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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map_implicit_function_eq (hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) :
∀ᶠ (p : F × (ker f')) in 𝓝 (f a, 0), f (hf.implicit_function f f' hf' p.1 p.2) = p.1 | by apply map_implicit_function_of_complemented_eq | lemma | has_strict_fderiv_at.map_implicit_function_eq | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | `implicit_function` sends `(z, y)` to a point in `f ⁻¹' z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
implicit_function_apply_image (hf : has_strict_fderiv_at f f' a)
(hf' : range f' = ⊤) :
hf.implicit_function f f' hf' (f a) 0 = a | by apply implicit_function_of_complemented_apply_image | lemma | has_strict_fderiv_at.implicit_function_apply_image | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_implicit_function (hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) :
∀ᶠ x in 𝓝 a, hf.implicit_function f f' hf' (f x)
(hf.implicit_to_local_homeomorph f f' hf' x).snd = x | by apply eq_implicit_function_of_complemented | lemma | has_strict_fderiv_at.eq_implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | Any point in some neighborhood of `a` can be represented as `implicit_function`
of some point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_implicit_function (hf : has_strict_fderiv_at f f' a) (hf' : range f' = ⊤) :
has_strict_fderiv_at (hf.implicit_function f f' hf' (f a))
(ker f').subtypeL 0 | by apply to_implicit_function_of_complemented | lemma | has_strict_fderiv_at.to_implicit_function | analysis.calculus | src/analysis/calculus/implicit.lean | [
"analysis.calculus.inverse",
"analysis.normed_space.complemented"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approximates_linear_on (f : E → F) (f' : E →L[𝕜] F) (s : set E) (c : ℝ≥0) : Prop | ∀ (x ∈ s) (y ∈ s), ‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖ | def | approximates_linear_on | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [] | We say that `f` approximates a continuous linear map `f'` on `s` with constant `c`,
if `‖f x - f y - f' (x - y)‖ ≤ c * ‖x - y‖` whenever `x, y ∈ s`.
This predicate is defined to facilitate the splitting of the inverse function theorem into small
lemmas. Some of these lemmas can be useful, e.g., to prove that the inver... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
approximates_linear_on_empty (f : E → F) (f' : E →L[𝕜] F) (c : ℝ≥0) :
approximates_linear_on f f' ∅ c | by simp [approximates_linear_on] | lemma | approximates_linear_on_empty | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_num (hc : c ≤ c') (hf : approximates_linear_on f f' s c) :
approximates_linear_on f f' s c' | λ x hx y hy, le_trans (hf x hx y hy) (mul_le_mul_of_nonneg_right hc $ norm_nonneg _) | theorem | approximates_linear_on.mono_num | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono_set (hst : s ⊆ t) (hf : approximates_linear_on f f' t c) :
approximates_linear_on f f' s c | λ x hx y hy, hf x (hst hx) y (hst hy) | theorem | approximates_linear_on.mono_set | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approximates_linear_on_iff_lipschitz_on_with
{f : E → F} {f' : E →L[𝕜] F} {s : set E} {c : ℝ≥0} :
approximates_linear_on f f' s c ↔ lipschitz_on_with c (f - f') s | begin
have : ∀ x y, f x - f y - f' (x - y) = (f - f') x - (f - f') y,
{ assume x y, simp only [map_sub, pi.sub_apply], abel },
simp only [this, lipschitz_on_with_iff_norm_sub_le, approximates_linear_on],
end | lemma | approximates_linear_on.approximates_linear_on_iff_lipschitz_on_with | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"lipschitz_on_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz_sub (hf : approximates_linear_on f f' s c) :
lipschitz_with c (λ x : s, f x - f' x) | begin
refine lipschitz_with.of_dist_le_mul (λ x y, _),
rw [dist_eq_norm, subtype.dist_eq, dist_eq_norm],
convert hf x x.2 y y.2 using 2,
rw [f'.map_sub], abel
end | lemma | approximates_linear_on.lipschitz_sub | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"lipschitz_with",
"subtype.dist_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lipschitz (hf : approximates_linear_on f f' s c) :
lipschitz_with (‖f'‖₊ + c) (s.restrict f) | by simpa only [restrict_apply, add_sub_cancel'_right]
using (f'.lipschitz.restrict s).add hf.lipschitz_sub | lemma | approximates_linear_on.lipschitz | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"lipschitz_with"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (hf : approximates_linear_on f f' s c) :
continuous (s.restrict f) | hf.lipschitz.continuous | lemma | approximates_linear_on.continuous | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on (hf : approximates_linear_on f f' s c) :
continuous_on f s | continuous_on_iff_continuous_restrict.2 hf.continuous | lemma | approximates_linear_on.continuous_on | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_closed_ball_of_nonlinear_right_inverse
(hf : approximates_linear_on f f' s c) (f'symm : f'.nonlinear_right_inverse)
{ε : ℝ} {b : E} (ε0 : 0 ≤ ε) (hε : closed_ball b ε ⊆ s) :
surj_on f (closed_ball b ε) (closed_ball (f b) (((f'symm.nnnorm : ℝ)⁻¹ - c) * ε)) | begin
assume y hy,
cases le_or_lt (f'symm.nnnorm : ℝ) ⁻¹ c with hc hc,
{ refine ⟨b, by simp [ε0], _⟩,
have : dist y (f b) ≤ 0 :=
(mem_closed_ball.1 hy).trans (mul_nonpos_of_nonpos_of_nonneg (by linarith) ε0),
simp only [dist_le_zero] at this,
rw this },
have If' : (0 : ℝ) < f'symm.nnnorm,
... | theorem | approximates_linear_on.surj_on_closed_ball_of_nonlinear_right_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"cauchy_seq",
"cauchy_seq_of_le_geometric",
"cauchy_seq_tendsto_of_complete",
"continuous_linear_map.nonlinear_right_inverse.right_inv",
"dist_comm",
"dist_le_zero",
"dist_nonneg",
"dist_triangle",
"div_le_iff",
"div_mul_eq_mul_div",
"ge_iff_le",
"inv_eq_one_div",
... | If a function is linearly approximated by a continuous linear map with a (possibly nonlinear)
right inverse, then it is locally onto: a ball of an explicit radius is included in the image
of the map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_image (hf : approximates_linear_on f f' s c) (f'symm : f'.nonlinear_right_inverse)
(hs : is_open s) (hc : subsingleton F ∨ c < f'symm.nnnorm⁻¹) : is_open (f '' s) | begin
cases hc with hE hc, { resetI, apply is_open_discrete },
simp only [is_open_iff_mem_nhds, nhds_basis_closed_ball.mem_iff, ball_image_iff] at hs ⊢,
intros x hx,
rcases hs x hx with ⟨ε, ε0, hε⟩,
refine ⟨(f'symm.nnnorm⁻¹ - c) * ε, mul_pos (sub_pos.2 hc) ε0, _⟩,
exact (hf.surj_on_closed_ball_of_nonlinear_... | lemma | approximates_linear_on.open_image | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open",
