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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iterated_fderiv_within_inter_open {n : ℕ} (hu : is_open u) (hx : x ∈ u) :
iterated_fderiv_within 𝕜 n f (s ∩ u) x = iterated_fderiv_within 𝕜 n f s x | iterated_fderiv_within_inter (hu.mem_nhds hx) | lemma | iterated_fderiv_within_inter_open | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"is_open",
"iterated_fderiv_within",
"iterated_fderiv_within_inter"
] | The iterated differential within a set `s` at a point `x` is not modified if one intersects
`s` with an open set containing `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_zero :
cont_diff_on 𝕜 0 f s ↔ continuous_on f s | begin
refine ⟨λ H, H.continuous_on, λ H, _⟩,
assume x hx m hm,
have : (m : ℕ∞) = 0 := le_antisymm hm bot_le,
rw this,
refine ⟨insert x s, self_mem_nhds_within, ftaylor_series_within 𝕜 f s, _⟩,
rw has_ftaylor_series_up_to_on_zero_iff,
exact ⟨by rwa insert_eq_of_mem hx, λ x hx, by simp [ftaylor_series_with... | lemma | cont_diff_on_zero | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"bot_le",
"cont_diff_on",
"continuous_on",
"ftaylor_series_within",
"has_ftaylor_series_up_to_on_zero_iff",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_zero (hx : x ∈ s) :
cont_diff_within_at 𝕜 0 f s x ↔ ∃ u ∈ 𝓝[s] x, continuous_on f (s ∩ u) | begin
split,
{ intros h,
obtain ⟨u, H, p, hp⟩ := h 0 (by norm_num),
refine ⟨u, _, _⟩,
{ simpa [hx] using H },
{ simp only [with_top.coe_zero, has_ftaylor_series_up_to_on_zero_iff] at hp,
exact hp.1.mono (inter_subset_right s u) } },
{ rintros ⟨u, H, hu⟩,
rw ← cont_diff_within_at_inter' H... | lemma | cont_diff_within_at_zero | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"cont_diff_within_at_inter'",
"continuous_on",
"has_ftaylor_series_up_to_on_zero_iff",
"mem_of_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on
(h : has_ftaylor_series_up_to_on n f p s)
{m : ℕ} (hmn : (m : ℕ∞) ≤ n) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s) :
p x m = iterated_fderiv_within 𝕜 m f s x | begin
induction m with m IH generalizing x,
{ rw [h.zero_eq' hx, iterated_fderiv_within_zero_eq_comp] },
{ have A : (m : ℕ∞) < n := lt_of_lt_of_le (with_top.coe_lt_coe.2 (lt_add_one m)) hmn,
have : has_fderiv_within_at (λ (y : E), iterated_fderiv_within 𝕜 m f s y)
(continuous_multilinear_map.curry_left... | theorem | has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_map.curry_left",
"continuous_multilinear_map.uncurry_curry_left",
"function.comp_apply",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_eq_comp_left",
"iterated_fderiv_within_zero_eq_comp",
"lt_add_one",
"unique... | On a set with unique differentiability, any choice of iterated differential has to coincide
with the one we have chosen in `iterated_fderiv_within 𝕜 m f s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.ftaylor_series_within
(h : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) :
has_ftaylor_series_up_to_on n f (ftaylor_series_within 𝕜 f s) s | begin
split,
{ assume x hx,
simp only [ftaylor_series_within, continuous_multilinear_map.uncurry0_apply,
iterated_fderiv_within_zero_apply] },
{ assume m hm x hx,
rcases (h x hx) m.succ (enat.add_one_le_of_lt hm) with ⟨u, hu, p, Hp⟩,
rw insert_eq_of_mem hx at hu,
rcases mem_nhds_wit... | theorem | cont_diff_on.ftaylor_series_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont",
"cont_diff_on",
"continuous_multilinear_map.uncurry0_apply",
"continuous_on_of_locally_continuous_on",
"enat.add_one_le_of_lt",
"fderiv_within",
"ftaylor_series_within",
"has_fderiv_within_at_inter",
"has_ftaylor_series_up_to_on",
"is_open.mem_nhds",
"iterated_fderiv_within",
"iterated... | When a function is `C^n` in a set `s` of unique differentiability, it admits
`ftaylor_series_within 𝕜 f s` as a Taylor series up to order `n` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_of_continuous_on_differentiable_on
(Hcont : ∀ (m : ℕ), (m : ℕ∞) ≤ n →
continuous_on (λ x, iterated_fderiv_within 𝕜 m f s x) s)
(Hdiff : ∀ (m : ℕ), (m : ℕ∞) < n →
differentiable_on 𝕜 (λ x, iterated_fderiv_within 𝕜 m f s x) s) :
cont_diff_on 𝕜 n f s | begin
assume x hx m hm,
rw insert_eq_of_mem hx,
refine ⟨s, self_mem_nhds_within, ftaylor_series_within 𝕜 f s, _⟩,
split,
{ assume y hy,
simp only [ftaylor_series_within, continuous_multilinear_map.uncurry0_apply,
iterated_fderiv_within_zero_apply] },
{ assume k hk y hy,
convert (Hdi... | lemma | cont_diff_on_of_continuous_on_differentiable_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"coe_fn_coe_base",
"cont_diff_on",
"continuous_linear_equiv.coe_apply",
"continuous_linear_map.curry_uncurry_left",
"continuous_multilinear_map.uncurry0_apply",
"continuous_on",
"differentiable_on",
"ftaylor_series_within",
"has_fderiv_within_at",
"iterated_fderiv_within",
"iterated_fderiv_withi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_of_differentiable_on
(h : ∀(m : ℕ), (m : ℕ∞) ≤ n → differentiable_on 𝕜 (iterated_fderiv_within 𝕜 m f s) s) :
cont_diff_on 𝕜 n f s | cont_diff_on_of_continuous_on_differentiable_on
(λ m hm, (h m hm).continuous_on) (λ m hm, (h m (le_of_lt hm))) | lemma | cont_diff_on_of_differentiable_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_of_continuous_on_differentiable_on",
"continuous_on",
"differentiable_on",
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.continuous_on_iterated_fderiv_within {m : ℕ}
(h : cont_diff_on 𝕜 n f s) (hmn : (m : ℕ∞) ≤ n) (hs : unique_diff_on 𝕜 s) :
continuous_on (iterated_fderiv_within 𝕜 m f s) s | (h.ftaylor_series_within hs).cont m hmn | lemma | cont_diff_on.continuous_on_iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont",
"cont_diff_on",
"continuous_on",
"iterated_fderiv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.differentiable_on_iterated_fderiv_within {m : ℕ}
(h : cont_diff_on 𝕜 n f s) (hmn : (m : ℕ∞) < n) (hs : unique_diff_on 𝕜 s) :
differentiable_on 𝕜 (iterated_fderiv_within 𝕜 m f s) s | λ x hx, ((h.ftaylor_series_within hs).fderiv_within m hmn x hx).differentiable_within_at | lemma | cont_diff_on.differentiable_on_iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"differentiable_on",
