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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|---|---|---|---|---|---|---|---|---|---|---|
_root_.is_const_of_deriv_eq_zero (hf : differentiable 𝕜 f) (hf' : ∀ x, deriv f x = 0)
(x y : 𝕜) :
f x = f y | is_const_of_fderiv_eq_zero hf (λ z, by { ext, simp [← deriv_fderiv, hf'] }) _ _ | theorem | is_const_of_deriv_eq_zero | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"deriv_fderiv",
"differentiable",
"is_const_of_fderiv_eq_zero"
] | If `f : 𝕜 → G`, `𝕜 = R` or `𝕜 = ℂ`, is differentiable everywhere and its derivative equal zero,
then it is a constant function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ratio_has_deriv_at_eq_ratio_slope :
∃ c ∈ Ioo a b, (g b - g a) * f' c = (f b - f a) * g' c | begin
let h := λ x, (g b - g a) * f x - (f b - f a) * g x,
have hI : h a = h b,
{ simp only [h], ring },
let h' := λ x, (g b - g a) * f' x - (f b - f a) * g' x,
have hhh' : ∀ x ∈ Ioo a b, has_deriv_at h (h' x) x,
from λ x hx, ((hff' x hx).const_mul (g b - g a)).sub ((hgg' x hx).const_mul (f b - f a)),
h... | lemma | exists_ratio_has_deriv_at_eq_ratio_slope | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"exists_has_deriv_at_eq_zero",
"has_deriv_at",
"ring"
] | Cauchy's **Mean Value Theorem**, `has_deriv_at` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ratio_has_deriv_at_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
(hff' : ∀ x ∈ Ioo a b, has_deriv_at f (f' x) x) (hgg' : ∀ x ∈ Ioo a b, has_deriv_at g (g' x) x)
(hfa : tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : tendsto g (𝓝[>] a) (𝓝 lga))
(hfb : tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : tendsto g (𝓝[<] b) (𝓝 lgb)) :
∃ ... | begin
let h := λ x, (lgb - lga) * f x - (lfb - lfa) * g x,
have hha : tendsto h (𝓝[>] a) (𝓝 $ lgb * lfa - lfb * lga),
{ have : tendsto h (𝓝[>] a)(𝓝 $ (lgb - lga) * lfa - (lfb - lfa) * lga) :=
(tendsto_const_nhds.mul hfa).sub (tendsto_const_nhds.mul hga),
convert this using 2,
ring },
have hhb ... | lemma | exists_ratio_has_deriv_at_eq_ratio_slope' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"exists_has_deriv_at_eq_zero'",
"has_deriv_at",
"ring"
] | Cauchy's **Mean Value Theorem**, extended `has_deriv_at` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_has_deriv_at_eq_slope : ∃ c ∈ Ioo a b, f' c = (f b - f a) / (b - a) | begin
rcases exists_ratio_has_deriv_at_eq_ratio_slope f f' hab hfc hff'
id 1 continuous_id.continuous_on (λ x hx, has_deriv_at_id x) with ⟨c, cmem, hc⟩,
use [c, cmem],
simp only [_root_.id, pi.one_apply, mul_one] at hc,
rw [← hc, mul_div_cancel_left],
exact ne_of_gt (sub_pos.2 hab)
end | lemma | exists_has_deriv_at_eq_slope | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"exists_ratio_has_deriv_at_eq_ratio_slope",
"has_deriv_at_id",
"mul_div_cancel_left",
"mul_one",
"pi.one_apply"
] | Lagrange's Mean Value Theorem, `has_deriv_at` version | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ratio_deriv_eq_ratio_slope :
∃ c ∈ Ioo a b, (g b - g a) * (deriv f c) = (f b - f a) * (deriv g c) | exists_ratio_has_deriv_at_eq_ratio_slope f (deriv f) hab hfc
(λ x hx, ((hfd x hx).differentiable_at $ is_open.mem_nhds is_open_Ioo hx).has_deriv_at)
g (deriv g) hgc $
λ x hx, ((hgd x hx).differentiable_at $ is_open.mem_nhds is_open_Ioo hx).has_deriv_at | lemma | exists_ratio_deriv_eq_ratio_slope | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable_at",
"exists_ratio_has_deriv_at_eq_ratio_slope",
"has_deriv_at",
"is_open.mem_nhds",
"is_open_Ioo"
] | Cauchy's Mean Value Theorem, `deriv` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_ratio_deriv_eq_ratio_slope' {lfa lga lfb lgb : ℝ}
(hdf : differentiable_on ℝ f $ Ioo a b) (hdg : differentiable_on ℝ g $ Ioo a b)
(hfa : tendsto f (𝓝[>] a) (𝓝 lfa)) (hga : tendsto g (𝓝[>] a) (𝓝 lga))
(hfb : tendsto f (𝓝[<] b) (𝓝 lfb)) (hgb : tendsto g (𝓝[<] b) (𝓝 lgb)) :
∃ c ∈ Ioo a b, (lgb - lga... | exists_ratio_has_deriv_at_eq_ratio_slope' _ _ hab _ _
(λ x hx, ((hdf x hx).differentiable_at $ Ioo_mem_nhds hx.1 hx.2).has_deriv_at)
(λ x hx, ((hdg x hx).differentiable_at $ Ioo_mem_nhds hx.1 hx.2).has_deriv_at)
hfa hga hfb hgb | lemma | exists_ratio_deriv_eq_ratio_slope' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"Ioo_mem_nhds",
"deriv",
"differentiable_at",
"differentiable_on",
"exists_ratio_has_deriv_at_eq_ratio_slope'",
"has_deriv_at"
] | Cauchy's Mean Value Theorem, extended `deriv` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_deriv_eq_slope : ∃ c ∈ Ioo a b, deriv f c = (f b - f a) / (b - a) | exists_has_deriv_at_eq_slope f (deriv f) hab hfc
(λ x hx, ((hfd x hx).differentiable_at $ is_open.mem_nhds is_open_Ioo hx).has_deriv_at) | lemma | exists_deriv_eq_slope | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable_at",
"exists_has_deriv_at_eq_slope",
"has_deriv_at",
"is_open.mem_nhds",
"is_open_Ioo"
] | Lagrange's **Mean Value Theorem**, `deriv` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mul_sub_lt_image_sub_of_lt_deriv {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (hf'_gt : ∀ x ∈ interior D, C < deriv f x) :
∀ x y ∈ D, x < y → C * (y - x) < f y - f x | begin
assume x hx y hy hxy,
have hxyD : Icc x y ⊆ D, from hD.ord_connected.out hx hy,
have hxyD' : Ioo x y ⊆ interior D,
from subset_sUnion_of_mem ⟨is_open_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩,
obtain ⟨a, a_mem, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x),
from exists_deriv_eq_slope f ... | theorem | convex.mul_sub_lt_image_sub_of_lt_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"differentiable_on",
"exists_deriv_eq_slope",
"interior",
"lt_div_iff"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `C < f'`, then
`f` grows faster than `C * x` on `D`, i.e., `C * (y - x) < f y - f x` whenever `x, y ∈ D`,
`x < y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_sub_lt_image_sub_of_lt_deriv {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (hf'_gt : ∀ x, C < deriv f x) ⦃x y⦄ (hxy : x < y) :
C * (y - x) < f y - f x | convex_univ.mul_sub_lt_image_sub_of_lt_deriv hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf'_gt x) x trivial y trivial hxy | theorem | mul_sub_lt_image_sub_of_lt_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable"
] | Let `f : ℝ → ℝ` be a differentiable function. If `C < f'`, then `f` grows faster than
`C * x`, i.e., `C * (y - x) < f y - f x` whenever `x < y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.mul_sub_le_image_sub_of_le_deriv {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (hf'_ge : ∀ x ∈ interior D, C ≤ deriv f x) :
∀ x y ∈ D, x ≤ y → C * (y - x) ≤ f y - f x | begin
assume x hx y hy hxy,
cases eq_or_lt_of_le hxy with hxy' hxy', by rw [hxy', sub_self, sub_self, mul_zero],
have hxyD : Icc x y ⊆ D, from hD.ord_connected.out hx hy,
have hxyD' : Ioo x y ⊆ interior D,
from subset_sUnion_of_mem ⟨is_open_Ioo, subset.trans Ioo_subset_Icc_self hxyD⟩,
obtain ⟨a, a_mem, ha... | theorem | convex.mul_sub_le_image_sub_of_le_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"differentiable_on",
"eq_or_lt_of_le",
"exists_deriv_eq_slope",
"interior",
"le_div_iff",
"mul_zero"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `C ≤ f'`, then
`f` grows at least as fast as `C * x` on `D`, i.e., `C * (y - x) ≤ f y - f x` whenever `x, y ∈ D`,
`x ≤ y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_sub_le_image_sub_of_le_deriv {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (hf'_ge : ∀ x, C ≤ deriv f x) ⦃x y⦄ (hxy : x ≤ y) :
C * (y - x) ≤ f y - f x | convex_univ.mul_sub_le_image_sub_of_le_deriv hf.continuous.continuous_on hf.differentiable_on
(λ x _, hf'_ge x) x trivial y trivial hxy | theorem | mul_sub_le_image_sub_of_le_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable"
] | Let `f : ℝ → ℝ` be a differentiable function. If `C ≤ f'`, then `f` grows at least as fast
as `C * x`, i.e., `C * (y - x) ≤ f y - f x` whenever `x ≤ y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.image_sub_lt_mul_sub_of_deriv_lt {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (lt_hf' : ∀ x ∈ interior D, deriv f x < C) :
∀ x y ∈ D, x < y → f y - f x < C * (y - x) | begin
assume x hx y hy hxy,
have hf'_gt : ∀ x ∈ interior D, -C < deriv (λ y, -f y) x,
{ assume x hx,
rw [deriv.neg, neg_lt_neg_iff],
exact lt_hf' x hx },
simpa [-neg_lt_neg_iff]
using neg_lt_neg (hD.mul_sub_lt_image_sub_of_lt_deriv hf.neg hf'.neg hf'_gt x hx y hy hxy)
end | theorem | convex.image_sub_lt_mul_sub_of_deriv_lt | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"deriv.neg",
"differentiable_on",
"interior"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f' < C`, then
`f` grows slower than `C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x, y ∈ D`,
`x < y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_sub_lt_mul_sub_of_deriv_lt {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (lt_hf' : ∀ x, deriv f x < C) ⦃x y⦄ (hxy : x < y) :
f y - f x < C * (y - x) | convex_univ.image_sub_lt_mul_sub_of_deriv_lt hf.continuous.continuous_on hf.differentiable_on
(λ x _, lt_hf' x) x trivial y trivial hxy | theorem | image_sub_lt_mul_sub_of_deriv_lt | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable"
] | Let `f : ℝ → ℝ` be a differentiable function. If `f' < C`, then `f` grows slower than
`C * x` on `D`, i.e., `f y - f x < C * (y - x)` whenever `x < y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.image_sub_le_mul_sub_of_deriv_le {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
{C} (le_hf' : ∀ x ∈ interior D, deriv f x ≤ C) :
∀ x y ∈ D, x ≤ y → f y - f x ≤ C * (y - x) | begin
assume x hx y hy hxy,
have hf'_ge : ∀ x ∈ interior D, -C ≤ deriv (λ y, -f y) x,
{ assume x hx,
rw [deriv.neg, neg_le_neg_iff],
exact le_hf' x hx },
simpa [-neg_le_neg_iff]
using neg_le_neg (hD.mul_sub_le_image_sub_of_le_deriv hf.neg hf'.neg hf'_ge x hx y hy hxy)
end | theorem | convex.image_sub_le_mul_sub_of_deriv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"deriv.neg",
"differentiable_on",
"interior"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f' ≤ C`, then
`f` grows at most as fast as `C * x` on `D`, i.e., `f y - f x ≤ C * (y - x)` whenever `x, y ∈ D`,
`x ≤ y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_sub_le_mul_sub_of_deriv_le {f : ℝ → ℝ} (hf : differentiable ℝ f)
{C} (le_hf' : ∀ x, deriv f x ≤ C) ⦃x y⦄ (hxy : x ≤ y) :
f y - f x ≤ C * (y - x) | convex_univ.image_sub_le_mul_sub_of_deriv_le hf.continuous.continuous_on hf.differentiable_on
(λ x _, le_hf' x) x trivial y trivial hxy | theorem | image_sub_le_mul_sub_of_deriv_le | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable"
] | Let `f : ℝ → ℝ` be a differentiable function. If `f' ≤ C`, then `f` grows at most as fast
as `C * x`, i.e., `f y - f x ≤ C * (y - x)` whenever `x ≤ y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.strict_mono_on_of_deriv_pos {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : ∀ x ∈ interior D, 0 < deriv f x) :
strict_mono_on f D | begin
rintro x hx y hy,
simpa only [zero_mul, sub_pos] using hD.mul_sub_lt_image_sub_of_lt_deriv hf _ hf' x hx y hy,
exact λ z hz, (differentiable_at_of_deriv_ne_zero (hf' z hz).ne').differentiable_within_at,
end | theorem | convex.strict_mono_on_of_deriv_pos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_within_at",
"interior",
"strict_mono_on",
"zero_mul"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is positive, then
`f` is a strictly monotone function on `D`.
