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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fderiv_within (f : E → F) (s : set E) (x : E) : E →L[𝕜] F | if h : ∃f', has_fderiv_within_at f f' s x then classical.some h else 0 | def | fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | If `f` has a derivative at `x` within `s`, then `fderiv_within 𝕜 f s x` is such a derivative.
Otherwise, it is set to `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv (f : E → F) (x : E) : E →L[𝕜] F | if h : ∃f', has_fderiv_at f f' x then classical.some h else 0 | def | fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at"
] | If `f` has a derivative at `x`, then `fderiv 𝕜 f x` is such a derivative. Otherwise, it is
set to `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_on (f : E → F) (s : set E) | ∀x ∈ s, differentiable_within_at 𝕜 f s x | def | differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | `differentiable_on 𝕜 f s` means that `f` is differentiable within `s` at any point of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable (f : E → F) | ∀x, differentiable_at 𝕜 f x | def | differentiable | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at"
] | `differentiable 𝕜 f` means that `f` is differentiable at any point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_within_zero_of_not_differentiable_within_at
(h : ¬ differentiable_within_at 𝕜 f s x) : fderiv_within 𝕜 f s x = 0 | have ¬ ∃ f', has_fderiv_within_at f f' s x, from h,
by simp [fderiv_within, this] | lemma | fderiv_within_zero_of_not_differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_zero_of_not_differentiable_at (h : ¬ differentiable_at 𝕜 f x) : fderiv 𝕜 f x = 0 | have ¬ ∃ f', has_fderiv_at f f' x, from h,
by simp [fderiv, this] | lemma | fderiv_zero_of_not_differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"fderiv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.lim (h : has_fderiv_within_at f f' s x) {α : Type*} (l : filter α)
{c : α → 𝕜} {d : α → E} {v : E} (dtop : ∀ᶠ n in l, x + d n ∈ s)
(clim : tendsto (λ n, ‖c n‖) l at_top)
(cdlim : tendsto (λ n, c n • d n) l (𝓝 v)) :
tendsto (λn, c n • (f (x + d n) - f x)) l (𝓝 (f' v)) | begin
have tendsto_arg : tendsto (λ n, x + d n) l (𝓝[s] x),
{ conv in (𝓝[s] x) { rw ← add_zero x },
rw [nhds_within, tendsto_inf],
split,
{ apply tendsto_const_nhds.add (tangent_cone_at.lim_zero l clim cdlim) },
{ rwa tendsto_principal } },
have : (λ y, f y - f x - f' (y - x)) =o[𝓝[s] x] (λ y, ... | theorem | has_fderiv_within_at.lim | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"filter",
"has_fderiv_within_at",
"nhds_within",
"smul_add",
"smul_sub",
"tangent_cone_at.lim_zero"
] | If a function f has a derivative f' at x, a rescaled version of f around x converges to f',
i.e., `n (f (x + (1/n) v) - f x)` converges to `f' v`. More generally, if `c n` tends to infinity
and `c n * d n` tends to `v`, then `c n * (f (x + d n) - f x)` tends to `f' v`. This lemma expresses
this fact, for functions havi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_within_at.unique_on (hf : has_fderiv_within_at f f' s x)
(hg : has_fderiv_within_at f f₁' s x) :
eq_on f' f₁' (tangent_cone_at 𝕜 s x) | λ y ⟨c, d, dtop, clim, cdlim⟩,
tendsto_nhds_unique (hf.lim at_top dtop clim cdlim) (hg.lim at_top dtop clim cdlim) | theorem | has_fderiv_within_at.unique_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"tangent_cone_at",
"tendsto_nhds_unique"
] | If `f'` and `f₁'` are two derivatives of `f` within `s` at `x`, then they are equal on the
tangent cone to `s` at `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_within_at.eq (H : unique_diff_within_at 𝕜 s x)
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at f f₁' s x) : f' = f₁' | continuous_linear_map.ext_on H.1 (hf.unique_on hg) | theorem | unique_diff_within_at.eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_linear_map.ext_on",
"has_fderiv_within_at",
"unique_diff_within_at"
] | `unique_diff_within_at` achieves its goal: it implies the uniqueness of the derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on.eq (H : unique_diff_on 𝕜 s) (hx : x ∈ s)
(h : has_fderiv_within_at f f' s x) (h₁ : has_fderiv_within_at f f₁' s x) : f' = f₁' | (H x hx).eq h h₁ | theorem | unique_diff_on.eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_iff_tendsto :
has_fderiv_at_filter f f' x L ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) | have h : ∀ x', ‖x' - x‖ = 0 → ‖f x' - f x - f' (x' - x)‖ = 0, from λ x' hx',