"is_open_discrete",
"is_open_iff_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_nhds (hf : approximates_linear_on f f' s c) (f'symm : f'.nonlinear_right_inverse)
{x : E} (hs : s ∈ 𝓝 x) (hc : subsingleton F ∨ c < f'symm.nnnorm⁻¹) :
f '' s ∈ 𝓝 (f x) | begin
obtain ⟨t, hts, ht, xt⟩ : ∃ t ⊆ s, is_open t ∧ x ∈ t := _root_.mem_nhds_iff.1 hs,
have := is_open.mem_nhds ((hf.mono_set hts).open_image f'symm ht hc) (mem_image_of_mem _ xt),
exact mem_of_superset this (image_subset _ hts),
end | lemma | approximates_linear_on.image_mem_nhds | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_eq (hf : approximates_linear_on f f' s c) (f'symm : f'.nonlinear_right_inverse)
{x : E} (hs : s ∈ 𝓝 x) (hc : subsingleton F ∨ c < f'symm.nnnorm⁻¹) :
map f (𝓝 x) = 𝓝 (f x) | begin
refine le_antisymm ((hf.continuous_on x (mem_of_mem_nhds hs)).continuous_at hs)
(le_map (λ t ht, _)),
have : f '' (s ∩ t) ∈ 𝓝 (f x) := (hf.mono_set (inter_subset_left s t)).image_mem_nhds
f'symm (inter_mem hs ht) hc,
exact mem_of_superset this (image_subset _ (inter_subset_right _ _)),
end | lemma | approximates_linear_on.map_nhds_eq | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"continuous_at",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
antilipschitz (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
antilipschitz_with (N⁻¹ - c)⁻¹ (s.restrict f) | begin
cases hc with hE hc,
{ haveI : subsingleton s := ⟨λ x y, subtype.eq $ @subsingleton.elim _ hE _ _⟩,
exact antilipschitz_with.of_subsingleton },
convert (f'.antilipschitz.restrict s).add_lipschitz_with hf.lipschitz_sub hc,
simp [restrict]
end | lemma | approximates_linear_on.antilipschitz | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"antilipschitz_with",
"antilipschitz_with.of_subsingleton",
"approximates_linear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
injective (s.restrict f) | (hf.antilipschitz hc).injective | lemma | approximates_linear_on.injective | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_on (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
inj_on f s | inj_on_iff_injective.2 $ hf.injective hc | lemma | approximates_linear_on.inj_on | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective [complete_space E]
(hf : approximates_linear_on f (f' : E →L[𝕜] F) univ c) (hc : subsingleton E ∨ c < N⁻¹) :
surjective f | begin
cases hc with hE hc,
{ haveI : subsingleton F := (equiv.subsingleton_congr f'.to_linear_equiv.to_equiv).1 hE,
exact surjective_to_subsingleton _ },
{ apply forall_of_forall_mem_closed_ball (λ (y : F), ∃ a, f a = y) (f 0) _,
have hc' : (0 : ℝ) < N⁻¹ - c, by { rw sub_pos, exact hc },
let p : ℝ → P... | lemma | approximates_linear_on.surjective | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"complete_space",
"equiv.subsingleton_congr",
"set.range",
"surjective_to_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_equiv (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) : local_equiv E F | (hf.inj_on hc).to_local_equiv _ _ | def | approximates_linear_on.to_local_equiv | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"local_equiv"
] | A map approximating a linear equivalence on a set defines a local equivalence on this set.
Should not be used outside of this file, because it is superseded by `to_local_homeomorph` below.
This is a first step towards the inverse function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inverse_continuous_on (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
continuous_on (hf.to_local_equiv hc).symm (f '' s) | begin
apply continuous_on_iff_continuous_restrict.2,
refine ((hf.antilipschitz hc).to_right_inv_on' _ (hf.to_local_equiv hc).right_inv').continuous,
exact (λ x hx, (hf.to_local_equiv hc).map_target hx)
end | lemma | approximates_linear_on.inverse_continuous_on | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"continuous",
"continuous_on"
] | The inverse function is continuous on `f '' s`. Use properties of `local_homeomorph` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_inv (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) :
approximates_linear_on (hf.to_local_equiv hc).symm (f'.symm : F →L[𝕜] E) (f '' s)
(N * (N⁻¹ - c)⁻¹ * c) | begin
assume x hx y hy,
set A := hf.to_local_equiv hc with hA,
have Af : ∀ z, A z = f z := λ z, rfl,
rcases (mem_image _ _ _).1 hx with ⟨x', x's, rfl⟩,
rcases (mem_image _ _ _).1 hy with ⟨y', y's, rfl⟩,
rw [← Af x', ← Af y', A.left_inv x's, A.left_inv y's],
calc ‖x' - y' - (f'.symm) (A x' - A y')‖
≤... | lemma | approximates_linear_on.to_inv | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"continuous_linear_equiv.apply_symm_apply",
"continuous_linear_equiv.map_sub",
"mul_le_mul_of_nonneg_left",
"nnreal.coe_nonneg",
"nonneg.coe_mul",
"ring"
] | The inverse function is approximated linearly on `f '' s` by `f'.symm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_homeomorph (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) : local_homeomorph E F | { to_local_equiv := hf.to_local_equiv hc,
open_source := hs,
open_target := hf.open_image f'.to_nonlinear_right_inverse hs
(by rwa f'.to_linear_equiv.to_equiv.subsingleton_congr at hc),
continuous_to_fun := hf.continuous_on,
continuous_inv_fun := hf.inverse_continuous_on hc } | def | approximates_linear_on.to_local_homeomorph | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open",
"local_homeomorph"
] | Given a function `f` that approximates a linear equivalence on an open set `s`,
returns a local homeomorph with `to_fun = f` and `source = s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_homeomorph (hf : approximates_linear_on f (f' : E →L[𝕜] F) univ c)
(hc : subsingleton E ∨ c < N⁻¹) :
E ≃ₜ F | begin
refine (hf.to_local_homeomorph _ _ hc is_open_univ).to_homeomorph_of_source_eq_univ_target_eq_univ
rfl _,
change f '' univ = univ,
rw [image_univ, range_iff_surjective],
exact hf.surjective hc,
end | def | approximates_linear_on.to_homeomorph | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open_univ"
] | A function `f` that approximates a linear equivalence on the whole space is a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_homeomorph_extension {E : Type*} [normed_add_comm_group E] [normed_space ℝ E]
{F : Type*} [normed_add_comm_group F] [normed_space ℝ F] [finite_dimensional ℝ F]
{s : set E} {f : E → F} {f' : E ≃L[ℝ] F} {c : ℝ≥0}
(hf : approximates_linear_on f (f' : E →L[ℝ] F) s c)
(hc : subsingleton E ∨ lipschitz_extensio... | begin
-- the difference `f - f'` is Lipschitz on `s`. It can be extended to a Lipschitz function `u`
-- on the whole space, with a slightly worse Lipschitz constant. Then `f' + u` will be the
-- desired homeomorphism.
