics.is_O.const_mul_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"
] | [
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_mul_right_iff' {g : α → R} {c : R} (hc : is_unit c) :
f =O[l] (λ x, c * g x) ↔ f =O[l] g | ⟨λ h, h.of_const_mul_right, λ h, h.const_mul_right' hc⟩ | theorem | asymptotics.is_O_const_mul_right_iff' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_mul_right_iff {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) :
f =O[l] (λ x, c * g x) ↔ f =O[l] g | is_O_const_mul_right_iff' $ is_unit.mk0 c hc | theorem | asymptotics.is_O_const_mul_right_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.of_const_mul_right {g : α → R} {c : R} (h : f =o[l] (λ x, c * g x)) :
f =o[l] g | h.trans_is_O (is_O_const_mul_self c g l) | theorem | asymptotics.is_o.of_const_mul_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.const_mul_right' {g : α → R} {c : R} (hc : is_unit c) (h : f =o[l] g) :
f =o[l] (λ x, c * g x) | h.trans_is_O (is_O_self_const_mul' hc g l) | theorem | asymptotics.is_o.const_mul_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"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.const_mul_right {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) (h : f =o[l] g) :
f =o[l] (λ x, c * g x) | h.const_mul_right' $ is_unit.mk0 c hc | theorem | asymptotics.is_o.const_mul_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"
] | [
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_mul_right_iff' {g : α → R} {c : R} (hc : is_unit c) :
f =o[l] (λ x, c * g x) ↔ f =o[l] g | ⟨λ h, h.of_const_mul_right, λ h, h.const_mul_right' hc⟩ | theorem | asymptotics.is_o_const_mul_right_iff' | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_mul_right_iff {g : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) :
f =o[l] (λ x, c * g x) ↔ f =o[l] g | is_o_const_mul_right_iff' $ is_unit.mk0 c hc | theorem | asymptotics.is_o_const_mul_right_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.mul {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} {c₁ c₂ : ℝ}
(h₁ : is_O_with c₁ l f₁ g₁) (h₂ : is_O_with c₂ l f₂ g₂) :
is_O_with (c₁ * c₂) l (λ x, f₁ x * f₂ x) (λ x, g₁ x * g₂ x) | begin
unfold is_O_with at *,
filter_upwards [h₁, h₂] with _ hx₁ hx₂,
apply le_trans (norm_mul_le _ _),
convert mul_le_mul hx₁ hx₂ (norm_nonneg _) (le_trans (norm_nonneg _) hx₁) using 1,
rw [norm_mul, mul_mul_mul_comm]
end | theorem | asymptotics.is_O_with.mul | 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_le_mul",
"mul_mul_mul_comm",
"norm_mul",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.mul {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (h₁ : f₁ =O[l] g₁) (h₂ : f₂ =O[l] g₂) :
(λ x, f₁ x * f₂ x) =O[l] (λ x, g₁ x * g₂ x) | let ⟨c, hc⟩ := h₁.is_O_with, ⟨c', hc'⟩ := h₂.is_O_with in (hc.mul hc').is_O | theorem | asymptotics.is_O.mul | 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.mul_is_o {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜}
(h₁ : f₁ =O[l] g₁) (h₂ : f₂ =o[l] g₂) :
(λ x, f₁ x * f₂ x) =o[l] (λ x, g₁ x * g₂ x) | begin
unfold is_o at *,
intros c cpos,
rcases h₁.exists_pos with ⟨c', c'pos, hc'⟩,
exact (hc'.mul (h₂ (div_pos cpos c'pos))).congr_const (mul_div_cancel' _ (ne_of_gt c'pos))
end | theorem | asymptotics.is_O.mul_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.mul_is_O {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =O[l] g₂) :
(λ x, f₁ x * f₂ x) =o[l] (λ x, g₁ x * g₂ x) | begin
unfold is_o at *,
intros c cpos,
rcases h₂.exists_pos with ⟨c', c'pos, hc'⟩,
exact ((h₁ (div_pos cpos c'pos)).mul hc').congr_const (div_mul_cancel _ (ne_of_gt c'pos))
end | theorem | asymptotics.is_o.mul_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_mul_cancel",
"div_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.mul {f₁ f₂ : α → R} {g₁ g₂ : α → 𝕜} (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =o[l] g₂) :
(λ x, f₁ x * f₂ x) =o[l] (λ x, g₁ x * g₂ x) | h₁.mul_is_O h₂.is_O | theorem | asymptotics.is_o.mul | 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.pow' {f : α → R} {g : α → 𝕜} (h : is_O_with c l f g) :
∀ n : ℕ, is_O_with (nat.cases_on n ‖(1 : R)‖ (λ n, c ^ (n + 1))) l (λ x, f x ^ n) (λ x, g x ^ n) | | 0 := by simpa using is_O_with_const_const (1 : R) (one_ne_zero' 𝕜) l
| 1 := by simpa
| (n + 2) := by simpa [pow_succ] using h.mul (is_O_with.pow' (n + 1)) | theorem | asymptotics.is_O_with.pow' | 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_ne_zero'",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.pow [norm_one_class R] {f : α → R} {g : α → 𝕜} (h : is_O_with c l f g) :
∀ n : ℕ, is_O_with (c ^ n) l (λ x, f x ^ n) (λ x, g x ^ n) | | 0 := by simpa using h.pow' 0