"differentiable_within_at",
"fderiv_within",
"iterated_fderiv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.differentiable_within_at_iterated_fderiv_within {m : ℕ}
(h : cont_diff_within_at 𝕜 n f s x) (hmn : (m : ℕ∞) < n)
(hs : unique_diff_on 𝕜 (insert x s)) :
differentiable_within_at 𝕜 (iterated_fderiv_within 𝕜 m f s) s x | begin
rcases h.cont_diff_on' (enat.add_one_le_of_lt hmn) with ⟨u, uo, xu, hu⟩,
set t := insert x s ∩ u,
have A : t =ᶠ[𝓝[≠] x] s,
{ simp only [set_eventually_eq_iff_inf_principal, ← nhds_within_inter'],
rw [← inter_assoc, nhds_within_inter_of_mem', ← diff_eq_compl_inter, insert_diff_of_mem,
diff_eq_co... | lemma | cont_diff_within_at.differentiable_within_at_iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"differentiable_within_at",
"differentiable_within_at_congr_set'",
"enat.add_one_le_of_lt",
"inf_le_left",
"iterated_fderiv_within",
"iterated_fderiv_within_eventually_congr_set'",
"mem_nhds_within_of_mem_nhds",
"nhds_within_inter'",
"nhds_within_inter_of_mem'",
"unique_di... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_iff_continuous_on_differentiable_on
(hs : unique_diff_on 𝕜 s) :
cont_diff_on 𝕜 n f s ↔
(∀ (m : ℕ), (m : ℕ∞) ≤ n →
continuous_on (λ x, iterated_fderiv_within 𝕜 m f s x) s)
∧ (∀ (m : ℕ), (m : ℕ∞) < n →
differentiable_on 𝕜 (λ x, iterated_fderiv_within 𝕜 m f s x) s) | ⟨λ h, ⟨λ m hm, h.continuous_on_iterated_fderiv_within hm hs,
λ m hm, h.differentiable_on_iterated_fderiv_within hm hs⟩,
λ h, cont_diff_on_of_continuous_on_differentiable_on h.1 h.2⟩ | lemma | cont_diff_on_iff_continuous_on_differentiable_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_of_continuous_on_differentiable_on",
"continuous_on",
"differentiable_on",
"iterated_fderiv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_succ_of_fderiv_within {n : ℕ} (hf : differentiable_on 𝕜 f s)
(h : cont_diff_on 𝕜 n (λ y, fderiv_within 𝕜 f s y) s) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f s | begin
intros x hx,
rw [cont_diff_within_at_succ_iff_has_fderiv_within_at, insert_eq_of_mem hx],
exact ⟨s, self_mem_nhds_within, fderiv_within 𝕜 f s,
λ y hy, (hf y hy).has_fderiv_within_at, h x hx⟩
end | lemma | cont_diff_on_succ_of_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_within_at_succ_iff_has_fderiv_within_at",
"differentiable_on",
"fderiv_within",
"has_fderiv_within_at",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_succ_iff_fderiv_within {n : ℕ} (hs : unique_diff_on 𝕜 s) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f s ↔
differentiable_on 𝕜 f s ∧ cont_diff_on 𝕜 n (λ y, fderiv_within 𝕜 f s y) s | begin
refine ⟨λ H, _, λ h, cont_diff_on_succ_of_fderiv_within h.1 h.2⟩,
refine ⟨H.differentiable_on (with_top.coe_le_coe.2 (nat.le_add_left 1 n)), λ x hx, _⟩,
rcases cont_diff_within_at_succ_iff_has_fderiv_within_at.1 (H x hx)
with ⟨u, hu, f', hff', hf'⟩,
rcases mem_nhds_within.1 hu with ⟨o, o_open, xo, ho⟩... | theorem | cont_diff_on_succ_iff_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_succ_of_fderiv_within",
"cont_diff_within_at_inter'",
"differentiable_on",
"fderiv_within",
"fderiv_within_inter",
"filter.eventually_eq_of_mem",
"is_open.mem_nhds",
"mem_nhds_within_of_mem_nhds",
"unique_diff_on"
] | A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is
differentiable there, and its derivative (expressed with `fderiv_within`) is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_succ_iff_has_fderiv_within {n : ℕ} (hs : unique_diff_on 𝕜 s) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f s ↔ ∃ (f' : E → (E →L[𝕜] F)),
cont_diff_on 𝕜 n f' s ∧ ∀ x, x ∈ s → has_fderiv_within_at f (f' x) s x | begin
rw cont_diff_on_succ_iff_fderiv_within hs,
refine ⟨λ h, ⟨fderiv_within 𝕜 f s, h.2, λ x hx, (h.1 x hx).has_fderiv_within_at⟩, λ h, _⟩,
rcases h with ⟨f', h1, h2⟩,
refine ⟨λ x hx, (h2 x hx).differentiable_within_at, λ x hx, _⟩,
exact (h1 x hx).congr' (λ y hy, (h2 y hy).fderiv_within (hs y hy)) hx,
end | lemma | cont_diff_on_succ_iff_has_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_within",
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_succ_iff_fderiv_of_open {n : ℕ} (hs : is_open s) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f s ↔
differentiable_on 𝕜 f s ∧ cont_diff_on 𝕜 n (λ y, fderiv 𝕜 f y) s | begin
rw cont_diff_on_succ_iff_fderiv_within hs.unique_diff_on,
congrm _ ∧ _,
apply cont_diff_on_congr,
assume x hx,
exact fderiv_within_of_open hs hx
end | theorem | cont_diff_on_succ_iff_fderiv_of_open | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_congr",
"cont_diff_on_succ_iff_fderiv_within",
"differentiable_on",
"fderiv",
"fderiv_within_of_open",
"is_open"
] | A function is `C^(n + 1)` on an open domain if and only if it is
differentiable there, and its derivative (expressed with `fderiv`) is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_top_iff_fderiv_within (hs : unique_diff_on 𝕜 s) :
cont_diff_on 𝕜 ∞ f s ↔
differentiable_on 𝕜 f s ∧ cont_diff_on 𝕜 ∞ (λ y, fderiv_within 𝕜 f s y) s | begin
split,
{ assume h,
refine ⟨h.differentiable_on le_top, _⟩,
apply cont_diff_on_top.2 (λ n, ((cont_diff_on_succ_iff_fderiv_within hs).1 _).2),
exact h.of_le le_top },
{ assume h,
refine cont_diff_on_top.2 (λ n, _),
have A : (n : ℕ∞) ≤ ∞ := le_top,
apply ((cont_diff_on_succ_iff_fderiv_w... | theorem | cont_diff_on_top_iff_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_within",
"differentiable_on",
"fderiv_within",
"le_top",
"unique_diff_on"
] | A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
there, and its derivative (expressed with `fderiv_within`) is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_top_iff_fderiv_of_open (hs : is_open s) :
cont_diff_on 𝕜 ∞ f s ↔
differentiable_on 𝕜 f s ∧ cont_diff_on 𝕜 ∞ (λ y, fderiv 𝕜 f y) s | begin
rw cont_diff_on_top_iff_fderiv_within hs.unique_diff_on,
congrm _ ∧ _,
apply cont_diff_on_congr,
assume x hx,
exact fderiv_within_of_open hs hx
end | theorem | cont_diff_on_top_iff_fderiv_of_open | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_congr",
"cont_diff_on_top_iff_fderiv_within",
"differentiable_on",
"fderiv",
"fderiv_within_of_open",
"is_open"