obtain ⟨u, hu, uf⟩ : ∃ (u : E → F), lipschitz_with (lipschitz_extension_constant F * c) u
... | lemma | approximates_linear_on.exists_homeomorph_extension | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"continuous_linear_equiv.coe_coe",
"finite_dimensional",
"lipschitz_extension_constant",
"lipschitz_on_univ",
"lipschitz_with",
"normed_add_comm_group",
"normed_space"
] | In a real vector space, a function `f` that approximates a linear equivalence on a subset `s`
can be extended to a homeomorphism of the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_homeomorph_coe (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) :
(hf.to_local_homeomorph f s hc hs : E → F) = f | rfl | lemma | approximates_linear_on.to_local_homeomorph_coe | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_homeomorph_source (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) :
(hf.to_local_homeomorph f s hc hs).source = s | rfl | lemma | approximates_linear_on.to_local_homeomorph_source | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_homeomorph_target (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) :
(hf.to_local_homeomorph f s hc hs).target = f '' s | rfl | lemma | approximates_linear_on.to_local_homeomorph_target | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_subset_target (hf : approximates_linear_on f (f' : E →L[𝕜] F) s c)
(hc : subsingleton E ∨ c < N⁻¹) (hs : is_open s) {b : E} (ε0 : 0 ≤ ε) (hε : closed_ball b ε ⊆ s) :
closed_ball (f b) ((N⁻¹ - c) * ε) ⊆ (hf.to_local_homeomorph f s hc hs).target | (hf.surj_on_closed_ball_of_nonlinear_right_inverse f'.to_nonlinear_right_inverse
ε0 hε).mono hε (subset.refl _) | lemma | approximates_linear_on.closed_ball_subset_target | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approximates_deriv_on_nhds {f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : has_strict_fderiv_at f f' a) {c : ℝ≥0} (hc : subsingleton E ∨ 0 < c) :
∃ s ∈ 𝓝 a, approximates_linear_on f f' s c | begin
cases hc with hE hc,
{ refine ⟨univ, is_open.mem_nhds is_open_univ trivial, λ x hx y hy, _⟩,
simp [@subsingleton.elim E hE x y] },
have := hf.def hc,
rw [nhds_prod_eq, filter.eventually, mem_prod_same_iff] at this,
rcases this with ⟨s, has, hs⟩,
exact ⟨s, has, λ x hx y hy, hs (mk_mem_prod hx hy)⟩
... | lemma | has_strict_fderiv_at.approximates_deriv_on_nhds | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"filter.eventually",
"has_strict_fderiv_at",
"is_open.mem_nhds",
"is_open_univ",
"nhds_prod_eq"
] | If `f` has derivative `f'` at `a` in the strict sense and `c > 0`, then `f` approximates `f'`
with constant `c` on some neighborhood of `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_nhds_eq_of_surj [complete_space E] [complete_space F]
{f : E → F} {f' : E →L[𝕜] F} {a : E}
(hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) (h : linear_map.range f' = ⊤) :
map f (𝓝 a) = 𝓝 (f a) | begin
let f'symm := f'.nonlinear_right_inverse_of_surjective h,
set c : ℝ≥0 := f'symm.nnnorm⁻¹ / 2 with hc,
have f'symm_pos : 0 < f'symm.nnnorm := f'.nonlinear_right_inverse_of_surjective_nnnorm_pos h,
have cpos : 0 < c, by simp [hc, half_pos, inv_pos, f'symm_pos],
obtain ⟨s, s_nhds, hs⟩ : ∃ s ∈ 𝓝 a, approxi... | lemma | has_strict_fderiv_at.map_nhds_eq_of_surj | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"complete_space",
"half_pos",
"has_strict_fderiv_at",
"inv_pos",
"linear_map.range",
"nnreal.half_lt_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approximates_deriv_on_open_nhds (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
∃ (s : set E) (hs : a ∈ s ∧ is_open s),
approximates_linear_on f (f' : E →L[𝕜] F) s (‖(f'.symm : F →L[𝕜] E)‖₊⁻¹ / 2) | begin
refine ((nhds_basis_opens a).exists_iff _).1 _,
exact (λ s t, approximates_linear_on.mono_set),
exact (hf.approximates_deriv_on_nhds $ f'.subsingleton_or_nnnorm_symm_pos.imp id $
λ hf', half_pos $ inv_pos.2 hf')
end | lemma | has_strict_fderiv_at.approximates_deriv_on_open_nhds | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on",
"approximates_linear_on.mono_set",
"half_pos",
"has_strict_fderiv_at",
"is_open",
"nhds_basis_opens"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_homeomorph (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) : local_homeomorph E F | approximates_linear_on.to_local_homeomorph f
(classical.some hf.approximates_deriv_on_open_nhds)
(classical.some_spec hf.approximates_deriv_on_open_nhds).snd
(f'.subsingleton_or_nnnorm_symm_pos.imp id $ λ hf', nnreal.half_lt_self $ ne_of_gt $
inv_pos.2 hf')
(classical.some_spec hf.approximates_deriv_on_open... | def | has_strict_fderiv_at.to_local_homeomorph | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"approximates_linear_on.to_local_homeomorph",
"has_strict_fderiv_at",
"local_homeomorph",
"nnreal.half_lt_self"
] | Given a function with an invertible strict derivative at `a`, returns a `local_homeomorph`
with `to_fun = f` and `a ∈ source`. This is a part of the inverse function theorem.