| (n + 1) := h.pow' (n + 1) | theorem | asymptotics.is_O_with.pow | 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_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.of_pow {n : ℕ} {f : α → 𝕜} {g : α → R} (h : is_O_with c l (f ^ n) (g ^ n))
(hn : n ≠ 0) (hc : c ≤ c' ^ n) (hc' : 0 ≤ c') : is_O_with c' l f g | is_O_with.of_bound $ (h.weaken hc).bound.mono $ λ x hx,
le_of_pow_le_pow n (mul_nonneg hc' $ norm_nonneg _) hn.bot_lt $
calc ‖f x‖ ^ n = ‖(f x) ^ n‖ : (norm_pow _ _).symm
... ≤ c' ^ n * ‖(g x) ^ n‖ : hx
... ≤ c' ^ n * ‖g x‖ ^ n :
mul_le_mul_of_nonneg_left (norm_pow_le' _ hn.bot... | theorem | asymptotics.is_O_with.of_pow | 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_of_pow_le_pow",
"mul_le_mul_of_nonneg_left",
"mul_pow",
"norm_pow",
"norm_pow_le'",
"pow_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.pow {f : α → R} {g : α → 𝕜} (h : f =O[l] g) (n : ℕ) :
(λ x, f x ^ n) =O[l] (λ x, g x ^ n) | let ⟨C, hC⟩ := h.is_O_with in is_O_iff_is_O_with.2 ⟨_, hC.pow' n⟩ | theorem | asymptotics.is_O.pow | 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_pow {f : α → 𝕜} {g : α → R} {n : ℕ} (hn : n ≠ 0) (h : (f ^ n) =O[l] (g ^ n)) :
f =O[l] g | begin
rcases h.exists_pos with ⟨C, hC₀, hC⟩,
obtain ⟨c, hc₀, hc⟩ : ∃ c : ℝ, 0 ≤ c ∧ C ≤ c ^ n,
from ((eventually_ge_at_top _).and $ (tendsto_pow_at_top hn).eventually_ge_at_top C).exists,
exact (hC.of_pow hn hc hc₀).is_O
end | theorem | asymptotics.is_O.of_pow | 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.pow {f : α → R} {g : α → 𝕜} (h : f =o[l] g) {n : ℕ} (hn : 0 < n) :
(λ x, f x ^ n) =o[l] (λ x, g x ^ n) | begin
cases n, exact hn.false.elim, clear hn,
induction n with n ihn, { simpa only [pow_one] },
convert h.mul ihn; simp [pow_succ]
end | theorem | asymptotics.is_o.pow | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"pow_one",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.of_pow {f : α → 𝕜} {g : α → R} {n : ℕ} (h : (f ^ n) =o[l] (g ^ n)) (hn : n ≠ 0) :
f =o[l] g | is_o.of_is_O_with $ λ c hc, (h.def' $ pow_pos hc _).of_pow hn le_rfl hc.le | theorem | asymptotics.is_o.of_pow | 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_rfl",
"pow_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : is_O_with c l f g)
(h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : is_O_with c l (λ x, (g x)⁻¹) (λ x, (f x)⁻¹) | begin
refine is_O_with.of_bound (h.bound.mp (h₀.mono $ λ x h₀ hle, _)),
cases eq_or_ne (f x) 0 with hx hx,
{ simp only [hx, h₀ hx, inv_zero, norm_zero, mul_zero] },
{ have hc : 0 < c, from pos_of_mul_pos_left ((norm_pos_iff.2 hx).trans_le hle) (norm_nonneg _),
replace hle := inv_le_inv_of_le (norm_pos_iff.2... | theorem | asymptotics.is_O_with.inv_rev | 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",
"div_le_iff",
"eq_or_ne",
"inv_le_inv_of_le",
"inv_zero",
"mul_inv",
"mul_zero",
"norm_inv",
"pos_of_mul_pos_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : f =O[l] g)
(h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : (λ x, (g x)⁻¹) =O[l] (λ x, (f x)⁻¹) | let ⟨c, hc⟩ := h.is_O_with in (hc.inv_rev h₀).is_O | theorem | asymptotics.is_O.inv_rev | 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.inv_rev {f : α → 𝕜} {g : α → 𝕜'} (h : f =o[l] g)
(h₀ : ∀ᶠ x in l, f x = 0 → g x = 0) : (λ x, (g x)⁻¹) =o[l] (λ x, (f x)⁻¹) | is_o.of_is_O_with $ λ c hc, (h.def' hc).inv_rev h₀ | theorem | asymptotics.is_o.inv_rev | 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.const_smul_left (h : is_O_with c l f' g) (c' : 𝕜) :
is_O_with (‖c'‖ * c) l (λ x, c' • f' x) g | is_O_with.of_norm_left $
by simpa only [← norm_smul, norm_norm] using h.norm_left.const_mul_left (‖c'‖) | theorem | asymptotics.is_O_with.const_smul_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",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.const_smul_left (h : f' =O[l] g) (c : 𝕜) : (c • f') =O[l] g | let ⟨b, hb⟩ := h.is_O_with in (hb.const_smul_left _).is_O | lemma | asymptotics.is_O.const_smul_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.const_smul_left (h : f' =o[l] g) (c : 𝕜) : (c • f') =o[l] g | is_o.of_norm_left $ by simpa only [← norm_smul] using h.norm_left.const_mul_left (‖c‖) | lemma | asymptotics.is_o.const_smul_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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_smul_left {c : 𝕜} (hc : c ≠ 0) :
(λ x, c • f' x) =O[l] g ↔ f' =O[l] g | begin
have cne0 : ‖c‖ ≠ 0, from mt norm_eq_zero.mp hc,
rw [←is_O_norm_left], simp only [norm_smul],
rw [is_O_const_mul_left_iff cne0, is_O_norm_left],
end | theorem | asymptotics.is_O_const_smul_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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_smul_left {c : 𝕜} (hc : c ≠ 0) :
