statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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ball_norm_mul_subset {p : seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r | begin
rcases eq_or_ne k 0 with (rfl | hk),
{ rw [norm_zero, zero_mul, ball_eq_emptyset _ le_rfl],
exact empty_subset _ },
{ intro x,
rw [set.mem_smul_set, seminorm.mem_ball_zero],
refine λ hx, ⟨k⁻¹ • x, _, _⟩,
{ rwa [seminorm.mem_ball_zero, map_smul_eq_mul, norm_inv,
←(mul_lt_mul_left $ nor... | lemma | seminorm.ball_norm_mul_subset | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"div_eq_mul_inv",
"div_self",
"eq_or_ne",
"le_rfl",
"mul_lt_mul_left",
"norm_inv",
"one_mul",
"one_smul",
"seminorm",
"seminorm.mem_ball_zero",
"set.mem_smul_set",
"smul_eq_mul",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball_zero {p : seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : k ≠ 0) :
k • p.ball 0 r = p.ball 0 (‖k‖ * r) | begin
ext,
rw [mem_smul_set_iff_inv_smul_mem₀ hk, p.mem_ball_zero, p.mem_ball_zero, map_smul_eq_mul,
norm_inv, ← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hk), mul_comm]
end | lemma | seminorm.smul_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"div_eq_inv_mul",
"div_lt_iff",
"mul_comm",
"norm_inv",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball_subset {p : seminorm 𝕜 E} {k : 𝕜} {r : ℝ} :
k • p.closed_ball 0 r ⊆ p.closed_ball 0 (‖k‖ * r) | begin
rintros x ⟨y, hy, h⟩,
rw [seminorm.mem_closed_ball_zero, ←h, map_smul_eq_mul],
rw seminorm.mem_closed_ball_zero at hy,
exact mul_le_mul_of_nonneg_left hy (norm_nonneg _)
end | lemma | seminorm.smul_closed_ball_subset | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"mul_le_mul_of_nonneg_left",
"seminorm",
"seminorm.mem_closed_ball_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball_zero {p : seminorm 𝕜 E} {k : 𝕜} {r : ℝ} (hk : 0 < ‖k‖) :
k • p.closed_ball 0 r = p.closed_ball 0 (‖k‖ * r) | begin
refine subset_antisymm smul_closed_ball_subset _,
intro x,
rw [set.mem_smul_set, seminorm.mem_closed_ball_zero],
refine λ hx, ⟨k⁻¹ • x, _, _⟩,
{ rwa [seminorm.mem_closed_ball_zero, map_smul_eq_mul, norm_inv, ←(mul_le_mul_left hk),
←mul_assoc, ←(div_eq_mul_inv ‖k‖ ‖k‖), div_self (ne_of_gt hk), one_... | lemma | seminorm.smul_closed_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"div_eq_mul_inv",
"div_self",
"mul_le_mul_left",
"norm_inv",
"one_mul",
"one_smul",
"seminorm",
"seminorm.mem_closed_ball_zero",
"set.mem_smul_set",
"smul_eq_mul",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_zero_absorbs_ball_zero (p : seminorm 𝕜 E) {r₁ r₂ : ℝ} (hr₁ : 0 < r₁) :
absorbs 𝕜 (p.ball 0 r₁) (p.ball 0 r₂) | begin
rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩,
refine ⟨r, hr₀, λ a ha x hx, _⟩,
rw [smul_ball_zero (norm_pos_iff.1 $ hr₀.trans_le ha), p.mem_ball_zero],
rw p.mem_ball_zero at hx,
exact hx.trans (hr.trans_le $ mul_le_mul_of_nonneg_right ha hr₁.le)
end | lemma | seminorm.ball_zero_absorbs_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbs",
"exists_pos_lt_mul",
"mul_le_mul_of_nonneg_right",
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent_ball_zero (hr : 0 < r) : absorbent 𝕜 (ball p (0 : E) r) | absorbent_iff_forall_absorbs_singleton.2 $ λ x, (p.ball_zero_absorbs_ball_zero hr).mono_right $
singleton_subset_iff.2 $ p.mem_ball_zero.2 $ lt_add_one _ | lemma | seminorm.absorbent_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbent",
"absorbent_ball_zero",
"lt_add_one"
] | Seminorm-balls at the origin are absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absorbent_closed_ball_zero (hr : 0 < r) : absorbent 𝕜 (closed_ball p (0 : E) r) | (p.absorbent_ball_zero hr).subset (p.ball_subset_closed_ball _ _) | lemma | seminorm.absorbent_closed_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbent"
] | Closed seminorm-balls at the origin are absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absorbent_ball (hpr : p x < r) : absorbent 𝕜 (ball p x r) | begin
refine (p.absorbent_ball_zero $ sub_pos.2 hpr).subset (λ y hy, _),
rw p.mem_ball_zero at hy,
exact p.mem_ball.2 ((map_sub_le_add p _ _).trans_lt $ add_lt_of_lt_sub_right hy),
end | lemma | seminorm.absorbent_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbent",
"absorbent_ball"
] | Seminorm-balls containing the origin are absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absorbent_closed_ball (hpr : p x < r) : absorbent 𝕜 (closed_ball p x r) | begin
refine (p.absorbent_closed_ball_zero $ sub_pos.2 hpr).subset (λ y hy, _),
rw p.mem_closed_ball_zero at hy,
exact p.mem_closed_ball.2 ((map_sub_le_add p _ _).trans $ add_le_of_le_sub_right hy),
end | lemma | seminorm.absorbent_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbent"
] | Seminorm-balls containing the origin are absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symmetric_ball_zero (r : ℝ) (hx : x ∈ ball p 0 r) : -x ∈ ball p 0 r | balanced_ball_zero p r (-1) (by rw [norm_neg, norm_one]) ⟨x, hx, by rw [neg_smul, one_smul]⟩ | lemma | seminorm.symmetric_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"balanced_ball_zero",
"neg_smul",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_ball (p : seminorm 𝕜 E) (r : ℝ) (x : E) :
-ball p x r = ball p (-x) r | by { ext, rw [mem_neg, mem_ball, mem_ball, ←neg_add', sub_neg_eq_add, map_neg_eq_map] } | lemma | seminorm.neg_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball_preimage (p : seminorm 𝕜 E) (y : E) (r : ℝ) (a : 𝕜) (ha : a ≠ 0) :
((•) a) ⁻¹' p.ball y r = p.ball (a⁻¹ • y) (r / ‖a‖) | set.ext $ λ _, by rw [mem_preimage, mem_ball, mem_ball,
lt_div_iff (norm_pos_iff.mpr ha), mul_comm, ←map_smul_eq_mul p, smul_sub, smul_inv_smul₀ ha] | lemma | seminorm.smul_ball_preimage | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"lt_div_iff",
"mul_comm",
"seminorm",
"set.ext",
"smul_inv_smul₀",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
convex_on : convex_on ℝ univ p | begin