Note that we don't require differentiability explicitly as it already implied by the derivative
b... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_of_deriv_pos {f : ℝ → ℝ} (hf' : ∀ x, 0 < deriv f x) : strict_mono f | strict_mono_on_univ.1 $ convex_univ.strict_mono_on_of_deriv_pos
(λ z _, (differentiable_at_of_deriv_ne_zero (hf' z).ne').differentiable_within_at
.continuous_within_at)
(λ x _, hf' x) | theorem | strict_mono_of_deriv_pos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_within_at",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_within_at",
"strict_mono"
] | Let `f : ℝ → ℝ` be a differentiable function. If `f'` is positive, then
`f` is a strictly monotone function.
Note that we don't require differentiability explicitly as it already implied by the derivative
being strictly positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.monotone_on_of_deriv_nonneg {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_nonneg : ∀ x ∈ interior D, 0 ≤ deriv f x) :
monotone_on f D | λ x hx y hy hxy, by simpa only [zero_mul, sub_nonneg]
using hD.mul_sub_le_image_sub_of_le_deriv hf hf' hf'_nonneg x hx y hy hxy | theorem | convex.monotone_on_of_deriv_nonneg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"differentiable_on",
"interior",
"monotone_on",
"zero_mul"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonnegative, then
`f` is a monotone function on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_of_deriv_nonneg {f : ℝ → ℝ} (hf : differentiable ℝ f) (hf' : ∀ x, 0 ≤ deriv f x) :
monotone f | monotone_on_univ.1 $ convex_univ.monotone_on_of_deriv_nonneg hf.continuous.continuous_on
hf.differentiable_on (λ x _, hf' x) | theorem | monotone_of_deriv_nonneg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"deriv",
"differentiable",
"monotone"
] | Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonnegative, then
`f` is a monotone function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.strict_anti_on_of_deriv_neg {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : ∀ x ∈ interior D, deriv f x < 0) :
strict_anti_on f D | λ x hx y, by simpa only [zero_mul, sub_lt_zero]
using hD.image_sub_lt_mul_sub_of_deriv_lt hf
(λ z hz, (differentiable_at_of_deriv_ne_zero (hf' z hz).ne).differentiable_within_at) hf' x hx y | theorem | convex.strict_anti_on_of_deriv_neg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_within_at",
"interior",
"strict_anti_on",
"zero_mul"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is negative, then
`f` is a strictly antitone function on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_of_deriv_neg {f : ℝ → ℝ} (hf' : ∀ x, deriv f x < 0) :
strict_anti f | strict_anti_on_univ.1 $ convex_univ.strict_anti_on_of_deriv_neg
(λ z _, (differentiable_at_of_deriv_ne_zero (hf' z).ne).differentiable_within_at
.continuous_within_at)
(λ x _, hf' x) | theorem | strict_anti_of_deriv_neg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_within_at",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_within_at",
"strict_anti"
] | Let `f : ℝ → ℝ` be a differentiable function. If `f'` is negative, then
`f` is a strictly antitone function.
Note that we don't require differentiability explicitly as it already implied by the derivative
being strictly negative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.antitone_on_of_deriv_nonpos {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0) :
antitone_on f D | λ x hx y hy hxy, by simpa only [zero_mul, sub_nonpos]
using hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy | theorem | convex.antitone_on_of_deriv_nonpos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"antitone_on",
"continuous_on",
"convex",
"deriv",
"differentiable_on",
"interior",
"zero_mul"
] | Let `f` be a function continuous on a convex (or, equivalently, connected) subset `D`
of the real line. If `f` is differentiable on the interior of `D` and `f'` is nonpositive, then
`f` is an antitone function on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_of_deriv_nonpos {f : ℝ → ℝ} (hf : differentiable ℝ f) (hf' : ∀ x, deriv f x ≤ 0) :
antitone f | antitone_on_univ.1 $ convex_univ.antitone_on_of_deriv_nonpos hf.continuous.continuous_on
hf.differentiable_on (λ x _, hf' x) | theorem | antitone_of_deriv_nonpos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"antitone",
"deriv",
"differentiable"
] | Let `f : ℝ → ℝ` be a differentiable function. If `f'` is nonpositive, then
`f` is an antitone function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.convex_on_of_deriv {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'_mono : monotone_on (deriv f) (interior D)) :
convex_on ℝ D f | convex_on_of_slope_mono_adjacent hD
begin
intros x y z hx hz hxy hyz,
-- First we prove some trivial inclusions
have hxzD : Icc x z ⊆ D, from hD.ord_connected.out hx hz,
have hxyD : Icc x y ⊆ D, from subset.trans (Icc_subset_Icc_right $ le_of_lt hyz) hxzD,
have hxyD' : Ioo x y ⊆ interior D,
from subset_sU... | theorem | monotone_on.convex_on_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"convex_on",
"convex_on_of_slope_mono_adjacent",
"deriv",
"differentiable_on",
"exists_deriv_eq_slope",
"interior",
"monotone_on"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
and `f'` is monotone on the interior, then `f` is convex on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone_on.concave_on_of_deriv {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(h_anti : antitone_on (deriv f) (interior D)) :
concave_on ℝ D f | begin
have : monotone_on (deriv (-f)) (interior D),
{ intros x hx y hy hxy,
convert neg_le_neg (h_anti hx hy hxy);
convert deriv.neg },
exact neg_convex_on_iff.mp (this.convex_on_of_deriv hD hf.neg hf'.neg),
end | theorem | antitone_on.concave_on_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"antitone_on",
"concave_on",
"continuous_on",
"convex",
"deriv",
"deriv.neg",
"differentiable_on",
"interior",
"monotone_on"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ`, is differentiable on its interior,
and `f'` is antitone on the interior, then `f` is concave on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono_on.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ}
(hf : continuous_on f (Icc x y)) (hxy : x < y)
(hf'_mono : strict_mono_on (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a | begin
have A : differentiable_on ℝ f (Ioo x y),
from λ w wmem, (differentiable_at_of_deriv_ne_zero (h w wmem)).differentiable_within_at,
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x),
from exists_deriv_eq_slope f hxy hf A,
rcases nonempty_Ioo.2 hay with ⟨b, ⟨hab, hby⟩⟩,
r... | lemma | strict_mono_on.exists_slope_lt_deriv_aux | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_on",
"differentiable_within_at",
"exists_deriv_eq_slope",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.exists_slope_lt_deriv {x y : ℝ} {f : ℝ → ℝ}
(hf : continuous_on f (Icc x y)) (hxy : x < y)
(hf'_mono : strict_mono_on (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a | begin
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0,
{ apply strict_mono_on.exists_slope_lt_deriv_aux hf hxy hf'_mono h },
{ push_neg at h,
rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩,
obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), (f w - f x) / (w - x) < deriv f a,
{ apply strict_mono_on.exists_slope_lt... | lemma | strict_mono_on.exists_slope_lt_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"deriv",
"div_lt_iff",
"le_rfl",
"mul_lt_mul",
"strict_mono_on",
"strict_mono_on.exists_slope_lt_deriv_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.exists_deriv_lt_slope_aux {x y : ℝ} {f : ℝ → ℝ}
(hf : continuous_on f (Icc x y)) (hxy : x < y)
(hf'_mono : strict_mono_on (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | begin
have A : differentiable_on ℝ f (Ioo x y),
from λ w wmem, (differentiable_at_of_deriv_ne_zero (h w wmem)).differentiable_within_at,
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x),
from exists_deriv_eq_slope f hxy hf A,
rcases nonempty_Ioo.2 hxa with ⟨b, ⟨hxb, hba⟩⟩,
r... | lemma | strict_mono_on.exists_deriv_lt_slope_aux | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_on",
"differentiable_within_at",
"exists_deriv_eq_slope",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.exists_deriv_lt_slope {x y : ℝ} {f : ℝ → ℝ}
(hf : continuous_on f (Icc x y)) (hxy : x < y)
(hf'_mono : strict_mono_on (deriv f) (Ioo x y)) :
∃ a ∈ Ioo x y, deriv f a < (f y - f x) / (y - x) | begin
by_cases h : ∀ w ∈ Ioo x y, deriv f w ≠ 0,
{ apply strict_mono_on.exists_deriv_lt_slope_aux hf hxy hf'_mono h },
{ push_neg at h,
rcases h with ⟨w, ⟨hxw, hwy⟩, hw⟩,
obtain ⟨a, ⟨hxa, haw⟩, ha⟩ : ∃ (a : ℝ) (H : a ∈ Ioo x w), deriv f a < (f w - f x) / (w - x),
{ apply strict_mono_on.exists_deriv_lt... | lemma | strict_mono_on.exists_deriv_lt_slope | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"deriv",
"le_rfl",
"lt_div_iff",
"mul_lt_mul",
"strict_mono_on",
"strict_mono_on.exists_deriv_lt_slope_aux"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on.strict_convex_on_of_deriv {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : strict_mono_on (deriv f) (interior D)) :
strict_convex_on ℝ D f | strict_convex_on_of_slope_strict_mono_adjacent hD
begin
intros x y z hx hz hxy hyz,
-- First we prove some trivial inclusions
have hxzD : Icc x z ⊆ D, from hD.ord_connected.out hx hz,
have hxyD : Icc x y ⊆ D, from subset.trans (Icc_subset_Icc_right $ le_of_lt hyz) hxzD,
have hxyD' : Ioo x y ⊆ interior D,
... | lemma | strict_mono_on.strict_convex_on_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"interior",
"strict_convex_on",
"strict_convex_on_of_slope_strict_mono_adjacent",
"strict_mono_on",
"strict_mono_on.exists_deriv_lt_slope",
"strict_mono_on.exists_slope_lt_deriv"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ`, and `f'` is strictly monotone on the
interior, then `f` is strictly convex on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti_on.strict_concave_on_of_deriv {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (h_anti : strict_anti_on (deriv f) (interior D)) :
strict_concave_on ℝ D f | begin
have : strict_mono_on (deriv (-f)) (interior D),
{ intros x hx y hy hxy,
convert neg_lt_neg (h_anti hx hy hxy);
convert deriv.neg },
exact neg_strict_convex_on_iff.mp (this.strict_convex_on_of_deriv hD hf.neg),
end | lemma | strict_anti_on.strict_concave_on_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"deriv.neg",
"interior",
"strict_anti_on",
"strict_concave_on",
"strict_mono_on"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f'` is strictly antitone on the
interior, then `f` is strictly concave on `D`.
Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict antitonicity of `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.convex_on_univ_of_deriv {f : ℝ → ℝ} (hf : differentiable ℝ f)
(hf'_mono : monotone (deriv f)) : convex_on ℝ univ f | (hf'_mono.monotone_on _).convex_on_of_deriv convex_univ hf.continuous.continuous_on
hf.differentiable_on | theorem | monotone.convex_on_univ_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"convex_on",
"convex_univ",
"deriv",
"differentiable",
"monotone"
] | If a function `f` is differentiable and `f'` is monotone on `ℝ` then `f` is convex. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
antitone.concave_on_univ_of_deriv {f : ℝ → ℝ} (hf : differentiable ℝ f)
(hf'_anti : antitone (deriv f)) : concave_on ℝ univ f | (hf'_anti.antitone_on _).concave_on_of_deriv convex_univ hf.continuous.continuous_on
hf.differentiable_on | theorem | antitone.concave_on_univ_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"antitone",
"concave_on",
"convex_univ",
"deriv",
"differentiable"
] | If a function `f` is differentiable and `f'` is antitone on `ℝ` then `f` is concave. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_mono.strict_convex_on_univ_of_deriv {f : ℝ → ℝ} (hf : continuous f)
(hf'_mono : strict_mono (deriv f)) : strict_convex_on ℝ univ f | (hf'_mono.strict_mono_on _).strict_convex_on_of_deriv convex_univ hf.continuous_on | lemma | strict_mono.strict_convex_on_univ_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous",
"convex_univ",
"deriv",
"strict_convex_on",
"strict_mono"
] | If a function `f` is continuous and `f'` is strictly monotone on `ℝ` then `f` is strictly
convex. Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict monotonicity of `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_anti.strict_concave_on_univ_of_deriv {f : ℝ → ℝ} (hf : continuous f)
(hf'_anti : strict_anti (deriv f)) : strict_concave_on ℝ univ f | (hf'_anti.strict_anti_on _).strict_concave_on_of_deriv convex_univ hf.continuous_on | lemma | strict_anti.strict_concave_on_univ_of_deriv | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous",
"convex_univ",
"deriv",
"strict_anti",
"strict_concave_on"
] | If a function `f` is continuous and `f'` is strictly antitone on `ℝ` then `f` is strictly
concave. Note that we don't require differentiability, since it is guaranteed at all but at most
one point by the strict antitonicity of `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_of_deriv2_nonneg {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'' : differentiable_on ℝ (deriv f) (interior D))
(hf''_nonneg : ∀ x ∈ interior D, 0 ≤ (deriv^[2] f x)) :
convex_on ℝ D f | (hD.interior.monotone_on_of_deriv_nonneg hf''.continuous_on (by rwa interior_interior)
$ by rwa interior_interior).convex_on_of_deriv hD hf hf' | theorem | convex_on_of_deriv2_nonneg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"convex_on",
"deriv",
"differentiable_on",
"interior",
"interior_interior"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonnegative on the interior, then `f` is convex on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on_of_deriv2_nonpos {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf' : differentiable_on ℝ f (interior D))
(hf'' : differentiable_on ℝ (deriv f) (interior D))
(hf''_nonpos : ∀ x ∈ interior D, deriv^[2] f x ≤ 0) :
concave_on ℝ D f | (hD.interior.antitone_on_of_deriv_nonpos hf''.continuous_on (by rwa interior_interior)
$ by rwa interior_interior).concave_on_of_deriv hD hf hf' | theorem | concave_on_of_deriv2_nonpos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"concave_on",
"continuous_on",
"convex",
"deriv",
"differentiable_on",
"interior",
"interior_interior"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ`, is twice differentiable on its
interior, and `f''` is nonpositive on the interior, then `f` is concave on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on_of_deriv2_pos {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf'' : ∀ x ∈ interior D, 0 < (deriv^[2] f) x) :
strict_convex_on ℝ D f | (hD.interior.strict_mono_on_of_deriv_pos (λ z hz,
(differentiable_at_of_deriv_ne_zero (hf'' z hz).ne').differentiable_within_at
.continuous_within_at) $ by rwa interior_interior).strict_convex_on_of_deriv hD hf | lemma | strict_convex_on_of_deriv2_pos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"continuous_within_at",
"convex",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_within_at",
"interior",
"interior_interior",
"strict_convex_on"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on the
interior, then `f` is strictly convex on `D`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly positive, except at at most one point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on_of_deriv2_neg {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf : continuous_on f D) (hf'' : ∀ x ∈ interior D, deriv^[2] f x < 0) :
strict_concave_on ℝ D f | (hD.interior.strict_anti_on_of_deriv_neg (λ z hz,
(differentiable_at_of_deriv_ne_zero (hf'' z hz).ne).differentiable_within_at
.continuous_within_at) $ by rwa interior_interior).strict_concave_on_of_deriv hD hf | lemma | strict_concave_on_of_deriv2_neg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"continuous_within_at",
"convex",
"deriv",
"differentiable_at_of_deriv_ne_zero",
"differentiable_within_at",
"interior",
"interior_interior",
"strict_concave_on"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on the
interior, then `f` is strictly concave on `D`.
Note that we don't require twice differentiability explicitly as it already implied by the second
derivative being strictly negative, except at at most one point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_of_deriv2_nonneg' {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf' : differentiable_on ℝ f D) (hf'' : differentiable_on ℝ (deriv f) D)
(hf''_nonneg : ∀ x ∈ D, 0 ≤ (deriv^[2] f) x) : convex_on ℝ D f | convex_on_of_deriv2_nonneg hD hf'.continuous_on (hf'.mono interior_subset)
(hf''.mono interior_subset) (λ x hx, hf''_nonneg x (interior_subset hx)) | theorem | convex_on_of_deriv2_nonneg' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"convex",
"convex_on",
"convex_on_of_deriv2_nonneg",
"deriv",
"differentiable_on",
"interior_subset"
] | If a function `f` is twice differentiable on a open convex set `D ⊆ ℝ` and
`f''` is nonnegative on `D`, then `f` is convex on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on_of_deriv2_nonpos' {D : set ℝ} (hD : convex ℝ D) {f : ℝ → ℝ}
(hf' : differentiable_on ℝ f D) (hf'' : differentiable_on ℝ (deriv f) D)
(hf''_nonpos : ∀ x ∈ D, deriv^[2] f x ≤ 0) : concave_on ℝ D f | concave_on_of_deriv2_nonpos hD hf'.continuous_on (hf'.mono interior_subset)
(hf''.mono interior_subset) (λ x hx, hf''_nonpos x (interior_subset hx)) | theorem | concave_on_of_deriv2_nonpos' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"concave_on",
"concave_on_of_deriv2_nonpos",
"convex",
"deriv",
"differentiable_on",
"interior_subset"
] | If a function `f` is twice differentiable on an open convex set `D ⊆ ℝ` and
`f''` is nonpositive on `D`, then `f` is concave on `D`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on_of_deriv2_pos' {D : set ℝ} (hD : convex ℝ D)
{f : ℝ → ℝ} (hf : continuous_on f D) (hf'' : ∀ x ∈ D, 0 < (deriv^[2] f) x) :
strict_convex_on ℝ D f | strict_convex_on_of_deriv2_pos hD hf $ λ x hx, hf'' x (interior_subset hx) | lemma | strict_convex_on_of_deriv2_pos' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"interior_subset",
"strict_convex_on",
"strict_convex_on_of_deriv2_pos"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly positive on `D`,
then `f` is strictly convex on `D`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly positive, except at at most one point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on_of_deriv2_neg' {D : set ℝ} (hD : convex ℝ D)
{f : ℝ → ℝ} (hf : continuous_on f D) (hf'' : ∀ x ∈ D, deriv^[2] f x < 0) :
strict_concave_on ℝ D f | strict_concave_on_of_deriv2_neg hD hf $ λ x hx, hf'' x (interior_subset hx) | lemma | strict_concave_on_of_deriv2_neg' | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_on",
"convex",
"deriv",
"interior_subset",
"strict_concave_on",
"strict_concave_on_of_deriv2_neg"
] | If a function `f` is continuous on a convex set `D ⊆ ℝ` and `f''` is strictly negative on `D`,
then `f` is strictly concave on `D`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly negative, except at at most one point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex_on_univ_of_deriv2_nonneg {f : ℝ → ℝ} (hf' : differentiable ℝ f)
(hf'' : differentiable ℝ (deriv f)) (hf''_nonneg : ∀ x, 0 ≤ (deriv^[2] f) x) :
convex_on ℝ univ f | convex_on_of_deriv2_nonneg' convex_univ hf'.differentiable_on
hf''.differentiable_on (λ x _, hf''_nonneg x) | theorem | convex_on_univ_of_deriv2_nonneg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"convex_on",
"convex_on_of_deriv2_nonneg'",
"convex_univ",
"deriv",
"differentiable"
] | If a function `f` is twice differentiable on `ℝ`, and `f''` is nonnegative on `ℝ`,
then `f` is convex on `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
concave_on_univ_of_deriv2_nonpos {f : ℝ → ℝ} (hf' : differentiable ℝ f)
(hf'' : differentiable ℝ (deriv f)) (hf''_nonpos : ∀ x, deriv^[2] f x ≤ 0) :
concave_on ℝ univ f | concave_on_of_deriv2_nonpos' convex_univ hf'.differentiable_on
hf''.differentiable_on (λ x _, hf''_nonpos x) | theorem | concave_on_univ_of_deriv2_nonpos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"concave_on",
"concave_on_of_deriv2_nonpos'",
"convex_univ",
"deriv",
"differentiable"
] | If a function `f` is twice differentiable on `ℝ`, and `f''` is nonpositive on `ℝ`,
then `f` is concave on `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_convex_on_univ_of_deriv2_pos {f : ℝ → ℝ} (hf : continuous f)
(hf'' : ∀ x, 0 < (deriv^[2] f) x) :
strict_convex_on ℝ univ f | strict_convex_on_of_deriv2_pos' convex_univ hf.continuous_on $ λ x _, hf'' x | lemma | strict_convex_on_univ_of_deriv2_pos | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous",
"convex_univ",
"deriv",
"strict_convex_on",
"strict_convex_on_of_deriv2_pos'"
] | If a function `f` is continuous on `ℝ`, and `f''` is strictly positive on `ℝ`,
then `f` is strictly convex on `ℝ`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly positive, except at at most one point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
strict_concave_on_univ_of_deriv2_neg {f : ℝ → ℝ} (hf : continuous f)
(hf'' : ∀ x, deriv^[2] f x < 0) :
strict_concave_on ℝ univ f | strict_concave_on_of_deriv2_neg' convex_univ hf.continuous_on $ λ x _, hf'' x | lemma | strict_concave_on_univ_of_deriv2_neg | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous",
"convex_univ",
"deriv",
"strict_concave_on",
"strict_concave_on_of_deriv2_neg'"
] | If a function `f` is continuous on `ℝ`, and `f''` is strictly negative on `ℝ`,
then `f` is strictly concave on `ℝ`.