(λ x, c • f' x) =o[l] g ↔ f' =o[l] g | begin
have cne0 : ‖c‖ ≠ 0, from mt norm_eq_zero.mp hc,
rw [←is_o_norm_left], simp only [norm_smul],
rw [is_o_const_mul_left_iff cne0, is_o_norm_left]
end | theorem | asymptotics.is_o_const_smul_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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_smul_right {c : 𝕜} (hc : c ≠ 0) :
f =O[l] (λ x, c • f' x) ↔ f =O[l] f' | begin
have cne0 : ‖c‖ ≠ 0, from mt norm_eq_zero.mp hc,
rw [←is_O_norm_right], simp only [norm_smul],
rw [is_O_const_mul_right_iff cne0, is_O_norm_right]
end | theorem | asymptotics.is_O_const_smul_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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_smul_right {c : 𝕜} (hc : c ≠ 0) :
f =o[l] (λ x, c • f' x) ↔ f =o[l] f' | begin
have cne0 : ‖c‖ ≠ 0, from mt norm_eq_zero.mp hc,
rw [←is_o_norm_right], simp only [norm_smul],
rw [is_o_const_mul_right_iff cne0, is_o_norm_right]
end | theorem | asymptotics.is_o_const_smul_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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.smul (h₁ : is_O_with c l k₁ k₂) (h₂ : is_O_with c' l f' g') :
is_O_with (c * c') l (λ x, k₁ x • f' x) (λ x, k₂ x • g' x) | by refine ((h₁.norm_norm.mul h₂.norm_norm).congr rfl _ _).of_norm_norm;
by intros; simp only [norm_smul] | theorem | asymptotics.is_O_with.smul | 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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.smul (h₁ : k₁ =O[l] k₂) (h₂ : f' =O[l] g') :
(λ x, k₁ x • f' x) =O[l] (λ x, k₂ x • g' x) | by refine ((h₁.norm_norm.mul h₂.norm_norm).congr _ _).of_norm_norm;
by intros; simp only [norm_smul] | theorem | asymptotics.is_O.smul | 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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.smul_is_o (h₁ : k₁ =O[l] k₂) (h₂ : f' =o[l] g') :
(λ x, k₁ x • f' x) =o[l] (λ x, k₂ x • g' x) | by refine ((h₁.norm_norm.mul_is_o h₂.norm_norm).congr _ _).of_norm_norm;
by intros; simp only [norm_smul] | theorem | asymptotics.is_O.smul_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"
] | [
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.smul_is_O (h₁ : k₁ =o[l] k₂) (h₂ : f' =O[l] g') :
(λ x, k₁ x • f' x) =o[l] (λ x, k₂ x • g' x) | by refine ((h₁.norm_norm.mul_is_O h₂.norm_norm).congr _ _).of_norm_norm;
by intros; simp only [norm_smul] | theorem | asymptotics.is_o.smul_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"
] | [
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.smul (h₁ : k₁ =o[l] k₂) (h₂ : f' =o[l] g') :
(λ x, k₁ x • f' x) =o[l] (λ x, k₂ x • g' x) | by refine ((h₁.norm_norm.mul h₂.norm_norm).congr _ _).of_norm_norm;
by intros; simp only [norm_smul] | theorem | asymptotics.is_o.smul | 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_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.sum (h : ∀ i ∈ s, is_O_with (C i) l (A i) g) :
is_O_with (∑ i in s, C i) l (λ x, ∑ i in s, A i x) g | begin
induction s using finset.induction_on with i s is IH,
{ simp only [is_O_with_zero', finset.sum_empty, forall_true_iff] },
{ simp only [is, finset.sum_insert, not_false_iff],
exact (h _ (finset.mem_insert_self i s)).add (IH (λ j hj, h _ (finset.mem_insert_of_mem hj))) }
end | theorem | asymptotics.is_O_with.sum | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"forall_true_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.sum (h : ∀ i ∈ s, A i =O[l] g) :
(λ x, ∑ i in s, A i x) =O[l] g | begin
unfold is_O at *,
choose! C hC using h,
exact ⟨_, is_O_with.sum hC⟩,
end | theorem | asymptotics.is_O.sum | 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.sum (h : ∀ i ∈ s, (A i) =o[l] g') :
(λ x, ∑ i in s, A i x) =o[l] g' | begin
induction s using finset.induction_on with i s is IH,
{ simp only [is_o_zero, finset.sum_empty, forall_true_iff] },
{ simp only [is, finset.sum_insert, not_false_iff],
exact (h _ (finset.mem_insert_self i s)).add (IH (λ j hj, h _ (finset.mem_insert_of_mem hj))) }
end | theorem | asymptotics.is_o.sum | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"forall_true_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.tendsto_div_nhds_zero {f g : α → 𝕜} (h : f =o[l] g) :
tendsto (λ x, f x / (g x)) l (𝓝 0) | (is_o_one_iff 𝕜).mp $
calc (λ x, f x / g x) =o[l] (λ x, g x / g x) :
by simpa only [div_eq_mul_inv] using h.mul_is_O (is_O_refl _ _)
... =O[l] (λ x, (1 : 𝕜)) :
is_O_of_le _ (λ x, by simp [div_self_le_one]) | theorem | asymptotics.is_o.tendsto_div_nhds_zero | 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_mul_inv",
"div_self_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.tendsto_inv_smul_nhds_zero [normed_space 𝕜 E'] {f : α → E'} {g : α → 𝕜} {l : filter α}
