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_fst_prod : f' =O[l] (λ x, (f' x, g' x)) | is_O_with_fst_prod.is_O | lemma | asymptotics.is_O_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_snd_prod : g' =O[l] (λ x, (f' x, g' x)) | is_O_with_snd_prod.is_O | lemma | asymptotics.is_O_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 | |
is_O_fst_prod' {f' : α → E' × F'} : (λ x, (f' x).1) =O[l] f' | by simpa [is_O, is_O_with] using is_O_fst_prod | lemma | asymptotics.is_O_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_snd_prod' {f' : α → E' × F'} : (λ x, (f' x).2) =O[l] f' | by simpa [is_O, is_O_with] using is_O_snd_prod | lemma | asymptotics.is_O_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 | |
is_O_with.prod_rightl (h : is_O_with c l f g') (hc : 0 ≤ c) :
is_O_with c l f (λ x, (g' x, k' x)) | (h.trans is_O_with_fst_prod hc).congr_const (mul_one c) | lemma | asymptotics.is_O_with.prod_rightl | 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.prod_rightl (h : f =O[l] g') : f =O[l] (λ x, (g' x, k' x)) | let ⟨c, cnonneg, hc⟩ := h.exists_nonneg in (hc.prod_rightl k' cnonneg).is_O | lemma | asymptotics.is_O.prod_rightl | 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.prod_rightl (h : f =o[l] g') : f =o[l] (λ x, (g' x, k' x)) | is_o.of_is_O_with $ λ c cpos, (h.forall_is_O_with cpos).prod_rightl k' cpos.le | lemma | asymptotics.is_o.prod_rightl | 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.prod_rightr (h : is_O_with c l f g') (hc : 0 ≤ c) :
is_O_with c l f (λ x, (f' x, g' x)) | (h.trans is_O_with_snd_prod hc).congr_const (mul_one c) | lemma | asymptotics.is_O_with.prod_rightr | 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.prod_rightr (h : f =O[l] g') : f =O[l] (λ x, (f' x, g' x)) | let ⟨c, cnonneg, hc⟩ := h.exists_nonneg in (hc.prod_rightr f' cnonneg).is_O | lemma | asymptotics.is_O.prod_rightr | 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.prod_rightr (h : f =o[l] g') : f =o[l] (λx, (f' x, g' x)) | is_o.of_is_O_with $ λ c cpos, (h.forall_is_O_with cpos).prod_rightr f' cpos.le | lemma | asymptotics.is_o.prod_rightr | 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.prod_left_same (hf : is_O_with c l f' k') (hg : is_O_with c l g' k') :
is_O_with c l (λ x, (f' x, g' x)) k' | by rw is_O_with_iff at *; filter_upwards [hf, hg] with x using max_le | lemma | asymptotics.is_O_with.prod_left_same | 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.prod_left (hf : is_O_with c l f' k') (hg : is_O_with c' l g' k') :
is_O_with (max c c') l (λ x, (f' x, g' x)) k' | (hf.weaken $ le_max_left c c').prod_left_same (hg.weaken $ le_max_right c c') | lemma | asymptotics.is_O_with.prod_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.prod_left_fst (h : is_O_with c l (λ x, (f' x, g' x)) k') :
is_O_with c l f' k' | (is_O_with_fst_prod.trans h zero_le_one).congr_const $ one_mul c | lemma | asymptotics.is_O_with.prod_left_fst | 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",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.prod_left_snd (h : is_O_with c l (λ x, (f' x, g' x)) k') :
is_O_with c l g' k' | (is_O_with_snd_prod.trans h zero_le_one).congr_const $ one_mul c | lemma | asymptotics.is_O_with.prod_left_snd | 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",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_prod_left :