refine ⟨convex_univ, λ x _ y _ a b ha hb hab, _⟩,
calc p (a • x + b • y) ≤ p (a • x) + p (b • y) : map_add_le_add p _ _
... = ‖a • (1 : 𝕜)‖ * p x + ‖b • (1 : 𝕜)‖ * p y
: by rw [←map_smul_eq_mul p, ←map_smul_eq_mul p, smul_one_smul, smul_one_smul]
... = a * p x + b * p y
: by rw [norm... | lemma | seminorm.convex_on | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"convex_on",
"mul_one",
"norm_smul",
"real.norm_of_nonneg",
"smul_one_smul"
] | A seminorm is convex. Also see `convex_on_norm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_ball : convex ℝ (ball p x r) | begin
convert (p.convex_on.translate_left (-x)).convex_lt r,
ext y,
rw [preimage_univ, sep_univ, p.mem_ball, sub_eq_add_neg],
refl,
end | lemma | seminorm.convex_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"convex",
"convex_ball"
] | Seminorm-balls are convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_closed_ball : convex ℝ (closed_ball p x r) | begin
rw closed_ball_eq_bInter_ball,
exact convex_Inter₂ (λ _ _, convex_ball _ _ _)
end | lemma | seminorm.convex_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"convex",
"convex_Inter₂",
"convex_ball",
"convex_closed_ball"
] | Closed seminorm-balls are convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
restrict_scalars (p : seminorm 𝕜' E) :
seminorm 𝕜 E | { smul' := λ a x, by rw [← smul_one_smul 𝕜' a x, p.smul', norm_smul, norm_one, mul_one],
..p } | def | seminorm.restrict_scalars | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"mul_one",
"norm_smul",
"restrict_scalars",
"seminorm",
"smul_one_smul"
] | Reinterpret a seminorm over a field `𝕜'` as a seminorm over a smaller field `𝕜`. This will
typically be used with `is_R_or_C 𝕜'` and `𝕜 = ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict_scalars (p : seminorm 𝕜' E) :
(p.restrict_scalars 𝕜 : E → ℝ) = p | rfl | lemma | seminorm.coe_restrict_scalars | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_ball (p : seminorm 𝕜' E) :
(p.restrict_scalars 𝕜).ball = p.ball | rfl | lemma | seminorm.restrict_scalars_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_scalars_closed_ball (p : seminorm 𝕜' E) :
(p.restrict_scalars 𝕜).closed_ball = p.closed_ball | rfl | lemma | seminorm.restrict_scalars_closed_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_zero' [topological_space E] [has_continuous_const_smul 𝕜 E] {p : seminorm 𝕜 E}
{r : ℝ} (hr : 0 < r) (hp : p.closed_ball 0 r ∈ (𝓝 0 : filter E)) :
continuous_at p 0 | begin
refine metric.nhds_basis_closed_ball.tendsto_right_iff.mpr _,
intros ε hε,
rw map_zero,
suffices : p.closed_ball 0 ε ∈ (𝓝 0 : filter E),
{ rwa seminorm.closed_ball_zero_eq_preimage_closed_ball at this },
rcases exists_norm_lt 𝕜 (div_pos hε hr) with ⟨k, hk0, hkε⟩,
have hk0' := norm_pos_iff.mp hk0,
... | lemma | seminorm.continuous_at_zero' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous_at",
"div_mul_cancel",
"div_nonneg",
"div_pos",
"filter",
"filter.mem_of_superset",
"has_continuous_const_smul",
"map_nonneg",
"mul_le_mul",
"seminorm",
"seminorm.closed_ball_zero_eq_preimage_closed_ball",
"set_smul_mem_nhds_zero_iff",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_zero [topological_space E] [has_continuous_const_smul 𝕜 E] {p : seminorm 𝕜 E}
{r : ℝ} (hr : 0 < r) (hp : p.ball 0 r ∈ (𝓝 0 : filter E)) :
continuous_at p 0 | continuous_at_zero' hr (filter.mem_of_superset hp $ p.ball_subset_closed_ball _ _) | lemma | seminorm.continuous_at_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous_at",
"filter",
"filter.mem_of_superset",
"has_continuous_const_smul",
"seminorm",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_of_continuous_at_zero [uniform_space E] [uniform_add_group E]
{p : seminorm 𝕝 E} (hp : continuous_at p 0) :
uniform_continuous p | begin
have hp : filter.tendsto p (𝓝 0) (𝓝 0) := map_zero p ▸ hp,
rw [uniform_continuous, uniformity_eq_comap_nhds_zero_swapped,
metric.uniformity_eq_comap_nhds_zero, filter.tendsto_comap_iff],
exact tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds
(hp.comp filter.tendsto_comap) (λ xy, dist... | lemma | seminorm.uniform_continuous_of_continuous_at_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous_at",
"dist_nonneg",
"filter.tendsto",
"filter.tendsto_comap",
"filter.tendsto_comap_iff",
"metric.uniformity_eq_comap_nhds_zero",
"seminorm",
"tendsto_const_nhds",
"tendsto_of_tendsto_of_tendsto_of_le_of_le",
"uniform_add_group",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_continuous_at_zero [topological_space E] [topological_add_group E]
{p : seminorm 𝕝 E} (hp : continuous_at p 0) :
continuous p | begin
letI := topological_add_group.to_uniform_space E,
haveI : uniform_add_group E := topological_add_comm_group_is_uniform,
exact (seminorm.uniform_continuous_of_continuous_at_zero hp).continuous
end | lemma | seminorm.continuous_of_continuous_at_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous",
"continuous_at",
"seminorm",
"seminorm.uniform_continuous_of_continuous_at_zero",
"topological_add_group",
"topological_space",
"uniform_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous [uniform_space E] [uniform_add_group E]
[has_continuous_const_smul 𝕜 E] {p : seminorm 𝕜 E} {r : ℝ} (hr : 0 < r)
(hp : p.ball 0 r ∈ (𝓝 0 : filter E)) : uniform_continuous p | seminorm.uniform_continuous_of_continuous_at_zero (continuous_at_zero hr hp) | lemma | seminorm.uniform_continuous | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"filter",
"has_continuous_const_smul",
"seminorm",
"seminorm.uniform_continuous_of_continuous_at_zero",
"uniform_add_group",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous' [uniform_space E] [uniform_add_group E]
[has_continuous_const_smul 𝕜 E] {p : seminorm 𝕜 E} {r : ℝ} (hr : 0 < r)
(hp : p.closed_ball 0 r ∈ (𝓝 0 : filter E)) : uniform_continuous p | seminorm.uniform_continuous_of_continuous_at_zero (continuous_at_zero' hr hp) | lemma | seminorm.uniform_continuous' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"filter",