The other part `has_strict_fderiv_at.to_local_inverse` states that the inverse function
of this `local_homeomorph` has derivative `f'.symm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_homeomorph_coe (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
(hf.to_local_homeomorph f : E → F) = f | rfl | lemma | has_strict_fderiv_at.to_local_homeomorph_coe | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_local_homeomorph_source (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
a ∈ (hf.to_local_homeomorph f).source | (classical.some_spec hf.approximates_deriv_on_open_nhds).fst.1 | lemma | has_strict_fderiv_at.mem_to_local_homeomorph_source | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_to_local_homeomorph_target (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
f a ∈ (hf.to_local_homeomorph f).target | (hf.to_local_homeomorph f).map_source hf.mem_to_local_homeomorph_source | lemma | has_strict_fderiv_at.image_mem_to_local_homeomorph_target | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_eq_of_equiv (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
map f (𝓝 a) = 𝓝 (f a) | (hf.to_local_homeomorph f).map_nhds_eq hf.mem_to_local_homeomorph_source | lemma | has_strict_fderiv_at.map_nhds_eq_of_equiv | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) : F → E | (hf.to_local_homeomorph f).symm | def | has_strict_fderiv_at.local_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | Given a function `f` with an invertible derivative, returns a function that is locally inverse
to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_inverse_def (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
hf.local_inverse f _ _ = (hf.to_local_homeomorph f).symm | rfl | lemma | has_strict_fderiv_at.local_inverse_def | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_left_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
∀ᶠ x in 𝓝 a, hf.local_inverse f f' a (f x) = x | (hf.to_local_homeomorph f).eventually_left_inverse hf.mem_to_local_homeomorph_source | lemma | has_strict_fderiv_at.eventually_left_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_inverse_apply_image (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
hf.local_inverse f f' a (f a) = a | hf.eventually_left_inverse.self_of_nhds | lemma | has_strict_fderiv_at.local_inverse_apply_image | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_right_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
∀ᶠ y in 𝓝 (f a), f (hf.local_inverse f f' a y) = y | (hf.to_local_homeomorph f).eventually_right_inverse' hf.mem_to_local_homeomorph_source | lemma | has_strict_fderiv_at.eventually_right_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_inverse_continuous_at (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
continuous_at (hf.local_inverse f f' a) (f a) | (hf.to_local_homeomorph f).continuous_at_symm hf.image_mem_to_local_homeomorph_target | lemma | has_strict_fderiv_at.local_inverse_continuous_at | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"continuous_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_inverse_tendsto (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
tendsto (hf.local_inverse f f' a) (𝓝 $ f a) (𝓝 a) | (hf.to_local_homeomorph f).tendsto_symm hf.mem_to_local_homeomorph_source | lemma | has_strict_fderiv_at.local_inverse_tendsto | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_inverse_unique (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) {g : F → E}
(hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
∀ᶠ y in 𝓝 (f a), g y = local_inverse f f' a hf y | eventually_eq_of_left_inv_of_right_inv hg hf.eventually_right_inverse $
(hf.to_local_homeomorph f).tendsto_symm hf.mem_to_local_homeomorph_source | lemma | has_strict_fderiv_at.local_inverse_unique | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) :
has_strict_fderiv_at (hf.local_inverse f f' a) (f'.symm : F →L[𝕜] E) (f a) | (hf.to_local_homeomorph f).has_strict_fderiv_at_symm hf.image_mem_to_local_homeomorph_target $
by simpa [← local_inverse_def] using hf | theorem | has_strict_fderiv_at.to_local_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | If `f` has an invertible derivative `f'` at `a` in the sense of strict differentiability `(hf)`,
then the inverse function `hf.local_inverse f` has derivative `f'.symm` at `f a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_left_inverse (hf : has_strict_fderiv_at f (f' : E →L[𝕜] F) a) {g : F → E}
(hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
has_strict_fderiv_at g (f'.symm : F →L[𝕜] E) (f a) | hf.to_local_inverse.congr_of_eventually_eq $ (hf.local_inverse_unique hg).mono $ λ _, eq.symm | theorem | has_strict_fderiv_at.to_local_left_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_fderiv_at"
] | If `f : E → F` has an invertible derivative `f'` at `a` in the sense of strict differentiability
and `g (f x) = x` in a neighborhood of `a`, then `g` has derivative `f'.symm` at `f a`.
For a version assuming `f (g y) = y` and continuity of `g` at `f a` but not `[complete_space E]`
see `of_local_left_inverse`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_map_of_strict_fderiv_equiv [complete_space E] {f : E → F} {f' : E → E ≃L[𝕜] F}
(hf : ∀ x, has_strict_fderiv_at f (f' x : E →L[𝕜] F) x) :
is_open_map f | is_open_map_iff_nhds_le.2 $ λ x, (hf x).map_nhds_eq_of_equiv.ge | lemma | open_map_of_strict_fderiv_equiv | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"complete_space",
"has_strict_fderiv_at",
"is_open_map"
] | If a function has an invertible strict derivative at all points, then it is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_inverse : 𝕜 → 𝕜 | (hf.has_strict_fderiv_at_equiv hf').local_inverse _ _ _ | def | has_strict_deriv_at.local_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [] | A function that is inverse to `f` near `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_nhds_eq : map f (𝓝 a) = 𝓝 (f a) | (hf.has_strict_fderiv_at_equiv hf').map_nhds_eq_of_equiv | lemma | has_strict_deriv_at.map_nhds_eq | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_inverse : has_strict_deriv_at (hf.local_inverse f f' a hf') f'⁻¹ (f a) | (hf.has_strict_fderiv_at_equiv hf').to_local_inverse | theorem | has_strict_deriv_at.to_local_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_left_inverse {g : 𝕜 → 𝕜} (hg : ∀ᶠ x in 𝓝 a, g (f x) = x) :
has_strict_deriv_at g f'⁻¹ (f a) | (hf.has_strict_fderiv_at_equiv hf').to_local_left_inverse hg | theorem | has_strict_deriv_at.to_local_left_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_map_of_strict_deriv [complete_space 𝕜] {f f' : 𝕜 → 𝕜}
(hf : ∀ x, has_strict_deriv_at f (f' x) x) (h0 : ∀ x, f' x ≠ 0) :
is_open_map f | is_open_map_iff_nhds_le.2 $ λ x, ((hf x).map_nhds_eq (h0 x)).ge | lemma | open_map_of_strict_deriv | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"complete_space",
"has_strict_deriv_at",
"is_open_map"
] | If a function has a non-zero strict derivative at all points, then it is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_homeomorph