] | A function is `C^∞` on an open domain if and only if it is differentiable there, and its
derivative (expressed with `fderiv`) is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.fderiv_within
(hf : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (λ y, fderiv_within 𝕜 f s y) s | begin
cases m,
{ change ∞ + 1 ≤ n at hmn,
have : n = ∞, by simpa using hmn,
rw this at hf,
exact ((cont_diff_on_top_iff_fderiv_within hs).1 hf).2 },
{ change (m.succ : ℕ∞) ≤ n at hmn,
exact ((cont_diff_on_succ_iff_fderiv_within hs).1 (hf.of_le hmn)).2 }
end | lemma | cont_diff_on.fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_within",
"cont_diff_on_top_iff_fderiv_within",
"fderiv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.fderiv_of_open
(hf : cont_diff_on 𝕜 n f s) (hs : is_open s) (hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (λ y, fderiv 𝕜 f y) s | (hf.fderiv_within hs.unique_diff_on hmn).congr (λ x hx, (fderiv_within_of_open hs hx).symm) | lemma | cont_diff_on.fderiv_of_open | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"fderiv",
"fderiv_within_of_open",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.continuous_on_fderiv_within
(h : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hn : 1 ≤ n) :
continuous_on (λ x, fderiv_within 𝕜 f s x) s | ((cont_diff_on_succ_iff_fderiv_within hs).1 (h.of_le hn)).2.continuous_on | lemma | cont_diff_on.continuous_on_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_within",
"continuous_on",
"fderiv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.continuous_on_fderiv_of_open
(h : cont_diff_on 𝕜 n f s) (hs : is_open s) (hn : 1 ≤ n) :
continuous_on (λ x, fderiv 𝕜 f x) s | ((cont_diff_on_succ_iff_fderiv_of_open hs).1 (h.of_le hn)).2.continuous_on | lemma | cont_diff_on.continuous_on_fderiv_of_open | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_of_open",
"continuous_on",
"fderiv",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to (n : ℕ∞)
(f : E → F) (p : E → formal_multilinear_series 𝕜 E F) : Prop | (zero_eq : ∀ x, (p x 0).uncurry0 = f x)
(fderiv : ∀ (m : ℕ) (hm : (m : ℕ∞) < n), ∀ x,
has_fderiv_at (λ y, p y m) (p x m.succ).curry_left x)
(cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), continuous (λ x, p x m)) | structure | has_ftaylor_series_up_to | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont",
"continuous",
"fderiv",
"formal_multilinear_series",
"has_fderiv_at"
] | `has_ftaylor_series_up_to n f p` registers the fact that `p 0 = f` and `p (m+1)` is a
derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
`has_fderiv_at` but for higher order derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to.zero_eq'
(h : has_ftaylor_series_up_to n f p) (x : E) :
p x 0 = (continuous_multilinear_curry_fin0 𝕜 E F).symm (f x) | by { rw ← h.zero_eq x, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ } | lemma | has_ftaylor_series_up_to.zero_eq' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin0",
"continuous_multilinear_map.uncurry0_curry0",
"has_ftaylor_series_up_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on_univ_iff :
has_ftaylor_series_up_to_on n f p univ ↔ has_ftaylor_series_up_to n f p | begin
split,
{ assume H,
split,
{ exact λ x, H.zero_eq x (mem_univ x) },
{ assume m hm x,
rw ← has_fderiv_within_at_univ,
exact H.fderiv_within m hm x (mem_univ x) },
{ assume m hm,
rw continuous_iff_continuous_on_univ,
exact H.cont m hm } },
{ assume H,
split,
{ ex... | lemma | has_ftaylor_series_up_to_on_univ_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_iff_continuous_on_univ",
"has_fderiv_within_at_univ",
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to.has_ftaylor_series_up_to_on
(h : has_ftaylor_series_up_to n f p) (s : set E) :
has_ftaylor_series_up_to_on n f p s | (has_ftaylor_series_up_to_on_univ_iff.2 h).mono (subset_univ _) | lemma | has_ftaylor_series_up_to.has_ftaylor_series_up_to_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to.of_le
(h : has_ftaylor_series_up_to n f p) (hmn : m ≤ n) :
has_ftaylor_series_up_to m f p | by { rw ← has_ftaylor_series_up_to_on_univ_iff at h ⊢, exact h.of_le hmn } | lemma | has_ftaylor_series_up_to.of_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to.continuous
(h : has_ftaylor_series_up_to n f p) : continuous f | begin
rw ← has_ftaylor_series_up_to_on_univ_iff at h,
rw continuous_iff_continuous_on_univ,
exact h.continuous_on
end | lemma | has_ftaylor_series_up_to.continuous | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_zero_iff :
has_ftaylor_series_up_to 0 f p ↔ continuous f ∧ (∀ x, (p x 0).uncurry0 = f x) | by simp [has_ftaylor_series_up_to_on_univ_iff.symm, continuous_iff_continuous_on_univ,
has_ftaylor_series_up_to_on_zero_iff] | lemma | has_ftaylor_series_up_to_zero_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous",
"continuous_iff_continuous_on_univ",
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_top_iff : has_ftaylor_series_up_to ∞ f p ↔
∀ (n : ℕ), has_ftaylor_series_up_to n f p | by simp only [← has_ftaylor_series_up_to_on_univ_iff, has_ftaylor_series_up_to_on_top_iff] | lemma | has_ftaylor_series_up_to_top_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_top_iff",
"has_ftaylor_series_up_to_on_univ_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_top_iff' : has_ftaylor_series_up_to ∞ f p ↔
(∀ x, (p x 0).uncurry0 = f x) ∧
(∀ (m : ℕ) x, has_fderiv_at (λ y, p y m) (p x m.succ).curry_left x) | by simp only [← has_ftaylor_series_up_to_on_univ_iff, has_ftaylor_series_up_to_on_top_iff',
mem_univ, forall_true_left, has_fderiv_within_at_univ] | lemma | has_ftaylor_series_up_to_top_iff' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"forall_true_left",
"has_fderiv_at",
"has_fderiv_within_at_univ",
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_top_iff'",
"has_ftaylor_series_up_to_on_univ_iff"
] | In the case that `n = ∞` we don't need the continuity assumption in
`has_ftaylor_series_up_to`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to.has_fderiv_at
(h : has_ftaylor_series_up_to n f p) (hn : 1 ≤ n) (x : E) :
has_fderiv_at f (continuous_multilinear_curry_fin1 𝕜 E F (p x 1)) x | begin