"has_continuous_const_smul",
"seminorm",
"seminorm.uniform_continuous_of_continuous_at_zero",
"uniform_add_group",
"uniform_continuous",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous [topological_space E] [topological_add_group E]
[has_continuous_const_smul 𝕜 E] {p : seminorm 𝕜 E} {r : ℝ} (hr : 0 < r)
(hp : p.ball 0 r ∈ (𝓝 0 : filter E)) : continuous p | seminorm.continuous_of_continuous_at_zero (continuous_at_zero hr hp) | lemma | seminorm.continuous | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous",
"filter",
"has_continuous_const_smul",
"seminorm",
"seminorm.continuous_of_continuous_at_zero",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous' [topological_space E] [topological_add_group E]
[has_continuous_const_smul 𝕜 E] {p : seminorm 𝕜 E} {r : ℝ} (hr : 0 < r)
(hp : p.closed_ball 0 r ∈ (𝓝 0 : filter E)) : continuous p | seminorm.continuous_of_continuous_at_zero (continuous_at_zero' hr hp) | lemma | seminorm.continuous' | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous",
"filter",
"has_continuous_const_smul",
"seminorm",
"seminorm.continuous_of_continuous_at_zero",
"topological_add_group",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_le [topological_space E] [topological_add_group E]
[has_continuous_const_smul 𝕜 E] {p q : seminorm 𝕜 E} (hq : continuous q) (hpq : p ≤ q) :
continuous p | begin
refine seminorm.continuous one_pos (filter.mem_of_superset
(is_open.mem_nhds _ $ q.mem_ball_self zero_lt_one) (ball_antitone hpq)),
rw ball_zero_eq,
exact is_open_lt hq continuous_const
end | lemma | seminorm.continuous_of_le | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"continuous",
"continuous_const",
"filter.mem_of_superset",
"has_continuous_const_smul",
"is_open.mem_nhds",
"is_open_lt",
"seminorm",
"seminorm.continuous",
"topological_add_group",
"topological_space",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_seminorm : seminorm 𝕜 E | { smul' := norm_smul,
..(norm_add_group_seminorm E)} | def | norm_seminorm | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"norm_smul",
"seminorm"
] | The norm of a seminormed group as a seminorm. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_norm_seminorm : ⇑(norm_seminorm 𝕜 E) = norm | rfl | lemma | coe_norm_seminorm | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"norm_seminorm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_norm_seminorm : (norm_seminorm 𝕜 E).ball = metric.ball | by { ext x r y, simp only [seminorm.mem_ball, metric.mem_ball, coe_norm_seminorm, dist_eq_norm] } | lemma | ball_norm_seminorm | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"coe_norm_seminorm",
"metric.ball",
"metric.mem_ball",
"norm_seminorm",
"seminorm.mem_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
absorbent_ball_zero (hr : 0 < r) : absorbent 𝕜 (metric.ball (0 : E) r) | by { rw ←ball_norm_seminorm 𝕜, exact (norm_seminorm _ _).absorbent_ball_zero hr } | lemma | absorbent_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbent",
"metric.ball",
"norm_seminorm"
] | Balls at the origin are absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
absorbent_ball (hx : ‖x‖ < r) : absorbent 𝕜 (metric.ball x r) | by { rw ←ball_norm_seminorm 𝕜, exact (norm_seminorm _ _).absorbent_ball hx } | lemma | absorbent_ball | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"absorbent",
"metric.ball",
"norm_seminorm"
] | Balls containing the origin are absorbent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
balanced_ball_zero : balanced 𝕜 (metric.ball (0 : E) r) | by { rw ←ball_norm_seminorm 𝕜, exact (norm_seminorm _ _).balanced_ball_zero r } | lemma | balanced_ball_zero | analysis | src/analysis/seminorm.lean | [
"data.real.pointwise",
"analysis.convex.function",
"analysis.locally_convex.basic",
"analysis.normed.group.add_torsor"
] | [
"balanced",
"metric.ball",
"norm_seminorm"
] | Balls at the origin are balanced. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subadditive (u : ℕ → ℝ) : Prop | ∀ m n, u (m + n) ≤ u m + u n | def | subadditive | analysis | src/analysis/subadditive.lean | [
"topology.instances.real",
"order.filter.archimedean"
] | [] | A real-valued sequence is subadditive if it satisfies the inequality `u (m + n) ≤ u m + u n`
for all `m, n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim | Inf ((λ (n : ℕ), u n / n) '' (Ici 1)) | def | subadditive.lim | analysis | src/analysis/subadditive.lean | [
"topology.instances.real",
"order.filter.archimedean"
] | [
"lim"
] | The limit of a bounded-below subadditive sequence. The fact that the sequence indeed tends to
this limit is given in `subadditive.tendsto_lim` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lim_le_div (hbdd : bdd_below (range (λ n, u n / n))) {n : ℕ} (hn : n ≠ 0) :
h.lim ≤ u n / n | begin
rw subadditive.lim,
apply cInf_le _ _,
{ rcases hbdd with ⟨c, hc⟩,
exact ⟨c, λ x hx, hc (image_subset_range _ _ hx)⟩ },
{ apply mem_image_of_mem,
exact zero_lt_iff.2 hn }
end | lemma | subadditive.lim_le_div | analysis | src/analysis/subadditive.lean | [
"topology.instances.real",
"order.filter.archimedean"
] | [
"bdd_below",
"cInf_le",
"subadditive.lim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_mul_add_le (k n r) : u (k * n + r) ≤ k * u n + u r | begin
induction k with k IH, { simp only [nat.cast_zero, zero_mul, zero_add] },
calc
u ((k+1) * n + r)
= u (n + (k * n + r)) : by { congr' 1, ring }
... ≤ u n + u (k * n + r) : h _ _
... ≤ u n + (k * u n + u r) : add_le_add_left IH _
... = (k+1 : ℕ) * u n + u r : by simp; ring
end | lemma | subadditive.apply_mul_add_le | analysis | src/analysis/subadditive.lean | [
"topology.instances.real",
"order.filter.archimedean"
] | [
"nat.cast_zero",
"ring",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_div_lt_of_div_lt {L : ℝ} {n : ℕ} (hn : n ≠ 0) (hL : u n / n < L) :
∀ᶠ p in at_top, u p / p < L | begin