by { rw [sub_eq_zero.1 (norm_eq_zero.1 hx')], simp },
begin
unfold has_fderiv_at_filter,
rw [←is_o_norm_left, ←is_o_norm_right, is_o_iff_tendsto h],
exact tendsto_congr (λ _, div_eq_inv_mul _ _),
end | theorem | has_fderiv_at_filter_iff_tendsto | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"div_eq_inv_mul",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_iff_tendsto : has_fderiv_within_at f f' s x ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝[s] x) (𝓝 0) | has_fderiv_at_filter_iff_tendsto | theorem | has_fderiv_within_at_iff_tendsto | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter_iff_tendsto",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_iff_tendsto : has_fderiv_at f f' x ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) (𝓝 x) (𝓝 0) | has_fderiv_at_filter_iff_tendsto | theorem | has_fderiv_at_iff_tendsto | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_iff_is_o_nhds_zero : has_fderiv_at f f' x ↔
(λ h : E, f (x + h) - f x - f' h) =o[𝓝 0] (λh, h) | begin
rw [has_fderiv_at, has_fderiv_at_filter, ← map_add_left_nhds_zero x, is_o_map],
simp [(∘)]
end | theorem | has_fderiv_at_iff_is_o_nhds_zero | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.le_of_lip' {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : has_fderiv_at f f' x₀)
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) : ‖f'‖ ≤ C | begin
refine le_of_forall_pos_le_add (λ ε ε0, op_norm_le_of_nhds_zero _ _),
exact add_nonneg hC₀ ε0.le,
rw [← map_add_left_nhds_zero x₀, eventually_map] at hlip,
filter_upwards [is_o_iff.1 (has_fderiv_at_iff_is_o_nhds_zero.1 hf) ε0, hlip] with y hy hyC,
rw add_sub_cancel' at hyC,
calc ‖f' y‖ ≤ ‖f (x₀ + y) -... | lemma | has_fderiv_at.le_of_lip' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at"
] | Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
on a neighborhood of `x₀` then it its derivative at `x₀` has norm bounded by `C`. This version
only assumes that `‖f x - f x₀‖ ≤ C * ‖x - x₀‖` in a neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at.le_of_lip {f : E → F} {f' : E →L[𝕜] F} {x₀ : E} (hf : has_fderiv_at f f' x₀)
{s : set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : lipschitz_on_with C f s) : ‖f'‖ ≤ C | begin
refine hf.le_of_lip' C.coe_nonneg _,
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs),
end | lemma | has_fderiv_at.le_of_lip | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"lipschitz_on_with",
"mem_of_mem_nhds"
] | Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
on a neighborhood of `x₀` then it its derivative at `x₀` has norm bounded by `C`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_filter.mono (h : has_fderiv_at_filter f f' x L₂) (hst : L₁ ≤ L₂) :
has_fderiv_at_filter f f' x L₁ | h.mono hst | theorem | has_fderiv_at_filter.mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.mono_of_mem (h : has_fderiv_within_at f f' t x) (hst : t ∈ 𝓝[s] x) :
has_fderiv_within_at f f' s x | h.mono $ nhds_within_le_iff.mpr hst | theorem | has_fderiv_within_at.mono_of_mem | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.mono (h : has_fderiv_within_at f f' t x) (hst : s ⊆ t) :
has_fderiv_within_at f f' s x | h.mono $ nhds_within_mono _ hst | theorem | has_fderiv_within_at.mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.has_fderiv_at_filter (h : has_fderiv_at f f' x) (hL : L ≤ 𝓝 x) :
has_fderiv_at_filter f f' x L | h.mono hL | theorem | has_fderiv_at.has_fderiv_at_filter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.has_fderiv_within_at
(h : has_fderiv_at f f' x) : has_fderiv_within_at f f' s x | h.has_fderiv_at_filter inf_le_left | theorem | has_fderiv_at.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_within_at",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.differentiable_within_at (h : has_fderiv_within_at f f' s x) :
differentiable_within_at 𝕜 f s x | ⟨f', h⟩ | lemma | has_fderiv_within_at.differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.differentiable_at (h : has_fderiv_at f f' x) : differentiable_at 𝕜 f x | ⟨f', h⟩ | lemma | has_fderiv_at.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_univ :
has_fderiv_within_at f f' univ x ↔ has_fderiv_at f f' x | by { simp only [has_fderiv_within_at, nhds_within_univ], refl } | lemma | has_fderiv_within_at_univ | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_within_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_insert {y : E} :