is_O_with c l (λ x, (f' x, g' x)) k' ↔ is_O_with c l f' k' ∧ is_O_with c l g' k' | ⟨λ h, ⟨h.prod_left_fst, h.prod_left_snd⟩, λ h, h.1.prod_left_same h.2⟩ | lemma | asymptotics.is_O_with_prod_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.prod_left (hf : f' =O[l] k') (hg : g' =O[l] k') : (λ x, (f' x, g' x)) =O[l] k' | let ⟨c, hf⟩ := hf.is_O_with, ⟨c', hg⟩ := hg.is_O_with in (hf.prod_left hg).is_O | lemma | asymptotics.is_O.prod_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.prod_left_fst : (λ x, (f' x, g' x)) =O[l] k' → f' =O[l] k' | is_O.trans is_O_fst_prod | lemma | asymptotics.is_O.prod_left_fst | 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.prod_left_snd : (λ x, (f' x, g' x)) =O[l] k' → g' =O[l] k' | is_O.trans is_O_snd_prod | lemma | asymptotics.is_O.prod_left_snd | 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_prod_left : (λ x, (f' x, g' x)) =O[l] k' ↔ f' =O[l] k' ∧ g' =O[l] k' | ⟨λ h, ⟨h.prod_left_fst, h.prod_left_snd⟩, λ h, h.1.prod_left h.2⟩ | lemma | asymptotics.is_O_prod_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.prod_left (hf : f' =o[l] k') (hg : g' =o[l] k') : (λ x, (f' x, g' x)) =o[l] k' | is_o.of_is_O_with $ λ c hc, (hf.forall_is_O_with hc).prod_left_same (hg.forall_is_O_with hc) | lemma | asymptotics.is_o.prod_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.prod_left_fst : (λ x, (f' x, g' x)) =o[l] k' → f' =o[l] k' | is_O.trans_is_o is_O_fst_prod | lemma | asymptotics.is_o.prod_left_fst | 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.prod_left_snd : (λ x, (f' x, g' x)) =o[l] k' → g' =o[l] k' | is_O.trans_is_o is_O_snd_prod | lemma | asymptotics.is_o.prod_left_snd | 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_prod_left : (λ x, (f' x, g' x)) =o[l] k' ↔ f' =o[l] k' ∧ g' =o[l] k' | ⟨λ h, ⟨h.prod_left_fst, h.prod_left_snd⟩, λ h, h.1.prod_left h.2⟩ | lemma | asymptotics.is_o_prod_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.eq_zero_imp (h : is_O_with c l f'' g'') : ∀ᶠ x in l, g'' x = 0 → f'' x = 0 | eventually.mono h.bound $ λ x hx hg, norm_le_zero_iff.1 $ by simpa [hg] using hx | lemma | asymptotics.is_O_with.eq_zero_imp | 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.eq_zero_imp (h : f'' =O[l] g'') : ∀ᶠ x in l, g'' x = 0 → f'' x = 0 | let ⟨C, hC⟩ := h.is_O_with in hC.eq_zero_imp | lemma | asymptotics.is_O.eq_zero_imp | 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.add (h₁ : is_O_with c₁ l f₁ g) (h₂ : is_O_with c₂ l f₂ g) :
is_O_with (c₁ + c₂) l (λ x, f₁ x + f₂ x) g | by rw is_O_with at *; filter_upwards [h₁, h₂] with x hx₁ hx₂ using
calc ‖f₁ x + f₂ x‖ ≤ c₁ * ‖g x‖ + c₂ * ‖g x‖ : norm_add_le_of_le hx₁ hx₂
... = (c₁ + c₂) * ‖g x‖ : (add_mul _ _ _).symm | theorem | asymptotics.is_O_with.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.add (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g) : (λ x, f₁ x + f₂ x) =O[l] g | let ⟨c₁, hc₁⟩ := h₁.is_O_with, ⟨c₂, hc₂⟩ := h₂.is_O_with in (hc₁.add hc₂).is_O | theorem | asymptotics.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.add (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g) : (λ x, f₁ x + f₂ x) =o[l] g | is_o.of_is_O_with $ λ c cpos, ((h₁.forall_is_O_with $ half_pos cpos).add