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
local_homeomorph E' F' | (hf.has_strict_fderiv_at' hf' hn).to_local_homeomorph f | def | cont_diff_at.to_local_homeomorph | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at",
"local_homeomorph"
] | Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible
derivative at `a`, returns a `local_homeomorph` with `to_fun = f` and `a ∈ source`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_local_homeomorph_coe
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
(hf.to_local_homeomorph f hf' hn : E' → F') = f | rfl | lemma | cont_diff_at.to_local_homeomorph_coe | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_to_local_homeomorph_source
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
a ∈ (hf.to_local_homeomorph f hf' hn).source | (hf.has_strict_fderiv_at' hf' hn).mem_to_local_homeomorph_source | lemma | cont_diff_at.mem_to_local_homeomorph_source | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_mem_to_local_homeomorph_target
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
f a ∈ (hf.to_local_homeomorph f hf' hn).target | (hf.has_strict_fderiv_at' hf' hn).image_mem_to_local_homeomorph_target | lemma | cont_diff_at.image_mem_to_local_homeomorph_target | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_inverse
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
F' → E' | (hf.has_strict_fderiv_at' hf' hn).local_inverse f f' a | def | cont_diff_at.local_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at"
] | Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
at `a`, returns a function that is locally inverse to `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_inverse_apply_image
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
hf.local_inverse hf' hn (f a) = a | (hf.has_strict_fderiv_at' hf' hn).local_inverse_apply_image | lemma | cont_diff_at.local_inverse_apply_image | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_local_inverse
{n : ℕ∞} (hf : cont_diff_at 𝕂 n f a) (hf' : has_fderiv_at f (f' : E' →L[𝕂] F') a)
(hn : 1 ≤ n) :
cont_diff_at 𝕂 n (hf.local_inverse hf' hn) (f a) | begin
have := hf.local_inverse_apply_image hf' hn,
apply (hf.to_local_homeomorph f hf' hn).cont_diff_at_symm
(image_mem_to_local_homeomorph_target hf hf' hn),
{ convert hf' },
{ convert hf }
end | lemma | cont_diff_at.to_local_inverse | analysis.calculus | src/analysis/calculus/inverse.lean | [
"analysis.calculus.cont_diff",
"analysis.normed_space.banach"
] | [
"cont_diff_at",
"has_fderiv_at"
] | Given a `cont_diff` function over `𝕂` (which is `ℝ` or `ℂ`) with an invertible derivative
at `a`, the inverse function (produced by `cont_diff.to_local_homeomorph`) is
also `cont_diff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv (n : ℕ) (f : 𝕜 → F) (x : 𝕜) : F | (iterated_fderiv 𝕜 n f x : ((fin n) → 𝕜) → F) (λ(i : fin n), 1) | def | iterated_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_fderiv"
] | The `n`-th iterated derivative of a function from `𝕜` to `F`, as a function from `𝕜` to `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_within (n : ℕ) (f : 𝕜 → F) (s : set 𝕜) (x : 𝕜) : F | (iterated_fderiv_within 𝕜 n f s x : ((fin n) → 𝕜) → F) (λ(i : fin n), 1) | def | iterated_deriv_within | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_fderiv_within"
] | The `n`-th iterated derivative of a function from `𝕜` to `F` within a set `s`, as a function
from `𝕜` to `F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_within_univ :
iterated_deriv_within n f univ = iterated_deriv n f | by { ext x, rw [iterated_deriv_within, iterated_deriv, iterated_fderiv_within_univ] } | lemma | iterated_deriv_within_univ | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv",
"iterated_deriv_within",
"iterated_fderiv_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_within_eq_iterated_fderiv_within :
iterated_deriv_within n f s x
= (iterated_fderiv_within 𝕜 n f s x : ((fin n) → 𝕜) → F) (λ(i : fin n), 1) | rfl | lemma | iterated_deriv_within_eq_iterated_fderiv_within | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv_within",
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_within_eq_equiv_comp :
iterated_deriv_within n f s
= (continuous_multilinear_map.pi_field_equiv 𝕜 (fin n) F).symm ∘
(iterated_fderiv_within 𝕜 n f s) | by { ext x, refl } | lemma | iterated_deriv_within_eq_equiv_comp | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.pi_field_equiv",
"iterated_deriv_within",
"iterated_fderiv_within"
] | Write the iterated derivative as the composition of a continuous linear equiv and the iterated
Fréchet derivative | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_eq_equiv_comp :
iterated_fderiv_within 𝕜 n f s
= (continuous_multilinear_map.pi_field_equiv 𝕜 (fin n) F) ∘ (iterated_deriv_within n f s) | by rw [iterated_deriv_within_eq_equiv_comp, ← function.comp.assoc,
linear_isometry_equiv.self_comp_symm, function.left_id] | lemma | iterated_fderiv_within_eq_equiv_comp | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.pi_field_equiv",
"iterated_deriv_within",
"iterated_deriv_within_eq_equiv_comp",
"iterated_fderiv_within",
"linear_isometry_equiv.self_comp_symm"
] | Write the iterated Fréchet derivative as the composition of a continuous linear equiv and the
iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_apply_eq_iterated_deriv_within_mul_prod {m : (fin n) → 𝕜} :
(iterated_fderiv_within 𝕜 n f s x : ((fin n) → 𝕜) → F) m
= (∏ i, m i) • iterated_deriv_within n f s x | begin
rw [iterated_deriv_within_eq_iterated_fderiv_within, ← continuous_multilinear_map.map_smul_univ],
simp
end | lemma | iterated_fderiv_within_apply_eq_iterated_deriv_within_mul_prod | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.map_smul_univ",
"iterated_deriv_within",
"iterated_deriv_within_eq_iterated_fderiv_within",
"iterated_fderiv_within"
] | The `n`-th Fréchet derivative applied to a vector `(m 0, ..., m (n-1))` is the derivative
multiplied by the product of the `m i`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_within_eq_norm_iterated_deriv_within :
‖iterated_fderiv_within 𝕜 n f s x‖ = ‖iterated_deriv_within n f s x‖ | by rw [iterated_deriv_within_eq_equiv_comp, linear_isometry_equiv.norm_map] | lemma | norm_iterated_fderiv_within_eq_norm_iterated_deriv_within | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv_within_eq_equiv_comp",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_within_zero :
iterated_deriv_within 0 f s = f | by { ext x, simp [iterated_deriv_within] } | lemma | iterated_deriv_within_zero | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_within_one {x : 𝕜} (h : unique_diff_within_at 𝕜 s x):
iterated_deriv_within 1 f s x = deriv_within f s x | by { simp only [iterated_deriv_within, iterated_fderiv_within_one_apply h], refl } | lemma | iterated_deriv_within_one | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv_within",
"iterated_deriv_within",
"iterated_fderiv_within_one_apply",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_of_continuous_on_differentiable_on_deriv {n : ℕ∞}