has_fderiv_within_at f f' (insert y s) x ↔ has_fderiv_within_at f f' s x | begin
rcases eq_or_ne x y with rfl|h,
{ simp_rw [has_fderiv_within_at, has_fderiv_at_filter],
apply asymptotics.is_o_insert,
simp only [sub_self, map_zero] },
refine ⟨λ h, h.mono $ subset_insert y s, λ hf, hf.mono_of_mem _⟩,
simp_rw [nhds_within_insert_of_ne h, self_mem_nhds_within]
end | lemma | has_fderiv_within_at_insert | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"asymptotics.is_o_insert",
"eq_or_ne",
"has_fderiv_at_filter",
"has_fderiv_within_at",
"nhds_within_insert_of_ne",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.insert (h : has_fderiv_within_at f f' s x) :
has_fderiv_within_at f f' (insert x s) x | h.insert' | lemma | has_fderiv_within_at.insert | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_diff_singleton (y : E) :
has_fderiv_within_at f f' (s \ {y}) x ↔ has_fderiv_within_at f f' s x | by rw [← has_fderiv_within_at_insert, insert_diff_singleton, has_fderiv_within_at_insert] | lemma | has_fderiv_within_at_diff_singleton | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"has_fderiv_within_at_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.is_O_sub (hf : has_strict_fderiv_at f f' x) :
(λ p : E × E, f p.1 - f p.2) =O[𝓝 (x, x)] (λ p : E × E, p.1 - p.2) | hf.is_O.congr_of_sub.2 (f'.is_O_comp _ _) | lemma | has_strict_fderiv_at.is_O_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.is_O_sub (h : has_fderiv_at_filter f f' x L) :
(λ x', f x' - f x) =O[L] (λ x', x' - x) | h.is_O.congr_of_sub.2 (f'.is_O_sub _ _) | lemma | has_fderiv_at_filter.is_O_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.has_fderiv_at (hf : has_strict_fderiv_at f f' x) :
has_fderiv_at f f' x | begin
rw [has_fderiv_at, has_fderiv_at_filter, is_o_iff],
exact (λ c hc, tendsto_id.prod_mk_nhds tendsto_const_nhds (is_o_iff.1 hf hc))
end | lemma | has_strict_fderiv_at.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter",
"has_strict_fderiv_at",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.differentiable_at (hf : has_strict_fderiv_at f f' x) :
differentiable_at 𝕜 f x | hf.has_fderiv_at.differentiable_at | lemma | has_strict_fderiv_at.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.exists_lipschitz_on_with_of_nnnorm_lt (hf : has_strict_fderiv_at f f' x)
(K : ℝ≥0) (hK : ‖f'‖₊ < K) : ∃ s ∈ 𝓝 x, lipschitz_on_with K f s | begin
have := hf.add_is_O_with (f'.is_O_with_comp _ _) hK,
simp only [sub_add_cancel, is_O_with] at this,
rcases exists_nhds_square this with ⟨U, Uo, xU, hU⟩,
exact ⟨U, Uo.mem_nhds xU, lipschitz_on_with_iff_norm_sub_le.2 $
λ x hx y hy, hU (mk_mem_prod hx hy)⟩
end | lemma | has_strict_fderiv_at.exists_lipschitz_on_with_of_nnnorm_lt | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"exists_nhds_square",
"has_strict_fderiv_at",
"lipschitz_on_with"
] | If `f` is strictly differentiable at `x` with derivative `f'` and `K > ‖f'‖₊`, then `f` is
`K`-Lipschitz in a neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at.exists_lipschitz_on_with (hf : has_strict_fderiv_at f f' x) :
∃ K (s ∈ 𝓝 x), lipschitz_on_with K f s | (exists_gt _).imp hf.exists_lipschitz_on_with_of_nnnorm_lt | lemma | has_strict_fderiv_at.exists_lipschitz_on_with | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at",
"lipschitz_on_with"
] | If `f` is strictly differentiable at `x` with derivative `f'`, then `f` is Lipschitz in a
neighborhood of `x`. See also `has_strict_fderiv_at.exists_lipschitz_on_with_of_nnnorm_lt` for a
more precise statement. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at.lim (hf : has_fderiv_at f f' x) (v : E) {α : Type*} {c : α → 𝕜}
{l : filter α} (hc : tendsto (λ n, ‖c n‖) l at_top) :
tendsto (λ n, (c n) • (f (x + (c n)⁻¹ • v) - f x)) l (𝓝 (f' v)) | begin
refine (has_fderiv_within_at_univ.2 hf).lim _ univ_mem hc _,
assume U hU,
refine (eventually_ne_of_tendsto_norm_at_top hc (0:𝕜)).mono (λ y hy, _),
convert mem_of_mem_nhds hU,
dsimp only,
rw [← mul_smul, mul_inv_cancel hy, one_smul]
end | lemma | has_fderiv_at.lim | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"filter",
"has_fderiv_at",
"lim",
"mem_of_mem_nhds",
"mul_inv_cancel",
"one_smul"
] | Directional derivative agrees with `has_fderiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at.unique
(h₀ : has_fderiv_at f f₀' x) (h₁ : has_fderiv_at f f₁' x) : f₀' = f₁' | begin