have I : ∀ (i : ℕ), 0 < i → (i : ℝ) ≠ 0,
{ assume i hi, simp only [hi.ne', ne.def, nat.cast_eq_zero, not_false_iff] },
obtain ⟨w, nw, wL⟩ : ∃ w, u n / n < w ∧ w < L := exists_between hL,
obtain ⟨x, hx⟩ : ∃ x, ∀ i < n, u i - i * w ≤ x,
{ obtain ⟨x, hx⟩ : bdd_above (↑(finset.image (λ i, u i - i * w) (fins... | lemma | subadditive.eventually_div_lt_of_div_lt | analysis | src/analysis/subadditive.lean | [
"topology.instances.real",
"order.filter.archimedean"
] | [
"and_imp",
"bdd_above",
"div_le_div_of_le_of_nonneg",
"exists_between",
"finset.bdd_above",
"finset.coe_image",
"finset.image",
"finset.mem_coe",
"finset.mem_range",
"finset.range",
"forall_apply_eq_imp_iff₂",
"forall_exists_index",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"nat.cast_add"... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_lim (hbdd : bdd_below (range (λ n, u n / n))) :
tendsto (λ n, u n / n) at_top (𝓝 h.lim) | begin
refine tendsto_order.2 ⟨λ l hl, _, λ L hL, _⟩,
{ refine eventually_at_top.2
⟨1, λ n hn, hl.trans_le (h.lim_le_div hbdd ((zero_lt_one.trans_le hn).ne'))⟩ },
{ obtain ⟨n, npos, hn⟩ : ∃ (n : ℕ), 0 < n ∧ u n / n < L,
{ rw subadditive.lim at hL,
rcases exists_lt_of_cInf_lt (by simp) hL with ⟨x, h... | theorem | subadditive.tendsto_lim | analysis | src/analysis/subadditive.lean | [
"topology.instances.real",
"order.filter.archimedean"
] | [
"bdd_below",
"exists_lt_of_cInf_lt",
"subadditive.lim"
] | Fekete's lemma: a subadditive sequence which is bounded below converges. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.integral_le_sum (hf : antitone_on f (Icc x₀ (x₀ + a))) :
∫ x in x₀..(x₀ + a), f x ≤ ∑ i in finset.range a, f (x₀ + i) | begin
have hint : ∀ (k : ℕ), k < a → interval_integrable f volume (x₀+k) (x₀ + (k + 1 : ℕ)),
{ assume k hk,
refine (hf.mono _).interval_integrable,
rw uIcc_of_le,
{ apply Icc_subset_Icc,
{ simp only [le_add_iff_nonneg_right, nat.cast_nonneg] },
{ simp only [add_le_add_iff_left, nat.cast_le, ... | lemma | antitone_on.integral_le_sum | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"antitone_on",
"finset.range",
"interval_integrable",
"interval_integral.integral_mono_on",
"interval_integral.sum_integral_adjacent_intervals",
"nat.cast_le",
"nat.cast_nonneg",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.integral_le_sum_Ico (hab : a ≤ b) (hf : antitone_on f (set.Icc a b)) :
∫ x in a..b, f x ≤ ∑ x in finset.Ico a b, f x | begin
rw [(nat.sub_add_cancel hab).symm, nat.cast_add],
conv { congr, congr, skip, skip, rw add_comm, skip, skip, congr, congr, rw ←zero_add a, },
rw [← finset.sum_Ico_add, nat.Ico_zero_eq_range],
conv { to_rhs, congr, skip, funext, rw nat.cast_add, },
apply antitone_on.integral_le_sum,
simp only [hf, hab, ... | lemma | antitone_on.integral_le_sum_Ico | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"antitone_on",
"antitone_on.integral_le_sum",
"finset.Ico",
"nat.Ico_zero_eq_range",
"nat.cast_add",
"nat.cast_sub",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.sum_le_integral (hf : antitone_on f (Icc x₀ (x₀ + a))) :
∑ i in finset.range a, f (x₀ + (i + 1 : ℕ)) ≤ ∫ x in x₀..(x₀ + a), f x | begin
have hint : ∀ (k : ℕ), k < a → interval_integrable f volume (x₀+k) (x₀ + (k + 1 : ℕ)),
{ assume k hk,
refine (hf.mono _).interval_integrable,
rw uIcc_of_le,
{ apply Icc_subset_Icc,
{ simp only [le_add_iff_nonneg_right, nat.cast_nonneg] },
{ simp only [add_le_add_iff_left, nat.cast_le, ... | lemma | antitone_on.sum_le_integral | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"antitone_on",
"finset.range",
"interval_integrable",
"interval_integral.integral_mono_on",
"interval_integral.sum_integral_adjacent_intervals",
"nat.cast_le",
"nat.cast_nonneg",
"nat.cast_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antitone_on.sum_le_integral_Ico (hab : a ≤ b) (hf : antitone_on f (set.Icc a b)) :
∑ i in finset.Ico a b, f (i + 1 : ℕ) ≤ ∫ x in a..b, f x | begin
rw [(nat.sub_add_cancel hab).symm, nat.cast_add],
conv { congr, congr, congr, rw ← zero_add a, skip, skip, skip, rw add_comm, },
rw [← finset.sum_Ico_add, nat.Ico_zero_eq_range],
conv { to_lhs, congr, congr, skip, funext, rw [add_assoc, nat.cast_add], },
apply antitone_on.sum_le_integral,
simp only [h... | lemma | antitone_on.sum_le_integral_Ico | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"antitone_on",
"antitone_on.sum_le_integral",
"finset.Ico",
"nat.Ico_zero_eq_range",
"nat.cast_add",
"nat.cast_sub",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.sum_le_integral (hf : monotone_on f (Icc x₀ (x₀ + a))) :
∑ i in finset.range a, f (x₀ + i) ≤ ∫ x in x₀..(x₀ + a), f x | begin
rw [← neg_le_neg_iff, ← finset.sum_neg_distrib, ← interval_integral.integral_neg],
exact hf.neg.integral_le_sum,
end | lemma | monotone_on.sum_le_integral | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"finset.range",
"interval_integral.integral_neg",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.sum_le_integral_Ico (hab : a ≤ b) (hf : monotone_on f (set.Icc a b)) :
∑ x in finset.Ico a b, f x ≤ ∫ x in a..b, f x | begin
rw [← neg_le_neg_iff, ← finset.sum_neg_distrib, ← interval_integral.integral_neg],
exact hf.neg.integral_le_sum_Ico hab,
end | lemma | monotone_on.sum_le_integral_Ico | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"finset.Ico",
"interval_integral.integral_neg",
"monotone_on",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.integral_le_sum (hf : monotone_on f (Icc x₀ (x₀ + a))) :
∫ x in x₀..(x₀ + a), f x ≤ ∑ i in finset.range a, f (x₀ + (i + 1 : ℕ)) | begin
rw [← neg_le_neg_iff, ← finset.sum_neg_distrib, ← interval_integral.integral_neg],
exact hf.neg.sum_le_integral,
end | lemma | monotone_on.integral_le_sum | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"finset.range",
"interval_integral.integral_neg",
"monotone_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_on.integral_le_sum_Ico (hab : a ≤ b) (hf : monotone_on f (set.Icc a b)) :