(h₂.forall_is_O_with $ half_pos cpos)).congr_const (add_halves c) | theorem | asymptotics.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"
] | [
"add_halves",
"half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.add_add (h₁ : f₁ =o[l] g₁) (h₂ : f₂ =o[l] g₂) :
(λ x, f₁ x + f₂ x) =o[l] (λ x, ‖g₁ x‖ + ‖g₂ x‖) | by refine (h₁.trans_le $ λ x, _).add (h₂.trans_le _);
simp [abs_of_nonneg, add_nonneg] | theorem | asymptotics.is_o.add_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"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.add_is_o (h₁ : f₁ =O[l] g) (h₂ : f₂ =o[l] g) : (λ x, f₁ x + f₂ x) =O[l] g | h₁.add h₂.is_O | theorem | asymptotics.is_O.add_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.add_is_O (h₁ : f₁ =o[l] g) (h₂ : f₂ =O[l] g) : (λ x, f₁ x + f₂ x) =O[l] g | h₁.is_O.add h₂ | theorem | asymptotics.is_o.add_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.add_is_o (h₁ : is_O_with c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂) :
is_O_with c₂ l (λx, f₁ x + f₂ x) g | (h₁.add (h₂.forall_is_O_with (sub_pos.2 hc))).congr_const (add_sub_cancel'_right _ _) | theorem | asymptotics.is_O_with.add_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.add_is_O_with (h₁ : f₁ =o[l] g) (h₂ : is_O_with c₁ l f₂ g) (hc : c₁ < c₂) :
is_O_with c₂ l (λx, f₁ x + f₂ x) g | (h₂.add_is_o h₁ hc).congr_left $ λ _, add_comm _ _ | theorem | asymptotics.is_o.add_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.sub (h₁ : is_O_with c₁ l f₁ g) (h₂ : is_O_with c₂ l f₂ g) :
is_O_with (c₁ + c₂) l (λ x, f₁ x - f₂ x) g | by simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left | theorem | asymptotics.is_O_with.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.sub_is_o (h₁ : is_O_with c₁ l f₁ g) (h₂ : f₂ =o[l] g) (hc : c₁ < c₂) :
is_O_with c₂ l (λ x, f₁ x - f₂ x) g | by simpa only [sub_eq_add_neg] using h₁.add_is_o h₂.neg_left hc | theorem | asymptotics.is_O_with.sub_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.sub (h₁ : f₁ =O[l] g) (h₂ : f₂ =O[l] g) : (λ x, f₁ x - f₂ x) =O[l] g | by simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left | theorem | asymptotics.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.sub (h₁ : f₁ =o[l] g) (h₂ : f₂ =o[l] g) : (λ x, f₁ x - f₂ x) =o[l] g | by simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left | theorem | asymptotics.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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with.symm (h : is_O_with c l (λ x, f₁ x - f₂ x) g) :
is_O_with c l (λ x, f₂ x - f₁ x) g | h.neg_left.congr_left $ λ x, neg_sub _ _ | theorem | asymptotics.is_O_with.symm | 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_comm :