(h : f =o[l] g) : tendsto (λ x, (g x)⁻¹ • f x) l (𝓝 0) | by simpa only [div_eq_inv_mul, ← norm_inv, ← norm_smul,
← tendsto_zero_iff_norm_tendsto_zero] using h.norm_norm.tendsto_div_nhds_zero | theorem | asymptotics.is_o.tendsto_inv_smul_nhds_zero | 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",
"filter",
"norm_inv",
"norm_smul",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_iff_tendsto' {f g : α → 𝕜} (hgf : ∀ᶠ x in l, g x = 0 → f x = 0) :
f =o[l] g ↔ tendsto (λ x, f x / (g x)) l (𝓝 0) | ⟨is_o.tendsto_div_nhds_zero, λ h,
(((is_o_one_iff _).mpr h).mul_is_O (is_O_refl g l)).congr'
(hgf.mono $ λ x, div_mul_cancel_of_imp) (eventually_of_forall $ λ x, one_mul _)⟩ | theorem | asymptotics.is_o_iff_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"
] | [
"div_mul_cancel_of_imp",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_iff_tendsto {f g : α → 𝕜} (hgf : ∀ x, g x = 0 → f x = 0) :
f =o[l] g ↔ tendsto (λ x, f x / (g x)) l (𝓝 0) | is_o_iff_tendsto' (eventually_of_forall hgf) | theorem | asymptotics.is_o_iff_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_left_of_ne {c : E''} (hc : c ≠ 0) :
(λ x, c) =o[l] g ↔ tendsto (λ x, ‖g x‖) l at_top | begin
simp only [← is_o_one_left_iff ℝ],
exact ⟨(is_O_const_const (1 : ℝ) hc l).trans_is_o, (is_O_const_one ℝ c l).trans_is_o⟩
end | lemma | asymptotics.is_o_const_left_of_ne | 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_const_left {c : E''} :
(λ x, c) =o[l] g'' ↔ c = 0 ∨ tendsto (norm ∘ g'') l at_top | begin
rcases eq_or_ne c 0 with rfl | hc,
{ simp only [is_o_zero, eq_self_iff_true, true_or] },
{ simp only [hc, false_or, is_o_const_left_of_ne hc] }
end | lemma | asymptotics.is_o_const_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"
] | [
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_const_iff [ne_bot l] {d : E''} {c : F''} :
(λ x, d) =o[l] (λ x, c) ↔ d = 0 | have ¬tendsto (function.const α ‖c‖) l at_top,
from not_tendsto_at_top_of_tendsto_nhds tendsto_const_nhds,
by simp [function.const, this] | theorem | asymptotics.is_o_const_const_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"not_tendsto_at_top_of_tendsto_nhds",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pure {x} : f'' =o[pure x] g'' ↔ f'' x = 0 | calc f'' =o[pure x] g'' ↔ (λ y : α, f'' x) =o[pure x] (λ _, g'' x) : is_o_congr rfl rfl
... ↔ f'' x = 0 : is_o_const_const_iff | lemma | asymptotics.is_o_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_const_id_comap_norm_at_top (c : F'') : (λ x : E'', c) =o[comap norm at_top] id | is_o_const_left.2 $ or.inr tendsto_comap | lemma | asymptotics.is_o_const_id_comap_norm_at_top | 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_const_id_at_top (c : E'') : (λ x : ℝ, c) =o[at_top] id | is_o_const_left.2 $ or.inr tendsto_abs_at_top_at_top | lemma | asymptotics.is_o_const_id_at_top | 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_const_id_at_bot (c : E'') : (λ x : ℝ, c) =o[at_bot] id | is_o_const_left.2 $ or.inr tendsto_abs_at_bot_at_top | lemma | asymptotics.is_o_const_id_at_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.eventually_mul_div_cancel (h : is_O_with c l u v) :
(u / v) * v =ᶠ[l] u | eventually.mono h.bound (λ y hy, div_mul_cancel_of_imp $ λ hv, by simpa [hv] using hy) | lemma | asymptotics.is_O_with.eventually_mul_div_cancel | 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_of_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.eventually_mul_div_cancel (h : u =O[l] v) : (u / v) * v =ᶠ[l] u | let ⟨c, hc⟩ := h.is_O_with in hc.eventually_mul_div_cancel | lemma | asymptotics.is_O.eventually_mul_div_cancel | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [] | If `u = O(v)` along `l`, then `(u / v) * v = u` eventually at `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o.eventually_mul_div_cancel (h : u =o[l] v) : (u / v) * v =ᶠ[l] u | (h.forall_is_O_with zero_lt_one).eventually_mul_div_cancel | lemma | asymptotics.is_o.eventually_mul_div_cancel | 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"
] | If `u = o(v)` along `l`, then `(u / v) * v = u` eventually at `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_with_of_eq_mul (φ : α → 𝕜) (hφ : ∀ᶠ x in l, ‖φ x‖ ≤ c) (h : u =ᶠ[l] φ * v) :
is_O_with c l u v | begin
unfold is_O_with,
refine h.symm.rw (λ x a, ‖a‖ ≤ c * ‖v x‖) (hφ.mono $ λ x hx, _),
simp only [norm_mul, pi.mul_apply],
exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
end | lemma | asymptotics.is_O_with_of_eq_mul | 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_le_mul_of_nonneg_right",
"norm_mul",
"pi.mul_apply"
] | If `‖φ‖` is eventually bounded by `c`, and `u =ᶠ[l] φ * v`, then we have `is_O_with c u v l`.