rw ← has_fderiv_within_at_univ at h₀ h₁,
exact unique_diff_within_at_univ.eq h₀ h₁
end | theorem | has_fderiv_at.unique | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_inter' (h : t ∈ 𝓝[s] x) :
has_fderiv_within_at f f' (s ∩ t) x ↔ has_fderiv_within_at f f' s x | by simp [has_fderiv_within_at, nhds_within_restrict'' s h] | lemma | has_fderiv_within_at_inter' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"nhds_within_restrict''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_inter (h : t ∈ 𝓝 x) :
has_fderiv_within_at f f' (s ∩ t) x ↔ has_fderiv_within_at f f' s x | by simp [has_fderiv_within_at, nhds_within_restrict' s h] | lemma | has_fderiv_within_at_inter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"nhds_within_restrict'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.union (hs : has_fderiv_within_at f f' s x)
(ht : has_fderiv_within_at f f' t x) :
has_fderiv_within_at f f' (s ∪ t) x | begin
simp only [has_fderiv_within_at, nhds_within_union],
exact hs.sup ht,
end | lemma | has_fderiv_within_at.union | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"nhds_within_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.nhds_within (h : has_fderiv_within_at f f' s x)
(ht : s ∈ 𝓝[t] x) : has_fderiv_within_at f f' t x | (has_fderiv_within_at_inter' ht).1 (h.mono (inter_subset_right _ _)) | lemma | has_fderiv_within_at.nhds_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"has_fderiv_within_at_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.has_fderiv_at (h : has_fderiv_within_at f f' s x) (hs : s ∈ 𝓝 x) :
has_fderiv_at f f' x | by rwa [← univ_inter s, has_fderiv_within_at_inter hs, has_fderiv_within_at_univ] at h | lemma | has_fderiv_within_at.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_within_at",
"has_fderiv_within_at_inter",
"has_fderiv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.differentiable_at
(h : differentiable_within_at 𝕜 f s x) (hs : s ∈ 𝓝 x) : differentiable_at 𝕜 f x | h.imp (λ f' hf', hf'.has_fderiv_at hs) | lemma | differentiable_within_at.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.has_fderiv_within_at (h : differentiable_within_at 𝕜 f s x) :
has_fderiv_within_at f (fderiv_within 𝕜 f s x) s x | begin
dunfold fderiv_within,
dunfold differentiable_within_at at h,
rw dif_pos h,
exact classical.some_spec h
end | lemma | differentiable_within_at.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.has_fderiv_at (h : differentiable_at 𝕜 f x) :
has_fderiv_at f (fderiv 𝕜 f x) x | begin
dunfold fderiv,
dunfold differentiable_at at h,
rw dif_pos h,
exact classical.some_spec h
end | lemma | differentiable_at.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"fderiv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.has_fderiv_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
has_fderiv_at f (fderiv 𝕜 f x) x | ((h x (mem_of_mem_nhds hs)).differentiable_at hs).has_fderiv_at | lemma | differentiable_on.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"differentiable_on",
"fderiv",
"has_fderiv_at",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.differentiable_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
differentiable_at 𝕜 f x | (h.has_fderiv_at hs).differentiable_at | lemma | differentiable_on.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.eventually_differentiable_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
∀ᶠ y in 𝓝 x, differentiable_at 𝕜 f y | (eventually_eventually_nhds.2 hs).mono $ λ y, h.differentiable_at | lemma | differentiable_on.eventually_differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.fderiv (h : has_fderiv_at f f' x) : fderiv 𝕜 f x = f' | by { ext, rw h.unique h.differentiable_at.has_fderiv_at } | lemma | has_fderiv_at.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_eq {f' : E → E →L[𝕜] F} (h : ∀ x, has_fderiv_at f (f' x) x) : fderiv 𝕜 f = f' | funext $ λ x, (h x).fderiv | lemma | fderiv_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_at.le_of_lip {f : E → F} {x₀ : E} (hf : differentiable_at 𝕜 f x₀)
{s : set E} (hs : s ∈ 𝓝 x₀) {C : ℝ≥0} (hlip : lipschitz_on_with C f s) : ‖fderiv 𝕜 f x₀‖ ≤ C | hf.has_fderiv_at.le_of_lip hs hlip | lemma | fderiv_at.le_of_lip | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"lipschitz_on_with"
] | Converse to the mean value inequality: if `f` is differentiable at `x₀` and `C`-lipschitz
on a neighborhood of `x₀` then it its derivative at `x₀` has norm bounded by `C`.