is_O_with c l (λ x, f₁ x - f₂ x) g ↔ is_O_with c l (λ x, f₂ x - f₁ x) g | ⟨is_O_with.symm, is_O_with.symm⟩ | theorem | asymptotics.is_O_with_comm | 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.symm (h : (λ x, f₁ x - f₂ x) =O[l] g) : (λ x, f₂ x - f₁ x) =O[l] g | h.neg_left.congr_left $ λ x, neg_sub _ _ | theorem | asymptotics.is_O.symm | 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_comm : (λ x, f₁ x - f₂ x) =O[l] g ↔ (λ x, f₂ x - f₁ x) =O[l] g | ⟨is_O.symm, is_O.symm⟩ | theorem | asymptotics.is_O_comm | 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.symm (h : (λ x, f₁ x - f₂ x) =o[l] g) : (λ x, f₂ x - f₁ x) =o[l] g | by simpa only [neg_sub] using h.neg_left | theorem | asymptotics.is_o.symm | 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_comm : (λ x, f₁ x - f₂ x) =o[l] g ↔ (λ x, f₂ x - f₁ x) =o[l] g | ⟨is_o.symm, is_o.symm⟩ | theorem | asymptotics.is_o_comm | 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.triangle (h₁ : is_O_with c l (λ x, f₁ x - f₂ x) g)
(h₂ : is_O_with c' l (λ x, f₂ x - f₃ x) g) :
is_O_with (c + c') l (λ x, f₁ x - f₃ x) g | (h₁.add h₂).congr_left $ λ x, sub_add_sub_cancel _ _ _ | theorem | asymptotics.is_O_with.triangle | 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.triangle (h₁ : (λ x, f₁ x - f₂ x) =O[l] g) (h₂ : (λ x, f₂ x - f₃ x) =O[l] g) :
(λ x, f₁ x - f₃ x) =O[l] g | (h₁.add h₂).congr_left $ λ x, sub_add_sub_cancel _ _ _ | theorem | asymptotics.is_O.triangle | 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.triangle (h₁ : (λ x, f₁ x - f₂ x) =o[l] g) (h₂ : (λ x, f₂ x - f₃ x) =o[l] g) :
(λ x, f₁ x - f₃ x) =o[l] g | (h₁.add h₂).congr_left $ λ x, sub_add_sub_cancel _ _ _ | theorem | asymptotics.is_o.triangle | 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_of_sub (h : (λ x, f₁ x - f₂ x) =O[l] g) : f₁ =O[l] g ↔ f₂ =O[l] g | ⟨λ h', (h'.sub h).congr_left (λ x, sub_sub_cancel _ _),
λ h', (h.add h').congr_left (λ x, sub_add_cancel _ _)⟩ | theorem | asymptotics.is_O.congr_of_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o.congr_of_sub (h : (λ x, f₁ x - f₂ x) =o[l] g) : f₁ =o[l] g ↔ f₂ =o[l] g | ⟨λ h', (h'.sub h).congr_left (λ x, sub_sub_cancel _ _),
λ h', (h.add h').congr_left (λ x, sub_add_cancel _ _)⟩ | theorem | asymptotics.is_o.congr_of_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_zero : (λ x, (0 : E')) =o[l] g' | is_o.of_bound $ λ c hc, univ_mem' $ λ x,
by simpa using mul_nonneg hc.le (norm_nonneg $ g' x) | theorem | asymptotics.is_o_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_zero (hc : 0 ≤ c) : is_O_with c l (λ x, (0 : E')) g' | is_O_with.of_bound $ univ_mem' $ λ x, by simpa using mul_nonneg hc (norm_nonneg $ g' x) | theorem | asymptotics.is_O_with_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_zero' : is_O_with 0 l (λ x, (0 : E')) g | is_O_with.of_bound $ univ_mem' $ λ x, by simp | theorem | asymptotics.is_O_with_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_zero : (λ x, (0 : E')) =O[l] g | is_O_iff_is_O_with.2 ⟨0, is_O_with_zero' _ _⟩ | theorem | asymptotics.is_O_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_refl_left : (λ x, f' x - f' x) =O[l] g' | (is_O_zero g' l).congr_left $ λ x, (sub_self _).symm | theorem | asymptotics.is_O_refl_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_refl_left : (λ x, f' x - f' x) =o[l] g' | (is_o_zero g' l).congr_left $ λ x, (sub_self _).symm | theorem | asymptotics.is_o_refl_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_zero_right_iff :
is_O_with c l f'' (λ x, (0 : F')) ↔ f'' =ᶠ[l] 0 | by simp only [is_O_with, exists_prop, true_and, norm_zero, mul_zero, norm_le_zero_iff,