This does not require any assumptions on `c`, which is why we keep this version along with
`is_O_with_iff_exists_eq_mul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_with_iff_exists_eq_mul (hc : 0 ≤ c) :
is_O_with c l u v ↔ ∃ (φ : α → 𝕜) (hφ : ∀ᶠ x in l, ‖φ x‖ ≤ c), u =ᶠ[l] φ * v | begin
split,
{ intro h,
use (λ x, u x / v x),
refine ⟨eventually.mono h.bound (λ y hy, _), h.eventually_mul_div_cancel.symm⟩,
simpa using div_le_of_nonneg_of_le_mul (norm_nonneg _) hc hy },
{ rintros ⟨φ, hφ, h⟩,
exact is_O_with_of_eq_mul φ hφ h }
end | lemma | asymptotics.is_O_with_iff_exists_eq_mul | 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_le_of_nonneg_of_le_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.exists_eq_mul (h : is_O_with c l u v) (hc : 0 ≤ c) :
∃ (φ : α → 𝕜) (hφ : ∀ᶠ x in l, ‖φ x‖ ≤ c), u =ᶠ[l] φ * v | (is_O_with_iff_exists_eq_mul hc).mp h | lemma | asymptotics.is_O_with.exists_eq_mul | 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_exists_eq_mul :
u =O[l] v ↔ ∃ (φ : α → 𝕜) (hφ : l.is_bounded_under (≤) (norm ∘ φ)), u =ᶠ[l] φ * v | begin
split,
{ rintros h,
rcases h.exists_nonneg with ⟨c, hnnc, hc⟩,
rcases hc.exists_eq_mul hnnc with ⟨φ, hφ, huvφ⟩,
exact ⟨φ, ⟨c, hφ⟩, huvφ⟩ },
{ rintros ⟨φ, ⟨c, hφ⟩, huvφ⟩,
exact is_O_iff_is_O_with.2 ⟨c, is_O_with_of_eq_mul φ hφ huvφ⟩ }
end | lemma | asymptotics.is_O_iff_exists_eq_mul | 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_exists_eq_mul :
u =o[l] v ↔ ∃ (φ : α → 𝕜) (hφ : tendsto φ l (𝓝 0)), u =ᶠ[l] φ * v | begin
split,
{ exact λ h, ⟨λ x, u x / v x, h.tendsto_div_nhds_zero, h.eventually_mul_div_cancel.symm⟩ },
{ unfold is_o, rintros ⟨φ, hφ, huvφ⟩ c hpos,
rw normed_add_comm_group.tendsto_nhds_zero at hφ,
exact is_O_with_of_eq_mul _ ((hφ c hpos).mono $ λ x, le_of_lt) huvφ }
end | lemma | asymptotics.is_o_iff_exists_eq_mul | 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 | |
div_is_bounded_under_of_is_O {α : Type*} {l : filter α}
{f g : α → 𝕜} (h : f =O[l] g) :
is_bounded_under (≤) l (λ x, ‖f x / g x‖) | begin
obtain ⟨c, h₀, hc⟩ := h.exists_nonneg,
refine ⟨c, eventually_map.2 (hc.bound.mono (λ x hx, _))⟩,
rw [norm_div],
exact div_le_of_nonneg_of_le_mul (norm_nonneg _) h₀ hx,
end | theorem | asymptotics.div_is_bounded_under_of_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_le_of_nonneg_of_le_mul",
"filter",
"norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_iff_div_is_bounded_under {α : Type*} {l : filter α}
{f g : α → 𝕜} (hgf : ∀ᶠ x in l, g x = 0 → f x = 0) :
f =O[l] g ↔ is_bounded_under (≤) l (λ x, ‖f x / g x‖) | begin
refine ⟨div_is_bounded_under_of_is_O, λ h, _⟩,
obtain ⟨c, hc⟩ := h,
simp only [eventually_map, norm_div] at hc,
refine is_O.of_bound c (hc.mp $ hgf.mono (λ x hx₁ hx₂, _)),
by_cases hgx : g x = 0,
{ simp [hx₁ hgx, hgx] },
{ exact (div_le_iff (norm_pos_iff.2 hgx)).mp hx₂ },
end | theorem | asymptotics.is_O_iff_div_is_bounded_under | 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_le_iff",
"filter",
"norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_of_div_tendsto_nhds {α : Type*} {l : filter α}
{f g : α → 𝕜} (hgf : ∀ᶠ x in l, g x = 0 → f x = 0)
(c : 𝕜) (H : filter.tendsto (f / g) l (𝓝 c)) :
f =O[l] g | (is_O_iff_div_is_bounded_under hgf).2 $ H.norm.is_bounded_under_le | theorem | asymptotics.is_O_of_div_tendsto_nhds | 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",
"filter.tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.tendsto_zero_of_tendsto {α E 𝕜 : Type*} [normed_add_comm_group E] [normed_field 𝕜]
{u : α → E} {v : α → 𝕜} {l : filter α} {y : 𝕜} (huv : u =o[l] v) (hv : tendsto v l (𝓝 y)) :
tendsto u l (𝓝 0) | begin
suffices h : u =o[l] (λ x, (1 : 𝕜)),
{ rwa is_o_one_iff at h },
exact huv.trans_is_O (hv.is_O_one 𝕜),
end | lemma | asymptotics.is_o.tendsto_zero_of_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",
"normed_add_comm_group",
"normed_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pow_pow {m n : ℕ} (h : m < n) :
(λ x : 𝕜, x ^ n) =o[𝓝 0] (λ x, x ^ m) | begin
rcases lt_iff_exists_add.1 h with ⟨p, hp0 : 0 < p, rfl⟩,
suffices : (λ x : 𝕜, x ^ m * x ^ p) =o[𝓝 0] (λ x, x ^ m * 1 ^ p),
by simpa only [pow_add, one_pow, mul_one],
exact is_O.mul_is_o (is_O_refl _ _) (is_o.pow ((is_o_one_iff _).2 tendsto_id) hp0)