Version using `fderiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_within_at.fderiv_within
(h : has_fderiv_within_at f f' s x) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 f s x = f' | (hxs.eq h h.differentiable_within_at.has_fderiv_within_at).symm | lemma | has_fderiv_within_at.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"has_fderiv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_of_not_mem_closure (h : x ∉ closure s) :
has_fderiv_within_at f f' s x | begin
simp only [mem_closure_iff_nhds_within_ne_bot, ne_bot_iff, ne.def, not_not] at h,
simp [has_fderiv_within_at, has_fderiv_at_filter, h, is_o, is_O_with],
end | lemma | has_fderiv_within_at_of_not_mem_closure | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"closure",
"has_fderiv_at_filter",
"has_fderiv_within_at",
"mem_closure_iff_nhds_within_ne_bot",
"not_not"
] | If `x` is not in the closure of `s`, then `f` has any derivative at `x` within `s`,
as this statement is empty. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_within_at.mono (h : differentiable_within_at 𝕜 f t x) (st : s ⊆ t) :
differentiable_within_at 𝕜 f s x | begin
rcases h with ⟨f', hf'⟩,
exact ⟨f', hf'.mono st⟩
end | lemma | differentiable_within_at.mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.mono_of_mem (h : differentiable_within_at 𝕜 f s x) {t : set E}
(hst : s ∈ 𝓝[t] x) :
differentiable_within_at 𝕜 f t x | (h.has_fderiv_within_at.mono_of_mem hst).differentiable_within_at | lemma | differentiable_within_at.mono_of_mem | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_univ :
differentiable_within_at 𝕜 f univ x ↔ differentiable_at 𝕜 f x | by simp only [differentiable_within_at, has_fderiv_within_at_univ, differentiable_at] | lemma | differentiable_within_at_univ | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"differentiable_within_at",
"has_fderiv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_inter (ht : t ∈ 𝓝 x) :
differentiable_within_at 𝕜 f (s ∩ t) x ↔ differentiable_within_at 𝕜 f s x | by simp only [differentiable_within_at, has_fderiv_within_at_inter ht] | lemma | differentiable_within_at_inter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"has_fderiv_within_at_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_inter' (ht : t ∈ 𝓝[s] x) :
differentiable_within_at 𝕜 f (s ∩ t) x ↔ differentiable_within_at 𝕜 f s x | by simp only [differentiable_within_at, has_fderiv_within_at_inter' ht] | lemma | differentiable_within_at_inter' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"has_fderiv_within_at_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.differentiable_within_at
(h : differentiable_at 𝕜 f x) : differentiable_within_at 𝕜 f s x | (differentiable_within_at_univ.2 h).mono (subset_univ _) | lemma | differentiable_at.differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.differentiable_at (h : differentiable 𝕜 f) :
differentiable_at 𝕜 f x | h x | lemma | differentiable.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.fderiv_within
(h : differentiable_at 𝕜 f x) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x | h.has_fderiv_at.has_fderiv_within_at.fderiv_within hxs | lemma | differentiable_at.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"fderiv",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.mono (h : differentiable_on 𝕜 f t) (st : s ⊆ t) :
differentiable_on 𝕜 f s | λ x hx, (h x (st hx)).mono st | lemma | differentiable_on.mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_univ :
differentiable_on 𝕜 f univ ↔ differentiable 𝕜 f | by simp only [differentiable_on, differentiable, differentiable_within_at_univ, mem_univ,
forall_true_left] | lemma | differentiable_on_univ | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable",
"differentiable_on",
"differentiable_within_at_univ",
"forall_true_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.differentiable_on (h : differentiable 𝕜 f) : differentiable_on 𝕜 f s | (differentiable_on_univ.2 h).mono (subset_univ _) | lemma | differentiable.differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_of_locally_differentiable_on
(h : ∀x∈s, ∃u, is_open u ∧ x ∈ u ∧ differentiable_on 𝕜 f (s ∩ u)) : differentiable_on 𝕜 f s | begin
assume x xs,
rcases h x xs with ⟨t, t_open, xt, ht⟩,
exact (differentiable_within_at_inter (is_open.mem_nhds t_open xt)).1 (ht x ⟨xs, xt⟩)