rw [← has_fderiv_within_at_univ],
exact (has_ftaylor_series_up_to_on_univ_iff.2 h).has_fderiv_within_at hn (mem_univ _)
end | lemma | has_ftaylor_series_up_to.has_fderiv_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin1",
"has_fderiv_at",
"has_fderiv_within_at",
"has_fderiv_within_at_univ",
"has_ftaylor_series_up_to"
] | If a function has a Taylor series at order at least `1`, then the term of order `1` of this
series is a derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to.differentiable
(h : has_ftaylor_series_up_to n f p) (hn : 1 ≤ n) : differentiable 𝕜 f | λ x, (h.has_fderiv_at hn x).differentiable_at | lemma | has_ftaylor_series_up_to.differentiable | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"differentiable",
"differentiable_at",
"has_ftaylor_series_up_to"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_succ_iff_right {n : ℕ} :
has_ftaylor_series_up_to ((n + 1) : ℕ) f p ↔
(∀ x, (p x 0).uncurry0 = f x)
∧ (∀ x, has_fderiv_at (λ y, p y 0) (p x 1).curry_left x)
∧ has_ftaylor_series_up_to n
(λ x, continuous_multilinear_curry_fin1 𝕜 E F (p x 1)) (λ x, (p x).shift) | by simp only [has_ftaylor_series_up_to_on_succ_iff_right, ← has_ftaylor_series_up_to_on_univ_iff,
mem_univ, forall_true_left, has_fderiv_within_at_univ] | theorem | has_ftaylor_series_up_to_succ_iff_right | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin1",
"forall_true_left",
"has_fderiv_at",
"has_fderiv_within_at_univ",
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_succ_iff_right",
"has_ftaylor_series_up_to_on_univ_iff"
] | `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
for `p 1`, which is a derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at (n : ℕ∞) (f : E → F) (x : E) : Prop | cont_diff_within_at 𝕜 n f univ x | def | cont_diff_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | A function is continuously differentiable up to `n` at a point `x` if, for any integer `k ≤ n`,
there is a neighborhood of `x` where `f` admits derivatives up to order `n`, which are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_univ :
cont_diff_within_at 𝕜 n f univ x ↔ cont_diff_at 𝕜 n f x | iff.rfl | theorem | cont_diff_within_at_univ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_top :
cont_diff_at 𝕜 ∞ f x ↔ ∀ (n : ℕ), cont_diff_at 𝕜 n f x | by simp [← cont_diff_within_at_univ, cont_diff_within_at_top] | lemma | cont_diff_at_top | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"cont_diff_within_at_top",
"cont_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.cont_diff_within_at
(h : cont_diff_at 𝕜 n f x) : cont_diff_within_at 𝕜 n f s x | h.mono (subset_univ _) | lemma | cont_diff_at.cont_diff_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.cont_diff_at
(h : cont_diff_within_at 𝕜 n f s x) (hx : s ∈ 𝓝 x) :
cont_diff_at 𝕜 n f x | by rwa [cont_diff_at, ← cont_diff_within_at_inter hx, univ_inter] | lemma | cont_diff_within_at.cont_diff_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"cont_diff_within_at",
"cont_diff_within_at_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.congr_of_eventually_eq
(h : cont_diff_at 𝕜 n f x) (hg : f₁ =ᶠ[𝓝 x] f) :
cont_diff_at 𝕜 n f₁ x | h.congr_of_eventually_eq' (by rwa nhds_within_univ) (mem_univ x) | lemma | cont_diff_at.congr_of_eventually_eq | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.of_le
(h : cont_diff_at 𝕜 n f x) (hmn : m ≤ n) :
cont_diff_at 𝕜 m f x | h.of_le hmn | lemma | cont_diff_at.of_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.continuous_at
(h : cont_diff_at 𝕜 n f x) : continuous_at f x | by simpa [continuous_within_at_univ] using h.continuous_within_at | lemma | cont_diff_at.continuous_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"continuous_at",
"continuous_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.differentiable_at
(h : cont_diff_at 𝕜 n f x) (hn : 1 ≤ n) : differentiable_at 𝕜 f x | by simpa [hn, differentiable_within_at_univ] using h.differentiable_within_at | lemma | cont_diff_at.differentiable_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"differentiable_at",
"differentiable_within_at_univ"
] | If a function is `C^n` with `n ≥ 1` at a point, then it is differentiable there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at_succ_iff_has_fderiv_at {n : ℕ} :
cont_diff_at 𝕜 ((n + 1) : ℕ) f x
↔ (∃ f' : E → E →L[𝕜] F, (∃ u ∈ 𝓝 x, ∀ x ∈ u, has_fderiv_at f (f' x) x)
∧ cont_diff_at 𝕜 n f' x) | begin
rw [← cont_diff_within_at_univ, cont_diff_within_at_succ_iff_has_fderiv_within_at],
simp only [nhds_within_univ, exists_prop, mem_univ, insert_eq_of_mem],
split,
{ rintros ⟨u, H, f', h_fderiv, h_cont_diff⟩,
rcases mem_nhds_iff.mp H with ⟨t, htu, ht, hxt⟩,
refine ⟨f', ⟨t, _⟩, h_cont_diff.cont_diff_... | theorem | cont_diff_at_succ_iff_has_fderiv_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"cont_diff_within_at_succ_iff_has_fderiv_within_at",
"cont_diff_within_at_univ",
"exists_prop",
"has_fderiv_at",
"has_fderiv_within_at",
"nhds_within_univ"
] | A function is `C^(n + 1)` at a point iff locally, it has a derivative which is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.eventually {n : ℕ} (h : cont_diff_at 𝕜 n f x) :
∀ᶠ y in 𝓝 x, cont_diff_at 𝕜 n f y | by simpa [nhds_within_univ] using h.eventually | theorem | cont_diff_at.eventually | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff (n : ℕ∞) (f : E → F) : Prop | ∃ p : E → formal_multilinear_series 𝕜 E F, has_ftaylor_series_up_to n f p | def | cont_diff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"formal_multilinear_series",
"has_ftaylor_series_up_to"
] | A function is continuously differentiable up to `n` if it admits derivatives up to
order `n`, which are continuous. Contrary to the case of definitions in domains (where derivatives
might not be unique) we do not need to localize the definition in space or time. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to.cont_diff {f' : E → formal_multilinear_series 𝕜 E F}
(hf : has_ftaylor_series_up_to n f f') : cont_diff 𝕜 n f | ⟨f', hf⟩ | lemma | has_ftaylor_series_up_to.cont_diff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"formal_multilinear_series",