∫ x in a..b, f x ≤ ∑ i in finset.Ico a b, f (i + 1 : ℕ) | begin
rw [← neg_le_neg_iff, ← finset.sum_neg_distrib, ← interval_integral.integral_neg],
exact hf.neg.sum_le_integral_Ico hab,
end | lemma | monotone_on.integral_le_sum_Ico | analysis | src/analysis/sum_integral_comparisons.lean | [
"measure_theory.integral.interval_integral",
"data.set.function",
"analysis.special_functions.integrals"
] | [
"finset.Ico",
"interval_integral.integral_neg",
"monotone_on",
"set.Icc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum (p : formal_multilinear_series 𝕜 E F) (x : E) : F | ∑' n : ℕ , p n (λ i, x) | def | formal_multilinear_series.sum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series"
] | Given a formal multilinear series `p` and a vector `x`, then `p.sum x` is the sum `Σ pₙ xⁿ`. A
priori, it only behaves well when `‖x‖ < p.radius`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_sum (p : formal_multilinear_series 𝕜 E F) (n : ℕ) (x : E) : F | ∑ k in finset.range n, p k (λ(i : fin k), x) | def | formal_multilinear_series.partial_sum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"finset.range",
"formal_multilinear_series"
] | Given a formal multilinear series `p` and a vector `x`, then `p.partial_sum n x` is the sum
`Σ pₖ xᵏ` for `k ∈ {0,..., n-1}`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_sum_continuous (p : formal_multilinear_series 𝕜 E F) (n : ℕ) :
continuous (p.partial_sum n) | by continuity | lemma | formal_multilinear_series.partial_sum_continuous | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"continuity",
"continuous",
"formal_multilinear_series"
] | The partial sums of a formal multilinear series are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radius (p : formal_multilinear_series 𝕜 E F) : ℝ≥0∞ | ⨆ (r : ℝ≥0) (C : ℝ) (hr : ∀ n, ‖p n‖ * r ^ n ≤ C), (r : ℝ≥0∞) | def | formal_multilinear_series.radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series"
] | The radius of a formal multilinear series is the largest `r` such that the sum `Σ ‖pₙ‖ ‖y‖ⁿ`
converges for all `‖y‖ < r`. This implies that `Σ pₙ yⁿ` converges for all `‖y‖ < r`, but these
definitions are *not* equivalent in general. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_radius_of_bound (C : ℝ) {r : ℝ≥0} (h : ∀ (n : ℕ), ‖p n‖ * r^n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius | le_supr_of_le r $ le_supr_of_le C $ (le_supr (λ _, (r : ℝ≥0∞)) h) | lemma | formal_multilinear_series.le_radius_of_bound | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"le_supr",
"le_supr_of_le"
] | If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_radius_of_bound_nnreal (C : ℝ≥0) {r : ℝ≥0} (h : ∀ (n : ℕ), ‖p n‖₊ * r^n ≤ C) :
(r : ℝ≥0∞) ≤ p.radius | p.le_radius_of_bound C $ λ n, by exact_mod_cast (h n) | lemma | formal_multilinear_series.le_radius_of_bound_nnreal | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [] | If `‖pₙ‖ rⁿ` is bounded in `n`, then the radius of `p` is at least `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_radius_of_is_O (h : (λ n, ‖p n‖ * r^n) =O[at_top] (λ n, (1 : ℝ))) : ↑r ≤ p.radius | exists.elim (is_O_one_nat_at_top_iff.1 h) $ λ C hC, p.le_radius_of_bound C $
λ n, (le_abs_self _).trans (hC n) | lemma | formal_multilinear_series.le_radius_of_is_O | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"le_abs_self"
] | If `‖pₙ‖ rⁿ = O(1)`, as `n → ∞`, then the radius of `p` is at least `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_radius_of_eventually_le (C) (h : ∀ᶠ n in at_top, ‖p n‖ * r ^ n ≤ C) : ↑r ≤ p.radius | p.le_radius_of_is_O $ is_O.of_bound C $ h.mono $ λ n hn, by simpa | lemma | formal_multilinear_series.le_radius_of_eventually_le | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_radius_of_summable_nnnorm (h : summable (λ n, ‖p n‖₊ * r ^ n)) : ↑r ≤ p.radius | p.le_radius_of_bound_nnreal (∑' n, ‖p n‖₊ * r ^ n) $ λ n, le_tsum' h _ | lemma | formal_multilinear_series.le_radius_of_summable_nnnorm | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"le_tsum'",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_radius_of_summable (h : summable (λ n, ‖p n‖ * r ^ n)) : ↑r ≤ p.radius | p.le_radius_of_summable_nnnorm $ by { simp only [← coe_nnnorm] at h, exact_mod_cast h } | lemma | formal_multilinear_series.le_radius_of_summable | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_eq_top_of_forall_nnreal_is_O
(h : ∀ r : ℝ≥0, (λ n, ‖p n‖ * r^n) =O[at_top] (λ n, (1 : ℝ))) : p.radius = ∞ | ennreal.eq_top_of_forall_nnreal_le $ λ r, p.le_radius_of_is_O (h r) | lemma | formal_multilinear_series.radius_eq_top_of_forall_nnreal_is_O | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.eq_top_of_forall_nnreal_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_eq_top_of_eventually_eq_zero (h : ∀ᶠ n in at_top, p n = 0) : p.radius = ∞ | p.radius_eq_top_of_forall_nnreal_is_O $
λ r, (is_O_zero _ _).congr' (h.mono $ λ n hn, by simp [hn]) eventually_eq.rfl | lemma | formal_multilinear_series.radius_eq_top_of_eventually_eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_eq_top_of_forall_image_add_eq_zero (n : ℕ) (hn : ∀ m, p (m + n) = 0) : p.radius = ∞ | p.radius_eq_top_of_eventually_eq_zero $ mem_at_top_sets.2
⟨n, λ k hk, tsub_add_cancel_of_le hk ▸ hn _⟩ | lemma | formal_multilinear_series.radius_eq_top_of_forall_image_add_eq_zero | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"tsub_add_cancel_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_formal_multilinear_series_radius {v : F} :
(const_formal_multilinear_series 𝕜 E v).radius = ⊤ | (const_formal_multilinear_series 𝕜 E v).radius_eq_top_of_forall_image_add_eq_zero 1
(by simp [const_formal_multilinear_series]) | lemma | formal_multilinear_series.const_formal_multilinear_series_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"const_formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_o_of_lt_radius (h : ↑r < p.radius) :