end | lemma | differentiable_on_of_locally_differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on",
"differentiable_within_at_inter",
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_of_mem (st : t ∈ 𝓝[s] x) (ht : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f t x) :
fderiv_within 𝕜 f s x = fderiv_within 𝕜 f t x | ((differentiable_within_at.has_fderiv_within_at h).mono_of_mem st).fderiv_within ht | lemma | fderiv_within_of_mem | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"differentiable_within_at.has_fderiv_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_subset (st : s ⊆ t) (ht : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f t x) :
fderiv_within 𝕜 f s x = fderiv_within 𝕜 f t x | fderiv_within_of_mem (nhds_within_mono _ st self_mem_nhds_within) ht h | lemma | fderiv_within_subset | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_of_mem",
"nhds_within_mono",
"self_mem_nhds_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_inter (ht : t ∈ 𝓝 x) :
fderiv_within 𝕜 f (s ∩ t) x = fderiv_within 𝕜 f s x | by simp only [fderiv_within, has_fderiv_within_at_inter ht] | lemma | fderiv_within_inter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"has_fderiv_within_at_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_of_mem_nhds (h : s ∈ 𝓝 x) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x | by simp only [fderiv, fderiv_within, has_fderiv_at, has_fderiv_within_at, nhds_within_eq_nhds.2 h] | lemma | fderiv_within_of_mem_nhds | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"fderiv_within",
"has_fderiv_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_univ : fderiv_within 𝕜 f univ = fderiv 𝕜 f | funext $ λ _, fderiv_within_of_mem_nhds univ_mem | lemma | fderiv_within_univ | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"fderiv_within",
"fderiv_within_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_of_open (hs : is_open s) (hx : x ∈ s) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x | fderiv_within_of_mem_nhds (hs.mem_nhds hx) | lemma | fderiv_within_of_open | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"fderiv_within",
"fderiv_within_of_mem_nhds",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_eq_fderiv (hs : unique_diff_within_at 𝕜 s x) (h : differentiable_at 𝕜 f x) :
fderiv_within 𝕜 f s x = fderiv 𝕜 f x | begin
rw ← fderiv_within_univ,
exact fderiv_within_subset (subset_univ _) hs h.differentiable_within_at
end | lemma | fderiv_within_eq_fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"fderiv",
"fderiv_within",
"fderiv_within_subset",
"fderiv_within_univ",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_mem_iff {f : E → F} {s : set (E →L[𝕜] F)} {x : E} :
fderiv 𝕜 f x ∈ s ↔ (differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ s) ∨
(¬differentiable_at 𝕜 f x ∧ (0 : E →L[𝕜] F) ∈ s) | by by_cases hx : differentiable_at 𝕜 f x; simp [fderiv_zero_of_not_differentiable_at, *] | lemma | fderiv_mem_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"fderiv",
"fderiv_zero_of_not_differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_mem_iff {f : E → F} {t : set E} {s : set (E →L[𝕜] F)} {x : E} :
fderiv_within 𝕜 f t x ∈ s ↔ (differentiable_within_at 𝕜 f t x ∧ fderiv_within 𝕜 f t x ∈ s) ∨
(¬differentiable_within_at 𝕜 f t x ∧ (0 : E →L[𝕜] F) ∈ s) | by by_cases hx : differentiable_within_at 𝕜 f t x;
simp [fderiv_within_zero_of_not_differentiable_within_at, *] | lemma | fderiv_within_mem_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
asymptotics.is_O.has_fderiv_within_at {s : set E} {x₀ : E} {n : ℕ}
(h : f =O[𝓝[s] x₀] λ x, ‖x - x₀‖^n) (hx₀ : x₀ ∈ s) (hn : 1 < n) :
has_fderiv_within_at f (0 : E →L[𝕜] F) s x₀ | by simp_rw [has_fderiv_within_at, has_fderiv_at_filter,
h.eq_zero_of_norm_pow_within hx₀ $ zero_lt_one.trans hn, zero_apply, sub_zero,
h.trans_is_o ((is_o_pow_sub_sub x₀ hn).mono nhds_within_le_nhds)] | lemma | asymptotics.is_O.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter",
"has_fderiv_within_at",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
asymptotics.is_O.has_fderiv_at {x₀ : E} {n : ℕ}
(h : f =O[𝓝 x₀] λ x, ‖x - x₀‖^n) (hn : 1 < n) :
has_fderiv_at f (0 : E →L[𝕜] F) x₀ | begin
rw [← nhds_within_univ] at h,
exact (h.has_fderiv_within_at (mem_univ _) hn).has_fderiv_at_of_univ
end | lemma | asymptotics.is_O.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.is_O {f : E → F} {s : set E} {x₀ : E} {f' : E →L[𝕜] F}