"has_ftaylor_series_up_to"
] | If `f` has a Taylor series up to `n`, then it is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_univ : cont_diff_on 𝕜 n f univ ↔ cont_diff 𝕜 n f | begin
split,
{ assume H,
use ftaylor_series_within 𝕜 f univ,
rw ← has_ftaylor_series_up_to_on_univ_iff,
exact H.ftaylor_series_within unique_diff_on_univ },
{ rintros ⟨p, hp⟩ x hx m hm,
exact ⟨univ, filter.univ_sets _, p, (hp.has_ftaylor_series_up_to_on univ).of_le hm⟩ }
end | theorem | cont_diff_on_univ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on",
"ftaylor_series_within",
"has_ftaylor_series_up_to_on_univ_iff",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_iff_cont_diff_at : cont_diff 𝕜 n f ↔ ∀ x, cont_diff_at 𝕜 n f x | by simp [← cont_diff_on_univ, cont_diff_on, cont_diff_at] | lemma | cont_diff_iff_cont_diff_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_at",
"cont_diff_on",
"cont_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.cont_diff_at (h : cont_diff 𝕜 n f) : cont_diff_at 𝕜 n f x | cont_diff_iff_cont_diff_at.1 h x | lemma | cont_diff.cont_diff_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.cont_diff_within_at (h : cont_diff 𝕜 n f) : cont_diff_within_at 𝕜 n f s x | h.cont_diff_at.cont_diff_within_at | lemma | cont_diff.cont_diff_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_top : cont_diff 𝕜 ∞ f ↔ ∀ (n : ℕ), cont_diff 𝕜 n f | by simp [cont_diff_on_univ.symm, cont_diff_on_top] | lemma | cont_diff_top | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_all_iff_nat : (∀ n, cont_diff 𝕜 n f) ↔ (∀ n : ℕ, cont_diff 𝕜 n f) | by simp only [← cont_diff_on_univ, cont_diff_on_all_iff_nat] | lemma | cont_diff_all_iff_nat | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_all_iff_nat",
"cont_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.cont_diff_on (h : cont_diff 𝕜 n f) : cont_diff_on 𝕜 n f s | (cont_diff_on_univ.2 h).mono (subset_univ _) | lemma | cont_diff.cont_diff_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_zero : cont_diff 𝕜 0 f ↔ continuous f | begin
rw [← cont_diff_on_univ, continuous_iff_continuous_on_univ],
exact cont_diff_on_zero
end | lemma | cont_diff_zero | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_univ",
"cont_diff_on_zero",
"continuous",
"continuous_iff_continuous_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_zero : cont_diff_at 𝕜 0 f x ↔ ∃ u ∈ 𝓝 x, continuous_on f u | by { rw ← cont_diff_within_at_univ, simp [cont_diff_within_at_zero, nhds_within_univ] } | lemma | cont_diff_at_zero | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_at",
"cont_diff_within_at_univ",
"cont_diff_within_at_zero",
"continuous_on",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_one_iff : cont_diff_at 𝕜 1 f x ↔
∃ f' : E → (E →L[𝕜] F), ∃ u ∈ 𝓝 x, continuous_on f' u ∧ ∀ x ∈ u, has_fderiv_at f (f' x) x | by simp_rw [show (1 : ℕ∞) = (0 + 1 : ℕ), from (zero_add 1).symm,
cont_diff_at_succ_iff_has_fderiv_at, show ((0 : ℕ) : ℕ∞) = 0, from rfl,
cont_diff_at_zero, exists_mem_and_iff antitone_bforall antitone_continuous_on, and_comm] | theorem | cont_diff_at_one_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"antitone_continuous_on",
"cont_diff_at",
"cont_diff_at_succ_iff_has_fderiv_at",
"cont_diff_at_zero",
"continuous_on",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.of_le (h : cont_diff 𝕜 n f) (hmn : m ≤ n) : cont_diff 𝕜 m f | cont_diff_on_univ.1 $ (cont_diff_on_univ.2 h).of_le hmn | lemma | cont_diff.of_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.of_succ {n : ℕ} (h : cont_diff 𝕜 (n + 1) f) : cont_diff 𝕜 n f | h.of_le $ with_top.coe_le_coe.mpr le_self_add | lemma | cont_diff.of_succ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.one_of_succ {n : ℕ} (h : cont_diff 𝕜 (n + 1) f) : cont_diff 𝕜 1 f | h.of_le $ with_top.coe_le_coe.mpr le_add_self | lemma | cont_diff.one_of_succ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.continuous (h : cont_diff 𝕜 n f) : continuous f | cont_diff_zero.1 (h.of_le bot_le) | lemma | cont_diff.continuous | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"bot_le",
"cont_diff",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.differentiable (h : cont_diff 𝕜 n f) (hn : 1 ≤ n) : differentiable 𝕜 f | differentiable_on_univ.1 $ (cont_diff_on_univ.2 h).differentiable_on hn | lemma | cont_diff.differentiable | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"differentiable",
"differentiable_on"
] | If a function is `C^n` with `n ≥ 1`, then it is differentiable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_iff_forall_nat_le :
cont_diff 𝕜 n f ↔ ∀ m : ℕ, ↑m ≤ n → cont_diff 𝕜 m f | by { simp_rw [← cont_diff_on_univ], exact cont_diff_on_iff_forall_nat_le } | lemma | cont_diff_iff_forall_nat_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_iff_forall_nat_le",
"cont_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_succ_iff_has_fderiv {n : ℕ} : cont_diff 𝕜 ((n + 1) : ℕ) f ↔
∃ (f' : E → (E →L[𝕜] F)), cont_diff 𝕜 n f' ∧ ∀ x, has_fderiv_at f (f' x) x | by simp only [← cont_diff_on_univ, ← has_fderiv_within_at_univ,
cont_diff_on_succ_iff_has_fderiv_within (unique_diff_on_univ), set.mem_univ, forall_true_left] | lemma | cont_diff_succ_iff_has_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_succ_iff_has_fderiv_within",
"cont_diff_on_univ",
"forall_true_left",
"has_fderiv_at",
"has_fderiv_within_at_univ",
"set.mem_univ",
"unique_diff_on_univ"
] | A function is `C^(n+1)` iff it has a `C^n` derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv (n : ℕ) (f : E → F) :
E → (E [×n]→L[𝕜] F) | nat.rec_on n
(λ x, continuous_multilinear_map.curry0 𝕜 E (f x))
(λ n rec x, continuous_linear_map.uncurry_left (fderiv 𝕜 rec x)) | def | iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_linear_map.uncurry_left",
"continuous_multilinear_map.curry0",
"fderiv"
] | The `n`-th derivative of a function, as a multilinear map, defined inductively. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ftaylor_series (f : E → F) (x : E) : formal_multilinear_series 𝕜 E F | λ n, iterated_fderiv 𝕜 n f x | def | ftaylor_series | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"formal_multilinear_series",