eventually_eq, pi.zero_apply] | theorem | asymptotics.is_O_with_zero_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"
] | [
"exists_prop",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_zero_right_iff : f'' =O[l] (λ x, (0 : F')) ↔ f'' =ᶠ[l] 0 | ⟨λ h, let ⟨c, hc⟩ := h.is_O_with in is_O_with_zero_right_iff.1 hc,
λ h, (is_O_with_zero_right_iff.2 h : is_O_with 1 _ _ _).is_O⟩ | theorem | asymptotics.is_O_zero_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_zero_right_iff :
f'' =o[l] (λ x, (0 : F')) ↔ f'' =ᶠ[l] 0 | ⟨λ h, is_O_zero_right_iff.1 h.is_O, λ h, is_o.of_is_O_with $ λ c hc, is_O_with_zero_right_iff.2 h⟩ | theorem | asymptotics.is_o_zero_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : filter α) :
is_O_with (‖c‖ / ‖c'‖) l (λ x : α, c) (λ x, c') | begin
unfold is_O_with,
apply univ_mem',
intro x,
rw [mem_set_of_eq, div_mul_cancel],
rwa [ne.def, norm_eq_zero]
end | theorem | asymptotics.is_O_with_const_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"
] | [
"div_mul_cancel",
"filter",
"norm_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_const (c : E) {c' : F''} (hc' : c' ≠ 0) (l : filter α) :
(λ x : α, c) =O[l] (λ x, c') | (is_O_with_const_const c hc' l).is_O | theorem | asymptotics.is_O_const_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"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_const_iff {c : E''} {c' : F''} (l : filter α) [l.ne_bot] :
(λ x : α, c) =O[l] (λ x, c') ↔ (c' = 0 → c = 0) | begin
rcases eq_or_ne c' 0 with rfl|hc',
{ simp [eventually_eq] },
{ simp [hc', is_O_const_const _ hc'] }
end | 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"
] | [
"eq_or_ne",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_pure {x} : f'' =O[pure x] g'' ↔ (g'' x = 0 → 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
... ↔ g'' x = 0 → 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_with_top : is_O_with c ⊤ f g ↔ ∀ x, ‖f x‖ ≤ c * ‖g x‖ | by rw is_O_with; refl | lemma | asymptotics.is_O_with_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_top : f =O[⊤] g ↔ ∃ C, ∀ x, ‖f x‖ ≤ C * ‖g x‖ | by rw is_O_iff; refl | lemma | asymptotics.is_O_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_top : f'' =o[⊤] g'' ↔ ∀ x, f'' x = 0 | begin
refine ⟨_, λ h, (is_o_zero g'' ⊤).congr (λ x, (h x).symm) (λ x, rfl)⟩,
simp only [is_o_iff, eventually_top],
refine λ h x, norm_le_zero_iff.1 _,
have : tendsto (λ c : ℝ, c * ‖g'' x‖) (𝓝[>] 0) (𝓝 0) :=
((continuous_id.mul continuous_const).tendsto' _ _ (zero_mul _)).mono_left inf_le_left,
exact le_... | lemma | asymptotics.is_o_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"
] | [
"continuous_const",
"inf_le_left",
"le_of_tendsto_of_tendsto",
"tendsto_const_nhds",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_principal {s : set α} :
is_O_with c (𝓟 s) f g ↔ ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖ | by rw is_O_with; refl | lemma | asymptotics.is_O_with_principal | 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_principal {s : set α} : f =O[𝓟 s] g ↔ ∃ c, ∀ x ∈ s, ‖f x‖ ≤ c * ‖g x‖ | by rw is_O_iff; refl | lemma | asymptotics.is_O_principal | 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_one (c : E) (l : filter α) : is_O_with ‖c‖ l (λ x : α, c) (λ x, (1 : F)) | by simp [is_O_with_iff] | theorem | asymptotics.is_O_with_const_one | 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_const_one (c : E) (l : filter α) : (λ x : α, c) =O[l] (λ x, (1 : F)) | (is_O_with_const_one F c l).is_O | theorem | asymptotics.is_O_const_one | 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_const_iff_is_o_one {c : F''} (hc : c ≠ 0) :