(h : has_fderiv_within_at f f' s x₀) :
(λ x, f x - f x₀) =O[𝓝[s] x₀] λ x, x - x₀ | by simpa only [sub_add_cancel] using h.is_O.add (is_O_sub f' (𝓝[s] x₀) x₀) | lemma | has_fderiv_within_at.is_O | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.is_O {f : E → F} {x₀ : E} {f' : E →L[𝕜] F} (h : has_fderiv_at f f' x₀) :
(λ x, f x - f x₀) =O[𝓝 x₀] λ x, x - x₀ | by simpa only [sub_add_cancel] using h.is_O.add (is_O_sub f' (𝓝 x₀) x₀) | lemma | has_fderiv_at.is_O | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.tendsto_nhds
(hL : L ≤ 𝓝 x) (h : has_fderiv_at_filter f f' x L) :
tendsto f L (𝓝 (f x)) | begin
have : tendsto (λ x', f x' - f x) L (𝓝 0),
{ refine h.is_O_sub.trans_tendsto (tendsto.mono_left _ hL),
rw ← sub_self x, exact tendsto_id.sub tendsto_const_nhds },
have := tendsto.add this tendsto_const_nhds,
rw zero_add (f x) at this,
exact this.congr (by simp only [sub_add_cancel, eq_self_iff_true... | theorem | has_fderiv_at_filter.tendsto_nhds | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"forall_const",
"has_fderiv_at_filter",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.continuous_within_at
(h : has_fderiv_within_at f f' s x) : continuous_within_at f s x | has_fderiv_at_filter.tendsto_nhds inf_le_left h | theorem | has_fderiv_within_at.continuous_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_within_at",
"has_fderiv_at_filter.tendsto_nhds",
"has_fderiv_within_at",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.continuous_at (h : has_fderiv_at f f' x) :
continuous_at f x | has_fderiv_at_filter.tendsto_nhds le_rfl h | theorem | has_fderiv_at.continuous_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_at",
"has_fderiv_at",
"has_fderiv_at_filter.tendsto_nhds",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.continuous_within_at (h : differentiable_within_at 𝕜 f s x) :
continuous_within_at f s x | let ⟨f', hf'⟩ := h in hf'.continuous_within_at | lemma | differentiable_within_at.continuous_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_within_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.continuous_at (h : differentiable_at 𝕜 f x) : continuous_at f x | let ⟨f', hf'⟩ := h in hf'.continuous_at | lemma | differentiable_at.continuous_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_at",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.continuous_on (h : differentiable_on 𝕜 f s) : continuous_on f s | λx hx, (h x hx).continuous_within_at | lemma | differentiable_on.continuous_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_on",
"continuous_within_at",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.continuous (h : differentiable 𝕜 f) : continuous f | continuous_iff_continuous_at.2 $ λx, (h x).continuous_at | lemma | differentiable.continuous | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous",
"continuous_at",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.continuous_at (hf : has_strict_fderiv_at f f' x) :
continuous_at f x | hf.has_fderiv_at.continuous_at | lemma | has_strict_fderiv_at.continuous_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.is_O_sub_rev {f' : E ≃L[𝕜] F}
(hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) x) :
(λ p : E × E, p.1 - p.2) =O[𝓝 (x, x)](λ p : E × E, f p.1 - f p.2) | ((f'.is_O_comp_rev _ _).trans (hf.trans_is_O (f'.is_O_comp_rev _ _)).right_is_O_add).congr
(λ _, rfl) (λ _, sub_add_cancel _ _) | lemma | has_strict_fderiv_at.is_O_sub_rev | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.is_O_sub_rev (hf : has_fderiv_at_filter f f' x L) {C}
(hf' : antilipschitz_with C f') :
(λ x', x' - x) =O[L] (λ x', f x' - f x) | have (λ x', x' - x) =O[L] (λ x', f' (x' - x)),
from is_O_iff.2 ⟨C, eventually_of_forall $ λ x',
zero_hom_class.bound_of_antilipschitz f' hf' _⟩,
(this.trans (hf.trans_is_O this).right_is_O_add).congr (λ _, rfl) (λ _, sub_add_cancel _ _) | lemma | has_fderiv_at_filter.is_O_sub_rev | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"antilipschitz_with",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
has_fderiv_within_at f f' s x ↔ has_fderiv_within_at f f' t x | calc has_fderiv_within_at f f' s x ↔ has_fderiv_within_at f f' (s \ {y}) x :
(has_fderiv_within_at_diff_singleton _).symm
... ↔ has_fderiv_within_at f f' (t \ {y}) x :
suffices 𝓝[s \ {y}] x = 𝓝[t \ {y}] x, by simp only [has_fderiv_within_at, this],