"iterated_fderiv"
] | Formal Taylor series associated to a function within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_zero_apply (m : (fin 0) → E) :
(iterated_fderiv 𝕜 0 f x : ((fin 0) → E) → F) m = f x | rfl | lemma | iterated_fderiv_zero_apply | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_zero_eq_comp :
iterated_fderiv 𝕜 0 f = (continuous_multilinear_curry_fin0 𝕜 E F).symm ∘ f | rfl | lemma | iterated_fderiv_zero_eq_comp | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin0",
"iterated_fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_zero :
‖iterated_fderiv 𝕜 0 f x‖ = ‖f x‖ | by rw [iterated_fderiv_zero_eq_comp, linear_isometry_equiv.norm_map] | lemma | norm_iterated_fderiv_zero | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_zero_eq_comp",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_with_zero_eq :
iterated_fderiv_within 𝕜 0 f s = iterated_fderiv 𝕜 0 f | by { ext, refl } | lemma | iterated_fderiv_with_zero_eq | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv",
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_succ_apply_left {n : ℕ} (m : fin (n + 1) → E):
(iterated_fderiv 𝕜 (n + 1) f x : (fin (n + 1) → E) → F) m
= (fderiv 𝕜 (iterated_fderiv 𝕜 n f) x : E → (E [×n]→L[𝕜] F)) (m 0) (tail m) | rfl | lemma | iterated_fderiv_succ_apply_left | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv",
"iterated_fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_succ_eq_comp_left {n : ℕ} :
iterated_fderiv 𝕜 (n + 1) f =
(continuous_multilinear_curry_left_equiv 𝕜 (λ (i : fin (n + 1)), E) F)
∘ (fderiv 𝕜 (iterated_fderiv 𝕜 n f)) | rfl | lemma | iterated_fderiv_succ_eq_comp_left | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_left_equiv",
"fderiv",
"iterated_fderiv"
] | Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
and the derivative of the `n`-th derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_iterated_fderiv {n : ℕ} :
fderiv 𝕜 (iterated_fderiv 𝕜 n f) =
(continuous_multilinear_curry_left_equiv 𝕜 (λ (i : fin (n + 1)), E) F).symm
∘ (iterated_fderiv 𝕜 (n + 1) f) | begin
rw iterated_fderiv_succ_eq_comp_left,
ext1 x,
simp only [function.comp_app, linear_isometry_equiv.symm_apply_apply],
end | lemma | fderiv_iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_left_equiv",
"fderiv",
"iterated_fderiv",
"iterated_fderiv_succ_eq_comp_left",
"linear_isometry_equiv.symm_apply_apply"
] | Writing explicitly the derivative of the `n`-th derivative as the composition of a currying
linear equiv, and the `n + 1`-th derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_compact_support.iterated_fderiv (hf : has_compact_support f) (n : ℕ) :
has_compact_support (iterated_fderiv 𝕜 n f) | begin
induction n with n IH,
{ rw [iterated_fderiv_zero_eq_comp],
apply hf.comp_left,
exact linear_isometry_equiv.map_zero _ },
{ rw iterated_fderiv_succ_eq_comp_left,
apply (IH.fderiv 𝕜).comp_left,
exact linear_isometry_equiv.map_zero _ }
end | lemma | has_compact_support.iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv",
"iterated_fderiv_succ_eq_comp_left",
"iterated_fderiv_zero_eq_comp",
"linear_isometry_equiv.map_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_fderiv_iterated_fderiv {n : ℕ} :
‖fderiv 𝕜 (iterated_fderiv 𝕜 n f) x‖ = ‖iterated_fderiv 𝕜 (n + 1) f x‖ | by rw [iterated_fderiv_succ_eq_comp_left, linear_isometry_equiv.norm_map] | lemma | norm_fderiv_iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv",
"iterated_fderiv_succ_eq_comp_left",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_univ {n : ℕ} :
iterated_fderiv_within 𝕜 n f univ = iterated_fderiv 𝕜 n f | begin
induction n with n IH,
{ ext x, simp },
{ ext x m,
rw [iterated_fderiv_succ_apply_left, iterated_fderiv_within_succ_apply_left, IH,
fderiv_within_univ] }
end | lemma | iterated_fderiv_within_univ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv_within_univ",
"iterated_fderiv",
"iterated_fderiv_succ_apply_left",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_apply_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_of_is_open (n : ℕ) (hs : is_open s) :
eq_on (iterated_fderiv_within 𝕜 n f s) (iterated_fderiv 𝕜 n f) s | begin
induction n with n IH,
{ assume x hx,
ext1 m,
simp only [iterated_fderiv_within_zero_apply, iterated_fderiv_zero_apply] },
{ assume x hx,
rw [iterated_fderiv_succ_eq_comp_left, iterated_fderiv_within_succ_eq_comp_left],
dsimp,
congr' 1,
rw fderiv_within_of_open hs hx,
apply filte... | lemma | iterated_fderiv_within_of_is_open | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv_within_of_open",
"filter.eventually_eq.fderiv_eq",
"is_open",
"iterated_fderiv",
"iterated_fderiv_succ_eq_comp_left",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_eq_comp_left",
"iterated_fderiv_within_zero_apply",
"iterated_fderiv_zero_apply"
] | In an open set, the iterated derivative within this set coincides with the global iterated
derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ftaylor_series_within_univ :
ftaylor_series_within 𝕜 f univ = ftaylor_series 𝕜 f | begin
ext1 x, ext1 n,
change iterated_fderiv_within 𝕜 n f univ x = iterated_fderiv 𝕜 n f x,
rw iterated_fderiv_within_univ
end | lemma | ftaylor_series_within_univ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"ftaylor_series",
"ftaylor_series_within",
"iterated_fderiv",
"iterated_fderiv_within",
"iterated_fderiv_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_succ_apply_right {n : ℕ} (m : fin (n + 1) → E) :
(iterated_fderiv 𝕜 (n + 1) f x : (fin (n + 1) → E) → F) m
= iterated_fderiv 𝕜 n (λy, fderiv 𝕜 f y) x (init m) (m (last n)) | begin
rw [← iterated_fderiv_within_univ, ← iterated_fderiv_within_univ, ← fderiv_within_univ],
exact iterated_fderiv_within_succ_apply_right unique_diff_on_univ (mem_univ _) _
end | theorem | iterated_fderiv_succ_apply_right | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv",
"fderiv_within_univ",
"iterated_fderiv",
"iterated_fderiv_within_succ_apply_right",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_succ_eq_comp_right {n : ℕ} :
iterated_fderiv 𝕜 (n + 1) f x =
((continuous_multilinear_curry_right_equiv' 𝕜 n E F)