Note that we don't require twice differentiability explicitly as it is already implied by the second
derivative being strictly negative, except at at most one point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
domain_mvt
{f : E → ℝ} {s : set E} {x y : E} {f' : E → (E →L[ℝ] ℝ)}
(hf : ∀ x ∈ s, has_fderiv_within_at f (f' x) s x) (hs : convex ℝ s) (xs : x ∈ s) (ys : y ∈ s) :
∃ z ∈ segment ℝ x y, f y - f x = f' z (y - x) | begin
have hIccIoo := @Ioo_subset_Icc_self ℝ _ 0 1,
-- parametrize segment
set g : ℝ → E := λ t, x + t • (y - x),
have hseg : ∀ t ∈ Icc (0:ℝ) 1, g t ∈ segment ℝ x y,
{ rw segment_eq_image',
simp only [mem_image, and_imp, add_right_inj],
intros t ht, exact ⟨t, ht, rfl⟩ },
have hseg' : Icc 0 1 ⊆ g ⁻¹' s... | theorem | domain_mvt | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"Icc_mem_nhds",
"and_imp",
"continuous_within_at",
"convex",
"exists_has_deriv_at_eq_slope",
"has_deriv_at",
"has_deriv_at_id",
"has_deriv_within_at",
"has_fderiv_within_at",
"one_smul",
"segment",
"segment_eq_image'"
] | Lagrange's Mean Value Theorem, applied to convex domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at
(hder : ∀ᶠ y in 𝓝 x, has_fderiv_at f (f' y) y) (hcont : continuous_at f' x) :
has_strict_fderiv_at f (f' x) x | begin
-- turn little-o definition of strict_fderiv into an epsilon-delta statement
refine is_o_iff.mpr (λ c hc, metric.eventually_nhds_iff_ball.mpr _),
-- the correct ε is the modulus of continuity of f'
rcases metric.mem_nhds_iff.mp (inter_mem hder (hcont $ ball_mem_nhds _ hc)) with ⟨ε, ε0, hε⟩,
refine ⟨ε, ε0, _... | lemma | has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"ball_prod_same",
"continuous_at",
"convex_ball",
"has_fderiv_at",
"has_fderiv_within_at",
"has_strict_fderiv_at",
"normed_space"
] | Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_of_has_deriv_at_of_continuous_at {f f' : 𝕜 → G} {x : 𝕜}
(hder : ∀ᶠ y in 𝓝 x, has_deriv_at f (f' y) y) (hcont : continuous_at f' x) :
has_strict_deriv_at f (f' x) x | has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at (hder.mono (λ y hy, hy.has_fderiv_at)) $
(smul_rightL 𝕜 𝕜 G 1).continuous.continuous_at.comp hcont | lemma | has_strict_deriv_at_of_has_deriv_at_of_continuous_at | analysis.calculus | src/analysis/calculus/mean_value.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.local_extr",
"analysis.convex.slope",
"analysis.convex.normed",
"data.is_R_or_C.basic",
"topology.instances.real_vector_space"
] | [
"continuous_at",
"has_deriv_at",
"has_strict_deriv_at",
"has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at"
] | Over the reals or the complexes, a continuously differentiable function is strictly
differentiable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_apply_add_mul_sq_div_sub {f : ℝ → ℝ} {x a c d : ℝ} {l : filter ℝ} (hl : l ≤ 𝓝[≠] x)
(hf : tendsto (λ y, (f y - d) / (y - x)) l (𝓝 a))
(h' : tendsto (λ y, y + c * (y-x)^2) l l) :
tendsto (λ y, (f (y + c * (y-x)^2) - d) / (y - x)) l (𝓝 a) | begin
have L : tendsto (λ y, (y + c * (y - x)^2 - x) / (y - x)) l (𝓝 1),
{ have : tendsto (λ y, (1 + c * (y - x))) l (𝓝 (1 + c * (x - x))),
{ apply tendsto.mono_left _ (hl.trans nhds_within_le_nhds),
exact ((tendsto_id.sub_const x).const_mul c).const_add 1 },
simp only [_root_.sub_self, add_zero, mu... | lemma | tendsto_apply_add_mul_sq_div_sub | analysis.calculus | src/analysis/calculus/monotone.lean | [
"analysis.calculus.deriv.slope",
"measure_theory.covering.one_dim",
"order.monotone.extension"
] | [
"filter",
"mul_one",
"mul_zero",
"nhds_within_le_nhds",
"ring",
"self_mem_nhds_within"
] | If `(f y - f x) / (y - x)` converges to a limit as `y` tends to `x`, then the same goes if
`y` is shifted a little bit, i.e., `f (y + (y-x)^2) - f x) / (y - x)` converges to the same limit.
This lemma contains a slightly more general version of this statement (where one considers
convergence along some subfilter, typic... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stieltjes_function.ae_has_deriv_at (f : stieltjes_function) :
∀ᵐ x, has_deriv_at f (rn_deriv f.measure volume x).to_real x | begin
/- Denote by `μ` the Stieltjes measure associated to `f`.
The general theorem `vitali_family.ae_tendsto_rn_deriv` ensures that `μ [x, y] / (y - x)` tends
to the Radon-Nikodym derivative as `y` tends to `x` from the right. As `μ [x, y] = f y - f (x^-)`
and `f (x^-) = f x` almost everywhere, this gives diff... | lemma | stieltjes_function.ae_has_deriv_at | analysis.calculus | src/analysis/calculus/monotone.lean | [
"analysis.calculus.deriv.slope",
"measure_theory.covering.one_dim",
"order.monotone.extension"
] | [
"Ioo_mem_nhds_within_Iio",
"div_le_div_of_nonpos_of_le",
"div_neg",
"div_nonneg",
"ennreal.of_real_div_of_pos",
"ennreal.tendsto_to_real",
"ennreal.to_real_of_real",
"has_deriv_at",
"has_deriv_at_iff_tendsto_slope",
"neg_div'",
"nhds_left'_le_nhds_ne",
"nhds_left'_sup_nhds_right'",
"nhds_wit... | A Stieltjes function is almost everywhere differentiable, with derivative equal to the
Radon-Nikodym derivative of the associated Stieltjes measure with respect to Lebesgue. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.ae_has_deriv_at {f : ℝ → ℝ} (hf : monotone f) :
∀ᵐ x, has_deriv_at f (rn_deriv hf.stieltjes_function.measure volume x).to_real x | begin
/- We already know that the Stieltjes function associated to `f` (i.e., `g : x ↦ f (x^+)`) is
differentiable almost everywhere. We reduce to this statement by sandwiching values of `f` with
values of `g`, by shifting with `(y - x)^2` (which has no influence on the relevant
scale `y - x`.)-/
filter_upwar... | lemma | monotone.ae_has_deriv_at | analysis.calculus | src/analysis/calculus/monotone.lean | [
"analysis.calculus.deriv.slope",
"measure_theory.covering.one_dim",
"order.monotone.extension"
] | [
"Ioo_mem_nhds_within_Iio",
"Ioo_mem_nhds_within_Ioi",
"div_le_div_of_le_of_nonneg",
"div_le_div_of_nonpos_of_le",
"has_deriv_at",
"has_deriv_at_iff_tendsto_slope",
"monotone",
"nhds_left'_le_nhds_ne",
"nhds_left'_sup_nhds_right'",
"nhds_right'_le_nhds_ne",
"nhds_within_le_nhds",
"not_not",
"... | A monotone function is almost everywhere differentiable, with derivative equal to the
Radon-Nikodym derivative of the associated Stieltjes measure with respect to Lebesgue. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone.ae_differentiable_at {f : ℝ → ℝ} (hf : monotone f) :
∀ᵐ x, differentiable_at ℝ f x | by filter_upwards [hf.ae_has_deriv_at] with x hx using hx.differentiable_at | theorem | monotone.ae_differentiable_at | analysis.calculus | src/analysis/calculus/monotone.lean | [
"analysis.calculus.deriv.slope",
"measure_theory.covering.one_dim",
"order.monotone.extension"
] | [
"differentiable_at",
"monotone"
] | A monotone real function is differentiable Lebesgue-almost everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.ae_differentiable_within_at_of_mem
{f : ℝ → ℝ} {s : set ℝ} (hf : monotone_on f s) :
∀ᵐ x, x ∈ s → differentiable_within_at ℝ f s x | begin
/- We use a global monotone extension of `f`, and argue that this extension is differentiable
almost everywhere. Such an extension need not exist (think of `1/x` on `(0, +∞)`), but it exists
if one restricts first the function to a compact interval `[a, b]`. -/
apply ae_of_mem_of_ae_of_mem_inter_Ioo,
as... | theorem | monotone_on.ae_differentiable_within_at_of_mem | analysis.calculus | src/analysis/calculus/monotone.lean | [
"analysis.calculus.deriv.slope",
"measure_theory.covering.one_dim",
"order.monotone.extension"
] | [
"Ioo_mem_nhds",
"ae_of_mem_of_ae_of_mem_inter_Ioo",
"differentiable_within_at",
"monotone",
"monotone_on",
"nhds_within_le_nhds",
"self_mem_nhds_within"
] | A real function which is monotone on a set is differentiable Lebesgue-almost everywhere on
this set. This version does not assume that `s` is measurable. For a formulation with
`volume.restrict s` assuming that `s` is measurable, see `monotone_on.ae_differentiable_within_at`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monotone_on.ae_differentiable_within_at
{f : ℝ → ℝ} {s : set ℝ} (hf : monotone_on f s) (hs : measurable_set s) :
∀ᵐ x ∂(volume.restrict s), differentiable_within_at ℝ f s x | begin
rw ae_restrict_iff' hs,
exact hf.ae_differentiable_within_at_of_mem
end | theorem | monotone_on.ae_differentiable_within_at | analysis.calculus | src/analysis/calculus/monotone.lean | [
"analysis.calculus.deriv.slope",
"measure_theory.covering.one_dim",
"order.monotone.extension"
] | [
"differentiable_within_at",
"measurable_set",
"monotone_on"
] | A real function which is monotone on a set is differentiable Lebesgue-almost everywhere on
this set. This version assumes that `s` is measurable and uses `volume.restrict s`.