∃ a ∈ Ioo (0 : ℝ) 1, (λ n, ‖p n‖ * r ^ n) =o[at_top] (pow a) | begin
rw (tfae_exists_lt_is_o_pow (λ n, ‖p n‖ * r ^ n) 1).out 1 4,
simp only [radius, lt_supr_iff] at h,
rcases h with ⟨t, C, hC, rt⟩,
rw [ennreal.coe_lt_coe, ← nnreal.coe_lt_coe] at rt,
have : 0 < (t : ℝ), from r.coe_nonneg.trans_lt rt,
rw [← div_lt_one this] at rt,
refine ⟨_, rt, C, or.inr zero_lt_one, ... | lemma | formal_multilinear_series.is_o_of_lt_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"abs_mul",
"div_lt_one",
"div_nonneg",
"ennreal.coe_lt_coe",
"lt_supr_iff",
"mul_le_mul_of_nonneg_right",
"mul_right_comm",
"nnreal.coe_lt_coe",
"pow_nonneg",
"tfae_exists_lt_is_o_pow",
"zero_lt_one"
] | For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1`, `‖p n‖ rⁿ = o(aⁿ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_o_one_of_lt_radius (h : ↑r < p.radius) :
(λ n, ‖p n‖ * r ^ n) =o[at_top] (λ _, 1 : ℕ → ℝ) | let ⟨a, ha, hp⟩ := p.is_o_of_lt_radius h in
hp.trans $ (is_o_pow_pow_of_lt_left ha.1.le ha.2).congr (λ n, rfl) one_pow | lemma | formal_multilinear_series.is_o_one_of_lt_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"is_o_pow_pow_of_lt_left",
"one_pow"
] | For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ = o(1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mul_pow_le_mul_pow_of_lt_radius (h : ↑r < p.radius) :
∃ (a ∈ Ioo (0 : ℝ) 1) (C > 0), ∀ n, ‖p n‖ * r^n ≤ C * a^n | begin
rcases ((tfae_exists_lt_is_o_pow (λ n, ‖p n‖ * r ^ n) 1).out 1 5).mp (p.is_o_of_lt_radius h)
with ⟨a, ha, C, hC, H⟩,
exact ⟨a, ha, C, hC, λ n, (le_abs_self _).trans (H n)⟩
end | lemma | formal_multilinear_series.norm_mul_pow_le_mul_pow_of_lt_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"le_abs_self",
"tfae_exists_lt_is_o_pow"
] | For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` tends to zero exponentially:
for some `0 < a < 1` and `C > 0`, `‖p n‖ * r ^ n ≤ C * a ^ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_radius_of_is_O (h₀ : r ≠ 0) {a : ℝ} (ha : a ∈ Ioo (-1 : ℝ) 1)
(hp : (λ n, ‖p n‖ * r ^ n) =O[at_top] (pow a)) :
↑r < p.radius | begin
rcases ((tfae_exists_lt_is_o_pow (λ n, ‖p n‖ * r ^ n) 1).out 2 5).mp ⟨a, ha, hp⟩
with ⟨a, ha, C, hC, hp⟩,
rw [← pos_iff_ne_zero, ← nnreal.coe_pos] at h₀,
lift a to ℝ≥0 using ha.1.le,
have : (r : ℝ) < r / a :=
by simpa only [div_one] using (div_lt_div_left h₀ zero_lt_one ha.1).2 ha.2,
norm_cast a... | lemma | formal_multilinear_series.lt_radius_of_is_O | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"div_le_iff",
"div_lt_div_left",
"div_one",
"div_pow",
"ennreal.coe_lt_coe",
"le_abs_self",
"lift",
"mul_div_assoc",
"nnreal.coe_div",
"nnreal.coe_pos",
"pow_pos",
"tfae_exists_lt_is_o_pow",
"zero_lt_one"
] | If `r ≠ 0` and `‖pₙ‖ rⁿ = O(aⁿ)` for some `-1 < a < 1`, then `r < p.radius`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_mul_pow_le_of_lt_radius (p : formal_multilinear_series 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ * r^n ≤ C | let ⟨a, ha, C, hC, h⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h
in ⟨C, hC, λ n, (h n).trans $ mul_le_of_le_one_right hC.lt.le (pow_le_one _ ha.1.le ha.2.le)⟩ | lemma | formal_multilinear_series.norm_mul_pow_le_of_lt_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"mul_le_of_le_one_right",
"pow_le_one"
] | For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_le_div_pow_of_pos_of_lt_radius (p : formal_multilinear_series 𝕜 E F) {r : ℝ≥0}
(h0 : 0 < r) (h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖ ≤ C / r ^ n | let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h in
⟨C, hC, λ n, iff.mpr (le_div_iff (pow_pos h0 _)) (hp n)⟩ | lemma | formal_multilinear_series.norm_le_div_pow_of_pos_of_lt_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"le_div_iff",
"pow_pos"
] | For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_mul_pow_le_of_lt_radius (p : formal_multilinear_series 𝕜 E F) {r : ℝ≥0}
(h : (r : ℝ≥0∞) < p.radius) : ∃ C > 0, ∀ n, ‖p n‖₊ * r^n ≤ C | let ⟨C, hC, hp⟩ := p.norm_mul_pow_le_of_lt_radius h
in ⟨⟨C, hC.lt.le⟩, hC, by exact_mod_cast hp⟩ | lemma | formal_multilinear_series.nnnorm_mul_pow_le_of_lt_radius | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series"
] | For `r` strictly smaller than the radius of `p`, then `‖pₙ‖ rⁿ` is bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_radius_of_tendsto (p : formal_multilinear_series 𝕜 E F) {l : ℝ}
(h : tendsto (λ n, ‖p n‖ * r^n) at_top (𝓝 l)) : ↑r ≤ p.radius | p.le_radius_of_is_O (h.is_O_one _) | lemma | formal_multilinear_series.le_radius_of_tendsto | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_radius_of_summable_norm (p : formal_multilinear_series 𝕜 E F)
(hs : summable (λ n, ‖p n‖ * r^n)) : ↑r ≤ p.radius | p.le_radius_of_tendsto hs.tendsto_at_top_zero | lemma | formal_multilinear_series.le_radius_of_summable_norm | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_summable_norm_of_radius_lt_nnnorm (p : formal_multilinear_series 𝕜 E F) {x : E}
(h : p.radius < ‖x‖₊) : ¬ summable (λ n, ‖p n‖ * ‖x‖^n) | λ hs, not_le_of_lt h (p.le_radius_of_summable_norm hs) | lemma | formal_multilinear_series.not_summable_norm_of_radius_lt_nnnorm | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"not_le_of_lt",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_norm_mul_pow (p : formal_multilinear_series 𝕜 E F)
{r : ℝ≥0} (h : ↑r < p.radius) :
summable (λ n : ℕ, ‖p n‖ * r ^ n) | begin
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, hC : 0 < C, hp⟩ := p.norm_mul_pow_le_mul_pow_of_lt_radius h,
exact summable_of_nonneg_of_le (λ n, mul_nonneg (norm_nonneg _) (pow_nonneg r.coe_nonneg _)) hp
((summable_geometric_of_lt_1 ha.1.le ha.2).mul_left _),
end | lemma | formal_multilinear_series.summable_norm_mul_pow | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"pow_nonneg",
"summable",
"summable_geometric_of_lt_1",
"summable_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_norm_apply (p : formal_multilinear_series 𝕜 E F)