∘ (iterated_fderiv 𝕜 n (λy, fderiv 𝕜 f y))) x | by { ext m, rw iterated_fderiv_succ_apply_right, refl } | lemma | iterated_fderiv_succ_eq_comp_right | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_right_equiv'",
"fderiv",
"iterated_fderiv",
"iterated_fderiv_succ_apply_right"
] | Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
and the `n`-th derivative of the derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_fderiv {n : ℕ} :
‖iterated_fderiv 𝕜 n (fderiv 𝕜 f) x‖ = ‖iterated_fderiv 𝕜 (n + 1) f x‖ | by rw [iterated_fderiv_succ_eq_comp_right, linear_isometry_equiv.norm_map] | lemma | norm_iterated_fderiv_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv",
"iterated_fderiv_succ_eq_comp_right",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_one_apply (m : (fin 1) → E) :
(iterated_fderiv 𝕜 1 f x : ((fin 1) → E) → F) m
= (fderiv 𝕜 f x : E → F) (m 0) | by { rw [iterated_fderiv_succ_apply_right, iterated_fderiv_zero_apply], refl } | lemma | iterated_fderiv_one_apply | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv",
"iterated_fderiv",
"iterated_fderiv_succ_apply_right",
"iterated_fderiv_zero_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_iff_ftaylor_series :
cont_diff 𝕜 n f ↔ has_ftaylor_series_up_to n f (ftaylor_series 𝕜 f) | begin
split,
{ rw [← cont_diff_on_univ, ← has_ftaylor_series_up_to_on_univ_iff,
← ftaylor_series_within_univ],
exact λ h, cont_diff_on.ftaylor_series_within h unique_diff_on_univ },
{ assume h, exact ⟨ftaylor_series 𝕜 f, h⟩ }
end | theorem | cont_diff_iff_ftaylor_series | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on.ftaylor_series_within",
"cont_diff_on_univ",
"ftaylor_series",
"ftaylor_series_within_univ",
"has_ftaylor_series_up_to",
"has_ftaylor_series_up_to_on_univ_iff",
"unique_diff_on_univ"
] | When a function is `C^n` in a set `s` of unique differentiability, it admits
`ftaylor_series_within 𝕜 f s` as a Taylor series up to order `n` in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_iff_continuous_differentiable :
cont_diff 𝕜 n f ↔
(∀ (m : ℕ), (m : ℕ∞) ≤ n → continuous (λ x, iterated_fderiv 𝕜 m f x))
∧ (∀ (m : ℕ), (m : ℕ∞) < n → differentiable 𝕜 (λ x, iterated_fderiv 𝕜 m f x)) | by simp [cont_diff_on_univ.symm, continuous_iff_continuous_on_univ,
differentiable_on_univ.symm, iterated_fderiv_within_univ,
cont_diff_on_iff_continuous_on_differentiable_on unique_diff_on_univ] | lemma | cont_diff_iff_continuous_differentiable | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_iff_continuous_on_differentiable_on",
"continuous",
"continuous_iff_continuous_on_univ",
"differentiable",
"iterated_fderiv",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.continuous_iterated_fderiv {m : ℕ} (hm : (m : ℕ∞) ≤ n)
(hf : cont_diff 𝕜 n f) : continuous (λ x, iterated_fderiv 𝕜 m f x) | (cont_diff_iff_continuous_differentiable.mp hf).1 m hm | lemma | cont_diff.continuous_iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"continuous",
"iterated_fderiv"
] | If `f` is `C^n` then its `m`-times iterated derivative is continuous for `m ≤ n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.differentiable_iterated_fderiv {m : ℕ} (hm : (m : ℕ∞) < n)
(hf : cont_diff 𝕜 n f) : differentiable 𝕜 (λ x, iterated_fderiv 𝕜 m f x) | (cont_diff_iff_continuous_differentiable.mp hf).2 m hm | lemma | cont_diff.differentiable_iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"differentiable",
"iterated_fderiv"
] | If `f` is `C^n` then its `m`-times iterated derivative is differentiable for `m < n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_of_differentiable_iterated_fderiv
(h : ∀(m : ℕ), (m : ℕ∞) ≤ n → differentiable 𝕜 (iterated_fderiv 𝕜 m f)) :
cont_diff 𝕜 n f | cont_diff_iff_continuous_differentiable.2
⟨λ m hm, (h m hm).continuous, λ m hm, (h m (le_of_lt hm))⟩ | lemma | cont_diff_of_differentiable_iterated_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"continuous",
"differentiable",
"iterated_fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_succ_iff_fderiv {n : ℕ} :
cont_diff 𝕜 ((n + 1) : ℕ) f ↔
differentiable 𝕜 f ∧ cont_diff 𝕜 n (λ y, fderiv 𝕜 f y) | by simp only [← cont_diff_on_univ, ← differentiable_on_univ, ← fderiv_within_univ,
cont_diff_on_succ_iff_fderiv_within unique_diff_on_univ] | theorem | cont_diff_succ_iff_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_succ_iff_fderiv_within",
"cont_diff_on_univ",
"differentiable",
"differentiable_on_univ",
"fderiv",
"fderiv_within_univ",
"unique_diff_on_univ"
] | A function is `C^(n + 1)` if and only if it is differentiable,
and its derivative (formulated in terms of `fderiv`) is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_one_iff_fderiv :
cont_diff 𝕜 1 f ↔ differentiable 𝕜 f ∧ continuous (fderiv 𝕜 f) | cont_diff_succ_iff_fderiv.trans $ iff.rfl.and cont_diff_zero | theorem | cont_diff_one_iff_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_zero",
"continuous",
"differentiable",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_top_iff_fderiv :
cont_diff 𝕜 ∞ f ↔
differentiable 𝕜 f ∧ cont_diff 𝕜 ∞ (λ y, fderiv 𝕜 f y) | begin
simp only [← cont_diff_on_univ, ← differentiable_on_univ, ← fderiv_within_univ],
rw cont_diff_on_top_iff_fderiv_within unique_diff_on_univ,
end | theorem | cont_diff_top_iff_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_on_top_iff_fderiv_within",
"cont_diff_on_univ",
"differentiable",
"differentiable_on_univ",
"fderiv",
"fderiv_within_univ",
"unique_diff_on_univ"
] | A function is `C^∞` if and only if it is differentiable,
and its derivative (formulated in terms of `fderiv`) is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.continuous_fderiv
(h : cont_diff 𝕜 n f) (hn : 1 ≤ n) :
continuous (λ x, fderiv 𝕜 f x) | ((cont_diff_succ_iff_fderiv).1 (h.of_le hn)).2.continuous | lemma | cont_diff.continuous_fderiv | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"cont_diff_succ_iff_fderiv",
"continuous",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.continuous_fderiv_apply
(h : cont_diff 𝕜 n f) (hn : 1 ≤ n) :
continuous (λp : E × E, (fderiv 𝕜 f p.1 : E → F) p.2) | have A : continuous (λq : (E →L[𝕜] F) × E, q.1 q.2) := is_bounded_bilinear_map_apply.continuous,