For a formulation without measurability assumption,
see `monotone_on.ae_differentiable_within_at_of_mem`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_integral_of_dominated_loc_of_lip' {F : H → α → E} {F' : α → (H →L[𝕜] E)}
{x₀ : H} {bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ x ∈ ball x₀ ε, ae_strongly_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_strongly_measurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖... | begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := λ x, inv_nonneg.mpr (norm_nonneg _) ,
set b : α → ℝ := λ a, |bound a|,
have b_int : integrable b μ := bound_integrable.norm,
have b_nonneg : ∀ a, 0 ≤ b a := λ a, abs_nonneg _,
replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball ... | lemma | has_fderiv_at_integral_of_dominated_loc_of_lip' | analysis.calculus | src/analysis/calculus/parametric_integral.lean | [
"analysis.calculus.mean_value",
"measure_theory.integral.set_integral"
] | [
"abs_nonneg",
"bound",
"continuous_linear_map.integral_apply",
"has_fderiv_at",
"has_fderiv_at_iff_tendsto",
"le_abs_self",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"norm_smul_of_nonneg",
"smul_sub"
] | Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming `F x₀` is
integrable, `‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖` for `x` in a ball around `x₀` for ae `a` with
integrable Lipschitz bound `bound` (with a ball radius independent of `a`), and `F x` is
ae-measurable for `x` in the same ball. See... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_integral_of_dominated_loc_of_lip {F : H → α → E} {F' : α → (H →L[𝕜] E)} {x₀ : H}
{bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_strongly_measurable F' μ)
(h_lip : ∀ᵐ a ∂μ, lipschitz_on_with (real.... | begin
obtain ⟨δ, δ_pos, hδ⟩ : ∃ δ > 0, ∀ x ∈ ball x₀ δ, ae_strongly_measurable (F x) μ ∧ x ∈ ball x₀ ε,
from eventually_nhds_iff_ball.mp (hF_meas.and (ball_mem_nhds x₀ ε_pos)),
choose hδ_meas hδε using hδ,
replace h_lip : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ δ, ‖F x a - F x₀ a‖ ≤ |bound a| * ‖x - x₀‖,
from h_lip.... | lemma | has_fderiv_at_integral_of_dominated_loc_of_lip | analysis.calculus | src/analysis/calculus/parametric_integral.lean | [
"analysis.calculus.mean_value",
"measure_theory.integral.set_integral"
] | [
"bound",
"has_fderiv_at",
"has_fderiv_at_integral_of_dominated_loc_of_lip'",
"lipschitz_on_with",
"real.nnabs"
] | Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
(with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
for `x` in a possibly smaller neighborhood of `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_integral_of_dominated_of_fderiv_le {F : H → α → E} {F' : H → α → (H →L[𝕜] E)}
{x₀ : H} {bound : α → ℝ}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_strongly_measurable (F' x₀) μ)
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball ... | begin
letI : normed_space ℝ H := normed_space.restrict_scalars ℝ 𝕜 H,
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have diff_x₀ : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (F' x₀ a) x₀ :=
h_diff.mono (λ a ha, ha x₀ x₀_in),
have : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs (bound a)) (λ x, F x a) (ball x₀ ε),
... | lemma | has_fderiv_at_integral_of_dominated_of_fderiv_le | analysis.calculus | src/analysis/calculus/parametric_integral.lean | [
"analysis.calculus.mean_value",
"measure_theory.integral.set_integral"
] | [
"bound",
"convex_ball",
"has_fderiv_at",
"has_fderiv_at_integral_of_dominated_loc_of_lip",
"has_fderiv_within_at",
"le_abs_self",
"lipschitz_on_with",
"nnreal.coe_le_coe",
"normed_space",
"normed_space.restrict_scalars",
"real.coe_nnabs",
"real.nnabs"
] | Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
and `F x` is ae-measurable for `x` in a possibly ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_integral_of_dominated_loc_of_lip {F : 𝕜 → α → E} {F' : α → E} {x₀ : 𝕜}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_strongly_measurable F' μ) {bound : α → ℝ}
(h_lipsch : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs $ b... | begin
set L : E →L[𝕜] (𝕜 →L[𝕜] E) := (continuous_linear_map.smul_rightL 𝕜 𝕜 E 1),
replace h_diff : ∀ᵐ a ∂μ, has_fderiv_at (λ x, F x a) (L (F' a)) x₀ :=
h_diff.mono (λ x hx, hx.has_fderiv_at),
have hm : ae_strongly_measurable (L ∘ F') μ := L.continuous.comp_ae_strongly_measurable hF'_meas,
cases has_fde... | lemma | has_deriv_at_integral_of_dominated_loc_of_lip | analysis.calculus | src/analysis/calculus/parametric_integral.lean | [
"analysis.calculus.mean_value",
"measure_theory.integral.set_integral"
] | [
"bound",
"continuous_linear_map.coe_restrict_scalarsL'",
"continuous_linear_map.comp_apply",
"continuous_linear_map.integral_comp_comm",
"continuous_linear_map.norm_restrict_scalars",
"continuous_linear_map.norm_smul_rightL_apply",
"continuous_linear_map.smul_rightL",
"has_deriv_at",
"has_deriv_at_i... | Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
assuming `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
(with ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is
ae-measurable for `x` in a possibly smaller ne... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → α → E} {F' : 𝕜 → α → E} {x₀ : 𝕜}
{ε : ℝ} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) μ)
(hF_int : integrable (F x₀) μ)
(hF'_meas : ae_strongly_measurable (F' x₀) μ)
{bound : α → ℝ}
(h_bound : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ... | begin
have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos,
have diff_x₀ : ∀ᵐ a ∂μ, has_deriv_at (λ x, F x a) (F' x₀ a) x₀ :=
h_diff.mono (λ a ha, ha x₀ x₀_in),
have : ∀ᵐ a ∂μ, lipschitz_on_with (real.nnabs (bound a)) (λ (x : 𝕜), F x a) (ball x₀ ε),
{ apply (h_diff.and h_bound).mono,
rintros a ⟨ha_deriv,... | lemma | has_deriv_at_integral_of_dominated_loc_of_deriv_le | analysis.calculus | src/analysis/calculus/parametric_integral.lean | [
"analysis.calculus.mean_value",
"measure_theory.integral.set_integral"
] | [
"bound",
"convex_ball",
"has_deriv_at",
"has_deriv_at_integral_of_dominated_loc_of_lip",
"has_deriv_within_at",
"le_abs_self",
"lipschitz_on_with",
"nnreal.coe_le_coe",
"real.coe_nnabs",
"real.nnabs"
] | Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : ℝ`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with derivative uniformly bounded by an integrable
function, and `F x` is ae-measurable for `x` in a possibly... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_integral_of_dominated_loc_of_lip {F : H → ℝ → E} {F' : ℝ → (H →L[𝕜] E)} {x₀ : H}
(ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) (μ.restrict (Ι a b)))
(hF_int : interval_integrable (F x₀) μ a b)
(hF'_meas : ae_strongly_measurable F' (μ.restrict (Ι a b)))
(h_lip : ∀ᵐ t ∂μ,... | begin
simp only [interval_integrable_iff, interval_integral_eq_integral_uIoc,
← ae_restrict_iff' measurable_set_uIoc] at *,
have := has_fderiv_at_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas h_lip
bound_integrable h_diff,
exact ⟨this.1, this.2.const_smul _⟩
end | lemma | interval_integral.has_fderiv_at_integral_of_dominated_loc_of_lip | analysis.calculus | src/analysis/calculus/parametric_interval_integral.lean | [
"analysis.calculus.parametric_integral",
"measure_theory.integral.interval_integral"
] | [
"bound",
"has_fderiv_at",
"has_fderiv_at_integral_of_dominated_loc_of_lip",
"interval_integrable",
"interval_integrable_iff",
"lipschitz_on_with",
"measurable_set_uIoc",
"real.nnabs"
] | Differentiation under integral of `x ↦ ∫ t in a..b, F x t` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
(with a ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is ae-measurable
for `x` in a possibly smaller neighborhoo... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_integral_of_dominated_of_fderiv_le {F : H → ℝ → E} {F' : H → ℝ → (H →L[𝕜] E)}
{x₀ : H} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) (μ.restrict (Ι a b)))
(hF_int : interval_integrable (F x₀) μ a b)
(hF'_meas : ae_strongly_measurable (F' x₀) (μ.restrict (Ι a b)))
(h_bou... | begin
simp only [interval_integrable_iff, interval_integral_eq_integral_uIoc,
← ae_restrict_iff' measurable_set_uIoc] at *,
exact (has_fderiv_at_integral_of_dominated_of_fderiv_le ε_pos hF_meas hF_int hF'_meas h_bound
bound_integrable h_diff).const_smul _
end | lemma | interval_integral.has_fderiv_at_integral_of_dominated_of_fderiv_le | analysis.calculus | src/analysis/calculus/parametric_interval_integral.lean | [
"analysis.calculus.parametric_integral",
"measure_theory.integral.interval_integral"
] | [
"bound",
"has_fderiv_at",
"has_fderiv_at_integral_of_dominated_of_fderiv_le",
"interval_integrable",
"interval_integrable_iff",
"measurable_set_uIoc"