{x : E} (hx : x ∈ emetric.ball (0 : E) p.radius) :
summable (λ n : ℕ, ‖p n (λ _, x)‖) | begin
rw mem_emetric_ball_zero_iff at hx,
refine summable_of_nonneg_of_le (λ _, norm_nonneg _) (λ n, ((p n).le_op_norm _).trans_eq _)
(p.summable_norm_mul_pow hx),
simp
end | lemma | formal_multilinear_series.summable_norm_apply | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball",
"formal_multilinear_series",
"summable",
"summable_of_nonneg_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_nnnorm_mul_pow (p : formal_multilinear_series 𝕜 E F)
{r : ℝ≥0} (h : ↑r < p.radius) :
summable (λ n : ℕ, ‖p n‖₊ * r ^ n) | by { rw ← nnreal.summable_coe, push_cast, exact p.summable_norm_mul_pow h } | lemma | formal_multilinear_series.summable_nnnorm_mul_pow | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"nnreal.summable_coe",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable [complete_space F]
(p : formal_multilinear_series 𝕜 E F) {x : E} (hx : x ∈ emetric.ball (0 : E) p.radius) :
summable (λ n : ℕ, p n (λ _, x)) | summable_of_summable_norm (p.summable_norm_apply hx) | lemma | formal_multilinear_series.summable | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"complete_space",
"emetric.ball",
"formal_multilinear_series",
"summable",
"summable_of_summable_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_eq_top_of_summable_norm (p : formal_multilinear_series 𝕜 E F)
(hs : ∀ r : ℝ≥0, summable (λ n, ‖p n‖ * r^n)) : p.radius = ∞ | ennreal.eq_top_of_forall_nnreal_le (λ r, p.le_radius_of_summable_norm (hs r)) | lemma | formal_multilinear_series.radius_eq_top_of_summable_norm | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.eq_top_of_forall_nnreal_le",
"formal_multilinear_series",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_eq_top_iff_summable_norm (p : formal_multilinear_series 𝕜 E F) :
p.radius = ∞ ↔ ∀ r : ℝ≥0, summable (λ n, ‖p n‖ * r^n) | begin
split,
{ intros h r,
obtain ⟨a, ha : a ∈ Ioo (0 : ℝ) 1, C, hC : 0 < C, hp⟩ :=
p.norm_mul_pow_le_mul_pow_of_lt_radius
(show (r:ℝ≥0∞) < p.radius, from h.symm ▸ ennreal.coe_lt_top),
refine (summable_of_norm_bounded (λ n, (C : ℝ) * a ^ n)
((summable_geometric_of_lt_1 ha.1.le ha.2).mul_le... | lemma | formal_multilinear_series.radius_eq_top_iff_summable_norm | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.coe_lt_top",
"formal_multilinear_series",
"pow_nonneg",
"real.norm_of_nonneg",
"summable",
"summable_geometric_of_lt_1",
"summable_of_norm_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_mul_pow_of_radius_pos (p : formal_multilinear_series 𝕜 E F) (h : 0 < p.radius) :
∃ C r (hC : 0 < C) (hr : 0 < r), ∀ n, ‖p n‖ ≤ C * r ^ n | begin
rcases ennreal.lt_iff_exists_nnreal_btwn.1 h with ⟨r, r0, rlt⟩,
have rpos : 0 < (r : ℝ), by simp [ennreal.coe_pos.1 r0],
rcases norm_le_div_pow_of_pos_of_lt_radius p rpos rlt with ⟨C, Cpos, hCp⟩,
refine ⟨C, r ⁻¹, Cpos, by simp [rpos], λ n, _⟩,
convert hCp n,
exact inv_pow _ _,
end | lemma | formal_multilinear_series.le_mul_pow_of_radius_pos | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"inv_pow"
] | If the radius of `p` is positive, then `‖pₙ‖` grows at most geometrically. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
min_radius_le_radius_add (p q : formal_multilinear_series 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius | begin
refine ennreal.le_of_forall_nnreal_lt (λ r hr, _),
rw lt_min_iff at hr,
have := ((p.is_o_one_of_lt_radius hr.1).add (q.is_o_one_of_lt_radius hr.2)).is_O,
refine (p + q).le_radius_of_is_O ((is_O_of_le _ $ λ n, _).trans this),
rw [← add_mul, norm_mul, norm_mul, norm_norm],
exact mul_le_mul_of_nonneg_rig... | lemma | formal_multilinear_series.min_radius_le_radius_add | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.le_of_forall_nnreal_lt",
"formal_multilinear_series",
"le_abs_self",
"lt_min_iff",
"mul_le_mul_of_nonneg_right",
"norm_mul",
"norm_norm"
] | The radius of the sum of two formal series is at least the minimum of their two radii. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
radius_neg (p : formal_multilinear_series 𝕜 E F) : (-p).radius = p.radius | by simp [radius] | lemma | formal_multilinear_series.radius_neg | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum [complete_space F]
(p : formal_multilinear_series 𝕜 E F) {x : E} (hx : x ∈ emetric.ball (0 : E) p.radius) :
has_sum (λ n : ℕ, p n (λ _, x)) (p.sum x) | (p.summable hx).has_sum | lemma | formal_multilinear_series.has_sum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"complete_space",
"emetric.ball",
"formal_multilinear_series",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
radius_le_radius_continuous_linear_map_comp
(p : formal_multilinear_series 𝕜 E F) (f : F →L[𝕜] G) :
p.radius ≤ (f.comp_formal_multilinear_series p).radius | begin
refine ennreal.le_of_forall_nnreal_lt (λ r hr, _),
apply le_radius_of_is_O,
apply (is_O.trans_is_o _ (p.is_o_one_of_lt_radius hr)).is_O,
refine is_O.mul (@is_O_with.is_O _ _ _ _ _ (‖f‖) _ _ _ _) (is_O_refl _ _),
apply is_O_with.of_bound (eventually_of_forall (λ n, _)),
simpa only [norm_norm] using f.n... | lemma | formal_multilinear_series.radius_le_radius_continuous_linear_map_comp | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"ennreal.le_of_forall_nnreal_lt",
"formal_multilinear_series",
"norm_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball
(f : E → F) (p : formal_multilinear_series 𝕜 E F) (x : E) (r : ℝ≥0∞) : Prop | (r_le : r ≤ p.radius)
(r_pos : 0 < r)
(has_sum : ∀ {y}, y ∈ emetric.ball (0 : E) r → has_sum (λn:ℕ, p n (λ(i : fin n), y)) (f (x + y))) | structure | has_fpower_series_on_ball | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball",
"formal_multilinear_series",
"has_sum"
] | Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series on the ball of radius `r > 0` around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `‖y‖ < r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at (f : E → F) (p : formal_multilinear_series 𝕜 E F) (x : E) | ∃ r, has_fpower_series_on_ball f p x r | def | has_fpower_series_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_on_ball"