by simpa only [set_eventually_eq_iff_inf_principal, ← nhds_with... | lemma | has_fderiv_within_at_congr_set' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"has_fderiv_within_at_diff_singleton",
"nhds_within_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_congr_set (h : s =ᶠ[𝓝 x] t) :
has_fderiv_within_at f f' s x ↔ has_fderiv_within_at f f' t x | has_fderiv_within_at_congr_set' x $ h.filter_mono inf_le_left | lemma | has_fderiv_within_at_congr_set | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at",
"has_fderiv_within_at_congr_set'",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
differentiable_within_at 𝕜 f s x ↔ differentiable_within_at 𝕜 f t x | exists_congr $ λ _, has_fderiv_within_at_congr_set' _ h | lemma | differentiable_within_at_congr_set' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"has_fderiv_within_at_congr_set'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_congr_set (h : s =ᶠ[𝓝 x] t) :
differentiable_within_at 𝕜 f s x ↔ differentiable_within_at 𝕜 f t x | exists_congr $ λ _, has_fderiv_within_at_congr_set h | lemma | differentiable_within_at_congr_set | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"has_fderiv_within_at_congr_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
fderiv_within 𝕜 f s x = fderiv_within 𝕜 f t x | by simp only [fderiv_within, has_fderiv_within_at_congr_set' y h] | lemma | fderiv_within_congr_set' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"has_fderiv_within_at_congr_set'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_congr_set (h : s =ᶠ[𝓝 x] t) :
fderiv_within 𝕜 f s x = fderiv_within 𝕜 f t x | fderiv_within_congr_set' x $ h.filter_mono inf_le_left | lemma | fderiv_within_congr_set | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"fderiv_within_congr_set'",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
fderiv_within 𝕜 f s =ᶠ[𝓝 x] fderiv_within 𝕜 f t | (eventually_nhds_nhds_within.2 h).mono $ λ _, fderiv_within_congr_set' y | lemma | fderiv_within_eventually_congr_set' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"fderiv_within_congr_set'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_eventually_congr_set (h : s =ᶠ[𝓝 x] t) :
fderiv_within 𝕜 f s =ᶠ[𝓝 x] fderiv_within 𝕜 f t | fderiv_within_eventually_congr_set' x $ h.filter_mono inf_le_left | lemma | fderiv_within_eventually_congr_set | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"fderiv_within_eventually_congr_set'",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.has_strict_fderiv_at_iff
(h : f₀ =ᶠ[𝓝 x] f₁) (h' : ∀ y, f₀' y = f₁' y) :
has_strict_fderiv_at f₀ f₀' x ↔ has_strict_fderiv_at f₁ f₁' x | begin
refine is_o_congr ((h.prod_mk_nhds h).mono _) (eventually_of_forall $ λ _, rfl),
rintros p ⟨hp₁, hp₂⟩,
simp only [*]
end | theorem | filter.eventually_eq.has_strict_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.congr_of_eventually_eq (h : has_strict_fderiv_at f f' x)
(h₁ : f =ᶠ[𝓝 x] f₁) : has_strict_fderiv_at f₁ f' x | (h₁.has_strict_fderiv_at_iff (λ _, rfl)).1 h | theorem | has_strict_fderiv_at.congr_of_eventually_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.has_fderiv_at_filter_iff
(h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x) (h₁ : ∀ x, f₀' x = f₁' x) :
has_fderiv_at_filter f₀ f₀' x L ↔ has_fderiv_at_filter f₁ f₁' x L | is_o_congr (h₀.mono $ λ y hy, by simp only [hy, h₁, hx]) (eventually_of_forall $ λ _, rfl) | theorem | filter.eventually_eq.has_fderiv_at_filter_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.congr_of_eventually_eq (h : has_fderiv_at_filter f f' x L)
(hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x) : has_fderiv_at_filter f₁ f' x L | (hL.has_fderiv_at_filter_iff hx $ λ _, rfl).2 h | lemma | has_fderiv_at_filter.congr_of_eventually_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.has_fderiv_at_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
has_fderiv_at f₀ f' x ↔ has_fderiv_at f₁ f' x | h.has_fderiv_at_filter_iff h.eq_of_nhds (λ _, rfl) | theorem | filter.eventually_eq.has_fderiv_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.differentiable_at_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
differentiable_at 𝕜 f₀ x ↔ differentiable_at 𝕜 f₁ x | exists_congr $ λ f', h.has_fderiv_at_iff | theorem | filter.eventually_eq.differentiable_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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