end | theorem | asymptotics.is_o_pow_pow | 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",
"one_pow",
"pow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_norm_pow_norm_pow {m n : ℕ} (h : m < n) :
(λ x : E', ‖x‖^n) =o[𝓝 0] (λ x, ‖x‖^m) | (is_o_pow_pow h).comp_tendsto tendsto_norm_zero | theorem | asymptotics.is_o_norm_pow_norm_pow | 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_pow_id {n : ℕ} (h : 1 < n) :
(λ x : 𝕜, x^n) =o[𝓝 0] (λ x, x) | by { convert is_o_pow_pow h, simp only [pow_one] } | theorem | asymptotics.is_o_pow_id | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_norm_pow_id {n : ℕ} (h : 1 < n) :
(λ x : E', ‖x‖^n) =o[𝓝 0] (λ x, x) | by simpa only [pow_one, is_o_norm_right] using @is_o_norm_pow_norm_pow E' _ _ _ h | theorem | asymptotics.is_o_norm_pow_id | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.eq_zero_of_norm_pow_within {f : E'' → F''} {s : set E''} {x₀ : E''} {n : ℕ}
(h : f =O[𝓝[s] x₀] λ x, ‖x - x₀‖ ^ n) (hx₀ : x₀ ∈ s) (hn : 0 < n) : f x₀ = 0 | mem_of_mem_nhds_within hx₀ h.eq_zero_imp $ by simp_rw [sub_self, norm_zero, zero_pow hn] | lemma | asymptotics.is_O.eq_zero_of_norm_pow_within | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"mem_of_mem_nhds_within",
"zero_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.eq_zero_of_norm_pow {f : E'' → F''} {x₀ : E''} {n : ℕ}
(h : f =O[𝓝 x₀] λ x, ‖x - x₀‖ ^ n) (hn : 0 < n) : f x₀ = 0 | by { rw [← nhds_within_univ] at h, exact h.eq_zero_of_norm_pow_within (mem_univ _) hn } | lemma | asymptotics.is_O.eq_zero_of_norm_pow | 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_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pow_sub_pow_sub (x₀ : E') {n m : ℕ} (h : n < m) :
(λ x, ‖x - x₀‖ ^ m) =o[𝓝 x₀] λ x, ‖x - x₀‖^n | begin
have : tendsto (λ x, ‖x - x₀‖) (𝓝 x₀) (𝓝 0),
{ apply tendsto_norm_zero.comp,
rw ← sub_self x₀,
exact tendsto_id.sub tendsto_const_nhds },
exact (is_o_pow_pow h).comp_tendsto this
end | lemma | asymptotics.is_o_pow_sub_pow_sub | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pow_sub_sub (x₀ : E') {m : ℕ} (h : 1 < m) :
(λ x, ‖x - x₀‖^m) =o[𝓝 x₀] λ x, x - x₀ | by simpa only [is_o_norm_right, pow_one] using is_o_pow_sub_pow_sub x₀ h | lemma | asymptotics.is_o_pow_sub_sub | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.right_le_sub_of_lt_1 {f₁ f₂ : α → E'} (h : is_O_with c l f₁ f₂) (hc : c < 1) :
is_O_with (1 / (1 - c)) l f₂ (λx, f₂ x - f₁ x) | is_O_with.of_bound $ mem_of_superset h.bound $ λ x hx,
begin
simp only [mem_set_of_eq] at hx ⊢,
rw [mul_comm, one_div, ← div_eq_mul_inv, le_div_iff, mul_sub, mul_one, mul_comm],
{ exact le_trans (sub_le_sub_left hx _) (norm_sub_norm_le _ _) },
{ exact sub_pos.2 hc }
end | theorem | asymptotics.is_O_with.right_le_sub_of_lt_1 | 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_mul_inv",
"le_div_iff",
"mul_comm",
"mul_one",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.right_le_add_of_lt_1 {f₁ f₂ : α → E'} (h : is_O_with c l f₁ f₂) (hc : c < 1) :
is_O_with (1 / (1 - c)) l f₂ (λx, f₁ x + f₂ x) | (h.neg_right.right_le_sub_of_lt_1 hc).neg_right.of_neg_left.congr rfl (λ x, rfl)
(λ x, by rw [neg_sub, sub_neg_eq_add]) | theorem | asymptotics.is_O_with.right_le_add_of_lt_1 | 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.right_is_O_sub {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
f₂ =O[l] (λx, f₂ x - f₁ x) | ((h.def' one_half_pos).right_le_sub_of_lt_1 one_half_lt_one).is_O | theorem | asymptotics.is_o.right_is_O_sub | 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_half_lt_one",
"one_half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.right_is_O_add {f₁ f₂ : α → E'} (h : f₁ =o[l] f₂) :
f₂ =O[l] (λx, f₁ x + f₂ x) | ((h.def' one_half_pos).right_le_add_of_lt_1 one_half_lt_one).is_O | theorem | asymptotics.is_o.right_is_O_add | 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_half_lt_one",
"one_half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bound_of_is_O_cofinite (h : f =O[cofinite] g'') :
∃ C > 0, ∀ ⦃x⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖ | begin
rcases h.exists_pos with ⟨C, C₀, hC⟩,
rw [is_O_with, eventually_cofinite] at hC,
rcases (hC.to_finset.image (λ x, ‖f x‖ / ‖g'' x‖)).exists_le with ⟨C', hC'⟩,
have : ∀ x, C * ‖g'' x‖ < ‖f x‖ → ‖f x‖ / ‖g'' x‖ ≤ C', by simpa using hC',