f =o[l] (λ x, c) ↔ f =o[l] (λ x, (1 : F)) | ⟨λ h, h.trans_is_O_with (is_O_with_const_one _ _ _) (norm_pos_iff.2 hc),
λ h, h.trans_is_O $ is_O_const_const _ hc _⟩ | theorem | asymptotics.is_o_const_iff_is_o_one | 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_one_iff : f' =o[l] (λ x, 1 : α → F) ↔ tendsto f' l (𝓝 0) | by simp only [is_o_iff, norm_one, mul_one, metric.nhds_basis_closed_ball.tendsto_right_iff,
metric.mem_closed_ball, dist_zero_right] | theorem | asymptotics.is_o_one_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"
] | [
"metric.mem_closed_ball",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_one_iff : f =O[l] (λ x, 1 : α → F) ↔ is_bounded_under (≤) l (λ x, ‖f x‖) | by { simp only [is_O_iff, norm_one, mul_one], refl } | theorem | asymptotics.is_O_one_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_one_left_iff : (λ x, 1 : α → F) =o[l] f ↔ tendsto (λ x, ‖f x‖) l at_top | calc (λ x, 1 : α → F) =o[l] f ↔ ∀ n : ℕ, ∀ᶠ x in l, ↑n * ‖(1 : F)‖ ≤ ‖f x‖ :
is_o_iff_nat_mul_le_aux $ or.inl $ λ x, by simp only [norm_one, zero_le_one]
... ↔ ∀ n : ℕ, true → ∀ᶠ x in l, ‖f x‖ ∈ Ici (n : ℝ) :
by simp only [norm_one, mul_one, true_implies_iff, mem_Ici]
... ↔ tendsto (λ x, ‖f x‖) l at_top : at_top_co... | theorem | asymptotics.is_o_one_left_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",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.filter.tendsto.is_O_one {c : E'} (h : tendsto f' l (𝓝 c)) :
f' =O[l] (λ x, 1 : α → F) | h.norm.is_bounded_under_le.is_O_one F | theorem | filter.tendsto.is_O_one | 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_tendsto_nhds (hfg : f =O[l] g') {y : F'} (hg : tendsto g' l (𝓝 y)) :
f =O[l] (λ x, 1 : α → F) | hfg.trans $ hg.is_O_one F | theorem | asymptotics.is_O.trans_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_const_iff {c : F''} (hc : c ≠ 0) :
f'' =o[l] (λ x, c) ↔ tendsto f'' l (𝓝 0) | (is_o_const_iff_is_o_one ℝ hc).trans (is_o_one_iff _) | theorem | asymptotics.is_o_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_id_const {c : F''} (hc : c ≠ 0) :
(λ (x : E''), x) =o[𝓝 0] (λ x, c) | (is_o_const_iff hc).mpr (continuous_id.tendsto 0) | lemma | asymptotics.is_o_id_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 | |
_root_.filter.is_bounded_under.is_O_const (h : is_bounded_under (≤) l (norm ∘ f))
{c : F''} (hc : c ≠ 0) : f =O[l] (λ x, c) | (h.is_O_one ℝ).trans (is_O_const_const _ hc _) | theorem | filter.is_bounded_under.is_O_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_const_of_tendsto {y : E''} (h : tendsto f'' l (𝓝 y)) {c : F''} (hc : c ≠ 0) :
f'' =O[l] (λ x, c) | h.norm.is_bounded_under_le.is_O_const hc | theorem | asymptotics.is_O_const_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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O.is_bounded_under_le {c : F} (h : f =O[l] (λ x, c)) :