have B : continuous (λp : E × E, (fderiv 𝕜 f p.1, p.2)) :=
((h.continuous_fderiv hn).comp continuous_fst).prod_mk continuous_snd,
A.comp B | lemma | cont_diff.continuous_fderiv_apply | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff",
"continuous",
"continuous_fst",
"continuous_snd",
"fderiv"
] | If a function is at least `C^1`, its bundled derivative (mapping `(x, v)` to `Df(x) v`) is
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_has_deriv_within_at_eq_of_gt_of_lt
(hab : a ≤ b) (hf : ∀ x ∈ (Icc a b), has_deriv_within_at f (f' x) (Icc a b) x)
{m : ℝ} (hma : f' a < m) (hmb : m < f' b) :
m ∈ f' '' Ioo a b | begin
rcases hab.eq_or_lt with rfl | hab',
{ exact (lt_asymm hma hmb).elim },
set g : ℝ → ℝ := λ x, f x - m * x,
have hg : ∀ x ∈ Icc a b, has_deriv_within_at g (f' x - m) (Icc a b) x,
{ intros x hx,
simpa using (hf x hx).sub ((has_deriv_within_at_id x _).const_mul m) },
obtain ⟨c, cmem, hc⟩ : ∃ c ∈ Icc ... | theorem | exists_has_deriv_within_at_eq_of_gt_of_lt | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"continuous_linear_map.map_sub",
"continuous_within_at",
"has_deriv_at",
"has_deriv_within_at",
"has_deriv_within_at_id",
"interior_Icc",
"is_min_on",
"mem_interior_iff_mem_nhds",
"mem_pos_tangent_cone_at_of_segment_subset",
"nonneg_of_mul_nonneg_right",
"nonpos_of_mul_nonneg_right",
"not_le_o... | **Darboux's theorem**: if `a ≤ b` and `f' a < m < f' b`, then `f' c = m` for some
`c ∈ (a, b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_has_deriv_within_at_eq_of_lt_of_gt
(hab : a ≤ b) (hf : ∀ x ∈ (Icc a b), has_deriv_within_at f (f' x) (Icc a b) x)
{m : ℝ} (hma : m < f' a) (hmb : f' b < m) :
m ∈ f' '' Ioo a b | let ⟨c, cmem, hc⟩ := exists_has_deriv_within_at_eq_of_gt_of_lt hab (λ x hx, (hf x hx).neg)
(neg_lt_neg hma) (neg_lt_neg hmb)
in ⟨c, cmem, neg_injective hc⟩ | theorem | exists_has_deriv_within_at_eq_of_lt_of_gt | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"exists_has_deriv_within_at_eq_of_gt_of_lt",
"has_deriv_within_at"
] | **Darboux's theorem**: if `a ≤ b` and `f' b < m < f' a`, then `f' c = m` for some `c ∈ (a, b)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.ord_connected.image_has_deriv_within_at {s : set ℝ} (hs : ord_connected s)
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) :
ord_connected (f' '' s) | begin
apply ord_connected_of_Ioo,
rintros _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ - m ⟨hma, hmb⟩,
cases le_total a b with hab hab,
{ have : Icc a b ⊆ s, from hs.out ha hb,
rcases exists_has_deriv_within_at_eq_of_gt_of_lt hab
(λ x hx, (hf x $ this hx).mono this) hma hmb
with ⟨c, cmem, hc⟩,
exact ⟨c, th... | theorem | set.ord_connected.image_has_deriv_within_at | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"exists_has_deriv_within_at_eq_of_gt_of_lt",
"exists_has_deriv_within_at_eq_of_lt_of_gt",
"has_deriv_within_at"
] | **Darboux's theorem**: the image of an `ord_connected` set under `f'` is an `ord_connected`
set, `has_deriv_within_at` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.ord_connected.image_deriv_within {s : set ℝ} (hs : ord_connected s)
(hf : differentiable_on ℝ f s) :
ord_connected (deriv_within f s '' s) | hs.image_has_deriv_within_at $ λ x hx, (hf x hx).has_deriv_within_at | theorem | set.ord_connected.image_deriv_within | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"deriv_within",
"differentiable_on",
"has_deriv_within_at"
] | **Darboux's theorem**: the image of an `ord_connected` set under `f'` is an `ord_connected`
set, `deriv_within` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.ord_connected.image_deriv {s : set ℝ} (hs : ord_connected s)
(hf : ∀ x ∈ s, differentiable_at ℝ f x) :
ord_connected (deriv f '' s) | hs.image_has_deriv_within_at $ λ x hx, (hf x hx).has_deriv_at.has_deriv_within_at | theorem | set.ord_connected.image_deriv | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"deriv",
"differentiable_at",
"has_deriv_at.has_deriv_within_at"
] | **Darboux's theorem**: the image of an `ord_connected` set under `f'` is an `ord_connected`
set, `deriv` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.image_has_deriv_within_at {s : set ℝ} (hs : convex ℝ s)
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) :
convex ℝ (f' '' s) | (hs.ord_connected.image_has_deriv_within_at hf).convex | theorem | convex.image_has_deriv_within_at | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"convex",
"has_deriv_within_at"
] | **Darboux's theorem**: the image of a convex set under `f'` is a convex set,
`has_deriv_within_at` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.image_deriv_within {s : set ℝ} (hs : convex ℝ s)
(hf : differentiable_on ℝ f s) :
convex ℝ (deriv_within f s '' s) | (hs.ord_connected.image_deriv_within hf).convex | theorem | convex.image_deriv_within | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"convex",
"deriv_within",
"differentiable_on"
] | **Darboux's theorem**: the image of a convex set under `f'` is a convex set,
`deriv_within` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
convex.image_deriv {s : set ℝ} (hs : convex ℝ s)
(hf : ∀ x ∈ s, differentiable_at ℝ f x) :
convex ℝ (deriv f '' s) | (hs.ord_connected.image_deriv hf).convex | theorem | convex.image_deriv | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"convex",
"deriv",
"differentiable_at"
] | **Darboux's theorem**: the image of a convex set under `f'` is a convex set,
`deriv` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_has_deriv_within_at_eq_of_ge_of_le
(hab : a ≤ b) (hf : ∀ x ∈ (Icc a b), has_deriv_within_at f (f' x) (Icc a b) x)
{m : ℝ} (hma : f' a ≤ m) (hmb : m ≤ f' b) :
m ∈ f' '' Icc a b | (ord_connected_Icc.image_has_deriv_within_at hf).out
(mem_image_of_mem _ (left_mem_Icc.2 hab)) (mem_image_of_mem _ (right_mem_Icc.2 hab)) ⟨hma, hmb⟩ | theorem | exists_has_deriv_within_at_eq_of_ge_of_le | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"has_deriv_within_at"
] | **Darboux's theorem**: if `a ≤ b` and `f' a ≤ m ≤ f' b`, then `f' c = m` for some
`c ∈ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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