(Hcont : ∀ (m : ℕ), (m : ℕ∞) ≤ n →
continuous_on (λ x, iterated_deriv_within m f s x) s)
(Hdiff : ∀ (m : ℕ), (m : ℕ∞) < n →
differentiable_on 𝕜 (λ x, iterated_deriv_within m f s x) s) :
cont_diff_on 𝕜 n f s | begin
apply cont_diff_on_of_continuous_on_differentiable_on,
{ simpa [iterated_fderiv_within_eq_equiv_comp, linear_isometry_equiv.comp_continuous_on_iff] },
{ simpa [iterated_fderiv_within_eq_equiv_comp, linear_isometry_equiv.comp_differentiable_on_iff] }
end | lemma | cont_diff_on_of_continuous_on_differentiable_on_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff_on",
"cont_diff_on_of_continuous_on_differentiable_on",
"continuous_on",
"differentiable_on",
"iterated_deriv_within",
"iterated_fderiv_within_eq_equiv_comp",
"linear_isometry_equiv.comp_continuous_on_iff",
"linear_isometry_equiv.comp_differentiable_on_iff"
] | If the first `n` derivatives within a set of a function are continuous, and its first `n-1`
derivatives are differentiable, then the function is `C^n`. This is not an equivalence in general,
but this is an equivalence when the set has unique derivatives, see
`cont_diff_on_iff_continuous_on_differentiable_on_deriv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_of_differentiable_on_deriv {n : ℕ∞}
(h : ∀(m : ℕ), (m : ℕ∞) ≤ n → differentiable_on 𝕜 (iterated_deriv_within m f s) s) :
cont_diff_on 𝕜 n f s | begin
apply cont_diff_on_of_differentiable_on,
simpa only [iterated_fderiv_within_eq_equiv_comp,
linear_isometry_equiv.comp_differentiable_on_iff]
end | lemma | cont_diff_on_of_differentiable_on_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff_on",
"cont_diff_on_of_differentiable_on",
"differentiable_on",
"iterated_deriv_within",
"iterated_fderiv_within_eq_equiv_comp",
"linear_isometry_equiv.comp_differentiable_on_iff"
] | To check that a function is `n` times continuously differentiable, it suffices to check that its
first `n` derivatives are differentiable. This is slightly too strong as the condition we
require on the `n`-th derivative is differentiability instead of continuity, but it has the
advantage of avoiding the discussion of c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.continuous_on_iterated_deriv_within {n : ℕ∞} {m : ℕ}
(h : cont_diff_on 𝕜 n f s) (hmn : (m : ℕ∞) ≤ n) (hs : unique_diff_on 𝕜 s) :
continuous_on (iterated_deriv_within m f s) s | by simpa only [iterated_deriv_within_eq_equiv_comp, linear_isometry_equiv.comp_continuous_on_iff]
using h.continuous_on_iterated_fderiv_within hmn hs | lemma | cont_diff_on.continuous_on_iterated_deriv_within | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff_on",
"continuous_on",
"iterated_deriv_within",
"iterated_deriv_within_eq_equiv_comp",
"linear_isometry_equiv.comp_continuous_on_iff",
"unique_diff_on"
] | On a set with unique derivatives, a `C^n` function has derivatives up to `n` which are
continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.differentiable_within_at_iterated_deriv_within {n : ℕ∞} {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_deriv_within m f s) s x | by simpa only [iterated_deriv_within_eq_equiv_comp,
linear_isometry_equiv.comp_differentiable_within_at_iff]
using h.differentiable_within_at_iterated_fderiv_within hmn hs | lemma | cont_diff_within_at.differentiable_within_at_iterated_deriv_within | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff_within_at",
"differentiable_within_at",
"iterated_deriv_within",
"iterated_deriv_within_eq_equiv_comp",
"linear_isometry_equiv.comp_differentiable_within_at_iff",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.differentiable_on_iterated_deriv_within {n : ℕ∞} {m : ℕ}
(h : cont_diff_on 𝕜 n f s) (hmn : (m : ℕ∞) < n) (hs : unique_diff_on 𝕜 s) :
differentiable_on 𝕜 (iterated_deriv_within m f s) s | λ x hx, (h x hx).differentiable_within_at_iterated_deriv_within hmn $ by rwa [insert_eq_of_mem hx] | lemma | cont_diff_on.differentiable_on_iterated_deriv_within | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff_on",
"differentiable_on",
"iterated_deriv_within",
"unique_diff_on"
] | On a set with unique derivatives, a `C^n` function has derivatives less than `n` which are
differentiable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_iff_continuous_on_differentiable_on_deriv {n : ℕ∞}
(hs : unique_diff_on 𝕜 s) :
cont_diff_on 𝕜 n f s ↔
(∀m:ℕ, (m : ℕ∞) ≤ n → continuous_on (iterated_deriv_within m f s) s)
∧ (∀m:ℕ, (m : ℕ∞) < n → differentiable_on 𝕜 (iterated_deriv_within m f s) s) | by simp only [cont_diff_on_iff_continuous_on_differentiable_on hs,
iterated_fderiv_within_eq_equiv_comp, linear_isometry_equiv.comp_continuous_on_iff,
linear_isometry_equiv.comp_differentiable_on_iff] | lemma | cont_diff_on_iff_continuous_on_differentiable_on_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff_on",
"cont_diff_on_iff_continuous_on_differentiable_on",
"continuous_on",
"differentiable_on",
"iterated_deriv_within",
"iterated_fderiv_within_eq_equiv_comp",
"linear_isometry_equiv.comp_continuous_on_iff",
"linear_isometry_equiv.comp_differentiable_on_iff",
"unique_diff_on"
] | The property of being `C^n`, initially defined in terms of the Fréchet derivative, can be
reformulated in terms of the one-dimensional derivative on sets with unique derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_within_succ {x : 𝕜} (hxs : unique_diff_within_at 𝕜 s x) :
iterated_deriv_within (n + 1) f s x = deriv_within (iterated_deriv_within n f s) s x | begin
rw [iterated_deriv_within_eq_iterated_fderiv_within, iterated_fderiv_within_succ_apply_left,
iterated_fderiv_within_eq_equiv_comp, linear_isometry_equiv.comp_fderiv_within _ hxs,
deriv_within],
change ((continuous_multilinear_map.mk_pi_field 𝕜 (fin n)
((fderiv_within 𝕜 (iterated_deriv_within... | lemma | iterated_deriv_within_succ | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.mk_pi_field",
"deriv_within",
"fderiv_within",
"iterated_deriv_within",
"iterated_deriv_within_eq_iterated_fderiv_within",
"iterated_fderiv_within_eq_equiv_comp",
"iterated_fderiv_within_succ_apply_left",
"linear_isometry_equiv.comp_fderiv_within",
"unique_diff_within_at"... | The `n+1`-th iterated derivative within a set with unique derivatives can be obtained by
differentiating the `n`-th iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_within_eq_iterate {x : 𝕜} (hs : unique_diff_on 𝕜 s) (hx : x ∈ s) :
iterated_deriv_within n f s x = ((λ (g : 𝕜 → F), deriv_within g s)^[n]) f x | begin
induction n with n IH generalizing x,
{ simp },
{ rw [iterated_deriv_within_succ (hs x hx), function.iterate_succ'],
exact deriv_within_congr (λ y hy, IH hy) (IH hx) }
end | lemma | iterated_deriv_within_eq_iterate | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv_within",
"deriv_within_congr",
"function.iterate_succ'",
"iterated_deriv_within",
"iterated_deriv_within_succ",
"unique_diff_on"
] | The `n`-th iterated derivative within a set with unique derivatives can be obtained by