] | Given a function `f : E → F` and a formal multilinear series `p`, we say that `f` has `p` as
a power series around `x` if `f (x + y) = ∑' pₙ yⁿ` for all `y` in a neighborhood of `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_at (f : E → F) (x : E) | ∃ (p : formal_multilinear_series 𝕜 E F), has_fpower_series_at f p x | def | analytic_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"formal_multilinear_series",
"has_fpower_series_at"
] | Given a function `f : E → F`, we say that `f` is analytic at `x` if it admits a convergent power
series expansion around `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_on (f : E → F) (s : set E) | ∀ x, x ∈ s → analytic_at 𝕜 f x | def | analytic_on | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at"
] | Given a function `f : E → F`, we say that `f` is analytic on a set `s` if it is analytic around
every point of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.has_fpower_series_at (hf : has_fpower_series_on_ball f p x r) :
has_fpower_series_at f p x | ⟨r, hf⟩ | lemma | has_fpower_series_on_ball.has_fpower_series_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.analytic_at (hf : has_fpower_series_at f p x) : analytic_at 𝕜 f x | ⟨p, hf⟩ | lemma | has_fpower_series_at.analytic_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.analytic_at (hf : has_fpower_series_on_ball f p x r) :
analytic_at 𝕜 f x | hf.has_fpower_series_at.analytic_at | lemma | has_fpower_series_on_ball.analytic_at | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"analytic_at",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.congr (hf : has_fpower_series_on_ball f p x r)
(hg : eq_on f g (emetric.ball x r)) :
has_fpower_series_on_ball g p x r | { r_le := hf.r_le,
r_pos := hf.r_pos,
has_sum := λ y hy,
begin
convert hf.has_sum hy,
apply hg.symm,
simpa [edist_eq_coe_nnnorm_sub] using hy,
end } | lemma | has_fpower_series_on_ball.congr | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball",
"has_fpower_series_on_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.comp_sub (hf : has_fpower_series_on_ball f p x r) (y : E) :
has_fpower_series_on_ball (λ z, f (z - y)) p (x + y) r | { r_le := hf.r_le,
r_pos := hf.r_pos,
has_sum := λ z hz, by { convert hf.has_sum hz, abel } } | lemma | has_fpower_series_on_ball.comp_sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball",
"has_sum"
] | If a function `f` has a power series `p` around `x`, then the function `z ↦ f (z - y)` has the
same power series around `x + y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_on_ball.has_sum_sub (hf : has_fpower_series_on_ball f p x r) {y : E}
(hy : y ∈ emetric.ball x r) :
has_sum (λ n : ℕ, p n (λ i, y - x)) (f y) | have y - x ∈ emetric.ball (0 : E) r, by simpa [edist_eq_coe_nnnorm_sub] using hy,
by simpa only [add_sub_cancel'_right] using hf.has_sum this | lemma | has_fpower_series_on_ball.has_sum_sub | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball",
"has_fpower_series_on_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.radius_pos (hf : has_fpower_series_on_ball f p x r) :
0 < p.radius | lt_of_lt_of_le hf.r_pos hf.r_le | lemma | has_fpower_series_on_ball.radius_pos | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.radius_pos (hf : has_fpower_series_at f p x) :
0 < p.radius | let ⟨r, hr⟩ := hf in hr.radius_pos | lemma | has_fpower_series_at.radius_pos | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.mono
(hf : has_fpower_series_on_ball f p x r) (r'_pos : 0 < r') (hr : r' ≤ r) :
has_fpower_series_on_ball f p x r' | ⟨le_trans hr hf.1, r'_pos, λ y hy, hf.has_sum (emetric.ball_subset_ball hr hy)⟩ | lemma | has_fpower_series_on_ball.mono | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball_subset_ball",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.congr (hf : has_fpower_series_at f p x) (hg : f =ᶠ[𝓝 x] g) :
has_fpower_series_at g p x | begin
rcases hf with ⟨r₁, h₁⟩,
rcases emetric.mem_nhds_iff.mp hg with ⟨r₂, h₂pos, h₂⟩,
exact ⟨min r₁ r₂, (h₁.mono (lt_min h₁.r_pos h₂pos) inf_le_left).congr
(λ y hy, h₂ (emetric.ball_subset_ball inf_le_right hy))⟩
end | lemma | has_fpower_series_at.congr | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball_subset_ball",
"has_fpower_series_at",
"inf_le_left",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.eventually (hf : has_fpower_series_at f p x) :
∀ᶠ r : ℝ≥0∞ in 𝓝[>] 0, has_fpower_series_on_ball f p x r | let ⟨r, hr⟩ := hf in
mem_of_superset (Ioo_mem_nhds_within_Ioi (left_mem_Ico.2 hr.r_pos)) $
λ r' hr', hr.mono hr'.1 hr'.2.le | lemma | has_fpower_series_at.eventually | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"Ioo_mem_nhds_within_Ioi",
"has_fpower_series_at",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.eventually_has_sum (hf : has_fpower_series_on_ball f p x r) :
∀ᶠ y in 𝓝 0, has_sum (λn:ℕ, p n (λ(i : fin n), y)) (f (x + y)) | by filter_upwards [emetric.ball_mem_nhds (0 : E) hf.r_pos] using λ _, hf.has_sum | lemma | has_fpower_series_on_ball.eventually_has_sum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"emetric.ball_mem_nhds",
"has_fpower_series_on_ball",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.eventually_has_sum (hf : has_fpower_series_at f p x) :
∀ᶠ y in 𝓝 0, has_sum (λn:ℕ, p n (λ(i : fin n), y)) (f (x + y)) | let ⟨r, hr⟩ := hf in hr.eventually_has_sum | lemma | has_fpower_series_at.eventually_has_sum | analysis.analytic | src/analysis/analytic/basic.lean | [
"analysis.calculus.formal_multilinear_series",
"analysis.specific_limits.normed",
"logic.equiv.fin",
"topology.algebra.infinite_sum.module"
] | [
"has_fpower_series_at",
"has_sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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