refine ⟨max C C', lt_max_iff.2 (or.inl C₀), λ x h₀, _⟩,
rw [max_mul_... | theorem | asymptotics.bound_of_is_O_cofinite | 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_le_iff",
"le_max_iff",
"max_mul_of_nonneg",
"or_iff_not_imp_left"
] | If `f x = O(g x)` along `cofinite`, then there exists a positive constant `C` such that
`‖f x‖ ≤ C * ‖g x‖` whenever `g x ≠ 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_cofinite_iff (h : ∀ x, g'' x = 0 → f'' x = 0) :
f'' =O[cofinite] g'' ↔ ∃ C, ∀ x, ‖f'' x‖ ≤ C * ‖g'' x‖ | ⟨λ h', let ⟨C, C₀, hC⟩ := bound_of_is_O_cofinite h' in
⟨C, λ x, if hx : g'' x = 0 then by simp [h _ hx, hx] else hC hx⟩,
λ h, (is_O_top.2 h).mono le_top⟩ | theorem | asymptotics.is_O_cofinite_iff | 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 | |
bound_of_is_O_nat_at_top {f : ℕ → E} {g'' : ℕ → E''} (h : f =O[at_top] g'') :
∃ C > 0, ∀ ⦃x⦄, g'' x ≠ 0 → ‖f x‖ ≤ C * ‖g'' x‖ | bound_of_is_O_cofinite $ by rwa nat.cofinite_eq_at_top | theorem | asymptotics.bound_of_is_O_nat_at_top | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"nat.cofinite_eq_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_nat_at_top_iff {f : ℕ → E''} {g : ℕ → F''} (h : ∀ x, g x = 0 → f x = 0) :
f =O[at_top] g ↔ ∃ C, ∀ x, ‖f x‖ ≤ C * ‖g x‖ | by rw [← nat.cofinite_eq_at_top, is_O_cofinite_iff h] | theorem | asymptotics.is_O_nat_at_top_iff | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"nat.cofinite_eq_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_one_nat_at_top_iff {f : ℕ → E''} :
f =O[at_top] (λ n, 1 : ℕ → ℝ) ↔ ∃ C, ∀ n, ‖f n‖ ≤ C | iff.trans (is_O_nat_at_top_iff (λ n h, (one_ne_zero h).elim)) $
by simp only [norm_one, mul_one] | theorem | asymptotics.is_O_one_nat_at_top_iff | 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",
"one_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_pi {ι : Type*} [fintype ι] {E' : ι → Type*} [Π i, normed_add_comm_group (E' i)]
{f : α → Π i, E' i} {C : ℝ} (hC : 0 ≤ C) :
is_O_with C l f g' ↔ ∀ i, is_O_with C l (λ x, f x i) g' | have ∀ x, 0 ≤ C * ‖g' x‖, from λ x, mul_nonneg hC (norm_nonneg _),
by simp only [is_O_with_iff, pi_norm_le_iff_of_nonneg (this _), eventually_all] | theorem | asymptotics.is_O_with_pi | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"fintype",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_pi {ι : Type*} [fintype ι] {E' : ι → Type*} [Π i, normed_add_comm_group (E' i)]
{f : α → Π i, E' i} :
f =O[l] g' ↔ ∀ i, (λ x, f x i) =O[l] g' | begin
simp only [is_O_iff_eventually_is_O_with, ← eventually_all],
exact eventually_congr (eventually_at_top.2 ⟨0, λ c, is_O_with_pi⟩)
end | theorem | asymptotics.is_O_pi | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"fintype",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_pi {ι : Type*} [fintype ι] {E' : ι → Type*} [Π i, normed_add_comm_group (E' i)]
{f : α → Π i, E' i} :
f =o[l] g' ↔ ∀ i, (λ x, f x i) =o[l] g' | begin
simp only [is_o, is_O_with_pi, le_of_lt] { contextual := tt },
exact ⟨λ h i c hc, h hc i, λ h c hc i, h i hc⟩
end | theorem | asymptotics.is_o_pi | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"fintype",
"normed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_is_O {ι E} [normed_add_comm_group E] [complete_space E] {f : ι → E} {g : ι → ℝ}
(hg : summable g) (h : f =O[cofinite] g) : summable f | let ⟨C, hC⟩ := h.is_O_with in
summable_of_norm_bounded_eventually (λ x, C * ‖g x‖) (hg.abs.mul_left _) hC.bound | lemma | summable_of_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"
] | [
"complete_space",
"normed_add_comm_group",
"summable",
"summable_of_norm_bounded_eventually"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_is_O_nat {E} [normed_add_comm_group E] [complete_space E] {f : ℕ → E} {g : ℕ → ℝ}
(hg : summable g) (h : f =O[at_top] g) : summable f | summable_of_is_O hg $ nat.cofinite_eq_at_top.symm ▸ h | lemma | summable_of_is_O_nat | analysis.asymptotics | src/analysis/asymptotics/asymptotics.lean | [
"analysis.normed.group.infinite_sum",
"analysis.normed_space.basic",
"topology.algebra.order.liminf_limsup",
"topology.local_homeomorph"
] | [
"complete_space",
"normed_add_comm_group",
"summable",
"summable_of_is_O"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_congr (e : local_homeomorph α β) {b : β} (hb : b ∈ e.target)