is_bounded_under (≤) l (norm ∘ f) | let ⟨c', hc'⟩ := h.bound in ⟨c' * ‖c‖, eventually_map.2 hc'⟩ | lemma | asymptotics.is_O.is_bounded_under_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_const_of_ne {c : F''} (hc : c ≠ 0) :
f =O[l] (λ x, c) ↔ is_bounded_under (≤) l (norm ∘ f) | ⟨λ h, h.is_bounded_under_le, λ h, h.is_O_const hc⟩ | theorem | asymptotics.is_O_const_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_iff {c : F''} :
f'' =O[l] (λ x, c) ↔ (c = 0 → f'' =ᶠ[l] 0) ∧ is_bounded_under (≤) l (λ x, ‖f'' x‖) | begin
refine ⟨λ h, ⟨λ hc, is_O_zero_right_iff.1 (by rwa ← hc), h.is_bounded_under_le⟩, _⟩,
rintro ⟨hcf, hf⟩,
rcases eq_or_ne c 0 with hc|hc,
exacts [(hcf hc).trans_is_O (is_O_zero _ _), hf.is_O_const hc]
end | theorem | asymptotics.is_O_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"
] | [
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_iff_is_bounded_under_le_div (h : ∀ᶠ x in l, g'' x ≠ 0) :
f =O[l] g'' ↔ is_bounded_under (≤) l (λ x, ‖f x‖ / ‖g'' x‖) | begin
simp only [is_O_iff, is_bounded_under, is_bounded, eventually_map],
exact exists_congr (λ c, eventually_congr $ h.mono $
λ x hx, (div_le_iff $ norm_pos_iff.2 hx).symm)
end | theorem | asymptotics.is_O_iff_is_bounded_under_le_div | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_left_iff_pos_le_norm {c : E''} (hc : c ≠ 0) :
(λ x, c) =O[l] f' ↔ ∃ b, 0 < b ∧ ∀ᶠ x in l, b ≤ ‖f' x‖ | begin
split,
{ intro h,
rcases h.exists_pos with ⟨C, hC₀, hC⟩,
refine ⟨‖c‖ / C, div_pos (norm_pos_iff.2 hc) hC₀, _⟩,
exact hC.bound.mono (λ x, (div_le_iff' hC₀).2) },
{ rintro ⟨b, hb₀, hb⟩,
refine is_O.of_bound (‖c‖ / b) (hb.mono $ λ x hx, _),
rw [div_mul_eq_mul_div, mul_div_assoc],
exact ... | lemma | asymptotics.is_O_const_left_iff_pos_le_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"
] | [
"div_le_iff'",
"div_mul_eq_mul_div",
"div_pos",
"le_mul_of_one_le_right",
"mul_div_assoc",
"one_le_div"
] | `(λ x, c) =O[l] f` if and only if `f` is bounded away from zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O.trans_tendsto (hfg : f'' =O[l] g'') (hg : tendsto g'' l (𝓝 0)) :
tendsto f'' l (𝓝 0) | (is_o_one_iff ℝ).1 $ hfg.trans_is_o $ (is_o_one_iff ℝ).2 hg | theorem | asymptotics.is_O.trans_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.trans_tendsto (hfg : f'' =o[l] g'') (hg : tendsto g'' l (𝓝 0)) :
tendsto f'' l (𝓝 0) | hfg.is_O.trans_tendsto hg | theorem | asymptotics.is_o.trans_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_with_const_mul_self (c : R) (f : α → R) (l : filter α) :
is_O_with ‖c‖ l (λ x, c * f x) f | is_O_with_of_le' _ $ λ x, norm_mul_le _ _ | theorem | asymptotics.is_O_with_const_mul_self | 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",
"norm_mul_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_mul_self (c : R) (f : α → R) (l : filter α) : (λ x, c * f x) =O[l] f | (is_O_with_const_mul_self c f l).is_O | theorem | asymptotics.is_O_const_mul_self | 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.const_mul_left {f : α → R} (h : is_O_with c l f g) (c' : R) :
is_O_with (‖c'‖ * c) l (λ x, c' * f x) g | (is_O_with_const_mul_self c' f l).trans h (norm_nonneg c') | theorem | asymptotics.is_O_with.const_mul_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_mul_left {f : α → R} (h : f =O[l] g) (c' : R) :