] | Differentiation under integral of `x ↦ ∫ F x a` at a given point `x₀`, assuming
`F x₀` is integrable, `x ↦ F x a` is differentiable on a ball around `x₀` for ae `a` with
derivative norm uniformly bounded by an integrable function (the ball radius is independent of `a`),
and `F x` is ae-measurable for `x` in a possibly ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_integral_of_dominated_loc_of_lip {F : 𝕜 → ℝ → E} {F' : ℝ → E} {x₀ : 𝕜}
(ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) (μ.restrict (Ι a b)))
(hF_int : interval_integrable (F x₀) μ a b)
(hF'_meas : ae_strongly_measurable F' (μ.restrict (Ι a b)))
(h_lipsch : ∀ᵐ t ∂μ, t ∈ Ι ... | begin
simp only [interval_integrable_iff, interval_integral_eq_integral_uIoc,
← ae_restrict_iff' measurable_set_uIoc] at *,
have := has_deriv_at_integral_of_dominated_loc_of_lip ε_pos hF_meas hF_int hF'_meas h_lipsch
bound_integrable h_diff,
exact ⟨this.1, this.2.const_smul _⟩
end | lemma | interval_integral.has_deriv_at_integral_of_dominated_loc_of_lip | analysis.calculus | src/analysis/calculus/parametric_interval_integral.lean | [
"analysis.calculus.parametric_integral",
"measure_theory.integral.interval_integral"
] | [
"bound",
"has_deriv_at",
"has_deriv_at_integral_of_dominated_loc_of_lip",
"interval_integrable",
"interval_integrable_iff",
"lipschitz_on_with",
"measurable_set_uIoc",
"real.nnabs"
] | Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
assuming `F x₀` is integrable, `x ↦ F x a` is locally Lipschitz on a ball around `x₀` for ae `a`
(with ball radius independent of `a`) with integrable Lipschitz bound, and `F x` is
ae-measurable for `x` in a possibly smaller ne... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_integral_of_dominated_loc_of_deriv_le {F : 𝕜 → ℝ → E} {F' : 𝕜 → ℝ → E} {x₀ : 𝕜}
(ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, ae_strongly_measurable (F x) (μ.restrict (Ι a b)))
(hF_int : interval_integrable (F x₀) μ a b)
(hF'_meas : ae_strongly_measurable (F' x₀) (μ.restrict (Ι a b)))
(h_bound : ∀... | begin
simp only [interval_integrable_iff, interval_integral_eq_integral_uIoc,
← ae_restrict_iff' measurable_set_uIoc] at *,
have := has_deriv_at_integral_of_dominated_loc_of_deriv_le ε_pos hF_meas hF_int hF'_meas h_bound
bound_integrable h_diff,
exact ⟨this.1, this.2.const_smul _⟩
end | lemma | interval_integral.has_deriv_at_integral_of_dominated_loc_of_deriv_le | analysis.calculus | src/analysis/calculus/parametric_interval_integral.lean | [
"analysis.calculus.parametric_integral",
"measure_theory.integral.interval_integral"
] | [
"bound",
"has_deriv_at",
"has_deriv_at_integral_of_dominated_loc_of_deriv_le",
"interval_integrable",
"interval_integrable_iff",
"measurable_set_uIoc"
] | Derivative under integral of `x ↦ ∫ F x a` at a given point `x₀ : 𝕜`, `𝕜 = ℝ` or `𝕜 = ℂ`,
assuming `F x₀` is integrable, `x ↦ F x a` is differentiable on an interval around `x₀` for ae `a`
(with interval radius independent of `a`) with derivative uniformly bounded by an integrable
function, and `F x` is ae-measurabl... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_tsum {f : α → β → F} (hu : summable u) {s : set β}
(hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
tendsto_uniformly_on (λ (t : finset α), (λ x, ∑ n in t, f n x)) (λ x, ∑' n, f n x) at_top s | begin
refine tendsto_uniformly_on_iff.2 (λ ε εpos, _),
filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_at_top_zero u)).2 _ εpos] with t ht x hx,
have A : summable (λ n, ‖f n x‖),
from summable_of_nonneg_of_le (λ n, norm_nonneg _) (λ n, hfu n x hx) hu,
rw [dist_eq_norm, ← sum_add_tsum_subtype_compl (sum... | lemma | tendsto_uniformly_on_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"finset",
"norm_tsum_le_tsum_norm",
"sum_add_tsum_subtype_compl",
"summable",
"summable_of_nonneg_of_le",
"summable_of_summable_norm",
"tendsto_tsum_compl_at_top_zero",
"tendsto_uniformly_on",
"tsum_le_tsum"
] | An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
Version relative to a set, with general index set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : summable u) {s : set β}
(hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
tendsto_uniformly_on (λ N, (λ x, ∑ n in finset.range N, f n x)) (λ x, ∑' n, f n x) at_top s | λ v hv, tendsto_finset_range.eventually (tendsto_uniformly_on_tsum hu hfu v hv) | lemma | tendsto_uniformly_on_tsum_nat | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"finset.range",
"summable",
"tendsto_uniformly_on",
"tendsto_uniformly_on_tsum"
] | An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
Version relative to a set, with index set `ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_tsum {f : α → β → F} (hu : summable u)
(hfu : ∀ n x, ‖f n x‖ ≤ u n) :
tendsto_uniformly (λ (t : finset α), (λ x, ∑ n in t, f n x)) (λ x, ∑' n, f n x) at_top | by { rw ← tendsto_uniformly_on_univ, exact tendsto_uniformly_on_tsum hu (λ n x hx, hfu n x) } | lemma | tendsto_uniformly_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"finset",
"summable",
"tendsto_uniformly",
"tendsto_uniformly_on_tsum",
"tendsto_uniformly_on_univ"
] | An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
Version with general index set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_tsum_nat {f : ℕ → β → F} {u : ℕ → ℝ} (hu : summable u)
(hfu : ∀ n x, ‖f n x‖ ≤ u n) :
tendsto_uniformly (λ N, (λ x, ∑ n in finset.range N, f n x)) (λ x, ∑' n, f n x) at_top | λ v hv, tendsto_finset_range.eventually (tendsto_uniformly_tsum hu hfu v hv) | lemma | tendsto_uniformly_tsum_nat | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"finset.range",
"summable",
"tendsto_uniformly",
"tendsto_uniformly_tsum"
] | An infinite sum of functions with summable sup norm is the uniform limit of its partial sums.
Version with index set `ℕ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_tsum [topological_space β]
{f : α → β → F} {s : set β} (hf : ∀ i, continuous_on (f i) s) (hu : summable u)
(hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
continuous_on (λ x, ∑' n, f n x) s | begin
classical,
refine (tendsto_uniformly_on_tsum hu hfu).continuous_on (eventually_of_forall _),
assume t,
exact continuous_on_finset_sum _ (λ i hi, hf i),
end | lemma | continuous_on_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"continuous_on",
"summable",
"tendsto_uniformly_on_tsum",
"topological_space"
] | An infinite sum of functions with summable sup norm is continuous on a set if each individual
function is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_tsum [topological_space β]
{f : α → β → F} (hf : ∀ i, continuous (f i)) (hu : summable u)
(hfu : ∀ n x, ‖f n x‖ ≤ u n) :
continuous (λ x, ∑' n, f n x) | begin
simp_rw [continuous_iff_continuous_on_univ] at hf ⊢,
exact continuous_on_tsum hf hu (λ n x hx, hfu n x),
end | lemma | continuous_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"continuous_on_tsum",
"summable",
"topological_space"
] | An infinite sum of functions with summable sup norm is continuous if each individual
function is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_of_summable_has_fderiv_at_of_is_preconnected
(hu : summable u) (hs : is_open s) (h's : is_preconnected s)
(hf : ∀ n x, x ∈ s → has_fderiv_at (f n) (f' n x) x)
(hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n)
(hx₀ : x₀ ∈ s) (hf0 : summable (λ n, f n x₀)) {x : E} (hx : x ∈ s) :
summable (λ n, f n x) | begin
rw summable_iff_cauchy_seq_finset at hf0 ⊢,
have A : uniform_cauchy_seq_on (λ (t : finset α), (λ x, ∑ i in t, f' i x)) at_top s,
from (tendsto_uniformly_on_tsum hu hf').uniform_cauchy_seq_on,
apply cauchy_map_of_uniform_cauchy_seq_on_fderiv hs h's A (λ t y hy, _) hx₀ hx hf0,
exact has_fderiv_at.sum (λ... | lemma | summable_of_summable_has_fderiv_at_of_is_preconnected | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"cauchy_map_of_uniform_cauchy_seq_on_fderiv",
"finset",
"has_fderiv_at",
"has_fderiv_at.sum",
"is_open",
"is_preconnected",
"summable",
"summable_iff_cauchy_seq_finset",
"tendsto_uniformly_on_tsum",
"uniform_cauchy_seq_on"
] | Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
at a point, and all functions in the series are differentiable with a summable bound on the
derivatives, then the series converges everywhere on the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_tsum_of_is_preconnected
(hu : summable u) (hs : is_open s) (h's : is_preconnected s)
(hf : ∀ n x, x ∈ s → has_fderiv_at (f n) (f' n x) x)
(hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n)
(hx₀ : x₀ ∈ s) (hf0 : summable (λ n, f n x₀)) (hx : x ∈ s) :
has_fderiv_at (λ y, ∑' n, f n y) (∑' n, f' n x) x | begin
classical,
have A : ∀ (x : E), x ∈ s → tendsto (λ (t : finset α), ∑ n in t, f n x) at_top (𝓝 (∑' n, f n x)),
{ assume y hy,
apply summable.has_sum,