iterating `n` times the differentiation operation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_within_succ' {x : 𝕜} (hxs : unique_diff_on 𝕜 s) (hx : x ∈ s) :
iterated_deriv_within (n + 1) f s x = (iterated_deriv_within n (deriv_within f s) s) x | by { rw [iterated_deriv_within_eq_iterate hxs hx, iterated_deriv_within_eq_iterate hxs hx], refl } | lemma | iterated_deriv_within_succ' | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv_within",
"iterated_deriv_within",
"iterated_deriv_within_eq_iterate",
"unique_diff_on"
] | The `n+1`-th iterated derivative within a set with unique derivatives can be obtained by
taking the `n`-th derivative of the derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_eq_iterated_fderiv :
iterated_deriv n f x
= (iterated_fderiv 𝕜 n f x : ((fin n) → 𝕜) → F) (λ(i : fin n), 1) | rfl | lemma | iterated_deriv_eq_iterated_fderiv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv",
"iterated_fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_eq_equiv_comp :
iterated_deriv n f
= (continuous_multilinear_map.pi_field_equiv 𝕜 (fin n) F).symm ∘ (iterated_fderiv 𝕜 n f) | by { ext x, refl } | lemma | iterated_deriv_eq_equiv_comp | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.pi_field_equiv",
"iterated_deriv",
"iterated_fderiv"
] | Write the iterated derivative as the composition of a continuous linear equiv and the iterated
Fréchet derivative | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_eq_equiv_comp :
iterated_fderiv 𝕜 n f
= (continuous_multilinear_map.pi_field_equiv 𝕜 (fin n) F) ∘ (iterated_deriv n f) | by rw [iterated_deriv_eq_equiv_comp, ← function.comp.assoc, linear_isometry_equiv.self_comp_symm,
function.left_id] | lemma | iterated_fderiv_eq_equiv_comp | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.pi_field_equiv",
"iterated_deriv",
"iterated_deriv_eq_equiv_comp",
"iterated_fderiv",
"linear_isometry_equiv.self_comp_symm"
] | Write the iterated Fréchet derivative as the composition of a continuous linear equiv and the
iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_apply_eq_iterated_deriv_mul_prod {m : (fin n) → 𝕜} :
(iterated_fderiv 𝕜 n f x : ((fin n) → 𝕜) → F) m = (∏ i, m i) • iterated_deriv n f x | by { rw [iterated_deriv_eq_iterated_fderiv, ← continuous_multilinear_map.map_smul_univ], simp } | lemma | iterated_fderiv_apply_eq_iterated_deriv_mul_prod | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_map.map_smul_univ",
"iterated_deriv",
"iterated_deriv_eq_iterated_fderiv",
"iterated_fderiv"
] | The `n`-th Fréchet derivative applied to a vector `(m 0, ..., m (n-1))` is the derivative
multiplied by the product of the `m i`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_eq_norm_iterated_deriv :
‖iterated_fderiv 𝕜 n f x‖ = ‖iterated_deriv n f x‖ | by rw [iterated_deriv_eq_equiv_comp, linear_isometry_equiv.norm_map] | lemma | norm_iterated_fderiv_eq_norm_iterated_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv_eq_equiv_comp",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_zero :
iterated_deriv 0 f = f | by { ext x, simp [iterated_deriv] } | lemma | iterated_deriv_zero | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"iterated_deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_one :
iterated_deriv 1 f = deriv f | by { ext x, simp [iterated_deriv], refl } | lemma | iterated_deriv_one | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv",
"iterated_deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_iff_iterated_deriv {n : ℕ∞} :
cont_diff 𝕜 n f ↔
(∀m:ℕ, (m : ℕ∞) ≤ n → continuous (iterated_deriv m f))
∧ (∀m:ℕ, (m : ℕ∞) < n → differentiable 𝕜 (iterated_deriv m f)) | by simp only [cont_diff_iff_continuous_differentiable, iterated_fderiv_eq_equiv_comp,
linear_isometry_equiv.comp_continuous_iff, linear_isometry_equiv.comp_differentiable_iff] | lemma | cont_diff_iff_iterated_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff",
"cont_diff_iff_continuous_differentiable",
"continuous",
"differentiable",
"iterated_deriv",
"iterated_fderiv_eq_equiv_comp",
"linear_isometry_equiv.comp_continuous_iff",
"linear_isometry_equiv.comp_differentiable_iff"
] | The property of being `C^n`, initially defined in terms of the Fréchet derivative, can be
reformulated in terms of the one-dimensional derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_of_differentiable_iterated_deriv {n : ℕ∞}
(h : ∀(m : ℕ), (m : ℕ∞) ≤ n → differentiable 𝕜 (iterated_deriv m f)) :
cont_diff 𝕜 n f | cont_diff_iff_iterated_deriv.2
⟨λ m hm, (h m hm).continuous, λ m hm, (h m (le_of_lt hm))⟩ | lemma | cont_diff_of_differentiable_iterated_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff",
"continuous",
"differentiable",
"iterated_deriv"
] | To check that a function is `n` times continuously differentiable, it suffices to check that its
first `n` derivatives are differentiable. This is slightly too strong as the condition we
require on the `n`-th derivative is differentiability instead of continuity, but it has the
advantage of avoiding the discussion of c... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.continuous_iterated_deriv {n : ℕ∞} (m : ℕ)
(h : cont_diff 𝕜 n f) (hmn : (m : ℕ∞) ≤ n) :
continuous (iterated_deriv m f) | (cont_diff_iff_iterated_deriv.1 h).1 m hmn | lemma | cont_diff.continuous_iterated_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff",
"continuous",
"iterated_deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.differentiable_iterated_deriv {n : ℕ∞} (m : ℕ)
(h : cont_diff 𝕜 n f) (hmn : (m : ℕ∞) < n) :
differentiable 𝕜 (iterated_deriv m f) | (cont_diff_iff_iterated_deriv.1 h).2 m hmn | lemma | cont_diff.differentiable_iterated_deriv | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"cont_diff",
"differentiable",
"iterated_deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_deriv_succ : iterated_deriv (n + 1) f = deriv (iterated_deriv n f) | begin
ext x,
rw [← iterated_deriv_within_univ, ← iterated_deriv_within_univ, ← deriv_within_univ],
exact iterated_deriv_within_succ unique_diff_within_at_univ,
end | lemma | iterated_deriv_succ | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv",
"deriv_within_univ",
"iterated_deriv",
"iterated_deriv_within_succ",
"iterated_deriv_within_univ",
"unique_diff_within_at_univ"
] | The `n+1`-th iterated derivative can be obtained by differentiating the `n`-th
iterated derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_eq_iterate : iterated_deriv n f = (deriv^[n]) f | begin
ext x,
rw [← iterated_deriv_within_univ],
convert iterated_deriv_within_eq_iterate unique_diff_on_univ (mem_univ x),
simp [deriv_within_univ]
end | lemma | iterated_deriv_eq_iterate | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv",
"deriv_within_univ",
"iterated_deriv",
"iterated_deriv_within_eq_iterate",
"iterated_deriv_within_univ",
"unique_diff_on_univ"
] | The `n`-th iterated derivative can be obtained by iterating `n` times the