{f : β → E} {g : β → F} {C : ℝ} :
is_O_with C (𝓝 b) f g ↔ is_O_with C (𝓝 (e.symm b)) (f ∘ e) (g ∘ e) | ⟨λ h, h.comp_tendsto $
by { convert e.continuous_at (e.map_target hb), exact (e.right_inv hb).symm },
λ h, (h.comp_tendsto (e.continuous_at_symm hb)).congr' rfl
((e.eventually_right_inverse hb).mono $ λ x hx, congr_arg f hx)
((e.eventually_right_inverse hb).mono $ λ x hx, congr_arg g hx)⟩ | lemma | local_homeomorph.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"
] | [
"local_homeomorph"
] | Transfer `is_O_with` over a `local_homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_congr (e : local_homeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E} {g : β → F} :
f =O[𝓝 b] g ↔ (f ∘ e) =O[𝓝 (e.symm b)] (g ∘ e) | by { unfold is_O, exact exists_congr (λ C, e.is_O_with_congr hb) } | lemma | local_homeomorph.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"
] | [
"local_homeomorph"
] | Transfer `is_O` over a `local_homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_congr (e : local_homeomorph α β) {b : β} (hb : b ∈ e.target) {f : β → E} {g : β → F} :
f =o[𝓝 b] g ↔ (f ∘ e) =o[𝓝 (e.symm b)] (g ∘ e) | by { unfold is_o, exact forall₂_congr (λ c hc, e.is_O_with_congr hb) } | lemma | local_homeomorph.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",
"local_homeomorph"
] | Transfer `is_o` over a `local_homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_with_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} {C : ℝ} :
is_O_with C (𝓝 b) f g ↔ is_O_with C (𝓝 (e.symm b)) (f ∘ e) (g ∘ e) | e.to_local_homeomorph.is_O_with_congr trivial | lemma | homeomorph.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"
] | [] | Transfer `is_O_with` over a `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :
f =O[𝓝 b] g ↔ (f ∘ e) =O[𝓝 (e.symm b)] (g ∘ e) | by { unfold is_O, exact exists_congr (λ C, e.is_O_with_congr) } | lemma | homeomorph.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"
] | [] | Transfer `is_O` over a `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_congr (e : α ≃ₜ β) {b : β} {f : β → E} {g : β → F} :
f =o[𝓝 b] g ↔ (f ∘ e) =o[𝓝 (e.symm b)] (g ∘ e) | by { unfold is_o, exact forall₂_congr (λ c hc, e.is_O_with_congr) } | lemma | homeomorph.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"
] | Transfer `is_o` over a `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalent (l : filter α) (u v : α → β) | (u - v) =o[l] v | def | asymptotics.is_equivalent | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"filter"
] | Two functions `u` and `v` are said to be asymptotically equivalent along a filter `l` when
`u x - v x = o(v x)` as x converges along `l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalent.is_o (h : u ~[l] v) : (u - v) =o[l] v | h | lemma | asymptotics.is_equivalent.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.is_O (h : u ~[l] v) : u =O[l] v | (is_O.congr_of_sub h.is_O.symm).mp (is_O_refl _ _) | lemma | asymptotics.is_equivalent.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.is_O_symm (h : u ~[l] v) : v =O[l] u | begin
convert h.is_o.right_is_O_add,
ext,
simp
end | lemma | asymptotics.is_equivalent.is_O_symm | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.refl : u ~[l] u | begin
rw [is_equivalent, sub_self],
exact is_o_zero _ _
end | lemma | asymptotics.is_equivalent.refl | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.symm (h : u ~[l] v) : v ~[l] u | (h.is_o.trans_is_O h.is_O_symm).symm | lemma | asymptotics.is_equivalent.symm | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent.trans {l : filter α} {u v w : α → β}
(huv : u ~[l] v) (hvw : v ~[l] w) : u ~[l] w | (huv.is_o.trans_is_O hvw.is_O).triangle hvw.is_o | lemma | asymptotics.is_equivalent.trans | 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.congr_left {u v w : α → β} {l : filter α} (huv : u ~[l] v)
(huw : u =ᶠ[l] w) : w ~[l] v | huv.congr' (huw.sub (eventually_eq.refl _ _)) (eventually_eq.refl _ _) | lemma | asymptotics.is_equivalent.congr_left | analysis.asymptotics | src/analysis/asymptotics/asymptotic_equivalent.lean | [
"analysis.asymptotics.asymptotics",
"analysis.normed.order.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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