(λ x, c' * f x) =O[l] g | let ⟨c, hc⟩ := h.is_O_with in (hc.const_mul_left c').is_O | theorem | asymptotics.is_O.const_mul_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_self_const_mul' (u : Rˣ) (f : α → R) (l : filter α) :
is_O_with ‖(↑u⁻¹:R)‖ l f (λ x, ↑u * f x) | (is_O_with_const_mul_self ↑u⁻¹ _ l).congr_left $ λ x, u.inv_mul_cancel_left (f x) | theorem | asymptotics.is_O_with_self_const_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"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_with_self_const_mul (c : 𝕜) (hc : c ≠ 0) (f : α → 𝕜) (l : filter α) :
is_O_with ‖c‖⁻¹ l f (λ x, c * f x) | (is_O_with_self_const_mul' (units.mk0 c hc) f l).congr_const $
norm_inv c | theorem | asymptotics.is_O_with_self_const_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"
] | [
"filter",
"norm_inv",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_self_const_mul' {c : R} (hc : is_unit c) (f : α → R) (l : filter α) :
f =O[l] (λ x, c * f x) | let ⟨u, hu⟩ := hc in hu ▸ (is_O_with_self_const_mul' u f l).is_O | theorem | asymptotics.is_O_self_const_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"
] | [
"filter",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_self_const_mul (c : 𝕜) (hc : c ≠ 0) (f : α → 𝕜) (l : filter α) :
f =O[l] (λ x, c * f x) | is_O_self_const_mul' (is_unit.mk0 c hc) f l | theorem | asymptotics.is_O_self_const_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"
] | [
"filter",
"is_unit.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_const_mul_left_iff' {f : α → R} {c : R} (hc : is_unit c) :
(λ x, c * f x) =O[l] g ↔ f =O[l] g | ⟨(is_O_self_const_mul' hc f l).trans, λ h, h.const_mul_left c⟩ | theorem | asymptotics.is_O_const_mul_left_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_left_iff {f : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) :
(λ x, c * f x) =O[l] g ↔ f =O[l] g | is_O_const_mul_left_iff' $ is_unit.mk0 c hc | theorem | asymptotics.is_O_const_mul_left_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.const_mul_left {f : α → R} (h : f =o[l] g) (c : R) : (λ x, c * f x) =o[l] g | (is_O_const_mul_self c f l).trans_is_o h | theorem | asymptotics.is_o.const_mul_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_mul_left_iff' {f : α → R} {c : R} (hc : is_unit c) :
(λ x, c * f x) =o[l] g ↔ f =o[l] g | ⟨(is_O_self_const_mul' hc f l).trans_is_o, λ h, h.const_mul_left c⟩ | theorem | asymptotics.is_o_const_mul_left_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_left_iff {f : α → 𝕜} {c : 𝕜} (hc : c ≠ 0) :
(λ x, c * f x) =o[l] g ↔ f =o[l] g | is_o_const_mul_left_iff' $ is_unit.mk0 c hc | theorem | asymptotics.is_o_const_mul_left_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.of_const_mul_right {g : α → R} {c : R} (hc' : 0 ≤ c')
(h : is_O_with c' l f (λ x, c * g x)) :
is_O_with (c' * ‖c‖) l f g | h.trans (is_O_with_const_mul_self c g l) hc' | theorem | asymptotics.is_O_with.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.of_const_mul_right {g : α → R} {c : R} (h : f =O[l] (λ x, c * g x)) :
f =O[l] g | let ⟨c, cnonneg, hc⟩ := h.exists_nonneg in (hc.of_const_mul_right cnonneg).is_O | 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 |
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