exact summable_of_summable_has_fderiv_at_of_is_preconnected hu hs h's hf hf' hx₀ hf0 hy },
apply has_fderiv_at_of_tendsto_uniformly_on hs
(tendst... | lemma | has_fderiv_at_tsum_of_is_preconnected | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"finset",
"has_fderiv_at",
"has_fderiv_at.sum",
"has_fderiv_at_of_tendsto_uniformly_on",
"is_open",
"is_preconnected",
"summable",
"summable.has_sum",
"summable_of_summable_has_fderiv_at_of_is_preconnected",
"tendsto_uniformly_on_tsum"
] | Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
at a point, and all functions in the series are differentiable with a summable bound on the
derivatives, then the series is differentiable on the set and its derivative is the sum of the
derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_of_summable_has_fderiv_at
(hu : summable u) (hf : ∀ n x, has_fderiv_at (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
(hf0 : summable (λ n, f n x₀)) (x : E) :
summable (λ n, f n x) | begin
letI : normed_space ℝ E, from normed_space.restrict_scalars ℝ 𝕜 _,
apply summable_of_summable_has_fderiv_at_of_is_preconnected hu is_open_univ
is_connected_univ.is_preconnected (λ n x hx, hf n x)
(λ n x hx, hf' n x) (mem_univ _) hf0 (mem_univ _),
end | lemma | summable_of_summable_has_fderiv_at | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"has_fderiv_at",
"is_open_univ",
"normed_space",
"normed_space.restrict_scalars",
"summable",
"summable_of_summable_has_fderiv_at_of_is_preconnected"
] | Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series converges everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_tsum
(hu : summable u) (hf : ∀ n x, has_fderiv_at (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
(hf0 : summable (λ n, f n x₀)) (x : E) :
has_fderiv_at (λ y, ∑' n, f n y) (∑' n, f' n x) x | begin
letI : normed_space ℝ E, from normed_space.restrict_scalars ℝ 𝕜 _,
exact has_fderiv_at_tsum_of_is_preconnected hu is_open_univ
is_connected_univ.is_preconnected (λ n x hx, hf n x)
(λ n x hx, hf' n x) (mem_univ _) hf0 (mem_univ _),
end | lemma | has_fderiv_at_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"has_fderiv_at",
"has_fderiv_at_tsum_of_is_preconnected",
"is_open_univ",
"normed_space",
"normed_space.restrict_scalars",
"summable"
] | Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series is differentiable and its derivative is the sum of the derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_tsum
(hu : summable u) (hf : ∀ n x, has_fderiv_at (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n) :
differentiable 𝕜 (λ y, ∑' n, f n y) | begin
by_cases h : ∃ x₀, summable (λ n, f n x₀),
{ rcases h with ⟨x₀, hf0⟩,
assume x,
exact (has_fderiv_at_tsum hu hf hf' hf0 x).differentiable_at },
{ push_neg at h,
have : (λ x, ∑' n, f n x) = 0,
{ ext1 x, exact tsum_eq_zero_of_not_summable (h x) },
rw this,
exact differentiable_const 0 ... | lemma | differentiable_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"differentiable",
"differentiable_at",
"differentiable_const",
"has_fderiv_at",
"has_fderiv_at_tsum",
"summable",
"tsum_eq_zero_of_not_summable"
] | Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_tsum_apply
(hu : summable u) (hf : ∀ n, differentiable 𝕜 (f n)) (hf' : ∀ n x, ‖fderiv 𝕜 (f n) x‖ ≤ u n)
(hf0 : summable (λ n, f n x₀)) (x : E) :
fderiv 𝕜 (λ y, ∑' n, f n y) x = ∑' n, fderiv 𝕜 (f n) x | (has_fderiv_at_tsum hu (λ n x, (hf n x).has_fderiv_at) hf' hf0 _).fderiv | lemma | fderiv_tsum_apply | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"differentiable",
"fderiv",
"has_fderiv_at",
"has_fderiv_at_tsum",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_tsum
(hu : summable u) (hf : ∀ n, differentiable 𝕜 (f n)) (hf' : ∀ n x, ‖fderiv 𝕜 (f n) x‖ ≤ u n)
{x₀ : E} (hf0 : summable (λ n, f n x₀)) :
fderiv 𝕜 (λ y, ∑' n, f n y) = (λ x, ∑' n, fderiv 𝕜 (f n) x) | by { ext1 x, exact fderiv_tsum_apply hu hf hf' hf0 x} | lemma | fderiv_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"differentiable",
"fderiv",
"fderiv_tsum_apply",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_tsum
(hf : ∀ i, cont_diff 𝕜 N (f i)) (hv : ∀ (k : ℕ), (k : ℕ∞) ≤ N → summable (v k))
(h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iterated_fderiv 𝕜 k (f i) x‖ ≤ v k i)
{k : ℕ} (hk : (k : ℕ∞) ≤ N) :
iterated_fderiv 𝕜 k (λ y, ∑' n, f n y) = (λ x, ∑' n, iterated_fderiv 𝕜 k (f n) x) | begin
induction k with k IH,
{ ext1 x,
simp_rw [iterated_fderiv_zero_eq_comp],
exact (continuous_multilinear_curry_fin0 𝕜 E F).symm.to_continuous_linear_equiv.map_tsum },
{ have h'k : (k : ℕ∞) < N,
from lt_of_lt_of_le (with_top.coe_lt_coe.2 (nat.lt_succ_self _)) hk,
have A : summable (λ n, iter... | lemma | iterated_fderiv_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"cont_diff",
"continuous_multilinear_curry_fin0",
"continuous_multilinear_curry_left_equiv",
"fderiv_tsum",
"iterated_fderiv",
"iterated_fderiv_succ_eq_comp_left",
"iterated_fderiv_zero_eq_comp",
"linear_isometry_equiv.norm_map",
"summable",
"summable_of_norm_bounded"
] | Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_tsum_apply
(hf : ∀ i, cont_diff 𝕜 N (f i)) (hv : ∀ (k : ℕ), (k : ℕ∞) ≤ N → summable (v k))
(h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iterated_fderiv 𝕜 k (f i) x‖ ≤ v k i)
{k : ℕ} (hk : (k : ℕ∞) ≤ N) (x : E) :
iterated_fderiv 𝕜 k (λ y, ∑' n, f n y) x = ∑' n, iterated_fderiv 𝕜 k (f n) ... | by rw iterated_fderiv_tsum hf hv h'f hk | lemma | iterated_fderiv_tsum_apply | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"cont_diff",
"iterated_fderiv",
"iterated_fderiv_tsum",
"summable"
] | Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_tsum
(hf : ∀ i, cont_diff 𝕜 N (f i)) (hv : ∀ (k : ℕ), (k : ℕ∞) ≤ N → summable (v k))
(h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iterated_fderiv 𝕜 k (f i) x‖ ≤ v k i) :
cont_diff 𝕜 N (λ x, ∑' i, f i x) | begin
rw cont_diff_iff_continuous_differentiable,
split,
{ assume m hm,
rw iterated_fderiv_tsum hf hv h'f hm,
refine continuous_tsum _ (hv m hm) _,
{ assume i,
exact cont_diff.continuous_iterated_fderiv hm (hf i) },
{ assume n x,
exact h'f _ _ _ hm } },
{ assume m hm,
have h'm : ... | lemma | cont_diff_tsum | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"cont_diff",
"cont_diff.continuous_iterated_fderiv",
"cont_diff.differentiable_iterated_fderiv",
"cont_diff_iff_continuous_differentiable",
"continuous_tsum",
"differentiable_at.has_fderiv_at",
"differentiable_tsum",
"enat.add_one_le_of_lt",
"enat.coe_add",
"enat.coe_one",
"fderiv",
"fderiv_it... | Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N`. Then the series is also `C^N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_tsum_of_eventually
(hf : ∀ i, cont_diff 𝕜 N (f i)) (hv : ∀ (k : ℕ), (k : ℕ∞) ≤ N → summable (v k))
(h'f : ∀ (k : ℕ), (k : ℕ∞) ≤ N → ∀ᶠ i in (filter.cofinite : filter α), ∀ (x : E),
‖iterated_fderiv 𝕜 k (f i) x‖ ≤ v k i) :
cont_diff 𝕜 N (λ x, ∑' i, f i x) | begin
classical,
apply cont_diff_iff_forall_nat_le.2 (λ m hm, _),
let t : set α :=
{i : α | ¬∀ (k : ℕ), k ∈ finset.range (m + 1) → ∀ x, ‖iterated_fderiv 𝕜 k (f i) x‖ ≤ v k i},
have ht : set.finite t,
{ have A : ∀ᶠ i in (filter.cofinite : filter α), ∀ (k : ℕ), k ∈ finset.range (m+1) →
∀ (x : E), ‖it... | lemma | cont_diff_tsum_of_eventually | analysis.calculus | src/analysis/calculus/series.lean | [
"analysis.calculus.uniform_limits_deriv",
"analysis.calculus.cont_diff",
"data.nat.cast.with_top"
] | [
"cont_diff",
"cont_diff.sum",
"cont_diff_tsum",
"exists_prop",
"filter",
"filter.cofinite",
"finset",
"finset.mem_range",
"finset.mem_range_succ_iff",
"finset.range",
"norm_iterated_fderiv_zero",
"not_and",
"not_exists",
"not_forall",
"set.finite",
"sum_add_tsum_subtype_compl",
"summ... | Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tangent_cone_at (s : set E) (x : E) : set E | {y : E | ∃(c : ℕ → 𝕜) (d : ℕ → E), (∀ᶠ n in at_top, x + d n ∈ s) ∧
(tendsto (λn, ‖c n‖) at_top at_top) ∧ (tendsto (λn, c n • d n) at_top (𝓝 y))} | def | tangent_cone_at | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [] | The set of all tangent directions to the set `s` at the point `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_within_at (s : set E) (x : E) : Prop | (dense_tangent_cone : dense ((submodule.span 𝕜 (tangent_cone_at 𝕜 s x)) : set E))
(mem_closure : x ∈ closure s) | structure | unique_diff_within_at | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure",
"dense",
"submodule.span",
"tangent_cone_at"
] | A property ensuring that the tangent cone to `s` at `x` spans a dense subset of the whole space.
The main role of this property is to ensure that the differential within `s` at `x` is unique,
hence this name. The uniqueness it asserts is proved in `unique_diff_within_at.eq` in `fderiv.lean`.