differentiation operation. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_deriv_succ' : iterated_deriv (n + 1) f = iterated_deriv n (deriv f) | by { rw [iterated_deriv_eq_iterate, iterated_deriv_eq_iterate], refl } | lemma | iterated_deriv_succ' | analysis.calculus | src/analysis/calculus/iterated_deriv.lean | [
"analysis.calculus.deriv.comp",
"analysis.calculus.cont_diff_def"
] | [
"deriv",
"iterated_deriv",
"iterated_deriv_eq_iterate"
] | The `n+1`-th iterated derivative can be obtained by taking the `n`-th derivative of the
derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on.range_ne_top_of_has_strict_fderiv_at
(hextr : is_local_extr_on φ {x | f x = f x₀} x₀) (hf' : has_strict_fderiv_at f f' x₀)
(hφ' : has_strict_fderiv_at φ φ' x₀) :
linear_map.range (f'.prod φ') ≠ ⊤ | begin
intro htop,
set fφ := λ x, (f x, φ x),
have A : map φ (𝓝[f ⁻¹' {f x₀}] x₀) = 𝓝 (φ x₀),
{ change map (prod.snd ∘ fφ) (𝓝[fφ ⁻¹' {p | p.1 = f x₀}] x₀) = 𝓝 (φ x₀),
rw [← map_map, nhds_within, map_inf_principal_preimage,
(hf'.prod hφ').map_nhds_eq_of_surj htop],
exact map_snd_nhds_within _ },... | lemma | is_local_extr_on.range_ne_top_of_has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/lagrange_multipliers.lean | [
"analysis.calculus.inverse",
"linear_algebra.dual"
] | [
"has_strict_fderiv_at",
"is_local_extr_on",
"linear_map.range",
"map_snd_nhds_within",
"nhds_within"
] | Lagrange multipliers theorem: if `φ : E → ℝ` has a local extremum on the set `{x | f x = f x₀}`
at `x₀`, both `f : E → F` and `φ` are strictly differentiable at `x₀`, and the codomain of `f` is
a complete space, then the linear map `x ↦ (f' x, φ' x)` is not surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on.exists_linear_map_of_has_strict_fderiv_at
(hextr : is_local_extr_on φ {x | f x = f x₀} x₀) (hf' : has_strict_fderiv_at f f' x₀)
(hφ' : has_strict_fderiv_at φ φ' x₀) :
∃ (Λ : module.dual ℝ F) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ ∀ x, Λ (f' x) + Λ₀ • φ' x = 0 | begin
rcases submodule.exists_le_ker_of_lt_top _
(lt_top_iff_ne_top.2 $ hextr.range_ne_top_of_has_strict_fderiv_at hf' hφ') with ⟨Λ', h0, hΛ'⟩,
set e : ((F →ₗ[ℝ] ℝ) × ℝ) ≃ₗ[ℝ] (F × ℝ →ₗ[ℝ] ℝ) :=
((linear_equiv.refl ℝ (F →ₗ[ℝ] ℝ)).prod (linear_map.ring_lmap_equiv_self ℝ ℝ ℝ).symm).trans
(linear_map.cop... | lemma | is_local_extr_on.exists_linear_map_of_has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/lagrange_multipliers.lean | [
"analysis.calculus.inverse",
"linear_algebra.dual"
] | [
"algebra.id.smul_eq_mul",
"continuous_linear_map.coe_coe",
"continuous_linear_map.coe_prod",
"continuous_linear_map.to_linear_map_eq_coe",
"has_strict_fderiv_at",
"is_local_extr_on",
"linear_equiv.prod_apply",
"linear_equiv.refl",
"linear_equiv.refl_apply",
"linear_equiv.trans_apply",
"linear_ma... | Lagrange multipliers theorem: if `φ : E → ℝ` has a local extremum on the set `{x | f x = f x₀}`
at `x₀`, both `f : E → F` and `φ` are strictly differentiable at `x₀`, and the codomain of `f` is
a complete space, then there exist `Λ : dual ℝ F` and `Λ₀ : ℝ` such that `(Λ, Λ₀) ≠ 0` and
`Λ (f' x) + Λ₀ • φ' x = 0` for all ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on.exists_multipliers_of_has_strict_fderiv_at_1d
{f : E → ℝ} {f' : E →L[ℝ] ℝ}
(hextr : is_local_extr_on φ {x | f x = f x₀} x₀) (hf' : has_strict_fderiv_at f f' x₀)
(hφ' : has_strict_fderiv_at φ φ' x₀) :
∃ (a b : ℝ), (a, b) ≠ 0 ∧ a • f' + b • φ' = 0 | begin
obtain ⟨Λ, Λ₀, hΛ, hfΛ⟩ := hextr.exists_linear_map_of_has_strict_fderiv_at hf' hφ',
refine ⟨Λ 1, Λ₀, _, _⟩,
{ contrapose! hΛ,
simp only [prod.mk_eq_zero] at ⊢ hΛ,
refine ⟨linear_map.ext (λ x, _), hΛ.2⟩,
simpa [hΛ.1] using Λ.map_smul x 1 },
{ ext x,
have H₁ : Λ (f' x) = f' x * Λ 1,
{ si... | lemma | is_local_extr_on.exists_multipliers_of_has_strict_fderiv_at_1d | analysis.calculus | src/analysis/calculus/lagrange_multipliers.lean | [
"analysis.calculus.inverse",
"linear_algebra.dual"
] | [
"algebra.id.smul_eq_mul",
"has_strict_fderiv_at",
"is_local_extr_on",
"mul_comm",
"mul_one"
] | Lagrange multipliers theorem: if `φ : E → ℝ` has a local extremum on the set `{x | f x = f x₀}`
at `x₀`, and both `f : E → ℝ` and `φ` are strictly differentiable at `x₀`, then there exist
`a b : ℝ` such that `(a, b) ≠ 0` and `a • f' + b • φ' = 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on.exists_multipliers_of_has_strict_fderiv_at {ι : Type*} [fintype ι]
{f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ}
(hextr : is_local_extr_on φ {x | ∀ i, f i x = f i x₀} x₀)
(hf' : ∀ i, has_strict_fderiv_at (f i) (f' i) x₀)
(hφ' : has_strict_fderiv_at φ φ' x₀) :
∃ (Λ : ι → ℝ) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ ∑ i,... | begin
letI := classical.dec_eq ι,
replace hextr : is_local_extr_on φ {x | (λ i, f i x) = (λ i, f i x₀)} x₀,
by simpa only [function.funext_iff] using hextr,
rcases hextr.exists_linear_map_of_has_strict_fderiv_at
(has_strict_fderiv_at_pi.2 (λ i, hf' i)) hφ'
with ⟨Λ, Λ₀, h0, hsum⟩,
rcases (linear_equi... | lemma | is_local_extr_on.exists_multipliers_of_has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/lagrange_multipliers.lean | [
"analysis.calculus.inverse",
"linear_algebra.dual"
] | [
"classical.dec_eq",
"fintype",
"function.funext_iff",
"has_strict_fderiv_at",
"is_local_extr_on",
"linear_equiv.map_eq_zero_iff",
"linear_equiv.pi_ring",
"mul_comm",
"prod.ext_iff"
] | Lagrange multipliers theorem, 1d version. Let `f : ι → E → ℝ` be a finite family of functions.
Suppose that `φ : E → ℝ` has a local extremum on the set `{x | ∀ i, f i x = f i x₀}` at `x₀`.
Suppose that all functions `f i` as well as `φ` are strictly differentiable at `x₀`.
Then the derivatives `f' i : E → L[ℝ] ℝ` and `... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_local_extr_on.linear_dependent_of_has_strict_fderiv_at {ι : Type*} [finite ι]
{f : ι → E → ℝ} {f' : ι → E →L[ℝ] ℝ}
(hextr : is_local_extr_on φ {x | ∀ i, f i x = f i x₀} x₀)
(hf' : ∀ i, has_strict_fderiv_at (f i) (f' i) x₀)
(hφ' : has_strict_fderiv_at φ φ' x₀) :
¬linear_independent ℝ (option.elim φ' f' : op... | begin
casesI nonempty_fintype ι,
rw [fintype.linear_independent_iff], push_neg,
rcases hextr.exists_multipliers_of_has_strict_fderiv_at hf' hφ' with ⟨Λ, Λ₀, hΛ, hΛf⟩,
refine ⟨option.elim Λ₀ Λ, _, _⟩,
{ simpa [add_comm] using hΛf },
{ simpa [function.funext_iff, not_and_distrib, or_comm, option.exists] using... | lemma | is_local_extr_on.linear_dependent_of_has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/lagrange_multipliers.lean | [
"analysis.calculus.inverse",
"linear_algebra.dual"
] | [
"finite",
"fintype.linear_independent_iff",
"function.funext_iff",
"has_strict_fderiv_at",
"is_local_extr_on",
"linear_independent",
"nonempty_fintype",
"not_and_distrib",
"option.elim"
] | Lagrange multipliers theorem. Let `f : ι → E → ℝ` be a finite family of functions.
Suppose that `φ : E → ℝ` has a local extremum on the set `{x | ∀ i, f i x = f i x₀}` at `x₀`.
Suppose that all functions `f i` as well as `φ` are strictly differentiable at `x₀`.
Then the derivatives `f' i : E → L[ℝ] ℝ` and `φ' : E →L[ℝ]... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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