To avoid pathologies in dim... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on (s : set E) : Prop | ∀x ∈ s, unique_diff_within_at 𝕜 s x | def | unique_diff_on | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_within_at"
] | A property ensuring that the tangent cone to `s` at any of its points spans a dense subset of
the whole space. The main role of this property is to ensure that the differential along `s` is
unique, hence this name. The uniqueness it asserts is proved in `unique_diff_on.eq` in
`fderiv.lean`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tangent_cone_univ : tangent_cone_at 𝕜 univ x = univ | begin
refine univ_subset_iff.1 (λy hy, _),
rcases exists_one_lt_norm 𝕜 with ⟨w, hw⟩,
refine ⟨λn, w^n, λn, (w^n)⁻¹ • y, univ_mem' (λn, mem_univ _), _, _⟩,
{ simp only [norm_pow],
exact tendsto_pow_at_top_at_top_of_one_lt hw },
{ convert tendsto_const_nhds,
ext n,
have : w ^ n * (w ^ n)⁻¹ = 1,
... | lemma | tangent_cone_univ | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"mul_inv_cancel",
"norm_eq_zero",
"norm_pow",
"one_smul",
"pow_ne_zero",
"smul_smul",
"tangent_cone_at",
"tendsto_const_nhds",
"tendsto_pow_at_top_at_top_of_one_lt",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tangent_cone_mono (h : s ⊆ t) :
tangent_cone_at 𝕜 s x ⊆ tangent_cone_at 𝕜 t x | begin
rintros y ⟨c, d, ds, ctop, clim⟩,
exact ⟨c, d, mem_of_superset ds (λn hn, h hn), ctop, clim⟩
end | lemma | tangent_cone_mono | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"ctop",
"tangent_cone_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tangent_cone_at.lim_zero {α : Type*} (l : filter α) {c : α → 𝕜} {d : α → E}
(hc : tendsto (λn, ‖c n‖) l at_top) (hd : tendsto (λn, c n • d n) l (𝓝 y)) :
tendsto d l (𝓝 0) | begin
have A : tendsto (λn, ‖c n‖⁻¹) l (𝓝 0) := tendsto_inv_at_top_zero.comp hc,
have B : tendsto (λn, ‖c n • d n‖) l (𝓝 ‖y‖) :=
(continuous_norm.tendsto _).comp hd,
have C : tendsto (λn, ‖c n‖⁻¹ * ‖c n • d n‖) l (𝓝 (0 * ‖y‖)) := A.mul B,
rw zero_mul at C,
have : ∀ᶠ n in l, ‖c n‖⁻¹ * ‖c n • d n‖ = ‖d n... | lemma | tangent_cone_at.lim_zero | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"filter",
"inv_mul_cancel",
"mul_assoc",
"norm_eq_zero",
"norm_smul",
"one_mul",
"zero_mul"
] | Auxiliary lemma ensuring that, under the assumptions defining the tangent cone,
the sequence `d` tends to 0 at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tangent_cone_mono_nhds (h : 𝓝[s] x ≤ 𝓝[t] x) :
tangent_cone_at 𝕜 s x ⊆ tangent_cone_at 𝕜 t x | begin
rintros y ⟨c, d, ds, ctop, clim⟩,
refine ⟨c, d, _, ctop, clim⟩,
suffices : tendsto (λ n, x + d n) at_top (𝓝[t] x),
from tendsto_principal.1 (tendsto_inf.1 this).2,
refine (tendsto_inf.2 ⟨_, tendsto_principal.2 ds⟩).mono_right h,
simpa only [add_zero] using tendsto_const_nhds.add (tangent_cone_at.li... | lemma | tangent_cone_mono_nhds | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"ctop",
"tangent_cone_at",
"tangent_cone_at.lim_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tangent_cone_congr (h : 𝓝[s] x = 𝓝[t] x) :
tangent_cone_at 𝕜 s x = tangent_cone_at 𝕜 t x | subset.antisymm
(tangent_cone_mono_nhds $ le_of_eq h)
(tangent_cone_mono_nhds $ le_of_eq h.symm) | lemma | tangent_cone_congr | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"tangent_cone_at",
"tangent_cone_mono_nhds"
] | Tangent cone of `s` at `x` depends only on `𝓝[s] x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tangent_cone_inter_nhds (ht : t ∈ 𝓝 x) :
tangent_cone_at 𝕜 (s ∩ t) x = tangent_cone_at 𝕜 s x | tangent_cone_congr (nhds_within_restrict' _ ht).symm | lemma | tangent_cone_inter_nhds | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"nhds_within_restrict'",
"tangent_cone_at",
"tangent_cone_congr"
] | Intersecting with a neighborhood of the point does not change the tangent cone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_tangent_cone_prod_left {t : set F} {y : F} (ht : y ∈ closure t) :
linear_map.inl 𝕜 E F '' (tangent_cone_at 𝕜 s x) ⊆ tangent_cone_at 𝕜 (s ×ˢ t) (x, y) | begin
rintros _ ⟨v, ⟨c, d, hd, hc, hy⟩, rfl⟩,
have : ∀n, ∃d', y + d' ∈ t ∧ ‖c n • d'‖ < ((1:ℝ)/2)^n,
{ assume n,
rcases mem_closure_iff_nhds.1 ht _ (eventually_nhds_norm_smul_sub_lt (c n) y
(pow_pos one_half_pos n)) with ⟨z, hz, hzt⟩,
exact ⟨z - y, by simpa using hzt, by simpa using hz⟩ },
choose ... | lemma | subset_tangent_cone_prod_left | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure",
"eventually_nhds_norm_smul_sub_lt",
"linear_map.inl",
"one_half_lt_one",
"one_half_pos",
"pow_pos",
"tangent_cone_at",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | The tangent cone of a product contains the tangent cone of its left factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subset_tangent_cone_prod_right {t : set F} {y : F}
(hs : x ∈ closure s) :
linear_map.inr 𝕜 E F '' (tangent_cone_at 𝕜 t y) ⊆ tangent_cone_at 𝕜 (s ×ˢ t) (x, y) | begin
rintros _ ⟨w, ⟨c, d, hd, hc, hy⟩, rfl⟩,
have : ∀n, ∃d', x + d' ∈ s ∧ ‖c n • d'‖ < ((1:ℝ)/2)^n,
{ assume n,
rcases mem_closure_iff_nhds.1 hs _ (eventually_nhds_norm_smul_sub_lt (c n) x
(pow_pos one_half_pos n)) with ⟨z, hz, hzs⟩,
exact ⟨z - x, by simpa using hzs, by simpa using hz⟩ },
choose ... | lemma | subset_tangent_cone_prod_right | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure",
"eventually_nhds_norm_smul_sub_lt",
"linear_map.inr",
"one_half_lt_one",
"one_half_pos",
"pow_pos",
"tangent_cone_at",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | The tangent cone of a product contains the tangent cone of its right factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
maps_to_tangent_cone_pi {ι : Type*} [decidable_eq ι] {E : ι → Type*}
[Π i, normed_add_comm_group (E i)] [Π i, normed_space 𝕜 (E i)]
{s : Π i, set (E i)} {x : Π i, E i} {i : ι} (hi : ∀ j ≠ i, x j ∈ closure (s j)) :
maps_to (linear_map.single i : E i →ₗ[𝕜] Π j, E j) (tangent_cone_at 𝕜 (s i) (x i))
(tangent_c... | begin
rintros w ⟨c, d, hd, hc, hy⟩,
have : ∀ n (j ≠ i), ∃ d', x j + d' ∈ s j ∧ ‖c n • d'‖ < (1 / 2 : ℝ) ^ n,
{ assume n j hj,
rcases mem_closure_iff_nhds.1 (hi j hj) _ (eventually_nhds_norm_smul_sub_lt (c n) (x j)
(pow_pos one_half_pos n)) with ⟨z, hz, hzs⟩,
exact ⟨z - x j, by simpa using hzs, by si... | lemma | maps_to_tangent_cone_pi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure",
"em",
"eventually_nhds_norm_smul_sub_lt",
"linear_map.single",
"normed_add_comm_group",
"normed_space",
"one_half_lt_one",
"one_half_pos",
"pow_pos",
"set.pi",
"tangent_cone_at",
"tendsto_pow_at_top_nhds_0_of_lt_1"
] | The tangent cone of a product contains the tangent cone of each factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_tangent_cone_of_open_segment_subset {s : set G} {x y : G} (h : open_segment ℝ x y ⊆ s) :
y - x ∈ tangent_cone_at ℝ s x | begin
let c := λn:ℕ, (2:ℝ)^(n+1),
let d := λn:ℕ, (c n)⁻¹ • (y-x),
refine ⟨c, d, filter.univ_mem' (λn, h _), _, _⟩,
show x + d n ∈ open_segment ℝ x y,
{ rw open_segment_eq_image,
refine ⟨(c n)⁻¹, ⟨_, _⟩, _⟩,
{ rw inv_pos, apply pow_pos, norm_num },
{ apply inv_lt_one, apply one_lt_pow _ (nat.succ_n... | lemma | mem_tangent_cone_of_open_segment_subset | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"abs_of_nonneg",
"filter.at_top",
"filter.tendsto",
"filter.univ_mem'",
"inv_lt_one",
"inv_pos",
"mul_inv_cancel",
"one_lt_pow",
"one_smul",
"open_segment",
"open_segment_eq_image",
"pow_ne_zero",
"pow_nonneg",
"pow_pos",
"smul_smul",
"smul_sub",
"sub_smul",
"tangent_cone_at",
"t... | If a subset of a real vector space contains an open segment, then the direction of this
segment belongs to the tangent cone at its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_tangent_cone_of_segment_subset {s : set G} {x y : G} (h : segment ℝ x y ⊆ s) :
y - x ∈ tangent_cone_at ℝ s x | mem_tangent_cone_of_open_segment_subset ((open_segment_subset_segment ℝ x y).trans h) | lemma | mem_tangent_cone_of_segment_subset | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"mem_tangent_cone_of_open_segment_subset",
"open_segment_subset_segment",
"segment",
"tangent_cone_at"
] | If a subset of a real vector space contains a segment, then the direction of this
segment belongs to the tangent cone at its endpoints. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on.unique_diff_within_at {s : set E} {x} (hs : unique_diff_on 𝕜 s) (h : x ∈ s) :
unique_diff_within_at 𝕜 s x | hs x h | lemma | unique_diff_on.unique_diff_within_at | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_univ : unique_diff_within_at 𝕜 univ x | by { rw [unique_diff_within_at_iff, tangent_cone_univ], simp } | lemma | unique_diff_within_at_univ | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"tangent_cone_univ",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_univ : unique_diff_on 𝕜 (univ : set E) | λx hx, unique_diff_within_at_univ | lemma | unique_diff_on_univ | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_empty : unique_diff_on 𝕜 (∅ : set E) | λ x hx, hx.elim | lemma | unique_diff_on_empty | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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